Industrial Organization, Fall 2011: Midterm Exam Solutions and Comments Date: Wednesday October

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1 Industrial Organization, Fall 2011: Midterm Exam Solutions and Comments Date: Wednesday October Scores The exam was long. I know this. Final grades will definitely be curved. Here is a rough assessment of how I perceive those scores: Below 6: very poor. Between 6 and 10: from poor to fair. Between 10 and 15: good. Between 15 and 20: very good. Above 20: excellent. I would put the failing grade around 6. I encourage students with grades below or around this threshold to contact me, so that we discuss how they can improve their performance. 2 Some comments Marginal revenue. There seems to be a lot of confusion on the concept of marginal revenue. Let π(q) = p(q)q C(Q). Then, the marginal revenue with respect to Q is MR Q (Q) = d(p(q)q). The marginal profit with respect to Q is Mπ dq Q(Q) = dπ(q). Notice dq that MR Q and Mπ Q are usually different, since Mπ Q (Q) = MR Q (Q) C (Q). Profit maximization implies Mπ Q = 0, which is equivalent to MR Q = C = MC. One can define similar concepts when the firm sets p instead of Q. Let π(p) = pq(p) C(Q(p)). Then, the marginal revenue with respect to p is MR p (p) = d(pq(p)). The marginal dp profit with respect to p is Mπ p (p) = dπ(p). Again, MR dp p and Mπ p are usually different. Profit maximization implies Mπ p = 0, which is equivalent to MR p (p) Q (p)c (Q(p)) = 0. So profit maximization does not imply that MR p = MC, since, in general, Q (p) 1. Because of these kinds of mistakes, many students didn t get question 1.d) right. Welfare concepts. I was surprised to see that many students seem to ignore the definitions of consumer surplus and social welfare. Learn those definitions. You ll need them several times in this course, as well as in the final exam. 1

2 First-order conditions. Most of the time, when the objective function is clearly concave, it s okay to take first-order conditions without checking the second-order condition explicitly. In question 1.d), the objective function was not concave, so it was not enough to take firstorder conditions on (0, 8) and (8, 16). There were several ways to answer this question. One solution was to rigorously analyze the profit function, and to construct its table of variations. I offer another solution below. Game theory It s important to learn the names of equilibrium concepts, and to understand when they should be used. Likewise, it shouldn t be a problem for you to write a one-stage game in normal form. Bertrand equilibrium Those who did not get at least 1.25 points in question 1.d) should review their lecture notes and read the solution below carefully. The Bertrand equilibrium is a very important benchmark, and you should be able to prove its existence and uniqueness without blinking. 3 Static Monopoly and Price Discrimination (a) This is third-degree price discrimination. Examples: student discounts, car manufacturers setting different prices in different countries for the same model, etc. The following conditions should be satisfied for this kind of discrimination to be feasible. The firm should be able to identify which buyer belongs to which group. Arbitrage / resale across markets should not be feasible. (b) Profit is given by: π = (p A 2)(16 p A ) + (p B 2)(8 p B ). FOCs: π p A = 0 = 16 p A (p A 2) p D A = 9 π p B = 0 = 8 p B (p B 2) p D B = 5 (c) Equilibrium quantities: qa D = 7, qd B = 3. So πd = πa D + πd B Draw the usual triangle graph. Then, = = 58. CS D A = (16 p D A)q D A /2 = 7 7/2 = 49/2. Similarly, CS D B = 9/2, so CSD = 58/2 = 29. W D = CS D + π D = 87. 2

3 (d) Under non-discrimination, p A = p B = p. Firm s profit is given by: (p 2)(24 2p) if 0 p 8 π(p) = (p 2)(16 p) if 8 p 16 0 otherwise. Notice that π(p) is continuous on [0, ), but is kinked at point p = 8. π is strictly concave on [0, 8]. For all p (0, 8), π (p) = 24 2p 2(p 2) = 28 4p. π (p) = 0 if and only if p = 7( (0, 8)). By concavity and continuity, π is maximized on interval [0, 8] at point p = 7, with π(7) = 50. π is strictly concave on [8, 16]. For all p (8, 16), π (p) = 18 2p. π (p) = 0 if and only if p = 9( (8, 16)). By concavity and continuity, π is maximized on interval [8, 16] at point p = 9, with π(9) = 49. Since π(7) > π(9), and since π = 0 outside the interval (0, 16), it follows that π(.) reaches its global maximum at point p = p ND = 7. (e) We already know that π ND = 50. Equilibrium quantities are q ND A qb ND CS ND = 8 7 = 1. Draw the usual triangle graph. It follows that CSA ND B = 1/2, so CS ND = 41, and W ND = = 91. = 16 7 = 9 and = 81/2 and (f) π ND < π D. Under discrimination, the monopolist is always able to replicate the nondiscrimination outcome (i.e., he is always free to set p A = p B ). This implies that the monopolist should be at least weakly better off under discrimination. But it turns out that the monopolist chooses not to replicate the ND outcome, because demand elasticities differ across markets. By revealed profitability, it follows that the monopolist strictly prefers discrimination. (g) Since p D B < pnd < p D A, it follows immediately that consumers in market A benefit from a non-discrimination policy (CSA ND > CSA D ), whereas consumers in market B suffer from it. Intuitively, under discrimination, the monopolist sets a higher price in the less elastic market (market A). Under non-discrimination, the monopolist faces a total demand whose elasticity is half-way between market A s and market B s. This induces him to set a price between p D A and pd B. (h) CS ND > CS D. Intuitively, ND reduces the price in market A, and raises the price in market B. Since market B is smaller than market A, we should expect CS to go up (but we know that this is not a general result). (i) W ND > W D. Intuitively, ND reduces distortions in market A, and increases them in market B. Again, since market A is bigger, we should expect W to go up. Again, we know that this is not a general result. 3

4 (j) When q B = 3 p B, market B is much smaller than market A. It seems likely that the monopoly would simply give up on market B, and focus on market A. In this case, the monopoly s optimal price under non-discrimination would be p ND = 9(= p D A ). q B would be equal to zero, so CS B would drop to zero after price discrimination is banned. CS A would not be affected (since the price stays the same). Overall, CS would fall. Clearly, π would decrease, since the monopolist cannot supply market B anymore. Therefore, a ban on price discrimination would make everybody worse off, and, in particular, social welfare would fall. 4 Monopoly, Bertrand and Cournot (a) This doesn t matter. If the monopolist sets Q, then p is uniquely determined via p(q). If he sets p, then q is determined via Q(p). When the monopolist chooses a quantity, its profit function is: π(q) = (p(q) c 1 )Q. The first-order condition is: π (Q) = 0 = (P (Q) c 1 ) + P (Q)Q A perfectly competitive firm is price-taker, and therefore believes that P (Q) = 0. This means that the FOC of a perfectly competitive firm is given by P (Q) c 1 = 0. (b) Using the FOC, and rearranging terms, we get: L = P (Q) c 1 P (Q) = P (Q)Q P (Q) = dp/dq Q P (Q) 1 = ε d (p(q)). = ( ) 1 dq/q dp/p If ε d (p(q)) goes up, then the monopolist has less market power, and the markup, (P MC)/P decreases. (c) Define the game as follows. Players: firms 1 and 2. Action sets: for firm 1, p 1 [c 1, ); for firm 2, p 2 [c 2, ). Payoff functions: π 1 (p 1, p 2 ) = { (p1 c 1 )Q(p 1 ) if p 1 p 2 0 otherwise π 2 (p 1, p 2 ) = { (p2 c 2 )Q(p 2 ) if p 2 < p 1 0 otherwise 4

5 It s a one-stage game, players are moving simultaneously, so we will solve it using the Nash equilibrium concept. (d) There are several ways to prove this rigorously. You can prove existence and uniqueness sequentially, as we did in the lectures. Let me do this in a slightly different way. First, let us prove that, in a Nash equilibrium, p 2 = c 2. Assume, by contradiction, that there exists a Nash equilibrium, (p N 1, p N 2 ), in which p N 2 c 2. Given our restriction on action sets, it follows that p N 2 > c 2. If p N 1 c 2, then, firm 1 has a profitable deviation: increase its price. 1 Therefore, p N 1 > c 2. If p N 2 p N 1, then, firm 2 has a profitable deviation: set p 2 = p N 1 ε. Conversely, if p N 2 < p N 1, then, firm 1 has a profitable deviation: set p 1 = p N 2. Contradictions! Therefore, in a Nash equilibrium, p N 2 = c 2. Given the tie-breaking rule (firm 1 gets all the market demand when prices are the same), firm 1 s best response to p 2 = c 2 is to set p 1 = c 2. So we are left with a unique Nash equilibrium candidate: p B 1 = p B 2 = c 2, where firm 2 makes 0 profit. We already know that firm 1 does not want to deviate. By assumption, firm 2 cannot deviate downward. If it deviates upward, then it still makes 0 profit. We can conclude that the only Nash equilibrium is p B 1 = p B 2 = c 2. (e) If firms are allowed to price below costs, then it is easy to check that p B 1 = p B 2 = c 2 is still a Nash equilibrium. We also get a continuum of Nash equilibria, which can be described as follows: p 2 [c 1, c 2 ] and p 1 = p 2. As discussed in the lectures, equilibria in which firm 2 prices below c 2 do not make much sense, since if there is a small probability that firm 1 will tremble upwards, then firm 2 would be better off setting p 2 =. (f) Define the game as follows. Players: firms 1 and 2. Action sets: for firm i, q i [0, ). Firm i s payoff function: π i (q 1, q 2 ) = (10 (q 1 + q 2 ) c i )q i. It s again a one-stage game, players are moving simultaneously, so we will solve it using the Nash equilibrium concept. (g) Let i j in {1, 2}. Fix q j 0. Firm i s optimal quantity (given q j ) solves the following FOC: π i / q i = 0 = 10 2q i q j c i = 0. After some algebra, q i = R i (q j ) = 10 c i q j 2. Best responses are depicted on Figure 4. A few things that you should be careful with: best-response functions should be linear and negatively sloped and BR 1 should be steeper than BR 2. 1 Here, we assume implicitly that c 1 and c 2 are not too different, so that the monopoly price of firm 1 is 5

6 BR i < 0, i.e., quantities are strategic substitutes. Intuitively, when q j increases, the residual inverse demand faced by firm i shifts downward, and so does its residual marginal revenue. This induces firm i to set a lower quantity. (h) The Cournot equilibrium quantities should be best responses to each other, i.e., q c 1 = BR 1 (q c 2) and q c 2 = BR 2 (q c 1). So we need to solve the following system of equations: q 1 = 1 2 (10 c 1 q 2 ) q 2 = 1 2 (10 c 2 q 1 ) Substituting the second equation into the first one, we get: q 1 = 1 ( 10 c 1 1 ) 2 2 (10 c 2 q 1 ). After some algebra, we get q c 1 = 1 3 (10 2c 1+c 2 ) and, by symmetry, q c 2 = 1 3 (10 2c 2+c 1 ). Plugging these quantities into the inverse demand function, we get: P c = 10 q c 1 q c 2 = 1 3 (10 + c 1 + c 2 ). Notice that: P c > c c 1 + c 2 > 3c c 1 > 2c 2. By assumption, the left-hand side of the last inequality is > 10, whereas its righthand side is < 10. It follows that P c > c 2 = P B, which means that Cournot is less competitive than Bertrand. (i) When c 1 decreases, the equilibrium price goes down, and consumer surplus increases. above c 2. Denote the initial quantities by q 1 and q 2, and let ˆq 1 and ˆq 2 the new quantities (after c 1 decreases). Notice that ˆq 1 + ˆq 2 > q 1 + q 2, i.e., overall quantity increases. This is good for social welfare, since the initial quantity was too low, due to oligopoly markup. In addition, ˆq 2 < q 2 and ˆq 1 > q 1, which means that some output is reallocated from the less efficient firm to the more efficient firm. This is good for productive efficiency, and therefore for social welfare. Last, we have the direct effect of a cost decrease (even if quantities did not change, welfare would still go up), which is obviously positive. 6

7 At the end of the day, we have three (and only three) distinct effects on welfare. All of them are positive, so the overall effect is positive. (j) Let us introduce some notations: We have N firms indexed by i {1,..., N}. Each firm has a constant unit cost c. We denote the inverse demand function by P (Q) = a bq, where a > c. Q N i=1 q i and Q i j i q j. Let i, and fix (q j ) j i. Firm i s profit is given by: π(q i, Q i ) = (a b(q i + Q i ) c)q i. Firm i s optimal quantity solves the following first-order condition: a c 2bq i bq i = 0 Since firms are symmetric, we guess that the Nash equilibrium will be symmetric too: q i = q for all i. It follows that Q i = (N 1)q. The first-order condition can then be rewritten as follows: a c b(n 1)q 2bq = 0. Rearranging terms, we get: q = a c. The equilibrium market price is given by: b(n+1) P = a bnq = N a (a c) N + 1 a (a c) 1 = c N 7

8 q 2 BR 1 (q 2 ) q c 2 0 q c 1 BR 2 (q 1 ) q 1 8