ECO Chapter 9 Instructor: Lanlan Chu. Chapter 9 Basic Oligopoly Models


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1 Cournot Model Chapter 9 Basic Oligopoly Models A. Given a linear (inverse) demand function: P = a b(q 1 + Q 2 ) B. Cost functions: C 1 (Q 1 ) = c 1 Q 1 and C 2 (Q 2 ) = c 2 Q 2, C. So Marginal cost functions: MC 1 (Q 1 ) = c 1 and MC 2 (Q 2 ) = c 2 Derive their reaction functions: Derive Cournot Equilibrium: a situation in which neither firm has an incentive to change its output given the other firm s output EX1: Suppose the inverse demand function in a Cournot duopoly is given by P = 10 (Q 1 + Q 2 ) and their marginal costs are 2. a. What are the reaction functions for the two firms? b. What are the Cournot equilibrium outputs? c. What is the equilibrium price? d. What is the equilibrium profit for each firm? Cournot Oligopoly: Collusion EX2: Suppose the inverse demand function in a Cournot duopoly is given by P = 10 (Q 1 + Q 2 ) and their marginal costs are 2. Suppose they decide to collude, then a. What are the equilibrium outputs of each firm? b. What is the equilibrium price? c. What is the equilibrium profits for each firm? d. Compared to the profit in example1, which profit is higher?
2 Stackelberg Oligopoly Given a linear (inverse) demand function P = a b(q 1 + Q 2 ) and marginal cost functions are MC 1 (Q 1 ) = c 1 and MC 2 (Q 2 ) = c 2. Then, The follower sets output according to the reaction function Q 2 = r 2 (Q 1 ) = a c 2 1 Q 2b 2 1 and The leader s output is Q 1 = a+c 2 2c 1 2b EX3: Suppose the inverse demand function for two firms in a homogeneousproduct, Stackelberg oligopoly is given by P = 50 (Q 1 + Q 2 ) and their marginal costs are: MC 1 (Q 1 ) = MC 2 (Q 2 ) = 2 Firm 1 is the leader, and firm 2 is the follower. What is firm 2 s reaction function? What is firm 1 s output? What is firm 2 s output? What is the market price? Betrand Oligopoly EX4: Consider a Bertrand oligopoly consisting of four firms that produce an identical product at a marginal cost of $100.The inverse market demand for this product is P = 5002Q. a. Determine the equilibrium market price. b. Determine the equilibrium level of output in the market. c. Determine the profits of each firm.
3 EX5: The table below compares and contrasts the output levels and profits for the Cournot, Stackelberg, Bertrand and Collusion models. Fill in the table assuming that there are two firms in the market, the market demand is given by Q=1501/10 P, each firm has a marginal cost of $20, an average variable cost of 20, and fixed costs of zero. a. Cournot Firm One's Output Firm Two's Output b.stackelberg (leader) (follower) c. Bertrand d. Collusion Total Output Market Price Firm One's Profit Firm Two's Profit Answer: a. Cournot P= (q1+q2) For Firm one we have: MR1= q110q2 MC=20 MR1 =MC => q110q2=20 Solving for q1 we get the Firm One s best response function: q1=74 ½ q2 (1) Similarly, for Firm 2 we have that: R2=1500q210q2 210q1q2 MR 2 = q 210q 1 MC=20 MR 2 =MC => q 210q 1 =20 Solving for q2 we get the Firm Two s best response function: q2=74 ½ q1 (2) The equilibrium is the intersection of the two best response function, substituting (2) into (1) we get: q1=74 ½ (74 ½ q1) Solving for q1 we get q1= Substituting this into (2) we get q2= and Q= q1+q2=98.66 Substituting Q into the inverse demand equation we get P= (98.66) => P= With these values, the profits for both firms are $24,338
4 b. Stackelberg From the previous question Firm Two s best response function is given by: q2=74 ½ q1 Substituting this into the inverse demand function, we get: c. Bertrand d. Collusion P= q110(74 ½ q1) which simplifies to: P=7605q1 Thus, we have that the marginal revenue of the leader is: M R1= q1 MC1=20 From MR1= MC1 => q1=20 => q1=74 Substituting this into the best response function of the follower, we get: q2=37 and Q= q1+q2=111 Substituting Q into the inverse demand equation we get P= (111) => P=390 With these values, the profits for leader firm are $27,380 and the profits for the follower firm are $13,690. Firm set P=MC => P=20 => Q=20 =>Q=148 The firms split the total output in half => q1=q2=74 and both firms earn zero economic profits. When firms collude, they act as a multiplant monopoly. Because both firms have the same marginal cost, we can solve the problem for the regular monopolist and split the total output in half. P= Q For the monopolist we have: MR= Q MC=20 MR =MC => Q=20 Solving for Q we get the monopolist output: Q=74 => q1=q2=37 Substituting Q into the inverse demand equation we get P= (74) => P=760 The profits for both firms are $27,380 4
5 EX6. Consider a market consisting of two firms where the inverse demand curve is given by P = 5002Q12Q2. Each firm has a marginal cost of $50. Based on this information, we can conclude that equilibrium price in the different oligopoly models will follow which of the following orderings? A. P Bertrand < P Stackelberg < P Cournot < P Collusion B. P Stackelberg < P Collusion < P Cournot < P Bertrand C. P Collusion < P Cournot < P Stackelberg < P Bertrand D. P Bertrand < P Cournot < P Stackelberg < P Collusion EX7. Which of the following are quantitysetting oligopoly models? A. Stackelberg. B. Cournot. C. Bertrand. D. Stackelberg and Cournot. EX8. With linear demand and constant marginal cost, a Stackelberg leader's profits are the follower. A. less than B. equal to C. greater than D. either less than or greater than EX9. If firms compete in a Cournot fashion, then each firm views the: A. output of rivals as given. B. prices of rivals as given. C. profits of rivals as given. D. All of the statements associated with this question are correct. EX10. The Bertrand model of oligopoly reveals that: A. capacity constraints are not important in determining market performance. B. perfectly competitive prices can arise in markets with only a few firms. C. changes in marginal cost do not affect prices. D. All of the statements associated with this question are true. EX11. A new firm enters a market which is initially serviced by a Bertrand duopoly charging a price of $20. What will the new price be should the three firms coexist after the entry? A. $25 B. $20 C. $15 D. None of the answers is correct. 5
6 Summary of Four Market Structures: Perfect Competition Monopolistic Competition Oligopoly Monopoly # of firms Very many many few One Product identical differentiated (but highly substituable) idential or differentiated unique (no close substitutes) Entry/Exit Easy Easy difficult Nearly impossible Profit max rule MR=MC(or P=MC) MR=MC MR=MC MR=MC Short run profit π<, > or =0 π<, >, or =0 π<, > or =0 π<, > or =0 Long run profit π=0 π=0 π> or =0 π> or =0 Price(P) Quantity(Q) P pc <P mc <P o < Pm Q pc >Q mc >Q o >Q m 6
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