Solutions to problem set 11 (due Tuesday, April 29 1st, before class)

Transcription

1 Econ 30 Intermediate Microeconomics Prof. Marek Weretka Solutions to problem set (due Tuesday, April 9 st, before class) Problem (Oligopolistic Industry) a) Big four index (concentration ratio) of the beer industry in the USA is given by b) HHI index is given by CR = 36:8% + 9:% + 8:5% + 9:% = 83:6% HHI = = 7 c) Such industry is highly concentrated as HHI > 800 c)trade Commission is likely to block the merger to protect the consumers from the monopolistic behavior of the beer producers. Problem (Aircraft industry) a) Pro t function is given by and hence it has a shape of a parabola B (y B ) = (00 y A y B ) y B 0y B B (y B ) = 80y B (y B ) b) o, given y A = 00; the optimal production is y B = 40 jets. The best response of Boeing to Aribus production is y A R B c) Best response function is. R B (y A ) = 90 y A d) The best response function for Airbus is symmetric, and is given by R A (y B ) = 90 y B

2 e) The optimal production solves the two best response functions simultaneously y = y A +y B = 0 y A = y B = 60 p (0) = 00 0 = 80 and the individual pro ts in equilibrium are A = B = = = 3600 f) The deadweight loss (DWL) associated with oligopolistic trading by the two rms is DW L = (80 0) (80 0) = = = 800 g) If A and B form a cartel selling y jets, the pro t function will become Cartel = (00 y)y 0y and the optimal level of production is and the cartel price is y = 90 p (90) = = 0 Pro t per rm in cartel is A = B = Cartel = = = 4050 Therefore it is bene cial to form a cartel for both rms. h) A DWL in the case of collusion is DW L Cartel = (0 0) (80 90) = 4050 > DW LDuopoly = 800 and it is signi cantly greater than then one for duopoly. This is because collusion increases the market power of the two traders. i) If the interactions are only in a short run, cartel is not sustainable as each rm has incentives to "cheat" the other rm by unilaterally increasing a production above 45 jets. This is, because given the other rm produces 45 jets, producing more leads to the pro t that is even higher than in the case of the cartel. When interactions are repeated, the loss of reputation associated with cheating "today" makes the cooperation of the two rms tomorrow impossible. Such cost of cheating, might make the cooperation today sustainable. Problem 3 (Accounting & Audit services in the USA) a) Monopoly pro t is Monopoly = (000 y)y y i and optimal production is and the monopoly price is y = 495 p = = 505 Competitive rms supply can be found from the condition p = MC and hence p Competitive = 0 y Competitive = = 990

3 b) c) Consider rm i (you should think of i as a number identifying one rm) The pro t of such rm is X i = (000 y i y j )y i 0y i where symbol j6=i j6=i X y j = y + y + ::: + y i + y i+ + :::y denotes the level of production by all other rms but i: The optimality condition gives X 000 y i y j 0 = 0 j6=i or y i = 990 P j6=i yj Firms are symmetric therefore y i = y j for any j and hence we can write it as so the individual production is y i = and the price and aggregate level of production is 990 ( ) yj y i = y = y i = p = 000 y i = Finally the deadweight loss is DW L = = (990) ( ) d) The values of aggregate production y and p for = ; 5 and 0 are 5 0 y p

4 e) The price goes converges to a competitive one as the number of rms goes up. This is because and hence lim! + = lim p = 000 lim 990 = = 0!! + f) The limit production is lim y = lim 990 = 990 = 990!! + Hence output also converges to its competitive limit. g) Since therefore (990) lim DW L = lim!! hence DWL disappears. lim! + = 0 = (990) + lim! = (990) (0) = 0 + Problem 4 (Externality) a) Positive: Participation in a cellular network R&D by Google egative eighbor singing in the bathroom Too many cars in YC (congestion) b) Without regulation the outcome is likely to be Pareto ine cient. In case of positive externality in market interactions we will observe too little activity, as the positive e ect on others utility/pro t is not internalized. In case of negative externality we observe too much activity. c) In case of positive externality one could introduce a subsidy encouraging the action, while in case of negative one we could tax the activity. Problem 5 (Positive externality) a) Positive b) Given the price of dynamite is equal to one, a pro t function of the dynamite producer is given by d = d T C d (d; x) 4

5 hence The two secrets of happiness ( rst order conditions) are given d = = 0 = = 0 = ) d = = 0 ) x = The rst condition implies that the optimal level of dynamite production is and the optimal level of intensity is = = (x ) = 0 ) x = The maximal level of pro t is c) The marginal bene t is given by d = d T C d (d; x) = = = (x ) Given optimal level of x =, such bene t is equal to zero. Intuitively our produces increases the level of intensity x as long as the marginal revenue is positive as it increases its pro t. It stops when the last unit x does not add to the pro t. otice that for x = the parabola in the pro t function attains minimum. d) The farmer s pro t function is given by t = t t + t t = t t and the secret of happiness is and the maximal pro t of the farmer t = t = 0 ) t = e) The joint pro t is t = = d + t = f) Find the Pareto e cient level of production of d; t and use of nitrogen x: Compare these values to the ones obtained in points b and d: 5

6 Joint pro t is given = d d + (x ) + t t + t xt The three conditions for Pareto e ciency imply that From the second equation we get t = = 0 ) d = 0 ) 4 x + t = = 0 ) t + x = 4 and we plug it into the third one + 4 x + x = 0 ) x = 3 and t = In social optimum the intensity of used nitrogen is hither than the market outcome and the production of tomatoes is higher. The reason for that is the producer when choosing the optimal level of nitrogen is ignoring positive e ect on the production of tomatoes. Such e ect is taken into account by social planner. g) The marginal bene t is = (x ) = and it is negative: The reason for that is that the negative marginal bene t for the dynamite producer is compensated by the positive marginal bene t on the production of tomatoes. h) Yes, the market outcome is associated with x that is too small from the social point of view. 6