Once both firms have entered the market, they compete a la Cournot. The market inverse demand function is given by P(Q) = 8 − Q. Firm 2 now has a Research & Development department capable of reducing marginal costs—but at a cost. Total research and installation costs, given a choice of c2 ∈ [0, 3], are f(c2) = 6 − 2c2. So, given a choice of c2, total costs are C2(q) = f(c2) + c2q = 6 − 2c2 + c2q. That is, firm 2 can set up a plant with marginal cost c2 = 3 paying no fixed cost. Alternatively, if it wants to set up a plant with marginal cost 1, the fixed cost would be 4. Firm 1 still has the same technology, with total cost of C1(q) = 2 + 2q. The game has three stages. In the first, firm 1 enters and chooses the quantity q1. In stage 2, firm two enters, deciding its marginal cost c2. At the last stage, firm 2 chooses its quantity produced, and “the market” determines the price given the quantities produced by both firms. The “big picture” question is, what is the subgame perfect Nash equilibrium of this game? We will proceed step by step. 3. Solve the subgame at stage
1. That is: taking as given q1 and c2, what is the profit-maximizing quantity q2(q1, c2)? Hint: what is the best response function?
2. Given the previous answer, what is the profit anticipated by firm 2 at the second stage: that is, π2(q1, q2(q1, c2), c2)? Write that as a function of q1 and c2 only. Hint: remember to include the fixed cost!
3. What is the profit-maximizing marginal cost choice c2(q1)?
4.Given your last answer, what does firm 1 expect the quantity chosen by firm 2 to be, given its own choice of quantity? That is, what is the q2(q1) anticipated in the first stage?
5. What is the equilibrium quantity chosen by firm 1?
6. What are the equilibrium price, quantities, and profits in the market?
Consider the following repeated Prisoner’s Dilemma game. 
For the following parts, suppose this game is repeated infinitely. Both players have discount factor δ ∈ [0, 1).
3. Let both players adopt the following strategy: start with cooperation; cooperate as long as no one has ever defected before; otherwise, defect. Find the minimum value of δ such that this is a Subgame Perfect Nash Equilibria.
4. Suppose the payoff matrix for the original static game has been changed to: 
How does your answer to part 4 change? Is the new minimum value of δ larger or smaller? Explain.
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