Suppose there are two production periods (1 and 2). The firm is a price taker and faces a price of 12. The firm's total cost function in each period is given by TC(qt) = (1/2) * (qt)2 - (1/2) * qt * {cumulative past production}, where qt is production in period t. In the first period, the cumulative past production will be equal to zero. In the second period, the cumulative past production will be equal to period 1 production.
(a) Solve for the production level in period 1 that maximizes period 1 profits.
(b) Take your answer to (a) as the amount produced in period 1 and solve for the production level in period 2 that maximizes period 2 profits. What are the total profits for the two periods?
(c) What happens if, in period 1, the firm chooses its production level so as to maximize total profits (not just period 1 profits)? (Assume the future does not get discounted.) How does your solution compare to your answer in (b)? Explain why (no more than two sentences.)