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

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Overlapping Generations Financial Models for Pure Exchange Economies

Description

Some examples of overlapping generations models with financial instrument for pure exchange economies.

In these examples, there is a financial instrument (namely security) which serves as saving means and can be regarded as money, the shares of a firm, etc. Consumers use this security for saving, and this is the only use of the security. As Samuelson (1958) wrote, society by using money (i.e. security) will go from the non-optimal configuration to the optimal configuration.

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. Although CD-type nodes can be used instead of financial-type nodes in the consumer's demand structure tree, the use of financial-type nodes will make the demand structure tree easier to understand.

When there is a population growth, we will take the security-split assumption. That is, assume that in each period the security will be split just like share split, and the growth rate of the quantity of the security is equal to the growth rate of the population. 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 age1 (i.e. young) has one unit of labor and age2 (i.e. old) does not, then the optimal allocation can be obtained by introducing securities. Here it is assumed that each consumer consumes one unit of labor in total.

However, if age2 (i.e. adult) has one unit of labor and age1 (i.e. child) does not, we cannot get the optimal allocation by introducing securities. So we need the family system.

See Also

gemOLG_PureExchange

Examples

#### an OLGF pure exchange economy with two-period-lived consumers.
## Suppose each consumer has one unit of labor in her first period
## and she has a C-D intertemporal utility function
## (1 - beta) * log(c1) + beta * log(c2) and a constant saving rate beta.
beta <- 0.5
ratio.saving.consumption <- beta / (1 - beta)

dst.age1 <- node_new(
  "util",
  type = "FIN",
  rate = c(1, ratio.saving.consumption),
  "lab", "secy" # security, the financial instrument
)

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(
    1, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy"
)

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

#### population growth and demographic dividend.
## Suppose each consumer has a SCES intertemporal utility function.
pgr <- 0.03 # population growth rate

beta2 <- 0.4  # share parameters of the SCES function
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", # security, the financial instrument
  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(
    1, NA,
    NA, 1
  ), 2, 2, TRUE),
  names.commodity = c("lab", "secy"),
  names.agent = c("age1", "age2"),
  numeraire = "secy",
  policy = list(function(time, A, state) {
    state$S[1, 1] <- (1 + pgr)^time
    p.lab.current <- state$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 + pgr)^(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 + pgr)^(1 - es))

#### 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(
    1, 1,
    1, 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$p
ge$S
ge$D
ge$DV

#### 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 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

#### 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 Nov. 8, 2023, 9:07 a.m.