Eco 403: Industrial Organization Economics, Fall Dr. Abdel-Hameed H. Nawar. Oligopoly

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1 Eco 403: Industrial Organization Economics, Fall 2012 Dr. Abdel-Hameed H. Nawar Oligopoly Isoprofit Curve An isoprofit Curve of a firm,,, is defined to be the set of all possible combinations of each firm s output levels that give that firm the same level of profit..., :, =, =1,2 How is an isoprofit curve shaped? Property 1. An isoprofit curve is concave and reaches a maximum on the firm s reaction (best response) curve. Recall that for an -variable function, the hyperplane that is tangent to the function at a particular point lies above the function if it is concave and lies below the function if it is convex. For a single variable function, i.e. where =1, 1

2 At point E, firm 1 maximizes its profit given firm 2 s. Note that since at and, firm 2 has the same output level =, because, Since at point, firm 2 s output level is greater, at point at point and thus at point at point, then. Similarly, and 2

3 Property 2. A lower isoprofit curve has a greaterr level of profit. Property 3: Two each other. isoprofit curves of a firm cannot cross Proof. Consider points,, and. Since. By definition, = = =. Contradiction. 3

4 Property 4: An isoprofit curve of a firm cannot reaction curve more than once. cross its Proof. Consider points and. By Property 2,. However, for any given,. Contradiction. One Period Model of Collusion If firm 1 and firm 2 decide to collude to improve their profit, then this is possible by producing lower output level. What is the total output level that maximizes the joint profits? In collusion, Firm 1 and Firm 2 act as a monopolist and will produce the monopolist output level jointly. 4

5 Firm 1 s and Firm 2 s isoprofit functions are largest. = = = = = Note that: Is unilaterally deviation gainful? As we will see for firm 1, This is symmetric game: Deviation profit when the rival colludes: = Collusion profit when the rival deviates = Firm 1 Firm 2 Collude Deviate Collude,, Deviate,, firm 1 s optimal output when firm 2 produces Clearly, collusion cannot be sustained. 5

6 Collusion is not sustainable in one period model since cheater cannot be punished. Repeated Games Firms in some industries do not play one-shot game but rather strategically interact repeatedly. This may affect the equilibrium behavior. In particular, it opens possibilities for collusion in an industry. Repeated game is a game that is played over and over again. Repetition could be finite or infinite times and time is discrete. Due to the time-value of money, a 1 dollar earned during the first period is worth more than a dollar in later repetitions. Players must discount future payoffs when they make current decisions. An infinitely repeated game of Collusion Present Value = interest rate per period Period 1 Period 2 Period T

7 =,0 1 is a discount factor, which measures the time value of the money. The value today of a future payoff is called the present value (PV), which is an increasing function of. If approaches 0, the firm does not care about the future (impatient), If is close to 1, the firm really cares about the future (patient). Trigger Strategy Collusion could be achieved in an infinitely repeated game by using the following set of strategies: 1. Each firm produces the collusion level of output each period as long as its rival does the same. 2. If any firm produces a different level of output, then beginning from the next period and forever its rivals will punish the firm by playing the CN equilibrium. Collusion is sustainable if the PV(collusion) > PV(deviation once & then punished forever); or PV(punishment) > PV(gains from collusion) 7

8 Let PV(collusion) be denoted by and PV(deviation once & then punished forever) be denoted by = = = 1 1 = = = + 1 Collusion is sustainable if + Thus Thus 8

9 If firms care less about the future than the present, then deviation may be attractive. If firms care about future more, then collusion is sustainable. Collusion is sustainable if the firm is sufficiently patient (i.e. is close to 1). 9

10 Note: PV(gains from deviation) PV(punishment) = = + + = We can obtain PV(punishment) > PV(gains from deviation) which gives the same condition for sustainability: Numerical Example Inverse demand function: = Total cost: =7, =1,2 Monopoly Case max

11 FOC =0 =42 = = = 91 = = = 3528 In case of collusion we assume that each firm makes 1 2 of. = =21. = =1769 Duopoly Case Firm 1 s profit 11

12 max = = FOC Firm1 s best response curve: By symmetry, Solving for and, = =0 =42 =42.. = =28 By symmetry, =28 = = = 63 12

13 = = = 1568 By symmetry, =1568 Homework Exercise Draw firm 1's best response curve and the isoprofit curve corresponding to its profit in the Cournot-Nash equilibrium. Draw firm 2's best response curve and the isoprofit curve corresponding to its profit in the Cournot-Nash equilibrium. Hint: for a given level of profit, = +, hence =. Collusion If firm 1 thinks that form 2 will play the collusive level of output, firm 1 has an incentive to deviate taking firm 2 s output level as given and then maximizing its profit: 13

14 Using the reaction function =42, = 42 = = 31.5 = = = 70 = = = = = = 1323 The above results can be summarizes in as follows: Firm 2 Collude Deviate Firm 1 Collude 1764, , Deviate , ,1568 * 14

15 Clearly, (Deviate, Deviate) is the NE in a one period game. In an infinitely repeated game, collusion outcome (Collude, Collude) is sustainable if: and hence if.. = If the discount factor , then collusion is sustainable. CNE, Monopoly and Perfect Competition The -firm linear symmetric CNE For =1,2,, the following hold values = = = + as function of decreasing Increasing Decreasing 15

16 = = = Decreasing Increasing Increasing What are some of the insights? As the number of firms in the industry goes up, each individual firm produces less, but total output goes up, the market price decreases and the total surplus increases. Optimal Entry in a Cournot Nash World We assume: 16

17 Stage 1: Foreseeing a Cournot game, firms decide whether or not to enter at a cost of. Stage 2: the number of entrants (say ) is observed and the firms strategically choose Cournot Nash Equilibrium:,,, Backward Induction: in a multiple stage game, we solve the game backwards from the last stage. Stage 2:, the number of entrants = Stage 2: ntry will occur until profit equals entry cost: = where is the equilibrium number of firms +1 = +1 = 17

18 = 1 The social planner s decision problem: max = FOC = =0 = 1 In the linear symmetric Cournot game, the market provides too much entry. In general there are two extremes associated with entry. 1. Surplus appropriability: an entrant ignores an increase in consumer surplus due to entry 2. Business stealing: an entrant ignores declines of the profits of existing firms caused by entry (negative externality) 18