# Time-Circle Models (Closed Loop Overlapping Generations Models) 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). ```r gemOLG_TimeCircle(...) ``` ## Arguments - `...`: arguments to be passed to the function sdm2. ## Examples ```r #### 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$S #### a time-circle pure exchange economy with three-period-lived consumers # Suppose each consumer has a utility function log(c1) + log(c2) + log(c3). # See gemOLG_PureExchange. np <- 5 # the number of economic periods, commodity kinds and generations names.commodity <- c(paste0("payoff", 1:np)) names.agent <- paste0("gen", 1:np) index.comm <- c(1:np, 1:2) payoff <- c(50, 50, 0) # the payoffs of lifetime # the exogenous supply matrix. S0Exg <- matrix(0, np, np, dimnames = list(names.commodity, names.agent)) for (k in 1:np) { S0Exg[index.comm[k:(k + 2)], k] <- payoff } dstl.consumer <- list() for (k in 1:np) { dstl.consumer[[k]] <- node_new( "util", type = "CD", alpha = 1, beta = rep(1 / 3, 3), paste0("payoff", index.comm[k:(k + 2)]) ) } ge <- sdm2( A = dstl.consumer, B = matrix(0, np, np), S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "payoff1" ) ge$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$DV ## beta.consumer <- c(1 / 3, 1 / 3, 1 / 3) # the share parameter of consumers labor.supply <- c(50, 50, 0) # the labor supply of lifetime ge <- f() ge$D ge$DV ## gr.laborer <- 0 df <- (1 + gr.laborer)^-np beta.prod.firm <- 0.5 ge <- f() ge$z ``` ## 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. Diamond, Peter (1965) National Debt in a Neoclassical Growth Model. American Economic Review. 55 (5): 1126-1150. 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. Shell, Karl (1971) Notes on the Economics of Infinity. Journal of Political Economy. 79 (5): 1002–1011. ## See Also `gemOLG_PureExchange`

gemOLG_TimeCircle function