gemIntertemporal_Money_Dividend_Example7.5.1 {GE}

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

...

arguments to be passed to the function sdm2.

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