DCES {GE}

Displaced CES Utility Function and Displaced CES Demand Function

Description

The displaced CES utility function and the displaced CES demand function (Fullerton, 1989).

Usage

DCES(es, beta, xi, x)

DCES_demand(es, beta, xi, w, p)

DCES_compensated_demand(es, beta, xi, u, p)

DCES_indirect(es, beta, xi, w, p)

Arguments

es

a scalar indicating the elasticity of substitution.

beta

an n-vector consisting of the marginal expenditure share coefficients. The sum of all components of beta should be 1.

xi

an n-vector or a scalar. If xi is a scalar, it will be recycled to 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.

x

an n-vector consisting of the inputs.

w

a scalar indicating the income.

p

an n-vector indicating the prices.

u

a scalar indicating the utility level.

Value

The return values of these functions are as follows:
DCES: A scalar indicating the utility level.
DCES_demand: An n-vector indicating the demands.
DCES_compensated_demand: An n-vector indicating the compensated demands.
DCES_indirect: A scalar indicating the utility level.

Functions

• DCES: Compute the displaced CES utility function (Fullerton, 1989), e.g. (beta1 ^ (1 / es) * (x1 - xi1) ^ (1 - 1 / es) + beta2 ^ (1 / es) * (x2 - xi2) ^ (1 - 1 / es)) ^ (es / (es - 1) wherein beta1 + beta2 == 1.

  When es==1, the DCES utility function becomes the Stone-Geary utility function.

• DCES_demand: The displaced CES demand function (Fullerton, 1989).

• DCES_compensated_demand: The displaced CES compensated demand function (Fullerton, 1989).

• DCES_indirect: The displaced CES indirect utility function (Fullerton, 1989).

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

es <- 0.99
beta <- prop.table(1:5)
xi <- 0
w <- 500
p <- 2:6

x <- DCES_demand(
  es = es,
  beta = beta,
  xi = xi,
  w = w,
  p = p
)

DCES_demand(
  es = es,
  beta = prop.table(0:4),
  xi = 5:1,
  w = w,
  p = p
)

u <- DCES(
  es = es,
  beta = beta,
  xi = xi,
  x = x
)

SCES(
  es = es,
  alpha = 1,
  beta = beta,
  x = x
)

DCES_compensated_demand(
  es = es,
  beta = beta,
  xi = xi,
  u = u,
  p = p
)

DCES_compensated_demand(
  es = es,
  beta = beta,
  xi = seq(10, 50, 10),
  u = u,
  p = p
)

#### A 2-by-2 general equilibrium model
#### with a DCES utility function.
ge <- sdm2(
  A = function(state) {
    a.consumer <- DCES_demand(
      es = 2, beta = c(0.2, 0.8), xi = c(1000, 500),
      w = state$w[1], p = state$p
    )
    a.firm <- c(1.1, 0)
    cbind(a.consumer, a.firm)
  },
  B = diag(c(0, 1)),
  S0Exg = matrix(c(
    3500, NA,
    NA, NA
  ), 2, 2, TRUE),
  names.commodity = c("corn", "iron"),
  names.agent = c("consumer", "firm"),
  numeraire = "corn"
)

ge$p
ge$z
ge$A
ge$D

#### a 2-by-2 pure exchange economy
sdm2(
  A = function(state) {
    a1 <- CD_A(1, rbind(1 / 3, 2 / 3), state$p)
    a2 <- DCES_demand(
      es = 1, beta = c(0.4, 0.6), xi = c(0.1, 0.2),
      w = state$w[2], p = state$p
    )
    cbind(a1, a2)
  },
  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"
)

#### A 3-by-3 general equilibrium model
#### with a DCES utility function.
lab <- 1 # the amount of labor supplied by each laborer
n.laborer <- 100 # the number of laborers
ge <- sdm2(
  A = function(state) {
    a.firm.corn <- CD_A(alpha = 1, Beta = c(0, 0.5, 0.5), state$p)
    a.firm.iron <- CD_A(alpha = 5, Beta = c(0, 0.5, 0.5), state$p)
    a.laborer <- DCES_demand(
      es = 0, beta = c(0, 1, 0), xi = c(0.1, 0, 0),
      w = state$w[3] / n.laborer, p = state$p
    )

    cbind(a.firm.corn, a.firm.iron, a.laborer)
  },
  B = matrix(c(
    1, 0, 0,
    0, 1, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, NA, lab * n.laborer
  ), 3, 3, TRUE),
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "laborer"),
  numeraire = "lab",
  priceAdjustmentVelocity = 0.1
)

ge$z
ge$A
ge$D