# Some Examples of a 2-by-2 Time Circle Equilibrium Model Some examples of a 2-by-2 (intertemporal) time circle equilibrium model. In a time circle model, the economy borrows some resources from the outside in the beginning, and will repay it after the economy ends. In these examples, there is an np-period-lived consumer maximizing intertemporal utility, and there is a type of firm which produces from period 1 to np. There are two commodities, i.e. product and labor. Suppose the firm can borrow some product from outside in the first period and return them in the (np+1)-th period. And the supply of product in the first period can be regarded as the output of the firm in the (np+1)-th period. Hence the product supply in the first period is an endogenous variable. Suppose that the amount returned is zeta times the amount borrowed. ```r gemIntertemporal_TimeCircle_2_2(...) ``` ## Arguments - `...`: arguments to be passed to the function sdm2. ## Examples ```r #### an example with a Cobb-Douglas intertemporal utility function np <- 5 # the number of economic periods, firms. gr <- 0 # the growth rate of the labor supply zeta <- 1.25 # the ratio of repayments to loans # zeta <- (1 + gr)^np Gamma.beta <- 1 # the subjective discount factor n <- 2 * np # the number of commodity kinds m <- np + 1 # the number of agent kinds names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np)) names.agent <- c(paste0("firm", 1:np), "consumer") # the exogenous supply matrix. S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent)) S0Exg[paste0("lab", 1:np), "consumer"] <- 100 * (1 + gr)^(0:(np - 1)) # the supply of labor # 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 } B["prod1", paste0("firm", np)] <- 1 / zeta dstl.firm <- list() for (k in 1:np) { dstl.firm[[k]] <- node_new( "prod", type = "CD", alpha = 2, beta = c(0.5, 0.5), paste0("lab", k), paste0("prod", k) ) } dst.consumer <- node_new( "util", type = "CD", alpha = 1, beta = prop.table(Gamma.beta^(1:np)), paste0("prod", 1:np) ) ge <- sdm2( A = c(dstl.firm, dst.consumer), B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "prod1", ts = TRUE ) ge$p ge$z ge$D ge$S ge$DV ge$SV ## an example with a Leontief intertemporal utility function dst.consumer <- node_new( "util", type = "Leontief", a = rep(1, np), paste0("prod", 1:np) ) ge2 <- sdm2( A = c(dstl.firm, dst.consumer), B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "lab1", ts = TRUE ) ge2$p ge2$z ge2$D ge2$S ge2$DV ge2$SV ## Use a mean-value policy function to accelerate convergence. ge3 <- sdm2( A = c(dstl.firm, dst.consumer), B = B, S0Exg = S0Exg, names.commodity = c(paste0("prod", 1:np), paste0("lab", 1:np)), names.agent = c(paste0("firm", 1:np), "consumer"), numeraire = "lab1", ts = TRUE, policy = makePolicyMeanValue(30) ) #### an example with a linear intertemporal utility function (e.g. beta1 * x1 + beta2 * x2) ## The demand structure of the consumer will be adjusted sluggishly to accelerate convergence. np <- 5 # the number of economic periods, firms. rho <- 0.9 # the subjective discount factor beta.consumer <- rep(rho^(0:(np - 1))) zeta <- (1 / rho)^np n <- 2 * np # the number of commodity kinds m <- np + 1 # the number of agent kinds names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np)) names.agent <- c(paste0("firm", 1:np), "consumer") # the exogenous supply matrix. S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent)) S0Exg[paste0("lab", 1:np), "consumer"] <- 100 # the supply of labor # 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 } B["prod1", paste0("firm", np)] <- 1 / zeta dstl.firm <- list() for (k in 1:np) { dstl.firm[[k]] <- node_new( "prod", type = "CD", alpha = 2, beta = c(0.5, 0.5), paste0("lab", k), paste0("prod", k) ) } dst.consumer <- node_new( "util", type = "FUNC", last.a = rep(1, np), func = function(p) { value.marginal.utility <- beta.consumer / p ratio <- value.marginal.utility / mean(value.marginal.utility) a <- dst.consumer$last.a a <- prop.table(a * ratio_adjust(ratio, 0.15)) dst.consumer$last.a <- a a }, paste0("prod", 1:np) ) ge <- sdm2( A = c(dstl.firm, dst.consumer), B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "lab1", ts = TRUE, priceAdjustmentVelocity = 0.1 ) ge$p ge$z ge$D ge$S growth_rate(ge$p[1:np]) growth_rate(ge$p[(np + 1):(2 * np)]) ``` ## See Also `gemOLG_TimeCircle`

gemIntertemporal_TimeCircle_2_2 function