ECO 3320 Lanlan Chu Managerial Economics Chapter 9 Practice Question 1. The market for widgets consists of two firms that produce identical products. Competition in the market is such that each of the firms independently produces a quantity of output, and these quantities are then sold in the market at a price that is determined by the total amount produced by the two firms. Firm 2 is known to have a cost advantage over Firm 1. A recent study found that the (inverse) market demand curve faced by the two firms is P = 200 3(Q 1 + Q 2 ) and costs are C 1 (Q 1 ) = 26Q 1 and C 2 (Q 2 ) = 32Q 2. a. Determine the marginal revenue for each firm. b. Determine the reaction function for each firm. c. How much output will each firm produce in equilibrium? d. What are the equilibrium profits for each firm? 1
2. The inverse demand curve for a Stackelberg duopoly is P = 16,000 4Q. The leader s cost structure is C L (Q L ) = 4,000Q L. The follower s cost structure is C F (Q F ) = 6,000Q F. a. Determine the reaction function for the follower. b. Determine the equilibrium output levels for both the leader and the follower. c. Determine the equilibrium market price. d. What are the profits for the leader? For the follower? 2
3. 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 = 500-2Q. a. Determine the equilibrium market price. b. Determine the equilibrium level of output in the market. c. Determine the profits of each firm. 3
4. 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 QQ = 150 1 PP, each firm has a marginal cost of $20, an average variable cost 10 of 20, and fixed costs of zero. 4
4. 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 QQ = 150 1 PP, each firm has a 10 marginal cost of $20, an average variable cost of 20, and fixed costs of zero. Answer: a. Cournot Firm One's Output Firm Two's Output Total Output Market Price Firm One's Profit Firm Two's Profit b.stackelberg (leader) (follower) c. Bertrand d. Collusion a. Cournot P=1500-10(q1+q2) For Firm one we have: R1=1500q1-10q1 2-10q1q2 MR1=1500-20q1-10q2 MC=20 MR1 =MC => 1500-20q1-10q2=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=1500q2-10q2 2-10q1q2 MR1=1500-20q2-10q1 MC=20 MR1 =MC => 1500-20q2-10q1=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 q 1 we get q1=49.333. Substituting this into (2) we get q2=49.333 and Q= q1+q2=98.66 Substituting Q into the inverse demand equation we get P=1500-10(98.66) => P=513.33 With these values, the profits for both firms are $24,338 5
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: P=1500-10q1-10(74- ½ q1) which simplifies to: P=760-5q1 Thus, we have that the marginal revenue of the leader is: R1=P.q1 =760q1-5q1 2 M R1=760-10 q1 MC1=20 From MR1= MC1 => 760-10q1=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=1500-10(111) => P=390 With these values, the profits for leader firm are $27,380 and the profits for the follower firm are $13,690. c. Bertrand Firm set P=MC => P=20 => 1500-10Q=20 =>Q=148 The firms split the total output in half => q1=q2=74 and both firms earn zero economic profits. d. Collusion When firms collude, they act as a multi-plant 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=1500-10Q For the monopolist we have: R=1500Q-10Q 2 MR=1500-20Q MC=20 MR =MC => 1500-20Q=20 Solving for Q we get the monopolist output: Q=74 => q1=q2=37 Substituting Q into the inverse demand equation we get P=1500-10(74) => P=760 The profits for both firms are $27,380 6