STOCHASTIC

(f) Show that k > 0.
[Note (Exercise CR-10) that the sign of 8x/dw0 is positive, zero, or negative as the absolute
risk-aversion function — u"(w)/u'(w) is decreasing, constant, or increasing in w.]
(g) Show that x & 0, and decreasing absolute risk aversion implies that dx/da\,=0 is
positive.
(h) Indicate when 6xj8cn\,=0 is negative and interpret this result.
Consider a change in expectations such that all possible expectations of £ are multiplied by
a factor of X.
(i) Show that x = x/X satisfies the first- and second-order conditions for a maximum.
(j) Interpret the result of (i) and indicate how the result is useful for policy purposes
to encourage or discourage investment in the risky currency.
Consider a change in the dispersion of £ about a constant expected value. Let 0 = a+b£,
where a = 0, b = 1 indicates the initial position. Increasing b tends to stretch the distribution
of d; since all outcomes are multiplied by a number larger than 1. To keep the mean constant
a must be adjusted so that it satisfies
d& = da + db% = 0 or da/db = -|.
(k) Show that the first-order conditions become
E\—(a+b£)u'[(l + r)w0 + (a+b?)x] =0 when rf® = 0.
(1) Utilize the results of (e) and (k) to show that
(dxldb)imM = -x-^dx/dtx).
(m) Show that x and J have the same sign, and that if { > 0 and u has decreasing absolute
risk aversion, (dxjdb)? „„„, is negative.
(n) Indicate when {dxjdb)i const is positive and interpret this result.
4. Referring to the introduction to Exercise 3 suppose that the speculator may invest in
n forward exchange positions, i = 1,...,«, where xt refers to the investment level in currency i
and £[ to the random profit made per dollar invested in ;'. The problem is then to choose
x = (*!,...,x„) to maximize Eu(yv) = Eu[(l +r)w0 +£'*]•
(a) Show that first- and second-order conditions for expected utility maximization are
E[(u'(w)~\ = 0 (n equations) and E[££'u"(w)] negative definite. Show that these conditions
are sufficient for x to be the unique maximum.
Consider a shift in the mean of the /th return from |( to |, + a,.
(b) Utilize the first-order conditions to show that a small change in a, evaluated at
a, = 0 with expected utility remaining maximized leads to the first-order condition
£[f£'""(w)]( 0 */dai)«i=o = £[£«"( w)]*i + [m] (n equations),
where [m] is a column vector of zeros except for ~E[u'(w)] in the /th component.
(c) Utilize the first-order conditions to show that
E[tf'u"(w)][dxldw0] = £K«»](1+/-).
(d) Show that the results in (b) and (c) imply that
(0x,/0a,)„,=o = Su + Xl{dxjjSw0)l(\ + r),
where Su = —E[u'(wy]Dij/D, D is the determinant of the matrix E[(£'u"(w)], and Du
is the cofactor of the y th element of D.
184 PART II. QUALITATIVE ECONOMIC RESULTS