gem_3_2 {GE}

Some Simple 3-by-2 General Equilibrium Models

Description

Some simple 3-by-2 general equilibrium models with a firm and a consumer.

Usage

gem_3_2(...)

Arguments

...

arguments to be passed to the function sdm2.

References

http://www.econ.ucla.edu/riley/MAE/Course/SolvingForTheWE.pdf

He Zhangyong, Song Zheng (2010, ISBN: 9787040297270) Advanced Macroeconomics. Beijing: Higher Education Press.

Examples

ge.CD <- sdm2(
  A = function(state) {
    ## the vector of demand coefficients of the firm
    a1 <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), state$p)
    ## the vector of demand coefficients of the consumer
    a2 <- c(1, 0, 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge.CD$p
ge.CD$z
ge.CD$D
ge.CD$S

#### Example 2 in the ucla reference
## By introducing a new factor of production (called land here)
## a firm with diminishing returns to scale can be converted into
## a firm with constant returns to scale.
ge2.CD <- sdm2(
  A = function(state) {
    a.firm <- CD_A(alpha = 6, Beta = c(0, 0.5, 0.5), state$p)
    a.consumer <- CD_A(alpha = 1, Beta = c(0.2, 0.8, 0), state$p)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 81,
    NA, 1
  ), 3, 2, TRUE),
  names.commodity = c("prod", "lab", "land"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge2.CD$p
ge2.CD$z
ge2.CD$D
ge2.CD$S

####
ge.SCES <- sdm2(
  A = function(state) {
    a1 <- SCES_A(es = 0.5, alpha = 1, Beta = c(0, 0.5, 0.5), p = state$p)
    a2 <- c(1, 0, 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge.SCES$p
ge.SCES$z
ge.SCES$D
ge.SCES$S

####
ge2.SCES <- sdm2(
  A = function(state) {
    a1 <- SCES_A(es = 0.5, alpha = 1, Beta = c(0.2, 0.4, 0.4), p = state$p)
    a2 <- c(1, 0, 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge2.SCES$p
ge2.SCES$z
ge2.SCES$D
ge2.SCES$S

#### nested production function
dst.firm <- node_new(
  "prod",
  type = "Leontief",
  a = c(0.2, 0.8),
  "prod", "cc1"
)
node_set(dst.firm, "cc1",
         type = "SCES",
         es = 0.5, alpha = 1, beta = c(0.5, 0.5),
         "cap", "lab"
)

dst.consumer <- node_new(
  "util",
  type = "Leontief", a = 1,
  "prod"
)

ge3.SCES <- sdm2(
  A = list(dst.firm, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100,
    NA, 100
  ), 3, 2, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge3.SCES$p
ge3.SCES$z
ge3.SCES$D
ge3.SCES$S

#### a model with a quasilinear utility function (see He and Song, 2010, page 19).
alpha.firm <- 2
beta.cap.firm <- 0.6
beta.lab.firm <- 1 - beta.cap.firm
theta.consumer <- 0.8
lab.supply <- 2
cap.supply <- 1

ge <- sdm2(
  A = function(state) {
    a1 <- CD_A(alpha.firm, rbind(0, beta.lab.firm, beta.cap.firm), state$p)

    demand.lab.prod <- QL_demand(
      w = state$w[2], p = state$p[2:1], # the prices of lab and prod
      alpha = 1, beta = theta.consumer, type = "CRRA"
    )
    a2 <- c(demand.lab.prod[2:1], 0)
    cbind(a1, a2)
  },
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, lab.supply,
    NA, cap.supply
  ), 3, 2, TRUE),
  names.commodity = c("prod", "lab", "cap"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge$p
ge$z
ge$D
ge$S

# the equilibrium leisure
lab.supply - (beta.lab.firm * (alpha.firm * cap.supply^beta.cap.firm)^(1 - theta.consumer))^
  (1 / (beta.cap.firm + beta.lab.firm * theta.consumer))

# the equilibrium price of labor
w <- ((1 - beta.cap.firm)^(1 - beta.cap.firm) * (alpha.firm * cap.supply^beta.cap.firm))^
  (theta.consumer / (beta.cap.firm + (1 - beta.cap.firm) * theta.consumer))

# the equilibrium price of capital goods
beta.cap.firm * w^(1 / theta.consumer) / cap.supply