# Some Intertemporal (Timeline and Time-circle) Models with Land, Two Consumers, and Three Types of Firms Some intertemporal (timeline and time-circle) models with two consumers (i.e. a laborer and a landowner) and three types of firms (i.e. wheat producers, iron producers and iron leaser). Here the iron leasing firm is actually a quasi-firm, which does not require primary factors such as labor and land in its production process. There are four commodities in the model, namely wheat, iron, iron leased out as a capital good, labor and land. ```r gemIntertemporal_5_5(...) ``` ## Arguments - `...`: arguments to be passed to the function sdm2. ## Examples ```r #### a timeline model. np <- 15 # the number of economic periods gr <- 0 # the growth rate in the steady state equilibrium eis <- 1 # the elasticity of intertemporal substitution of consumers Gamma.beta <- 0.97 # the subjective discount factor of consumers last.beta.laborer <- 0 last.beta.landowner <- 0 depreciation.rate <- 0.06 alpha.firm.wheat <- rep(5, np - 1) alpha.firm.iron <- rep(5, np - 1) y1.wheat <- 200 y1.iron <- 100 names.commodity <- c( paste0("wheat", 1:np), paste0("iron", 1:np), paste0("cap", 1:(np - 1)), paste0("lab", 1:(np - 1)), paste0("land", 1:(np - 1)) ) names.agent <- c( paste0("firm.wheat", 1:(np - 1)), paste0("firm.iron", 1:(np - 1)), paste0("quasifirm.cap", 1:(np - 1)), # a quasifirm "laborer", "landowner" ) n <- length(names.commodity) # the number of commodity kinds, i.e. 5 * np - 3 m <- length(names.agent) # the number of agent kinds, i.e. 3 * np - 1 # the exogenous supply matrix. S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent)) S0Exg["wheat1", "laborer"] <- y1.wheat S0Exg["iron1", "landowner"] <- y1.iron S0Exg[paste0("lab", 1:(np - 1)), "laborer"] <- 100 * (1 + gr)^(0:(np - 2)) # the supply of labor S0Exg[paste0("land", 1:(np - 1)), "landowner"] <- 100 * (1 + gr)^(0:(np - 2)) # the supply of land # the output coefficient matrix. B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent)) for (k in 1:(np - 1)) { B[paste0("wheat", k + 1), paste0("firm.wheat", k)] <- 1 B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1 B[paste0("cap", k), paste0("quasifirm.cap", k)] <- 1 B[paste0("iron", k + 1), paste0("quasifirm.cap", k)] <- 1 - depreciation.rate } dstl.firm.wheat <- dstl.firm.iron <- dstl.quasifirm.cap <- list() for (k in 1:(np - 1)) { dstl.firm.wheat[[k]] <- node_new( "prod", type = "CES", es = 1, alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4), paste0("cap", k), paste0("lab", k), paste0("land", k) ) dstl.firm.iron[[k]] <- node_new( "prod", type = "CES", es = 1, alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2), paste0("cap", k), paste0("lab", k), paste0("land", k) ) dstl.quasifirm.cap[[k]] <- node_new( "output", type = "Leontief", a = 1, paste0("iron", k) ) } tmp.beta <- Gamma.beta^(1:(np - 1)) tmp.beta <- tmp.beta / tmp.beta[np - 1] tmp.beta <- c(tmp.beta, last.beta.laborer) dst.laborer <- node_new( "util", type = "CES", es = eis, alpha = 1, beta = prop.table(tmp.beta), paste0("cc", 1:(np - 1)), paste0("wheat", np) ) for (k in 1:(np - 1)) { node_set(dst.laborer, paste0("cc", k), type = "CES", es = 1, alpha = 1, beta = c(0.4, 0.4, 0.2), paste0("wheat", k), paste0("lab", k), paste0("land", k) ) } tmp.beta <- Gamma.beta^(1:(np - 1)) tmp.beta <- tmp.beta / tmp.beta[np - 1] tmp.beta <- c(tmp.beta, last.beta.landowner) dst.landowner <- node_new( "util", type = "CES", es = eis, alpha = 1, beta = prop.table(tmp.beta), paste0("cc", 1:(np - 1)), paste0("iron", np) ) for (k in 1:(np - 1)) { node_set(dst.landowner, paste0("cc", k), type = "CES", es = 1, alpha = 1, beta = c(0.2, 0.4, 0.4), paste0("wheat", k), paste0("lab", k), paste0("land", k) ) } ge <- sdm2( A = c( dstl.firm.wheat, dstl.firm.iron, dstl.quasifirm.cap, dst.laborer, dst.landowner ), B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "lab1", policy = makePolicyMeanValue(50), ts = TRUE, priceAdjustmentVelocity = 0.03 ) ge$p ge$z plot(ge$z[2 * (np - 1) + (1:(np - 1))], type = "b", pch = 20) lines(1:(np - 1), ge$z[1:(np - 1)], type = "b", pch = 21) lines(1:(np - 1), ge$z[np - 1 + (1:(np - 1))], type = "b", pch = 22) legend("topleft", c("cap","wheat", "iron"), pch = 20:22) #### a time-circle model. np <- 5 # the number of economic periods gr <- 0.03 # the growth rate in the steady state equilibrium eis <- 0.5 # the elasticity of intertemporal substitution of consumers Gamma.beta <- 0.97 # the subjective discount factor of consumers es.firm <- 1 depreciation.rate <- 0.06 zeta <- (1 + gr)^np # the ratio of repayments to loans yield.rate <- sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr, prepaid = TRUE) captial.share.laborer <- 0.7795 captial.share.landowner <- 1 - captial.share.laborer alpha.firm.wheat <- rep(5, np) alpha.firm.iron <- rep(5, np) names.commodity <- c( paste0("wheat", 1:np), paste0("iron", 1:np), paste0("cap", 1:np), paste0("lab", 1:np), paste0("land", 1:np), "claim" ) names.agent <- c( paste0("firm.wheat", 1:np), paste0("firm.iron", 1:np), paste0("quasifirm.cap", 1:np), # a quasifirm "laborer", "landowner" ) 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)) S0Exg[paste0("lab", 1:np), "laborer"] <- 100 * (1 + gr)^(0:(np - 1)) # the supply of labor S0Exg[paste0("land", 1:np), "landowner"] <- 100 * (1 + gr)^(0:(np - 1)) # the supply of land S0Exg["claim", "laborer"] <- 100 * captial.share.laborer S0Exg["claim", "landowner"] <- 100 * captial.share.landowner # the output coefficient matrix. B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent)) for (k in 1:(np - 1)) { B[paste0("wheat", k + 1), paste0("firm.wheat", k)] <- 1 B[paste0("iron", k + 1), paste0("firm.iron", k)] <- 1 B[paste0("cap", k), paste0("quasifirm.cap", k)] <- 1 B[paste0("iron", k + 1), paste0("quasifirm.cap", k)] <- 1 - depreciation.rate } B[paste0("wheat", 1), paste0("firm.wheat", np)] <- 1 / zeta B[paste0("iron", 1), paste0("firm.iron", np)] <- 1 / zeta B[paste0("cap", np), paste0("quasifirm.cap", np)] <- 1 B[paste0("iron", 1), paste0("quasifirm.cap", np)] <- (1 - depreciation.rate) / zeta dstl.firm.wheat <- dstl.firm.iron <- dstl.quasifirm.cap <- list() for (k in 1:(np - 1)) { dstl.firm.wheat[[k]] <- node_new( "prod", type = "CES", es = es.firm, alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4), paste0("cap", k), paste0("lab", k), paste0("land", k) ) dstl.firm.iron[[k]] <- node_new( "prod", type = "CES", es = es.firm, alpha = alpha.firm.iron[k], beta = c(0.4, 0.4, 0.2), paste0("cap", k), paste0("lab", k), paste0("land", k) ) dstl.quasifirm.cap[[k]] <- node_new( "output", type = "Leontief", a = 1, paste0("iron", k) ) } dstl.firm.wheat[[np]] <- node_new( "prod", type = "FIN", rate = c(1, (1 + yield.rate)^np - 1), "cc1", "claim" ) node_set(dstl.firm.wheat[[np]], "cc1", type = "CES", es = es.firm, alpha = alpha.firm.wheat[k], beta = c(0.2, 0.4, 0.4), paste0("cap", np), paste0("lab", np), paste0("land", np) ) dstl.firm.iron[[np]] <- node_new( "prod", type = "FIN", rate = c(1, (1 + yield.rate)^np - 1), "cc1", "claim" ) node_set(dstl.firm.iron[[np]], "cc1", type = "CES", es = es.firm, alpha = alpha.firm.wheat[k], beta = c(0.4, 0.4, 0.2), paste0("cap", np), paste0("lab", np), paste0("land", np) ) return.rate <- sserr(eis = eis, Gamma.beta = Gamma.beta, gr = gr) fund.occupancy.rate <- (1 - depreciation.rate) / (1 + return.rate) # The prepaid rent rate (i.e. the prepaid-rent-to-price ratio) of # the capital good is equal to 1 minus fund.occupancy.rate. dstl.quasifirm.cap[[np]] <- node_new( "prod", type = "FIN", rate = c(1, ((1 + yield.rate)^np - 1) * fund.occupancy.rate), "cc1", "claim" ) node_set(dstl.quasifirm.cap[[np]], "cc1", type = "Leontief", a = 1, paste0("iron", np) ) dst.laborer <- node_new( "util", type = "CES", es = eis, alpha = 1, beta = prop.table(Gamma.beta^(1:np)), paste0("cc", 1:np) ) for (k in 1:np) { node_set(dst.laborer, paste0("cc", k), type = "CES", es = 1, alpha = 1, beta = c(0.4, 0.4, 0.2), paste0("wheat", k), paste0("lab", k), paste0("land", k) ) } dst.landowner <- node_new( "util", type = "CES", es = eis, alpha = 1, beta = prop.table(Gamma.beta^(1:np)), paste0("cc", 1:np) ) for (k in 1:np) { node_set(dst.landowner, paste0("cc", k), type = "CES", es = 1, alpha = 1, beta = c(0.2, 0.4, 0.4), paste0("wheat", k), paste0("lab", k), paste0("land", k) ) } ge.tc <- sdm2( A = c( dstl.firm.wheat, dstl.firm.iron, dstl.quasifirm.cap, dst.laborer, dst.landowner ), B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "lab1", numberOfPeriods = 1000, priceAdjustmentVelocity = 0.03 ) ge.tc$p growth_rate(ge.tc$p) ge.tc$z growth_rate(ge.tc$z) ## the corresponding sequential model with the same steady-state equilibrium. dst.firm.wheat <- node_new("prod", type = "FIN", rate = c(1, yield.rate), "cc1", "equity.share.wheat" ) node_set(dst.firm.wheat, "cc1", type = "CES", es = es.firm, alpha = 5, beta = c(0.2, 0.4, 0.4), "cap", "lab", "land" ) dst.firm.iron <- node_new("prod", type = "FIN", rate = c(1, yield.rate), "cc1", "equity.share.iron" ) node_set(dst.firm.iron, "cc1", type = "CES", es = es.firm, alpha = 5, beta = c(0.4, 0.4, 0.2), "cap", "lab", "land" ) dst.quasifirm.cap <- node_new("prod", type = "FIN", rate = c(1, yield.rate * fund.occupancy.rate), "cc1", "equity.share.cap" ) node_set(dst.quasifirm.cap, "cc1", type = "Leontief", a = 1, "iron" ) dst.laborer <- node_new("util", type = "CES", es = 1, alpha = 1, beta = c(0.4, 0.4, 0.2), "wheat", "lab", "land" ) dst.landowner <- node_new("util", type = "CES", es = 1, alpha = 1, beta = c(0.2, 0.4, 0.4), "wheat", "lab", "land" ) ge.seq <- sdm2( A = list( dst.firm.wheat, dst.firm.iron, dst.quasifirm.cap, dst.laborer, dst.landowner ), B = matrix(c( 1, 0, 0, 0, 0, 0, 1, 1 - depreciation.rate, 0, 0, 0, 0, 1 + gr, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ), 8, 5, TRUE), S0Exg = matrix(c( NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 100, NA, NA, NA, NA, NA, 100, NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner, NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner, NA, NA, NA, 100 * captial.share.laborer, 100 * captial.share.landowner ), 8, 5, TRUE), names.commodity = c( "wheat", "iron", "cap", "lab", "land", "equity.share.wheat", "equity.share.iron", "equity.share.cap" ), names.agent = c("firm.wheat", "firm.iron", "quasifirm.cap", "laborer", "landowner"), numeraire = "lab", numberOfPeriods = 2000, priceAdjustmentVelocity = 0.03, GRExg = gr ) ge.seq$p ge.seq$z ge.tc$z ```

gemIntertemporal_5_5 function