Choice of strategic variables under relative profit maximization in asymmetric oligopoly

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1 Economics nd Business Letters () Choice of strtegic vriles under reltive profit mximiztion in symmetric oligopoly Atsuhiro Stoh Ysuhito Tnk * Fculty of Economics Doshish University Kyoto Jpn Received: Mrch 04 Revised: 4 My 04 Accepted: 0 My 04 Astrct We consider simple model of the choice of strtegic vriles under reltive profit mximiztion y firms in n symmetric oligopoly with differentited sustitutle goods such tht there re three firms Firm nd demnd functions re liner nd symmetric mrginl costs re constnt there is no fixed cost Firm nd hve the sme cost function ut Firm hs different cost function. In such model we show tht there re two pure strtegy su-gme perfect equiliri. One is such tht ll firms choose the outputs s their strtegic vriles nd the other is such tht Firm nd choose the outputs s their strtegic vriles nd Firm chooses the price s its strtegic vrile. Keywords: reltive profit mximiztion; symmetric oligopoly; choice of strtegic vriles JEL Clssifiction Codes: D4 L. Introduction In symmetric duopoly with differentited goods Tnk (0) hs shown the following results. If firms mximize their reltive profits the choice of strtegic vriles price or output is irrelevnt to the outcome of the gme in the sense tht the equilirium outputs prices nd profits of the firms re the sme in ll situtions. Thus ny comintion of strtegy choice y the firms constitutes su-gme perfect equilirium in two-stge gme such tht in the first stge the firms choose their strtegic vriles nd in the second stge they determine the vlues of their strtegic vriles. Symmetry mens tht demnd functions re symmetric nd firms hve the sme cost function. This conclusion cn e extended to symmetric oligopoly nd n symmetric duopoly. But it cnnot e extended to n symmetric oligopoly. Asymmetry mens tht demnd functions my e symmetric or firms my hve different cost functions. In recent yers mximizing reltive profit insted of solute profit hs roused the interest of * Corresponding uthor. E-mil: ysuhito@mil.doshish.c.jp. Cittion: Stoh A. nd Tnk Y. (04) Choice of strtegic vriles under reltive profit mximiztion in symmetric oligopoly Economics nd Business Letters () 5-6. Oviedo University Press 5 ISSN:

2 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly economists. Plese see Gions nd Murphy (990) Lu (0) Mtsumur Mtsushim nd Cto (0) Miller nd Pzgl (00) Veg-Redondo (997) nd Schffer (989). In Veg-Redondo (997) it is rgued tht in homogeneous good cse if firms mximize their reltive profits competitive equilirium cn e induced. But in the cse of differentited goods the result under reltive profit mximiztion is different from the competitive result. We think tht seeking for reltive profit or utility is sed on the nture of humn. Even if person erns ig money if his rother/sister or close friend erns igger money thn him he is not sufficiently hppy nd my e disppointed. On the other hnd even if he is very poor if his neighor is more poor he my e consoled y tht fct. In this pper we consider simple model of the choice of strtegic vriles under reltive profit mximiztion y firms in n symmetric oligopoly with differentited sustitutle goods such tht there re three firms Firm nd demnd functions re liner nd symmetric mrginl costs of the firms re constnt there is no fixed cost Firm nd hve the sme cost function ut Firm hs different cost function. In such model we show the following result. There re two pure strtegy su-gme perfect equiliri. One is such tht ll firms choose the outputs s their strtegic vriles nd the other is such tht Firm nd choose the outputs s their strtegic vriles nd Firm chooses the price s its strtegic vrile. In the next section we present model of this pper. In Section we nlyze equiliri in the second stge of the gme. In this stge (the mngers of) firms determine the vlues of their strtegic vriles so s to mximize their reltive profits. In Section 4 we investigte equiliri in the first stge of the gme. In this stge the owners of firms choose the strtegic vriles so s to mximize the reltive profits of the firms. In Section 5 we riefly mention the results when the owners of firms seek to mximize the solute profits of the firms nd we show the existence of four su-gme perfect equiliri. Miller nd Pzgl (00) presented similr study. They showed the equivlence of price strtegy nd output strtegy in delegtion gme of duopoly when owners of firms control mngers of firms seek to mximize n pproprite comintion of solute nd reltive profits. In their model the owner of ech firm determines the weight on the rivl firm s profit in the ojective function of its firm. We do not consider however such delegtion gme. In our nlysis owners of firms choose the strtegic vriles.. The model There re three firms. Firm nd. They produce differentited sustitutle goods. The outputs nd prices of the goods of the firms re denoted y x x x p p nd p. The inverse demnd functions re p = x x x () p = x x x () p = x x x. () We ssume > 0 nd 0 < <. From () nd () from () nd () nd from () nd () we hve p + p = ( + )( x + x ) x p + p = ( + )( x + x ) x p + p = ( + )( x + x) x. Sustituting them into () () nd () we otin the following ordinry demnd functions. 6

3 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly x = [( ) ( ) ] ( )( + ) + p + p + p (4) x = [( ) ( ) ] ( )( + ) + p + p + p (5) x = [( ) ( ) ] ( )( + ) + p + p + p. (6) The inverse nd ordinry demnd functions re symmetric. The constnt mrginl costs of Firm nd re c c nd c. There is no fixed cost. In Section nd 4 we ssume c = c nd c c. The reltive profit of firm is defined s the difference etween its (solute) profit nd the verge of the (solute) profits of the rivl firms. We consider two-stge gme. In the first stge (owners of) the firms choose their strtegic vriles price or output nd in the second stge (mngers of) the firms determine the vlues of their strtegic vriles. In Section 4 we consider cse where owners of firms choose the strtegic vriles to mximize the reltive profits nd in Section 5 we riefly mention cse where owners of firms choose the strtegic vriles to mximize the solute profits.. The second stge of the gme.. One firm is price setting firm In this susection we ssume tht one of the firms is price setting firm tht is it chooses the price s its strtegic vrile nd other two firms re quntity setting firms tht is they choose the outputs s their strtegic vriles. The price setting firm is Firm nd the quntity setting firms re Firm nd. The output nd the price of the good of Firm re denoted y x nd p nd so on. The ordinry demnd function for Firm is x = p x x. (7) From this nd () () nd () the inverse demnd functions for Firm nd re derived s follows. p = ( ) + p ( ) x ( ) x (8) nd p = ( ) + p ( ) x ( ) x. (9) Denote the reltive profits of Firm nd y nd. Then = [( ) + p ( ) x ( ) x] x cx {( p x x)( p c) + [( ) + ( ) ( ) ] p x x x cx nd = ( p x x )( p c ) [( ) + p ( ) x ( ) x ] x c x + [( ) + p ( ) x ( ) x ] x c x = [( ) + p ( ) x ( ) x] x cx {( p x x)( p c) + [( ) + ( ) ( ) ]. p x x x cx 7

4 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly The conditions of reltive profit mximiztion for Firm nd re ( ) + p ( ) x ( ) x c [ p ( ) x + c ] = 0 p x x + c ( x + x) = 0 nd ( ) + p ( ) x ( ) x c [ p ( ) x + c ] = 0. Arrnging the terms p + 4( ) x + ( ) x = ( ) c c (0) 4 p + x + x = + c () p + ( ) x + 4( ) x = ( ) c c () re derived. From (0) () () we get the equilirium price of Firm nd the equilirium outputs of Firm nd s follows. p ( )(4 ) + c + (4 + ) c + c p = ( + )(4 ) q ( )(4 )(4 + ) (6 7 ) c + ( )(4 + ) c + (4 + 5 ) c x = ( )( + )(4 )(4 + ) nd q ( )(4 )(4 + ) + (4 + 5 ) c + ( )(4 + ) c (6 7 ) c x =. ( )( + )(4 )(4 + ) A superscript p indictes price setting firm superscript q indictes quntity setting firm nd indictes the cse of this susection. Assuming tht c = c nd c c the reltive profits of Firm nd re written s q c c = [( )(4 ) (4 + ) ( )( + )(4 ) (4 + ) (6 7 )(8 + 4 ) c ( ) c ] 4 5 p c c = [( )(4 ) (4 + ) ( )( + )(4 ) (4 + ) c ) c ] 4 ( ) (8 64 q c c = [( )(4 ) (4 + ) ( )( + )(4 ) (4 + ) ( ) c ( ) c ] If Firm is price setting firm the reltive profits of the firms re written s follows. 8

5 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly q c c = [( )(4 ) (4 + ) ( )( + )(4 ) (4 + ) (6 7 )(8 + 4 ) c ( ) c ] 4 5 q c c = [( )(4 ) (4 + ) ( )( + )(4 ) (4 + ) c ) c ] 4 ( ) (8 64 p c c = [( )(4 ) (4 + ) ( )( + )(4 ) (4 + ) ( ) c ( ) c ] If Firm is price setting firm the reltive profits of the firms re written s follows. p c c = [(6 8 + ) (8 + ) c 8( ) c ] ( + )(4 ) q c c = [(6 8 + ) (8 + ) c 8( ) c] ( + )(4 ) q c c = [(6 8 + ) (8 + ) c 8( ) c]. ( + )(4 ).. Two firms re price setting firms In this susection we ssume tht one of the firms is quntity setting firm nd other two firms re price setting firms. The quntity setting firm is Firm nd the price setting firms re Firm nd. The inverse demnd function for Firm is p = [( ) ( )( + ) x + p + p ]. () + From this nd (4) (5) nd (6) the ordinry demnd functions for Firm nd re derived s follows. x = ( ) ( ) ( )( ) { + + p + [( ) ( )( + ) x + p + p ] + p } (4) + = {( ) p + p ( ) x } nd x = ( ) ( ) ( )( + ) { + p + [( ) ( )( + ) x + p + p ] + p } (5) + = {( ) + p p ( ) x }. The reltive profits of Firm nd re written s follows. 9

6 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly = ( ) p + p ( ) x ( p c) [( ) ( )( + ) x + p + p ] x c x + + ( ) + p p ( ) x ( p c) nd = [( ) ( )( + ) x + p + p ] x c x + ( ) p + p ( ) x ( p c ) + ( ) p + p ( ) x ( p c). = ( ) + p p ( ) x ( p c) ( ) + ( ) ( ) p p x p c + [( ) ( )( + ) x + p + p ] x cx + The conditions of reltive profit mximiztion for Firm nd re ( ) p + p ( ) x + c x + ( p c) = 0 + [( ) ( )( + ) x + p + p ] c + [ ( )( p c ) ( )( p c )] = 0. ( ) nd ( ) + p p ( ) x + c x + ( p c ) = 0 + Arrnging the terms ( ) 4 p + p ( ) x + c + c = 0 (6) ( ) 4( ) ( ) x ( ) p ( ) p (7) ( ) c ( )( + ) c ( ) c = 0 ( ) + p 4 p ( ) x + c + c = 0 (8) re derived. From (6) (7) (8) we get the equilirium output of Firm nd the equilirium prices of Firm nd s follows. 0

7 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly q ( )(4 + 5 ) + ( + ) c ( + )(4 ) c + ( + ) c x = ( )( + )(4 + 5 ) nd p = [( )(4 + )(4 + 5 ) + ( + )( ) c ( + )(4 + )(4 + 5 ) p + ( + )(4 + ) c + (4+ 7+ ) c ] + c p = [( )(4 + )(4 + 5 ) + ( ) c ( + )(4 + )(4 + 5 ) p + ( + )(4 + ) c + ( + )( ) c ] + c. A superscript indictes the cse of this susection. Assuming tht c = c nd c c the reltive profits of Firm nd re written s p c c = [( )(4 + ) (4 + 5 ) ( )( + )(4 + ) (4 + 5 ) + + c 4 5 ( ) ( ) c] + q c c = [( )(4 + ) (4 + 5 ) ( )( + )(4 + ) (4 + 5 ) 4 5 ( ) c + ( ) c] p c c = [( )(4 + ) (4 + 5 ) ( )( + )(4 + ) (4 + 5 ) 4 5 ( ) c ( ) c] If Firm is quntity setting firm the reltive profits of the firms re written s follows. p c c = [( )(4 + ) (4 + 5 ) ( )( + )(4 + ) (4 + 5 ) ( ) c ( ) c] + p c c = [( )(4 + ) (4 + 5 ) ( )( + )(4 + ) (4 + 5 ) ( ) c 4 5 ( ) c] + q c c = [( )(4 + ) (4 + 5 ) ( )( + )(4 + ) (4 + 5 ) ( ) c ( ) c] +. If Firm is quntity setting firm the reltive profits of the firms re written s follows.

8 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly q c c = [( )(4 + 5 ) ( )( + )(4 + 5 ) ( + )( ) c ( ) c ] p c c = [( )(4 + 5 ) ( )( + )(4 + 5 ) ( + )( ) c ( ) c ] p c c = [( )(4 + 5) ( )( + )(4 + 5 ) ( + )( ) c ( ) c ]... Cournot nd Bertrnd equiliri Consider the Cournot equilirium in which ll firms re quntity setting firms nd the Bertrnd equilirium in which ll firms re price setting firms. The equilirium outputs t the Cournot equilirium re otined s follows. C (4 ) (4 + ) c + c + c x = ( + )(4 ) C (4 ) + c (4 + ) c + c x = ( + )(4 ) nd C (4 ) + c + c (4 + ) c x =. ( + )(4 ) C indictes Cournot. Assuming c = c nd c c the reltive profits of Firm nd t the Cournot equilirium re written s C c c = [(4 ) (8 + ) c 8( ) c] ( + )(4 ) C c c = [(4 ) (8 + ) c 8( ) c] ( + )(4 ) C c c = [(4 ) (8 + ) c 8( ) c]. ( + )(4 ) The equilirium prices t the Bertrnd equilirium re B ( )(4 + 5 ) + ( ) c + ( + ) c + ( + ) c p = ( + )(4 + 5 ) B ( )(4 + 5 ) + ( + ) c + ( ) c + ( + ) c p = ( + )(4 + 5 ) nd B ( )(4 + 5 ) + ( + ) c + ( + ) c + ( ) c p =. ( + )(4 + 5 ) B indictes Bertrnd. Assuming tht c = c nd c c the reltive profits of Firm nd t the Bertrnd equilirium re

9 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly B c c = [( )(4 + 5 ) ( + )( ) c ( ) c] ( )( + )(4 + 5 ) B c c = [( )(4 + 5 ) ( + )( ) c ( ) c] ( )( + )(4 + 5 ) B c c = [( )(4 + 5 ) ( )( + )(4 + 5 ) ( + )( ) c ( ) c ]. 4. The first stge of the gme In this section we consider the cse tht the owner of ech firms seeks to mximize the reltive profits like the firm itself nd chooses the strtegic vrile price or output t the first stge of the gme. 4.. The est responses of Firm. Assume tht Firm nd re quntity setting firms. Compre the reltive profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve p C 9 (4 + )( c c) = ( )(4 ) (4 + ) = 0. (9) Thus price nd output re indifferent nd oth re est responses.. Assume tht Firm nd re price setting firms. Compre the reltive profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve B q 9 (4 + )( c c) = ( )(4 + ) (4 + 5 ) = 0. (0) Thus price nd output re indifferent nd oth re est responses.. Assume tht one of Firm nd is quntity setting firm nd the other is price setting firm. Compre the reltive profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve 7 p q 44 ( + )( c c) = ( )(4 ) (4 + ) (4 + ) (4 + 5 ) < 0. () Thus the output is the est response of Firm.. The est responses of Firm (or ) The sitution of Firm nd tht of Firm re symmetric.. Assume tht Firm nd re quntity setting firms. Compre the reltive profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then since c c nd c = c we hve p C 9 (4 + )( c c) = ( )(4 ) (4 + ) < 0. () Thus the output is the est response of Firm.. Assume tht Firm nd re price setting firms. Compre the reltive profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve

10 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly B q 9 (4 + )( c c) = ( )(4 + ) (4 + 5 ) < 0. () Thus the output is the est response of Firm.. Assume tht Firm is price setting firm nd Firm is quntity setting firm. Compre the reltive profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve p q 9 ( )( c c) = < 0. ( )(4 ) (4 + ) (4 + 5 ) (4) Thus the output is the est response of Firm. 4. Assume tht Firm is price setting firm nd Firm is quntity setting firm. Compre the reltive profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve p q 9 ( )( c c) = < 0. ( )(4 ) (4 + ) (4 + 5 ) (5) Thus the output is the est response of Firm. 4.. Su-gme perfect equiliri From these results we otin the following conclusion. Proposition. There re two pure strtegy su-gme perfect equiliri s follows.. All firms choose the outputs s their strtegic vriles.. Firm nd choose the outputs s their strtegic vriles nd Firm chooses the price s its strtegic vrile A note on symmetric cse If the oligopoly is symmetric tht is c = c = c ll of () () () (4) nd (5) re zero. Then output nd price re indifferent for ll firms in ll situtions nd so ny comintion of strtegies of the firms constitutes su-gme perfect equilirium. 5. Asolute profit mximizing owners It my e nturl tht the owners of firms seek to mximize the solute profits of their firms. In this section we riefly mention the results of tht cse. The equilirium outputs nd prices re the sme s those in the previous section. We consider the first stge of the gme. Denote the solute profit of Firm when it is price setting firm nd other firms re p quntity setting firms y π nd so on. 5.. The est responses of (the owner of) Firm. Assume tht Firm nd re quntity setting firms. Compre the solute profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve p C π π = 0. (6) Thus price nd output re indifferent nd oth re est responses.. Assume tht Firm nd re price setting firms. Compre the solute profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve 4

11 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly B q π π = 0. (7) Thus price nd output re indifferent nd oth re est responses.. Assume tht one of Firm nd is quntity setting firm nd the other is price setting firm. Compre the solute profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve 5 p q 48 ( + )(6 + 6 )( c c) π π = ( )(4 ) (4 + ) (4 + ) (4 + 5 ) > 0. (8) Thus the price is the est response of Firm. 5.. The est responses of (the owner of) Firm (or ) The sitution of Firm nd tht of Firm re symmetric.. Assume tht Firm nd re quntity setting firms. Compre the solute profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve p C π π = 0. (9) Thus price nd output re indifferent nd oth re est responses.. Assume tht Firm nd re price setting firms. Compre the solute profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve B q π π = 0. (0) Thus price nd output re indifferent nd oth re est responses.. Assume tht Firm is price setting firm nd Firm is quntity setting firm. Compre the solute profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve 4 p q 6 (6 4 )( c c) π π = ( )(4 ) (4 + ) (4 + 5 ) > 0. Thus the price is the est response of Firm. 4. Assume tht Firm is price setting firm nd Firm is quntity setting firm. Compre the solute profit of Firm when it chooses the price s strtegic vrile nd tht when it chooses the output s strtegic vrile. Then we hve 4 p q 6 ( )( c c) π π = ( )(4 ) (4 + ) (4 + 5 ) < 0. () Thus the output is the est response of Firm. From these results we otin the following conclusion. Proposition. There re the following four pure strtegy su-gme perfect equiliri.. All firms choose the outputs s their strtegic vriles.. All firms choose the prices s their strtegic vriles.. Firm nd choose the prices s their strtegic vriles nd Firm chooses the output s its strtegic vrile. 4. Firm nd choose the prices s their strtegic vriles nd Firm chooses the output s its strtegic vrile. 5

12 A. Stoh nd Y. Tnk Strtegic vriles in symmetric oligopoly 6. Conclusion We hve studied the choice of strtegic vriles under reltive profit mximiztion in n symmetric oligopoly with differentited sustitutle goods. We considered cse with three firms such tht two firms hve the sme cost functions nd the other one firm hs different cost function nd hve shown tht there re two pure strtegy su-gme perfect equiliri. In duopoly nd symmetric oligopoly the equivlence of price nd output strtegies under reltive profit mximiztion hve een proved. In n symmetric oligopoly however they re not equivlent. On the other hnd in the lst section we hve shown tht if the owners of firms seek to mximize the solute profits of their firms even though (the mngers of) firms mximize their reltive profits there re four su-gme perfect equiliri. Acknowledgments. The uthors would like to thnk the referee for his/her vlule comments which helped to improve the mnuscript. References Gions R. nd Murphy K.J. (990) Reltive performnce evlution for chief executive officers Industril nd Lor Reltions Review 4() 0-5. Lu Y. (0) The reltive-profit-mximiztion ojective of privte firms nd endogenous timing in mixed oligopoly The Singpore Economic Review 56() 0-. Mtsumur T. Mtsushim N. nd Cto S. (0) Competitiveness nd R&D competition revisited Economic Modeling Miller N.H. nd Pzgl A.I. (00) The equivlence of price nd quntity competition with delegtion Rnd Journl of Economics () Veg-Redondo F. (997) The evolution of Wlrsin ehvior Econometric Schffer M.E. (989) Are profit mximizers the est survivors? Journl of Economic Behvior nd Orgniztion () Tnk Y. (0) Irrelevnce of the choice of strtegic vriles in duopoly under reltive profit mximiztion Economics nd Business Letters ()