25. A fruit processing firm is introducing a new fruit drink, “Peach Passion,” into the domestic market. The firm faces uncertain output prices in the initial marketing period and intends to make a short-run decision by choosing the level of production that maximize the expected value of utility:

EðUðpÞÞ ¼ Eð Þ p avarðpÞ:

Profit is defined by p ¼ Pq CðqÞ; p is the price received for a unit of Peach Passion, U is utility, the cost function is defined by c(q) ¼ .5q2

, a  0 is a risk aversion parameter, and the probability density function of the uncertain output price is given by f(p) ¼ .048(5p p2

) I[0,5] (p).

(a) If the firm were risk neutral, i.e., a ¼ 0, find the level of production that maximizes expected utility.

(b) Now consider the case where the firm is risk averse, i.e., a > 0. Graph the relationship between the optimal level of output and the level of risk aversion (i.e., the level of a). How large does a have to be for optimal q ¼ 1?

(c) Assume that a ¼ 1. Suppose that the Dept. of Agriculture were to guarantee a price to the firm. What guaranteed price would induce the firm to produce the same level of output as in the case where price was uncertain?