# Price competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly]

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1 ECON9 (Spring 0) & 350 (Tutorial ) Chapter Monopolistic Competition and Oligopoly (Part ) Price competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly] Bertrand model: Oligopoly model in which firms produce a homogeneous good, each firm treats the price of its competitors as fixed, and all firms decide simultaneously what price to charge Each firm assumes its rivals will keep their price level constant when it changes its own price The model Residual demands for Firm : D ( p, p ) D( p) if p p D ( p, p ) D( p) if p p D ( p, p ) 0 if p p Residual demands for Firm : D ( p, p) D( p) if p p D ( p, p) D( p) if p p D ( p, p) 0 if p p The Bertrand equilibrium Bertrand equilibrium: if the two firms have identical marginal cost equal to c, then the Bertrand equilibrium price is equal to c p = p = c (stable equilibrium) Suppose p > p > c Firm captures the whole market and earns profits Firm will match or undercut firm s price p > p > c Firm captures the whole market and earns profits Firm will match or undercut firm s price p > p > c Firm will match or undercut firm s price The adjustment will continue until p = p = c Any pair of unequal prices cannot be the Bertrand equilibrium The situation p = p > c is not stable As long as there is profit for the one who can capture the whole market, both firms will have incentive to undercut the other firm s price The outcome in Bertrand equilibrium is the same as the perfectly competitive market No firm can earn profit

2 Representing the Bertrand duopoly game by algebra Market demand: p a q q ) ( Cost functions: c(q ) = cq, c(q ) = cq Marginal cost = c for both firms As p B = c in the equilibrium, p B a q q ) q q q q ( B a p a c Since the two firms will share the market when their prices are equal, therefore, B B a c q q B B 0 Price competition with differentiated products Market shares are now determined not just by prices, but by differences in the design, performance, and durability of each firm s product In these markets, more likely to compete using price instead of quantity Example: Duopoly with fixed costs of \$0 but zero variable costs Firms face the same demand curves: Firm s demand: q p p Firm s demand: q p p Quantity that each firm can sell decreases when it raises its own price but increases when its competitor charges a higher price Firm s maximizing problem: Max p p p p 0 0 p FOC: 4p p 0 p p (Firm s reaction function) 4 Firm s maximizing problem: Max p p p p 0 0 p FOC: 4p p 0 p p (Firm s reaction function) 4 Solving the two reaction functions for the equilibrium, it yields, p p 4 q q q Profit for Firm = Profit for Firm = q 8 6

3 Collusion in price competition They both decide to charge the same price that maximized both of their profits Firms will charge \$6 and will be better off colluding since they will earn a profit of \$6 (Try to verify!) Representing price competition (differentiated products) by algebra The two firms are facing the same demand curve: q = a bp + cp and q = a bp + cp Also assume MC = 0 Firm s maximizing problem: Max ap bp cp p 0 p FOC: a bp cp 0 p a cp (Firm s reaction function) b Firm s maximizing problem: Max ap bp cp p 0 p FOC: a bp cp 0 p a cp (Firm s reaction function) b Solving the two reaction functions for the equilibrium, it yields, a ab p p, q q b c b c, q q ab b c Representing the Collusion model in price competition Total profit: pq pq ap bp cp p ap bp cp p The maximizing problems of the two firms are as follows: Max pq pq ap bp cp p ap bp cp 0 p Firm : FOC: a bp cp cp 0 p Max pq pq ap bp cp p ap bp cp p p 0 Firm : FOC: a bp cp cp 0 a Solving for the equilibrium, it yields p p, ( b c) q a q, q q a 3

4 Competition VS Collusion: The Prisoner s Dilemma Dominate strategy: when one strategy is best for a player no matter what strategy the other player uses We will explain these concepts with the classic example of Prisoner s Dilemma Example: Prisoner s Dilemma The story: Ann and Bob have been caught stealing a car The police suspect that they have also robbed the bank, a more serious crime The police has no evidence for the robbery, and needs at least one person to confess to get a conviction Ann and Bob are separated and each told: (i) If each confesses, then each will get a 0 year sentence (ii) If one confesses, but the other denies, then he will get year and his accomplice will get yrs (iii) If neither confesses, then each will get a 3 year sentence for auto theft We will represent the prisoner s dilemma with normal form Ann Bob Confess Deny Confess -0, -0 -, - Deny -, - -3, -3 Is there any dominated strategy for Ann and Bob? Let s consider Ann, If Ann expects Bob to confess, then Ann should confess ( 0 ) If Ann expects Bob to deny, then Ann should confess ( 3) Ann gets a higher payoff with confess than deny no matter what she expects Bob to do If Ann is rational, she will confess Formally, we say that deny is strictly dominated by confess Or we say that confess is a dominant strategy for Ann By the same way, we can find that confess is a dominant strategy and deny is dominated strategy for Bob In the prisoner s dilemma, if both players are rational, they will choose to use their dominant strategies, Confess The Nash equilibrium for this game is (Confess, Confess) with a payoff of 0 for each player We find that the payoff for both players will be much better { 3, 3} if they both choose deny, however in the prisoner s dilemma the NE is (Confess, Confess) Individual rationality does not imply socially optimal outcome in the prisoner s dilemma 4

5 Nash equilibrium Nash equilibrium: a collection of strategies, one for each player, such that no player can improve his situation by choosing a different strategy that is available to him, given that all other players stay put In other words, the strategy (s, s ) constitutes a NE if given player s strategy s, player finds it optimal to choose s, and given player s strategy s, player finds it optimal to choose s (Best response) When the NE is reached, there is no incentive for any player to deviate from it No player can benefit or increase his/her payoff by deviating from the NE For example, Ann would not deviate if given Bob uses his dominate strategy confess Deviation would lower her payoff to given Bob stay puts Example: Cartel/ Collusion (Please refer to T0) Firm A Firm B Honor Agreement Break Agreement Honor Agreement 7, 7 54, 8 Break Agreement 8, 54 64, 64 In the duopoly model, price is lower than the monopoly price Incentive for the two firms to collude They get into agreement to set a higher price, and produce less (monopoly output) in order to have higher profit In this duopoly game, Break Agreement is a dominant strategy for both firms NE: (Break Agreement, Break Agreement) with profit of 64 to each firm This game is a prisoner dilemma They can both get a higher profit of 7 by following the cartel The cartel is not stable, they will have incentive to cheat and deviate from the agreement Both firms will cheat and ends up in the Cournot equilibrium 5

6 Chapter : Problem Consider two firms facing the demand curve P = 50-5Q, where Q = Q + Q The firms cost functions are C (Q ) = 0 + 0Q and C (Q ) = 0 + Q a Suppose both firms have entered the industry What is the joint profit-maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry? If both firms enter the market, and they collude, they will set MR = MC to determine the profit-maximizing output (MC < MC ) MR = 50-0Q = 0 = MC Q = 4, P = \$30 The question now is how the firms will divide the total output of 4 among themselves Since the two firms have different cost functions, it will not be optimal for them to split the output evenly between them The profit maximizing solution is for firm to produce all of the output so that The profit for Firm will be: = (30)(4) - (0 + (0)(4)) = \$60 The profit for Firm will be: = (30)(0) - (0 + ()(0)) = -\$0 Total industry profit will be: T = + = 60-0 = \$50 If they split the output evenly between them then total profit would be \$46 (\$0 for firm and \$6 for firm ) If firm preferred to earn a profit of \$6 as opposed to \$5 (\$50/) then firm could give \$ to firm and it would still have profit of \$4, which is higher than the \$0 it would earn if they split output Note that if firm supplied all the output then it would set marginal revenue equal to its marginal cost or and earn a profit of 6 In this case, firm would earn a profit of 0, so that total industry profit would be 4 If Firm were the only entrant, its profits would be \$60 and Firm s would be 0 If Firm were the only entrant, then it would equate marginal revenue with its marginal cost to determine its profit-maximizing quantity: 50-0Q =, or Q = 38 P = = \$3 The profits for Firm will be: = (3)(38) - (0 + ()(38)) = \$60 b What is each firm s equilibrium output and profit if they behave noncooperatively? Use the Cournot model Draw the firms reaction curves and show the equilibrium In the Cournot model, Firm takes Firm s output as given and maximizes profits The profit function derived in a becomes = (50-5Q - 5Q )Q - (0 + 0Q ), or 40Q 5Q 5Q Q 0 Firm s reaction function: Q = 40 0 Q Q - 5 Q = 0, or Q = 4-6

7 Similarly, Firm s reaction function is Q 38 Q Solving for the Cournot equilibrium, Q 4 38 Q, or Q 8and Q = 4 P = 50 5(8+4) = \$4 The profits for Firms and are equal to = (4)(8) - (0 + (0)(8)) = 90 = (4)(4) - (0 + ()(4)) = 880 c How much should Firm be willing to pay to purchase Firm if collusion is illegal but the takeover is not? In order to determine how much Firm will be willing to pay to purchase Firm, we must compare Firm s profits in the monopoly situation versus those in an oligopoly The difference between the two will be what Firm is willing to pay for Firm From part a, profit of firm when it set marginal revenue equal to its marginal cost was \$60 This is what the firm would earn if it was a monopolist From part b, profit was \$90 for firm Firm would therefore be willing to pay up to \$4080 for firm 7

8 Chapter : Problem 6 Suppose that two identical firms produce widgets and that they are the only firms in the market Their costs are given by C = 60Q and C = 60Q, where Q is the output of Firm and Q the output of Firm Price is determined by the following demand curve: P = 300 Q where Q = Q + Q a Find the Cournot-Nash equilibrium Calculate the profit of each firm at this equilibrium Firm s profit function: 300Q Q Q Q 60Q 40Q Q Q Q Q 40 Q Q = 0 Firm s reaction function: Q = 0-05Q Firm s reaction function: Q = 0-05Q Solving for the Cournot equilibrium, Q = 0 - (05)(0-05Q ), or Q = 80 Q = 80 P = = \$40 = (40)(80) - (60)(80) = \$6,400 and = (40)(80) - (60)(80) = \$6,400 b Suppose the two firms form a cartel to maximize joint profits How many widgets will be produced? Calculate each firm s profit Given the demand curve P = 300-Q MR=300-Q MR = 300 Q = 60 = MC Q = 0, P = 80 Each firm produces 60 Profit for each firm is: = 80(60)-60(60)=\$7,00 c Suppose Firm were the only firm in the industry How would the market output and Firm s profit differ from that found in part (b) above? MR = 300 Q = 60 = MC Q = 0, P = 80 Profit = \$4,400 d Returning to the duopoly of part (b), suppose Firm abides by the agreement, but Firm cheats by increasing production How many widgets will Firm produce? What will be each firm s profits? Assuming their agreement is to split the market equally, Firm produces 60 widgets Firm cheats by producing its profit-maximizing level, given Q = 60 Given Q = 60 into Firm s reaction function: Q Total industry output, Q T, is equal to Q plus Q : Q T = = 50 P = = \$50 = (50)(60) - (60)(60) = \$5,400 and = (50)(90) - (60)(90) = \$8,00 8

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