R/gemIntertemporal_EndogenousEquilibriumInterestRate.R

In GE: General Equilibrium Modeling

#' @export
#' @title Some Examples Illustrating Endogenous Equilibrium Interest Rates in (Timeline) Transitional Equilibrium Paths
#' @aliases gemIntertemporal_EndogenousEquilibriumInterestRate
#' @description These examples illustrate (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.
#'
#' @param ... arguments to be passed to the function sdm2.
#' @examples
#' \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
#' }

gemIntertemporal_EndogenousEquilibriumInterestRate <- function(...) sdm2(...)