1. Entrepreneurial Perks: A risk-averse entrepreneur is considering selling stock in her company to the public. She will continue to manage the rm after it " goes public". The entrepreneur gets utility from income, X with mean Y and variance V(X), and from the consumption of onthejob perks, c, according to the expected utility function 1;.(X, c) = Y V(X) + 100'/E. The uncertain prots of the rm are Y 0: each dollar spent on perks reduces prot by a dollar. The variance of Y, (and thus of Y c) is a2 = 2500. (a) The entrepreneur is currently the sole owner of the rm, receiving as income the rm's prot. What level of perks will she choose? (b) Now, suppose the entrepreneur sells a fraction or > 0 of the rm to risk-neutral investors, retaining (1 a) for herself. Thus, the entreprenuer receives as income whatever amount the investors pay for this fraction of the rm, say M(a), and then gets her share, (1 a)(Y c), of the random prot. The variance of her income is thus (1 00202. What is the relationship between a, the fraction of ownership the entrepreneur sells, and the level of perks 6(a) she will subsequently choose? Does (3(a) maximize total value (the expected utility of the entrepreneur plus that of the investor)? Explain. (c) Assume that competition among investors leads them to pay an amount for any given ownership share equal to the prots they expect to receive. How much will the enterpreneur receive from selling a fraction 0: of the rm if investors cor rectly anticipate the level of (3(a) that the entrepreneur will choose after selling that fraction of the rm? What is the realized level of expected utility for the entrepreneur from selling a fraction (1 of the rm if the investors have correct expectations? Show that the optimal level of a solves a = (1 (1)4 (so, ap- proximately a = 0.2755). Could the entrepreneur gain by binding herself not to increase 0 as a changes