ECOS3005 Mid-semester Test Answers Industrial Organisation Practice Section A: Answer all short answer questions in the booklet provided. Please be sure to show your working and explain your reasoning where appropriate. (Total 40 marks) 1. The demand for good A is given by Q(P) = 75 0.5P, where Q is the market quantity, and P is the market price. Production of good A involves costs of C(q) = 400+30q, where q is firm output. (a) Suppose a single firm operates in the market. Find the profit-maximising price and quantity of the monopolist. [4 marks] ANS: Rewrite the demand curve as P= 150 2Q. Monopoly profits are then given by pim = Q(150 2Q) 30Q 400. First order conditions yield: 150 4Q 30 = 0 4Q= 120 Q= 30. Substituting into the demand curve, we have P= 90. (b) Suppose two firms operate in the market. The firms engage in simultaneous quantity competition in a single period. i. Find the reaction function for each firm. [4 marks] ANS: Firm i chooses output to maximise profits given by pii = qi(150 2Q 30) 400. This leads to first order conditions 150 4qi 2q j 30 = 0 qi = 120 2q j 4 qi = 30 q j2 . This is the reaction function for each firm. ii. Find the Nash equilibrium outputs of both firms. [3 marks] ANS: Substituting firm 2’s reaction function into firm 1’s, we obtain q1 = 30 30 q1/22 3 4 q1 = 15 q1 = 20. Similarly, q2 = 20. Each firm produces an output of 20 in equilibrium. 1 (c) Suppose that two firms operate in the market. The firms engage in Stackelberg competition. Firm 1 chooses its output first, then Firm 2 chooses it’s output. Find the output of each firm. [4 marks] ANS: The Stackelberg leader (Firm 1) earns profits pi1 = q1 ( 150 2q1 2 ( 30 q1 2 ) 30 ) 400 = q1(60 q1) 400 Firm 1 then has FOCs 60 2q1 = 0 q1 = 30. Firm 1 produces an output of 6 and Firm 2 produces q2 = 30 q1/2 = 15 units. 2. Consider the following game. Firm 1 and Firm 2 have three strategies available, A, B, and C. The first entry in each cell contains the payoffs for Firm 1 and the second entry contains the payoffs for Firm 2. Both firms have the common discount factor, δ, where 0 < δ< 1. Firm 1 Firm 2 A B C A 0,0 6, 1 2, 2 B 1,6 4,4 2,2 C 2,2 2,2 3,3 (a) Suppose the game above is played once. Identify any Nash equilibria. Explain briefly. [4 marks] ANS: There are two Nash equilibria: (A,A) and (C,C). In each case, both players are choosing an optimal strategy given the strategy of their rival. That is, if Firm 1 chooses A, the best response of Firm 2 is also to play A. Similarly, if Firm 2 chooses A, the best response of Firm 1 is also to play A. The same reasoning applies to the Nash equilibrium at (C,C). (b) Suppose the game above is played twice. Consider the following strategy: in period 1: play B; in period 2: play C if both played B in period 1; otherwise play A. For what value of δ (if any) is there a subgame perfect Nash equilibria in which both players play the above strategy [6 marks] ANS: To identify a subgame perfect Nash equilibrium, we must ensure that the strategies consti- tute a Nash equilibrium to each subgame. We can sole this by backward induction. First, note that in stage 2, the strategies call on either both players to play A or both players to playC. From part (a), we know that (A,A) and (C,C) are Nash equilibria. Therefore, there is no incentive to deviate in stage 2 for either player. 2 Next, consider stage 1. Consider the perspective of Firm 1 (everything is symmetric for Firm 2). By playing B in stage 1, Firm 1 anticipates a total payoff of 4+δ×3. If instead, Firm 1 were to deviate, the optimal deviation involves playing A instead of B. This yields a payoff of 6+δ×0. There is no incentive for Firm 1 to deviate if 4+3δ≥ 6 δ≥ 2/3. Therefore, there is a subgame perfect Nash equilibrium with these strategies if δ≥ 2/3. 3. Two firms compete in the market for a homogeneous product. Each firm has a capacity of 200 units. Market demand is given by Q(p) = 500 10p, where Q= q1 +q2. Each firm has a constant marginal cost and no fixed costs: C(q) = 10q. The firms compete by choosing price. The lowest priced firm captures the whole market (up to their capacity constraints), while the higher priced firm serves any residual demand. The firms interact sequentially in a single period. First, Firm 1 chooses a price p1. Then, after observing p1, Firm 2 chooses a price p2. (a) Solve for the reaction function for Firm 2. [8 marks] ANS: Firm 2 has an incentive to undercut and steal market share if p1 is high, while Firm 2 has an incentive to relent if p1 is close to marginal cost. To identify Firm 2’s reaction function, we must find the point of indifference. If relenting, Firm 2 will choose p2 to maximise profits based on residual demand. Relenting profits are given by piR = (500 200 10p2)(p2 10). Maximising profits leads to the FOCs 300 10p2 10(p2 10) = 0 p2 = 20 Relenting profits are then piR = (300 10p2)(p2 10) = 100.10 = 1000. The optimal way to undercut for Firm 2 is to charge a price just below p1. Undercutting profits will then be given by piU ≈ 200(p1 10). 3 Firm 2 is indifferent between undercutting and relenting when piU = piR 200(p1 10) = 1000 p1 = 15. In fact, when p1 is exactly equal to 15, Firm 2 just prefers relenting to undercutting. This is because relenting gives a profit of 1000, while undercutting will give a profit just below 1000. Finally, for completeness, let us calculate the monopoly price. Monopoly profits are pi= (500 10p)(p 10). Solving first order conditions yields: 500 20p+100 = 0 pm = 30. Notice that the monopolist’s output at this price is Q= 200, the same as the firm’s capacity. Summarising, Firm 2’s reaction function is: p2 = R2(p1) = 30 if p1 > 30 p1 ε if 15 < p1 ≤ 30 20 if p1 ≤ 15. (1) (b) Solve for a subgame perfect Nash equilibrium to this game. Explain. [7 marks] ANS: Firm 1 will understand Firm 2’s reaction function when setting price. Suppose Firm 1 sets p1 > 15, but below the monopoly price. Then, Firm 2 will undercut, giving Firm 1 profits of pi1 = (500 200 10p1)(p1 10). Firm 1 could choose p1 to maximise these profits. Notice that this problem is exactly the same as Firm 2’s relenting problem above. Hence, p1 = 20 and pi1 = 1000. Suppose instead that Firm 1 sets p1 ≤ 15. Then, Firm 2 will relent to p2 = 20. Firm 1 would then earn profits of pi1 = 200(p1 10). This would be maximised (subject to the constraint that p1 ≤ 15) by setting p1 = 15. This leads to profits of pi1 = 200×5 = 1000. Hence, Firm 1 is indifferent between p1 = 20 and p1 = 15. There are 2 subgame perfect Nash equilibria. In the first, Firm 1 sets p1 = 15 and Firm 2 follows the reaction function from equation (1) above. In the second, Firm 2 sets p1 = 20 and Firm 2 follows the reaction function from equation (1) above. 4
