STOCHASTIC

(e) Suppose {i and £2 are independent. Show that a necessary and sufficient condition
for intermediation is that Ji > r > (2.
(f) Interpret the result in (e), noting that £,—r may be considered to be a risk premium
for security j.
It is said that £j and & are positively dependent if Zj(£k) is strictly increasing in £k.
(g) Show that Fj(0,xk) given positively dependent yields is greater than, equal to, or
less than Fj(0,xk) given independently distributed yields as xk is less than, equal to, or
greater than zero.
(h) Utilizing the result of (g) show that a sufficient but not a necessary condition for
intermediation is that Ji > r > |2.
(i) Develop restrictions on F and/or u that lead to a sufficient condition for intermediation
when Ji > r and £2 > r , and £1 and {2 are positively dependent.
7. Refer to the description of Exercise 6. Suppose that the firm has a utility function G
defined over mean fi and variance a 2 , which has the property that SG/fy > 0 and dG/8 0.
Show that a necessary and sufficient condition for intermediation is p£l'a2 > |2' 0.
J — 00
(a) Suppose that h = w and that F is symmetric. Show that Sh = \a 2 , where a 2 is the
variance of w.
Consider the utility function
u(w) = aw + 6[min{w—h,0} 2 ].
(b) Graph u and note that it is quadratic for w < h and linear for w^h. When is u
concave?
(c) Show that maximizing the expected utility is equivalent to maximizing aiv + bSh.
(d) Show that the indifference curves Sh = (/i) are strictly monotone increasing and
strictly concave if a > 0 and b < 0.
(e) Devise a method for computing the fi-St, efficient surface. [Hint: See Exercise 24.]
(f) Devise a method for computing the fi-sH efficient surface, where sh is the positive
square root of Sh.
MIND-EXPANDING EXERCISES 345