gemOLG_Basic: Some Examples of Basic (Timeline) OLG Models with Production

In GE: General Equilibrium Modeling

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 consumers and firms, and to allow consumers to live for more periods.

Usage

gemOLG_Basic(...)

Arguments

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

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(100, 100, 0)
labor.last <- c(100, 100)
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)]

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