# \donttest{
#### There are two types of economic agents in this example: firms and a consumer.
# Suppose the consumer needs to use money to buy products,
# and firms need to use money to buy labor.
set.seed(123)
eis <- 1 # the elasticity of intertemporal substitution
np <- 3 # the number of economic periods
alpha <- runif(np, 1, 3)
beta <- runif(np, 0.9, 1) |>
cumprod() |>
proportions()
f <- function(ir = rep(0.1, np), return.ge = FALSE) {
ir[np] <- 1e6
n <- 3 * np # the number of commodity kinds
m <- np + 1 # the number of agent kinds
names.commodity <- c(
paste0("prod", 1:np),
paste0("lab", 1:np),
paste0("money", 1:np)
)
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
S0Exg[paste0("money", 1:np), "consumer"] <- 1
# the output coefficient matrix.
B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent))
for (k in 1:np) {
B[paste0("prod", k), paste0("firm", k)] <- 1
}
dstl.firm <- list()
for (k in 1:np) {
dstl.firm[[k]] <- node_new(
"prod",
type = "FIN", rate = c(1, ir[k]),
"cc1", paste0("money", k)
)
node_set(dstl.firm[[k]], "cc1",
type = "Leontief", a = 1 / alpha[k],
paste0("lab", k)
)
}
dst.consumer <- node_new(
"util",
type = "SCES", es = eis,
alpha = 1, beta = beta,
paste0("cc", 1:np)
)
for (k in 1:np) {
node_set(dst.consumer, paste0("cc", k),
type = "FIN", rate = c(1, ir[k]),
paste0("prod", k), paste0("money", k)
)
}
ge <- sdm2(
A = c(
dstl.firm, dst.consumer
),
B = B,
S0Exg = S0Exg,
names.commodity = names.commodity,
names.agent = names.agent,
numeraire = "prod1",
)
tmp <- rowSums(ge$SV)
ts.trading.value <- tmp[paste0("prod", 1:np)] + tmp[paste0("lab", 1:np)] +
tmp[paste0("money", 1:np)]
ir.new <- ts.trading.value[1:(np - 1)] / ts.trading.value[2:np] - 1
ir.new <- pmax(1e-6, ir.new)
ir.new[np] <- 1e6
ir <- ir.new
cat("ir: ", ir, "\n")
if (return.ge) {
ge$ts.trading.value <- unname(ts.trading.value)
return(ge)
} else {
return(ir)
}
}
mat.ir <- iterate(rep(0.1, np), f, tol = 1e-3)
# When eis equals 1 and np equals 3, compute the
# interest rates using the closed-form formulas.
compute_ir <- function(beta) {
b1 <- beta[1]
b2 <- beta[2]
b3 <- beta[3]
A <- sqrt(b2^2 + 4 * b2 * b3)
B <- sqrt(b1^2 + 2 * b1 * (b2 + A))
r1 <- (b1 - b2 - A + B) / (b2 + A)
r2 <- (b2 - 2 * b3 + A) / (2 * b3)
c(r1 = r1, r2 = r2)
}
compute_ir(beta)
ge <- f(tail(mat.ir, 1), return.ge = TRUE)
ge$p
#### There are three types of economic agents in this example: firms, a laborer, and a money owner.
# Suppose the laborer and the money owner need to use money to buy products,
# and firms need to use money to buy products and labor.
# Formally, the money owner borrows money from himself and pays interest to himself.
eis <- 0.8 # the elasticity of intertemporal substitution
Gamma.beta <- 0.8 # the subjective discount factor
gr <- 0 # the steady-state growth rate
np <- 20 # the number of economic periods
f <- function(ir = rep(0.25, np - 1), return.ge = FALSE,
y1 = 20, # the product supply in the first period
alpha.firm = rep(2, np - 1) # the efficiency parameters of firms
) {
n <- 2 * np # the number of commodity kinds
m <- np + 1 # the number of agent kinds
names.commodity <- c(
paste0("prod", 1:np),
paste0("lab", 1:(np - 1)),
"money"
)
names.agent <- c(
paste0("firm", 1:(np - 1)),
"laborer", "moneyOwner"
)
# the exogenous supply matrix.
S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 * (1 + gr)^(0:(np - 2))
S0Exg["money", "moneyOwner"] <- 100
S0Exg["prod1", "laborer"] <- y1
# 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, ir[k]),
"cc1", "money"
)
node_set(dstl.firm[[k]], "cc1",
type = "CD", alpha = alpha.firm[k], beta = c(0.5, 0.5),
paste0("prod", k), paste0("lab", k)
)
}
dst.laborer <- node_new(
"util",
type = "CES", es = eis,
alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
paste0("cc", 1:(np - 1)), paste0("prod", np)
)
for (k in 1:(np - 1)) {
node_set(dst.laborer, paste0("cc", k),
type = "FIN", rate = c(1, ir[k]),
paste0("prod", k), "money"
)
}
dst.moneyOwner <- node_new(
"util",
type = "CES", es = eis,
alpha = 1, beta = prop.table(Gamma.beta^(1:(np - 1))),
paste0("cc", 1:(np - 1))
)
for (k in 1:(np - 1)) {
node_set(dst.moneyOwner, paste0("cc", k),
type = "FIN", rate = c(1, ir[k]),
paste0("prod", k), "money"
)
}
ge <- sdm2(
A = c(dstl.firm, dst.laborer, dst.moneyOwner),
B = B,
S0Exg = S0Exg,
names.commodity = names.commodity,
names.agent = names.agent,
numeraire = "prod1",
policy = makePolicyHeadTailAdjustment(gr = gr, np = np, type = c("tail"))
)
tmp <- rowSums(ge$SV)
ts.trading.value <- (tmp[paste0("prod", 1:(np - 1))] + tmp[paste0("lab", 1:(np - 1))]) * (1 + ir)
ir.new <- ts.trading.value[1:(np - 2)] / ts.trading.value[2:(np - 1)] - 1
ir.new <- pmax(1e-6, ir.new)
ir.new[np - 1] <- ir.new[np - 2]
ir <- ir.new
cat("ir: ", ir, "\n")
if (return.ge) {
ge$ts.trading.value <- ts.trading.value
return(ge)
} else {
return(ir)
}
}
## Calculate equilibrium interest rates.
## Warning: Running the program below may take several minutes.
# mat.ir <- iterate(rep(0.1, np - 1), f, tol = 1e-4)
# sserr(eis, Gamma.beta, gr, prepaid = TRUE)
## Below are the calculated equilibrium interest rates.
ir <- c(0.4297, 0.3449, 0.3014, 0.2782, 0.2656, 0.2587, 0.2548, 0.2527,
0.2515, 0.2508, 0.2505, 0.2503, 0.2501, 0.2501, 0.2500, 0.2500,
0.2500, 0.2500, 0.2500)
ge <- f(ir, TRUE)
plot(ge$z[1:(np - 1)], type = "o")
ge$ts.trading.value[1:(np - 2)] / ge$ts.trading.value[2:(np - 1)] - 1
ir
# }
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