Elements of Economic Analysis II Lecture XI: Oligopoly: Cournot and Bertrand Competition

Transcription

1 Elements of Economic Analysis II Lecture XI: Oligopoly: Cournot and Bertrand Competition Kai Hao Yang /2/207 In this lecture, we will apply the concepts in game theory to study oligopoly. In short, unlike a monopolist, firms in an oligopolistic market faces several competitors and will behave differently. However, on the other hand, such competition contrary to the competition in a perfectly competitive market where everyone takes prices as given is direct. Firms compete with each others by setting prices or choosing the amount of quantities to produce while maintaining market powers to manipulate market prices by changing quantities. We will look at two of the most widely known models for such competition competition on the quantity margin and competition on the price margin. Cournot Competition In the Cournot competition model, firms compete on the quantity margin. Consider an environment where there are two firms who are producing a homogeneous good in this market. Each firm can choose what quantity to produce and has a cost function. The firms, unlike the firms in a competitive market, can affect market prices through the quantities. However, unlike a monopolist, when making productions decisions, the firms have to take into account the other firm s decision. Department of Economics, University of Chicago; khyang@uchicago.edu

2 2 Formally, let D be a (differnetiable) demand function of this good and let P be the inverse demand. For each firm i {, 2}, i can choose to produce quantity q i 0. Given the produced quantities q, q 2, aggregate supply is Q = q + q 2 and therefore market price is given by P (q + q 2 ) and hence firm i {, 2} s profit is π C i (q i, q j ) = q i P (q i + q j ) c i (q i ), given that firm j i produces q j units, where c i ( ) is firm i s cost function. Using our definition, the model above is indeed a strategic form game. To see this, the set of players is {, 2}. Each player s strategy space is R +, where the quantities q i R + can be chosen. Finally, the payoff functions are given by π C and π C 2, respectively. We use the solution concept of the Nash equilibrium as defined before. That is, a profile of quantities (q i, q 2) is a Nash equilibrium if and only if for all i {, 2}, j i, for all q i 0 q i P (q i + q j ) c i (q i ) q ip (q i + q j ) c i (q i). Using our language of best response correspondences, for each firm i {, 2}, given firm j i s quantity q j, best response correspondence is β i (q j ) = argmax q i P (q i + q j ) c i (q i ). q i 0 If, furthermore, cost function is convex and the demand function is such that marginal revenue is decreasing, we know that the solution to each firm s profit maximization problem is unique and thus the best response correspondence β i is essentially a function. In this case, solving for Nash equilibrium is equivalent to solving a system of equations: q i = β i (q j ) q j = β j (q i ) In this case, first order condition is sufficient. Thus, β i (q j ) is given by the solution of As such, in equilibrium, P (q i + q j ) + q i P (q i + q j ) c i(q i ) = 0. P (q i + q j ) + q i P (q i + q j ) c i (q i ) = 0

3 3 for both i {, 2}. Notice that if both firms have the same cost function c( ), the equilibrium is symmetric. That is, q = q2 and therefore P (2q ) + q P (2q ) = c (q ). Rearranging, we have, P (2q ) + (2q )P (2q ) = c (q ) + q P (2q ). Moreover, since the demand is downward-sloping, and cost function is convex, c (q ) + q P (2q ) < c (q ) < c (2q ). Together, we have P (2q ) + (2q )P (2q ) < c (2q ). Let q M be the optimal quantity that a monopolist will produce when facing the inverse demand P and cost function c. Since the marginal revenue is assumed to be strictly decreasing and marginal cost is assumed to be strictly increasing, we have P (q M ) + q M P (q M ) = c (q M ). Together, we know that q M < 2q. That is, when there are two firms in the market competing with each other, the aggregate quantity produced, 2q, is greater than that when there is only one monopolist, which makes the outcome more efficient. Intuitively, strategic interaction between two firms forces them to produce in a more efficient way, although it is not to their best interest. (To their best interest, they should split q M equally.) However, on the other hand, since P < 0, in this equilibrium, P (2q ) < P (2q ) + q P (2q ) = c (q ) and therefore the equilibrium market price is still less than the marginal cost, meaning that the market outcome is still inefficient.

4 4 2 Bertrand Competition As the Cournot competition model studies the firms competition on the quantity margin, the Bertrand competition model focuses on the firm s competition on the price margin. Specifically, consider an environment where there are two firms who are producing a homogeneous good. For simplicity, suppose that each firm has a constant marginal cost of production c 0 and that each firm can set a price for their commodity. If one firm has a lower price, then that firm wins over the whole market and the consumers will only buy from the winning firm at the posted price. The losing firm earns nothing. Formally, let D be the market demand. For any firm i {, 2}, this firm can choose a price p i 0. If the posted price is lower, p i < p j, j i, then firm i wins the market and thus the quantity traded is D(p i ). If the posted price is higher, p i > p j, firm i gets nothing. If the posted price is the same as the opponent s, both firm divide the market equally. Together, given the opponent s price, firm i s profit is πi B (p i, p j ) = (p i c)d(p i ), if p i > p j 2 i c)d(p i ), if p i = p j. 0, if p i < p j Again, according to the definition, this model is a strategic form game. The set of players is {, 2}, strategy spaces are R + for each player and the payoff functions are given by π B, π B 2. As such, a pricing strategy (p, p 2) is a Nash equilibrium if and only if for any i {, 2}, for any p i 0, π B (p i, p j) π B i (p i, p j). It turns out that in this model, both firms setting prices at the marginal cost c is always a Nash equilibrium. To see this, it suffices to show that neither firm would want to deviate from p = c when the opponent is setting prices at c as well. Notice that when both firms set a price p = c, profit is given by 2 (p c)d(p ) = 0. On the other hand, if a firm deviates to a price p > p, this firm loses the whole market and thus will still have a profit 0. If a firm deviates to a price p < p, this firm will win the whole

5 5 market but the profit becomes (p c)d(p) < 0. Together, there is no incentives for both firms to deviate from p = c and hence both firms setting p = c is indeed a Nash equilibrium. In fact, the above equilibrium is the unique pure strategy equilibrium in this model. This means that in the pure strategy equilibrium, competition on price margin drives the firms to set a price as if the market were competitive.