STOCHASTIC

(d) By differentiating (cl) and (c2) with respect to pu show that
= afe /fo' = (b'(.Pi)lb(Pi))-(VPi) u'(wr+k)lu\wr + k)
dkldPl -6'((Pi)/6(Pi))-(l/P2) ' u'{w-k)lu\wr-k)'
differentiating (bl) and (b2) with respect to w, show that
u'(wr + k)lu"(w+k) _ 1 r + (8kldw)
u'(wr-k)lu"(w-k) ~ p r-(8k/8w) '
Thus, show that dk/dw = c(pur) (some constant); hence k = cw+f.
(e) From (cl), show that
hence,
(f)
b_+bXp})
u"(wr+cw+f)[c'(Pl)w+fXPl)] = | -T—2 + -£r)uXwy,
2rPl 2 2rp .),,
«'(*) 1
u"(x) c'w+f
Show that
uXw)
u"(w)
aipi)-
Pi \-
1
-(bXPl)lb(Pl))'
r + c(Pi)
-Pi(b'(Pl)/b(.Pl))
where x = wr + wc + f.
[cXPi)w+fXPi)l
u'{W)
77^-= aw + fl, a, j8 constant. (fl)
u (w)
(g) Show that the solution to (f 1) is of the form «'(>") = A(w + a) 1 , where / = 1/a and
a = fila. For/and c, as in (d), show that we must have a = (a—f)l(r — c) = («+/)/(/ + c);
hence r = 1 or a = /= 0.
This proves the necessity part of the theorem. Now prove sufficiency.
Sufficiency: In the general case, Eu(R) = E[u(wr + xh)] is maximized for x = x*,
satisfying E[uXR)h] = 0.
(h) Assuming — uXR*)l"'XR*) = P+aR*, show that x* = c(f)+arw), where c is a constant.
[Hint: -«'(#*) = (P+OLR*)U'XR*). Multiply both sides by h and take
expectations.]
(i) Show that EuXR*) = buXrw), where b is a constant. Hint: Show that
„ 1 dR* 1 dR* uXrw)
r dw r dw u (rw)
(j) Show that if r — 1 or /} = 0, U is equivalent to u.
(k) Prove the following:
Theorem If the expected utility maximization problem has an interior solution x* s (0, w),
a necessary and sufficient condition for U(w) to be equivalent to u{rw) is that
u'Xi)
In the context of a multiperiod investment problem, the equivalence of U(w) and u{rw) is
MIND-EXPANDING EXERCISES 423