The milk-production of each goes down. Each cow costs the farmer c to buy, and assume that cows are perfectly divisible. The price of milk is p per unit.
Farmers simultaneously (and non-cooperatively) each decide how many cows to purchase; all purchased cows will graze on the common, as there is no other grassy area near the village. Thus N = ∑Ii=1ni where ni is the number of cows bought by farmer i. Each farmer is interested in maximizing her profit i.e. her revenue from selling milk minus the cost of buying cows.
(i) Write this situation as a game in strategic form i.e. identify the set of players, their strategy spaces and payoffs.
(ii) Find the Nash equilibrium of this game i.e. how many cows would each farmer buy, and what would be their profits.
(iii) Suppose instead that the village-headman decided on the total number of cows so as to maximize total surplus of the village i.e. total production minus total cost. How many cows would he choose? How does this compare with what you found in the non-cooperative equilibrium in part (ii)?