# The Identical Steady-state Equilibrium: Four Models Illustrating Dividend and Technological Progress Four models with labor-saving technological progress are presented to illustrate dividend, which have the same steady-state equilibrium. These models are as follows: (1) a real timeline model with head-tail adjustment; (2) a financial timeline model with dividend and head-tail adjustment; (3) a financial sequential model with dividend; (4) a financial time-circle model with dividend. ```r gemIntertemporal_Dividend_TechnologicalProgress(...) ``` ## Arguments - `...`: arguments to be passed to the function sdm2. ## Examples ```r #### (1) a real timeline model with head-tail adjustment. eis <- 0.8 # the elasticity of intertemporal substitution Gamma.beta <- 0.8 # the subjective discount factor gr.tech <- 0.02 # the technological progress rate gr.lab <- 0.03 # the growth rate of labor supply gr <- (1 + gr.lab) * (1 + gr.tech) - 1 # the growth rate np <- 4 # the number of economic periods 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 * (1 + gr.lab)^(0:(np - 2)) S0Exg["prod1", "consumer"] <- 140 # the product supply in the first period, which will be adjusted. # 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), "cc1" ) node_set(dstl.firm[[k]], "cc1", type = "Leontief", a = 1 / (1 + gr.tech)^(k - 1), paste0("lab", k) ) } node_plot(dstl.firm[[np - 1]], TRUE) 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", policy = makePolicyHeadTailAdjustment(gr = gr, np = np) ) sserr(eis, Gamma.beta, gr) # the steady-state equilibrium return rate ge$p[1:(np - 1)] / ge$p[2:np] - 1 # the steady-state equilibrium return rate ge$z growth_rate(ge$z) ## (2) a financial timeline model with dividend and head-tail adjustment. yield.rate <- sserr(eis, Gamma.beta, gr, prepaid = TRUE) n <- 2 * np # the number of commodity kinds m <- np # the number of agent kinds names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:(np - 1)), "claim") 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 * (1 + gr.lab)^(0:(np - 2)) S0Exg["claim", "consumer"] <- 100 S0Exg["prod1", "consumer"] <- 140 # the product supply in the first period, which will be adjusted. # 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 = "FIN", rate = c(1, yield.rate), "cc1", "claim" ) node_set(dstl.firm[[k]], "cc1", type = "CD", alpha = 2, beta = c(0.5, 0.5), paste0("prod", k), "cc1.1" ) node_set(dstl.firm[[k]], "cc1.1", type = "Leontief", a = 1 / (1 + gr.tech)^(k - 1), paste0("lab", k) ) } dst.consumer <- node_new( "util", type = "CES", es = 1, alpha = 1, beta = prop.table(rep(1, np)), # 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", policy = makePolicyHeadTailAdjustment(gr = gr, np = np) ) ge$z ## (3) a financial sequential model with dividend. dst.firm <- node_new("output", type = "FIN", rate = c(1, dividend.rate = yield.rate), "cc1", "equity.share" ) node_set(dst.firm, "cc1", type = "CD", alpha = 2, beta = c(0.5, 0.5), "prod", "cc1.1" ) node_set(dst.firm, "cc1.1", type = "Leontief", a = 1, "lab" ) node_plot(dst.firm, TRUE) dst.laborer <- node_new("util", type = "Leontief", a = 1, "prod" ) dst.shareholder <- Clone(dst.laborer) ge <- sdm2( A = list(dst.firm, dst.laborer, dst.shareholder), B = diag(c(1, 0, 0)), S0Exg = { S0Exg <- matrix(NA, 3, 3) S0Exg[2, 2] <- 100 / (1 + gr.lab) S0Exg[3, 3] <- 100 S0Exg }, names.commodity = c("prod", "lab", "equity.share"), names.agent = c("firm", "laborer", "shareholder"), numeraire = "equity.share", maxIteration = 1, numberOfPeriods = 20, z0 = c(143.1811, 0, 0), policy = list(policy.technology <- function(time, A, state) { node_set(A[[1]], "cc1.1", a = 1 / (1 + gr.tech)^(time - 1) ) state$S[2, 2] <- 100 * (1 + gr.lab)^(time - 1) state }, policyMarketClearingPrice), ts = TRUE ) ge$ts.z[, 1] growth_rate(ge$ts.z[, 1]) ## (4) a financial time-circle model with dividend. zeta <- (1 + gr)^np # the ratio of repayments to loans n <- 2 * np + 1 # the number of commodity kinds m <- np + 1 # the number of agent kinds names.commodity <- c(paste0("prod", 1:np), paste0("lab", 1:np), "claim") 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.lab)^(0:(np - 1)) S0Exg["claim", "consumer"] <- 100 # 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("output", type = "FIN", rate = c(1, yield.rate), "cc1", "claim" ) node_set(dstl.firm[[k]], "cc1", type = "CD", alpha = 2, beta = c(0.5, 0.5), paste0("prod", k), "cc1.1" ) node_set(dstl.firm[[k]], "cc1.1", type = "Leontief", a = 1 / (1 + gr.tech)^(k - 1), paste0("lab", k) ) } dst.consumer <- node_new( "util", type = "CES", es = 1, alpha = 1, beta = prop.table(rep(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$z growth_rate(ge$z[1:np]) ``` ## See Also `gemIntertemporal_Dividend`

gemIntertemporal_Dividend_TechnologicalProgress function