gemIntertemporal_EndogenousEquilibriumInterestRate {GE}

An Example Illustrating Endogenous Equilibrium Interest Rates in a (Timeline) Transitional Equilibrium Path

Description

This example illustrates (endogenous) equilibrium primitive interest rates in a transitional equilibrium path, which is an intertemporal path distinct from a steady-state equilibrium. Assume that the velocity of money is equal to one, that is, money circulates once per period.

The interest rate calculated here is adjusted from the nominal interest rate based on the growth rate of the money supply, which is equal to the nominal interest rate when the money stock remains unchanged. We refer to this kind of interest rate as the primitive interest rate, which usually differs from the real interest rate obtained by adjusting the nominal rate based on the inflation rate.

There are three types of economic agents in the model: 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.

Usage

gemIntertemporal_EndogenousEquilibriumInterestRate(...)

Arguments

...

arguments to be passed to the function sdm2.

Examples

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 = 10, # 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.exchange.value <- tmp[paste0("prod", 1:(np - 1))] + tmp[paste0("lab", 1:(np - 1))]
  ir.new <- ts.exchange.value[1:(np - 2)] / ts.exchange.value[2:(np - 1)] - 1
  ir.new <- pmax(1e-6, ir.new)
  ir.new[np - 1] <- ir.new[np - 2]

  ir <- c(ir * ratio_adjust(ir.new / ir, 0.3))
  cat("ir: ", ir, "\n")

  if (return.ge) {
    ge$ts.exchange.value <- ts.exchange.value
    return(ge)
  } else {
    return(ir)
  }
}

## Calculate equilibrium interest rates.
## Warning: Running the program below takes about 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 <- rep(0.25, np - 1)
ir[1:14] <- c(0.4301, 0.3443, 0.3007, 0.2776, 0.2652, 0.2584, 0.2546,
  0.2526, 0.2514, 0.2508, 0.2504, 0.2502, 0.2501, 0.2501)

ge <- f(ir, TRUE)

plot(ge$z[1:(np - 1)], type = "o")
ge$ts.exchange.value[1:(np - 2)] / ge$ts.exchange.value[2:(np - 1)] - 1
ir

## Calculate the growth rate of the money supply and the equilibrium nominal
## interest rate when the current price of the product remains constant.
price.money <- 1 / c(1, cumprod(ir + 1))
currentPrice.prod <- ge$p[1:np] / price.money
gr.moneySupply <- unname(growth_rate(1 / currentPrice.prod))
(ir + 1) * (gr.moneySupply[2:np] + 1) - 1

## the corresponding sequential model with the same steady-state equilibrium.
np <- 5
ge.ss <- f(return.ge = TRUE, y1 = 128)

dividend.rate <- ir <- sserr(eis, Gamma.beta, prepaid = TRUE)

dst.firm <- node_new("prod",
                     type = "FIN", rate = c(1, ir, (1 + ir) * dividend.rate),
                     "cc1", "money", "equity.share"
)
node_set(dst.firm, "cc1",
         type = "CD",
         alpha = 2, beta = c(0.5, 0.5),
         "prod", "lab"
)

dst.laborer <- node_new("util",
                        type = "FIN", rate = c(1, ir),
                        "prod", "money"
)

dst.moneyOwner <- node_new("util",
                           type = "FIN", rate = c(1, ir),
                           "prod", "money"
)

ge2 <- sdm2(
  A = list(dst.firm, dst.laborer, dst.moneyOwner),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 4, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100,
    NA, 100, NA
  ), 4, 3, TRUE),
  names.commodity = c(
    "prod", "lab", "money", "equity.share"
  ),
  names.agent = c("firm", "laborer", "moneyOwner"),
  numeraire = "prod"
)

ge2$p
ge.ss$z[np - 1]
ge2$z
ge.ss$D[paste0("prod", np - 1), c("laborer", "moneyOwner")]
ge2$D

## a technology shock.
## Warning: Running the program below takes about several minutes.
# np <- 50
# f2 <- function(x) {
#  f(
#     ir = x, return.ge = FALSE,
#     y1 = 128, alpha.firm = {
#       tmp <- rep(2, np - 1)
#       tmp[25] <- 1.8
#       tmp
#     }
#   )
# }
#
# mat.ir <- iterate(rep(0.25, np - 1), f2, tol = 1e-4)
# tail(mat.ir, 1) # the equilibrium interest rates

## Calculate equilibrium interest rates.
## Warning: Running the program below takes about several minutes.
# np <- 20
# gr <- 0.03
# mat.ir <- iterate(rep(0.1, np - 1), f, tol = 1e-4)
# sserr(eis, Gamma.beta, gr, prepaid = TRUE)