sdm2: Structural Dynamic Model (alias Structural Growth Model)...

In GE: General Equilibrium Modeling

Structural Dynamic Model (alias Structural Growth Model) Version 2

Description

A new version of the sdm function in the package CGE. Now the parameter A can be a demand structure tree list. Hence we actually no longer need the function sdm_dstl. Some rarely used parameters in the function sdm have been deleted. This function is the core of this package.

Usage

sdm2(
  A,
  B,
  S0Exg = matrix(NA, nrow(B), ncol(B)),
  names.commodity = paste("comm", 1:nrow(B), sep = ""),
  names.agent = paste("agt", 1:ncol(B), sep = ""),
  p0 = matrix(1, nrow = nrow(B), ncol = 1),
  z0 = matrix(100, nrow = ncol(B), ncol = 1),
  GRExg = NA,
  pExg = NULL,
  numeraire = NULL,
  tolCond = 1e-05,
  maxIteration = 200,
  numberOfPeriods = 300,
  depreciationCoef = 0.8,
  priceAdjustmentFunction = NULL,
  priceAdjustmentVelocity = 0.15,
  trace = TRUE,
  ts = FALSE,
  policy = NULL,
  exchangeFunction = F_Z
)

Arguments

A	a demand structure tree list (i.e. dstl, see demand_coefficient), a demand coefficient n-by-m matrix (alias demand structure matrix) or a function A(state) which returns an n-by-m matrix. n is the number of commodity types. m is the number of economic agents. The argument state is a list consisting of time (the current time), p (the current price vector), last.z (the output and utility vector of the previous period), w (the current wealth vector) and last.A (the demand coefficient matrix of the previous period).
B	an n-by-m matrix containing of the output coefficients (i.e. yield coefficients) of producers. Each producer produces one or more commodities. The output of each producer is equal to its activity level multiplied by the output coefficients. Columns corresponding to consumers are usually zeros. If the (i,j)-th element of S0Exg is not NA, the value of the (i,j)-th element of B will be useless and ignored.
S0Exg	an initial exogenous supply n-by-m matrix. If the (i,j)-th element of S0Exg is zero, it means there is no supply, and NA means the exogenous part of the supply is zero and there may be an endogenous supply part. In most cases, this matrix contains NA values but no zeros.
names.commodity	names of commodities. If the parameter A is a demand structure tree list, the values in names.commodity should be the names of those leaf nodes.
names.agent	names of agents.
p0	an initial price n-vector.
z0	an m-vector consisting of the initial exchange levels (i.e. activity levels, production levels or utility levels).
GRExg	an exogenous growth rate of the exogenous supplies in S0Exg. If GRExg is NA and some commodities have exogenous supply, then GRExg will be set to 0.
pExg	an n-vector indicating the exogenous prices (if any).
numeraire	the name, index or price of the numeraire commodity. If it is a character string, then it is assumed to be the name of the numeraire commodity. If it is a number without a name, then it is assumed to be the index of the numeraire commodity. If it is a number with a name, e.g. c("lab" = 0.5), then the name is assumed to be the name of the numeraire commodity and the number is assumed to be the price of the numeraire commodity, even though the price of the numeraire commodity usually is 1.
tolCond	the relative tolerance condition.
maxIteration	the maximum number of (outer) iterations. If the main purpose of running this function is to do simulation instead of calculating equilibrium, then maxIteration should be set to 1.
numberOfPeriods	the period number (i.e. the number of inner iterations) in each (outer) iteration, which should not be less than 20.
depreciationCoef	the depreciation coefficient (i.e. 1 minus the depreciation rate) of the unsold products.
priceAdjustmentFunction	the price adjustment function. The arguments are a price n-vector p and a sales rate n-vector q. The return value is a price n-vector. The default price adjustment method is p * (1 - priceAdjustmentVelocity * (1 - q)).
priceAdjustmentVelocity	a scalar or an n-vector specifying the price adjustment velocity.
trace	if TRUE, information is printed during the running of sdm2.
ts	if TRUE, the time series of the last outer iteration are returned.
policy	a policy function or a list consisting of policy functions and/or policy function lists. A policy function has the following optional parameters: time - the current time. A - the same as the parameter A of sdm2. When A is a demand structure tree list, it needs not be returned after it is adjusted. state - the current state, which is a list. state$p is the current price vector with names. state$S is the current supply matrix. state$last.z is the output and utility vector of the previous period. state$B is the current supply coefficient matrix. state$last.A is the demand coefficient matrix of the previous period. state$names.commodity contains the names of commodities. state$names.agent contains the names of agents. state.history - the state history, which is a list consisting of the time series of p, S, q, and z. The return value of the policy function other than a list will be ignored. If the return value is a list, it should have elements p, S and B which specify the prices, supplies and supply coefficient matrix after adjustment. A vector with the name current.policy.data can be put into the state list as well, which will be put into the return value of the sdm2.
exchangeFunction	the exchange function.

Details

In each period of the structural dynamic model, the economy runs as follows.
Firstly, the new price vector emerges on the basis of the price vector and sales rates of the previous period, which indicates the current market prices.
Secondly, outputs and depreciated inventories of the previous period constitute the current supplies.
Thirdly, policy functions (if any) are implemented.
Fourthly, the current input coefficient matrix is computed and the supplies are exchanged under market prices. The exchange vector and sales rate vector are obtained. Unsold goods constitute the inventories, which will undergo depreciation and become a portion of the supplies of the next period. The exchange vector determines the current outputs and utility levels.

Value

A list usually containing the following components:

• tolerance - the relative tolerance of the results.

• p - equilibrium prices.

• z- equilibrium exchange levels (i.e. activity levels, output levels or utility levels).

• S - the equilibrium supply matrix at the initial period.

• growthRate - the endogenous equilibrium growth rate in a pure production economy.

• A - the equilibrium demand coefficient matrix.

• B - the supply coefficient matrix.

• S0Exg - the initial exogenous supply n-by-m matrix.

• D - the demand matrix.

• DV - the demand value matrix.

• SV - the supply value matrix.

• ts.p - the time series of prices in the last outer iteration.

• ts.z - the time series of exchange levels (i.e. activity levels, production levels or utility levels) in the last outer iteration.

• ts.S - the time series of supply matrix in the last outer iteration.

• ts.q - the time series of sales rates in the last outer iteration.

• policy.data - the policy data.

Note

In the package CGE, the temporary equilibrium path (alias market clearing path, instantaneous equilibrium path) is computed by the function iep. In this package, the temporary equilibrium path can be computed by the function sdm2 with the parameter policy equal to policyMarketClearingPrice.

The order of implementation of various policies is critical. When a policy list contains a supply policy, a technology (i.e. dstl) policy, a price policy (e.g. a market-clearing-price policy) and a B policy (i.e. a policy adjusting the argument B), both the supply policy and the technology policy should be placed before the price policy, and the B policy should be placed after the price policy. The reason is that the calculation of the current prices may require the use of supply and technology, while the calculation of B may require the use of the current prices.

References

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

Examples

dst.firm <- node_new("output",
  type = "Leontief", a = c(0.5, 1),
  "prod", "lab"
)

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

dstl <- list(dst.firm, dst.consumer)

B <- matrix(c(
  1, 0,
  0, 0
), 2, 2, TRUE)
S0Exg <- matrix(c(
  NA, NA,
  NA, 100
), 2, 2, TRUE)

## variable dst and technology progress
policy.TP <- function(time, state, A) {
  if (time >= 200) {
    A[[1]]$a <- c(0.5, 0.8)
  } else {
    A[[1]]$a <- c(0.5, 1)
  }
  state
}

ge.TP <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.TP,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000
)
matplot(ge.TP$ts.z, type = "l")
plot(ge.TP$ts.p[, 1] / ge.TP$ts.p[, 2], type = "l")

## variable supply coefficient matrix and technology progress
policy.TP <- function(time, state) {
  if (time >= 200) {
    state$B[1, 1] <- 2
  } else {
    state$B[1, 1] <- 1
  }
  state
}

ge.TP <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.TP,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000
)
matplot(ge.TP$ts.z, type = "l")
plot(ge.TP$ts.p[, 1] / ge.TP$ts.p[, 2], type = "l")

## variable dst and disequilibrium
policy.DE <- function(time, A) {
  if (time >= 200) {
    A[[1]]$a[2] <- A[[1]]$a[2] * 0.999
  } else {
    A[[1]]$a[2] <- 1
  }
}

ge.DE <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.DE,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000
)
matplot(ge.DE$ts.z, type = "l")
plot(ge.DE$ts.p[, 1] / ge.DE$ts.p[, 2], type = "l")


## structural equilibria and structural transition
policy.SE <- function(time, state, A) {
  A[[1]]$a[2] <- structural_function(state$last.z[1], c(105, 125), 1, 0.5)
}

ge.low.level <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.SE,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000,
  z0 = c(100, 0)
)
matplot(ge.low.level$ts.z, type = "l")

ge.high.level <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.SE,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000,
  z0 = c(150, 0)
)
matplot(ge.high.level$ts.z, type = "l")

policy.ST <- function(time, state, A) {
  A[[1]]$a[2] <- structural_function(state$last.z[1], c(105, 125), 1, 0.5)
  if (time >= 200 && time <= 210) state$S[2, 2] <- 125 # Introduce foreign labor.
  state
}

ge.ST <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  policy = policy.ST,
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 1000,
  z0 = c(100, 0)
)
matplot(ge.ST$ts.z, type = "l")

#### economic cycles and an interest rate policy for the firm
dst.firm <- node_new("cc", # composite commodity
  type = "FIN",
  rate = c(1, 0.25),
  "cc1", "money"
)
node_set(dst.firm, "cc1",
  type = "Leontief",
  a = c(0.5, 0.5),
  "wheat", "labor"
)

dst.laborer <- Clone(dst.firm)
dst.money.lender <- Clone(dst.firm)

dstl <- list(dst.firm, dst.laborer, dst.money.lender)

policy.interest.rate <- function(time, state, A, state.history) {
  upsilon <- NA
  if (time >= 600) {
    upsilon <- state.history$z[time - 1, 1] / mean(state.history$z[(time - 50):(time - 1), 1])
    A[[1]]$rate[2] <- max(0.25 + 0.5 * log(upsilon), 0)
  } else {
    A[[1]]$rate[2] <- 0.25
  }

  state$current.policy.data <- c(time, A[[1]]$rate[2], upsilon)
  state
}

B <- matrix(0, 3, 3)
B[1, 1] <- 1

S0Exg <- matrix(NA, 3, 3)
S0Exg[2, 2] <- 100
S0Exg[3, 3] <- 100

de <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("wheat", "labor", "money"),
  names.agent = c("firm", "laborer", "money.lender"),
  p0 = rbind(0.625, 0.375, 0.25),
  z0 = rbind(95, 0, 0),
  priceAdjustmentVelocity = 0.3,
  numberOfPeriods = 1000,
  maxIteration = 1,
  trace = FALSE,
  ts = TRUE
)
matplot(de$ts.z, type = "l")

ge.policy <- sdm2(
  A = dstl, B = B, S0Exg = S0Exg,
  names.commodity = c("wheat", "labor", "money"),
  names.agent = c("firm", "laborer", "money.lender"),
  p0 = rbind(0.625, 0.375, 0.25),
  z0 = rbind(95, 0, 0),
  priceAdjustmentVelocity = 0.3,
  numberOfPeriods = 1000,
  maxIteration = 1,
  trace = FALSE,
  ts = TRUE,
  policy = policy.interest.rate
)
matplot(ge.policy$ts.z, type = "l")

#### Example 9.3 in Li (2019): fixed-ratio price adjustment method
#### and disequilibrium (business cycles) in a pure production economy
fixedRatioPriceAdjustmentFunction <- function(p, q) {
  thresholdForPriceAdjustment <- 0.99
  priceAdjustmentVelocity <- 0.02
  result <- ifelse(q <= thresholdForPriceAdjustment,
    p * (1 - priceAdjustmentVelocity),
    p
  )
  return(prop.table(result))
}

de.Sraffa <- sdm2(
  A = matrix(c(
    56 / 115, 6,
    12 / 575, 2 / 5
  ), 2, 2, TRUE),
  B = diag(2),
  maxIteration = 1,
  numberOfPeriods = 100,
  p0 = rbind(1 / 15, 1),
  z0 = rbind(575, 20),
  priceAdjustmentFunction = fixedRatioPriceAdjustmentFunction,
  ts = TRUE
)
matplot(growth_rate(de.Sraffa$ts.z), type = "l")

GE documentation built on Nov. 8, 2023, 9:07 a.m.