Subsidization in Mixed Oligopoly with. Managerial Delegation: Price Competition

Transcription

1 G-COE GLOPE II Workng Paper Seres Subsdzaton n Mxed Olgopoly wth Manageral Delegaton: Prce Competton Yoshhro Tomaru, Yasuhko Nakamura, and Masayuk Sato Workng Paper No.35 If you have any comment or queston on the workng paper seres, please contact each author. When makng a copy or reproducton of the content, please contact us n advance to request permsson. The source should explctly be credted. GLOPEⅡ Web Ste:

2 Subsdzaton n Mxed Olgopoly wth Manageral Delegaton: Prce Competton Yoshhro Tomaru Faculty of Economcs, Toyo Unversty, Japan Yasuhko Nakamura Graduate School of Economcs, Waseda Unversty, Japan Masayuk Sato Graduate School of Economcs, Waseda Unversty, Japan Abstract Ths paper studes optmal producton subsdy n a mxed duopoly wth separaton between ownershp and management. Although many exstng lterature shows that the frst best producton allocaton s acheved under several economc envronments (so-called rrelevance result), we fnd that ths result holds n the followng two cases. (1) The owners smultaneously/sequentally decde ther sales weght n the manageral delegaton contracts. (2) The managers smultaneously/sequentally choose ther quantty levels. Therefore, the rrelevance result s robust aganst the ntroducton of sales delegaton wth separaton between ownershp and management, even f the frms smultaneously/sequentally determne ther sales delegaton weghts or quantty levels. JEL Classfcaton: D21, L13, L33 Keywords: Mxed Duopoly, Manageral Delegaton, Prce Compatton, Subsdzaton, Prvatzaton Correspondng author s address: Graduate School of Economcs, Waseda Unversty, Nsh-waseda, Shnjuku-ku, Tokyo , Japan; E-mal: yasu-net.system@asag.waseda.jp; Tel:

3 INTRODUCTION Ths paper presents a theoretcal analyss of the optmal subsdzaton ssue n a mxed duopolstc ndustry wth Bertrand competton style one welfare-maxmzng publc frm and one proftmaxmzng prvate frm. In partcular, we consder the stuaton wheren both the frms are manageral,.e., the owner of each frm enters nto a manageral delegaton contract wth the frm s manager. Thus, we conduct an analyss where both the optmal producton subsdy and each frm s delegaton parameter are taken nto account. 1 More precsely, we scrutnze the mpact of producton subsdy on the delegaton contract and on the equlbrum market outcomes. Many works have studed subsdzaton ssue n a mxed olgopoly. Whte (1996), consdered to be semnal work n ths feld, showed that the frst best allocaton wth respect to socal welfare s acheved n a smultaneous settng of a mxed olgopoly contanng of one publc frm and n prvate frms, f the government frst commts to the producton subsdy. Further, he found that such a frst best allocaton s observed agan after the prvatzaton of the publc frm. 2 After Whte (1996), research on subsdzaton n mxed olgopoly has focused on confrmng the robustness of Whte (1996) s result aganst several types of changes n the economc envronment: Poyago-Theotoky (2001) and Myles (2002) showed that Whte 1996 s result holds n a Stackelberg competton where the publc frm becomes the leader. Hashmzade, Khodavas and Myles (2007) showed that the rrelevance result extends to both quantty and prce compettons wth product dfferentaton. 3 However, the above papers assumed that both the publc frm and the prvate frms are assumed to be entrepreneural,.e., every manageral decson s made by the owner. Thus, n ther models, the separaton between ownershp and management s gnored. In ths paper, we consder subsdzaton n a mxed duopoly where each frm s manageral,.e., each frm s ownershp and management are separable. Past lterature on manageral delegaton challenged the tradtonal assumpton that the objec- 1 As n ths paper, Tomaru, Nakamura and Sato (2009) tackled the optmal subsdzaton problem wth manageral delegaton n a quantty settng mxed olgopoly, and showed that the optmal subsdes yeld the frst-best allocaton n Cournot mxed and prvate duopoles, and mxed duopoles wth publc output and delegaton leadershp. 2 Ths result s referred to the an rrelevance result n Poyago-Theotoky (2001). 3 Other works on subsdzaton n mxed olgopoly Tomaru (2006), Kato and Tomaru (2007), and Tomaru and Sato (2009) nvestgated the robustness of Whte (1996) s result. More precsely, Tomaru (2006) showed that the rrelevance result holds under Matsumura (1998) s partal prvatzaton. Kato and Tomaru (2007) confrmed that a result smlar to that n Whte (1996) s also satsfed even f the prvate frms maxmze an objectve functon slghtly dfferent from ther own profts: the weghted average of ther profts and revenues (or negatve cost). More recently, Tomaru and Sato (2009) showed that the rrelevance result also holds n mxed and prvate duopoles even f each frm s producton tmng s endogenzed. 1

4 tve functon of a (prvate) frm s to solely maxmze ts own proft, and ntroduced several types of delegaton contracts whch owners provde to managers n a prncpal-agent model, wth the most well-known delegaton regme beng the sales delegaton case presented n Fershtman and Judd (1987), Sklvas (1987), and Vckers (1985) (F JS V-fashoned contract case): the owner of each frm provdes the manager wth an ncentve contract on the bass of the frm s proft and sales. Ths stuaton s studed n the context of varous economc envronments. 4 In partcular, n the lterature on the standard mxed olgopoly, Barros (1995) and Whte (2001) nvestgated the nfluence of the use of FJS V delegaton contract on socal welfare before and after prvatzaton. Nshmor and Ogawa (2005) analyzed the nteracton between the length of F JS V ncentve contracts and market behavor. More recently, Heywood and Ye (2009) reconsdered the effect of the FJS V delegaton on welfare, and dentfed whether or not prvatzng the publc frm usng an optmal ncentve contract reduces welfare. 5 However, there s no pror lterature on optmal producton subsdy wth manageral delegaton n all frms. Ours s the frst paper to study ths problem n a prce-settng mxed olgopoly. The man purpose of ths paper s to check the rrelevance result aganst the ntroducton of each frm s FJS V contract n a mxed olgopoly. In ths paper, we consder the followng two cases. () Delegaton sequencng case the owners smultaneously or sequentally select the content of ther delegaton contracts whch are provded to the managers, and the managers smultaneously determne ther prce levels. () Competton sequencng case the managers determne 4 Many works on manageral delegaton that study dfferent delegaton regmes other than the FJS V contract, we can exemplfy the followng fve papers. Salas Fumas (1992) and Mller and Pazgal (2001) consdered the stuaton wheren the owner provdes the manager wth an ncentve contract based on a weghted sum of the frm s own proft and the rval s proft (whch they refer to as the market share case). Mller and Pazgal (2002) appled the relatve performance regme n varous competton types and found that managers atttudes can be used as a strategc commtment devce that can ncrease the frms profts n certan envronments. More recently, Jansen, Ler and van Wtteloostujn (2007) and Rtz (2008) together ntroduced the market-share case wheren the delegaton contract s a combnaton of the frm s own proft and market share. On the other hand, the FJS V contract has been used n many areas of appled economcs. Zhang and Zhang (1997) analyzed the beneft of strategc delegaton n an R&D settng wth spllovers, and Kopel and Regler (2006) reconsdered such a problem to clarfy the correct nfluences of delegatng the producton and R&D decsons to managers on owners benefts. Subsequently, Goerng (2007) consdered the mplcatons of F JS V contract for non-proft frms. Furthermore, n the context of nternatonal trade, Colle (1997) and Das (1997) explored the relaton between sales delegaton wthn a frm and strategc trade polcy. Furthermore, Colle (2003), González-Maestre and López-Cuñat (2001), Straume (2006), Zss (2001), etc., studed the some wth regard to managers. 5 Further, the followng works study manageral delegaton n the context of mxed duopoly: Bárcena-Ruz (2009), Nakamura (2008), Nakamura and Inoue (2007), Nakamura and Inoue (2009), and Saha and Sensarma (2008). 2

5 ther prces smultaneously or sequentally, whle the owners smultaneously select the content of ther delegaton contracts. For case (), we show that when the delegaton contract s beng determned, the sum of each frm s equlbrum producton subsdy and delegaton parameter, and the equlbrum market outcomes concde n the smultaneous move case and the two types of sequental move cases (the Stackelberg leadershp cases), resultng n the achevement of the frst best producton allocaton. Furthermore, for case (), as n case (), we fnd that the frst best producton allocaton s acheved from a vewpont of socal welfare n the smultaneous move case and the two types of sequental move cases. On the other hand, we obtan the result that the equlbrum producton subsdy of each frm dffers for the smultaneous move and the two types of sequental move cases under both cases () and (). More precsely, n case (), the government ncurs a hgher subsdy the owner of the publc frm becomes a follower, whle, n case (), t must pay the hghest subsdy when the manager of the publc frm s the leader. Thus, n mxed olgopolstc ndustres, where all frms are manageral, the government have to pay hgher producton subsdy n order to attan the frst best level of socal welfare, as compared to the subsdy pad n a standard mxed olgopoly wthout manageral delegaton. The remander of ths paper s organzed as follows. In Secton 2, we formulate the basc settng of the mxed duopolstc model consdered n ths paper. In Secton 3, we nvestgate the delegaton sequencng case where the two owners determne the content of ther FJS V contracts smultaneously or sequentally, whle the manager smultaneously choose ther prces. In Secton 4, we presents the analyss of the competton sequencng case where the managers choose ther prce levels smultaneously or sequentally, whle the owners determne the content of ther F JS V contracts. Secton 5 provdes the concludng remarks. A detaled descrpton of each frm s delegaton parameter n all cases (gven the value of each frm s producton subsdy as consdered n sectons 3 and 4) s relegated to the Appendx. MODEL We formulate a dfferentated products mxed duopoly wth lnear demand and quadratc cost. The basc structure of the model follows Dxt (1979) and Sngh and Vves (1984). We, thus, have an economy contanng a monopolstc sector wth a publc frm and a prvate frm, and a compettve sector producng the numerare good. The two frms produce dfferentated goods n a monopolstc sector. Henceforth, we refer to the publc frm as frm 0 (prvate frm as frm 1) and the owner of the publc frm as owner 0 (owner of prvate frm as owner 1). On the demand sde of the market, the representatve consumer maxmzes U (q 0, q 1 ) p 0 q 0 p 1 q 1, where q 0 3

6 s the amount of the good that frm produces and p s ts prce ( = 0, 1). We assume that the functon U (q 0, q 1 ) s quadratc, strctly concave, and symmetrc n q 0 and q 1 : U (q 0, q 1 ) = a (q 0 + q 1 ) 1 2 ( q bq 0q 1 + q 2 1), a > 0, b (0, 1), (1) where b represents the degree of product dfferentaton. The specfcaton mples the followng drect demand functon: q (p 0, p 1 ) = a (1 b) p + bp j 1 b 2,, j = 0, 1, j. (2) We assume that the technologes of both frms are specfed by quadratc functons of ther own outputs,.e., C (q ) = kq 2 /2. The proft functon of the frm s as follows: Π = p q 1 2 kq2 + sq,, j = 0, 1, j, (3) where s s the producton subsdy gven by the government and as such s assumed to be nonnegatve. Usually, socal welfare, denoted by W, s measured as the sum of consumer surplus (CS ) and producer surplus (PS ): where PS = Π 1 + Π 2, and CS s gven by W = CS + PS s (q 0 + q 1 ), (4) CS = U (q 0, q 1 ) p 0 q 0 p 1 q 1 = a2 (1 b) + p 2 0 2bp 0p 1 + p 2 1 2a (1 b) (p 0 + p 1 ) 2 ( 1 b2). (5) Owner 0 s assumed to be a welfare maxmzer, whle owner 1 s assumed to maxmze her/hs own proft. In addton, ths paper focuses on the manageral aspects of the frms. As such, we consder the stuaton where the frms owners decde to delegate control to managers. To formalze manageral delegaton, we assume that each frm s owned by a sngle agent, and follow Lambertn (2000), Nakamura (2008), and Nakamura and Inoue (2009). In each frm, the owner provdes the followng type of of ncentve contract V (Π, q ) to her/hs manger: V (Π (q ), q ) = Π (q ) + θ q, θ R, = 0, 1, (6) where parameter θ measures the relevance of the sales of frm, ( = 0, 1). 6 Alternatvely, θ can be nterpreted as a varant of the subsdes provded by the owner of each frm to the assocated 6 Queston arses as to why the manager of frm selects the prce such that the followng functon s maxmzed: V (Π (p ), p ) = Π (p ) + θ p, θ R, = 0, 1. (7) 4

7 manager. Note that ths parameter s not a physcal subsdy, that s, θ does not mply a drect transfer from the owner to the manager. The manager of frm can maxmze her/hs payoff by choosng a prce p that maxmzes U ( = 0, 1). Ths can be supported by the assumpton that the payoff to the manager of frm s represented as λ +µ V for some real number λ and some postve number µ ( = 0, 1). Smlar to many exstng works, we assume that the payoffs to the managers are neglgble as compared to the profts because we emphasze the mpact of manageral delegaton on the equlbrum outcomes. In ths model, we consder that each manager receves two types of subsdes from the two agents; the producton subsdy by the government and the level of the delegaton parameter by her/hs owner. Therefore, n the followng analyss, we defne the effectve subsdy as the sum of the producton subsdy and the level of each frm s delegaton parameter. We should note that V can be redefned usng net proft and effectve subsdy: V (p 0, p 1, u ) = (p q (p 0, p 1 ) 12 ) kq (p 0, p 1 ) 2 + u q (p 0, p 1 ), where u = θ + s, ( = 0, 1). In the followng two sectons, we wll present the analyss for the () Delegaton sequencng case and () Competton sequencng case, respectvely. DELEGATION SEQUENCING In ths secton, we nvestgate the followng three stage game. In the frst stage, the government sets the subsdy level of each frm whle tryng maxmze of socal welfare. In the second stage, the owner decdes the delegaton parameter θ of frm ( = 0, 1). 7 Then, we consder the followng three cases: In the frst case, denoted by the superscrpt db, the two owners smultaneously decde the delegaton parameters. In the second case, whch s denoted by the superscrpt dl, owner 0 frst decdes her/hs level of the delegaton parameter and s then followed by owner 1. In the thrd case, whch s denoted by the superscrpt df, wheren the frst mover s owner 1 and the second mover s the owner 0. Note that n vrtualty, both owners control varables can be regarded However, we obtan the same equlbrum market outcomes for both cases (6) and (7), snce each prce has a one-toone correspondence wth each quantty under a gven opponent s prce as per the drect demand functon as n (2). Thus, n what follows, as n Lambertn (2000) and Nakamura (2008) we only consder the case wheren the objectve functon of each manager of the frm, ( = 0, 1) s represented as (6). The above fact s also ndcated n Nakamura and Inoue (2009). 7 Note that n ths paper, we dentfy the owner 0 wth the government. 5

8 as effectve subsdes, u, gven the level of physcal subsdy s. Hence, we presume that owners decde ther effectve subsdes n the above three cases; ths enables us to analyze our model far more easly. In the fnal and thrd stage, the managers smultaneously choose ther prce levels. We obtan the reacton functons of the owners n the second stage: u 0 = R 0 (u 1 ) = a(1 b2 )( 2 + b + b 2 k) 2 + b [ b 2 b 4 + k(2 + k) ] u 1 (1 + k)(2 + k) 2 + b 4 (3 + k) b 2 (7 + 7k + 2k 2 ) u 1 = R 1 (u 0, s), and = ab2 [2+b 3 +b 4 +3k+k 2 b(1+k) b 2 (3+2k)]+[ b 6 +(2+k) 3 +b 4 (7+3k) b 2 (14+13k+3k 2 )]s+b 3 (1 b 2 +k)u 0. (2 + k) 3 + b 4 (4 + k) 2b 2 (6 + 5k + k 2 ) Prameters u ( = 0, 1) are strategc complement for both owner 0 and owner 1. Note that the reacton functon of owner 0, R 0, s ndependent of s, whereas that of owner 1, R 1, s dependent. The government s not nterested n ncome transfer through physcal subsdes snce these eventually return to the government. Hence, s does not affect the reacton of owner 0. On the other hand, an ncrease n subsdy s rases u 1 and thus shfts the reacton curve of the owner 1 outward. As s ncreases, owner 1 selects a hgher u 1, whch encourages hs manager to produce more, n order to receve more benefts from the hgher subsdy. Further, note that f the government owns both frms 0 and 1 and proposes manageral contracts to managers, the frst best allocaton can be attaned. Ths s because t s possble for the government to adjust u such that both managers maxmze socal welfare. Fgure 1 demonstrates the attanment of the frst best and the three cases (smultaneous move and the two Stackeleberg leadershps). Curves R 0 R 0 and R 1R 1 are reacton curves of the owners 0 and 1, respectvely. AA depcts the output profle satsfyng u 0 = argmax u0 W(p 0 (u 0, u 1 ), p 1 (u 0, u 1 )). R 0 R 0 and AA ntersect at FB, whch s the frst best allocaton. Further, R 0 R 0 ntersects R 1 R 1 at S, whch s the equlbrum pont n the smultaneous move case. As descrbed above, the subsdy shfts the prvate owner s reacton curve outwards. Thus, the government can acheve the frst best allocaton by adjustng subsdy rate s such that S overlaps FB. Lkewse, n the case where the government s a Stackleberg leader, ths allocaton s acheved at the same rate, snce the government can always choose S. Unfortunately, ths subsdy level does not yeld the frst best allocaton n the case of prvate leadershp, whch s shown n Fgure 1. The so-proft curve Π 1 Π 1, whch touches the reacton curve of the government at F, s depcted n ths fgure. The fgure ndcates that the prvate owner 1 commts to a lower u 1. Ths stems from avodng stff competton through both owners provdng effectve subsdes for ther managers excessvely. Accordngly, a hgher s s requred to tally F and FB. 6

9 Ù ½ Ê ¼ ¼ Ï ½ Ï ¼ Ê ¼ ½ Ê ½ Ë Ä Ç ¼ ½ Ê ¼ ¼ Ù ¼ Fgure 1: Delegaton sequencng 7

10 These results are summarzed n the followng proposton. Proposton 1. When the government provdes output subsdzaton, the frst best allocatons are acheved n the followng three cases: the smultaneous move (db) and the two sequental move cases (dl and df). Further, the effectve subsdy,.e., the sum of the equlbrum subsdy and the delegaton parameter s the same for all the three cases: u d j := s d + θ d j = a ( ) 1 b 2, = S, L, F, j = 0, 1. (8) 1 + b + k Moreover, the optmal subsdy rates and the relatonshps among them are as follows: a ( 2 2b 2 + k ) () s db = s dl = ( 2 b2 + k ) and (1 + b + k) a [ 2 + b 4 + 3k + k 2 b 2 (3 + k) ] s df = (2 + k) [ and b 3 + b (1 + k) b 2 (1 + k) + (1 + k) 2], () s db = s dl < s df. Ths proposton states that even f there s separaton between the ownershp and management wthn each frm, and each owner enters nto a F JS V-fashoned contract her/hs manager, the rrelevance result stll holds rrespectve of whether the two frms move smultaneously or one acts as the leader and the other as the follower wth regard to the decson of the delegaton parameter. More precsely, n a mxed market wth manageral delegaton, snce the effectve subsdy,.e., the sum of the producton subsdy and each frm s delegaton parameter plays the same role as the sole producton subsdy n the case where each frm s entrepreneural (every manageral decson s made by the owner of the frm), the two market falures under-producton resultng from mperfect competton and the dstorton of outputs (addtonally, cost neffcency) occurrng because of frms havng dfferent objectve functons are resolved, and nstead, cost effcency s restored by forcng all frms to produce the same amount of output at equal cost. 8 It s noteworthy that equlbrum delegaton parameters n all the delegaton sequencng regmes are negatve. θ db θ df ab ( 2 1 b 2 + k ) = θ dl = 2 b 3 + 3k + k 2 b 2 (1 + k) + b (2 + k) < 0 and [ = ab k + k 2 b 2 (1 + k) ] ( 1 b2 + k ) < 0, = 0, 1. (2 + k) (1 + b + k) 8 The equlbrum value of the producton subsdy n the entrepreneural case concdes wth that of the effectve subsdy n the manageral case nvestgated n ths paper. 8

11 Put dfferently, owners 0 and 1 mpose a non-physcal tax on ther managers. Although these results are ntrgung, they are not very surprsng. The welfare-maxmzng government has an ncentve to provde physcal subsdes to both frms excessvely because t fears that loose competton prevals n the market. To refran from ncreased producton because of such excessve subsdes, both owners offer negatve delegaton parameters. We should also note that the equlbrum profts for db and dl are equal and smaller than that for df. In fact, [ (2 + k) 2 b 2 (4 + k) ] Π db and = Π dl Π df = a2 2 ( 2 b 2 + k ) (1 + b + k), 2 ΠdF Π db = = a2 [ 2b 4 + (1 + k) (2 + k) 2 b 2 ( 6 + 4k + k 2)] 2 ( 1 b 2 + k ) (2 + k) (1 + b + k) 2, a 2 b 2 ( [ b 2 b 4 + k(2 + k) ] (2 + k)(1 + b + k) [ ] > 0, = 0, b 4 + 3k + k 2 b 2 (3 + 2k) However, the net proft, Π sq, s equal for all the delegaton sequencng structures. Π d j s d q d j = a 2 k, = B, L, F j = 0, 1. 2(1 + b + k) 2 COMPETITION SEQUENCING In ths secton, somewhat unlke n the delegaton sequencng case, we nvestgate the followng three stage game. In the frst stage, the government sets the optmum producton subsdy level for each frm n terms of socal welfare. In the second stage, the owners smultaneously choose ther delegaton parameter values. In the thrd stage, we consder the followng three cases. In the frst case whch s denoted by the superscrpt cb, the managers smultaneously decde the prces of ther frms. In the second case, whch s denoted by the superscrpt cl, frst the maneger of frm 0 decdes her/hs prce and s then followed by the manager of frm 1. In the thrd case whch s denoted by the superscrpt cf, the frst mover s the manager of frm 1 and the second mover s the manager of frm 0. 9 As n the delegaton case consdered n the prevous secton, we obtan the result that the frst best producton allocaton s acheved n all the three game structures. Proposton 2. When the government provdes output subsdy, the frst best allocatons are acheved n the followng three cases: the smultaneous move (cb) and the two sequental move cases 9 Obvously, the equlbrum market outcomes n case cb concde wth those n case db snce the game structure s the same n both the cases. 9

12 (cl and cf). Then, the effectve subsdes,.e., the sum of the equlbrum subsdes and the delegaton parameters n all the three cases are gven as follows: () u cb 0 = u cb 1 = u cl 1 = u cf 0 = a ( ) 1 b 2 a ( 2 2b 2 + k ), and ucl 0 = u cf 1 = ( 1 + b + k 2 b2 + k ) (1 + b + k), and () u cb 0 = u cb 1 = u cl 1 = u cf 0 > u cl 0 = u cf 1, where u c j = s c j + θ c j among them are as follows: ( = 0, 1, j = B, L, F). Moreover, the optmal subsdes and the relatonshps () s cl = a [ b 4 + (2 + k) 2 b 2 (5 + 2k) ] (1 + b + k) [ (2 + k) 2 b 2 (3 + k) ] and scb = s cf = () s cl > s cb = s cf. a ( 2 2b 2 + k ) ( 2 b2 + k ) (1 + b + k), and It s relatvely clear that the frst best allocaton s achevable n all the competton regmes, because two control varables, s and θ 0, are avalable to the government to correct dstorton due to shortage n supples and to correct the competton structures (or the leadershps of publc and prvate frms). Thus, n what follows, we concentrate on explanng why the effectve subsdes n equlbrum dffer for the three competton regmes. For ths purpose, we present the delegaton parameters n equlbrum: ab ( 4 1 b 2 + k ) 0 = b(2+k) 3 +(1+k)(2+k) 3 +b 5 (3+k)+b 4 (3+4k+k 2 ) b 3 (10+9k+2k 2 ) b 2 (10+19k+11k 2 +2k 3 ) < 0, θ cl ab ( 2 1 b 2 + k ) (2 + k) θ cl 1 = b (2 + k) 2 + (1 + k) (2 + k) 2 b 3 (3 + k) b ( k + k 2) < 0, ab ( 2 1 b 2 + k ) θ cf 0 = θ cb = < 0, = 0, 1, and 2 b 3 + 3k + k 2 b 2 θcf 1 = 0, (1 + k) + b (2 + k) b (0, 1) and k > 0. Frst, we demonstrate the case of cf. In the case of cb, owner 1 sets θ 1 such that her/hs manager exhbts the same behavor as the owner who s a leader n the prce settng game wthout any separaton between management and ownershp. In the case of cf, however, the manager behaves as a leader wthout any delegaton contracts; thus, owner 1 sets θ 1 = 0 regardless of the delegaton parameter of the publc frm θ Nevertheless, snce the frst best allocaton s acheved, 10 Thus, n the case of cf, the objectve functon of owner 1,.e., her/hs sole proft concdes wth that of her/hs correspondng manager. 10

13 the reacton curve of the publc frm s manager should go through the frst best allocaton pont and the so-proft curve of the prvate frm should touch ths curve at that pont. These hold only when u 0 = u cb 0,.e., when ucf 0 = u cb 0. Moreover, takng nto account the fact that the tangency between the reacton curve of the publc frm s manager and the so-proft curve at the frst best allocaton s owng to the proft maxmzaton by the prvate owner under a gven subsdy rate, we mmedately fnd that s cf = s cb. Ths mples that u cf 1 s less than u cb 1 by θ cb 1. Next, let us proceed to the case of cl. Of course, the frst-best allocaton s attaned n ths case, too. As n the case of cf, the so-payoff curve of the publc frm s manager as a leader should be tangent to the reacton curve of the prvate frm s manager at the frst best allocaton pont, and as such, u cl 1 = u cb 1. Here, recall that the thrd stage s parallel n all the competton sequencng regmes, n that both the managers have dentcal objectves,.e., maxmzng the sum of proft and effectve subsdy. Further, recall that u cb 1 = u cf 0. Thus, f ucl 1 = u cf 0, then ucl 0 = u cf 1 n order to ensure the frst best allocaton. However, the optmal subsdy s cl s larger than the other ones s cb = s cf. In the case of cl, the government should encourage the managers to produce more, because not only does the prvate frm s manager behave less aggressvely owng to the manageral contract offered by the prvate owner but also the publc frm s manager, as a Stackelberg leader, s less aggressvely. Fnally, we make some remarks. The frst relates to the profts of the frms n competton sequencng that are as follows: [ 2b 4 + (2 + k) 3 b ( k + k 2)] Π cl Π cl = a2 2 (1 + b + k) [ 2 (2 + k) 2 b 2 (3 + k) ], ΠcB Π cb = = Π cf a 2 b 4 (1 b 2 + k) (2 b 2 + k)(1 + b + k) [ ] > 0, = 0, 1. 2 (2 + k) 2 b 2 (3 + k) [ = a2 (2 + k) 2 b 2 (4 + k) ] 2 ( 2 b 2 + k ) (1 + b + k), and 2 As n the delegaton sequencng case, n competton sequencng, the proft of the frm that receves the hghest subsdy rate s the largest n all the three cases. Lkewse, the net profts are dentcal: Π c j s c q c j = a 2 k, = B, L, F, j = 0, 1. 2(1 + b + k) 2 The second remark s on the mportance of Proposton 2, whch n our case s hgher as compared to n prevous lterature on mxed olgopoly wth manageral delegaton. If the producton subsdy s not taken nto account, analogous to ths paper s case where the producton subsdy s consdered, the equlbrum delegaton parameter of frm 1, as n Nakamura (2008) and Nakamura and Inoue (2009), s negatve for an arbtrary degree of producton dfferentaton n both 11

14 smultaneous and sequental move cases. Both Nakamura (2008) and Nakamura and Inoue (2009) consder a prce settng mxed olgopoly wth quadratc cost functons and constant margnal cost. However, n ths paper, n both the delegaton sequencng and competton sequencng cases, the delegaton parameter of frm 0 s also negatve for all b (0, 1) and k > 0. Ths s qute unlke n Nakamura (2008) and Nakamura and Inoue (2009). The fnal remark s on the relatonshp between our propostons and the results n the exstng works on mxed olgopoly wth subsdzaton. Hashmzade et al. (2007) showed that the frst best allocaton can be acheved by provdng equal subsdy under both cases: the smultaneous move case and the case where there s publc leadershp n mxed olgopoly wth prce competton and no separaton between the management and the ownershp. One of the contrbutons of ths paper s to examne the robustness of ther results when manageral delegaton s consdered. Propostons 1 and 2 state that ther results hold only n delegaton sequencng. Surprsngly, we found that, by focusng on the effectve subsdes, ther results can be extended to the case where the publc frm s a follower n delegaton sequencng. Further, as ndcated n Tomaru and Sato (2009) and Zkos (2007), a mxed olgopoly wth both subsdzaton and prvate leadershp requres a smaller subsdy n quantty competton and larger subsdy n prce competton, as compared to the subsdy requred n the smultaneous move case to acheve the frst best allocaton. Once we take nto account the separaton of the management and ownershp, these results are altered. CONCLUSION Ths paper nvestgated the optmal subsdzaton problem n a mxed duopoly where the owners of both the publc frm and the prvate frm enter nto a sales delegaton contract wth ther managers as n Fershtman and Judd (1987), Sklvas (1987), and Vckers (1985) (the so-called FJS V contract). For ths purpose, we formulated a game wth the followng three stages. In the frst stage, the government sets each frm s producton subsdy at the optmum level wth respect to socal welfare. In the second stage, the owner of each frm decdes her/hs delegaton parameter for the sales delegaton contract, and subsequently, n the thrd stage, each frm determnes ts prce n the dfferentated goods market. Then, we analyze the followng two cases. () Delegaton sequencng case: where the frms smultaneously/sequentally decde ther delegaton parameters n the second stage, and the managers smultaneously choose ther prces n the thrd stage. () Competton sequencng case: the case where the frms set ther prces sequentally/smultaneously n the thrd stage, and the owners smultaneously decde ther delegaton parameters n the second stage. 12

15 In both the cases, for all types of moves, we showed that the frst best producton allocaton wth respect to socal welfare s acheved, and the equlbrum delegaton parameter of each frm s negatve for any degree of producton dfferentaton. The equlbrum delegaton parameter of the publc frm beng negatve was qute unexpected: on the other hand, the prvate frm s delegaton parameter beng negatve was expected and n lne wth Nakamura (2008) and Nakamura and Inoue (2009) who consder a mxed duopolstc ndustry wthout subsdzaton. More concretely, n case (), the effectve subsdy of each frm,.e., the sum of producton subsdy and the delegaton parameter of each frm, concde for all the three types of moves. Furthermore, when the owner of the publc frm s the follower, n order to ncrease the equlbrum total output n the market, the government gves the hghest producton subsdy to both the frms n all the three cases because the total output of the market tends to stay low as the prvate frm (the market leader) s less aggressve. On the other hand, n case (), we fnd that the government provdes excess subsdy, when the publc frm s the leader. However, the frst best allocaton s acheved for all the three moves. Thus, we realze that the move that the government payng the hghest producton subsdy s dfferent for cases () and (). Ths result mples that n a mxed olgopolstc ndustry of manageral frms, the government must set the value of each frm s producton subsdy takng nto account ther moves. Ths leads to the equlbrum level of each frm s producton subsdy beng larger as compared to the equlbrum (optmal) producton subsdy obtaned n a standard mxed olgopoly wthout manageral delegaton. Consequently, the achevement of the frst best allocaton usng the optmal producton subsdy n mxed olgopoly, the so-called rrelevance result s robust aganst the ntroducton of sales delegaton n prce competton. Ths s smlar to the quantty competton consdered n Tomaru et al. (2009). In addton, even f the market contans two prvate frms whose objectve functons are ther profts and whose owners enter nto a FJS V delegaton contract wth ther managers,.e., prvate duopoly wth sales delegaton, the frst best allocaton, as n the above two propostons n ths paper, holds as long as the owners smultaneously decde ther delegaton parameters n the second stage, and ther prces smultaneously n the thrd stage. 11 We seek to extend our analyss of mxed duopoly to general mxed olgopoly wth one publc frm and n ( 2) prvate frms n the future, and thus follow exstng works on subsdzaton n a mxed olgopolstc ndustry, such as Whte (1996), Poyago-Theotoky (2001), and Kato and Tomaru (2007). However, we consder that the result obtaned n ths model holds n a general mxed olgopoly model as long as multple prvate frms move smultaneously. Furthermore, although we addressed sales delegaton as n Fershtman and Judd (1987) Sklvas (1987), and 11 The equlbrum market outcomes for ths case (case (dpb)) are presented n the Appendx. 13

16 Vckers (1985), as an ncentve contract wthn a frm, we must clarfy the robustness of our result aganst other delegaton schemes, such as the market share delegaton of Jansen et al. (2007) and the relatve performance case of Mller and Pazgal (2001). The extensons are left for future research. References Bárcena-Ruz JC The decson to hre managers n mxed markets under Bertrand competton. Japanese Economc Revew (forthcomng). Barros F Incentve schemes as strategc varables: An applcaton to a mxed duopoly. Internatonal Journal of Industral Organzaton 13: Colle DR Delegaton and Strategc Trade Polcy. Internatonal Economc Journal 11: Colle DR Mergers and Trade Polcy under Olgopoly. Revew of Internatonal Economcs 11: Das SP Strategc Manageral Delegaton and Trade Polcy. Journal of Internatonal Economcs 43: Dxt A A Model of Duopoly Suggestng a Theory of Entry Barrers. Bell Journal of Economcs 10: Fershtman C, Judd K Equlbrum Incentves n Olgopoly. Amercan Economc Revew 77: Goerng GE The Strategc Use of Manageral Incentves n a Non-Proft Frm Mxed Duopoly Manageral and Decson Economcs 28: González-Maestre M, López-Cuñat J Delegaton and mergers n olgopoly. Internatonal Journal of Industral Organzaton 19: Hashmzade N, Khodavas H, Myles G An rrelevance result wth dfferentated goods. Economcs Bulletn 8(2): 1-7. Heywood JS. Ye G Delegaton n a Mxed Olgopoly: The Case of Multple Prvate Frms. Manageral and Decson Economcs 30,

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19 Zhang J, Zhang Z R&D n a Strategc Delegaton Game. Manageral and Decson Economcs 18: Zkos V Stackelberg mxed olgopoly wth asymmetrc subsdes. Economcs Bulletn 12(13): 1-5. Zss S Horzontal Mergers and Delegaton. Internatonal Journal of Industral Organzaton 19: Appendx Prvate Frms n the smultaneous move case Here, we present the equlbrum market outcomes n the case (case dpb) where both the frms are prvate and proft-maxmzers, and smultaneously decde ther delegaton parameters, and prce levels. θ dpb = p dpb = ab 2 ( 1 b 2 + k ) 2 b 3 + 3k + k 2 b 2 (1 + k) + b (2 + k), sdpb = ak 1 + b + k, qdpb = a, and ΠdpB 1 + b + k and CS dpb = a2 (1 + b) (1 + b + k) 2, WdpB = = a2 a ( 2 2b 2 + k ) 2 b 3 + 3k + k 2 b 2 (1 + k) + b (2 + k), [ (2 + k) 2 b 2 (4 + k) ] 2 ( 2 b 2 + k ), ( = 0, 1), 2 (1 + b + k) a b + k. (9) 17