This is intermediate microeconomics problem. Consider the following duopoly game

Question

This is intermediate microeconomics problem.

Consider the following duopoly game: each of 2 firms can produce as much of a good as he wishes at marginal cost 1 (i.e., producing q units costs 1q = q in total), and the market demand curve is given by Q(p) = 10p - p.

(a) Find the inverse market demand curve, specifying the price at which Q units can be sold.

(b) Write out each firms profit expression, if firm 1 produces quantity q1 and firm 2 produces quantity q2.

(c) We found in class that the NE of this game is for each firm to produce 3 units. If you did (c) correctly, you will see that this yields a profit of 9 to each firm. However, if firms could agree at the start to each produce only 2 units, they would both earn a higher profit, namely 10. Explain why such an agreement might not work. the profit-maximizing quantity, Q.

Explanation / Answer

(a) p = Q/9

(b) Profit for firm 1 = (Q/9)q1 - q1

Profit for firm 1 = (Q/9)q2 - q2

(c) If firm 1 decides to choose to produce 3 units, then firm 2 is certainly better off choosing to produce 3 units, since then 2 will make a profit of 9. Similarly, if firm 1 chooses to produce 2 units, then 2 will be better off choosing to produce 3 units, since then 2 will get a profit of 10 rather than a payoff of 0. Thus, whatever firm 1 does, firm 2 is better off choosing to produce 3 units. The same thing goes for firm 1, it is better off producing 3 units as well.

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