gemIntertemporal_TimeCircle_2_2 {GE}

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

...

arguments to be passed to the function sdm2.

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)])