Suppose the production possibility frontier for an economy that produces one public good (y) and one private good (x) is given by
This economy is populated by 100 identical individuals, each with a utility function of the form
where xi 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.