R/gemQuasilinearPureExchange_2_2.R

In GE: General Equilibrium Modeling

#' @export
#' @title A Pure Exchange Economy with a Quasilinear Utility Function
#' @aliases gemQuasilinearPureExchange_2_2
#' @description An example of a pure exchange economy with a quasilinear utility function (Karaivanov, see the reference).
#' @details Suppose there are only two goods (bananas and fish) and 2 consumers (Annie and Ben) in an exchange economy.
#' Annie has a utility function x_1^(1/3) * x_2^(2/3) where x_1 is the amount of fish she eats and x_2 is the amount of
#' bananas she eats.
#' Annie has an endowment of 3 kilos of fish and 7 bananas.
#' Ben has a utility function x_1 + 1.5 * log(x_2) and endowments of 4 kilos of fish and 0 bananas.
#' Assume the price of bananas is 1.
#' See the reference for more details.
#' @param A a demand structure tree list, a demand coefficient 2-by-2 matrix (alias demand structure matrix)
#' or a function A(state) which returns a 2-by-2 matrix (see \code{\link{sdm2}}).
#' @param Endowment a 2-by-2 matrix.
#' @param policy a policy function (see \code{\link{sdm2}}).
#' @return A general equilibrium.
#' @references http://www.sfu.ca/~akaraiva/CE_example.pdf
#' @examples
#' \donttest{
#' demand_consumer2 <- function(w, p) {
#'   QL_demand(w = w, p = p, alpha = 1.5, type = "log")
#' }
#'
#' A <- function(state) {
#'   a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
#'   a2 <- demand_consumer2(state$w[2], state$p)
#'   cbind(a1, a2)
#' }
#'
#' ge.mat <- gemQuasilinearPureExchange_2_2(A = A)
#' ge.mat
#'
#' ## Use a dstl and a policy function to compute the general equilibrium above.
#' dst.consumer1 <- node_new("util",
#'                           type = "CD", alpha = 1, beta = c(1 / 3, 2 / 3),
#'                           "fish", "banana"
#' )
#' dst.consumer2 <- node_new("util",
#'                           type = "Leontief", a = c(1, 1),
#'                           "fish", "banana"
#' )
#'
#' dstl <- list(dst.consumer1, dst.consumer2)
#'
#' policy.quasilinear <- function(A, state) {
#'   wealth <- t(state$p) %*% state$S
#'   A[[2]]$a <- demand_consumer2(wealth[2], state$p)
#' }
#'
#' ge.dstl <- gemQuasilinearPureExchange_2_2(
#'   A = dstl,
#'   policy = policy.quasilinear
#' )
#' ge.dstl
#'
#' #### Another example. Now Ben has a utility function x_1 + sqrt(x_2).
#' demand_consumer2 <- function(w, p) {
#'   QL_demand(w = w, p = p, alpha = 1, beta = 0.5, type = "power")
#' }
#'
#' A <- function(state) {
#'   a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
#'   a2 <- demand_consumer2(state$w[2], state$p)
#'   cbind(a1, a2)
#' }
#'
#' ge.2_2 <- gemQuasilinearPureExchange_2_2(A = A)
#' ge.2_2
#'
#' ## another computation method for the economy above
#' A <- function(state) {
#'   a1 <- CD_A(1, rbind(1 / 3, 2 / 3, 0, 0), state$p)
#'   a2 <- c(0, 0, 1, 0)
#'   a3 <- c(1, 0, 0, 0) # firm 1
#'   a4 <- CD_A(1, rbind(0, 1 / 2, 0, 1 / 2), state$p) # firm 2
#'   cbind(a1, a2, a3, a4)
#' }
#'
#' ge.4_4 <- sdm2(
#'   A = A,
#'   B = {
#'     B <- matrix(0, 4, 4)
#'     B[3, 3] <- 1
#'     B[3, 4] <- 1
#'     B
#'   },
#'   S0Exg = {
#'     S0Exg <- matrix(NA, 4, 4)
#'     S0Exg[1:2, 1] <- c(3, 7)
#'     S0Exg[1:2, 2] <- c(4, 0)
#'     S0Exg[4, 1:2] <- c(0, 1)
#'     S0Exg
#'   },
#'   names.commodity = c("fish", "banana", "util2", "land"),
#'   names.agent = c("Annie", "Ben", "firm1", "firm2"),
#'   numeraire = "banana"
#' )
#' ge.4_4
#'
#' #### another example
#' n.fish.demander <- 21
#' wealth <- 20 # the wealth (or income) of each fish demander
#' fish.supply <- 12
#' aggregare.demand <- function(p) {
#'   result <- 0
#'   for (alpha in seq(5, 15, length.out = n.fish.demander)) {
#'     result <- result + QL_demand(w = wealth, p = p, alpha = alpha, beta = 1, type = "min")
#'   }
#'   result
#' }
#'
#' ge <- sdm2(
#'   A = function(state) {
#'     a1 <- aggregare.demand(state$p / state$p[1])
#'     a2 <- c(1, 0)
#'     cbind(a1, a2)
#'   },
#'   B = matrix(0, 2, 2),
#'   S0Exg = matrix(c(
#'     n.fish.demander * wealth, 0,
#'     0, fish.supply
#'   ), 2, 2, TRUE),
#'   names.commodity = c("gold", "fish"),
#'   names.agent = c("fish.demander", "fish.supplier"),
#'   numeraire = "gold",
#'   p0 = c(1, 1) # p0 = c(1, 9.25)
#' )
#'
#' ge$p
#' ge$z
#' ge$D
#' ge$S
#'
#' aggregare.demand.fish <- c()
#' p2.set <- seq(0, 16, 0.01)
#' for (p2 in p2.set) {
#'   aggregare.demand.fish <- c(
#'     aggregare.demand.fish,
#'     aggregare.demand(c(1, p2))[2]
#'   )
#' }
#'
#' plot(aggregare.demand.fish,
#'      p2.set,
#'      xlab = "demand for fish", ylab = "price of fish", pch = 20
#' )
#' abline(v = fish.supply)
#' grid()
#' points(ge$D[2, 1], ge$p[2], pch = 8, col = "red")
#' }

gemQuasilinearPureExchange_2_2 <- function(A,
                                           Endowment = matrix(c(
                                             3, 4,
                                             7, 0
                                           ), 2, 2, TRUE),
                                           policy = NULL) {
  ge <- sdm2(
    A = A,
    B = matrix(0, 2, 2),
    S0Exg = Endowment,
    names.commodity = c("fish", "banana"),
    names.agent = c("Annie", "Ben"),
    numeraire = "banana",
    policy = policy
  )
}