STOCHASTIC

the decision problem is to maximize
£«(w) = (l-6)[*i(l-0fi + *2&]
-*[[*, (1 -/) Ji + x2|2] 2 + [x^l-OW + xa 2 ira a ]],
subject to ^1+^2 = M, where it is assumed that it and £2 have independent distributions
having means d g JT, £ 0, X2 g 0}.
6. Refer to Leland's paper. Find utility functions possessing the properties U'(c) > 0
and U"(c) < 0 for all c e C that are
(a) bounded from above and below;
(b) bounded from above but not from below; and
(c) bounded from below but not from above.
(d) Construct a St. Petersburg paradox for the utility function in (c), i.e., a gamble
having infinite expected utility for which the probability that one makes a positive gain
goes to zero as the price of the gamble increases.
(e) How might one construct a St. Petersburg paradox for the utility function in (b) ?
(f) Does an optimal policy always exist for the utility function in (a) ?
(g) Suppose that the utility function is
lyKy+1) if y^0,
1-J-/0-+D if ^ < 0
332 PART III STATIC PORTFOLIO SELECTION MODELS