PSET 6 Answers - Econ 100A – Microeconomics ...

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Unformatted text preview: Econ 100A – Microeconomics Professor Reynolds Department of Economics University of California, Berkeley Problem Set 6 – Suggested Solutions 1. Two firms compete for consumers who have the aggregate demand curve x=100 2p. Both firms have constant marginal cost, with MC1=1 and the other firm has MC2 equal to some constant c>1. a. Using Bertrand price competition, illustrate both firms’ best response functions & indicate what the outcome is for each firms’ production & profit. P2 c 1 1 c p1 Firm 1’s response function is in red, Firm 2’s is in blue and the dotted line represents where p1=p2. The equilibrium is where the two response functions intersect. As we can see from the illustration, Firm 1 will charge a price just below c and Firm 2 will choose c. Since Firm 2 cannot price below its MC of c, it will not be able to compete with Firm 1’s lower price and will be driven out of business. In the long ­run, Firm 2 exits the market and Firm 1 will be a monopoly. But Firm 1 will not be able to charge the monopoly price – it will continue to charge a price just below c. The threat of Firm 2 re ­ entering the market will probably limit the profits of Firm 1. We would expect Firm 1 to have positive profits, but smaller profits than if it were a monopoly. (Unless c>monopoly price, in which case the threat of Firm 2 re ­entering the market is non ­binding.) b. Using Cournot quantity competition, illustrate both firms’ best response functions. For firm 1: The Residual Demand for firm 1 is: Firm 1’s Marginal Revenue line has twice the slope of the inverse, residual demand line: Firm 1 sets MR equal to MC, which is 1 for them. MR = MC = 1 So Firm 1’s reaction function is: For firm 2: Similarly, the Marginal Revenue line can be derived from the Residual Demand for firm 2 is: Firm 2 sets MR equal to MC, which is c for them. MR = MC = c Looking at the equations, it is clear that Firm 2 will always produce less than Firm 1 (since c>1). The greater c is, the more that Firm 1 will produce relative to Firm 2. Both will have positive profits, though. Graphically, the result is this: x2 98 49 5 0 -c 49 100-2 c 98 x1 c. Suppose firm 2 is the “smarter” firm and is able to tell what firm 1’s best response function is. Using Stackleberg competition, what is the highest value of c such that firm two has more profits than firm one? Firm 2 is the lead firm and firm 1 is the follower. So firm 2 will decide how much to produce by plugging firm 1’s reaction function into its residual demand function. is firm 1’s reaction function from Part B. Substituting firm 1’s reaction function into firm 2’s residual demand: ! To profit maximize, firm 2 will set MR = MC. So, we have to find MR. First, find the inverse demand "#!$%#&'(!)*+')',-.!&'%)!/!0'11!2-(!34!5!367!8#.!0-!9*:-!(#!&';<!347!='%2(.!&';<!(9-!';:-%2-!<-)*;<! function: &>;?('#;@! ! Then find the MR function and set it equal to MC = c. "9-;!&';<!(9-!34!&>;?('#;!*;<!2-(!'(!-A>*1!(#!36!5!?7!! ! ! ! Given firm 2’s response function, this means that firm 1’s response function is: B':-;!&'%)!/C2!%-2$#;2-!&>;?('#;.!(9'2!)-*;2!(9*(!&'%)!DC2!%-2$#;2-!&>;?('#;!'2@! ! ! To answer the second question, we need to first find p in terms of c. "#!*;20-%!(9-!2-?#;<!A>-2('#;.!0-!;--<!(#!&'%2(!&';<!$!';!(-%)2!#&!?7! ! So: 8#@! xx==x1 ++ x == 23.5 + c + 51 " 2c x1 x 2 2 23.5 + c + 51 − 2c 74.5 −" c== 100 " 2 p ! 74.5 c 100 − 2 p pp== 12.75 + 0.5c 12.75 + 0.5c ! Now we can write the profit equations for each firm only in terms of c. E#0!0-!?*;!0%'(-!(9-!$%#&'(!-A>*('#;2!&#%!-*?9!&'%)!#;1F!';!(-%)2!#&!?7! ! € ! ! ! ! "#!&';<!(9-!)*+')>)!?!09-%-!&'%)!/!'2!)#%-!$%#&'(*G1-.!0-!?*;!H>2(!2-(I>$!(9-!';-A>*1'(F #(9-%! To find the maximum c where firm 2 is more profitable, we can just set ­up the inequality other *;<!2#1:-!&#%!?!J;#(';K!(9*(!$%#&'(2!)>2(!G-!$#2'(':-L7!!"9-!%-2>1(!'2!(9*(!?!5!MN7/!'2!(9-!)*+')>)!:*1>-! and solve for c (noting that profits must be positive). The result is that c = $5.2 is the maximum value 09-%-!='%)!DC2!$%#&'(2!*%-!K%-*(-%!#%!-A>*1!(#!='%)!DC27!! where Firm 1’s profits are greater or equal to Firm 1’s. ! !" #$%&'()%*!+",*-..*/-&0)*1&23*45*-6*7*1&23*85*9*:*9*;*7%.2<*=12&*!+",8-6*1(>?*/@(*->?*/@AB* 2. E*xercise 26.4 all parts from A; a, b from B; & g & h below (for 26.4Ba, find p*i and p*j) 8C)(>%))*->?*D2.('E*4//.('-0(2>F*G%&:%&)*->?*4>0(0&C)0*D2.('E*(>*H%.-0%?*D&2?C'0*G-&I%0)F*J%*>2<* Business and Policy Application: Mergers and (>02*-*)(>:.%*1(&3*(>*)C';*%>K(&2>3%>0)*L*->?*0;%*.%K%.*21* (>K%)0(:-0%*0;%*(>'%>0(K%)*12&*1(&3)*02*3%&:%* Antitrust Policy in Related Product Markets: We now investigate the incentives for firms to merge into a single firm in such environments — and the level of '2>'%&>*0;-0*0;()*3(:;0*&-()%*-32>:*->0(0&C)0*&%:C.-02&)"* concern that this might raise among antitrust regulators. * 4F*M>%*<-E*02*0;(>I*-72C0*1(&3)*0;-0*'23/%0%*(>*&%.-0%?*3-&I%0)*()*02*0;(>I*21*0;%*%$0%&>-.(0E*0;%E*%-';* A: One way to think about firms that compete in related markets is to think of the externality they each (3/2)%*2>*0;%*20;%&*-)*0;%E*)%0*/&('%"*N2&*(>)0->'%6*(1*0;%*0<2*1(&3)*/&2?C'%*&%.-0(K%.E*)C7)0(0C0-7.%*:22?)* impose on the other as they set price. For instance, if the two firms produce relatively substitutable goods =-)*?%)'&(7%?*(>*=-B*7%.2<B6*1(&3*O*/&2K(?%)*-*/2)(0(K%*%$0%&>-.(0E*02*1(&3*!*<;%>*(0*&-()%)*/O*7%'-C)%*(0* (as described in (a) below), firm 1 provides a positive externality to firm 2 when it raises p1 because it &-()%)*1(&3*!P)*?%3->?*<;%>*(0*&-()%)*(0)*2<>*/&('%"* raises firm 2’s demand when it raises its own price. * =-B*QC//2)%*0;-0*0<2*1(&3)*/&2?C'%*:22?)*0;-0*-&%*&%.-0(K%.E*)C7)0(0C0-7.%*(>*0;%*)%>)%*0;-06*<;%>*0;%* (a) Suppose that two firms produce goods that are relatively substitutable in the sense that, when the /&('%*21*2>%*1(&3P)*:22?*:2%)*C/6*0;()*(>'&%-)%)*0;%*?%3->?*12&*0;%*20;%&*1(&3P)*:22?)"*R1*0;%)%*0<2* price of one firm’s good goes up, this increases the demand for the other firm’s goods. If these two 1(&3)*3%&:%?6*<2C.?*E2C*%$/%'0*0;%*&%)C.0(>:*32>2/2.E*1(&3*02*';-&:%*;(:;%&*2&*.2<%&*/&('%)*12&*0;%* firms merged, would you expect the resulting monopoly firm to charge higher or lower prices for the :22?)*/&%K(2C).E*/&2?C'%?*7E*0;%*'23/%0(>:*1(&3)S*=T;(>I*21*0;%*%$0%&>-.(0E*0;-0*()*>20*7%(>:*0-I%>* (>02*-''2C>0*7E*0;%*0<2*1(&3)*-)*0;%E*'23/%0%"B* goods previously produced by the competing firms? (Think of the externality that is not being taken into account by the two firms as they compete.) As firm 1 raises its price, it only considers its own profit and not firm 2’s profit. But as firm 1 raises price, it is raising demand — and thus profit — for firm 2. This is a positive externality that firm 1 is not taking into account — and, as a result, it will set “too low” a price relative to what the firms would do if they could make joint decisions (as they can if they merge into a single monopoly). Thus, if the two firms merge, the price of both goods will increase. (b) Next, suppose that the two firms produce goods that are relatively complementary in the sense that an increase in the price of one firm’s good decreases the demand for the other firm’s good. How is the externality now different? When firm 1 raises price, it now lowers demand (and profit) for firm 2 — thus emitting a negative (rather than a positive) externality that it is not taking into account when it sets its own price. (c) When the two firms in (b) merge, would you now expect price to increase or decrease? Since firm i is not taking into account the damage it does to firm j when it raises price, firm i will raise its price “too high” relative to what the two firms would decide jointly. Thus, as the firms merge, I would expect prices to fall. (d) If you were an antitrust regulator, which merger would you be worried about: The one in (a) or the one in (b)? You would worry about the merger in (a) — the merger of firms that produce relatively substitutable goods. In those cases, we determined that the merged firm will raise price — whereas in the case of complements we determined the merged firm will lower price. In both cases, the firms exercise market power when they merge into a monopoly — but in the case of complements, they are (by merging) eliminating an externality that resulted in too high a price. Anti ­trust regulators worry about mergers resulting in higher prices as firms collude — and would probably not be concerned about mergers that result in a reduction in output price. (e) Suppose that instead the firms were producing goods in unrelated markets (with the price of one firm not affecting the demand for the goods produced by the other firm). What would you expect to happen to price if the two firms merge? If the two markets are unrelated and a change in price in one market has no impact on the demand for goods in the other, then the initial firms were already two separate monopolists. Without any relationship between the markets, a merger does not change this — the merged firm would continue to behave as a monopolist in the individual markets. Thus, in this case, we would not expect price to change. (f ) Why are the positive externalities we encountered in this exercise good for society? The positive externality we encountered was for the case of competing oligopolistic firms in markets that produce relative substitutes. Here, neither firm took into account the “benefit” it creates for the other firm’s profits as it raises price and thereby increases demand for the other firm. Because of this, each firm will set price lower than it would if it were colluding with the other firm — and this is good for consumers (which are the rest of society in this example). B: Suppose we have two firms — firm 1 and 2 — competing on price. The demand for firm i is given by: x i ( pi , p j ) = 1000 − 10 pi + βp j . (a) Calculate the equilibrium price p* as a function of β. € Let’s assume that marginal costs are the same for both firms and equal to some constant, c, as in our other examples. Using the result from your textbook (26.5 on page 1005) or from lecture, we can simply plug in the values for A and alpha, so that: p* = pi = p j = 1000 + 10c 20 − β (b) Suppose that the two firms merged into one firm that now maximized overall profit. Derive the prices for the two goods (in terms of β) that the new monopolist will charge — keeping in mind that the monopolist now solves a single optimization problem to set the two prices. (Given the symmetry of the demands, you € should of course get that the monopolist will charge the same price for both goods). Let’s assume that marginal costs are the same for both firms. The monopolist then solves the problem: max π = ( pi − c )(1000 − 10 pi + βp j ) + ( p j − c )(1000 − 10 p j + βpi ) pi , p j From the two first order conditions we get: € pi = 1000 + 2 βp j + 10c − βc 1000 + 2 βpi + 10c − βc pj = 20 20 and Substituting the latter into the former and solving for p1, we get: pi = € 1000 + 10c − βc β ȹ 1000 + 2 βpi + 10c − βc ȹ + ȹ ȹ € Ⱥ 20 10 ȹ 20 c βc β 2 pi βc β 2c pi = 50 + − + 5β + + − 2 20 100 20 200 c β 2 pi β 2c pi = 50 + + 5 β + − 2 100 200 100 ȹ c β 2c ȹ pi = ȹ 50 + + 5 β − ȹ 100 − β 2 ȹ 2 200 Ⱥ 5000 + 50c + 500 β − .5 β 2c 100 − β 2 (500 + 5c − .5 βc )(10 + β) 500 + 5c − .5 βc pi = = (10 − β)(10 + β) 10 − β pi = Substituting this back into our equation for pj, we get the same for pi. Thus, pi = p j = pm = € g. € 500 + 5c − .5 βc 10 − β Are there any values of β for which the monopolist prices for both goods are smaller than the oligopoly prices? If so, explain how this can be profitable for the monopolist. Yes. By merging, they can internalize the externality that keeps them from jointly attaining the maximum profit when they are in the oligopoly setting — and this is true whether the firms are creating positive or negative externalities for one another. Recall that, β represents the responsiveness of demand for one good relative to demand for the other good. As a result, they will price “too low” relative to the monopoly outcome in the case of substitutes and “too high” in the case of complements. This implies that price will rise as a result of a merger if the two goods are substitutes — and it will fall when they are complements. β > 0 represents cases where the two firms produce relative substitutes and with β < 0 implying they produce relative complements. To see this algebraically, let’s explore where monopolist prices for the goods are smaller than oligopoly prices by finding where the answer in Part B. b. is smaller than in Part B. a. Simply set up the inequality: 500 + 5c − .5 βc 1000 + 10c < 10 − β 20 − β Solving for β: 10000 + 100c − 10 βc − 500 β − 5 βc + .5 β 2c < 10000 − 1000 β + 100c − 10cβ € 500 β − 5 βc + .5 β 2c < 0 β(500 − 5c + .5 βc ) < 0 € It is clear now that when β < 0, then the monopolist’s prices will be smaller than the oligopolists’ prices. h. In terms of β, what are the socially optimal prices? The social optimum occurs where MC = MB. That means price, which represents social marginal benefit in the demand function, is equal to the marginal cost of c. If we set our initial definition of price for the two firms equal to c and then solve to get c only in terms of β: 1000 + 10c 20 − β 10c 1000 p* = c − = 20 − β 20 − β 1000 p* = c = 10 − β p* = c = € We find that when the marginal cost is equal to 1000/(10 ­β), then the firms will set the optimal prices for both goods without any intervention. 3. I am going on vacation in the summer and I would like to hire a student to run my cookie baking business. Profits p are based on the student’s effort e and equal to 2e + . The student’s cost of effort is e2. If I do not hire the student, the student will work at as dorm clerk, exert zero effort, and make $3. Help me determine the salary scheme: s=a+b p. a. What is the student’s participation constraint? The student’s participation constraint is the condition that the student will only take the job if the expected gain from it exceeds the expected opportunity cost. In this case, the expected gain is the student’s salary minus the cost of their effort and the opportunity cost is earning $3 and expending 0 effort as a dorm clerk. Therefore, the student’s participation constraint is: E ( s) − c (e) ≥ 3 E (a + b * π ) − e 2 ≥ 3 E ( a + b(2e + ε )) − e 2 ≥ 3 a + 2be − e 2 ≥ 3 a ≥ 3 − 2be + e 2 b. What is the incentive constraint? € The incentive constraint is the condition that the student will choose e to suit herself rather the owner, who cannot observe e. So, assuming the student takes the contract, she will choose an e that maximizes her own salary. Thus, she will max.: max E ( a + b(2e + ε )) − e 2 e max a + 2be − e 2 e Taking the first order condition, setting it equal to 0 and solving for e, we get: 2b − 2e = 0 € b=e c. What is my optimization problem? My goal is to maximize my surplus. Since I’m sharing my profits, the surplus is equal to the profits € minus the compensation I give the student. As result, my optimization is: max E (2e + ε − s) a ,b max E (2e + ε − ( a + b(2e + ε )) a ,b max 2e − a − 2be a ,b Plugging in that, to be compatible, e(b) = b and a max(1 − b)2b − ( 3 − 2b * b − b 2 ) a ,b Taking the first order conditions : 2 − 4 b + 4 b − 2b = 0 b =1 d. € What will I pay the student & how much effort will be exerted? You will pay the student $0 and the student will exert 0 effort because student won’t take the job. Taking our answer in c and substituting back into the incentive compatible constraint, the optimal level of effort is 1. From there, looking at the participation constraint, the student will only take the contract if a = $2. I’m not earning enough profit to pay them $2 + $2, since my expected profit is only $2. e. What will be my profits? Do you think I should skip my vacation to stay home & monitor the student? (My vacation would provide me with $7 net utility.) N/A. Since you will never hirer the student, this question isn’t relevant. In general, though, you should recognize that by staying you might be able to move from the second ­best effort to first ­best by monitoring the student. f. What is the socially efficient level of effort? The socially efficient level of effort is 0, since the student can earn $3 by taking the job in the dorm and exerting 0 effort. If the reservation utility was 0 and the student took the job, then it would be where the marginal cost of their effort equals the marginal social benefit of their effort. The socially efficient level of effort, assuming the student took the job, would be where the marginal cost of their effort equals the marginal social benefit of their effort. Setting marginal cost equal to marginal benefit: SMC = SMB dC dπ = de de e =2 Note that the result here would necessarily be $0 economic profit since the total cost of the effort ($4) would equal the expected value of the total benefit ($4). € ...
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This note was uploaded on 09/11/2011 for the course ECON 100A taught by Professor Woroch during the Spring '08 term at University of California, Berkeley.

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