gemIntertemporal_Dividend_TechnologicalProgress: The Identical Steady-state Equilibrium: Four Models...

In GE: General Equilibrium Modeling

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

...	arguments to be passed to the function sdm2.

See Also

gemIntertemporal_Dividend

Examples

#### (1) a real timeline model with head-tail adjustment.
eis <- 0.8 # the elasticity of intertemporal substitution
rho.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(rho.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, rho.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 <- (1 + gr)^(1 / eis - 1) / rho.beta - 1

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),
    "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(rho.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),
                     "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),
                             "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])

GE documentation built on Nov. 8, 2023, 9:07 a.m.