marginal_utility {GE}

Marginal Utility

Description

If the argument price is null, this function computes the (delta) marginal utility. By default, delta is set to 1e-10. Otherwise this function computes the (delta) value marginal utility. For a utility function U(x), two vector x, y and a scalar price, the marginal utility is (U(x + delta * y) - U(x)) / delta, and the value marginal utility is (U(x + delta * y / price) - U(x)) / delta. For a marginal utility function M(x), three vector x, y, wt and a scalar price, the marginal utility is sum(M(x) * y * wt), and the value marginal utility is sum(M(x) * y * wt / price).

Usage

marginal_utility(x, y, uf, price = NULL, delta = 1e-10, muf = NULL)

Arguments

x

a numeric k-by-m matrix or a numeric vector (i.e. a k-by-1 matrix).

y

a numeric k-by-n matrix or a numeric vector (i.e. a k-by-1 matrix).

uf

a utility function or a list consisting of m utility functions.

price

NULL or a numeric n-vector consisting of prices of each column of y.

delta

a scalar.

muf

a marginal utility function or a list consisting of m marginal utility functions. A marginal utility function is the gradient of a utility function. When the components in the marginal utility vector are the same, a marginal utility function may calculate only one of the components.

Value

An n-by-m marginal utility matrix. Its (i,j)-th element corresponds to the i-th column of y and the j-th column of x.

References

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

Examples

marginal_utility(1:2, cbind(1:2, 1:1), AMV)
marginal_utility(1:2, cbind(1:2, 1:1), AMV, delta = 100)
marginal_utility(cbind(1:2, 3:4), cbind(1:2, 1:1), AMV)
marginal_utility(
  cbind(1:2, 3:4), cbind(1:2, 1:1),
  list(AMV, function(x) AMV(x, gamma = 0.5))
)

####
wt <- 1:2
uf <- function(x) (x - x^2 / 400) %*% wt
muf <- function(x) (1 - 1 / 200 * x) * wt
marginal_utility(1:2, cbind(1:2, 1:1), uf)
marginal_utility(1:2, cbind(1:2, 1:1), muf = muf)

####
marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x, gamma = 1, p = rep(1, length(x))) CRRA(x, gamma, p)$CE
)
marginal_utility(1:2, cbind(1:2, 1:1), function(x) sqrt(prod(x)))

gamma <- 0.8
wt <- c(0.25, 0.75)
marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x) CRRA(x, gamma = gamma, prob = wt)$CE
)
## the same as above. CRRA and CES utility funcitons are essentially the same.
es <- 1 / gamma
beta <- wt^es
marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x) CES(x = x, sigma = 1 - 1 / es, alpha = 1, beta = wt)
)

prop.table(marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x) CRRA(x, gamma = gamma, prob = wt)$CE
))
prop.table(marginal_utility(
  1:2, cbind(1:2, 1:1),
  function(x) CRRA(x, gamma = gamma, prob = wt)$u
))