R/gemOLG_TimeCircle.R

In GE: General Equilibrium Modeling

#' @export
#' @title Time-Circle Models (Closed Loop Overlapping Generations Models)
#' @aliases gemOLG_TimeCircle
#' @description Some examples of time-circle models, which usually contain overlapping generations.
#' When we connect together the head and tail of time of a dynamic model, we get a time-circle model which
#' treats time as a circle of finite length instead of a straight line of infinite length.
#' The (discounted) output of the final period will enter the utility function and the production function
#' of the first period, which implies the influence mechanism of the past on the present is
#' just like the influence mechanism of the present on the future.
#' A time-circle OLG model may be called a reincarnation model.
#'
#' In a time-circle model, time can be thought of as having no beginning and no end.
#' And we can assume all souls meet in a single market (see Shell, 1971),
#' or there are implicit financial instruments that facilitate transactions between adjacent generations.
#'
#' As in the Arrow–Debreu approach, in the following examples with production, we assume that firms operate independently and are not owned by consumers.
#' That is, there is no explicit property of firms as the firms do not make pure profit at equilibrium (constant return to scale)
#' (see de la Croix and Michel, 2002, page 292).
#' @param ... arguments to be passed to the function sdm2.
#' @references de la Croix, David and Philippe Michel (2002, ISBN: 9780521001151) A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations. Cambridge University Press.
#' @references Diamond, Peter (1965) National Debt in a Neoclassical Growth Model. American Economic Review. 55 (5): 1126-1150.
#' @references Samuelson, P. A. (1958) An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money. Journal of Political Economy, vol. 66(6): 467-482.
#' @references Shell, Karl (1971) Notes on the Economics of Infinity. Journal of Political Economy. 79 (5): 1002–1011.
#' @seealso {
#' \code{\link{gemOLG_PureExchange}}
#' }
#' @examples
#' \donttest{
#' #### a time-circle pure exchange economy with two-period-lived consumers.
#' # In this example, each agent sells some payoff to the previous
#' # generation and buys some payoff from the next generation.
#'
#' # Here np can tend to infinity.
#' np <- 4 # the number of economic periods, commodity kinds and generations
#'
#' names.commodity <- c(paste0("payoff", 1:np))
#' names.agent <- paste0("gen", 1:np)
#'
#' index.comm <- c(1:np, 1)
#' payoff <- c(100, 0) # the payoffs of lifetime
#' # the exogenous supply matrix.
#' S0Exg <- matrix(0, np, np, dimnames = list(names.commodity, names.agent))
#' for (k in 1:np) {
#'   S0Exg[index.comm[k:(k + 1)], k] <- payoff
#' }
#'
#' # Suppose each consumer has a utility function log(c1) + log(c2).
#' beta.consumer <- c(0.5, 0.5)
#' index.comm <- c(1:np, 1)
#' dstl.consumer <- list()
#' for (k in 1:np) {
#'   dstl.consumer[[k]] <- node_new(
#'     "util",
#'     type = "CD", alpha = 1,
#'     beta = beta.consumer,
#'     paste0("payoff", index.comm[k]), paste0("payoff", index.comm[k + 1])
#'   )
#' }
#'
#' ge <- sdm2(
#'   A = dstl.consumer,
#'   B = matrix(0, np, np),
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "payoff1"
#' )
#'
#' ge$p
#' ge$D
#' ge$S
#'
#' ## Introduce population growth into the above pure exchange economy.
#' GRExg <- 0.03 # the population growth rate
#'
#' S0Exg <- S0Exg %*% diag((1 + GRExg)^(0:(np - 1)))
#' S0Exg[1, np] <- S0Exg[1, np] * (1 + GRExg)^-np
#'
#' dstl.consumer[[np]] <- node_new(
#'   "util",
#'   type = "CD", alpha = 1,
#'   beta = beta.consumer,
#'   paste0("payoff", np), "cc1"
#' )
#' node_set(dstl.consumer[[np]],
#'   "cc1",
#'   type = "Leontief", a = (1 + GRExg)^(-np), # a discounting factor
#'   "payoff1"
#' )
#'
#' ge2 <- sdm2(
#'   A = dstl.consumer,
#'   B = matrix(0, np, np),
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "payoff1"
#' )
#'
#' ge2$p
#' ge2$D
#' ge2$DV
#' ge2$S
#'
#' #### a time-circle pure exchange economy with three-period-lived consumers
#' # Suppose each consumer has a utility function log(c1) + log(c2) + log(c3).
#' # See gemOLG_PureExchange.
#' np <- 5 # the number of economic periods, commodity kinds and generations
#'
#' names.commodity <- c(paste0("payoff", 1:np))
#' names.agent <- paste0("gen", 1:np)
#'
#' index.comm <- c(1:np, 1:2)
#' payoff <- c(100, 100, 0) # the payoffs of lifetime
#' # the exogenous supply matrix.
#' S0Exg <- matrix(0, np, np, dimnames = list(names.commodity, names.agent))
#' for (k in 1:np) {
#'   S0Exg[index.comm[k:(k + 2)], k] <- payoff
#' }
#'
#' dstl.consumer <- list()
#' for (k in 1:np) {
#'   dstl.consumer[[k]] <- node_new(
#'     "util",
#'     type = "CD", alpha = 1,
#'     beta = rep(1 / 3, 3),
#'     paste0("payoff", index.comm[k:(k + 2)])
#'   )
#' }
#'
#' ge <- sdm2(
#'   A = dstl.consumer,
#'   B = matrix(0, np, np),
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "payoff1"
#' )
#'
#' ge$p
#' ge$D
#'
#' ## Introduce population growth into the above pure exchange economy.
#' GRExg <- 0.03 # the population growth rate
#' df <- (1 + GRExg)^-np # a discounting factor
#'
#' # the exogenous supply matrix.
#' S0Exg <- matrix(0, np, np, dimnames = list(names.commodity, names.agent))
#' for (k in 1:np) {
#'   S0Exg[paste0("payoff", index.comm[k:(k + 2)]), paste0("gen", k)] <-
#'     payoff * (1 + GRExg)^(k - 1)
#' }
#' S0Exg[paste0("payoff", 1:2), paste0("gen", np)] <-
#'   S0Exg[paste0("payoff", 1:2), paste0("gen", np)] * df
#' S0Exg[paste0("payoff", 1), paste0("gen", np - 1)] <-
#'   S0Exg[paste0("payoff", 1), paste0("gen", np - 1)] * df
#'
#' dstl.consumer[[np - 1]] <- node_new(
#'   "util",
#'   type = "CD", alpha = 1,
#'   beta = rep(1 / 3, 3),
#'   paste0("payoff", (np - 1):(np + 1))
#' )
#' node_set(dstl.consumer[[np - 1]], paste0("payoff", np + 1),
#'   type = "Leontief", a = df,
#'   "payoff1"
#' )
#'
#' dstl.consumer[[np]] <- node_new(
#'   "util",
#'   type = "CD", alpha = 1,
#'   beta = rep(1 / 3, 3),
#'   paste0("payoff", np:(np + 2))
#' )
#' node_set(dstl.consumer[[np]], paste0("payoff", np + 1),
#'   type = "Leontief", a = df,
#'   "payoff1"
#' )
#' node_set(dstl.consumer[[np]], paste0("payoff", np + 2),
#'   type = "Leontief", a = df,
#'   "payoff2"
#' )
#'
#' ge2 <- sdm2(
#'   A = dstl.consumer,
#'   B = matrix(0, np, np),
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "payoff1"
#' )
#'
#' ge2$p
#' growth_rate(ge2$p) + 1 # 1 / (1 + GRExg)
#' ge2$D
#' ge2$DV
#'
#' #### a time-circle model with production and two-period-lived consumers
#' # Suppose each consumer has a utility function log(c1) + log(c2).
#' np <- 5 # the number of economic periods, consumers and firms.
#' names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
#' names.agent <- c(paste0("firm", 1:np), paste0("consumer", 1:np))
#'
#' index.comm <- c(1:np, 1)
#' labor.supply <- c(100, 0) # the labor supply of lifetime
#' # the exogenous supply matrix.
#' S0Exg <- matrix(NA, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
#' for (k in 1:np) {
#'   S0Exg[paste0("lab", index.comm[k:(k + 1)]), paste0("consumer", k)] <- labor.supply
#' }
#'
#' B <- matrix(0, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
#' for (k in 1:np) {
#'   B[paste0("prod", index.comm[k + 1]), paste0("firm", k)] <- 1
#' }
#'
#' dstl.firm <- list()
#' for (k in 1:np) {
#'   dstl.firm[[k]] <- node_new(
#'     "prod",
#'     type = "CD", alpha = 5,
#'     beta = c(1 / 3, 2 / 3),
#'     paste0("lab", k), paste0("prod", k)
#'   )
#' }
#'
#' dstl.consumer <- list()
#' for (k in 1:np) {
#'   dstl.consumer[[k]] <- node_new(
#'     "util",
#'     type = "CD", alpha = 1,
#'     beta = c(1 / 2, 1 / 2),
#'     paste0("prod", index.comm[k:(k + 1)])
#'   )
#' }
#'
#' ge <- sdm2(
#'   A = c(dstl.firm, dstl.consumer),
#'   B = B,
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "lab1"
#' )
#'
#' ge$p
#' ge$D
#'
#' ## Introduce population growth into the above economy.
#' GRExg <- 0.03 # the population growth rate
#' df <- (1 + GRExg)^-np # a discounting factor
#'
#' for (k in 1:np) {
#'   S0Exg[paste0("lab", index.comm[k:(k + 1)]), paste0("consumer", k)] <-
#'     labor.supply * (1 + GRExg)^(k - 1)
#' }
#'
#' B[1, np] <- df
#'
#' dstl.consumer[[np]] <- node_new(
#'   "util",
#'   type = "CD", alpha = 1,
#'   beta = c(1 / 2, 1 / 2),
#'   paste0("prod", np), "cc1"
#' )
#' node_set(dstl.consumer[[np]], "cc1",
#'   type = "Leontief", a = df,
#'   "prod1"
#' )
#'
#' ge2 <- sdm2(
#'   A = c(dstl.firm, dstl.consumer),
#'   B = B,
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "lab1",
#'   policy = makePolicyMeanValue(30),
#'   maxIteration = 1,
#'   numberOfPeriods = 600,
#'   ts = TRUE
#' )
#'
#' ge2$p
#' growth_rate(ge2$p[1:np]) + 1 # 1 / (1 + GRExg)
#' growth_rate(ge2$p[(np + 1):(2 * np)]) + 1 # 1 / (1 + GRExg)
#' ge2$D
#' ge2$DV
#'
#' #### a time-circle model with production and one-period-lived consumers.
#' # These consumers also can be regarded as infinite-lived agents maximizing
#' # their per period utility subject to their disposable income per period.
#' np <- 5 # the number of economic periods, consumers and firms.
#' GRExg <- 0.03 # the population growth rate
#' # df <- (1 + GRExg)^-np # a discounting factor
#' df <- 0.5
#'
#' names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
#' names.agent <- c(paste0("firm", 1:np), paste0("consumer", 1:np))
#'
#' # the exogenous supply matrix.
#' S0Exg <- matrix(NA, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
#' for (k in 1:np) {
#'   S0Exg[paste0("lab", k), paste0("consumer", k)] <- 100 * (1 + GRExg)^(k - 1)
#' }
#'
#' B <- matrix(0, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
#' for (k in 1:(np - 1)) {
#'   B[paste0("prod", index.comm[k + 1]), paste0("firm", k)] <- 1
#' }
#' B["prod1", paste0("firm", np)] <- df
#'
#' beta.firm <- c(1 / 3, 2 / 3)
#' beta.consumer <- c(1 / 2, 1 / 2)
#'
#' dstl.firm <- list()
#' for (k in 1:np) {
#'   dstl.firm[[k]] <- node_new(
#'     "prod",
#'     type = "CD", alpha = 5,
#'     beta = beta.firm,
#'     paste0("lab", k), paste0("prod", k)
#'   )
#' }
#'
#' dstl.consumer <- list()
#' for (k in 1:np) {
#'   dstl.consumer[[k]] <- node_new(
#'     "util",
#'     type = "CD", alpha = 1,
#'     beta = beta.consumer,
#'     paste0("lab", k), paste0("prod", k)
#'   )
#' }
#'
#' ge <- sdm2(
#'   A = c(dstl.firm, dstl.consumer),
#'   B = B,
#'   S0Exg = S0Exg,
#'   names.commodity = names.commodity,
#'   names.agent = names.agent,
#'   numeraire = "lab1",
#'   policy = makePolicyMeanValue(30),
#'   maxIteration = 1,
#'   numberOfPeriods = 600,
#'   ts = TRUE
#' )
#'
#' ge$p
#' growth_rate(ge$p[1:np]) + 1 # 1 / (1 + GRExg)
#' growth_rate(ge$p[(np + 1):(2 * np)]) + 1 # 1 / (1 + GRExg)
#' ge$D
#'
#' ## the sequential form of the above model.
#' dst.firm <- node_new(
#'   "prod",
#'   type = "CD", alpha = 5,
#'   beta = beta.firm,
#'   "lab", "prod"
#' )
#'
#' dst.consumer <- node_new(
#'   "util",
#'   type = "CD", alpha = 1,
#'   beta = beta.consumer,
#'   "lab", "prod"
#' )
#'
#' ge2 <- sdm2(
#'   A = list(
#'     dst.firm,
#'     dst.consumer
#'   ),
#'   B = matrix(c(
#'     1, 0,
#'     0, 0
#'   ), 2, 2, TRUE),
#'   S0Exg = matrix(c(
#'     NA, NA,
#'     NA, 100 / (1 + GRExg)
#'   ), 2, 2, TRUE),
#'   names.commodity = c("prod", "lab"),
#'   names.agent = c("firm", "consumer"),
#'   numeraire = "lab",
#'   z0 = c(ge$S[1, np], 0),
#'   GRExg = GRExg,
#'   policy = policyMarketClearingPrice,
#'   maxIteration = 1,
#'   numberOfPeriods = 20,
#'   ts = TRUE
#' )
#'
#' ge2$p
#' ge2$D
#' ge2$ts.z[, 1]
#' ge$z
#'
#' #### a time-circle OLG model with production and three-period-lived consumers.
#' np <- 6 # the number of economic periods, consumers and firms
#' gr.laborer <- 0.03 # the population growth rate
#' df <- (1 + gr.laborer)^-np # a discounting factor
#' alpha.firm <- 2 # the efficient parameter of firms
#' beta.prod.firm <- 0.4 # the product (i.e. capital) share parameter of firms
#' beta.consumer <- c(0, 0.8, 0.2) # the share parameter of consumers
#' labor.supply <- c(100, 0, 0) # the labor supply of lifetime
#'
#' f <- function() {
#'   names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np))
#'   names.agent <- c(paste0("firm", 1:np), paste0("consumer", 1:np))
#'
#'   index.comm <- c(1:np, 1:2)
#'
#'   # the exogenous supply matrix.
#'   S0Exg <- matrix(NA, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
#'   for (k in 1:np) {
#'     S0Exg[paste0("lab", index.comm[k:(k + 2)]), paste0("consumer", k)] <-
#'       labor.supply * (1 + gr.laborer)^(k - 1)
#'   }
#'   S0Exg[paste0("lab", 1:2), paste0("consumer", np)] <-
#'     S0Exg[paste0("lab", 1:2), paste0("consumer", np)] * df
#'   S0Exg[paste0("lab", 1), paste0("consumer", np - 1)] <-
#'     S0Exg[paste0("lab", 1), paste0("consumer", np - 1)] * df
#'
#'   B <- matrix(0, 2 * np, 2 * np, dimnames = list(names.commodity, names.agent))
#'   for (k in 1:np) {
#'     B[paste0("prod", index.comm[k + 1]), paste0("firm", k)] <- 1
#'   }
#'   B["prod1", paste0("firm", np)] <- df
#'
#'
#'   dstl.firm <- list()
#'   for (k in 1:np) {
#'     dstl.firm[[k]] <- node_new(
#'       "prod",
#'       type = "CD", alpha = alpha.firm,
#'       beta = c(beta.prod.firm, 1 - beta.prod.firm),
#'       paste0("prod", k), paste0("lab", k)
#'     )
#'   }
#'
#'   dstl.consumer <- list()
#'   for (k in 1:np) {
#'     dstl.consumer[[k]] <- node_new(
#'       "util",
#'       type = "CD", alpha = 1,
#'       beta = beta.consumer,
#'       paste0("prod", k:(k + 2))
#'     )
#'   }
#'
#'   node_set(dstl.consumer[[np - 1]], paste0("prod", np + 1),
#'            type = "Leontief", a = df,
#'            "prod1"
#'   )
#'
#'   node_set(dstl.consumer[[np]], paste0("prod", np + 1),
#'            type = "Leontief", a = df,
#'            "prod1"
#'   )
#'   node_set(dstl.consumer[[np]], paste0("prod", np + 2),
#'            type = "Leontief", a = df,
#'            "prod2"
#'   )
#'
#'   ge <- sdm2(
#'     A = c(dstl.firm, dstl.consumer),
#'     B = B,
#'     S0Exg = S0Exg,
#'     names.commodity = names.commodity,
#'     names.agent = names.agent,
#'     numeraire = "lab1"
#'   )
#'   invisible(ge)
#' }
#'
#' ge <- f()
#' growth_rate(ge$p[1:np]) + 1 # 1 / (1 + gr.laborer)
#' growth_rate(ge$p[(np + 1):(2 * np)]) + 1 # 1 / (1 + gr.laborer)
#' ge$D
#' ge$DV
#'
#' ##
#' beta.consumer <- c(1 / 3, 1 / 3, 1 / 3) # the share parameter of consumers
#' labor.supply <- c(100, 100, 0) # the labor supply of lifetime
#' ge <- f()
#' ge$D
#' ge$DV
#' }

gemOLG_TimeCircle <- function(...) sdm2(...)