A small manufacturing firm produces and sells a product in a market where output prices are uncertain. The owner of the firm wishes to make a short run production decision that will maximize her expected utility, defined by

\(\mathrm{E}(U(\pi))=\mathrm{E}(\pi)-\alpha[\operatorname{var}(\pi)]\)

where \(U\) is utility, \(\pi=P q-c(q)\) is profit, \(\mathrm{q}\) is measured in 1,000 's of units, \(P\) is the uncertain price received for a unit of the product, the cost function is defined by \(c(q)=.5 q\) \(+.1 q^{2}, \alpha \geq 0\), is a "risk aversion" parameter, and the probability density function of the uncertain output price is given by

\(f(p)=.5 e^{-.5 p} I_{[0, \infty)}(p)\)

(a) If the owner were risk neutral, i.e., \(\alpha=0\), find the level of production that maximizes expected utility.

(b) Now consider the case where the owner is risk averse, i.e., \(\alpha>0\). Graph the relationship between the optimal level of output and the level of risk aversion (i.e., the level of \(\alpha\) ). How large does \(\alpha\) have to be for optimal \(q=5\) ?

(c) Assume that \(\alpha=.1\). Suppose that the government were to guarantee a price to the owner. What guaranteed level of price would induce the owner to produce the same level of output as in the case where price was uncertain?