Consumer surplus is zero and the outcome is Pareto efficient since there is no deadweight loss.

Transcription

1 Problem Set : Solutions ECO 30: Intermediate Microeconomics Prof. Marek Weretka Problem (Price Discrimination) (a) If Microsoft can perfectl price discriminate, its profit (and the producer surplus P S) is the total surplus area (accounting for the fixed costs of F =,000) that we found in Problem (a): P S = π = T S = F = 5,000,000 = 4,000. Consumer surplus is zero and the outcome is Pareto efficient since there is no deadweight loss. (b) To find the aggregate demand with these two segments, we sum over quantit. (ote that if the firm does not price discriminate, it must charge the same price so p = p I = p F.) AGG (p) = I (p) + F (p) = ( p) + (50 p) = 00 p 5 which gives us the inverse aggregate demand we had in Problem : p AGG () = 00. (c) With third-degree price discrimination, Microsoft charges price p I to the individual buers market segment and price p F to the firm buers. We can simpl solve the profit maximizing problem for each segment separatel. Individual Buers: The inverse demand curve in this market is p I () = I and so the marginal revenue in the individual buers market is MR I () = I. Profit maximizing condition MR( I ) = MC( I ) = I = 0 = I = 5.

2 Price is then p I (5) = 3 4. Total revenue in the individual segment is then T RI = 5 4 =,78 4, and consumer surplus is CSI = ( ) 5 = Firm Buers: The inverse demand curve in this market is p F () = 50 5 F and so the marginal revenue in the firm buers market is MR F () = 50 0 F. Profit maximizing condition MR( F ) = MC( F ) = 50 0 F = 0 = F = 5. Price is then p F (5) = 5. Total revenue in the firm segment is then T R F = 5 5 = 3,5, and consumer surplus is CS I = (50 5) 5 =,56. To find Microsoft s total profit (which corresponds to producer surplus here), we add the total revenue from each segment and subtract the total cost of F =,000: π = T R I T R F T C =, ,5,000 = 3,906 4 (It would also be fine to assume that the fixed cost of F =,000 is incurred in each market, giving instead profit π =,906 4.) (d) Here we compare consumer and producer surplus in the three scenarios. Uniform pricing: CS=,50 PS=,500 Perfect price discrimination (first-degree): CS=0 PS=4,000 Third-degree price discrimination: CS= ,56 =,953 8 PS=3,906 4 (or,906 4 ) otice that the producer surplus is the greatest with first-degree price discrimination and lowest with uniform pricing.

3 Problem (Demand Elasticit) (a) The inverse demand curve, p() =, is shown below: p() "(0) = elastic "(.5) = inelastic "() =0 (b) For this demand function, we have ε = (p) p() = ( ), so ε(0) = 0 0 = ε(.5) =.5.5 = ε() = = 0 These points are marked on the graph in part (a). (c) When T C() = c, we have constant marginal cost of MC() = T C () = c. Total revenue is T R() = p() and so marginal revenue is MR() = T R () = p() + p () (product rule). Since the profit-maximizing firm chooses output such that MR() = MC(), we have MR() = MC() = p() + p () = c. Changing notation so that p() = p and p () = dp, we have d p + dp d = c = p + p Recognizing that /ε = dp, we have d p c = p + ]. ε ] dp = c. d 3

4 Then since c > 0, we must have p + ε] > 0. Since p > 0, we can solve for ε to get p + ] ε = + ε > 0 = ε > = ε < = ε > and so the optimal output is on the elastic portion of the demand curve for an MC = c > 0. p() elastic inelastic MC > 0 for which MR = MC As we can see in the figure, whenever MC > 0, output occurs where MR=MC which is necessaril on the elastic region of the demand curve. (d) Equating MR() = MC(), we get p + ] = MC = p = ε ] so the price markup over marginal cost is. Problem 3 (Oligopolistic Industr) +/ε + /ε ] MC (a) The big four concentration ratio of the light beer industr in the U.S. is (b) The HHI is given b CR = 36.8% + 9.% + 8.5% + 9.% = 83.6%. HHI = ,7. (c) This industr can be considered highl concentrated since HHI >,800. 4

5 (d) The FTC would likel attempt to block the merger as a monopol would result from it. Problem 4 (Aircraft Industr) (a) Boeing s profit function is π B ( B ) = T R T C = (00 A B ) B 0 B and given that Airbus is production at A = 00 this reduces to π B ( B ) = 80 B ( B ) B ( B ) 40 B This attains an optimum at B = 40 (using first-order condition π B ( B) = 0). (b) o, we found in part (a) that Boeing s best response to Airbus producing A = 00 is B = 40 (not B = 00). For A = 50, Boing s profit-maximizing output is B = 65; when A = 0, we have B = 90. (You can verif this b going through the steps in part (a).) (c) The best response function for Boeing, which tells us their optimal output for an level of production Airbus might choose, we take the first-order condition of Boeing s profit function which is π B ( B ) = (00 A B ) B 0 B, π B B = 0 = 00 A B 0 = 0. Solving this for B, we find B = 80 A, which is what gives us the best response function, so R B ( A ) = 80 A. 5

6 B A This is shown in the graph below: B R A ( B ) E R B ( A ) (d) The best response function for Airbus is smmetric (the have the same cost functions). To find Airbus best response function in general though we take the same steps as we did to find Boeing s. We take the profit function A and the first-order condition π A ( A ) = (00 A B ) A 0 A π A A = 0 = 00 A B 0 = 0. Solving this for A, we find A = 80 B, which is what gives us the best response function R A ( B ) = 80 B. This is shown on the graph above along with Boeing s best response function R B ( A ). 6

7 (e) The Cournot-ash equilibrium level of output produced b each firm solves the best response functions simultaneousl. You can set A = R B ( A ) and B = R A ( B ) to get A = B = 60 so total output in the market is = A + B = 0. Turning to the inverse demand curve, this output is associated with a market price of and the profits for each firm are p(0) = 00 0 = 80 π A = π B = = 3,600. (f) To find the DWL, we first need to find the Pareto efficient output (where p = MC): In this problem MC = 0 and p = MC = 00 = 0 = = 80. The DWL then is the triangular area DWL Duopol = (80 0)(80 0) =,800. (g) If the two firms were to form a cartel, the would determine output as a monopolist, maximizing cartel profit π Cartel = (00 ) 0. Using the first order condition we get a profit-maximizing output of = 90 and a cartel price of p(90) = = 0. Each of them produce half of the optimal output, so A = B = 45, and the split the profits: π A = π B = π Cartel = = 4,050. This gives a higher level of profit than what we found with Cournot competition, so the cartel would be beneficial to both firms. (h) The deadweight loss here is now the triangular area DWL Cartel = (0 0)(80 90) = 4,050 > DWLDuopol. (i) If the interactions are onl in a short run, cartel is not sustainable as each firm has incentives to cheat the other firm b unilaterall increasing a production above 45 jets; ou can see this b checking what their best response is to the other firm producing 45. This 7

8 is, because given the other firm produces 45 jets, producing more leads to the firm that is even higher than in the case of the cartel. When interactions are repeated however, the loss of reputation associated with cheating toda ma make the cooperation of the two firms tomorrow possible. If the firm cheats toda, it gains some profit toda but loses out on all of the cartel profit it could make in the future if ti cooperates (which is alwas higher than the duopol profit it gets if it cheats). So such costs of cheating (i.e., the loss of future cartel profits), might make the cooperation sustainable. Problem 5 (Accounting and Audit Services in the U.S.) (a) Monolop: Monopol profit is π Monopol = (,000 ) 0 and taking the first-order condition we get optimal output = 495 and price p(495) =, = 505. Perfect Competition: With perfect competition, we will have that p = MC = 0 and so output is (turning to demand function (p) = 000 p) then (0) = 990 (b) The two points from part (a) are shown below along the inverse demand curve: p() 505 Monopol Perfect Competition 0 MC (c) Let j i j represent the output of all of the firms other than firm i. (So that j i j = i + i+ + +.) In Cournot-ash competition, profit for firm i is π i = (,000 i j ) i 0 i. j i 8

9 The first-order optimalit condition gives,000 i j i j 0 = 0 or i = 990 j i j Since the firms are smmetric, i = j and so we can write this as and solving for i we get Aggregate production with firms is i = 990 ( ) i i = = i = This gives price p() =, The DWL will depend on the number of firms in the market and is given b DWL = (,000 ] ] 990) ( ) = 990 ( ) + (d) We have the following aggregate output and prices p for =, 5, and 0: p These are shown on the graph below: (e) As the number of firms becomes large, the market looks more like a perfectl competitive market. Since lim =, we can see that + lim p = lim, ] ( ) =, lim =, () =

10 p() Monopol = 340 =5 75 = 0 Perfect Competition 00 MC p p() 0 MC (f) As increases, we will get that output also converges to its competitive limit: lim = lim 990 = () (g) Again, here we have that as the number of firms becomes ver large, the DWL converges 0

11 to its competitive limit. Since lim = 0, we have that + ( ) 990 lim DWL = lim = 990 (0) = 0. +