STOCHASTIC

and where Z, Ylt...,Ym are independently but identically distributed random variables
with the standardized Laplace density
iexp{-b|}.
Let Y?= i^i = l.^i^0, and let U be the investor's utility of wealth function. Investigate
what restrictions are required on U in order that
1 / m \
is quadratic in (Ai,..., Am). [Assume, without loss of generality that £/(l) = 0]. Find the
limit when it exists. [Hint: Consider first the case when U is bounded on (0, oo), and
show that extremely mild conditions then are required on U. For the case where U is
unbounded, consult Ohlson's paper. Note that all moments of EW k , k = 1,2,..., do not
exist, regardless of how small / is; hence, EU(W) can never be expressed as a linear function
of all moments EW k , k = 1,2,.... (Compare this with Samuelson's paper, in which it is
required that all moments exist.)]
32. Suppose an investor has a monotone, nondecreasing, differentiable, concave utility
function u(w) over wealth w. Assume further that w & 0. We are concerned with the question
of when the expected utility will be finite, given that u may be unbounded.
(a) Suppose «'(0) and E(w) are finite. Show that expected utility is finite. [Hint: First
establish that u'(W) is uniformly bounded, then utilize the differential definition of a concave
function.]
It is appropriate to consider weakening the assumptions on the finiteness of marginal
utility at zero wealth and/or the finiteness of the mean wealth level.
(b) Suppose M is a polynomial of degree n. Show that expected utility is never finite if
E(w) is infinite. [Hint: Begin by showing that if E(w) is infinite, then so is E(w") for
all n > 1).]
(c) Show that the result in (b) also holds for any nonconstant, monotone, nondecreasing,
concave utility function. Hence it is not possible to relax the assumption concerning the
finiteness of the mean vector.
(d) Suppose that £(w) is finite and H(0) is finite with no specification regarding the
finiteness of w'(0) and u(w0) is finite for some w0 > 0. Show that expected utility is finite.
(e) Extend the result in (d) to the case w £ A for A > — oo.
Exercise Source Notes
Exercise 1 was adapted from Freund (1956); Exercise 2 was adapted from Borch (1968);
Exercise 4 was adapted from Madansky (1962) and Ziemba (1971); Exercises 6 and 7 were
adapted from Pyle (1971); portions of Exercise 8 were adapted from Mao (1970); Exercises
9 and 10 were adapted from Press (1972); Exercise 11 was based on Pye (1974); Exercise 12
was based on Arrow (1971); portions of Exercises 13-15 were adapted from Ziemba
(1972a,c); Exercise 17 was adapted from Stone (1970) utilizing notes written by Professor
K. Nagatani; Exercise 20 was adapted from Aitchison and Brown (1966); Exercise 21 was
adapted from notes provided by Professor J. Ohlson and Exercises 22 and 23 were adapted
from Ohlson (1972b); Exercise 25 was adapted from Hogan and Warren (1972); Exercise 26
was adapted from Parikh (1968); Exercises 27 and 28 were adapted from Brumelle (1974);
Exercise 29 was developed by Professor J. Ohlson; Exercise 30 was developed in collaboration
with Professor W. E. Diewert; and Exercise 31 was developed by Professor J. Ohlson, and
Exercise 32 was adapted from Arrow (1974) and Ryan (1974).
364 PART III STATIC PORTFOLIO SELECTION MODELS