# Some Examples Illustrating Endogenous Equilibrium Interest Rates and Foreign Exchange Rates in an Intertemporal Pure Exchange Economy with Two Currencies These examples illustrate (endogenous) equilibrium primitive interest rates and foreign exchange rates in an intertemporal pure exchange economy with two currencies. Assume that the velocity of money is equal to one, that is, money circulates once per period. ```r gemIntertemporal_EndogenousEquilibriumInterestRate_ForeignExchangeRate(...) ``` ## Arguments - `...`: arguments to be passed to the function sdm2. ## Examples ```r f <- function(ir, es = 1, beta, beta_prime = beta, return.ge = FALSE) { np <- length(beta) / 2 n <- 4 * np # the number of commodity kinds m <- 2 # the number of agent kinds ir[c(np, 2 * np)] <- last.ir <- 1000 dst.consumer1 <- node_new( "util", type = "SCES", alpha = 1, es = es, beta = beta, paste0("cc", 1:(2 * np)) ) for (k in 1:(2 * np)) { node_set( dst.consumer1, paste0("cc", k), type = "FIN", rate = ir[k], paste0("lab", k), paste0("money", k) ) } dst.consumer2 <- Clone(dst.consumer1) dst.consumer2$beta <- beta_prime names.commodity <- c( paste0("lab", 1:(2 * np)), paste0("money", 1:(2 * np)) ) names.agent <- paste0("consumer", 1:m) ## the exogenous supply matrix. S0Exg <- matrix(NA, n, m, dimnames = list(names.commodity, names.agent)) S0Exg[paste0("lab", 1:np), "consumer1"] <- S0Exg[paste0("lab", (np + 1):(2 * np)), "consumer2"] <- 100 S0Exg[paste0("money", 1:np), "consumer1"] <- S0Exg[paste0("money", (np + 1):(2 * np)), "consumer2"] <- 1 # the output coefficient matrix. B <- matrix(0, n, m, dimnames = list(names.commodity, names.agent)) ge <- sdm2( A = list(dst.consumer1, dst.consumer2), B = B, S0Exg = S0Exg, names.commodity = names.commodity, names.agent = names.agent, numeraire = "lab1", ts = TRUE ) tmp <- rowSums(ge$SV) ts.trading.value <- tmp[paste0("lab", 1:(2 * np))] * (1 + ir) ir[1:(np - 1)] <- ts.trading.value[1:(np - 1)] / ts.trading.value[2:np] - 1 ir[(np + 1):(2 * np - 1)] <- ts.trading.value[(np + 1):(2 * np - 1)] / ts.trading.value[(np + 2):(2 * np)] - 1 ir <- pmax(1e-6, ir) cat("ir: ", ir, "\n") cat("foreign.exchange.rate: ", ts.trading.value[(np + 1):(2 * np)] / ts.trading.value[1:np], "\n") if (return.ge) { ge$ts.trading.value <- ts.trading.value return(ge) } else { return(ir) } } ## np <- 4 # the number of economic periods beta <- proportions(c(0.4, 0.3, 0.2, 0.1, 0.5, 0.25, 0.15, 0.1)) mat.ir <- iterate(rep(0.1, 2 * np), f, tol = 1e-4, beta = beta) beta[1:(np - 1)] / beta[2:np] - 1 # domestic interest rates beta[(np + 1):(2 * np - 1)] / beta[(np + 2):(2 * np)] - 1 # foreign interest rates beta[(np + 1):(2 * np)] / beta[1:np] # foreign exchange rates ge <- f(ir = tail(mat.ir, 1), es = 1.1, beta = beta, return.ge = TRUE) ## Assume that the two consumers have different preferences. random.beta <- function(np) { beta <- runif(np, 0.9, 1) |> cumprod() |> proportions() } set.seed(1) np <- 5 beta <- proportions(c(random.beta(np), random.beta(np))) beta_prime <- proportions(c(random.beta(np), random.beta(np))) mat.ir <- iterate(rep(0.1, 2 * np), f, tol = 1e-4, beta = beta, beta_prime = beta_prime) # Calculate the equilibrium exchange rates and # interest rates based on closed-form solutions. compute_fx_ir <- function(beta, beta_prime) { np <- length(beta) / 2 xi1 <- sum(beta_prime[1:np]) xi2 <- sum(beta[(np + 1):(2 * np)]) v_tilde <- xi2 * beta_prime + xi1 * beta names(v_tilde) <- paste0("v", seq_len(2 * np)) foreign.exchange.rate <- v_tilde[(np + 1):(2 * np)] / v_tilde[1:np] names(foreign.exchange.rate) <- paste0("epsilon", 1:np) r <- rep(NA_real_, 2 * np) for (i in 1:(2 * np - 1)) r[i] <- v_tilde[i] / v_tilde[i + 1] - 1 r[np] <- Inf r[2 * np] <- Inf names(r) <- paste0("r", 1:(2 * np)) list( ir = r, foreign.exchange.rate = foreign.exchange.rate ) } compute_fx_ir(beta, beta_prime) ``` ## See Also `gemIntertemporal_EndogenousEquilibriumInterestRate`

gemIntertemporal_EndogenousEquilibriumInterestRate_ForeignExchangeRate function