Cost arrangement and welfare in a multi-product Cournot oligopoly

Transcription

1 Eoomis Worig Papers (00 016) Eoomis Cost arragemet ad welfare i a multi-produt Courot oligopol Harve E. Lapa Iowa State Uiversit, hlapa@iastate.edu David A. Heess Iowa State Uiversit Follow this ad additioal wors at: Part of the Eoomis Commos Reommeded Citatio Lapa, Harve E. ad Heess, David A., "Cost arragemet ad welfare i a multi-produt Courot oligopol" (004). Eoomis Worig Papers (00 016) This Worig Paper is brought to ou for free ad ope aess b the Eoomis at Iowa State Uiversit Digital Repositor. It has bee aepted for ilusio i Eoomis Worig Papers (00 016) b a authoried admiistrator of Iowa State Uiversit Digital Repositor. For more iformatio, please otat digirep@iastate.edu.

2 Cost arragemet ad welfare i a multi-produt Courot oligopol Abstrat Welfare i a two-produt Courot oligopol is show to irease (derease) with a irease i orrelatio betwee uit osts whe the outputs omplemet (substitute) i demad. A more qualified orrelatio struture is required for the result to appl i a three-produt Courot oligopol whe produts omplemet i demad. Kewords omplemetarit, arragemet ireasig, ivariae, Courot oligopol Disiplies Eoomis This worig paper is available at Iowa State Uiversit Digital Repositor:

3 IOWA STATE UIVERSITY Cost Arragemet ad Welfare i a Multi-Produt Courot Oligopol Harve E. Lapa, David A. Heess Otober 004 Worig Paper # 0407 Departmet of Eoomis Worig Papers Series Ames, Iowa Iowa State Uiversit does ot disrimiate o the basis of rae, olor, age, atioal origi, seual orietatio, se, marital status, disabilit or status as a U.S. Vietam Era Vetera. A persos havig iquiries oerig this ma otat the Diretor of Equal Opportuit ad Diversit, 3680 Beardshear Hall,

4 Cost Arragemet ad Welfare i a Multi-Produt Courot Oligopol b Otober 004 Harve E. Lapa David A. Heess Uiversit Professor Professor Departmet of Eoomis Departmet of Eoomis 83 Head Hall 578C Head Hall Iowa State Uiversit Iowa State Uiversit Ames, IA Ames, IA Abstrat Welfare i a two-produt Courot oligopol is show to irease (derease) with a irease i orrelatio betwee uit osts whe the outputs omplemet (substitute) i demad. A more qualified orrelatio struture is required for the result to appl i a three-produt Courot oligopol whe produts omplemet i demad. Kewords: arragemet ireasig, omplemetarit, ivariae JEL Classifiatio umbers: C6; D43 Eletroi iformatio for orrespodee with otat author David A. Heess Ph: US , Fa: US , heess@iastate.edu

5 It has log bee ow that soial welfare i a homogeeous good Courot oligopol will irease whe firms have more heterogeeous osts (Bergstrom ad Varia, 1985; Février ad Liemer). Reet researh has demostrated the relevae of ost dispersio i this maret struture whe the firms ooperate at a earlier stage (Salat ad Shaffer, 1999; Va Log ad Soubera, 001). Iitiall smmetri firms have ietives to ooperate at a earlier stage i order to reate later-stage ost asmmetries so that total profit to oligopolists is maimied. I this lass of models, earl ooperatio will also irease soial welfare beause aggregate output is ivariat to ost dispersio. This letter eteds the aalsis o soial welfare to oligopol i plural marets. I a two-produt Courot oligopol, we show how orrelatio betwee uit osts matters whe the outputs are omplemets or substitutes i demad. The same essetial isights appl for a three-produt oligopol uder omplemetarit i demad, but the rearragemet i ost struture eeds to be more qualified i order to avoid iteratio effets. 1. Two produts A fied set of firms, deoted as {0,1,,..., 1} =Ω, eah produe stritl positive amouts of goods ad i a two-maret Courot oligopol. The th firm produes quatit of good at ostat uit ost substitutes. The th firm produes of good at uit ost, ad firm outputs i this maret are perfet ad firm outputs i this maret are also perfet substitutes. Aggregate outputs are X = ad Ω Y =. Ω With liear-i-iome ( I ) ad oave preferees for a represetative osumer, the utilit futio ma be speified as uxyi (,, ) = I+ UXY (, ). Iverse demads for the goods are give as otiuousl differetiable futios P = U ( X, Y); P = U ( X, Y); (1) where the law of demad requires U ( ) 0 ad U () 0. The solutios to (1) are demad

6 D D futios X ( P, P ) ad Y ( P, P ). The th firm s profit is π = U () () + U, () with private optimalit derivatives U () + U () + U () = ; U () + U () + U () =. (3) We assume uique iterior solutios throughout, with firm values ( ˆ, ˆ ), Ω. Write C = ad Ω C =, the sum (3) aross firms to observe that Ω C ad C are suffiiet statistis for idetifig how aggregate outputs X ˆ = ˆ ad Y ˆ = ˆ Ω Ω relate to osts. Cosumer surplus will ot hage so log as all firms produe while uit ost sums are C ad C. With = U U > from the preferee struture, write (3) at ash solutios i [ U] 0 matri form as U U ˆ U ˆ 1 U U U. U U ˆ = U = ˆ U U U (4) With represetig the ordered set of osts (,,...,,,,..., ), aggregate profit is U [ U C ] V() = π = UC UC + U U UU Ω U [ U C ] + UC UC + UU UU U U U ( ) + ( ). Ω Ω Ω (5) Ol the last three summatio terms deped o ost statistis other tha Followig Lapa ad Heess (00), assert C ad C. Defiitio 1. Cosider the futio f( 0, 1,..., 1; 0, 1,..., 1) :. Let operatio (0,1,..., 1) τ map (0,1,..., 1) oto ( τ (0), τ(1),..., τ( 1)), a permutatio

7 o Ω. The f ( ) is said to be permutatio ivariat if f( 0,..., 1; 0,..., 1) f( τ(0),..., τ( 1) ; τ(0),..., τ( 1) ) for all permutatios o Ω. Defiitio. The futio f () i defiitio 1 is said to ehibit permutatio order if f(,..., ;,..., ) f(,..., ;,..., ) for all permutatios τ o Ω 0 1 τ(0) τ( 1) that a) iterhage two idies ad for whih ( )( ) 0, ad b) leave all other idies uhaged. Defiitio 3. The futio f ( ) i defiitio 1 is said to be arragemet ireasig (AI) if it ehibits permutatio ivariae ad permutatio order. It is said to be arragemet dereasig (AD) if f ( ) is AI. The simplest AI futio is f () =, where = τ( ) τ( ) Ω demostrates permutatio ivariae ad + + uder ( )( ) 0 demostrates permutatio order. Turig to demad-side iteratios, we sa that ad D D omplemet (substitute) i demad if X ( P, P ) / P Y ( P, P ) / P ( ) 0. It follows Ω Ω from (1) that the goods omplemet (substitute) i demad wheever U ( )0. Propositio 1. If ad are omplemets (substitutes) i demad, the soial welfare is arragemet ireasig (dereasig) i uit osts. Ω ad Proof. A sequee of iterhages satisfig defiitio leaves the values of ( ) Ω ( ) ivariat. But the sequee ireases the value of Ω as it is a AI futio. Give omplemetarit i demad, 1 U Ω is AI ad ireases alog the sequee. All other terms o the right-had side i (5) do ot hage i value beause C ad 3

8 C do ot hage. So the value of V() at ash equilibrium ireases uder the sequee of iterhages. The values of ˆX ad ˆ Y do ot hage ad either does osumer surplus. Soial welfare, the sum of idustr profits ad osumer surplus, must irease. Give substitutio i demad, the 1 U Ω is AD ad similar logi applies. G The ost rearragemet maes the firms that are most ompetitive (low ost) i oe maret more ompetitive i the other too. Viewig (3), whe there is omplemetarit i demad the a stregtheig of the ost orrelatio helps maret ietives to guide produtio toward lower ost firms. Maret power aross produts ad firms will be reassiged, but the overall effet of maret power i the maretplae will be uaffeted i the sese that maret outputs do ot hage uder the AI ost rearragemet. So idustr ost delies while outputs do ot hage. Whe there is substitutio i demad, the firms with low uit osts i oe produt have a reveue-side ietive to be low output i the seod produt. A firm with low uit osts i both outputs will have strog ietives to uder-produe relative to the firm s omparative ost advatages, so maret power ietives ted to distort produtio toward higher-ost firms. We olude this setio with a alterative perspetive o ireasig orrelatio. Whe uit osts orrelate positivel, orrelatio should irease if we irease the magitude of a produt s high uit ost ad ompesate b dereasig the magitude of a low uit ost for that produt. From (5) we ow that soial welfare ireases i the two produt model if the ostat sums ost struture hage ireases the value of U ( ) U Ω ( ) U. Without loss of geeralit, let produt uit osts be ordered b the firm Ω Ω ide, i.e.,.... Additioal struture is required o how uit osts are arraged Defiitio 4. Uit osts ad m, < m, are said to be positivel arraged if ad m egativel arraged if >. m 4

9 Fiig the values of the, osider the followig rearragemet of the : 1) Ω,, m; ) ε ad + ε where ε > 0 ad < m. 1 m m m Sums of uit osts do ot hage after this hage i uit osts for produt. Uder 1) ad ), the differee i oligopol profits satisfies sig m m V( ) V( ) = ( ) U ( ε + ) U. (6) m Uder ), if produts omplemet i demad, the ( ) U 0. If uit osts are positivel m m arraged, the ( ε + ) U 0 while ( ) U 0 whe produts substitute i demad. m m Fiall, ( ε + ) U 0 whe uit osts are egativel arraged ad ε <. Thus, a small irease i dispersio of uit osts for either produt ireases soial welfare whe a) the goods omplemet i demad ad uit osts are positivel arraged, or b) the goods substitute i demad ad uit osts are egativel arraged.. Three produt model I this ase the firms produe a third good,, with quatit at uit ost for the th firm. Aggregate output is Z =, ad the produt s sum of uit osts is Ω C =. Ω With a represetative osumer, quasi-liear ad oave preferees uxyzi (,,, ) U( X, Y, Z ), the iverse demads are = I+ P = U ( X, Y, Z); P = U ( X, Y, Z); P = U ( X, Y, Z). (7) The th firm s profit is π = U + U + U. (8) 1 This trasfer approah a be show to be equivalet to a irease i variae of the, ad also to domiae i the maoriatio pre-orderig. See pp. 3-1 i Marshall ad Oli (1979). 5

10 Write the private optimalit derivatives i matri form as ˆ UU [ U] UU UU UU UU U 1 ˆ = UU UU UU [ U] UU UU U, Γ ˆ [ ] UU UU UU UU UU U U (9) where Γ< 0 due to oavit. Isert solutios ito (8) to see that aggregate profit depeds diretl o { UU [ U] } ( ) + { UU [ U] } ( ) Ω Ω + { [ ] } ( ) + { } Ω Ω + { } + { } U U U U U U U U U U U U U U U. Ω Ω (10) From differetiatig (7), assert that,, ad are mutual omplemets if UU UU, UU UU, ad UU UU. The all three oeffiiets o the iteratio sums i (10) are positive. Uder mutual omplemetarit, therefore, the effet of some ost rearragemet o soial welfare is determied b the effets o the ovariabilit summatios Ω, Ω, ad Ω. Without loss of geeralit, suppose that firms are ordered i asedig order of margial osts for produtio of, i.e.,.... Cosider the followig hage i the speifiatio of margial osts. Case i) If ad for some {0,1,..., 1}, the map =, =, =, =, ad leave other osts uhaged. Case ii) If ad > for some {0,1,..., 1}, the map =, =, ad leave other osts uhaged. Case iii) If ad > for some {0,1,..., 1}, the map =, A geeraliatio of the vetor rearragemet we are about to eplai was developed i Bolad 6

11 =, ad leave other osts uhaged. If osts hage i this maer, the we sa that osts are better ordered i the trivariate arragemet ireasig (TAI) sese. Eample 1. A four-firm three-maret oligopol has ost struture { } { } C = (,,, ),(,,, ),(,,, ) = (1,3, 4,6),(7,6,5, 4),(4,6,,3). (11) ow ad 1 4, so that ase i) applies. We must iterhage both if we are to 1 4 iterhage either. Write the rearraged ost struture as 1 C = {(1,3, 4,6),(4,6,5,7),(3,6,, 4)}. But the 6 ad 5 i the seod vetor ad the 6 ad i the third vetor are ot properl aliged with the 3 ad 4 i the first vetor, aother eample of ase i). A further rearragemet osistet with the idea of TAI is to iterhage both at the same time. Doig so, we obtai C = {(1,3, 4,6),(4,5,6,7),(3,,6, 4)}. The ad uit produt osts are aliged. But produt osts are ot aliged with those of the other produts, a eample of ase iii). Swith the 6 with the 4 i that third vetor; {(1,3,4,6),(4,5,6,7),(3,,4,6)}. Oe additioal ase iii) rearragemet brigs osts i lie C 3 = aross all produts, i.e., write C 4 = {(1,3, 4,6),(4,5,6,7),(,3, 4,6)}. Eah amog this sequee of rearragemets weal ireases the values of Ω, Ω, ad Ω. that Regardless of the umber of firms i the idustr, a TAI ost rearragemet is desiged so Ω, Ω, ad Ω irease. Ispetio of (10) the establishes Propositio. Let,, ad be mutual omplemets i demad. The soial welfare is larger after a TAI i uit osts ours. ad Prosha (1988). See also Heess ad Lapa (003). 7

12 Whe oe or more of wea iequalities UU UU, UU UU, ad UU U U fail the the result does ot eessaril appl beause a firm s output epasio ietives are ot well-aliged aross produts. Maret power osideratios mea that lower ost firms might ot have strog ietives to produe larger quatities of all three ommodities. The propositio 1 fidig o two-produt oligopol uder substitutio does ot eted to multiprodut oligopol. There is a uique wa to rearrage uit osts i duopol so that that the are less well aliged, ad this allows for a substitutio result. There is ot a uique wa to rearrage uit osts aross three or more produts so that uit osts are less well aliged. 3. Colusio Whe there is omplemetarit i demad, ituitio might suggest that more strogl orrelated uit osts would eourage low ost firms to epad ad high ost firms to otrat. As low ost firms produe more, a ost rearragemet that favors these firms should redue idustr osts ad irease soial welfare. I otrollig for maret power effets, our model provides formal ofirmatio for this ituitio. Referees Bergstrom, T.C., Varia, H.R., Whe are ash equilibria idepedet of the distributio of agets' harateristis? Review of Eoomi Studies 5 (Otober, 4), Bolad, P.J., Prosha, F., Multivariate arragemet ireasig futios with appliatios i probabilit ad statistis. Joural of Multivariate Aalsis 5 (Ma, ), Février, P., Liemer, L., 004. Idiosrati shos i a asmmetri Courot oligopol. Iteratioal Joural of Idustrial Orgaiatio (Jue, 6), Heess, D.A., Lapa, H.E., 003. A defiitio of more sstemati ris with some welfare impliatios. Eoomia 70 (August, 79),

13 Lapa, H.E., Heess, D.A., 00. Smmetr ad order i the portfolio alloatio problem. Eoomi Theor 19 (Jue, 4), Marshall, A.W., Oli, I., Iequalities: Theor of Maoriatio ad Its Appliatios. Aademi Press, Sa Diego. Salat, S.W., Shaffer, G., Uequal treatmet of idetial agets i Courot equilibrium. Ameria Eoomi Review 89 (Jue, 3), Va Log.-V., Soubera, A., 001. Cost maipulatio games i oligopol, with osts of maipulatig. Iteratioal Eoomi Review 4 (Ma, ),