5. A risk-averse entrepreneur is considering selling stock in his or her company to the public. He or she will continue to manage the firm after it "goes public." The entrepreneur gets utility from income, X, and from the consumption of on-the-job perquisites, c, according to the utility function u(x, c) = 7 - Avar(x) + 100c, where i is the mean of the income x and var(x) is its variance. The uncertain profits of the firm are Y - c each dollar spent on perquisites reduces profit by a dollar. The variance of Y, (and thus of (Y c), is o?, which we assume to be greater than 2,500. The entrepreneur is currently the sole owner of the firm, receiving as income the firm's profit. What level of perquisites will he or she choose? Now, suppose the entrepreneur sells a fraction a of the firm to risk-neutral investors, retaining (1 - a) for him- of herself. Thus, the entrepreneur receives as income whatever amount the investors pay for this fraction of the firm, say Ma), and then gets his or her share, (1 - XY - c), of the random profit.' The variance of his or her income is thus (1 - a)o? What is the relationship between a, the fraction of ownership the entrepreneur sells, and the level of c he or she will subsequently choose? Does this choice maximize total value (the expected utility of the entrepreneur plus that of the investors)? What will be the expected profit as a function of c? Assume that competition among investors leads them to pay an amount for any given ownership share equal to the profits they expect to receive. How much will the entrepreneur receive from selling a fraction a of the firm if investors correctly anticipate the level of c that the entrepreneur will choose after selling that fraction of the firm? What is the realized level of expected utility for the entrepreneur from selling a fraction a of the firm if the investors have correct expectations? What is the best level of a for him or her to pick? Could the entrepreneur gain by binding hirn- or herself not to increase c as a changes