Additive-Mean-Variance Utility Function and Additive-Mean-Standard-Deviation Utility Function

Description

Compute the utility function mean(x) - (gamma * sd.p(x))^theta / theta or weighted.mean(x, wt) - (gamma * sd.p(x, wt))^theta / theta.

Usage

AMSD(x, gamma = 1, wt = NULL, theta = 1)

AMV(x, gamma = 1, wt = NULL)

Arguments

Value

A scalar indicating the utility level.

Functions

• AMSD(): Computes the utility function mean(x) - (gamma * sd.p(x))^theta / theta or weighted.mean(x, wt) - (gamma * sd.p(x, wt))^theta / theta. When theta == 2, it is the additive mean-variance utility function (i.e. the function AMV). When theta == 1 (the default value), it is the additive mean and standard deviation utility function.

• AMV(): Compute the additive mean-variance utility function mean(x) - 0.5 * gamma * var.p(x) or weighted.mean(x, wt) - 0.5 * gamma * var.p(x, wt).

References

Nakamura, Yutaka (2015). Mean-Variance Utility. Journal of Economic Theory, 160: 536-556.

Examples

AMSD(1:2, gamma = 0.05)
AMSD(1:2, gamma = 1, theta = 2)

marginal_utility(
  c(1, 1.001),
  c(0, 1), function(x) AMSD(x, gamma = 0.5)
)
marginal_utility(
  c(1.001, 1),
  c(0, 1), function(x) AMSD(x, gamma = 0.5)
)

Additive-Mean-Standard-Deviation Portfolio Utility Function

Description

Compute the utility function x %*% mp - gamma^theta * (t(x) %*% Cov %*% x)^(0.5 * theta) / theta for a portfolio x.

Usage

AMSDP(x, mp, Cov, gamma = 1, theta = 1)

Arguments

Value

A scalar indicating the utility level.

References

Danthine, J. P., Donaldson, J. (2005, ISBN: 9780123693808) Intermediate Financial Theory. Elsevier Academic Press.

Nakamura, Yutaka (2015) Mean-Variance Utility. Journal of Economic Theory, 160: 536-556.

Sharpe, William F (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

Xu Gao (2018, ISBN: 9787300258232) Twenty-five Lectures on Financial Economics. Beijing: China Renmin University Press. (In Chinese)

See Also

AMSD

Examples

UAP <- matrix(c(
  0, 1, 1,
  0, 2, 1,
  1, 1, 1,
  1, 2, 1,
  2, 0, 1
), nrow = 5, byrow = TRUE)

portfolio <- c(1.977, 1.183, 3.820)

AMSDP(portfolio, colMeans(UAP),
  cov.wt(UAP, method = "ML")$cov,
  gamma = 1, theta = 1
)

AMSD(UAP %*% portfolio, gamma = 1, theta = 1)

Constant Absolute Risk Aversion (CARA) Utility Function

Description

Compute the value and the certainty equivalent of the CARA utility function, i.e. -exp(-gamma*x). In general equilibrium analysis, the CARA utility function has an interval scale like temperature.

Usage

CARA(x, gamma, prob = rep(1/length(x), length(x)))

Arguments

Value

A list containing the following components:

• u: the utility level.

• CE: the certainty equivalent.

Examples

mu <- 5 # mu <- 8
a <- 1
x <- c(mu - a, mu + a)
gamma <- 0.8
mu - CARA(x, gamma)$CE

####
gamma <- 0.8
mu <- 2
sigma <- 2
x <- seq(mu - 5 * sigma, mu + 5 * sigma, length.out = 10000)
# two CE calculation methods for random variables of normal distribution
CARA(x, gamma, dnorm(x, mean = mu, sd = sigma))
mu - gamma * sigma^2 / 2

CES Function

Description

CES function, e.g. alpha * (beta1 * (x1 / theta1)^sigma + beta2 * (x2 / theta2)^sigma)^(1 / sigma).

Usage

CES(sigma = 1 - 1/es, alpha, beta, x, theta = rep(1, length(beta)), es = NA)

Arguments

Value

A scalar indicating the output or utility level.

Examples

CES(0.5, 1, c(0.4, 0.6), c(1, 1), c(0.4, 0.6))
CES(0.5, 1, c(0.4, 0.6), c(1, 2))

Demand coefficients of the CESAK Production Function

Description

Computing the demand coefficients of the CESAK production function alpha * (betaK * x1^((es - 1) / es) + (1 - betaK) * x2^((es - 1) / es))^(es / (es - 1)) + alphaK * x1. When es==1, the CESAK production function becomes the CDAK production function alpha * x1^betaK * x2^(1 - betaK) + alphaK * x1.

Usage

CESAK_dc(alpha, betaK, alphaK, p, es = 1)

Arguments

Value

A 2-vector indicating the demand coefficients.

Examples

CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 1 - 0.06, p = c(10, 1))
CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 1 - 0.06, p = c(1, 10))
CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 2, p = c(1, 1))

CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 1 - 0.06, p = c(1, 1), es = 0.5)
CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 0.1, p = c(1, 10), es = 0.5)
CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 1.5, p = c(1, 10), es = 0.5)

Constant Relative Risk Aversion (CRRA) Utility Function

Description

Compute the value and the certainty equivalent of the CRRA utility function.

Usage

CRRA(x, gamma, prob = rep(1/length(x), length(x)))

Arguments

Value

A list containing the following components:

• u: the utility level.

• CE: the certainty equivalent.

Examples

csv <- 0.05 # coefficient of standard deviation
mu <- 90 # mu <- 100
sigma <- mu * csv
x <- seq(mu - 5 * sigma, mu + 5 * sigma, length.out = 10000)
pd <- dnorm(x, mean = mu, sd = sigma)
gamma <- 0.8
# the ratio of risk premium to expected return (i.e. the relative risk premium).
(mu - CRRA(x, gamma, pd)$CE) / mu

####
df <- apply_expand.grid(
  function(arg) {
    CRRA(arg["x"], arg["gamma"])$u
  },
  x = seq(0.5, 3, 0.1),
  gamma = c(0.5, 1, 2, 3)
)
coplot(result ~ x | as.factor(gamma), data = df)

Displaced CES Utility Function and Displaced CES Demand Function

Description

The displaced CES utility function and the displaced CES demand function (Fullerton, 1989).

Usage

DCES(es, beta, xi, x)

DCES_demand(es, beta, xi, w, p)

DCES_compensated_demand(es, beta, xi, u, p)

DCES_indirect(es, beta, xi, w, p)

Arguments

Value

The return values of these functions are as follows:
DCES: A scalar indicating the utility level.
DCES_demand: An n-vector indicating the demands.
DCES_compensated_demand: An n-vector indicating the compensated demands.
DCES_indirect: A scalar indicating the utility level.

Functions

• DCES(): Compute the displaced CES utility function (Fullerton, 1989), e.g. (beta1 ^ (1 / es) * (x1 - xi1) ^ (1 - 1 / es) + beta2 ^ (1 / es) * (x2 - xi2) ^ (1 - 1 / es)) ^ (es / (es - 1) wherein beta1 + beta2 == 1.

  When es==1, the DCES utility function becomes the Stone-Geary utility function.

• DCES_demand(): The displaced CES demand function (Fullerton, 1989).

• DCES_compensated_demand(): The displaced CES compensated demand function (Fullerton, 1989).

• DCES_indirect(): The displaced CES indirect utility function (Fullerton, 1989).

References

Acemoglu, D. (2009, ISBN: 9780691132921) Introduction to Modern Economic Growth. Princeton University Press.

Fullerton, D. (1989) Notes on Displaced CES Functional Forms. Available at: https://works.bepress.com/don_fullerton/39/

Examples

es <- 0.99
beta <- prop.table(1:5)
xi <- 0
w <- 500
p <- 2:6

x <- DCES_demand(
  es = es,
  beta = beta,
  xi = xi,
  w = w,
  p = p
)

DCES_demand(
  es = es,
  beta = prop.table(0:4),
  xi = 5:1,
  w = w,
  p = p
)

u <- DCES(
  es = es,
  beta = beta,
  xi = xi,
  x = x
)

SCES(
  es = es,
  alpha = 1,
  beta = beta,
  x = x
)

DCES_compensated_demand(
  es = es,
  beta = beta,
  xi = xi,
  u = u,
  p = p
)

DCES_compensated_demand(
  es = es,
  beta = beta,
  xi = seq(10, 50, 10),
  u = u,
  p = p
)

#### A 2-by-2 general equilibrium model
#### with a DCES utility function.
ge <- sdm2(
  A = function(state) {
    a.consumer <- DCES_demand(
      es = 2, beta = c(0.2, 0.8), xi = c(1000, 500),
      w = state$w[1], p = state$p
    )
    a.firm <- c(1.1, 0)
    cbind(a.consumer, a.firm)
  },
  B = diag(c(0, 1)),
  S0Exg = matrix(c(
    3500, NA,
    NA, NA
  ), 2, 2, TRUE),
  names.commodity = c("corn", "iron"),
  names.agent = c("consumer", "firm"),
  numeraire = "corn"
)

ge$p
ge$z
ge$A
ge$D

#### a 2-by-2 pure exchange economy
sdm2(
  A = function(state) {
    a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
    a2 <- DCES_demand(
      es = 1, beta = c(0.4, 0.6), xi = c(0.1, 0.2),
      w = state$w[2], p = state$p
    )
    cbind(a1, a2)
  },
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    3, 4,
    7, 0
  ), 2, 2, TRUE),
  names.commodity = c("fish", "banana"),
  names.agent = c("Annie", "Ben"),
  numeraire = "banana"
)

#### A 3-by-3 general equilibrium model
#### with a DCES utility function.
lab <- 1 # the amount of labor supplied by each laborer
n.laborer <- 100 # the number of laborers
ge <- sdm2(
  A = function(state) {
    a.firm.corn <- CD_A(alpha = 1, Beta = c(0, 0.5, 0.5), state$p)
    a.firm.iron <- CD_A(alpha = 5, Beta = c(0, 0.5, 0.5), state$p)
    a.laborer <- DCES_demand(
      es = 0, beta = c(0, 1, 0), xi = c(0.1, 0, 0),
      w = state$w[3] / n.laborer, p = state$p
    )

    cbind(a.firm.corn, a.firm.iron, a.laborer)
  },
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, NA, lab * n.laborer
  ), 3, 3, TRUE),
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "laborer"),
  numeraire = "lab",
  priceAdjustmentVelocity = 0.1
)

ge$z
ge$A
ge$D

Modified Displaced CES Demand Function

Description

Compute the modified displaced CES demand function. Firstly, the (unmodified) DCES demand vector and the (unmodified) utility level are computed under the given income and prices. Secondly, the modified beta and es are computed under the unmodified utility level. Finally, the DCES demand vector (namely the modified DCES demand vector) and the utility level (namely the modified DCES utility) are computed under the modified beta, the modified es, the given income and prices.

Usage

MDCES_demand(es, beta, xi, w, p, betaMod = NULL, esMod = NULL, detail = FALSE)

Arguments

References

Acemoglu, D. (2009, ISBN: 9780691132921) Introduction to Modern Economic Growth. Princeton University Press.

Fullerton, D. (1989) Notes on Displaced CES Functional Forms. Available at: https://works.bepress.com/don_fullerton/39/

Examples

MDCES_demand(
  es = 1.7, beta = c(0.9, 0.1), xi = c(12, 48),
  w = 24, p = c(1, 1 / 400),
  betaMod = function(u.unmod) {
    beta2 <- min(0.1, 10 / u.unmod)
    c(1 - beta2, beta2)
  },
  detail = TRUE
)

#### An example of computing the daily
#### labor supply at various wage rates.
result <- c()

for (real.wage in 4:400) {
  x <- MDCES_demand(
    es = 1.7, beta = c(0.9, 0.1),
    xi = c(12, 48), w = 24,
    p = c(1, 1 / real.wage),
    betaMod = function(u.unmod) {
      beta2 <- min(0.1, 10 / u.unmod)
      c(1 - beta2, beta2)
    },
    detail = TRUE
  )

  lab.supply <- unname(24 - x[1])
  result <- rbind(
    result,
    c(real.wage, lab.supply, x)
  )
}

plot(result[, 1:2],
  type = "o", pch = 20,
  xlab = "hourly real wage",
  ylab = "daily labor supply"
)

#### A 2-by-2 general equilibrium model
#### with a MDCES demand function
ge <- sdm2(
  A = function(state) {
    a.firm <- CD_A(alpha = 5, Beta = c(0.5, 0.5), state$p)
    a.consumer <-
      MDCES_demand(
        es = 1, beta = c(0.5, 0.5), xi = c(0, 0), w = state$w[2], p = state$p,
        betaMod = function(u.unmod) {
          beta2 <- 0.95 * plogis(u.unmod, location = 2, scale = 2)
          c(1 - beta2, beta2)
        }
      )
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab"
)

ge$z
ge$D
MDCES_demand(
  es = 1, beta = c(0.5, 0.5), xi = c(0, 0),
  w = 1, p = ge$p,
  betaMod = function(u.unmod) {
    beta2 <- 0.95 * plogis(u.unmod, location = 2, scale = 2)
    c(1 - beta2, beta2)
  }
)

Quasilinear Demand Functions

Description

Some quasilinear demand functions. The corresponding utility functions are as follows:
power: x1 + alpha * x2^beta, wherein alpha>0, 0<beta<1.
log: x1 + alpha * log(x2), wherein alpha>0.
quadratic1: x1 + alpha * x2 - 0.5 * beta * x2^2, wherein alpha>0, beta>0.
quadratic2: x1 + beta * (alpha * x2 - 0.5 * x2^2), wherein alpha>0, beta>0.
min: x1 + alpha * min(x2, beta), wherein alpha>0, beta>0.
CRRA: x1 + alpha * (x2^(1 - beta) - 1) / (1 - beta), wherein alpha>0, beta>0. If beta==1, the function becomes x1 + alpha * log(x2).

Usage

QL_demand(
  w,
  p,
  alpha,
  beta,
  type = c("power", "log", "quadratic1", "quadratic2", "min", "CRRA")
)

Arguments

Value

A 2-by-1 matrix indicating demands.

Examples

QL_demand(w = 0.5, p = c(1, 1), alpha = 1, type = "log")
QL_demand(w = 2, p = c(1, 1), alpha = 1, type = "log")

QL_demand(w = 1, p = c(1, 5), alpha = 2, beta = 0.5)

Standard CES Function

Description

Standard CES function, e.g. alpha * (beta1 * (x1 / beta1)^sigma + beta2 * (x2 / beta2)^sigma)^(1 / sigma) wherein beta1 + beta2 == 1.

Usage

SCES(sigma = 1 - 1/es, alpha, beta, x, es = NA)

Arguments

Value

The output or utility level.

Examples

beta <- c(0.6, 0.4)
SCES(alpha = 1, beta = beta, x = beta, es = 0.5)

Standard CES Demand Coefficient Matrix

Description

This function computes the standard CES demand coefficient matrix (i.e. Theta==Beta), which is a wrapper of CES_A of CGE package.

Usage

SCES_A(sigma = 1 - 1/es, alpha, Beta, p, es = NA)

Arguments

Value

A demand coefficient n-by-m matrix.

Examples

SCES_A(-1, 1, c(0.9, 0.1), c(1, 1))
SCES_A(alpha = 1, Beta = c(0.9, 0.1), p = c(1, 1), es = 0.5)
SCES_A(0, 1, c(0.9, 0.1), c(1, 1))
beta <- c(0.9, 0.1)
CD_A(prod(beta^-beta), c(0.9, 0.1), c(1, 1))

####
SCES_A(0, 1, c(0.9, 0.1, 0), c(1, 1, 1))

####
input <- matrix(c(
  200, 300, 100,
  150, 320, 530,
  250, 380, 0
), 3, 3, TRUE)
Beta <- prop.table(input, 2)
SCES_A(sigma = rep(0, 3), alpha = c(1, 1, 1), Beta = Beta, p = c(1, 1, 1))
SCES_A(sigma = rep(-Inf, 3), alpha = c(1, 1, 1), Beta = Beta, p = c(1, 1, 1))

Applying a Function to All Combinations of the Supplied Vectors

Description

A wrapper of the functions apply and expand.grid. Returns a data frame of values obtained by applying a function to all combinations of the supplied vectors. Firstly, the function expand.grid will be used for the supplied vectors in ... and we will get a data frame containing one row for each combination of the supplied vectors. Then the function will be applied to each row of the data frame. The values of the data frame will also be included in the returned data frame.

Usage

apply_expand.grid(FUN, ...)

Arguments

Value

A data frame.

Examples

apply_expand.grid(prod, a = 1:9, b = 1:9)

####
f <- function(x) c(r1 = sum(x), r2 = unname(x["b"] - x["a"]))
apply_expand.grid(f, a = c(1, 2), b = c(3, 4))

####
f <- function(x) list(list(sum(x)), prod(x))
apply_expand.grid(f, a = c(1, 2), b = c(3, 4))

####
f <- function(x) {
  result <- SCES_A(alpha = 1, Beta = c(0.5, 0.5), p = c(x["p1"], 1), es = x["es"])
  names(result) <- c("dc1", "dc2")
  result
}

apply_expand.grid(f, p1 = seq(0.1, 10, 0.1), es = c(0.3, 0.5, 1))

Convert between (Primitive) Period Interest Rates and (Primitive) Intraperiod Interest Rates

Description

This function converts between (primitive) period interest rates and (primitive) intraperiod interest rates. Here, a period is considered as the shortest term for monetary lending.

The primitive interest rate is defined as the interest rate when the money stock is adjusted to be constant under the assumption of monetary neutrality.

In the structural dynamic model, the period interest rate refers to the interest rate at which principal and interest are repaid at the beginning of the next period after borrowing money in the current period, while the intraperiod interest rate refers to the interest rate at which the principal and interest are repaid during the current period after borrowing money in the current period.

When the velocity of money is equal to one, these two types of interest rates are the same. When the velocity of money, namely vm, is greater than one, the intraperiod interest will be repaid in vm installments within the period, and there is usually a difference between the two types of interest rates.

Usage

convert_ir(ir, vm, to = c("period", "intraperiod"))

Arguments

Value

A vector consisting of intraperiod interest rates or period interest rates.

Examples

ir <- seq(0, 1, 0.1)
plot(ir, convert_ir(ir, 2, "period"), "b")
plot(ir, convert_ir(ir, 2, "intraperiod"), "b")

A Disequilibrium Model with Credit

Description

These are some examples to illustrate that credit policies may lead to business cycles. When the firm's profit rate is high, the laborer lends labor or labor income to the firm; when the firm's profit rate is low, the firm repays the loan with products.

Usage

demCreditPolicy(...)

Arguments

Examples

dst.firm <- node_new("output",
  type = "CD", alpha = 1.2,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new("utility",
  type = "Leontief", a = 1,
  "prod"
)

f <- function(policy = NULL) {
  ge <- sdm2(
    A = list(dst.firm, dst.consumer),
    B = matrix(c(
      1, 0,
      0, 1
    ), 2, 2, TRUE),
    S0Exg = matrix(c(
      NA, NA,
      NA, 100
    ), 2, 2, TRUE),
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "consumer"),
    ts = TRUE,
    policy = policy,
    numberOfPeriods = 200,
    maxIteration = 1,
    priceAdjustmentVelocity = 0.05
  )

  matplot(ge$ts.z, type = "o", pch = 20)
  ge
}

## no credit policy
ge <- f()

## credit policy
policy.credit <- function(time, state) {
  profit.rate <- state$p[1] / sum(state$last.A[, 1] * state$p) - 1

  if (profit.rate > 0.01) {
    state$S[2, 2] <- 50
    state$S[2, 1] <- 50
  } else if (profit.rate < -0.01) {
    state$S[1, 2] <- state$S[1, 1] * 0.5
    state$S[1, 1] <- state$S[1, 1] * 0.5
  }

  state
}

de <- f(policy = policy.credit)

#### an example with 3 firms.
policy.credit <- function(time, state) {
  if (time <= 10) {
    return(state)
  }
  profit.rate <- state$p[1] / sum(state$last.A[, 1] * state$p) - 1

  if (profit.rate > 0.01) {
    state$S[3, 1] <- 30
    # state$S[3, 1:3] <- 10
    state$S[3, 4] <- 70
  } else if (profit.rate < -0.01) {
    state$S[1, 4] <- state$S[1, 1] * 0.3
    state$S[1, 1] <- state$S[1, 1] * 0.7
  }

  state
}

f <- function(policy = NULL,
              numberOfPeriods = 50) {
  ge <- sdm2(
    A = function(state) {
      a.firm.prod <- CD_A(alpha = 1, Beta = c(0, 0.5, 0.5, 0), state$p)
      a.firm.cap1 <- c(1, 0, 0.1, 0)
      a.firm.cap2 <- c(0, 0, 0.1, 1)
      a.consumer <- c(1, 0, 0, 0)
      cbind(a.firm.prod, a.firm.cap1, a.firm.cap2, a.consumer)
    },
    B = matrix(c(
      1, 0, 0, 0,
      0, 0, 1, 0,
      0, 0, 0, 0,
      0, 1, 0, 0
    ), 4, 4, TRUE),
    S0Exg = {
      tmp <- matrix(NA, 4, 4)
      tmp[3, 4] <- 100
      tmp
    },
    names.commodity = c("prod", "cap2", "lab", "cap1"),
    names.agent = c("firm.prod", "firm.cap1", "firm.cap2", "consumer"),
    numeraire = "lab",
    maxIteration = 1,
    numberOfPeriods = numberOfPeriods,
    ts = TRUE,
    p0 = c(4.191, 4.391, 1, 4.291),
    # The equilibrium output of firm.prod is 45.64.
    z0 = c(50, 21.78, 21.78, 23.86),
    policy = policy
  )

  matplot(ge$ts.z, type = "o", pch = 20)
  ge
}

## a disequilibrium path
de <- f(numberOfPeriods = 500)

## a spot equilibrium path converging to the steady-state equilibrium
ge <- f(
  policy = policyMarketClearingPrice,
  numberOfPeriods = 40
)
ge$p
ge$z

## a spot equilibrium path with persisting fluctuations
de <- f(policy = list(
  policy.credit,
  policyMarketClearingPrice
))

A Disequilibrium Model Illustrating Insufficient Demand (Supply-demand Structural Mismatch)

Description

A disequilibrium model illustrating supply-demand structural mismatch and insufficient demand. Assume that from the 5th period, the producer expects the sales rate of products to decline, so he reduces investment in production and increases the demand for value storage means (such as foreign assets, gold, etc.); the laborer expects the unemployment rate to rise, so he reduces consumption and increases the demand for value storage means.

Here the supplier of value storage means is referred to as ROW (the rest of the world).

Usage

demInsufficientDemand_3_3(...)

Arguments

Examples

dst.firm <- node_new("output",
  type = "FIN", rate = c(1, 0),
  "cc1", "store of value"
)
node_set(dst.firm, "cc1",
  type = "CD", alpha = 2,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.laborer <- node_new("util",
  type = "FIN", rate = c(1, 0),
  "cc1", "store of value"
)
node_set(dst.laborer, "cc1",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.ROW <- node_new("util",
  type = "Leontief", a = 1,
  "lab"
)

policy.demand <- function(time, A, state) {
  if (time >= 5) {
    A[[1]]$rate <- c(1, 0.25)
    A[[2]]$rate <- c(1, 0.25)
  } else {
    A[[1]]$rate <- c(1, 0)
    A[[2]]$rate <- c(1, 0)
  }

  state
}

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.ROW),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "store of value"),
  names.agent = c("firm", "laborer", "ROW"),
  ts = TRUE,
  policy = policy.demand,
  numberOfPeriods = 100,
  maxIteration = 1,
  numeraire = "prod",
  z0 = c(100, 0, 0),
  p0 = c(1, 1, 1),
  pExg = c(1, NA, 1)
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)
matplot(ge$ts.q, type = "o", pch = 20)

Compute Demand Coefficients of an Agent with a Demand Structural Tree

Description

Given a price vector, this function computes the demand coefficients of an agent with a demand structural tree. The class of a demand structural tree is Node defined by the package data.tree.

Usage

demand_coefficient(node, p, trace = FALSE)

Arguments

Details

Demand coefficients often indicate the quantity of various commodities needed by an economic agent in order to obtain a unit of output or utility, and these commodities can include both real commodities and financial instruments such as tax receipts, stocks, bonds and currency.
The demand for various commodities by an economic agent can be expressed by a demand structure tree. Each non-leaf node can be regarded as the output of all its child nodes. Each node can be regarded as an input of its parent node. In other words, the commodity represented by each non-leaf node is a composite commodity composed of the commodities represented by its child nodes. Each non-leaf node usually has an attribute named type. This attribute describes the input-output relationship between the child nodes and the parent node. This relationship can sometimes be represented by a production function or a utility function. The type attribute of each non-leaf node can take the following values.

• SCES. In this case, this node also has parameters alpha, beta and es (or sigma = 1 - 1 / es). alpha and es are scalars. beta is a vector. These parameters are parameters of a standard CES function (see SCES and SCES_A).

• CES. In this case, this node also has parameters alpha, beta, theta (optional) and es (or sigma = 1 - 1 / es) (see CGE::CES_A).

• Leontief. In this case, this node also has the parameter a, which is a vector and is the parameter of a Leontief function.

• CD. CD is Cobb-Douglas. In this case, this node also has parameters alpha and beta, which are parameters of a Cobb-Douglas function.

• CESAK. In this case, this node also has parameters es, alpha, betaK and alphaK, which are parameters of the CESAK function (see CESAK_dc). Moreover, the first child node should represent capital goods.

• FIN. That is the financial type. In this case, this node also has the parameter rate or beta. If the parameter beta is not NULL, then the parameter rate will be ignored. The parameter rate applies to all situations, while the parameter beta only applies for some special cases. For FIN nodes, the first child node should represent for a physical commodity or a composite commodity containing a physical commodity, and other child nodes represent for financial instruments. The parameter beta indicates the proportion of each child node's expenditure. The parameter rate indicates the expenditure ratios between financial-instrument-type child nodes and the first child node. The first element of the parameter rate indicates the amount of the first child node needed to get a unit of output.

• FUNC. That is the function type. In this case, this node also has an attribute named func. The value of that attribute is a function which calculates the demand coefficient for the child nodes. The argument of that function is a price vector. The length of that price vector is equal to the number of the child nodes.

• StickyLinear or SL. That is the sticky linear type. In this case, this node also has an attribute named beta that contains the coefficients of the linear utility or production function. In order to avoid too drastic changes in the demand structure, the adjustment process of the demand structure has a certain stickiness when prices change.

Value

A vector consisting of demand coefficients.

Examples

#### a Leontief-type node
dst <- node_new("firm",
  type = "Leontief", a = c(0.5, 0.1),
  "wheat", "iron"
)
print(dst, "type")
node_print(dst)
plot(dst)
node_plot(dst, TRUE)

demand_coefficient(dst, p = c(wheat = 1, iron = 2)) # the same as a = c(0.5, 0.1)

#### a CD-type node
dst <- node_new("firm",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "wheat", "iron"
)

demand_coefficient(dst, p = c(wheat = 1, iron = 2))
# the same as the following
CD_A(1, c(0.5, 0.5), c(1, 2))

#### a SCES-type node
dst <- node_new("firm",
  type = "SCES",
  alpha = 2, beta = c(0.8, 0.2), es = 0.5,
  "wheat", "iron"
)

demand_coefficient(dst, p = c(wheat = 1, iron = 2))

# the same as the following
SCES_A(alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), es = 0.5)
CES_A(sigma = 1 - 1 / 0.5, alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), Theta = c(0.8, 0.2))

#### a FUNC-type node
dst <- node_new("firm",
  type = "FUNC",
  func = function(p) {
    CES_A(
      sigma = -1, alpha = 2,
      Beta = c(0.8, 0.2), p,
      Theta = c(0.8, 0.2)
    )
  },
  "wheat", "iron"
)

demand_coefficient(dst, p = c(wheat = 1, iron = 2))

# the same as the following
CES_A(sigma = -1, alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), Theta = c(0.8, 0.2))

####
p <- c(wheat = 1, iron = 3, labor = 2, capital = 4)
dst <- node_new("firm 1",
  type = "SCES", sigma = -1, alpha = 1, beta = c(1, 1),
  "cc1", "cc2"
)
node_set(dst, "cc1",
  type = "Leontief", a = c(0.6, 0.4),
  "wheat", "iron"
)
node_set(dst, "cc2",
  type = "SCES", sigma = -1, alpha = 1, beta = c(1, 1),
  "labor", "capital"
)

node_plot(dst)
demand_coefficient(dst, p)

####
p <- c(product = 1, labor = 1, money = 1)
dst <- node_new("firm",
  type = "FIN", rate = c(0.75, 1 / 3),
  "cc1", "money"
) # a financial-type node
node_set(dst, "cc1",
  type = "Leontief", a = c(0.8, 0.2),
  "product", "labor"
)

node_plot(dst)
demand_coefficient(dst, p)

#### the same as above
p <- c(product = 1, labor = 1, money = 1)
dst <- node_new("firm",
  type = "Leontief", a = c(0.8, 0.2),
  "cc1", "cc2"
)
node_set(dst, "cc1",
  type = "FIN", rate = c(0.75, 1 / 3),
  "product", "money"
)

node_set(dst, "cc2",
  type = "FIN", rate = c(0.75, 1 / 3),
  "labor", "money"
)
node_plot(dst)
demand_coefficient(dst, p)

#### the same as above
p <- c(product = 1, labor = 1, money = 1)
dst <- node_new("firm",
  type = "FIN", rate = c(1, 1 / 3),
  "cc1", "money"
) # Financial-type Demand Structure
node_set(dst, "cc1",
  type = "Leontief", a = c(0.6, 0.15),
  "product", "labor"
)

node_plot(dst)
demand_coefficient(dst, p)

Tidy a General Equilibrium

Description

Add names to the matrices and vectors of a general equilibrium, and add demand matrix, demand value matrix and supply value matrix to it.

Usage

ge_tidy(ge, names.commodity, names.agent)

Arguments

Value

A tidied general equilibrium.

An Example Illustrating the Matthew Effect of Asset Exchange

Description

This is an example that illustrates the Matthew effect of asset exchange, wherein the wealth gap between two traders widens after the exchange process. Initially, these traders had a relatively small wealth (i.e. expected average payoff) gap. However, the exchange leads to an expansion of the wealth gap. This outcome can be attributed to the fact that a trader's risk aversion coefficient is affected by his level of wealth. When traders have less wealth their risk aversion coefficient is higher. Consequently, a trader with less wealth tends to acquire more low-risk, low-average-payoff assets through trading. As a result, the expected average payoff of a trader with less wealth may decrease after the exchange. Conversely, a trader with more wealth may hold more high-risk, high-average-payoff assets after trading.

Usage

gemAssetExchange_MatthewEffect_2_2(...)

Arguments

See Also

gemAssetPricing_PUF.

Examples

#### Matthew effect
asset1 <- c(40, 200)
asset2 <- c(100, 100)

# unit asset payoff matrix.
UAP <- cbind(asset1, asset2)

S <- matrix(c(
  0.49, 0.51,
  0.49, 0.51
), 2, 2, TRUE)

ge <- sdm2(
  A = function(state) {
    Portfolio <- state$last.A %*% dg(state$last.z)

    Payoff <- UAP %*% Portfolio
    payoff.average <- colMeans(Payoff)

    # the risk aversion coefficients.
    rac <- ifelse(payoff.average > mean(UAP) * 1.02, 0.5, 1)
    rac <- ifelse(payoff.average < mean(UAP) / 1.02, 2, rac)

    uf1 <- function(portfolio) {
      payoff <- UAP %*% portfolio
      CES(alpha = 1, beta = c(0.5, 0.5), x = payoff, es = 1 / rac[1])
    }

    uf2 <- function(portfolio) {
      payoff <- UAP %*% portfolio
      CES(alpha = 1, beta = c(0.5, 0.5), x = payoff, es = 1 / rac[2])
    }

    VMU <- marginal_utility(Portfolio, diag(2), list(uf1, uf2), state$p)
    VMU <- pmax(VMU, 1e-10)

    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")
    A <- state$last.A * ratio_adjust(Ratio, coef = 0.1, method = "linear")

    prop.table(A, 2)
  },
  B = matrix(0, 2, 2),
  S0Exg = S,
  names.commodity = c("asset1", "asset2"),
  numeraire = 2,
  maxIteration = 1,
  numberOfPeriods = 1000,
  policy = makePolicyMeanValue(50),
  ts = TRUE
)

matplot(ge$ts.p, type = "l")
ge$p
ge$z
ge$D

(Payoff.S <- UAP %*% S)
colMeans(Payoff.S)

(Payoff.D <- UAP %*% ge$D)
colMeans(Payoff.D)

## Calculate the equilibrium under the fixed risk aversion coefficients.
rac <- c(2, 0.5)

uf <- list()
uf[[1]] <- function(portfolio) {
  payoff <- UAP %*% portfolio
  CES(alpha = 1, beta = c(0.5, 0.5), x = payoff, es = 1 / rac[1])
}

uf[[2]] <- function(portfolio) {
  payoff <- UAP %*% portfolio
  CES(alpha = 1, beta = c(0.5, 0.5), x = payoff, es = 1 / rac[2])
}

ge <- gemAssetPricing_PUF(
  S = S,
  uf = uf,
  policy = makePolicyMeanValue(50)
)

ge$p
addmargins(ge$D, 2)
addmargins(ge$S, 2)
ge$VMU

(Payoff <- UAP %*% ge$D)
colMeans(Payoff)

Some Examples of Asset Pricing

Description

These examples illustrate how to find the equilibrium of an asset market by the function sdm2 and by computing marginal utility of assets (see Sharpe, 2008).

Usage

gemAssetPricingExample(...)

Arguments

Value

A general equilibrium.

References

Danthine, J. P., Donaldson, J. (2005, ISBN: 9780123693808) Intermediate Financial Theory. Elsevier Academic Press.

Sharpe, William F. (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

Xu Gao (2018, ISBN: 9787300258232) Twenty-five Lectures on Financial Economics. Beijing: China Renmin University Press. (In Chinese)

https://web.stanford.edu/~wfsharpe/apsim/index.html

See Also

gemAssetPricing_CUF.

Examples

#### an example of Danthine and Donaldson (2005, section 8.3).
uf <- function(x) 0.5 * x[1] + 0.9 * (1 / 3 * log(x[2]) + 2 / 3 * log(x[3]))

ge <- sdm2(
  A = function(state) {
    VMU <- marginal_utility(state$last.A %*% dg(state$last.z), diag(3), uf, state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * Ratio
    prop.table(A, 2)
  },
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    10, 5,
    1, 4,
    2, 6
  ), 3, 2, TRUE),
  names.commodity = c("asset1", "asset2", "asset3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "asset1",
  ts = TRUE
)

ge$p

#### an example of Sharpe (2008, chapter 2)
asset1 <- c(1, 0, 0, 0, 0)
asset2 <- c(0, 1, 1, 1, 1)
asset3 <- c(0, 5, 3, 8, 4) - 3 * asset2
asset4 <- c(0, 3, 5, 4, 8) - 3 * asset2
# unit asset payoff matrix
UAP <- cbind(asset1, asset2, asset3, asset4)

prob <- c(0.15, 0.25, 0.25, 0.35)
wt <- prop.table(c(1, 0.96 * prob)) # weights

gamma.agt1 <- 1.5
gamma.agt2 <- 2.5

ge <- sdm2(
  A = function(state) {
    Payoff <- UAP %*% (state$last.A %*% dg(state$last.z))

    VMU <- marginal_utility(Payoff, UAP, list(
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt1),
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt2)
    ), price = state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * ratio_adjust(Ratio, coef = 0.05, method = "linear")

    A <- prop.table(A, 2)
  },
  B = matrix(0, 4, 2),
  S0Exg = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  names.commodity = c("asset1", "asset2", "asset3", "asset4"),
  names.agent = c("agt1", "agt2"),
  numeraire = "asset1"
)

ge$p
ge$p[3:4] + 3 * ge$p[2]

#### an example of Xu (2018, section 10.4, P151)
asset1 <- c(1, 0, 0)
asset2 <- c(0, 1, 0)
asset3 <- c(0, 0, 1)
prob <- c(0.5, 0.5)
wt <- c(1, prob)
UAP <- cbind(asset1, asset2, asset3)

gamma.agt1 <- 1
gamma.agt2 <- 0.5

ge <- sdm2(
  A = function(state) {
    Payoff <- UAP %*% (state$last.A %*% dg(state$last.z))

    VMU <- marginal_utility(Payoff, UAP, list(
      # Here CRRA(...)$u, CRRA(...)$CE and CES functions are interexchangeable.
      function(x) CRRA(x, gamma = gamma.agt1, p = wt)$u,
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt2)
    ), state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * Ratio
    prop.table(A, 2)
  },
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  names.commodity = c("asset1", "asset2", "asset3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "asset1",
  maxIteration = 1,
  ts = TRUE
)

ge$p #c(1, (1 + sqrt(5)) / 4, (1 + sqrt(17)) / 16)

## the same as above.
dst.agt1 <- node_new("util",
  type = "CD", alpha = 1, beta = c(0.5, 0.25, 0.25),
  "asset1", "asset2", "asset3"
)

dst.agt2 <- node_new("util",
  type = "CES", alpha = 1, beta = c(2, 1, 1), sigma = 0.5,
  "asset1", "asset2", "asset3"
)

ge <- sdm2(
  A = list(dst.agt1, dst.agt2),
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  names.commodity = c("asset1", "asset2", "asset3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "asset1",
  maxIteration = 1,
  ts = TRUE
)

ge$p

#### an example with production.
asset1 <- c(1, 0, 0, 0, 0, 0)
asset2 <- c(0, 1, 0, 0, 0, 0)
asset3 <- c(0, 0, 1, 3, 1, 2)
asset4 <- c(0, 0, 4, 2, 6, 2)
asset5 <- c(0, 0, 1, 0, 2, 0)

# unit asset payoff matrix
UAP <- cbind(asset1, asset2, asset3, asset4, asset5)

muf1 <- function(x) 1 / x
muf2 <- function(x) 1 / x * c(0.4, 0.1, 0.2, 0.05, 0.2, 0.05)

ge <- sdm2(
  A = function(state) {
    Payoff <- UAP %*% (state$last.A[, 1:2] %*% dg(state$last.z[1:2]))

    VMU <- marginal_utility(Payoff, UAP, muf = list(muf1, muf2), price = state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A[, 1:2] * ratio_adjust(Ratio, coef = 0.15, method = "linear")
    A <- prop.table(A, 2)

    a.firm <- CD_A(alpha = 4, Beta = c(0.5, 0.5, 0, 0, 0), state$p)
    A <- cbind(A, a.firm)
  },
  B = matrix(c(
    0, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 1
  ), 5, 3, TRUE),
  S0Exg = matrix(c(
    1, 1, NA,
    1, 2, NA,
    1, NA, NA,
    NA, 1, NA,
    NA, NA, NA
  ), 5, 3, TRUE),
  names.commodity = c("asset1", "asset2", "asset3", "asset4", "asset5"),
  names.agent = c("consumer1", "consumer2", "firm"),
  numeraire = "asset1"
)

ge$p
ge$z

#### an example with demand structure trees.
asset1 <- c(1, 0, 0, 0, 0)
asset2 <- c(0, 1, 3, 1, 2)
asset3 <- c(0, 2, 1, 3, 1)

# the asset unit payoff matrix.
UAP <- cbind(asset1, asset2, asset3)

dst.consumer1 <- node_new("util",
                          type = "CES", es = 0.5, alpha = 1, beta = c(0.5, 0.5),
                          "x1", "u2"
)
node_set(dst.consumer1, "u2",
         type = "CES", es = 0.8, alpha = 1, beta = c(0.6, 0.4),
         "u2.1", "u2.2"
)
node_set(dst.consumer1, "u2.1",
         type = "CES", es = 1, alpha = 1, beta = c(0.8, 0.2),
         "x2", "x3"
)
node_set(dst.consumer1, "u2.2",
         type = "CES", es = 1, alpha = 1, beta = c(0.8, 0.2),
         "x4", "x5"
)

dst.consumer2 <- node_new("util",
                          type = "CES", es = 0.5, alpha = 1, beta = c(0.5, 0.5),
                          "x1", "u2"
)
node_set(dst.consumer2, "u2",
         type = "CES", es = 0.8, alpha = 1, beta = c(0.6, 0.4),
         "u2.1", "u2.2"
)
node_set(dst.consumer2, "u2.1",
         type = "CES", es = 1, alpha = 1, beta = c(0.2, 0.8),
         "x2", "x3"
)
node_set(dst.consumer2, "u2.2",
         type = "CES", es = 1, alpha = 1, beta = c(0.2, 0.8),
         "x4", "x5"
)

uf1 <- function(x) {
  names(x) <- paste0("x", seq_along(x))
  output(dst.consumer1, x)
}

uf2 <- function(x) {
  names(x) <- paste0("x", seq_along(x))
  output(dst.consumer2, x)
}

ge <- gemAssetPricing_CUF(
  S = matrix(c(
    3, 3,
    1, 0,
    0, 2
  ), 3, 2, TRUE),
  UAP = UAP,
  uf = list(uf1, uf2)
)

ge$p
ge$z

Compute Asset Market Equilibria with Commodity Utility Functions for Some Simple Cases

Description

Compute the equilibrium of an asset market by the function sdm2 and by computing marginal utility of assets (see Sharpe, 2008). The argument of the utility function used in the calculation is the commodity vector (i.e. payoff vector).

Usage

gemAssetPricing_CUF(
  S = diag(2),
  UAP = diag(nrow(S)),
  uf = NULL,
  muf = NULL,
  ratio_adjust_coef = 0.05,
  numeraire = 1,
  ...
)

Arguments

Value

A general equilibrium containing a value marginal utility matrix (VMU).

References

Danthine, J. P., Donaldson, J. (2005, ISBN: 9780123693808) Intermediate Financial Theory. Elsevier Academic Press.

Sharpe, William F. (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

Wang Jiang (2006, ISBN: 9787300073477) Financial Economics. Beijing: China Renmin University Press. (In Chinese)

Xu Gao (2018, ISBN: 9787300258232) Twenty-five Lectures on Financial Economics. Beijing: China Renmin University Press. (In Chinese)

https://web.stanford.edu/~wfsharpe/apsim/index.html

See Also

gemAssetPricingExample.

Examples

gemAssetPricing_CUF(muf = function(x) 1 / x)

gemAssetPricing_CUF(
  S = cbind(c(1, 0), c(0, 2)),
  muf = function(x) 1 / x
)

gemAssetPricing_CUF(
  UAP = cbind(c(1, 0), c(0, 2)),
  muf = function(x) 1 / x
)

#### an example of Danthine and Donaldson (2005, section 8.3).
ge <- gemAssetPricing_CUF(
  S = matrix(c(
    10, 5,
    1, 4,
    2, 6
  ), 3, 2, TRUE),
  uf = function(x) 0.5 * x[1] + 0.9 * (1 / 3 * log(x[2]) + 2 / 3 * log(x[3]))
)

ge$p

#### an example of Sharpe (2008, chapter 2, case 1)
asset1 <- c(1, 0, 0, 0, 0)
asset2 <- c(0, 1, 1, 1, 1)
asset3 <- c(0, 5, 3, 8, 4) - 3 * asset2
asset4 <- c(0, 3, 5, 4, 8) - 3 * asset2
# unit asset payoff matrix
UAP <- cbind(asset1, asset2, asset3, asset4)

prob <- c(0.15, 0.25, 0.25, 0.35)
wt <- prop.table(c(1, 0.96 * prob)) # weights

geSharpe1 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  uf = list(
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5)
  )
)
geSharpe1$p
geSharpe1$p[3:4] + 3 * geSharpe1$p[2]

## an example of Sharpe (2008, chapter 3, case 2)
geSharpe2 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49, 98, 98,
    30, 30, 60, 60,
    10, 0, 20, 0,
    0, 10, 0, 20
  ), 4, 4, TRUE),
  UAP = UAP,
  uf = list(
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5)
  )
)

geSharpe2$p
geSharpe2$p[3:4] + 3 * geSharpe2$p[2]
geSharpe2$D

## an example of Sharpe (2008, chapter 3, case 3)
geSharpe3 <- gemAssetPricing_CUF(UAP,
  uf = function(x) (x - x^2 / 400) %*% wt,
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE)
)
geSharpe3$p
geSharpe3$p[3:4] + 3 * geSharpe3$p[2]

# the same as above
geSharpe3b <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  muf = function(x) (1 - x / 200) * wt
)

geSharpe3b$p
geSharpe3b$p[3:4] + 3 * geSharpe3b$p[2]

## an example of Sharpe (2008, chapter 3, case 4)
geSharpe4 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE),
  UAP,
  muf = function(x) abs((x - 20)^(-1)) * wt,
  maxIteration = 100,
  numberOfPeriods = 300,
  ts = TRUE
)

geSharpe4$p
geSharpe4$p[3:4] + 3 * geSharpe4$p[2]

## an example of Sharpe (2008, chapter 6, case 14)
prob1 <- c(0.15, 0.26, 0.31, 0.28)
wt1 <- prop.table(c(1, 0.96 * prob1))
prob2 <- c(0.08, 0.23, 0.28, 0.41)
wt2 <- prop.table(c(1, 0.96 * prob2))

uf1 <- function(x) CES(alpha = 1, beta = wt1, x = x, es = 1 / 1.5)
uf2 <- function(x) CES(alpha = 1, beta = wt2, x = x, es = 1 / 2.5)
geSharpe14 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  uf = list(uf1,uf2)
)

geSharpe14$D
geSharpe14$p
geSharpe14$p[3:4] + 3 * geSharpe14$p[2]
mu <- marginal_utility(geSharpe14$Payoff, diag(5),uf=list(uf1,uf2))
mu[,1]/mu[1,1]
mu[,2]/mu[1,2]

#### an example of Wang (2006, example 10.1, P146)
geWang <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 2,
    0, 1
  ), 3, 2, TRUE),
  muf = list(
    function(x) 1 / x * c(0.5, 0.25, 0.25),
    function(x) 1 / sqrt(x) * c(0.5, 0.25, 0.25)
  )
)

geWang$p # c(1, (1 + sqrt(17)) / 16)

# the same as above
geWang.b <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 2,
    0, 1
  ), 3, 2, TRUE),
  uf = list(
    function(x) log(x) %*% c(0.5, 0.25, 0.25),
    function(x) 2 * sqrt(x) %*% c(0.5, 0.25, 0.25)
  )
)

geWang.b$p

#### an example of Xu (2018, section 10.4, P151)
wt <- c(1, 0.5, 0.5)
ge <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  uf = list(
    function(x) CRRA(x, gamma = 1, prob = wt)$u,
    function(x) CRRA(x, gamma = 0.5, prob = wt)$u
  )
)

ge$p # c(1, (1 + sqrt(5)) / 4, (1 + sqrt(17)) / 16)

#### an example of incomplete market
ge <- gemAssetPricing_CUF(
  UAP = cbind(c(1, 1), c(2, 1)),
  uf = list(
    function(x) sum(log(x)) / 2,
    function(x) sum(sqrt(x))
  ),
  ratio_adjust_coef = 0.1,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(span = 100),
  maxIteration = 1,
  numberOfPeriods = 2000,
)

ge$p

## the same as above
ge.b <- gemAssetPricing_CUF(
  UAP = cbind(c(1, 1), c(2, 1)),
  muf = list(
    function(x) 1 / x * c(0.5, 0.5),
    function(x) 1 / sqrt(x) * c(0.5, 0.5)
  ),
  ratio_adjust_coef = 0.1,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(span = 100),
  maxIteration = 1,
  numberOfPeriods = 2000,
  ts = TRUE
)

ge.b$p
matplot(ge.b$ts.p, type = "l")

#### an example with outside position.
asset1 <- c(1, 0, 0)
asset2 <- c(0, 1, 1)

# unit (asset) payoff matrix
UAP <- cbind(asset1, asset2)
wt <- c(0.5, 0.25, 0.25) # weights

uf1 <- function(x) prod((x + c(0, 0, 2))^wt)
uf2 <- function(x) prod(x^wt)

ge <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 1,
    0, 2
  ), 2, 2, TRUE),
  UAP = UAP,
  uf = list(uf1, uf2),
  numeraire = 1
)

ge$p
ge$z
uf1(ge$Payoff[,1])
uf2(ge$Payoff[,2])

Compute Asset Market Equilibria with Portfolio Utility Functions for Some Simple Cases

Description

Compute the equilibrium of an asset market by the function sdm2 and by computing marginal utility of assets. The argument of the utility function used in the calculation is the asset vector (i.e. portfolio).

Usage

gemAssetPricing_PUF(S, uf, numeraire = nrow(S), ratio_adjust_coef = 0.1, ...)

Arguments

Value

A general equilibrium containing a value marginal utility matrix (VMU).

References

Danthine, J. P., Donaldson, J. (2005, ISBN: 9780123693808) Intermediate Financial Theory. Elsevier Academic Press.

Sharpe, William F. (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

https://web.stanford.edu/~wfsharpe/apsim/index.html

See Also

gemAssetPricing_CUF.

Examples

#### an example of Danthine and Donaldson (2005, section 8.3).
ge <- gemAssetPricing_PUF(
  S = matrix(c(
    10, 5,
    1, 4,
    2, 6
  ), 3, 2, TRUE),
  uf = function(x) 0.5 * x[1] + 0.9 * (1 / 3 * log(x[2]) + 2 / 3 * log(x[3])),
  maxIteration = 1,
  numberOfPeriods = 500,
  ts = TRUE
)
matplot(ge$ts.p, type = "l")
ge$p

#### an example of Sharpe (2008, chapter 2, case 1)
asset1 <- c(1, 0, 0, 0, 0)
asset2 <- c(0, 1, 1, 1, 1)
asset3 <- c(0, 5, 3, 8, 4) - 3 * asset2
asset4 <- c(0, 3, 5, 4, 8) - 3 * asset2
# the unit asset payoff matrix
UAP <- cbind(asset1, asset2, asset3, asset4)

prob <- c(0.15, 0.25, 0.25, 0.35)
wt <- prop.table(c(1, 0.96 * prob)) # weights

ge <- gemAssetPricing_PUF(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  uf = list(
    function(portfolio) CES(alpha = 1, beta = wt, x = UAP %*% portfolio, es = 1 / 1.5),
    function(portfolio) CES(alpha = 1, beta = wt, x = UAP %*% portfolio, es = 1 / 2.5)
  ),
  maxIteration = 1,
  numberOfPeriods = 1000,
  numeraire = 1,
  ts = TRUE
)
matplot(ge$ts.p, type = "l")
ge$p
ge$p[3:4] + 3 * ge$p[2]

#### a 3-by-2 example of asset pricing with two heterogeneous agents who
## have different beliefs and predict different payoff vectors.
## the predicted payoff vectors of agent 1 on the two assets.
asset1.1 <- c(1, 2, 2, 0)
asset2.1 <- c(2, 2, 0, 2)

## the predicted payoff vectors of agent 2 on the two assets.
asset1.2 <- c(1, 0, 2, 0)
asset2.2 <- c(2, 1, 0, 2)

asset3 <- c(1, 1, 1, 1)

## the unit asset payoff matrix of agent 1.
UAP1 <- cbind(asset1.1, asset2.1, asset3)

## the unit asset payoff matrix of agent 2.
UAP2 <- cbind(asset1.2, asset2.2, asset3)

mp1 <- colMeans(UAP1)
Cov1 <- cov.wt(UAP1, method = "ML")$cov

mp2 <- colMeans(UAP2)
Cov2 <- cov.wt(UAP2, method = "ML")$cov

ge <- gemAssetPricing_PUF(
  S = matrix(c(
    1, 5,
    2, 5,
    3, 5
  ), 3, 2, TRUE),
  uf = list(
    # the utility function of agent 1.
    function(x) AMSDP(x, mp1, Cov1, gamma = 0.2, theta = 2),
    function(x) AMSDP(x, mp2, Cov2) # the utility function of agent 2
  ),
  maxIteration = 1,
  numberOfPeriods = 1000,
  ts = TRUE
)
matplot(ge$ts.p, type = "l")
ge$p
ge$VMU

#### another 3-by-2 example.
asset1.1 <- c(0, 0, 1, 1, 2)
asset2.1 <- c(1, 2, 1, 2, 0)
asset3.1 <- c(1, 1, 1, 1, 1)

asset1.2 <- c(0, 0, 1, 2)
asset2.2 <- c(1, 2, 2, 1)
asset3.2 <- c(1, 1, 1, 1)

## the unit asset payoff matrix of agent 1.
UAP1 <- cbind(asset1.1, asset2.1, asset3.1)

## the unit asset payoff matrix of agent 2.
UAP2 <- cbind(asset1.2, asset2.2, asset3.2)

mp1 <- colMeans(UAP1)
Cov1 <- cov.wt(UAP1, method = "ML")$cov

mp2 <- colMeans(UAP2)
Cov2 <- cov.wt(UAP2, method = "ML")$cov

ge <- gemAssetPricing_PUF(
  S = matrix(c(
    1, 5,
    2, 5,
    3, 5
  ), 3, 2, TRUE),
  uf = list(
    function(x) AMSDP(x, mp1, Cov1), # the utility function of agent 1.
    function(x) AMSDP(x, mp2, Cov2) # the utility function of agent 2.
  ),
  maxIteration = 1,
  numberOfPeriods = 3000,
  ts = TRUE
)

ge$p
ge$D

#### a 5-by-3 example.
set.seed(1)
n <- 5 # the number of asset types
m <- 3 # the number of agents
Supply <- matrix(runif(n * m, 10, 100), n, m)

# the risk aversion coefficients of agents.
gamma <- runif(m, 0.25, 1)

# the predicted mean payoffs, which may be gross return rates, price indices or prices.
PMP <- matrix(runif(n * m, min = 0.8, max = 1.5), n, m)

# the predicted standard deviations of payoffs.
PSD <- matrix(runif(n * m, min = 0.01, max = 0.2), n, m)
PSD[n, ] <- 0

# Suppose the predicted payoff correlation matrices of agents are the same.
Cor <- cor(matrix(runif(2 * n^2), 2 * n, n))
Cor[, n] <- Cor[n, ] <- 0
Cor[n, n] <- 1

# the list of utility functions.
lst.uf <- list()

make.uf <- function(mp, Cov, gamma) {
  force(mp)
  force(Cov)
  force(gamma)
  function(x) {
    AMSDP(x, mp = mp, Cov = Cov, gamma = gamma, theta = 1)
  }
}

for (k in 1:m) {
  sigma <- PSD[, k]
  if (is.matrix(Cor)) {
    Cov <- dg(sigma) %*% Cor %*% dg(sigma)
  } else {
    Cov <- dg(sigma) %*% Cor[[k]] %*% dg(sigma)
  }

  lst.uf[[k]] <- make.uf(mp = PMP[, k], Cov = Cov, gamma = gamma[k])
}

ge <- gemAssetPricing_PUF(
  S = Supply, uf = lst.uf,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(100),
  ts = TRUE,
  tolCond = 1e-04
)

ge$p
round(addmargins(ge$D, 2), 3)
round(addmargins(ge$S, 2), 3)
ge$VMU

#### a 3-by-2 example.
asset1 <- c(1, 0, 0)
asset2 <- c(0, 0, 2)
asset3 <- c(0, 1, 1)

# the unit asset payoff matrix.
UAP <- cbind(asset1, asset2, asset3)
wt <- c(0.5, 0.25, 0.25) # weights

uf <- function(portfolio) {
  payoff <- UAP %*% portfolio
  prod(payoff^wt)
}

ge <- gemAssetPricing_PUF(
  matrix(c(
    1, 1,
    1, 0,
    0, 2
  ), 3, 2, TRUE),
  uf = uf,
  numeraire = 1
)

ge$p
ge$z
ge$A
addmargins(ge$D, 2)
addmargins(UAP %*% ge$D, 2)
ge$VMU

## a price-control stationary state.
pcss <- gemAssetPricing_PUF(
  matrix(c(
    1, 1,
    1, 0,
    0, 2
  ), 3, 2, TRUE),
  uf = uf,
  numeraire = 1,
  pExg = c(1, 2, 1),
  maxIteration = 1,
  numberOfPeriods = 300,
  ts = TRUE
)

matplot(pcss$ts.q, type = "l")
tail(pcss$ts.q, 3)
addmargins(round(pcss$D, 4), 2)
pcss$VMU

#### a 2-by-2 example with outside position.
asset1 <- c(1, 0, 0)
asset2 <- c(0, 1, 1)

# the unit asset payoff matrix
UAP <- cbind(asset1, asset2)
wt <- c(0.5, 0.25, 0.25) # weights

uf1 <- function(portfolio) prod((UAP %*% portfolio + c(0, 0, 2))^wt)
uf2 <- function(portfolio) prod((UAP %*% portfolio)^wt)

ge <- gemAssetPricing_PUF(
  S = matrix(c(
    1, 1,
    0, 2
  ), 2, 2, TRUE),
  uf = list(uf1, uf2),
  numeraire = 1
)

ge$p
ge$z
uf1(ge$D[,1])
uf2(ge$D[,2])

Some Examples of Balanced Growth Paths

Description

Some examples of spot equilibrium paths that converge to balanced growth paths.

Usage

gemBalancedGrowthPath(...)

Arguments

Examples

#### an example with a firm and a laborer
dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 5, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

dstl <- list(dst.firm, dst.consumer)

ge <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  z0 = c(1, 1),
  ts = TRUE,
  policy = policyMarketClearingPrice,
  numberOfPeriods = 40,
  maxIteration = 1,
  GRExg = 0.03
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(growth_rate(ge$ts.z), type = "o", pch = 20)

#### an example with two firms and a laborer
dst.firm.corn <- node_new(
  "corn",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "iron", "lab"
)

dst.firm.iron <- node_new(
  "iron",
  type = "CD", alpha = 5, beta = c(0.5, 0.5),
  "iron", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "corn"
)

ge <- sdm2(
  A = list(dst.firm.corn, dst.firm.iron, dst.consumer),
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer"),
  numeraire = "lab",
  ts = TRUE,
  policy = policyMarketClearingPrice,
  numberOfPeriods = 30,
  maxIteration = 1,
  GRExg = 0.03
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(growth_rate(ge$ts.z), type = "o", pch = 20)

#### another example with two firms and a laborer
dst.manu <- node_new("manu",
                     type = "SCES", es = 1, alpha = 1,
                     beta = c(0.6, 0.4),
                     "manu", "lab"
)

dst.serv <- node_new("serv",
                     type = "SCES", es = 1, alpha = 1,
                     beta = c(0.4, 0.6),
                     "manu", "lab"
)

dst.consumer <- node_new("util",
                         type = "SCES", es = 1, alpha = 1,
                         beta = c(0.4, 0.6),
                         "manu", "serv"
)

dstl <- list(dst.manu, dst.serv, dst.consumer)

S0Exg <- matrix(NA, 3, 3)
S0Exg[3, 3] <- 100

ge <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = S0Exg,
  names.commodity = c("manu", "serv", "lab"),
  names.agent = c("manu", "serv", "consumer"),
  numeraire = c("manu"),
  ts = TRUE,
  policy = list(
    function(time, state) {
      if (time >= 5) {
        state$S[3, 3] <- 100 * 1.03^(time - 4)
      }
      state
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 20,
  maxIteration = 1,
  z0 = c(160, 60, 100),
  p0 = c(1, 1, 1)
)

ge$p
ge$D
ge$S

matplot(ge$ts.z, type = "o", pch = 20)
matplot(growth_rate(ge$ts.z), type = "o", pch = 20)

Some Timeline Equilibrium Models with CESAK Production Function

Description

Some timeline general equilibrium models with CESAK production function.

Usage

gemCESAK_Timeline_2_2(...)

Arguments

Examples

np <- 5
gr <- 0
initial.product.supply <- 20
beta.prod.firm <- 0.35
eis <- 1
Gamma.beta <- 0.97
gr <- 0.05
alphaK <- 1.05

S0Exg <- matrix(NA, 2 * np - 1, np)
S0Exg[(np + 1):(2 * np - 1), np] <- 100 * (1 + gr)^(0:(np - 2))
S0Exg[1, np] <- initial.product.supply

B <- matrix(0, 2 * np - 1, np)
B[2:np, 1:(np - 1)] <- diag(np - 1)

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new("output",
                             type = "CESAK", es = 1,
                             alpha = 0.1, betaK = beta.prod.firm, alphaK = alphaK,
                             paste0("prod", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = c(paste0("prod", 1:np), paste0("lab", 1:(np - 1))),
  names.agent = c(paste0("firm", 1:(np - 1)), "consumer"),
  numeraire = "lab1",
  priceAdjustmentVelocity = 0.03
)

growth_rate(ge$p[1:np])
growth_rate(ge$z[1:np])
ge$D

#### a AK model
np <- 5
initial.product.supply <- 20
eis <- 1
Gamma.beta <- 1
alphaK <- 1.25

S0Exg <- matrix(NA, np, np)
S0Exg[1, np] <- initial.product.supply

B <- matrix(0, np, np)
B[2:np, 1:(np - 1)] <- diag(np - 1)

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new("output",
                             type = "Leontief", a = 1 / alphaK,
                             paste0("prod", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = paste0("prod", 1:np),
  names.agent = c(paste0("firm", 1:(np - 1)), "consumer"),
  numeraire = "prod1",
  priceAdjustmentVelocity = 0.03,
  policy = makePolicyMeanValue(50),
)

growth_rate(ge$p)
growth_rate(ge$z[1:(np - 1)])
ge$D

## Simplify the production function in the model above.
dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new("output",
                             type = "Leontief", a = 1 / alphaK^(k),
                             "prod1"
  )
}

ge2 <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = paste0("prod", 1:np),
  names.agent = c(paste0("firm", 1:(np - 1)), "consumer"),
  numeraire = "prod1",
  priceAdjustmentVelocity = 0.03,
  policy = makePolicyMeanValue(50),
)

growth_rate(ge2$p)
growth_rate(ge2$z[1:(np - 1)])
ge2$D

A Canonical Dynamic Macroeconomic General Equilibrium Model (see Torres, 2016)

Description

This is a function used to calculate the steady state of a canonical dynamic macroeconomic general equilibrium model (see Torres, 2016, Table 2.1 and Table 2.2). The model consists of three commodities (i.e., product, labor, and equity shares) and two agents (i.e., a firm and a consumer), with labor serving as the numeraire. It can also compute the spot equilibrium path based on the steady-state return rate. The spot equilibrium path includes only spot market transactions, with no intertemporal transactions.

Usage

gemCanonicalDynamicMacroeconomic_3_2(
  discount.factor = 0.97,
  depreciation.rate = 0.06,
  beta.prod.firm = 0.35,
  beta.prod.consumer = 0.4,
  policy.supply = NULL,
  policy.technology = NULL,
  policy.price = NULL,
  ...
)

Arguments

Value

A general equilibrium (see sdm2).

References

Torres, Jose L. (2016, ISBN: 9781622730452) Introduction to Dynamic Macroeconomic General Equilibrium Models (Second Edition). Vernon Press.

Li Xiangyang (2018, ISBN: 9787302497745) Dynamic Stochastic General Equilibrium (DSGE) Model: Theory, Methodology, and Dynare Practice. Tsinghua University Press. (In Chinese)

See Also

The spot equilibrium path can be computed with the function policyMarketClearingPrice.

Examples

gemCanonicalDynamicMacroeconomic_3_2()

#### (A) A spot equilibrium path.
ge <- gemCanonicalDynamicMacroeconomic_3_2(
  policy.price = policyMarketClearingPrice,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 100,
  z0 = c(50, 100)
)

par(mfrow = c(1, 2))
matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

#### (B) Technology change in a spot equilibrium path.
policyTechnologyChange <- function(time, A) {
  alpha <- 1.2 # The original value is 1.
  time.win <- c(50, 50)
  discount.factor <- 0.97
  depreciation.rate <- 0.06
  beta.prod.firm  <- 0.35
  return.rate <- 1 / discount.factor - 1

  if (time >= time.win[1] && time <= time.win[2]) {
    A[[1]]$func <- function(p) {
      result <- CD_A(
        alpha, rbind(beta.prod.firm , 1 - beta.prod.firm , 0),
        c(p[1] * (return.rate + depreciation.rate), p[2:3])
      )
      result[3] <- p[1] * result[1] * return.rate / p[3]
      result
    }
  }
}

ge <- gemCanonicalDynamicMacroeconomic_3_2(
  policy.technology = policyTechnologyChange,
  policy.price = policyMarketClearingPrice,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 100,
  z0 = c(50, 100)
)

par(mfrow = c(1, 2))
matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

#### (C) An example on page 46 in Li Xiangyang (2018).
ge <- gemCanonicalDynamicMacroeconomic_3_2(
  discount.factor = 0.99,
  depreciation.rate = 0.025,
  beta.prod.firm  = 0.36,
  beta.prod.consumer = 1
)

A Canonical Dynamic Macroeconomic General Equilibrium Model (see Torres, 2016)

Description

This is a function used to calculate the steady state of a canonical dynamic macroeconomic general equilibrium model (see Torres, 2016, Table 2.1 and Table 2.2). The model consists of four commodities (i.e., product, labor, capital, and equity shares) and three agents (i.e., a production firm, a consumer, and a capital-leasing firm). It can also compute the spot equilibrium path based on the steady-state return rate.

Usage

gemCanonicalDynamicMacroeconomic_4_3(
  discount.factor = 0.97,
  depreciation.rate = 0.06,
  beta.prod.firm = 0.35,
  beta.prod.consumer = 0.4,
  ...
)

Arguments

Value

A general equilibrium (see sdm2)

References

Torres, Jose L. (2016, ISBN: 9781622730452) Introduction to Dynamic Macroeconomic General Equilibrium Models (Second Edition). Vernon Press.

Examples

#### (A) A spot equilibrium path that converges to the steady-state equilibrium.
ge <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = 100,
  policy = policyMarketClearingPrice
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

## population growth: a spot equilibrium path
## that converges to a balanced growth path
ge <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = 100,
  GRExg = 0.01,
  policy = policyMarketClearingPrice
)

matplot((ge$ts.p), type = "l")
matplot((ge$ts.z), type = "l")
matplot(growth_rate(ge$ts.z), type = "l")

#### (B) A disequilibrium path and the steady-state equilibrium.
ge <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = 5000,
  priceAdjustmentVelocity = 0.03,
)

ge$p
ge$z
matplot(ge$ts.z, type = "l")
node_plot(ge$dstl[[3]], param = TRUE)

## a small disturbance to the product supply
ge <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = 4000,
  priceAdjustmentVelocity = 0.03,
  policy = function(time, state) {
    if (time == 1500) {
      state$S[1, 1] <- state$S[1, 1] * 0.999
    }
    state
  }
)

#### (C) Business cycles.
de <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = 1000,
  priceAdjustmentVelocity = 0.15
)

## A tax rate policy is implemented from the 600th period to stabilize the economy.
ge <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = 1500,
  priceAdjustmentVelocity = 0.15,
  policy = Example9.10.policy.tax
)

matplot(ge$ts.z, type = "l")
plot(ge$policy.data, type = "l") # tax rates

#### (D) A spot equilibrium path with a productivity shock.
nPeriod <- 100 # the number of periods of the spot equilibrium path
set.seed(1)
alpha.shock <- rep(1, nPeriod)
alpha.shock[11] <- exp(0.01)
for (t in 12:nPeriod) {
  alpha.shock[t] <- exp(0.95 * log(alpha.shock[t - 1]))
}
plot(alpha.shock)

ge <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = nPeriod,
  p0 = c(1, 1.34312, 0.09093, 0.08865),
  z0 = c(74.47, 61.20, 286.65),
  policy = list(
    function(time, A) {
      A[[1]]$alpha <- alpha.shock[time]
    },
    policyMarketClearingPrice
  )
)

matplot(ge$ts.z[, 1], type = "o", pch = 20)

A Canonical Dynamic Macroeconomic General Equilibrium Model in Sequential Form under the Wage Postpayment Assumption (see Torres, 2016)

Description

A canonical dynamic macroeconomic general equilibrium model in sequential form under the wage postpayment assumption (see Torres, 2016, Table 2.1 and 2.2). In this model, there are two firms and one consumer. Under the wage postpayment assumption, the consumer actually consumes a kind of labor (that is, leisure) and the products produced by this labor at the same time. Firm 1 is a regular production firm. Firm 2 can store labor from one period to the next period for consumption by the consumer.

Usage

gemCanonicalDynamicMacroeconomic_Sequential_WagePostpayment_4_3(
  alpha.firm = 1,
  es.prod.lab.firm = 1,
  beta.prod.firm = 0.35,
  depreciation.rate = 0.06,
  eis = 1,
  Gamma.beta = 0.97,
  es.prod.lab.consumer = 1,
  beta.prod.consumer = 0.4,
  gr = 0,
  ...
)

Arguments

Value

A general equilibrium (see sdm2).

See Also

gemCanonicalDynamicMacroeconomic_Timeline_2_2,
gemCanonicalDynamicMacroeconomic_TimeCircle_2_2,
gemDynamicMacroeconomic_SpotTrading_3_2.

Examples

gemCanonicalDynamicMacroeconomic_Sequential_WagePostpayment_4_3()

####
eis <- 0.8
Gamma.beta <- 0.97
gr <- 0.03
ge <- gemCanonicalDynamicMacroeconomic_Sequential_WagePostpayment_4_3(
  es.prod.lab.firm = 0.8,
  eis = eis, Gamma.beta = Gamma.beta, es.prod.lab.consumer = 0.8,
  gr = gr
)

ge$p
ge$p[1] * (sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr, prepaid = TRUE) + 1)
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
ge$S[1, 1] * (1 + gr)

A Canonical Dynamic Macroeconomic General Equilibrium Model in Time-circle Form (see Torres, 2016)

Description

A canonical dynamic macroeconomic general equilibrium model in time-circle form (see Torres, 2016, Table 2.1 and 2.2).

Usage

gemCanonicalDynamicMacroeconomic_TimeCircle_2_2(
  alpha.firm = rep(1, 3),
  es.prod.lab.firm = 1,
  beta.prod.firm = 0.35,
  depreciation.rate = 0.06,
  eis = 1,
  Gamma.beta = 0.97,
  beta.prod.consumer = 0.4,
  es.prod.lab.consumer = 1,
  gr = 0,
  wage.payment = "post",
  ...
)

Arguments

Value

A general equilibrium (see sdm2).

References

Torres, Jose L. (2016, ISBN: 9781622730452) Introduction to Dynamic Macroeconomic General Equilibrium Models (Second Edition). Vernon Press.

See Also

gemCanonicalDynamicMacroeconomic_Timeline_2_2,
gemDynamicMacroeconomic_SpotTrading_3_2,
gemCanonicalDynamicMacroeconomic_Sequential_WagePostpayment_4_3.

Examples

#### Take the wage postpayment assumption.
ge <- gemCanonicalDynamicMacroeconomic_TimeCircle_2_2()
np <- 3
eis <- 1
Gamma.beta <- 0.97
gr <- 0
ge$p
growth_rate(ge$p[1:np])
1 / (1 + sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr)) - 1
ge$z
growth_rate(ge$z[1:np])
ge$D
ge$S

##  Take the wage postpayment assumption.
eis <- 0.8
Gamma.beta <- 0.97
gr <- 0.03
ge <- gemCanonicalDynamicMacroeconomic_TimeCircle_2_2(
  es.prod.lab.firm = 0.8,
  eis = eis, Gamma.beta = Gamma.beta, es.prod.lab.consumer = 0.8,
  gr = gr
)

ge$p
growth_rate(ge$p[1:np])
1 / (1 + sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr)) - 1
ge$z
growth_rate(ge$z[1:np])
ge$D
ge$S

#### an anticipated technology shock.
## Warning: Running the program below takes about 4 minutes.
# np <- 120
# alpha.firm <- rep(1, np)
# alpha.firm[40] <- 1.05
# ge <- gemCanonicalDynamicMacroeconomic_TimeCircle_2_2(alpha.firm = alpha.firm)

## The steady state product supply is 343.92.
## the (economic) time series of product supply
# plot(ge$z[1:np] / 343.92 - 1, type = "o", pch = 20)
## The steady state product consumption is 57.27.
## the (economic) time series of product consumption
# plot(ge$D[2:np, np + 1] / 57.27 - 1, type = "o", pch = 20)

#### Take the wage prepayment assumption.
ge <- gemCanonicalDynamicMacroeconomic_TimeCircle_2_2(wage.payment = "pre")
np <- 3
eis <- 1
Gamma.beta <- 0.97
gr <- 0
ge$p
growth_rate(ge$p[1:np])
1 / (1 + sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr)) - 1
ge$z
growth_rate(ge$z[1:np])
ge$D
ge$S

##  Take the wage prepayment assumption.
eis <- 0.8
Gamma.beta <- 0.97
gr <- 0.03

ge <- gemCanonicalDynamicMacroeconomic_TimeCircle_2_2(
  es.prod.lab.firm = 0.8,
  eis = eis, es.prod.lab.consumer = 0.8,
  Gamma.beta = Gamma.beta, gr = gr,
  wage.payment = "pre"
)

ge$p
growth_rate(ge$p[1:np])
1 / (1 + sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr)) - 1
ge$z
growth_rate(ge$z[1:np])
ge$D
ge$S

A Canonical Dynamic Macroeconomic General Equilibrium Model in Timeline Form (see Torres, 2016)

Description

A canonical dynamic macroeconomic general equilibrium model in timeline form (see Torres, 2016, Table 2.1 and 2.2). The firm has a CESAK production function.

Usage

gemCanonicalDynamicMacroeconomic_Timeline_2_2(
  alpha.firm = rep(1, 4),
  es.prod.lab.firm = 1,
  beta.prod.firm = 0.35,
  depreciation.rate = 0.06,
  eis = 1,
  Gamma.beta = 0.97,
  beta.prod.consumer = 0.4,
  es.prod.lab.consumer = 1,
  gr = 0,
  initial.product.supply = 200,
  head.tail.adjustment = "both",
  wage.payment = "post",
  beta.consumer = NULL,
  ...
)

Arguments

References

Torres, Jose L. (2016, ISBN: 9781622730452) Introduction to Dynamic Macroeconomic General Equilibrium Models (Second Edition). Vernon Press.

See Also

gemCanonicalDynamicMacroeconomic_TimeCircle_2_2,
gemDynamicMacroeconomic_SpotTrading_3_2,
gemCanonicalDynamicMacroeconomic_Sequential_WagePostpayment_4_3.

Examples

#### Take the wage postpayment assumption.
ge <- gemCanonicalDynamicMacroeconomic_Timeline_2_2()
np <- 5
eis <- 1
Gamma.beta <- 0.97
gr <- 0
ge$p
ge$p[1:(np - 1)] / ge$p[2:np] - 1
ge$p[(np + 1):(2 * np - 2)] / ge$p[(np + 2):(2 * np - 1)] - 1
sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr) # the steady-state equilibrium return rate
ge$z
ge$D
node_plot(ge$dst.consumer, TRUE)

#### Take the wage postpayment assumption.
eis <- 0.8
Gamma.beta <- 0.97
gr <- 0.03
ge <- gemCanonicalDynamicMacroeconomic_Timeline_2_2(
  es.prod.lab.firm = 0.8,
  eis = eis, Gamma.beta = Gamma.beta, es.prod.lab.consumer = 0.8,
  gr = gr
)

np <- 5
ge$p
growth_rate(ge$p[1:np])
1 / (1 + sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr)) - 1
ge$z
growth_rate(ge$z[1:(np - 1)])
ge$D
ge$S

##### a fully anticipated technology shock.
## Warning: Running the program below takes several minutes.
# np <- 120
# alpha.firm <- rep(1, np - 1)
# alpha.firm[40] <- 1.05
# ge <- gemCanonicalDynamicMacroeconomic_Timeline_2_2(alpha.firm = alpha.firm)
#
## The steady state product supply is 343.92.
## the (economic) time series of product supply.
# plot(ge$z[1:(np - 1)] / 343.92 - 1, type = "o", pch = 20)
## The steady state product consumption is 57.27.
## the (economic) time series of product consumption.
# plot(ge$D[2:(np - 1), np] / 57.27 - 1, type = "o", pch = 20)
# plot(growth_rate(ge$p[1:(np)]), type = "o", pch = 20)
# plot(growth_rate(ge$p[(np + 1):(2 * np)]), type = "o", pch = 20)
#
##### an unanticipated technology shock.
# np <- 50
# alpha.firm <- rep(1, np - 1)
# alpha.firm[1] <- 1.05
# ge <- gemCanonicalDynamicMacroeconomic_Timeline_2_2(
#   alpha.firm = alpha.firm,
#   initial.product.supply = 286.6341, # the steady state value
#   head.tail.adjustment = "tail"
# )
#
## The steady state product supply is 343.92.
## the (economic) time series of product supply.
# plot(ge$z[1:(np - 1)] / 343.92 - 1, type = "o", pch = 20)
## The steady state product consumption is 57.27.
## the (economic) time series of product consumption.
# plot(ge$D[2:(np - 1), np] / 57.27 - 1, type = "o", pch = 20)
# plot(growth_rate(ge$p[1:(np)]), type = "o", pch = 20)
# plot(growth_rate(ge$p[(np + 1):(2 * np)]), type = "o", pch = 20)
#
### a technology shock anticipated several periods in advance.
# np <- 50
# alpha.firm <- rep(1, np - 1)
# alpha.firm[5] <- 1.05
# ge5 <- gemCanonicalDynamicMacroeconomic_Timeline_2_2(
#   alpha.firm = alpha.firm,
#   initial.product.supply = 286.6341, # the steady state value
#   head.tail.adjustment = "tail"
# )
#
## The steady state product supply is 343.92.
## the (economic) time series of product supply
# plot(ge5$z[1:(np - 1)] / 343.92 - 1, type = "o", pch = 20)
## The steady state product consumption is 57.27.
## the (economic) time series of product consumption
# plot(ge5$D[2:(np - 1), np] / 57.27 - 1, type = "o", pch = 20)
# plot(growth_rate(ge5$p[1:(np)]), type = "o", pch = 20)
# plot(growth_rate(ge5$p[(np + 1):(2 * np)]), type = "o", pch = 20)
#
# alpha.firm <- rep(1, np - 1)
# alpha.firm[10] <- 1.05
# ge10 <- gemCanonicalDynamicMacroeconomic_Timeline_2_2(
#   alpha.firm = alpha.firm,
#   initial.product.supply = 286.6341, # the steady state value
#   head.tail.adjustment = "tail"
# )
# plot(ge$z[1:(np - 1)] / 343.92 - 1, type = "o", pch = 20, ylim = c(-0.005, 0.017))
# lines(ge5$z[1:(np - 1)] / 343.92 - 1, type = "o", pch = 21)
# lines(ge10$z[1:(np - 1)] / 343.92 - 1, type = "o", pch = 22)

##### an unanticipated technology shock.
## Warning: Running the program below takes several minutes.
# np <- 100
# alpha.firm <- exp(0.01)
# for (t in 2:(np - 1)) {
#   alpha.firm[t] <- exp(0.9 * log(alpha.firm[t - 1]))
# }
# plot(alpha.firm)
#
# ge <- gemCanonicalDynamicMacroeconomic_Timeline_2_2(
#   alpha.firm = alpha.firm,
#   initial.product.supply = 286.6341, # the steady state value
#   head.tail.adjustment = "tail"
# )
#
## The steady state product supply is 343.92.
## the (economic) time series of product supply
# plot(ge$z[1:(np - 1)] / 343.92 - 1, type = "o", pch = 20)
## The steady state product consumption is 57.27.
## the (economic) time series of product consumption
# plot(ge$D[2:(np - 1), np] / 57.27 - 1, type = "o", pch = 20)
# plot(growth_rate(ge$p[1:(np)]), type = "o", pch = 20)
# plot(growth_rate(ge$p[(np + 1):(2 * np)]), type = "o", pch = 20)

#### Take the wage prepayment assumption.
ge <- gemCanonicalDynamicMacroeconomic_Timeline_2_2(wage.payment = "pre")
np <- 5
eis <- 1
Gamma.beta <- 0.97
gr <- 0
ge$p
ge$p[1:(np - 1)] / ge$p[2:np] - 1
ge$p[(np + 1):(2 * np - 2)] / ge$p[(np + 2):(2 * np - 1)] - 1
sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr) # the steady-state equilibrium return rate
ge$z
ge$D
node_plot(ge$dst.consumer, TRUE)

#### Take the wage prepayment assumption.
np <- 5
eis <- 0.8
Gamma.beta <- 0.97
gr <- 0.03
ge <- gemCanonicalDynamicMacroeconomic_Timeline_2_2(
  es.prod.lab.firm = 0.8,
  eis = eis, Gamma.beta = Gamma.beta, es.prod.lab.consumer = 0.8,
  gr = gr,
  wage.payment = "pre"
)

ge$p
growth_rate(ge$p[1:np])
1 / (1 + sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr)) - 1
ge$z
growth_rate(ge$z[1:(np - 1)])
ge$D
ge$S

Some Examples of Spot Equilibrium Paths with Capital Accumulation

Description

Some examples of spot equilibrium paths with capital accumulation. The economy contains a production firm, a capital-leasing firm and a consumer.

Usage

gemCapitalAccumulation(...)

Arguments

See Also

gemPersistentTechnologicalProgress

Examples

#### a 3-by-3 example
dst.firm1 <- node_new(
  "prod",
  type = "CD", alpha = 2, beta = c(0.5, 0.5),
  "cap", "cc1"
)
node_set(dst.firm1, "cc1",
  type = "Leontief", a = 1,
  "lab"
)

dst.consumer <- dst.firm2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

dstl <- list(dst.firm1, dst.consumer, dst.firm2)

B <- matrix(c(
  1, 0, 0.5,
  0, 0, 1,
  0, 0, 0
), 3, 3, TRUE)

S0Exg <- matrix(c(
  NA, NA, NA,
  NA, NA, NA,
  NA, 100, NA
), 3, 3, TRUE)

ge <- sdm2(
  A = dstl,
  B = B,
  S0Exg = S0Exg,
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm1", "laborer", "firm2"),
  numeraire = "prod",
  z0 = c(100, 0, 50),
  policy = policyMarketClearingPrice,
  maxIteration = 1,
  numberOfPeriods = 30,
  ts = TRUE
)

matplot((ge$ts.z), type = "o", pch = 20)
matplot((ge$ts.p), type = "o", pch = 20)

## a MCP with labor supply change
ge <- sdm2(
  A = dstl,
  B = B,
  S0Exg = S0Exg,
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm1", "laborer", "firm2"),
  numeraire = "prod",
  z0 = c(400, 200, 400),
  policy = list(
    function(time, state) {
      if (time >= 5) state$S[3, 2] <- 150
      state
    },
    policyMarketClearingPrice
  ),
  maxIteration = 1,
  numberOfPeriods = 30,
  ts = TRUE
)

matplot((ge$ts.z), type = "o", pch = 20)
matplot((ge$ts.p), type = "o", pch = 20)

## a MCP with transitory technological progress
ge <- sdm2(
  A = dstl,
  B = B,
  S0Exg = S0Exg,
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm1", "laborer", "firm2"),
  numeraire = "prod",
  z0 = c(400, 200, 400),
  policy = list(
    function(time, A) {
      if (time == 5) {
        A[[1]]$alpha <- 3
      } else {
        A[[1]]$alpha <- 2
      }
    },
    policyMarketClearingPrice
  ),
  maxIteration = 1,
  numberOfPeriods = 30,
  ts = TRUE
)

matplot((ge$ts.z), type = "o", pch = 20)
matplot((ge$ts.p), type = "o", pch = 20)

## a MCP with permanent technological progress
ge <- sdm2(
  A = dstl,
  B = B,
  S0Exg = S0Exg,
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm1", "laborer", "firm2"),
  numeraire = "prod",
  z0 = c(400, 200, 400),
  policy = list(
    function(time, A) {
      if (time >= 5) {
        A[[1]]$alpha <- 3
      } else {
        A[[1]]$alpha <- 2
      }
    },
    policyMarketClearingPrice
  ),
  maxIteration = 1,
  numberOfPeriods = 30,
  ts = TRUE
)

matplot((ge$ts.z), type = "o", pch = 20)
matplot((ge$ts.p), type = "o", pch = 20)

#### A 4-by-4 example wherein the capital goods
#### have a useful life of two periods.
ge <- sdm2(
  A = function(state) {
    a.firm1 <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5, 0), state$p)
    a.consumer <- c(1, 0, 0, 0)
    a.firm2 <- c(1, 0, 0, 0)
    a.firm3 <- c(0, 0, 0, 1)
    cbind(a.firm1, a.consumer, a.firm2, a.firm3)
  },
  B = matrix(c(
    1, 0, 0, 0,
    0, 0, 1, 1,
    0, 0, 0, 0,
    0, 0, 1, 0
  ), 4, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, 100, NA, NA,
    NA, NA, NA, NA
  ), 4, 4, TRUE),
  names.commodity = c("prod", "cap", "lab", "prod.used"),
  names.agent = c("firm1", "consumer", "firm2", "firm3"),
  numeraire = "prod",
  policy = policyMarketClearingPrice,
  z0 = c(100, 100, 100, 100),
  ts = TRUE,
  numberOfPeriods = 30,
  maxIteration = 1
)

matplot(ge$ts.z, type = "o", pch = 20)

Coffee Problem: Some Examples of Equilibrium and Disequilibrium Pure Exchange Economies

Description

Some examples of equilibrium and disequilibrium pure exchange economies.

Usage

gemCoffeeProblem_3_3(...)

Arguments

References

Bapat, R. B., Raghavan, T. E. S. (1997, ISBN: 9780521571678) Nonnegative Matrices and Applications. Cambridge University Press.

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

Examples

#### the equilibrium coffee problem (Bapat, Raghavan, 1997, example 7.1; Li, 2019, example 8.1)
ge <- sdm2(
  A = matrix(c(
    0.05, 0.05, 0.1,
    0.1, 0, 0.1,
    0, 0.15, 0.05
  ), 3, 3, TRUE),
  B = matrix(0, 3, 3),
  S0Exg = diag(3),
  names.commodity = c("coffee powder", "milk", "sugar"),
  names.agent = c("consumer1", "consumer2", "consumer3"),
  numeraire = "sugar"
)

ge$p

#### the disequilibrium coffee problem with exogenous prices (Li, 2019, example 8.3).
## Computing the price-control stationary state.
pcss <- sdm2(
  A = matrix(c(
    0.05, 0.05, 0.1,
    0.1, 0, 0.1,
    0, 0.15, 0.05
  ), 3, 3, TRUE),
  B = matrix(0, 3, 3),
  S0Exg = diag(3),
  names.commodity = c("coffee powder", "milk", "sugar"),
  names.agent = c("consumer1", "consumer2", "consumer3"),
  pExg = c(1, 1, 1),
  maxIteration = 1,
  numberOfPeriods = 50,
  ts = TRUE
)

pcss$z
addmargins(pcss$D, 2)
addmargins(pcss$S, 2)

matplot(pcss$ts.z, type = "o", pch = 20)
matplot(pcss$ts.q, type = "o", pch = 20)

Constant Growth Paths with Technology Progress

Description

This is an example of a spot equilibrium path that converges to a constant growth path. In a constant growth path, the supply of each commodity grows at a constant rate. The balanced growth path is a special case of the constant growth path.

Usage

gemConstantGrowthPath_TechnologyProgress_3_3(...)

Arguments

Examples

dst.firm1 <- node_new(
  "output",
  type = "CD", alpha = 1, beta = c(0.35, 0.65),
  "prod1", "lab"
)

dst.firm2 <- node_new(
  "output",
  type = "CD", alpha = 1, beta = c(0.4, 0.6),
  "prod1", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod2"
)

ge <- sdm2(
  A = list(dst.firm1, dst.firm2, dst.consumer),
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, NA, 1
  ), 3, 3, TRUE),
  names.commodity = c("prod1", "prod2", "lab"),
  names.agent = c("firm1", "firm2", "consumer"),
  numeraire = "lab",
  z0 = c(0.2, 0.2, 1),
  ts = TRUE,
  policy = list(
    function(time, A, state) {
      A[[1]]$alpha <- exp(time * 0.01)
      A[[2]]$alpha <- exp(time * 0.01)
      state$S[3, 3] <- exp(time * 0.01)
      state
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 20,
  maxIteration = 1
)

matplot(ge$ts.z, type = "l")
matplot(log(ge$ts.z[, 1:2]), type = "l")
matplot(growth_rate(ge$ts.z[, 1:2], log = TRUE), type = "o", pch = 20)
matplot(growth_rate(ge$ts.p[, 1:2], log = TRUE), type = "o", pch = 20)

A Model with a Displaced CES Utility Function

Description

A model with a displaced CES utility function (Zhang, 2008, page 134; Li, 2019, example 3.12, page 130).

Usage

gemDCES_5_3(...)

Arguments

References

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

Zhang Jinshui (2008, ISBN: 9787040224818) Mathematical Economics. Beijing: Higher Education Press. (In Chinese)

Examples

ge <- sdm2(
  A = function(state) {
    a.firm1 <- CD_A(alpha = 1, Beta = c(0, 0, 0.5, 0.5, 0), state$p)
    a.firm2 <- CD_A(alpha = 2, Beta = c(0, 0, 0.5, 0, 0.5), state$p)
    a.consumer <- DCES_demand(
      es = 1,
      beta = c(1 / 3, 1 / 3, 1 / 3, 0, 0),
      xi = c(0, 0, 0.4, 0, 0),
      w = state$w[3] / 10^4,
      p = state$p
    )
    cbind(a.firm1, a.firm2, a.consumer)
  },
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 5, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, NA, 10000,
    NA, NA, 1,
    NA, NA, 1
  ), 5, 3, TRUE),
  names.commodity = c("prod1", "prod2", "lab", "land1", "land2"),
  names.agent = c("firm1", "firm2", "consumer"),
  numeraire = "lab"
)

ge$p
ge$z

General Equilibrium Models and Linear Programming Problems (see Winston, 2003)

Description

Some examples illustrating the relationship between general equilibrium problems and (dual) linear programming problems. Some linear programming problems can be transformed into general equilibrium problems and vice versa.

Usage

gemDualLinearProgramming(...)

Arguments

Details

These examples are similar and let us explain briefly the first example (Winston, 2003).
The Dakota Furniture Company manufactures desks, tables, and chairs. The manufacture of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry. The amount of each resource needed to make each type of furniture is as follows:
desk: (8, 4, 2)
table: (6, 2, 1.5)
chair: (1, 1.5, 0.5)
Currently, 48 board feet of lumber, 20 finishing hours, and 8 carpentry hours are available. A desk sells for $60, a table for $30, and a chair for $20. Because the available resources have already been purchased, Dakota wants to maximize total revenue. This problem can be solved by the linear programming method.
Now let us regard the problem above as a general equilibrium problem. The Dakota Furniture Company can be regarded as a consumer who obtains 1 unit of utility from 1 dollar and owns lumber and two types of skilled labor. There are four commodities (i.e. dollar, lumber and two types of skilled labor) and four agents (i.e. a desk producer, a table producer, a chair producer and the consumer Dakota) in this problem. We need to compute the equilibrium activity levels and the equilibrium prices, which are also the solutions of the (dual) linear programming problems (i.e. the utility-maximizing problem of the consumer and the cost-minimizing problem of the producers).

Value

A general equilibrium.

Note

Below is a simplified form of the von Neumann general equilibrium model (von Neumann, 1945; Kemeny, Morgenstern, Thompson, 1956):

\mathbf p^{T}\mathbf{A} \geq \rho \mathbf p^{T}\mathbf{B}

\mathbf{Az} \leq \rho \mathbf {Bz}

The above model can be extended to the following general equilibrium model, namely the structural equilibrium model (Li, 2019, section 3.4):

\mathbf p^{T}\mathbf{A(p,u,z)} \geq \rho \mathbf p^{T}\mathbf{B(p,u,z)}

\mathbf{A(p,u,z)z} \leq \rho \mathbf {B(p,u,z)z}

We explain the structural equilibrium model as follows:
(i) The vectors \mathbf p and \mathbf z reflect the prices of various commodities and the activity levels of various economic agents, respectively.
(ii) The vector \mathbf u reflects the utility levels of various consumers. In this model, the matrices \mathbf A and \mathbf B are functions of prices, utilities, and activity levels.
(iii) When describing a static general equilibrium and a steady-state equilibrium without intertemporal decisions, the structural equilibrium model usually does not explicitly include time, while when describing an intertemporal general equilibrium, variables such as prices and activity levels explicitly include time, that is, they are labeled with time.
(iv) In a time-independent model, \rho is the discount factor \frac{1}{1+\gamma} corresponding to the steady-state growth rate \gamma. In a time-dependent model, \rho is usually equal to 1.
(v) The unit demand matrix \mathbf{A(p,u,z)}, the unit supply matrix \mathbf {B(p,u,z)} and the activity level vector \mathbf z in the structural equilibrium model are different from the input coefficient matrix A, the output coefficient matrix B and the purchase level vector z in the structural dynamic model. The input coefficient matrix A is equivalent to the unit demand matrix with utility levels equal to 1. The output coefficient matrix B, unlike the unit supply matrix, does not contain the exogenous supplies. In the structural equilibrium model, the elements corresponding to consumers in \mathbf z usually reflect the number of consumers, while in the structural dynamic model, they usually reflect the utility levels.

Now consider the following linear programming problem:

\max \quad \mathbf b^T\mathbf z \quad \text{s.t.} \quad \mathbf{Az\le e},\quad \mathbf{z \ge 0}

The dual linear programming problem is

\min \quad \mathbf p^T\mathbf e \quad \text{s.t.} \quad \mathbf p^T \mathbf{A\ge b},\quad \mathbf{p \ge 0}

In the example of Winston (2003), we have \mathbf e=(48,20,8)^T, \mathbf b=(60,30,20)^T and

\mathbf A=\left[\begin{matrix} 8 &6& 1\\ 4& 2& 1.5 \\ 2& 1.5& 0.5\\ \end{matrix}\right]

The corresponding structural equilibrium model is

\mathbf p^{T}\mathbf A(u) \geq \mathbf p^{T}\mathbf B

\mathbf A(u) \mathbf z \leq \mathbf {Bz}

wherein \mathbf p=(1,p_2,p_3,p_4)^T, \mathbf z=(z_1,z_2,z_3,1)^T,

\mathbf A(u)=\left[\begin{matrix} 0& 0& 0& u\\ 8 &6& 1&0 \\ 4& 2& 1.5&0 \\ 2& 1.5& 0.5&0\\ \end{matrix}\right]

and

\mathbf B=\left[\begin{matrix} 60& 30& 20& 0\\ 0 & 0& 0&48 \\ 0 & 0& 0&20 \\ 0 & 0& 0&8\\ \end{matrix}\right]

The following results are obtained by solving the above structural equilibrium model:

\mathbf p^*=(1, 0, 10, 10)^T, \quad \mathbf z^*=(2, 0, 8, 1)^T, \quad u^*=280

References

Kemeny, J. G., O. Morgenstern and G. L. Thompson (1956) A Generalization of the von Neumann Model of an Expanding Economy, Econometrica, 24, pp. 115-135.

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

von Neumann, J. (1945) A Model of General Economic Equilibrium. The Review of Economic Studies, 13. pp. 1-9.

Winston, Wayne L. (2003, ISBN: 9780534380588) Operations Research: Applications and Algorithms. Cengage Learning.

Examples

#### the Dakota example of Winston (2003, section 6.3, 6.6 and 6.8)
A <- matrix(c(
  0, 0, 0, 1,
  8, 6, 1, 0,
  4, 2, 1.5, 0,
  2, 1.5, 0.5, 0
), 4, 4, TRUE)
B <- matrix(c(
  60, 30, 20, 0,
  0, 0, 0, 0,
  0, 0, 0, 0,
  0, 0, 0, 0
), 4, 4, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 4)
  S0Exg[2:4, 4] <- c(48, 20, 8)
  S0Exg
}

## Compute the equilibrium by the function CGE::sdm.
ge <- CGE::sdm(A = A, B = B, S0Exg = S0Exg)

ge$p / ge$p[1]
ge$z

## Compute the equilibrium by the function sdm2.
## The function policyMeanValue is used to accelerate convergence.
ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  names.commodity = c("dollar", "lumber", "lab1", "lab2"),
  names.agent = c("desk producer", "table producer", "chair producer", "consumer"),
  numeraire = "dollar"
)

ge$z
ge$p

#### an example at http://web.mit.edu/15.053/www/AMP-Chapter-04.pdf.
A <- matrix(c(
  0, 0, 0, 1,
  0.5, 2, 1, 0,
  1, 2, 4, 0
), 3, 4, TRUE)
B <- matrix(c(
  6, 14, 13, 0,
  0, 0, 0, 0,
  0, 0, 0, 0
), 3, 4, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 3, 4)
  S0Exg[2:3, 4] <- c(24, 60)
  S0Exg
}

ge <- CGE::sdm(
  A = A, B = B, S0Exg = S0Exg
)

ge$z
ge$p / ge$p[1]

#### an example at https://web.stanford.edu/~ashishg/msande111/notes/chapter4.pdf.
A <- matrix(c(
  0, 0, 1,
  4.44, 0, 0,
  0, 6.67, 0,
  4, 2.86, 0,
  3, 6, 0
), 5, 3, TRUE)
B <- matrix(c(
  3, 2.5, 0,
  0, 0, 0,
  0, 0, 0,
  0, 0, 0,
  0, 0, 0
), 5, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 5, 3)
  S0Exg[2:5, 3] <- 100
  S0Exg
}

ge <- CGE::sdm(
  A = A, B = B, S0Exg = S0Exg
)

ge$z
ge$p / ge$p[1]

#### an example at https://utw11041.utweb.utexas.edu/ORMM/supplements/methods/lpmethod/S3_dual.pdf.
A <- matrix(c(
  0, 0, 1,
  0, 1, 0,
  1, 3, 0,
  1, 0, 0
), 4, 3, TRUE)
B <- matrix(c(
  2, 3, 0,
  1, 0, 0,
  0, 0, 0,
  0, 0, 0
), 4, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 3)
  S0Exg[2:4, 3] <- c(5, 35, 20)
  S0Exg
}

ge <- CGE::sdm(
  A = A, B = B, S0Exg = S0Exg
)

ge$z
ge$p / ge$p[1]

#### the Giapetto example of Winston (2003, section 3.1)
A <- matrix(c(
  0, 0, 1,
  2, 1, 0,
  1, 1, 0,
  1, 0, 0
), 4, 3, TRUE)
B <- matrix(c(
  27 - 10 - 14, 21 - 9 - 10, 0,
  0, 0, 0,
  0, 0, 0,
  0, 0, 0
), 4, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 3)
  S0Exg[2:4, 3] <- c(100, 80, 40)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  numeraire = 1
)

ge$z
ge$p

#### the Dorian example (a minimization problem) of Winston (2003, section 3.2)
A <- matrix(c(
  0, 0, 1,
  7, 2, 0,
  2, 12, 0
), 3, 3, TRUE)
B <- matrix(c(
  28, 24, 0,
  0, 0, 0,
  0, 0, 0
), 3, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 3, 3)
  S0Exg[2:3, 3] <- c(50, 100)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  numeraire = 1
)

ge$p
ge$z

#### the diet example (a minimization problem) of Winston (2003, section 3.4)
A <- matrix(c(
  0, 0, 0, 0, 1,
  400, 3, 2, 2, 0,
  200, 2, 2, 4, 0,
  150, 0, 4, 1, 0,
  500, 0, 4, 5, 0
), 5, 5, TRUE)
B <- matrix(c(
  500, 6, 10, 8, 0,
  0, 0, 0, 0, 0,
  0, 0, 0, 0, 0,
  0, 0, 0, 0, 0,
  0, 0, 0, 0, 0
), 5, 5, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 5, 5)
  S0Exg[2:5, 5] <- c(50, 20, 30, 80)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  numeraire = 1
)

ge$p
ge$z

#### An example of Elizabeth Stapel (Linear Programming: Introduction. Purplemath.
## Available from https://www.purplemath.com/modules/linprog.htm):
## Find the maximal value of 3x + 4y subject to the following constraints:
## x + 2y <= 14, 3x - y >= 0, x - y <= 2, x >= 0, y >= 0

A <- matrix(c(
  0, 0, 1,
  1, 2, 0,
  0, 1, 0,
  1, 0, 0
), 4, 3, TRUE)
B <- matrix(c(
  3, 4, 0,
  0, 0, 0,
  3, 0, 0,
  0, 1, 0
), 4, 3, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 3)
  S0Exg[2:4, 3] <- c(14, 0, 2)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  policy = policyMeanValue,
  priceAdjustmentVelocity = 0.03,
  numeraire = 1
)

ge$z
ge$p

A Dynamic Macroeconomic General Equilibrium Model with Spot Trading

Description

A dynamic macroeconomic general equilibrium model in sequential form.

Usage

gemDynamicMacroeconomic_SpotTrading_3_2(
  alpha.firm = 1,
  es.prod.lab.firm = 1,
  beta.prod.firm = 0.35,
  depreciation.rate = 0.06,
  eis = 1,
  Gamma.beta = 0.97,
  es.prod.lab.consumer = 1,
  beta.prod.consumer = 0.4,
  gr = 0,
  wage.payment = "post",
  ...
)

Arguments

Value

A general equilibrium (see sdm2).

Note

In the timeline model and the time-circle model, we refer to the labor provided in period t as labor t, and the product produced using labor t as product t+1. When the consumer's period utility function simultaneously includes both labor (or leisure) and product, we can adopt one of two assumptions: either we assume that the period utility function of the consumer in period t includes labor t and product t, or we assume it includes labor t and product t+1. These two assumptions are referred to as the wage prepayment assumption and the wage postpayment assumption, respectively.

See Also

gemCanonicalDynamicMacroeconomic_Timeline_2_2,
gemCanonicalDynamicMacroeconomic_TimeCircle_2_2,
gemCanonicalDynamicMacroeconomic_Sequential_WagePostpayment_4_3.

Examples

#### Take the wage postpayment assumption.
ge <- gemDynamicMacroeconomic_SpotTrading_3_2()
ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

#### Take the wage prepayment assumption.
ge <- gemDynamicMacroeconomic_SpotTrading_3_2(wage.payment  = "pre")
ge$p
ge$z
ge$D
ge$S

#### Take the wage prepayment assumption.
ge <- gemDynamicMacroeconomic_SpotTrading_3_2(
  es.prod.lab.firm = 0.8,
  eis = 0.8, es.prod.lab.consumer = 0.8, gr = 0.03,
  wage.payment  = "pre"
)
ge$p
ge$z
ge$D
ge$S

#### an example of steady-state equilibrium at
# http://gecon.r-forge.r-project.org/models/rbc.pdf
ge <- gemDynamicMacroeconomic_SpotTrading_3_2(
  beta.prod.firm = 0.36,
  depreciation.rate = 0.025,
  Gamma.beta = 0.99,
  eis = 0.5,
  beta.prod.consumer = 0.3,
)

ge$p / ge$p[1]
addmargins(ge$D, 2)
addmargins(ge$S, 2)

A General Equilibrium Model with Equity Shares

Description

A general equilibrium model with equity shares and dividend.

Usage

gemEquityShare_3_3(...)

Arguments

Examples

dst.firm <- node_new("output",
                     type = "FIN",
                     rate = c(1, dividend.rate = 0.25),
                     "cc1", "equity.share"
)
node_set(dst.firm, "cc1",
         type = "CD",
         alpha = 2, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.laborer <- node_new("util",
                        type = "Leontief",  a = 1,
                        "prod"
)

dst.shareholder <- Clone(dst.laborer)

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.shareholder),
  B = diag(c(1, 0, 0)),
  S0Exg = {
    S0Exg <- matrix(NA, 3, 3)
    S0Exg[2, 2] <- S0Exg[3, 3] <- 100
    S0Exg
  },
  names.commodity = c("prod", "lab", "equity.share"),
  names.agent = c("firm", "laborer", "shareholder"),
  numeraire = "prod"
)

ge$p # The third component is the dividend per unit of share.
ge$DV
ge$SV

## Set the growth rate to 0.03.
ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.shareholder),
  B = diag(c(1, 0, 0)),
  S0Exg = {
    S0Exg <- matrix(NA, 3, 3)
    S0Exg[2, 2] <- S0Exg[3, 3] <- 100
    S0Exg
  },
  names.commodity = c("prod", "lab", "equity.share"),
  names.agent = c("firm", "laborer", "shareholder"),
  numeraire = "prod",
  GRExg = 0.03
)

ge$z
ge$p

#### an equivalent intertemporal model.
gr <- 0.03 # the growth rate of the labor supply
Gamma.beta <- 0.8 # the subjective discount factor
np <- 5 # the number of economic periods
y1 <- 100 # the initial product supply

n <- 2 * np - 1 # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)))
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100 * (1 + gr)^(0:(np - 2))
S0Exg["prod1", "consumer"] <- y1

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD",
    alpha = 2, beta = c(0.5, 0.5),
    paste0("prod", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = 1,
  alpha = 1,  beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  policy = makePolicyHeadTailAdjustment(gr = gr, np = np)
)

ge$z
growth_rate(ge$z[1:4])
ge$p[6:9] / ge$p[1:4]
addmargins(ge$D, 2)
addmargins(ge$S, 2)

A General Equilibrium Model with Equity Shares and Bond

Description

A general equilibrium model with equity shares and bond.

Usage

gemEquityShare_Bond_4_4(...)

Arguments

Examples

dst.firm <- node_new("output",
                     type = "FIN",
                     rate = c(1, dividend.rate = 0.15, bond.yield.rate = 0.1),
                     "cc1", "equity.share", "bond"
)
node_set(dst.firm, "cc1",
         type = "CD",
         alpha = 2, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.laborer <- dst.shareholder <- dst.bondholder <-
  node_new("util",
           type = "Leontief",  a = 1,
           "prod"
  )

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.shareholder, dst.bondholder),
  B = diag(c(1, 0, 0, 0)),
  S0Exg = {
    S0Exg <- matrix(NA, 4, 4)
    S0Exg[2, 2] <- S0Exg[3, 3] <-
      S0Exg[4, 4] <- 100
    S0Exg
  },
  names.commodity = c("prod", "lab", "equity.share", "bond"),
  names.agent = c("firm", "laborer", "shareholder", "bondholder"),
  numeraire = "prod"
)

ge$p
ge$DV
ge$SV

Some Examples with Exogenous Price (Price Control)

Description

Some examples with exogenous price (i.e. price control, price regulation). When a price control policy is imposed in a structural dynamic model, the economy may converge to a steady state where the market does not clear.

Usage

gemExogenousPrice(...)

Arguments

References

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

See Also

gemExogenousPrice_EndogenousLaborSupply_3_3

Examples

dst.firm <- node_new("output",
  type = "CD", alpha = 5,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new("utility",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

f <- function(pExg = NULL, policy = NULL) {
  pcss <- sdm2(
    A = list(dst.firm, dst.consumer),
    B = diag(c(1, 0)),
    S0Exg = {
      S0Exg <- matrix(NA, 2, 2)
      S0Exg[2, 2] <- 100
      S0Exg
    },
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "consumer"),
    numeraire = "lab",
    maxIteration = 1,
    numberOfPeriods = 100,
    p0 = c(0.16, 1),
    ts = TRUE,
    pExg = pExg,
    policy = policy
  )

  print(pcss$p)
  print(pcss$z)
  par(mfrow = c(1, 2))
  matplot(pcss$ts.p, type = "l")
  matplot(pcss$ts.z, type = "l")
  invisible(pcss)
}

## No price control policy.
f()

## Set the market prices to the steady-state equilibrium prices from the beginning.
## The labor market keeps oversupplied.
result <- f(pExg = c(0.16, 1))
matplot(result$ts.q, type = "l") # sale rates

## the same as above
f(policy = function(state) {
  state$p <- c(0.16, 1)
  state
})

## The price control policy is implemented from the 10th period.
f(policy = function(time, state) {
  if (time >= 10) state$p <- c(0.16, 1)
  state
})

## The price control policy is implemented from the 30th period.
f(policy = function(time, state) {
  if (time >= 30) state$p <- c(0.16, 1)
  state
})

## price ceil
f(policy = function(time, state) {
  if (time >= 30) {
    state$p <- state$p / state$p[2]
    if (state$p[1] > 0.15) state$p[1] <- 0.15
  }
  state
})

##
pcss <- f(policy = function(time, state) {
  if (time >= 30) state$p <- c(0.17, 1)
  state
})

tail(pcss$ts.q)

#### another 2-by-2 example.
f <- function(GRExg = 0, pExg = c(2, 1)) {
  pcss <- sdm(
    A = matrix(c(
      0, 1,
      1, 0
    ), 2, 2, TRUE),
    B = matrix(c(
      1, 0,
      0, 0
    ), 2, 2, TRUE),
    S0Exg = matrix(c(
      NA, NA,
      NA, 100
    ), 2, 2, TRUE),
    GRExg = GRExg,
    pExg = pExg,
    maxIteration = 1,
    numberOfPeriods = 300,
    depreciationCoef = 0,
    z0 = c(100, 0),
    ts = TRUE
  )
  matplot(pcss$ts.z, type = "l")
  print("pcss$z:")
  pcss$z
  print("tail(pcss$ts.q, 3)")
  print(tail(pcss$ts.q, 3))
  invisible(pcss)
}

f()
f(GRExg = 0.01)
f(pExg = c(1, 2))

#### Example 9.5 in Li (2019).
f <- function(GRExg = 0, pExg = c(1, NA, 0.625)) {
  pcss <- sdm(
    A = function(state) {
      alpha <- rbind(1, 1, 1)
      Beta <- matrix(c(
        0, 1, 1,
        0.5, 0, 0,
        0.5, 0, 0
      ), 3, 3, TRUE)
      CD_A(alpha, Beta, state$p)
    },
    B = diag(c(1, 0, 0)),
    S0Exg = matrix(c(
      NA, NA, NA,
      NA, 100, NA,
      NA, NA, 100
    ), 3, 3, TRUE),
    GRExg = GRExg,
    pExg = pExg,
    maxIteration = 1,
    numberOfPeriods = 300,
    depreciationCoef = 0,
    z0 = c(100, 0, 0),
    ts = TRUE
  )
  matplot(pcss$ts.z, type = "l")
  print("pcss$z:")
  pcss$z
  print("tail(pcss$ts.q, 3)")
  print(tail(pcss$ts.q, 3))
  invisible(pcss)
}

f()
f(GRExg = 0.01)
f(pExg = c(1, 0.25, 0.25))
f(pExg = c(1, 0.2, 0.25))

An Example of Price Regulation and Endogenous Labor Supply (Example 9.5 of Li, 2019)

Description

This is an example of price regulation and endogenous labor supply. See CGE::Example9.5.

Usage

gemExogenousPrice_EndogenousLaborSupply_3_3(...)

Arguments

See Also

gemExogenousPrice

Examples

## the exogenous labor price with product as numeraire.
p.labor <- 0.625

dst.firm <- node_new("output",
  type = "CD",
  alpha = 1,  beta = c(0.5, 0.5),
  "land", "lab"
)

dst.landowner <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

dst.laborer <- Clone(dst.landowner)

ge <- sdm2(
  A = list(
    dst.firm,
    dst.landowner,
    dst.laborer
  ),
  B = diag(3),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  GRExg = 0,
  names.commodity = c("prod", "land", "lab"),
  names.agent = c("firm", "landowner", "laborer"),
  maxIteration = 1,
  numberOfPeriods = 200,
  depreciationCoef = 0,
  numeraire = "prod",
  ts = TRUE,
  policy = function(time, state, state.history) {
    if (time > 1) {
      ratio <- state$p[3] / state$p[1] / p.labor
      last.labor.supply <- state.history$S[3, 3, time - 1]
      state$S[3, 3] <- last.labor.supply * ratio
    }

    state
  }
)

matplot(ge$ts.p, type = "l")
tail(ge$ts.S[3, 3, ])
plot(ge$ts.S[3, 3, ], type = "l")

Some Examples with Exogenous Utility Level and Endogenous Labor Supply

Description

Some examples with exogenous utility level and endogenous labor supply.

Usage

gemExogenousUtilityLevel_EndogenousLaborSupply_3_3(...)

Arguments

Examples

## a spot equilibrium path
utility.level.laborer <- 0.625

dst.firm <- node_new("output",
  type = "CD",
  alpha = 1,
  beta = c(0.5, 0.5),
  "land", "lab"
)

dst.landowner <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

dst.laborer <- Clone(dst.landowner)

dstl <- list(dst.firm, dst.landowner, dst.laborer)

ge <- sdm2(
  A = dstl,
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  GRExg = 0,
  names.commodity = c("prod", "land", "lab"),
  names.agent = c("firm", "landowner", "laborer"),
  maxIteration = 1,
  numberOfPeriods = 30,
  numeraire = "prod",
  ts = TRUE,
  policy = list(
    function(time, state, state.history) {
      if (time > 1) {
        last.labor.supply <- state.history$S[3, 3, time - 1]

        ratio <- state$last.z[3] / last.labor.supply / utility.level.laborer
        state$S[3, 3] <- last.labor.supply * ratio
      }
      state
    },
    policyMarketClearingPrice
  )
)

matplot(ge$ts.p, type = "l")
plot(ge$ts.S[3, 3, ], type = "l")
ge$S


## Regard the laborer as a firm.
dstl[[3]] <- node_new(
  "lab",
  type = "Leontief", a = utility.level.laborer,
  "prod"
)

ge <- sdm2(
  A = dstl,
  B = diag(c(1, 0, 1)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, NA
  ), 3, 3, TRUE),
  names.commodity = c("prod", "land", "lab"),
  names.agent = c("firm", "landowner", "laborer"),
  maxIteration = 1,
  numberOfPeriods = 30,
  numeraire = "prod",
  ts = TRUE,
  policy = policyMarketClearingPrice
)

matplot(ge$ts.p, type = "l")
plot(ge$ts.S[3, 3, ], type = "l")
ge$p
ge$S

Some Examples Illustrating Negative Externality

Description

Some examples illustrating negative externality.

Usage

gemExternality_Negative(...)

Arguments

Examples

#### negative externality of consumption to consumption
# Here the distortion means that an agent
# will use environmental resources for free.
dst.consumer1.distorted <- node_new("util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "lab", "land"
)

dst.consumer1 <- node_new("util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "lab", "cc1"
)
node_set(dst.consumer1, "cc1",
  type = "Leontief",
  a = c(1, 1),
  "land", "env"
)

dst.consumer2 <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = c(0.4, 0.4, 0.2),
  "lab", "land", "env"
)

ge.externality <- sdm2(
  A = list(dst.consumer1.distorted, dst.consumer2),
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    50, 0,
    0, 50,
    0, 0
  ), 3, 2, TRUE),
  names.commodity = c("lab", "land", "env"),
  names.agent = c("consumer1.distorted", "consumer2"),
  numeraire = "lab",
  policy = function(state) {
    last.D <- state$last.A %*% dg(state$last.z)
    state$S[3, 2] <- 100 - last.D[2, 1]
    state
  }
)

ge.externality$p
ge.externality$z
addmargins(ge.externality$D, 2)
addmargins(ge.externality$S, 2)

ge <- sdm2(
  A = list(dst.consumer1, dst.consumer2),
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    50, 0,
    0, 50,
    13, 87
  ), 3, 2, TRUE),
  names.commodity = c("lab", "land", "env"),
  names.agent = c("consumer1", "consumer2"),
  numeraire = "lab"
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

## A corrective tax is imposed on distorted consumer 1.
# 54% of tax revenue is allocated to distorted consumers 1, 46% to consumers 2.
dst.consumer1.distorted.taxed <- node_new("util",
                                          type = "CD",
                                          alpha = 1, beta = c(0.5, 0.5),
                                          "lab", "cc1"
)
node_set(dst.consumer1.distorted.taxed, "cc1",
         type = "FIN", rate = c(1, ge$DV[3, 1] / ge$DV[2, 1]),
         "land", "tax"
)

ge.corrective.tax <- sdm2(
  A = list(dst.consumer1.distorted.taxed, dst.consumer2),
  B = matrix(0, 4, 2),
  S0Exg = matrix(c(
    50, 0,
    0, 50,
    0, 0,
    54, 46
  ), 4, 2, TRUE),
  names.commodity = c("lab", "land", "env", "tax"),
  names.agent = c("consumer1.distorted", "consumer2"),
  numeraire = "lab",
  policy = function(state) {
    last.D <- state$last.A %*% dg(state$last.z)
    state$S[3, 2] <- 100 - last.D[2, 1]
    state
  }
)

ge.corrective.tax$z
addmargins(ge.corrective.tax$D, 2)
addmargins(ge.corrective.tax$S, 2)

## negative externality of production to consumption
dst.firm1.distorted <- dst.consumer1.distorted
dst.firm1 <- dst.consumer1

dst.consumer1.Leontief <- node_new(
  "util",
  type = "Leontief",
  a = 1,
  "prod1"
)

ge.externality <- sdm2(
  A = list(dst.firm1.distorted, dst.consumer1.Leontief, dst.consumer2),
  B = diag(c(1, 0, 0), 4, 3),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 50, NA,
    NA, NA, 50,
    NA, NA, NA
  ), 4, 3, TRUE),
  names.commodity = c("prod1", "lab", "land", "env"),
  names.agent = c("firm1", "consumer1.Leontief", "consumer2"),
  numeraire = "lab",
  policy = function(state) {
    last.D <- state$last.A %*% dg(state$last.z)
    state$S[4, 3] <- 100 - last.D[3, 1]
    state
  }
)

ge.externality$p
ge.externality$z
addmargins(ge.externality$D, 2)
addmargins(ge.externality$S, 2)

ge <- sdm2(
  A = list(dst.firm1, dst.consumer1.Leontief, dst.consumer2),
  B = diag(c(1, 0, 0), 4, 3),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 50, NA,
    NA, NA, 50,
    NA, 13, 87
  ), 4, 3, TRUE),
  names.commodity = c("prod1", "lab", "land", "env"),
  names.agent = c("firm1", "consumer1.Leontief", "consumer2"),
  numeraire = "lab"
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

#### negative externality of consumption to production and consumption
dst.firm2 <- dst.consumer2
dst.consumer2.Leontief <- node_new(
  "util",
  type = "Leontief",
  a = 1,
  "prod2"
)

ge.externality <- sdm2(
  A = list(dst.firm2, dst.consumer1.distorted, dst.consumer2.Leontief),
  B = diag(c(1, 0, 0), 4, 3),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 50, NA,
    NA, NA, 50,
    NA, NA, NA
  ), 4, 3, TRUE),
  names.commodity = c("prod2", "lab", "land", "env"),
  names.agent = c("firm2", "consumer1", "consumer2.Leontief"),
  numeraire = "lab",
  policy = function(state) {
    last.D <- state$last.A %*% dg(state$last.z)
    state$S[4, 3] <- 100 - last.D[3, 2]
    state
  }
)

ge.externality$p
ge.externality$z
addmargins(ge.externality$D, 2)
addmargins(ge.externality$S, 2)

ge <- sdm2(
  A = list(dst.firm2, dst.consumer1, dst.consumer2.Leontief),
  B = diag(c(1, 0, 0), 4, 3),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 50, NA,
    NA, NA, 50,
    NA, 13, 87
  ), 4, 3, TRUE),
  names.commodity = c("prod2", "lab", "land", "env"),
  names.agent = c("firm2", "consumer1", "consumer2.Leontief"),
  numeraire = "lab"
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

## negative externality of production to production and consumption
ge.externality <- sdm2(
  A = list(dst.firm1.distorted, dst.firm2, dst.consumer1.Leontief, dst.consumer2.Leontief),
  B = diag(c(1, 1, 0, 0), 5, 4),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, 50, NA,
    NA, NA, NA, 50,
    NA, NA, NA, NA
  ), 5, 4, TRUE),
  names.commodity = c("prod1", "prod2", "lab", "land", "env"),
  names.agent = c("firm1", "firm2", "consumer1.Leontief", "consumer2.Leontief"),
  numeraire = "lab",
  policy = function(state) {
    last.D <- state$last.A %*% dg(state$last.z)
    state$S[5, 4] <- 100 - last.D[4, 1]
    state
  }
)

ge.externality$p
ge.externality$z
addmargins(ge.externality$D, 2)
addmargins(ge.externality$S, 2)

ge <- sdm2(
  A = list(dst.firm1, dst.firm2, dst.consumer1.Leontief, dst.consumer2.Leontief),
  B = diag(c(1, 1, 0, 0), 5, 4),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, 50, NA,
    NA, NA, NA, 50,
    NA, NA, 13, 87
  ), 5, 4, TRUE),
  names.commodity = c("prod1", "prod2", "lab", "land", "env"),
  names.agent = c("firm1", "firm2", "consumer1.Leontief", "consumer2.Leontief"),
  numeraire = "lab"
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

## negative externality of consumption to production
dst.firm2.distorted <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "lab", "land"
)

ge.externality <- sdm2(
  A = list(dst.firm2.distorted, dst.consumer1.distorted, dst.consumer2.Leontief),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 50, NA,
    NA, NA, 50
  ), 3, 3, TRUE),
  names.commodity = c("prod2", "lab", "land"),
  names.agent = c("firm2.distorted", "consumer1", "consumer2.Leontief"),
  numeraire = "lab",
  policy = function(A, state) {
    last.D <- state$last.A %*% dg(state$last.z)
    state$S[1, 1] <- (100 - last.D[3, 2])^0.2 * state$S[1, 1]^0.8
    state
  }
)

ge.externality$p
ge.externality$z
addmargins(ge.externality$D, 2)
addmargins(ge.externality$S, 2)

Some Examples Illustrating Positive Externality

Description

Some examples illustrating positive externality.

Usage

gemExternality_Positive(...)

Arguments

Examples

#### positive externality of consumption to consumption
dst.consumer1 <- node_new("util",
  type = "Leontief", a = 1,
  "lab"
)

dst.consumer2 <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = c(0.2, 0.8), # c(0.8, 0.2),
  "lab", "byproduct"
)

ge.externality <- sdm2(
  A = list(dst.consumer1, dst.consumer2),
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    50, 50,
    0, 0
  ), 2, 2, TRUE),
  names.commodity = c("lab", "byproduct"),
  names.agent = c("consumer1", "consumer2"),
  numeraire = "lab",
  policy = function(state) {
    state$S[2, 2] <- state$last.z[1]
    state
  }
)

ge.externality$p
ge.externality$z
addmargins(ge.externality$D, 2)
addmargins(ge.externality$S, 2)

ge <- sdm2(
  A = list(dst.consumer1, dst.consumer2),
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    50, 50,
    0, 0
  ), 2, 2, TRUE),
  names.commodity = c("lab", "byproduct"),
  names.agent = c("consumer1", "consumer2"),
  numeraire = "lab",
  policy = function(state) {
    state$S[2, 1] <- state$last.z[1]
    state
  }
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

## positive externality of production to consumption
dst.firm1 <- node_new("prod1",
  type = "Leontief", a = 1,
  "lab"
)

dst.consumer3 <- node_new("util",
  type = "Leontief", a = 1,
  "prod1"
)

dst.consumer2 <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = c(0.2, 0.8),
  "lab", "byproduct"
)

ge.externality <- sdm2(
  A = list(dst.firm1, dst.consumer3, dst.consumer2),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 50, 50,
    NA, NA, NA
  ), 3, 3, TRUE),
  names.commodity = c("prod1", "lab", "byproduct"),
  names.agent = c("firm1", "consumer3", "consumer2"),
  numeraire = "lab",
  policy = function(state) {
    state$S[3, 3] <- state$last.z[1]
    state
  }
)

ge.externality$p
ge.externality$z
addmargins(ge.externality$D, 2)
addmargins(ge.externality$S, 2)

ge <- sdm2(
  A = list(dst.firm1, dst.consumer3, dst.consumer2),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    1, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 50, 50,
    NA, NA, NA
  ), 3, 3, TRUE),
  names.commodity = c("prod1", "lab", "byproduct"),
  names.agent = c("firm1", "consumer3", "consumer2"),
  numeraire = "lab"
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

## positive externality of consumption to production
dst.consumer1 <- node_new("util",
  type = "Leontief", a = 1,
  "lab"
)

dst.firm2.distorted <- node_new(
  "prod2",
  type = "Leontief", a = 1,
  "lab"
)

dst.consumer2.Leontief <- node_new("util",
  type = "Leontief", a = 1,
  "prod2"
)

ge.externality <- sdm2(
  A = list(dst.firm2.distorted, dst.consumer1, dst.consumer2.Leontief),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0
  ), 2, 3, TRUE),
  S0Exg = matrix(c(
    0, 0, 0,
    0, 50, 50
  ), 2, 3, TRUE),
  names.commodity = c("prod2", "lab"),
  names.agent = c("firm2.distorted", "consumer1", "dst.consumer2.Leontief"),
  numeraire = "lab",
  policy = function(state) {
    state$S[1, 1] <- state$last.z[1]^0.2 * state$last.z[2]^0.8
    state
  }
)

ge.externality$p
ge.externality$z
addmargins(ge.externality$D, 2)
addmargins(ge.externality$S, 2)

Some Examples of Treating Firms as Consumer-Type Agents

Description

Some examples of equilibrium models wherein firms are treated as consumer-type agents instead of producer-type agents.

Usage

gemFirmAsConsumer(...)

Arguments

See Also

gemIntertemporal_2_2

Examples

#### an intertemporal model with firm
## (see gemIntertemporal_2_2)
np <- 3 # the number of economic periods

S0Exg <- matrix(c(
  0, 0, 150,
  1000, 0, 0,
  0, 1000, 0,
  0, 0, 100,
  0, 0, 100
), 5, 3, TRUE)

B <- matrix(0, 5, 3, TRUE)

dst.firm1 <- node_new("util",
  type = "StickyLinear",
  beta = c(1, 1),
  "prod2", "cc1"
)
node_set(dst.firm1, "cc1",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "prod1", "lab1"
)

dst.firm2 <- node_new("util",
  type = "StickyLinear",
  beta = c(1, 1),
  "prod3", "cc1"
)
node_set(dst.firm2, "cc1",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "prod2", "lab2"
)

dst.consumer <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = prop.table(rep(1, np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = list(dst.firm1, dst.firm2, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = c(paste0("prod", 1:np), paste0("lab", 1:(np - 1))),
  names.agent = c(paste0("firm", 1:(np - 1)), "consumer"),
  numeraire = "prod1",
  ts = TRUE
)

#### an intertemporal model with bank
igr <- 1.1
beta.bank <- c(1, 1 / igr, 1 / igr^2)

dst.bank <- node_new(
  "output",
  type = "StickyLinear",
  beta = beta.bank,
  "payoff1", "payoff2", "payoff3"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 2, 1 / 6, 1 / 3),
  "payoff1", "payoff2", "payoff3"
)

ge <- sdm2(
  A = list(dst.bank, dst.consumer),
  B = matrix(0, 3, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    100, 2,
    100, 1
  ), 3, 2, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("bank", "consumer"),
  numeraire = "payoff1"
)

#### a spot sequential model
dst.firm <- node_new("output",
  type = "StickyLinear",
  beta = c(1, 1),
  "prod", "cc1"
)
node_set(dst.firm, "cc1",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "cap", "lab"
)

dst.consumer <- node_new("util",
  type = "Leontief",
  a = 1,
  "prod"
)

ge <- sdm2(
  A = list(dst.firm, dst.consumer),
  B = matrix(0, 3, 2, TRUE),
  S0Exg = matrix(c(
    1000, 0,
    0, 50,
    0, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)
ge$p
ge$z
ge$D

## the corresponding model treating a firm as a producer-type agent
ge <- sdm2(
  A = function(state) {
    a1 <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), p = state$p)
    a2 <- c(1, 0, 0)
    cbind(a1, a2)
  },
  B = diag(c(1, 0), 3, 2),
  S0Exg = matrix(c(
    NA, NA,
    NA, 50,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)
ge$p
ge$z
ge$D

Spot equilibrium paths with Heterogeneous Firms

Description

This is an example of spot equilibrium paths with heterogeneous firms.

Usage

gemHeterogeneousFirms_2_3(...)

Arguments

Examples

dst.firm1 <- node_new(
  "output",
  type = "CD", alpha = 1, beta = c(0.35, 0.65),
  "prod", "lab"
)

dst.firm2 <- node_new(
  "output",
  type = "CD", alpha = 1.3, beta = c(0.9, 0.1),
  "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge <- sdm2(
  A = list(dst.firm1, dst.firm2, dst.consumer),
  B = matrix(c(
    1, 1, 0,
    0, 0, 0
  ), 2, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, 100
  ), 2, 3, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm1", "firm2", "consumer"),
  numeraire = "lab",
  z0 = c(1, 1, 1),
  ts = TRUE,
  policy = policyMarketClearingPrice,
  numberOfPeriods = 200,
  maxIteration = 1
)

matplot(ge$ts.z, type = "l")

An Example Illustrating Product Quality Information

Description

An examples illustrating product quality information.

Usage

gemInformation_ProductQuality(...)

Arguments

Examples

dst.firm.normal <- node_new("normal prod",
  type = "Leontief", a = 1,
  "lab"
)

dst.firm.inferior <- node_new("inferior prod",
  type = "Leontief", a = 0.5,
  "lab"
)

dst.quasifirm <- node_new("normal prod",
  type = "Leontief", a = 1,
  # a = 1 means that consumers cannot distinguish between normal and inferior products.
  # In this case, the calculated consumer utility is nominal.
  # The real utility of the consumer is lower than the nominal utility.
  # a = 10 is the opposite.
  "inferior prod"
)

dst.consumer <- node_new("util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "lab", "normal prod"
)

ge <- sdm2(
  A = list(dst.firm.normal, dst.firm.inferior, dst.quasifirm, dst.consumer),
  B = matrix(c(
    1, 0, 1, 0,
    0, 1, 0, 0,
    0, 0, 0, 0
  ), 3, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA,  NA, NA,
    NA, NA,  NA, NA,
    NA, NA,  NA, 100
  ), 3, 4, TRUE),
  names.commodity = c("normal prod", "inferior prod", "lab"),
  names.agent = c("normal firm", "inferior firm", "quasifirm", "consumer"),
  numeraire = "lab",
  maxIteration = 1,
  numberOfPeriods = 800
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

A General Equilibrium Model based on a 2×2 (Unbalanced) Input-Output Table

Description

A general equilibrium model based on a 2×2 (unbalanced) input-output table (unit: yuan).

Usage

gemInputOutputTable_2_2(...)

Arguments

Examples

names.commodity <- c("prod", "lab")
names.agent <- c("firm", "laborer")

IT <- matrix(c(
  40, 40,
  40, 60
), 2, 2, TRUE)

OT <- matrix(c(
  100, 0,
  0, 100
), 2, 2, TRUE)

dimnames(IT) <- dimnames(OT) <- list(names.commodity, names.agent)

addmargins(IT)
addmargins(OT)

#### the model
dst.firm <- node_new(
  "prod",
  type = "SCES",
  es = 1,
  alpha = 1.25, # 100 / (40 + 40)
  beta = prop.table(c(40, 40)),
  "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "SCES",
  es = 1, alpha = 1,
  beta = prop.table(c(40, 60)),
  "prod", "lab"
)

dstl <- list(dst.firm, dst.consumer)

ge.benchmark <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab"
)

ge.benchmark$p
ge.benchmark$D
ge.benchmark$S

addmargins(ge.benchmark$DV)
addmargins(ge.benchmark$SV)

## the same as above
ge <- sdm2(
  A = function(state) {
    a.firm <- SCES_A(es = 1, alpha = 1.25, Beta = prop.table(c(40, 40)), p = state$p)
    a.consumer <- SCES_A(es = 1, alpha = 1.25, Beta = prop.table(c(40, 60)), p = state$p)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab"
)

## technology progress
dstl[[1]]$alpha <- 2.5

ge.TP <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab"
)

ge.TP$p
ge.TP$D
ge.TP$S

addmargins(ge.TP$DV)
addmargins(ge.TP$SV)

A Two-Country General Equilibrium Model

Description

A two-country general equilibrium model. This general equilibrium model is based on a two-country (i.e. CHN and ROW) input-output table consisting of an input part and an output part. Each country contains 2 sectors and 7 commodities (or subjects). The 2 sectors are firm and household. The 7 commodities (or subjects) are product, labor, capital goods, bond, tax, dividend, tariff. Hence the input-output table has 14 rows and 4 columns.

Usage

gemInputOutputTable_2_7_2(
  IT,
  OT,
  es.DIProduct.firm.CHN = 3,
  es.DIProduct.firm.ROW = 3,
  es.laborCapital.firm.CHN = 0.75,
  es.laborCapital.firm.ROW = 0.75,
  es.household.CHN = 3,
  es.household.ROW = 3,
  return.dstl = FALSE,
  ...
)

Arguments

Value

A general equilibrium, which usually is a list with the following elements:

• p - the price vector with CHN labor as numeraire.

• dstl - the demand structure tree list of sectors if return.dstl == TRUE.

• ... - some elements returned by the function sdm2.

Examples

IT <- matrix(c(
  142, 84, 13, 4.1,
  47, 0, 0, 0,
  13, 0, 0, 0,
  0, 0, 0, 3.4,
  9.3, 0, 0, 0,
  22, 0, 0, 0,
  0.15, 0.091, 0, 0,
  10, 6, 381, 451,
  0, 0, 252, 0,
  0, 0, 81, 0,
  0, 4.9, 0, 0,
  0, 0, 26, 0,
  0, 0, 92, 0,
  0, 0, 1.9, 0.59
), 14, 4, TRUE)

OT <- matrix(c(
  244, 0, 0, 0,
  0, 47, 0, 0,
  0, 13, 0, 0,
  0, 3.4, 0, 0,
  0, 9.3, 0, 0,
  0, 22, 0, 0,
  0, 0.24, 0, 0,
  0, 0, 849, 0,
  0, 0, 0, 252,
  0, 0, 0, 81,
  0, 0, 0, 4.9,
  0, 0, 0, 26,
  0, 0, 0, 92,
  0, 0, 0, 2.5
), 14, 4, TRUE)

dimnames(IT) <- dimnames(OT) <- list(
  c(
    "product.CHN", "labor.CHN", "capital.CHN", "bond.CHN",
    "tax.CHN", "dividend.CHN", "tariff.CHN",
    "product.ROW", "labor.ROW", "capital.ROW", "bond.ROW",
    "tax.ROW", "dividend.ROW", "tariff.ROW"
  ),
  c(
    "firm.CHN", "household.CHN",
    "firm.ROW", "household.ROW"
  )
)

ge <- gemInputOutputTable_2_7_2(IT, OT, return.dstl = TRUE)
ge$p
ge$z
node_plot(ge$dstl[[1]])
ge$dstl[[1]]$a

## tariff rate change in CHN
dstl <- lapply(ge$dstl, Clone)
tmp <- node_set(dstl[[1]], "cc1.1")
tmp$beta[2] <- tmp$beta[2] * 10

ge.TRC <- sdm2(
  A = dstl, B = ge$B, S0Exg = ge$S0Exg,
  names.commodity = rownames(ge$B),
  names.agent = colnames(ge$B),
  numeraire = "labor.CHN"
)

ge.TRC$p
ge.TRC$z
#### technology progress in CHN
OT.TP <- OT
OT.TP["product.CHN", "firm.CHN"] <- OT["product.CHN", "firm.CHN"] * 1.2

ge.TP <- gemInputOutputTable_2_7_2(IT, OT.TP, return.dstl = TRUE)
ge.TP$p
ge.TP$z
ge.TP$dstl[[1]]$a

#### capital accumulation in CHN
OT.CA <- OT
OT.CA["capital.CHN", "household.CHN"] <- OT["capital.CHN", "household.CHN"] * 2

ge.CA <- gemInputOutputTable_2_7_2(IT, OT.CA)
ge.CA$p
ge.CA$z

#### labor supply change in CHN
OT.LSC <- OT
OT.LSC["labor.CHN", "household.CHN"] <- OT["labor.CHN", "household.CHN"] * 0.5

ge.LSC <- gemInputOutputTable_2_7_2(IT, OT.LSC)
ge.LSC$p
ge.LSC$z

A Two-Country General Equilibrium Model

Description

A two-country general equilibrium model. This general equilibrium model is based on a two-country (i.e. CHN and ROW) input-output table consisting of an input part and an output part. Each country contains 4 sectors and 7 commodities (or subjects). The 4 sectors are production, consumption, investment and foreign trade. The 7 commodities (or subjects) are product, labor, capital goods, bond, tax, dividend, imported product. Hence the input-output table has 14 rows and 8 columns.

Usage

gemInputOutputTable_2_7_4(
  IT,
  OT,
  es.DIProduct.production.CHN = 3,
  es.DIProduct.production.ROW = 3,
  es.laborCapital.production.CHN = 0.75,
  es.laborCapital.production.ROW = 0.75,
  es.consumption.CHN = 3,
  es.consumption.ROW = 3,
  es.investment.CHN = 3,
  es.investment.ROW = 3,
  return.dstl = FALSE,
  ...
)

Arguments

Value

A general equilibrium, which usually is a list with the following elements:

• p - the price vector with CHN labor as numeraire.

• dstl - the demand structure tree list of sectors if return.dstl == TRUE.

• ... - some elements returned by the function sdm2.

Examples

IT <- matrix(c(
  30, 12, 9, 0, 0, 0, 0, 13,
  15, 0, 0, 0, 0, 0, 0, 0,
  2, 0, 0, 0, 0, 0, 0, 0,
  0, 9, 0, 0, 0, 2, 0, 0,
  3, 0, 0, 1, 0, 0, 0, 0,
  6, 0, 0, 0, 0, 0, 0, 0,
  8, 3, 3, 0, 0, 0, 0, 0,
  0, 0, 0, 13, 150, 316, 258, 0,
  0, 0, 0, 0, 288, 0, 0, 0,
  0, 0, 0, 0, 92, 0, 0, 0,
  0, 2, 0, 0, 0, 269, 0, 0,
  0, 0, 0, 0, 35, 0, 0, 1,
  0, 0, 0, 0, 172, 0, 0, 0,
  0, 0, 0, 0, 1, 5, 13, 0
), 14, 8, TRUE)

OT <- matrix(c(
  64, 0, 0, 0, 0, 0, 0, 0,
  0, 15, 0, 0, 0, 0, 0, 0,
  0, 2, 0, 0, 0, 0, 0, 0,
  0, 0, 11, 0, 0, 0, 0, 0,
  0, 3, 0, 0, 0, 0, 0, 0,
  0, 6, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 13, 0, 0, 0, 0,
  0, 0, 0, 0, 738, 0, 0, 0,
  0, 0, 0, 0, 0, 288, 0, 0,
  0, 0, 0, 0, 0, 92, 0, 0,
  0, 0, 0, 0, 0, 0, 271, 0,
  0, 0, 0, 0, 0, 36, 0, 0,
  0, 0, 0, 0, 0, 172, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 14
), 14, 8, TRUE)

dimnames(IT) <- dimnames(OT) <- list(
  c(
    "product.CHN", "labor.CHN", "capital.CHN", "bond.CHN",
    "tax.CHN", "dividend.CHN", "imported.product.CHN",
    "product.ROW", "labor.ROW", "capital.ROW", "bond.ROW",
    "tax.ROW", "dividend.ROW", "imported.product.ROW"
  ),
  c(
    "production.CHN", "consumption.CHN", "investment.CHN", "foreign.trade.CHN",
    "production.ROW", "consumption.ROW", "investment.ROW", "foreign.trade.ROW"
  )
)

ge <- gemInputOutputTable_2_7_4(IT, OT, return.dstl = TRUE)
ge$p
ge$z
node_plot(ge$dstl[[1]])
ge$dstl[[1]]$a

#### technology progress in CHN
OT.TP <- OT
OT.TP["product.CHN", "production.CHN"] <- OT["product.CHN", "production.CHN"] * 1.2

ge.TP <- gemInputOutputTable_2_7_4(IT, OT.TP, return.dstl = TRUE)
ge.TP$p
ge.TP$z
ge.TP$dstl[[1]]$a

#### capital accumulation in CHN
OT.CA <- OT
OT.CA["capital.CHN", "consumption.CHN"] <- OT["capital.CHN", "consumption.CHN"] * 2

ge.CA <- gemInputOutputTable_2_7_4(IT, OT.CA)
ge.CA$p
ge.CA$z

#### labor supply change in CHN
OT.LSC <- OT
OT.LSC["labor.CHN", "consumption.CHN"] <- OT["labor.CHN", "consumption.CHN"] * 0.5

ge.LSC <- gemInputOutputTable_2_7_4(IT, OT.LSC)
ge.LSC$p
ge.LSC$z

#### tariff rate change in CHN
IT.TRC <- IT
IT.TRC["tax.CHN","foreign.trade.CHN"] <- IT.TRC["tax.CHN","foreign.trade.CHN"] * 1.2
ge.TRC <- gemInputOutputTable_2_7_4(IT.TRC, OT)
ge.TRC$p
ge.TRC$z

A Two-Country General Equilibrium Model with Money

Description

A two-country general equilibrium model with money. This general equilibrium model is based on a two-country (i.e. CHN and ROW) input-output table. Each country contains four sectors and eight commodities (or subjects). The four sectors are production, consumption, investment and foreign trade. The eight commodities (or subjects) are product, labor, capital goods, bond, tax, dividend, imported product and money interest. Hence the input-output table has 16 rows and 8 columns.

Usage

gemInputOutputTable_2_8_4(
  IT,
  product.output.CHN = sum(IT[, "production.CHN"]),
  product.output.ROW = sum(IT[, "production.ROW"]),
  labor.supply.CHN = sum(IT["labor.CHN", ]),
  labor.supply.ROW = sum(IT["labor.ROW", ]),
  capital.supply.CHN = sum(IT["capital.CHN", ]),
  capital.supply.ROW = sum(IT["capital.ROW", ]),
  money.interest.supply.CHN = 5,
  money.interest.supply.ROW = 30,
  es.DIProduct.production.CHN = 0.5,
  es.DIProduct.production.ROW = 0.5,
  es.laborCapital.production.CHN = 0.75,
  es.laborCapital.production.ROW = 0.75,
  es.consumption.CHN = 0.5,
  es.consumption.ROW = 0.5,
  es.investment.CHN = 0.9,
  es.investment.ROW = 0.9,
  interest.rate.CHN = NA,
  interest.rate.ROW = NA,
  return.dstl = FALSE,
  ...
)

Arguments

Details

If interest.rate.CHN is NA or interest.rate.CHN is NA, they are assumed to be equal. And in this case, the exchange rate is determined by the ratio of the interest of unit currency of the two countries. In this model, the ratio of a sector's monetary interest expenditure to its transaction value may not be equal to the interest rate because the ratio is not only affected by the interest rate, but also by the sector's currency circulation velocity and other factors.

Value

A general equilibrium, which usually is a list with the following elements:

• p - the price vector with CHN labor as numeraire, wherein the price of a currency is the interest per unit of currency.

• D - the demand matrix, also called the input table. Wherein the benchmark prices are used.

• DV - the demand value matrix, also called the value input table. Wherein the current price is used.

• SV - the supply value matrix, also called the value output table. Wherein the current price is used.

• eri.CHN - the exchange rate index of CHN currency.

• eri.ROW - the exchange rate index of ROW currency.

• p.money - the price vector with CHN money as numeraire if both interest.rate.CHN and interest.rate.CHN are not NA.

• dstl - the demand structure tree list of sectors if return.dstl == TRUE.

• ... - some elements returned by the function sdm2.

Examples

ITExample <- matrix(0, 16, 8, dimnames = list(
  c(
    "product.CHN", "labor.CHN", "capital.CHN", "bond.CHN",
    "tax.CHN", "dividend.CHN", "imported.product.CHN", "money.interest.CHN",
    "product.ROW", "labor.ROW", "capital.ROW", "bond.ROW",
    "tax.ROW", "dividend.ROW", "imported.product.ROW", "money.interest.ROW"
  ),
  c(
    "production.CHN", "consumption.CHN", "investment.CHN", "foreign.trade.CHN",
    "production.ROW", "consumption.ROW", "investment.ROW", "foreign.trade.ROW"
  )
))

production.CHN <- c(
  product.CHN = 140, labor.CHN = 40, capital.CHN = 10,
  tax.CHN = 10, dividend.CHN = 20, imported.product.CHN = 5, money.interest.CHN = 5
)
production.ROW <- c(
  product.ROW = 840, labor.ROW = 240, capital.ROW = 60,
  tax.ROW = 60, dividend.ROW = 120, imported.product.ROW = 6, money.interest.ROW = 30
)

consumption.CHN <- c(
  product.CHN = 40, bond.CHN = 30, imported.product.CHN = 5, money.interest.CHN = 2
)

consumption.ROW <- c(
  product.ROW = 240, bond.ROW = 180, imported.product.ROW = 6, money.interest.ROW = 12
)

investment.CHN <- c(
  product.CHN = 30,
  imported.product.CHN = 4, money.interest.CHN = 1,
  bond.ROW = 1,
  money.interest.ROW = 0.02
)

investment.ROW <- c(
  bond.CHN = 1,
  money.interest.CHN = 0.02,
  product.ROW = 180,
  imported.product.ROW = 4, money.interest.ROW = 6
)


foreign.trade.CHN <- c(
  product.ROW = 13,
  tax.CHN = 0.65,
  money.interest.ROW = 0.26
)

foreign.trade.ROW <- c(
  product.CHN = 15,
  tax.ROW = 0.75,
  money.interest.CHN = 0.3
)

ITExample <- matrix_add_by_name(
  ITExample, production.CHN, consumption.CHN, investment.CHN, foreign.trade.CHN,
  production.ROW, consumption.ROW, investment.ROW, foreign.trade.ROW
)

ge <- gemInputOutputTable_2_8_4(
  IT = ITExample,
  return.dstl = TRUE
)
ge$eri.CHN
ge$p
node_plot(ge$dstl[[4]], TRUE)

ge2 <- gemInputOutputTable_2_8_4(
  IT = ge$DV,
  money.interest.supply.CHN = sum(ge$DV["money.interest.CHN", ]),
  money.interest.supply.ROW = sum(ge$DV["money.interest.ROW", ]),
  return.dstl = TRUE
)
ge2$eri.CHN
ge2$p


#### technology progress in CHN
ITTmp <- ITExample
ITTmp["labor.CHN", "production.CHN"] <- ITTmp["labor.CHN", "production.CHN"] * 0.8
geTmp <- gemInputOutputTable_2_8_4(
  IT = ITTmp,
  product.output.CHN = sum(ITExample[, "production.CHN"]),
  return.dstl = TRUE
)
geTmp$eri.CHN


#### increased demand for imported product in CHN
ITTmp <- ITExample
ITTmp["imported.product.CHN", "production.CHN"] <-
  ITTmp["imported.product.CHN", "production.CHN"] * 1.2
geTmp <- gemInputOutputTable_2_8_4(
  IT = ITTmp,
  return.dstl = TRUE
)
geTmp$eri.CHN


#### capital accumulation in CHN
geTmp <- gemInputOutputTable_2_8_4(
  IT = ITExample,
  capital.supply.CHN = sum(ITExample["capital.CHN", ]) * 1.2,
  return.dstl = TRUE
)
geTmp$eri.CHN

##
geTmp <- gemInputOutputTable_2_8_4(
  IT = ITExample,
  capital.supply.CHN = sum(ITExample["capital.CHN", ]) * 1.2,
  es.DIProduct.production.CHN = 0.3,
  return.dstl = TRUE
)
geTmp$eri.CHN

A General Equilibrium Model based on a 5×4 Input-Output Table (see Zhang Xin, 2017, Table 8.6.1)

Description

This is a general equilibrium model based on a 5×4 input-output table (see Zhang Xin, 2017, Table 8.6.1).

Usage

gemInputOutputTable_5_4(
  dstl,
  supply.labor = 850,
  supply.capital = 770,
  names.commodity = c("agri", "manu", "serv", "lab", "cap"),
  names.agent = c("agri", "manu", "serv", "hh")
)

Arguments

Details

Given a 5×4 input-output table (e.g., see Zhang Xin, 2017, Table 8.6.1), this model calculates the corresponding general equilibrium. This input-output table contains 3 production sectors and one household. The household consumes products and supplies labor and capital.

Value

A general equilibrium which is a list with the following elements:

• D - the demand matrix, also called the input table. Wherein the benchmark prices are used.

• DV - the demand value matrix, also called the value input table. Wherein the current price is used.

• SV - the supply value matrix, also called the value output table. Wherein the current price is used.

• ... - some elements returned by the CGE::sdm function

References

Zhang Xin (2017, ISBN: 9787543227637) Principles of Computable General Equilibrium Modeling and Programming (Second Edition). Shanghai: Gezhi Press. (In Chinese)

Examples

es.agri <- 0.2 # the elasticity of substitution
es.manu <- 0.3
es.serv <- 0.1

es.VA.agri <- 0.25
es.VA.manu <- 0.5
es.VA.serv <- 0.8

d.agri <- c(260, 345, 400, 200, 160)
d.manu <- c(320, 390, 365, 250, 400)
d.serv <- c(150, 390, 320, 400, 210)
d.hh <- c(635, 600, 385, 0, 0)
# d.hh <- c(635, 600, 100, 0, 0)

IT <- cbind(d.agri, d.manu, d.serv, d.hh)
OT <- matrix(c(
  1365, 0, 0, 0,
  0, 1725, 0, 0,
  0, 0, 1470, 0,
  0, 0, 0, 850,
  0, 0, 0, 770
), 5, 4, TRUE)

dimnames(IT) <- dimnames(OT) <-
  list(
    c("agri", "manu", "serv", "lab", "cap"),
    c("agri", "manu", "serv", "hh")
  )

addmargins(IT)
addmargins(OT)

dst.agri <- node_new("sector.agri",
                     type = "SCES", es = es.agri,
                     alpha = 1,
                     beta = prop.table(
                       c(sum(d.agri[1:3]), sum(d.agri[4:5]))
                     ),
                     "cc1.agri", "cc2.agri"
)
node_set(dst.agri, "cc1.agri",
         type = "Leontief",
         a = prop.table(d.agri[1:3]),
         "agri", "manu", "serv"
)
node_set(dst.agri, "cc2.agri",
         type = "SCES", es = es.VA.agri,
         alpha = 1,
         beta = prop.table(d.agri[4:5]),
         "lab", "cap"
)


dst.manu <- node_new("sector.manu",
                     type = "SCES", es = es.manu,
                     alpha = 1,
                     beta = prop.table(
                       c(sum(d.manu[1:3]), sum(d.manu[4:5]))
                     ),
                     "cc1.manu", "cc2.manu"
)
node_set(dst.manu, "cc1.manu",
         type = "Leontief",
         a = prop.table(d.manu[1:3]),
         "agri", "manu", "serv"
)
node_set(dst.manu, "cc2.manu",
         type = "SCES", es = es.VA.manu,
         alpha = 1,
         beta = prop.table(d.manu[4:5]),
         "lab", "cap"
)

dst.serv <- node_new("sector.serv",
                     type = "SCES", es = es.serv,
                     alpha = 1,
                     beta = prop.table(
                       c(sum(d.serv[1:3]), sum(d.serv[4:5]))
                     ),
                     "cc1.serv", "cc2.serv"
)
node_set(dst.serv, "cc1.serv",
         type = "Leontief",
         a = prop.table(d.serv[1:3]),
         "agri", "manu", "serv"
)
node_set(dst.serv, "cc2.serv",
         type = "SCES", es = es.VA.serv,
         alpha = 1,
         beta = prop.table(d.serv[4:5]),
         "lab", "cap"
)

##
dst.hh <- node_new("sector.hh",
                   type = "SCES", es = 0.5,
                   alpha = 1,
                   beta = prop.table(d.hh[1:3]),
                   "agri", "manu", "serv"
)

dstl <- list(dst.agri, dst.manu, dst.serv, dst.hh)

ge <- gemInputOutputTable_5_4(dstl)

#### labor supply increase
geLSI <- gemInputOutputTable_5_4(dstl, supply.labor = 850 * 1.08)
geLSI$p
geLSI$z / ge$z

## capital supply change
ge.CSC <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 1, 0,
    0, 0, 0, 1,
    0, 0, 0, 1
  ), 5, 4, TRUE),
  S0Exg = {
    tmp <- matrix(NA, 5, 4)
    tmp[4, 4] <- 850
    tmp[5, 4] <- 770
    tmp
  },
  names.commodity = c("agri", "manu", "serv", "lab", "cap"),
  names.agent = c("agri", "manu", "serv", "hh"),
  numeraire = "lab",
  ts = TRUE,
  numberOfPeriods = 100,
  maxIteration = 1,
  z0 = c(1365, 1725, 1470, 1620),
  p0 = rep(1, 5),
  policy = function(time, state) {
    if (time >= 5) {
      state$S[5, 4] <- 880
    }
    state
  }
)

matplot(ge.CSC$ts.p, type = "l")
matplot(ge.CSC$ts.z, type = "l")

## economic fluctuation: a sticky-price path
de <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 1, 0,
    0, 0, 0, 1,
    0, 0, 0, 1
  ), 5, 4, TRUE),
  S0Exg = {
    tmp <- matrix(NA, 5, 4)
    tmp[4, 4] <- 850
    tmp[5, 4] <- 770
    tmp
  },
  names.commodity = c("agri", "manu", "serv", "lab", "cap"),
  names.agent = c("agri", "manu", "serv", "hh"),
  numeraire = "lab",
  ts = TRUE,
  numberOfPeriods = 50,
  maxIteration = 1,
  z0 = c(1365, 1725, 1470, 1620),
  p0 = rep(1, 5),
  policy = list(
    function(time, state) {
      if (time >= 5) {
        state$S[5, 4] <- 880
      }
      state
    },
    makePolicyStickyPrice(0.5)
  ),
  priceAdjustmentVelocity = 0
)

matplot(de$ts.p, type = "o", pch = 20)
matplot(de$ts.z, type = "o", pch = 20)

General Equilibrium Models based on a 5×5 Input-Output Table (see Zhang Xin, 2017, Table 3.2.1)

Description

Some general equilibrium models based on a 5×5 input-output table (see Zhang Xin, 2017, Table 3.2.1).

Usage

gemInputOutputTable_5_5(...)

Arguments

References

Zhang Xin (2017, ISBN: 9787543227637) Principles of Computable General Equilibrium Modeling and Programming (Second Edition). Shanghai: Gezhi Press. (In Chinese)

Examples

names.commodity <- c("agri", "manu", "serv", "lab", "cap")
names.agent <- c("agri", "manu", "serv", "consumer", "investor")

IT <- matrix(c(
  200, 300, 150, 280, 70,
  80, 400, 250, 550, 320,
  30, 420, 240, 350, 110,
  500, 250, 330, 0, 0,
  190, 230, 180, 0, 0
), 5, 5, TRUE)

OT <- matrix(c(
  1000, 0, 0, 0, 0,
  0, 1600, 0, 0, 0,
  0, 0, 1150, 0, 0,
  0, 0, 0, 758.5714, 321.4286,
  0, 0, 0, 421.4286, 178.5714
), 5, 5, TRUE)

dimnames(IT) <- dimnames(OT) <- list(names.commodity, names.agent)

addmargins(IT)
addmargins(OT)

#### a model with non-nested production functions (demand structure trees)
dst.agri <- node_new("output",
  type = "SCES", es = 1, alpha = 1,
  beta = prop.table(c(200, 80, 30, 500, 190)),
  "agri", "manu", "serv", "lab", "cap"
)

dst.manu <- node_new("output",
  type = "SCES", es = 1, alpha = 1,
  beta = prop.table(c(300, 400, 420, 250, 230)),
  "agri", "manu", "serv", "lab", "cap"
)

dst.serv <- node_new("output",
  type = "SCES", es = 1, alpha = 1,
  beta = prop.table(c(150, 250, 240, 330, 180)),
  "agri", "manu", "serv", "lab", "cap"
)

dst.consumer <- node_new("util",
  type = "SCES", es = 0.5, alpha = 1,
  beta = prop.table(c(280, 550, 350)),
  "agri", "manu", "serv"
)

dst.investor <- node_new("util",
  type = "SCES", es = 0.5, alpha = 1,
  beta = prop.table(c(70, 320, 110)),
  "agri", "manu", "serv"
)

ge1.benchmark <- sdm2(list(dst.agri, dst.manu, dst.serv, dst.consumer, dst.investor),
  B = matrix(c(
    1, 0, 0, 0, 0,
    0, 1, 0, 0, 0,
    0, 0, 1, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0
  ), 5, 5, TRUE),
  S0Exg = {
    S0Exg <- matrix(NA, 5, 5)
    S0Exg[4:5, 4] <- c(1080, 600) * (1180 / (1180 + 500))
    S0Exg[4:5, 5] <- c(1080, 600) * (500 / (1180 + 500))
    S0Exg
  },
  names.commodity = c("agri", "manu", "serv", "lab", "cap"),
  names.agent = c("agri", "manu", "serv", "consumer", "investor"),
  numeraire = c("lab")
)

addmargins(ge1.benchmark$D)
addmargins(ge1.benchmark$S)

#### a model with nested production functions (demand structure trees)
dst.agri <- node_new("output",
  type = "SCES", es = 0, alpha = 1,
  beta = prop.table(c(200 + 80 + 30, 500 + 190)),
  "cc.II", "cc.VA"
)
node_set(dst.agri, "cc.II",
  type = "SCES", es = 0, alpha = 1,
  beta = prop.table(c(200, 80, 30)),
  "agri", "manu", "serv"
)
node_set(dst.agri, "cc.VA",
  type = "SCES", es = 0.5, alpha = 1,
  beta = prop.table(c(500, 190)),
  "lab", "cap"
)

dst.manu <- node_new("output",
  type = "SCES", es = 0, alpha = 1,
  beta = prop.table(c(300 + 400 + 420, 250 + 230)),
  "cc.II", "cc.VA"
)
node_set(dst.manu, "cc.II",
  type = "SCES", es = 0, alpha = 1,
  beta = prop.table(c(300, 400, 420)),
  "agri", "manu", "serv"
)
node_set(dst.manu, "cc.VA",
  type = "SCES", es = 0.5, alpha = 1,
  beta = prop.table(c(250, 230)),
  "lab", "cap"
)

dst.serv <- node_new("output",
  type = "SCES", es = 0, alpha = 1,
  beta = prop.table(c(150 + 250 + 240, 330 + 180)),
  "cc.II", "cc.VA"
)
node_set(dst.serv, "cc.II",
  type = "SCES", es = 0, alpha = 1,
  beta = prop.table(c(150, 250, 240)),
  "agri", "manu", "serv"
)
node_set(dst.serv, "cc.VA",
  type = "SCES", es = 0.5, alpha = 1,
  beta = prop.table(c(330, 180)),
  "lab", "cap"
)

dst.consumer <- node_new("util",
  type = "SCES", es = 0.5, alpha = 1,
  beta = prop.table(c(280, 550, 350)),
  "agri", "manu", "serv"
)

dst.investor <- node_new("util",
  type = "SCES", es = 0.5, alpha = 1,
  beta = prop.table(c(70, 320, 110)),
  "agri", "manu", "serv"
)

ge2.benchmark <- sdm2(list(dst.agri, dst.manu, dst.serv, dst.consumer, dst.investor),
  B = matrix(c(
    1, 0, 0, 0, 0,
    0, 1, 0, 0, 0,
    0, 0, 1, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0
  ), 5, 5, TRUE),
  S0Exg = {
    S0Exg <- matrix(NA, 5, 5)
    S0Exg[4:5, 4] <- c(1080, 600) * (1180 / (1180 + 500))
    S0Exg[4:5, 5] <- c(1080, 600) * (500 / (1180 + 500))
    S0Exg
  },
  names.commodity = c("agri", "manu", "serv", "lab", "cap"),
  names.agent = c("agri", "manu", "serv", "consumer", "investor"),
  numeraire = c("lab")
)

addmargins(ge2.benchmark$D)
addmargins(ge2.benchmark$S)

A General Equilibrium Model based on a 7×4 (Standard) Input-Output Table

Description

This is a general equilibrium model based on a 7×4 standard input-output table. There is no negative number in this standard input-output table, and both the input and output parts are 7×4 matrices. The standard input-output table consists of input and output parts with the same dimensions.

Usage

gemInputOutputTable_7_4(
  IT,
  OT,
  es.agri = 0,
  es.manu = 0,
  es.serv = 0,
  es.hh = 0,
  es.VA.agri = 0.25,
  es.VA.manu = 0.5,
  es.VA.serv = 0.8,
  ...
)

Arguments

Details

Given a 7×4 input-output table, this model calculates the corresponding general equilibrium. This input-output table contains 3 production sectors and 1 household. The household consumes products and supplies labor, capital, stock and tax receipt. Generally speaking, the value of the elasticity of substitution in this model should be between 0 and 1.

Value

A general equilibrium, which is a list with the following elements:

• p - the price vector with labor as numeraire.

• D - the demand matrix, also called the input table. Wherein the benchmark prices are used.

• DV - the demand value matrix, also called the value input table. Wherein the current price is used.

• SV - the supply value matrix, also called the value output table. Wherein the current price is used.

• value.added - the value-added of the three production sectors.

• dstl - the demand structure tree list of sectors.

• ... - some elements returned by the sdm2 function.

Examples

IT2017 <- matrix(c(
  1.47, 6.47, 0.57, 2.51,
  2.18, 76.32, 12.83, 44.20,
  0.82, 19.47, 23.33, 35.61,
  6.53, 13.92, 21.88, 0,
  0.23, 4.05, 6.76, 0,
  0, 6.43, 3.40, 0,
  0.13, 8.87, 10.46, 0
), 7, 4, TRUE)

OT2017 <- matrix(c(
  11.02, 0, 0, 0,
  0, 135.53, 0, 0,
  0, 0, 79.23, 0,
  0, 0, 0, 42.33,
  0, 0, 0, 11.04,
  0.34, 0, 0, 9.49,
  0, 0, 0, 19.46
), 7, 4, TRUE)

rownames(IT2017) <- rownames(OT2017) <-
  c("agri", "manu", "serv", "lab", "cap", "tax", "dividend")
colnames(IT2017) <- colnames(OT2017) <-
  c("sector.agri", "sector.manu", "sector.serv", "sector.hh")

ge <- gemInputOutputTable_7_4(
  IT = IT2017,
  OT = OT2017
)

#### labor supply reduction
OTLSR <- OT2017
OTLSR["lab", "sector.hh"] <- OTLSR["lab", "sector.hh"] * 0.9
geLSR <- gemInputOutputTable_7_4(
  IT = IT2017,
  OT = OTLSR
)

geLSR$z / ge$z
geLSR$p / ge$p

#### capital accumulation
OTCA <- OT2017
OTCA["cap", "sector.hh"] <- OTCA["cap", "sector.hh"] * 1.1
geCA <- gemInputOutputTable_7_4(
  IT = IT2017,
  OT = OTCA
)

geCA$z / ge$z
geCA$p / ge$p

#### technology progress
IT.TP <- IT2017
IT.TP ["lab", "sector.manu"] <-
  IT.TP ["lab", "sector.manu"] * 0.9

geTP <- gemInputOutputTable_7_4(
  IT = IT.TP,
  OT = OT2017
)

geTP$z / ge$z
geTP$p / ge$p

##
IT.TP2 <- IT.TP
IT.TP2 ["cap", "sector.manu"] <-
  IT.TP2["cap", "sector.manu"] * 1.02
geTP2 <- gemInputOutputTable_7_4(
  IT = IT.TP2,
  OT = OT2017
)

geTP2$z / ge$z
geTP2$p / ge$p

##
IT.TP3 <- IT2017
IT.TP3 ["lab", "sector.manu"] <-
  IT.TP3 ["lab", "sector.manu"] * 0.9
IT.TP3 ["lab", "sector.agri"] <-
  IT.TP3 ["lab", "sector.agri"] * 0.8

geTP3 <- gemInputOutputTable_7_4(
  IT = IT.TP3,
  OT = OT2017
)

geTP3$value.added / ge$value.added
prop.table(geTP3$value.added) - prop.table(ge$value.added)

#### demand structure change
IT.DSC <- IT2017
IT.DSC["serv", "sector.hh"] <- IT.DSC ["serv", "sector.hh"] * 1.2

geDSC <- gemInputOutputTable_7_4(
  IT = IT.DSC,
  OT = OT2017
)

geDSC$z[1:3] / ge$z[1:3]
geDSC$p / ge$p

#### tax change
OT.TC <- OT2017
OT.TC["tax", "sector.agri"] <- OT.TC["tax", "sector.agri"] * 2

geTC <- gemInputOutputTable_7_4(
  IT = IT2017,
  OT = OT.TC
)

geTC$z / ge$z
geTC$p / ge$p

##
IT.TC2 <- IT2017
IT.TC2["tax", "sector.manu"] <- IT.TC2["tax", "sector.manu"] * 0.8

geTC2 <- gemInputOutputTable_7_4(
  IT = IT.TC2,
  OT = OT2017
)

geTC2$z / ge$z
geTC2$p / ge$p

A General Equilibrium Model based on an 8×8 Input-Output Table

Description

This is a general equilibrium model based on a 8×8 input-output table.

Usage

gemInputOutputTable_8_8(
  IT,
  OT,
  es.agri = 0,
  es.manu = 0,
  es.serv = 0,
  es.CI = 0,
  es.FT = 0,
  es.VA.agri = 0.25,
  es.VA.manu = 0.5,
  es.VA.serv = 0.8,
  es.prodDM = 0.5,
  ...
)

Arguments

Details

Given an 8×8 input-output table, this model calculates the corresponding general equilibrium. This input-output table contains 3 production sectors, 1 consumption and (temporarily unproductive) investment sector (CI sector), 1 foreign trade sector importing agriculture goods, 1 foreign trade sector importing manufacturing goods, 1 foreign trade sector importing service, 1 foreign trade sector importing bond. There are 8 kinds of commodities (or subjects) in the table, i.e. agriculture product, manufacturing product, service, labor, capital goods, tax, dividend and bond of ROW (i.e. the rest of the world). The CI sector uses products and supplies labor, capital, stock and tax receipt. Generally speaking, the value of the elasticity of substitution in this model should be between 0 and 1.

Value

A general equilibrium, which is a list with the following elements:

• p - the price vector with labor as numeraire.

• D - the demand matrix, also called the input table. Wherein the benchmark prices are used.

• DV - the demand value matrix, also called the value input table. Wherein the current price is used.

• SV - the supply value matrix, also called the value output table. Wherein the current price is used.

• value.added - the value-added of the three production sectors.

• dstl - the demand structure tree list of sectors.

• ... - some elements returned by the CGE::sdm function

Examples

IT17 <- matrix(c(
  1.47, 6.47, 0.57, 2.99, 0.12 * 0.60 / (0.60 + 12.10 + 2.23 + 1.45),
  0.12 * 12.10 / (0.60 + 12.10 + 2.23 + 1.45),
  0.12 * 2.23 / (0.60 + 12.10 + 2.23 + 1.45),
  0.12 * 1.45 / (0.60 + 12.10 + 2.23 + 1.45),

  2.18, 76.32, 12.83, 43, 13.30 * 0.60 / (0.60 + 12.10 + 2.23 + 1.45),
  13.30 * 12.10 / (0.60 + 12.10 + 2.23 + 1.45),
  13.30 * 2.23 / (0.60 + 12.10 + 2.23 + 1.45),
  13.30 * 1.45 / (0.60 + 12.10 + 2.23 + 1.45),


  0.82, 19.47, 23.33, 34.88, 2.96 * 0.60 / (0.60 + 12.10 + 2.23 + 1.45),
  2.96 * 12.10 / (0.60 + 12.10 + 2.23 + 1.45),
  2.96 * 2.23 / (0.60 + 12.10 + 2.23 + 1.45),
  2.96 * 1.45 / (0.60 + 12.10 + 2.23 + 1.45),

  6.53, 13.92, 21.88, 0, 0, 0, 0, 0,
  0.23, 4.05, 6.76, 0, 0, 0, 0, 0,
  0, 6.43, 3.40, 0, 0, 0, 0, 0,
  0.13, 8.87, 10.46, 0, 0, 0, 0, 0,
  0, 0, 0, 1.45, 0, 0, 0, 0
), 8, 8, TRUE)

OT17 <- matrix(c(
  11.02, 0, 0, 0, 0.60, 0, 0, 0,
  0, 135.53, 0, 0, 0, 12.10, 0, 0,
  0, 0, 79.23, 0, 0, 0, 2.23, 0,
  0, 0, 0, 42.33, 0, 0, 0, 0,
  0, 0, 0, 11.04, 0, 0, 0, 0,
  0.34, 0, 0, 9.49, 0, 0, 0, 0,
  0, 0, 0, 19.46, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 1.45
), 8, 8, TRUE)

rownames(IT17) <- rownames(OT17) <-
  c("agri", "manu", "serv", "lab", "cap", "tax", "dividend", "bond.ROW")
colnames(IT17) <- colnames(OT17) <- c(
  "sector.agri", "sector.manu", "sector.serv", "sector.CI",
  "sector.FT.agri", "sector.FT.manu", "sector.FT.serv", "sector.FT.bond.ROW"
)

# the benchmark equilibrium.
ge <- gemInputOutputTable_8_8(
  IT = IT17,
  OT = OT17
)

#### technology progress.
IT.TP <- IT17
IT.TP ["lab", "sector.manu"] <-
  IT.TP ["lab", "sector.manu"] * 0.9

geTP <- gemInputOutputTable_8_8(
  IT = IT.TP,
  OT = OT17
)

geTP$z / ge$z
geTP$p / ge$p
geTP$value.added
prop.table(geTP$value.added) - prop.table(ge$value.added)

#### capital accumulation.
OT.CA <- OT17
OT.CA["cap", "sector.CI"] <- OT.CA["cap", "sector.CI"] * 1.1
geCA <- gemInputOutputTable_8_8(
  IT = IT17,
  OT = OT.CA
)

geCA$z / ge$z
geCA$p / ge$p
geCA$p
geCA$value.added
prop.table(geCA$value.added) - prop.table(ge$value.added)

#### tax change.
OT.TC <- OT17
OT.TC["tax", "sector.agri"] <- OT.TC["tax", "sector.agri"] * 2

geTC <- gemInputOutputTable_8_8(
  IT = IT17,
  OT = OT.TC
)

geTC$z / ge$z
geTC$p / ge$p

##
IT.TC2 <- IT17
IT.TC2["tax", "sector.manu"] <- IT.TC2["tax", "sector.manu"] * 0.8

geTC2 <- gemInputOutputTable_8_8(
  IT = IT.TC2,
  OT = OT17
)

geTC2$z / ge$z
geTC2$p / ge$p

A Leontief-type General Equilibrium Model based on a 3×3 Input-Output Table

Description

Given a 3×3 input-output table (e.g., see Zhang Xin, 2017, Table 2.2.2), this model can be used to calculate the corresponding equilibrium. This input-output table contains two firms and one household. The household consumes products and supplies labor.

Usage

gemInputOutputTable_Leontief_3_3(
  input = matrix(c(200, 300, 100, 150, 320, 530, 250, 380, 0), 3, 3, TRUE),
  output = c(600, 1000, 630)
)

Arguments

Value

A general equilibrium, which is a list with the following elements:

• p - the price vector with labor as numeraire.

• D - the demand matrix, also called the input table. Wherein the benchmark prices are used.

• DV - the demand value matrix, also called the value input table. Wherein the current price is used.

• SV - the supply value matrix, also called the value output table. Wherein the current price is used.

• ... - some elements returned by the CGE::sdm function

References

Zhang Xin. (2017, ISBN: 9787543227637). Principles of Computable General Equilibrium Modeling and Programming (Second Edition). Shanghai: Gezhi Press. (In Chinese)

Examples

x <- 75
gemInputOutputTable_Leontief_3_3(
  input = matrix(c(
    200, 300, 100,
    x, 320, 530,
    250, 380, 0
  ), 3, 3, TRUE),
  output = c(600, 1000, 630)
)

A SCES-type General Equilibrium Model based on an Input-Output Table.

Description

Given a 3×3 input-output table (e.g., see Zhang Xin, 2017, Table 2.2.2), this model can be used to calculate the corresponding equilibrium. This input-output table contains two firms and one household. The household consumes products and supplies labor.

Usage

gemInputOutputTable_SCES_3_3(
  input = matrix(c(200, 300, 100, 150, 320, 530, 250, 380, 0), 3, 3, TRUE),
  output = c(600, 1000, 630),
  es = 0
)

Arguments

Value

A general equilibrium, which is a list with the following elements:

• p - the price vector with labor as numeraire.

• D - the demand matrix, also called the input table. Wherein the benchmark prices are used.

• DV - the demand value matrix, also called the value input table. Wherein the current price is used.

• SV - the supply value matrix, also called the value output table. Wherein the current price is used.

• ... - some elements returned by the CGE::sdm function

References

Zhang Xin. (2017, ISBN: 9787543227637). Principles of Computable General Equilibrium Modeling and Programming (Second Edition). Shanghai: Gezhi Press. (In Chinese)

Examples

x <- 75
gemInputOutputTable_SCES_3_3(
  input = matrix(c(
    200, 300, 100,
    x, 320, 530,
    250, 380, 0
  ), 3, 3, TRUE),
  output = c(600, 1000, 630),
  es = 0.5
)

An Easy General Equilibrium Model based on a 5×4 Input-Output Table (see Zhang Xin, 2017, Table 8.6.1)

Description

This is a general equilibrium model based on a 5×4 input-output table (see Zhang Xin, 2017, Table 8.6.1).

Usage

gemInputOutputTable_easy_5_4(
  IT = cbind(sector.agri = c(agri = 260, manu = 345, serv = 400, lab = 200, cap = 160),
    sector.manu = c(agri = 320, manu = 390, serv = 365, lab = 250, cap = 400),
    sector.serv = c(agri = 150, manu = 390, serv = 320, lab = 400, cap = 210), sector.hh
    = c(agri = 635, manu = 600, serv = 385, lab = 0, cap = 0)),
  supply.labor = 850,
  supply.capital = 770,
  es.agri = 0.2,
  es.manu = 0.3,
  es.serv = 0.1,
  es.VA.agri = 0.25,
  es.VA.manu = 0.5,
  es.VA.serv = 0.8
)

Arguments

Details

Given a 5×4 input-output table (e.g., see Zhang Xin, 2017, Table 8.6.1), this model calculates the corresponding general equilibrium. This input-output table contains 3 production sectors and one household. The household consumes products and supplies labor and capital.

Value

A general equilibrium, which is a list with the following elements:

• p - the price vector with labor as numeraire.

• D - the demand matrix, also called the input table. Wherein the benchmark prices are used.

• DV - the demand value matrix, also called the value input table. Wherein the current price is used.

• SV - the supply value matrix, also called the value output table. Wherein the current price is used.

• ... - some elements returned by the CGE::sdm function

References

Zhang Xin (2017, ISBN: 9787543227637) Principles of Computable General Equilibrium Modeling and Programming (Second Edition). Shanghai: Gezhi Press. (In Chinese)

Examples

sector.agri <- c(260, 345, 400, 200, 160)
sector.manu <- c(320, 390, 365, 250, 400)
sector.serv <- c(150, 390, 320, 400, 210)
sector.hh <- c(635, 600, 100, 0, 0)

IT <- cbind(sector.agri, sector.manu, sector.serv, sector.hh)
rownames(IT) <- c("agri", "manu", "serv", "lab", "cap")

ge <- gemInputOutputTable_easy_5_4(IT)

####
ge <- gemInputOutputTable_easy_5_4(supply.capital = 1870)
prop.table(ge$z[1:3])

An Intertemporal Stochastic Model with a Consumer and Some Banks

Description

An intertemporal stochastic model with a consumer and some banks. In the model the consumer will live for three periods. There is one natural state in the first period, and two natural states in the second and third period.

Usage

gemIntertemporalStochastic_Bank_ThreePeriods(...)

Arguments

Examples

dst.bank1a <- node_new(
  "payoff2.1&2.2",
  type = "Leontief", a = 1,
  "payoff1"
)

dst.bank1b <- node_new(
  "payoff3.1&3.2",
  type = "Leontief", a = 1,
  "payoff1"
)

dst.bank2.1 <- node_new(
  "payoff3.1",
  type = "Leontief", a = 1,
  "payoff2.1"
)

dst.bank2.2 <- node_new(
  "payoff3.2",
  type = "Leontief", a = 1,
  "payoff2.2"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1/3, 1/6, 1/6, 1/6, 1/6),
  "payoff1", "payoff2.1", "payoff2.2", "payoff3.1", "payoff3.2"
)

ge <- sdm2(
  A = list(dst.bank1a, dst.bank1b,dst.bank2.1,dst.bank2.2,dst.consumer),
  B = matrix(c(
    0,   0,0, 0,0,
    1.1, 0,0, 0,0,
    1.1, 0,0, 0, 0,
    0,   1.5, 1.1, 0,0,
    0,   1.5, 0, 1.1,0
  ), 5, 5, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, 1,
    NA, NA, NA, NA, 1,
    NA, NA, NA, NA, 0,
    NA, NA, NA, NA, 0,
    NA, NA, NA, NA, 0
  ), 5, 5, TRUE),
  names.commodity = c("payoff1", "payoff2.1", "payoff2.2", "payoff3.1", "payoff3.2"),
  names.agent = c("bank1a","bank1b","bank2.1","bank2.2", "consumer"),
  numeraire = "payoff1"
)

ge$p
round(ge$D, 4)
round(ge$S, 4)

#### the general equilibrium in the first natural state in period 2
dst.bank2.1 <- node_new(
  "payoff3.1",
  type = "Leontief", a = 1,
  "payoff2.1"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "payoff2.1", "payoff3.1"
)

ge2.1 <- sdm2(
  A = list(dst.bank2.1,
           dst.consumer),
  B = matrix(c(
    0,   0,
    1.1, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, 1.3084,
    NA, 0.4599
  ), 2, 2, TRUE),
  names.commodity = c("payoff2.1", "payoff3.1"),
  names.agent = c("bank2.1", "consumer"),
  numeraire = "payoff2.1"
)

ge2.1$p
round(ge2.1$D, 4)
round(ge2.1$S, 4)

## the general equilibrium in an unanticipated natural state in period 2
ge2.3 <- sdm2(
  A = list(dst.bank2.1,
           dst.consumer),
  B = matrix(c(
    0,   0,
    1.1, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, 0.4,
    NA, 0.4599
  ), 2, 2, TRUE),
  names.commodity = c("payoff2.1", "payoff3.1"),
  names.agent = c("bank2.1", "consumer"),
  numeraire = "payoff2.1"
)

ge2.3$p
round(ge2.3$D, 4)
round(ge2.3$S, 4)

An Intertemporal Stochastic Model with a Consumer and a Bank

Description

An intertemporal stochastic model with a consumer and a bank. In this model the consumer will live for two periods. There is one natural state in the first period, and two natural states in the second period.

The consumer has an intertemporal stochastic utility function of the Cobb-Douglas (CD) type, x_1^{1/2} x_2^{1/6} x_3^{1/3}, where x_1, x_2, and x_3 represent the payoffs in three different states of nature, respectively. The ratio of the share coefficients for the two future states of nature is equal to the ratio of their corresponding probabilities. The share coefficient for the present is the same as that for the future.

Usage

gemIntertemporalStochastic_Bank_TwoPeriods(...)

Arguments

Examples

#### (A) A savings bank.
Ra <- 1.2 # the interest rate coefficient in the first natural state in the future
Rb <- 1.1 # the interest rate coefficient in the second natural state in the future

# When the savings bank invests one unit of payoff 1, it can
# produce Ra units of payoff 2 and Rb units of payoff 3.
dst.bank <- node_new(
  "output",
  type = "Leontief", a = 1,
  "payoff1"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 2, 1 / 6, 1 / 3),
  "payoff1", "payoff2", "payoff3"
)

ge <- sdm2(
  A = list(dst.bank, dst.consumer),
  B = matrix(c(
    0, 0,
    Ra, 0,
    Rb, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, 1,
    NA, 0,
    NA, 2
  ), 3, 2, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("bank", "consumer"),
  numeraire = "payoff1",
)

ge$p
unname(Ra * ge$p[2] + Rb * ge$p[3])
# The amount of savings in equilibrium is 0.2995.
addmargins(ge$D, 2)
addmargins(ge$S, 2)

## Solve with the optimization method and Rsolnp package.
# library(Rsolnp)
# Ra <- 1.2
# Rb <- 1.1
# payoff <- c(1, 0, 2)
# # The loss function is used to calculate the loss (i.e.,
# # the negative utility) for a given amount of savings.
# loss <- function(savings) {
#   utility <- prod(c(payoff[1] - savings, payoff[2] +
#                       Ra * savings, payoff[3] + Rb * savings)^wt)
#   return(-utility)
# }
#
# result <- solnp(0.5, loss, LB = 0, UB = 1)
# result$pars
#
# x <- rbind(payoff[1] - result$pars, payoff[2] + Ra * result$pars, payoff[3] + Rb * result$pars)
# mu <- marginal_utility(x, diag(3), uf = function(x) prod(x^wt))
# mu / mu[1]

#### (B) A lending bank.
Ra <- 1.2
Rb <- 1.1

# To produce one unit of payoff 1, the lending bank needs
# to invest Ra units of payoff 2 and Rb units of payoff 3.
dst.bank <- node_new(
  "payoff1",
  type = "Leontief", a = c(Ra, Rb),
  "payoff2", "payoff3"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 2, 1 / 6, 1 / 3),
  "payoff1", "payoff2", "payoff3"
)

ge <- sdm2(
  A = list(dst.bank, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, 0,
    NA, 1,
    NA, 2
  ), 3, 2, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("bank", "consumer"),
  numeraire = "payoff1"
)

ge$p
unname(Ra * ge$p[2] + Rb * ge$p[3])
addmargins(ge$D, 2)
# In equilibrium, the bank lends out 0.5645 units of payoff in period 1.
addmargins(ge$S, 2)

## Solve with the optimization method and Rsolnp package.
# library(Rsolnp)
# Ra <- 1.2
# Rb <- 1.1
# payoff <- c(0, 1, 2)
# loss <- function(savings) {
#   utility <- prod(c(payoff[1] - savings, payoff[2] +
#                       Ra * savings, payoff[3] + Rb * savings)^wt)
#   return(-utility)
# }
#
# result <- solnp(-0.5, loss, LB = -2, UB = 0)
# result$pars
#
# x <- rbind(payoff[1] - result$pars, payoff[2] + Ra * result$pars, payoff[3] + Rb * result$pars)
# mu <- marginal_utility(x, diag(3), uf = function(x) prod(x^wt))
# mu / mu[1]

A Three-Period Intertemporal Stochastic Equilibrium Model with a Consumer and a Type of Firm

Description

An intertemporal stochastic equilibrium model of three periods with a consumer and a type of firm. The consumer will live for three periods and has a von Neumann-Morgenstern expected utility function. There is one natural state in the first period, two natural states in the second period and two natural states in the third period.

Usage

gemIntertemporalStochastic_ThreePeriods_2_2(...)

Arguments

Examples

dst.firm1 <- node_new(
  "prod2",
  type = "CD", alpha = 2,
  beta = c(0.5, 0.5),
  "lab1", "prod1"
)

dst.firm2.1 <- node_new(
  "prod3.1",
  type = "CD", alpha = 2,
  beta = c(0.5, 0.5),
  "prod2.1", "lab2.1"
)

dst.firm2.2 <- node_new(
  "prod3.2",
  type = "CD", alpha = 1,
  beta = c(0.4, 0.6),
  "prod2.2", "lab2.2"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = rep(1 / 5, 5),
  "prod1", "prod2.1", "prod2.2",
  "prod3.1", "prod3.2"
)

ge <- sdm2(
  A = c(
    dst.firm1, dst.firm2.1, dst.firm2.2,
    dst.consumer
  ),
  B = matrix(c(
    0, 0, 0, 0,
    1, 0, 0, 0,
    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 1, 0,
    0, 0, 0, 0,
    0, 0, 0, 0,
    0, 0, 0, 0
  ), 8, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, 50,
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, NA, 100,
    NA, NA, NA, 100,
    NA, NA, NA, 100
  ), 8, 4, TRUE),
  names.commodity = c(
    "prod1", "prod2.1", "prod2.2",
    "prod3.1", "prod3.2",
    "lab1", "lab2.1", "lab2.2"
  ),
  names.agent = c(
    "firm1", "firm2.1", "firm2.2",
    "consumer"
  ),
  numeraire = "lab1",
  policy = makePolicyMeanValue(30),
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

Some Examples of a Two-Period Intertemporal Stochastic Equilibrium Model

Description

Some examples of a two-period intertemporal equilibrium model with two types of commodities (i.e. product and labor) and one firm. In the second period there are two states of nature, in which the firm has different productivity.

Usage

gemIntertemporalStochastic_TwoPeriods(...)

Arguments

Examples

#### an examples with a consumer and a firm.
alpha1 <- 1
alpha2 <- 2

supply.lab <- 100
supply.prod1 <- 30

dst.firm <- node_new(
  "prod2",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "prod1", "lab1"
)

dst.consumer <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.25, 0.25),
  "prod1", "prod2.1", "prod2.2"
)

ge <- sdm2(
  A = c(dst.firm, dst.consumer),
  B = matrix(c(
    0, 0,
    0, 0,
    alpha1, 0,
    alpha2, 0
  ), 4, 2, TRUE),
  S0Exg = matrix(c(
    NA, supply.prod1,
    NA, supply.lab,
    NA, NA,
    NA, NA
  ), 4, 2, TRUE),
  names.commodity = c("prod1", "lab1", "prod2.1", "prod2.2"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod1"
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

#### an examples with two types of consumer and a firm.
dst.firm <- node_new(
  "prod2",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "prod1", "lab1"
)

dst.consumer1 <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.4, 0.1),
  "prod1", "prod2.1", "prod2.2"
)

dst.consumer2 <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.25, 0.25),
  "prod1", "prod2.1", "prod2.2"
)

ge <- sdm2(
  A = c(dst.firm, dst.consumer1, dst.consumer2),
  B = matrix(c(
    0, 0, 0,
    0, 0, 0,
    1, 0, 0,
    2, 0, 0
  ), 4, 3, TRUE),
  S0Exg = matrix(c(
    NA, 30, 30,
    NA, 100, 100,
    NA, NA, NA,
    NA, NA, NA
  ), 4, 3, TRUE),
  names.commodity = c("prod1", "lab1", "prod2.1", "prod2.2"),
  names.agent = c("firm", "consumer1", "consumer2"),
  numeraire = "prod1"
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

An Example of a 1-by-2 Intertemporal Equilibrium Model

Description

An example of an intertemporal equilibrium model with one type of commodity (i.e., product) and two types of agents (i.e., a firm with an AK production function and a consumer).

Usage

gemIntertemporal_1_2(...)

Arguments

Examples

eis <- 0.5 # the elasticity of intertemporal substitution of the consumer
Gamma.beta <- 0.97 # the subjective discount factor of the consumer
alphaK <- 1.1 # the parameter of the AK production function

np <- 5 # the number of economic periods

n <- np # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- paste0("prod", 1:np)
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("prod", 1:np), "consumer"] <- 100

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "Leontief",
    a = 1 / alphaK,
    paste0("prod", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1"
)

ge$p
ge$z
ge$D
ge$S
growth_rate(ge$D[, m])
(Gamma.beta * alphaK)^eis - 1

Some Examples of a 2-by-2 Intertemporal Equilibrium Model

Description

Some examples of an intertemporal equilibrium model with two types of commodities and two types of agents.

In these examples, there is an np-period-lived consumer maximizing intertemporal utility, and there is a type of firm which produces from period 1 to np-1. There are two types of commodities, i.e. product and labor. Assume the consumer has some product in the first period. That is, the product supply in the first period is an exogenous variable.

Usage

gemIntertemporal_2_2(...)

Arguments

Examples

#### an example with a Cobb-Douglas intertemporal utility function
np <- 5 # the number of economic periods
y1 <- 150 # the initial product supply

n <- 2 * np - 1 # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)))
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100
S0Exg["prod1", "consumer"] <- y1

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD",
    alpha = 2, beta = c(0.5, 0.5),
    paste0("prod", k), paste0("lab", k)
  )
}

dst.consumer.CD <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = prop.table(rep(1, np)),
  paste0("prod", 1:np)
)

f <- function(dstl) {
  sdm2(
    A = dstl,
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "prod1",
    ts = TRUE
  )
}

ge <- f(c(dstl.firm, dst.consumer.CD))

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

## an example with a Leontief intertemporal utility function
dst.consumer.Leontief <- node_new(
  "util",
  type = "Leontief",
  a = rep(1, np),
  paste0("prod", 1:np)
)

ge2 <- f(c(dstl.firm, dst.consumer.Leontief))

ge2$p
ge2$z
ge2$D
ge2$S
ge2$DV
ge2$SV

## Assume that the consumer has a CES (i.e. CRRA) intertemporal utility function.
# eis is the elasticity of intertemporal substitution.
# Gamma.beta is the subjective discount factor.
f2 <- function(eis = 1, Gamma.beta = 1, head.tail.adjustment = "none") {
  dst.consumer <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
    paste0("prod", 1:np)
  )

  ge <- sdm2(
    A = c(dstl.firm, dst.consumer),
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "prod1",
    ts = TRUE,
    policy = makePolicyHeadTailAdjustment(head.tail.adjustment, np = np)
  )

  list(
    p = ge$p, z = ge$z,
    D = addmargins(ge$D, 2), S = addmargins(ge$S, 2),
    DV = addmargins(ge$DV), SV = addmargins(ge$SV)
  )
}

f2(Gamma.beta = 0.9)
f2(Gamma.beta = 0.9, head.tail.adjustment = "both") # the steady state in the worldsheet
f2(Gamma.beta = 1.25, head.tail.adjustment = "both") # the steady state in the worldsheet
f2(eis = 2, Gamma.beta = 0.9)

Some Examples of Intertemporal Models with One Consumer and Two Types of Firms

Description

Some examples of intertemporal models with one consumer and two types of firms. There are three types of commodities (i.e. corn, iron and labor). The consumer may consume corn and iron in each period, and may have a nested intertemporal utility function.

Usage

gemIntertemporal_3_3(...)

Arguments

References

Zen Xiangjin (1995, ISBN: 7030046560). Basics of Economic Cybernetics. Beijing: Science Press. (In Chinese)

Examples

#### an example with a nested intertemporal utility function
np <- 5 # the number of economic periods

n <- 3 * np - 1 # the number of commodity kinds
m <- 2 * (np - 1) + 1 # the number of agent kinds

names.commodity <- c(
  paste0("corn", 1:np),
  paste0("iron", 1:np),
  paste0("lab", 1:(np - 1))
)
names.agent <- c(
  paste0("firm.corn", 1:(np - 1)),
  paste0("firm.iron", 1:(np - 1)),
  "consumer"
)

## the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100
S0Exg["corn1", "consumer"] <- 25
S0Exg["iron1", "consumer"] <- 100

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("corn", k + 1), paste0("firm.corn", k)] <-
    B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
}

dstl.firm.corn <- dstl.firm.iron <- list()
for (k in 1:(np - 1)) {
  dstl.firm.corn[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 1, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )

  dstl.firm.iron[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 2, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = prop.table(rep(1, np)),
  paste0("cc", 1:np)
)
for (k in 1:np) {
  node_set(
    dst.consumer,
    paste0("cc", k),
    type = "CD", alpha = 1, beta = c(0.5, 0.5),
    paste0("corn", k), paste0("iron", k)
  )
}

ge <- sdm2(
  A = c(dstl.firm.corn, dstl.firm.iron, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

#### an example with a non-nested intertemporal utility function
np <- 3 # the number of economic periods

## There are np types of corn, np-1 types of iron and np-1 types of labor.
## There are np-1 corn firms, np-2 iron firms and one consumer.
n <- 3 * np - 2
m <- 2 * np - 2

names.commodity <- c(
  paste0("corn", 1:np),
  paste0("iron", 1:(np - 1)),
  paste0("lab", 1:(np - 1))
)
names.agent <- c(
  paste0("firm.corn", 1:(np - 1)),
  paste0("firm.iron", 1:(np - 2)),
  "consumer"
)

## the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100
S0Exg["corn1", "consumer"] <- 25
S0Exg["iron1", "consumer"] <- 100

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("corn", k + 1), paste0("firm.corn", k)] <- 1
}
for (k in 1:(np - 2)) {
  B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
}

dstl.firm.corn <- dstl.firm.iron <- list()
for (k in 1:(np - 1)) {
  dstl.firm.corn[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 1, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )
}

for (k in seq_along(np:(2 * np - 3))) {
  dstl.firm.iron[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 2, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = prop.table(rep(1, np)),
  paste0("corn", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm.corn, dstl.firm.iron, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

#### an example of Zeng (1995, page 227)
ic1 <- 1 / 10 # input coefficient
ic2 <- 1 / 7
dc1 <- 2 / 3 # depreciation coefficient
dc2 <- 9 / 10

ge <- sdm2(
  A = {
    # corn, iron1, iron2, iron3, iron4
    a1.1 <- c(0, ic1, 0, 0, 0)
    a1.2 <- c(0, ic2, 0, 0, 0)
    a2.1 <- c(0, 0, ic1, 0, 0)
    a2.2 <- c(0, 0, ic2, 0, 0)
    a3.1 <- c(0, 0, 0, ic1, 0)
    a3.2 <- c(0, 0, 0, ic2, 0)
    a4.1 <- c(0, 0, 0, 0, ic1)
    a4.2 <- c(0, 0, 0, 0, ic2)

    a.consumer <- c(1, 0, 0, 0, 0)

    cbind(a1.1, a1.2, a2.1, a2.2, a3.1, a3.2, a4.1, a4.2, a.consumer)
  },
  B = {
    b1.1 <- c(1, 0, ic1 * dc1, 0, 0)
    b1.2 <- c(1, 0, ic2 * dc2, 0, 0)
    b2.1 <- c(1, 0, 0, ic1 * dc1, 0)
    b2.2 <- c(1, 0, 0, ic2 * dc2, 0)
    b3.1 <- c(1, 0, 0, 0, ic1 * dc1)
    b3.2 <- c(1, 0, 0, 0, ic2 * dc2)
    b4.1 <- c(1, 0, 0, 0, 0)
    b4.2 <- c(1, 0, 0, 0, 0)
    b.consumer <- c(0, 0, 0, 0, 0)

    cbind(b1.1, b1.2, b2.1, b2.2, b3.1, b3.2, b4.1, b4.2, b.consumer)
  },
  S0Exg = {
    tmp <- matrix(NA, 5, 9)
    tmp[2, 9] <- 100
    tmp
  },
  names.commodity = c("corn", paste0("iron", 1:4)),
  names.agent = c(paste0("firm", 1:8), "consumer"),
  numeraire = "corn",
  policy = makePolicyMeanValue(30),
  priceAdjustmentVelocity = 0.05,
  maxIteration = 1,
  numberOfPeriods = 1000,
  ts = TRUE
)

matplot(ge$ts.z, type = "l")

ge$p
ge$z
ge$D
ge$S

An Intertemporal Model with Two Consumers and Two Types of Firms

Description

An intertemporal (timeline) model with two consumers and two types of firms.

Usage

gemIntertemporal_3_4(...)

Arguments

Examples

#### an example with Cobb-Douglas production functions and intertemporal utility functions.
np <- 5 # the number of economic periods

n <- 3 * np - 1 # the number of commodity kinds
m <- 2 * (np - 1) + 2 # the number of agent kinds

names.commodity <- c(
  paste0("corn", 1:np),
  paste0("iron", 1:np),
  paste0("lab", 1:(np - 1))
)
names.agent <- c(
  paste0("firm.corn", 1:(np - 1)),
  paste0("firm.iron", 1:(np - 1)),
  "consumer1", "consumer2"
)

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), c("consumer1", "consumer2")] <- 100
S0Exg["corn1", c("consumer1", "consumer2")] <- 25
S0Exg["iron1", c("consumer1", "consumer2")] <- 100

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("corn", k + 1), paste0("firm.corn", k)] <-
    B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
}

dstl.firm.corn <- dstl.firm.iron <- list()
for (k in 1:(np - 1)) {
  dstl.firm.corn[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 1, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )

  dstl.firm.iron[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 2, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )
}

dst.consumer1 <- node_new(
  "util",
  type = "CD", alpha = 1, beta = prop.table(rep(1, np)),
  paste0("corn", 1:np)
)

dst.consumer2 <- node_new(
  "util",
  type = "CD", alpha = 1, beta = prop.table(rep(1, np)),
  paste0("cc", 1:np)
)
for (k in 1:np) {
  node_set(
    dst.consumer2,
    paste0("cc", k),
    type = "CD", alpha = 1, beta = c(0.5, 0.5),
    paste0("corn", k), paste0("iron", k)
  )
}

ge <- sdm2(
  A = c(dstl.firm.corn, dstl.firm.iron, dst.consumer1, dst.consumer2),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

An Intertemporal Model with Land, Two Consumers and Two Types of Firms

Description

An (intertemporal) timeline model with two consumers (i.e. a laborer and a landowner) and two types of firms (i.e. wheat producers and iron producers). There are four commodities in the model, namely wheat, iron, labor and land.

Usage

gemIntertemporal_4_4(...)

Arguments

Examples

np <- 15 # the number of economic periods
alpha.firm.wheat <- rep(5, np - 1)
alpha.firm.iron <- rep(5, np - 1)

Gamma.beta <- 0.97 # 1, 1.03 # the subjective discount factor of consumers
eis <- 0.5 # the elasticity of intertemporal substitution  of consumers
y1.wheat <- 100 # 126, 129.96
y1.iron <- 30 # 40.59, 43.47

gr <- 0 # the growth rate in the steady state equilibrium

last.beta.laborer <- 0
last.beta.landowner <- 0

names.commodity <- c(
  paste0("wheat", 1:np),
  paste0("iron", 1:np),
  paste0("lab", 1:(np - 1)),
  paste0("land", 1:(np - 1))
)

names.agent <- c(
  paste0("firm", 1:(np - 1), ".wheat"), paste0("firm", 1:(np - 1), ".iron"),
  "laborer", "landowner"
)

f <- function(policy = NULL) {
  n <- length(names.commodity) # the number of commodity kinds
  m <- length(names.agent) # the number of agent kinds

  # the exogenous supply matrix.
  S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
  S0Exg["wheat1", "laborer"] <- y1.wheat
  S0Exg["iron1", "landowner"] <- y1.iron
  S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 * (1 + gr)^(0:(np - 2)) # the supply of labor
  S0Exg[paste0("land", 1:(np - 1)), "landowner"] <- 100 * (1 + gr)^(0:(np - 2)) # the supply of land

  # the output coefficient matrix.
  B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
  for (k in 1:(np - 1)) {
    B[paste0("wheat", k + 1), paste0("firm", k, ".wheat")] <- 1
    B[paste0("iron", k + 1), paste0("firm", k, ".iron")] <- 1
  }

  dstl.firm.wheat <- dstl.firm.iron <- list()
  for (k in 1:(np - 1)) {
    dstl.firm.wheat[[k]] <- node_new(
      "prod",
      type = "CES", es = 0.8,
      alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
      paste0("iron", k), paste0("lab", k), paste0("land", k)
    )

    dstl.firm.iron[[k]] <- node_new(
      "prod",
      type = "CES", es = 0.8,
      alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2),
      paste0("iron", k), paste0("lab", k), paste0("land", k)
    )
  }

  tmp.beta <- Gamma.beta^(1:(np - 1))
  tmp.beta <- tmp.beta / tmp.beta[np - 1]
  tmp.beta <- c(tmp.beta, last.beta.laborer)
  dst.laborer <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(tmp.beta),
    paste0("cc", 1:(np - 1)), paste0("wheat", np)
  )
  for (k in 1:(np - 1)) {
    node_set(dst.laborer, paste0("cc", k),
             type = "CES", es = 1,
             alpha = 1, beta = c(0.4, 0.4, 0.2),
             paste0("wheat", k), paste0("lab", k), paste0("land", k)
    )
  }

  tmp.beta <- Gamma.beta^(1:(np - 1))
  tmp.beta <- tmp.beta / tmp.beta[np - 1]
  tmp.beta <- c(tmp.beta, last.beta.landowner)
  dst.landowner <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(tmp.beta),
    paste0("cc", 1:(np - 1)), paste0("iron", np)
  )
  for (k in 1:(np - 1)) {
    node_set(dst.landowner, paste0("cc", k),
             type = "CES", es = 1,
             alpha = 1, beta = c(0.2, 0.4, 0.4),
             paste0("wheat", k), paste0("lab", k), paste0("land", k)
    )
  }
  ge <- sdm2(
    A = c(dstl.firm.wheat, dstl.firm.iron, Clone(dst.laborer), Clone(dst.landowner)),
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "lab1",
    policy = policy,
    ts = TRUE,
    maxIteration = 1,
    numberOfPeriods = 1000,
    priceAdjustmentVelocity = 0.05
  )

  plot(ge$z[1:(np - 1)],
       type = "o", pch = 20, ylab = "production level",
       xlab = "time", ylim = range(ge$z[1:(2 * np - 2)])
  )
  lines(ge$z[np:(2 * np - 2)], type = "o", pch = 21)
  legend("bottom", c("wheat", "iron"), pch = 20:21)

  invisible(ge)
}

ge <- f()
plot(2:(np - 1), ge$z[1:(np - 2)],
     type = "o", pch = 20, ylab = "production output",
     xlab = "time", ylim = range(ge$z[1:(2 * np - 2)])
)
lines(2:(np - 1), ge$z[np:(2 * np - 3)], type = "o", pch = 21)
legend("bottom", c("wheat", "iron"), pch = 20:21)

## Compute the steady-state equilibrium based on head and tail adjustments.
policyHeadAdjustment <- makePolicyHeadAdjustment(
  ind = rbind(
    c(
      which(names.commodity == "wheat1"), which(names.agent == "laborer"),
      which(names.commodity == "wheat2"), which(names.agent == "firm1.wheat")
    ),
    c(
      which(names.commodity == "iron1"), which(names.agent == "landowner"),
      which(names.commodity == "iron2"), which(names.agent == "firm1.iron")
    )
  ),
  gr = gr
)
policyTailAdjustment <- makePolicyTailAdjustment(
  ind = rbind(
    c(which(names.agent == paste0("firm", np - 1, ".wheat")), which(names.agent == "laborer")),
    c(which(names.agent == paste0("firm", np - 1, ".iron")), which(names.agent == "landowner"))
  ),
  gr = gr
)

f(list(policyHeadAdjustment, policyTailAdjustment))$z

## the corresponding sequential model with the same steady-state equilibrium.
dividend.rate <- sserr(eis, Gamma.beta, prepaid = TRUE)

dst.firm.wheat <- node_new("prod",
                           type = "FIN", rate = c(1, dividend.rate),
                           "cc1", "equity.share.wheat"
)
node_set(dst.firm.wheat, "cc1",
         type = "CES", es = 0.8,
         alpha = 5, beta = c(0.2, 0.4, 0.4),
         "iron", "lab", "land"
)

dst.firm.iron <- node_new("prod",
                          type = "FIN", rate = c(1, dividend.rate),
                          "cc1", "equity.share.iron"
)
node_set(dst.firm.iron, "cc1",
         type = "CES", es = 0.8,
         alpha = 5, beta = c(0.4, 0.4, 0.2),
         "iron", "lab", "land"
)

dst.laborer <- node_new("util",
                        type = "CES", es = 1,
                        alpha = 1, beta = c(0.4, 0.4, 0.2),
                        "wheat", "lab", "land"
)

dst.landowner <- node_new("util",
                          type = "CES", es = 1,
                          alpha = 1, beta = c(0.2, 0.4, 0.4),
                          "wheat", "lab", "land"
)

ge <- sdm2(
  A = list(dst.firm.wheat, dst.firm.iron, dst.laborer, dst.landowner),
  B = matrix(c(
    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 0, 0,
    0, 0, 0, 0,
    0, 0, 0, 0,
    0, 0, 0, 0
  ), 6, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, 100, NA,
    NA, NA, NA, 100,
    NA, NA, 100, NA,
    NA, NA, NA, 100
  ), 6, 4, TRUE),
  names.commodity = c(
    "wheat", "iron", "lab", "land",
    "equity.share.wheat", "equity.share.iron"
  ),
  names.agent = c("firm.wheat", "firm.iron", "laborer", "landowner"),
  numeraire = "lab"
)

ge$p
ge$z
ge$D
ge$S

# f(policyTailAdjustment)

## an anticipated technological shock
# np <- 50 # the number of economic periods
# alpha.firm.wheat <- rep(5, np - 1)
# alpha.firm.iron <- rep(5, np - 1)
# alpha.firm.iron[25] <- 10
# names.commodity <- c(
#   paste0("wheat", 1:np),
#   paste0("iron", 1:np),
#   paste0("lab", 1:(np - 1)),
#   paste0("land", 1:(np - 1))
# )
# names.agent <- c(
#   paste0("firm", 1:(np - 1), ".wheat"), paste0("firm", 1:(np - 1), ".iron"),
#   "laborer", "landowner"
# )
#
# ge <- f()
# plot(2:(np - 1), ge$z[1:(np - 2)],
#      type = "o", pch = 20, ylab = "production output",
#      xlab = "time", ylim = range(ge$z[1:(2 * np - 2)])
# )
# lines(2:(np - 1), ge$z[np:(2 * np - 3)], type = "o", pch = 21)
# legend("bottom", c("wheat", "iron"), pch = 20:21)
# grid()

# #### a structural transformation path
# np <- 50
# tax.rate <- 0.1 # the tax rate imposed on income from land and labor income.
# tax.time <- 1 # tax.time <- 20
#
# alpha.firm.wheat <- rep(5, np - 1)
# # Suppose the tax rate is high enough so that the iron
# # producer's efficiency coefficient immediately rises to 10.
# alpha.firm.iron <- c()
# for (k in 1:(np - 1)) {
#   alpha.firm.iron[k] <- ifelse(k <= tax.time, 5, 10)
# }
#
# Gamma.beta <- 0.97 # 1, 1.03 # the subjective discount factor of consumers
# eis <- 0.5 # the elasticity of intertemporal substitution  of consumers
# y1.wheat <- 100
# y1.iron <- 30
# last.beta.laborer <- 0
# last.beta.landowner <- 0
#
# names.commodity <- c(
#   paste0("wheat", 1:np),
#   paste0("iron", 1:np),
#   paste0("lab", 1:(np - 1)),
#   paste0("land", 1:(np - 1))
# )
# names.agent <- c(
#   paste0("firm", 1:(np - 1), ".wheat"), paste0("firm", 1:(np - 1), ".iron"),
#   "laborer", "landowner"
# )
#
# n <- length(names.commodity) # the number of commodity kinds
# m <- length(names.agent) # the number of agent kinds
#
# # the exogenous supply matrix.
# S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
# S0Exg["wheat1", "laborer"] <- y1.wheat
# S0Exg["iron1", "landowner"] <- y1.iron
# S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 # the supply of labor
# S0Exg[paste0("land", 1:(np - 1)), "landowner"] <- 100 # the supply of land
#
# S0Exg[paste0("lab", tax.time), paste0("firm", tax.time, ".iron")] <-
#   S0Exg[paste0("lab", tax.time), "laborer"] * tax.rate
# S0Exg[paste0("land", tax.time), paste0("firm", tax.time, ".iron")] <-
#   S0Exg[paste0("land", tax.time), "landowner"] * tax.rate
#
# S0Exg[paste0("lab", tax.time), "laborer"] <-
#   S0Exg[paste0("lab", tax.time), "laborer"] * (1 - tax.rate)
# S0Exg[paste0("land", tax.time), "landowner"] <-
#   S0Exg[paste0("land", tax.time), "landowner"] * (1 - tax.rate)
#
# # the output coefficient matrix.
# B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
# for (k in 1:(np - 1)) {
#   B[paste0("wheat", k + 1), paste0("firm", k, ".wheat")] <- 1
#   B[paste0("iron", k + 1), paste0("firm", k, ".iron")] <- 1
# }
#
# dstl.firm.wheat <- dstl.firm.iron <- list()
# for (k in 1:(np - 1)) {
#   dstl.firm.wheat[[k]] <- node_new(
#     "prod",
#     type = "CES", es = 0.8,
#     alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
#     paste0("iron", k), paste0("lab", k), paste0("land", k)
#   )
#
#   dstl.firm.iron[[k]] <- node_new(
#     "prod",
#     type = "CES", es = 0.8,
#     alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2),
#     paste0("iron", k), paste0("lab", k), paste0("land", k)
#   )
# }
#
# tmp.beta <- Gamma.beta^(1:(np - 1))
# tmp.beta <- tmp.beta / tmp.beta[np - 1]
# tmp.beta <- c(tmp.beta, last.beta.laborer)
# dst.laborer <- node_new(
#   "util",
#   type = "CES", es = eis,
#   alpha = 1, beta = prop.table(tmp.beta),
#   paste0("cc", 1:(np - 1)), paste0("wheat", np)
# )
# for (k in 1:(np - 1)) {
#   node_set(dst.laborer, paste0("cc", k),
#            type = "CES", es = 1,
#            alpha = 1, beta = c(0.4, 0.4, 0.2),
#            paste0("wheat", k), paste0("lab", k), paste0("land", k)
#   )
# }
#
# tmp.beta <- Gamma.beta^(1:(np - 1))
# tmp.beta <- tmp.beta / tmp.beta[np - 1]
# tmp.beta <- c(tmp.beta, last.beta.landowner)
# dst.landowner <- node_new(
#   "util",
#   type = "CES", es = eis,
#   alpha = 1, beta = prop.table(tmp.beta),
#   paste0("cc", 1:(np - 1)), paste0("iron", np)
# )
# for (k in 1:(np - 1)) {
#   node_set(dst.landowner, paste0("cc", k),
#            type = "CES", es = 1,
#            alpha = 1, beta = c(0.2, 0.4, 0.4),
#            paste0("wheat", k), paste0("lab", k), paste0("land", k)
#   )
# }
# ge <- sdm2(
#   A = c(dstl.firm.wheat, dstl.firm.iron, Clone(dst.laborer), Clone(dst.landowner)),
#   B = B,
#   S0Exg = S0Exg,
#   names.commodity = names.commodity,
#   names.agent = names.agent,
#   numeraire = "lab1",
#   ts = TRUE,
#   maxIteration = 1,
#   numberOfPeriods = 1000,
#   priceAdjustmentVelocity = 0.05
# )
#
# plot(2:(np - 1), ge$z[1:(np - 2)],
#      type = "o", pch = 20, ylab = "production output",
#      xlab = "time", ylim = range(ge$z[1:(2 * np - 2)])
# )
# lines(2:(np - 1), ge$z[np:(2 * np - 3)], type = "o", pch = 21)
# legend("bottom", c("wheat", "iron"), pch = 20:21)

Some Intertemporal (Timeline and Time-circle) Models with Land, Two Consumers, and Three Types of Firms

Description

Some intertemporal (timeline and time-circle) models with two consumers (i.e. a laborer and a landowner) and three types of firms (i.e. wheat producers, iron producers and iron leaser). Here the iron leasing firm is actually a quasi-firm, which does not require primary factors such as labor and land in its production process. There are four commodities in the model, namely wheat, iron, iron leased out as a capital good, labor and land.

Usage

gemIntertemporal_5_5(...)

Arguments

Examples

#### a timeline model.
np <- 15 # the number of economic periods
gr <- 0 # the growth rate in the steady state equilibrium
eis <- 1 # the elasticity of intertemporal substitution  of consumers
Gamma.beta <- 0.97 # the subjective discount factor of consumers
last.beta.laborer <- 0
last.beta.landowner <- 0
depreciation.rate <- 0.06

alpha.firm.wheat <- rep(5, np - 1)
alpha.firm.iron <- rep(5, np - 1)

y1.wheat <- 200
y1.iron <- 100

names.commodity <- c(
  paste0("wheat", 1:np),
  paste0("iron", 1:np),
  paste0("cap", 1:(np - 1)),
  paste0("lab", 1:(np - 1)),
  paste0("land", 1:(np - 1))
)
names.agent <- c(
  paste0("firm.wheat", 1:(np - 1)), paste0("firm.iron", 1:(np - 1)),
  paste0("quasifirm.cap", 1:(np - 1)), # a quasifirm
  "laborer", "landowner"
)

n <- length(names.commodity) # the number of commodity kinds, i.e. 5 * np - 3
m <- length(names.agent) # the number of agent kinds, i.e. 3 * np - 1

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg["wheat1", "laborer"] <- y1.wheat
S0Exg["iron1", "landowner"] <- y1.iron
S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 * (1 + gr)^(0:(np - 2)) # the supply of labor
S0Exg[paste0("land", 1:(np - 1)), "landowner"] <- 100 * (1 + gr)^(0:(np - 2)) # the supply of land

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("wheat", k + 1), paste0("firm.wheat", k)] <- 1
  B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
  B[paste0("cap", k), paste0("quasifirm.cap", k)] <- 1
  B[paste0("iron", k + 1), paste0("quasifirm.cap", k)] <- 1 - depreciation.rate
}

dstl.firm.wheat <- dstl.firm.iron <- dstl.quasifirm.cap <- list()
for (k in 1:(np - 1)) {
  dstl.firm.wheat[[k]] <- node_new(
    "prod",
    type = "CES", es = 1,
    alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
    paste0("cap", k), paste0("lab", k), paste0("land", k)
  )

  dstl.firm.iron[[k]] <- node_new(
    "prod",
    type = "CES", es = 1,
    alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2),
    paste0("cap", k), paste0("lab", k), paste0("land", k)
  )

  dstl.quasifirm.cap[[k]] <- node_new(
    "output",
    type = "Leontief", a = 1,
    paste0("iron", k)
  )
}

tmp.beta <- Gamma.beta^(1:(np - 1))
tmp.beta <- tmp.beta / tmp.beta[np - 1]
tmp.beta <- c(tmp.beta, last.beta.laborer)
dst.laborer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(tmp.beta),
  paste0("cc", 1:(np - 1)), paste0("wheat", np)
)
for (k in 1:(np - 1)) {
  node_set(dst.laborer, paste0("cc", k),
    type = "CES", es = 1,
    alpha = 1, beta = c(0.4, 0.4, 0.2),
    paste0("wheat", k), paste0("lab", k), paste0("land", k)
  )
}

tmp.beta <- Gamma.beta^(1:(np - 1))
tmp.beta <- tmp.beta / tmp.beta[np - 1]
tmp.beta <- c(tmp.beta, last.beta.landowner)
dst.landowner <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(tmp.beta),
  paste0("cc", 1:(np - 1)), paste0("iron", np)
)
for (k in 1:(np - 1)) {
  node_set(dst.landowner, paste0("cc", k),
    type = "CES", es = 1,
    alpha = 1, beta = c(0.2, 0.4, 0.4),
    paste0("wheat", k), paste0("lab", k), paste0("land", k)
  )
}

ge <- sdm2(
  A = c(
    dstl.firm.wheat, dstl.firm.iron, dstl.quasifirm.cap,
    dst.laborer, dst.landowner
  ),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  policy = makePolicyMeanValue(50),
  ts = TRUE,
  priceAdjustmentVelocity = 0.03
)

ge$p
ge$z
plot(ge$z[2 * (np - 1) + (1:(np - 1))], type = "b", pch = 20)
lines(1:(np - 1), ge$z[1:(np - 1)], type = "b", pch = 21)
lines(1:(np - 1), ge$z[np - 1 + (1:(np - 1))], type = "b", pch = 22)
legend("topleft", c("cap","wheat", "iron"), pch = 20:22)

#### a time-circle model.
np <- 5 # the number of economic periods
gr <- 0.03 # the growth rate in the steady state equilibrium
eis <- 0.5 # the elasticity of intertemporal substitution  of consumers
Gamma.beta <- 0.97 # the subjective discount factor of consumers
es.firm <- 1
depreciation.rate <- 0.06
zeta <- (1 + gr)^np # the ratio of repayments to loans
yield.rate <- sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr, prepaid = TRUE)
captial.share.laborer <- 0.7795
captial.share.landowner <- 1 - captial.share.laborer

alpha.firm.wheat <- rep(5, np)
alpha.firm.iron <- rep(5, np)

names.commodity <- c(
  paste0("wheat", 1:np),
  paste0("iron", 1:np),
  paste0("cap", 1:np),
  paste0("lab", 1:np),
  paste0("land", 1:np),
  "claim"
)
names.agent <- c(
  paste0("firm.wheat", 1:np), paste0("firm.iron", 1:np),
  paste0("quasifirm.cap", 1:np), # a quasifirm
  "laborer", "landowner"
)
n <- length(names.commodity) # the number of commodity kinds
m <- length(names.agent) # the number of agent kinds

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "laborer"] <- 100 * (1 + gr)^(0:(np - 1)) # the supply of labor
S0Exg[paste0("land", 1:np), "landowner"] <- 100 * (1 + gr)^(0:(np - 1)) # the supply of land
S0Exg["claim", "laborer"] <- 100 * captial.share.laborer
S0Exg["claim", "landowner"] <- 100 * captial.share.landowner

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("wheat", k + 1), paste0("firm.wheat", k)] <- 1
  B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
  B[paste0("cap", k), paste0("quasifirm.cap", k)] <- 1
  B[paste0("iron", k + 1), paste0("quasifirm.cap", k)] <- 1 - depreciation.rate
}
B[paste0("wheat", 1), paste0("firm.wheat", np)] <- 1 / zeta
B[paste0("iron", 1), paste0("firm.iron", np)] <- 1 / zeta
B[paste0("cap", np), paste0("quasifirm.cap", np)] <- 1
B[paste0("iron", 1), paste0("quasifirm.cap", np)] <- (1 - depreciation.rate) / zeta

dstl.firm.wheat <- dstl.firm.iron <- dstl.quasifirm.cap <- list()
for (k in 1:(np - 1)) {
  dstl.firm.wheat[[k]] <- node_new(
    "prod",
    type = "CES", es = es.firm,
    alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
    paste0("cap", k), paste0("lab", k), paste0("land", k)
  )

  dstl.firm.iron[[k]] <- node_new(
    "prod",
    type = "CES", es = es.firm,
    alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2),
    paste0("cap", k), paste0("lab", k), paste0("land", k)
  )

  dstl.quasifirm.cap[[k]] <- node_new(
    "output",
    type = "Leontief", a = 1,
    paste0("iron", k)
  )
}

dstl.firm.wheat[[np]] <- node_new(
  "prod",
  type = "FIN", rate = c(1, (1 + yield.rate)^np - 1),
  "cc1", "claim"
)
node_set(dstl.firm.wheat[[np]], "cc1",
         type = "CES", es = es.firm,
         alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
         paste0("cap", np), paste0("lab", np), paste0("land", np)
)

dstl.firm.iron[[np]] <- node_new(
  "prod",
  type = "FIN", rate = c(1, (1 + yield.rate)^np - 1),
  "cc1", "claim"
)
node_set(dstl.firm.iron[[np]], "cc1",
         type = "CES", es = es.firm,
         alpha = alpha.firm.wheat[k], beta = c(0.4, 0.4, 0.2),
         paste0("cap", np), paste0("lab", np), paste0("land", np)
)

return.rate <- sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr)
fund.occupancy.rate <- (1 - depreciation.rate) / (1 + return.rate)
# The prepaid rent rate (i.e. the prepaid-rent-to-price ratio) of
# the capital good is equal to 1 minus fund.occupancy.rate.

dstl.quasifirm.cap[[np]] <- node_new(
  "prod",
  type = "FIN", rate = c(1, ((1 + yield.rate)^np - 1) * fund.occupancy.rate),
  "cc1", "claim"
)
node_set(dstl.quasifirm.cap[[np]], "cc1",
         type = "Leontief", a = 1,
         paste0("iron", np)
)

dst.laborer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("cc", 1:np)
)
for (k in 1:np) {
  node_set(dst.laborer, paste0("cc", k),
           type = "CES", es = 1,
           alpha = 1, beta = c(0.4, 0.4, 0.2),
           paste0("wheat", k), paste0("lab", k), paste0("land", k)
  )
}

dst.landowner <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("cc", 1:np)
)
for (k in 1:np) {
  node_set(dst.landowner, paste0("cc", k),
           type = "CES", es = 1,
           alpha = 1, beta = c(0.2, 0.4, 0.4),
           paste0("wheat", k), paste0("lab", k), paste0("land", k)
  )
}

ge.tc <- sdm2(
  A = c(
    dstl.firm.wheat, dstl.firm.iron, dstl.quasifirm.cap,
    dst.laborer, dst.landowner
  ),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  numberOfPeriods = 1000,
  priceAdjustmentVelocity = 0.03
)

ge.tc$p
growth_rate(ge.tc$p)
ge.tc$z
growth_rate(ge.tc$z)

## the corresponding sequential model with the same steady-state equilibrium.
dst.firm.wheat <- node_new("prod",
                           type = "FIN", rate = c(1, yield.rate),
                           "cc1", "equity.share.wheat"
)
node_set(dst.firm.wheat, "cc1",
         type = "CES", es = es.firm,
         alpha = 5, beta = c(0.2, 0.4, 0.4),
         "cap", "lab", "land"
)

dst.firm.iron <- node_new("prod",
                          type = "FIN", rate = c(1, yield.rate),
                          "cc1", "equity.share.iron"
)
node_set(dst.firm.iron, "cc1",
         type = "CES", es = es.firm,
         alpha = 5, beta = c(0.4, 0.4, 0.2),
         "cap", "lab", "land"
)

dst.quasifirm.cap <- node_new("prod",
                              type = "FIN", rate = c(1, yield.rate * fund.occupancy.rate),
                              "cc1", "equity.share.cap"
)
node_set(dst.quasifirm.cap, "cc1",
         type = "Leontief", a = 1,
         "iron"
)

dst.laborer <- node_new("util",
                        type = "CES", es = 1,
                        alpha = 1, beta = c(0.4, 0.4, 0.2),
                        "wheat", "lab", "land"
)

dst.landowner <- node_new("util",
                          type = "CES", es = 1,
                          alpha = 1, beta = c(0.2, 0.4, 0.4),
                          "wheat", "lab", "land"
)

ge.seq <- sdm2(
  A = list(
    dst.firm.wheat, dst.firm.iron, dst.quasifirm.cap,
    dst.laborer, dst.landowner
  ),
  B = matrix(c(
    1, 0, 0, 0, 0,
    0, 1, 1 - depreciation.rate, 0, 0,
    0, 0, 1 + gr, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0
  ), 8, 5, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA,
    NA, NA, NA, 100, NA,
    NA, NA, NA, NA, 100,
    NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner,
    NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner,
    NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner
  ), 8, 5, TRUE),
  names.commodity = c(
    "wheat", "iron", "cap", "lab", "land",
           "equity.share.wheat", "equity.share.iron", "equity.share.cap"
  ),
  names.agent = c("firm.wheat", "firm.iron", "quasifirm.cap", "laborer", "landowner"),
  numeraire = "lab",
  numberOfPeriods = 2000,
  priceAdjustmentVelocity = 0.03,
  GRExg = gr
)

ge.seq$p
ge.seq$z
ge.tc$z

Some Intertemporal Models with Ad Valorem Claim

Description

Some intertemporal models with ad valorem claim. Ad valorem claims may be equities, bonds, ad valorem taxation rights (ad valorem tax receipt), fiat money etc, which can be treated in the same way in models.

Usage

gemIntertemporal_AdValoremClaim(...)

Arguments

Examples

#### a model with tax.
np <- 5 # the number of economic periods
gr.lab <- 0.03 # the growth rate of the labor supply
tax.rate <- 0.25

n <- 2 * np + 1 # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np), "tax receipt")
names.agent <- c(paste0("firm", 1:(np - 1)), "laborer", "government")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "laborer"] <- 100 * (1 + gr.lab)^(0:(np - 1)) # the labor supply
S0Exg["prod1", "laborer"] <- 10 # the product supply in the first period
S0Exg["tax receipt", "government"] <- np * 100 # the supply of tax receipt (i.e. ad valorem claim)

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "FIN", rate = c(1, tax.rate),
    "cc1", "tax receipt"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "CD",
           alpha = 1, beta = c(0.5, 0.5),
           paste0("prod", k), paste0("lab", k)
  )
}

dst.laborer <- node_new(
  "util",
  type = "FIN", rate = c(1, tax.rate),
  "cc1", "tax receipt"
)
node_set(dst.laborer, "cc1",
         type = "CES", es = 0.5,
         alpha = 1, beta = rep(1 / np, np),
         paste0("cc1.", 1:np)
)
for (k in 1:np) {
  node_set(dst.laborer, paste0("cc1.", k),
           type = "CD", alpha = 1, beta = c(0.5, 0.5),
           paste0("prod", k), paste0("lab", k)
  )
}

dst.government <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = rep(1 / np, np),
  paste0("cc1.", 1:np)
)
for (k in 1:np) {
  node_set(dst.government, paste0("cc1.", k),
           type = "CD", alpha = 1, beta = c(0.5, 0.5),
           paste0("prod", k), paste0("lab", k)
  )
}
node_plot(dst.government, TRUE)

ge <- sdm2(
  A = c(dstl.firm, dst.laborer, dst.government),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  # policy = makePolicyHeadTailAdjustment(gr = gr.lab, np = np)
)

ge$D
ge$z
ge$DV

#### a pure exchange model with money.
np <- 3 # the number of economic periods
gr.lab <- 0.03 # the growth rate of the labor supply
eis <- 0.8 # the elasticity of intertemporal substitution
Gamma.beta <- 0.8 # the subjective discount factor
interest.rate <- sserr(eis, Gamma.beta, gr.lab, prepaid = TRUE) # 0.2593

dst.laborer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = Gamma.beta^(0:(np - 1)),
  paste0("cc", 1:np)
)

for (k in 1:np) {
  node_set(dst.laborer, paste0("cc", k),
    type = "FIN",
    rate = c(1, interest.rate),
    paste0("lab", k), paste0("money", k)
  )
}

node_plot(dst.laborer, TRUE)

dst.moneyOwner <- Clone(dst.laborer)

ge <- sdm2(
  A = list(dst.laborer, dst.moneyOwner),
  B = matrix(0, 2 * np, 2),
  S0Exg = {
    tmp <- matrix(0, 2 * np, 2)
    tmp[1:np, 1] <- 100 * (1 + gr.lab)^(0:(np - 1))
    tmp[(np + 1):(2 * np), 2] <- 200
    tmp
  },
  names.commodity = c(paste0("lab", 1:np), paste0("money", 1:np)),
  names.agent = c("laborer", "moneyOwner"),
  numeraire = c(money1 = interest.rate)
)

ge$p
growth_rate(ge$p[1:3]) + 1
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

## In the following program, the periods to which
## the money belongs are not distinguished.
dst.laborer <- node_new(
  "util",
  type = "FIN",
  rate = c(1, interest.rate),
  "cc1", "money"
)
node_set(dst.laborer, "cc1",
  type = "CES", es = eis,
  alpha = 1, beta = Gamma.beta^(0:(np - 1)),
  paste0("lab", 1:np)
)

dst.moneyOwner <- Clone(dst.laborer)

ge <- sdm2(
  A = list(dst.laborer, dst.moneyOwner),
  B = matrix(0, np + 1, 2),
  S0Exg = {
    tmp <- matrix(0, np + 1, 2)
    tmp[1:np, 1] <- 100 * (1 + gr.lab)^(0:(np - 1))
    tmp[np + 1, 2] <- 100
    tmp
  },
  names.commodity = c(paste0("lab", 1:np), "money"),
  names.agent = c("laborer", "moneyOwner"),
  numeraire = c(money = interest.rate)
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

#### a two-period model with production and money.
interest.rate1 <- 0.25
interest.rate2 <- 0.1

dst.firm <- node_new(
  "prod",
  type = "FIN",
  rate = c(1, interest.rate1),
  "cc1", "money1"
)
node_set(dst.firm, "cc1",
  type = "CES",
  es = 1, alpha = 2, beta = c(0.5, 0.5),
  "prod1", "lab1"
)

dst.laborer <- node_new(
  "util",
  type = "CES",
  es = 0.5, alpha = 1, beta = c(2 / 3, 1 / 3),
  "cc1", "cc2"
)
node_set(dst.laborer, "cc1",
  type = "FIN",
  rate = c(1, interest.rate1),
  "prod1", "money1"
)

node_set(dst.laborer, "cc2",
  type = "FIN",
  rate = c(1, interest.rate2),
  "prod2", "money2"
)

dst.moneyOwner <- Clone(dst.laborer)

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.moneyOwner),
  B = matrix(c(
    0, 0, 0,
    0, 0, 0,
    0, 0, 0,
    1, 0, 0,
    0, 0, 0
  ), 5, 3, TRUE),
  S0Exg = matrix(c(
    NA, 200, NA,
    NA, 100, NA,
    NA, NA, 100,
    NA, NA, NA,
    NA, NA, 100
  ), 5, 3, TRUE),
  names.commodity = c("prod1", "lab1", "money1", "prod2", "money2"),
  names.agent = c("firm", "laborer", "moneyOwner"),
  numeraire = c(money1 = interest.rate1)
)

ge$p
ge$DV

Some Examples of an Intertemporal Model with a Consumer and a Type of Bank

Description

Some examples of an intertemporal model with a consumer and a type of bank. These models can be used to solve some intertemporal savings problems. Below is an example.

A np-period-lived consumer has some payoff (or cash, exhaustible resource etc.) in each period. In each period the consumer can use payoff for consumption or save payoff into bank. The interest rate is given. The consumer has a SCES intertemporal utility function and attempts to maximize intertemporal utility by saving.

Usage

gemIntertemporal_Bank_1_2(...)

Arguments

Examples

#### an example with a 5-period-lived consumer
np <- 5 # the number of economic periods

interest.rate <- 0.1

n <- np # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- paste0("payoff", 1:np)
names.agent <- c(paste0("bank", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("payoff", 1:np), "consumer"] <- 100 / (np:1)

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("payoff", k + 1), paste0("bank", k)] <- 1
}

dstl.bank <- list()
for (k in 1:(np - 1)) {
  dstl.bank[[k]] <- node_new("output",
                             type = "Leontief", a = 1 / (1 + interest.rate),
                             paste0("payoff", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "SCES", es = 1, alpha = 1, beta = prop.table(1:np),
  paste0("payoff", 1:np)
)

ge <- sdm2(
  A = c(dstl.bank, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1",
  policy = makePolicyMeanValue(30),
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV
growth_rate(ge$p)

##
dst.consumer$es <- 0
dst.consumer$beta <- rep(1 / np, np)
S0Exg[paste0("payoff", 1:np), "consumer"] <- 100 / (1:np)
ge <- sdm2(
  A = c(dstl.bank, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1",
  policy = makePolicyMeanValue(30),
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

Some Examples of an Intertemporal Model with Two Consumers and a Type of Bank

Description

Some examples of an intertemporal model with two consumers and a type of bank.

Usage

gemIntertemporal_Bank_1_3(...)

Arguments

See Also

gemIntertemporal_Bank_1_2

Examples

#### an example with a 5-period-lived consumer
np <- 5 # the number of economic periods
interest.rate <- 0.1

n <- np # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- paste0("payoff", 1:np)
names.agent <- c(paste0("bank", 1:(np - 1)), "consumer1", "consumer2")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("payoff", 1:np), "consumer1"] <- 100 / (np:1)
S0Exg[paste0("payoff", 1:np), "consumer2"] <- 100 / (1:np)

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("payoff", k + 1), paste0("bank", k)] <- 1
}

dstl.bank <- list()
for (k in 1:(np - 1)) {
  dstl.bank[[k]] <- node_new(
    "output",
    type = "Leontief", a = 1 / (1 + interest.rate),
    paste0("payoff", k)
  )
}

dst.consumer1 <- node_new(
  "util",
  type = "SCES",
  es = 1, alpha = 1, beta = prop.table(1:np),
  paste0("payoff", 1:np)
)

dst.consumer2 <- node_new(
  "util",
  type = "SCES",
  es = 1, alpha = 1, beta = prop.table(np:1),
  paste0("payoff", 1:np)
)

ge <- sdm2(
  A = c(dstl.bank, dst.consumer1, dst.consumer2),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1",
  policy = makePolicyMeanValue(30),
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV
growth_rate(ge$p)

##
dst.consumer1$es <- 0
dst.consumer1$beta <- rep(1 / np, np)

ge <- sdm2(
  A = c(dstl.bank, dst.consumer1, dst.consumer2),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1",
  policy = makePolicyMeanValue(30),
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV
growth_rate(ge$p)

The Identical Steady-state Equilibrium: Four Models Illustrating Dividend

Description

Four models are presented to illustrate dividend, which have the same steady-state equilibrium.

These models are as follows: (1) a real timeline model with head-tail adjustment; (2) a financial timeline model with dividend and head-tail adjustment; (3) a financial sequential model with dividend; (4) a financial time-circle model with dividend.

Usage

gemIntertemporal_Dividend(...)

Arguments

Examples

#### (1) a real timeline model with head-tail adjustment.
eis <- 0.8 # the elasticity of intertemporal substitution
Gamma.beta <- 0.8 # the subjective discount factor
gr <- 0.03 # the growth rate
np <- 5 # the number of economic periods

n <- 2 * np - 1 # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)))
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100 * (1 + gr)^(0:(np - 2))
S0Exg["prod1", "consumer"] <- 140 # the product supply in the first period, which will be adjusted.

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD",
    alpha = 2, beta = c(0.5, 0.5),
    paste0("prod", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge.tl <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  policy = makePolicyHeadTailAdjustment(gr = gr, np = np)
)

sserr(eis, Gamma.beta, gr) # the steady-state equilibrium return rate, 0.2970
ge.tl$p[1:(np - 1)] / ge.tl$p[2:np] - 1
ge.tl$z

## (2) a financial timeline model with dividend and head-tail adjustment.
yield.rate <- sserr(
  eis = eis, Gamma.beta = Gamma.beta,
  gr = gr, prepaid = TRUE
) # the prepaid steady-state equilibrium return rate, 0.2593

n <- 2 * np # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)), "claim")
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100 * (1 + gr)^(0:(np - 2))
S0Exg["claim", "consumer"] <- 100
S0Exg["prod1", "consumer"] <- 140 # the product supply in the first period, which will be adjusted.

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "FIN", rate = c(1, yield.rate),
    "cc1", "claim"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "CD", alpha = 2, beta = c(0.5, 0.5),
           paste0("prod", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = 1,
  alpha = 1, beta = prop.table(rep(1, np)), # prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge.ftl <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  policy = makePolicyHeadTailAdjustment(gr = gr, np = np)
)

ge.ftl$z

## (3) a financial sequential model with dividend.
dst.firm <- node_new("output",
                     type = "FIN",
                     rate = c(1, dividend.rate = yield.rate),
                     "cc1", "equity.share"
)
node_set(dst.firm, "cc1",
         type = "CD",
         alpha = 2, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.laborer <- node_new("util",
                        type = "Leontief", a = 1,
                        "prod"
)

dst.shareholder <- Clone(dst.laborer)

ge.fs <- sdm2(
  A = list(dst.firm, dst.laborer, dst.shareholder),
  B = diag(c(1, 0, 0)),
  S0Exg = {
    S0Exg <- matrix(NA, 3, 3)
    S0Exg[2, 2] <- S0Exg[3, 3] <- 100
    S0Exg
  },
  names.commodity = c("prod", "lab", "equity.share"),
  names.agent = c("firm", "laborer", "shareholder"),
  numeraire = "prod",
  GRExg = gr
)

ge.fs$z

# a steady-state path.
ge2.fs <- sdm2(
  A = list(dst.firm, dst.laborer, dst.shareholder),
  B = diag(c(1, 0, 0)),
  S0Exg = {
    S0Exg <- matrix(NA, 3, 3)
    S0Exg[2, 2] <- S0Exg[3, 3] <- 100
    S0Exg
  },
  names.commodity = c("prod", "lab", "equity.share"),
  names.agent = c("firm", "laborer", "shareholder"),
  numeraire = "prod",
  GRExg = gr,
  maxIteration = 1,
  numberOfPeriods = 20,
  z0 = ge.fs$z,
  policy = policyMarketClearingPrice,
  ts = TRUE
)

ge2.fs$ts.z[, 1]
growth_rate(ge2.fs$ts.z[, 1])

## (4) a financial time-circle model with dividend.
np <- 5
zeta <- (1 + gr)^np # the ratio of repayments to loans

n <- 2 * np + 1 # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np), "claim")
names.agent <- c(paste0("firm", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 * (1 + gr)^(0:(np - 1))
S0Exg["claim", "consumer"] <- 100

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}
B["prod1", paste0("firm", np)] <- 1 / zeta

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new("output",
                             type = "FIN", rate = c(1, yield.rate),
                             "cc1", "claim"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "CD", alpha = 2,
           beta = c(0.5, 0.5),
           paste0("lab", k), paste0("prod", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = 1,
  alpha = 1, beta = prop.table(rep(1, np)),
  paste0("prod", 1:np)
)

ge.ftc <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  ts = TRUE
)

ge.ftc$z

##
ge.tc <- gemCanonicalDynamicMacroeconomic_TimeCircle_2_2(
  alpha.firm = rep(2, 5),
  es.prod.lab.firm = 1,
  beta.prod.firm = 0.5,
  depreciation.rate = 1,
  eis = 0.8,
  Gamma.beta = 0.8,
  beta.prod.consumer = 1,
  es.prod.lab.consumer = 1,
  gr = 0.03,
  wage.payment = "pre"
)

ge.tc$z

The Identical Steady-state Equilibrium: Four Models Illustrating Dividend and Technological Progress

Description

Four models with labor-saving technological progress are presented to illustrate dividend, which have the same steady-state equilibrium.

These models are as follows: (1) a real timeline model with head-tail adjustment; (2) a financial timeline model with dividend and head-tail adjustment; (3) a financial sequential model with dividend; (4) a financial time-circle model with dividend.

Usage

gemIntertemporal_Dividend_TechnologicalProgress(...)

Arguments

See Also

gemIntertemporal_Dividend

Examples

#### (1) a real timeline model with head-tail adjustment.
eis <- 0.8 # the elasticity of intertemporal substitution
Gamma.beta <- 0.8 # the subjective discount factor
gr.tech <- 0.02 # the technological progress rate
gr.lab <- 0.03 # the growth rate of labor supply
gr <- (1 + gr.lab) * (1 + gr.tech) - 1 # the growth rate
np <- 4 # the number of economic periods

n <- 2 * np - 1 # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)))
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100 * (1 + gr.lab)^(0:(np - 2))
S0Exg["prod1", "consumer"] <- 140 # the product supply in the first period, which will be adjusted.

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD",
    alpha = 2, beta = c(0.5, 0.5),
    paste0("prod", k), "cc1"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "Leontief", a = 1 / (1 + gr.tech)^(k - 1),
           paste0("lab", k)
  )
}

node_plot(dstl.firm[[np - 1]], TRUE)

dst.consumer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  policy = makePolicyHeadTailAdjustment(gr = gr, np = np)
)

sserr(eis, Gamma.beta, gr) # the steady-state equilibrium return rate
ge$p[1:(np - 1)] / ge$p[2:np] - 1 # the steady-state equilibrium return rate
ge$z
growth_rate(ge$z)

## (2) a financial timeline model with dividend and head-tail adjustment.
yield.rate <- sserr(eis, Gamma.beta, gr, prepaid = TRUE)

n <- 2 * np # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)), "claim")
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100 * (1 + gr.lab)^(0:(np - 2))
S0Exg["claim", "consumer"] <- 100
S0Exg["prod1", "consumer"] <- 140 # the product supply in the first period, which will be adjusted.

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "FIN", rate = c(1, yield.rate),
    "cc1", "claim"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "CD", alpha = 2, beta = c(0.5, 0.5),
           paste0("prod", k), "cc1.1"
  )
  node_set(dstl.firm[[k]], "cc1.1",
           type = "Leontief", a = 1 / (1 + gr.tech)^(k - 1),
           paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = 1,
  alpha = 1, beta = prop.table(rep(1, np)), # prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  policy = makePolicyHeadTailAdjustment(gr = gr, np = np)
)

ge$z

## (3) a financial sequential model with dividend.
dst.firm <- node_new("output",
                     type = "FIN",
                     rate = c(1, dividend.rate = yield.rate),
                     "cc1", "equity.share"
)
node_set(dst.firm, "cc1",
         type = "CD",
         alpha = 2, beta = c(0.5, 0.5),
         "prod", "cc1.1"
)
node_set(dst.firm, "cc1.1",
         type = "Leontief", a = 1,
         "lab"
)

node_plot(dst.firm, TRUE)

dst.laborer <- node_new("util",
                        type = "Leontief", a = 1,
                        "prod"
)

dst.shareholder <- Clone(dst.laborer)

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.shareholder),
  B = diag(c(1, 0, 0)),
  S0Exg = {
    S0Exg <- matrix(NA, 3, 3)
    S0Exg[2, 2] <- 100 / (1 + gr.lab)
    S0Exg[3, 3] <- 100
    S0Exg
  },
  names.commodity = c("prod", "lab", "equity.share"),
  names.agent = c("firm", "laborer", "shareholder"),
  numeraire = "equity.share",
  maxIteration = 1,
  numberOfPeriods = 20,
  z0 = c(143.1811, 0, 0),
  policy = list(policy.technology <- function(time, A, state) {
    node_set(A[[1]], "cc1.1",
             a = 1 / (1 + gr.tech)^(time - 1)
    )
    state$S[2, 2] <- 100 * (1 + gr.lab)^(time - 1)

    state
  }, policyMarketClearingPrice),
  ts = TRUE
)

ge$ts.z[, 1]
growth_rate(ge$ts.z[, 1])

## (4) a financial time-circle model with dividend.
zeta <- (1 + gr)^np # the ratio of repayments to loans

n <- 2 * np + 1 # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np), "claim")
names.agent <- c(paste0("firm", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 * (1 + gr.lab)^(0:(np - 1))
S0Exg["claim", "consumer"] <- 100

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}
B["prod1", paste0("firm", np)] <- 1 / zeta

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new("output",
                             type = "FIN", rate = c(1, yield.rate),
                             "cc1", "claim"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "CD", alpha = 2,
           beta = c(0.5, 0.5),
           paste0("prod", k), "cc1.1"
  )
  node_set(dstl.firm[[k]], "cc1.1",
           type = "Leontief", a = 1 / (1 + gr.tech)^(k - 1),
           paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = 1,
  alpha = 1, beta = prop.table(rep(1, np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  ts = TRUE
)

ge$z
growth_rate(ge$z[1:np])

An Example Illustrating Endogenous Equilibrium Interest Rates in a (Timeline) Transitional Equilibrium Path

Description

This example illustrates (endogenous) equilibrium primitive interest rates in a transitional equilibrium path, which is an intertemporal path distinct from a steady-state equilibrium. Assume that the velocity of money is equal to one, that is, money circulates once per period.

The interest rate calculated here is adjusted from the nominal interest rate based on the growth rate of the money supply, which is equal to the nominal interest rate when the money stock remains unchanged. We refer to this kind of interest rate as the primitive interest rate, which usually differs from the real interest rate obtained by adjusting the nominal rate based on the inflation rate.

There are three types of economic agents in the model: firms, a laborer, and a money owner. Suppose the laborer and the money owner need to use money to buy products, and firms need to use money to buy products and labor. Formally, the money owner borrows money from himself and pays interest to himself.

Usage

gemIntertemporal_EndogenousEquilibriumInterestRate(...)

Arguments

Examples

eis <- 0.8 # the elasticity of intertemporal substitution
Gamma.beta <- 0.8 # the subjective discount factor
gr <- 0 # the steady-state growth rate
np <- 20 # the number of economic periods

f <- function(ir = rep(0.25, np - 1), return.ge = FALSE,
              y1 = 10, # the product supply in the first period
              alpha.firm = rep(2, np - 1) # the efficiency parameters of firms
) {
  n <- 2 * np # the number of commodity kinds
  m <- np + 1 # the number of agent kinds

  names.commodity <- c(
    paste0("prod", 1:np),
    paste0("lab", 1:(np - 1)),
    "money"
  )
  names.agent <- c(
    paste0("firm", 1:(np - 1)),
    "laborer", "moneyOwner"
  )

  # the exogenous supply matrix.
  S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
  S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 * (1 + gr)^(0:(np - 2))
  S0Exg["money", "moneyOwner"] <- 100
  S0Exg["prod1", "laborer"] <- y1

  # the output coefficient matrix.
  B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
  for (k in 1:(np - 1)) {
    B[paste0("prod", k + 1), paste0("firm", k)] <- 1
  }

  dstl.firm <- list()
  for (k in 1:(np - 1)) {
    dstl.firm[[k]] <- node_new(
      "prod",
      type = "FIN", rate = c(1, ir[k]),
      "cc1", "money"
    )
    node_set(dstl.firm[[k]], "cc1",
             type = "CD", alpha = alpha.firm[k], beta = c(0.5, 0.5),
             paste0("prod", k), paste0("lab", k)
    )
  }

  dst.laborer <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
    paste0("cc", 1:(np - 1)), paste0("prod", np)
  )

  for (k in 1:(np - 1)) {
    node_set(dst.laborer, paste0("cc", k),
             type = "FIN", rate = c(1, ir[k]),
             paste0("prod", k), "money"
    )
  }

  dst.moneyOwner <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:(np - 1))),
    paste0("cc", 1:(np - 1))
  )
  for (k in 1:(np - 1)) {
    node_set(dst.moneyOwner, paste0("cc", k),
             type = "FIN", rate = c(1, ir[k]),
             paste0("prod", k), "money"
    )
  }

  ge <- sdm2(
    A = c(dstl.firm, dst.laborer, dst.moneyOwner),
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "prod1",
    policy = makePolicyHeadTailAdjustment(gr = gr, np = np, type = c("tail"))
  )

  tmp <- rowSums(ge$SV)
  ts.exchange.value <- tmp[paste0("prod", 1:(np - 1))] + tmp[paste0("lab", 1:(np - 1))]
  ir.new <- ts.exchange.value[1:(np - 2)] / ts.exchange.value[2:(np - 1)] - 1
  ir.new <- pmax(1e-6, ir.new)
  ir.new[np - 1] <- ir.new[np - 2]

  ir <- c(ir * ratio_adjust(ir.new / ir, 0.3))
  cat("ir: ", ir, "\n")

  if (return.ge) {
    ge$ts.exchange.value <- ts.exchange.value
    return(ge)
  } else {
    return(ir)
  }
}

## Calculate equilibrium interest rates.
## Warning: Running the program below takes about several minutes.
# mat.ir <- iterate(rep(0.1, np - 1), f, tol = 1e-4)
# sserr(eis, Gamma.beta, gr, prepaid = TRUE)

## Below are the calculated equilibrium interest rates.
ir <- rep(0.25, np - 1)
ir[1:14] <- c(0.4301, 0.3443, 0.3007, 0.2776, 0.2652, 0.2584, 0.2546,
  0.2526, 0.2514, 0.2508, 0.2504, 0.2502, 0.2501, 0.2501)

ge <- f(ir, TRUE)

plot(ge$z[1:(np - 1)], type = "o")
ge$ts.exchange.value[1:(np - 2)] / ge$ts.exchange.value[2:(np - 1)] - 1
ir

## Calculate the growth rate of the money supply and the equilibrium nominal
## interest rate when the current price of the product remains constant.
price.money <- 1 / c(1, cumprod(ir + 1))
currentPrice.prod <- ge$p[1:np] / price.money
gr.moneySupply <- unname(growth_rate(1 / currentPrice.prod))
(ir + 1) * (gr.moneySupply[2:np] + 1) - 1

## the corresponding sequential model with the same steady-state equilibrium.
np <- 5
ge.ss <- f(return.ge = TRUE, y1 = 128)

dividend.rate <- ir <- sserr(eis, Gamma.beta, prepaid = TRUE)

dst.firm <- node_new("prod",
                     type = "FIN", rate = c(1, ir, (1 + ir) * dividend.rate),
                     "cc1", "money", "equity.share"
)
node_set(dst.firm, "cc1",
         type = "CD",
         alpha = 2, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.laborer <- node_new("util",
                        type = "FIN", rate = c(1, ir),
                        "prod", "money"
)

dst.moneyOwner <- node_new("util",
                           type = "FIN", rate = c(1, ir),
                           "prod", "money"
)

ge2 <- sdm2(
  A = list(dst.firm, dst.laborer, dst.moneyOwner),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100,
    NA, 100, NA
  ), 4, 3, TRUE),
  names.commodity = c(
    "prod", "lab", "money", "equity.share"
  ),
  names.agent = c("firm", "laborer", "moneyOwner"),
  numeraire = "prod"
)

ge2$p
ge.ss$z[np - 1]
ge2$z
ge.ss$D[paste0("prod", np - 1), c("laborer", "moneyOwner")]
ge2$D

## a technology shock.
## Warning: Running the program below takes about several minutes.
# np <- 50
# f2 <- function(x) {
#  f(
#     ir = x, return.ge = FALSE,
#     y1 = 128, alpha.firm = {
#       tmp <- rep(2, np - 1)
#       tmp[25] <- 1.8
#       tmp
#     }
#   )
# }
#
# mat.ir <- iterate(rep(0.25, np - 1), f2, tol = 1e-4)
# tail(mat.ir, 1) # the equilibrium interest rates

## Calculate equilibrium interest rates.
## Warning: Running the program below takes about several minutes.
# np <- 20
# gr <- 0.03
# mat.ir <- iterate(rep(0.1, np - 1), f, tol = 1e-4)
# sserr(eis, Gamma.beta, gr, prepaid = TRUE)

An Example Illustrating Endogenous Equilibrium Interest Rates and Foreign Exchange Rates in a Two-country (Timeline) Transitional Equilibrium Path

Description

This example illustrates (endogenous) equilibrium primitive interest rates and foreign exchange rates in a two-country transitional equilibrium path. Assume that the velocity of money is equal to one, that is, money circulates once per period.

Usage

gemIntertemporal_EndogenousEquilibriumInterestRate_ForeignExchangeRate(...)

Arguments

See Also

gemIntertemporal_EndogenousEquilibriumInterestRate

Examples

eis <- 0.8 # the elasticity of intertemporal substitution
Gamma.beta <- 10 / 11 # the subjective discount factor
gr <- 0 # the steady-state growth rate
money1.supply <- 600
money2.supply <- 100
np <- 20 # the number of economic periods

sserr(eis, Gamma.beta, gr, prepaid = TRUE)

f <- function(ir = rep(0.1, 2 * np - 2), return.ge = FALSE,
              y1.wheat = 10, # 49.24 #49.79 the wheat supply in the first period
              y1.iron = 5 # 41.32 #45.45 the iron supply in the first period
) {
  ir1 <- head(ir, np - 1)
  ir2 <- tail(ir, np - 1)

  n <- 2 * np + 2 * (np - 1) + 2 # the number of commodity kinds
  m <- 2 * (np - 1) + 4 # the number of agent kinds

  names.commodity <- c(
    paste0("wheat", 1:np),
    paste0("lab1.", 1:(np - 1)),
    "money1",
    paste0("iron", 1:np),
    paste0("lab2.", 1:(np - 1)),
    "money2"
  )

  names.agent <- c(
    paste0("firm.wheat", 1:(np - 1)),
    "laborer1", "moneyOwner1",
    paste0("firm.iron", 1:(np - 1)),
    "laborer2", "moneyOwner2"
  )

  # the exogenous supply matrix.
  S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
  S0Exg[paste0("lab1.", 1:(np - 1)), "laborer1"] <- 100 * (1 + gr)^(0:(np - 2))
  S0Exg["money1", "moneyOwner1"] <- money1.supply
  S0Exg["wheat1", "laborer1"] <- y1.wheat

  S0Exg[paste0("lab2.", 1:(np - 1)), "laborer2"] <- 100 * (1 + gr)^(0:(np - 2))
  S0Exg["money2", "moneyOwner2"] <- money2.supply
  S0Exg["iron1", "laborer2"] <- y1.iron

  # the output coefficient matrix.
  B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
  for (k in 1:(np - 1)) {
    B[paste0("wheat", k + 1), paste0("firm.wheat", k)] <- 1
    B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
  }

  dstl.firm.wheat <- dstl.firm.iron <- list()
  for (k in 1:(np - 1)) {
    dstl.firm.wheat[[k]] <- node_new(
      "prod",
      type = "FIN", rate = c(1, ir1[k]),
      "cc1", "money1"
    )
    node_set(dstl.firm.wheat[[k]], "cc1",
      type = "CD", alpha = 1, beta = c(0.5, 0.5),
      paste0("iron", k), paste0("lab1.", k)
    )

    dstl.firm.iron[[k]] <- node_new(
      "prod",
      type = "FIN", rate = c(1, ir2[k]),
      "cc1", "money2"
    )
    node_set(dstl.firm.iron[[k]], "cc1",
      type = "CD", alpha = 1, beta = c(0.5, 0.5),
      paste0("iron", k), paste0("lab2.", k)
    )
  }

  dst.laborer1 <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
    paste0("cc", 1:(np - 1)), paste0("wheat", np)
  )
  for (k in 1:(np - 1)) {
    node_set(dst.laborer1, paste0("cc", k),
      type = "FIN", rate = c(1, ir1[k]),
      paste0("wheat", k), "money1"
    )
  }

  dst.moneyOwner1 <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:(np - 1))),
    paste0("cc", 1:(np - 1))
  )
  for (k in 1:(np - 1)) {
    node_set(dst.moneyOwner1, paste0("cc", k),
      type = "FIN", rate = c(1, ir1[k]),
      paste0("wheat", k), "money1"
    )
  }

  dst.laborer2 <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
    paste0("cc", 1:(np - 1)), paste0("iron", np)
  )

  for (k in 1:(np - 1)) {
    node_set(dst.laborer2, paste0("cc", k),
      type = "FIN", rate = c(1, ir2[k]),
      paste0("wheat", k), "money2"
    )
  }

  dst.moneyOwner2 <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:(np - 1))),
    paste0("cc", 1:(np - 1))
  )
  for (k in 1:(np - 1)) {
    node_set(dst.moneyOwner2, paste0("cc", k),
      type = "FIN", rate = c(1, ir2[k]),
      paste0("wheat", k), "money2"
    )
  }

  ge <- sdm2(
    A = c(
      dstl.firm.wheat, dst.laborer1, dst.moneyOwner1,
      dstl.firm.iron, dst.laborer2, dst.moneyOwner2
    ),
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "wheat1",
    policy = makePolicyTailAdjustment(
      ind = rbind(
        c(np - 1, np),
        c(2 * np, 2 * (np - 1) + 3)
      ),
      gr = gr
    )
  )

  tmp <- rowSums(ge$SV)

  ts1.exchange.value <- tmp[paste0("wheat", 1:(np - 1))] + tmp[paste0("lab1.", 1:(np - 1))]
  ir1.new <- ts1.exchange.value[1:(np - 2)] / ts1.exchange.value[2:(np - 1)] - 1
  ir1.new <- pmax(1e-6, ir1.new)
  ir1.new[np - 1] <- ir1.new[np - 2]

  ir1 <- c(ir1 * ratio_adjust(ir1.new / ir1, 0.3))
  cat("ir1: ", ir1, "\n")


  ts2.exchange.value <- tmp[paste0("iron", 1:(np - 1))] + tmp[paste0("lab2.", 1:(np - 1))]
  ir2.new <- ts2.exchange.value[1:(np - 2)] / ts2.exchange.value[2:(np - 1)] - 1
  ir2.new <- pmax(1e-6, ir2.new)
  ir2.new[np - 1] <- ir2.new[np - 2]

  ir2 <- c(ir2 * ratio_adjust(ir2.new / ir2, 0.3))
  cat("ir2: ", ir2, "\n")

  if (return.ge) {
    ge$ts1.exchange.value <- unname(ts1.exchange.value)
    ge$ts2.exchange.value <- unname(ts2.exchange.value)
    ge$ts.forex <- unname((ge$ts2.exchange.value / money2.supply) /
      (ge$ts1.exchange.value / money1.supply))
    return(ge)
  } else {
    return(c(ir1, ir2))
  }
}

## Calculate equilibrium interest rates.
## Warning: Running the program below takes about several minutes.
# mat.ir <- iterate(rep(0.1, 2*np - 2), f, tol = 1e-4)
# sserr(eis, Gamma.beta, gr, prepaid = TRUE)

## Below are the calculated equilibrium interest rates.
ir1 <- c(
  0.2218, 0.1888, 0.1455, 0.1228, 0.1115, 0.1058, 0.1029, 0.1015, 0.1008,
  0.1004, 0.1002, 0.1001, 0.1001, 0.1000, 0.1000, 0.1000, 0.1000, 0.1000, 0.1000
)
ir2 <- c(
  0.1292, 0.1080, 0.1037, 0.1019, 0.1010, 0.1005, 0.1003, 0.1001, 0.1001,
  0.1000, 0.1000, 0.1000, 0.1000, 0.1000, 0.1000, 0.1000, 0.1000, 0.1000, 0.1000
)

ge <- f(c(ir1, ir2), return.ge = TRUE)
plot(ge$z[1:(np - 1)], type = "o", ylab = "wheat output")
ge$ts.forex

## the corresponding sequential model with the same steady-state equilibrium.
np <- 5
ge.ss <- f(return.ge = TRUE, y1.wheat = 49.24, y1.iron = 41.32)

ir <- dividend.rate <- 0.1

dst.firm.wheat <- node_new("output",
  type = "FIN", rate = c(1, ir, (1 + ir) * dividend.rate),
  "cc1", "money1", "equity.share.wheat"
)
node_set(dst.firm.wheat, "cc1",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "iron", "lab1"
)

dst.firm.iron <- node_new("output",
  type = "FIN", rate = c(1, ir, (1 + ir) * dividend.rate),
  "cc1", "money2", "equity.share.iron"
)
node_set(dst.firm.iron, "cc1",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "iron", "lab2"
)

dst.laborer1 <- node_new("util",
  type = "FIN", rate = c(1, interest.rate = 0.1),
  "cc1", "money1"
)
node_set(dst.laborer1, "cc1",
  type = "Leontief", a = 1,
  "wheat"
)

dst.moneyOwner1 <- Clone(dst.laborer1)

dst.laborer2 <- Clone(dst.laborer1)
node_replace(dst.laborer2, "money1", "money2")

dst.moneyOwner2 <- Clone(dst.laborer2)

ge <- sdm2(
  A = list(
    dst.firm.wheat, dst.laborer1, dst.moneyOwner1,
    dst.firm.iron, dst.laborer2, dst.moneyOwner2
  ),
  B = matrix(c(
    1, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0,
    0, 0, 0, 1, 0, 0,
    0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0
  ), 8, 6, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA, NA,
    NA, 100, NA, NA, NA, NA,
    NA, NA, 600, NA, NA, NA,
    NA, 100, NA, NA, NA, NA,
    NA, NA, NA, NA, NA, NA,
    NA, NA, NA, NA, 100, NA,
    NA, NA, NA, NA, NA, 100,
    NA, NA, NA, NA, 100, NA
  ), 8, 6, TRUE),
  names.commodity = c(
    "wheat", "lab1", "money1", "equity.share.wheat",
    "iron", "lab2", "money2", "equity.share.iron"
  ),
  names.agent = c(
    "firm1", "laborer1", "moneyOwner1",
    "firm2", "laborer2", "moneyOwner2"
  ),
  numeraire = c("money1" = 0.1) # interest.rate
)

ge.ss$ts.forex
ge$p["money2"] / ge$p["money1"] # the foreign exchange rate

## Calculate equilibrium interest rates.
## Warning: Running the program below takes about several minutes.
# np <- 20
# gr <- 0.03
# mat.ir <- iterate(rep(0.1, 2*np - 2), f, tol = 1e-4)
# sserr(eis, Gamma.beta, gr, prepaid = TRUE)

## a steady-state equilibrium with an exogenous interest rate 0.1.
## The endogenous equilibrium interest rate and dividend rate are 0.
## See also CGE::Example7.6.
eis <- 1 # the elasticity of intertemporal substitution
Gamma.beta <- 1 # the subjective discount factor
gr <- 0 # the steady-state growth rate
money1.supply <- 600
money2.supply <- 100
np <- 20 # the number of economic periods

np <- 5
ge.ss <- f(return.ge = TRUE, y1.wheat = 49.79, y1.iron = 45.45)
ge.ss$z
ge.ss$ts.forex

An Example Illustrating Endogenous Equilibrium Interest Rates and the Velocity of Money in a (Timeline) Transitional Equilibrium Path

Description

This example illustrates (endogenous) equilibrium primitive interest rates and the velocity of money in a transitional equilibrium path. There are two types of interest rates involved here, namely the (primitive) period interest rate and the (primitive) intraperiod interest rate (see convert_ir).

Usage

gemIntertemporal_EndogenousEquilibriumInterestRate_MoneyVelocity(...)

Arguments

See Also

gemIntertemporal_EndogenousEquilibriumInterestRate

Examples

vm <- 4 # the velocity of money
eis <- 0.8 # the elasticity of intertemporal substitution
Gamma.beta <- 0.8 # the subjective discount factor
gr <- 0 # the steady-state growth rate
np <- 20 # the number of economic periods

f <- function(ir.intraperiod = rep(0.25, np - 1), return.ge = FALSE,
              y1 = 10, # the product supply in the first period
              alpha.firm = rep(2, np - 1) # the efficiency parameters of firms
) {
  n <- 2 * np # the number of commodity kinds
  m <- np + 1 # the number of agent kinds

  names.commodity <- c(
    paste0("prod", 1:np),
    paste0("lab", 1:(np - 1)),
    "money"
  )
  names.agent <- c(
    paste0("firm", 1:(np - 1)),
    "laborer", "moneyOwner"
  )

  # the exogenous supply matrix.
  S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
  S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 * (1 + gr)^(0:(np - 2))
  S0Exg["money", "moneyOwner"] <- 100
  S0Exg["prod1", "laborer"] <- y1

  # the output coefficient matrix.
  B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
  for (k in 1:(np - 1)) {
    B[paste0("prod", k + 1), paste0("firm", k)] <- 1
  }

  dstl.firm <- list()
  for (k in 1:(np - 1)) {
    dstl.firm[[k]] <- node_new(
      "prod",
      type = "FIN", rate = c(1, ir.intraperiod[k] / vm),
      "cc1", "money"
    )
    node_set(dstl.firm[[k]], "cc1",
      type = "CD", alpha = alpha.firm[k], beta = c(0.5, 0.5),
      paste0("prod", k), paste0("lab", k)
    )
  }

  dst.laborer <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
    paste0("cc", 1:(np - 1)), paste0("prod", np)
  )

  for (k in 1:(np - 1)) {
    node_set(dst.laborer, paste0("cc", k),
      type = "FIN", rate = c(1, ir.intraperiod[k] / vm),
      paste0("prod", k), "money"
    )
  }

  dst.moneyOwner <- node_new(
    "util",
    type = "CES", es = eis,
    alpha = 1, beta = prop.table(Gamma.beta^(1:(np - 1))),
    paste0("cc", 1:(np - 1))
  )
  for (k in 1:(np - 1)) {
    node_set(dst.moneyOwner, paste0("cc", k),
      type = "FIN", rate = c(1, ir.intraperiod[k] / vm),
      paste0("prod", k), "money"
    )
  }

  ge <- sdm2(
    A = c(dstl.firm, dst.laborer, dst.moneyOwner),
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "prod1",
    policy = makePolicyHeadTailAdjustment(gr = gr, np = np, type = c("tail"))
  )

  tmp <- rowSums(ge$SV)
  ts.exchange.value <- tmp[paste0("prod", 1:(np - 1))] + tmp[paste0("lab", 1:(np - 1))]
  # The period interest rate equals the yield rate of money.
  ir.period <- ts.exchange.value[1:(np - 2)] / ts.exchange.value[2:(np - 1)] - 1
  ir.period <- pmax(1e-6, ir.period)
  ir.period[np - 1] <- ir.period[np - 2]
  ir.intraperiod.new <- convert_ir(ir.period, vm, "intraperiod")

  ir.intraperiod <- c(ir.intraperiod * ratio_adjust(ir.intraperiod.new / ir.intraperiod, 0.3))
  cat("ir.intraperiod: ", ir.intraperiod, "\n")

  if (return.ge) {
    ge$ts.exchange.value <- ts.exchange.value
    return(ge)
  } else {
    return(ir.intraperiod)
  }
}

## Calculate equilibrium intraperiod interest rates.
## Warning: Running the program below takes about several minutes.
## mat.ir.intraperiod <- iterate(rep(0.1, np - 1), f, tol = 1e-4)
## # the period interest rate in the steady-state equilibrium.
## ir.period.steadyState <- sserr(eis, Gamma.beta, gr, prepaid = TRUE)
## # the intraperiod interest rate in the steady-state equilibrium.
## convert_ir(ir.period.steadyState, vm, "intraperiod")

## Below are the calculated equilibrium intraperiod interest rates.
(ir.intraperiod <- c(
  0.3723, 0.2981, 0.2580, 0.2363, 0.2245, 0.2181, 0.2147,
  0.2128, 0.2118, 0.2112, 0.2109, 0.2107, 0.2106, 0.2106,
  0.2106, 0.2105, 0.2105, 0.2106, 0.2106
))

ge <- f(ir.intraperiod, TRUE)

plot(ge$z[1:(np - 1)], type = "o")

# The yield rate of money is equal to the primitive period interest rate.
(ir.period <- ge$ts.exchange.value[1:(np - 2)] / ge$ts.exchange.value[2:(np - 1)] - 1)
convert_ir(ir.period, vm, "intraperiod") # consistent with ir.intraperiod
# consistent with ir.period.
convert_ir(ir.intraperiod, vm, "period")

## Calculate the growth rate of the money supply and the equilibrium nominal
## period interest rates when the current price of the product remains constant.
price.money <- 1 / c(1, cumprod(ir.intraperiod + 1))
currentPrice.prod <- ge$p[1:np] / price.money
gr.moneySupply <- unname(growth_rate(1 / currentPrice.prod))
# the equilibrium nominal period interest rates
(1 + ir.period) * (gr.moneySupply[2:(np - 1)] + 1) - 1
(1 + convert_ir(ir.intraperiod, vm, "period")) * (gr.moneySupply[2:np] + 1) - 1

## the corresponding sequential model with the same steady-state equilibrium.
dividend.rate <- sserr(eis, Gamma.beta, prepaid = TRUE)
ir.intraperiod <- convert_ir(dividend.rate, vm, "intraperiod")

np <- 5
ge.ss <- f(ir.intraperiod = rep(ir.intraperiod, np - 1), return.ge = TRUE, y1 = 152)

dst.firm <- node_new("prod",
  type = "FIN", rate = c(1, ir.intraperiod / vm, (1 + ir.intraperiod / vm) * dividend.rate),
  "cc1", "money", "equity.share"
)
node_set(dst.firm, "cc1",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.laborer <- node_new("util",
  type = "FIN", rate = c(1, ir.intraperiod / vm),
  "prod", "money"
)

dst.moneyOwner <- node_new("util",
  type = "FIN", rate = c(1, ir.intraperiod / vm),
  "prod", "money"
)

ge2 <- sdm2(
  A = list(dst.firm, dst.laborer, dst.moneyOwner),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100,
    NA, 100, NA
  ), 4, 3, TRUE),
  names.commodity = c(
    "prod", "lab", "money", "equity.share"
  ),
  names.agent = c("firm", "laborer", "moneyOwner"),
  numeraire = c("money" = ir.intraperiod)
)

ge2$p
ge.ss$p
ge2$p / ge2$p["prod"]

ge.ss$z[np - 1]
ge2$z

ge.ss$D[paste0("prod", np - 1), c("laborer", "moneyOwner")]
ge2$D

# The total value of the product and labor is equal to 400, which
# is equal to the money stock multiplied by the velocity of money.
addmargins(ge2$SV)
addmargins(ge2$DV)

The Identical Steady-state Equilibrium: Three Models with Money and Dividend

Description

Three steady-state-identical models with money and dividend as follows: (1) a sequential model (Li, 2019, example 7.5); (2) a time-circle model; (3) a timeline model with head-tail adjustment.

Stocks, fiat currencies, bonds, and taxes, etc. can be collectively referred to as ad valorem claims. Sometimes we do not need to differentiate between these financial instruments when modeling. Furthermore, sometimes we do not need to consider which period these financial instruments belong to.

Usage

gemIntertemporal_Money_Dividend_Example7.5.1(...)

Arguments

References

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

Examples

#### (1) a sequential model. See the first part of example 7.5 in Li (2019).
dividend.rate <- 0.25
ir <- 0.25 # the interest rate.

dst.firm <- node_new(
  "output",
  type = "FIN", rate = c(1, dividend.rate),
  "cc1", "dividend"
)
node_set(dst.firm, "cc1",
         type = "FIN", rate = c(1, ir),
         "cc1.1", "money"
)
node_set(dst.firm, "cc1.1",
         type = "CD", alpha = 1, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "FIN", rate = c(1, ir),
  "cc1", "money"
)
node_set(dst.consumer, "cc1",
         type = "CD", alpha = 1, beta = c(0.5, 0.5),
         "prod", "lab"
)

ge.seq <- sdm2(
  A = list(
    dst.firm, dst.consumer, dst.consumer, dst.consumer
  ),
  B = diag(c(1, 0, 0, 0)),
  S0Exg = {
    tmp <- matrix(NA, 4, 4)
    tmp[2, 2] <- tmp[3, 3] <- tmp[4, 4] <- 100
    tmp
  },
  names.commodity = c("prod", "lab", "money", "dividend"),
  names.agent = c("firm", "laborer", "moneyOwner", "shareholder"),
  numeraire = "prod",
  GRExg = 0.1,
  z0 = c(9.30909, 0, 0, 0),
  policy = policyMarketClearingPrice,
  maxIteration = 1,
  numberOfPeriods = 20,
  ts = TRUE
)

matplot(ge.seq$ts.z, type = "o", pch = 20)
ge.seq$D
ge.seq$S
ge.seq$ts.z[,1]
growth_rate(ge.seq$ts.z[,1])

#### (2) a time-circle model.
np <- 5 # the number of economic periods
gr <- 0.1 # the growth rate.
dividend.rate <- 0.25
ir <- 0.25
zeta <- (1 + gr)^np # the ratio of repayments to loans

n <- 2 * np + 1 # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np), "claim")
names.agent <- c(paste0("firm", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 * (1 + gr)^(0:(np - 1)) # the labor supply.
S0Exg["claim", "consumer"] <- np * 100 # the ad valorem claim supply.

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}
B["prod1", paste0("firm", np)] <- 1 / zeta

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "FIN", rate = c(1, (1 + ir) * (1 + dividend.rate) - 1),
    "cc1", "claim"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "CD", alpha = 1, beta = c(0.5, 0.5),
           paste0("prod", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "FIN", rate = c(1, ir),
  "cc1", "claim"
)
node_set(dst.consumer, "cc1",
         # type = "CES", es = 1,
         type = "CD",
         alpha = 1, beta = rep(1 / np, np),
         paste0("cc1.", 1:np)
)
for (k in 1:np) {
  node_set(dst.consumer, paste0("cc1.", k),
           type = "CD", alpha = 1, beta = c(0.5, 0.5),
           paste0("prod", k), paste0("lab", k)
  )
}
node_plot(dst.consumer, TRUE)

ge.tc <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1"
)

ge.tc$D
ge.tc$z

#### (3) a timeline model with head-tail adjustment.
np <- 5 # the number of economic periods
gr <- 0.1
dividend.rate <- 0.25
ir <- 0.25

n <- 2 * np + 1 # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np), "claim")
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 * (1 + gr)^(0:(np - 1))
S0Exg["claim", "consumer"] <- np * 100
S0Exg["prod1", "consumer"] <- 10 # the product supply in the first period, which will be adjusted.

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "FIN", rate = c(1, (1 + ir) * (1 + dividend.rate) - 1),
    "cc1", "claim"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "CD", alpha = 1, beta = c(0.5, 0.5),
           paste0("prod", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "FIN", rate = c(1, ir),
  "cc1", "claim"
)
node_set(dst.consumer, "cc1",
         type = "CD",
         alpha = 1, beta = rep(1 / np, np),
         paste0("cc1.", 1:np)
)
for (k in 1:np) {
  node_set(dst.consumer, paste0("cc1.", k),
           type = "CD", alpha = 1, beta = c(0.5, 0.5),
           paste0("prod", k), paste0("lab", k)
  )
}

ge.tl <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  policy = makePolicyHeadTailAdjustment(gr = gr, np = np)
)

node_plot(dst.consumer, TRUE)
ge.tl$D
ge.tl$z

Some Examples of Intertemporal (Timeline) Models with Production and Public Firms

Description

Some examples of intertemporal (timeline) models with production and public firms. A public producer is akin to a consumer with an infinite lifespan. The public producer owns the products it manufactures. In each period, it exchanges the products it has produced for the inputs required for production. In intertemporal models, a public producer can be treated as multiple public firms that each only produces for a single period. Each public firm hands over its products to the public firm of the next period, which in turn uses these products for trading.

Usage

gemIntertemporal_PublicFirm(...)

Arguments

Examples

np <- 15 # the number of economic periods, firms.
gr <- 0  # the growth rate of the labor supply
eis <- 0.5 # the elasticity of intertemporal substitution
Gamma.beta <- 0.9 # the subjective discount factor
y1 <- 100 # the initial product supply

n <- 2 * np # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
names.agent <- c(paste0("firm", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
# the supply of labor.
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 * (1 + gr)^(0:(np - 1))
S0Exg["prod1", "firm1"] <- y1

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 2, beta = c(0.5, 0.5),
    paste0("lab", k), paste0("prod", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

policy.PublicFirm <- function(state) {
  for (k in 1:(np - 1)) {
    state$S[k + 1, k + 1] <- state$S[k + 1, k]
    state$S[k + 1, k] <- 0
  }
  state
}

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  policy=policy.PublicFirm
)

ge$p
ge$z[1:15]

#### the sequential form of the above model.
dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 2, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

dstl <- list(dst.firm, dst.consumer)

ge.seq <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  z0 = c(100, 1),
  ts = TRUE,
  policy = policyMarketClearingPrice,
  numberOfPeriods = 20,
  maxIteration = 1
)

growth_rate(ge$p[paste0("prod", 1:np)]) + 1
growth_rate(ge$p[paste0("lab", 1:np)]) + 1
1 / (1 + sserr(eis, Gamma.beta, gr))
ge.seq$ts.z[, 1]

Some Examples of a 2-by-2 Time Circle Equilibrium Model

Description

Some examples of a 2-by-2 (intertemporal) time circle equilibrium model. In a time circle model, the economy borrows some resources from the outside in the beginning, and will repay it after the economy ends.

In these examples, there is an np-period-lived consumer maximizing intertemporal utility, and there is a type of firm which produces from period 1 to np. There are two commodities, i.e. product and labor. Suppose the firm can borrow some product from outside in the first period and return them in the (np+1)-th period. And the supply of product in the first period can be regarded as the output of the firm in the (np+1)-th period. Hence the product supply in the first period is an endogenous variable. Suppose that the amount returned is zeta times the amount borrowed.

Usage

gemIntertemporal_TimeCircle_2_2(...)

Arguments

See Also

gemOLG_TimeCircle

Examples

#### an example with a Cobb-Douglas intertemporal utility function
np <- 5 # the number of economic periods, firms.
gr <- 0 # the growth rate of the labor supply
zeta <- 1.25 # the ratio of repayments to loans
# zeta <- (1 + gr)^np
Gamma.beta <- 1 # the subjective discount factor

n <- 2 * np # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
names.agent <- c(paste0("firm", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 * (1 + gr)^(0:(np - 1)) # the supply of labor

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}
B["prod1", paste0("firm", np)] <- 1 / zeta

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 2, beta = c(0.5, 0.5),
    paste0("lab", k), paste0("prod", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

## an example with a Leontief intertemporal utility function
dst.consumer <- node_new(
  "util",
  type = "Leontief", a = rep(1, np),
  paste0("prod", 1:np)
)

ge2 <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  ts = TRUE
)

ge2$p
ge2$z
ge2$D
ge2$S
ge2$DV
ge2$SV

## Use a mean-value policy function to accelerate convergence.
ge3 <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = c(paste0("prod", 1:np), paste0("lab", 1:np)),
  names.agent = c(paste0("firm", 1:np), "consumer"),
  numeraire = "lab1",
  ts = TRUE,
  policy = makePolicyMeanValue(30)
)

#### an example with a linear intertemporal utility function (e.g. beta1 * x1 + beta2 * x2)
## The demand structure of the consumer will be adjusted sluggishly to accelerate convergence.
np <- 5 # the number of economic periods, firms.
rho <- 0.9 # the subjective discount factor

beta.consumer <- rep(rho^(0:(np - 1)))
zeta <- (1 / rho)^np

n <- 2 * np # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
names.agent <- c(paste0("firm", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 # the supply of labor

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}
B["prod1", paste0("firm", np)] <- 1 / zeta

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 2, beta = c(0.5, 0.5),
    paste0("lab", k), paste0("prod", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "FUNC",
  last.a = rep(1, np),
  func = function(p) {
    value.marginal.utility <- beta.consumer / p
    ratio <- value.marginal.utility / mean(value.marginal.utility)
    a <- dst.consumer$last.a
    a <- prop.table(a * ratio_adjust(ratio, 0.15))
    dst.consumer$last.a <- a
    a
  },
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  ts = TRUE,
  priceAdjustmentVelocity = 0.1
)

ge$p
ge$z
ge$D
ge$S
growth_rate(ge$p[1:np])
growth_rate(ge$p[(np + 1):(2 * np)])

A Time Circle Model with One Consumer and Two Types of Firms

Description

An (intertemporal) time circle model with one consumer and two types of firms.

Usage

gemIntertemporal_TimeCircle_3_3(...)

Arguments

Examples

#### an example with a Cobb-Douglas intertemporal utility function
np <- 5 # the number of economic periods, firms.
zeta <- 1.25 # the ratio of repayments to loans

n <- 3 * np # the number of commodity kinds
m <- 2 * np + 1 # the number of agent kinds

names.commodity <- c(
  paste0("corn", 1:np),
  paste0("iron", 1:np),
  paste0("lab", 1:np)
)
names.agent <- c(
  paste0("firm.corn", 1:np),
  paste0("firm.iron", 1:np),
  "consumer"
)

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 # the supply of labor

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("corn", k + 1), paste0("firm.corn", k)] <- 1
  B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
}
B["corn1", paste0("firm.corn", np)] <- 1 / zeta
B["iron1", paste0("firm.iron", np)] <- 1 / zeta

dstl.firm.corn <- dstl.firm.iron <- list()
for (k in 1:np) {
  dstl.firm.corn[[k]] <- node_new(
    "prod",
    type = "CD",
    alpha = 1, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )

  dstl.firm.iron[[k]] <- node_new(
    "prod",
    type = "CD",
    alpha = 2, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = prop.table(rep(1, np)),
  paste0("corn", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm.corn, dstl.firm.iron, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

A Time Circle Model with Two Consumers and Two Types of Firms

Description

An (intertemporal) time circle model with two consumers and two types of firms. There are three commodities in the model, namely wheat, iron, and labor.

Usage

gemIntertemporal_TimeCircle_3_4(...)

Arguments

Examples

#### an example with a Cobb-Douglas intertemporal utility function
np <- 5 # the number of economic periods, firms.
zeta <- 1.25 # the ratio of repayments to loans

n <- 3 * np # the number of commodity kinds
m <- 2 * np + 2 # the number of agent kinds

names.commodity <- c(
  paste0("wheat", 1:np),
  paste0("iron", 1:np),
  paste0("lab", 1:np)
)
names.agent <- c(
  paste0("firm.wheat", 1:np), paste0("firm.iron", 1:np),
  "consumer1", "consumer2"
)

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), c("consumer1", "consumer2")] <- 100 # the supply of labor

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("wheat", k + 1), paste0("firm.wheat", k)] <- 1
  B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
}
B["wheat1", paste0("firm.wheat", np)] <- 1 / zeta
B["iron1", paste0("firm.iron", np)] <- 1 / zeta

dstl.firm.wheat <- dstl.firm.iron <- list()
for (k in 1:np) {
  dstl.firm.wheat[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 1, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )

  dstl.firm.iron[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 2, beta = c(0.5, 0.5),
    paste0("iron", k), paste0("lab", k)
  )
}

dst.consumer1 <- node_new(
  "util",
  type = "CD", alpha = 1, beta = prop.table(rep(1, np)),
  paste0("wheat", 1:np)
)

dst.consumer2 <- node_new(
  "util",
  type = "CD", alpha = 1, beta = prop.table(rep(1, np)),
  paste0("cc", 1:np)
)
for (k in 1:np) {
  node_set(
    dst.consumer2, paste0("cc", k),
    type = "CD", alpha = 1, beta = c(0.5, 0.5),
    paste0("wheat", k), paste0("iron", k)
  )
}

ge <- sdm2(
  A = c(dstl.firm.wheat, dstl.firm.iron, dst.consumer1, dst.consumer2),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

Some Examples of a Time Circle Model with a Consumer and a Type of Bank

Description

Some examples of a time circle model with a consumer and a type of bank. These models can be used to solve some intertemporal savings problems.

In these example, an np-period-lived consumer gets some payoff (or cash, exhaustible resource etc.) in each period. In each period the consumer can use payoff for consumption, save payoff into bank or get a loan from the bank. The interest rate is given. The consumer has a CES intertemporal utility function and attempts to maximize intertemporal utility by saving and borrowing.

Usage

gemIntertemporal_TimeCircle_Bank_1_2(...)

Arguments

Examples

#### an example with a 5-period-lived consumer (see Zhang, 2008, section 1.3)
np <- 5 # the number of economic periods
interest.rate <- 0.1
zeta <- (1 + interest.rate)^np # the ratio of repayments to loans

n <- np # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- paste0("payoff", 1:np)
names.agent <- c(paste0("bank", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("payoff", 1:np), "consumer"] <- 100 / (1:np)

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("payoff", k + 1), paste0("bank", k)] <- 1
}
B["payoff1", paste0("bank", np)] <- 1 / zeta

dstl.bank <- list()
for (k in 1:np) {
  dstl.bank[[k]] <- node_new(
    "output",
    type = "Leontief", a = 1 / (1 + interest.rate),
    paste0("payoff", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = 0.5, alpha = 1, beta = prop.table(1:np),
  paste0("payoff", 1:np)
)

ge <- sdm2(
  A = c(dstl.bank, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1",
  ts = TRUE,
  policy = makePolicyMeanValue(30)
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV
growth_rate(ge$p)

####
dst.consumer$es <- 0

ge <- sdm2(
  A = c(dstl.bank, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1",
  ts = TRUE,
  policy = makePolicyMeanValue(30)
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV
growth_rate(ge$p)

Some 2-by-2 Time Circle Models with Uncertainty

Description

Some time circle models with uncertainty. In these models, there is a consumer who will live for two periods and has a von Neumann-Morgenstern expected utility function. There is one natural state in the first period, and two natural states in the second period. In the economy, there are two types of commodities: product and labor. In the first period, the economy can borrow a certain amount of product from an external source, such as a bank, and repay it after the economic operation is complete. The amount of product to be repaid is zeta times the amount borrowed. zeta is an exogenous variable.

Usage

gemIntertemporal_TimeCircle_Stochastic_2_2(...)

Arguments

Examples

#### Assume that the consumer supplies labor only in the first period,
## and the firm produces only in the first period.
zeta <- 1.25 # the ratio of repayments to loans
dst.firm <- node_new(
  "prod2",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "prod1", "lab1"
)

dst.bank <- node_new(
  "prod1",
  type = "Leontief",
  a = c(1, 1) * zeta,
  "prod2.1", "prod2.2"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.25, 0.25),
  "prod1", "prod2.1", "prod2.2"
)

ge <- sdm2(
  A = c(dst.firm, dst.bank, dst.consumer),
  B = matrix(c(
    0, 1, 0,
    2, 0, 0,
    1, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, NA, NA,
    NA, NA, 100
  ), 4, 3, TRUE),
  names.commodity = c("prod1", "prod2.1", "prod2.2", "lab1"),
  names.agent = c("firm", "bank", "consumer"),
  numeraire = "lab1",
  policy = makePolicyMeanValue(30),
  ts = TRUE
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

#### Assume that the consumer supplies labor in both periods and
## firms produce in both periods.
zeta <- 1.25 # the ratio of repayments to loans
dst.firm1 <- node_new(
  "prod2",
  type = "CD", alpha = 2,
  beta = c(0.5, 0.5),
  "lab1", "prod1"
)

dst.firm2.1 <- node_new(
  "prod3.1",
  type = "CD", alpha = 2,
  beta = c(0.5, 0.5),
  "lab2.1", "prod2.1"
)

dst.firm2.2 <- node_new(
  "prod3.2",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "lab2.2", "prod2.2"
)

dst.bank <- node_new(
  "prod1",
  type = "Leontief",
  a = c(1, 1) * zeta,
  "prod3.1", "prod3.2"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = c(1 / 3, 1 / 3, 1 / 3),
  "prod1", "prod2.1", "prod2.2"
)

ge <- sdm2(
  A = c(
    dst.firm1, dst.firm2.1, dst.firm2.2,
    dst.bank, dst.consumer
  ),
  B = matrix(c(
    0, 0, 0, 1, 0,
    1, 0, 0, 0, 0,
    1, 0, 0, 0, 0,
    0, 1, 0, 0, 0,
    0, 0, 1, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0
  ), 8, 5, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, 100,
    NA, NA, NA, NA, 100,
    NA, NA, NA, NA, 100
  ), 8, 5, TRUE),
  names.commodity = c(
    "prod1", "prod2.1", "prod2.2",
    "prod3.1", "prod3.2",
    "lab1", "lab2.1", "lab2.2"
  ),
  names.agent = c(
    "firm1", "firm2.1", "firm2.2",
    "bank", "consumer"
  ),
  numeraire = "lab1",
  policy = makePolicyMeanValue(30),
  ts = TRUE
)

ge$p
ge$z
ge$D
ge$S
ge$DV
ge$SV

Some Examples of Spot Equilibrium Paths Involving Land and Labor

Description

Some examples of spot equilibrium paths involving land and labor. The labor supply may begin to increase starting from the fifth period.

Usage

gemLand_Labor(...)

Arguments

Examples

#### a 3-by-3 economy
f <- function(GRLabor = 0,
              es.land.labor = 1) {
  dst.firm <- node_new("output",
                       type = "SCES",
                       es = es.land.labor, alpha = 1,
                       beta = c(0.5, 0.5),
                       "land", "lab"
  )

  dst.landowner <- node_new(
    "util",
    type = "Leontief", a = 1,
    "prod"
  )

  dst.laborer <- Clone(dst.landowner)

  dstl <- list(dst.firm, dst.landowner, dst.laborer)

  ge <- sdm2(
    A = dstl,
    B = diag(c(1, 0, 0)),
    S0Exg = matrix(c(
      NA, NA, NA,
      NA, 100, NA,
      NA, NA, 100
    ), 3, 3, TRUE),
    names.commodity = c("prod", "land", "lab"),
    names.agent = c("firm", "landowner", "laborer"),
    maxIteration = 1,
    numberOfPeriods = 30,
    numeraire = "lab",
    ts = TRUE,
    policy = list(
      function(time, state) {
        if (time >= 5) {
          state$S[3, 3] <- 100 * (1 + GRLabor)^(time - 4)
        }
        state
      },
      policyMarketClearingPrice
    ),
    z0 = c(200, 100, 100),
    p0 = c(1, 1, 1)
  )

  par(mfrow = c(1, 2))
  matplot(growth_rate(ge$ts.p), type = "o", pch = 20)
  matplot(growth_rate(ge$ts.z), type = "o", pch = 20)

  ge
}

ge <- f()
ge$p
ge$z


f(GRLabor = 0.03)
f(GRLabor = -0.03)
f(GRLabor = 0.03, es.land.labor = 0.5)
f(GRLabor = 0.03, es.land.labor = 1.5)

#### a 4-by-3 economy
GRLabor <- 0.03

dst.agri <- node_new("agri",
                     type = "SCES", es = 0.5, alpha = 1,
                     beta = c(0.75, 0.25),
                     "land", "lab"
)

dst.manu <- node_new("manu",
                     type = "SCES", es = 0.5, alpha = 1,
                     beta = c(0.25, 0.75),
                     "land", "lab"
)

dst.consumer <- node_new("util",
                         type = "SCES", es = 0.5, alpha = 1,
                         beta = c(0.5, 0.5),
                         "agri", "manu"
)

dstl <- list(dst.agri, dst.manu, dst.consumer)

ge <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = {
    S0Exg <- matrix(NA, 4, 3)
    S0Exg[3:4, 3] <- 100
    S0Exg
  },
  names.commodity = c("agri", "manu", "land", "lab"),
  names.agent = c("agri", "manu", "consumer"),
  numeraire = c("manu"),
  ts = TRUE,
  policy = list(
    function(time, state) {
      if (time >= 5) {
        state$S[4, 3] <- 100 * (1 + GRLabor)^(time - 4)
      }
      state
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 40,
  maxIteration = 1,
  z0 = c(100, 100, 200),
  p0 = c(1, 1, 1, 1)
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(growth_rate(ge$ts.z), type = "o", pch = 20)
matplot(growth_rate(ge$ts.p), type = "o", pch = 20)

Some Examples of Spot Equilibrium Paths Involving Land, Labor and Capital

Description

Some examples of spot equilibrium paths involving land, labor and capital.

Usage

gemLand_Labor_Capital_4_3(...)

Arguments

Examples

depreciation.rate <- 0.05

dst.firm.production <- node_new("prod",
  type = "CD",
  alpha = 1, beta = c(0.4, 0.4, 0.2),
  "lab", "cap", "land"
)

dst.firm.capital.leasing <- node_new("cap",
  type = "Leontief", a = 1,
  "prod"
)

dst.consumer <- node_new("util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "prod", "lab"
)

dstl <- list(dst.firm.production, dst.consumer, dst.firm.capital.leasing)
f <- function(policy = policyMarketClearingPrice,
              p0 = c(1, 1, 1, 1),
              z0 = c(10, 10, 10),
              numberOfPeriods = 100) {
  sdm2(
    A = dstl,
    B = matrix(c(
      1, 0, 1 - depreciation.rate,
      0, 0, 0,
      0, 0, 1,
      0, 0, 0
    ), 4, 3, TRUE),
    S0Exg = {
      S0Exg <- matrix(NA, 4, 3)
      S0Exg[2, 2] <- S0Exg[4, 2] <- 1
      S0Exg
    },
    names.commodity = c("prod", "lab", "cap", "land"),
    names.agent = c("firm.production", "consumer", "firm.capital.leasing"),
    numeraire = "prod",
    maxIteration = 1,
    numberOfPeriods = numberOfPeriods,
    p0 = p0,
    z0 = z0,
    policy = policy,
    ts = TRUE
  )
}

ge1 <- f()
ge1$p
ge1$DV
ge1$SV
matplot(ge1$ts.z, type = "l")

## a spot equilibrium path with population growth
policy.population.growth <- function(time, state) {
  if (time >= 5) {
    state$S[2, 2] <- 1.01^(time - 4)
  }
  state
}

ge2 <- f(
  policy = list(
    policy.population.growth,
    policyMarketClearingPrice
  ),
  p0 = ge1$p, z0 = ge1$z,
  numberOfPeriods = 30
)
matplot(ge2$ts.z, type = "o", pch = 40)
matplot(growth_rate(ge2$ts.z), type = "o", pch = 20)

## a spot equilibrium path with technology progress
policy.technology.progress <- function(time, A) {
  if (time >= 5) {
    A[[1]]$alpha <- 1.02^(time - 4)
  }
}

ge3 <- f(
  policy = list(
    policy.technology.progress,
    policyMarketClearingPrice
  ),
  p0 = ge1$p, z0 = ge1$z,
  numberOfPeriods = 30
)

matplot(ge3$ts.z, type = "o", pch = 20)
matplot(growth_rate(ge3$ts.z), type = "o", pch = 20)

## a spot equilibrium path with population growth and technology progress
ge4 <- f(
  policy = list(
    policy.population.growth,
    policy.technology.progress,
    policyMarketClearingPrice
  ),
  p0 = ge1$p, z0 = ge1$z,
  numberOfPeriods = 30
)

matplot(ge4$ts.z, type = "o", pch = 20)
matplot(growth_rate(ge4$ts.z), type = "o", pch = 20)

Some Examples of Spot Equilibrium Paths

Description

Some examples of zero-dividend spot equilibrium paths, involving a firm and a laborer (consumer)

Usage

gemMarketClearingPath_2_2(...)

Arguments

Examples

## the benchmark equilibrium
dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 5, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

dstl <- list(dst.firm, dst.consumer)

f <- function(policy = NULL) {
  sdm2(
    A = dstl,
    B = matrix(c(
      1, 0,
      0, 0
    ), 2, 2, TRUE),
    S0Exg = matrix(c(
      NA, NA,
      NA, 1
    ), 2, 2, TRUE),
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "consumer"),
    numeraire = "lab",
    z0 = c(1, 1),
    ts = TRUE,
    policy = policy,
    numberOfPeriods = 40,
    maxIteration = 1
  )
}

ge <- f(policy = policyMarketClearingPrice)
matplot(ge$ts.S[1, 1, ], type = "o", pch = 20)
matplot(ge$ts.z, type = "o", pch = 20)

## labor supply change
ge.LSC <- f(policy = list(
  function(time, state) {
    if (time >= 21) state$S[2, 2] <- state$S[2, 2] * 2
    state
  },
  policyMarketClearingPrice
))

matplot(ge.LSC$ts.z, type = "o", pch = 20)

## technology progress
ge.TP <- f(policy = list(
  makePolicyTechnologyChange(
    adjumentment.ratio = 2,
    agent = "firm",
    time.win = c(21, 21)
  ),
  policyMarketClearingPrice
))

matplot(ge.TP$ts.z, type = "o", pch = 20)

## the same as above
ge.TP2 <- f(policy = list(
  function(time, A) {
    if (time >= 21) {
      A[[1]]$alpha <- 10
    } else {
      A[[1]]$alpha <- 5
    }
  },
  policyMarketClearingPrice
))

matplot(ge.TP2$ts.z, type = "o", pch = 20)

#### A timeline model, the equilibrium of which is the same as the benchmark equilibrium.
# In this model, in terms of form, firms are treated as consumer-type agents rather than
# producer-type agents. Firms hold products. The utility level of each firm determines
# the quantity of the product that the firm owns in the subsequent economic period.
np <- 5 # the number of economic periods
y1 <- 1 # the initial product supply
eis <- 1 # elasticity of intertemporal substitution
Gamma.beta <- 1 # the subjective discount factor

n <- 2 * np # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
names.agent <- c(paste0("firm", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 1
for (k in 1:np) {
  S0Exg[paste0("prod", k), paste0("firm", k)] <- y1
}

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD",
    alpha = 5, beta = c(0.5, 0.5),
    paste0("prod", k), paste0("lab", k)
  )
}

dst.consumer.CD <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = prop.table(rep(1, np)),
  paste0("prod", 1:np)
)

dst.consumer <- node_new(
  "util",
  type = "CES", es = eis,
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
  paste0("prod", 1:np)
)

ge.timeline <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = matrix(0, n, m),
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  ts = TRUE,
  policy = function(time, state) {
    names(state$last.z) <- state$names.agent
    dimnames(state$S) <- list(names.commodity, names.agent)

    for (k in 2:np) {
      state$S[paste0("prod", k), paste0("firm", k)] <- state$last.z[paste0("firm", k - 1)]
    }

    return(state)
  }
)

head(ge.timeline$p, np) / tail(ge.timeline$p, np)
ge$ts.p[1:5, 1] # the same as above

ge.timeline$z[1:np]
ge$ts.z[1:np, 1] # the same as above

ge.timeline$D
ge.timeline$S

A General Equilibrium Model with Money

Description

A general equilibrium model with money as a medium of exchange and a means of payment.

Usage

gemMoney_3_2(
  dstl,
  supply.labor = 100,
  supply.money = 300,
  names.commodity = c("product", "labor", "money"),
  names.agent = c("firm", "household"),
  ...
)

Arguments

Details

A general equilibrium model with 3 commodities (i.e. product, labor, and money) and 2 agents (i.e. a firm and a household). To produce, the firm needs product, labor and money. The household only consumes the product. But money is also needed to buy the product. The household supplies labor and money.

In the calculation results, the price of the currency is the interest per unit of currency (i.e., the rental price). It should be noted that the unit of currency can be arbitrarily selected. For example, a unit of currency may be two dollars or ten dollars. The rental price divided by the interest rate is the asset price of 1 unit of the currency.

Value

A general equilibrium (see sdm2)

Examples

#### Leontief-type firm
interest.rate <- 0.25
vm <- 1 # the velocity of money

dst.Leontief.firm <- node_new("output",
  type = "FIN", rate = c(1, interest.rate / vm),
  "cc1", "money"
)
node_set(dst.Leontief.firm, "cc1",
  type = "Leontief", a = c(0.6, 0.2),
  "product", "labor"
)

dst.household <- node_new("utility",
  type = "FIN", rate = c(1, interest.rate / vm),
  "product", "money"
)

dstl.Leontief <- list(dst.Leontief.firm, dst.household)

ge.Leontief <- gemMoney_3_2(dstl.Leontief)
ge.Leontief$p

## SCES-type firm
dst.SCES.firm <- Clone(dst.Leontief.firm)
node_set(dst.SCES.firm, "cc1",
  type = "SCES", alpha = 1, beta = c(0.6, 0.2),
  es = 0 # es is the elasticity of substitution.
)

node_plot(dst.SCES.firm, TRUE)

dstl.SCES <- list(dst.SCES.firm, dst.household)

ge.SCES <- gemMoney_3_2(dstl.SCES)
ge.SCES$p
p.money <- ge.SCES$p
p.money["money"] <- p.money["money"] / interest.rate
p.money <- p.money / p.money["money"] # prices in terms of the asset price of the currency
p.money

## The price of money is the interest rate.
## The other prices are in terms of the asset price of the currency.
gemMoney_3_2(dstl.SCES,
             numeraire = c("money" = interest.rate)
)

Some 3-by-3 General Equilibrium Models with Money and Exogenous Interest Rate

Description

Some 3-by-3 general equilibrium models with money as a medium of exchange and a means of payment. Here, the interest rate is an exogenous variable.

In these examples, the price of money refers to its rental price, which is the interest amount generated per unit of money. The value of a unit of currency (i.e., its selling price or asset price) is its rental price divided by the interest rate. When the rental price of money equals the interest rate, the value of the currency equals 1, which implies that money is used as the numeraire (i.e. the unit of account).

Usage

gemMoney_3_3(...)

Arguments

References

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

Examples

#### a monetary pure exchange model
interest.rate <- 0.25
vm <- 1 # the velocity of money

dst.consumer1 <- node_new("util",
                          type = "CD",
                          alpha = 1, beta = c(0.5, 0.5),
                          "cc1", "wheat"
)
node_set(dst.consumer1, "cc1",
         type = "FIN", rate = c(1, interest.rate / vm),
         "iron", "money"
)

dst.consumer2 <- node_new("util",
                          type = "CD",
                          alpha = 1, beta = c(0.5, 0.5),
                          "cc1", "iron"
)
node_set(dst.consumer2, "cc1",
         type = "FIN", rate = c(1, interest.rate / vm),
         "wheat", "money"
)

dst.consumer3 <- node_new("util",
                          type = "FIN", rate = c(1, interest.rate / vm),
                          "cc1", "money"
)
node_set(dst.consumer3, "cc1",
         type = "CD",
         alpha = 1, beta = c(0.5, 0.5),
         "wheat", "iron"
)

ge <- sdm2(
  A = list(dst.consumer1, dst.consumer2, dst.consumer3),
  B = matrix(0, 3, 3),
  S0Exg = matrix(c(
    100, 0, 0,
    0, 100, 0,
    0, 0, 100
  ), 3, 3, TRUE),
  names.commodity = c("wheat", "iron", "money"),
  names.agent = c("consumer1", "consumer2", "consumer3"),
  numeraire = c(money = interest.rate)
)
ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$DV)
addmargins(ge$SV)

## Here are a few examples of calculating demand coefficients.
node_plot(dst.consumer1, TRUE)

# In the following example, the rental price of money is 0.25,
# which equals the interest rate of the money,
# thus it is known that the value of one unit of money is 1.
demand_coefficient(dst.consumer1, p = c(wheat = 1, iron = 1, money = 0.25))

demand_coefficient(dst.consumer1, p = c(wheat = 1, iron = 2, money = 0.25))

# In the following example,  the rental price of money is 0.5,
# and the value of one unit of money is 0.5/0.25=2.
demand_coefficient(dst.consumer1, p = c(wheat = 1, iron = 2, money = 0.5))

#### a monetary model with production
interest.rate <- 0.25
vm <- 1 # the velocity of money

dst.firm <- node_new("prod",
                     type = "FIN", rate = c(1, interest.rate / vm),
                     "cc1", "money"
)
node_set(dst.firm, "cc1",
         type = "CD", alpha = 2, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.laborer <- dst.moneyOwner <-
  node_new("util",
           type = "FIN", rate = c(1, interest.rate / vm),
           "prod", "money"
  )

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.moneyOwner),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "money"),
  names.agent = c("firm", "laborer", "moneyOwner"),
  numeraire = "prod"
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

## Take money as numeraire, that is, let the asset price of money equal to 1,
## and let the interest per unit of money equal to the exogenous interest rate.
ge2 <- sdm2(
  A = list(dst.firm, dst.laborer, dst.moneyOwner),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "money"),
  names.agent = c("firm", "laborer", "moneyOwner"),
  numeraire = c(money = interest.rate)
)

ge2$p
ge2$z
addmargins(ge2$D, 2)
addmargins(ge2$S, 2)
addmargins(ge2$DV)
addmargins(ge2$SV)

#### another model (Li, 2019, example 7.2)
interest.rate <- 0.25
vm <- 1 # the velocity of money

dst <- node_new("demand",
                type = "FIN", rate = c(1, interest.rate / vm),
                "cc1", "money"
)
node_set(dst, "cc1",
         type = "CD", alpha = 1, beta = c(0.5, 0.5),
         "prod", "lab"
)

ge <- sdm2(
  A = list(dst, dst, dst),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "money"),
  names.agent = c("firm", "laborer", "money.lender"),
  numeraire = c(money = interest.rate)
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

Some Examples Illustrating Non-excludability

Description

Some examples illustrating non-rival goods with non-excludability.

Usage

gemNonexcludability(...)

Arguments

See Also

gemNonrivalry_Uncongestiblity

Examples

dst.firm0 <- node_new(
  "non-rival services",
  type = "Leontief", a = 1,
  "labor"
)

dst.consumer1 <- node_new(
  "util",
  type = "SCES", es = 1, # es = 0
  alpha = 1, beta = c(0.75, 0.25),
  "serv1", "labor"
)

dst.consumer2 <- node_new(
  "util",
  type = "SCES", es = 1, # es = 0
  alpha = 1, beta = c(0.5, 0.5),
  "serv2", "labor"
)

f.CD <- function(policy = NULL) {
  ge <- sdm2(
    A = list(dst.firm0, dst.consumer1, dst.consumer2),
    B = matrix(c(
      1, 0, 0,
      1, 0, 0,
      0, 0, 0
    ), 3, 3, TRUE),
    S0Exg = matrix(c(
      NA, NA, NA,
      NA, NA, NA,
      NA, 60, 60
    ), 3, 3, TRUE),
    names.commodity = c("serv1", "serv2", "labor"),
    names.agent = c("firm", "consumer1", "consumer2"),
    numeraire = "labor",
    policy = policy
  )

  cat("ge$p:\n")
  print(round(ge$p, 5))
  cat("ge$z:\n")
  print(round(ge$z, 5))
  cat("ge$D:\n")
  print(addmargins(round(ge$D, 5), 2))
  cat("ge$S:\n")
  print(addmargins(round(ge$S, 5), 2))
}

f.CD()

# Suppose consumer 2 is a free rider.
policy.nonexcludability <- function(state) {
  state$S[2, 3] <- state$S[2, 1]
  state$S[2, 1] <- 0
  state
}
f.CD(policy.nonexcludability)

## Assume that both consumers have the same linear utility
# function x1 + 1.25 * x2, wherein x1 is the quantity
# of service and x2 is the quantity of labor.

dst.firm1 <- node_new(
  "serv1",
  type = "Leontief", a = 0.8,
  "labor"
)

dst.firm2 <- node_new(
  "serv2",
  type = "Leontief", a = 0.8,
  "labor"
)

dst.consumer1 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "serv1"
)

dst.consumer2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "serv2"
)

f.linear <- function(policy = NULL) {
  ge <- sdm2(
    A = list(
      dst.firm0, dst.consumer1, dst.consumer2,
      dst.firm1, dst.firm2
    ),
    B = matrix(c(
      1, 0, 0, 1, 0,
      1, 0, 0, 0, 1,
      0, 0, 0, 0, 0
    ), 3, 5, TRUE),
    S0Exg = matrix(c(
      NA, NA, NA, NA, NA,
      NA, NA, NA, NA, NA,
      NA, 60, 60, NA, NA
    ), 3, 5, TRUE),
    names.commodity = c("serv1", "serv2", "labor"),
    names.agent = c("firm.public", "consumer1", "consumer2", "firm1", "firm2"),
    numeraire = "labor",
    policy = policy
  )

  cat("ge$p:\n")
  print(round(ge$p, 5))
  cat("ge$z:\n")
  print(round(ge$z, 5))
  cat("ge$D:\n")
  print(addmargins(round(ge$D, 5), 2))
  cat("ge$S:\n")
  print(addmargins(round(ge$S, 5), 2))
}

f.linear()

# Suppose consumer 2 is a free rider.
f.linear(policy.nonexcludability)

Some Examples Illustrating Congestible Non-rival Goods

Description

Some examples illustrating congestible non-rival goods (or services).

Usage

gemNonrivalry_Congestiblity(...)

Arguments

See Also

gemNonrivalry_Uncongestiblity

Examples

## The firm supplies non-rival services.
dst.firm <- node_new(
  "non-rival services",
  type = "Leontief", a = 1,
  "labor"
)

dst.consumer1 <- node_new(
  "util",
  type = "SCES", es = 1, # es = 0
  alpha = 1, beta = c(0.75, 0.25),
  "serv1", "labor"
)

dst.consumer2 <- node_new(
  "util",
  type = "SCES", es = 1, # es = 0
  alpha = 1, beta = c(0.5, 0.5),
  "serv2", "labor"
)

dst.consumer3 <- node_new(
  "util",
  type = "SCES", es = 1,
  alpha = 1, beta = c(0.1, 0.9),
  "serv3", "labor"
)

efficient.coef <- 0.6 # 0.8, 0.7

ge <- sdm2(
  A = list(
    dst.firm, dst.firm, dst.firm, dst.firm,
    dst.consumer1, dst.consumer2, dst.consumer3
  ),
  B = matrix(c(
    1, 1, 0, efficient.coef, 0, 0, 0,
    1, 0, 1, efficient.coef, 0, 0, 0,
    0, 1, 1, efficient.coef, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0
  ), 4, 7, TRUE),
  S0Exg = {
    tmp <- matrix(NA, 4, 7)
    tmp[4, 5:7] <- 100
    tmp
  },
  names.commodity = c(paste0("serv", 1:3), "labor"),
  names.agent = c(paste0("firm", 1:4), paste0("consumer", 1:3)),
  numeraire = "labor"
)

ge$p
round(ge$z, 5)
round(ge$D, 5)
round(ge$S, 5)

##
efficient.coef <- 0.2

ge <- sdm2(
  A = list(
    dst.firm, dst.firm, dst.firm,
    dst.consumer1, dst.consumer2, dst.consumer3
  ),
  B = matrix(c(
    1, efficient.coef, 0, 0, 0, 0,
    1, efficient.coef, 0, 0, 0, 0,
    0, efficient.coef, 0.5, 0, 0, 0,
    0, 0, 0, 0, 0, 0
  ), 4, 6, TRUE),
  S0Exg = {
    tmp <- matrix(NA, 4, 6)
    tmp[4, 4:6] <- 100
    tmp
  },
  names.commodity = c(paste0("serv", 1:3), "labor"),
  names.agent = c(paste0("firm", 1:3), paste0("consumer", 1:3)),
  numeraire = "labor"
)

ge$p
round(ge$z, 5)
round(ge$D, 5)
round(ge$S, 5)

## congested land services.
efficient.coef <- 0.6 # 0.8, 0.7

dst.firm <- node_new(
  "non-rival services",
  type = "Leontief", a = 1,
  "land"
)

ge <- sdm2(
  A = list(
    dst.firm, dst.firm, dst.firm, dst.firm,
    dst.consumer1, dst.consumer2, dst.consumer3
  ),
  B = matrix(c(
    1, 1, 0, efficient.coef, 0, 0, 0,
    1, 0, 1, efficient.coef, 0, 0, 0,
    0, 1, 1, efficient.coef, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0
  ), 5, 7, TRUE),
  S0Exg = {
    tmp <- matrix(NA, 5, 7)
    tmp[4, 5:7] <- 55
    tmp[5, 5:7] <- 45
    tmp
  },
  names.commodity = c(paste0("serv", 1:3), "labor", "land"),
  names.agent = c(paste0("firm", 1:4), paste0("consumer", 1:3)),
  numeraire = "labor"
)

ge$p
round(ge$z, 5)
round(ge$D, 5)
round(ge$S, 5)

##
efficient.coef <- 0.2

ge <- sdm2(
  A = list(
    dst.firm, dst.firm, dst.firm,
    dst.consumer1, dst.consumer2, dst.consumer3
  ),
  B = matrix(c(
    1, efficient.coef, 0, 0, 0, 0,
    1, efficient.coef, 0, 0, 0, 0,
    0, efficient.coef, 0.5, 0, 0, 0,
    0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0
  ), 5, 6, TRUE),
  S0Exg = {
    tmp <- matrix(NA, 5, 6)
    tmp[4, 4:6] <- 55
    tmp[5, 4:6] <- 45
    tmp
  },
  names.commodity = c(paste0("serv", 1:3), "labor", "land"),
  names.agent = c(paste0("firm", 1:3), paste0("consumer", 1:3)),
  numeraire = "labor"
)

ge$p
round(ge$z, 5)
round(ge$D, 5)
round(ge$S, 5)

Some Examples Illustrating Uncongestible Non-rival Goods

Description

Some examples illustrating (uncongestible) non-rival goods (or services), Lindahl prices and the uniform price. In general equilibrium models, non-rival services can be regarded as personalized services, which are joint products of a production process (see Mas-Colell, Whinston, and Green, 1995, section 16.G).

Usage

gemNonrivalry_Uncongestiblity(...)

Arguments

References

Mas-Colell, Andreu and Whinston, Michael Dennis and Green, Jerry R. (1995, ISBN: 0195073401) Microeconomic Theory. Oxford University Press (New York).

Examples

## The firm supplies non-rival services.
dst.firm <- node_new(
  "non-rival services",
  type = "Leontief", a = 1,
  "labor"
)

dst.consumer1 <- node_new(
  "util",
  type = "SCES", es = 1, # es = 0
  alpha = 1, beta = c(0.75, 0.25),
  "serv1", "labor"
)

dst.consumer2 <- node_new(
  "util",
  type = "SCES", es = 1, # es = 0
  alpha = 1, beta = c(0.5, 0.5),
  "serv2", "labor"
)

ge <- sdm2(
  A = list(dst.firm, dst.consumer1, dst.consumer2),
  B = matrix(c(
    1, 0, 0,
    1, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, 60, 60
  ), 3, 3, TRUE),
  names.commodity = c("serv1", "serv2", "labor"),
  names.agent = c("firm", "consumer1", "consumer2"),
  numeraire = "labor"
)

ge$p # Lindahl prices
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)

## Computing the uniform price of the non-rival services
## by transfer payment between consumers.
ge <- sdm2(
  A = list(dst.firm, dst.consumer1, dst.consumer2),
  B = matrix(c(
    1, 0, 0,
    1, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, 60, 60
  ), 3, 3, TRUE),
  names.commodity = c("serv1", "serv2", "labor"),
  names.agent = c("firm", "consumer1", "consumer2"),
  numeraire = "labor",
  policy = function(A, state) {
    # A[[1]]$last.s is the previous labor supply of consumer1.
    if (is.null(A[[1]]$last.s)) A[[1]]$last.s <- 60

    p <- state$p / state$p[3]
    last.DV <- dg(p) %*% state$last.A %*% dg(state$last.z)
    transfer.payment <- last.DV[1, 2] - mean(c(last.DV[1, 2], last.DV[2, 3]))

    A[[1]]$last.s <- state$S[3, 2] <- A[[1]]$last.s *
      ratio_adjust((60 + transfer.payment) / A[[1]]$last.s, 0.2)
    state$S[3, 3] <- 120 - state$S[3, 2]

    state
  }
)

# Taking transfer payment into account, the uniform price of the non-rival services is 0.5.
ge$p
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)

ge2 <- sdm2(
  A = list(dst.firm, dst.consumer1, dst.consumer2),
  B = matrix(c(
    1, 0, 0,
    1, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, 80, 40
  ), 3, 3, TRUE),
  names.commodity = c("serv1", "serv2", "labor"),
  names.agent = c("firm", "consumer1", "consumer2"),
  numeraire = "labor"
)

ge2$p
addmargins(ge2$D, 2)
addmargins(ge2$S, 2)
addmargins(ge2$DV)

## Calculate a stationary state with price regulation.
## Both services have the same price and service 2 is oversupplied.
pcss <- sdm2(
  A = list(dst.firm, dst.consumer1, dst.consumer2),
  B = matrix(c(
    1, 0, 0,
    1, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, 60, 60
  ), 3, 3, TRUE),
  names.commodity = c("serv1", "serv2", "labor"),
  names.agent = c("firm", "consumer1", "consumer2"),
  numeraire = "labor",
  policy = function(state) {
    state$p[2] <- state$p[1]
    state
  },
  maxIteration = 1,
  numberOfPeriods = 200,
  depreciationCoef = 0,
  ts = TRUE
)

pcss$p
addmargins(pcss$D, 2)
addmargins(pcss$S, 2)
matplot(pcss$ts.q, type = "l")
matplot(pcss$ts.z, type = "l")
matplot(pcss$ts.p, type = "l")

##
pcss <- sdm2(
  A = list(dst.firm, dst.consumer1, dst.consumer2),
  B = matrix(c(
    1, 0, 0,
    1, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, 50, 50
  ), 3, 3, TRUE),
  names.commodity = c("serv1", "serv2", "labor"),
  names.agent = c("firm", "consumer1", "consumer2"),
  numeraire = "labor",
  policy = list(
    function(state) {
      state$p[1:2] <- sum(state$p[1:2] * c(0.8, 0.2))
      state
    },
    makePolicyMeanValue()
  ),
  maxIteration = 1,
  numberOfPeriods = 1000,
  ts = TRUE
)

pcss$p
addmargins(pcss$D, 2)
addmargins(pcss$S, 2)
addmargins(pcss$DV)
addmargins(pcss$SV)
matplot(pcss$ts.q, type = "l")
matplot(pcss$ts.z, type = "l")
matplot(pcss$ts.p, type = "l")

Overlapping Generations Financial Sequential Models with One Firm

Description

Some examples of overlapping generations financial sequential models with one firm.

When there is a population growth, we will take the security-split assumption (see gemOLGF_PureExchange).

Usage

gemOLGF_OneFirm(...)

Arguments

References

Samuelson, P. A. (1958) An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money. Journal of Political Economy, vol. 66(6): 467-482.

de la Croix, David and Philippe Michel (2002, ISBN: 9780521001151) A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations. Cambridge University Press.

See Also

gemOLG_PureExchange gemOLG_TimeCircle

Examples

#### an OLGF economy with a firm and two-period-lived consumers
beta.firm <- c(1 / 3, 2 / 3)
# the population growth rate
GRExg <- 0.03
saving.rate <- 0.5
ratio.saving.consumption <- saving.rate / (1 - saving.rate)

dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 5,
  beta = beta.firm,
  "lab", "prod"
)

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption),
  "prod", "secy" # security, the financial instrument
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge <- sdm2(
  A = list(
    dst.firm, dst.age1, dst.age2
  ),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 1, NA,
    NA, NA, 1
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "secy"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "lab",
  GRExg = GRExg,
  maxIteration = 1,
  ts = TRUE
)

ge$p
matplot(ge$ts.p, type = "l")
matplot(growth_rate(ge$ts.z), type = "l") # GRExg
addmargins(ge$D, 2) # the demand matrix of the current period
addmargins(ge$S, 2) # the supply matrix of the current period
addmargins(ge$S * (1 + GRExg), 2) # the supply matrix of the next period
addmargins(ge$DV)
addmargins(ge$SV)

## Suppose consumers consume product and labor (i.e. service) and
## age1 and age2 may have different period utility functions.
dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption),
  "cc1", "secy" # security, the financial instrument
)
node_set(dst.age1,  "cc1",
  type = "Leontief",
  a = c(0.5, 0.5),
  "prod", "lab"
)
node_plot(dst.age1)

dst.age2 <- node_new("util",
  type = "Leontief",
  a = c(0.2, 0.8),
  "prod", "lab"
)

ge <- sdm2(
  A = list(
    dst.firm, dst.age1, dst.age2
  ),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 1, NA,
    NA, NA, 1
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "secy"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "lab",
  GRExg = GRExg,
  priceAdjustmentVelocity = 0.05
)

ge$p
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

## Aggregate the above consumers into one infinite-lived consumer,
## who always spends the same amount on cc1 and cc2.
dst.consumer <- node_new("util",
                         type = "CD", alpha = 1,
                         beta = c(0.5, 0.5),
                         "cc1", "cc2"
)
node_set(dst.consumer, "cc1",
         type = "Leontief",
         a = c(0.5, 0.5),
         "prod", "lab"
)
node_set(dst.consumer,  "cc2",
         type = "Leontief",
         a = c(0.2, 0.8),
         "prod", "lab"
)

ge <- sdm2(
  A = list(
    dst.firm, dst.consumer
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  GRExg = GRExg,
  priceAdjustmentVelocity = 0.05
)

ge$p
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

#### an OLGF economy with a firm and two-period-lived consumers
## Suppose each consumer has a Leontief-type utility function min(c1, c2/a).
beta.firm <- c(1 / 3, 2 / 3)
# the population growth rate, the equilibrium interest rate and profit rate
GRExg <- 0.03
rho <- 1 / (1 + GRExg)
a <- 0.9

dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 5,
  beta = beta.firm,
  "lab", "prod"
)

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = a * rho),
  "prod", "secy" # security, the financial instrument
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge <- sdm2(
  A = list(
    dst.firm, dst.age1, dst.age2
  ),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 1, NA,
    NA, NA, 1
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "secy"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "lab",
  GRExg = GRExg,
  maxIteration = 1,
  ts = TRUE
)

ge$p
matplot(ge$ts.p, type = "l")
matplot(growth_rate(ge$ts.z), type = "l") # GRExg
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

## the corresponding time-cycle model
n <- 5 # the number of periods, consumers and firms.
S <- matrix(NA, 2 * n, 2 * n)

S.lab.consumer <- diag((1 + GRExg)^(0:(n - 1)), n)
S[(n + 1):(2 * n), (n + 1):(2 * n)] <- S.lab.consumer

B <- matrix(0, 2 * n, 2 * n)
B[1:n, 1:n] <- diag(n)[, c(2:n, 1)]
B[1, n] <- rho^n

dstl.firm <- list()
for (k in 1:n) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 5,
    beta = beta.firm,
    paste0("lab", k), paste0("prod", k)
  )
}

dstl.consumer <- list()
for (k in 1:(n - 1)) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "FIN",
    rate = c(1, ratio.saving.consumption = a * rho),
    paste0("prod", k), paste0("prod", k + 1)
  )
}

dstl.consumer[[n]] <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = a * rho),
  paste0("prod", n), "cc1"
)
node_set(dstl.consumer[[n]], "cc1",
         type = "Leontief", a = rho^n, # a discounting factor
         "prod1"
)

ge2 <- sdm2(
  A = c(dstl.firm, dstl.consumer),
  B = B,
  S0Exg = S,
  names.commodity = c(paste0("prod", 1:n), paste0("lab", 1:n)),
  names.agent = c(paste0("firm", 1:n), paste0("consumer", 1:n)),
  numeraire = "lab1",
  policy = makePolicyMeanValue(30),
  maxIteration = 1,
  numberOfPeriods = 600,
  ts = TRUE
)

ge2$p
growth_rate(ge2$p[1:n]) + 1 # rho
growth_rate(ge2$p[(n + 1):(2 * n)]) + 1 # rho
ge2$D

Overlapping Generations Financial Sequential Models for Pure Exchange Economies

Description

Some examples of overlapping generations sequential models with security for pure exchange economies.

In these examples, a security (e.g., fiat money, public bond, shares of a firm, etc) serves as a means of saving (see Samuelson, 1958). Consumers use these securities to transfer value across time, enabling intertemporal allocation of resources.

Here financial demand structure trees are used, which contain financial nodes. A financial demand structure tree reflects the demand structure of a consumer who has a demand for financial instruments.

When experiencing population growth, we adopt the security-split assumption. Specifically, we assume that in each period, the security undergoes a split similar to a stock split, with the quantity of the security growing at a rate equal to the population growth rate. Obviously, this assumption will not affect the calculation results essentially. And with this assumption, the equilibrium price vector can keep constant in each period, and the nominal rates of profit and interest will equal the real rates of profit and interest (i.e. the population growth rate). In contrast, in the time circle model the nominal rates of profit and interest equal zero and the real rates of profit and interest equal the population growth rate.

Usage

gemOLGF_PureExchange(...)

Arguments

Note

As can be seen from the first example below, in a pure exchange economy with two-period-lived consumers, if the young (i.e., age 1) has labor and the old (i.e., age 2) does not, then the optimal steady-state allocation can be obtained by introducing securities as a store of value.

However, if the old (i.e., age 2) has 1 unit of labor and the young (i.e., age 1) does not, then the optimal steady-state allocation cannot be obtained by introducing securities as a store of value. Therefore, we need to introduce the family system, or in other words, grant the young the right to tax the old.

See Also

gemOLG_PureExchange

Examples

#### (A) An OLG pure exchange economy with two-period-lived consumers.
## To simplify the analysis, we assume that the savings rate of the
## young consumer is an exogenous value. The consumer's utility
## function that corresponds to this savings rate can be constructed
## based on the steady-state equilibrium results.

sr <- 0.5

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption <- sr / (1 - sr)),
  "lab", "secy"
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "lab"
)

ge <- sdm2(
  A = list(
    dst.age1, dst.age2
  ),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    0, 100
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy"
)

ge$p
ge$D
ge$DV
ge$S

##
sr <- 0.375
dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption <- sr / (1 - sr)),
  "lab", "secy"
)

ge <- sdm2(
  A = list(
    dst.age1, dst.age2
  ),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    80, 20,
    0, 100
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy"
)

ge$p
ge$D
ge$DV
ge$S

##
sr <- 0.4
dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption <- sr / (1 - sr)),
  "lab", "secy"
)

ge <- sdm2(
  A = list(
    dst.age1, dst.age2
  ),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    100, 20,
    0, 100
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy"
)

ge$p
ge$D
ge$DV
ge$S

####  (B) The population growth and demographic dividend.
## Suppose each consumer has a SCES intertemporal utility function.
gr.lab <- 0.03 # the growth rate of population and labor supply

# share parameters of the SCES function
beta2 <- 0.4
beta1 <- 1 - beta2
es <- 0.5 # the elasticity of substitution in the SCES function

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = 0.1),
  "lab", "secy",
  p.lab.last = 1,
  p.lab.ratio.predicted.last = 1,
  ts.saving.rate = numeric(0)
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "lab"
)

ge <- sdm2(
  A = list(
    dst.age1, dst.age2
  ),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    100, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy",
  policy = list(function(time, A, state) {
    state$S[1, 1] <- 100 * (1 + gr.lab)^time
    p.lab.current <- state$p[1] / state$p[2]

    lambda <- 0.6
    p.lab.ratio.predicted <- p.lab.current / A[[1]]$p.lab.last * lambda +
      A[[1]]$p.lab.ratio.predicted.last * (1 - lambda)
    A[[1]]$p.lab.last <- p.lab.current
    A[[1]]$p.lab.ratio.predicted.last <- p.lab.ratio.predicted

    ratio.saving.consumption <- beta2 / beta1 * (p.lab.ratio.predicted)^(1 - es)
    A[[1]]$rate <- c(1, ratio.saving.consumption)
    A[[1]]$ts.saving.rate <- c(A[[1]]$ts.saving.rate, ratio.saving.consumption /
                                 (1 + ratio.saving.consumption))

    state
  }, policyMarketClearingPrice),
  maxIteration = 1,
  numberOfPeriods = 50,
  ts = TRUE
)

matplot(growth_rate(ge$ts.p), type = "o", pch = 20)
matplot(growth_rate(ge$ts.z), type = "o", pch = 20)
ge$p
dst.age1$rate[2] # beta2 / beta1 * (1 + gr.lab)^(es - 1)
dst.age1$p.lab.ratio.predicted.last

plot(dst.age1$ts.saving.rate, type = "o", pch = 20)
tail(dst.age1$ts.saving.rate,1) # beta2 / (beta2 + beta1 * (1 + gr.lab)^(1 - es))

#### (C) The basic overlapping generations (inefficient) exchange model.
## Here the lab2 is regarded as a financial instrument (saving instrument).
## See gemOLG_PureExchange.
dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.totalSaving.consumption = 2),
  "lab1", "lab2"
)

dst.age2 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = 1),
  "lab1", "lab2"
)

ge <- sdm2(
  A = list(dst.age1, dst.age2),
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    50, 50,
    50, 0
  ), 2, 2, TRUE),
  names.commodity = c("lab1", "lab2"),
  names.agent = c("age1", "age2"),
  numeraire = "lab1",
  policy = function(time, state) {
    pension <- (state$last.A[, 2] * state$last.z[2])[2]
    if (time > 1) state$S[1, 2] <- 1 - pension
    state
  }
)

ge$p
ge$S
ge$D
ge$DV

#### (D) the basic financial overlapping generations exchange model (see Samuelson, 1958).
## Suppose each consumer has a utility function log(c1) + log(c2) + log(c3).
GRExg <- 0.03 # the population growth rate
rho <- 1 / (1 + GRExg)

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = {
    saving.rate <- (2 - rho) / 3
    c(1, ratio.saving.consumption = saving.rate / (1 - saving.rate))
  },
  "lab", "secy"
)

dst.age2 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = 1),
  "lab", "secy"
)

dst.age3 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "lab"
)

ge <- sdm2(
  A = list(dst.age1, dst.age2, dst.age3),
  B = matrix(0, 2, 3),
  S0Exg = matrix(c(
    1 + GRExg, 1, 0,
    0, 0.5, 0.5
  ), 2, 3, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2", "age3"),
  numeraire = "lab",
  policy = function(time, state) {
    # Assume that unsold public bond will be void.
    last.Demand <- state$last.A %*% dg(state$last.z)
    secy.holding <- prop.table(last.Demand[2, ])
    if (time > 1) {
      state$S[2, 2:3] <- secy.holding[1:2]
    }
    state
  }
)

ge$p
ge$S
ge$D

#### (E) a pure exchange economy with three-period-lived consumers.
## Suppose each consumer has a Leontief-type utility function min(c1, c2, c3).
GRExg <- 0.03 # the population growth rate
igr <- 1 + GRExg

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = {
    saving.rate <- 1 / (1 + igr + igr^2)
    c(1, ratio.saving.consumption = saving.rate / (1 - saving.rate))
  },
  "lab", "secy"
)

dst.age2 <- node_new(
  "util",
  type = "FIN",
  rate = {
    saving.rate <- 1 / (1 + igr)
    c(1, ratio.saving.consumption = saving.rate / (1 - saving.rate))
  },
  "lab", "secy"
)

dst.age3 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "lab"
)

ge <- sdm2(
  A = list(dst.age1, dst.age2, dst.age3),
  B = matrix(0, 2, 3),
  S0Exg = matrix(c(
    1 + GRExg, 1, 0,
    0, 0.5, 0.5
  ), 2, 3, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2", "age3"),
  numeraire = "lab",
  policy = function(time, state) {
    # Assume that unsold security will be void.
    last.Demand <- state$last.A %*% dg(state$last.z)
    secy.holding <- prop.table(last.Demand[2, ])
    if (time > 1) {
      state$S[2, 2:3] <- secy.holding[1:2]
    }
    state
  }
)

ge$p
ge$S
ge$D

## Assume that the unsold security of age3 will be void.
## The calculation results are the same as above.
ge <- sdm2(
  A = list(dst.age1, dst.age2, dst.age3),
  B = matrix(0, 2, 3),
  S0Exg = matrix(c(
    1 + GRExg, 1, 0,
    0, 0.5, 0.5
  ), 2, 3, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2", "age3"),
  numeraire = "lab",
  policy = function(time, state, state.history) {
    secy.unsold <- state.history$S[2, , time - 1] * (1 - state.history$q[time - 1, 2])
    last.Demand <- state$last.A %*% dg(state$last.z)
    secy.purchased <- last.Demand[2, ]

    if (time > 1) {
      # Assume that the unsold security of age3 will be void.
      state$S[2, 2:3] <- prop.table(secy.purchased[1:2] + secy.unsold[1:2])
    }
    state
  },
  maxIteration = 1
)

ge$p
ge$S
ge$D

Overlapping Generations Financial Sequential Models with Two Firms

Description

Some examples of overlapping generations financial sequential models with two firms.

Usage

gemOLGF_TwoFirms(...)

Arguments

See Also

gemOLGF_PureExchange

Examples

#### an example with two-period-lived consumers
dst.firm.corn <- node_new(
  "corn",
  type = "CD", alpha = 5,
  beta = c(1 / 2, 1 / 2),
  "iron", "lab"
)

dst.firm.iron <- node_new(
  "iron",
  type = "CD", alpha = 5,
  beta = c(1 / 2, 1 / 2),
  "iron", "lab"
)

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = 1),
  "corn", "secy" # security, the financial instrument
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "corn"
)

ge <- sdm2(
  A = list(
    dst.firm.corn, dst.firm.iron, dst.age1, dst.age2
  ),
  B = matrix(c(
    1, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 0, 0,
    0, 0, 0, 0
  ), 4, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, 1, NA,
    NA, NA, NA, 1
  ), 4, 4, TRUE),
  names.commodity = c("corn", "iron", "lab", "secy"),
  names.agent = c("firm.corn", "firm.iron", "age1", "age2"),
  numeraire = "lab"
)

ge$p
ge$D
ge$DV
ge$S

## an example with three-period-lived consumers
dst.age1$rate <- c(1, ratio.saving.consumption = 1 / 2)

dst.age3 <- Clone(dst.age2)

dst.age2 <- Clone(dst.age1)
dst.age2$rate <- c(1, ratio.saving.consumption = 1)

ge <- sdm2(
  A = list(
    dst.firm.corn, dst.firm.iron, dst.age1, dst.age2, dst.age3
  ),
  B = matrix(c(
    1, 0, 0, 0, 0,
    0, 1, 0, 0, 0,
    0, 0, 0, 0, 0,
    0, 0, 0, 0, 0
  ), 4, 5, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA,
    NA, NA, 1, 1, NA,
    NA, NA, NA, 1, 1
  ), 4, 5, TRUE),
  names.commodity = c("corn", "iron", "lab", "secy"),
  names.agent = c("firm.corn", "firm.iron", "age1", "age2", "age3"),
  numeraire = "lab",
  policy = function(time, state) {
    # Assume that unsold security will be void.
    last.Demand <- state$last.A %*% dg(state$last.z)
    secy.holding <- prop.table(last.Demand[4, ])
    if (time > 1) {
      state$S[4, 4:5] <- secy.holding[3:4]
    }
    state
  }
)

ge$p
ge$D
ge$DV
ge$S

Some Examples of Basic (Timeline) OLG Models with Production

Description

Some examples of basic (timeline) OLG models with production. In these models there are two types of commodities (i.e., labor and product) and two types of economic agents (i.e., laborers and firms). Laborers (i.e., consumers) live for two or three periods. These models can be easily extended to include more types of commodities, consumers and firms, and to allow consumers to live for more periods.

Usage

gemOLG_Basic(...)

Arguments

Examples

## When beta.consumer[1]==0, beta.consumer[2:3]>0, labor.first[1]>0, labor.first[2:3]==0,
## this model is actually the Diamond model. However, the division of periods is
## slightly different from the Diamond model.
ng <- 15 # the number of generations
alpha.firm <- 2 # the efficient parameter of firms
beta.prod.firm <- 0.4 # the product (i.e. capital) share parameter of firms
beta.consumer <- c(0, 0.8, 0.2) # the share parameter of consumers
gr.laborer <- 0.03 # the population growth rate
labor.first <- c(100, 0, 0) # the labor supply of the first generation
labor.last <- 100 * (1 + gr.laborer)^((ng - 1):ng) # the labor supply of the last generation
y1 <- 8 # the initial product supply

f <- function() {
  names.commodity <- c(paste0("prod", 1:(ng + 2)), paste0("lab", 1:(ng + 1)))
  names.agent <- c(paste0("firm", 1:(ng + 1)), paste0("consumer", 1:ng))

  n <- length(names.commodity) # the number of commodity kinds
  m <- length(names.agent) # the number of agent kinds

  # the exogenous supply matrix.
  S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
  for (k in 1:(ng - 1)) {
    S0Exg[paste0("lab", k:(k + 2)), paste0("consumer", k)] <- labor.first * (1 + gr.laborer)^(k - 1)
  }
  S0Exg[paste0("lab", ng:(ng + 1)), paste0("consumer", ng)] <- labor.last
  S0Exg["prod1", "consumer1"] <- y1

  B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
  for (k in 1:(ng + 1)) {
    B[paste0("prod", k + 1), paste0("firm", k)] <- 1
  }

  dstl.consumer <- list()
  for (k in 1:ng) {
    dstl.consumer[[k]] <- node_new(
      "util",
      type = "CD", alpha = 1,
      beta = beta.consumer,
      paste0("prod", k:(k + 2))
    )
  }


  dstl.firm <- list()
  for (k in 1:(ng + 1)) {
    dstl.firm[[k]] <- node_new(
      "prod",
      type = "CD", alpha = alpha.firm,
      beta = c(beta.prod.firm, 1 - beta.prod.firm),
      paste0("prod", k), paste0("lab", k)
    )
  }

  ge <- sdm2(
    A = c(dstl.firm, dstl.consumer),
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "lab1",
    priceAdjustmentVelocity = 0.05
  )

  cat("ge$p:\n")
  print(ge$p)
  cat("ge$z:\n")
  print(ge$z)
  invisible(ge)
}

ge <- f()

# the growth rates of prices
growth_rate(ge$p[paste0("prod", 1:ng)]) + 1
growth_rate(ge$p[paste0("lab", 1:ng)]) + 1
# the steady-state growth rate of prices in the Diamond model
beta.consumer[3] * (1 - beta.prod.firm) / beta.prod.firm / (1 + gr.laborer)

# the output-labor ratios
ge$z[paste0("firm", 1:ng)] / rowSums(ge$S)[paste0("lab", 1:ng)]
# the steady-state output-labor ratio in the Diamond model
alpha.firm^(1 / (1 - beta.prod.firm)) * (beta.consumer[3] * (1 - beta.prod.firm) /
                                           (1 + gr.laborer))^(beta.prod.firm / (1 - beta.prod.firm))

# the captial-labor ratios
rowSums(ge$D[paste0("prod", 1:ng), paste0("firm", 1:ng)]) /
  rowSums(ge$S[paste0("lab", 1:ng), paste0("consumer", 1:ng)])
# the steady-state captial-labor ratio in the Diamond model
alpha.firm^(1 / (1 - beta.prod.firm)) * (beta.consumer[3] * (1 - beta.prod.firm)
                                         / (1 + gr.laborer))^(1 / (1 - beta.prod.firm))

##
ng <- 15
alpha.firm <- 2
beta.prod.firm <- 0.5
beta.consumer <- c(1, 1, 1) / 3
labor.first <- c(50, 50, 0)
labor.last <- c(50, 50)
y1 <- 8
gr.laborer <- 0
f()

#### Assume that consumers live for two periods and consume labor (i.e., leisure).
ng <- 10 # the number of generations
alpha.firm <- 2 # the efficient parameter of firms
beta.prod.firm <- 0.5 # the product (i.e. capital) share parameter of firms
beta.consumer <- prop.table(c(
  lab1 = 1, lab2 = 1,
  prod1 = 1, prod2 = 1
)) # the share parameter of consumers
labor <- c(100, 0) # the labor supply of each generation
y1 <- 8 # the initial product supply

names.commodity <- c(paste0("lab", 1:(ng + 1)), paste0("prod", 1:(ng + 1)))
names.agent <- c(paste0("consumer", 1:ng), paste0("firm", 1:ng))

n <- length(names.commodity) # the number of commodity kinds
m <- length(names.agent) # the number of agent kinds

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:ng) {
  S0Exg[paste0("lab", k:(k + 1)), paste0("consumer", k)] <- labor
}
if (labor[2] == 0) S0Exg[paste0("lab", ng + 1), paste0("consumer", ng)] <- labor[1]
S0Exg["prod1", "consumer1"] <- y1


B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:ng) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.consumer <- list()
for (k in 1:ng) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = beta.consumer, # prop.table(c(1e-5,1e-5,0.5,0.5)),
    paste0("lab", k:(k + 1)), paste0("prod", k:(k + 1))
  )
}

# Assume that consumers live for three periods.
# dstl.consumer <- list()
# for (k in 1:(ng - 1)) {
#   dstl.consumer[[k]] <- node_new(
#     "util",
#     type = "CD", alpha = 1,
#     beta = rep(1 / 6, 6),
#     paste0("lab", k:(k + 2)), paste0("prod", k:(k + 2))
#   )
# }
#
# dstl.consumer[[ng]] <- node_new(
#   "util",
#   type = "CD", alpha = 1,
#   beta = rep(1 / 4, 4),
#   paste0("lab", ng:(ng + 1)), paste0("prod", ng:(ng + 1))
# )

dstl.firm <- list()
for (k in 1:ng) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD", alpha = alpha.firm,
    beta = c(1 - beta.prod.firm, beta.prod.firm),
    paste0("lab", k), paste0("prod", k)
  )
}

ge <- sdm2(
  A = c(dstl.consumer, dstl.firm),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1"
)

ge$z
growth_rate(ge$p[paste0("prod", 1:ng)]) + 1
growth_rate(ge$p[paste0("lab", 1:ng)]) + 1
ge$p[paste0("prod", 1:ng)] / ge$p[paste0("lab", 1:ng)]

An Overlapping Generations Model with Land

Description

Some examples of an overlapping generations model with land.

Usage

gemOLG_Land_4_3(...)

Arguments

Details

In this model, consumers live for two periods. Age2 owns a unit of land and age1 owns a unit of labor. Here the land use rights and land ownership are regarded as two commodities. Age2 gets land rent by selling land use rights to the firm. Then land ownership is sold to age1. Age1 saves by purchasing land ownership.

Here, the ratio of land rent to wage (denoted as xi) is determined by the production function. No matter what the saving rate of age1 is, and at what price the land ownership is transferred to age1, the consumption ratio of age2 and age1 will not be less than xi.

When the consumer's intertemporal utility function is min(c1, c2), the intertemporal substitution elasticity is 0, and each consumer desires the same amount of consumption in the two periods. It can also be assumed that the consumer's intertemporal substitution elasticity is close to 0 rather than exactly zero.

If xi>1, then this economy will inevitably have efficiency loss (dynamic inefficiency). In this case, in order to achieve the optimal allocation, not only age2 should give land to age1 for free, but age2 also needs to distribute part of the land rent to age1. However, age2 has no incentive to do so.

References

Rhee, Changyong (1991) Dynamic Inefficiency in an Economy with Land. Review of Economic Studies. 58(4), pp:791-797.

See Also

gemOLG_PureExchange

Examples

saving.rate <- 0.001
ratio.saving.consumption <- saving.rate / (1 - saving.rate)

dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 5,
  beta = c(1 / 6, 2 / 6, 3 / 6),
  "lab", "prod", "land.use.rights"
)

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption),
  "prod", "land.ownership"
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge <- sdm2(
  A = list(
    dst.firm, dst.age1, dst.age2
  ),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 1, NA,
    NA, NA, 1,
    NA, NA, 1
  ), 4, 3, TRUE),
  names.commodity = c("prod", "lab", "land.use.rights", "land.ownership"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "lab"
)

ge$p
ge$D
ge$DV
ge$S
ge$SV

## Change the saving-consumption ratio.
dst.age1$rate <- c(1, ratio.saving.consumption = 99)

ge <- sdm2(
  A = list(
    dst.firm, dst.age1, dst.age2
  ),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 1, NA,
    NA, NA, 1,
    NA, NA, 1
  ), 4, 3, TRUE),
  names.commodity = c("prod", "lab", "land.use.rights", "land.ownership"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "lab"
)

ge$p
ge$D
ge$DV
ge$S
ge$SV

Overlapping Generations Models with Private Firm

Description

Some examples of overlapping generations models with private firm. A public (i.e. publicly held) firm exists permanently and operates independently. If a public firm ownership transfers between generations, this transfer will be done through the exchange of shares. In contrast, here a private firm is established by a consumer and only runs before she retires.

In the first example, there are some two-period-lived consumers and a private firm. Suppose age1 has a unit of labor and age2 has not. In each period age1 establishes a private firm and the firm gets some labor as investment from age1 and will sell it in the market to buy some inputs for production. In the next period, age2 (i.e. age1 in the previous period) gets the output of the firm. Age2 consumes product and labor (i.e. service). Hence age1 and the firm can sell labor to age2 and buy product from age2.

In the second example with three-period-lived consumers, there are two private firms (i.e. firm1 and firm2). In each period age1 establishes a new firm1 and age2 establishes a new firm2. Firm1 gets some labor as investment from age1 and firm2 gets some product and labor as investment from age2. The output of firm1 belongs to age2 in the next period and the output of firm2 belongs to age3 in the next period. In each period age2 (i.e. age1 in the previous period) takes away the output of firm1 and age3 (i.e. age2 in the previous period) takes away the output of firm2.

Usage

gemOLG_PrivateFirm(...)

Arguments

References

Acemoglu, D. (2009, ISBN: 9780691132921) Introduction to Modern Economic Growth. Princeton University Press.

See Also

gemOLGF_PureExchange

Examples

#### an example with a private firm and two-period-lived consumers
saving.rate <- 0.5
beta.consumer <-  c(1 / 2, 1 / 2) # c(9 / 10, 1 / 10)

dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 5,
  beta = c(2 / 3, 1 / 3),
  "prod", "lab"
)

dst.age1 <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = beta.consumer,
  "prod", "lab"
)

dst.age2 <- Clone(dst.age1)

ge <- sdm2(
  A = list(
    dst.firm, dst.age1, dst.age2
  ),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0
  ), 2, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 1, NA
  ), 2, 3, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "lab",
  policy = function(time, state) {
    if (time > 1) {
      supply.prod <- state$S[1, 1]
      supply.lab <- state$S[2, 2]
      state$S <- 0 * state$S
      state$S[1, 3] <- supply.prod # age2 supplies prod.
      state$S[2, 1] <- saving.rate * supply.lab # The firm gets investment from age1.
      state$S[2, 2] <- (1 - saving.rate) * supply.lab
    }
    state
  }
)

ge$p
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

#### an example with two private firm and three-period-lived consumers
saving.rate.age1 <- 1 / 3
saving.rate.age2 <- 0.95

dst.firm1 <- dst.firm2 <- node_new(
  "prod",
  type = "CD", alpha = 5,
  beta = c(2 / 3, 1 / 3),
  "prod", "lab"
)

dst.age1 <- dst.age2 <- dst.age3 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge <- sdm2(
  A = list(
    dst.firm1, dst.firm2, dst.age1, dst.age2, dst.age3
  ),
  B = matrix(c(
    1, 1, 0, 0, 0,
    0, 0, 0, 0, 0
  ), 2, 5, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA,
    NA, NA, 1, 1, NA
  ), 2, 5, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm1", "firm2", "age1", "age2", "age3"),
  numeraire = "lab",
  policy = function(time, state) {
    if (time > 1) {
      state$S[1, 5] <- state$S[1, 2] # Age3 takes away the output of firm2.
      state$S[1, 2] <- 0

      # Age2 takes away the output of firm1.
      prod.age2 <- state$S[1, 1]
      state$S[1, 1] <- 0
      # Age2 establishes a new firm2.
      lab.age2 <- state$S[2, 4]
      state$S[2, 2] <- lab.age2 * saving.rate.age2 # Firm2 sells labor.
      state$S[1, 2] <- prod.age2 * saving.rate.age2 # Firm2 sells product.

      state$S[2, 4] <- lab.age2 * (1 - saving.rate.age2) # Age2 sells labor.
      state$S[1, 4] <- prod.age2 * (1 - saving.rate.age2) # Age2 sells product.

      # Age1 establishes a new firm1.
      state$S[2, 1] <- state$S[2, 3] * saving.rate.age1 # Firm1 sells labor.
      state$S[2, 3] <- state$S[2, 3] * (1 - saving.rate.age1) # Age1 sells labor.
    }
    state
  }
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

Some Examples of (Timeline) OLG Models with Production and Public Firms

Description

Some examples of (timeline) OLG models with production and public firms (see gemIntertemporal_PublicFirm).

Usage

gemOLG_PublicFirm(...)

Arguments

See Also

gemIntertemporal_PublicFirm

Examples

ng <- 8 # the number of generations
alpha.firm <- 2 # the efficient parameter of firms
beta.prod.firm <- 0.5 # the product (i.e. capital) share parameter of firms
beta.consumer <- c(1 / 3, 1 / 3, 1 / 3) # the share parameter of consumers
gr.laborer <- 0 # the population growth rate
labor.first <- c(50, 50, 0) # the labor supply of the first generation
# the labor supply of the last generation.
labor.last <- 50 * (1 + gr.laborer)^((ng - 1):(ng + 1))
y1 <- 100 # the initial product supply

policy.PublicFirm <- function(state) {
  for (k in 1:(ng + 1)) {
    state$S[k + 1, k + 1] <- state$S[k + 1, k]
    state$S[k + 1, k] <- 0
  }
  state
}

f <- function(policy = policy.PublicFirm) {
  names.commodity <- c(paste0("prod", 1:(ng + 2)), paste0("lab", 1:(ng + 2)))
  names.agent <- c(paste0("firm", 1:(ng + 2)), paste0("consumer", 1:ng))

  n <- length(names.commodity) # the number of commodity kinds
  m <- length(names.agent) # the number of agent kinds

  # the exogenous supply matrix.
  S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
  for (k in 1:(ng - 1)) {
    S0Exg[paste0("lab", k:(k + 2)), paste0("consumer", k)] <-
      labor.first * (1 + gr.laborer)^(k - 1)
  }
  S0Exg[paste0("lab", ng:(ng + 2)), paste0("consumer", ng)] <- labor.last
  S0Exg["prod1", "firm1"] <- y1

  B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
  for (k in 1:(ng + 1)) {
    B[paste0("prod", k + 1), paste0("firm", k)] <- 1
  }

  dstl.consumer <- list()
  for (k in 1:ng) {
    dstl.consumer[[k]] <- node_new(
      "util",
      type = "CD", alpha = 1,
      beta = beta.consumer,
      paste0("prod", k:(k + 2))
    )
  }

  dstl.firm <- list()
  for (k in 1:(ng + 2)) {
    dstl.firm[[k]] <- node_new(
      "prod",
      type = "CD", alpha = alpha.firm,
      beta = c(beta.prod.firm, 1 - beta.prod.firm),
      paste0("prod", k), paste0("lab", k)
    )
  }

  ge <- sdm2(
    A = c(dstl.firm, dstl.consumer),
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "lab1",
    priceAdjustmentVelocity = 0.05,
    policy = policy
  )

  cat("ge$p:\n")
  print(ge$p)
  cat("ge$z:\n")
  print(ge$z)
  invisible(ge)
}

ge <- f()

# the growth rates of prices
growth_rate(ge$p[paste0("prod", 1:ng)]) + 1
growth_rate(ge$p[paste0("lab", 1:ng)]) + 1

##
labor.first <- c(100 / 3, 100 / 3, 100 / 3) # the labor supply of the first generation
ge <- f()

##
tax.rate <- 0.1
policy.PublicFirm.Tax <- function(state) {
  for (k in 1:(ng + 1)) {
    state$S[k + 1, k + 1] <- state$S[k + 1, k] * (1 - tax.rate)
    state$S[k + 1, k + ng] <- state$S[k + 1, k] * tax.rate
    state$S[k + 1, k] <- 0
  }
  state
}

ge <- f(policy.PublicFirm.Tax)

##
beta.consumer <- c(1 / 2, 1 / 2, 0) # the share parameter of consumers
labor.first <- c(90, 10, 0) # the labor supply of the first generation
ge <- f()

The Basic Overlapping Generations Pure Exchange Model (see Samuelson, 1958)

Description

This is the basic overlapping generations pure exchange model.

Usage

gemOLG_PureExchange(...)

Arguments

Details

As Samuelson (1958) wrote, break each life up into thirds. Agents get 50 units of payoff in period 1 and 50 units in period 2; in period 3 they retire and get nothing. Suppose there are three agents in each period, namely age1, age2 and age3. In the next period, the present age1 will become age2, the present age2 will become age3, the present age3 will disappear and a new age1 will appear. Let c1, c2 and c3 denote the consumption of an agent in each period. Suppose the utility function is (c1 * c2 * c3)^(1 / 3), which is actually the same as log(c1) + log(c2) + log(c3). In each period, age1 and age2 will exchange their payoffs of the present period and the next period. Age2 will sell some present payoff and buy some future payoff as pension, and for age1, it's the opposite. Age3 simply receives the pension and need not take part in the exchange. Hence only two agents participate in the pure exchange economy. In the exchange process, the utility function of age1 is c1^(1 / 3) * x2^(2 / 3), wherein x2 is the revenue of the next period, and the utility function of age2 is c2^(1 / 2) * c3^(1 / 2).

Note

We can also suppose only age2 gets payoff and age1 does not.

References

Samuelson, P. A. (1958) An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money. Journal of Political Economy, vol. 66(6): 467-482.

See Also

gemOLG_TimeCircle

Examples

#### the basic overlapping generations (inefficient) exchange model in sequential form.
dst.age1 <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 3, 2 / 3),
  "payoff1", "payoff2"
)

dst.age2 <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 2, 1 / 2),
  "payoff1", "payoff2"
)

policy.supply <- function(time, state) {
  pension <- (state$last.A[, 2] * state$last.z[2])[2]
  if (time > 1) state$S[1, 2] <- 50 - pension
  state
}

ge <- sdm2(
  A = list(dst.age1, dst.age2),
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    50, 50,
    50, 0
  ), 2, 2, TRUE),
  names.commodity = c("payoff1", "payoff2"),
  names.agent = c("age1", "age2"),
  numeraire = "payoff1",
  policy = list(policy.supply, policyMarketClearingPrice),
  maxIteration = 1,
  numberOfPeriods = 20,
  ts = TRUE
)

ge$p # c(1, 3 / 2 + sqrt(13) / 2)
ge$ts.p
ge$S
ge$D
ge$DV

#### the basic overlapping generations exchange model in timeline form.
m <- 15 # the number of generations
n <- m + 1 # the number of commodity kinds

names.commodity <- paste0("payoff", 1:n)
names.agent <- paste0("gen", 1:m)

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:m) {
  S0Exg[k:(k + 1), k] <- 50
}

dstl.consumer <- list()
for (k in 1:(m - 1)) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = rep(1 / 3, 3),
    paste0("payoff", k:(k + 2))
  )
}

dstl.consumer[[m]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  paste0("payoff", m:(m + 1))
)

ge <- sdm2(
  A = dstl.consumer,
  B = matrix(0, n, m),
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1"
)

round(addmargins(ge$D, 2), 2)
round(addmargins(ge$S, 2), 2)
growth_rate(ge$p) + 1 # 3 / 2 + sqrt(13) / 2

#### Assume that in the timeline model, each consumer lives for four periods or more.
nl <- 4 # the number of life periods
payoff <- c(rep(100 / (nl - 1), nl - 1), 1e-10) # the lifetime payoffs
m <- 20 # the number of generations
n <- m + nl - 1 # the number of commodity kinds

names.commodity <- paste0("payoff", 1:n)
names.agent <- paste0("gen", 1:m)

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:m) {
  S0Exg[k:(k + nl - 1), k] <- payoff
}

dstl.consumer <- list()
for (k in 1:m) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = rep(1 / nl, nl),
    paste0("payoff", k:(k + nl - 1))
  )
}

ge <- sdm2(
  A = dstl.consumer,
  B = matrix(0, n, m),
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1"
)

round(addmargins(ge$D, 2), 2)
round(addmargins(ge$S, 2), 2)
growth_rate(ge$p) + 1

Overlapping Generations Pure Exchange Models with Bank

Description

Some examples of overlapping generations pure exchange models with bank. Under a pay-as-you-go system, banks may only redistribute payoffs among consumers in each period. This is, in each period a bank can get a part of payoff of age1 and pay it to age2 immediately. From the consumer's point of view, the bank converts the current period's payoff into the next period's payoff. Each consumer only transacts with the bank, and she can assume that there are no other consumers.

Usage

gemOLG_PureExchange_Bank(...)

Arguments

See Also

gemOLG_PureExchange

Examples

#### an example with a two-period-lived consumer
dst.bank <- node_new(
  "payoff2",
  type = "Leontief", a = 1,
  "payoff1"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 2, 1 / 2),
  "payoff1", "payoff2"
)

ge <- sdm2(
  A = list(dst.bank, dst.consumer),
  B = matrix(c(
    0, 0,
    1, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, 100,
    NA, NA
  ), 2, 2, TRUE),
  names.commodity = c("payoff1", "payoff2"),
  names.agent = c("bank", "consumer"),
  numeraire = "payoff1"
)

ge$p
ge$D
ge$S

#### an example with a three-period-lived consumer.
dst.bank1 <- node_new(
  "payoff2",
  type = "Leontief", a = 1,
  "payoff1"
)

dst.bank2 <- node_new(
  "payoff3",
  type = "Leontief", a = 1,
  "payoff2"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 3, 1 / 3, 1 / 3),
  "payoff1", "payoff2", "payoff3"
)

ge <- sdm2(
  A = list(dst.bank1, dst.bank2, dst.consumer),
  B = matrix(c(
    0, 0, 0,
    1, 0, 0,
    0, 1, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, 50,
    NA, NA, 50,
    NA, NA, NA
  ), 3, 3, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("bank1", "bank2", "consumer"),
  numeraire = "payoff1"
)

ge$p
ge$S
ge$D

## Assume that banks can earn interest through foreign investment.
dst.bank1$a <- 0.8
dst.bank2$a <- 0.8
ge <- sdm2(
  A = list(dst.bank1, dst.bank2, dst.consumer),
  B = matrix(c(
    0, 0, 0,
    1, 0, 0,
    0, 1, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, 50,
    NA, NA, 50,
    NA, NA, NA
  ), 3, 3, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("bank1", "bank2", "consumer"),
  numeraire = "payoff1"
)

ge$p
ge$S
ge$D
ge$DV

A 3-by-3 OLG Stochastic Sequential General Equilibrium Model

Description

A 3-by-3 OLG stochastic sequential general equilibrium model. There are two-period lived consumers and a type of firm. There are three types of commodities, i.e. product, labor and security (i.e. paper asset, store of value).

In each period there are a young (i.e. age1), an old (i.e. age2) and a firm. The young supplies a unit of labor and the old supplies a unit of security.

Consumers only consume products. The young buys the security to save.

The firm inputs labor to produce the product. Productivity is random. The amount of labor required to produce 1 unit of product is equal to 0.1 or 0.2, both of which occur with equal probability.

All labor is used in production. Therefore, it can be known that the product output in each period (that is, the product supply in the next period) is equal to 10 or 5, that is to say, there are two natural states with equal probability of occurrence in each period: good state and bad state. The states are denoted as a and b, respectively. There are then four cases of natural states experienced by a consumer in two periods: aa, bb, ab and ba.

Take the product as numeraire. The ratio of the security price in the latter period to the previous period of the two periods is the gross rate of return. The gross rates of return on the savings of the young in the four cases is denoted as Raa, Rbb, Rab and Rba, respectively.

The utility function of the young is -0.5*x1^-1 - 0.25*x2^-1 - 0.25*x3^-1, where x1, x2 and x3 are the (expected) consumption in the current period and the next two natural states, respectively. It can be seen that the intertemporal substitution elasticity and the inter-natural-state substitution elasticity in the utility function are both 0.5, that is, the relative risk aversion coefficient is 2.

In each period, the young determines the saving rate based on the expected gross rates of return on the security and the utility function.

In each period the young adopts the following anticipation method for the gross rate of return of the savings: First, determine the two types of gross rates of return (Raa, Rab or Rbb, Rba) that need to be predicted according to the current state. Both interest rates are then forecasted using the adaptive expectation method.

A spot equilibrium path and a disequilibrium path will be calculated below. The spot equilibrium path will converge to the stochastic equilibrium consisting of two equilibrium states, and the disequilibrium paths not.

Usage

gemOLG_StochasticSequential_3_3(...)

Arguments

Examples

# a function to find the optimal saving rate given the expected gross rates of return.
osr <- function(Ra, Rb, es = 0.5, beta = c(0.5, 0.25, 0.25)) {
  sigma <- 1 - 1 / es
  ((beta[2] * Ra^sigma / beta[1] + beta[3] * Rb^sigma / beta[1])^-es + 1)^-1
}

dst.firm <- node_new(
  "prod",
  type = "Leontief", a = 0.2,
  "lab"
)

dst.age1 <-
  node_new("util",
    type = "FIN", rate = c(1, 1),
    "prod", "secy",
    last.price = c(1, 1, 1),
    last.output = 10,
    Raa = 1, Rbb = 1, Rab = 1, Rba = 1,
    REbb = 1, REba = 1, REab = 1, REaa = 1,
    sr.ts.1 = vector(),
    sr.ts.2 = vector()
  )

dst.age2 <-
  node_new("util",
    type = "Leontief", a = 1,
    "prod"
  )

policyStochasticTechnology <- function(time, A) {
  A[[1]]$a <- sample(c(0.1, 0.2), 1)
}

policySaving <- function(time, A, state) {
  output <- state$S[1, 1]

  lambda <- 0.9

  if (output < 8) {
    A[[2]]$REba <- lambda * tail(A[[2]]$Rba, 1) + (1 - lambda) * A[[2]]$REba
    A[[2]]$REbb <- lambda * tail(A[[2]]$Rbb, 1) + (1 - lambda) * A[[2]]$REbb

    saving.rate <- osr(A[[2]]$REba, A[[2]]$REbb)
    A[[2]]$sr.ts.1 <- c(A[[2]]$sr.ts.1, saving.rate)
  }

  if (output >= 8) {
    A[[2]]$REaa <- lambda * tail(A[[2]]$Raa, 1) + (1 - lambda) * A[[2]]$REaa
    A[[2]]$REab <- lambda * tail(A[[2]]$Rab, 1) + (1 - lambda) * A[[2]]$REab

    saving.rate <- osr(A[[2]]$REaa, A[[2]]$REab)
    A[[2]]$sr.ts.2 <- c(A[[2]]$sr.ts.2, saving.rate)
  }

  A[[2]]$rate <- c(1, saving.rate / (1 - saving.rate))
}


policyRecord <- function(time, A, state) {
  last.p <- A[[2]]$last.price
  p <- state$p / state$p[1]

  last.output <- A[[2]]$last.output
  output <- A[[2]]$last.output <- state$S[1, 1]

  if ((last.output < 8) && (output < 8)) A[[2]]$Rbb <- c(A[[2]]$Rbb, p[3] / last.p[3])
  if ((last.output < 8) && (output >= 8)) A[[2]]$Rba <- c(A[[2]]$Rba, p[3] / last.p[3])
  if ((last.output >= 8) && (output < 8)) A[[2]]$Rab <- c(A[[2]]$Rab, p[3] / last.p[3])
  if ((last.output >= 8) && (output >= 8)) A[[2]]$Raa <- c(A[[2]]$Raa, p[3] / last.p[3])

  A[[2]]$last.price <- p
}

dstl <- list(dst.firm, dst.age1, dst.age2)

B <- matrix(c(
  1, 0, 0,
  0, 0, 0,
  0, 0, 0
), 3, 3, TRUE)

S0Exg <- matrix(c(
  NA, NA, NA,
  NA, 1, NA,
  NA, NA, 1
), 3, 3, TRUE)

## a spot equilibrium path.
set.seed(1)
ge <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab", "secy"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "prod",
  policy = list(
    policyStochasticTechnology,
    policySaving,
    policyMarketClearingPrice,
    policyRecord
  ),
  z0 = c(5, 1, 1),
  maxIteration = 1,
  numberOfPeriods = 40,
  ts = TRUE
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

## a disequilibrium path.
set.seed(1)
de <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab", "secy"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "prod",
  policy = list(
    policyStochasticTechnology,
    policySaving,
    policyRecord
  ),
  maxIteration = 1,
  numberOfPeriods = 400,
  ts = TRUE
)

matplot(de$ts.z, type = "o", pch = 20)
matplot(de$ts.p, type = "o", pch = 20)

## an equilibrium model for solving the optimal saving
# rate based on the expected gross rates of return.
Ra <- 1
Rb <- 0.4

ge <- sdm2(
  A = function(state) {
    a.bank <- c(1, 0, 0)
    a.consumer <- CES_A(
      sigma = (1 - 1 / 0.5), alpha = 1,
      Beta = c(0.5, 0.25, 0.25), p = state$p
    )
    cbind(a.bank, a.consumer)
  },
  B = matrix(c(
    0, 0,
    Ra, 0,
    Rb, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, 1,
    NA, 0,
    NA, 0
  ), 3, 2, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("bank", "consumer"),
  numeraire = "payoff1",
)

ge$p
addmargins(ge$D, 2)
addmargins(ge$S, 2)

ge$z[1]
osr(Ra, Rb)

## a pure exchange model.
dst.age1 <- node_new("util",
  type = "FIN", rate = c(1, 1),
  "payoff", "secy",
  last.price = c(1, 1),
  last.payoff = 1,
  Rbb = 1, Rba = 1, Rab = 1, Raa = 1,
  REbb = 1, REba = 1, REab = 1, REaa = 1,
  sr.ts.1 = vector(),
  sr.ts.2 = vector()
)

dst.age2 <- node_new("util",
  type = "Leontief", a = 1,
  "payoff"
)

policyStochasticSupply <- function(state) {
  state$S[1, 1] <- sample(c(5, 10), 1)
  state
}

policySaving <- function(time, A, state) {
  payoff <- state$S[1, 1]

  lambda <- 0.9
  if (time >= 5) {
    if (payoff == 5) {
      A[[1]]$REba <- lambda * tail(A[[1]]$Rba, 1) + (1 - lambda) * A[[1]]$REba
      A[[1]]$REbb <- lambda * tail(A[[1]]$Rbb, 1) + (1 - lambda) * A[[1]]$REbb

      saving.rate <- osr(A[[1]]$REba, A[[1]]$REbb)
      A[[1]]$sr.ts.1 <- c(A[[1]]$sr.ts.1, saving.rate)
    }

    if (payoff == 10) {
      A[[1]]$REaa <- lambda * tail(A[[1]]$Raa, 1) + (1 - lambda) * A[[1]]$REaa
      A[[1]]$REab <- lambda * tail(A[[1]]$Rab, 1) + (1 - lambda) * A[[1]]$REab

      saving.rate <- osr(A[[1]]$REaa, A[[1]]$REab)
      A[[1]]$sr.ts.2 <- c(A[[1]]$sr.ts.2, saving.rate)
    }

    A[[1]]$rate <- c(1, saving.rate / (1 - saving.rate))
  }
}

policyRecord <- function(time, A, state) {
  last.p <- A[[1]]$last.price
  p <- state$p / state$p[1]

  last.payoff <- A[[1]]$last.payoff
  payoff <- state$S[1, 1]

  if ((last.payoff == 5) && (payoff == 5)) A[[1]]$Rbb <- c(A[[1]]$Rbb, p[2] / last.p[2])
  if ((last.payoff == 5) && (payoff == 10)) A[[1]]$Rba <- c(A[[1]]$Rba, p[2] / last.p[2])
  if ((last.payoff == 10) && (payoff == 5)) A[[1]]$Rab <- c(A[[1]]$Rab, p[2] / last.p[2])
  if ((last.payoff == 10) && (payoff == 10)) A[[1]]$Raa <- c(A[[1]]$Raa, p[2] / last.p[2])


  A[[1]]$last.price <- p
  A[[1]]$last.payoff <- state$S[1, 1]
}

set.seed(1)
ge <- sdm2(
  A = list(dst.age1, dst.age2),
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    1, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("payoff", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "payoff",
  policy = list(
    policyStochasticSupply,
    policySaving,
    policyMarketClearingPrice,
    policyRecord
  ),
  maxIteration = 1,
  numberOfPeriods = 40,
  ts = TRUE
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)
dst.age1$last.payoff
dst.age1$last.price
dst.age1$Rbb

An Example Illustrating OLG Stochastic Spot Equilibrium Paths

Description

An example illustrating OLG stochastic spot equilibrium paths. The economy includes consumers with a lifespan of two periods and a type of spot-trading firm (i.e., a spot-trading enterprise). The economy features three types of commodities: products, labor, and public bonds, with the latter serving as a means of storing value.

Consumers in the economy live for two periods: youth and old age. Consumers only consume the product.

Young people save by purchasing public bonds. The savings rate of young consumers is an exogenous variable. This savings rate determines how each young person's income is distributed between their youth and old age.

Young people supply a total of 100 units of labor each period; old people supply a total of 100 units of public bonds each period.

The spot-trading enterprise operates independent of consumers, requiring products and labor for production, with a production function of \alpha\sqrt{x_1 x_2}. Assume \alpha follows a first-order autoregressive process. The enterprise's objective is to maximize current profits, without making forecasts for the future.

Usage

gemOLG_StochasticSpotEquilibrium_3_3(...)

Arguments

Note

The savings rate of the youth in the program is set to 0.5, meaning the savings-to-consumption ratio is 1.

Public bonds are used as the numeraire.

The initial output (i.e., initial product supply) is set to 20 in the parameter z0.

Examples

set.seed(1)
dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = 1),
  "prod", "publicBond"
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

policy.technology <- function(time, A) {
  A[[1]]$alpha <- exp(0.95 * log(A[[1]]$alpha) +
                        rnorm(1, sd = 0.01))
}

set.seed(1)
ge <- sdm2(
  A = list(
    dst.firm, dst.age1, dst.age2
  ),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "publicBond"),
  names.agent = c("firm", "age1", "age2"),
  numeraire = "publicBond",
  maxIteration = 1,
  numberOfPeriods = 20,
  policy = list(
    policy.technology,
    policyMarketClearingPrice
  ),
  z0 = c(20, 1, 1),
  ts = TRUE
)

matplot(ge$ts.z, type = "o", pch = 20)

Time-Circle Models (Closed Loop Overlapping Generations Models)

Description

Some examples of time-circle models, which usually contain overlapping generations. When we connect together the head and tail of time of a dynamic model, we get a time-circle model which treats time as a circle of finite length instead of a straight line of infinite length. The (discounted) output of the final period will enter the utility function and the production function of the first period, which implies the influence mechanism of the past on the present is just like the influence mechanism of the present on the future. A time-circle OLG model may be called a reincarnation model.

In a time-circle model, time can be thought of as having no beginning and no end. And we can assume all souls meet in a single market (see Shell, 1971), or there are implicit financial instruments that facilitate transactions between adjacent generations.

As in the Arrow–Debreu approach, in the following examples with production, we assume that firms operate independently and are not owned by consumers. That is, there is no explicit property of firms as the firms do not make pure profit at equilibrium (constant return to scale) (see de la Croix and Michel, 2002, page 292).

Usage

gemOLG_TimeCircle(...)

Arguments

References

de la Croix, David and Philippe Michel (2002, ISBN: 9780521001151) A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations. Cambridge University Press.

Diamond, Peter (1965) National Debt in a Neoclassical Growth Model. American Economic Review. 55 (5): 1126-1150.

Samuelson, P. A. (1958) An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money. Journal of Political Economy, vol. 66(6): 467-482.

Shell, Karl (1971) Notes on the Economics of Infinity. Journal of Political Economy. 79 (5): 1002–1011.

See Also

gemOLG_PureExchange

Examples

#### a time-circle pure exchange economy with two-period-lived consumers.
# In this example, each agent sells some payoff to the previous
# generation and buys some payoff from the next generation.

# Here np can tend to infinity.
np <- 4 # the number of economic periods, commodity kinds and generations

names.commodity <- c(paste0("payoff", 1:np))
names.agent <- paste0("gen", 1:np)

index.comm <- c(1:np, 1)
payoff <- c(100, 0) # the payoffs of lifetime
# the exogenous supply matrix.
S0Exg <- matrix(0, np, np, dimnames = list(names.commodity, names.agent))
for (k in 1:np) {
  S0Exg[index.comm[k:(k + 1)], k] <- payoff
}

# Suppose each consumer has a utility function log(c1) + log(c2).
beta.consumer <- c(0.5, 0.5)
index.comm <- c(1:np, 1)
dstl.consumer <- list()
for (k in 1:np) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = beta.consumer,
    paste0("payoff", index.comm[k]), paste0("payoff", index.comm[k + 1])
  )
}

ge <- sdm2(
  A = dstl.consumer,
  B = matrix(0, np, np),
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1"
)

ge$p
ge$D
ge$S

## Introduce population growth into the above pure exchange economy.
GRExg <- 0.03 # the population growth rate

S0Exg <- S0Exg %*% diag((1 + GRExg)^(0:(np - 1)))
S0Exg[1, np] <- S0Exg[1, np] * (1 + GRExg)^-np

dstl.consumer[[np]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = beta.consumer,
  paste0("payoff", np), "cc1"
)
node_set(dstl.consumer[[np]],
  "cc1",
  type = "Leontief", a = (1 + GRExg)^(-np), # a discounting factor
  "payoff1"
)

ge2 <- sdm2(
  A = dstl.consumer,
  B = matrix(0, np, np),
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1"
)

ge2$p
ge2$D
ge2$DV
ge2$S

#### a time-circle pure exchange economy with three-period-lived consumers
# Suppose each consumer has a utility function log(c1) + log(c2) + log(c3).
# See gemOLG_PureExchange.
np <- 5 # the number of economic periods, commodity kinds and generations

names.commodity <- c(paste0("payoff", 1:np))
names.agent <- paste0("gen", 1:np)

index.comm <- c(1:np, 1:2)
payoff <- c(50, 50, 0) # the payoffs of lifetime
# the exogenous supply matrix.
S0Exg <- matrix(0, np, np, dimnames = list(names.commodity, names.agent))
for (k in 1:np) {
  S0Exg[index.comm[k:(k + 2)], k] <- payoff
}

dstl.consumer <- list()
for (k in 1:np) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = rep(1 / 3, 3),
    paste0("payoff", index.comm[k:(k + 2)])
  )
}

ge <- sdm2(
  A = dstl.consumer,
  B = matrix(0, np, np),
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1"
)

ge$p
ge$D

## Introduce population growth into the above pure exchange economy.
GRExg <- 0.03 # the population growth rate
df <- (1 + GRExg)^-np # a discounting factor

# the exogenous supply matrix.
S0Exg <- matrix(0, np, np, dimnames = list(names.commodity, names.agent))
for (k in 1:np) {
  S0Exg[paste0("payoff", index.comm[k:(k + 2)]), paste0("gen", k)] <-
    payoff * (1 + GRExg)^(k - 1)
}
S0Exg[paste0("payoff", 1:2), paste0("gen", np)] <-
  S0Exg[paste0("payoff", 1:2), paste0("gen", np)] * df
S0Exg[paste0("payoff", 1), paste0("gen", np - 1)] <-
  S0Exg[paste0("payoff", 1), paste0("gen", np - 1)] * df

dstl.consumer[[np - 1]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = rep(1 / 3, 3),
  paste0("payoff", (np - 1):(np + 1))
)
node_set(dstl.consumer[[np - 1]], paste0("payoff", np + 1),
  type = "Leontief", a = df,
  "payoff1"
)

dstl.consumer[[np]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = rep(1 / 3, 3),
  paste0("payoff", np:(np + 2))
)
node_set(dstl.consumer[[np]], paste0("payoff", np + 1),
  type = "Leontief", a = df,
  "payoff1"
)
node_set(dstl.consumer[[np]], paste0("payoff", np + 2),
  type = "Leontief", a = df,
  "payoff2"
)

ge2 <- sdm2(
  A = dstl.consumer,
  B = matrix(0, np, np),
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "payoff1"
)

ge2$p
growth_rate(ge2$p) + 1 # 1 / (1 + GRExg)
ge2$D
ge2$DV

#### a time-circle model with production and two-period-lived consumers
# Suppose each consumer has a utility function log(c1) + log(c2).
np <- 5 # the number of economic periods, consumers and firms.
names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
names.agent <- c(paste0("firm", 1:np), paste0("consumer", 1:np))

index.comm <- c(1:np, 1)
labor.supply <- c(100, 0) # the labor supply of lifetime
# the exogenous supply matrix.
S0Exg <- matrix(NA, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
for (k in 1:np) {
  S0Exg[paste0("lab", index.comm[k:(k + 1)]), paste0("consumer", k)] <- labor.supply
}

B <- matrix(0, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
for (k in 1:np) {
  B[paste0("prod", index.comm[k + 1]), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 5,
    beta = c(1 / 3, 2 / 3),
    paste0("lab", k), paste0("prod", k)
  )
}

dstl.consumer <- list()
for (k in 1:np) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = c(1 / 2, 1 / 2),
    paste0("prod", index.comm[k:(k + 1)])
  )
}

ge <- sdm2(
  A = c(dstl.firm, dstl.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1"
)

ge$p
ge$D

## Introduce population growth into the above economy.
GRExg <- 0.03 # the population growth rate
df <- (1 + GRExg)^-np # a discounting factor

for (k in 1:np) {
  S0Exg[paste0("lab", index.comm[k:(k + 1)]), paste0("consumer", k)] <-
    labor.supply * (1 + GRExg)^(k - 1)
}

B[1, np] <- df

dstl.consumer[[np]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = c(1 / 2, 1 / 2),
  paste0("prod", np), "cc1"
)
node_set(dstl.consumer[[np]], "cc1",
  type = "Leontief", a = df,
  "prod1"
)

ge2 <- sdm2(
  A = c(dstl.firm, dstl.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  policy = makePolicyMeanValue(30),
  maxIteration = 1,
  numberOfPeriods = 600,
  ts = TRUE
)

ge2$p
growth_rate(ge2$p[1:np]) + 1 # 1 / (1 + GRExg)
growth_rate(ge2$p[(np + 1):(2 * np)]) + 1 # 1 / (1 + GRExg)
ge2$D
ge2$DV

#### a time-circle model with production and one-period-lived consumers.
# These consumers also can be regarded as infinite-lived agents maximizing
# their per period utility subject to their disposable income per period.
np <- 5 # the number of economic periods, consumers and firms.
GRExg <- 0.03 # the population growth rate
# df <- (1 + GRExg)^-np # a discounting factor
df <- 0.5

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
names.agent <- c(paste0("firm", 1:np), paste0("consumer", 1:np))

# the exogenous supply matrix.
S0Exg <- matrix(NA, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
for (k in 1:np) {
  S0Exg[paste0("lab", k), paste0("consumer", k)] <- 100 * (1 + GRExg)^(k - 1)
}

B <- matrix(0, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", index.comm[k + 1]), paste0("firm", k)] <- 1
}
B["prod1", paste0("firm", np)] <- df

beta.firm <- c(1 / 3, 2 / 3)
beta.consumer <- c(1 / 2, 1 / 2)

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 5,
    beta = beta.firm,
    paste0("lab", k), paste0("prod", k)
  )
}

dstl.consumer <- list()
for (k in 1:np) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = beta.consumer,
    paste0("lab", k), paste0("prod", k)
  )
}

ge <- sdm2(
  A = c(dstl.firm, dstl.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "lab1",
  policy = makePolicyMeanValue(30),
  maxIteration = 1,
  numberOfPeriods = 600,
  ts = TRUE
)

ge$p
growth_rate(ge$p[1:np]) + 1 # 1 / (1 + GRExg)
growth_rate(ge$p[(np + 1):(2 * np)]) + 1 # 1 / (1 + GRExg)
ge$D

## the sequential form of the above model.
dst.firm <- node_new(
  "prod",
  type = "CD", alpha = 5,
  beta = beta.firm,
  "lab", "prod"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = beta.consumer,
  "lab", "prod"
)

ge2 <- sdm2(
  A = list(
    dst.firm,
    dst.consumer
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100 / (1 + GRExg)
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  z0 = c(ge$S[1, np], 0),
  GRExg = GRExg,
  policy = policyMarketClearingPrice,
  maxIteration = 1,
  numberOfPeriods = 20,
  ts = TRUE
)

ge2$p
ge2$D
ge2$ts.z[, 1]
ge$z

#### a time-circle OLG model with production and three-period-lived consumers.
np <- 6 # the number of economic periods, consumers and firms
gr.laborer <- 0.03 # the population growth rate
df <- (1 + gr.laborer)^-np # a discounting factor
alpha.firm <- 2 # the efficient parameter of firms
beta.prod.firm <- 0.4 # the product (i.e. capital) share parameter of firms
beta.consumer <- c(0, 0.8, 0.2) # the share parameter of consumers
labor.supply <- c(100, 0, 0) # the labor supply of lifetime

f <- function() {
  names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
  names.agent <- c(paste0("firm", 1:np), paste0("consumer", 1:np))

  index.comm <- c(1:np, 1:2)

  # the exogenous supply matrix.
  S0Exg <- matrix(NA, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
  for (k in 1:np) {
    S0Exg[paste0("lab", index.comm[k:(k + 2)]), paste0("consumer", k)] <-
      labor.supply * (1 + gr.laborer)^(k - 1)
  }
  S0Exg[paste0("lab", 1:2), paste0("consumer", np)] <-
    S0Exg[paste0("lab", 1:2), paste0("consumer", np)] * df
  S0Exg[paste0("lab", 1), paste0("consumer", np - 1)] <-
    S0Exg[paste0("lab", 1), paste0("consumer", np - 1)] * df

  B <- matrix(0, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
  for (k in 1:np) {
    B[paste0("prod", index.comm[k + 1]), paste0("firm", k)] <- 1
  }
  B["prod1", paste0("firm", np)] <- df


  dstl.firm <- list()
  for (k in 1:np) {
    dstl.firm[[k]] <- node_new(
      "prod",
      type = "CD", alpha = alpha.firm,
      beta = c(beta.prod.firm, 1 - beta.prod.firm),
      paste0("prod", k), paste0("lab", k)
    )
  }

  dstl.consumer <- list()
  for (k in 1:np) {
    dstl.consumer[[k]] <- node_new(
      "util",
      type = "CD", alpha = 1,
      beta = beta.consumer,
      paste0("prod", k:(k + 2))
    )
  }

  node_set(dstl.consumer[[np - 1]], paste0("prod", np + 1),
           type = "Leontief", a = df,
           "prod1"
  )

  node_set(dstl.consumer[[np]], paste0("prod", np + 1),
           type = "Leontief", a = df,
           "prod1"
  )
  node_set(dstl.consumer[[np]], paste0("prod", np + 2),
           type = "Leontief", a = df,
           "prod2"
  )

  ge <- sdm2(
    A = c(dstl.firm, dstl.consumer),
    B = B,
    S0Exg = S0Exg,
    names.commodity = names.commodity,
    names.agent = names.agent,
    numeraire = "lab1"
  )
  invisible(ge)
}

ge <- f()
growth_rate(ge$p[1:np]) + 1 # 1 / (1 + gr.laborer)
growth_rate(ge$p[(np + 1):(2 * np)]) + 1 # 1 / (1 + gr.laborer)
ge$D
ge$DV

##
beta.consumer <- c(1 / 3, 1 / 3, 1 / 3) # the share parameter of consumers
labor.supply <- c(50, 50, 0) # the labor supply of lifetime
ge <- f()
ge$D
ge$DV

##
gr.laborer <- 0
df <- (1 + gr.laborer)^-np
beta.prod.firm <- 0.5
ge <- f()
ge$z

A 4-by-4 Open Economy with Bond

Description

Some examples of a 4-by-4 open economy with bond.

Usage

gemOpenEconomy_4_4(...)

Arguments

Examples

#### an open economy with foreign bond (bond.ROW)
dst.firm <- node_new(
  "output",
  type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = 1,
  "prod.CHN", "lab.CHN"
)

dst.consumer <- node_new(
  "util",
  type = "FIN", beta = c(0.8, 0.2),
  "cc1", "bond.ROW"
)
node_set(dst.consumer, "cc1",
         type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = 1,
         "prod.CHN", "prod.ROW"
)

dst.FT <- node_new(
  "prod.ROW",
  type = "SCES", alpha = 1, beta = c(2/3, 1/3), es = 1,
  "prod.CHN", "lab.CHN"
)

dst.ROW <- node_new(
  "util",
  type = "SCES", alpha = 1, beta = c(2/3, 1/3), es = 1,
  "prod.CHN", "lab.CHN"
)

ge.open <- sdm2(
  A = list(dst.firm, dst.consumer, dst.FT, dst.ROW),
  B = matrix(c(
    1, 0, 0, 0,
    0, 0, 0, 0,
    0, 0, 1, 0,
    0, 0, 0, 0
  ), 4, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, 300, NA, NA,
    NA, NA, NA, NA,
    NA, NA, NA, 60
  ), 4, 4, TRUE),
  names.commodity = c("prod.CHN", "lab.CHN", "prod.ROW", "bond.ROW"),
  names.agent = c("firm", "consumer", "FT", "ROW"),
  numeraire = "lab.CHN"
)

ge.open$p
addmargins(ge.open$D, 2)
addmargins(ge.open$S, 2)

## a corresponding two-country model
dst.firm.CHN <- node_new(
  "output",
  type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = 1,
  "prod.CHN", "lab.CHN"
)

dst.consumer.CHN <- node_new(
  "util",
  type = "FIN", beta = c(0.8, 0.2),
  "cc1", "bond.ROW"
)
node_set(dst.consumer.CHN, "cc1",
         type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = 1,
         "prod.CHN", "prod.ROW"
)

dst.firm.ROW <- node_new(
  "prod.ROW",
  type = "SCES", alpha = 1, beta = c(0.25, 0.25, 0.5), es = 1,
  "prod.CHN", "prod.ROW", "lab.ROW"
)

dst.consumer.ROW <- node_new(
  "util",
  type = "SCES", alpha = 1, beta = c(0.25, 0.25, 0.25, 0.25), es = 1,
  "prod.CHN", "prod.ROW", "lab.CHN", "lab.ROW"
)

ge.TC <- sdm2(
  A = list(dst.firm.CHN, dst.consumer.CHN, dst.firm.ROW, dst.consumer.ROW),
  B = matrix(c(
    1, 0, 0, 0,
    0, 0, 0, 0,
    0, 0, 1, 0,
    0, 0, 0, 0,
    0, 0, 0, 0
  ), 5, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, 300, NA, NA,
    NA, NA, NA, NA,
    NA, NA, NA, 180,
    NA, NA, NA, 60
  ), 5, 4, TRUE),
  names.commodity = c("prod.CHN", "lab.CHN", "prod.ROW", "lab.ROW", "bond.ROW"),
  names.agent = c("firm.CHN", "consumer.CHN", "firm.ROW", "consumer.ROW"),
  numeraire = "lab.CHN"
)

ge.TC$p
addmargins(ge.TC$D, 2)
addmargins(ge.TC$S, 2)

#### an open economy with domestic bond (bond.CHN)
dst.firm <- node_new(
  "output",
  type = "CD", alpha = 5, beta = c(0.5, 0.5),
  "prod.CHN", "lab.CHN"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "prod.CHN", "prod.ROW"
)

dst.FT <- node_new(
  "prod.ROW",
  type = "Leontief", a = 2,
  "prod.CHN"
)

dst.ROW <- node_new(
  "prod.ROW",
  type = "Leontief", a = 1,
  "bond.CHN"
)

ge <- sdm2(
  A = list(dst.firm, dst.consumer, dst.FT, dst.ROW),
  B = matrix(c(
    1, 0, 0, 0,
    0, 0, 0, 0,
    0, 0, 1, 1,
    0, 0, 0, 0
  ), 4, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, 1, NA, NA,
    NA, NA, NA, NA,
    NA, 0.2, NA, NA
  ), 4, 4, TRUE),
  names.commodity = c("prod.CHN", "lab.CHN", "prod.ROW", "bond.CHN"),
  names.agent = c("firm", "consumer", "FT", "ROW"),
  numeraire = "lab.CHN"
)

ge$p
addmargins(ge$D, 2)
addmargins(ge$S, 2)

A 6-by-6 Open Economy with Bond

Description

Some examples of a 6-by-6 open economy with bond.

Usage

gemOpenEconomy_6_6(...)

Arguments

Examples

#### an open economy with foreign bond (bond.ROW)
dst.firm1 <- node_new(
  "prod1.CHN",
  type = "SCES", es=1, alpha = 1, beta = c(0.4, 0.4, 0.2),
  "prod2.CHN", "prod2.ROW", "lab.CHN"
)

dst.firm2 <- node_new(
  "prod2.CHN",
  type = "SCES", es=1, alpha = 1, beta = c(0.4, 0.4, 0.2),
  "prod2.CHN", "prod2.ROW", "lab.CHN"
)

dst.consumer <- node_new(
  "util",
  type = "FIN", beta = c(0.8, 0.2),
  "cc1", "bond.ROW"
)
node_set(dst.consumer, "cc1",
         type = "SCES", es=1, alpha = 1, beta = c(0.5, 0.5),
         "prod1.CHN", "prod1.ROW"
)

dst.FT1 <- node_new(
  "prod1.ROW",
  type = "SCES", es=1, alpha = 1, beta = c(0.5, 0.5),
  "prod1.CHN", "prod2.CHN"
)

dst.FT2 <- node_new(
  "prod2.ROW",
  type = "SCES", es=1, alpha = 1, beta = c(0.5, 0.5),
  "prod1.CHN", "prod2.CHN"
)

dst.Bond <- node_new(
  "util",
  type = "SCES", es=1, alpha = 1, beta = c(0.5, 0.5),
  "prod1.CHN", "prod2.CHN"
)

ge <- sdm2(
  A = list(dst.firm1, dst.firm2, dst.consumer, dst.FT1, dst.FT2, dst.Bond),
  B = matrix(c(
    1, 0, 0, 0, 0, 0,
    0, 1, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0,
    0, 0, 0, 1, 0, 0,
    0, 0, 0, 0, 1, 0,
    0, 0, 0, 0, 0, 0
  ), 6, 6, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA, NA,
    NA, NA, 100, NA, NA, NA,
    NA, NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA, 20
  ), 6, 6, TRUE),
  names.commodity = c(
    "prod1.CHN", "prod2.CHN", "lab.CHN",
    "prod1.ROW", "prod2.ROW", "bond.ROW"
  ),
  names.agent = c("firm1", "firm2", "consumer", "FT1", "FT2", "Bond"),
  numeraire = "lab.CHN"
)

ge$D
ge$p
ge$z

## Suppose the domestic consumer owns some foreign product by borrowing.
ge <- sdm2(
  A = list(dst.firm1, dst.firm2, dst.consumer, dst.FT1, dst.FT2, dst.Bond),
  B = matrix(c(
    1, 0, 0, 0, 0, 0,
    0, 1, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0,
    0, 0, 0, 1, 0, 0,
    0, 0, 0, 0, 1, 0,
    0, 0, 0, 0, 0, 0
  ), 6, 6, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA, NA,
    NA, NA, 100, NA, NA, NA,
    NA, NA, 10, NA, NA, NA,
    NA, NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA, 20
  ), 6, 6, TRUE),
  names.commodity = c(
    "prod1.CHN", "prod2.CHN", "lab.CHN",
    "prod1.ROW", "prod2.ROW", "bond.ROW"
  ),
  names.agent = c("firm1", "firm2", "consumer", "FT1", "FT2", "Bond"),
  numeraire = "lab.CHN"
)

ge$D
ge$p
ge$z

Some Examples of Spot Equilibrium Paths with Persistent Technological Progress

Description

Some examples of spot equilibrium paths with persistent technological progress. Technological progress starts from the fifth period.

Usage

gemPersistentTechnologicalProgress(...)

Arguments

See Also

gemCapitalAccumulation

Examples

#### a 2-by-2 example with labor-saving technological progress
tpr <- 0.03 # technological progress rate

dst.firm <- node_new(
  "prod",
  type = "SCES",
  es = 0.5, alpha = 1,
  beta = c(0.5, 0.5),
  "prod", "cc1"
)
node_set(dst.firm, "cc1",
         type = "Leontief", a = 1,
         "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

dstl <- list(dst.firm, dst.consumer)

ge <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod",
  ts = TRUE,
  policy = list(
    function(time, A) {
      if (time >= 5) {
        node_set(A[[1]], "cc1",
                 a = (1 + tpr)^-(time - 4)
        )
      }
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 40,
  maxIteration = 1,
  z0 = c(200, 100),
  p0 = c(1, 1)
)

matplot(growth_rate(ge$ts.z), type = "o", pch = 20)
matplot(growth_rate(ge$ts.p), type = "o", pch = 20)

#### a 3-by-3 example with labor-saving technological progress
tpr <- 0.03 # technological progress rate

dst.manu <- node_new("manu",
                     type = "SCES", es = 0.5, alpha = 1,
                     beta = c(0.6, 0.4),
                     "manu", "cc1"
)
node_set(dst.manu, "cc1",
         type = "Leontief", a = 1,
         "lab"
)

dst.serv <- node_new("serv",
                     type = "SCES", es = 0.5, alpha = 1,
                     beta = c(0.4, 0.6),
                     "manu", "lab"
)

dst.consumer <- node_new("util",
                         type = "SCES", es = 0.5, alpha = 1,
                         beta = c(0.4, 0.6),
                         "manu", "serv"
)

dstl <- list(dst.manu, dst.serv, dst.consumer)

ge <- sdm2(
  A = dstl,
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = {
    S0Exg <- matrix(NA, 3, 3)
    S0Exg[3, 3] <- 100
    S0Exg
  },
  names.commodity = c("manu", "serv", "lab"),
  names.agent = c("manu", "serv", "consumer"),
  numeraire = c("manu"),
  ts = TRUE,
  policy = list(
    function(time, A) {
      if (time >= 5) {
        node_set(A[[1]], "cc1",
                 a = (1 + tpr)^-(time - 4)
        )
      }
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 40,
  maxIteration = 1,
  z0 = c(160, 60, 100),
  p0 = c(1, 1, 1)
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(growth_rate(ge$ts.z), type = "o", pch = 20)
matplot(growth_rate(ge$ts.p), type = "o", pch = 20)

#### a 3-by-3 example with labor-saving technological
#### progress and capital accumulation
dst.firm1 <- node_new(
  "prod",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "cap", "cc1"
)
node_set(dst.firm1, "cc1",
         type="Leontief", a=1,
         "lab")

dst.consumer <- dst.firm2 <- node_new(
  "util",
  type = "Leontief",
  a= 1,
  "prod"
)

ge <- sdm2(
  A = list(dst.firm1, dst.consumer, dst.firm2),
  B = matrix(c(
    1, 0, 0.5,
    0, 0, 1,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, 100,NA
  ), 3, 3, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm1", "laborer","firm2"),
  numeraire = "prod",
  z0=c(400,200,400),
  policy = list(
    function(time, A) {
      if (time >= 5) {
        node_set(A[[1]],"cc1", a = (1 + 0.03)^-(time - 4))
      }
    },
    policyMarketClearingPrice
  ),
  maxIteration = 1,
  numberOfPeriods = 30,
  ts=TRUE
)

matplot(growth_rate(ge$ts.z), type="l")

Some Simple Pure Exchange Equilibrium Models

Description

Some simple pure exchange general equilibrium models.

Usage

gemPureExchange(...)

Arguments

Examples

dst.consumer1 <- dst.consumer2 <- node_new("util",
  type = "CD", alpha = 1, beta = c(0.5, 0.25, 0.25),
  "payoff1", "payoff2", "payoff3"
)

ge <- sdm2(
  A = list(dst.consumer1, dst.consumer2),
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    1, 1,
    0, 2,
    2, 2
  ), 3, 2, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("consumer1", "consumer2"),
  numeraire = "payoff1"
)

ge$p
ge$D

##
dst.consumer2 <- node_new("util",
  type = "CD", alpha = 1, beta = c(0.5, 0.1, 0.4),
  "payoff1", "payoff2", "payoff3"
)

ge <- sdm2(
  A = list(dst.consumer1, dst.consumer2),
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    1, 1,
    0, 2,
    2, 2
  ), 3, 2, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("consumer1", "consumer2"),
  numeraire = "payoff1"
)

ge$p
ge$D

####
dst.consumer1 <- node_new("util",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "cc1", "cc2"
)
node_set(dst.consumer1, "cc2",
  type = "CD", alpha = 1,
  beta = c(0.2, 0.8),
  "cc2.1", "cc2.2"
)
node_set(dst.consumer1, "cc1",
  type = "Leontief",
  a = c(0.5, 0.5),
  "corn1", "iron1"
)

node_set(dst.consumer1, "cc2.1",
  type = "Leontief", a = c(0.5, 0.5),
  "corn2.1", "iron2.1"
)
node_set(dst.consumer1, "cc2.2",
  type = "Leontief", a = c(0.5, 0.5),
  "corn2.2", "iron2.2"
)

dst.consumer2 <- node_new("util",
  type = "CD", alpha = 1,
  beta = prop.table(c(
    0.5 * c(1, 1),
    0.5 * 0.2 * c(1, 1), 0.5 * 0.8 * c(1, 1)
  )),
  "corn1", "iron1", "corn2.1",
  "iron2.1", "corn2.2", "iron2.2"
)

node_plot(dst.consumer1, TRUE)

ge <- sdm2(
  A = list(dst.consumer1, dst.consumer2),
  B = matrix(0, 6, 2),
  S0Exg = matrix(c(
    2, 2,
    2, 2,
    3, 2,
    1, 2,
    1, 2,
    3, 2
  ), 6, 2, TRUE),
  names.commodity = c(
    "corn1", "iron1", "corn2.1",
    "iron2.1", "corn2.2", "iron2.2"
  ),
  names.agent = c("consumer1", "consumer2"),
  numeraire = "corn1"
)

ge$D
ge$DV

A Pure Exchange Economy with a Quasilinear Utility Function

Description

An example of a pure exchange economy with a quasilinear utility function (Karaivanov, see the reference).

Usage

gemQuasilinearPureExchange_2_2(
  A,
  Endowment = matrix(c(3, 4, 7, 0), 2, 2, TRUE),
  policy = NULL
)

Arguments

Details

Suppose there are only two goods (bananas and fish) and 2 consumers (Annie and Ben) in an exchange economy. Annie has a utility function x_1^(1/3) * x_2^(2/3) where x_1 is the amount of fish she eats and x_2 is the amount of bananas she eats. Annie has an endowment of 3 kilos of fish and 7 bananas. Ben has a utility function x_1 + 1.5 * log(x_2) and endowments of 4 kilos of fish and 0 bananas. Assume the price of bananas is 1. See the reference for more details.

Value

A general equilibrium.

References

http://www.sfu.ca/~akaraiva/CE_example.pdf

Examples

demand_consumer2 <- function(w, p) {
  QL_demand(w = w, p = p, alpha = 1.5, type = "log")
}

A <- function(state) {
  a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
  a2 <- demand_consumer2(state$w[2], state$p)
  cbind(a1, a2)
}

ge.mat <- gemQuasilinearPureExchange_2_2(A = A)
ge.mat

## Use a dstl and a policy function to compute the general equilibrium above.
dst.consumer1 <- node_new("util",
                          type = "CD", alpha = 1, beta = c(1 / 3, 2 / 3),
                          "fish", "banana"
)
dst.consumer2 <- node_new("util",
                          type = "Leontief", a = c(1, 1),
                          "fish", "banana"
)

dstl <- list(dst.consumer1, dst.consumer2)

policy.quasilinear <- function(A, state) {
  wealth <- t(state$p) %*% state$S
  A[[2]]$a <- demand_consumer2(wealth[2], state$p)
}

ge.dstl <- gemQuasilinearPureExchange_2_2(
  A = dstl,
  policy = policy.quasilinear
)
ge.dstl

#### Another example. Now Ben has a utility function x_1 + sqrt(x_2).
demand_consumer2 <- function(w, p) {
  QL_demand(w = w, p = p, alpha = 1, beta = 0.5, type = "power")
}

A <- function(state) {
  a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
  a2 <- demand_consumer2(state$w[2], state$p)
  cbind(a1, a2)
}

ge.2_2 <- gemQuasilinearPureExchange_2_2(A = A)
ge.2_2

## another computation method for the economy above
A <- function(state) {
  a1 <- CD_A(1, rbind(1 / 3, 2 / 3, 0, 0), state$p)
  a2 <- c(0, 0, 1, 0)
  a3 <- c(1, 0, 0, 0) # firm 1
  a4 <- CD_A(1, rbind(0, 1 / 2, 0, 1 / 2), state$p) # firm 2
  cbind(a1, a2, a3, a4)
}

ge.4_4 <- sdm2(
  A = A,
  B = {
    B <- matrix(0, 4, 4)
    B[3, 3] <- 1
    B[3, 4] <- 1
    B
  },
  S0Exg = {
    S0Exg <- matrix(NA, 4, 4)
    S0Exg[1:2, 1] <- c(3, 7)
    S0Exg[1:2, 2] <- c(4, 0)
    S0Exg[4, 1:2] <- c(0, 1)
    S0Exg
  },
  names.commodity = c("fish", "banana", "util2", "land"),
  names.agent = c("Annie", "Ben", "firm1", "firm2"),
  numeraire = "banana"
)
ge.4_4

#### another example
n.fish.demander <- 21
wealth <- 20 # the wealth (or income) of each fish demander
fish.supply <- 12
aggregare.demand <- function(p) {
  result <- 0
  for (alpha in seq(5, 15, length.out = n.fish.demander)) {
    result <- result + QL_demand(w = wealth, p = p, alpha = alpha, beta = 1, type = "min")
  }
  result
}

ge <- sdm2(
  A = function(state) {
    a1 <- aggregare.demand(state$p / state$p[1])
    a2 <- c(1, 0)
    cbind(a1, a2)
  },
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    n.fish.demander * wealth, 0,
    0, fish.supply
  ), 2, 2, TRUE),
  names.commodity = c("gold", "fish"),
  names.agent = c("fish.demander", "fish.supplier"),
  numeraire = "gold",
  p0 = c(1, 1) # p0 = c(1, 9.25)
)

ge$p
ge$z
ge$D
ge$S

aggregare.demand.fish <- c()
p2.set <- seq(0, 16, 0.01)
for (p2 in p2.set) {
  aggregare.demand.fish <- c(
    aggregare.demand.fish,
    aggregare.demand(c(1, p2))[2]
  )
}

plot(aggregare.demand.fish,
     p2.set,
     xlab = "demand for fish", ylab = "price of fish", pch = 20
)
abline(v = fish.supply)
grid()
points(ge$D[2, 1], ge$p[2], pch = 8, col = "red")

Some Examples of Spot Equilibrium Paths Illustrating Research and Development Intensity

Description

Some examples of spot equilibrium paths illustrating R&D intensity. R&D intensity of a firm is the ratio of expenditures by the firm on R&D to the firm's sales.

Usage

gemResearchDevelopmentIntensity(...)

Arguments

Details

The first example contains two kinds of commodities (namely product and labor) and three economic agents (namely a firm, an R&D center of the firm and a laborer). Since the R&D center does not produce products, the R&D center is regarded as a consumer-type agent in the model. The utility level of the R&D center (that is, the R&D level) will affect the technological progress rate of the firm. In the model, the firm allocates part of its output to the R&D centers for sale according to a given R&D intensity, which is equivalent to allocating part of the firm's sales revenue to the R&D center. Initially, the economy is in steady-state equilibrium without R&D activity. R&D activities begin in the fifth period.

Value

A spot equilibrium path.

Examples

#### a 2-by-3 example.
RDIntensity <- 0.3
RDEffectivenessCoefficient <- 0.001

dst.firm <- node_new(
  "prod",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "prod", "cc1"
)
node_set(dst.firm, "cc1",
  type = "Leontief", a = 1,
  "lab"
)

dst.RDCenter <- node_new(
  "util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.laborer <- node_new(
  "util",
  type = "Leontief",
  a = 1,
  "prod"
)

# a function calculating the rate of technological progress according to the level of R&D.
f.TPR <- function(RDLevel) RDEffectivenessCoefficient * RDLevel

f <- function() {
  sdm2(
    A = list(dst.firm, dst.RDCenter, dst.laborer),
    B = matrix(c(
      1, 0, 0,
      0, 0, 0
    ), 2, 3, TRUE),
    S0Exg = matrix(c(
      NA, NA, NA,
      NA, NA, 100
    ), 2, 3, TRUE),
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "RDCenter", "laborer"),
    numeraire = "prod",
    z0 = c(200, 0, 100),
    policy = list(
      function(time, A, state) {
        if (time >= 5) {
          state$S[1, 2] <- state$S[1, 1] * RDIntensity
          state$S[1, 1] <- state$S[1, 1] * (1 - RDIntensity)
          last.a <- node_set(A[[1]], "cc1")$a
          last.RDLevel <- state$last.z[2]
          technology.progress.rate <- f.TPR(last.RDLevel)
          node_set(A[[1]], "cc1", a = last.a / (1 + technology.progress.rate))
        }

        state
      },
      policyMarketClearingPrice
    ),
    maxIteration = 1,
    numberOfPeriods = 50,
    ts = TRUE
  )
}

ge <- f()
matplot((ge$ts.z), type = "o", pch = 20)
ge$z

## change the R&D intensity.
node_set(dst.firm, "cc1", a = 1)
RDIntensity <- 0.8
ge <- f()
matplot((ge$ts.z), type = "o", pch = 20)
ge$z

## random rate of technological progress.
set.seed(1)
RDIntensity <- 0.3
node_set(dst.firm, "cc1", a = 1)
f.TPR <- function(RDLevel) max(0, rnorm(1, RDEffectivenessCoefficient * RDLevel,
  sqrt(RDEffectivenessCoefficient * RDLevel)))
ge <- f()
matplot((ge$ts.z), type = "o", pch = 20)
ge$z

## two firms with different R&D intensity.
node_set(dst.firm, "cc1", a = 1)
RDIntensity1 <- 0.1
RDIntensity2 <- 0.05
RDEffectivenessCoefficient <- 0.002

dst.firm2 <- Clone(dst.firm)
dst.RDCenter2 <- Clone(dst.RDCenter)
ge <- sdm2(
  A = list(dst.firm, dst.RDCenter, dst.laborer, dst.firm2, dst.RDCenter2),
  B = matrix(c(
    1, 0, 0, 1, 0,
    0, 0, 0, 0, 0
  ), 2, 5, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA,
    NA, NA, 200, NA, NA
  ), 2, 5, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm1", "RDCenter1", "laborer", "firm2", "RDCenter2"),
  numeraire = "prod",
  z0 = c(200, 0, 200, 200, 0),
  policy = list(
    function(time, A, state) {
      if (time >= 5) {
        state$S[1, 2] <- state$S[1, 1] * RDIntensity1
        state$S[1, 1] <- state$S[1, 1] * (1 - RDIntensity1)
        last.a1 <- node_set(A[[1]], "cc1")$a
        last.RDLevel1 <- state$last.z[2]
        technology.progress.rate1 <- RDEffectivenessCoefficient * last.RDLevel1
        node_set(A[[1]], "cc1", a = last.a1 / (1 + technology.progress.rate1))

        state$S[1, 5] <- state$S[1, 4] * RDIntensity2
        state$S[1, 4] <- state$S[1, 4] * (1 - RDIntensity2)
        last.a2 <- node_set(A[[4]], "cc1")$a
        last.RDLevel2 <- state$last.z[5]
        technology.progress.rate2 <- RDEffectivenessCoefficient * last.RDLevel2
        node_set(A[[4]], "cc1", a = last.a2 / (1 + technology.progress.rate2))
      }

      state
    },
    policyMarketClearingPrice
  ),
  maxIteration = 1,
  numberOfPeriods = 50,
  ts = TRUE
)

matplot((ge$ts.z), type = "o", pch = 20)
ge$z

## Assume that the R&D center is owned by the government and
## receives revenue through taxation on firms.
## The technologies developed by the R&D center are public goods.
node_set(dst.firm, "cc1", a = 1)
RDIntensity <- 0.1
RDEffectivenessCoefficient <- 0.002

dst.firm2 <- Clone(dst.firm)

ge <- sdm2(
  A = list(dst.firm, dst.RDCenter, dst.laborer, dst.firm2),
  B = matrix(c(
    1, 0, 0, 1,
    0, 0, 0, 0
  ), 2, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, 200, NA
  ), 2, 4, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm1", "RDCenter", "laborer", "firm2"),
  numeraire = "prod",
  z0 = c(200, 0, 200, 200),
  policy = list(
    function(time, A, state) {
      if (time >= 5) {
        last.RDLevel <- state$last.z[2]
        technology.progress.rate <- RDEffectivenessCoefficient * last.RDLevel

        state$S[1, 2] <- (state$S[1, 1] + state$S[1, 4]) * RDIntensity
        state$S[1, 1] <- state$S[1, 1] * (1 - RDIntensity)
        state$S[1, 4] <- state$S[1, 4] * (1 - RDIntensity)
        last.a1 <- node_set(A[[1]], "cc1")$a
        node_set(A[[1]], "cc1", a = last.a1 / (1 + technology.progress.rate))

        last.a2 <- node_set(A[[4]], "cc1")$a
        node_set(A[[4]], "cc1", a = last.a2 / (1 + technology.progress.rate))
      }

      state
    },
    policyMarketClearingPrice
  ),
  maxIteration = 1,
  numberOfPeriods = 30,
  ts = TRUE
)

matplot((ge$ts.z), type = "o", pch = 20)
ge$z

A Robinson Crusoe Economy

Description

Compute the general equilibrium of a Robinson Crusoe economy.

Usage

gemRobinson_3_2(dstl, endowment)

Arguments

Details

A general equilibrium model with 3 commodities (i.e. product, labor, and land) and 2 agents (i.e. a firm and a consumer). The numeraire is labor.

Value

A general equilibrium.

References

http://essentialmicroeconomics.com/ChapterY5/SlideChapter5-1.pdf

http://homepage.ntu.edu.tw/~josephw/MicroTheory_Lecture_11a_RobinsonCrusoeEconomy.pdf

Examples

#### a general equilibrium model with 2 basic commodities (i.e. labor and land)
#### and 1 agent (see the first reference)
dst.Robinson <- node_new("util",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "prod", "lab"
)
node_set(dst.Robinson, "prod",
  type = "CD", alpha = 8, beta = c(0.5, 0.5),
  "lab", "land"
)

node_plot(dst.Robinson)

ge <- sdm2(
  A = list(dst.Robinson),
  names.commodity = c("lab", "land"),
  names.agent = c("Robinson"),
  B = matrix(0, 2, 1),
  S0Exg = matrix(c(
    12,
    1
  ), 2, 1, TRUE),
  numeraire = "lab"
)
ge


## the same economy as above
dst.Robinson <- node_new("util",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "prod", "lab"
)
dst.firm <- node_new("output",
  type = "CD", alpha = 8, beta = c(0.5, 0.5),
  "lab", "land"
)

dstl <- list(dst.firm, dst.Robinson)

ge <- gemRobinson_3_2(dstl, endowment = c(0, 12, 1))
ge

## another example (see the second reference)
dst.firm$alpha <- 1

ge <- gemRobinson_3_2(dstl, endowment = c(3, 144, 1))
ge

#### a Robinson Crusoe economy with labor and two types of land
dst.Robinson <- node_new("util",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "prod1", "prod2"
)
node_set(dst.Robinson, "prod1",
  type = "CD", alpha = 1, beta = c(0.2, 0.8),
  "lab", "land1"
)
node_set(dst.Robinson, "prod2",
  type = "CD", alpha = 1, beta = c(0.8, 0.2),
  "lab", "land2"
)
node_plot(dst.Robinson)

dstl <- list(dst.Robinson)

ge.3_1 <- sdm2(dstl,
  names.commodity = c("lab", "land1", "land2"),
  names.agent = c("Robinson"),
  B = matrix(0, 3, 1),
  S0Exg = matrix(c(
    100,
    100,
    100
  ), 3, 1, TRUE),
  numeraire = "lab"
)
ge.3_1

#### the same economy as above
ge.5_3 <- sdm2(
  A = list(
    dst.firm1 = node_new("output",
                         type = "CD", alpha = 1, beta = c(0.2, 0.8),
                         "lab", "land1"
    ),
    dst.firm2 = node_new("output",
                         type = "CD", alpha = 1, beta = c(0.8, 0.2),
                         "lab", "land2"
    ),
    dst.Robinson = node_new("util",
                            type = "CD", alpha = 1, beta = c(0.5, 0.5),
                            "prod1", "prod2"
    )
  ),
  names.commodity = c("prod1", "prod2", "lab", "land1", "land2"),
  names.agent = c("firm1", "firm2", "Robinson"),
  B = {
    B <- matrix(0, 5, 3)
    B[1, 1] <- B[2, 2] <- 1
    B
  },
  S0Exg = {
    S0Exg <- matrix(NA, 5, 3)
    S0Exg[3:5, 3] <- 100
    S0Exg
  },
  numeraire = "lab"
)
ge.5_3

Some Examples Illustrating Short-Term Investment

Description

Some examples illustrating short-term investment.

Usage

gemShortTermInvestment_2_3(...)

Arguments

Examples

#### an example with an exogenous investment rate
investment.rate <- 0.2
dst.firm <- node_new("output",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.laborer <- node_new("laborer.util",
  type = "CD",
  alpha = 1, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.investor <- node_new("investor.util",
  type = "Leontief",
  a = 1,
  "prod"
)

dstl <- list(dst.firm, dst.laborer, dst.investor)

ge <- sdm2(
  A = dstl,
  B = diag(c(1, 0), 2, 3),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100 * (1 - investment.rate), 100 * investment.rate
  ), 2, 3, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer", "investor"),
  numeraire = "prod"
)

addmargins(ge$D, 2)

## an example with an exogenous investment level
dst.investor$current.investment.rate <- 0.5
investment.level <- 20

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.investor),
  B = diag(c(1, 0), 2, 3),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, 0
  ), 2, 3, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer", "investor"),
  numeraire = "prod",
  policy = function(time, A, state) {
    A[[3]]$current.investment.rate <- A[[3]]$current.investment.rate *
      (investment.level / state$last.z[3])
    state$S[2, 2] <- 100 * (1 - A[[3]]$current.investment.rate)
    state$S[2, 3] <- 100 * A[[3]]$current.investment.rate
    state
  }
)

addmargins(ge$D, 2)

Some General Equilibrium Models with Skill (i.e. Human Capital)

Description

Some general equilibrium models with skill (i.e. human capital).

Usage

gemSkill(...)

Arguments

Examples

depreciation.rate <- 0.05 # the depreciation rate of skill
skill.density <- 10
relative.efficiency.coef <- 2
efficiency.coef <- skill.density * relative.efficiency.coef

dst.efficiency.unit <- node_new("efficiency unit",
  type = "Leontief",
  a =  1 / efficiency.coef,
  "complex labor"
)

dst.firm <- node_new(
  "product",
  type = "SCES", alpha = 1,
  beta = c(0.4, 0.6), es = 0.5,
  "product", "labor"
)
node_set(dst.firm, "labor",
  type = "SCES", alpha = 1,
  beta = c(0.5, 0.5), es = 1.5,
  "simple labor", dst.efficiency.unit
)

dst.school <- node_new(
  "skill",
  type = "Leontief",
  a = c(0.1, 1, 0.1),
  "product", "simple labor", dst.efficiency.unit
)

dst.complex.laborer <- node_new(
  "complex labor",
  type = "Leontief", a = c(skill.density, 1),
  "skill", "simple labor"
)

dst.simple.laborer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "product"
)

ge <- sdm2(
  A = list(dst.firm, dst.school, dst.complex.laborer, dst.simple.laborer),
  B = matrix(c(
    1, 0, 0, 0,
    0, 1, skill.density * (1 - depreciation.rate), 0,
    0, 0, 1, 0,
    0, 0, 0, 0
  ), 4, 4, TRUE),
  S0Exg = {
    tmp <- matrix(NA, 4, 4)
    tmp[4, 4] <- 100
    tmp
  },
  names.commodity = c("product", "skill", "complex labor", "simple labor"),
  names.agent = c("firm", "school", "complex laborer", "simple laborer"),
  numeraire = "simple labor",
  policy = makePolicyMeanValue(50),
  priceAdjustmentVelocity = 0.05,
  maxIteration = 1,
  numberOfPeriods = 1000,
  ts = TRUE
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
matplot(ge$ts.p, type = "l")

#### Assumed that the amount of education and training purchased by laborers is
## determined primarily by their preferences rather than their investment motives.
depreciation.rate <- 0.05
skill.density <- 10
relative.efficiency.coef <- 2
efficiency.coef <- skill.density * relative.efficiency.coef

dst.efficiency.unit <- node_new("efficiency unit",
  type = "Leontief",
  a = c(skill.density / efficiency.coef, 1 / efficiency.coef),
  "skill service", "simple labor"
)

dst.firm <- node_new(
  "product",
  type = "SCES", alpha = 1,
  beta = c(0.4, 0.6), es = 0.5,
  "product", "labor"
)
node_set(dst.firm, "labor",
  type = "SCES", alpha = 1,
  beta = c(0.5, 0.5), es = 1.5,
  "simple labor", dst.efficiency.unit
)

dst.school <- node_new(
  "skill",
  type = "Leontief",
  a = c(0.1, 1, 0.1),
  "product", "simple labor", dst.efficiency.unit
)

dst.laborer <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = c(0.7887, 0.2113),
  # beta <- c(0.9, 0.1),
  # beta <- c(0.6, 0.4),
  "product", "skill",
  skill.stock = 0
)

ge <- sdm2(
  A = list(dst.firm, dst.school, dst.laborer),
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = {
    tmp <- matrix(NA, 4, 3)
    tmp[4, 3] <- 100
    tmp
  },
  names.commodity = c("product", "skill", "skill service", "simple labor"),
  names.agent = c("firm", "school", "laborer"),
  numeraire = "simple labor",
  policy = function(A, state) {
    last.D <- state$last.A %*% dg(state$last.z)
    new.skill <- last.D[2, 3]
    state$S[3, 3] <- A[[3]]$skill.stock <-
      A[[3]]$skill.stock * (1 - depreciation.rate) + new.skill
    state
  },
  priceAdjustmentVelocity = 0.05,
  maxIteration = 1,
  numberOfPeriods = 1000,
  ts = TRUE
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
matplot(log(ge$ts.p), type = "l")

Some Examples of Spot Equilibrium Paths with Sticky Decisions

Description

Some examples of spot equilibrium paths with sticky decisions of a firm, that is, the firm sluggishly adjusts its technology in response to price changes.

Under the assumption of (complete) rationality, economic agents will make decisions that are most beneficial to them based on the information they have. If the information does not change, then the decision will not change. However, under the assumption of bounded rationality, the decisions made by economic agents may not be optimal. They may follow some simple rules-of-thumb, and might adjust their previous decisions sluggishly according to the changes in information, even though they have the capability to adjust flexibly, so that the new decisions are better than the old ones under the new information. Hence the current decision is not necessarily the optimal decision. Even if the information does not change, it is still possible for agents to make further improvements to this decision in the next period. It can also be said that in this case, the decision maker's decision is sticky, that is, it only makes limited improvements to the previous decision based on new information, rather than directly adjusting to the optimal decision.

Usage

gemSpotEquilibriumPath_StickyDecisions(...)

Arguments

See Also

policyMarketClearingPrice

Examples

f <- function(stickiness.firm = 0) {
  dst.firm <- node_new("output",
    type = "Leontief", a = c(1 - stickiness.firm, stickiness.firm),
    "cc1", "cc2"
  )
  node_set(dst.firm, "cc1",
    type = "CD", alpha = 5,
    beta = c(0.5, 0.5),
    "prod", "lab"
  )
  node_set(dst.firm, "cc2",
    type = "CD", alpha = 5,
    beta = c(0.5, 0.5),
    "prod", "lab"
  )

  dst.consumer <- node_new("utility",
    type = "CD", alpha = 1,
    beta = c(0.5, 0.5),
    "prod", "lab"
  )

  ge <- sdm2(
    A = list(dst.firm, dst.consumer),
    B = diag(c(1, 0)),
    S0Exg = {
      S0Exg <- matrix(NA, 2, 2)
      S0Exg[2, 2] <- 100
      S0Exg
    },
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "consumer"),
    numeraire = "lab",
    maxIteration = 1,
    numberOfPeriods = 20,
    policy = list(
      function(time, A, state) {
        if (time > 1) {
          node_set(A[[1]], "cc2",
            type = "Leontief", a = state$last.A[, 1]
          )
        }
      },
      policyMarketClearingPrice
    ),
    ts = TRUE
  )

  print(ge$p)
  print(ge$z)
  par(mfrow = c(1, 2))
  matplot(ge$ts.p, type = "l")
  matplot(ge$ts.z, type = "l")
}

f()
f(stickiness.firm = 0.8)

An Example Illustrating the Sticky-Decision Path and Business Cycles

Description

This is an 2-by-2 example that illustrates the sticky-decision path and business cycles. Assume that the consumer has a linear utility function x1 + 0.8 * x2.

Usage

gemStickyDecisionPath_2_2(...)

Arguments

Examples

stickiness <- 0.1 # 0.5
ge <- sdm2(
  A = function(state) {
    a.firm <- CD_A(alpha = 2, Beta = c(0.5, 0.5), state$p)
    if (1.25 * state$p[2] < state$p[1]) {
      a.consumer <- c(0, 1)
    } else {
      a.consumer <- c(1, 0)
    }

    a.consumer <- state$last.A[, 2] * stickiness + a.consumer * (1 - stickiness)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod",
  z0 = c(80, 100),
  maxIteration = 1,
  numberOfPeriods = 100,
  ts = TRUE
)

matplot(ge$ts.z, type = "o", pch = 20)

Some Examples Illustrating the Sticky-Price Path and Business Cycles

Description

These are some examples that illustrate the sticky-price path and business cycles.

Usage

gemStickyPricePath_2_2(...)

Arguments

Examples

stickiness <- 0.7

dst.firm <- node_new(
  "output",
  type = "CD", alpha = 2, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge <- sdm2(
  A = list(dst.firm, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 1
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  z0 = c(100, 100),
  p0 = c(1, 1),
  ts = TRUE,
  policy = list(
    makePolicyTechnologyChange(
      adjumentment.ratio = 2,
      agent = "firm",
      time.win = c(30, 30)
    ),
    makePolicyStickyPrice(stickiness = stickiness)
  ),
  priceAdjustmentVelocity = 0,
  numberOfPeriods = 60,
  maxIteration = 1
)

matplot(ge$ts.z, type = "b", pch = 20)

#### another example.
## When the stickiness is 0, there will be business cycles.
stickiness <- 0.5

dst.firm <- node_new(
  "output",
  type = "Leontief", a = 0.2,
  "lab"
)

dst.consumer <- node_new(
  "util",
  type = "CES", es = 0.3, alpha = 1, beta = c(0.5, 0.5),
  "prod", "lab"
)

ge <- sdm2(
  A = list(dst.firm, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  z0 = c(100, 100),
  p0 = c(1, 1),
  ts = TRUE,
  policy = makePolicyStickyPrice(stickiness	= stickiness),
  priceAdjustmentVelocity = 0,
  numberOfPeriods = 40,
  maxIteration = 1
)

matplot(ge$ts.z, type = "b", pch = 20)

Some General Equilibrium Models with Tax

Description

Some general equilibrium models with tax.

Usage

gemTax_3_3(...)

Arguments

References

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

Examples

#### turnover tax.
dst.firm <- node_new("prod",
                     type = "FIN",
                     rate = c(1, tax.rate = 0.25),
                     "cc1", "tax"
)
node_set(dst.firm, "cc1",
         type = "CD",
         alpha = 2, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.laborer <- dst.government <-
  node_new("util",
    type = "Leontief",
    a = 1,
    "prod"
  )

ge.TT <- sdm2(
  A = list(dst.firm, dst.laborer, dst.government),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "tax"),
  names.agent = c("firm", "laborer", "government"),
  numeraire = "prod"
)

ge.TT$p
ge.TT$z
ge.TT$D
ge.TT$S

#### product tax.
dst.taxed.prod <- node_new("taxed.prod",
                           type = "FIN",
                           rate = c(1, tax.rate = 0.25),
                           "prod", "tax"
)

dst.firm <- node_new("prod",
                     type = "CD",
                     alpha = 2, beta = c(0.5, 0.5),
                     dst.taxed.prod, "lab"
)

dst.laborer <- dst.government <-
  node_new("util",
    type = "Leontief",
    a = 1,
    dst.taxed.prod
  )

ge.PT <- sdm2(
  A = list(dst.firm, dst.laborer, dst.government),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "tax"),
  names.agent = c("firm", "laborer", "government"),
  numeraire = "prod"
)

ge.PT$p
ge.PT$z
ge.PT$D
ge.PT$S

#### consumption tax.
dst.firm <- node_new("output",
                     type = "CD", alpha = 2,
                     beta = c(0.5, 0.5),
                     "prod", "lab"
)

dst.laborer <- node_new("util",
                        type = "FIN",
                        rate = c(1, consumption.tax.rate = 1/3),
                        "prod", "tax"
)

dst.government <- node_new("utility",
                           type = "Leontief",
                           a = 1,
                           "prod"
)

ge.CT <- sdm2(
  A = list(dst.firm, dst.laborer, dst.government),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "tax"),
  names.agent = c("firm", "laborer", "government"),
  numeraire = "prod"
)

ge.CT$p
ge.CT$z
ge.CT$D
ge.CT$S

#### value added tax.
dst.firm <- node_new("output",
  type = "CD", alpha = 2,
  beta = c(0.5, 0.5),
  "prod", "taxed.lab"
)
node_set(dst.firm, "taxed.lab",
  type = "FIN",
  rate = c(1, vat.rate = 1/3),
  "lab", "tax"
)

dst.laborer <- dst.government <-
  node_new("util",
    type = "Leontief",
    a = 1,
    "prod"
  )

ge.VAT <- sdm2(
  A = list(dst.firm, dst.laborer, dst.government),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "tax"),
  names.agent = c("firm", "laborer", "government"),
  numeraire = "prod"
)

ge.VAT$p
ge.VAT$z
ge.VAT$D
ge.VAT$S

#### income tax.
income.tax.rate <- 1 / 4

dst.firm <- node_new("output",
                     type = "CD",
                     alpha = 2, beta = c(0.5, 0.5),
                     "prod", "lab"
)

dst.laborer <- dst.government <-
  node_new("util",
    type = "Leontief",
    a = 1,
    "prod"
  )

ge.IT <- sdm2(
  A = list(dst.firm, dst.laborer, dst.government),
  B <- diag(c(1, 0), 2, 3),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100 * (1 - income.tax.rate), 100 * income.tax.rate
  ), 2, 3, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer", "government"),
  numeraire = "prod"
)

ge.IT$p
ge.IT$z
ge.IT$D
ge.IT$S

#### turnover tax (Li, 2019, example 4.11).
dst.firm <- node_new("output",
                     type = "FIN",
                     rate = c(1, turnover.tax.rate = 1),
                     "cc1", "tax"
)
node_set(dst.firm, "cc1",
         type = "CD",
         alpha = 1, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.laborer1 <- node_new("util",
                          type = "CD",
                          alpha = 1, beta = c(0.5, 0.5),
                          "prod", "lab"
)

dst.laborer2 <- node_new("utility",
                          type = "Leontief",
                          a = 1,
                          "prod"
)

ge.TT2 <- sdm2(
  A = list(dst.firm, dst.laborer1, dst.laborer2),
  B = diag(c(1, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "tax"),
  names.agent = c("firm", "laborer1", "laborer2"),
  numeraire = "lab"
)

ge.TT2$p
ge.TT2$z

#### commodity tax in a pure exchange economy.
tax.rate <- 0.25
es.consumer1 <- 0.5
es.consumer2 <- 2

dst.consumer1 <- node_new("util",
                          type = "SCES", es = es.consumer1,
                          alpha = 1, beta = c(0.5, 0.5),
                          "comm1", "comm2"
)

dst.consumer2 <- node_new("util",
                          type = "SCES", es = es.consumer2,
                          alpha = 1, beta = c(0.5, 0.5),
                          "taxed.comm1", "comm2"
)
node_set(dst.consumer2, "taxed.comm1",
         type = "FIN",
         rate = c(1, tax.rate = tax.rate),
         "comm1", "tax"
)

dst.gov <- node_new("util",
                    type = "SCES", es = 0,
                    alpha = 1, beta = c(0.5, 0.5),
                    "comm1", "comm2"
)

ge.CT <- sdm2(
  A = list(dst.consumer1, dst.consumer2, dst.gov),
  B = matrix(0, 3, 3),
  S0Exg = matrix(c(
    100, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("comm1", "comm2", "tax"),
  names.agent = c("consumer1", "consumer2", "gov"),
  numeraire = "comm1"
)

ge.CT$p
ge.CT$z
ge.CT$D

Some General Equilibrium Models with Endogenous Tax Rates

Description

Some general equilibrium models with endogenous tax rates.

Usage

gemTax_4_4(...)

Arguments

Examples

ge <- sdm2(
  A = function(state) {
    a.firm <- CD_A(alpha = 2, Beta = c(0.5, 0.5, 0, 0), state$p)
    a.laborer <- CD_A(alpha = 1, Beta = c(0.75, 0.25, 0, 0), state$p)
    a.government <- c(1, 0, 0, 0)
    a.planner <- c(0, 0, 2, 1)
    cbind(a.firm, a.laborer, a.government, a.planner)
  },
  B = matrix(c(
    1, 0, 0, 0,
    0, 0, 0, 0,
    0, 1, 0, 0,
    0, 0, 1, 0
  ), 4, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, 100,
    NA, NA, NA, NA,
    NA, NA, NA, NA
  ), 4, 4, TRUE),
  names.commodity = c("prod", "lab", "util1", "util2"),
  names.agent = c("firm", "laborer", "government", "planner"),
  numeraire = "prod"
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

#### an equivalent 2-by-2 model.
dst.firm <- node_new("output",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.planner <- node_new("util",
  type = "Leontief",
  a = c(2, 1),
  "util1", "util2"
)

node_set(dst.planner, "util1",
  type = "CD",
  alpha = 1, beta = c(0.75, 0.25),
  "prod", "lab"
)
node_set(dst.planner, "util2",
  type = "Leontief",
  a = 1,
  "prod"
)

ge <- sdm2(
  A = list(dst.firm, dst.planner),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "planner"),
  numeraire = "prod"
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)
addmargins(ge$DV)
addmargins(ge$SV)

A General Equilibrium Model with Tax (see Cardenete et al., 2012).

Description

A general equilibrium model with tax (see chapter 4, Cardenete et al., 2012), wherein there are 5 commodities (i.e. product 1, product 2, labor, capital goods, and tax receipt) and 4 agents (i.e. 2 firms and 2 consumers).

Usage

gemTax_5_4(
  dstl,
  names.commodity = c("prod1", "prod2", "lab", "cap", "tax"),
  names.agent = c("taxed.firm1", "taxed.firm2", "consumer1", "consumer2"),
  delta = 1,
  supply.lab.consumer1 = 30,
  supply.cap.consumer1 = 20,
  supply.lab.consumer2 = 20,
  supply.cap.consumer2 = 5,
  policy.tax = NULL
)

Arguments

Value

A general equilibrium (see sdm2), wherein labor is the numeraire.

References

Manuel Alejandro Cardenete, Ana-Isabel Guerra, Ferran Sancho (2012, ISBN: 9783642247453) Applied General Equilibrium: An Introduction. Springer-Verlag Berlin Heidelberg.

Examples

dst.consumer1 <- node_new("utility",
  type = "CD",
  alpha = 1,
  beta = c(0.3, 0.7),
  "prod1", "prod2"
)

dst.consumer2 <- Clone(dst.consumer1)
dst.consumer2$beta <- c(0.6, 0.4)

dst.firm1 <- node_new("output",
  type = "Leontief",
  a = c(0.5, 0.2, 0.3),
  "VA", "prod1", "prod2"
)
node_set(dst.firm1, "VA",
  type = "CD",
  alpha = 0.8^-0.8 * 0.2^-0.2,
  beta = c(0.8, 0.2),
  "lab", "cap"
)

dst.firm2 <- Clone(dst.firm1)
node_set(dst.firm2, "output",
  a = c(0.25, 0.5, 0.25)
)
node_set(dst.firm2, "VA",
  alpha = 0.4^-0.4 * 0.6^-0.6,
  beta = c(0.4, 0.6)
)

## no taxation
dstl <- list(dst.firm1, dst.firm2, dst.consumer1, dst.consumer2)
ge <- gemTax_5_4(dstl, delta = 1)

## ad valorem output tax (see Table 4.1)
output.tax.rate <- 0.1
dst.taxed.firm1 <- node_new("taxed.output",
  type = "FIN", rate = c(1, output.tax.rate),
  dst.firm1, "tax"
)
node_plot(dst.taxed.firm1)

dst.taxed.firm2 <- node_new("taxed.output",
  type = "FIN", rate = c(1, output.tax.rate),
  dst.firm2, "tax"
)
node_plot(dst.taxed.firm2)

dstl <- list(dst.taxed.firm1, dst.taxed.firm2, dst.consumer1, dst.consumer2)

ge.output.tax1 <- gemTax_5_4(dstl, delta = 1)
ge.output.tax2 <- gemTax_5_4(dstl, delta = 0.5)
ge.output.tax3 <- gemTax_5_4(dstl, delta = 0)

## labor tax (see Table 4.3)
lab.tax.rate <- 0.1

dst.taxed.lab <- node_new("taxed.lab",
  type = "FIN",
  rate = c(1, lab.tax.rate),
  "lab",
  "tax"
)

dst.labor.taxed.firm1 <- Clone(dst.firm1)
node_prune(dst.labor.taxed.firm1, "lab", "cap")
node_set(
  dst.labor.taxed.firm1, "VA",
  dst.taxed.lab,
  "cap"
)

dst.labor.taxed.firm2 <- Clone(dst.labor.taxed.firm1)
node_set(dst.labor.taxed.firm2, "output",
  a = c(0.25, 0.5, 0.25)
)
node_set(dst.labor.taxed.firm2, "VA",
  alpha = 0.4^-0.4 * 0.6^-0.6,
  beta = c(0.4, 0.6)
)

dstl.labor.tax <- list(dst.labor.taxed.firm1, dst.labor.taxed.firm2, dst.consumer1, dst.consumer2)

ge.lab.tax <- gemTax_5_4(dstl.labor.tax, delta = 0.5)

ge.lab.tax$p
ge.lab.tax$z / ge$z - 1

## income tax (see Table 4.3)
income.tax.rate <- 0.2
consumption.tax.rate <- income.tax.rate / (1 - income.tax.rate)
dst.taxed.consumer1 <- node_new("taxed.utility",
  type = "FIN",
  rate = c(1, consumption.tax.rate),
  dst.consumer1,
  "tax"
)

dst.taxed.consumer2 <- node_new("taxed.utility",
  type = "FIN",
  rate = c(1, consumption.tax.rate),
  dst.consumer2,
  "tax"
)

dstl <- list(dst.firm1, dst.firm2, dst.taxed.consumer1, dst.taxed.consumer2)

ge.income.tax <- gemTax_5_4(dstl, delta = 0.5)
ge.income.tax$z / ge$z - 1

## labor tax (see Table 4.3)
lab.tax.rate <- 0.3742
node_set(dst.labor.taxed.firm1, "taxed.lab",
  rate = c(1, lab.tax.rate)
)
node_set(dst.labor.taxed.firm2, "taxed.lab",
  rate = c(1, lab.tax.rate)
)

ge.lab.tax <- gemTax_5_4(list(
  dst.labor.taxed.firm1,
  dst.labor.taxed.firm2,
  dst.consumer1,
  dst.consumer2
), delta = 0.5)
ge.lab.tax$z / ge$z - 1

## variable labor tax rate
policy.var.tax.rate <- function(time, A, state) {
  current.tax.rate <- NA
  if (time >= 200) {
    tax.amount <- (state$p / state$p[3])[5]
    adjustment.ratio <- ratio_adjust(tax.amount / 18.7132504, coef = 0.1)
    last.tax.rate <- node_set(A[[1]], "taxed.lab")$rate[2]
    current.tax.rate <- last.tax.rate / adjustment.ratio
  } else {
    current.tax.rate <- 0.1
  }
  node_set(A[[1]], "taxed.lab", rate = c(1, current.tax.rate))
  node_set(A[[2]], "taxed.lab", rate = c(1, current.tax.rate))

  state$current.policy.data <- c(time, current.tax.rate)
  state
}

ge.var.lab.tax <- gemTax_5_4(dstl.labor.tax, policy = policy.var.tax.rate)
matplot(ge.var.lab.tax$ts.z, type = "l")
matplot(ge.var.lab.tax$ts.p / ge.var.lab.tax$p[3], type = "l")
plot(ge.var.lab.tax$policy.data[, 1], ge.var.lab.tax$policy.data[, 2],
  ylab = "labor tax rate"
)
ge.var.lab.tax$p / ge.var.lab.tax$p[3]

A General Equilibrium Model with Tax

Description

A general equilibrium model with tax. The model contains 5 types of commodities (i.e. prod1, prod2, labor, capital and tax payment receipts) and 5 agents (i.e. firm1, firm2, laborer, capital owner and government).

Usage

gemTax_5_5(...)

Arguments

Examples

tax.rate.cap1 <- 0.25
tax.rate.lab1 <- 0.25

tax.rate.cap2 <- 0.25
tax.rate.lab2 <- 0.25

es.prod <- 0.5
es.cap.lab <- 0.5

beta.firm1 <- c(0.2, 0.8)
beta.firm2 <- c(0.8, 0.2)

beta.laborer <- c(0.5, 0.5)
beta.capitalOwner <- c(0.5, 0.5)
beta.government <- c(0.8, 0.2)

dst.firm1 <- node_new("prod",
  type = "SCES",
  alpha = 1, beta = beta.firm1, es = es.cap.lab,
  "cc1", "cc2"
)
node_set(dst.firm1, "cc1",
  type = "FIN",
  rate = c(1, tax.rate = tax.rate.lab1),
  "lab", "tax"
)
node_set(dst.firm1, "cc2",
  type = "FIN",
  rate = c(1, tax.rate = tax.rate.cap1),
  "cap", "tax"
)

node_plot(dst.firm1, TRUE)

dst.firm2 <- node_new("prod",
  type = "SCES",
  alpha = 1, beta = beta.firm2, es = es.cap.lab,
  "cc1", "cc2"
)
node_set(dst.firm2, "cc1",
  type = "FIN",
  rate = c(1, tax.rate = tax.rate.lab2),
  "lab", "tax"
)
node_set(dst.firm2, "cc2",
  type = "FIN",
  rate = c(1, tax.rate = tax.rate.cap2),
  "cap", "tax"
)

dst.laborer <- node_new("util",
  type = "SCES",
  alpha = 1, beta = beta.laborer, es = es.prod,
  "prod1", "prod2"
)

dst.capitalOwner <- node_new("util",
  type = "SCES",
  alpha = 1, beta = beta.capitalOwner, es = es.prod,
  "prod1", "prod2"
)

dst.government <- node_new("util",
  type = "SCES",
  alpha = 1, beta = beta.government, es = es.prod,
  "prod1", "prod2"
)

ge <- sdm2(
  A = list(dst.firm1, dst.firm2, dst.laborer, dst.capitalOwner, dst.government),
  B = diag(c(1, 1, 0, 0, 0)),
  S0Exg = matrix(c(
    NA, NA, NA, NA, NA,
    NA, NA, NA, NA, NA,
    NA, NA, 100, NA, NA,
    NA, NA, NA, 100, NA,
    NA, NA, NA, NA, 100
  ), 5, 5, TRUE),
  names.commodity = c("prod1", "prod2", "lab", "cap", "tax"),
  names.agent = c("firm1", "firm2", "laborer", "capitalOwner", "government"),
  numeraire = "lab"
)

ge$p
ge$z
ge$D
ge$S
addmargins(ge$DV)
addmargins(ge$SV)

A General Equilibrium Model with Tax and Quasilinear Utility Functions.

Description

This model is essentially a pure exchange economy. The model contains 4 types of commodities (i.e. corn, iron, taxed iron and tax payment receipts) and 4 agents (i.e. consumer 1, consumer 2, a firm and the government). Consumer 1 has corn and the utility function is x1 + beta1 * (alpha1 * x3 - 0.5 * x3^2) wherein x1 is corn and x3 is taxed iron. Consumer 2 has iron and the utility function is x1 + beta2 * (alpha2 * x2 - 0.5 * x2^2) wherein x1 is corn and x2 is iron. Consumer 1 (i.e. the iron demander) wants to buy iron from consumer 2 (i.e. the iron supplier) and the government will tax the transaction. The firm (i.e. a tax agency) inputs iron and tax payment receipts (similar to tax stamp) to output taxed iron, and due to government taxation requirements consumer 1 have to buy taxed iron from the firm and consumer 2 have to sell iron through the firm. Government has tax payment receipts and the utility function is x1.

Usage

gemTax_QuasilinearPreference_4_4(...)

Arguments

Examples

tax.rate <- 1

beta1 <- 0.05
alpha1 <- 3 / beta1 + 60

iron.endowment <- 100
beta2 <- 0.05
alpha2 <- iron.endowment - 60 + (3 / beta2)

ge <- sdm2(
  A = function(state) {
    a1 <- QL_demand(
      w = state$w[1],
      p = c(state$p[1], state$p[3]),
      alpha = alpha1, beta = beta1,
      type = "quadratic2"
    )
    a1 <- c(a1[1], 0, a1[2], 0)

    a2 <- QL_demand(
      w = state$w[2],
      p = state$p[1:2],
      alpha = alpha2, beta = beta2,
      type = "quadratic2"
    )
    a2 <- c(a2, 0, 0)

    a.firm <- c(0, 1, 0, tax.rate * state$p[2] / state$p[4])

    a.gov <- c(1, 0, 0, 0)

    cbind(a1, a2, a.firm, a.gov)
  },
  B = matrix(c(
    0, 0, 0, 0,
    0, 0, 0, 0,
    0, 0, 1, 0,
    0, 0, 0, 0
  ), 4, 4, TRUE),
  S0Exg = matrix(c(
    1000, NA, NA, NA,
    NA, iron.endowment, NA, NA,
    NA, NA, NA, NA,
    NA, NA, NA, 1
  ), 4, 4, TRUE),
  names.commodity = c("corn", "iron", "taxed.iron", "tax"),
  names.agent = c("consumer1", "consumer2", "firm", "gov"),
  numeraire = "corn",
  priceAdjustmentVelocity = 0.05
)

ge$p
ge$D
ge$S
addmargins(ge$DV)
addmargins(ge$SV)

ge.x <- ge$D[3, 1]
ge.pl <- ge$p[2]
ge.ph <- ge$p[3]
plot(function(x) (alpha1 - x) * beta1, 0, alpha1,
  xlim = c(0, 100), ylim = c(0, 6), xlab = "iron", ylab = "price"
)
curve((alpha2 - iron.endowment + x) * beta2, 0,
  alpha1,
  add = TRUE
)
grid()
points(ge.x, ge.ph, col = "red", pch = 20) # pch=8
points(ge.x, ge.pl, col = "red", pch = 20)

polygon(c(0, ge.x, ge.x, 0), c(ge.ph, ge.ph, ge.pl, ge.pl))
segments(0, 3, x1 = 60, y1 = 3, col = "red")
text(c(0, ge.x, ge.x, 0) + 3, c(
  ge.ph + 0.3, ge.ph + 0.3,
  ge.pl - 0.3, ge.pl - 0.3
), c("A", "B", "C", "D"))
text(c(3, ge.x + 3, 60), 3.3, c("E", "F", "G"))

u.consumer1 <- function(x) x[1] + beta1 * (alpha1 * x[2] - 0.5 * x[2]^2)
u.consumer2 <- function(x) x[1] + beta2 * (alpha2 * x[2] - 0.5 * x[2]^2)

u.consumer1(ge$D[c(1, 3), 1]) + u.consumer2(ge$D[c(1:2), 2]) + ge$z[4]
# The value above is 1430 when the tax rate is 0.

A General Equilibrium Model with Value-added Tax and Income Tax

Description

A general equilibrium model with value-added tax and income tax.

Usage

gemTax_VAT_IncomeTax_5_4(...)

Arguments

Examples

vat.rate <- 0.2 # value-added tax rate
income.tax.rate <- 0.2

dst.manu <- node_new("output",
                     type = "FIN",
                     rate = c(5 / 6, vat.rate),
                     "cc1", "vat"
)
node_set(dst.manu, "cc1",
         type = "SCES",
         es = 0.5, alpha = 1,
         beta = c(0.25, 0.75),
         "lab", "cap"
)

dst.serv <- node_new("output",
                     type = "FIN",
                     rate = c(5 / 6, vat.rate),
                     "cc1", "vat"
)
node_set(dst.serv, "cc1",
         type = "SCES",
         es = 0.5, alpha = 1,
         beta = c(0.75, 0.25),
         "lab", "cap"
)

dst.household <- node_new("util",
                          type = "SCES",
                          es = 0.5, alpha = 1,
                          beta = c(0.2, 0.8),
                          "manu", "serv"
)

dst.government <- node_new("util",
                           type = "SCES",
                           es = 0.5, alpha = 1,
                           beta = c(0.5, 0.5),
                           "manu", "serv"
)

dstl <- list(dst.manu, dst.serv, dst.household, dst.government)

ge <- sdm2(
  A = dstl,
  B = diag(c(1, 1, 0, 0), 5, 4),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, 360 * (1 - income.tax.rate), 360 * income.tax.rate,
    NA, NA, 240 * (1 - income.tax.rate), 240 * income.tax.rate,
    NA, NA, NA, 120
  ), 5, 4, TRUE),
  names.commodity = c("manu", "serv", "lab", "cap", "vat"),
  names.agent = c("manu", "serv", "household", "government"),
  numeraire = "lab"
)

ge$p
ge$z
addmargins(ge$DV)
addmargins(ge$SV)

## VAT rate reduction
dst.manu$rate <- dst.serv$rate <- c(5 / 6, vat.rate = 0.1)

ge.new <- sdm2(
  A = dstl,
  B = diag(c(1, 1, 0, 0), 5, 4),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, NA, 360 * (1 - income.tax.rate), 360 * income.tax.rate,
    NA, NA, 240 * (1 - income.tax.rate), 240 * income.tax.rate,
    NA, NA, NA, 120
  ), 5, 4, TRUE),
  names.commodity = c("manu", "serv", "lab", "cap", "vat"),
  names.agent = c("manu", "serv", "household", "government"),
  numeraire = "lab"
)

ge.new$p
ge.new$z
addmargins(ge.new$DV)
addmargins(ge.new$SV)

Some General Equilibrium Models with Technology Progress and Population Growth

Description

Some examples illustrating technology Progress and population growth.

Usage

gemTechnologyProgress_PopulationGrowth(...)

Arguments

Examples

#### a financial sequential model
gr.e <- 0.03 # the population growth rate
tpr <- 0.02 # the rate of technological progress
gr <- (1 + gr.e) * (1 + tpr) - 1
eis <- 0.8 # the elasticity of intertemporal substitution
Gamma.beta <- 0.8 # the subjective discount factor
yield.rate <- (1 + gr)^(1 / eis - 1) / Gamma.beta - 1 # the dividend rate
y1 <- 143.18115 # the initial product supply

dst.firm <- node_new("output",
  type = "FIN",
  rate = c(1, dividend.rate = yield.rate),
  "cc1", "equity.share"
)
node_set(dst.firm, "cc1",
  type = "CD",
  alpha = 2, beta = c(0.5, 0.5),
  "prod", "cc1.1"
)
node_set(dst.firm, "cc1.1",
  type = "Leontief", a = 1,
  "lab"
)

dst.laborer <- node_new("util",
  type = "Leontief", a = 1,
  "prod"
)

dst.shareholder <- Clone(dst.laborer)

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.shareholder),
  B = diag(c(1, 0, 0)),
  S0Exg = {
    S0Exg <- matrix(NA, 3, 3)
    S0Exg[2, 2] <- S0Exg[3, 3] <- 100 / (1 + gr.e)
    S0Exg
  },
  names.commodity = c("prod", "lab", "equity.share"),
  names.agent = c("firm", "laborer", "shareholder"),
  numeraire = "prod",
  maxIteration = 1,
  numberOfPeriods = 20,
  policy = list(function(time, A) {
    node_set(A[[1]], "cc1.1", a = 1 / (1 + tpr)^(time - 1))
  }, policyMarketClearingPrice),
  z0 = c(y1, 0, 0),
  GRExg = gr.e,
  ts = TRUE
)

matplot(growth_rate(ge$ts.p), type = "l")
matplot(growth_rate(ge$ts.z), type = "l")
ge$ts.z

## a timeline model
np <- 5 # the number of economic periods.

n <- 2 * np - 1 # the number of commodity kinds
m <- np # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)))
names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100 * (1 + gr.e)^(0:(np - 2))
S0Exg["prod1", "consumer"] <- y1

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}

dstl.firm <- list()
for (k in 1:(np - 1)) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD",
    alpha = 2, beta = c(0.5, 0.5),
    paste0("prod", k), "cc1"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "Leontief", a = 1 / ((1 + tpr)^(k - 1)),
           paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES",
  alpha = 1, beta = prop.table(Gamma.beta^(1:np)), es = eis,
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  maxIteration = 1,
  numberOfPeriods = 40,
  ts = TRUE,
  policy = list(
    makePolicyTailAdjustment(ind = c(np - 1, np), gr = gr),
    policyMarketClearingPrice
  )
)

ge$z
ge$D
ge$S
ge$p[1:3] / ge$p[2:4] - 1 # the steady-state equilibrium return rate
sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr) # the steady-state equilibrium return rate

## a financial time-circle model
zeta <- (1 + gr)^np # the ratio of repayments to loans

n <- 2 * np + 1 # the number of commodity kinds
m <- np + 1 # the number of agent kinds

names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np), "claim")
names.agent <- c(paste0("firm", 1:np), "consumer")

# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:np), "consumer"] <- 100 * (1 + gr.e)^(0:(np - 1))
S0Exg["claim", "consumer"] <- 100

# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
  B[paste0("prod", k + 1), paste0("firm", k)] <- 1
}
B["prod1", paste0("firm", np)] <- 1 / zeta

dstl.firm <- list()
for (k in 1:np) {
  dstl.firm[[k]] <- node_new("output",
                             type = "FIN", rate = c(1, yield.rate),
                             "cc1", "claim"
  )
  node_set(dstl.firm[[k]], "cc1",
           type = "CD", alpha = 2,
           beta = c(0.5, 0.5),
           paste0("prod", k), "cc1.1"
  )
  node_set(dstl.firm[[k]], "cc1.1",
           type = "Leontief", a = 1 / ((1 + tpr)^(k - 1)),
           paste0("lab", k)
  )
}

dst.consumer <- node_new(
  "util",
  type = "CES", es = 1,
  alpha = 1, beta = prop.table(rep(1, np)),
  paste0("prod", 1:np)
)

ge <- sdm2(
  A = c(dstl.firm, dst.consumer),
  B = B,
  S0Exg = S0Exg,
  names.commodity = names.commodity,
  names.agent = names.agent,
  numeraire = "prod1",
  ts = TRUE
)

ge$z
growth_rate(ge$z)
ge$D
ge$S

Some Examples of Temporary Equilibrium Paths

Description

Some examples of temporary equilibrium paths. The temporary equilibrium path consists of a series of temporary equilibria. Each temporary equilibrium achieves market clearing and may involve futures markets in addition to spot markets. A spot equilibrium path is a temporary equilibrium path that only involves spot market transactions.

Usage

gemTemporaryEquilibriumPath(...)

Arguments

References

Grandmont, J.M. (1977). Temporary General Equilibrium Theory. Econometrica 45, 535-572.

See Also

policyMarketClearingPrice

Examples

#### A pure exchange economy.
## Consumers 1 and 2 each supply 50 units of payoff each period.
## Consumers have limited foresight into the future and always expect
## that the supply of consumer 1 will be 50 and the supply of
## consumer 2 will be 40 in the next period.
dst.consumer1 <- node_new("util",
  type = "SCES", es = 2,
  alpha = 1, beta = c(0.5, 0.5),
  "payoff1", "payoff2"
)

dst.consumer2 <- node_new("util",
  type = "SCES", es = 1,
  alpha = 1, beta = c(0.5, 0.5),
  "payoff1", "payoff2"
)

result <- list()
for (time in 1:20) {
  if (time == 1) {
    S0Exg <- matrix(c(
      50, 50,
      50, 40
    ), 2, 2, TRUE)
  } else {
    S0Exg <- rbind(ge$D[2, ] + c(0, 10), c(50, 40))
  }

  ge <- sdm2(
    A = list(dst.consumer1, dst.consumer2),
    B = matrix(0, 2, 2),
    S0Exg = S0Exg,
    names.commodity = c("payoff1", "payoff2"),
    names.agent = c("consumer1", "consumer2"),
    numeraire = "payoff1"
  )

  result[[time]] <- ge
}

sapply(result, function(x) x$p)
sapply(result, function(x) x$z)
# lapply(result, function(x) x$D)
# lapply(result, function(x) x$S)

#### An economy with production.
dst.consumer <- node_new("util",
  type = "CD",
  alpha = 1,
  beta = c(1 / 3, 2 / 3), # beta = c(1/2, 1/2)
  "prod1", "prod2"
)

dst.firm <- node_new("prod2",
  type = "CD", alpha = 2, beta = c(0.5, 0.5),
  "prod1", "lab"
)

result <- list()
for (time in 1:20) {
  if (time == 1) {
    S0Exg <- matrix(c(
      10, 0,
      0, NA,
      100, 0
    ), 3, 2, TRUE)
  } else {
    S0Exg <- matrix(c(
      ge$D[2, 1], 0,
      0, NA,
      100, 0
    ), 3, 2, TRUE)
  }

  ge <- sdm2(
    A = list(dst.consumer, dst.firm),
    B = matrix(c(
      0, 0,
      0, 1,
      0, 0
    ), 3, 2, TRUE),
    S0Exg = S0Exg,
    names.commodity = c("prod1", "prod2", "lab"),
    names.agent = c("consumer", "firm"),
    numeraire = "prod1"
  )

  result[[time]] <- ge
}


sapply(result, function(x) x$p)
sapply(result, function(x) x$z)
# lapply(result, function(x) x$D)
# lapply(result, function(x) x$S)

##
result <- list()
last.output <- 10
for (time in 1:30) {
  if (time == 1) {
    S0Exg <- matrix(c(
      10, 0,
      0, last.output,
      100, 0
    ), 3, 2, TRUE)
  } else {
    S0Exg <- rbind(
      c(ge$D[2, 1], max(ge$z[2] - last.output, 0)),
      c(0, last.output),
      c(100, 0)
    )

    last.output <- ge$z[2]
  }

  ge <- sdm2(
    A = list(dst.consumer, dst.firm),
    B = matrix(c(
      0, 0,
      0, 1,
      0, 0
    ), 3, 2, TRUE),
    # B = matrix(0, 3, 2),
    S0Exg = S0Exg,
    names.commodity = c("prod1", "prod2", "lab"),
    names.agent = c("consumer", "firm"),
    numeraire = "prod1"
  )

  result[[time]] <- ge
}
sapply(result, function(x) x$p)
sapply(result, function(x) x$z)
# lapply(result, function(x) x$D)
# lapply(result, function(x) x$S)

Example 7.6 (Foreign Exchange Rate) in Li (2019)

Description

This is Example 7.6 in Li (2019), which illustrates foreign exchange rates.

Usage

gemTwoCountryForeignExchangeRate_6_6(...)

Arguments

Examples

dst.firm1 <- node_new("output",
  type = "FIN", rate = c(1, interest.rate = 0.1),
  "cc1", "money1"
)
node_set(dst.firm1, "cc1",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "iron", "lab1"
)

dst.firm2 <- Clone(dst.firm1)
node_replace(dst.firm2, "money1", "money2")
node_replace(dst.firm2, "lab1", "lab2")
node_plot(dst.firm2)

dst.laborer1 <- node_new("util",
  type = "FIN", rate = c(1, interest.rate = 0.1),
  "cc1", "money1"
)
node_set(dst.laborer1, "cc1",
  type = "Leontief", a = 1,
  "wheat"
)

dst.moneyOwner1 <- Clone(dst.laborer1)

dst.laborer2 <- Clone(dst.laborer1)
node_replace(dst.laborer2, "money1", "money2")

dst.moneyOwner2 <- Clone(dst.laborer2)

ge <- sdm2(
  A = list(
    dst.firm1, dst.laborer1, dst.moneyOwner1,
    dst.firm2, dst.laborer2, dst.moneyOwner2
  ),
  B = diag(c(1, 0, 0, 1, 0, 0)),
  S0Exg = {
    tmp <- matrix(NA, 6, 6)
    tmp[2, 2] <- 100
    tmp[3, 3] <- 600
    tmp[5, 5] <- 100
    tmp[6, 6] <- 100
    tmp
  },
  names.commodity = c(
    "wheat", "lab1", "money1",
    "iron", "lab2", "money2"
  ),
  names.agent = c(
    "firm1", "laborer1", "moneyOwner1",
    "firm2", "laborer2", "moneyOwner2"
  ),
  numeraire = c("money1" = 0.1) # interest.rate
)

ge$p[6] / ge$p[3] # foreign exchange rate

Some Examples of Two-Country Pure Exchange Economy

Description

Some general equilibrium examples of two-country pure exchange economy.

Usage

gemTwoCountryPureExchange(...)

Arguments

Value

A general equilibrium.

Examples

es.DFProd <- 0.8 # substitution elasticity between domestic and foreign products
technology.level.CHN <- 0.9

dst.CHN <- node_new("util",
  type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = es.DFProd,
  "prod.CHN", "prod.USA"
)
node_set(dst.CHN, "prod.CHN",
  type = "Leontief", a = 1 / technology.level.CHN,
  "lab.CHN"
)
node_set(dst.CHN, "prod.USA",
  type = "Leontief", a = 1,
  "lab.USA"
)
node_plot(dst.CHN)

dst.USA <- Clone(dst.CHN)

dstl <- list(dst.CHN, dst.USA)

ge <- sdm2(dstl,
  names.commodity = c("lab.CHN", "lab.USA"),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    0, 100
  ), 2, 2, TRUE),
  numeraire = "lab.CHN"
)
ge$p[2]
# the same as above
technology.level.CHN^(1 / es.DFProd - 1)

## supply change
geSC <- sdm2(dstl,
  names.commodity = c("lab.CHN", "lab.USA"),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    200, 0,
    0, 100
  ), 2, 2, TRUE),
  numeraire = "lab.CHN"
)
geSC$p[2]

## preference change
dst.CHN$beta <- c(0.6, 0.4)
gePC <- sdm2(dstl,
  names.commodity = c("lab.CHN", "lab.USA"),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    0, 100
  ), 2, 2, TRUE),
  numeraire = "lab.CHN"
)

gePC$p[2]

#### Add currencies to the example above.
interest.rate <- 1e-4
es.DFProd <- 0.8
technology.level.CHN <- 0.9

prod_money.CHN <- node_new("prod_money.CHN",
  type = "FIN", rate = c(1, interest.rate),
  "prod.CHN", "money.CHN"
)
node_set(prod_money.CHN, "prod.CHN",
  type = "Leontief", a = 1 / technology.level.CHN,
  "lab.CHN"
)

prod_money.USA <- node_new("prod_money.USA",
  type = "FIN", rate = c(1, interest.rate),
  "prod.USA", "money.USA"
)
node_set(prod_money.USA, "prod.USA",
  type = "Leontief", a = 1,
  "lab.USA"
)

dst.CHN <- node_new("util",
  type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = es.DFProd,
  prod_money.CHN, prod_money.USA
)

dst.USA <- Clone(dst.CHN)

dstl <- list(dst.CHN, dst.USA)

ge <- sdm2(dstl,
  names.commodity = c(
    "lab.CHN", "money.CHN",
    "lab.USA", "money.USA"
  ),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 4, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    100, 0,
    0, 100,
    0, 100
  ), 4, 2, TRUE),
  numeraire = c("money.CHN" = interest.rate)
)

ge$p["money.USA"] / ge$p["money.CHN"] # the exchange rate

#### supply change
geSC <- sdm2(dstl,
  names.commodity = c(
    "lab.CHN", "money.CHN",
    "lab.USA", "money.USA"
  ),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 4, 2, TRUE),
  S0Exg = matrix(c(
    200, 0,
    100, 0,
    0, 100,
    0, 100
  ), 4, 2, TRUE),
  numeraire = c("money.CHN" = interest.rate)
)
geSC$p["money.USA"] / geSC$p["money.CHN"]

## preference change
dst.CHN$beta <- c(0.6, 0.4)
gePC <- sdm2(dstl,
  names.commodity = c(
    "lab.CHN", "money.CHN",
    "lab.USA", "money.USA"
  ),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 4, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    100, 0,
    0, 100,
    0, 100
  ), 4, 2, TRUE),
  numeraire = c("money.CHN" = interest.rate)
)

gePC$p["money.USA"] / gePC$p["money.CHN"]

#### the exchange rate under a high substitution elasticity
#### between domestic and foreign products.
interest.rate <- 1e-4
es.DFProd <- 3
technology.level.CHN <- 0.9

prod_money.CHN <- node_new("prod_money.CHN",
  type = "FIN", rate = c(1, interest.rate),
  "prod.CHN", "money.CHN"
)
node_set(prod_money.CHN, "prod.CHN",
  type = "Leontief", a = 1 / technology.level.CHN,
  "lab.CHN"
)

prod_money.USA <- node_new("prod_money.USA",
  type = "FIN", rate = c(1, interest.rate),
  "prod.USA", "money.USA"
)
node_set(prod_money.USA, "prod.USA",
  type = "Leontief", a = 1,
  "lab.USA"
)

dst.CHN <- node_new("util",
  type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = es.DFProd,
  prod_money.CHN, prod_money.USA
)

dst.USA <- Clone(dst.CHN)

dstl <- list(dst.CHN, dst.USA)

ge <- sdm2(dstl,
  names.commodity = c(
    "lab.CHN", "money.CHN",
    "lab.USA", "money.USA"
  ),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 4, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    100, 0,
    0, 100,
    0, 100
  ), 4, 2, TRUE),
  numeraire = c("money.CHN" = interest.rate)
)

ge$p["money.USA"] / ge$p["money.CHN"] # the exchange rate

## supply change and high substitution elasticity
geSC <- sdm2(dstl,
  names.commodity = c(
    "lab.CHN", "money.CHN",
    "lab.USA", "money.USA"
  ),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 4, 2, TRUE),
  S0Exg = matrix(c(
    200, 0,
    100, 0,
    0, 100,
    0, 100
  ), 4, 2, TRUE),
  numeraire = c("money.CHN" = interest.rate)
)
geSC$p["money.USA"] / geSC$p["money.CHN"]

Some Examples of Two-Country Pure Exchange Economy with Bond

Description

Some general equilibrium examples of two-country pure exchange economy with bond.

Usage

gemTwoCountryPureExchange_Bond(...)

Arguments

Value

A general equilibrium.

Examples

es.DFProd <- 0.8 # substitution elasticity between domestic and foreign products
technology.level.CHN <- 1
# the amount of outbound investment corresponding to each unit of consumption
outbound.investment.rate <- 0.1

dst.consumption <- node_new("consumption",
  type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = es.DFProd,
  "prod.CHN", "prod.USA"
)
node_set(dst.consumption, "prod.CHN",
  type = "Leontief", a = 1 / technology.level.CHN,
  "lab.CHN"
)
node_set(dst.consumption, "prod.USA",
  type = "Leontief", a = 1,
  "lab.USA"
)
dst.CHN <- node_new("CHN",
  type = "FIN", rate = c(1, outbound.investment.rate),
  dst.consumption, "bond.USA"
)
node_plot(dst.CHN)

dst.USA <- Clone(dst.consumption)

dstl <- list(dst.CHN, dst.USA)

ge <- sdm2(dstl,
  names.commodity = c("lab.CHN", "lab.USA", "bond.USA"),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 3, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    0, 100,
    0, 100
  ), 3, 2, TRUE),
  numeraire = "lab.CHN"
)
ge$p[2]

#### Add currencies to the example above.
es.DFProd <- 0.8
technology.level.CHN <- 1
outbound.investment.rate <- 0.1
interest.rate <- 1e-4

prod_money.CHN <- node_new("prod_money.CHN",
  type = "FIN", rate = c(1, interest.rate),
  "prod.CHN", "money.CHN"
)
node_set(prod_money.CHN, "prod.CHN",
  type = "Leontief", a = 1 / technology.level.CHN,
  "lab.CHN"
)

prod_money.USA <- node_new("prod_money.USA",
  type = "FIN", rate = c(1, interest.rate),
  "prod.USA", "money.USA"
)
node_set(prod_money.USA, "prod.USA",
  type = "Leontief", a = 1,
  "lab.USA"
)

dst.CHN <- node_new("CHN",
  type = "FIN",
  rate = c(
    1, outbound.investment.rate,
    outbound.investment.rate * interest.rate
  ),
  "consumption", "bond.USA", "money.USA"
)
node_set(dst.CHN, "consumption",
  type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = es.DFProd,
  prod_money.CHN, prod_money.USA
)

node_plot(dst.CHN)

dst.USA <- Clone(node_set(dst.CHN, "consumption"))
node_plot(dst.USA)

dstl <- list(dst.CHN, dst.USA)

ge <- sdm2(dstl,
  names.commodity = c(
    "lab.CHN", "money.CHN",
    "lab.USA", "money.USA",
    "bond.USA"
  ),
  names.agent = c("CHN", "USA"),
  B = matrix(0, 5, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    100, 0,
    0, 100,
    0, 100,
    0, 100
  ), 5, 2, TRUE),
  numeraire = c("money.CHN" = interest.rate)
)

ge$p["money.USA"] / ge$p["money.CHN"] # the exchange rate

An Example of Two-Country Economy with Bond

Description

A general equilibrium example of two-country economy with bond.

Usage

gemTwoCountry_Bond_7_4(...)

Arguments

Value

A general equilibrium.

See Also

gemTwoCountry_Tariff_9_5

Examples

es.DFProd <- 0.8 # the substitution elasticity between domestic and foreign products
es.CL <- 0.8 # the substitution elasticity between capital and labor

dst.firm.CHN <- node_new("output",
  type = "SCES", alpha = 1, beta = c(0.78, 0.22), es = es.CL,
  "lab.CHN", "cap.CHN"
)

dst.household.CHN <- node_new("util",
  type = "FIN", rate = c(1,  outbound.investment.rate = 0.028),
  "cc1", "bond.ROW"
)
node_set(dst.household.CHN, "cc1",
  type = "SCES", alpha = 1, beta = c(0.93, 0.07), es = es.DFProd,
  "prod.CHN", "prod.ROW"
)

node_plot(dst.household.CHN)

dst.firm.ROW <- node_new("output",
  type = "SCES", alpha = 1, beta = c(0.75, 0.25), es = es.CL,
  "lab.ROW", "cap.ROW"
)

dst.household.ROW <- node_new("util",
  type = "SCES", alpha = 1, beta = c(0.02, 0.98), es = es.DFProd,
  "prod.CHN", "prod.ROW"
)

dstl <- list(dst.firm.CHN, dst.household.CHN, dst.firm.ROW, dst.household.ROW)

ge <- sdm2(dstl,
  names.commodity = c(
    "prod.CHN", "lab.CHN", "cap.CHN",
    "prod.ROW", "lab.ROW", "cap.ROW", "bond.ROW"
  ),
  names.agent = c(
    "firm.CHN", "household.CHN",
    "firm.ROW", "household.ROW"
  ),
  B = {
    tmp <- matrix(0, 7, 4, TRUE)
    tmp[1, 1] <- tmp[4, 3] <- 1
    tmp
  },
  S0Exg = {
    tmp <- matrix(NA, 7, 4, TRUE)
    tmp[2, 2] <- 53 # the supply of lab.CHN
    tmp[3, 2] <- 15 # the supply of cap.CHN
    tmp[5, 4] <- 240 # the supply of lab.ROW
    tmp[6, 4] <- 77 # the supply of cap.ROW
    tmp[7, 4] <- 2 # the supply of bond.ROW
    tmp
  },
  numeraire = "lab.CHN"
)

ge$p
ge$z

# Determine the parameters in the
# example based on an input-output table.
IT <- matrix(c(
  0, 61.44, 0, 6.498,
  53, 0, 0, 0,
  14.94, 0, 0, 0,
  0, 4.647, 0, 320,
  0, 0, 242.9, 0,
  0, 0, 81.74, 0,
  0, 1.85, 0, 0
), 7, 4, TRUE)

OT <- matrix(c(
  67.94, 0, 0, 0,
  0, 53, 0, 0,
  0, 14.94, 0, 0,
  0, 0, 324.64, 0,
  0, 0, 0, 242.9,
  0, 0, 0, 81.74,
  0, 0, 0, 1.85
), 7, 4, TRUE)

dimnames(IT) <- dimnames(OT) <- list(
  c("prod.CHN", "lab.CHN", "cap.CHN", "prod.ROW", "lab.ROW", "cap.ROW", "bond.ROW"),
  c("firm.CHN", "household.CHN", "firm.ROW", "household.ROW")
)

es.DFProd <- 0.8 # the substitution elasticity between domestic and foreign products
es.CL <- 0.8 # the substitution elasticity between capital and labor

dst.firm.CHN <- node_new("output",
                         type = "SCES",
                         alpha = OT["prod.CHN", "firm.CHN"] /
                           sum(IT[c("lab.CHN", "cap.CHN"), "firm.CHN"]),
                         beta = prop.table(IT[c("lab.CHN", "cap.CHN"), "firm.CHN"]),
                         es = es.CL,
                         "lab.CHN", "cap.CHN"
)

# the amount of outbound investment corresponding to
# each unit of composite commodity 1 used by household.
outbound.investment.rate <- IT["bond.ROW", "household.CHN"] /
  sum(IT[c("prod.CHN", "prod.ROW"), "household.CHN"])

dst.household.CHN <- node_new("util",
                             type = "FIN",
                             rate = c(1, outbound.investment.rate),
                             "cc1", "bond.ROW"
)

node_set(dst.household.CHN, "cc1",
         type = "SCES", alpha = 1,
         beta = prop.table(IT[c("prod.CHN", "prod.ROW"), "household.CHN"]),
         es = es.DFProd,
         "prod.CHN", "prod.ROW"
)

dst.firm.ROW <- node_new("output",
                         type = "SCES", alpha = 1,
                         beta = prop.table(IT[c("lab.ROW", "cap.ROW"), "firm.ROW"]),
                         es = es.CL,
                         "lab.ROW", "cap.ROW"
)

dst.household.ROW <- node_new("util",
                             type = "SCES", alpha = 1,
                             beta = prop.table(IT[c("prod.CHN", "prod.ROW"), "household.ROW"]),
                             es = es.DFProd,
                             "prod.CHN", "prod.ROW"
)

dstl <- list(dst.firm.CHN, dst.household.CHN, dst.firm.ROW, dst.household.ROW)

ge <- sdm2(dstl,
  names.commodity = c(
    "prod.CHN", "lab.CHN", "cap.CHN",
    "prod.ROW", "lab.ROW", "cap.ROW", "bond.ROW"
  ),
  names.agent = c(
    "firm.CHN", "household.CHN",
    "firm.ROW", "household.ROW"
  ),
  B = {
    tmp <- matrix(0, 7, 4, TRUE)
    tmp[1, 1] <- tmp[4, 3] <- 1
    tmp
  },
  S0Exg = {
    tmp <- matrix(NA, 7, 4, TRUE)
    tmp[2, 2] <- OT["lab.CHN", "household.CHN"]
    tmp[3, 2] <- OT["cap.CHN", "household.CHN"]
    tmp[5, 4] <- OT["lab.ROW", "household.ROW"]
    tmp[6, 4] <- OT["cap.ROW", "household.ROW"]
    tmp[7, 4] <- OT["bond.ROW", "household.ROW"]
    tmp
  },
  numeraire = "lab.CHN"
)

ge$p
ge$z

Calculating Real Exchange Rate Index

Description

Some examples of calculating the real exchange rate index in a two-country economy.

Usage

gemTwoCountry_RealExchangeRateIndex_7_4(...)

Arguments

Value

A real exchange rate index.

See Also

gemTwoCountry_Bond_7_4

Examples

# es.DFProd is the substitution elasticity between domestic and foreign products.
makeDstl <- function(es.DFProd = 0.5,
                     alpha.firm.CHN = 1,
                     beta.household.CHN = c(0.75, 0.25),
                     outbound.investment.rate = 0.25) {
  es.CL <- 0.8 # substitution elasticity between capital and labor

  dst.firm.CHN <- node_new("output",
    type = "SCES", alpha = alpha.firm.CHN, beta = c(0.75, 0.25), es = es.CL,
    "lab.CHN", "cap.CHN"
  )

  dst.household.CHN <- node_new("util",
    type = "FIN", rate = c(1, outbound.investment.rate),
    "cc1", "bond.ROW"
  ) # 0.1 is the amount of foreign investment corresponding to
  # each unit of cc1 (i.e. composite commodity 1).

  node_set(dst.household.CHN, "cc1",
    type = "SCES", alpha = 1, beta = beta.household.CHN, es = es.DFProd,
    "prod.CHN", "prod.ROW"
  )

  node_plot(dst.household.CHN)

  dst.firm.ROW <- node_new("output",
    type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = es.CL,
    "lab.ROW", "cap.ROW"
  )

  dst.household.ROW <- node_new("util",
    type = "SCES", alpha = 1, beta = c(0.05, 0.95), es = es.DFProd,
    "prod.CHN", "prod.ROW"
  )

  list(dst.firm.CHN, dst.household.CHN, dst.firm.ROW, dst.household.ROW)
}

dstl <- makeDstl()

SExg <- {
  tmp <- matrix(NA, 7, 4, TRUE)
  tmp[2, 2] <- 30 # the supply of lab.CHN
  tmp[3, 2] <- 10 # the supply of cap.CHN
  tmp[5, 4] <- 156 # the supply of lab.ROW
  tmp[6, 4] <- 156 # the supply of cap.ROW
  tmp[7, 4] <- 8 # the supply of bond.ROW
  tmp
}

f <- function(A = dstl,
              S0Exg = SExg) {
  sdm2(
    A = A,
    names.commodity = c(
      "prod.CHN", "lab.CHN", "cap.CHN",
      "prod.ROW", "lab.ROW", "cap.ROW", "bond.ROW"
    ),
    names.agent = c(
      "firm.CHN", "household.CHN",
      "firm.ROW", "household.ROW"
    ),
    B = {
      tmp <- matrix(0, 7, 4, TRUE)
      tmp[1, 1] <- tmp[4, 3] <- 1
      tmp
    },
    S0Exg = S0Exg,
    numeraire = "lab.CHN"
  )
}

ge.benchmark <- f()

## real exchange rate index
reri <- function(ge.new, ge.benchmark) {
  weight.CHN <- rowSums(ge.benchmark$SV)[c("prod.CHN", "lab.CHN", "cap.CHN")]
  weight.ROW <- rowSums(ge.benchmark$SV)[c("prod.ROW", "lab.ROW", "cap.ROW")]

  weighted.mean(ge.new$p[c("prod.ROW", "lab.ROW", "cap.ROW")], weight.ROW) /
    weighted.mean(ge.new$p[c("prod.CHN", "lab.CHN", "cap.CHN")], weight.CHN)
}


## technology progress in CHN
reri(f(A = makeDstl(es.DFProd = 5, alpha.firm.CHN = 2)), ge.benchmark)
reri(f(A = makeDstl(es.DFProd = 0.5, alpha.firm.CHN = 2)), ge.benchmark)

## labor supply change in CHN
SExg.LSC <- SExg
SExg.LSC[2, 2] <- SExg.LSC[2, 2] * 2

reri(f(A = makeDstl(es.DFProd = 5), S0Exg = SExg.LSC), ge.benchmark)

reri(f(S0Exg = SExg.LSC), ge.benchmark)

## capital accumulation in CHN
SExg.CA <- SExg
SExg.CA[3, 2] <- SExg.CA[3, 2] * 3

reri(f(A = makeDstl(es.DFProd = 5), S0Exg = SExg.CA), ge.benchmark)

reri(f(S0Exg = SExg.CA), ge.benchmark)

## preference change in China
reri(f(A = makeDstl(es.DFProd = 5, beta.household.CHN = c(0.5, 0.5))), ge.benchmark)
reri(f(A = makeDstl(beta.household.CHN = c(0.5, 0.5))), ge.benchmark)

## outbound-investment-rate change in China
reri(f(A = makeDstl(es.DFProd = 5, outbound.investment.rate = 0.1)), ge.benchmark)
reri(f(A = makeDstl(outbound.investment.rate = 0.1)), ge.benchmark)

An Example of Two-Country Economy with Tariff

Description

A general equilibrium example of two-country economy with tariff.

Usage

gemTwoCountry_Tariff_9_5(...)

Arguments

Value

A general equilibrium.

See Also

gemTwoCountry_Bond_7_4

Examples

es.DFProd <- 0.8 # substitution elasticity between domestic and foreign products
es.CL <- 0.8 # substitution elasticity between capital and labor

dst.firm.CHN <- node_new("output",
  type = "SCES", alpha = 1, beta = c(0.78, 0.22), es = es.CL,
  "lab.CHN", "cap.CHN"
)

dst.household.CHN <- node_new("util",
  type = "FIN", rate = c(1, outbound.investment.rate = 0.028),
  "cc1", "bond.ROW"
)

node_set(dst.household.CHN, "cc1",
  type = "SCES", alpha = 1, beta = c(0.93, 0.07), es = es.DFProd,
  "prod.CHN", "imported.prod.CHN"
)

node_plot(dst.household.CHN)

dst.foreign.trade.CHN <- node_new("imported.product",
  type = "FIN",
  rate = c(1, 0.016),
  "prod.ROW", "tariff.CHN"
)

dst.firm.ROW <- node_new("output",
  type = "SCES", alpha = 1, beta = c(0.75, 0.25), es = es.CL,
  "lab.ROW", "cap.ROW"
)

dst.household.ROW <- node_new("util",
  type = "SCES", alpha = 1, beta = c(0.02, 0.98), es = es.DFProd,
  "prod.CHN", "prod.ROW"
)

dstl <- list(
  dst.firm.CHN, dst.household.CHN, dst.foreign.trade.CHN,
  dst.firm.ROW, dst.household.ROW
)

ge <- sdm2(dstl,
  names.commodity = c(
    "prod.CHN", "lab.CHN", "cap.CHN", "imported.prod.CHN", "tariff.CHN",
    "prod.ROW", "lab.ROW", "cap.ROW", "bond.ROW"
  ),
  names.agent = c(
    "firm.CHN", "household.CHN", "foreign.trade.CHN",
    "firm.ROW", "household.ROW"
  ),
  B = {
    tmp <- matrix(0, 9, 5, TRUE)
    tmp[1, 1] <- tmp[6, 4] <- 1
    tmp[4, 3] <- 1
    tmp
  },
  S0Exg = {
    tmp <- matrix(NA, 9, 5, TRUE)
    tmp[2, 2] <- 53 # the supply of lab.CHN
    tmp[3, 2] <- 15 # the supply of cap.CHN
    tmp[5, 2] <- 0.29 # the supply of tariff.CHN
    tmp[7, 5] <- 240 # the supply of lab.ROW
    tmp[8, 5] <- 77 # the supply of cap.ROW
    tmp[9, 5] <- 2 # the supply of bond.ROW
    tmp
  },
  numeraire = "lab.CHN"
)

ge$p
ge$z

A 4-by-3 Economy with Two Industries

Description

A 4-by-3 economy with two industries.

Usage

gemTwoIndustries_4_3(...)

Arguments

References

Ju, Jiandong, Justin Yifu Lin, Qing Liu, Kang Shi (2020) Structural Changes and the Real Exchange Rate Dynamics. Journal of International Money and Finance, Vol. 107, pp: 102192.

Examples

dst.manu <- node_new("output",
  type = "SCES", es = 1,
  alpha = 1, beta = c(0.2, 0.8),
  "lab", "cap"
)

dst.serv <- node_new("output",
  type = "SCES", es = 1,
  alpha = 1, beta = c(0.8, 0.2),
  "lab", "cap"
)

dst.consumer <- node_new("util",
  type = "SCES", es = 1,
  alpha = 1, beta = c(0.5, 0.5),
  "manu", "serv"
)

dstl <- list(dst.manu, dst.serv, dst.consumer)

S0Exg <- matrix(c(
  NA, NA, NA,
  NA, NA, NA,
  NA, NA, 100,
  NA, NA, 100
), 4, 3, TRUE)

f <- function(dstl, S0Exg) {
  sdm2(
    A = dstl,
    B = matrix(c(
      1, 0, 0,
      0, 1, 0,
      0, 0, 0,
      0, 0, 0
    ), 4, 3, TRUE),
    S0Exg = S0Exg,
    names.commodity = c("manu", "serv", "lab", "cap"),
    names.agent = c("manu", "serv", "consumer"),
    numeraire = c("manu")
  )
}

ge <- f(dstl = dstl, S0Exg = S0Exg)

ge$D
ge$p

##
dstl2 <- lapply(dstl, Clone)
dstl2[[1]]$alpha <- 2

ge <- f(dstl = dstl2, S0Exg = S0Exg)
ge$D
ge$p

##
S0Exg2 <- S0Exg
S0Exg2[3, 3] <- 200 # labor supply
ge <- f(dstl = dstl, S0Exg = S0Exg2)
ge$D
ge$p

##
S0Exg3 <- S0Exg
S0Exg3[4, 3] <- 200 # capital supply
ge <- f(dstl = dstl, S0Exg = S0Exg3)
ge$D
ge$p

##
dstl3 <- lapply(dstl, Clone)
dstl3[[3]]$beta <- c(0.2, 0.8)
ge <- f(dstl = dstl3, S0Exg = S0Exg)
ge$D
ge$p

## exogenous wage rate
S0Exg4 <- S0Exg
S0Exg4[3, 3] <- 300 # labor supply

# Compute the price-control stationary state.
pcss <- sdm2(
  A = dstl2,
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = S0Exg4,
  names.commodity = c("manu", "serv", "lab", "cap"),
  names.agent = c("manu", "serv", "consumer"),
  numeraire = c("manu"),
  pExg = c(1, NA, 1, NA),
  maxIteration = 1,
  ts = TRUE
)

matplot(pcss$ts.z, type = "l")
matplot(pcss$ts.q, type = "l")
tail(pcss$ts.q)
pcss$p

Some Simple 2-by-2 General Equilibrium Models

Description

Some simple 2-by-2 general equilibrium models with a firm and a laborer.

Usage

gem_2_2(...)

Arguments

References

http://www.econ.ucla.edu/riley/MAE/Course/SolvingForTheWE.pdf

Examples

#### a 2-by-2 general equilibrium model with a Leontief production function.
ge.Leontief <- sdm2(
  A = matrix(c(
    0.5, 1,
    0.5, 0
  ), 2, 2, TRUE),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge.Leontief$p
ge.Leontief$z
addmargins(ge.Leontief$D, 2)
addmargins(ge.Leontief$S, 2)

## the same as above.
ge2.Leontief <- sdm2(
  A = list(
    dst.firm = node_new(
      "output",
      type = "Leontief",
      a = c(0.5, 0.5),
      "prod", "lab"
    ),
    dst.consumer = node_new(
      "util",
      type = "Leontief", a = 1,
      "prod"
    )
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge2.Leontief$p
ge2.Leontief$z
addmargins(ge2.Leontief$D, 2)
addmargins(ge2.Leontief$S, 2)

#### a 2-by-2 general equilibrium model with a CD production function.
ge.CD <- sdm2(
  A = function(state) {
    ## the vector of demand coefficients of the firm
    a1 <- CD_A(alpha = 2, Beta = c(0.5, 0.5), state$p)
    ## the vector of demand coefficients of the laborer
    a2 <- c(1, 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge.CD$p
ge.CD$z
addmargins(ge.CD$D, 2)
addmargins(ge.CD$S, 2)

## the same as above.
ge2.CD <- sdm2(
  A = list(
    dst.firm = node_new(
      "output",
      type = "CD", alpha = 2, beta = c(0.5, 0.5),
      "prod", "lab"
    ),
    dst.consumer = node_new(
      "util",
      type = "Leontief", a = 1,
      "prod"
    )
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge2.CD$p
ge2.CD$z
addmargins(ge2.CD$D, 2)
addmargins(ge2.CD$S, 2)

#### a 2-by-2 general equilibrium model with a SCES production function.
ge.SCES <- sdm2(
  A = function(state) {
    a1 <- SCES_A(es = 0.5, alpha = 1, Beta = c(0.5, 0.5), p = state$p)
    a2 <- c(1, 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge.SCES$p
ge.SCES$z
addmargins(ge.SCES$D, 2)
addmargins(ge.SCES$S, 2)

## the same as above.
ge2.SCES <- sdm2(
  A = list(
    dst.firm = node_new(
      "output",
      type = "SCES",
      es = 0.5, alpha = 1, beta = c(0.5, 0.5),
      "prod", "lab"
    ),
    dst.consumer = node_new(
      "util",
      type = "Leontief", a = 1,
      "prod"
    )
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge2.SCES$p
ge2.SCES$z
addmargins(ge2.SCES$D, 2)
addmargins(ge2.SCES$S, 2)

#### a 2-by-2 general equilibrium model with a CESAK production function.
ge.CESAK <- sdm2(
  A = function(state) {
    a.firm <- CESAK_dc(alpha = 2, betaK = 0.5, alphaK = 0.5, p = state$p, es = 1)
    a.consumer <- c(1, 0)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge.CESAK$p
ge.CESAK$z
addmargins(ge.CESAK$D, 2)
addmargins(ge.CESAK$S, 2)

## the same as above.
ge2.CESAK <- sdm2(
  A = list(
    dst.firm = node_new(
      "output",
      type = "CESAK", es = 1,
      alpha = 2, betaK = 0.5, alphaK = 0.5,
      "prod", "lab"
    ),
    dst.consumer = node_new(
      "util",
      type = "Leontief", a = 1,
      "prod"
    )
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge2.CESAK$p
ge2.CESAK$z
addmargins(ge2.CESAK$D, 2)
addmargins(ge2.CESAK$S, 2)

#### Example 1 in the ucla reference.
ge3.SCES <- sdm2(
  A = function(state) {
    a.firm <- c(0, 0.25)
    a.consumer <- SCES_A(es = 0.5, alpha = 1, Beta = c(0.5, 0.5), p = state$p)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 30
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge3.SCES$p
ge3.SCES$z
ge3.SCES$D
ge3.SCES$S

#### The laborer has some product.
ge <- sdm2(
  A = function(state) {
    a.firm <- c(0, 1) # c(0, 2)
    a.consumer <- SCES_A(es = 0.5, alpha = 1, Beta = c(0.5, 0.5), p = state$p)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, 50, # 500
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge$p
ge$z
ge$D
ge$S

Some Simple 3-by-2 General Equilibrium Models

Description

Some simple 3-by-2 general equilibrium models with a firm and a consumer.

Usage

gem_3_2(...)

Arguments

References

http://www.econ.ucla.edu/riley/MAE/Course/SolvingForTheWE.pdf

He Zhangyong, Song Zheng (2010, ISBN: 9787040297270) Advanced Macroeconomics. Beijing: Higher Education Press.

Examples

ge.CD <- sdm2(
  A = function(state) {
    ## the vector of demand coefficients of the firm
    a1 <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), state$p)
    ## the vector of demand coefficients of the consumer
    a2 <- c(1, 0, 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge.CD$p
ge.CD$z
ge.CD$D
ge.CD$S

#### Example 2 in the ucla reference
## By introducing a new factor of production (called land here)
## a firm with diminishing returns to scale can be converted into
## a firm with constant returns to scale.
ge2.CD <- sdm2(
  A = function(state) {
    a.firm <- CD_A(alpha = 6, Beta = c(0, 0.5, 0.5), state$p)
    a.consumer <- CD_A(alpha = 1, Beta = c(0.2, 0.8, 0), state$p)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 81,
    NA, 1
  ), 3, 2, TRUE),
  names.commodity = c("prod", "lab", "land"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge2.CD$p
ge2.CD$z
ge2.CD$D
ge2.CD$S

####
ge.SCES <- sdm2(
  A = function(state) {
    a1 <- SCES_A(es = 0.5, alpha = 1, Beta = c(0, 0.5, 0.5), p = state$p)
    a2 <- c(1, 0, 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge.SCES$p
ge.SCES$z
ge.SCES$D
ge.SCES$S

####
ge2.SCES <- sdm2(
  A = function(state) {
    a1 <- SCES_A(es = 0.5, alpha = 1, Beta = c(0.2, 0.4, 0.4), p = state$p)
    a2 <- c(1, 0, 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge2.SCES$p
ge2.SCES$z
ge2.SCES$D
ge2.SCES$S

#### nested production function
dst.firm <- node_new(
  "prod",
  type = "Leontief",
  a = c(0.2, 0.8),
  "prod", "cc1"
)
node_set(dst.firm, "cc1",
         type = "SCES",
         es = 0.5, alpha = 1, beta = c(0.5, 0.5),
         "cap", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge3.SCES <- sdm2(
  A = list(dst.firm, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge3.SCES$p
ge3.SCES$z
ge3.SCES$D
ge3.SCES$S

#### a model with a quasilinear utility function (see He and Song, 2010, page 19).
alpha.firm <- 2
beta.cap.firm <- 0.6
beta.lab.firm <- 1 - beta.cap.firm
theta.consumer <- 0.8
lab.supply <- 2
cap.supply <- 1

ge <- sdm2(
  A = function(state) {
    a1 <- CD_A(alpha.firm, rbind(0, beta.lab.firm, beta.cap.firm), state$p)

    demand.lab.prod <- QL_demand(
      w = state$w[2], p = state$p[2:1], # the prices of lab and prod
      alpha = 1, beta = theta.consumer, type = "CRRA"
    )
    a2 <- c(demand.lab.prod[2:1], 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, lab.supply,
    NA, cap.supply
  ), 3, 2, TRUE),
  names.commodity = c("prod", "lab", "cap"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge$p
ge$z
ge$D
ge$S

# the equilibrium leisure
lab.supply - (beta.lab.firm * (alpha.firm * cap.supply^beta.cap.firm)^(1 - theta.consumer))^
  (1 / (beta.cap.firm + beta.lab.firm * theta.consumer))

# the equilibrium price of labor
w <- ((1 - beta.cap.firm)^(1 - beta.cap.firm) * (alpha.firm * cap.supply^beta.cap.firm))^
  (theta.consumer / (beta.cap.firm + (1 - beta.cap.firm) * theta.consumer))

# the equilibrium price of capital goods
beta.cap.firm * w^(1 / theta.consumer) / cap.supply

Some Simple 3-by-3 General Equilibrium Models

Description

Some simple 3-by-3 general equilibrium models with two firms and a consumer.

Usage

gem_3_3(...)

Arguments

See Also

gemCapitalAccumulation

Examples

####
ge <- sdm2(
  A = function(state) {
    ## the vector of demand coefficients of the firm
    a.firm.corn <- CD_A(alpha = 1, Beta = c(0, 0.5, 0.5), p = state$p)
    a.firm.iron <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), p = state$p)
    ## the vector of demand coefficients of the consumer
    a.consumer <- CD_A(alpha = 1, Beta = c(0.5, 0.5, 0), p = state$p)
    cbind(a.firm.corn, a.firm.iron, a.consumer)
  },
  B = diag(c(1, 1), 3),
  S0Exg = {
    tmp <- matrix(NA, 3, 3)
    tmp[3, 3] <- 100
    tmp
  },
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer"),
  numeraire = "lab"
)

ge$p
ge$z
ge$D
ge$S

####
ge <- sdm2(
  A = function(state) {
    ## the vector of demand coefficients of the firm
    a.firm.corn <-
      SCES_A(es = 1,
             alpha = 1,
             Beta = c(0, 0.5, 0.5),
             p = state$p)
    a.firm.iron <-
      SCES_A(es = 1,
             alpha = 2,
             Beta = c(0, 0.5, 0.5),
             p = state$p)
    ## the vector of demand coefficients of the consumer
    a.consumer <- CD_A(alpha = 1, Beta = c(0.5, 0.5, 0), p = state$p)
    cbind(a.firm.corn, a.firm.iron, a.consumer)
  },
  B = diag(c(1, 1), 3),
  S0Exg = {
    tmp <- matrix(NA, 3, 3)
    tmp[3, 3] <- 100
    tmp
  },
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer"),
  numeraire = "lab"
)

ge$p
ge$z
ge$D
ge$S

#### a general equilibrium model containing a production firm
#### and a capital-goods-leasing firm
ge <- sdm2(
  A = function(state) {
    a.firm1 <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), state$p)
    a.consumer <- c(1, 0, 0)
    a.firm2 <- c(1, 0, 0)
    cbind(a.firm1, a.consumer, a.firm2)
  },
  B = matrix(c(
    1, 0, 0.5,
    0, 0, 1,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, 100,NA
  ), 3, 3, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm1", "consumer","firm2"),
  numeraire = "prod",
  priceAdjustmentVelocity = 0.05
)
ge$p
ge$z
ge$D

Some Simple 3-by-4 General Equilibrium Models

Description

Some simple 3-by-4 general equilibrium models with two firms and two consumers.

Usage

gem_3_4(...)

Arguments

Examples

####
ge <- sdm2(
  A = function(state) {
    a.firm.corn <- CD_A(alpha = 1, Beta = c(0, 0.5, 0.5), state$p)
    a.firm.iron <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), state$p)
    a.consumer1 <- c(1, 0, 0)
    a.consumer2 <- CD_A(alpha = 1, Beta = c(0.5, 0.5, 0), state$p)

    cbind(a.firm.corn, a.firm.iron, a.consumer1, a.consumer2)
  },
  B = diag(c(1, 1, 0), 3, 4),
  S0Exg = {
    tmp <- matrix(NA, 3, 4)
    tmp[3, 3:4] <- 100
    tmp
  },
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer1", "consumer2"),
  numeraire = "lab"
)

ge$p
ge$z
ge$D
ge$S

####
ge <- sdm2(
  A = function(state) {
    a.firm.corn <-
      SCES_A(
        es = 1,
        alpha = 1,
        Beta = c(0, 0.5, 0.5),
        p = state$p
      )
    a.firm.iron <-
      SCES_A(
        es = 1,
        alpha = 2,
        Beta = c(0, 0.5, 0.5),
        p = state$p
      )
    a.consumer1 <- c(1, 0, 0)
    a.consumer2 <- CD_A(alpha = 1, Beta = c(0.5, 0.5, 0), state$p)

    cbind(a.firm.corn, a.firm.iron, a.consumer1, a.consumer2)
  },
  B = diag(c(1, 1, 0), 3, 4),
  S0Exg = {
    tmp <- matrix(NA, 3, 4)
    tmp[3, 3:4] <- 100
    tmp
  },
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer1", "consumer2"),
  numeraire = "lab"
)

ge$p
ge$z
ge$D
ge$S

#### an example at
## https://web.stanford.edu/~jdlevin/Econ%20202/General%20Equilibrium.pdf
ge <- sdm2(
  A = function(state) {
    a.firm.1 <- c(0, 1, 0)
    a.firm.2 <- c(0, 0, 1)
    a.consumer1 <- CD_A(alpha = 1, Beta = c(1 / 3, 1 / 3, 1 / 3), state$p)
    a.consumer2 <- CD_A(alpha = 1, Beta = c(1 / 3, 1 / 3, 1 / 3), state$p)

    cbind(a.firm.1, a.firm.2, a.consumer1, a.consumer2)
  },
  B = matrix(c(
    2, 0, 0, 0,
    0, 1, 0, 0,
    0.5, 0, 0, 0
  ), 3, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, 1, 2,
    NA, NA, 2, 2,
    NA, NA, 3, 2
  ), 3, 4, TRUE),
  names.agent = c("firm1", "firm2", "consumer1", "consumer2"),
  numeraire = 3
)

ge$p
ge$z
ge$D
ge$S

Some Simple 4-by-4 General Equilibrium Models

Description

Some simple 4-by-4 general equilibrium models.

Usage

gem_4_4(...)

Arguments

Examples

#### A general equilibrium model containing a capital good with service-life
# wear-and-tear. The new product can be used as a capital good with a service
# life of two periods, and the used old capital good is the old product.
ge <- sdm2(
  A = function(state) {
    a.firm1 <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5, 0), state$p)
    a.consumer <- c(1, 0, 0, 0)
    a.firm2 <- c(1, 0, 0, 0)
    a.firm3 <- c(0, 0, 0, 1)
    cbind(a.firm1, a.consumer, a.firm2, a.firm3)
  },
  B = matrix(c(
    1, 0, 0, 0,
    0, 0, 1, 1,
    0, 0, 0, 0,
    0, 0, 1, 0
  ), 4, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, 100, NA, NA,
    NA, NA, NA, NA
  ), 4, 4, TRUE),
  names.commodity = c("prod.new", "cap", "lab", "prod.old"),
  names.agent = c("firm1", "consumer", "firm2", "firm3"),
  numeraire = "prod.new",
  priceAdjustmentVelocity = 0.05
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

#### the Shoven-Whalley model at
## https://lexjansen.com/nesug/nesug03/st/st002.pdf
ge <- sdm2(
  A = function(state) {
    a.firm.corn <- CES_A(sigma = 1 - 1 / 2, alpha = 1.5, Beta = c(0, 0, 0.4, 0.6), state$p)
    a.firm.iron <- CES_A(sigma = 1 - 1 / 0.5, alpha = 2, Beta = c(0, 0, 0.3, 0.7), state$p)
    a.consumer1 <- SCES_A(alpha = 1, Beta = c(0.5, 0.5, 0, 0), es = 1.5, p = state$p)
    a.consumer2 <- SCES_A(alpha = 1, Beta = c(0.3, 0.7, 0, 0), es = 0.75, p = state$p)

    cbind(a.firm.corn, a.firm.iron, a.consumer1, a.consumer2)
  },
  B = diag(c(1, 1, 0, 0), 4, 4),
  S0Exg = {
    tmp <- matrix(NA, 4, 4)
    tmp[3, 3] <- 25
    tmp[4, 4] <- 60
    tmp
  },
  names.commodity = c("corn", "iron", "cap", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer1", "consumer2"),
  numeraire = "lab"
)

ge$p
ge$z
ge$D
ge$S

#### an n-by-n general equilibrium model with Cobb-Douglas functions.
f <- function(n, policy = NULL, z0 = rep(100 * n, n), numberOfPeriods = 30,
              Beta = matrix(1 / n, n, n), n.firm = n - 1) {
  ge <- sdm2(
    A = function(state) {
      CD_A(alpha = rep(n, n), Beta = Beta, p = state$p)
    },
    B = {
      tmp <- diag(n)
      tmp[, (n.firm + 1):n] <- 0
      tmp
    },
    S0Exg = {
      tmp <- matrix(NA, n, n)
      for (k in (n.firm + 1):n) tmp[k, k] <- 100 * n
      tmp
    },
    numeraire = n,
    policy = policy,
    z0 = z0,
    maxIteration = 1,
    numberOfPeriods = numberOfPeriods,
    names.agent = c(paste0("firm", 1:n.firm), paste0("consumer", 1:(n - n.firm))),
    ts = TRUE
  )
  print(ge$z)
  print(ge$p)
  invisible(ge)
}

n <- 4
f(n, n.firm = n - 2)
## a spot equilibrium path
ge <- f(n, policy = policyMarketClearingPrice, z0 = runif(n, 10 * n, 100 * n), n.firm = n - 2)
matplot(ge$ts.z, type = "b", pch = 20)
matplot(ge$ts.p, type = "b", pch = 20)

A General Equilibrium Model with Endogenous Labor Supply

Description

This is an example of the spot equilibrium path with endogenous labor supply. Assume that as the level of utility increases, laborer will purchase or receive more education and training, resulting in an increase in human capital, which can be regarded as an increase in labor supply. That is to say, the utility level as an endogenous variable will affect the supply of labor. Therefore, labor supply becomes an endogenous variable.

Usage

gemstEndogenousLaborSupply_2_2(...)

Arguments

Examples

f <- function(z0 = c(20, 1)) {
  ge <- sdm2(
    A = function(state) {
      a.firm <- CD_A(alpha = 5, Beta = c(0.5, 0.5), p = state$p)
      a.consumer <- c(1, 0)
      cbind(a.firm, a.consumer)
    },
    B = matrix(c(
      1, 0,
      0, 0
    ), 2, 2, TRUE),
    S0Exg = matrix(c(
      NA, NA,
      NA, 1
    ), 2, 2, TRUE),
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "consumer"),
    numeraire = "lab",
    z0 = z0,
    ts = TRUE,
    policy = list(
      function(state) {
        state$S[2, 2] <- structural_function(state$last.z[2], c(6.5, 7), 1, 2)
        state
      },
      policyMarketClearingPrice
    ),
    numberOfPeriods = 20,
    maxIteration = 1
  )
  matplot(ge$ts.z, type = "o", pch = 20)
  print(ge$z)
  print(ge$S)
}

f()

f(c(10,1))

A General Equilibrium Model with Endogenous Production Function

Description

These are examples of spot-trading dynamic equilibria with endogenous production functions. In these models, the parameters of the production functions vary with the output level in the previous period.

To address locally or globally increasing returns to scale, we can use an endogenous CES-type production function with constant returns to scale, instead of adopting a more complex functional form.

Usage

gemstEndogenousProductionFunction_2_2(...)

Arguments

See Also

gemstIntertemporal_EndogenousProductionFunction_2_2

Examples

dst.firm <- node_new(
  "output",
  type = "CD", alpha = NA, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge <- sdm2(
  A = list(dst.firm, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  z0 = c(1, 1),
  p0 = c(1, 1),
  ts = TRUE,
  policy = list(
    function(A, state) {
      A[[1]]$alpha <- 5 * state$last.z[1]^0.1
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 40,
  maxIteration = 1
)

matplot(ge$ts.z, type = "o", pch = 20)

# #### An example of a spot-trading dynamic equilibrium with corrective taxation.
# tau <- 0.2
#
# dst.firm <- node_new(
#   "output",
#   type = "CD", alpha = NA, beta = c(0.5, 0.5),
#   "prod", "lab"
# )
#
# dst.consumer <- node_new(
#   "util",
#   type = "Leontief", a = 1,
#   "prod"
# )
#
# ge <- sdm2(
#   A = list(dst.firm, dst.consumer),
#   B = matrix(c(
#     1, 0,
#     0, 0
#   ), 2, 2, TRUE),
#   S0Exg = matrix(c(
#     NA, NA,
#     100 * tau, 100 * (1 - tau)
#   ), 2, 2, TRUE),
#   names.commodity = c("prod", "lab"),
#   names.agent = c("firm", "consumer"),
#   numeraire = "lab",
#   z0 = c(100, 100),
#   p0 = c(1, 1),
#   ts = TRUE,
#   policy = list(
#     function(A, state) {
#       A[[1]]$alpha <- 2 * state$last.z[1]^0.1
#     },
#     policyMarketClearingPrice
#   ),
#   numberOfPeriods = 40,
#   maxIteration = 1
# )
#
# matplot(ge$ts.z, type = "o", pch = 20)
# ge$z

Some General Equilibrium Models with Endogenous Utility Function

Description

Some examples of the spot equilibrium path with an endogenous utility function. The parameters of the utility function will change with the utility level.

To deal with non-homothetic preferences, we can simply use an endogenous CES-type utility function instead of a utility function with a more complex form.

Usage

gemstEndogenousUtilityFunction(...)

Arguments

Examples

#### a 2-by-2 example
dst.firm <- node_new(
  "output",
  type = "CD", alpha = 5, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "prod", "lab"
)

ge <- sdm2(
  A = list(dst.firm, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  z0 = c(0.01, 1),
  p0 = c(1, 1),
  ts = TRUE,
  policy = list(
    function(A, state) {
      util <- state$last.z[2]
      beta2 <- 0.95 * plogis(util, location = 2, scale = 2)
      A[[2]]$beta <- c(1 - beta2, beta2)
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 20,
  maxIteration = 1
)

matplot(ge$ts.z, type = "o", pch = 20)
ge$z
dst.consumer$beta

#### a 3-by-3 example with 100 laborers
#### Assume that each laborer desires to consume one unit of
#### corn per period, regardless of her level of utility.
dst.firm.corn <- node_new(
  "corn",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "iron", "lab"
)

dst.firm.iron <- node_new(
  "iron",
  type = "CD", alpha = 5, beta = c(0.5, 0.5),
  "iron", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief",
  a = c(0.5, 0.5),
  "corn", "iron"
)

ge <- sdm2(
  A = list(dst.firm.corn, dst.firm.iron, dst.consumer),
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer"),
  numeraire = "lab",
  ts = TRUE,
  policy = list(
    function(A, state) {
      last.util <- state$last.z[3] / 100 # the previous utility level of each laborer
      a1 <- min(1 / last.util, 1)
      A[[3]]$a <- c(a1, 1 - a1)
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 40,
  maxIteration = 1
)

matplot(ge$ts.z, type = "o", pch = 20)
ge$z
ge$A
ge$D

#### a 4-by-4 example with 100 homogeneous laborers
dst.agri <- node_new(
  "output",
  type = "SCES",
  es = 0.5, alpha = 2, beta = c(0.2, 0.8),
  "manu", "lab"
)

dst.manu <- node_new(
  "output",
  type = "SCES",
  es = 0.5, alpha = 3, beta = c(0.6, 0.4),
  "manu", "lab"
)

dst.serv <- node_new(
  "output",
  type = "SCES",
  es = 0.5, alpha = 2, beta = c(0.4, 0.6),
  "manu", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(0.6, 0.3, 0.1),
  "agri", "manu", "serv"
)

ge <- sdm2(
  A = list(dst.agri, dst.manu, dst.serv, dst.consumer),
  B = diag(c(1, 1, 1, 0)),
  S0Exg = {
    tmp <- matrix(NA, 4, 4)
    tmp[4, 4] <- 100
    tmp
  },
  names.commodity = c("agri", "manu", "serv", "lab"),
  names.agent = c("agri", "manu", "serv", "consumer"),
  numeraire = "lab",
  z0 = c(1, 1, 1, 0),
  ts = TRUE,
  policy = list(
    function(A, state) {
      util <- state$last.z[4] / 100
      beta1 <- structural_function(util, c(1, 6), 0.6, 0.1)
      beta3 <- structural_function(util, c(1, 6), 0.1, 0.5)
      beta2 <- 1 - beta1 - beta3
      A[[4]]$beta <- c(beta1, beta2, beta3)
    },
    policyMarketClearingPrice
  ),
  numberOfPeriods = 20,
  maxIteration = 1
)

matplot(ge$ts.z, type = "o", pch = 20)
ge$z
dst.consumer$beta

An Intertemporal Equilibrium Model with Endogenous Production Function

Description

This is an example of an intertemporal equilibrium model with an endogenous production function.

Usage

gemstIntertemporal_EndogenousProductionFunction_2_2(...)

Arguments

See Also

gemstEndogenousProductionFunction_2_2

Examples

# np <- 50 # the number of economic periods
# y1 <- 100 # the initial product supply
#
# n <- 2 * np - 1 # the number of commodity kinds
# m <- np # the number of agent kinds
#
# names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)))
# names.agent <- c(paste0("firm", 1:(np - 1)), "consumer")
#
# # the exogenous supply matrix.
# S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
# S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100
# S0Exg["prod1", "consumer"] <- y1
#
# # the output coefficient matrix.
# B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
# for (k in 1:(np - 1)) {
#   B[paste0("prod", k + 1), paste0("firm", k)] <- 1
# }
#
# dstl.firm <- list()
# for (k in 1:(np - 1)) {
#   dstl.firm[[k]] <- node_new(
#     "prod",
#     type = "CD",
#     alpha = 2, beta = c(0.5, 0.5),
#     paste0("prod", k), paste0("lab", k)
#   )
# }
# dstl.firm[[1]]$alpha <- y1^0.1 * dstl.firm[[1]]$alpha
#
# dst.consumer.CD <- node_new(
#   "util",
#   type = "CD",
#   alpha = 1, beta = prop.table(rep(1, np)),
#   paste0("prod", 1:np)
# )
#
# ge <-sdm2(
#     A = c(dstl.firm, dst.consumer.CD),
#     B = B,
#     S0Exg = S0Exg,
#     names.commodity = names.commodity,
#     names.agent = names.agent,
#     numeraire = "prod1",
#     maxIteration = 1,
#     numberOfPeriods = 2000,
#     policy = list(
#       function(A, state) {
#         for (k in 2:(np - 1)) {
#           A[[k]]$alpha <- 2 * state$last.z[k - 1]^0.1
#         }
#       }
#     ),
#     ts = TRUE
#   )
#
# plot(ge$z[1:(np - 1)], type = "o", xlab = "time", ylab = "output")
# grid()

Structural Multiple Equilibria and Structural Transition Policy

Description

Some examples of structural multiple equilibria and structural transition policy. In these examples it is assumed that the firm has a structural production function (see Li, Fu, 2020), e.g.

structural_function(last.output, c(0.3, 0.4), 1, 2) * x1^0.35 * x2^0.65

wherein last.output is the output of the firm in the previous period.

Usage

gemstStructuralMultipleEquilibria_2_2(...)

Arguments

References

Li Wu, Fu Caihui (2020) A Simulation Study on the Economic Structure Transition Policy. Journal of Shanghai University (Social Sciences). 37(2), pp: 33-45. (In Chinese)

Examples

dst.firm <- node_new("output",
  type = "CD", alpha = 1,
  beta = c(0.35, 0.65),
  "prod", "lab"
)

dst.consumer <- node_new("utility",
  type = "CD", alpha = 1,
  beta = c(0.4, 0.6),
  "prod", "lab"
)

policy.technology <- function(time, state, A) {
  # state$last.z[1] is the previous output.
  A[[1]]$alpha <- structural_function(state$last.z[1], c(0.3, 0.4), 1, 2)
}

policy.tax <- function(time, state) {
  if ((time >= 15) && state$last.z[1] < 0.4) {
    state$S[2, 2] <- 0.8
    state$S[2, 1] <- 0.2
  } else {
    state$S[2, 2] <- 1
    state$S[2, 1] <- 0
  }

  state
}

f <- function(z0 = c(0.1, 1),
              policy = list(
                policy.technology,
                policyMarketClearingPrice
              )) {
  ge <- sdm2(
    A = list(dst.firm, dst.consumer),
    B = matrix(c(
      1, 0,
      0, 0
    ), 2, 2, TRUE),
    S0Exg = matrix(c(
      NA, NA,
      NA, 1
    ), 2, 2, TRUE),
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "consumer"),
    numeraire = "lab",
    z0 = z0,
    p0 = c(1, 1),
    maxIteration = 1,
    numberOfPeriods = 30,
    policy = policy,
    ts = TRUE
  )

  matplot(ge$ts.z, type = "o", pch = 20)
  invisible(ge)
}

geLow <- f()
geLow$z

geHigh <- f(z0 = c(0.5, 1))
geHigh$z

f(policy = list(
  policy.technology,
  policy.tax,
  policyMarketClearingPrice
))

#### structural transition: disequilibrium path and
## a spot equilibrium path
dst.firm <- node_new("output",
  type = "CD", alpha = 5,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new("utility",
  type = "Leontief", a = 1,
  "prod"
)

policy.technology <- function(time, state, A) {
  # state$last.z[1] is last output.
  A[[1]]$alpha <- structural_function(state$last.z[1], c(15, 20), 5, 15)
  return(NULL)
}

policy.tax <- function(time, state) {
  if ((time >= 100) && (time <= 109)) {
    state$S[2, 2] <- 0.6
    state$S[2, 1] <- 0.4
  } else {
    state$S[2, 2] <- 1
    state$S[2, 1] <- 0
  }

  state
}

f <- function(z0 = c(1, 1),
              p0 = c(1, 1),
              policy = policy.technology) {
  ge <- sdm2(
    A = list(dst.firm, dst.consumer),
    B = matrix(c(
      1, 0,
      0, 1
    ), 2, 2, TRUE),
    S0Exg = matrix(c(
      NA, NA,
      NA, 1
    ), 2, 2, TRUE),
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "consumer"),
    numeraire = "lab",
    z0 = z0,
    p0 = p0,
    maxIteration = 1,
    numberOfPeriods = 200,
    policy = policy,
    priceAdjustmentVelocity = 0.4,
    ts = TRUE
  )

  matplot(ge$ts.z, type = "l", pch = 20)
  invisible(ge)
}

geLow <- f()
geLow$z

geHigh <- f(z0 = c(18, 1), p0 = c(1, 9))
geHigh$z

## structural transition: a disequilibrium path
f(policy = list(
  policy.technology,
  policy.tax
))$z


## structural transition: a spot equilibrium path
f(policy = list(
  policy.technology,
  policy.tax,
  policyMarketClearingPrice
))$z

## structural transition through foreign aid
policy.foreign_aid <- function(time, state) {
  if ((time >= 100) && (time <= 109)) {
    state$S[2, 2] <- 3
  } else {
    state$S[2, 2] <- 1
  }

  state
}

f(policy = list(
  function(time, state, A) { # technology policy
    # state$last.z[1] is last output.
    A[[1]]$alpha <- structural_function(state$last.z[1], c(30, 35), 5, 15)
  },
  policy.foreign_aid
))

#### another example
dst.firm <- node_new("prod",
  type = "CD", alpha = 2,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new("util",
  type = "Leontief", a = 1,
  "prod"
)

policy.technology <- function(time, state, A) {
  # state$last.z[1] is the previous output.
  A[[1]]$alpha <- structural_function(state$last.z[1], c(220, 250), 2, 4)
}

policy.tax <- function(time, state) {
  if ((time >= 15) && state$last.z[1] < 240) {
    state$S[2, 2] <- 80
    state$S[2, 1] <- 20
  } else {
    state$S[2, 2] <- 100
    state$S[2, 1] <- 0
  }

  state
}

ge <- sdm2(
  A = list(dst.firm, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  z0 = c(100, 100),
  maxIteration = 1,
  numberOfPeriods = 30,
  policy = list(
    policy.technology,
    policy.tax,
    policyMarketClearingPrice
  ),
  ts = TRUE
)

matplot(ge$ts.z, type = "b", pch = 20)

Compute the Growth Rate

Description

Compute the growth rates for a vector or each column of a matrix.

Usage

growth_rate(x, log = FALSE, first.na = TRUE)

Arguments

Value

a vector or a matrix consisting of growth rates.

Examples

x <- matrix(1:8, 4, 2)
growth_rate(x)

Iteration Function

Description

Iteration function

Usage

iterate(x, f, times = 100, tol = NA, ...)

Arguments

Value

A matrix consisting of state vectors.

Examples

x <- c(1, 2)
f <- function(x, a) prop.table(c(sum(x), a * prod(x)^(1 / 2)))
iterate(x, f, 100, a = 3)
iterate(x, f, 100, tol = 1e-5, a = 3)

#### Heron's method for finding square roots
x <- 1
f <- function(x, n) (x + n / x) / 2
iterate(x, f, 10, n = 5)

#### Find a root of the equation x^3-x-1==0.
x <- 1.5
f <- function(x) (x + 1)^(1 / 3)
iterate(x, f, 10)

####
x <- c(1, 2, 3)
f <- function(x) {
  n <- length(x)
  sigma <- seq(-1, 1, length.out = n)
  result <- rep(NA, n)
  for (k in 1:n) result[k] <- CES(sigma[k], 1, rep(1 / n, n), x, rep(1 / n, n))
  prop.table(result)
}
iterate(x, f, 100)

Make a Countercyclical Product Tax Policy Function

Description

This function returns a countercyclical product tax policy function to accelerate convergence when calculating general equilibrium. In some cases this tax policy with variable tax rates can stabilize the economy (see Li, 2019, section 9.4.5.4) . When a firm's output is higher than the average output in previous periods, a tax is imposed on the firm to reduce the output of the product. Tax revenue will be used for implicit public spending. The way of taxation is to directly deduct a part of the supply of the firm.

Usage

makeCountercyclicalProductTax(agent = 1, time.win = c(100, Inf), span = 50)

Arguments

Value

A countercyclical product tax policy function.

See Also

CGE::Example9.10.policy.tax

Examples

ge <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = 1000
)

ge <- gemCanonicalDynamicMacroeconomic_4_3(
  numberOfPeriods = 1000,
  policy = makeCountercyclicalProductTax(time.win = c(500, Inf))
)

Make a Policy of Head Adjustment for a Timeline Model

Description

Make a policy of head adjustment for a timeline model. Head adjustment refers to the adjustment of the initial product supply to a steady-state value.

Usage

makePolicyHeadAdjustment(ind, gr = 0)

Arguments

Value

A policy, which is often used as an argument of the function sdm2.

See Also

makePolicyHeadTailAdjustment; makePolicyTailAdjustment

Make a Policy of Head and/or Tail Adjustment for a Timeline Model

Description

Make a policy of head and/or tail adjustment for a timeline model. A timeline model is an intertemporal non-sequential model that includes production and a given initial product supply. Head adjustment refers to the adjustment of the initial product supply to a steady-state value. Similarly, tail adjustment refers to the adjustment of the share coefficient of the last period of the consumer in the timeline model in order to let the model run in a steady-state equilibrium path.

Usage

makePolicyHeadTailAdjustment(
  type = c("both", "tail", "head", "none"),
  gr = 0,
  np
)

Arguments

Value

A policy, which is often used as an argument of the function sdm2.

Note

The statement policy = makePolicyHeadTailAdjustment(gr = gr, np = np) is equivalent to policy = list( makePolicyHeadAdjustment(ind = c(1, np, 2, 1), gr = gr), makePolicyTailAdjustment(ind = c(np - 1, np), gr = gr) ).

See Also

gemIntertemporal_Dividend; gemIntertemporal_Money_Dividend_Example7.5.1

Make a Policy of Income Tax

Description

This function returns a policy function that redistributes the supplies of economic agents, and the effect is equivalent to the collection of income tax.

Usage

makePolicyIncomeTax(agent, tax.rate, redistribution, time.win = c(1, Inf))

Arguments

Value

A policy function, which is often used as an argument of the function sdm2.

References

Manuel Alejandro Cardenete, Ana-Isabel Guerra, Ferran Sancho (2012, ISBN: 9783642247453) Applied General Equilibrium: An Introduction. Springer-Verlag Berlin Heidelberg.

See Also

gemTax_5_4

Examples

## an exmaple of income tax (see Cardenete et al., 2012, Table 4.3)
dst.consumer1 <- node_new("utility",
                          type = "CD",
                          alpha = 1,
                          beta = c(0.3, 0.7),
                          "prod1", "prod2"
)

dst.consumer2 <- Clone(dst.consumer1)
dst.consumer2$beta <- c(0.6, 0.4)

dst.firm1 <- node_new("output",
                      type = "Leontief",
                      a = c(0.5, 0.2, 0.3),
                      "VA", "prod1", "prod2"
)
node_set(dst.firm1, "VA",
         type = "CD",
         alpha = 0.8^-0.8 * 0.2^-0.2,
         beta = c(0.8, 0.2),
         "lab", "cap"
)

dst.firm2 <- Clone(dst.firm1)
node_set(dst.firm2, "output",
         a = c(0.25, 0.5, 0.25)
)
node_set(dst.firm2, "VA",
         alpha = 0.4^-0.4 * 0.6^-0.6,
         beta = c(0.4, 0.6)
)
dstl <- list(dst.firm1, dst.firm2, dst.consumer1, dst.consumer2)
ge <- sdm2(dstl,
  names.commodity = c("prod1", "prod2", "lab", "cap"),
  names.agent = c("firm1", "firm2", "consumer1", "consumer2"),
  numeraire = "lab",
  B = {
    tmp <- matrix(0, 4, 4)
    tmp[1, 1] <- 1
    tmp[2, 2] <- 1
    tmp
  },
  S0Exg = {
    tmp <- matrix(NA, 4, 4)
    tmp[3:4, 3] <- c(30, 20)
    tmp[3:4, 4] <- c(20, 5)
    tmp
  },
  maxIteration = 1,
  policy = makePolicyIncomeTax(
    agent = c(3, 4),
    tax.rate = c(0.2, 0.2),
    redistribution = c(0.5, 0.5)
  )
)

Make a Mean Value Policy Function

Description

This function returns a mean value policy function with a given span to accelerate convergence when calculating general equilibrium. We can observe the number of periods included in the economic cycle of the time series, and then set the number of periods as the parameter (i.e. span) of this function. See policyMeanValue

Usage

makePolicyMeanValue(span = 200)

Arguments

Value

A mean value policy function.

See Also

policyMeanValue gemDualLinearProgramming.

Examples

## See the function gemDualLinearProgramming.
A <- matrix(c(
  0, 0, 0, 1,
  8, 6, 1, 0,
  4, 2, 1.5, 0,
  2, 1.5, 0.5, 0
), 4, 4, TRUE)
B <- matrix(c(
  60, 30, 20, 0,
  0, 0, 0, 0,
  0, 0, 0, 0,
  0, 0, 0, 0
), 4, 4, TRUE)
S0Exg <- {
  S0Exg <- matrix(NA, 4, 4)
  S0Exg[2:4, 4] <- c(48, 20, 8)
  S0Exg
}

ge <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  maxIteration = 1,
  numberOfPeriods = 1000,
  ts = TRUE
)
matplot(ge$ts.q, type = "l")

ge2 <- sdm2(
  A = A, B = B, S0Exg = S0Exg,
  maxIteration = 1,
  numberOfPeriods = 1000,
  ts = TRUE,
  policy = makePolicyMeanValue(150)
)
matplot(ge2$ts.q, type = "l")

Make a Policy of Sticky Price

Description

Given a stickiness value and a time window vector, this function returns a policy function that sets the current prices equal to the weighted mean of the market-clearing prices and the current prices during this time window. When the stickiness value is 0, the prices will be set to the market-clearing prices. When the stickiness value is 1, the current prices will keep unchanged.

Usage

makePolicyStickyPrice(stickiness = 0.5, time.win = c(1, Inf), tolCond = 1e-06)

Arguments

Value

A policy function, which is often used as an argument of the function sdm2.

Note

Three major price adjustment methods can be used in the structural dynamic model. The corresponding three kinds of prices are exploratory prices (the default case), market clearing prices, and sticky prices. The exploratory prices are computed based on the prices and sales rates of the previous period. In economic reality, the market clearing prices are unknown, so exploratory prices are more realistic.

When the stickiness value is positive and the priceAdjustmentVelocity parameter in sdm2 is set to 0 (indicating that current prices remain unchanged from the previous period), implementing the sticky-price policy results in current prices that are the weighted average of the market-clearing prices and the prices from the previous period. Typically, this function should be utilized in this manner.

See Also

sdm2

Examples

InitialEndowments <- {
  tmp <- matrix(0, 3, 2)
  tmp[1, 1] <- 0.01
  tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

ge <- gemCanonicalDynamicMacroeconomic_3_2(
  priceAdjustmentVelocity = 0,
  policy.supply = makePolicySupply(InitialEndowments),
  policy.price = makePolicyStickyPrice(stickiness = 0.5),
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 50
)

par(mfrow = c(1, 2))
matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

Make a Policy of Supply

Description

Given a supply matrix and a time window vector, this function returns a policy function that sets the supply during this time window. By default, the time window of this function is c(1, 1), which means that this function will set the initial supply.

Usage

makePolicySupply(S, time.win = c(1, 1))

Arguments

Value

A policy function, which is often used as an argument of the function sdm2.

See Also

sdm2

Examples

InitialEndowments <- {
  tmp <- matrix(0, 3, 2)
  tmp[1, 1] <- 0.01
  tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

ge <- gemCanonicalDynamicMacroeconomic_3_2(
  policy.supply = makePolicySupply(InitialEndowments),
  policy.price = policyMarketClearingPrice,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 50
)

par(mfrow = c(1, 2))
matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

Make a Policy of Tail Adjustment for a Timeline Model

Description

Make a policy of tail adjustment for a timeline model. Tail adjustment refers to the adjustment of the share coefficient of the last period of the consumer in the timeline model in order to let the model run in a steady-state equilibrium path.

Usage

makePolicyTailAdjustment(ind, gr = 0)

Arguments

Value

A policy, which is often used as an argument of the function sdm2.

See Also

makePolicyHeadTailAdjustment; makePolicyHeadAdjustment

Make a Policy of Technology Change

Description

This function returns a policy function that changes the attributes alpha and a of the demand structure trees of agents specified. An attribute alpha is usually a parameter of a CES or CD function. An attribute a is usually a parameter of a Leontief function. For demand structure trees that do not contain these two attributes, this function has no effect.

Usage

makePolicyTechnologyChange(
  adjumentment.ratio = 1.1,
  agent = 1,
  time.win = c(20, 20)
)

Arguments

Value

A policy function, which is often used as an argument of the function sdm2.

See Also

sdm2

Examples

dst.firm <- node_new("output",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.consumer <- node_new("utility",
  type = "Leontief", a = 1, "prod"
)

B <- matrix(c(
  1, 0,
  0, 0
), 2, 2, TRUE)
S0Exg <- matrix(c(
  NA, NA,
  NA, 100
), 2, 2, TRUE)

ge <- sdm2(
  A = list(dst.firm, dst.consumer), B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  priceAdjustmentVelocity = 0,
  policy = list(
    makePolicyTechnologyChange(agent = "firm"),
    makePolicyStickyPrice(stickiness = 0, time.win = c(1, 20)),
    makePolicyStickyPrice(stickiness = 0.9, time.win = c(20, Inf))
  ),
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 40
)

par(mfrow = c(1, 2))
matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

Marginal Utility

Description

If the argument price is null, this function computes the (delta) marginal utility. By default, delta is set to 1e-10. Otherwise this function computes the (delta) value marginal utility. For a utility function U(x), two vector x, y and a scalar price, the marginal utility is (U(x + delta * y) - U(x)) / delta, and the value marginal utility is (U(x + delta * y / price) - U(x)) / delta. For a marginal utility function M(x), three vector x, y, wt and a scalar price, the marginal utility is sum(M(x) * y * wt), and the value marginal utility is sum(M(x) * y * wt / price).

Usage

marginal_utility(x, y, uf, price = NULL, delta = 1e-10, muf = NULL)

Arguments

Value

An n-by-m marginal utility matrix. Its (i,j)-th element corresponds to the i-th column of y and the j-th column of x.

References

Sharpe, William F. (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

Examples

marginal_utility(1:2, cbind(1:2, 1:1), AMV)
marginal_utility(1:2, cbind(1:2, 1:1), AMV, delta = 100)
marginal_utility(cbind(1:2, 3:4), cbind(1:2, 1:1), AMV)
marginal_utility(
  cbind(1:2, 3:4), cbind(1:2, 1:1),
  list(AMV, function(x) AMV(x, gamma = 0.5))
)

####
wt <- 1:2
uf <- function(x) (x - x^2 / 400) %*% wt
muf <- function(x) (1 - 1 / 200 * x) * wt
marginal_utility(1:2, cbind(1:2, 1:1), uf)
marginal_utility(1:2, cbind(1:2, 1:1), muf = muf)

####
marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x, gamma = 1, p = rep(1, length(x))) CRRA(x, gamma, p)$CE
)
marginal_utility(1:2, cbind(1:2, 1:1), function(x) sqrt(prod(x)))

gamma <- 0.8
wt <- c(0.25, 0.75)
marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x) CRRA(x, gamma = gamma, prob = wt)$CE
)
## the same as above. CRRA and CES utility funcitons are essentially the same.
es <- 1 / gamma
beta <- wt^es
marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x) CES(x = x, sigma = 1 - 1 / es, alpha = 1, beta = wt)
)

prop.table(marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x) CRRA(x, gamma = gamma, prob = wt)$CE
))
prop.table(marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x) CRRA(x, gamma = gamma, prob = wt)$u
))

Add Matrices by Names of Columns and Rows

Description

Add together some matrices by names of columns and rows. Those matrices may have distinct sizes. All matrices should not have column names and row names other than those of the first matrix.

Usage

matrix_add_by_name(M, ...)

Arguments

Value

A matirx.

Examples

M <- matrix(0, 5, 5)
colnames(M) <- paste("c", 1:5, sep = "")
rownames(M) <- paste("r", 1:5, sep = "")

M2 <- matrix(1:9, 3, 3)
colnames(M2) <- c("c2", "c3", "c5")
rownames(M2) <- c("r1", "r2", "r4")

matrix_add_by_name(M, M2)

c1 <- c(r1 = 1, r3 = 2)
matrix_add_by_name(M, c1)
matrix_add_by_name(M, c1, M2)

Aggregate Some Rows and Columns of a Matrix

Description

Aggregate some rows and columns of a matrix to obtain a matrix with smaller dimensions. This function can be used for aggregating some rows and columns of an input-output table.

Usage

matrix_aggregate(
  M,
  row.index = NULL,
  col.index = NULL,
  row.name = NULL,
  col.name = NULL
)

Arguments

Examples

M <- matrix(1:16,4,4,TRUE)
colnames(M) <- paste0("c",1:4)
rownames(M) <- paste0("r",1:4)
addmargins(M)

M2 <- matrix_aggregate(M, list(c(1,3),c(2, 4)), 2:3)
addmargins(M2)

M3 <- matrix_aggregate(M, row.name = list(c("r1","r3"),c("r2","r4")), col.name = c("c2","c3"))
addmargins(M3)

Convert a Matrix into a Demand Structural Tree List

Description

Convert a demand coefficient matrix into a demand structural tree list.

Usage

matrix_to_dstl(
  x,
  names.commodity = paste("comm", 1:nrow(x), sep = ""),
  names.agent = paste("agt", 1:ncol(x), sep = "")
)

Arguments

Value

A demand structural tree list.

Examples

A <- matrix(c(
  0, 0, 0, 1,
  8, 6, 1, 0,
  4, 2, 1.5, 0,
  2, 1.5, 0.5, 0
), 4, 4, TRUE)

dstl <- matrix_to_dstl(A)
node_print(dstl[[1]])

Insert Nodes into a Tree

Description

Scan the tree and insert nodes before the first non-root node having the name specified.
This function is based on the package data.tree and has side-effects. It modifies the tree given by the argument (see the package data.tree).

Usage

node_insert(tree, node.name, ...)

Arguments

Value

Invisibly returns the parent node of those new nodes.

Examples

dst.firm <- node_new(
  "output",
  "prod1",  "prod2"
)
plot(dst.firm)

dst.VA <- node_new(
  "VA",
  "lab",  "cap"
)

node_insert(
  dst.firm, "prod1",
  dst.VA, "prod3"
)
node_set(
  dst.firm, "output",
  "prod4"
)
plot(dst.firm)

Create a Tree

Description

Create a tree by the node_set function and the package data.tree.

As the package data.tree says:

"One of most important things to note about data.tree is that it exhibits reference semantics. In a nutshell, this means that you can modify your tree along the way, without having to reassign it to a variable after each modification. By and large, this is a rather exceptional behavior in R, where value-semantics is king most of the time."

Usage

node_new(root.name, ...)

Arguments

Value

A tree (i.e. a Node object).

Examples

#### create a tree
dst1 <- node_new("firm1")
print(dst1)

## create a tree with children
dst <- node_new(
  "firm",
  "lab", "cap", dst1
)
print(dst)

# the same as above
dst <- node_new(
  "firm",
  c("lab", "cap"), dst1
)
print(dst)

#### create a tree with attributes
dst <- node_new("firm",
  type = "CD", alpha = 1, beta = c(0.5, 0.5)
)
node_print(dst)

#### create a tree with attributes and children
dst <- node_new("firm",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "lab", "cap"
)
node_plot(dst)
node_plot(dst, TRUE)

Plot a Tree and Show the Type Attribute

Description

A wrapper of the function plot.Node of the packages data.tree. If a non-leaf node has a type attribute, then the attribute will be shown.

Usage

node_plot(node, param = FALSE, ...)

Arguments

See Also

demand_coefficient

Print a Tree and Its Fields

Description

A wrapper of the function print.Node of the package data.tree. Print a tree and its fields except the func field.

Usage

node_print(node, ...)

Arguments

Examples

dst <- node_new("firm",
                type = "SCES",
                alpha = 2, beta = c(0.8, 0.2),
                es = 0.5,
                "wheat", "iron"
)

node_print(dst)

####
dst <- node_new("firm",
                type = "FUNC",
                func = min,
                "wheat", "iron"
)

node_print(dst)

Prune Nodes off a Tree by Names

Description

A wrapper of data.tree::Prunes. Prune nodes off a tree by names. This function has side-effects, it modifies the tree given by the argument (see the package data.tree).

Usage

node_prune(tree, ...)

Arguments

Value

Invisibly returns the tree.

Examples

dst <- node_new(
  "firm",
  "lab", "cap", "land"
)
node_prune(
  dst,
  "cap", "land"
)
plot(dst)

Replace a Node of a Tree

Description

Scan the tree and replace the first non-root node having the name specified.
This function is based on the package data.tree and has side-effects. It modifies the tree given by the argument (see the package data.tree).

Usage

node_replace(tree, node.name, ...)

Arguments

Value

Invisibly returns the parent node of those new nodes.

Examples

dst.firm <- node_new(
  "output",
  "prod1", "prod2"
)
plot(dst.firm)

dst.VA <- node_new(
  "VA",
  "lab", "cap"
)

node_replace(
  dst.firm, "prod2",
  dst.VA, "prod3"
)
plot(dst.firm)

node_replace(
  dst.firm, "lab",
  "labor"
)
plot(dst.firm)

node_replace(
  dst.firm, "VA",
  "prod2"
)
plot(dst.firm)

Create a Tree or Set Attributes for a Node

Description

Create a tree or set attributes for a node by the package data.tree. This function can also be used to add child nodes to a node. This function has side-effects, it modifies the tree given by the argument (see the package data.tree).

Usage

node_set(tree, node.name = NA, ...)

Arguments

Value

Invisibly returns the node.

See Also

node_new

Examples

#### create a tree
dst1 <- node_set("firm1")
print(dst1)

## create a tree with children
dst <- node_set(
  "firm", NA,
  "lab", "cap", dst1
)
print(dst)

# the same as above
dst <- node_set(
  "firm", NA,
  c("lab", "cap"), dst1
)
print(dst)

#### create a tree with attributes
dst <- node_set("firm", NA,
  type = "CD", alpha = 1, beta = c(0.5, 0.5)
)
print(dst, "type", "alpha", "beta")

#### create a tree with attributes and children
dst <- node_set("firm", NA,
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "lab", "cap"
)
print(dst, "type", "alpha", "beta")

#### set attributes for a node
dst.firm <- node_set("firm", NA, "VA")
node_set(dst.firm, "VA",
  type = "CD",
  alpha = 0.8^-0.8 * 0.2^-0.2,
  beta = c(0.8, 0.2),
  "lab",
  "cap"
)
print(dst.firm, "alpha", "beta")

## set attributes and add a child for a node
node_set(dst.firm, "VA",
  type = "SCES",
  alpha = 1,
  beta = c(0.1, 0.8, 0.1),
  es = 0,
  "land"
)
print(dst.firm, "type", "alpha", "beta", "es")

## find a node
x <- node_set(dst.firm, "VA")
node_print(x)

Compute the Utility of a Consumer or the Output of a Firm by the Demand Structural Tree

Description

Given a demand structural tree and an input vector, this function computes the utility of a consumer or the output of a firm. If the demand structural tree has a FUNC-type node, the node should has an attribute named fun that is a function computing the output.

Usage

output(node, input)

Arguments

Value

A scalar.

Examples

dst <- node_new("output",
                type = "SCES", es = 0, alpha = 1, beta = c(0.5, 0.5),
                "cc1", "cc2"
)
node_set(dst, "cc1",
         type = "Leontief", a = c(0.6, 0.4),
         "wheat", "iron"
)
node_set(dst, "cc2",
         type = "SCES", sigma = -1, alpha = 1, beta = c(0.5, 0.5),
         "labor", "capital"
)

node_plot(dst, TRUE)

p <- c(wheat = 1, iron = 3, labor = 2, capital = 4)
x <-demand_coefficient(dst, p)
output(dst, x)

output(dst, c(wheat = 3, iron = 3, labor = 3, capital = 3))
SCES(
  es = 0, alpha = 1, beta = c(0.5, 0.5),
  x = c(
    min(3 / 0.6, 3 / 0.4),
    SCES(es = 0.5, alpha = 1, beta = c(0.5, 0.5), x = c(3, 3))
  )
)

Market-Clearing-Price Policy Function

Description

This policy is to make the market clear every period. In this case, the path of the economy is the spot equilibrium path. Generally, this function is passed to the function sdm2 as an argument to compute the spot equilibrium path.

Usage

policyMarketClearingPrice(time, A, state, ...)

Arguments

Value

A list consisting of p, S and B which specify the prices, supplies and supply coefficient matrix after adjustment.

References

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

Grandmont, J.M. (1977). Temporary General Equilibrium Theory. Econometrica 45, 535-572.

See Also

CGE::iep and sdm2, gemTemporaryEquilibriumPath. The market clearing prices are the prices with a stickiness value equal to zero. Therefore, this function can actually be replaced by makePolicyStickyPrice in the calculation.

Examples

#### an iep of the example (see Table 2.1 and 2.2) of the canonical dynamic
#### macroeconomic general equilibrium model in Torres (2016).
ge <- gemCanonicalDynamicMacroeconomic_3_2(
  policy.price = policyMarketClearingPrice,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 50,
  z0 = c(0.5, 1)
)

par(mfrow = c(1, 2))
matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

#### the same as above
ge <- gemCanonicalDynamicMacroeconomic_3_2(
  policy.price = makePolicyStickyPrice(stickiness = 0),
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 50,
  z0 = c(0.5, 1)
)

par(mfrow = c(1, 2))
matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

#### TFP shock in the economy above (see Torres, 2016, section 2.8).
numberOfPeriods <- 200

discount.factor <- 0.97
depreciation.rate <- 0.06
beta1.firm <- 0.35
return.rate <- 1 / discount.factor - 1

set.seed(1)
alpha.shock <- rep(1, 100)
alpha.shock[101] <- exp(0.01)
for (t in 102:numberOfPeriods) {
  alpha.shock[t] <- exp(0.95 * log(alpha.shock[t - 1]))
}

policyTechnologyChange <- function(time, A) {
  A[[1]]$func <- function(p) {
    result <- CD_A(
      alpha.shock[time], rbind(beta1.firm, 1 - beta1.firm, 0),
      c(p[1] * (return.rate + depreciation.rate), p[2:3])
    )
    result[3] <- p[1] * result[1] * return.rate / p[3]
    result
  }
}

InitialEndowments <- {
  tmp <- matrix(0, 3, 2)
  tmp[1, 1] <- tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

ge <- gemCanonicalDynamicMacroeconomic_3_2(
  policy.supply = makePolicySupply(InitialEndowments),
  policy.technology = policyTechnologyChange,
  policy.price = policyMarketClearingPrice,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 200
)

c <- ge$A[1, 2] * ge$ts.z[, 2] # consumption
par(mfrow = c(2, 2))
matplot(ge$ts.z, type = "l")
x <- 100:140
plot(x, ge$ts.z[x, 1] / ge$ts.z[x[1], 1], type = "o", pch = 20)
plot(x, ge$ts.z[x, 2] / ge$ts.z[x[1], 2], type = "o", pch = 20)
plot(x, c[x] / c[x[1]], type = "o", pch = 20)

#### an iep of example 7.2 (a monetary economy) in Li (2019). See CGE::Example7.2.
interest.rate <- 0.25
dst.firm <- node_new("cc", #composite commodity
                     type = "FIN",
                     rate = c(1, interest.rate),
                     "cc1", "money"
)
node_set(dst.firm, "cc1",
         type = "CD", alpha = 1, beta = c(0.5, 0.5),
         "wheat", "labor"
)

dst.laborer <- Clone(dst.firm)
dst.money.lender <- Clone(dst.firm)

dstl <- list(dst.firm, dst.laborer, dst.money.lender)

B <- matrix(0, 3, 3)
B[1, 1] <- 1

S0Exg <- matrix(NA, 3, 3)
S0Exg[2, 2] <- 100
S0Exg[3, 3] <- 100

InitialEndowments <- {
  tmp <- matrix(0, 3, 3)
  tmp[1, 1] <- 10
  tmp[2, 2] <- tmp[3, 3] <- 100
  tmp
}

ge <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("wheat", "labor", "money"),
  names.agent = c("firm", "laborer", "money.lender"),
  numeraire = c(money = interest.rate),
  numberOfPeriods = 20,
  maxIteration = 1,
  ts = TRUE,
  policy = list(
    makePolicySupply(S = InitialEndowments),
    policyMarketClearingPrice
  )
)

par(mfrow = c(1, 2))
matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)

Mean Value Policy Function

Description

When the time index is an integer multiple of 200, this policy sets the current prices and supplies to the averages of the previous 199 periods. This policy function is mainly used as an argument of the function sdm2 in order to accelerate convergence when calculating general equilibrium.

Usage

policyMeanValue(time, state, state.history)

Arguments

Value

A list consisting of p, S and B which specify the prices, supplies and supply coefficient matrix after adjustment.

See Also

makePolicyMeanValue sdm2 gemDualLinearProgramming.

Conversion between a Rate Vector and a Beta Vector

Description

Conversion between an expenditure rate vector and a beta vector (i.e. an expenditure proportion vector). For an economic agent, the rate vector indicates the ratios between expenditures on financial instruments and the physical commodity. The first element of the rate vector indicates the quantity of the physical commodity needed to obtain a unit of output. Other elements indicate the ratio of expenditures on various financial instruments to that of the physical commodity, which may be equal to the interest rate, the tax rate, the dividend rate, etc. The beta vector indicates the proportions of expenditures on various commodities.

Usage

rate_to_beta(x)

beta_to_rate(x)

Arguments

Value

A numeric vector.

Functions

• rate_to_beta(): Convert a rate vector to a beta vector.

• beta_to_rate(): Convert a beta vector to a rate vector. When converting the beta vector into a rate vector, it will be assumed that the first element of these two vectors is the same.

See Also

demand_coefficient

Examples

rate_to_beta(c(1, 1 / 3, 1 / 4))
rate_to_beta(c(0.5, 1 / 3, 1 / 4))

x <- beta_to_rate(c(0.7, 0.1, 0.2))
rate_to_beta(x)

Ratio Adjustment

Description

Adjust ratios to new values.

Usage

ratio_adjust(
  ratio,
  coef = 0.8,
  method = c("log", "left.linear", "trunc.log", "linear")
)

Arguments

Details

For a positive ratio and the following methods, the return values are as follows:

• log : coef * log(ratio) + 1, if ratio >= 1; 1 / (coef * log(1 / ratio) + 1), if ratio < 1.

• left.linear : 1 / (coef * (1 / ratio - 1) + 1), if ratio >= 1; 1 + coef * (ratio - 1), if ratio < 1.

• trunc.log : max(coef * log(ratio) + 1, 0).

• linear : coef * (ratio - 1) + 1.

Value

A vector or a matrix with dimensions the same as the argument ratio.

Examples

ratio_adjust(10, 0.8)
ratio_adjust(0.1, 0.8)

x <- seq(0.01, 2, 0.01)
plot(x, x, type = "l")
lines(x, ratio_adjust(x, 0.8, method = "log"), col = "red")
lines(x, ratio_adjust(x, 0.8, method = "left.linear"), col = "blue")
lines(x, ratio_adjust(x, 0.8, method = "trunc.log"), col = "green")

X <- replicate(3, x)
Y <- ratio_adjust(X, c(0.8, 1, 1.2))
matplot(x, Y, type = "l")

Structural Dynamic Model (alias Structural Growth Model) Version 2

Description

A new version of the sdm function in the package CGE. Now the parameter A can be a demand structure tree list. Hence we actually no longer need the function sdm_dstl. Some rarely used parameters in the function sdm have been deleted. This function is the core of this package.

Usage

sdm2(
  A,
  B,
  S0Exg = matrix(NA, nrow(B), ncol(B)),
  names.commodity = paste("comm", 1:nrow(B), sep = ""),
  names.agent = paste("agt", 1:ncol(B), sep = ""),
  p0 = matrix(1, nrow = nrow(B), ncol = 1),
  z0 = matrix(100, nrow = ncol(B), ncol = 1),
  GRExg = NA,
  pExg = NULL,
  numeraire = NULL,
  tolCond = 1e-05,
  maxIteration = 200,
  numberOfPeriods = 300,
  depreciationCoef = 0.8,
  priceAdjustmentFunction = NULL,
  priceAdjustmentVelocity = 0.15,
  trace = TRUE,
  ts = FALSE,
  policy = NULL,
  exchangeFunction = F_Z
)

Arguments

Details

In each period of the structural dynamic model, the economy runs as follows.
Firstly, the new price vector emerges on the basis of the price vector and sales rates of the previous period, which indicates the current market prices.
Secondly, outputs and depreciated inventories of the previous period constitute the current supplies.
Thirdly, policy functions (if any) are implemented.
Fourthly, the current input coefficient matrix is computed and the supplies are exchanged under market prices. The exchange vector and sales rate vector are obtained. Unsold goods constitute the inventories, which will undergo depreciation and become a portion of the supplies of the next period. The exchange vector determines the current outputs and utility levels.

Value

A list usually containing the following components:

• tolerance - the relative tolerance of the results.

• p - equilibrium prices.

• z - equilibrium purchase levels (i.e. production levels or utility levels).

• S - the equilibrium supply matrix at the initial period.

• growthRate - the endogenous equilibrium growth rate in a pure production economy.

• A - the equilibrium demand coefficient matrix.

• B - the supply coefficient matrix.

• S0Exg - the initial exogenous supply n-by-m matrix.

• D - the demand matrix.

• DV - the demand value matrix.

• SV - the supply value matrix.

• ts.p - the time series of prices in the last outer iteration.

• ts.z - the time series of purchase levels (i.e. production levels or utility levels) in the last outer iteration.

• ts.S - the time series of supply matrix in the last outer iteration.

• ts.q - the time series of sales rates in the last outer iteration.

• policy.data - the policy data.

Note

In the package CGE, the spot equilibrium path (alias instantaneous equilibrium path) is computed by the function iep. The spot equilibrium path includes only spot market transactions, with no intertemporal transactions. In this package, the spot equilibrium path can be computed by the function sdm2 with the parameter policy equal to policyMarketClearingPrice.

The order of implementation of various policies is critical. When a policy list contains a supply policy, a technology (i.e. dstl) policy, a price policy (e.g. a market-clearing-price policy) and a B policy (i.e. a policy adjusting the argument B), both the supply policy and the technology policy should be placed before the price policy, and the B policy should be placed after the price policy. The reason is that the calculation of the current prices may require the use of supply and technology, while the calculation of B may require the use of the current prices.

In general equilibrium models, decreasing returns to scale can be transformed into constant returns to scale; therefore, we usually assume constant returns to scale. The decreasing-returns-to-scale feature of the technology is presumably due to the presence of a fixed factor. This factor is often referred to as "land" and measured in units so that the total amount of land is 1 (Varian, 1992, p. 353).

References

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

LI Wu (2010) A Structural Growth Model and its Applications to Sraffa's System. http://www.iioa.org/conferences/18th/papers/files/104_20100729011_AStructuralGrowthModelanditsApplicationstoSraffasSstem.pdf

Varian, Hal R. (1992, ISBN: 0393957357) Microeconomic Analysis. W. W. Norton & Company.

Examples

dst.firm <- node_new("output",
  type = "Leontief", a = c(0.5, 1),
  "prod", "lab"
)

dst.consumer <- node_new("utility",
  type = "Leontief", a = 1, "prod"
)

dstl <- list(dst.firm, dst.consumer)

B <- matrix(c(
  1, 0,
  0, 0
), 2, 2, TRUE)
S0Exg <- matrix(c(
  NA, NA,
  NA, 100
), 2, 2, TRUE)

## variable dst and technology progress
policy.TP <- function(time, state, A) {
  if (time >= 200) {
    A[[1]]$a <- c(0.5, 0.8)
  } else {
    A[[1]]$a <- c(0.5, 1)
  }
  state
}

ge.TP <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.TP,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000
)
matplot(ge.TP$ts.z, type = "l")
plot(ge.TP$ts.p[, 1] / ge.TP$ts.p[, 2], type = "l")

## variable supply coefficient matrix and technology progress
policy.TP <- function(time, state) {
  if (time >= 200) {
    state$B[1, 1] <- 2
  } else {
    state$B[1, 1] <- 1
  }
  state
}

ge.TP <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.TP,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000
)
matplot(ge.TP$ts.z, type = "l")
plot(ge.TP$ts.p[, 1] / ge.TP$ts.p[, 2], type = "l")

## variable dst and disequilibrium
policy.DE <- function(time, A) {
  if (time >= 200) {
    A[[1]]$a[2] <- A[[1]]$a[2] * 0.999
  } else {
    A[[1]]$a[2] <- 1
  }
}

ge.DE <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.DE,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000
)
matplot(ge.DE$ts.z, type = "l")
plot(ge.DE$ts.p[, 1] / ge.DE$ts.p[, 2], type = "l")


## structural equilibria and structural transition
policy.SE <- function(time, state, A) {
  A[[1]]$a[2] <- structural_function(state$last.z[1], c(105, 125), 1, 0.5)
}

ge.low.level <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.SE,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000,
  z0 = c(100, 0)
)
matplot(ge.low.level$ts.z, type = "l")

ge.high.level <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.SE,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000,
  z0 = c(150, 0)
)
matplot(ge.high.level$ts.z, type = "l")

policy.ST <- function(time, state, A) {
  A[[1]]$a[2] <- structural_function(state$last.z[1], c(105, 125), 1, 0.5)
  if (time >= 200 && time <= 210) state$S[2, 2] <- 125 # Introduce foreign labor.
  state
}

ge.ST <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.ST,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000,
  z0 = c(100, 0)
)
matplot(ge.ST$ts.z, type = "l")

#### economic cycles and an interest rate policy for the firm
dst.firm <- node_new("cc", # composite commodity
  type = "FIN",
  rate = c(1, 0.25),
  "cc1", "money"
)
node_set(dst.firm, "cc1",
  type = "Leontief",
  a = c(0.5, 0.5),
  "wheat", "labor"
)

dst.laborer <- Clone(dst.firm)
dst.money.lender <- Clone(dst.firm)

dstl <- list(dst.firm, dst.laborer, dst.money.lender)

policy.interest.rate <- function(time, state, A, state.history) {
  upsilon <- NA
  if (time >= 600) {
    upsilon <- state.history$z[time - 1, 1] / mean(state.history$z[(time - 50):(time - 1), 1])
    A[[1]]$rate[2] <- max(0.25 + 0.5 * log(upsilon), 0)
  } else {
    A[[1]]$rate[2] <- 0.25
  }

  state$current.policy.data <- c(time, A[[1]]$rate[2], upsilon)
  state
}

B <- matrix(0, 3, 3)
B[1, 1] <- 1

S0Exg <- matrix(NA, 3, 3)
S0Exg[2, 2] <- 100
S0Exg[3, 3] <- 100

de <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("wheat", "labor", "money"),
  names.agent = c("firm", "laborer", "money.lender"),
  p0 = rbind(0.625, 0.375, 0.25),
  z0 = rbind(95, 0, 0),
  priceAdjustmentVelocity = 0.3,
  numberOfPeriods = 1000,
  maxIteration = 1,
  trace = FALSE,
  ts = TRUE
)
matplot(de$ts.z, type = "l")

ge.policy <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("wheat", "labor", "money"),
  names.agent = c("firm", "laborer", "money.lender"),
  p0 = rbind(0.625, 0.375, 0.25),
  z0 = rbind(95, 0, 0),
  priceAdjustmentVelocity = 0.3,
  numberOfPeriods = 1000,
  maxIteration = 1,
  trace = FALSE,
  ts = TRUE,
  policy = policy.interest.rate
)
matplot(ge.policy$ts.z, type = "l")

#### Example 9.3 in Li (2019): fixed-ratio price adjustment method
#### and disequilibrium (business cycles) in a pure production economy
fixedRatioPriceAdjustmentFunction <- function(p, q) {
  thresholdForPriceAdjustment <- 0.99
  priceAdjustmentVelocity <- 0.02
  result <- ifelse(q <= thresholdForPriceAdjustment,
    p * (1 - priceAdjustmentVelocity),
    p
  )
  return(prop.table(result))
}

de.Sraffa <- sdm2(
  A = matrix(c(
    56 / 115, 6,
    12 / 575, 2 / 5
  ), 2, 2, TRUE),
  B = diag(2),
  maxIteration = 1,
  numberOfPeriods = 100,
  p0 = rbind(1 / 15, 1),
  z0 = rbind(575, 20),
  priceAdjustmentFunction = fixedRatioPriceAdjustmentFunction,
  ts = TRUE
)
matplot(growth_rate(de.Sraffa$ts.z), type = "l")

Structural Dynamic Model (alias Structural Growth Model) with a Demand Structure Tree List

Description

This is a wrapper of the function CGE::sdm. The parameter A of CGE::sdm is replaced with a demand structure tree list. This function can be replaced by the more comprehensive function sdm2, so it is not recommended.

Usage

sdm_dstl(dstl, names.commodity, names.agent, ...)

Arguments

Value

A general equilibrium, which is a list with the following elements:

• D - the demand matrix.

• DV - the demand value matrix.

• SV - the supply value matrix.

• ... - some elements returned by the CGE::sdm function

References

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

LI Wu (2010) A Structural Growth Model and its Applications to Sraffa's System. http://www.iioa.org/conferences/18th/papers/files/104_20100729011_AStructuralGrowthModelanditsApplicationstoSraffasSstem.pdf

Manuel Alejandro Cardenete, Ana-Isabel Guerra, Ferran Sancho (2012, ISBN: 9783642247453) Applied General Equilibrium: An Introduction. Springer-Verlag Berlin Heidelberg.

Torres, Jose L. (2016, ISBN: 9781622730452) Introduction to Dynamic Macroeconomic General Equilibrium Models (Second Edition). Vernon Press.

See Also

sdm2

Examples

#### a pure exchange economy with two agents and two commodities
dst.CHN <- node_new("util.CHN",
                    type = "SCES", alpha = 1, beta = c(0.8, 0.2), es = 2,
                    "lab.CHN", "lab.ROW"
)
node_plot(dst.CHN)

dst.ROW <- node_new("util.ROW",
                    type = "SCES", alpha = 1, beta = c(0.05, 0.95), es = 2,
                    "lab.CHN", "lab.ROW"
)

dstl <- list(dst.CHN, dst.ROW)

ge <- sdm_dstl(dstl,
               names.commodity = c("lab.CHN", "lab.ROW"),
               names.agent = c("CHN", "ROW"),
               B = matrix(0, 2, 2, TRUE),
               S0Exg = matrix(c(
                 100, 0,
                 0, 600
               ), 2, 2, TRUE)
)

## supply change
geSC <- sdm_dstl(dstl,
                 names.commodity = c("lab.CHN", "lab.ROW"),
                 names.agent = c("CHN", "ROW"),
                 B = matrix(0, 2, 2, TRUE),
                 S0Exg = matrix(c(
                   200, 0,
                   0, 600
                 ), 2, 2, TRUE)
)

geSC$p / ge$p

## preference change
dst.CHN$beta <- c(0.9, 0.1)
gePC <- sdm_dstl(dstl,
                 names.commodity = c("lab.CHN", "lab.ROW"),
                 names.agent = c("CHN", "ROW"),
                 B = matrix(0, 2, 2, TRUE),
                 S0Exg = matrix(c(
                   100, 0,
                   0, 600
                 ), 2, 2, TRUE)
)

gePC$p / ge$p


#### a pure exchange economy with two agents and four basic commodities
prod.CHN <- node_new("prod.CHN",
                     type = "SCES", alpha = 1, beta = c(0.5, 0.5), es = 0.75,
                     "lab.CHN", "cap.CHN"
)

node_plot(prod.CHN)

prod.ROW <- node_new("prod.ROW",
                     type = "SCES", alpha = 2, beta = c(0.4, 0.6), es = 0.75,
                     "lab.ROW", "cap.ROW"
)

dst.CHN <- node_new("CHN",
                    type = "SCES", alpha = 1, beta = c(0.8, 0.2), es = 2,
                    prod.CHN, prod.ROW
)

node_plot(dst.CHN)
node_print(dst.CHN)
p <- c("lab.CHN" = 1, "cap.CHN" = 1, "lab.ROW" = 1, "cap.ROW" = 1)
demand_coefficient(dst.CHN, p)


dst.ROW <- node_new("ROW",
                    type = "SCES", alpha = 1, beta = c(0.05, 0.95), es = 2,
                    prod.CHN, prod.ROW
)

node_plot(dst.ROW)
node_print(dst.ROW)

dstl <- list(dst.CHN, dst.ROW)

ge <- sdm_dstl(dstl,
               names.commodity = c("lab.CHN", "cap.CHN", "lab.ROW", "cap.ROW"),
               names.agent = c("CHN", "ROW"),
               B = matrix(0, 4, 2, TRUE),
               S0Exg = matrix(c(
                 100, 0,
                 100, 0,
                 0, 600,
                 0, 800
               ), 4, 2, TRUE)
)


## Add currencies to the example above.
prod_money.CHN <- node_new("prod_money.CHN",
                           type = "FIN", rate = c(1, 0.1), # 0.1 is the interest rate.
                           prod.CHN, "money.CHN"
)

prod_money.ROW <- node_new("prod_money.ROW",
                           type = "FIN", rate = c(1, 0.1),
                           prod.ROW, "money.ROW"
)

dst.CHN <- node_new("util.CHN",
                    type = "SCES", alpha = 1, beta = c(0.8, 0.2), es = 2,
                    prod_money.CHN, prod_money.ROW
)

dst.ROW <- node_new("util.ROW",
                    type = "SCES", alpha = 1, beta = c(0.05, 0.95), es = 2,
                    prod_money.CHN, prod_money.ROW
)

dstl <- list(dst.CHN, dst.ROW)

ge <- sdm_dstl(dstl,
               names.commodity = c(
                 "lab.CHN", "cap.CHN", "money.CHN",
                 "lab.ROW", "cap.ROW", "money.ROW"
               ),
               names.agent = c("CHN", "ROW"),
               B = matrix(0, 6, 2, TRUE),
               S0Exg = matrix(c(
                 100, 0,
                 100, 0,
                 100, 0,
                 0, 600,
                 0, 800,
                 0, 100
               ), 6, 2, TRUE)
)

ge$p["money.ROW"] / ge$p["money.CHN"] # the exchange rate


#### Example 7.6 in Li (2019), which illustrates foreign exchange rates.
interest.rate.CHN <- 0.1
interest.rate.ROW <- 0.1

firm.CHN <- node_new("output.CHN",
                     type = "FIN", rate = c(1, interest.rate.CHN),
                     "cc1.CHN", "money.CHN"
)
node_set(firm.CHN, "cc1.CHN",
         type = "CD", alpha = 1, beta = c(0.5, 0.5),
         "lab.CHN", "iron"
)

household.CHN <- node_new("util",
                          type = "FIN", rate = c(1, interest.rate.CHN),
                          "wheat", "money.CHN"
)

moneylender.CHN <- Clone(household.CHN)


firm.ROW <- node_new("output.ROW",
                     type = "FIN", rate = c(1, interest.rate.ROW),
                     "cc1.ROW", "money.ROW"
)
node_set(firm.ROW, "cc1.ROW",
         type = "CD", alpha = 1, beta = c(0.5, 0.5),
         "iron", "lab.ROW"
)

household.ROW <- node_new("util",
                          type = "FIN", rate = c(1, interest.rate.ROW),
                          "wheat", "money.ROW"
)

moneylender.ROW <- Clone(household.ROW)


dstl <- list(
  firm.CHN, household.CHN, moneylender.CHN,
  firm.ROW, household.ROW, moneylender.ROW
)

ge <- sdm_dstl(dstl,
               names.commodity = c(
                 "wheat", "lab.CHN", "money.CHN",
                 "iron", "lab.ROW", "money.ROW"
               ),
               names.agent = c(
                 "firm.CHN", "household.CHN", "moneylender.CHN",
                 "firm.ROW", "household.ROW", "moneylender.ROW"
               ),
               B = {
                 tmp <- matrix(0, 6, 6)
                 tmp[1, 1] <- 1
                 tmp[4, 4] <- 1
                 tmp
               },
               S0Exg = {
                 tmp <- matrix(NA, 6, 6)
                 tmp[2, 2] <- 100
                 tmp[3, 3] <- 600
                 tmp[5, 5] <- 100
                 tmp[6, 6] <- 100
                 tmp
               }
)

ge$p.money <- ge$p
ge$p.money["money.CHN"] <- ge$p["money.CHN"] / interest.rate.CHN
ge$p.money["money.ROW"] <- ge$p["money.ROW"] / interest.rate.ROW
ge$p.money <- ge$p.money / ge$p.money["money.CHN"]

ge$p.money["money.ROW"] / ge$p.money["money.CHN"] # the exchange rate


#### the example (see Table 2.1 and 2.2) of the canonical dynamic
#### macroeconomic general equilibrium model in Torres (2016).
discount.factor <- 0.97
return.rate <- 1 / discount.factor - 1
depreciation.rate <- 0.06

production.firm <- node_new("output",
                            type = "CD", alpha = 1, beta = c(0.65, 0.35),
                            "labor", "capital.goods"
)

household <- node_new("util",
                      type = "CD", alpha = 1, beta = c(0.4, 0.6),
                      "product", "labor"
)

leasing.firm <- node_new("output",
                         type = "FIN", rate = c(1, return.rate),
                         "product", "dividend"
)

dstl <- list(
  production.firm, household, leasing.firm
)

ge <- sdm_dstl(dstl,
               names.commodity = c("product", "labor", "capital.goods", "dividend"),
               names.agent = c("production.firm", "household", "leasing.firm"),
               B = matrix(c(
                 1, 0, 1 - depreciation.rate,
                 0, 1, 0,
                 0, 0, 1,
                 0, 1, 0
               ), 4, 3, TRUE),
               S0Exg = {
                 tmp <- matrix(NA, 4, 3)
                 tmp[2, 2] <- 1
                 tmp[4, 2] <- 1
                 tmp
               },
               priceAdjustmentVelocity = 0.03,
               maxIteration = 1,
               numberOfPeriods = 15000,
               ts = TRUE
)

ge$D # the demand matrix
ge$p / ge$p[1]

plot(ge$ts.z[, 1], type = "l")


#### an example of applied general equilibrium (see section 3.4, Cardenete et al., 2012).
dst.consumer1 <- node_new("util",
                          type = "CD", alpha = 1, beta = c(0.3, 0.7),
                          "prod1", "prod2"
)

dst.consumer2 <- node_new("util",
                          type = "CD", alpha = 1, beta = c(0.6, 0.4),
                          "prod1", "prod2"
)

dst.firm1 <- node_new("output",
                      type = "Leontief", a = c(0.5, 0.2, 0.3),
                      "VA", "prod1", "prod2"
)
node_set(dst.firm1, "VA",
         type = "CD",
         alpha = 0.8^-0.8 * 0.2^-0.2, beta = c(0.8, 0.2),
         "lab", "cap"
)

dst.firm2 <- Clone(dst.firm1)
dst.firm2$a <- c(0.25, 0.5, 0.25)
node_set(dst.firm2, "VA",
         alpha = 0.4^-0.4 * 0.6^-0.6, beta = c(0.4, 0.6)
)

node_print(dst.firm2)

dstl <- list(dst.firm1, dst.firm2, dst.consumer1, dst.consumer2)

ge <- sdm_dstl(dstl,
               names.commodity = c("prod1", "prod2", "lab", "cap"),
               names.agent = c("firm1", "firm2", "consumer1", "consumer2"),
               B = {
                 tmp <- matrix(0, 4, 4)
                 tmp[1, 1] <- 1
                 tmp[2, 2] <- 1
                 tmp
               },
               S0Exg = {
                 tmp <- matrix(NA, 4, 4)
                 tmp[3, 3] <- 30
                 tmp[4, 3] <- 20
                 tmp[3, 4] <- 20
                 tmp[4, 4] <- 5
                 tmp
               }
)

Compute the Return Rate in the Steady State Equilibrium

Description

Compute the (postpaid) return rate in the steady state equilibrium.

Usage

sserr(eis, Gamma.beta, gr = 0, type = "CES", prepaid = FALSE)

Arguments

Examples

sserr(eis = 1, Gamma.beta = 0.97, gr = 0)
sserr(eis = 1, Gamma.beta = 1.25, gr = 0)
sserr(eis = 1, Gamma.beta = 0.97, gr = 0, type = "SCES")

sserr(eis = 0.5, Gamma.beta = 0.97, gr = 0)
sserr(eis = 0.5, Gamma.beta = 0.97, gr = 0, type = "SCES")

Structural Function

Description

A structured function is a kind of kinked (piecewise) function generated by connecting two functions through a transition region. This function calculates the value of a structured function.

Usage

structural_function(theta, transition.interval, f1, f2, ...)

Arguments

Value

The value of the structural function.

Examples

x <- seq(1, 5, 0.1)
y <- c()
for (theta in x) y <- c(y, structural_function(theta, c(2, 3), log, sqrt, theta))
plot(x, y)
lines(x, log(x), col = "blue")
lines(x, sqrt(x), col = "red")

####
f <- function(theta) {
  p <- c(1, 1)
  structural_function(
    theta,
    c(15, 20),
    function(p) CD_A(alpha = 5, Beta = c(0.6, 0.4), p),
    function(p) CD_A(alpha = 15, Beta = c(0.3, 0.7), p),
    p
  )
}

tmp <- sapply(1:25, f)
matplot(t(tmp), type = "l")

Population Variance and Population Standard Deviation

Description

The function var.p computes a population variance. The function sd.p computes a population standard deviation.

Usage

var.p(x, wt = rep(1, length(x)), na.rm = FALSE)

sd.p(x, wt = rep(1, length(x)), na.rm = FALSE)

Arguments

Functions

• var.p(): Population variance.

• sd.p(): Population standard deviation.

Examples

var.p(1:6)

var.p(x = 1:5, wt = 6:10)
var.p(x = rep(1:5, 6:10))

sd.p(x = 1:5, wt = 6:10)
sd.p(x = rep(1:5, 6:10))