R/gemIntertemporal_5_5.R

In GE: General Equilibrium Modeling

#' @export
#' @title Some Intertemporal (Timeline and Time-circle) Models with Land, Two Consumers, and Three Types of Firms
#' @aliases gemIntertemporal_5_5
#' @description Some intertemporal (timeline and time-circle) models with two consumers (i.e. a laborer and a landowner) and three types of firms
#' (i.e. wheat producers, iron producers and iron leaser).
#' Here the iron leasing firm is actually a quasi-firm, which does not require primary factors such as labor and land in its production process.
#' There are four commodities in the model, namely wheat, iron, iron leased out as a capital good, labor and land.
#' @param ... arguments to be passed to the function sdm2.
#' @examples
#' \donttest{
#' #### a timeline model.
#' np <- 15 # the number of economic periods
#' gr <- 0 # the growth rate in the steady state equilibrium
#' eis <- 1 # the elasticity of intertemporal substitution  of consumers
#' Gamma.beta <- 0.97 # the subjective discount factor of consumers
#' last.beta.laborer <- 0
#' last.beta.landowner <- 0
#' depreciation.rate <- 0.06
#'
#' alpha.firm.wheat <- rep(5, np - 1)
#' alpha.firm.iron <- rep(5, np - 1)
#'
#' y1.wheat <- 200
#' y1.iron <- 100
#'
#' names.commodity <- c(
#'   paste0("wheat", 1:np),
#'   paste0("iron", 1:np),
#'   paste0("cap", 1:(np - 1)),
#'   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)),
#'   paste0("quasifirm.cap", 1:(np - 1)), # a quasifirm
#'   "laborer", "landowner"
#' )
#'
#' n <- length(names.commodity) # the number of commodity kinds, i.e. 5 * np - 3
#' m <- length(names.agent) # the number of agent kinds, i.e. 3 * np - 1
#'
#' # 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
#'   B[paste0("cap", k), paste0("quasifirm.cap", k)] <- 1
#'   B[paste0("iron", k + 1), paste0("quasifirm.cap", k)] <- 1 - depreciation.rate
#' }
#'
#' dstl.firm.wheat <- dstl.firm.iron <- dstl.quasifirm.cap <- list()
#' for (k in 1:(np - 1)) {
#'   dstl.firm.wheat[[k]] <- node_new(
#'     "prod",
#'     type = "CES", es = 1,
#'     alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
#'     paste0("cap", k), paste0("lab", k), paste0("land", k)
#'   )
#'
#'   dstl.firm.iron[[k]] <- node_new(
#'     "prod",
#'     type = "CES", es = 1,
#'     alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2),
#'     paste0("cap", k), paste0("lab", k), paste0("land", k)
#'   )
#'
#'   dstl.quasifirm.cap[[k]] <- node_new(
#'     "output",
#'     type = "Leontief", a = 1,
#'     paste0("iron", 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, dstl.quasifirm.cap,
#'     dst.laborer, dst.landowner
#'   ),
#'   B = B,
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "lab1",
#'   policy = makePolicyMeanValue(50),
#'   ts = TRUE,
#'   priceAdjustmentVelocity = 0.03
#' )
#'
#' ge$p
#' ge$z
#' plot(ge$z[2 * (np - 1) + (1:(np - 1))], type = "b", pch = 20)
#' lines(1:(np - 1), ge$z[1:(np - 1)], type = "b", pch = 21)
#' lines(1:(np - 1), ge$z[np - 1 + (1:(np - 1))], type = "b", pch = 22)
#' legend("topleft", c("cap","wheat", "iron"), pch = 20:22)
#'
#' #### a time-circle model.
#' np <- 5 # the number of economic periods
#' gr <- 0.03 # the growth rate in the steady state equilibrium
#' eis <- 0.5 # the elasticity of intertemporal substitution  of consumers
#' Gamma.beta <- 0.97 # the subjective discount factor of consumers
#' es.firm <- 1
#' depreciation.rate <- 0.06
#' zeta <- (1 + gr)^np # the ratio of repayments to loans
#' yield.rate <- sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr, prepaid = TRUE)
#' captial.share.laborer <- 0.7795
#' captial.share.landowner <- 1 - captial.share.laborer
#'
#' alpha.firm.wheat <- rep(5, np)
#' alpha.firm.iron <- rep(5, np)
#'
#' names.commodity <- c(
#'   paste0("wheat", 1:np),
#'   paste0("iron", 1:np),
#'   paste0("cap", 1:np),
#'   paste0("lab", 1:np),
#'   paste0("land", 1:np),
#'   "claim"
#' )
#' names.agent <- c(
#'   paste0("firm.wheat", 1:np), paste0("firm.iron", 1:np),
#'   paste0("quasifirm.cap", 1:np), # a quasifirm
#'   "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[paste0("lab", 1:np), "laborer"] <- 100 * (1 + gr)^(0:(np - 1)) # the supply of labor
#' S0Exg[paste0("land", 1:np), "landowner"] <- 100 * (1 + gr)^(0:(np - 1)) # the supply of land
#' S0Exg["claim", "laborer"] <- 100 * captial.share.laborer
#' S0Exg["claim", "landowner"] <- 100 * captial.share.landowner
#'
#' # 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
#'   B[paste0("cap", k), paste0("quasifirm.cap", k)] <- 1
#'   B[paste0("iron", k + 1), paste0("quasifirm.cap", k)] <- 1 - depreciation.rate
#' }
#' B[paste0("wheat", 1), paste0("firm.wheat", np)] <- 1 / zeta
#' B[paste0("iron", 1), paste0("firm.iron", np)] <- 1 / zeta
#' B[paste0("cap", np), paste0("quasifirm.cap", np)] <- 1
#' B[paste0("iron", 1), paste0("quasifirm.cap", np)] <- (1 - depreciation.rate) / zeta
#'
#' dstl.firm.wheat <- dstl.firm.iron <- dstl.quasifirm.cap <- list()
#' for (k in 1:(np - 1)) {
#'   dstl.firm.wheat[[k]] <- node_new(
#'     "prod",
#'     type = "CES", es = es.firm,
#'     alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
#'     paste0("cap", k), paste0("lab", k), paste0("land", k)
#'   )
#'
#'   dstl.firm.iron[[k]] <- node_new(
#'     "prod",
#'     type = "CES", es = es.firm,
#'     alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2),
#'     paste0("cap", k), paste0("lab", k), paste0("land", k)
#'   )
#'
#'   dstl.quasifirm.cap[[k]] <- node_new(
#'     "output",
#'     type = "Leontief", a = 1,
#'     paste0("iron", k)
#'   )
#' }
#'
#' dstl.firm.wheat[[np]] <- node_new(
#'   "prod",
#'   type = "FIN", rate = c(1, (1 + yield.rate)^np - 1),
#'   "cc1", "claim"
#' )
#' node_set(dstl.firm.wheat[[np]], "cc1",
#'          type = "CES", es = es.firm,
#'          alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4),
#'          paste0("cap", np), paste0("lab", np), paste0("land", np)
#' )
#'
#' dstl.firm.iron[[np]] <- node_new(
#'   "prod",
#'   type = "FIN", rate = c(1, (1 + yield.rate)^np - 1),
#'   "cc1", "claim"
#' )
#' node_set(dstl.firm.iron[[np]], "cc1",
#'          type = "CES", es = es.firm,
#'          alpha = alpha.firm.wheat[k], beta = c(0.4, 0.4, 0.2),
#'          paste0("cap", np), paste0("lab", np), paste0("land", np)
#' )
#'
#' return.rate <- sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr)
#' fund.occupancy.rate <- (1 - depreciation.rate) / (1 + return.rate)
#' # The prepaid rent rate (i.e. the prepaid-rent-to-price ratio) of
#' # the capital good is equal to 1 minus fund.occupancy.rate.
#'
#' dstl.quasifirm.cap[[np]] <- node_new(
#'   "prod",
#'   type = "FIN", rate = c(1, ((1 + yield.rate)^np - 1) * fund.occupancy.rate),
#'   "cc1", "claim"
#' )
#' node_set(dstl.quasifirm.cap[[np]], "cc1",
#'          type = "Leontief", a = 1,
#'          paste0("iron", np)
#' )
#'
#' dst.laborer <- node_new(
#'   "util",
#'   type = "CES", es = eis,
#'   alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
#'   paste0("cc", 1:np)
#' )
#' for (k in 1:np) {
#'   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)
#'   )
#' }
#'
#' dst.landowner <- node_new(
#'   "util",
#'   type = "CES", es = eis,
#'   alpha = 1, beta = prop.table(Gamma.beta^(1:np)),
#'   paste0("cc", 1:np)
#' )
#' for (k in 1:np) {
#'   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.tc <- sdm2(
#'   A = c(
#'     dstl.firm.wheat, dstl.firm.iron, dstl.quasifirm.cap,
#'     dst.laborer, dst.landowner
#'   ),
#'   B = B,
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "lab1",
#'   numberOfPeriods = 1000,
#'   priceAdjustmentVelocity = 0.03
#' )
#'
#' ge.tc$p
#' growth_rate(ge.tc$p)
#' ge.tc$z
#' growth_rate(ge.tc$z)
#'
#' ## the corresponding sequential model with the same steady-state equilibrium.
#' dst.firm.wheat <- node_new("prod",
#'                            type = "FIN", rate = c(1, yield.rate),
#'                            "cc1", "equity.share.wheat"
#' )
#' node_set(dst.firm.wheat, "cc1",
#'          type = "CES", es = es.firm,
#'          alpha = 5, beta = c(0.2, 0.4, 0.4),
#'          "cap", "lab", "land"
#' )
#'
#' dst.firm.iron <- node_new("prod",
#'                           type = "FIN", rate = c(1, yield.rate),
#'                           "cc1", "equity.share.iron"
#' )
#' node_set(dst.firm.iron, "cc1",
#'          type = "CES", es = es.firm,
#'          alpha = 5, beta = c(0.4, 0.4, 0.2),
#'          "cap", "lab", "land"
#' )
#'
#' dst.quasifirm.cap <- node_new("prod",
#'                               type = "FIN", rate = c(1, yield.rate * fund.occupancy.rate),
#'                               "cc1", "equity.share.cap"
#' )
#' node_set(dst.quasifirm.cap, "cc1",
#'          type = "Leontief", a = 1,
#'          "iron"
#' )
#'
#' 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.seq <- sdm2(
#'   A = list(
#'     dst.firm.wheat, dst.firm.iron, dst.quasifirm.cap,
#'     dst.laborer, dst.landowner
#'   ),
#'   B = matrix(c(
#'     1, 0, 0, 0, 0,
#'     0, 1, 1 - depreciation.rate, 0, 0,
#'     0, 0, 1 + gr, 0, 0,
#'     0, 0, 0, 0, 0,
#'     0, 0, 0, 0, 0,
#'     0, 0, 0, 0, 0,
#'     0, 0, 0, 0, 0,
#'     0, 0, 0, 0, 0
#'   ), 8, 5, TRUE),
#'   S0Exg = matrix(c(
#'     NA, NA, NA, NA, NA,
#'     NA, NA, NA, NA, NA,
#'     NA, NA, NA, NA, NA,
#'     NA, NA, NA, 100, NA,
#'     NA, NA, NA, NA, 100,
#'     NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner,
#'     NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner,
#'     NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner
#'   ), 8, 5, TRUE),
#'   names.commodity = c(
#'     "wheat", "iron", "cap", "lab", "land",
#'            "equity.share.wheat", "equity.share.iron", "equity.share.cap"
#'   ),
#'   names.agent = c("firm.wheat", "firm.iron", "quasifirm.cap", "laborer", "landowner"),
#'   numeraire = "lab",
#'   numberOfPeriods = 2000,
#'   priceAdjustmentVelocity = 0.03,
#'   GRExg = gr
#' )
#'
#' ge.seq$p
#' ge.seq$z
#' ge.tc$z
#' }

gemIntertemporal_5_5 <- function(...) sdm2(...)