STOCHASTIC

MIND-EXPANDING EXERCISES
1. Consider a two-period problem and let the decision vectors in periods 1 and 2 be xt
and x2, where these vectors must be chosen from the convex sets Ki and K2 (•, •) respectively.
Suppose that £i and {2 have a known joint distribution and that the preference functions in
periods 1 and 2 are/i and f2, where/I(JCI, Oand/jC*!,;^,', •) are finite and convex, respectively.
Assume that all maxima and expectations indicated below are finite. The problem is
min E(l
Mxl,Z1)+ min \EiMJ2(xux2,Zu£2)
X2 € K2(Xl.5l)
Let |i be the mean of {t and |2(^i) be the mean of £2 conditional on £i, and denote by
(xi, x2) a solution to
min [fi(xuZi)+Mxi,X2,ZuZi(£i)y] = ZL.
X! 6 Kl,X2 € K2(*li?i>
(a) Show that Zt g Z*. [#i7rt: Utilize Jensen's inequality.]
(b) Under what assumptions is
Z* g £i,{/i(*i,«+£i1|tl/i(*i.*a,«i,«}?
(c) Compute these bounds when /i (•,•) = (*i-£i) 4 > /2(-, •,-,) = (JC2-I?2), 4
if, = tf2(-, •) = 0| 1 g x^0, x s R],
^Ki=0] = i,
/',K2 = 2|«1=0] = i,
/ , ,K» = 4|f, = l]-i,
PrKl = 1] = i,
^[«2 = 3|^1 = 0] = i,
i*,Ki = 2|«l = 2] = i,
/",Ki-2]=i,
i*rK2-3|«i-l] = i,
PfK2 = 4|^=2] = l.
(d) Suppose that £1 and £2 have general discrete distributions. Show that the random
problem is equivalent to a deterministic problem.
(e) Show that the deterministic problem in (d) is convex.
(f) Illustrate the problem of (c) via the result of (d). Solve this problem and indicate
how sharp the bounds are.
2. This problem concerns the effect of uncertainty on savings decisions when the consumer
has uncertainty concerning either his future income or his yield on capital investments.
Suppose that the consumer's utility function over present and future consumption
(Ci, C2) is C(Ci, C2), where u is strictly monotone, concave, and three times continuously
differentiable. Equation (2.8) in the Dreze-Modigliani article develops the risk-aversion
function, which may be written as
_2 -U22(CUC2)
h* P £/2(C„C2)
for equally likely gambles (Ci,C2 — h) and (C1,C2 + h), where p > 0 is the risk premium
(as discussed in Pratt's paper in Part II). Assume that the risk-aversion function is increasing
in d and decreasing in C2; that is, there is decreasing temporal risk aversion.
(a) Graphically interpret the concept of decreasing temporal risk aversion.
(b) Suppose the consumer is endowed with consumption (Ci, C2) and is offered a gamble
having outcomes ± h of future consumption. Show that the consumer will accept the
gamble only when the probability of a gain of A, say n(h), is greater than \.
(c) Show that n is an increasing function of Ct.
(d) Show that n will fall with a simultaneous increase in C2 and decrease in Ct.
MIND-EXPANDING EXERCISES 677