Lecture 9: Basic Oligopoly Models Managerial Economics November 16, 2012 Prof. Dr. Sebastian Rausch Centre for Energy Policy and Economics Department of Management, Technology and Economics ETH Zürich
Four Basic Models of Market Structure Perfect competition Market is characterized by many firms, each of which is small relative to the entire market. Firms have access to same technology and produce similar products. Firm do not have market power, i.e. no individual firm has a perceivable impact on the market price. Monopoly A firm (monopolist) is the sole producer of a good. Monopolistic competition Market is characterized by many firms as in perfect competition. However, unlike in perfect competition, each firm produces a good that is slightly different from products produced by other firms. Firms have some control over price. Oligopoly A few large firms tend to dominate the market. When one firm in an oligopolistic markets changes its price or output, it affects its own and other firms profits. This interdependence of profits gives rise to strategic interaction among firms.
Overview I. Conditions for Oligopoly? II. Role of Strategic Interdependence III. Profit Maximization in Three Oligopoly Settings Cournot Model Stackelberg Model Bertrand Model
Oligopoly A monopoly is an industry consisting of a single firm. A duopoly is an industry consisting of two firms. An oligopoly is an industry consisting of a few firms. Firms decisions impact one another: each firm s own price or output decisions affect its competitors profits. Many different strategic variables are modeled: No single oligopoly model. We will focus here on the following strategic variables: price and output.
Oligopoly How do we analyze markets in which the supplying industry is oligopolistic? How will rivals respond to your actions? Consider the duopolistic case of two firms supplying the same product. Although we focus on duopolies, our basic results will also apply to markets with more than two firms.
Quantity Competition Assume that firms compete by choosing output levels (firms produce either differentiated or homogeneous products). If firm 1 produces units and firm 2 produces y 2 units then total quantity supplied is y T = + y 2. The market price will be p( + y 2 ). The firms total cost functions are c 1 ( ) and c 2 (y 2 ). Each firm believes their rivals will hold output constant if it changes its own output (the output of rivals is viewed as given or fixed ). Barriers to entry exist.
Quantity Competition Suppose firm 1 takes firm 2 s output level choice y 2 as given. Then firm 1 sees its profit function as ( y ; y ) = p( y + y ) y c ( y ). Π 1 1 2 1 2 1 1 1 Given y 2, what output level maximizes firm 1 s profit?
Quantity Competition: An Example Suppose that the market inverse demand function is p( y ) = 60 T and that the firms total cost functions are c1( y1) = y1 2 and c2( y2) = 15y2 + y2 2. y T
Quantity Competition: An Example Then, for given y 2, firm 1 s profit function is 1 2 1 2 1 1 2 Π( y ; y ) = ( 60 y y ) y y. So, given y 2, firm 1 s profit-maximizing output level solves Π y 1 = 60 2y y 2y = 0. 1 2 1 Firm 1 s best response or reaction to y 2 is therefore given by 1 y1 = R1( y2) = 15 y2. 4
Quantity Competition: An Example y 2 60 Firm 1 s reaction curve 1 y1 = R1( y2) = 15 y2. 4 15
Quantity Competition: An Example Similarly, given, firm 2 s profit function is Π( y ; y ) = ( 60 y y ) y 15y y. So, given, firm 2 s profit-maximizing output level solves Π = 60 y1 2y2 15 2y2 = 0. y2 Firm 2 s best response to is 45 y y2 = R2( y 1 1) =. 4 2 1 1 2 2 2 2 2
Quantity Competition; An Example y 2 45/4 Firm 2 s reaction curve 45 y y2 = R2( y 1 1) =. 4 45
Quantity Competition: An Example An equilibrium is when each firm s output level is a best response to the other firm s output level, for then neither wants to deviate from its output level. A pair of output levels ( *,y 2 *) is a Cournot- Nash equilibrium if * y = R1( y2) * 1 and * 2 2 * 1 y = R ( y ).
* * 1 * y1 = R1( y2) = 15 y * * 45 y 2 and y R y 1 4 2 = 2( 1) =. 4 Substitute for y 2 * to get y * 1 = 15 1 4 45 4 * 1 y 4 Hence * 45 13 y 2 = * 1 y = = 8. So the Cournot-Nash equilibrium is * * ( y1, y2) = ( 13, 8). 13 *
y 2 60 Firm 1 s reaction curve 1 y1 = R1( y2) = 15 y2. 4 Firm 2 s reaction curve 45 y y2 = R2( y 1 1) =. 4 45/4 15 45
y 2 60 8 Firm 1 s reaction curve 1 y1 = R1( y2) = 15 y2. 4 Firm 2 s reaction curve 45 y y2 = R2( y 1 1) =. 4 Cournot-Nash equilibrium ( y * y * ) = ( ) 1, 2 13, 8. 13 48
y Π 1 1 Quantity Competition Generally, given firm 2 s chosen output level y 2, firm 1 s profit function is ( y ; y ) = p( y + y ) y c ( y ) Π 1 1 2 1 2 1 1 1 and the profit-maximizing value of solves = p( y + y ) + y 1 2 1 p( y1 + y2) y The solution, = R 1 (y 2 ), is firm 1 s Cournot- Nash reaction to y 2. 1 c 1 y1 = 0 ( ).
Similarly, given firm 1 s chosen output level, firm 2 s profit function is ( ; ) ( ) ( ) y Π 2 2 Π 2 y 2 = p + y 2 y 2 c 2 y 2 and the profit-maximizing value of y 2 solves = p( y + y ) + y 1 2 2 p( y1 + y2) y The solution, y 2 = R 2 ( ), is firm 2 s Cournot- Nash reaction to. 2 c ( ). 2 y2 = 0
y 2 y 2 * Firm 1 s reaction curve y1 = R1( y2). Firm 1 s reaction curve y2 = R2( y1). Cournot-Nash equilibrium * = R 1 (y 2 *) and y 2 * = R 2 ( *) *
Summary of Cournot Equilibrium Output of each firm maximizes its own profits, given what other firms produce. Neither firm has an incentive to change its output, given the output of the rival. Beliefs are consistent: In equilibrium, each firm thinks rivals will stick to their current output and they do. Cournot-Nash equilibrium: Each firm is doing the best it can given what its competitors are doing. Recall in market equilibrium in competitive and monopolistic markets: When a market is in equilibrium, firms are doing the best they can and have no reason to change their price or output.
Iso-Profit Curves For firm 1, an iso-profit curve contains all the output pairs (,y 2 ) giving firm 1 the same profit level Π 1. What do iso-profit curves look like?
Iso-Profit Curves for Firm 1 y 2 With fixed, firm 1 s profit increases as y 2 decreases.
Iso-Profit Curves for Firm 1 y 2 y 2 Q: Firm 2 chooses y 2 = y 2. Where along the line y 2 = y 2 is the output level that maximizes firm 1 s profit? A: The point attaining the highest iso-profit curve for firm 1. is firm 1 s best response to y 2 = y 2.
Iso-Profit Curves for Firm 1 y 2 y 2 y 2 R 1 (y 2 ) R 1 (y 2 )
Iso-Profit Curves for Firm 1 y 2 y 2 Firm 1 s reaction curve passes through the tops of firm 1 s iso-profit curves. y 2 R 1 (y 2 ) R 1 (y 2 )
Iso-Profit Curves for Firm 2 y 2 Increasing profit for firm 2.
Iso-Profit Curves for Firm 2 y 2 Firm 2 s reaction curve passes through the tops of firm 2 s iso-profit curves. y 2 = R 2 ( )
Collusion Q: Are the Cournot-Nash equilibrium profits the largest that the firms can earn in total?
y 2 y 2 * ( *,y 2 *) is the Cournot-Nash equilibrium. Are there other output level pairs (,y 2 ) that give higher profits to both firms? *
y 2 y 2 * ( *,y 2 *) is the Cournot-Nash equilibrium. Are there other output level pairs (,y 2 ) that give higher profits to both firms? *
y 2 y 2 * ( *,y 2 *) is the Cournot-Nash equilibrium. Are there other output level pairs (,y 2 ) that give higher profits to both firms? *
y 2 ( *,y 2 *) is the Cournot-Nash equilibrium. Higher Π 2 y 2 * Higher Π 1 *
y 2 Higher Π 2 y 2 y 2 * Higher Π 1 *
y 2 Higher Π 2 y 2 y 2 * Higher Π 1 *
y 2 y 2 y 2 * Higher Π 2 (,y 2 ) earns higher profits for both firms than does ( *,y 2 *). Higher Π 1 *
Collusion So there are profit incentives for both firms to cooperate by lowering their output levels. This is collusion. Firms that collude are said to have formed a cartel. If firms form a cartel, how should they do it?
Collusion Suppose the two firms want to maximize their total profit and divide it between them. Their goal is to choose cooperatively output levels and y 2 that maximize Π m ( y, y ) = p( y + y )( y + y ) c ( y ) c ( y ). 1 2 1 2 1 2 1 1 2 2
Collusion The firms cannot do worse by colluding since they can cooperatively choose their Cournot- Nash equilibrium output levels and so earn their Cournot-Nash equilibrium profits. So collusion must provide profits at least as large as their Cournot-Nash equilibrium profits.
y 2 y 2 y 2 * Higher Π 2 (,y 2 ) earns higher profits for both firms than does ( *,y 2 *). Higher Π 1 *
y 2 y 2 y 2 * y 2 Higher Π 2 (,y 2 ) earns higher profits for both firms than does ( *,y 2 *). Higher Π 1 * (,y 2 ) earns still higher profits for both firms.
y 2 y 2 * (y ~ 1,y ~ 2 ) maximizes firm 1 s profit while leaving firm 2 s profit at the Cournot-Nash equilibrium level. ~ y 2 ~ *
_ y 2 y 2 y 2 * ~ y 2 _ y 2 (y ~ 1,y ~ 2 ) maximizes firm 1 s profit while leaving firm 2 s profit at the Cournot-Nash equilibrium level. ~ * (,y 2 ) maximizes firm 2 s profit while leaving firm 1 s profit at the Cournot-Nash equilibrium level.
_ y 2 y 2 y 2 * ~ y 2 The path of output pairs that maximize one firm s profit while giving the other firm at least its C-N equilibrium profit. One of these output pairs must maximize the cartel s joint profit. _ y 2 ~ *
y 2 y 2 * y 2 m m * (m,y 2m ) denotes the output levels that maximize the cartel s total profit. What would be quantity chosen if either firm 1 or firm 2 are a monopolist? Firms divide market among each other by ensuring that they maximize total profits, i.e. they set total output = monopoly output. Cartel quantities where iso-profit curves of both firms are tangential.
Collusion Is such a cartel stable? Does one firm have an incentive to cheat on the other? I.e., if firm 1 continues to produce y m 1 units, is it profit-maximizing for firm 2 to continue to produce y m 2 units?
Collusion Firm 2 s profit-maximizing response to = m is y 2 = R 2 (m ).
y 2 = R 1 (y 2 ), firm 1 s reaction curve R 2 (m ) y 2 = R 2 (m ) is firm 2 s best response to firm 1 choosing = m. y 2 m y 2 = R 2 ( ), firm 2 s reaction curve m
Collusion Firm 2 s profit-maximizing response to = y m 1 is y 2 = R 2 (m ) > y 2m. Firm 2 s profit increases if it cheats on firm 1 by increasing its output level from y m 2 to R 2 (m ).
Collusion Similarly, firm 1 s profit increases if it cheats on firm 2 by increasing its output level from m to R 1 (y 2m ).
y 2 = R 1 (y 2 ), firm 1 s reaction curve y 2 = R 2 (m ) is firm 2 s best response to firm 1 choosing = m. y 2 m y 2 = R 2 ( ), firm 2 s reaction curve m R 1 (y 2m )
Collusion So a profit-seeking cartel in which firms cooperatively set their output levels is fundamentally unstable. E.g., OPEC s broken agreements. But is the cartel unstable if the game is repeated many times, instead of being played only once? Then there is an opportunity to punish a cheater.
Stackelberg Model Environment Few firms serving many consumers. As in Cournot model, firms compete over quantity (and firms either produce differentiated or homogeneous products). Barriers to entry. Firm 1 is the leader. The leader commits to an output before all other firms ( first-mover advantage ). Remaining firms are followers. They choose their outputs so as to maximize profits, given the leader s output.
Stackelberg Equilibrium Q 2 r 1 π 2 C Follower s Profits Decline π F S Stackelberg Equilibrium Q 2 C Q 2 S π L S π 1 C r 2 Q 1 C Q 1 S Q 1 M Q 1
The Algebra of the Stackelberg Model Since the follower reacts to the leader s output, the follower s output is determined by its reaction function. The Stackelberg leader uses this reaction function to determine its profit maximizing output level. Example: Linear demand function and assume that cost of both firms are zero. Algebra?
Stackelberg Summary Stackelberg model illustrates how commitment can enhance profits in strategic environments. Leader produces more than the Cournot equilibrium output. Larger market share, higher profits. First-mover advantage. Follower produces less than the Cournot equilibrium output. Smaller market share, lower profits.
Bertrand Model Environment Few firms that sell to many consumers. Firms produce identical products at constant marginal cost. Each firm independently sets its price in order to maximize profits (price is each firms control variable). Barriers to entry exist. Consumers enjoy Perfect information. Zero transaction costs.
Bertrand Equilibrium Firms set P 1 = P 2 = MC. Why? Suppose MC < P 1 < P 2. Firm 1 earns (P 1 - MC) on each unit sold, while firm 2 earns nothing. Firm 2 has an incentive to slightly undercut firm 1 s price to capture the entire market. Firm 1 then has an incentive to undercut firm 2 s price. This undercutting continues... Equilibrium: Each firm charges P 1 = P 2 = MC. By changing the strategic choice variable from quantity to price, we get a dramatically different outcome: In Cournot model, firms make positive profits. Profits of each firm are zero under Bertrand competition.
Conclusion Different oligopoly scenarios give rise to different optimal strategies and different outcomes. Optimal price and output decisions depend on: Beliefs about the reactions of rivals. Your choice variable (P or Q) and the nature of the product market (differentiated or homogeneous products). Your ability to credibly commit prior to your rivals.