Wage-rse contract and endogenous tmng n nternatonal med duopoly Kazuhro Ohnsh Osaka Unversty, Ph. D. July 007 Abstract The study of Matsumura (003) nvestgates a med duopoly model, where a domestc publc frm and a foregn prvate frm frst choose the tmng for choosng ther quanttes and shows that, n contrast to Pal (998) dscussng a case of domestc compettors, the publc frm becomes the leader. Ths paper eamnes an nternatonal med duopoly model, where a domestc publc frm competes aganst a foregn prvate frm. Each frm frst chooses the tmng for adoptng a wage-rse contract as a strategc commtment. The paper shows the equlbrum of the nternatonal med model. Keywords: Wage-rse contract, Endogenous tmng, Internatonal med duopoly, Domestc publc frm, Foregn prvate frm JEL classfcaton: F3, H4, L30 Correspondng author. Phone/fa: +8-7-7-8638. Emal: ohnsh@e.people.or.jp
. Introducton The study of Hamlton and Slutsky (990) eamnes the novel ssue of endogenous tmng n two-payoff-mamzng player games, wth mportant modellng mplcatons for several model n ndustral economcs. In a preplay stage, players decde whether to select actons n the basc game at the frst opportunty or to wat untl observng ther rvals frst perod actons. In one etended game, players frst decde when to select actons wthout commttng to actons n the basc game. The equlbrum has a smultaneous play subgame unless payoffs n a sequental play subgame Pareto domnate those payoffs. In another etended game, decdng to select at the frst turn requres commttng to an acton. They show that both sequental play outcomes are the equlbra only n undomnated strateges. Amr (995) relates to the study of Hamlton and Slutsky (990) on endogenous tmng (wth observable delay). He shows, va countereample, that monotoncty of the best-response functons n a two-player game s not suffcent to derve predctons about the order of moves and ths requres, addtonally, the monotoncty of each payoff n the other player s actons. Pal (998) analyses the subgame perfect Nash equlbrum of a med market, where one socal-surplus-mamzng publc and N proft-mamzng prvate frms frst choose the tmng for selectng ther quanttes, and shows that the results are strkngly dfferent from those obtaned n a correspondng quantty-settng olgopoly wth all proft-mamzng frms. Furthermore, Matsumura (003) eamnes a Stackelberg med duopoly where a socal-surplus-mamzng domestc publc frm competes aganst a proft-mamzng foregn prvate frm and shows that, n contrast to Pal (998) dscussng a case of domestc compettors, the publc frm should be the leader. We study the behavours of a socal-surplus-mamzng domestc publc frm and a proft-mamzng foregn prvate frm n an nternatonal med duopoly model. Each frm frst chooses the tmng for adoptng the wage-rse-contract polcy (henceforth Followng the poneerng work of Merrll and Schneder (966), the analyss of med market models that ncorporate socal-welfare-mamzng publc frms has receved ncreasng attenton n recent year. See, for nstance, Bös (986, 00), Vckers and Yarrow (988), Cremer, Marchand and Thsse (989) and Nett (993) for ecellent surveys.
WRCP). We consder the followng stuaton. In the frst stage, each frm ndependently chooses stage or stage 3. At the end of the frst stage, each frm observes the behavour of the rval. In the second stage, the frm choosng stage adopts WRCP n ths stage. At the end of the second stage, each frm observes the behavour of the rval. In the thrd stage, the frm choosng stage 3 adopts WRCP n ths stage. At the end of the thrd stage, each frm observes the behavour of the rval. At the end of the game, each frm ndependently chooses and sells ts actual output. We dscuss the equlbrum of the nternatonal med duopoly model. The purpose of ths paper s to show the role of WRCP as a strategc commtment n the nternatonal med model where the domestc publc frm competes aganst foregn prvate frm. The paper s organzed as follows. In Secton, we formulate the nternatonal med model. Secton 3 dscusses the equlbrum of the nternatonal med model. Secton 4 concludes the paper. All proofs of Lemmas and Propostons are presented n the Append.. The model Let us consder an nternatonal med duopoly model wth one socal-surplus-mamzng domestc publc frm (frm ) and one proft-mamzng foregn prvate frm (frm ), producng perfectly substtutable goods. For the remander of ths paper, when and j are used to refer to frms n an epresson, they should be understood to refer to and wth j. There s no possblty of entry or et. The market prce s determned by the nverse demand functon p( X ), where denotes the aggregate quantty. We assume that p ' < 0 and p '' < 0. The tmng of the game runs as follows. In the frst stage, each frm ndependently chooses t (,3), where t ndcates when to adopt WRCP. That s, t = mples that frm can adopt WRCP n the second stage, and t = 3 mples that t can adopt WRCP n the thrd stage. At the end of the frst stage, each frm observes t and t. In the second stage, frm choosng t = can adopt WRCP n ths stage. At the end of X = = For detals see Ohnsh (003). 3
the second stage, each frm observes the behavour of the rval. In the thrd stage, frm choosng t = 3 can adopt WRCP n ths stage. At the end of the thrd stage, each frm observes the behavour of the rval. At the end of the game, each frm ndependently chooses and sells ts actual 0. If frm adopts WRCP, then t chooses an output level 0 and a wage premum rate w > 0, and agrees to pay each employee a wage premum unformly f t actually produces more than. Hence, frm s proft s gven by px ( ) c( ) f, π = () px ( ) c( ) ( ) w f, where c ( ) s the cost functon. We assume that frms have dentcal technologes wth c ' > 0 and c '' > 0. 3 The objectve of frm s to mamze ts own proft. Domestc socal surplus (S), whch s the sum of consumers surplus and frm s proft, s gven by X pqdq ( ) c( ) px ( ) 0 f, S = X pqdq ( ) c f. ( ) ( ) w px ( ) 0 The objectve of frm s to mamze domestc socal surplus. WRCP specfes a hgher margnal cost for output, and frm s margnal cost ehbts a dscontnuty at =. In ths paper, we wll dscuss subgame perfect equlbra n undomnated strateges. 4 We derve both frms reacton functons n quanttes. If frm s margnal cost s c ', ts reacton functon s defned by ( ) argma X R = pqdq ( ) c ( ) px ( ) { 0}. (3) 0 () 3 The assumpton of ncreasng margnal cost s often used n lterature studyng med markets. See, for nstance, Harrs and Wens (980), Ware (986), Delbono and Scarpa (995), Fjell and Pal (996), Whte (996), Pal and Whte (998), Bárcena-Ruz and Garzón (003) and Matsumura and Kanda (005). If c " 0, then frm mamzes socal surplus by supplyng monopolstcally n the market. Ths assumpton s made to elmnate such a trval soluton. 4 Ths s also used n Hamlton and Slutsky (990) and Matsumura (003). 4
If frm s margnal cost s c' + w, ts reacton functon s defned by ( ) argma X w R = pqdq ( ) c ( ) ( ) ( ) w px { 0}. (4) 0 Therefore, f frm offers WRCP, then ts best response s R( ) f <, W R ( ) = f =, w R ( ) f >. The equlbrum occurs where each frm mamzes ts objectve wth respect to ts own output level, gven the output level of ts rval. Frm ams to mamze socal surplus wth respect to, gven. The equlbrum must satsfy the followng condtons. If frm s margnal cost s c ', the frst-order condton s p c ' p' = 0. (6) If frm s margnal cost s c' + w, the frst-order condton s p c ' w p' = 0. (7) The second-order condton s p' c " p" < 0. (8) Furthermore, we have R '( ) = R '( ) = w p" p ' p" (5). (9) w Thus, R( ) and R ( ) are upward slopng. If frm s margnal cost s c ', ts reacton functon s defned by [ ] R ( ) = argma p( X) c ( ). (0) { 0} If frm s margnal cost s c' + w, ts reacton functon s defned by R ( ) = argma p( X) c ( ) ( ) w. () w { 0} Therefore, f frm offers WRCP, then ts best response s R( ) f <, W R ( ) = f =, w R ( ) f >. Frm ams to mamze ts proft wth respect to, gven. The equlbrum must satsfy the followng condtons. If frm s margnal cost s () c ', the frst-order condton 5
s p ' + p c' = 0. (3) If ts margnal cost s c' + w, the frst-order condton s p ' + p c' w= 0, (4) The second-order condton s p' + p'' c " < 0. (5) Furthermore, we have w p' + p" R'( ) = R '( ) =. (6) p ' + p" c " w Thus, R( ) and R ( ) are downward slopng. 3. Equlbrum In ths secton, we state the equlbrum of the nternatonal med model. Frst of all, we wll present the net two lemmas. Lemma : If frm adopts WRCP, then n equlbrum =. Lemma : Frm s optmal output s smaller when t adopts WRCP than when t does not. These lemmas provde characterzatons of WRCP as a strategc commtment. Lemma means that n equlbrum frm does not pay ts employees wage premums. Lemma means that f frm offers WRCP, then ts optmum output decreases. Now, we wll dscuss the followng three cases: Case. t = and t = 3 Case. t = 3 and t = Case 3. t = t = ( t = t = 3 ) We wll dscuss these cases n order. 6
Case. t = and t = 3 In ths case, frst frm moves, then frm observes frm s move, then frm moves. In the second stage, frm can adopt WRCP. At the end of the second stage, frm observes frm s behavour. In the thrd stage, frm can adopt WRCP. At the end of the thrd stage, frm observes frm s behavour. At the end of the game, each frm ndependently chooses ts actual output, and both socal surplus and frm s proft are decded. If unlaterally frm offers WRCP, then ts margnal cost ncreases and thus t decreases ts output. Gven the output level of frm, decreasng frm s output decreases the total market output and rases the market prce. Hence, frm s proft ncreases. Furthermore, frm ncreases ts output because of strategc substtutes. Hence, socal surplus may ncrease. The equlbrum outcome can be stated as follows. Proposton : Consder the game where t = and t = 3. Then there ests an equlbrum whch concdes wth the Stackelberg soluton where frm leads and frm L C L C follows. Furthermore, S > S and π > π. The superscrpt L denotes the equlbrum outcome where frm s the Stackelberg leader, and the superscrpt C the equlbrum outcome of the Cournot game wthout WRCP. Proposton means that n the second stage, frm chooses L and w correspondng to ts Stackelberg leader soluton and agrees to pay each employee a wage premum unformly f t actually produces more than. L L Case. t = 3 and t = In ths case, frst frm moves. In the second stage, frm can adopt WRCP. At the end of the second stage, frm observes frm s behavour. In the thrd stage, frm can adopt WRCP. At the end of the thrd stage, frm observes frm s behavour. At the end of the game, each frm ndependently chooses ts actual output, and both socal surplus and frm s proft are decded. If frm unlaterally offers WRCP, then ts margnal cost ncreases and thus t decreases ts output. Gven the output level of frm, decreasng frm s output decreases the total market output and rases the market prce. Furthermore, frm decreases ts output because of strategc complements. Gven the output level of frm, decreasng frm s output decreases the total market output and 7
rases the market prce. Hence, frm s proft ncreases, whle socal surplus decreases. The equlbrum outcome can be stated as follows. Proposton : Consder the game where t = 3 and t =. Then there ests an equlbrum whch concdes wth the Stackelberg soluton where frm leads and frm F C F C follows. Furthermore, π > π and S < S. The superscrpt F denotes the equlbrum outcome where frm s the Stackelberg follower. Proposton means that n the second stage, frm chooses F and w correspondng to ts Stackelberg leader soluton and agrees to pay each employee a wage premum unformly f t actually produces more than. F F Case 3. t = t = ( t = t = 3 ) Case 3 s the case n whch both frms choose the same stage n the frst stage. If each frm chooses t =, ths case runs as follows. In the second stage, each frm can ndependently offer WRCP. At the end of the second stage, each frm observes the behavour of the rval. In the thrd stage, nether frm acts. At the end of the game, each frm ndependently chooses ts actual output, and both socal surplus and frm s proft are decded. In Matsumura s (003) model, f both frms choose the same stage n the frst stage, the equlbrum occurs at the Cournot soluton. However, n our model, f frm offers WRCP, then ts best response changes as defned by (5) and (). Hence, even f both frms choose the same stage n the frst stage, the equlbrum does not occur at the Cournot soluton. If frm unlaterally offers WRCP, then both socal surplus and frm s proft ncrease. Therefore, frm hopes that frm wll offer WRCP. On the other hand, f frm unlaterally offers WRCP, then frm s proft ncreases, whle socal surplus decreases. Therefore, frm hopes that frm wll not offer WRCP. The equlbrum outcome can be stated as follows. Proposton 3: Consder the game where both frms choose the same stage n the frst stage. If π π, there ests an equlbrum whch concdes wth the Stackelberg F L soluton where frm s the leader, whle f π > π, there ests an equlbrum whch 8
concdes wth the Stackelberg soluton where frm s the leader. The man result of ths study s descrbed by the followng proposton. Proposton 4: In the nternatonal med model, there est two equlbra: () t = and t = and () t = and t = 3. 4. Concluson We have eamned an nternatonal med model, where a socal-surplus-mamzng domestc publc frm and a proft-mamzng foregn prvate frm frst choose the tmng for adoptng WRCP. We have shown that there est two equlbra: () stage for the domestc publc frm and stage for the foregn prvate frm and () stage for the domestc publc frm and stage 3 for the foregn prvate frm. Matsumura (003) eamnes a Stackelberg med duopoly where a socal-surplus-mamzng domestc publc frm competes aganst a proft-mamzng foregn prvate frm and shows that there ests a unque equlbrum: stage for the domestc publc frm and stage 3 for the foregn prvate frm. Our result s dfferent from that of Matsumura (003). It s thought that WRCP s effectve for the foregn prvate frm. Append Proof of Lemma Frst, we prove that f frm adopts WRCP, then n equlbrum possblty that > n equlbrum. From (), socal surplus s =. Consder the Here, f X W S = pqdq ( ) c( ) ( ) w px ( ). 0 >, frm must pay ts employees wage premums can mprove socal surplus by rsng. Hence, > does not result n an equlbrum. Consder the possblty that 9 ( ) w. That s, frm, and the equlbrum pont does not change n < n equlbrum. From (), frm s margnal cost s
c '. It s mpossble for frm to change ts output n equlbrum because such a strategy s not credble. That s, f <, WRCP does not functon as a strategc commtment. Net, we prove that f frm adopts WRCP, then n equlbrum possblty that Here, f > n equlbrum. From (), frm s proft s W π = p( X) c( ) ( ) w. >, frm must pay ts employees wage premums frm can mprove ts proft by rsng. Hence, > does not result n an equlbrum. =. Consder the ( ) w. That s,, and the equlbrum pont does not change n Consder the possblty that < n equlbrum. From (), frm s margnal cost s c '. It s mpossble for frm to change ts output n equlbrum because such a strategy s not credble. That s, f <, WRCP does not functon as a strategc commtment. Q.E.D. Proof of Lemma Frst, frm s socal-surplus-mamzng output s smaller when t adopts WRCP than when t does not. From (), we see that WRCP wll never decrease the margnal cost of frm. If frm s margnal cost s c ', the frst-order condton s (6), and f ts margnal cost s c ' + w, the frst-order condton s (7). Here, w s postve. To satsfy (7), p c ' p' must be postve. Thus, frm s optmum output s smaller when ts margnal cost s c' + w than when ts margnal cost s c '. Net, frm s proft-mamzng output s smaller when t adopts WRCP than when t does not. From (), we see that WRCP wll never decrease the margnal cost of frm. If frm s margnal cost s c ', the frst-order condton s (3), and f ts margnal cost s c' + w, the frst-order condton s (4). Here, w s postve. To satsfy (4), p ' + p c ' must be postve. Thus, frm s optmum output s smaller when ts margnal cost s c' + w than when ts margnal cost s c '. Q.E.D. Proof of Proposton We consder frm s Stackelberg leader output when each frm s margnal cost s Frm selects, and frm selects after observng. If frm s the Stackelberg leader, then t mamzes socal surplus S (, R( )) wth respect to. Therefore, frm s Stackelberg leader output satsfes the frst-order condton: 0 c '.
p c' p ' pr ' ' = 0. (7) From p', R ' < 0, to satsfy (7), p c' p' must be postve. Hence, frm s Stackelberg leader output s smaller than ts Cournot output. Lemma shows that socal-surplus-mamzng output s smaller when frm adopts WRCP than when t does not. From (7), we see that a decrease n frm s output s decded by the value of w. Let w be a varable that can take any value more than zero. Thus, the equlbrum concdes wth the Stackelberg soluton where frm s the leader. In R, the Stackelberg leader (frm ) mamzes domestc socal surplus and can choose ts Cournot output. Hence, we obtan S L S C. We show that S L S C. Let X ( ) ( ) ( ) be contnuous and concave wth respect to 0 S= pqdq c px. In R, socal surplus s the hghest at frm s Stackelberg leader pont, and the further the pont on R gets from frm s Stackelberg leader pont, the more socal surplus decreases. Frm s Stackelberg leader output s smaller than ts Cournot output. Lemma shows L C that n equlbrum =. Thus, S > S. L C Fnally, we show that π > π. Frm s Stackelberg leader output s smaller than ts Cournot output. Snce π / = p' < 0, decreasng ncreases π gven. L C Frm s optmal strategy must yeld at least ths proft. Thus, π > π. Q.E.D. Proof of Proposton We consder frm s Stackelberg leader output when each frm s margnal cost s Frm selects, and frm selects after observng. If frm s the Stackelberg leader, then t mamzes ts proft π (, R( )) wth respect to. Therefore, frm s Stackelberg leader output satsfes the frst-order condton: p c ' + p' + p' R ' = 0. (8) From p ' < 0 and R ' > 0, to satsfy (8), p c' + p' must be postve. Hence, frm s Stackelberg leader output s smaller than ts Cournot output. Lemma shows that frm s proft mamzng output s smaller when frm adopts WRCP than when t does not. From (4), we see that a decrease n frm s output s decded by the value of w. Let w be a varable that can take any value more than zero. Thus, the equlbrum concdes wth the Stackelberg soluton where frm s the leader. In R, the Stackelberg leader (frm ) mamzes domestc ts proft and can choose ts c '.
F C F C Cournot output. Hence, we obtan π π. We show that π π. Let π = p( X) c( ) be contnuous and concave wth respect to. In R, frm s proft s the hghest at ts Stackelberg leader pont, and the further the pont on R gets from frm s Stackelberg leader pont, the more frm s proft decreases. Frm s Stackelberg leader output s smaller than ts Cournot output. Lemma shows that n F C equlbrum =. Thus, π > π. F C Fnally, we show that S < S. Consder the game where frm s the Stackelberg follower. Snce S(, R( ))/ = p' > 0, S(, R( )) s ncreasng n. Frm s Stackelberg leader output s smaller than ts Cournot output. Thus, S S(, R ( )) < S(, R ( )) S. Q.E.D. F F F C C C Proof of Proposton 3 Frst, suppose that π > π. Then frm hopes that frm wll adopt WRCP. Snce S > S, frm chooses L and w correspondng to ts Stackelberg leader soluton L and adopts WRCP. Suppose that frm also adopts WRCP. Lemma shows that n equlbrum =. From (5) and (), we see that each frm s reacton functon has a zero slope at =. Ths mples that even f j s ncreased, s constant. Hence, frm can ncrease ts proft by ncreasng and. Frm mamzes ts proft by ncreasng and to a pont of R. Thus, the equlbrum concdes wth the Stackelberg soluton where frm s the leader. F F Second, suppose that π < π, so that frm chooses and w correspondng to ts Stackelberg leader soluton and adopts WRCP. Suppose that frm also adopts WRCP. Lemma shows that n equlbrum =. From (5) and (), we see that each frm s reacton functon has a zero slope at =. Ths mples that even f j s ncreased, s constant. Hence, frm can ncrease socal surplus by ncreasng and. Frm mamzes socal welfare by ncreasng and to a pont of R. Thus, the equlbrum concdes wth the Stackelberg soluton where frm s the leader. Thrd, suppose that π = π. If frm adopts WRCP, then the equlbrum concdes wth the Stackelberg soluton where frm s the leader, and socal surplus decreases. On the other hand, f frm adopts WRCP, then the equlbrum concdes wth the Stackelberg soluton where frm s the leader, and socal surplus ncreases. Snce S > S, frm chooses L and w correspondng to ts Stackelberg leader soluton L
and adopts WRCP. Suppose that frm also adopts WRCP. Lemma shows that n equlbrum =. From (5) and (), we see that each frm s reacton functon has a zero slope at =. Ths mples that even f j s ncreased, s constant. Hence, frm can ncrease ts proft by ncreasng and. Frm mamzes ts proft by ncreasng and to a pont of R. Thus, the proposton follows. Q.E.D. Proof of Proposton 4 From Propostons -3, we can consder the followng two matrces: () for π < π, Frm Stage Stage 3 Frm Stage S, π S, π F F L L Stage 3 S, π S, π F F F F and () for π π, Frm Stage Stage 3 Frm Stage S, π S, π L L L L Stage 3 S, π S, π From Propostons -, we see that F F L L S > S. In (), the equlbrum occurs at stage for frm and stage for frm. In (), the equlbrum occurs at stage for frm and stage 3 for frm. Q.E.D. References Amr, R., 995. Endogenous tmng n two-player games: a countereample. Games and Economc Behavor 9, 34-37. Bárcena-Ruz, J. C., Garzón, M. B., 003. Med duopoly, merger and multproduct frms. Journal of Economcs/Zetschrft für Natonalökonome 80, 7-4. Bös, D., 986. Publc Enterprse Economcs. North-Holland, Amsterdam. Bös, D., 00. Prvatzaton: A Theoretcal Treatment. Clarendon Press, Oford. Cooper, T. E., 986. Most-favored-customer prcng and tact colluson. Rand Journal of Economcs 7, 377-388. Cremer, H., Marchand, M., Thsse, J.-F., 989. The publc frm as an nstrument for 3
regulatng an olgopolstc market. Oford Economc Papers 4, 83-30. Cremer, H., Marchand, M., Thsse, J.-F., 99. Med olgopoly wth dfferentated products. Internatonal Journal of Industral Organzaton 9, 43-53. Delbono, F., Scarpa, C., 995. Upward-slopng reacton functons under quantty competton n med olgopoles. Bulletn of Economc Research 47, 34-346. Fjell, K., Pal, D., 996. A med olgopoly n the presence of foregn prvate frms. Canadan Journal of Economcs 9, 737-743. George, K., La Manna, M., 996. Med duopoly, neffcency, and publc ownershp. Revew of Industral Organzaton, 853-860. Harrs, R. G., Wens, E, G., 980. Government enterprse: an nstrument for nternal regulaton of ndustry. Canadan Journal of Economcs 3, 5-3. Hamlton, J. H., Slutsky, S. M., 990. Endogenous tmng n duopoly games: Stackelberg or Cournot equlbra. Games and Economc Behavor, 9-46. Matsumura, T., 003. Stackelberg med duopoly wth a foregn compettor. Bulletn of Economc Research 55, 75-87. Matsumura, T., Kanda, O., 005. Med olgopoly at free entry markets. Journal of Economcs/Zetschrft für Natonalökonome 84, 7-48. Merrll, W., Schneder, N., 966. Government frms n olgopoly ndustres: a short-run Analyss. Quarterly Journal of Economcs 80, 400-4. Mujumdar, S., Pal, D., 998. Effects of ndrect taaton n a med olgopoly. Economcs Letters 58, 99-04. Nelson, W. S., Wnter, H., 99. Unlateral most-favored-customer prcng: a comparson wth Stackelberg. Economcs Letters 38, 9-3. Nett, L., 993. Med olgopoly wth homogeneous goods. Annals of Publc and Cooperatve Economcs 64, 367-393. Nett, L., 994. Why prvate frms are more nnovatve than publc frms. European Journal of Poltcal Economy 0, 639-653. Ohnsh, K., 003. A model of a prce-settng duopoly wth a wage-rse contract. Australan Economc Papers 4, 49-57. Ohnsh, K., 006. A med duopoly wth a lfetme employment contract as a strategc commtment. FnanzArchv 6, 08-3. Pal, D., 998. Endogenous tmng n a med olgopoly. Economcs Letters 6, 8-85. 4
Pal, D., Whte, M. D., 998. Med olgopoly, prvatzaton, and strategc trade polcy. Southern Economc Journal 65, 64-8. Vckers, J., Yarrow, G.., 988. Prvatzaton: An Economc Analyss. MIT Press, Cambrdge MA. Ware, R., 986. A model of publc enterprse wth entry. Canadan Journal Economcs 9, 64-655. Whte, M. D., 996. Med olgopoly, prvatzaton and subsdzaton. Economcs Letters 53, 89-95. 5