Economics of Strategy (ECON 4550) Maymester 2015 Game Theoretic Models of Oligopoly

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1 Eonomis of trategy (ECN 55) Maymester 5 Game Theoreti Models of ligooly Reading: The Right Game: Use Game Theory to hae trategy (ECN 55 Courseak, Page 5) and Partsometer Priing (ECN 55 Courseak, Page ) Definitions and Conets: ertrand Model a model of ometition in whih firms interat by simultaneously hoosing ries Cournot Model a model of ometition in whih firms interat by simultaneously hoosing uantities of outut

2 Two examles of rodut develoment deisions: Examle : Toyota and Ford must eah deide to either develo or not develo a new omat hybrid ar uose the ayoffs are: Ford Develo Hybrid Not Develo Develo Hybrid, 95, Toyota Not Develo 5, 5 5, 9 tart by reognizing that, for the given ayoffs, neither layer has a dominant strategy Ford Develo Hybrid Not Develo Develo Hybrid, 95, Toyota Not Develo 5, 5 5, 9 There are two Pure trategy Nash Euilibria (one in whih Toyota develos the hybrid and Ford does not, and another in whih Ford develos the hybrid and Toyota does not) Mixed Extension: Let denote the robability with whih Toyota hooses Develo (so that denotes the robability with whih Toyota hooses Not Develo ) Let denote the robability with whih Ford hooses Develo (so that denotes the robability with whih Ford hooses Not Develo ) Deriving the est Resonse Corresondene for Toyota, reognize that Develo (i.e., ) is the stritly better hoie if and only if ( ) ( ) 5( ) 5( ) 5( ) imilarly, deriving the est Rely Corresondene for Ford, reognize that Develo (i.e., ) is the stritly better hoie if and only if 95( ) 5( ) ( ) 9( ) 5( )

3 Visually R T ( ).75 R F ().75 The grah above illustrates all three euilibria of this game: i. Pure trategy Nash Euilibrium in whih Toyota hooses Develo and Ford hooses Not Develo ( and ) ii. Pure trategy Nash Euilibrium in whih Toyota hooses Not Develo and Ford hooses Develo ( and ) iii. Mixed trategy Nash Euilibrium in whih Toyota hooses Develo with robability. 75 and Ford hooses Develo with robability. 75 Examle : ( at and mouse game ) uose Lexus and Hyundai must hoose a design for their model, either sleek or boxy Hyundai wants their ar to look like the Lexus, while Lexus wants their ar to look different than the Hyundai Hyundai leek Hyundai oxy Hyundai leek Lexus 6,, 9 Lexus oxy Lexus 7,, No Pure trategy Nash Euilibrium => onsider the Mixed Extension of the game Let denote the robability with whih Hyundai hooses leek (so that denotes the robability with whih Hyundai hooses oxy ) Let denote the robability with whih Lexus hooses leek (so that denotes the robability with whih Lexus hooses oxy ) Deriving the est Resonse Corresondene for Hyundai, reognize that leek (i.e., ) is the stritly better hoie if and only if ( ) ( ) 9( ) ( ) 5 ( ).6 6 6

4 imilarly, deriving the est Rely Corresondene for Lexus, reognize that leek (i.e., ) is the stritly better hoie if and only if 6( ) ( ) 7( ) ( ) 7( ) Visually R H ( ).75 6 R L () The grah above illustrates the uniue euilibrium of this game => a Mixed trategy Nash 7 Euilibrium in whih Hyundai hooses leek with robability and Lexus hooses leek with robability 6

5 asi ertrand Model: Market demand of D ( ) Two firms selling idential roduts => onsumers base their urhasing deisions simly uon rie If firms set different ries, then all onsumers will buy from the lower rie firm If firms set the same rie, they slit the market evenly and eah sell D( ) uose the firms eah have onstant marginal osts of MC and MC (and no Fixed Costs of rodution) asi ertrand Model with Idential Costs uose Reognize that neither firm would ever want to harge a rie below (sine doing so only ever leads to a negative rofit or rofit of zero) If Firm were to hoose a rie of (with ), what is the best rely of Firm? (harge a higher rie than Firm ) => => (harge the same rie as Firm ) => D ) => D( ) D( ) D( ) D( ( (slightly underut the rie of Firm ) => D ) => For small enough, ) ( Thus, when Firm harges a rie of, the est Rely of Firm is to hoose (i.e., slightly underut the rie of Firm in order to get the entire market) ut, when both firms behave in this way, the only stable air of ries is and => Marginal Cost Priing When firms hoose these ries Eah firm sells D ( ) units and earns zero rofit There is zero Deadweight-Loss (all units for whih uyer s Reservation Prie is greater than Marginal Costs are rodued) two firms is enough for ometition asi ertrand Model with Different Costs uose Muh of the same logi used for the ase of idential osts an be alied Doing so, it is lear that there an never be an euilibrium in whih Firm (the higher ost firm ) sells any ositive uantity, sine at suh any outome, Firm ould do better by slightly underutting the rie of Firm Thus, in euilibrium Firm will set a rie eual to marginal osts ( ) Firm will set a rie below the marginal osts of Firm and ature the entire market ut, regarding the otimal hoie of Firm, we have one of two ossible ases, deending uon whether there is a slight differene or a big differene in osts

6 If there is only a slight differene in osts, then Firm will harge (i.e., slightly underut Firm, but not move any further down the market demand urve) and sell D ) units => visually (with linear demand) ( rie Market Demand Demand Faing Firm Marginal Cost of Firm uantity If instead there is a big differene in osts, then the rofit maximizing rie is signifiantly below the Marginal Cost of Firm (i.e., do not simly slightly underut Firm, but rather move further down the market demand urve) Choie by Firm is the standard Monooly solution => visually (with linear demand): D( ) Marginal Revenue of Firm rie Market Demand Demand Faing Firm Marginal Cost of Firm uantity D ( ) Marginal Revenue of Firm Intuition: when the differene in osts is big enough, then the osts of Firm are so high (relative to the osts of Firm ) so that Firm is not a relevant ometitor to Firm => the otimal hoie of Firm is to simly behave as a monoolist utome for ertrand with different osts (with either a slight differene or big differene in osts) is ineffiient sine there are some units that are not traded for whih buyer s reservation rie is greater than marginal osts => Deadweight-Loss due to too little trade

7 ertrand Model with differentiated roduts: Critial assumtion of the basi ertrand model: all onsumers will urhase the rodut from the firm harging the lower rie (imliitly assumes that onsumers view the roduts as idential to eah other) If instead the roduts are differentiated from one another, then Firm an harge a slightly higher rie than Firm and still retain some ustomers ut, if the two goods are substitutes for eah other, then Firm should exet to lose some ustomers as Firm lowers its rie (and vie versa) Examle: versus => Gasmi, Vuong, and Laffont, Eonometri nalysis of Collusive ehavior in a oft-drink Market, Journal of Eonomis and Management trategy (ummer 99): 77-. Used sohistiated eonometri tehniues to estimate rodution osts and residual demand for and for the eriod from 96 through 96 ll ries are inflation adjusted and exressed in dollars er unit (a unit is ases, with twelve -oune ans in eah ase), while uantities are exressed in millions of units of ola Rounding the estimated values to the nearest integers Residual demands of: 6 for 5 5 for Marginal Costs of Prodution of: MC 5 for MC for If the firms omete by simultaneously hoosing ries, what ries should they hoose? tart by analyzing the roblem from the ersetive of Profit an be exressed as MC F MC 6 F 56 F has diret ontrol over its own rie, but not over s rie Conetually think about identifying the value of to maximize, for any arbitrary value of => this derivation would give us s est Rely Funtion Partial differentiation of with reset to yields 6 5 6

8 ( ertrand Model with differentiated roduts ontinued) In order to maximize rofit, would want to oerate where this exression is eual to zero R This final exression is s est Rely Funtion, whih seifies the otimal hoie of rie by as a funtion of the rie hosen by imilarly, for Profit an be exressed as 5 5 F Conetually think about identifying the value of to maximize, for any arbitrary value of => this derivation would give us s est Rely Funtion Partial differentiation of with reset to yields In order to maximize rofit, would want to oerate where this exression is eual to zero R 7 This final exression is s est Rely Funtion, whih seifies the otimal hoie of rie by as a funtion of the rie hosen by

9 ( ertrand Model with differentiated roduts ontinued) t a Nash Euilibrium, eah layer must be hoosing a strategy that is a best rely to the strategy hosen by her rival In terms of the resent game, this ours at the intersetion of the two est Rely Funtions. Visually R 7 7 R.5 lgebraially, the euilibrium ries are the uniue air for whih the following two R R onditions hold simultaneously and That is, we have a system of two euations with two unknowns: 7 y 7 x i. ii. x y ne way to obtain the solution Take Condition (ii) and lug it into Condition (i) : y 7 y Then solve for y: y 7 y y y Finally, lug this bak into Condition (ii) to obtain the orresonding value of x: y 9 x x,6, That is, the reditions of the ertrand Model are:. 6,. 56,. 6, and. During this time-eriod (96-6), the atual real world average values were: atual atual atual atual. 6,. 96,. 7, and. Thus, when alied to the demand and ost funtions estimated by Gasmi, Vuong, and Laffont, the ertrand model with differentiated roduts does a fairly good job of mathing the atual riing behavior of and

10 Cournot Model: Market demand summarized by the inverse funtion P D (Q) Two firms simultaneously hoose uantities of and => market uantity sulied of Q results in a rie of (Q) P D s an examle, onsider: P D ( Q) Q uose the firms eah have onstant marginal osts of MC and MC (and no Fixed Costs of rodution) Under these assumtions, firm rofits are P D ( Q) P D ( Q) and P D ( Q) P D ( Q) Firm gets to hoose, but not tart by deriving the est Rely Funtion for Firm Partial differentiation of with reset to yields etting this eual to zero and solving for : R ( ) R imilarly, for Firm we would have (by symmetry) ( ) Euilibrium of this simultaneous move game is the uniue air of uantities for whih R ( ) and R ( ) Grahially: R ( ) ) R (

11 ( Cournot Model ontinued) lgebraially, we again have a system of two euations with two unknowns: (i) and (ii) Plugging Euation (ii) into Euation (i) and solving for imilarly, for Firm we would obtain Total industry outut is Q Thus, rie is: ) ( D Q P P Firm rofits are: P and P From here we an easily obtain the intuitive results that an inrease in would result in: o a derease in, Q, and o an inrease in, P, and an inrease in would result in: o a derease in, Q, and o an inrease in, P, and

12 Multile Choie Questions:. In the Cournot Model of ometition, firms omete by. seuentially hoosing uantities of outut.. simultaneously hoosing uantities of outut. C. seuentially hoosing ries. D. simultaneously hoosing ries.. Consider a market in whih Firm and Firm omete by simultaneously hoosing uantities of outut ( and resetively). The est Rely Funtion for Firm is R R ( ), and the est Rely Funtion for Firm is ( ) 5. It follows that at the Nash Euilibrium, Firm will rodue units of outut and Firm will rodue units of outut.. ;.. ;. C. ; 5. D. ; 7. For uestions and, onsider the following senario. Firms and oerate in a market with demand of D( ), 9. They omete by simultaneously setting ries. Consumers make no distintion between the outut of Firm and the outut of Firm (and will therefore simly buy from the firm offering the lower rie). Firm has rodution osts of C ( ), and Firm has rodution osts of C ( ).. If, then in euilibrium. eah firm will set a rie of $.. there will be a ositive Deadweight-Loss, due to not enough trade. C. eah firm will earn zero rofit. D. More than one (erhas all) of the above answers is orret.. If and, then in euilibrium. both firms will sell a ositive amount of outut.. both firms will earn a stritly ositive rofit. C. Deadweight-Loss will be eual to zero. D. None of the above answers are orret.

13 Problem olving or hort nswer Questions:. Consider a market in whih Firm and Firm omete by simultaneously hoosing uantities of outut (denoted and resetively). Market demand is given by the inverse funtion P D ( Q) Q, where Q. Firm has osts of C ( ),, while Firm has osts of C ( ), 5.. Derive and grahially illustrate the est Rely Funtion of eah firm.. uose 7 for the remainder of the uestion. till allowing to take on any arbitrary value, determine exressions for the otimal uantity of Firm and of Firm (eah as a funtion of ). Clearly exlain how the otimal level of outut of eah firm deends uon the value of. C. uose 9. Determine the otimal level of outut of eah firm, along with the resulting market uantity of outut, market rie, and rofit of eah firm. D. uose 6. Determine the otimal level of outut of eah firm, along with the resulting market uantity of outut, market rie, and rofit of eah firm. E. ased uon your answers above, learly exlain how the otimal level of outut of eah firm, market uantity of outut, market rie, and rofit of eah firm eah hanged as the marginal osts of Firm dereased from 9 to 6.. te Right and diles omete with eah other in the market for hildren s footwear by simultaneously hoosing ries. Residual demand for te Right s shoes is given by 6, and residual demand for diles shoes is given by 6 5. te Right has osts of C ) 6 C ) 9. (, and diles has osts of (. ssume throughout that.. Derive and grahially illustrate the est Rely Funtions for both te Right and diles.. uose. Determine the euilibrium rie that eah firm will set. Determine the orresonding uantity sold and rofit for eah firm. Whih firm harges a higher rie? Whih firm sells a greater uantity of outut? C. uose. Determine the euilibrium rie that eah firm will set. Determine the orresonding uantity sold and rofit for eah firm. Whih firm harges a higher rie? Whih firm sells a greater uantity of outut? D. llowing to take on any arbitrary value, determine the otimal rie of eah firm as a funtion of. Determine the orresonding uantity sold and rofit of eah firm (again as funtions of ). E. Exlain how eah of the funtions in art (D) behaves as the value of is inreased. F. Determine the range of for whih diles sets a higher rie than te Right. G. Determine the range of for whih diles sells more outut than te Right.

14 . Consider a market in whih Firm and Firm omete by simultaneously hoosing ries. Consumers make no distintion between the outut of Firm and the outut of Firm (and therefore base their urhasing deision uon only rie). Market Demand is given by D( ) 6,. Firm has rodution osts of C ( ), while Firm has rodution osts of C ( ). ssume throughout that.. uose. Determine the euilibrium rie of eah firm. Determine the resulting uantity sold by eah firm, rofit of eah firm, and Deadweight-Loss in this market.. uose. Determine the euilibrium rie of eah firm. Determine the resulting uantity sold by eah firm, rofit of eah firm, and Deadweight-Loss in this market. C. Will Firm ever want to underut the rie of Firm by more than? If so, determine the range of for whih Firm would want to do so.. Golden Fleee and Jonhawn are two firms that rodue men s lothing. They must simultaneously hoose their rodut lines for next year. They eah broadly have a hoie of either a traditional line or a trendy line. Historially, the lothing of Golden Fleee has tended to aeal to onsumers with onservative tastes, while the lothing of Jonhawn has tended to aeal to those onsumers who want to be on the utting edge of fashion. If the two firms hoose similar lines, then their roduts will be less differentiated from one another. s a result, they would be ometing for essentially the same segment of onsumers, making joint rofits lower. More reisely, the rofits of the firms will be: GF 95 and J 7, if both introdue a traditional line ; GF 5 and J, if both introdue a trendy line ; GF 55 and J, if Golden Fleee introdues a Trendy line and Jonhawn introdues a traditional line ; and GF 5 and J, if Golden Fleee introdues a traditional line and Jonhawn introdues a trendy line.. Illustrate the interation between these two firms by way of a ayoff matrix.. Identify all Pure trategy Nash Euilibria of this game. C. Considering the Mixed Extension of the game (in whih Golden Fleee hooses traditional line with robability and Jonhawn hooses traditional line with robability ), grahially illustrate the est Rely Corresondene of eah layer. ased uon this grah, identify any Mixed trategy Nash Euilibria of this game. nswers to Multile Choie Questions:... C. D

15 nswers to Problem olving or hort nswer Questions:. Profit of Firm an be exressed as,. Partial differentiation with reset to gives us. etting this R exression eual to zero and solving for, we obtain ( ),. R imilarly, for Firm we would obtain ( ),. Grahially,,, R ( ), R ( ),,, R. uosing 7 R, we have ( ), and ( ),. The euilibrium levels of outut must simultaneously satisfy (i), and (ii),. This system of two euations with two unknowns an be solved as follows:,,,, 5 and 5,,,, From these exressions of and, it is lear that is inreasing in (i.e., if the marginal ost of Firm were larger, then Firm would rodue a greater uantity of outut), while is dereasing in (i.e., if the marginal ost of Firm were larger, then Firm would rodue a smaller uantity of outut).

16 C. For 9, we have, and, 6. Thus, Q, 6,6, for whih P P ( Q) (,6). s a result, D ( P ), ( 7)(,),, and ( P ),5 ( 9)(6),5. D. For 6, we have and,,. Thus, Q,,, for whih P PD ( Q) (,). s a result, ( P ), ( 7)(),, and ( P ),5 ( 6)(,),5,5. E. etween arts (C) and (D) above, the value of Firm s Marginal Costs dereased from 9 to 6 (with all other fators fixed). This derease ultimately resulted in Firm roduing more outut and Firm roduing less outut. Further, total industry outut inreased as a result of this derease in rodution osts. Conseuently, rie dereased. Finally, the rofit of Firm dereased and the rofit of Firm inreased. ll of the results are intuitive and exeted.. te Right s rofit an be exressed as: ( 6) ( 6) 6. imilarly, diles rofit an be exressed as: ( ) ( ) 6 5. Partial differentiation of with reset to yields 6 ( 6), and artial differentiation of with reset to yields 6 5 5( ) 6 5 etting and solving for, we get: R ( ) 5 imilarly, setting and solving for, we get:

17 The funtions illustrated as: 6 5 R ( ) 6 5. R ( ) 5 and ( ) 6 an be R 5 R ( ) 5 6 R ( ) R. With, diles est Rely funtion is ( ) 5. The uniue euilibrium illustrated in the grah above ours at the uniue air of ries for whih 5 and 5. olving this system of two euations for the two,5, unknowns, we obtain and The resulting,96,7 uantities are 5. 6 and Profits are: 9 96,,67,,5 6. and,5,. 9. Thus, diles harges a higher rie and sells a greater uantity of outut. R C. With, diles est Rely funtion is ( ) 5. The uniue euilibrium illustrated in the grah above ours at the uniue air of ries for whih 5 and 5. olving this system of two euations for the two 5 unknowns, we obtain. and The resulting uantities 5 5,76 are 9. and.. Profits are: 6. and,5 9,.9. Thus, diles still sells a greater uantity of outut, but now te Right harges a higher rie. R D. For an arbitrary, diles est Rely Funtion is ( ) 6 5. Using 5 this funtion (along with 5 ), the euilibrium ries are,

18 and, 9 9. The orresonding uantities of outut are 9, and. The resulting rofits are: and E. s is inreased,, and 9 9 both inrease. Further,,, 95 inreases, while dereases. Finally, inreases, while 5 dereases. 9 9 F. 9 9 is greater than, if and only if: 7. 7., 95, G. is greater than if and only if: Throughout the uestion we are assuming (i.e., the Marginal Costs of Firm are a onstant $ er unit). If we additionally have, then the firms have idential osts. The only euilibrium when the firms hoose ries simultaneously is for both firms to set rie eual to $ that is,. t this rie a total of D ( ) 6, ()() 7, units are sold, with the firms slitting the market evenly (i.e., eah selling,5 units). This outome is effiient, sine all units for whih Marginal Costs are below uyer s Reservation Prie are indeed traded. Thus, Deadweight-Loss is eual to zero. Finally, eah firm earns zero rofit, sine rie is eual to the onstant valued Marginal Costs of rodution (and Fixed Costs are zero).. If instead, it is now the ase that Firm has stritly lower Marginal Costs than Firm. In suh instanes, the euilibrium is suh that Firm will set, while Firm will set rie stritly below this level. For suh ries, Firm does not sell any outut, while Firm sells to all onsumers who urhase the good. It is neessary to determine if Firm wants to simly underut the rie of Firm by an infinitesimal amount (i.e., harge a rie of with ) or underut the rie of Firm by a more substantial amount. To address this issue, it is helful to grahially

19 illustrate market demand and the demand faing Firm when Firm sets. These funtions an be illustrated as: rie 5 5 Market Demand (blue line) Demand Faing Firm (red line) 6 uantity 6, D ( ) D() 7,, Marginal Revenue of Firm (green line) Reognize that with Market Demand of D( ) 6,, Inverse Market Demand is P D ( Q) 5 Q. Thus, a firm serving the entire market (as Firm would do if harging a rie below $) would have Marginal Revenue of MR( Q) 5 Q. It 6 follows that at a uantity of 7,, MR ( 7,) 5 (7,) 6, as illustrated above. 6 For, we have: rie From the grah above, we an infer that the best hoie by Firm is. That is, Firm will just slightly underut the rie of Firm. When and, it follows that Firm does not sell any outut and Firm sells D D() 7, units of outut. The rofit of Firm is learly zero, Market Demand (blue line) Demand Faing Firm (red line) Marginal Cost of Firm (orange line) uantity 6, D ( ) D() 7,, Marginal Revenue of Firm (green line)

20 while the rofit of Firm is ( )( ) ( )(7,), 6. However, note that Firm is roduing less than the effiient uantity of the good. The effiient uantity is the uniue uantity at whih Marginal Cost of Firm intersets Market Demand. lgebraially this uantity an be identified as: 5 E ( )(6),. With linear demand and onstant Marginal Costs for Firm, it follows that the area of the resulting Deadweight-Loss is a triangle with area of DWL (, 7,)( ) (,)(),56. C. Firm would want to underut the rie of Firm by more than just if the Marginal Costs of Firm are lower than the value of Marginal Revenue at D ( ) 7,. s noted above, MR ( 7,) 5 (7,) 6. Thus, Firm would hoose a rie 6 stritly below when 6. This insight is intuitive, sine this is reisely the range of for whih the otimal monooly rie is less than $.. Given the desrition of the game, the resulting ayoff matrix is: Golden Fleee. Drawing the est Rely rrows as below: Golden Fleee Jonhawn Traditional Trendy Traditional 95, 7 5, Trendy 55, 5, Jonhawn Traditional Trendy Traditional 95, 7 5, Trendy 55, 5, We see that there are two Pure trategy Nash Euilibrum, one in whih Golden Fleee hooses Trendy and Jonhawn hooses Traditional, and one in whih Golden Fleee hooses Traditional and Jonhawn hooses Trendy. C. The est Rely Corresondene for Golden Fleee an be derived by first reognizing when Golden Fleee would have a strit referene for hoosing (i.e., always hoosing a traditional line). This will be the ase so long as: Trad Trend GF 95 5( ) 55 5( ) GF ( ) It follows that hoosing (i.e., always hoosing a trendy line) is the best rely if Finally, if 5. 65, then any value of between zero and one is a best rely. imilarly, for Jonhawn, we have that (i.e., always hoosing a traditional line) is the uniue best rely if: Trad Trend J 7 ( ) ( ) J ( ) 5

21 . It follows that hoosing (i.e., always hoosing a trendy line) is the best rely if.. Finally, if., then any value of between zero and one is a best rely. Grahially: R ( ).65 () R For the mixed extension of the game, we have a Nash Euilibrium wherever the two est Rely Corresondenes interset. Thus, the grah above identifies not just the two aforementioned Pure trategy Nash Euilibria (of (, ) (, ) and (, ) (,) ), but also 5 (, ),. reveals the existene of a Mixed trategy Nash Euilibrium with