MDCES_demand {GE}

Modified Displaced CES Demand Function

Description

Compute the modified displaced CES demand function. Firstly, the (unmodified) DCES demand vector and the (unmodified) utility level are computed under the given income and prices. Secondly, the modified beta and es are computed under the unmodified utility level. Finally, the DCES demand vector (namely the modified DCES demand vector) and the utility level (namely the modified DCES utility) are computed under the modified beta, the modified es, the given income and prices.

Usage

MDCES_demand(es, beta, xi, w, p, betaMod = NULL, esMod = NULL, detail = FALSE)

Arguments

es

the elasticity of substitution.

beta

an n-vector consisting of the marginal expenditure share coefficients (Fullerton, 1989).

xi

an n-vector. Each element of xi parameterizes whether the particular good is a necessity for the household (Acemoglu, 2009, page 152). For example, xi[i] > 0 may mean that the household needs to consume at least a certain amount of good i to survive.

w

a scalar indicating the income.

p

an n-vector indicating the prices.

betaMod

a function with the unmodified utility level u.unmod as the argument.

esMod

a function with the unmodified utility level u.unmod as the argument.

detail

If detail==FALSE, the modified demand vector is returned. If detail==TRUE, the returned vector consists of the modified demand vector, the modified utility, the modified es, the modified beta, the unmodified utility and the unmodified demand vector.

References

Acemoglu, D. (2009, ISBN: 9780691132921) Introduction to Modern Economic Growth. Princeton University Press.

Fullerton, D. (1989) Notes on Displaced CES Functional Forms. Available at: https://works.bepress.com/don_fullerton/39/

Examples

MDCES_demand(
  es = 1.7, beta = c(0.9, 0.1), xi = c(12, 48),
  w = 24, p = c(1, 1 / 400),
  betaMod = function(u.unmod) {
    beta2 <- min(0.1, 10 / u.unmod)
    c(1 - beta2, beta2)
  },
  detail = TRUE
)

#### An example of computing the daily
#### labor supply at various wage rates.
result <- c()

for (real.wage in 4:400) {
  x <- MDCES_demand(
    es = 1.7, beta = c(0.9, 0.1),
    xi = c(12, 48), w = 24,
    p = c(1, 1 / real.wage),
    betaMod = function(u.unmod) {
      beta2 <- min(0.1, 10 / u.unmod)
      c(1 - beta2, beta2)
    },
    detail = TRUE
  )

  lab.supply <- unname(24 - x[1])
  result <- rbind(
    result,
    c(real.wage, lab.supply, x)
  )
}

plot(result[, 1:2],
  type = "o", pch = 20,
  xlab = "hourly real wage",
  ylab = "daily labor supply"
)

#### A 2-by-2 general equilibrium model
#### with a MDCES demand function
ge <- sdm2(
  A = function(state) {
    a.firm <- CD_A(alpha = 5, Beta = c(0.5, 0.5), state$p)
    a.consumer <-
      MDCES_demand(
        es = 1, beta = c(0.5, 0.5), xi = c(0, 0), w = state$w[2], p = state$p,
        betaMod = function(u.unmod) {
          beta2 <- 0.95 * plogis(u.unmod, location = 2, scale = 2)
          c(1 - beta2, beta2)
        }
      )
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "lab"
)

ge$z
ge$D
MDCES_demand(
  es = 1, beta = c(0.5, 0.5), xi = c(0, 0),
  w = 1, p = ge$p,
  betaMod = function(u.unmod) {
    beta2 <- 0.95 * plogis(u.unmod, location = 2, scale = 2)
    c(1 - beta2, beta2)
  }
)