R/gemIntertemporal_4_4.R

In GE: General Equilibrium Modeling

#' @export
#' @title An Intertemporal Model with Land, Two Consumers and Two Types of Firms
#' @aliases gemIntertemporal_4_4
#' @description An (intertemporal) timeline model with two consumers (i.e. a laborer and a landowner) and two types of firms
#' (i.e. wheat producers and iron producers).
#' There are four commodities in the model, namely wheat, iron, labor and land.
#' @param ... arguments to be passed to the function sdm2.
#' @examples
#' \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
#'
#' 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", 1:(np - 1), ".wheat"), paste0("firm", 1:(np - 1), ".iron"),
#'   "laborer", "landowner"
#' )
#'
#' f <- function(policy = NULL) {
#'   n <- length(names.commodity) # the number of commodity kinds
#'   m <- length(names.agent) # the number of agent kinds
#'
#'   # 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", k, ".wheat")] <- 1
#'     B[paste0("iron", k + 1), paste0("firm", k, ".iron")] <- 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 <- makePolicyHeadAdjustment(
#'   ind = rbind(
#'     c(
#'       which(names.commodity == "wheat1"), which(names.agent == "laborer"),
#'       which(names.commodity == "wheat2"), which(names.agent == "firm1.wheat")
#'     ),
#'     c(
#'       which(names.commodity == "iron1"), which(names.agent == "landowner"),
#'       which(names.commodity == "iron2"), which(names.agent == "firm1.iron")
#'     )
#'   ),
#'   gr = gr
#' )
#' policyTailAdjustment <- makePolicyTailAdjustment(
#'   ind = rbind(
#'     c(which(names.agent == paste0("firm", np - 1, ".wheat")), which(names.agent == "laborer")),
#'     c(which(names.agent == paste0("firm", np - 1, ".iron")), which(names.agent == "landowner"))
#'   ),
#'   gr = gr
#' )
#'
#' 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$S
#'
#' # f(policyTailAdjustment)
#'
#' ## an anticipated technological shock
#' # np <- 50 # the number of economic periods
#' # alpha.firm.wheat <- rep(5, np - 1)
#' # alpha.firm.iron <- rep(5, np - 1)
#' # alpha.firm.iron[25] <- 10
#' # names.commodity <- c(
#' #   paste0("wheat", 1:np),
#' #   paste0("iron", 1:np),
#' #   paste0("lab", 1:(np - 1)),
#' #   paste0("land", 1:(np - 1))
#' # )
#' # names.agent <- c(
#' #   paste0("firm", 1:(np - 1), ".wheat"), paste0("firm", 1:(np - 1), ".iron"),
#' #   "laborer", "landowner"
#' # )
#' #
#' # ge <- f()
#' # plot(2:(np - 1), ge$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 # tax.time <- 20
#' #
#' # 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
#' #
#' # 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", 1:(np - 1), ".wheat"), paste0("firm", 1:(np - 1), ".iron"),
#' #   "laborer", "landowner"
#' # )
#' #
#' # n <- length(names.commodity) # the number of commodity kinds
#' # m <- length(names.agent) # the number of agent kinds
#' #
#' # # 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", tax.time, ".iron")] <-
#' #   S0Exg[paste0("lab", tax.time), "laborer"] * tax.rate
#' # S0Exg[paste0("land", tax.time), paste0("firm", tax.time, ".iron")] <-
#' #   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", k, ".wheat")] <- 1
#' #   B[paste0("iron", k + 1), paste0("firm", k, ".iron")] <- 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)
#' }

gemIntertemporal_4_4 <- function(...) sdm2(...)