GE (version 0.3.0)

AMSD: Additive-Mean-Variance Utility Function and Additive-Mean-Standard-Deviation Utility Function

Description

Compute the utility function mean(x) - 0.5 * gamma * sd.p(x)^theta or weighted.mean(x, wt) - 0.5 * gamma * sd.p(x, wt)^theta.

Usage

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

Arguments

x

a numeric vector.

gamma

the risk aversion coefficient.

wt

a numeric vector of weights (or probability). If wt is NULL, all elements of x are given the same weight.

theta

a non-negative scalar with a default value of 1.

Functions

• AMSD: Computes the utility function mean(x) - 0.5 * gamma * sd.p(x)^theta or weighted.mean(x, wt) - 0.5 * gamma * sd.p(x, wt)^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

# \donttest{
AMSD(1:2, gamma = 0.1)
AMSD(1:2, gamma = 1, theta = 2)

marginal_utility(
  c(1, 1.001),
  c(0, 1), AMSD
)
marginal_utility(
  c(1.001, 1),
  c(0, 1), AMSD
)
####
## This example computes the equilibrium prices of securities under the additive
## mean-variance utility function.In the calculation, the demand structure of each
## agent is adjusted according to the marginal utility and the prices of securities.
secy1 <- c(1, 0)
secy2 <- c(0, 1)
prob <- c(0.5, 0.5)
UP <- cbind(secy1, secy2) # unit (security) payoff matrix

fun <- function(gamma.agt1 = 1, gamma.agt2 = 1,
                S0Exg = matrix(c(
                  0.5, 0.5,
                  1, 1
                ), 2, 2, TRUE)) {
  sdm2(
    A = function(state) {
      Payoff <- UP %*% (state$last.A %*% dg(state$last.z))

      VMU <- marginal_utility(Payoff, UP, list(
        function(x) AMV(x, gamma = gamma.agt1, prob),
        function(x) AMV(x, gamma = gamma.agt2, prob)
      ), state$p)

      A <- state$last.A * pmax(VMU, 0)
      prop.table(A, 2)
    },
    B = matrix(0, 2, 2),
    S0Exg = S0Exg,
    names.commodity = c("secy1", "secy2"),
    names.agent = c("agt1", "agt2")
  )
}

fun()$p
Payoff <- UP %*% c(0.5, 1)
marginal_utility(
  Payoff, UP,
  function(x) AMV(x, gamma = 1, prob)
)

fun(S0Exg = matrix(c(
  1, 0,
  0, 2
), 2, 2, TRUE))$p

fun(gamma.agt1 = 0.5, gamma.agt2 = 1)$p

####
## This example computes the equilibrium prices of securities under
## the additive mean and standard deviation utility function.
secy1 <- c(1, 0)
secy2 <- c(0, 1)
prob <- c(0.5, 0.5)
UP <- cbind(secy1, secy2) # security structure matrix

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

    VMU <- marginal_utility(
      Payoff, UP,
      function(x) AMSD(x, gamma = 0.1, prob), state$p
    )
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * ratio_adjust(Ratio, coef = 0.3, method = "linear")
    prop.table(A, 2)
  },
  B = matrix(0, 2, 2),
  S0Exg = matrix(c(
    1, 0,
    0, 4
  ), 2, 2, TRUE),
  names.commodity = c("secy1", "secy2"),
  names.agent = c("agt1", "agt2"),
  ts = TRUE,
  maxIteration = 1,
  numberOfPeriods = 2000,
  policy = policyMeanValue
)

matplot(tail(ge$ts.q, 100), type = "l")

ge$D
ge$p
Payoff <- UP %*% ge$D
marginal_utility(
  Payoff, UP,
  function(x) AMSD(x, gamma = 0.1, prob)
)

marginal_utility(
  c(1,4), UP,
  function(x) AMSD(x, gamma = 0.1, prob)
)
# }