Package 'GE'

Description

Compute the utility function mean(x) - (gamma * sd.p(x))^theta / theta or weighted.mean(x, wt) - (gamma * sd.p(x, wt))^theta / theta.

Usage

AMSD(x, gamma = 1, wt = NULL, theta = 1)

AMV(x, gamma = 1, wt = NULL)

Arguments

Value

A scalar indicating the utility level.

Functions

• AMSD: Computes the utility function mean(x) - (gamma * sd.p(x))^theta / theta or weighted.mean(x, wt) - (gamma * sd.p(x, wt))^theta / theta. When theta == 2, it is the additive mean-variance utility function (i.e. the function AMV). When theta == 1 (the default value), it is the additive mean and standard deviation utility function.

• AMV: Compute the additive mean-variance utility function mean(x) - 0.5 * gamma * var.p(x) or weighted.mean(x, wt) - 0.5 * gamma * var.p(x, wt).

References

Nakamura, Yutaka (2015). Mean-Variance Utility. Journal of Economic Theory, 160: 536-556.

Examples

AMSD(1:2, gamma = 0.05)
AMSD(1:2, gamma = 1, theta = 2)

marginal_utility(
  c(1, 1.001),
  c(0, 1), function(x) AMSD(x, gamma = 0.5)
)
marginal_utility(
  c(1.001, 1),
  c(0, 1), function(x) AMSD(x, gamma = 0.5)
)

Description

Compute the utility function x %*% mp - gamma^theta * (t(x) %*% Cov %*% x)^(0.5 * theta) / theta for a portfolio x.

Usage

AMSDP(x, mp, Cov, gamma = 1, theta = 1)

Arguments

Value

A scalar indicating the utility level.

References

Danthine, J. P., Donaldson, J. (2005, ISBN: 9780123693808) Intermediate Financial Theory. Elsevier Academic Press.

Nakamura, Yutaka (2015) Mean-Variance Utility. Journal of Economic Theory, 160: 536-556.

Sharpe, William F (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

Xu Gao (2018, ISBN: 9787300258232) Twenty-five Lectures on Financial Economics. Beijing: China Renmin University Press. (In Chinese)

See Also

AMSD

Examples

UAP <- matrix(c(
  0, 1, 1,
  0, 2, 1,
  1, 1, 1,
  1, 2, 1,
  2, 0, 1
), nrow = 5, byrow = TRUE)

portfolio <- c(1.977, 1.183, 3.820)

AMSDP(portfolio, colMeans(UAP),
  cov.wt(UAP, method = "ML")$cov,
  gamma = 1, theta = 1
)

AMSD(UAP %*% portfolio, gamma = 1, theta = 1)

Description

A wrapper of the functions apply and expand.grid. Returns a data frame of values obtained by applying a function to all combinations of the supplied vectors. Firstly, the function expand.grid will be used for the supplied vectors in ... and we will get a data frame containing one row for each combination of the supplied vectors. Then the function will be applied to each row of the data frame. The values of the data frame will also be included in the returned data frame.

Usage

apply_expand.grid(FUN, ...)

Arguments

Value

A data frame.

Examples

apply_expand.grid(prod, a = 1:9, b = 1:9)

####
f <- function(x) c(r1 = sum(x), r2 = unname(x["b"] - x["a"]))
apply_expand.grid(f, a = c(1, 2), b = c(3, 4))

####
f <- function(x) list(list(sum(x)), prod(x))
apply_expand.grid(f, a = c(1, 2), b = c(3, 4))

####
f <- function(x) {
  result <- SCES_A(alpha = 1, Beta = c(0.5, 0.5), p = c(x["p1"], 1), es = x["es"])
  names(result) <- c("dc1", "dc2")
  result
}

apply_expand.grid(f, p1 = seq(0.1, 10, 0.1), es = c(0.3, 0.5, 1))

Description

Compute the value and the certainty equivalent of the CARA utility function, i.e. -exp(-gamma*x). In general equilibrium analysis, the CARA utility function has an interval scale like temperature.

Usage

CARA(x, gamma, prob = rep(1/length(x), length(x)))

Arguments

Value

A list containing the following components:

• u: the utility level.

• CE: the certainty equivalent.

Examples

mu <- 5 # mu <- 8
a <- 1
x <- c(mu - a, mu + a)
gamma <- 0.8
mu - CARA(x, gamma)$CE

####
gamma <- 0.8
mu <- 2
sigma <- 2
x <- seq(mu - 5 * sigma, mu + 5 * sigma, length.out = 10000)
# two CE calculation methods for random variables of normal distribution
CARA(x, gamma, dnorm(x, mean = mu, sd = sigma))
mu - gamma * sigma^2 / 2

Description

CES function, e.g. alpha * (beta1 * (x1 / theta1)^sigma + beta2 * (x2 / theta2)^sigma)^(1 / sigma).

Usage

CES(sigma = 1 - 1/es, alpha, beta, x, theta = rep(1, length(beta)), es = NA)

Arguments

Value

A scalar indicating the output or utility level.

Examples

CES(0.5, 1, c(0.4, 0.6), c(1, 1), c(0.4, 0.6))
CES(0.5, 1, c(0.4, 0.6), c(1, 2))

Description

Computing the demand coefficients of the CESAK production function alpha * (betaK * x1^((es - 1) / es) + (1 - betaK) * x2^((es - 1) / es))^(es / (es - 1)) + alphaK * x1. When es==1, the CESAK production function becomes the CDAK production function alpha * x1^betaK * x2^(1 - betaK) + alphaK * x1.

Usage

CESAK_dc(alpha, betaK, alphaK, p, es = 1)

Arguments

Value

A 2-vector indicating the demand coefficients.

Examples

CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 1 - 0.06, p = c(10, 1))
CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 1 - 0.06, p = c(1, 10))
CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 2, p = c(1, 1))

CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 1 - 0.06, p = c(1, 1), es = 0.5)
CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 0.1, p = c(1, 10), es = 0.5)
CESAK_dc(alpha = 1, betaK = 0.35, alphaK = 1.5, p = c(1, 10), es = 0.5)

Description

Compute the value and the certainty equivalent of the CRRA utility function.

Usage

CRRA(x, gamma, prob = rep(1/length(x), length(x)))

Arguments

Value

A list containing the following components:

• u: the utility level.

• CE: the certainty equivalent.

Examples

csv <- 0.05 # coefficient of standard deviation
mu <- 90 # mu <- 100
sigma <- mu * csv
x <- seq(mu - 5 * sigma, mu + 5 * sigma, length.out = 10000)
pd <- dnorm(x, mean = mu, sd = sigma)
gamma <- 0.8
# the ratio of risk premium to expected return (i.e. the relative risk premium).
(mu - CRRA(x, gamma, pd)$CE) / mu

####
df <- apply_expand.grid(
  function(arg) {
    CRRA(arg["x"], arg["gamma"])$u
  },
  x = seq(0.5, 3, 0.1),
  gamma = c(0.5, 1, 2, 3)
)
coplot(result ~ x | as.factor(gamma), data = df)

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

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

Description

Given a price vector, this function computes the demand coefficients of an agent with a demand structural tree. The class of a demand structural tree is Node defined by the package data.tree.

Usage

demand_coefficient(node, p, trace = FALSE)

Arguments

Details

Demand coefficients often indicate the quantity of various commodities needed by an economic agent in order to obtain a unit of output or utility, and these commodities can include both real commodities and financial instruments such as tax receipts, stocks, bonds and currency.
The demand for various commodities by an economic agent can be expressed by a demand structure tree. Each non-leaf node can be regarded as the output of all its child nodes. Each node can be regarded as an input of its parent node. In other words, the commodity represented by each non-leaf node is a composite commodity composed of the commodities represented by its child nodes. Each non-leaf node usually has an attribute named type. This attribute describes the input-output relationship between the child nodes and the parent node. This relationship can sometimes be represented by a production function or a utility function. The type attribute of each non-leaf node can take the following values.

• SCES. In this case, this node also has parameters alpha, beta and es (or sigma = 1 - 1 / es). alpha and es are scalars. beta is a vector. These parameters are parameters of a standard CES function (see SCES and SCES_A).

• CES. In this case, this node also has parameters alpha, beta, theta (optional) and es (or sigma = 1 - 1 / es) (see CGE::CES_A).

• Leontief. In this case, this node also has the parameter a, which is a vector and is the parameter of a Leontief function.

• CD. CD is Cobb-Douglas. In this case, this node also has parameters alpha and beta, which are parameters of a Cobb-Douglas function.

• CESAK. In this case, this node also has parameters es, alpha, betaK and alphaK, which are parameters of the CESAK function (see CESAK_dc). Moreover, the first child node should represent capital goods.

• FIN. That is the financial type. In this case, this node also has the parameter rate or beta. If the parameter beta is not NULL, then the parameter rate will be ignored. The parameter rate applies to all situations, while the parameter beta only applies for some special cases. For FIN nodes, the first child node should represent for a physical commodity or a composite commodity containing a physical commodity, and other child nodes represent for financial instruments. The parameter beta indicates the proportion of each child node's expenditure. The parameter rate indicates the expenditure ratios between financial-instrument-type child nodes and the first child node. The first element of the parameter rate indicates the amount of the first child node needed to get a unit of output.

• FUNC. That is the function type. In this case, this node also has an attribute named func. The value of that attribute is a function which calculates the demand coefficient for the child nodes. The argument of that function is a price vector. The length of that price vector is equal to the number of the child nodes.

• StickyLinear or SL. That is the sticky linear type. In this case, this node also has an attribute named beta that contains the coefficients of the linear utility or production function. In order to avoid too drastic changes in the demand structure, the adjustment process of the demand structure has a certain stickiness when prices change.

Value

A vector consisting of demand coefficients.

Examples

#### a Leontief-type node
dst <- node_new("firm",
  type = "Leontief", a = c(0.5, 0.1),
  "wheat", "iron"
)
print(dst, "type")
node_print(dst)
plot(dst)
node_plot(dst, TRUE)

demand_coefficient(dst, p = c(wheat = 1, iron = 2)) # the same as a = c(0.5, 0.1)

#### a CD-type node
dst <- node_new("firm",
  type = "CD", alpha = 1, beta = c(0.5, 0.5),
  "wheat", "iron"
)

demand_coefficient(dst, p = c(wheat = 1, iron = 2))
# the same as the following
CD_A(1, c(0.5, 0.5), c(1, 2))

#### a SCES-type node
dst <- node_new("firm",
  type = "SCES",
  alpha = 2, beta = c(0.8, 0.2), es = 0.5,
  "wheat", "iron"
)

demand_coefficient(dst, p = c(wheat = 1, iron = 2))

# the same as the following
SCES_A(alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), es = 0.5)
CES_A(sigma = 1 - 1 / 0.5, alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), Theta = c(0.8, 0.2))

#### a FUNC-type node
dst <- node_new("firm",
  type = "FUNC",
  func = function(p) {
    CES_A(
      sigma = -1, alpha = 2,
      Beta = c(0.8, 0.2), p,
      Theta = c(0.8, 0.2)
    )
  },
  "wheat", "iron"
)

demand_coefficient(dst, p = c(wheat = 1, iron = 2))

# the same as the following
CES_A(sigma = -1, alpha = 2, Beta = c(0.8, 0.2), p = c(1, 2), Theta = c(0.8, 0.2))

####
p <- c(wheat = 1, iron = 3, labor = 2, capital = 4)
dst <- node_new("firm 1",
  type = "SCES", sigma = -1, alpha = 1, beta = c(1, 1),
  "cc1", "cc2"
)
node_set(dst, "cc1",
  type = "Leontief", a = c(0.6, 0.4),
  "wheat", "iron"
)
node_set(dst, "cc2",
  type = "SCES", sigma = -1, alpha = 1, beta = c(1, 1),
  "labor", "capital"
)

node_plot(dst)
demand_coefficient(dst, p)

####
p <- c(product = 1, labor = 1, money = 1)
dst <- node_new("firm",
  type = "FIN", rate = c(0.75, 1 / 3),
  "cc1", "money"
) # a financial-type node
node_set(dst, "cc1",
  type = "Leontief", a = c(0.8, 0.2),
  "product", "labor"
)

node_plot(dst)
demand_coefficient(dst, p)

#### the same as above
p <- c(product = 1, labor = 1, money = 1)
dst <- node_new("firm",
  type = "Leontief", a = c(0.8, 0.2),
  "cc1", "cc2"
)
node_set(dst, "cc1",
  type = "FIN", rate = c(0.75, 1 / 3),
  "product", "money"
)

node_set(dst, "cc2",
  type = "FIN", rate = c(0.75, 1 / 3),
  "labor", "money"
)
node_plot(dst)
demand_coefficient(dst, p)

#### the same as above
p <- c(product = 1, labor = 1, money = 1)
dst <- node_new("firm",
  type = "FIN", rate = c(1, 1 / 3),
  "cc1", "money"
) # Financial-type Demand Structure
node_set(dst, "cc1",
  type = "Leontief", a = c(0.6, 0.15),
  "product", "labor"
)

node_plot(dst)
demand_coefficient(dst, p)

Description

These are some examples to illustrate that credit policies may lead to business cycles. When the firm's profit rate is high, the laborer lends labor or labor income to the firm; when the firm's profit rate is low, the firm repays the loan with products.

Usage

demCreditPolicy(...)

Arguments

Examples

dst.firm <- node_new("output",
  type = "CD", alpha = 1.2,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

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

f <- function(policy = NULL) {
  ge <- sdm2(
    A = list(dst.firm, dst.consumer),
    B = matrix(c(
      1, 0,
      0, 1
    ), 2, 2, TRUE),
    S0Exg = matrix(c(
      NA, NA,
      NA, 100
    ), 2, 2, TRUE),
    names.commodity = c("prod", "lab"),
    names.agent = c("firm", "consumer"),
    ts = TRUE,
    policy = policy,
    numberOfPeriods = 200,
    maxIteration = 1,
    priceAdjustmentVelocity = 0.05
  )

  matplot(ge$ts.z, type = "o", pch = 20)
  ge
}

## no credit policy
ge <- f()

## credit policy
policy.credit <- function(time, state) {
  profit.rate <- state$p[1] / sum(state$last.A[, 1] * state$p) - 1

  if (profit.rate > 0.01) {
    state$S[2, 2] <- 50
    state$S[2, 1] <- 50
  } else if (profit.rate < -0.01) {
    state$S[1, 2] <- state$S[1, 1] * 0.5
    state$S[1, 1] <- state$S[1, 1] * 0.5
  }

  state
}

de <- f(policy = policy.credit)

#### an example with 3 firms.
policy.credit <- function(time, state) {
  if (time <= 10) {
    return(state)
  }
  profit.rate <- state$p[1] / sum(state$last.A[, 1] * state$p) - 1

  if (profit.rate > 0.01) {
    state$S[3, 1] <- 30
    # state$S[3, 1:3] <- 10
    state$S[3, 4] <- 70
  } else if (profit.rate < -0.01) {
    state$S[1, 4] <- state$S[1, 1] * 0.3
    state$S[1, 1] <- state$S[1, 1] * 0.7
  }

  state
}

f <- function(policy = NULL,
              numberOfPeriods = 50) {
  ge <- sdm2(
    A = function(state) {
      a.firm.prod <- CD_A(alpha = 1, Beta = c(0, 0.5, 0.5, 0), state$p)
      a.firm.cap1 <- c(1, 0, 0.1, 0)
      a.firm.cap2 <- c(0, 0, 0.1, 1)
      a.consumer <- c(1, 0, 0, 0)
      cbind(a.firm.prod, a.firm.cap1, a.firm.cap2, a.consumer)
    },
    B = matrix(c(
      1, 0, 0, 0,
      0, 0, 1, 0,
      0, 0, 0, 0,
      0, 1, 0, 0
    ), 4, 4, TRUE),
    S0Exg = {
      tmp <- matrix(NA, 4, 4)
      tmp[3, 4] <- 100
      tmp
    },
    names.commodity = c("prod", "cap2", "lab", "cap1"),
    names.agent = c("firm.prod", "firm.cap1", "firm.cap2", "consumer"),
    numeraire = "lab",
    maxIteration = 1,
    numberOfPeriods = numberOfPeriods,
    ts = TRUE,
    p0 = c(4.191, 4.391, 1, 4.291),
    # The equilibrium output of firm.prod is 45.64.
    z0 = c(50, 21.78, 21.78, 23.86),
    policy = policy
  )

  matplot(ge$ts.z, type = "o", pch = 20)
  ge
}

## a disequilibrium path
de <- f(numberOfPeriods = 500)

## a spot market clearing path converging to equilibrium
ge <- f(
  policy = policyMarketClearingPrice,
  numberOfPeriods = 40
)
ge$p
ge$z

## a spot market clearing path with persisting fluctuations
de <- f(policy = list(
  policy.credit,
  policyMarketClearingPrice
))

Description

A disequilibrium model illustrating supply-demand structural mismatch and insufficient effective demand. Assume that from the 5th period, the producer expects the sales rate of products to decline, so he reduces investment in production and increases the demand for value storage means (such as foreign assets, gold, etc.); the laborer expects the unemployment rate to rise, so he reduces consumption and increases the demand for value storage means.

Here the supplier of value storage means is referred to as ROW (the rest of the world).

Usage

demInsufficientEffectiveDemand_3_3(...)

Arguments

Examples

dst.firm <- node_new("output",
  type = "FIN", rate = c(1, 0),
  "cc1", "store of value"
)
node_set(dst.firm, "cc1",
  type = "CD", alpha = 2,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.laborer <- node_new("util",
  type = "FIN", rate = c(1, 0),
  "cc1", "store of value"
)
node_set(dst.laborer, "cc1",
  type = "CD", alpha = 1,
  beta = c(0.5, 0.5),
  "prod", "lab"
)

dst.ROW <- node_new("util",
  type = "Leontief", a = 1,
  "lab"
)

policy.demand <- function(time, A, state) {
  if (time >= 5) {
    A[[1]]$rate <- c(1, 0.25)
    A[[2]]$rate <- c(1, 0.25)
  } else {
    A[[1]]$rate <- c(1, 0)
    A[[2]]$rate <- c(1, 0)
  }

  state
}

ge <- sdm2(
  A = list(dst.firm, dst.laborer, dst.ROW),
  B = matrix(c(
    1, 0, 0,
    0, 0, 0,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, 100, NA,
    NA, NA, 100
  ), 3, 3, TRUE),
  names.commodity = c("prod", "lab", "store of value"),
  names.agent = c("firm", "laborer", "ROW"),
  ts = TRUE,
  policy = policy.demand,
  numberOfPeriods = 100,
  maxIteration = 1,
  numeraire = "prod",
  z0 = c(100, 0, 0),
  p0 = c(1, 1, 1),
  pExg = c(1, NA, 1)
)

matplot(ge$ts.z, type = "o", pch = 20)
matplot(ge$ts.p, type = "o", pch = 20)
matplot(ge$ts.q, type = "o", pch = 20)

Description

Add names to the matrices and vectors of a general equilibrium, and add demand matrix, demand value matrix and supply value matrix to it.

Usage

ge_tidy(ge, names.commodity, names.agent)

Arguments

Value

A tidied general equilibrium.

Description

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

Usage

gem_2_2(...)

Arguments

References

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

Examples

#### a 2-by-2 general equilibrium model with a Leontief production function.
ge.Leontief <- sdm2(
  A = matrix(c(
    0.5, 1,
    0.5, 0
  ), 2, 2, TRUE),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge.Leontief$p
ge.Leontief$z
addmargins(ge.Leontief$D, 2)
addmargins(ge.Leontief$S, 2)

## the same as above.
ge2.Leontief <- sdm2(
  A = list(
    dst.firm = node_new(
      "output",
      type = "Leontief",
      a = c(0.5, 0.5),
      "prod", "lab"
    ),
    dst.consumer = node_new(
      "util",
      type = "Leontief", a = 1,
      "prod"
    )
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge2.Leontief$p
ge2.Leontief$z
addmargins(ge2.Leontief$D, 2)
addmargins(ge2.Leontief$S, 2)

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

ge.CD$p
ge.CD$z
addmargins(ge.CD$D, 2)
addmargins(ge.CD$S, 2)

## the same as above.
ge2.CD <- sdm2(
  A = list(
    dst.firm = node_new(
      "output",
      type = "CD", alpha = 2, beta = c(0.5, 0.5),
      "prod", "lab"
    ),
    dst.consumer = node_new(
      "util",
      type = "Leontief", a = 1,
      "prod"
    )
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge2.CD$p
ge2.CD$z
addmargins(ge2.CD$D, 2)
addmargins(ge2.CD$S, 2)

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

ge.SCES$p
ge.SCES$z
addmargins(ge.SCES$D, 2)
addmargins(ge.SCES$S, 2)

## the same as above.
ge2.SCES <- sdm2(
  A = list(
    dst.firm = node_new(
      "output",
      type = "SCES",
      es = 0.5, alpha = 1, beta = c(0.5, 0.5),
      "prod", "lab"
    ),
    dst.consumer = node_new(
      "util",
      type = "Leontief", a = 1,
      "prod"
    )
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

ge2.SCES$p
ge2.SCES$z
addmargins(ge2.SCES$D, 2)
addmargins(ge2.SCES$S, 2)

#### a 2-by-2 general equilibrium model with a CESAK production function.
ge.CESAK <- sdm2(
  A = function(state) {
    a.firm <- CESAK_dc(alpha = 2, betaK = 0.5, alphaK = 0.5, p = state$p, es = 1)
    a.consumer <- c(1, 0)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge.CESAK$p
ge.CESAK$z
addmargins(ge.CESAK$D, 2)
addmargins(ge.CESAK$S, 2)

## the same as above.
ge2.CESAK <- sdm2(
  A = list(
    dst.firm = node_new(
      "output",
      type = "CESAK", es = 1,
      alpha = 2, betaK = 0.5, alphaK = 0.5,
      "prod", "lab"
    ),
    dst.consumer = node_new(
      "util",
      type = "Leontief", a = 1,
      "prod"
    )
  ),
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "consumer"),
  numeraire = "prod"
)

ge2.CESAK$p
ge2.CESAK$z
addmargins(ge2.CESAK$D, 2)
addmargins(ge2.CESAK$S, 2)

#### Example 1 in the ucla reference.
ge3.SCES <- sdm2(
  A = function(state) {
    a.firm <- c(0, 0.25)
    a.consumer <- SCES_A(es = 0.5, alpha = 1, Beta = c(0.5, 0.5), p = state$p)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, NA,
    NA, 30
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

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

#### The laborer has some product.
ge <- sdm2(
  A = function(state) {
    a.firm <- c(0, 1) # c(0, 2)
    a.consumer <- SCES_A(es = 0.5, alpha = 1, Beta = c(0.5, 0.5), p = state$p)
    cbind(a.firm, a.consumer)
  },
  B = matrix(c(
    1, 0,
    0, 0
  ), 2, 2, TRUE),
  S0Exg = matrix(c(
    NA, 50, # 500
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("prod", "lab"),
  names.agent = c("firm", "laborer"),
  numeraire = "prod"
)

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

Description

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

Usage

gem_3_2(...)

Arguments

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

Description

Some simple 3-by-3 general equilibrium models with two firms and a consumer.

Usage

gem_3_3(...)

Arguments

See Also

gemCapitalAccumulation

Examples

####
ge <- sdm2(
  A = function(state) {
    ## the vector of demand coefficients of the firm
    a.firm.corn <- CD_A(alpha = 1, Beta = c(0, 0.5, 0.5), p = state$p)
    a.firm.iron <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), p = state$p)
    ## the vector of demand coefficients of the consumer
    a.consumer <- CD_A(alpha = 1, Beta = c(0.5, 0.5, 0), p = state$p)
    cbind(a.firm.corn, a.firm.iron, a.consumer)
  },
  B = diag(c(1, 1), 3),
  S0Exg = {
    tmp <- matrix(NA, 3, 3)
    tmp[3, 3] <- 100
    tmp
  },
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer"),
  numeraire = "lab"
)

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

####
ge <- sdm2(
  A = function(state) {
    ## the vector of demand coefficients of the firm
    a.firm.corn <-
      SCES_A(es = 1,
             alpha = 1,
             Beta = c(0, 0.5, 0.5),
             p = state$p)
    a.firm.iron <-
      SCES_A(es = 1,
             alpha = 2,
             Beta = c(0, 0.5, 0.5),
             p = state$p)
    ## the vector of demand coefficients of the consumer
    a.consumer <- CD_A(alpha = 1, Beta = c(0.5, 0.5, 0), p = state$p)
    cbind(a.firm.corn, a.firm.iron, a.consumer)
  },
  B = diag(c(1, 1), 3),
  S0Exg = {
    tmp <- matrix(NA, 3, 3)
    tmp[3, 3] <- 100
    tmp
  },
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer"),
  numeraire = "lab"
)

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

#### a general equilibrium model containing a production firm
#### and a capital-goods-leasing firm
ge <- sdm2(
  A = function(state) {
    a.firm1 <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), state$p)
    a.consumer <- c(1, 0, 0)
    a.firm2 <- c(1, 0, 0)
    cbind(a.firm1, a.consumer, a.firm2)
  },
  B = matrix(c(
    1, 0, 0.5,
    0, 0, 1,
    0, 0, 0
  ), 3, 3, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA,
    NA, NA, NA,
    NA, 100,NA
  ), 3, 3, TRUE),
  names.commodity = c("prod", "cap", "lab"),
  names.agent = c("firm1", "consumer","firm2"),
  numeraire = "prod",
  priceAdjustmentVelocity = 0.05
)
ge$p
ge$z
ge$D

Description

Some simple 3-by-4 general equilibrium models with two firms and two consumers.

Usage

gem_3_4(...)

Arguments

Examples

####
ge <- sdm2(
  A = function(state) {
    a.firm.corn <- CD_A(alpha = 1, Beta = c(0, 0.5, 0.5), state$p)
    a.firm.iron <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5), state$p)
    a.consumer1 <- c(1, 0, 0)
    a.consumer2 <- CD_A(alpha = 1, Beta = c(0.5, 0.5, 0), state$p)

    cbind(a.firm.corn, a.firm.iron, a.consumer1, a.consumer2)
  },
  B = diag(c(1, 1, 0), 3, 4),
  S0Exg = {
    tmp <- matrix(NA, 3, 4)
    tmp[3, 3:4] <- 100
    tmp
  },
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer1", "consumer2"),
  numeraire = "lab"
)

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

####
ge <- sdm2(
  A = function(state) {
    a.firm.corn <-
      SCES_A(
        es = 1,
        alpha = 1,
        Beta = c(0, 0.5, 0.5),
        p = state$p
      )
    a.firm.iron <-
      SCES_A(
        es = 1,
        alpha = 2,
        Beta = c(0, 0.5, 0.5),
        p = state$p
      )
    a.consumer1 <- c(1, 0, 0)
    a.consumer2 <- CD_A(alpha = 1, Beta = c(0.5, 0.5, 0), state$p)

    cbind(a.firm.corn, a.firm.iron, a.consumer1, a.consumer2)
  },
  B = diag(c(1, 1, 0), 3, 4),
  S0Exg = {
    tmp <- matrix(NA, 3, 4)
    tmp[3, 3:4] <- 100
    tmp
  },
  names.commodity = c("corn", "iron", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer1", "consumer2"),
  numeraire = "lab"
)

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

#### an example at
## https://web.stanford.edu/~jdlevin/Econ%20202/General%20Equilibrium.pdf
ge <- sdm2(
  A = function(state) {
    a.firm.1 <- c(0, 1, 0)
    a.firm.2 <- c(0, 0, 1)
    a.consumer1 <- CD_A(alpha = 1, Beta = c(1 / 3, 1 / 3, 1 / 3), state$p)
    a.consumer2 <- CD_A(alpha = 1, Beta = c(1 / 3, 1 / 3, 1 / 3), state$p)

    cbind(a.firm.1, a.firm.2, a.consumer1, a.consumer2)
  },
  B = matrix(c(
    2, 0, 0, 0,
    0, 1, 0, 0,
    0.5, 0, 0, 0
  ), 3, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, 1, 2,
    NA, NA, 2, 2,
    NA, NA, 3, 2
  ), 3, 4, TRUE),
  names.agent = c("firm1", "firm2", "consumer1", "consumer2"),
  numeraire = 3
)

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

Description

Some simple 4-by-4 general equilibrium models.

Usage

gem_4_4(...)

Arguments

Examples

#### A general equilibrium model containing a capital good with service-life
# wear-and-tear. The new product can be used as a capital good with a service
# life of two periods, and the used old capital good is the old product.
ge <- sdm2(
  A = function(state) {
    a.firm1 <- CD_A(alpha = 2, Beta = c(0, 0.5, 0.5, 0), state$p)
    a.consumer <- c(1, 0, 0, 0)
    a.firm2 <- c(1, 0, 0, 0)
    a.firm3 <- c(0, 0, 0, 1)
    cbind(a.firm1, a.consumer, a.firm2, a.firm3)
  },
  B = matrix(c(
    1, 0, 0, 0,
    0, 0, 1, 1,
    0, 0, 0, 0,
    0, 0, 1, 0
  ), 4, 4, TRUE),
  S0Exg = matrix(c(
    NA, NA, NA, NA,
    NA, NA, NA, NA,
    NA, 100, NA, NA,
    NA, NA, NA, NA
  ), 4, 4, TRUE),
  names.commodity = c("prod.new", "cap", "lab", "prod.old"),
  names.agent = c("firm1", "consumer", "firm2", "firm3"),
  numeraire = "prod.new",
  priceAdjustmentVelocity = 0.05
)

ge$p
ge$z
addmargins(ge$D, 2)
addmargins(ge$S, 2)

#### the Shoven-Whalley model at
## https://lexjansen.com/nesug/nesug03/st/st002.pdf
ge <- sdm2(
  A = function(state) {
    a.firm.corn <- CES_A(sigma = 1 - 1 / 2, alpha = 1.5, Beta = c(0, 0, 0.4, 0.6), state$p)
    a.firm.iron <- CES_A(sigma = 1 - 1 / 0.5, alpha = 2, Beta = c(0, 0, 0.3, 0.7), state$p)
    a.consumer1 <- SCES_A(alpha = 1, Beta = c(0.5, 0.5, 0, 0), es = 1.5, p = state$p)
    a.consumer2 <- SCES_A(alpha = 1, Beta = c(0.3, 0.7, 0, 0), es = 0.75, p = state$p)

    cbind(a.firm.corn, a.firm.iron, a.consumer1, a.consumer2)
  },
  B = diag(c(1, 1, 0, 0), 4, 4),
  S0Exg = {
    tmp <- matrix(NA, 4, 4)
    tmp[3, 3] <- 25
    tmp[4, 4] <- 60
    tmp
  },
  names.commodity = c("corn", "iron", "cap", "lab"),
  names.agent = c("firm.corn", "firm.iron", "consumer1", "consumer2"),
  numeraire = "lab"
)

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

#### an n-by-n general equilibrium model with Cobb-Douglas functions.
f <- function(n, policy = NULL, z0 = rep(100 * n, n), numberOfPeriods = 30,
              Beta = matrix(1 / n, n, n), n.firm = n - 1) {
  ge <- sdm2(
    A = function(state) {
      CD_A(alpha = rep(n, n), Beta = Beta, p = state$p)
    },
    B = {
      tmp <- diag(n)
      tmp[, (n.firm + 1):n] <- 0
      tmp
    },
    S0Exg = {
      tmp <- matrix(NA, n, n)
      for (k in (n.firm + 1):n) tmp[k, k] <- 100 * n
      tmp
    },
    numeraire = n,
    policy = policy,
    z0 = z0,
    maxIteration = 1,
    numberOfPeriods = numberOfPeriods,
    names.agent = c(paste0("firm", 1:n.firm), paste0("consumer", 1:(n - n.firm))),
    ts = TRUE
  )
  print(ge$z)
  print(ge$p)
  invisible(ge)
}

n <- 4
f(n, n.firm = n - 2)
## a spot market clearing path (alias instantaneous equilibrium path)
ge <- f(n, policy = policyMarketClearingPrice, z0 = runif(n, 10 * n, 100 * n), n.firm = n - 2)
matplot(ge$ts.z, type = "b", pch = 20)
matplot(ge$ts.p, type = "b", pch = 20)

Description

This is an example that illustrates the Matthew effect of asset exchange, wherein the wealth gap between two traders widens after the exchange process. Initially, these traders had a relatively small wealth (i.e. expected average payoff) gap. However, the exchange leads to an expansion of the wealth gap. This outcome can be attributed to the fact that a trader's risk aversion coefficient is affected by his level of wealth. When traders have less wealth their risk aversion coefficient is higher. Consequently, a trader with less wealth tends to acquire more low-risk, low-average-payoff assets through trading. As a result, the expected average payoff of a trader with less wealth may decrease after the exchange. Conversely, a trader with more wealth may hold more high-risk, high-average-payoff assets after trading.

Usage

gemAssetExchange_MatthewEffect_2_2(...)

Arguments

See Also

gemAssetPricing_PUF.

Examples

#### Matthew effect
asset1 <- c(40, 200)
asset2 <- c(100, 100)

# unit asset payoff matrix.
UAP <- cbind(asset1, asset2)

S <- matrix(c(
  0.49, 0.51,
  0.49, 0.51
), 2, 2, TRUE)

ge <- sdm2(
  A = function(state) {
    Portfolio <- state$last.A %*% dg(state$last.z)

    Payoff <- UAP %*% Portfolio
    payoff.average <- colMeans(Payoff)

    # the risk aversion coefficients.
    rac <- ifelse(payoff.average > mean(UAP) * 1.02, 0.5, 1)
    rac <- ifelse(payoff.average < mean(UAP) / 1.02, 2, rac)

    uf1 <- function(portfolio) {
      payoff <- UAP %*% portfolio
      CES(alpha = 1, beta = c(0.5, 0.5), x = payoff, es = 1 / rac[1])
    }

    uf2 <- function(portfolio) {
      payoff <- UAP %*% portfolio
      CES(alpha = 1, beta = c(0.5, 0.5), x = payoff, es = 1 / rac[2])
    }

    VMU <- marginal_utility(Portfolio, diag(2), list(uf1, uf2), state$p)
    VMU <- pmax(VMU, 1e-10)

    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")
    A <- state$last.A * ratio_adjust(Ratio, coef = 0.1, method = "linear")

    prop.table(A, 2)
  },
  B = matrix(0, 2, 2),
  S0Exg = S,
  names.commodity = c("asset1", "asset2"),
  numeraire = 2,
  maxIteration = 1,
  numberOfPeriods = 1000,
  policy = makePolicyMeanValue(50),
  ts = TRUE
)

matplot(ge$ts.p, type = "l")
ge$p
ge$z
ge$D

(Payoff.S <- UAP %*% S)
colMeans(Payoff.S)

(Payoff.D <- UAP %*% ge$D)
colMeans(Payoff.D)

## Calculate the equilibrium under the fixed risk aversion coefficients.
rac <- c(2, 0.5)

uf <- list()
uf[[1]] <- function(portfolio) {
  payoff <- UAP %*% portfolio
  CES(alpha = 1, beta = c(0.5, 0.5), x = payoff, es = 1 / rac[1])
}

uf[[2]] <- function(portfolio) {
  payoff <- UAP %*% portfolio
  CES(alpha = 1, beta = c(0.5, 0.5), x = payoff, es = 1 / rac[2])
}

ge <- gemAssetPricing_PUF(
  S = S,
  uf = uf,
  policy = makePolicyMeanValue(50)
)

ge$p
addmargins(ge$D, 2)
addmargins(ge$S, 2)
ge$VMU

(Payoff <- UAP %*% ge$D)
colMeans(Payoff)

Description

Compute the equilibrium of an asset market by the function sdm2 and by computing marginal utility of assets (see Sharpe, 2008). The argument of the utility function used in the calculation is the commodity vector (i.e. payoff vector).

Usage

gemAssetPricing_CUF(
  S = diag(2),
  UAP = diag(nrow(S)),
  uf = NULL,
  muf = NULL,
  ratio_adjust_coef = 0.05,
  numeraire = 1,
  ...
)

Arguments

Value

A general equilibrium containing a value marginal utility matrix (VMU).

References

Danthine, J. P., Donaldson, J. (2005, ISBN: 9780123693808) Intermediate Financial Theory. Elsevier Academic Press.

Sharpe, William F. (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

Wang Jiang (2006, ISBN: 9787300073477) Financial Economics. Beijing: China Renmin University Press. (In Chinese)

Xu Gao (2018, ISBN: 9787300258232) Twenty-five Lectures on Financial Economics. Beijing: China Renmin University Press. (In Chinese)

https://web.stanford.edu/~wfsharpe/apsim/index.html

See Also

gemAssetPricingExample.

Examples

gemAssetPricing_CUF(muf = function(x) 1 / x)

gemAssetPricing_CUF(
  S = cbind(c(1, 0), c(0, 2)),
  muf = function(x) 1 / x
)

gemAssetPricing_CUF(
  UAP = cbind(c(1, 0), c(0, 2)),
  muf = function(x) 1 / x
)

#### an example of Danthine and Donaldson (2005, section 8.3).
ge <- gemAssetPricing_CUF(
  S = matrix(c(
    10, 5,
    1, 4,
    2, 6
  ), 3, 2, TRUE),
  uf = function(x) 0.5 * x[1] + 0.9 * (1 / 3 * log(x[2]) + 2 / 3 * log(x[3]))
)

ge$p

#### an example of Sharpe (2008, chapter 2, case 1)
asset1 <- c(1, 0, 0, 0, 0)
asset2 <- c(0, 1, 1, 1, 1)
asset3 <- c(0, 5, 3, 8, 4) - 3 * asset2
asset4 <- c(0, 3, 5, 4, 8) - 3 * asset2
# unit asset payoff matrix
UAP <- cbind(asset1, asset2, asset3, asset4)

prob <- c(0.15, 0.25, 0.25, 0.35)
wt <- prop.table(c(1, 0.96 * prob)) # weights

geSharpe1 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  uf = list(
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5)
  )
)
geSharpe1$p
geSharpe1$p[3:4] + 3 * geSharpe1$p[2]

## an example of Sharpe (2008, chapter 3, case 2)
geSharpe2 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49, 98, 98,
    30, 30, 60, 60,
    10, 0, 20, 0,
    0, 10, 0, 20
  ), 4, 4, TRUE),
  UAP = UAP,
  uf = list(
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5)
  )
)

geSharpe2$p
geSharpe2$p[3:4] + 3 * geSharpe2$p[2]
geSharpe2$D

## an example of Sharpe (2008, chapter 3, case 3)
geSharpe3 <- gemAssetPricing_CUF(UAP,
  uf = function(x) (x - x^2 / 400) %*% wt,
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE)
)
geSharpe3$p
geSharpe3$p[3:4] + 3 * geSharpe3$p[2]

# the same as above
geSharpe3b <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  muf = function(x) (1 - x / 200) * wt
)

geSharpe3b$p
geSharpe3b$p[3:4] + 3 * geSharpe3b$p[2]

## an example of Sharpe (2008, chapter 3, case 4)
geSharpe4 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE),
  UAP,
  muf = function(x) abs((x - 20)^(-1)) * wt,
  maxIteration = 100,
  numberOfPeriods = 300,
  ts = TRUE
)

geSharpe4$p
geSharpe4$p[3:4] + 3 * geSharpe4$p[2]

## an example of Sharpe (2008, chapter 6, case 14)
prob1 <- c(0.15, 0.26, 0.31, 0.28)
wt1 <- prop.table(c(1, 0.96 * prob1))
prob2 <- c(0.08, 0.23, 0.28, 0.41)
wt2 <- prop.table(c(1, 0.96 * prob2))

uf1 <- function(x) CES(alpha = 1, beta = wt1, x = x, es = 1 / 1.5)
uf2 <- function(x) CES(alpha = 1, beta = wt2, x = x, es = 1 / 2.5)
geSharpe14 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  uf = list(uf1,uf2)
)

geSharpe14$D
geSharpe14$p
geSharpe14$p[3:4] + 3 * geSharpe14$p[2]
mu <- marginal_utility(geSharpe14$Payoff, diag(5),uf=list(uf1,uf2))
mu[,1]/mu[1,1]
mu[,2]/mu[1,2]

#### an example of Wang (2006, example 10.1, P146)
geWang <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 2,
    0, 1
  ), 3, 2, TRUE),
  muf = list(
    function(x) 1 / x * c(0.5, 0.25, 0.25),
    function(x) 1 / sqrt(x) * c(0.5, 0.25, 0.25)
  )
)

geWang$p # c(1, (1 + sqrt(17)) / 16)

# the same as above
geWang.b <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 2,
    0, 1
  ), 3, 2, TRUE),
  uf = list(
    function(x) log(x) %*% c(0.5, 0.25, 0.25),
    function(x) 2 * sqrt(x) %*% c(0.5, 0.25, 0.25)
  )
)

geWang.b$p

#### an example of Xu (2018, section 10.4, P151)
wt <- c(1, 0.5, 0.5)
ge <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  uf = list(
    function(x) CRRA(x, gamma = 1, prob = wt)$u,
    function(x) CRRA(x, gamma = 0.5, prob = wt)$u
  )
)

ge$p # c(1, (1 + sqrt(5)) / 4, (1 + sqrt(17)) / 16)

#### an example of incomplete market
ge <- gemAssetPricing_CUF(
  UAP = cbind(c(1, 1), c(2, 1)),
  uf = list(
    function(x) sum(log(x)) / 2,
    function(x) sum(sqrt(x))
  ),
  ratio_adjust_coef = 0.1,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(span = 100),
  maxIteration = 1,
  numberOfPeriods = 2000,
)

ge$p

## the same as above
ge.b <- gemAssetPricing_CUF(
  UAP = cbind(c(1, 1), c(2, 1)),
  muf = list(
    function(x) 1 / x * c(0.5, 0.5),
    function(x) 1 / sqrt(x) * c(0.5, 0.5)
  ),
  ratio_adjust_coef = 0.1,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(span = 100),
  maxIteration = 1,
  numberOfPeriods = 2000,
  ts = TRUE
)

ge.b$p
matplot(ge.b$ts.p, type = "l")

#### an example with outside position.
asset1 <- c(1, 0, 0)
asset2 <- c(0, 1, 1)

# unit (asset) payoff matrix
UAP <- cbind(asset1, asset2)
wt <- c(0.5, 0.25, 0.25) # weights

uf1 <- function(x) prod((x + c(0, 0, 2))^wt)
uf2 <- function(x) prod(x^wt)

ge <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 1,
    0, 2
  ), 2, 2, TRUE),
  UAP = UAP,
  uf = list(uf1, uf2),
  numeraire = 1
)

ge$p
ge$z
uf1(ge$Payoff[,1])
uf2(ge$Payoff[,2])