Lecture Note 3. Oligopoly

Transcription

1 Lecture Note 3. Oligopoly 1. Competition by Quantity? Or by Price? By what do firms compete with each other? Competition by price seems more reasonable. However, the Bertrand model (by price) does not explain the competition outcome very well, while the Cournot model (by quantity) explains relatively well. What is wrong with the Bertrand model? What supports the Cournot model? 2. Bertrand model Assumptions Homogeneous goods Firms choose their price. Firms can produce however many units they want. The marginal cost is the same across firms. Outcome Since goods are homogeneous, consumers would purchase the cheapest one. Firms are better off by setting their price a little less than competitiors price, as long as their price is still no less than the marginal cost. Firms keep cutting the price until it reaches the marginal cost. So the outcome is such that P = MC for every firm. The outcome does not depend on the number of firms as long as it is at least as many as 2. Problem Not very likely that 2 firms compete as competitively as under perfect competition. Maybe some assumptions are misleading. Assume nonidentical marginal cost. This is not successful. The outcome is that all the firms choose the price at the second lowest marginal cost among those of all firms, and that the firm with the lowest marginal cost produces all. The other firms do not have an incentive to produce at the price less than their marginal cost. Yong Yang 1

2 Assume firms have a capacity constraint. Firms may not be able to lower the price too much. They choose their capacity first, and then price next subject to the capacity constraint. The outcome is the same with that of the Cournot model! Assume firms produce differentiated commodities. Now consumers may not appreciate only the price. Firms with higher price have nonzero demand. The demand function facing a firm is decreasing in its own price, but increasing in its competitors price. The Hotelling s model is the simplest example. The monopolistic competition model is a general form of model which takes this approach. 3. Cournot model Assumptions Homogeneous goods Firms choose their quantity. The price is determined by the demand function, given the quantity choices of the firms. The outcome with the same marginal cost Suppose n firms, MC is constant at c, and the inverse demand function is P = a bq. The outcome is q 1 = = q n = a c (n + 1)b, P = a + nc n + 1, π (a c)2 1 = = π n = (n + 1) 2 b The profit is strictly less than that under simple monopoly. In fact, the sum of all firms profit is also strictly less than that of the monopolist, which gives firms an incentive to form a cartel and divide the monopolist s profit. The above outcome implies that as the number of firms increases, P = a + nc n + 1 c n n(a c) Q = q i = (n + 1)b a c b i=1 n n(a c)2 Π = π i = (n + 1) 2 b 0 i=1 This means that the Cournot competition converges to the perfect competition outcome. But they are never the same as long as the number of firms is finite. Yong Yang 2

3 4. Bertrand Model with Capacity Constraint Assumptions Homogeneous goods Firms choose their price simultaneously under the constraint that they can produce only the amount less than or equal to the capacity. The marginal cost is the same across firms. Solution: Solve by backward induction. Suppose firms have their capacities set at k 1,, k n. The equilibrium price is p 1 = = p n = P where P = a b(k k n ). To roughly prove the claim, note that if it is not the case, we would have either (1) that all the prices are less than or equal to P with at least one strictly less than P, or (2) that at least one price is greater than P. In the former case, there is an excess demand. Firms would be able to sell their commodities out at a higher price, so would like to increase the price. In the latter case, the firm with a price greater than P is not selling up to its capacity. It would want to decrease the price so that it sells more and increase its revenue. How do the firms set their capacity? Note that given the set of capacities, they know how much they earn. The revenue is R i = P k i = ak i b(k k n )k i Suppose the marginal cost to increase its capacity is constant at c and no fixed cost is incurred. Each firm would maximize max π i = R i ck i = ak i b(k k n )k i ck i Solve the system of equations, then k 1 = = k n = a c (n + 1)b, P = a + nc n + 1, π (a c)2 1 = = π n = (n + 1) 2 b The solution here exactly coincides with the Cournot outcome. The firms get positive profit, the price is above the marginal cost, and the total quantity produced is strictly less than that under perfect competition. This sounds reasonable, and actually supports the Cournot model, not in their assumption, but in their result. Yong Yang 3

4 5. Hotelling s Model Model Suppose a line, whose left-end is 0 and right-end is 1. Each point is assigned with a number which is the distance from the left-end divided by the whole length of the line. Each endpoint stands for a store. Both of them sell homogeneous goods. Consumers are uniformly located between the two endpoints. Each consumer has to choose a store, travel there and buy a good. Traveling costs in proportion to the traveling distance. Denote the marginal cost by t. This is one of the simplest models reflecting the heterogeneity of the commodity. Goods are homogeneous themselves. The location of the stores is different, so buying at store 0 is different from buying at store 1 across consumers. Each consumer compares the final utility considering the traveling cost and chooses a store. Demand Suppose store 0 sets its price at p 0, and store 1 at p 1. Suppose a consumer located at θ wants to buy a good. This consumer would get the following utility, when she wants to buy at store 0, or at store 1, respectively. U 0 = a p 0 tθ U 1 = a p 1 t(1 θ) where a is the (monetary) benefit she gets when she buys one. She buys at store 0 if U 0 U 1, or equivalently if θ < t (p 1 p 0 ) Suppose there are M consumers, then the number of consumers who want to buy at store 0 is D 0 = M 2t (p 1 p 0 ) In the same way, the number of consumers who want to buy at store 1 is D 1 = M 2t (p 0 p 1 ) Yong Yang 4

5 Competition Suppose no production cost. The marginal cost is 0. Store 0 s profit is Maxmizing its profit, π 0 = p 0 D 0 = p 0 M 2t (p 1 p 0 ) which we call store 0 s best response function. p 0 = 1 2 (t + p 1) Store 1 s profit is Maximizing its profit π 1 = p 1 D 1 = p 1 M p 1 = 1 2 (t + p 0) 2t (p 0 p 1 ) which we call store 1 s best response function. The solution of the two equations is The quantity they produce is and the profit is p 0 = p 1 = t D 0 = D 1 = 1 2 M π 0 = π 1 = 1 2 Mt Yong Yang 5

6 6. More General Differentiated Product Model There are many ways to model how the price of one good affects that of other goods. Here we want to directly model such a situation. Suppose two firms have the following demand function. q 1 = 1 ap 1 + bp 2 q 2 = 1 ap 2 + bp 1 The above functional from reflects that an increase in the price of the good reduces its own demand, but increases its competitor s demand. Solution Suppose there is no cost. Firm 1 wants to maximize max π 1 = p 1 q 1 = p 1 ap bp 1 p 2 Firm 2 wants to maximize max π 2 = p 2 q 2 = p 2 ap bp 1 p 2 Solve the two equations simultaneously, then p 1 = p 2 = 1 2a b, q 1 = q 2 = a 2a b, π a 1 = π 2 = (2a b) 2 The outcome is different from that of the Bertrand model. It is not easy to compare this result to that of the Cournot model, nor to that of the perfect competition, because goods are not homogeneous. Yong Yang 6