Question: Suppose there are I farmers, each of whom has the right to grace cows on the village common. The amount of milk a cow produces depends on the total number of cows, N, grazing on the green. The revenue produced by ni cows is niv(N) for N <>, and v(N) a 0 for N ≥ N, where r(0) > 0. v'< 0,="" and="" v"="">≤ 0. Each cow costs c, and cows are perfectly divisible. Suppose v(0) > c. Farmers simultaneously decide how many cows to purchase; all purchased cows will graze on the common.
(a) Write this as a game in strategic form.
(b) Find the Nash equilibrium, and compare it against the social optimum.
(c) Discuss the relationship between this game and the Cournot oligopoly model.
(This exercise, constructed by R. Gibbons, is based on a discussion in Hume 1739.)