GE (version 0.4.8)

gemIntertemporalStochastic_Bank_TwoPeriods: An Intertemporal Stochastic Model with a Consumer and a Bank

Description

An intertemporal stochastic model with a consumer and a bank. In this model the consumer will live for two periods. There is one natural state in the first period, and two natural states in the second period.

The consumer has an intertemporal stochastic utility function of the Cobb-Douglas (CD) type, \(x_1^{1/2} x_2^{1/6} x_3^{1/3}\), where \(x_1\), \(x_2\), and \(x_3\) represent the payoffs in three different states of nature, respectively. The ratio of the share coefficients for the two future states of nature is equal to the ratio of their corresponding probabilities. The share coefficient for the present is the same as that for the future.

Usage

gemIntertemporalStochastic_Bank_TwoPeriods(...)

Arguments

...

arguments to be passed to the function sdm2.

Examples

# \donttest{
#### (A) A savings bank.
Ra <- 1.2 # the interest rate coefficient in the first natural state in the future
Rb <- 1.1 # the interest rate coefficient in the second natural state in the future

# When the savings bank invests one unit of payoff 1, it can
# produce Ra units of payoff 2 and Rb units of payoff 3.
dst.bank <- node_new(
  "output",
  type = "Leontief", a = 1,
  "payoff1"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 2, 1 / 6, 1 / 3),
  "payoff1", "payoff2", "payoff3"
)

ge <- sdm2(
  A = list(dst.bank, dst.consumer),
  B = matrix(c(
    0, 0,
    Ra, 0,
    Rb, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, 1,
    NA, 0,
    NA, 2
  ), 3, 2, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("bank", "consumer"),
  numeraire = "payoff1",
)

ge$p
unname(Ra * ge$p[2] + Rb * ge$p[3])
# The amount of savings in equilibrium is 0.2995.
addmargins(ge$D, 2)
addmargins(ge$S, 2)

## Solve with the optimization method and Rsolnp package.
# library(Rsolnp)
# Ra <- 1.2
# Rb <- 1.1
# payoff <- c(1, 0, 2)
# # The loss function is used to calculate the loss (i.e.,
# # the negative utility) for a given amount of savings.
# loss <- function(savings) {
#   utility <- prod(c(payoff[1] - savings, payoff[2] +
#                       Ra * savings, payoff[3] + Rb * savings)^wt)
#   return(-utility)
# }
#
# result <- solnp(0.5, loss, LB = 0, UB = 1)
# result$pars
#
# x <- rbind(payoff[1] - result$pars, payoff[2] + Ra * result$pars, payoff[3] + Rb * result$pars)
# mu <- marginal_utility(x, diag(3), uf = function(x) prod(x^wt))
# mu / mu[1]

#### (B) A lending bank.
Ra <- 1.2
Rb <- 1.1

# To produce one unit of payoff 1, the lending bank needs
# to invest Ra units of payoff 2 and Rb units of payoff 3.
dst.bank <- node_new(
  "payoff1",
  type = "Leontief", a = c(Ra, Rb),
  "payoff2", "payoff3"
)

dst.consumer <- node_new(
  "util",
  type = "CD", alpha = 1, beta = c(1 / 2, 1 / 6, 1 / 3),
  "payoff1", "payoff2", "payoff3"
)

ge <- sdm2(
  A = list(dst.bank, dst.consumer),
  B = matrix(c(
    1, 0,
    0, 0,
    0, 0
  ), 3, 2, TRUE),
  S0Exg = matrix(c(
    NA, 0,
    NA, 1,
    NA, 2
  ), 3, 2, TRUE),
  names.commodity = c("payoff1", "payoff2", "payoff3"),
  names.agent = c("bank", "consumer"),
  numeraire = "payoff1"
)

ge$p
unname(Ra * ge$p[2] + Rb * ge$p[3])
addmargins(ge$D, 2)
# In equilibrium, the bank lends out 0.5645 units of payoff in period 1.
addmargins(ge$S, 2)

## Solve with the optimization method and Rsolnp package.
# library(Rsolnp)
# Ra <- 1.2
# Rb <- 1.1
# payoff <- c(0, 1, 2)
# loss <- function(savings) {
#   utility <- prod(c(payoff[1] - savings, payoff[2] +
#                       Ra * savings, payoff[3] + Rb * savings)^wt)
#   return(-utility)
# }
#
# result <- solnp(-0.5, loss, LB = -2, UB = 0)
# result$pars
#
# x <- rbind(payoff[1] - result$pars, payoff[2] + Ra * result$pars, payoff[3] + Rb * result$pars)
# mu <- marginal_utility(x, diag(3), uf = function(x) prod(x^wt))
# mu / mu[1]
# }