gemOLGF_PureExchange: Overlapping Generations Financial Sequential Models for Pure...

In GE: General Equilibrium Modeling

Overlapping Generations Financial Sequential Models for Pure Exchange Economies

Description

Some examples of overlapping generations sequential models with security for pure exchange economies.

In these examples, a security (e.g., fiat money, public bond, shares of a firm, etc) serves as a means of saving (see Samuelson, 1958). Consumers use these securities to transfer value across time, enabling intertemporal allocation of resources.

Here financial demand structure trees are used, which contain financial nodes. A financial demand structure tree reflects the demand structure of a consumer who has a demand for financial instruments.

When experiencing population growth, we adopt the security-split assumption. Specifically, we assume that in each period, the security undergoes a split similar to a stock split, with the quantity of the security growing at a rate equal to the population growth rate. Obviously, this assumption will not affect the calculation results essentially. And with this assumption, the equilibrium price vector can keep constant in each period, and the nominal rates of profit and interest will equal the real rates of profit and interest (i.e. the population growth rate). In contrast, in the time circle model the nominal rates of profit and interest equal zero and the real rates of profit and interest equal the population growth rate.

Usage

gemOLGF_PureExchange(...)

Arguments

...	arguments to be passed to the function sdm2.

Note

As can be seen from the first example below, in a pure exchange economy with two-period-lived consumers, if the young (i.e., age 1) has labor and the old (i.e., age 2) does not, then the optimal steady-state allocation can be obtained by introducing securities as a store of value.

However, if the old (i.e., age 2) has 1 unit of labor and the young (i.e., age 1) does not, then the optimal steady-state allocation cannot be obtained by introducing securities as a store of value. Therefore, we need to introduce the family system, or in other words, grant the young the right to tax the old.

See Also

gemOLG_PureExchange

Examples

#### (A) An OLG pure exchange economy with two-period-lived consumers.
## To simplify the analysis, we assume that the savings rate of the
## young consumer is an exogenous value. The consumer's utility
## function that corresponds to this savings rate can be constructed
## based on the steady-state equilibrium results.

sr <- 0.5

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption <- sr / (1 - sr)),
  "lab", "secy"
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "lab"
)

ge <- sdm2(
  A = list(
    dst.age1, dst.age2
  ),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    100, 0,
    0, 100
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy"
)

ge$p
ge$D
ge$DV
ge$S

##
sr <- 0.375
dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption <- sr / (1 - sr)),
  "lab", "secy"
)

ge <- sdm2(
  A = list(
    dst.age1, dst.age2
  ),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    80, 20,
    0, 100
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy"
)

ge$p
ge$D
ge$DV
ge$S

##
sr <- 0.4
dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption <- sr / (1 - sr)),
  "lab", "secy"
)

ge <- sdm2(
  A = list(
    dst.age1, dst.age2
  ),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    100, 20,
    0, 100
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy"
)

ge$p
ge$D
ge$DV
ge$S

####  (B) The population growth and demographic dividend.
## Suppose each consumer has a SCES intertemporal utility function.
gr.lab <- 0.03 # the growth rate of population and labor supply

# share parameters of the SCES function
beta2 <- 0.4
beta1 <- 1 - beta2
es <- 0.5 # the elasticity of substitution in the SCES function

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = 0.1),
  "lab", "secy",
  p.lab.last = 1,
  p.lab.ratio.predicted.last = 1,
  ts.saving.rate = numeric(0)
)

dst.age2 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "lab"
)

ge <- sdm2(
  A = list(
    dst.age1, dst.age2
  ),
  B = matrix(0, 2, 2, TRUE),
  S0Exg = matrix(c(
    100, NA,
    NA, 100
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy",
  policy = list(function(time, A, state) {
    state$S[1, 1] <- 100 * (1 + gr.lab)^time
    p.lab.current <- state$p[1] / state$p[2]

    lambda <- 0.6
    p.lab.ratio.predicted <- p.lab.current / A[[1]]$p.lab.last * lambda +
      A[[1]]$p.lab.ratio.predicted.last * (1 - lambda)
    A[[1]]$p.lab.last <- p.lab.current
    A[[1]]$p.lab.ratio.predicted.last <- p.lab.ratio.predicted

    ratio.saving.consumption <- beta2 / beta1 * (p.lab.ratio.predicted)^(1 - es)
    A[[1]]$rate <- c(1, ratio.saving.consumption)
    A[[1]]$ts.saving.rate <- c(A[[1]]$ts.saving.rate, ratio.saving.consumption /
                                 (1 + ratio.saving.consumption))

    state
  }, policyMarketClearingPrice),
  maxIteration = 1,
  numberOfPeriods = 50,
  ts = TRUE
)

matplot(growth_rate(ge$ts.p), type = "o", pch = 20)
matplot(growth_rate(ge$ts.z), type = "o", pch = 20)
ge$p
dst.age1$rate[2] # beta2 / beta1 * (1 + gr.lab)^(es - 1)
dst.age1$p.lab.ratio.predicted.last

plot(dst.age1$ts.saving.rate, type = "o", pch = 20)
tail(dst.age1$ts.saving.rate,1) # beta2 / (beta2 + beta1 * (1 + gr.lab)^(1 - es))

#### (C) The basic overlapping generations (inefficient) exchange model.
## Here the lab2 is regarded as a financial instrument (saving instrument).
## See gemOLG_PureExchange.
dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.totalSaving.consumption = 2),
  "lab1", "lab2"
)

dst.age2 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = 1),
  "lab1", "lab2"
)

ge <- sdm2(
  A = list(dst.age1, dst.age2),
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    50, 50,
    50, 0
  ), 2, 2, TRUE),
  names.commodity = c("lab1", "lab2"),
  names.agent = c("age1", "age2"),
  numeraire = "lab1",
  policy = function(time, state) {
    pension <- (state$last.A[, 2] * state$last.z[2])[2]
    if (time > 1) state$S[1, 2] <- 1 - pension
    state
  }
)

ge$p
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ge$D
ge$DV

#### (D) the basic financial overlapping generations exchange model (see Samuelson, 1958).
## Suppose each consumer has a utility function log(c1) + log(c2) + log(c3).
GRExg <- 0.03 # the population growth rate
rho <- 1 / (1 + GRExg)

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = {
    saving.rate <- (2 - rho) / 3
    c(1, ratio.saving.consumption = saving.rate / (1 - saving.rate))
  },
  "lab", "secy"
)

dst.age2 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption = 1),
  "lab", "secy"
)

dst.age3 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "lab"
)

ge <- sdm2(
  A = list(dst.age1, dst.age2, dst.age3),
  B = matrix(0, 2, 3),
  S0Exg = matrix(c(
    1 + GRExg, 1, 0,
    0, 0.5, 0.5
  ), 2, 3, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2", "age3"),
  numeraire = "lab",
  policy = function(time, state) {
    # Assume that unsold public bond will be void.
    last.Demand <- state$last.A %*% dg(state$last.z)
    secy.holding <- prop.table(last.Demand[2, ])
    if (time > 1) {
      state$S[2, 2:3] <- secy.holding[1:2]
    }
    state
  }
)

ge$p
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#### (E) a pure exchange economy with three-period-lived consumers.
## Suppose each consumer has a Leontief-type utility function min(c1, c2, c3).
GRExg <- 0.03 # the population growth rate
igr <- 1 + GRExg

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = {
    saving.rate <- 1 / (1 + igr + igr^2)
    c(1, ratio.saving.consumption = saving.rate / (1 - saving.rate))
  },
  "lab", "secy"
)

dst.age2 <- node_new(
  "util",
  type = "FIN",
  rate = {
    saving.rate <- 1 / (1 + igr)
    c(1, ratio.saving.consumption = saving.rate / (1 - saving.rate))
  },
  "lab", "secy"
)

dst.age3 <- node_new(
  "util",
  type = "Leontief", a = 1,
  "lab"
)

ge <- sdm2(
  A = list(dst.age1, dst.age2, dst.age3),
  B = matrix(0, 2, 3),
  S0Exg = matrix(c(
    1 + GRExg, 1, 0,
    0, 0.5, 0.5
  ), 2, 3, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2", "age3"),
  numeraire = "lab",
  policy = function(time, state) {
    # Assume that unsold security will be void.
    last.Demand <- state$last.A %*% dg(state$last.z)
    secy.holding <- prop.table(last.Demand[2, ])
    if (time > 1) {
      state$S[2, 2:3] <- secy.holding[1:2]
    }
    state
  }
)

ge$p
ge$S
ge$D

## Assume that the unsold security of age3 will be void.
## The calculation results are the same as above.
ge <- sdm2(
  A = list(dst.age1, dst.age2, dst.age3),
  B = matrix(0, 2, 3),
  S0Exg = matrix(c(
    1 + GRExg, 1, 0,
    0, 0.5, 0.5
  ), 2, 3, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2", "age3"),
  numeraire = "lab",
  policy = function(time, state, state.history) {
    secy.unsold <- state.history$S[2, , time - 1] * (1 - state.history$q[time - 1, 2])
    last.Demand <- state$last.A %*% dg(state$last.z)
    secy.purchased <- last.Demand[2, ]

    if (time > 1) {
      # Assume that the unsold security of age3 will be void.
      state$S[2, 2:3] <- prop.table(secy.purchased[1:2] + secy.unsold[1:2])
    }
    state
  },
  maxIteration = 1
)

ge$p
ge$S
ge$D

GE documentation built on April 4, 2025, 5:33 a.m.