# Some Examples of Basic (Timeline) OLG Models with Production Some examples of basic (timeline) OLG models with production. In these models there are two types of commodities (i.e., labor and product) and two types of economic agents (i.e., laborers and firms). Laborers (i.e., consumers) live for two or three periods. These models can be easily extended to include more types of commodities, consumers and firms, and to allow consumers to live for more periods. ```r gemOLG_Basic(...) ``` ## Arguments - `...`: arguments to be passed to the function sdm2. ## Examples ```r ## When beta.consumer[1]==0, beta.consumer[2:3]>0, labor.first[1]>0, labor.first[2:3]==0, ## this model is actually the Diamond model. However, the division of periods is ## slightly different from the Diamond model. ng <- 15 # the number of generations 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 gr.laborer <- 0.03 # the population growth rate labor.first <- c(100, 0, 0) # the labor supply of the first generation labor.last <- 100 * (1 + gr.laborer)^((ng - 1):ng) # the labor supply of the last generation y1 <- 8 # the initial product supply f <- function() { names.commodity <- c(paste0("prod", 1:(ng + 2)), paste0("lab", 1:(ng + 1))) names.agent <- c(paste0("firm", 1:(ng + 1)), paste0("consumer", 1:ng)) n <- length(names.commodity) # the number of commodity kinds m <- length(names.agent) # the number of agent kinds # the exogenous supply matrix. S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent)) for (k in 1:(ng - 1)) { S0Exg[paste0("lab", k:(k + 2)), paste0("consumer", k)] <- labor.first * (1 + gr.laborer)^(k - 1) } S0Exg[paste0("lab", ng:(ng + 1)), paste0("consumer", ng)] <- labor.last S0Exg["prod1", "consumer1"] <- y1 B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent)) for (k in 1:(ng + 1)) { B[paste0("prod", k + 1), paste0("firm", k)] <- 1 } dstl.consumer <- list() for (k in 1:ng) { dstl.consumer[[k]] <- node_new( "util", type = "CD", alpha = 1, beta = beta.consumer, paste0("prod", k:(k + 2)) ) } dstl.firm <- list() for (k in 1:(ng + 1)) { 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) ) } ge <- sdm2( A = c(dstl.firm, dstl.consumer), B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "lab1", priceAdjustmentVelocity = 0.05 ) cat("ge$p:\n") print(ge$p) cat("ge$z:\n") print(ge$z) invisible(ge) } ge <- f() # the growth rates of prices growth_rate(ge$p[paste0("prod", 1:ng)]) + 1 growth_rate(ge$p[paste0("lab", 1:ng)]) + 1 # the steady-state growth rate of prices in the Diamond model beta.consumer[3] * (1 - beta.prod.firm) / beta.prod.firm / (1 + gr.laborer) # the output-labor ratios ge$z[paste0("firm", 1:ng)] / rowSums(ge$S)[paste0("lab", 1:ng)] # the steady-state output-labor ratio in the Diamond model alpha.firm^(1 / (1 - beta.prod.firm)) * (beta.consumer[3] * (1 - beta.prod.firm) / (1 + gr.laborer))^(beta.prod.firm / (1 - beta.prod.firm)) # the captial-labor ratios rowSums(ge$D[paste0("prod", 1:ng), paste0("firm", 1:ng)]) / rowSums(ge$S[paste0("lab", 1:ng), paste0("consumer", 1:ng)]) # the steady-state captial-labor ratio in the Diamond model alpha.firm^(1 / (1 - beta.prod.firm)) * (beta.consumer[3] * (1 - beta.prod.firm) / (1 + gr.laborer))^(1 / (1 - beta.prod.firm)) ## ng <- 15 alpha.firm <- 2 beta.prod.firm <- 0.5 beta.consumer <- c(1, 1, 1) / 3 labor.first <- c(50, 50, 0) labor.last <- c(50, 50) y1 <- 8 gr.laborer <- 0 f() #### Assume that consumers live for two periods and consume labor (i.e., leisure). ng <- 10 # the number of generations alpha.firm <- 2 # the efficient parameter of firms beta.prod.firm <- 0.5 # the product (i.e. capital) share parameter of firms beta.consumer <- prop.table(c( lab1 = 1, lab2 = 1, prod1 = 1, prod2 = 1 )) # the share parameter of consumers labor <- c(100, 0) # the labor supply of each generation y1 <- 8 # the initial product supply names.commodity <- c(paste0("lab", 1:(ng + 1)), paste0("prod", 1:(ng + 1))) names.agent <- c(paste0("consumer", 1:ng), paste0("firm", 1:ng)) n <- length(names.commodity) # the number of commodity kinds m <- length(names.agent) # the number of agent kinds # the exogenous supply matrix. S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent)) for (k in 1:ng) { S0Exg[paste0("lab", k:(k + 1)), paste0("consumer", k)] <- labor } if (labor[2] == 0) S0Exg[paste0("lab", ng + 1), paste0("consumer", ng)] <- labor[1] S0Exg["prod1", "consumer1"] <- y1 B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent)) for (k in 1:ng) { B[paste0("prod", k + 1), paste0("firm", k)] <- 1 } dstl.consumer <- list() for (k in 1:ng) { dstl.consumer[[k]] <- node_new( "util", type = "CD", alpha = 1, beta = beta.consumer, # prop.table(c(1e-5,1e-5,0.5,0.5)), paste0("lab", k:(k + 1)), paste0("prod", k:(k + 1)) ) } # Assume that consumers live for three periods. # dstl.consumer <- list() # for (k in 1:(ng - 1)) { # dstl.consumer[[k]] <- node_new( # "util", # type = "CD", alpha = 1, # beta = rep(1 / 6, 6), # paste0("lab", k:(k + 2)), paste0("prod", k:(k + 2)) # ) # } # # dstl.consumer[[ng]] <- node_new( # "util", # type = "CD", alpha = 1, # beta = rep(1 / 4, 4), # paste0("lab", ng:(ng + 1)), paste0("prod", ng:(ng + 1)) # ) dstl.firm <- list() for (k in 1:ng) { dstl.firm[[k]] <- node_new( "prod", type = "CD", alpha = alpha.firm, beta = c(1 - beta.prod.firm, beta.prod.firm), paste0("lab", k), paste0("prod", k) ) } ge <- sdm2( A = c(dstl.consumer, dstl.firm), B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "lab1" ) ge$z growth_rate(ge$p[paste0("prod", 1:ng)]) + 1 growth_rate(ge$p[paste0("lab", 1:ng)]) + 1 ge$p[paste0("prod", 1:ng)] / ge$p[paste0("lab", 1:ng)] ```

gemOLG_Basic function