iep {CGE}

Compute Instantaneous Equilibrium Path (alias Market Clearing Path)

Description

This function computes the instantaneous equilibrium path (alias market clearing path).

Usage

iep(A.iep = NULL, A = NULL,  B.iep = NULL, B = NULL,
    SExg.iep, InitialEndowments, nPeriods.iep, ...)

Arguments

A.iep

A.iep(state.iep) is a function which returns a demand coefficient matrix or a function A(state). state.iep is a list consisting of time (the iep time), p (the price vector at the iep time), z (output and utility vector at the iep time).

A

a demand coefficient matrix or a function A(state) which returns a demand coefficient matrix. If A.iep is not NULL, A will be ignored.

B.iep

B.iep(state.iep) is a function which returns a supply coefficient matrix or a function B(state) at the iep time.

B

a supply coefficient matrix or a function B(state) which returns a supply coefficient matrix. If B.iep is not NULL, B will be ignored.

SExg.iep

an exogenous supply matrix or a function SExg.iep(state.iep) which returns an exogenous supply matrix at the iep time.

InitialEndowments

a matrix indicating the initial endowments.

nPeriods.iep

number of periods of the instantaneous equilibrium path.

...

parameters of the function sdm.

Details

This function computes the instantaneous equilibrium path (alias market clearing path) of a dynamic economy with the structural dynamic model (the sdm function).

Value

a list of general equilibria.

Author(s)

LI Wu <[e-mail]>

References

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

LI Wu (2019, ISBN: 9787521804225) General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics. Beijing: Economic Science Press. (In Chinese)

LI Wu (2010) A Structural Growth Model and its Applications to Sraffa's System. http://www.iioa.org/conferences/18th/papers/files/104_20100729011_AStructuralGrowthModelanditsApplicationstoSraffasSstem.pdf

Torres, Jose L. (2016, ISBN: 9781622730452). Introduction to Dynamic Macroeconomic General Equilibrium Models (Second Edition). Vernon Press.

See Also

sdm; Example7.2

Examples

## example 6.4 of Acemoglu (2009, page 206)
discount.factor <- 0.97
return.rate <- 1 / discount.factor - 1

A <- function(state) {
  a1 <- CD_A(
    1, rbind(0.35, 0.65, 0),
    c(state$p[1] * (1 + return.rate), state$p[2:3])
  )
  a2 <- c(1, 0, 0)
  a1[3] <- state$p[1] * a1[1] * return.rate / state$p[3]
  cbind(a1, a2)
}

B <- matrix(c(
  1, 0,
  0, 1,
  0, 1
), 3, 2, TRUE)


SExg.iep <- {
  tmp <- matrix(NA, 3, 2)
  tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

InitialEndowments <- {
  tmp <- matrix(0, 3, 2)
  tmp[1, 1] <- 0.01
  tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

ge.list <- iep(
  A = A, B = B, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = 50
)

z <- t(sapply(ge.list, function(x) x$z))
matplot(z, type = "l")

z[1:49, 1] * (1 - 0.97 * 0.35) # the same as z[-1,2] (i.e. consumption)

# stochastic (instantaneous) equilibrium path (SEP) in the economy above.
nPeriods.iep <-  150
set.seed(1)
alpha.SEP <- rep(1, 50)
for (t in 51:nPeriods.iep) {
  alpha.SEP[t] <- exp(0.95 * log(alpha.SEP[t - 1]) +
    rnorm(1, sd = 0.01))
}

A.iep <- function(state.iep) {
  A <- function(state) {
    a1 <- CD_A(
      alpha.SEP[state.iep$time],
      rbind(0.35, 0.65, 0),
      c(state$p[1] * (1 + return.rate), state$p[2:3])
    )
    a2 <- c(1, 0, 0)
    a1[3] <- state$p[1] * a1[1] * return.rate / state$p[3]
    cbind(a1, a2)
  }

  return(A)
}

ge.list <- iep(
  A.iep = A.iep, B = B, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = nPeriods.iep
)

z <- t(sapply(ge.list, function(x) x$z))
matplot(z, type = "l")

## an example with two firms
sigma <- 0 # 0 implies Cobb-Douglas production functions
gamma1 <- 0.01
gamma2 <- 0.01
gamma3 <- 0.01
beta1 <- 0.35
beta2 <- 0.4

A.iep <- function(state.iep) {
  A <- function(state) {
    a1 <- CES_A(sigma, exp(gamma1 * (state.iep$time - 1)), rbind(beta1, 0, 1 - beta1), state$p)
    a2 <- CES_A(sigma, exp(gamma2 * (state.iep$time - 1)), rbind(beta2, 0, 1 - beta2), state$p)
    a3 <- c(0, 1, 0)
    cbind(a1, a2, a3)
  }

  return(A)
}

B <- diag(3)

SExg.iep <- function(state.iep) {
  tmp <- matrix(NA, 3, 3)
  tmp[3, 3] <- exp(gamma3 * (state.iep$time - 1))
  tmp
}

InitialEndowments <- {
  tmp <- matrix(0, 3, 3)
  tmp[1, 1] <- 0.01
  tmp[2, 2] <- 0.02
  tmp[3, 3] <- 1
  tmp
}

ge.list <- iep(
  A.iep = A.iep, B = B, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = 100, trace = FALSE
)

z <- t(sapply(ge.list, function(x) x$z)) # outputs and utility
matplot(z, type = "l")

diff(log(z)) # logarithmic growth rate

## an example with heterogeneous firms
A <- function(state) {
  a1 <- CD_A(1, rbind(0.35, 0.65), state$p)
  a2 <- CD_A(1.3, rbind(0.9, 0.1), state$p)
  a3 <- c(1, 0)
  cbind(a1, a2, a3)
}

B <- matrix(c(
  1, 1, 0,
  0, 0, 1
), 2, 3, TRUE)

SExg.iep <- {
  tmp <- matrix(NA, 2, 3)
  tmp[2, 3] <- 1
  tmp
}

InitialEndowments <- {
  tmp <- matrix(0, 2, 3)
  tmp[1, 1] <- tmp[1, 2] <- 0.01
  tmp[2, 3] <- 1
  tmp
}

ge.list <- iep(
  A = A, B = B, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = 200, trace = FALSE
)

z <- t(sapply(ge.list, function(x) x$z))
matplot(z, type = "l")

## an iep of the example (see Table 2.1 and 2.2) of the canonical dynamic
## macroeconomic general equilibrium model in Torres (2016).
discount.factor <- 0.97
return.rate <- 1 / discount.factor - 1
depreciation.rate <- 0.06

A <- function(state) {
  a1 <- CD_A(1, rbind(0, 0.65, 0.35, 0), state$p)
  a2 <- CD_A(1, rbind(0.4, 1 - 0.4, 0, 0), state$p)
  a3 <- c(1, 0, 0, state$p[1] * return.rate / state$p[4])
  cbind(a1, a2, a3)
}

B <- matrix(c(
  1, 0, 1 - depreciation.rate,
  0, 1, 0,
  0, 0, 1,
  0, 1, 0
), 4, 3, TRUE)

SExg.iep <- {
  tmp <- matrix(NA, 4, 3)
  tmp[2, 2] <- tmp[4, 2] <- 1
  tmp
}

InitialEndowments <- {
  tmp <- matrix(0, 4, 3)
  tmp[1, 1] <- 0.01
  tmp[2, 2] <- tmp[4, 2] <- 1
  tmp[3, 3] <- 0.01
  tmp
}

ge.list <- iep(
  A = A, B = B, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = 200, trace = FALSE
)

z <- t(sapply(ge.list, function(x) x$z))
matplot(z, type = "l")

## another iep of the economy above
discount.factor <- 0.97
return.rate <- 1 / discount.factor - 1
depreciation.rate <- 0.06

A <- function(state) {
  a1 <- CD_A(
    1, rbind(0.35, 0.65, 0),
    c(state$p[1] * (return.rate + depreciation.rate), state$p[2:3])
  )
  a2 <- CD_A(1, rbind(0.4, 1 - 0.4, 0), state$p)
  a1[3] <- state$p[1] * a1[1] * return.rate / state$p[3]
  cbind(a1, a2)
}

B <- function(state) {
  tmp <- matrix(c(
    1, 0,
    0, 1,
    0, 1
  ), 3, 2, TRUE)

  tmp[1] <- tmp[1] + A(state)[1, 1] * (1 - depreciation.rate)
  tmp
}

SExg.iep <- {
  tmp <- matrix(NA, 3, 2)
  tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

InitialEndowments <- {
  tmp <- matrix(0, 3, 2)
  tmp[1, 1] <- 0.01
  tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

ge.list <- iep(
  A = A, B = B, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = 100, n = 3, m = 2, trace = FALSE
)

z <- t(sapply(ge.list, function(x) x$z))
matplot(z, type = "l")

## TFP shock in the economy above (see Torres, 2016, section 2.8).
nPeriods.iep <- 200

discount.factor <- 0.97
return.rate <- 1 / discount.factor - 1
depreciation.rate <- 0.06

set.seed(1)
alpha.shock <- rep(1, 100)
alpha.shock[101] <- exp(0.01)
for (t in 102:nPeriods.iep) {
  alpha.shock[t] <- exp(0.95 * log(alpha.shock[t - 1]))
}

A.iep <- function(state.iep) {
  A <- function(state) {
    a1 <- CD_A(
      alpha.shock[state.iep$time],
      rbind(0.35, 0.65, 0),
      c(state$p[1] * (return.rate + depreciation.rate), state$p[2:3])
    )
    a2 <- CD_A(1, rbind(0.4, 1 - 0.4, 0), state$p)
    a1[3] <- state$p[1] * a1[1] * return.rate / state$p[3]
    cbind(a1, a2)
  }

  return(A)
}

B.iep <- function(state.iep) {
  B <- function(state) {
    tmp <- matrix(c(
      1, 0,
      0, 1,
      0, 1
    ), 3, 2, TRUE)

    a1 <- CD_A(
      alpha.shock[state.iep$time],
      rbind(0.35, 0.65, 0),
      c(state$p[1] * (return.rate + depreciation.rate), state$p[2:3])
    )

    tmp[1] <- tmp[1] + a1[1] * (1 - depreciation.rate)
    tmp
  }

  return(B)
}

SExg.iep <- {
  tmp <- matrix(NA, 3, 2)
  tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

InitialEndowments <- {
  tmp <- matrix(0, 3, 2)
  tmp[1, 1] <- tmp[2, 2] <- tmp[3, 2] <- 1
  tmp
}

ge.list <- iep(
  A.iep = A.iep, B.iep = B.iep, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = nPeriods.iep, n = 3, m = 2, trace = FALSE
)

z <- t(sapply(ge.list, function(x) x$z))
c <- sapply(ge.list, function(x) x$A[1,2]*x$z[2]) #consumption

par(mfrow = c(2, 2))
matplot(z, type = "l")
x <- 100:140
plot(x, z[x, 1] / z[x[1], 1], type = "b", pch = 20)
plot(x, z[x, 2] / z[x[1], 2], type = "b", pch = 20)
plot(x, c[x] / c[x[1]], type = "b", pch = 20)

## an iep of example 7.2 (a monetary economy) in Li (2019).
A <- function(state) {
  alpha <- rbind(1, 1, 1)
  Beta <- matrix(c(
    0.5, 0.5, 0.5,
    0.5, 0.5, 0.5,
    -1, -1, -1
  ), 3, 3, TRUE)
  CD_mA(alpha, Beta, state$p)
}

B <- diag(3)

SExg.iep <- {
  tmp <- matrix(NA, 3, 3)
  tmp[2, 2] <- 100
  tmp[3, 3] <- 100
  tmp
}

InitialEndowments <- {
  tmp <- matrix(0, 3, 3)
  tmp[1, 1] <- 10
  tmp[2, 2] <- tmp[3, 3] <- 100
  tmp
}

ge.list <- iep(
  A = A, B = B, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = 20,
  moneyIndex = 3,
  moneyOwnerIndex = 3,
  pExg = rbind(NA, NA, 0.25)
)

par(mfrow = c(1, 2))
z <- t(sapply(ge.list, function(x) x$z))
matplot(z, type = "b", pch = 20)
p <- t(sapply(ge.list, function(x) x$p))
matplot(p, type = "b", pch = 20)

## an example of structural transition policy
A.iep <- function(state.iep) {
  a <- 15
  b <- 25
  A <- function(state) {
    alpha1 <- 5
    alpha2 <- 15

    if (state.iep$time == 1 || state.iep$z[1] <= a) {
      alpha <- alpha1
    } else if (state.iep$z[1] > b) {
      alpha <- alpha2
    } else {
      alpha <- (b - state.iep$z[1]) / (b - a) * alpha1 +
        (state.iep$z[1] - a) / (b - a) * alpha2
    }

    return(cbind(
      CD_A(alpha, c(0.5, 0.5), state$p),
      c(1, 0)
    ))
  }

  return(A)
}

B <- matrix(c(
  1, 0,
  0, 1
), 2, 2, TRUE)

SExg.iep <- function(state.iep) {
  if (state.iep$time >= 15 && state.iep$z[1] < 30) {
    result <- matrix(c(
      NA, NA,
      0.6, 0.4
    ), 2, 2, TRUE)
  } else {
    result <- matrix(c(
      NA, NA,
      0, 1
    ), 2, 2, TRUE)
  }

  return(result)
}

InitialEndowments <- {
  tmp <- matrix(0, 2, 2)
  tmp[1, 1] <- 1
  tmp[2, 2] <- 1
  tmp
}

ge.list <- iep(
  A.iep = A.iep, B = B, SExg.iep = SExg.iep,
  InitialEndowments = InitialEndowments,
  nPeriods.iep = 30, trace = FALSE
)

z <- t(sapply(ge.list, function(x) x$z))
matplot(z, type = "b", pch = 20)