1. "Strategic Investments" Consider two firms, A and B that compete in the potash market. Each firm simultaneously chooses quantities to produce.

Prices are given by P = 2 (1/3) [q_A + q_B]

where the quantities are given in million tons. Costs are given for firm A by c_Aq_A and for firm B by c_Bq_B, where c_A = c_B = $1 per ton.

(a) Find the Nash equilibrium of this Cournot game. What are the equilibrium prices, quantities and firms' profits? Now suppose that firm A has the opportunity to rent a new machine that only works with A's production process. The machine costs $0.7M to rent, and it would reduce firm A's variable costs to $0.5 per ton. That is A's total costs would be $0.7M + $0.5q_A. Firm A is considering renting the new machine.

(b) Firm A first consults an accountant who considers what would happen to A's costs if A rents the new machine and the two firms continue to produce the same quantities q_A and q_B as before. Would the accountant recommend renting the new machine?

(c) Firm A then consults a university-trained economist who assumes that, were firm A to rent the new machine, firm B will produce the same quantity as before, but firm A will produce more since firm A's marginal costs will be lower. Would the economist recommend renting the new machine?

(d) Finally, firm A consults an undergraduate who suggests that they find the Nash equilibrium of the Cournot subgame in which firm A does the new machine. Would the undergraduate recommend renting the new machine?