STOCHASTIC

38 LELAND
"no easy money" condition [2]: Any infinitely reproducible vector of
gambling positions must have a nonzero probability of loss.
Assumption U.\ states that a larger outcome (money return) is always
preferred, when that outcome occurs with certainty. If the utility function
is presumed to be derived from the ranking over X alone, clearly C
must be an interval or set of intervals. Assumption U.2 is equivalent to
risk aversion: an amount with certainty will always be preferred to a
gamble with equal expected value (see, e.g., Pratt [5]).
In the analysis which follows, we shall find it useful to define a function
V{x) = E{U[G(s)'x]}.
Assumptions U.l and U.2 imply V(x) is a continuous concave function
with continuous partial derivatives. 3
Economic problems which fall within the special class of problems
introduced above include the portfolio problem and the selection of
inputs and outputs by a perfectly competitive risk averse firm with constant
returns to scale, when prices of inputs and/or outputs are uncertain. 4
Indeed, if the production set can be imbedded in a weakly convex set,
the existence results derived in subsequent sections will hold.
III. THE EXISTENCE PROBLEM
Without further assumptions, there is no assurance that the class of
problems considered in Section II will possess a maximal element. To
prove this contention, consider the following example of portfolio choice.
Let asset 1 return r with certainty, asset 2 return a or b with probability
£ each, with a > r > b.
The investor seeks to
max £[[/(£,)] subject to 5, = (1 + r)x, + (1 + e2) x2,
where e2 = a w.p. -J (1)
= b w.p. J
x^ + xs = B0. (2)
3 "Note these statements are conditional on the existence of V(x). If £[#'(,$)] < co,
/ = l,...,w, there are no problems. If V(x) is to exist for any gamble with infinite
expected value, then l/must be bounded; see [1]. For the relation between the existence
of V(x) and the existence of optimal policies, see the Conclusion.
4 Risk aversion of utility over profit is justified on the basis that the firm is owned
by risk averse investors; see Sandmo [6] and Leland [4]. In the portfolio (firm) case,
X.3 implies there is no combination of assets (inputs and outputs) which yields a
positive return (profit) with probability one, and is infinitely reproducible.
270 PART III. STATIC PORTFOLIO SELECTION MODELS