gemOLGTimeCircle: A Time-Circle Model (Closed Loop Overlapping Generations Model)

Description

Some examples of a time-circle model, which usually contains 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.

As in the Arrow<U+2013>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

gemOLGTimeCircle(...)

Arguments

...

arguments to be passed to the function sdm2.

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.

Diamond, Peter (1965) National Debt in a Neoclassical Growth Model. American Economic Review. 55 (5): 1126-1150.

Examples

Run this code

# NOT RUN {
#### a time-circle pure exchange economy with three-period-lived consumers
## See gemOLGPureExchange_2_2.
n <- 5 # the number of periods and consumers (i.e. laborers).
lab.age1 <- 1 # labor endowment of age1
lab.age2 <- 1
lab.age3 <- 0
S <- lab.age1 * diag(n) + lab.age2 * diag(n)[, c(2:n, 1)] + lab.age3 * diag(n)[, c(3:n, 1:2)]

index <- c(1:n, 1, 2)
dstl.consumer <- list()
for (k in 1:n) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = c(1 / 6, 2 / 6, 3 / 6),
    paste0("lab", index[k]), paste0("lab", index[k + 1]), paste0("lab", index[k + 2])
  )
}

ge <- sdm2(
  A = dstl.consumer,
  B = matrix(0, n, n, TRUE),
  S0Exg = S,
  names.commodity = paste0("lab", 1:n),
  numeraire = "lab1"
)

ge$p
ge$D

#### Introduce population growth into the above economy.
n <- 5 # the number of periods and consumers.
lab.age1 <- 1
lab.age2 <- 1
lab.age3 <- 0
S <- lab.age1 * diag(n) + lab.age2 * diag(n)[, c(2:n, 1)] + lab.age3 * diag(n)[, c(3:n, 1:2)]

GRExg <- 0.03 # the population growth rate
S <- S %*% diag((1 + GRExg)^(0:(n - 1)))
S[1, n - 1] <- S[1, n - 1] * (1 + GRExg)^-n
S[1:2, n] <- S[1:2, n] * (1 + GRExg)^-n

dstl.consumer <- list()
beta.consumer <- c(1 / 6, 2 / 6, 3 / 6)
for (k in 1:(n - 2)) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = beta.consumer,
    paste0("lab", k), paste0("lab", k + 1), paste0("lab", k + 2)
  )
}

dstl.consumer[[n - 1]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = beta.consumer,
  paste0("lab", n - 1), paste0("lab", n), "cc1"
)
node_set(dstl.consumer[[n - 1]], "cc1",
         type = "Leontief", a = (1 + GRExg)^(-n), # a discounting factor
         "lab1"
)

dstl.consumer[[n]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = beta.consumer,
  paste0("lab", n), "cc1", "cc2"
)
node_set(dstl.consumer[[n]], "cc1",
         type = "Leontief", a = (1 + GRExg)^(-n), # a discounting factor
         "lab1"
)
node_set(dstl.consumer[[n]], "cc2",
         type = "Leontief", a = (1 + GRExg)^(-n), # a discounting factor
         "lab2"
)

ge <- sdm2(
  A = dstl.consumer,
  B = matrix(0, n, n, TRUE),
  S0Exg = S,
  names.commodity = paste0("lab", 1:n),
  numeraire = "lab1"
)

ge$p
diff(log(ge$p)) # -log(1 + GRExg)
ge$D

#### a time-circle model with production and two-period-lived consumers
n <- 5 # the number of periods, consumers and firms.
S <- matrix(NA, 2 * n, 2 * n)

S.lab.consumer <- diag(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)]

dstl.firm <- list()
for (k in 1:n) {
  dstl.firm[[k]] <- node_new(
    "prod",
    type = "CD", alpha = 5,
    beta = c(1 / 3, 2 / 3),
    paste0("lab", k), paste0("prod", k)
  )
}

index <- c(1:n, 1)
dstl.consumer <- list()
for (k in 1:n) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = c(1 / 3, 2 / 3),
    paste0("prod", index[k]), paste0("prod", index[k + 1])
  )
}

ge <- 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"
)

ge$p
ge$D

#### Introduce population growth into the above economy.
n <- 5 # the number of periods, consumers and firms.
GRExg <- 0.03 # the population growth rate
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] <- (1 + GRExg)^(-n)

beta.firm <- c(3 / 5, 2 / 5)
beta.consumer <- c(1 / 3, 2 / 3)

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 = "CD", alpha = 1,
    beta = beta.consumer,
    paste0("prod", k), paste0("prod", k + 1)
  )
}

dstl.consumer[[n]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = beta.consumer,
  paste0("prod", n),"cc1"
)
node_set(dstl.consumer[[n]], "cc1",
         type = "Leontief", a = (1 + GRExg)^(-n), # a discounting factor
         "prod1"
)

ge <- 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
)

ge$p
diff(log(ge$p[1:n]))
diff(log(ge$p[(n + 1):(2 * n)]))
ge$D

#### a time-circle model with production and one-period-lived consumers
## These consumers also can be regarded as infinite-lived carpe diem agents,
## who have endowments in each period, maximize the instantaneous utility and do not
## consider the future.
n <- 5 # the number of periods, consumers and firms.
GRExg <- 0.03 # the population growth rate
discount.factor.last.output <- (1 + GRExg)^(-n)
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] <- discount.factor.last.output

beta.firm <- c(3 / 5, 2 / 5)
beta.consumer <- c(1 / 3, 2 / 3)

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) {
  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 = 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
)

ge$p
diff(log(ge$p[1:n]))
diff(log(ge$p[(n + 1):(2 * n)]))
ge$D

## the reduced 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, 1
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab",
  GRExg = 0.03
)

ge2$p
ge2$D

#### a time-circle model with production and three-period-lived consumers
n <- 6 # the number of periods, consumers and firms.
GRExg <- 0.03 # the population growth rate
S <- matrix(NA, 2 * n, 2 * n)

S.lab.consumer <- rbind(diag((1 + GRExg)^(0:(n - 1)), n), 0) +
  rbind(0, diag((1 + GRExg)^(0:(n - 1)), n))
S.lab.consumer <- S.lab.consumer[1:n, ]
S.lab.consumer[1, 6] <- (1 + GRExg)^-1

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] <- (1 + GRExg)^(-n)

beta.firm <- c(2 / 5, 3 / 5)
beta.consumer <- c(1 / 6, 3 / 6, 2 / 6)

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 - 2)) {
  dstl.consumer[[k]] <- node_new(
    "util",
    type = "CD", alpha = 1,
    beta = beta.consumer,
    paste0("prod", k), paste0("prod", k + 1), paste0("prod", k + 2)
  )
}

dstl.consumer[[n - 1]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = beta.consumer,
  paste0("prod", n - 1), paste0("prod", n), "cc1"
)
node_set(dstl.consumer[[n - 1]], "cc1",
         type = "Leontief", a = (1 + GRExg)^(-n), # a discounting factor
         "prod1"
)

dstl.consumer[[n]] <- node_new(
  "util",
  type = "CD", alpha = 1,
  beta = beta.consumer,
  paste0("prod", n), "cc1", "cc2"
)
node_set(dstl.consumer[[n]], "cc1",
         type = "Leontief", a = (1 + GRExg)^(-n), # a discounting factor
         "prod1"
)
node_set(dstl.consumer[[n]], "cc2",
         type = "Leontief", a = (1 + GRExg)^(-n), # a discounting factor
         "prod2"
)

ge <- 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
)

ge$p
diff(log(ge$p[1:n]))
diff(log(ge$p[(n + 1):(2 * n)]))
ge$D
# }
# NOT RUN {
# }

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Description

Usage

Arguments

References

See Also

Examples