Problem Set 1a - Oligopoly

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1 Advaced Idustrial Ecoomics Sprig 2014 Joha Steek 6 may 2014 Problem Set 1a - Oligopoly 1

2 Table of Cotets 2 Price Competitio Courot Oligopoly with Homogeous Goods ad Differet Costs Bertrad Duopoly with Homogeous Goods ad Differet Costs Courot Oligopoly with Differetiated Goods Bertrad Oligopoly with Differetiated Goods * Comparig Courot ad Bertad... 7 * = Oly this questio is discussed i class 2

3 2 Price Competitio This problem set gives you some experiece i workig with formal oligopoly models. We will use the liear Bertrad ad the Courot models ad allow the firms to have differet margial costs. We will also cosider the case whe the firms produce differetiated products. Istead of workig with localized competitio a la Hotellig, where goods are described by their characteristics ad cosumers have differet prefereces over characteristics, we will here assume that all cosumers are idetical but that they love variety, ie that they like to cosume a little of may differet variats of the product rather tha a lot of just oe variat. 2.1 Courot Oligopoly with Homogeous Goods ad Differet Costs This exercise ivestigates whether productio costs are miimized i a Courot oligopoly market. Cosider a homogeous goods market with firms competig a la Courot. Let be firm i s quatity ad Q = q i the aggregate quatity. The price is give by the iverse demad fuctio, which is liear ad give by p = α Q. Firm i s margial cost is costat ad deoted. Let c = 1 c i be the average cost i the market. 1. Compute the equilibrium price. 2. How does the equilibrium price deped o the distributio of costs (mea, variace et.c.)? 3. Compute firm i s quatity. 4. Uder what coditios will firm i produce a positive output? 5. Keepig the umber of firms i the market fixed, what determies firm i s market share s i = q i Q? 6. Are productio costs miimized i the market? 3

4 2.2 Bertrad Duopoly with Homogeous Goods ad Differet Costs This exercise ivestigates whether productio costs are miimized i a Bertrad oligopoly market. Cosider a homogeous goods market with 2 firms competig a la Bertrad. Let be firm i s price. Assume that firm 1 s cost is ad that firm 2 s cost is higher c H > c L. Market demad is give by D( p). If the firms charge the same price, half of the cosumers buy from each firm. Let p 1 m deote the price that firm 1 would choose if it were a moopolist ad p 2 m the price that firm 2 would charge if it were a moopolist. Note that p 2 m > p 1 m sice firm 2 s cost is higher. 1. Compute firm 2 s best- reply fuctio. Display it i a diagram with p 1 o the x- axis ad p 2 o the y- axis. Hit: What price would firm 2 like to charge if firm 1 charges a price above p m 2? What price would firm 2 like to charge if firm 1 charges a price at or below c H? What price would firm 2 like to charge if firm 1 charges a price betwee firm 2 s moopoly price ad cost? 1 2. Compute firm 1 s best- reply fuctio. Display it i the same diagram as above. 3. Fid the Nash equilibrium. Hit: Use the diagram. 4. Is there ay market power ad, if so, what determies market power? 5. Are productio costs miimized i the market? 6. Explai the differece to Courot. 2.3 Courot Oligopoly with Differetiated Goods Cosider a differetiated goods market with firms competig a la Courot. The cosumer s utility is quadratic ad give by 1 To avoid some techical complicatios you should go back to see how we solved the Bertrad model with homogeous goods, but with idetical costs. Oe problem is that sometimes several differet prices may all give the highest possible profit. You may the select oe of these prices for the best- reply fuctio. Aother problem is that sometimes firms wat to udercut their rival s price with the smallest possible amout. You may the view p j ε as the best reply (despite the fact that ε could be made arbitrarily small). 4

5 U = α q i q i σ q i q j + y where deotes the quatity of firm i s product that the cosumer cosumes, is the quatity cosumed of a outside good (also servig as a umeraire) ad σ [ 0,1) is the degree of substitutio betwee the q- goods. Let be firm i s price. All firms have the same costat margial cost c, which is assumed to be lower tha the demad itercept, i.e. c < α. Firm i s iverse demad is the give by: p i = α q i σ q j. 1. What does σ = 0 sigify? What does σ = 1 sigify? 2. Compute the equilibrium prices. If you wish, you may assume that the equilibrium is symmetric, i.e. that the firms produce the same quatity q i = q. But, if you ca, try to solve the problem without assumig symmetry. 3. What determies the stregth of competitio? Hit: How does price deped o the umber of firms ad the degree of substitutability? 2.4 Bertrad Oligopoly with Differetiated Goods * Cosider a differetiated goods market with firms competig a la Bertrad. The cosumer s utility is quadratic ad give by U = α q i q i σ q i q j + y where deotes the quatity of firm i s product that the cosumer cosumes, is the quatity cosumed of a outside good (also servig as a umeraire) ad σ [ 0,1) is the degree of substitutio betwee the q- goods. Let be firm i s price. The idirect demad fuctios, used i the Courot model above, are easy to derive. To derive the demad fuctios, oe has to ivert the system of liear idirect demad fuctio. Oe may the show that firm i s demad is liear ad give by 5

6 q i = A Bp i + G 1 1 p j where ( 1 σ )α A = 1+ ( 2)σ ( 1)σ ( 2)σ B = 1+ ( 2)σ ( 1)σ 0 2. ( 1)σ G = 1+ ( 2)σ ( 1)σ 0 2 (If you wish, you may try to show this yourself!) Note that if the goods would be perfect substitutes, ie if σ = 1, the the above expressios would etail divisio by zero, 1+ ( 2)σ ( 1)σ 2 = 0, which is ot allowed. The reaso is that i this case we caot solve for separate demad fuctios for the firms. That is, the demad for the idividual goods caot be determied. Note also that if firm 1 would charge a very high price p 1 i compariso to the price charged by the competitors, firm 1 would ot be able to sell aythig, i.e. q 1 = 0. The demad fuctio seems to suggest, however, that the quatity could be egative. But this is just because we have take a short- cut here, simply assumig that the two prices will ot be too differet. Luckily, this simplificatio turs out to be okay i the aalysis we will do here. All firms have the same costat margial cost c, which is assumed to be lower tha the demad itercept, i.e. c < α. 1. Compute the equilibrium prices (cosiderig A, B ad G as parameters). You may assume the equilibrium to be symmetric, i.e. that all firms charge the same price ( p i = p ) i equilibrium. 2. Compute the equilibrium price as a fuctio of the deep parameters, i.e. α,σ,. 3. What determies the stregth of competitio (market power) i the market? 6

7 a. How does price deped o the umber of firms? What happes to price if the umber of firms is oe, two or teds to ifiity? b. How does price deped o the degree of substitutio? What happes to price if goods are urelated σ = 0 of almost perfect substitutes σ 1? 2.5 Comparig Courot ad Bertad Compare the Courot equilibrium price ad the Bertrad equilibrium price, whe firms produce differetiated products but have the same margial cost. Which is higher? 7

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