# OLIGOPOLY THEORY. 1. Cournot Oligopoly

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1 OLIGOPOLY THEORY 1. Cournot Oligopoly Here we are considering a generalized version of a simple example of the linear Cournot oligopoly in the market for some homogeneous good. We will suppose that there are N identical rms, that entry by additional rms is e ectively blocked, and that each rm has identical constant marginal costs. C (q i ) = cq i ; c 0; and i = 1; 2; :::; N: Firms sell output on a common market, so market price depends on the total quantity sold by all rms in the industry. Let the inverse market demand be the linear form, p = a b N P q i i=1 where a > 0; b > 0; and we will require that a > c. pro t for form i as We can write the! P i (; q 2 ; :::q N ) = a b N q j q i cq i j=1 We seek a vector of outputs (q1; q2; :::; qn ) such that each rm s output is pro t imizing given the best output choices choosen by the other rms. Such a vector of outputs is called a Cournot-Nash equilibrium. If (q 1; q 2; :::; q N ) is a Cournot-Nash equilibrium, q i must imize i (; q 2 ; :::q N ) when q j = q j for all j 6= i: Consequently, the derivative of i (; q 2 ; :::q N ) with respect to q j must be zero when q j = q j for all j = 1; 2; :::; N: Thus, multiplying q i to simplify the expression yields aq i bq i ( + q 2 + :::q i 1 + q i + q i+1 + :::q N 1 + q N ) cq i i (; q 2 ; :::q N ) = aq i bq 2 i bq i NP q j j6=i cq i 1

2 The marginal pro t is P = a q i b N q i q j j6=i c q j j6=i P q i = a bq i bq i b N c P = a bq i b N q i q j j=1 c 2 q 2 i = < 0 so the following optimal solution yields the imum value. We set the rst order condition to equal zero to nd the imized pro t. P a bq i b N q j c = 0 j=1 bq i = a c b N P q j j=1 Note that the right hand side is independent of which rm i we are considering. We conclude that all rms must produce the same amount of output in equilibrium. By letting q denote this common equilibrium output so that = q 2 = ::: = q N = q ; the previous equation can be written as bq = a c bnq bq (N + 1) = a Each rm s output q = a c b (N + 1) Industry s output Q = c N (a c) b (N + 1) Market Price P N (a c) = a b b (N + 1) = a N (a c) (N + 1) < a Each rm s pro t i = 2 (a c)2 b (N + 1) 2

3 Equilibrium in this Cournot oligopoly has some interesting features. We can calculate the deviation of price from marginal cost, P c = a c N + 1 > 0 and observe that equilibrium price will typically exceed the marginal cost of each identical rm. When N = 1, this coincides with the monopoly market and the deviation of price from marginal cost is greatest. At the other a c extreme, when the number of rms, N! 1;we see that lim N!1 N + 1 = 0: This tells us that price will approach marginal cost as the number of rms becomes large. This suggests that prefect competition can be viewed as a limiting case of imperfect competition when the number of rms becomes in nity. 2. Stackelberg Oligopoly We are considering the linear Stackelberg duopoly before moving to the case of N rms since this is a little bit trickier than Cournot with N identical rms. Assume that the market inverse demand function is P = a bq; where Q = + q 2 : We assume that both rm have constant marginal cost c < a; produce identical products, and that rm 1 is the Stackelberg leader. Firm 2 will chooses q 2 given : The rst order condtion is q 2 (a bq) q 2 cq 2 q 2 (a b bq 2 ) q 2 cq 2 q 2 aq 2 b q 2 bq 2 2 cq 2 2 q 2 2 q 2 = a c b q 2 = < 0! imum a c b q 2 = 0 q 2 = a c b : 3

4 This is called rm 2 s reaction function. Firm 2 thinks that rm 1 will choose its output as its best strategy so rm 2 chooses its output given what rm 2 thinks of rm 1 s best strategy. In the Stackelberg duopoly; however, rm 1 is a leader so it can choose its output based on what rm 2 thinks it will produce. That is, rm 1 thinks rm 2 will choose q 2 = a c b so we insert this expression into rm 1 s imization problem. (a bq) c (a c b bq 2 ) q 1 a c b b a c bq 1 a c b a c b 2 a c bq1 2 The rst order condition yields q1 = a c Note that the competitive output, where P = MC = c; is q c = a c ; b and the monopoly output where MR = MC, is q m = a c : Hence, the Stackelberg leader produces exactly as the monopolist and half of competitive output. We can then compute the quantity produced by the follower as q 2 = a c b = a c q1 2 = a c a c 4b = a c 4b = q c 4 : Therefore, the equilibrium quantity in the Stackerlberg duopoly is that q1 = a c and q2 = a c 4b : Now, we are examining the generalized case where there are N identical rms. We assume that each rm chooses output sequentially, in which rm 4

5 1 is the rst leader, rm 2 follows rm 1, until rm N follows rm N -1. How much does the leader produce? Again, the leader will produce q c 2, rm 2 will produce q c ; :::; and rm N 4 will produce q c : Thus, if we rank each rm by the order they are choosing 2N the output, we have that any rm j will produce exactly an amount of q c 2 for j j = 1; 2; :::; N: As N! 1; we see that the total industry output equals q c 2 + q c 4 +:::+ q c N +:::. This approaches q c so pro t will becomes zero and price will be reduced to equal marginal cost. How could this happen? If we work backwards, rm j s optimal output cuts pro t of each of its predecessors in half. Therefore, rm N-1 acts as residual monopolist, as do all predecessors. 3. Bertrand Oligopoly We are considering the price competition instead of quantity competition described previously. First, we will examine the case of Bertrand duopoly before generalizing the idea to the case of N competitors. There is one misleading point. Some authors summarize the di erences between Cournot Oligopoly and Bertrand Oligopoly by referring to the Cournot and Bertrand equilibria. Such usage refers to the di erence between the equilibrium behavior in these models, not to a di erence in the equilibrium concept used in the games. In both games, the equilibrium concept used is the Nash equilibrium. We rst translate the problem into a normal form game. There are again two players. This time, however, the strategies available to each rm are the di erent prices it might charge, rather than the di erent quantities it might produce. Hence, each rm s strategy space can be written as S i = [0; 1); i = 1; 2 The basic concepts of Bertrand duopoly is that rms choose prices p i simultaneously with homogeneous product. Both rms have identical constant marginal costs equal to c. Suppose that there is no xed cost from entering the industry. The market demand is given by the general function D (p). It is not hard to see that the pro t will be categorized as follows. 5

6 P MC 1 MC 1 +MC 2 Inverse Demand Q 8 < 0 i = D (p i ) (p i c) =2 : D (p i ) (p i c) if p i > p j if p i = p j if p i < p j The highest possible price that can be set is the monopoly price, p m ; which can be solved from di erentiating D (p i ) (p i c) with respect to p i : Typically, rm i has no best response for c < p i p m since i wants to undercut marginally, i.e., p i = p j " for any " in nitestimally more than zero (i.e., " = 10 1;000;000 ): If rm j reasons the same way, it will set price p 0 j = p i " = p j 2": This will continues until p i = p j = c: This is often called a Bertrand Paradox; even with only two rms, the price is similar as that of perfect competition. As an exercise, you are to show that this is a Nash Equilibrium, and show that it is the only pure strategy Nash Equilibrium. Bertrand Model of Di erentiated Products Although this topic is to be studied in the product di erentiation chapter, for the sake of writing the handout it is more appropriate to include some variations of Bertrand model here. We consider the case of di erentiated products. If rms 1 and 2, assuming they have the same constant marginal costs, choose prices p 1; and p 2 ; respectively, the quantities that consumers demand from each rm are = a p 1 + bp 2 q 2 = a p 2 + bp 1 6

7 We can see whether both goods are substitutes or complements from the sign of b. These are unrealistic demand functions because demand for each rm s product is positive even when it charges an arbitrarily high price, provided that another rm also charges high enough price. The pro ts for both rms are 1 (p 1 ; p 2 ) = (p 1 c) (p 1 ; p 2 ) = (p 1 c) (a p 1 + bp 2 ) 2 (p 1 ; p 2 ) = (p 2 c) q 2 (p 1 ; p 2 ) = (p 2 c) (a p 2 + bp 1 ) The objective function is quadratic. midpoint of its roots. A quadratic is imized at the Here, the roots for each function is, c; and a responses for each rm are bp j ; j = 1; 2: The best p 1 = a + bp 2 + c 2 p 2 = a + bp 1 + c 2 Solving these pair of equations yields the Nash Equilibrium: p 1 = p 2 = a + c 2 b : What can we conclude from the case of di erentiated products? We see that the introduction of product di erentiation results in price charged at higher level than the marginal cost, provided b < 2: The increased product di erentiation can result in higher prices does not imply that increased product di erentiation lowers social welfare, because in return for higher prices consumers receive increased variaety of products. Other variations of Bertrand Oligopoly Before moving to the N rms case, it is interesting to consider other variations in the Bertrand model. Suppose that rm 1 has higher marginal cost than that of rm 2, c 1 > c 2 : There are two cases. The boring case is 7

8 that c 1 > p m2 = arg D (p) (p c 2 ) : So, at rm 2 s monopoly price, rm p 1 is just not pro table and is out of the market. The interesting case is that when c 1 < p m2 : Mathematically, there is no equilibrium exists since there is no best response. Intuitively, if p 1 = c 1 ; then p 2 = c 1 " for any " > 0: Therefore, there are many Nash Equilibria such that p 1 = p and p 2 = p " for any c 2 < p c 1 : Although these are all Nash Equilibria, the strategy p 1 = c 1 weakly dominates all p 1 < c 1 for player 1. There is another interesting case if a rm faces an increasing marginal cost, as it should be due to the law of diminishing returns. Bertand is much tougher problem now since a lower priced rm may choose to serve only a part of quantity demanded at its price. From the gure in page 6, the rationing rule used by the lower priced rm can a ect the sales and pro ts of the other rm. For example, if rm 1 sells only to the most eager buyers (i.e. those who have highest reservation price), then rm 2 will sell nothing. Finally, we consider the case of N rms in Bertrand oligopoly. Assume for simplicity that they have identical constant marginal cost c. Now, there are in nitely many Nash Equilibria. All rms i will set prices p i c and at least 2 rms i will set price p i = c: It is left to the reader why this is so. (be aware that this might be on the midterm exam.) 4. Price Leadership Model This is probably one of the models that I see little bene t from it. Those of you who will pursue a graduate degree may plausibly not see this in your graduate study. Suppose that a market consists of one dominant rm that controls a large percentage of total industry output and a signi cant number of relatively small "fringe" rms. Suppose the total industry inverse demand is given by p = a bq = a bq d bq f ; where q d is the quantity demanded from the dominant rm and q f is the quantity demanded from the small fringe rms. We assume that the fringe s total inverse supply curve is given by p = c+dq f, and the dominant rm s marginal cost curve is given by MC d = c+eq d, where d > e; and a > c: To obtain the dominant rm s residual demand curve, we need to subtract the fringe supply curve from the total industry demand curve at every price greater than c. 8

9 dq f = p c! q f = p c d ; Q = a p b The residual demand, written as q d = R (p) is calculated from q d = R (p) = Q q f = a b p b p d + c d = bd () p bd () p bd = bd q d Inverse Residual Demand : p = Marginal Revenue : MR = d q d The dominant rm equate MR to equal MC, bd q d d q d = c + eq d c = eq d + d q d ad cd e () + d = q d = ad cd e () + d q d The dominant rm sets price from its inverse residual demand p = = 1 = 1 = 1 bd ad cd e () + d abd 2 + bcd 2 e () + d abde + ad 2 e + 2abd 2 + b 2 ce + bcde + 2 cd abd 2 + bcd 2 e () + d abde + ad 2 e + abd 2 + b 2 ce + bcde + 2 cd + bcd 2 e () + d 9

10 Now, I understand why there is no textbook explaining the dominant price leadership model in the parameterized form because of this tremendously intimidating,but little intuitive expression. The nal step is to nd the quantity produced by the small fringe rms. That is, they are behaving like competitive price takers so they equate this (ugly) price to their supply (or MC). 1 abde + ad 2 e + abd 2 + b 2 ce + bcde + 2 cd + bcd 2 = c + dq f e () + d 1 abde + ad 2 e + abd 2 + b 2 ce + bcde + 2 cd + bcd 2 c q f = bd + d 2 e () + d d As I expected, little insight would be gained from working the price leadership model in this general form. Perhaps you may better understand this model by consulting the Waldman and Jensen s textbook. 5. Collusive Oligopoly In this models we are exploring how collusive equilibrium could form, and why each oligopolist has an incentive to move away from the equilibrium. It the rms collude, then they will act like a single monopolist to set prices and outputs so as to imize total industry pro ts, not individual pro t. Such collusion is called cartel. Consider the simple duopoly where each rm acts to collude to set prices and quantities. the total industry pro ts could be written as total = ;q 2 p (Q) Q c (Q) = p ( + q 2 ) [ + q 2 ] c 1 ( ) c 2 (q 2 ) The rst order conditions are p (Q ) + p 0 (Q ) [Q ] c 0 1 (q 1) = 0 q 2 p (Q ) + p 0 (Q ) [Q ] c 0 2 (q 2) = 0 10

11 p (q 1 + q 2) + p 0 (Q ) [q 1 + q 2] = MC 1 (q 1) p (q 1 + q 2) + p 0 (Q ) [q 1 + q 2] = MC 2 (q 2) ) MC 1 (q 1) = MC 2 (q 2) Two marginal costs are equal in equilibrium. If one rm has a cost lower than that of another, it will necessarily produce more output in equilibrium in the cartel solution. The problem of forming a cartel is that there is always a temptatiuon to cheat. Suppose that two rms are producing at the imizing quantities q 1 and q 2: Firm 1 considers producing a little bit more output, by the amount d : That is, rm 1 is considering imizing its own pro t. 1 = p (Q ) c 1 ( ) 1 p (Q ) + p 0 (Q ) q 1 MC 1 (q 1) ; but p (Q ) + p 0 (Q ) q 1 + p 0 (Q ) q 2 = MC 1 (q 1) Therefore, 1 = p (Q ) + p 0 (Q ) q 1 MC 1 (q 1) = p 0 (Q ) q 2 > 0: The last inequality follows since p 0 (Q ) is negative due to negatively inverse demand slope. Hence, if rm 1 believes that rm 2 will keep its output xed, then it will believe that it can increase pro ts by increasing its own production since 1 > 0: Firm 2 also believes similarly. Thus, each rm will be tempted to increase its own pro ts by expanding its own output, and then the equilibrium will go to Counot Nash Equilibrium. Consider the case of linear collusive equilibrium. Suppose that the inverse demand function is p (Q) = a bq = a b ( + q 2 ) : The marginal costs for both rms are c 1 and c 2 q 2 : The total cartel pro t is 11

12 = [a b ( + q 2 )] ( + q 2 ) c 1 c 2 q 2 = a ( + q 2 ) b ( + q 2 ) 2 c 1 c 2 q 2 = a ( + q 2 ) c 1 = 0! q1 + q2 = a c 1 = a ( + q 2 ) c 2 = 0! q1 + q2 = a c 2 q 2 c 1 = c 2 If marginal costs are constant, both of them must be equal in the equilibrium. As you can see, the two equations are not independent so that we can only determine the total output. Nonetheless, the division of output between the two rm does not matter. We have seen that a cartel is fundamentally unstable in that it is always in the best interest of each rms to increase their production. Then, how could a collusive equilibrium be sustained? We will explore this matter in the Grim Trigger Strategy from the study of dynamic game of complete information. 12

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