Suppose the production possibility frontier for an economy that produces one public good (y) and one private good (x) is given by x + 100y = 5,000. This economy is populated by 100 identical individuals, each with a utility function of the form utility = xy, where x, is the individual's share of private good production (= x/100). Notice that the public good is nonexclusive and that everyone benefits equally from its level of production.

a. If the market for x and y were perfectly competitive, what levels of those goods would be produced? What would the typical individual's utility be in this situation?

b. What are the optimal production levels for x and y? What would the typical individual's utility level be? How should consumption of good x be taxed to achieve this result? Hint: The numbers in this problem do not come out evenly, and some approximations should suffice.