Suggested Solutions to Assignment 3 (Optional)

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1 EC 370 A & B Interediate Microeconoic II Intructor: Sharif F. Khan Departent of Econoic Wilfrid Laurier Univerit Winter 008 Suggeted Solution to Aignent 3 (Optional) Total Mark: 90 Proble Solving Quetion Read each part of the quetion ver carefull. Show all the tep of our calculation to get full ark. B3. [5 Mark] Two fir are copeting in an oligopolitic indutr. Fir, the larger of the two fir, i conteplating it capacit trateg, which could be either aggreive or paive. The aggreive trateg involve a large increae in capacit aied at increaing the fir arket hare, while the paive trateg involve no change in the fir capacit. Fir, the aller copetitor, i alo pondering it capacit expanion trateg; it will alo chooe between an aggreive trateg and a paive trateg. The table below how the profit aociated with each pair of choice. Fir Aggreive Paive Fir Aggreive 5, 9 33, 0 Paive 30, 3 36, (a) If both fir decide their trategie iultaneoul, what i the Nah equilibriu (or equilibria)? I there an ixed trateg Nah equilibriu in thi gae? If e, what i the ixed trateg Nah equilibriu (or equilibria)? If Fir chooe aggreive, the bet repone for Fir i to chooe paive. Becaue 30 > 5. If Fir chooe paive, the bet repone for Fir i to chooe paive. Becaue 36 > 33. Thi iplie that paive i a doinant trateg for Fir. However, there i no doinant trateg for Fir in thi gae. Fir will chooe it doinant trateg paive. Fir, knowing fir ha a doinant trateg, will pla it bet repone, aggreive. Thi i the onl Nah equilibriu in the iultaneou-ove gae. There i no ixed trateg Nah equilibriu becaue one of the plaer, Fir, ha a doinant trateg in thi gae. Given that fir i chooing paive with a 00 percent probabilit, the bet repone of fir i to chooe aggreive with a 00 percent probabilit rather than to chooe a randoized trateg over paive and aggreive, vice vera. Page of Page

2 Now aue that Fir can decide firt and can credibl coit to it capacit expanion trateg. (b) Draw the extenive for of thi equential gae. What i the ubgae perfect equilibriu? What i the final outcoe of thi gae? See Figure B3(b) for the extenive for repreentation of thi gae. A hown in Figure B3(b), if Fir can chooe firt, then if it chooe aggreive Fir will chooe paive and Fir will receive 33. If Fir intead chooe paive, then Fir will elect aggreive and Fir will receive a paoff of 30. Therefore, if Fir can ove firt, it doe bet to elect aggreive in which cae Fir will elect it bet repone paive earning Fir a paoff of 33 and Fir a paoff of 0. The Subgae perfect equilibriu: In period Fir will chooe paive if Fir chooe aggreive in period. In period Fir will chooe aggreive if Fir chooe paive in period. In period Fir will chooe aggreive. The final outcoe of thi gae: In period Fir will chooe aggreive and in period Fir will chooe paive earning Fir a paoff of 33 and Fir a paoff of 0. (c) If Fir threaten to pla aggreive if Fir pla aggreive, will it be credible to Fir? If thi threat i not credible to Fir, what could Fir do to ake it threat credible? If fir threaten to pla aggreive if Fir pla aggreive, it will NOT be credible to Fir. Becaue Fir know that if it pla aggreive in period, Fir bet repone in period will be paive. Becaue 0 > 9. Fir proble i that once Fir ha ade it choice, Fir expect Fir to do the rational thing. Fir can ake it threat credible if it could coit itelf to pla would be better off if it could coit itelf to pla aggreive if Fir pla aggreive. One wa for Fir to ake uch a coitent i to allow oeone ele to ake it choice. For exaple, Fir ight hire a lawer and intruct hi to pla aggreive if Fir pla aggreive. If Fir i aware of thee intruction, the ituation i radicall different fro it point of view. If Fir know about Fir intruction to it lawer, then it know that if it pla aggreive it will end up with a paoff of 5. Page of Page

3 So, the enible thing for Fir to do i to pla paive and get a relativel higher paoff of 30. See Figure B3(c) for an extenive for repreentation of thi cae. B5. [5 Mark] Suppoe deand for a coodit i given b 00 p. There are onl two poible factorie that can produce thi coodit, each with cot function: c j 50 + j, where j, denote the factor. The total arket output i the u of the output fro thee two plant. (a) Find the efficient level of output and price for thi arket. Alo, find the total profit of the two fir in thi ituation. The total arket output i the u of the output fro thee two plant. That i, + Market Deand: p 00 ( + ) Factor cot function: c 50 + c Factor arginal cot function: MC Factor cot function: c 50 + c Factor arginal cot function: MC At the efficient level of output of the following condition ha to be atified: P MC MC Thi ean that the following two equation have to be atified: 00 () and 00 () Since both factorie have the ae cot function, at the efficient level of production both ut produce exactl the ae level of output: e e Page 3 of Page

4 Setting into equation () we get: 00 e 00 5 e So, 5 e e e The efficient level of total output: π e e e e Factor profit: p 50 ( ) 50(5) 50 ( 5) 575 Factor profit: π e π e 575 e e e Total profit of the two factorie: π π + π (b) Suppoe the two fir for a cartel. Copute the profit axiizing total output, pice, profit, and deadweight lo of the cartel in thi ituation. The cartel proble i to axiize the joint profit b chooing and. π ax ( )( ) {, } Firt order condition: (3) () 00 Page of Page

5 Since both factorie have the ae cot function, at the joint-profit axiizing equilibriu the will produce the ae aount of output. Setting into equation (3) we get: So, The joint-profit axiizing total output of the cartel, The joint-profit axiizing total output of the cartel, p 00 ( + ) 00 (33.33) The axiized profit of the cartel: π p 00 ( ) 50 ( ) 50 ( ) 66.67(33.33) Deadweight lo of each factor under the cartel i the area between the downward loping arket deand curve and individual factor upward loping arginal cot curve over the quantitie between the joint-profit axiizing output and the efficient level of output of each factor (tr to viualize thi area b drawing a diagra). Deadweight lo of the cartel Deadweight lo of factor + Deadweight lo of factor (33.3)(8.33) Page 5 of Page

6 (c) Intead of a cartel, uppoe the two plant are owned b Cournot duopolit. Find the Cournot-Nah equilibriu output b each fir, the price, and the total profit. Alo copute the deadweight lo aociated with the Cournot duopol. You can olve the proble following the ethod ued in olving quetion B (a). You can find the deadweight uing the technique ued in B5 (b). You can our anwer with the following reult: The Cournot-Nah equilibriu output: c ( c ) ( 0,0), c The arket price at the Cournot-Nah equilibriu: p 60 c The total profit of the fir at the Cournot-Nah equilibriu: π 500 c The total deadweight lo aociated with the Cournot duopol: DWL 00 (d) Intead of the Cournot auption, uppoe that fir et it output before fir doe. Fir doe oberve the output choice of fir before it ake it own output choice. Find the Stackelberg equilibriu output produced b each fir, the price, and total profit. Alo copute the deadweight lo in thi ituation. In thi Stackelberg gae, Fir i the quantit leader and Fir i the quantit follower. We can olve thi equential gae b backward induction. That ean we have to firt olve the follower proble and then olve the leader proble. Fir (the follower ) proble in period i to axiize it profit b chooing for a given level of choen b Fir (the leader) in the firt period. Fir profit axiization proble: ax π (, ) [ 00 ( + )] ( ) 50 Page 6 of Page

7 Firt order condition: ( ) ( ) So, Fir bet repone to or Fir bet repone or reaction function i 00 R( ) (5) Fir (the leader ) proble in period i to axiize it profit b chooing and conidering the fact the bet repone function of Fir in period i given b equation Fir profit axiization proble: ax π ( ) [ ( )] ( ), [ 00 ( + R( )] 50 ( ) ( ) ( ) ( ) Firt order condition: S Subtituting. 3 into Fir bet repone or reaction function we get: (5) Page 7 of Page

8 Therefore, the Stackelberg equilibriu quantitie produced b each fir are:.3 and The arket price at the Stackelberg equilibriu: p 00 ( + ) 00 ( ) Fir profit at the Stackelberg equilibriu: (, ) p ( ) π 58.93(9.6)-50-(9.6) Fir profit at the Stackelberg equilibriu: (, ) p ( ) π 58.93(.3)-50-(.3) Total profit of the fir at the Stackelberg equilibriu: π ( ) π (, ) + π (, ) , Deadweight lo of each factor in the Stackelberg odel i the area between the downward loping arket deand curve and individual factor upward loping arginal cot curve over the quantitie between the Stackelberg equilibriu output and the efficient level of output of each factor (tr to viualize thi area b drawing a diagra). Total deadweight lo in the Stackelberg odel Deadweight lo of factor + Deadweight lo of factor ( )( 5.3) + ( )( 5 9.6) (e) Copare the reult ou found in (a), (b), (c), and (d). Tr to copare the reult on our own. Page 8 of Page

9 B6. [5 Mark] Two fir, Fir and Fir, are copeting in an oligopolitic indutr. The produce an identical product. But Fir doe it at a lower cot than Fir. Fir ha a contant arginal cot of $5 and fir ha a contant arginal cot of $30. The arket deand for the coodit i p 0, where i aggregate output. (a) Suppoe that fir chooe quantitie. Find both bet-repone function. Reeber, arginal cot are different, o the bet repone function will not be etric. Find the Cournot-Nah equilibriu quantitie. Illutrate our reult in a diagra. Suppoe fir take fir output choice a given. Then fir proble i to axiize it profit b chooing it output level. If fir produce unit and fir produce unit then total quantit upplied i +. Define +. The arket price will be P 0. Fir ha a contant arginal and average cot of $5: c 5. Fir ha a contant arginal and average cot of $30: c 30. Fir profit axiization proble: ax π (, ) [ 0 ( + )] 5 Firt order condition: ( + ) + ( ) So, Fir bet repone to or Fir bet repone or reaction function i : R( ) 05 () Page 9 of Page

10 Since the profit- axiization proble faced b the two fir are NOT etric in thi cae, we have to explicitl olve Fir proble to find it bet repone function or reaction function. Fir profit axiization proble: ax π (, ) [ 0 ( + )] 30 Firt order condition: ( + ) + ( ) So, Fir bet repone to or Fir bet repone or reaction function i : R( ) 90 () To find the Cournot-Nah equilibriu quantitie, we have to olve equation () and () iultaneoul for and. Subtituting () into (), Page 0 of Page

11 Subtituting 5 into (), So, the Cournot-Nah equilibriu quantitie are: (, ) ( 0,5) The arket price at the Cournot-Nah equilibriu i: P (b) Suppoe that the fir chooe price intead of quantitie and that price ut be announced in dollar and cent. (That i, $5.7, and $39.00 are periible price, but $5.975 i not.) What are the Bertrand equilibriu price? How uch doe each fir earn in the Bertrand equilibriu? Becaue of the price copetition between the two fir, the high-cot fir, Fir, will not be able to et a price higher than it arginal cot ($30). If Fir et a price higher than $30, the low-cot fir will undercut that price and capture the entire arket. Thu, price copetition will force the high-cot fir to et it price equal to 30, P 30. The low-cot fir, Fir, will undercut thi price b etting it price lightl le than 30 and capture the entire arket. It will et it price equal to 9.99, P So, the Bertrand equilibriu price are: P and P 30. At the equilibriu Fir ake all the ale. Total unit of output old b Fir: 0 P Fir profit: π ( P c ) (9.99 5) Fir profit: π 0 Page of Page