Suppose that two individuals, Jon and David, form a community and would like to construct a communal fort that would protect them from attacks. They consume both good X, a private good, and the protection from the fort, P. One unit of good X costs 1 unit while one unit of P costs 2 units, so the budget constraint for each is given by: X_i + 2P_i = 100. Both Jon and David have an income of 100 and a utility function of the form:

U = log(X_i) + 2log(P_j + P_d)

1. How much protection, P, will be privately provided? What is optimal consumption of X, the private good?
2. What are the socially optimal amounts of protection, P, and consumption, X, of the private good? How do the socially optimal amounts compare to that privately provided? Explain why