# \donttest{
np <- 15 # the number of economic periods
alpha.firm.wheat <- rep(5, np - 1)
alpha.firm.iron <- rep(5, np - 1)
Gamma.beta <- 0.97 # 1, 1.03 # the subjective discount factor of consumers
eis <- 0.5 # the elasticity of intertemporal substitution of consumers
y1.wheat <- 100 # 126, 129.96
y1.iron <- 30 # 40.59, 43.47
gr <- 0 # the growth rate in the steady state equilibrium
last.beta.laborer <- 0
last.beta.landowner <- 0
f <- function(policy = NULL) {
n <- 4 * np - 2 # the number of commodity kinds
m <- 2 * np # the number of agent kinds
names.commodity <- c(
paste0("wheat", 1:np),
paste0("iron", 1:np),
paste0("lab", 1:(np - 1)),
paste0("land", 1:(np - 1))
)
names.agent <- c(
paste0("firm.wheat", 1:(np - 1)), paste0("firm.iron", 1:(np - 1)),
"laborer", "landowner"
)
# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg["wheat1", "laborer"] <- y1.wheat
S0Exg["iron1", "landowner"] <- y1.iron
S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 * (1 + gr)^(0:(np - 2)) # the supply of labor
S0Exg[paste0("land", 1:(np - 1)), "landowner"] <- 100 * (1 + gr)^(0:(np - 2)) # the supply of land
# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:(np - 1)) {
B[paste0("wheat", k + 1), paste0("firm.wheat", k)] <- 1
B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
}
dstl.firm.wheat <- dstl.firm.iron <- list()
for (k in 1:(np - 1)) {
dstl.firm.wheat[[k]] <- node_new(
"prod",
type = "CES", es = 0.8,
alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
paste0("iron", k), paste0("lab", k), paste0("land", k)
)
dstl.firm.iron[[k]] <- node_new(
"prod",
type = "CES", es = 0.8,
alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2),
paste0("iron", k), paste0("lab", k), paste0("land", k)
)
}
tmp.beta <- Gamma.beta^(1:(np - 1))
tmp.beta <- tmp.beta / tmp.beta[np - 1]
tmp.beta <- c(tmp.beta, last.beta.laborer)
dst.laborer <- node_new(
"util",
type = "CES", es = eis,
alpha = 1, beta = prop.table(tmp.beta),
paste0("cc", 1:(np - 1)), paste0("wheat", np)
)
for (k in 1:(np - 1)) {
node_set(dst.laborer, paste0("cc", k),
type = "CES", es = 1,
alpha = 1, beta = c(0.4, 0.4, 0.2),
paste0("wheat", k), paste0("lab", k), paste0("land", k)
)
}
tmp.beta <- Gamma.beta^(1:(np - 1))
tmp.beta <- tmp.beta / tmp.beta[np - 1]
tmp.beta <- c(tmp.beta, last.beta.landowner)
dst.landowner <- node_new(
"util",
type = "CES", es = eis,
alpha = 1, beta = prop.table(tmp.beta),
paste0("cc", 1:(np - 1)), paste0("iron", np)
)
for (k in 1:(np - 1)) {
node_set(dst.landowner, paste0("cc", k),
type = "CES", es = 1,
alpha = 1, beta = c(0.2, 0.4, 0.4),
paste0("wheat", k), paste0("lab", k), paste0("land", k)
)
}
ge <- sdm2(
A = c(dstl.firm.wheat, dstl.firm.iron, Clone(dst.laborer), Clone(dst.landowner)),
B = B,
S0Exg = S0Exg,
names.commodity = names.commodity,
names.agent = names.agent,
numeraire = "lab1",
policy = policy,
ts = TRUE,
maxIteration = 1,
numberOfPeriods = 1000,
priceAdjustmentVelocity = 0.05
)
plot(ge$z[1:(np - 1)],
type = "o", pch = 20, ylab = "production level",
xlab = "time", ylim = range(ge$z[1:(2 * np - 2)])
)
lines(ge$z[np:(2 * np - 2)], type = "o", pch = 21)
legend("bottom", c("wheat", "iron"), pch = 20:21)
invisible(ge)
}
ge <- f()
plot(2:(np - 1), ge$z[1:(np - 2)],
type = "o", pch = 20, ylab = "production output",
xlab = "time", ylim = range(ge$z[1:(2 * np - 2)])
)
lines(2:(np - 1), ge$z[np:(2 * np - 3)], type = "o", pch = 21)
legend("bottom", c("wheat", "iron"), pch = 20:21)
## Compute the steady-state equilibrium based on head and tail adjustments.
policyHeadAdjustment <- function(time, state) {
m <- 2 * np # the number of agent kinds
if (time > 100) {
state$S[1, m - 1] <- state$last.z[round(np / 2)] / (1 + gr)^(round(np / 2))
state$S[np + 1, m] <- state$last.z[np - 1 + round(np / 2)] / (1 + gr)^(round(np / 2))
}
state
}
policyTailAdjustment <- function(A, state) {
m <- 2 * np # the number of agent kinds
# wheat
ratio.output.tail <- state$last.z[np - 1] / (state$last.z[np - 2] * (1 + gr))
tmp.node <- A[[m - 1]]
tmp.n <- length(tmp.node$beta)
tail.beta <- tmp.node$beta[tmp.n]
if (tail.beta == 0) tail.beta <- 1 / tmp.n
tail.beta <- tail.beta / ratio.output.tail
tmp.node$beta <- prop.table(c(tmp.node$beta[1:(tmp.n - 1)], tail.beta))
# iron
ratio.output.tail <- state$last.z[2 * np - 2] / (state$last.z[2 * np - 3] * (1 + gr))
tmp.node <- A[[m]]
tmp.n <- length(tmp.node$beta)
tail.beta <- tmp.node$beta[tmp.n]
if (tail.beta == 0) tail.beta <- 1 / tmp.n
tail.beta <- tail.beta / ratio.output.tail
tmp.node$beta <- prop.table(c(tmp.node$beta[1:(tmp.n - 1)], tail.beta))
}
f(list(policyHeadAdjustment, policyTailAdjustment))$z
## the corresponding sequential model with the same steady-state equilibrium.
dividend.rate <- sserr(eis, Gamma.beta, prepaid = TRUE)
dst.firm.wheat <- node_new("prod",
type = "FIN", rate = c(1, dividend.rate),
"cc1", "equity.share.wheat"
)
node_set(dst.firm.wheat, "cc1",
type = "CES", es = 0.8,
alpha = 5, beta = c(0.2, 0.4, 0.4),
"iron", "lab", "land"
)
dst.firm.iron <- node_new("prod",
type = "FIN", rate = c(1, dividend.rate),
"cc1", "equity.share.iron"
)
node_set(dst.firm.iron, "cc1",
type = "CES", es = 0.8,
alpha = 5, beta = c(0.4, 0.4, 0.2),
"iron", "lab", "land"
)
dst.laborer <- node_new("util",
type = "CES", es = 1,
alpha = 1, beta = c(0.4, 0.4, 0.2),
"wheat", "lab", "land"
)
dst.landowner <- node_new("util",
type = "CES", es = 1,
alpha = 1, beta = c(0.2, 0.4, 0.4),
"wheat", "lab", "land"
)
ge <- sdm2(
A = list(dst.firm.wheat, dst.firm.iron, dst.laborer, dst.landowner),
B = matrix(c(
1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 0, 0,
0, 0, 0, 0,
0, 0, 0, 0,
0, 0, 0, 0
), 6, 4, TRUE),
S0Exg = matrix(c(
NA, NA, NA, NA,
NA, NA, NA, NA,
NA, NA, 100, NA,
NA, NA, NA, 100,
NA, NA, 100, NA,
NA, NA, NA, 100
), 6, 4, TRUE),
names.commodity = c(
"wheat", "iron", "lab", "land",
"equity.share.wheat", "equity.share.iron"
),
names.agent = c("firm.wheat", "firm.iron", "laborer", "landowner"),
numeraire = "lab"
)
ge$p
ge$z
ge$D
ge$S
# f(policyHeadAdjustment)
# f(policyTailAdjustment)
## an anticipated technological shock
# np <- 50 # the number of economic periods
# alpha.firm.wheat <- rep(5, np - 1)
# alpha.firm.iron <- rep(5, np - 1)
# alpha.firm.iron[25] <- 10
# ge <- f()
# plot(2:(np - 1), ge$z[1:(np - 2)],
# type = "o", pch = 20, ylab = "production output",
# xlab = "time", ylim = range(ge$z[1:(2 * np - 2)])
# )
# lines(2:(np - 1), ge$z[np:(2 * np - 3)], type = "o", pch = 21)
# legend("bottom", c("wheat", "iron"), pch = 20:21)
# grid()
# #### a structural transformation path
# np <- 50
# tax.rate <- 0.1 # the tax rate imposed on income from land and labor income.
# tax.time <- 1
#
# alpha.firm.wheat <- rep(5, np - 1)
# # Suppose the tax rate is high enough so that the iron
# # producer's efficiency coefficient immediately rises to 10.
# alpha.firm.iron <- c()
# for (k in 1:(np - 1)) {
# alpha.firm.iron[k] <- ifelse(k <= tax.time, 5, 10)
# }
#
# Gamma.beta <- 0.97 # 1, 1.03 # the subjective discount factor of consumers
# eis <- 0.5 # the elasticity of intertemporal substitution of consumers
# y1.wheat <- 100
# y1.iron <- 30
# last.beta.laborer <- 0
# last.beta.landowner <- 0
#
# n <- 4 * np - 2 # the number of commodity kinds
# m <- 2 * np # the number of agent kinds
#
# names.commodity <- c(
# paste0("wheat", 1:np),
# paste0("iron", 1:np),
# paste0("lab", 1:(np - 1)),
# paste0("land", 1:(np - 1))
# )
#
# names.agent <- c(
# paste0("firm.wheat", 1:(np - 1)), paste0("firm.iron", 1:(np - 1)),
# "laborer", "landowner"
# )
#
# # the exogenous supply matrix.
# S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
# S0Exg["wheat1", "laborer"] <- y1.wheat
# S0Exg["iron1", "landowner"] <- y1.iron
# S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 # the supply of labor
# S0Exg[paste0("land", 1:(np - 1)), "landowner"] <- 100 # the supply of land
#
# S0Exg[paste0("lab", tax.time), paste0("firm.iron", tax.time)] <-
# S0Exg[paste0("lab", tax.time), "laborer"] * tax.rate
# S0Exg[paste0("land", tax.time), paste0("firm.iron", tax.time)] <-
# S0Exg[paste0("land", tax.time), "landowner"] * tax.rate
#
# S0Exg[paste0("lab", tax.time), "laborer"] <-
# S0Exg[paste0("lab", tax.time), "laborer"] * (1 - tax.rate)
# S0Exg[paste0("land", tax.time), "landowner"] <-
# S0Exg[paste0("land", tax.time), "landowner"] * (1 - tax.rate)
#
# # the output coefficient matrix.
# B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
# for (k in 1:(np - 1)) {
# B[paste0("wheat", k + 1), paste0("firm.wheat", k)] <- 1
# B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1
# }
#
# dstl.firm.wheat <- dstl.firm.iron <- list()
# for (k in 1:(np - 1)) {
# dstl.firm.wheat[[k]] <- node_new(
# "prod",
# type = "CES", es = 0.8,
# alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
# paste0("iron", k), paste0("lab", k), paste0("land", k)
# )
#
# dstl.firm.iron[[k]] <- node_new(
# "prod",
# type = "CES", es = 0.8,
# alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2),
# paste0("iron", k), paste0("lab", k), paste0("land", k)
# )
# }
#
# tmp.beta <- Gamma.beta^(1:(np - 1))
# tmp.beta <- tmp.beta / tmp.beta[np - 1]
# tmp.beta <- c(tmp.beta, last.beta.laborer)
# dst.laborer <- node_new(
# "util",
# type = "CES", es = eis,
# alpha = 1, beta = prop.table(tmp.beta),
# paste0("cc", 1:(np - 1)), paste0("wheat", np)
# )
# for (k in 1:(np - 1)) {
# node_set(dst.laborer, paste0("cc", k),
# type = "CES", es = 1,
# alpha = 1, beta = c(0.4, 0.4, 0.2),
# paste0("wheat", k), paste0("lab", k), paste0("land", k)
# )
# }
#
# tmp.beta <- Gamma.beta^(1:(np - 1))
# tmp.beta <- tmp.beta / tmp.beta[np - 1]
# tmp.beta <- c(tmp.beta, last.beta.landowner)
# dst.landowner <- node_new(
# "util",
# type = "CES", es = eis,
# alpha = 1, beta = prop.table(tmp.beta),
# paste0("cc", 1:(np - 1)), paste0("iron", np)
# )
# for (k in 1:(np - 1)) {
# node_set(dst.landowner, paste0("cc", k),
# type = "CES", es = 1,
# alpha = 1, beta = c(0.2, 0.4, 0.4),
# paste0("wheat", k), paste0("lab", k), paste0("land", k)
# )
# }
# ge <- sdm2(
# A = c(dstl.firm.wheat, dstl.firm.iron, Clone(dst.laborer), Clone(dst.landowner)),
# B = B,
# S0Exg = S0Exg,
# names.commodity = names.commodity,
# names.agent = names.agent,
# numeraire = "lab1",
# ts = TRUE,
# maxIteration = 1,
# numberOfPeriods = 1000,
# priceAdjustmentVelocity = 0.05
# )
#
# plot(2:(np - 1), ge$z[1:(np - 2)],
# type = "o", pch = 20, ylab = "production output",
# xlab = "time", ylim = range(ge$z[1:(2 * np - 2)])
# )
# lines(2:(np - 1), ge$z[np:(2 * np - 3)], type = "o", pch = 21)
# legend("bottom", c("wheat", "iron"), pch = 20:21)
# }
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