GE (version 0.2.5)

gemQuasilinearPureExchange_2_2: A Pure Exchange Economy with a Quasilinear Utility Function

Description

An example of a pure exchange economy with a quasilinear utility function (Karaivanov, see the reference).

Usage

gemQuasilinearPureExchange_2_2(
  A,
  Endowment = matrix(c(3, 4, 7, 0), 2, 2, TRUE),
  policy = NULL
)

Arguments

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 sdm2).

Endowment

a 2-by-2 matrix.

policy

a policy function (see sdm2).

Value

A general equilibrium.

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.

References

http://www.sfu.ca/~akaraiva/CE_example.pdf

Examples

# NOT RUN {
demand_consumer2 <- function(p, w) {
  a <- 1.5
  d <- rbind(0, 0)
  w <- w / p[1]
  p <- p / p[1] # normalize the wealth and prices
  d[2] <- a / p[2]
  if (d[2] * p[2] > w) {
    d[2] <- w / p[2]
    d[1] <- 0
  } else {
    d[1] <- w - d[2] * p[2]
  }

  d
}

A <- function(state) {
  a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
  a2 <- demand_consumer2(state$p, state$w[2])
  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(dstl, state) {
  wealth <- t(state$p) %*% state$S
  dstl[[2]]$a <- demand_consumer2(state$p, wealth[2])
}

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(p, w) {
  d <- rbind(0, 0)
  w <- w / p[1]
  p <- p / p[1] # normalize the wealth and prices
  d[2] <- (2 * p[2])^-2
  if (d[2] * p[2] > w) {
    d[2] <- w / p[2]
    d[1] <- 0
  } else {
    d[1] <- w - d[2] * p[2]
  }

  d
}

A <- function(state) {
  a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
  a2 <- demand_consumer2(state$p, state$w[2])
  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

#### an example with a LES (linear expenditure system) utility function
demand_LES <- function(p, w,
                       gamma = c(0.1, 0.2),
                       beta = c(0.4, 0.6)) {
  d <- c()
  discretionary.income <- w - sum(p * gamma)
  for (k in seq_along(gamma)) {
    d[k] <- gamma[k] + beta[k] * discretionary.income / p[k]
  }

  d
}

A <- function(state) {
  a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
  a2 <- demand_LES(state$p, state$w[2])
  cbind(a1, a2)
}

sdm2(
  A = A,
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    3, 4,
    7, 0
  ), 2, 2, TRUE),
  names.commodity = c("fish", "banana"),
  names.agent = c("Annie", "Ben"),
  numeraire = "banana"
)
# }