Exercise for Final Exam - MAN401

Transcription

1 Exercise for Final Exam - MAN Consider a firm with monopoly power that faces the demand curve P = 100 3Q + 4A 1/2 and has the total cost function T C = 4Q Q + A where A is the level of advertising expenditures, and P and Q are price and output. a. Find the values of A, Q and P that maximize this firm s profit. max π = (100 3Q + Q,A 4A1/2 )Q 4Q Q + A π/ Q = 100 6Q + 4A 1/2 8Q + 10 = 0 (1) π/ A = 2A 1/2 1 = 0 (2) when we solve equation (1) and (2) together, optimal advertising budget, A = 900, output, Q = 15 and price, P = 175. b. Calculate the profit margin at its profit-maximizing levels of A, Q and P. MC = 8Q + 10 = 8(15) + 10 = 130 so, Lerner = ( )/175 = Sal s satellite company broadcasts TV to subscribers in Los Angeles and New York. The demand functions for each of these groups are Q NY = 50 (1/3)P NY Q LA = 80 (2/3)P LA where Q is in thousands of subscribers per year, and P is the subscription price per year. The cost of providing Q units of service is given by T C = Q where Q = Q NY + Q LA. a. What are the profit-maximizing prices and quantities for the New York and Los Angeles markets? First, we write demand functions are inverse demand function, i.e, P = f(q) and then derive MRs for each location: From Q NY = 50 (1/3)P NY, we can find inverse demand as: P NY = 150 3Q NY and from Q LA = 80 (2/3)P LA, we can write P LA = 120 (2/3)P LA. Thus, MR NY = 150 Q NY = 30 = MC and Q NY = 20, P NY = 90 MR NY = 120 (3/2)Q LA = 30 = MC and Q LA = 30, P LA = 75 b. As a consequence of a new satellite that the government recently deployed, people in Los Angeles receive Sal s New York broadcast and people in New York

2 receives Sal s Los Angeles broadcasts. As a result, anyone in New York or Los Angeles can receive Sal s broadcasts by subscribing in either city. Sal can charge only a single price. What price should he charge, and what quantities will he sell in New York and Los Angeles? Q T = Q NY + Q LA = 50 (1/3)P + 80 (2/3)P = 130 P P = 130 Q MR = 130 2Q = 30 = MC and Q = 50, P = 80 Q NY = 50 (1/3)(80) = 69/3 Q LA = 80 (2/3)(80) = 78/3 c. In which of the above situations, (a) and (b), is Sal better off? In terms of consumer surplus, which situation do people in New York prefer and which do people in Los Angeles prefer? Why? Under market condition in (a) profit is equal to the sum of revenues from each market minus the cost of producing quantity for both markets: Q NY P NY + Q LA P LA T C(Q NY + Q LA ) = 20(90) + 30(75) ( ) = Under market condition in (b) profit is equal to the total revenue minus total cost: QP T C(Q) = 50(80) (50) = So, Sal makes more money when two markets are separated. 3. Your firm produces two products, the demands for which are independent. Both products at zero marginal cost. Yo face four consumers (or groups of customers) with following reservation prices: Consumer Good 1($) Good 2 ($) A B C D a. Consider three alternative pricing strategies: (i) selling the goods separately; (ii)pure bundling; (iii)mixed bundling. For each strategy, determine the optimal prices to be charged and the resulting profits. Which strategy is best? Bundled Price 1 Price 2 Price Profit Sell Separately Pure Bundling Mixed Bundling Pure bundling is the best strategy. Notice that in the mixed bundling, if single

3 prices are $70 or more, consumers A and D will prefer the bundle because by paying $100, they will purchase least wanted product with reservation price of $30 and the other less than its reservation value. b. Now suppose the production of each good entails a marginal of $35. How does this change your answers to (a)? Why is the optimal strategy now different? Bundled Price 1 Price 2 Price Profit Sell Separately Pure Bundling Mixed Bundling Notice that in the separate pricing, only customer types A and D purchases goods 2 and 1 respectively, so profit is (90-35)+(90-35)=110). In the pure bundling profit is (100-70)x4=120. Finally, in the mixed pricing profit is ( )+(100-70)+(100-70)+( )= Thus, mixed bundling is the best strategy. 4. Reebok produces and sells running shoes. It faces a market demand schedule P = Q s, where Q s is the number of pairs of shoes sold (in thousands) and P is the price in dollars per thousand pairs of shoes. Production of each pair of shoes requires 1 square meter of leather. The leather is shaped and cut by the Form Division of Reebok. The cost function for leather is T C L = 1 + Q L + 0.5Q 2 L, where Q L is the quantity of leather (in thousands of square meter) produced. The cost function for running shoes is (excluding leather) T C s = 2Q s. a. What is the optimal transfer price? Each shoe requires 1 square meter leather, Q s = Q L, so max π s Q s = (11 1.5Q s )Q s 2Q s (1 + Q s + 0.5Q 2 s) π s Q s = 11 3Q s 2 1 Q s = 0 (3) The solution of equation (3) yields Q s = 2. Thus, Q L = 2 sq meter and transfer price is $3 since MC(Q L = 2) = = $3. b. Leather can be bought and sold in a competitive market at the price of P = 1.5.

4 The transfer price is now $1.5, and the optimal number of shoes can be derived from max Q s π s = (11 1.5Q s )Q s 2Q s 1.5Q s π s Q s = 11 3Q s = 0 Q s = 2.5 and 2 square meter of it is purchased from the outside market. c. Now suppose that the leather is unique and of extremely high quality. Therefore the Form Division may act as a monopoly supplier to the outside market as well as a supplier to the downstream division. Suppose the outside demand for leather is given by P = 32 Q L. What is the optimal transfer price for the use of leather by the downstream division? At what price, if any, should leather be sold to the outside market? What quantity, if any, should leather be sold to the outside market? This time, number of shoe sales depends on the imperfectly competitive leather market. In particular, upstream division s pricing on leather determines the overall profit. Thus, the choice variables would be inside sale of leather (Q s = Q I L ) and outside sale of leather (QO L ): max π = (11 1.5Q I L)Q I Q I L 2Q I L + (32 Q O L )Q O L (1 + Q I L + Q O L + 0.5(Q I L + Q O L ) 2 ) L,QO L π Q I L π Q O L = 11 3Q I L 2 1 Q I L Q O L = 0 (4) = 32 2Q O L 1 Q I L Q O L = 0 (5) Solving (4) and (5) together, Q I L < 0 and QO L = sq meter. Thus, price is P = = $ Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C 1 = 30Q 1 and C 2 = 30Q 2 where Q 1 is the output of Firm 1 and Q 2 is the output of Firm 2. Price is determined by the following demand curve: where Q = Q 1 + Q 2. P = 150 Q a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.

5 Write profit functions for firm 1 and 2 respectively, max π 1 Q 1 = (150 Q 1 Q 2 )Q 1 30Q 1 π 1 Q 1 = 150 Q 1 Q 2 Q 2 30 = 0 (6) max π 2 Q 2 = (150 Q 1 Q 2 )Q 2 30Q 2 π 2 Q 2 = 150 Q 1 Q 2 Q 1 30 = 0 (7) Now, solve equations (6) and (7) together and find Q 1 = Q 2=40, P = = 70 and π 1 = π 2 = b. Suppose the two firms for a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm s profit. Now, Q 1 = Q 2 = Q and max π Q = (150 Q)Q 30Q π Q = 150 2Q 30 = 0 Q = 60, and price is P = = $90 and π = 90(60) 30(60) = $3600. Thus, Q 1 = Q 2=30, and π 1 = π 2 = By cooperating, both firms are better off. c. Suppose Firm 1 were the only firm in the industry. How would the market output and Firm s profit differ from that found in part (b) above? If firm 1 is the only firm, it would be same solution as in part b but Q 1 = 60 and π 1 = $ EGO decides to use peak-load pricing for electricity during the summer months. During the week between the hours of 8:00 a.m. and 5:00 p.m., the demand for electricity is high; during the off hours the demand is low. The respective demand functions can be given as: Q H = 400 5P (high demand) Q L = 50 2P (low demand) where P is the unit price of electricity per hour and Q is amount of electricity used each hour. The total cost of providing electricity is given by: T C = Q 2. a. What is the firms marginal cost function, and what does this equation imply about EGOs ability to provide electricity? Since Marginal cost is linear function of Q (MC = 4Q), more demand will increase cost of providing electricity service.

6 b. In order to divert electricity use to off-house, EGOs management uses peak-load pricing. Calculate the appropriate prices charged and determine the amount of electricity used for each time period. The optimization rule argues that we have to produce where MR = MC: MR H = Q = 4Q = MC Q H = 400 (P eakload) 22 MR L = 25 Q = 4Q = MC Q L = 5 (Off P eakload) and P H = 80 (1/5) (400/22) = $76.36 and P L = 25 (1/2)5 = $22.5. c. Calculate the own-price demand elasticity in each market. Do the numbers make sense in the context of this problem? ɛ H = d Q P d P Q = = 21 and ɛ L = = 9 5 These elasticities do not make sense because consumers who use the service at the peak load period are less price elastic but we found contradictory results. 7. In the inland waterways shipping industry, bulk carriers (barges) are chartered on an annual basis to haul grain, oil, and other bulk commodities. As far as the shippers are concerned, the service provided by barges of any given class are homogenous products. Industry demand for carriers vary over time, depending on grain and oil movements. At present, industry demand is estimated as: Q = P The industry consists of one large firm, Mississippi Barge Transport Company (MBT), and ten smaller firms of roughly equal size. MBT is the industry leader with regard to pricing decisions, and its marginal cost curve is given by: MC L = $ $6Q The following firms marginal cost curve, derived by summing the MC curves of the ten follower firms, is given by: MC F = $ $4Q a. Under this market structure, what price will MBT establish, and what will be its output at this price? First, we have to derive the supply function of the follower firms and subtract this amount from industry demand to get dominant firm s demand function. From MC relation of the follower firms, we can write down their supply function as MC = Q Q = MC since MC=P in the perfectly competitive firms,

7 the supply function is written as Q = 0.25P When we subtract this from industry demand: Q L = Q Q F = P (0.25P 11000) = P (=Demand for Leader). Thus, to derive marginal revenue for leader, we will write demand curve as inverse demand curve: P = 51000/0.45 (1/0.45)Q and MR L = Q = 6 = Q Q L = and P = 51000/0.45 (1/0.45) = $ b. How many units of output will the following firms supply? From P = MC,$ = Q Q F = c. Check whether industry supply equals to industry demand?