# Some Examples of a 2-by-2 Intertemporal Equilibrium Model Some examples of an intertemporal equilibrium model with two types of commodities and two types of agents. 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-1. There are two types of commodities, i.e. product and labor. Assume the consumer has some product in the first period. That is, the product supply in the first period is an exogenous variable. ```r gemIntertemporal_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 y1 <- 150 # the initial product supply n <- 2 * np - 1 # the number of commodity kinds m <- np # the number of agent kinds names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1))) names.agent <- c(paste0("firm", 1:(np - 1)), "consumer") # the exogenous supply matrix. S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent)) S0Exg[paste0("lab", 1:(np - 1)), "consumer"] <- 100 S0Exg["prod1", "consumer"] <- y1 # 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 = "CD", alpha = 2, beta = c(0.5, 0.5), paste0("prod", k), paste0("lab", k) ) } dst.consumer.CD <- node_new( "util", type = "CD", alpha = 1, beta = prop.table(rep(1, np)), paste0("prod", 1:np) ) f <- function(dstl) { sdm2( A = dstl, B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "prod1", ts = TRUE ) } ge <- f(c(dstl.firm, dst.consumer.CD)) ge$p ge$z ge$D ge$S ge$DV ge$SV ## an example with a Leontief intertemporal utility function dst.consumer.Leontief <- node_new( "util", type = "Leontief", a = rep(1, np), paste0("prod", 1:np) ) ge2 <- f(c(dstl.firm, dst.consumer.Leontief)) ge2$p ge2$z ge2$D ge2$S ge2$DV ge2$SV ## Assume that the consumer has a CES (i.e. CRRA) intertemporal utility function. # eis is the elasticity of intertemporal substitution. # Gamma.beta is the subjective discount factor. f2 <- function(eis = 1, Gamma.beta = 1, head.tail.adjustment = "none") { dst.consumer <- node_new( "util", type = "CES", es = eis, 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, policy = makePolicyHeadTailAdjustment(head.tail.adjustment, np = np) ) list( p = ge$p, z = ge$z, D = addmargins(ge$D, 2), S = addmargins(ge$S, 2), DV = addmargins(ge$DV), SV = addmargins(ge$SV) ) } f2(Gamma.beta = 0.9) f2(Gamma.beta = 0.9, head.tail.adjustment = "both") # the steady state in the worldsheet f2(Gamma.beta = 1.25, head.tail.adjustment = "both") # the steady state in the worldsheet f2(eis = 2, Gamma.beta = 0.9) ```

gemIntertemporal_2_2 function