STOCHASTIC

9. Let the partial relative risk-aversion function P(t; w) and relative risk-aversion function
R(J) be defined as in Exercise 8.
(a) Let w be fixed. Show that if P(t;w) is nonincreasing in / for t in some interval (0, /0)
with t0 > 0, then either
(i) P(t; w) = 0 [so «"(/+ w) = 0 for 0 < / < to], or
(ii) w = 0.
Let w > 0 be fixed, and suppose that t0 > 0.
(b) Show that if Pit; w) is monotone (strictly monotone) in t for 0 < / < t0, then it is
nondecreasing (strictly increasing) for 0 < t < t0.
(c) Show that if R(t) is nondecreasing, then either
(i) u"(t) = 0, or
(ii) P(t; w) is strictly increasing in t for each w.
10. Consider a strictly increasing utility function u{w) which is concave and bounded on
[), co). It is of interest to examine the behavior of R(w) = — wu"(w)/u'(w) for large and small
values of w. Suppose there exist numbers r and w0 > 0 such that R(w) ^r for w^w0.
(a) Show that
(b) Show that
u'(w) £ cw~' for w a w0, where c = H'(WO) Wo' > 0.
W(H>)
u(w0) + -^—[w i - r -wh~ r ], r*\
\—r
u{w0) + c[logH> — logWo], r — 1.
(c) Show that R(w) cannot converge to a limit which is less than 1 as w -> oo.
(d) Show that R(w) cannot converge to a limit which is greater than 1 as w-*0. This
shows that Riyv) must be increasing on the average, from values somewhat less than 1 for
small w to values somewhat greater than 1 for large w.
11. Suppose the random variables X and Y have distributions F and G, respectively, and
that
(i) /"^(G—F) du & 0 for all concave nondecreasing u, and
(ii) S-„(G—F)du^0 for all concave nonincreasing «.
(a) Show that Ji« [G0)-F(/)] dt^O for all x and J»M [G(/)-F(/)] dt = 0.
(b)* Either verify the converse of (a), i.e., that the conditions in (a) imply (i)-(ii), or
find a counterexample.
(c) Show that (i)-(ii) is equivalent to Eu(X) £ Eu(Y) for all concave u, provided all the
integrals involved exist and are finite.
12. Let X and Y be discrete random variables concentrated on the finite set of points
Q = {ai,a2, ...,«„} c \a,b~\. Let Xt be a discrete random variable concentrated on fi, and
let pi = P[X= a,], pi' = P[Xi = a,]. Xt is said to differ from X by a mean-preserving
spread (MPS) if
(i) pi — p^ except for four values ilt i2, '3, and i4, with a(l S a,-2 g a,3 g at4;
(«i) /»i',-^i, =PH~P'H S 0, p,3-pi3=Pu-Pu = °:
(iii) EJ 4 =1ai,(/>o-/>(j) = 0.
188 PART II. QUALITATIVE ECONOMIC RESULTS