INTMIC9.jpg Chapter 28 Oligopoly Oligopoly uA monopoly is an industry consisting a single firm. uA duopoly is an industry consisting of two firms. uAn oligopoly is an industry consisting of a few firms. Particularly, each firm’s own price or output decisions affect its competitors’ profits. Oligopoly uHow do we analyze markets in which the supplying industry is oligopolistic? uConsider the duopolistic case of two firms supplying the same product. Quantity Competition uAssume that firms compete by choosing output levels. uIf firm 1 produces y1 units and firm 2 produces y2 units then total quantity supplied is y1 + y2. The market price will be p(y1+ y2). uThe firms’ total cost functions are c1(y1) and c2(y2). Quantity Competition uSuppose firm 1 takes firm 2’s output level choice y2 as given. Then firm 1 sees its profit function as uGiven y2, what output level y1 maximizes firm 1’s profit? Quantity Competition; An Example uSuppose that the market inverse demand function is and that the firms’ total cost functions are and Quantity Competition; An Example Then, for given y2, firm 1’s profit function is Quantity Competition; An Example Then, for given y2, firm 1’s profit function is So, given y2, firm 1’s profit-maximizing output level solves Quantity Competition; An Example Then, for given y2, firm 1’s profit function is So, given y2, firm 1’s profit-maximizing output level solves I.e., firm 1’s best response to y2 is Quantity Competition; An Example y2 y1 60 15 Firm 1’s “reaction curve” Quantity Competition; An Example Similarly, given y1, firm 2’s profit function is Quantity Competition; An Example Similarly, given y1, firm 2’s profit function is So, given y1, firm 2’s profit-maximizing output level solves Quantity Competition; An Example Similarly, given y1, firm 2’s profit function is So, given y1, firm 2’s profit-maximizing output level solves I.e., firm 1’s best response to y2 is Quantity Competition; An Example y2 y1 Firm 2’s “reaction curve” 45/4 45 Quantity Competition; An Example uAn 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. uA pair of output levels (y1*,y2*) is a Cournot-Nash equilibrium if and Quantity Competition; An Example and Quantity Competition; An Example and Substitute for y2* to get Quantity Competition; An Example and Substitute for y2* to get Quantity Competition; An Example and Substitute for y2* to get Hence Quantity Competition; An Example and Substitute for y2* to get Hence So the Cournot-Nash equilibrium is Quantity Competition; An Example y2 y1 Firm 2’s “reaction curve” 60 15 Firm 1’s “reaction curve” 45/4 45 Quantity Competition; An Example y2 y1 Firm 2’s “reaction curve” 48 60 Firm 1’s “reaction curve” 8 13 Cournot-Nash equilibrium Quantity Competition Generally, given firm 2’s chosen output level y2, firm 1’s profit function is and the profit-maximizing value of y1 solves The solution, y1 = R1(y2), is firm 1’s Cournot- Nash reaction to y2. Quantity Competition Similarly, given firm 1’s chosen output level y1, firm 2’s profit function is and the profit-maximizing value of y2 solves The solution, y2 = R2(y1), is firm 2’s Cournot- Nash reaction to y1. Quantity Competition y2 y1 Firm 1’s “reaction curve” Firm 1’s “reaction curve” Cournot-Nash equilibrium y1* = R1(y2*) and y2* = R2(y1*) Iso-Profit Curves uFor firm 1, an iso-profit curve contains all the output pairs (y1,y2) giving firm 1 the same profit level P1. uWhat do iso-profit curves look like? y2 y1 Iso-Profit Curves for Firm 1 With y1 fixed, firm 1’s profit increases as y2 decreases. y2 y1 Increasing profit for firm 1. Iso-Profit Curves for Firm 1 y2 y1 Iso-Profit Curves for Firm 1 Q: Firm 2 chooses y2 = y2’. Where along the line y2 = y2’ is the output level that maximizes firm 1’s profit? y2’ y2 y1 Iso-Profit Curves for Firm 1 Q: Firm 2 chooses y2 = y2’. Where along the line y2 = y2’ is the output level that maximizes firm 1’s profit? A: The point attaining the highest iso-profit curve for firm 1. y2’ y1’ y2 y1 Iso-Profit Curves for Firm 1 Q: Firm 2 chooses y2 = y2’. Where along the line y2 = y2’ is the output level that maximizes firm 1’s profit? A: The point attaining the highest iso-profit curve for firm 1. y1’ is firm 1’s best response to y2 = y2’. y2’ y1’ y2 y1 Iso-Profit Curves for Firm 1 Q: Firm 2 chooses y2 = y2’. Where along the line y2 = y2’ is the output level that maximizes firm 1’s profit? A: The point attaining the highest iso-profit curve for firm 1. y1’ is firm 1’s best response to y2 = y2’. y2’ R1(y2’) y2 y1 y2’ R1(y2’) y2” R1(y2”) Iso-Profit Curves for Firm 1 y2 y1 y2’ y2” R1(y2”) R1(y2’) Firm 1’s reaction curve passes through the “tops” of firm 1’s iso-profit curves. Iso-Profit Curves for Firm 1 y2 y1 Iso-Profit Curves for Firm 2 Increasing profit for firm 2. y2 y1 Iso-Profit Curves for Firm 2 Firm 2’s reaction curve passes through the “tops” of firm 2’s iso-profit curves. y2 = R2(y1) Collusion uQ: Are the Cournot-Nash equilibrium profits the largest that the firms can earn in total? Collusion y2 y1 y1* y2* Are there other output level pairs (y1,y2) that give higher profits to both firms? (y1*,y2*) is the Cournot-Nash equilibrium. Collusion y2 y1 y1* y2* Are there other output level pairs (y1,y2) that give higher profits to both firms? (y1*,y2*) is the Cournot-Nash equilibrium. Collusion y2 y1 y1* y2* Are there other output level pairs (y1,y2) that give higher profits to both firms? (y1*,y2*) is the Cournot-Nash equilibrium. Collusion y2 y1 y1* y2* (y1*,y2*) is the Cournot-Nash equilibrium. Higher P2 Higher P1 Collusion y2 y1 y1* y2* Higher P2 Higher P1 y2’ y1’ Collusion y2 y1 y1* y2* y2’ y1’ Higher P2 Higher P1 Collusion y2 y1 y1* y2* y2’ y1’ Higher P2 Higher P1 (y1’,y2’) earns higher profits for both firms than does (y1*,y2*). Collusion uSo there are profit incentives for both firms to “cooperate” by lowering their output levels. uThis is collusion. uFirms that collude are said to have formed a cartel. uIf firms form a cartel, how should they do it? Collusion uSuppose the two firms want to maximize their total profit and divide it between them. Their goal is to choose cooperatively output levels y1 and y2 that maximize Collusion uThe 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. Collusion y2 y1 y1* y2* y2’ y1’ Higher P2 Higher P1 (y1’,y2’) earns higher profits for both firms than does (y1*,y2*). Collusion y2 y1 y1* y2* y2’ y1’ Higher P2 Higher P1 (y1’,y2’) earns higher profits for both firms than does (y1*,y2*). (y1”,y2”) earns still higher profits for both firms. y2” y1” Collusion y2 y1 y1* y2* y2 ~ y1 ~ (y1,y2) maximizes firm 1’s profit while leaving firm 2’s profit at the Cournot-Nash equilibrium level. ~ ~ Collusion y2 y1 y1* y2* y2 ~ y1 ~ (y1,y2) maximizes firm 1’s profit while leaving firm 2’s profit at the Cournot-Nash equilibrium level. ~ ~ y2 _ y2 _ (y1,y2) maximizes firm 2’s profit while leaving firm 1’s profit at the Cournot-Nash equilibrium level. _ _ Collusion y2 y1 y1* y2* y2 ~ y1 ~ y2 _ y2 _ The path of output pairs that maximize one firm’s profit while giving the other firm at least its C-N equilibrium profit. Collusion y2 y1 y1* y2* y2 ~ y1 ~ y2 _ y2 _ 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. Collusion y2 y1 y1* y2* y2m y1m (y1m,y2m) denotes the output levels that maximize the cartel’s total profit. Collusion uIs such a cartel stable? uDoes one firm have an incentive to cheat on the other? uI.e., if firm 1 continues to produce y1m units, is it profit-maximizing for firm 2 to continue to produce y2m units? Collusion uFirm 2’s profit-maximizing response to y1 = y1m is y2 = R2(y1m). Collusion y2 y1 y2m y1m y2 = R2(y1m) is firm 2’s best response to firm 1 choosing y1 = y1m. R2(y1m) y1 = R1(y2), firm 1’s reaction curve y2 = R2(y1), firm 2’s reaction curve Collusion uFirm 2’s profit-maximizing response to y1 = y1m is y2 = R2(y1m) > y2m. uFirm 2’s profit increases if it cheats on firm 1 by increasing its output level from y2m to R2(y1m). Collusion uSimilarly, firm 1’s profit increases if it cheats on firm 2 by increasing its output level from y1m to R1(y2m). Collusion y2 y1 y2m y1m y2 = R2(y1m) is firm 2’s best response to firm 1 choosing y1 = y1m. R1(y2m) y1 = R1(y2), firm 1’s reaction curve y2 = R2(y1), firm 2’s reaction curve Collusion uSo a profit-seeking cartel in which firms cooperatively set their output levels is fundamentally unstable. uE.g., OPEC’s broken agreements. Collusion uSo a profit-seeking cartel in which firms cooperatively set their output levels is fundamentally unstable. uE.g., OPEC’s broken agreements. uBut 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. Collusion & Punishment Strategies uTo determine if such a cartel can be stable we need to know 3 things: –(i) What is each firm’s per period profit in the cartel? –(ii) What is the profit a cheat earns in the first period in which it cheats? –(iii) What is the profit the cheat earns in each period after it first cheats? Collusion & Punishment Strategies uSuppose two firms face an inverse market demand of p(yT) = 24 – yT and have total costs of c1(y1) = y21 and c2(y2) = y22. Collusion & Punishment Strategies u(i) What is each firm’s per period profit in the cartel? up(yT) = 24 – yT , c1(y1) = y21 , c2(y2) = y22. uIf the firms collude then their joint profit function is pM(y1,y2) = (24 – y1 – y2)(y1 + y2) – y21 – y22. uWhat values of y1 and y2 maximize the cartel’s profit? Collusion & Punishment Strategies upM(y1,y2) = (24 – y1 – y2)(y1 + y2) – y21 – y22. uWhat values of y1 and y2 maximize the cartel’s profit? Solve Collusion & Punishment Strategies upM(y1,y2) = (24 – y1 – y2)(y1 + y2) – y21 – y22. uWhat values of y1 and y2 maximize the cartel’s profit? Solve uSolution is yM1 = yM2 = 4. Collusion & Punishment Strategies upM(y1,y2) = (24 – y1 – y2)(y1 + y2) – y21 – y22. uyM1 = yM2 = 4 maximizes the cartel’s profit. uThe maximum profit is therefore pM = $(24 – 8)(8) - $16 - $16 = $112. uSuppose the firms share the profit equally, getting $112/2 = $56 each per period. Collusion & Punishment Strategies u(iii) What is the profit the cheat earns in each period after it first cheats? uThis depends upon the punishment inflicted upon the cheat by the other firm. Collusion & Punishment Strategies u(iii) What is the profit the cheat earns in each period after it first cheats? uThis depends upon the punishment inflicted upon the cheat by the other firm. uSuppose the other firm punishes by forever after not cooperating with the cheat. uWhat are the firms’ profits in the noncooperative C-N equilibrium? Collusion & Punishment Strategies uWhat are the firms’ profits in the noncooperative C-N equilibrium? up(yT) = 24 – yT , c1(y1) = y21 , c2(y2) = y22. uGiven y2, firm 1’s profit function is p1(y1;y2) = (24 – y1 – y2)y1 – y21. Collusion & Punishment Strategies uWhat are the firms’ profits in the noncooperative C-N equilibrium? up(yT) = 24 – yT , c1(y1) = y21 , c2(y2) = y22. uGiven y2, firm 1’s profit function is p1(y1;y2) = (24 – y1 – y2)y1 – y21. uThe value of y1 that is firm 1’s best response to y2 solves Collusion & Punishment Strategies uWhat are the firms’ profits in the noncooperative C-N equilibrium? up1(y1;y2) = (24 – y1 – y2)y1 – y21. u uSimilarly, Collusion & Punishment Strategies uWhat are the firms’ profits in the noncooperative C-N equilibrium? up1(y1;y2) = (24 – y1 – y2)y1 – y21. u uSimilarly, uThe C-N equilibrium (y*1,y*2) solves y1 = R1(y2) and y2 = R2(y1) Þ y*1 = y*2 = 4×8. Collusion & Punishment Strategies uWhat are the firms’ profits in the noncooperative C-N equilibrium? up1(y1;y2) = (24 – y1 – y2)y1 – y21. u y*1 = y*2 = 4×8. uSo each firm’s profit in the C-N equilibrium is p*1 = p*2 = (14×4)(4×8) – 4×82 » $46 each period. Collusion & Punishment Strategies u(ii) What is the profit a cheat earns in the first period in which it cheats? uFirm 1 cheats on firm 2 by producing the quantity yCH1 that maximizes firm 1’s profit given that firm 2 continues to produce yM2 = 4. What is the value of yCH1? Collusion & Punishment Strategies u(ii) What is the profit a cheat earns in the first period in which it cheats? uFirm 1 cheats on firm 2 by producing the quantity yCH1 that maximizes firm 1’s profit given that firm 2 continues to produce yM2 = 4. What is the value of yCH1? uyCH1 = R1(yM2) = (24 – yM2)/4 = (24 – 4)/4 = 5. uFirm 1’s profit in the period in which it cheats is therefore pCH1 = (24 – 5 – 1)(5) – 52 = $65. Collusion & Punishment Strategies uTo determine if such a cartel can be stable we need to know 3 things: –(i) What is each firm’s per period profit in the cartel? $56. –(ii) What is the profit a cheat earns in the first period in which it cheats? $65. –(iii) What is the profit the cheat earns in each period after it first cheats? $46. Collusion & Punishment Strategies uEach firm’s periodic discount factor is 1/(1+r). uThe present-value of firm 1’s profits if it does not cheat is ?? Collusion & Punishment Strategies uEach firm’s periodic discount factor is 1/(1+r). uThe present-value of firm 1’s profits if it does not cheat is Collusion & Punishment Strategies uEach firm’s periodic discount factor is 1/(1+r). uThe present-value of firm 1’s profits if it does not cheat is uThe present-value of firm 1’s profit if it cheats this period is ?? Collusion & Punishment Strategies uEach firm’s periodic discount factor is 1/(1+r). uThe present-value of firm 1’s profits if it does not cheat is uThe present-value of firm 1’s profit if it cheats this period is Collusion & Punishment Strategies uSo the cartel will be stable if The Order of Play uSo far it has been assumed that firms choose their output levels simultaneously. uThe competition between the firms is then a simultaneous play game in which the output levels are the strategic variables. The Order of Play uWhat if firm 1 chooses its output level first and then firm 2 responds to this choice? uFirm 1 is then a leader. Firm 2 is a follower. uThe competition is a sequential game in which the output levels are the strategic variables. The Order of Play uSuch games are von Stackelberg games. uIs it better to be the leader? uOr is it better to be the follower? Stackelberg Games uQ: What is the best response that follower firm 2 can make to the choice y1 already made by the leader, firm 1? Stackelberg Games uQ: What is the best response that follower firm 2 can make to the choice y1 already made by the leader, firm 1? uA: Choose y2 = R2(y1). Stackelberg Games uQ: What is the best response that follower firm 2 can make to the choice y1 already made by the leader, firm 1? uA: Choose y2 = R2(y1). uFirm 1 knows this and so perfectly anticipates firm 2’s reaction to any y1 chosen by firm 1. Stackelberg Games uThis makes the leader’s profit function Stackelberg Games uThis makes the leader’s profit function uThe leader chooses y1 to maximize its profit. Stackelberg Games uThis makes the leader’s profit function uThe leader chooses y1 to maximize its profit. uQ: Will the leader make a profit at least as large as its Cournot-Nash equilibrium profit? Stackelberg Games uA: Yes. The leader could choose its Cournot-Nash output level, knowing that the follower would then also choose its C-N output level. The leader’s profit would then be its C-N profit. But the leader does not have to do this, so its profit must be at least as large as its C-N profit. Stackelberg Games; An Example uThe market inverse demand function is p = 60 - yT. The firms’ cost functions are c1(y1) = y12 and c2(y2) = 15y2 + y22. uFirm 2 is the follower. Its reaction function is Stackelberg Games; An Example The leader’s profit function is therefore Stackelberg Games; An Example The leader’s profit function is therefore For a profit-maximum for firm 1, Stackelberg Games; An Example Q: What is firm 2’s response to the leader’s choice Stackelberg Games; An Example Q: What is firm 2’s response to the leader’s choice A: Stackelberg Games; An Example Q: What is firm 2’s response to the leader’s choice A: The C-N output levels are (y1*,y2*) = (13,8) so the leader produces more than its C-N output and the follower produces less than its C-N output. This is true generally. Stackelberg Games y2 y1 y1* y2* (y1*,y2*) is the Cournot-Nash equilibrium. Higher P2 Higher P1 Stackelberg Games y2 y1 y1* y2* (y1*,y2*) is the Cournot-Nash equilibrium. Higher P1 Follower’s reaction curve Stackelberg Games y2 y1 y1* y2* (y1*,y2*) is the Cournot-Nash equilibrium. (y1S,y2S) is the Stackelberg equilibrium. Higher P1 y1S Follower’s reaction curve y2S Stackelberg Games y2 y1 y1* y2* (y1*,y2*) is the Cournot-Nash equilibrium. (y1S,y2S) is the Stackelberg equilibrium. y1S Follower’s reaction curve y2S Price Competition uWhat if firms compete using only price-setting strategies, instead of using only quantity-setting strategies? uGames in which firms use only price strategies and play simultaneously are Bertrand games. Bertrand Games uEach firm’s marginal production cost is constant at c. uAll firms set their prices simultaneously. uQ: Is there a Nash equilibrium? Bertrand Games uEach firm’s marginal production cost is constant at c. uAll firms set their prices simultaneously. uQ: Is there a Nash equilibrium? uA: Yes. Exactly one. Bertrand Games uEach firm’s marginal production cost is constant at c. uAll firms set their prices simultaneously. uQ: Is there a Nash equilibrium? uA: Yes. Exactly one. All firms set their prices equal to the marginal cost c. Why? Bertrand Games uSuppose one firm sets its price higher than another firm’s price. Bertrand Games uSuppose one firm sets its price higher than another firm’s price. uThen the higher-priced firm would have no customers. Bertrand Games uSuppose one firm sets its price higher than another firm’s price. uThen the higher-priced firm would have no customers. uHence, at an equilibrium, all firms must set the same price. Bertrand Games uSuppose the common price set by all firm is higher than marginal cost c. Bertrand Games uSuppose the common price set by all firm is higher than marginal cost c. uThen one firm can just slightly lower its price and sell to all the buyers, thereby increasing its profit. Bertrand Games uSuppose the common price set by all firm is higher than marginal cost c. uThen one firm can just slightly lower its price and sell to all the buyers, thereby increasing its profit. uThe only common price which prevents undercutting is c. Hence this is the only Nash equilibrium. Sequential Price Games uWhat if, instead of simultaneous play in pricing strategies, one firm decides its price ahead of the others. uThis is a sequential game in pricing strategies called a price-leadership game. uThe firm which sets its price ahead of the other firms is the price-leader. Sequential Price Games uThink of one large firm (the leader) and many competitive small firms (the followers). uThe small firms are price-takers and so their collective supply reaction to a market price p is their aggregate supply function Yf(p). Sequential Price Games uThe market demand function is D(p). uSo the leader knows that if it sets a price p the quantity demanded from it will be the residual demand uHence the leader’s profit function is Sequential Price Games uThe leader’s profit function is so the leader chooses the price level p* for which profit is maximized. uThe followers collectively supply Yf(p*) units and the leader supplies the residual quantity D(p*) - Yf(p*).