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#' @export
#' @title Some Intertemporal Models with Ad Valorem Claim
#' @aliases gemIntertemporal_AdValoremClaim
#' @description Some intertemporal models with ad valorem claim.
#' Ad valorem claims may be equities, bonds, ad valorem taxation rights (ad valorem tax receipt), fiat money etc,
#' which can be treated in the same way in models.
#' @param ... arguments to be passed to the function sdm2.
#' @examples
#' \donttest{
#' #### a model with tax.
#' np <- 5 # the number of economic periods
#' gr.lab <- 0.03 # the growth rate of the labor supply
#' tax.rate <- 0.25
#'
#' n <- 2 * np + 1 # the number of commodity kinds
#' m <- np + 1 # the number of agent kinds
#'
#' names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np), "tax receipt")
#' names.agent <- c(paste0("firm", 1:(np - 1)), "laborer", "government")
#'
#' # the exogenous supply matrix.
#' S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent))
#' S0Exg[paste0("lab", 1:np), "laborer"] <- 100 * (1 + gr.lab)^(0:(np - 1)) # the labor supply
#' S0Exg["prod1", "laborer"] <- 10 # the product supply in the first period
#' S0Exg["tax receipt", "government"] <- np * 100 # the supply of tax receipt (i.e. ad valorem claim)
#'
#' # 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, tax.rate),
#' "cc1", "tax receipt"
#' )
#' node_set(dstl.firm[[k]], "cc1",
#' type = "CD",
#' alpha = 1, beta = c(0.5, 0.5),
#' paste0("prod", k), paste0("lab", k)
#' )
#' }
#'
#' dst.laborer <- node_new(
#' "util",
#' type = "FIN", rate = c(1, tax.rate),
#' "cc1", "tax receipt"
#' )
#' node_set(dst.laborer, "cc1",
#' type = "CES", es = 0.5,
#' alpha = 1, beta = rep(1 / np, np),
#' paste0("cc1.", 1:np)
#' )
#' for (k in 1:np) {
#' node_set(dst.laborer, paste0("cc1.", k),
#' type = "CD", alpha = 1, beta = c(0.5, 0.5),
#' paste0("prod", k), paste0("lab", k)
#' )
#' }
#'
#' dst.government <- node_new(
#' "util",
#' type = "CD",
#' alpha = 1, beta = rep(1 / np, np),
#' paste0("cc1.", 1:np)
#' )
#' for (k in 1:np) {
#' node_set(dst.government, paste0("cc1.", k),
#' type = "CD", alpha = 1, beta = c(0.5, 0.5),
#' paste0("prod", k), paste0("lab", k)
#' )
#' }
#' node_plot(dst.government, TRUE)
#'
#' ge <- sdm2(
#' A = c(dstl.firm, dst.laborer, dst.government),
#' B = B,
#' S0Exg = S0Exg,
#' names.commodity = names.commodity,
#' names.agent = names.agent,
#' numeraire = "prod1",
#' # policy = makePolicyHeadTailAdjustment(gr = gr.lab, np = np)
#' )
#'
#' ge$D
#' ge$z
#' ge$DV
#'
#' #### a pure exchange model with money.
#' np <- 3 # the number of economic periods
#' gr.lab <- 0.03 # the growth rate of the labor supply
#' eis <- 0.8 # the elasticity of intertemporal substitution
#' Gamma.beta <- 0.8 # the subjective discount factor
#' interest.rate <- sserr(eis, Gamma.beta, gr.lab, prepaid = TRUE) # 0.2593
#'
#' dst.laborer <- node_new(
#' "util",
#' type = "CES", es = eis,
#' alpha = 1, beta = Gamma.beta^(0:(np - 1)),
#' paste0("cc", 1:np)
#' )
#'
#' for (k in 1:np) {
#' node_set(dst.laborer, paste0("cc", k),
#' type = "FIN",
#' rate = c(1, interest.rate),
#' paste0("lab", k), paste0("money", k)
#' )
#' }
#'
#' node_plot(dst.laborer, TRUE)
#'
#' dst.moneyOwner <- Clone(dst.laborer)
#'
#' ge <- sdm2(
#' A = list(dst.laborer, dst.moneyOwner),
#' B = matrix(0, 2 * np, 2),
#' S0Exg = {
#' tmp <- matrix(0, 2 * np, 2)
#' tmp[1:np, 1] <- 100 * (1 + gr.lab)^(0:(np - 1))
#' tmp[(np + 1):(2 * np), 2] <- 200
#' tmp
#' },
#' names.commodity = c(paste0("lab", 1:np), paste0("money", 1:np)),
#' names.agent = c("laborer", "moneyOwner"),
#' numeraire = c(money1 = interest.rate)
#' )
#'
#' ge$p
#' growth_rate(ge$p[1:3]) + 1
#' ge$z
#' addmargins(ge$D, 2)
#' addmargins(ge$S, 2)
#'
#' ## In the following program, the periods to which
#' ## the money belongs are not distinguished.
#' dst.laborer <- node_new(
#' "util",
#' type = "FIN",
#' rate = c(1, interest.rate),
#' "cc1", "money"
#' )
#' node_set(dst.laborer, "cc1",
#' type = "CES", es = eis,
#' alpha = 1, beta = Gamma.beta^(0:(np - 1)),
#' paste0("lab", 1:np)
#' )
#'
#' dst.moneyOwner <- Clone(dst.laborer)
#'
#' ge <- sdm2(
#' A = list(dst.laborer, dst.moneyOwner),
#' B = matrix(0, np + 1, 2),
#' S0Exg = {
#' tmp <- matrix(0, np + 1, 2)
#' tmp[1:np, 1] <- 100 * (1 + gr.lab)^(0:(np - 1))
#' tmp[np + 1, 2] <- 100
#' tmp
#' },
#' names.commodity = c(paste0("lab", 1:np), "money"),
#' names.agent = c("laborer", "moneyOwner"),
#' numeraire = c(money = interest.rate)
#' )
#'
#' ge$p
#' ge$z
#' addmargins(ge$D, 2)
#' addmargins(ge$S, 2)
#'
#' #### a two-period model with production and money.
#' interest.rate1 <- 0.25
#' interest.rate2 <- 0.1
#'
#' dst.firm <- node_new(
#' "prod",
#' type = "FIN",
#' rate = c(1, interest.rate1),
#' "cc1", "money1"
#' )
#' node_set(dst.firm, "cc1",
#' type = "CES",
#' es = 1, alpha = 2, beta = c(0.5, 0.5),
#' "prod1", "lab1"
#' )
#'
#' dst.laborer <- node_new(
#' "util",
#' type = "CES",
#' es = 0.5, alpha = 1, beta = c(2 / 3, 1 / 3),
#' "cc1", "cc2"
#' )
#' node_set(dst.laborer, "cc1",
#' type = "FIN",
#' rate = c(1, interest.rate1),
#' "prod1", "money1"
#' )
#'
#' node_set(dst.laborer, "cc2",
#' type = "FIN",
#' rate = c(1, interest.rate2),
#' "prod2", "money2"
#' )
#'
#' dst.moneyOwner <- Clone(dst.laborer)
#'
#' ge <- sdm2(
#' A = list(dst.firm, dst.laborer, dst.moneyOwner),
#' B = matrix(c(
#' 0, 0, 0,
#' 0, 0, 0,
#' 0, 0, 0,
#' 1, 0, 0,
#' 0, 0, 0
#' ), 5, 3, TRUE),
#' S0Exg = matrix(c(
#' NA, 200, NA,
#' NA, 100, NA,
#' NA, NA, 100,
#' NA, NA, NA,
#' NA, NA, 100
#' ), 5, 3, TRUE),
#' names.commodity = c("prod1", "lab1", "money1", "prod2", "money2"),
#' names.agent = c("firm", "laborer", "moneyOwner"),
#' numeraire = c(money1 = interest.rate1)
#' )
#'
#' ge$p
#' ge$DV
#' }
gemIntertemporal_AdValoremClaim <- function(...) sdm2(...)
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