STOCHASTIC

Show that the Kuhn-Tucker conditions are V/(*) + A' Vg(x) = 0, g(x) 5= 0, and A g 0.
[Hint: Investigate the direction gradients for the problem {min x 2 \ — x := 0}. ]
(b) Utilize the result in (a) to verify the following chart:
g^o
g^o
max/
AgO
Ag 0
min/
AgO
A^ 0
giving the sign of the multipliers in the Kuhn-Tucker conditions: V/(JC)+A' Vg(x) = 0,
g{x) g 0 or £ 0.
17. Suppose that an investor's preference ordering for proportional gambles on wealth
is independent of his wealth level.
(a) Show that
if and only if
/>i«[>(l+'"1)] + p2u[w(\ + r2y\ = 4i"l>(l+.Ji)] + q2u[w(l+s2)],
/>i«0+r1)+/>2«(l+r2) = qlu(l + sl) + q2u(l+s2)
for all/),, qt,rt,st, and w, with w > 0, pt, qt & 0, pt+p2 = qi +q2 = 1-
(b) Show that « must have the form w" or log w.
18. This problem presents an alternative to the expected utility approach based on the
assumption that the decision maker is an augmented income maximizer. What is altered
in this approach is not the shape of the utility function (assumed to be linear) but the
measure of gains and losses.
Let c be the cost of a lottery ticket or an insurance policy and rc the opportunity rate on c.
Suppose that the possible monetary outcomes are £i,...,£m having probabilities pt > 0
and that the opportunity interest rate on each {, is rt. Over a T-period horizon the augmented
cost is c„ = c(l + rc) T and the expected augmented gain (or loss) is ya = X