Workng Paper Seres No51, Faculty of Economcs, Ngata Unversty Export Subsdes and Tmng of Decson-Makng An Extenson to the Sequental-Move Game of Brander and Spencer (1985) Model Koun Hamada Seres No51 Address: 8050 Ikarash 2-no-cho, Ngata Cty, 950-2181 Japan Tel and fax: +81-25-262-6538 E-mal address: khamada@econngata-uacp
Export Subsdes and Tmng of Decson-Makng An Extenson to the Sequental-Move Game of Brander and Spencer (1985) Model Koun Hamada Abstract Ths paper examnes how the tmng of decson-makng affects strategc trade polcy In ths paper, we analyze the relatonshp between the dfferent tmng of decson-makng by exportng frms and ther subsdzng governments and ts mpact on export subsdy The paper ams to extend the analyss of Brander and Spencer (1985) to nclude the Stackelberg competton and the sequental-move decson on the subsdy choce by governments Some man results are presented as follows: Frst, when governments decde smultaneously the export subsdes n advance under the followng Stackelberg quantty competton, the orgnal leader frm produces as f t was the follower Dfferent from the Cournot model, under the Stackelberg model, the subsdy polcy by the government that can subsdze the leader frm does not work effectvely Second, under the sequental-move game n whch the government that can subsdze the leader frm decdes ts subsdy level frst, the proft of the leader frm s less than that of the follower n the Stackelberg model, although the frst-mover advantage of the government s mantaned The result proves that the tmng of decson-makng affects the results of the export subsdy polcy sgnfcantly JEL classfcaton: D43; F12; L13 Keywords: export subsdy; sequental-move game; Stackelberg competton Faculty of Economcs, Ngata Unversty, 8050, Ikarash 2-no-cho, Ngata Cty 950-2181, Japan Tel and fax: +81-25-262-6538 E-mal address: khamada@econngata-uacp The research for ths paper s supported by Grant-n-Ad for Scentfc Research (KAKENHI 16730095) from JSPS and MEXT of the Japanese Government 1
1 Introducton Ths paper examnes how the tmng of decson-makng affects the strategc trade polcy We analyze the relatonshp between the dfferent tmng of decson-makng by exportng frms and ther subsdzng governments and ts mpact on the export subsdy polcy Although WTO reorganzed from the GATT n 1995 and the FTAs have been concluded among many countres and tend to ncrease rapdly nowadays, the export subsdy polcy s stll beng practced n many countres as a strategc tool to nduce more domestc surplus from exportaton In the WTO agreements (Agreement on Subsdes and Countervalng Measures), the export subsdes to the manufactured products are prohbted per se, and a reducton of the subsdy rate for the agrcultural products s beng negotated by the WTO members Counterval and ant-dumpng measures are offcally enshrned n WTO rules; ths allows damaged government to counterval the export subsdes However, n realty, we can easly fnd many cases about dsputes between multnatonal frms on the export subsdes n the context of the nternatonal market competton From the theoretcal pont of vew, many studes about export subsdes have been done Snce Brander and Spencer (1985) elucdated the strategc effect of subsdy polcy n ther semnal paper on strategc trade polcy, many studes have been done analyzng the export subsdes n the context of the strategc trade polcy Usng the thrd-country model, Brander and Spencer (1985) analyzed the rent-shftng effect of the export subsdy and the strategc nteracton between the export subsdes They argued that the export subsdy effectvely rases domestc welfare, but t mples that the strategc subsdy choces of two governments n the exportng countres fall nto the suboptmal excessve competton such as a prsoner s dlemma 2
Another poneer work by Eaton and Grossman (1986) analyzed a more generalzed model They extended the model of Brander and Spencer (1985) to allow the dfferent conectural varatons from the Cournot case, that s, the dfferent competng envronments They showed that under the Bertrand conecture, the optmal trade polcy s the exportng tax mposton to the domestc exportng frm Although the two representatve papers mentoned above and ther successors dealt wth a general demand structure and llumnated the strategc aspects on the trade polcy, however, those papers have lmted ther analyss to the stuaton n whch the choces of the strategc varables are made smultaneously by the compettve frms For example, Brander and Spencer (1985) restrcted ther analyss to the Cournot quantty competton Although Eaton and Grossman (1986) generalzed the conectural varatons ncludng Cournot, Bertrand, and consstent conectures, these conectural varatons between frms are dentcal The exstng lterature has usually dealt wth only symmetrc case between frms, that s, only wth the smultaneous-move game on output choce We consder the Stackelberg leader-follower competton and deal wth the asymmetrc conecture as a result Extendng the smultaneous-move game on output choce and also subsdy choce to the sequental one, we present a new perspectve about the strategc subsdy polcy that s nfluenced by the tmng of decson-makng In the actual nternatonal trade polcy, we can magne many stuatons n whch the tmng of decson-makng about the trade polces by governments s dfferent For nstance, t may take place that the governments of developed countres determne the subsdy levels n advance of the governments of developng countres Because of the dfferent abltes of the governments to mplement and enforce the trade polcy, there exsts usually a tme lag on the subsdy decsons by governments On the one hand, whether or not a country has the leadng ndustry may affect 3
the speed of polcy determnaton postvely On the other hand, to facltate the nfant ndustry, the government may forestall the rval government and determne the subsdy level n advance The paper ntroduces the dfference on the tmng of decson-makng on output and subsdy levels n the model We examne how the dfferent tmng determnng strategc varables mpacts on the export subsdy polcy under mperfect compettve envronments The paper extends the analyss of the Cournot model by Brander and Spencer (1985) to the Stackelberg competton and the sequental-move game on the subsdy choce by governments Although the argument s lmted to the lnear demand and lnear cost model wth any loss of generalty, t s possble to make a comparatve statcs wth regard to subsdy, output, proft, and welfare levels, n order to clarfy the mpact of the tmng of decson-makng by exportng frms and ther governments Brander and Spencer (1985) presented a well-known result n Proposton 3 (p89) n ther paper: Proposton 3 The optmal export subsdy, s, moves the ndustry equlbrum to what would, n the absence of a subsdy, be the Stackelberg leader-follower poston n output space wth the domestc frm as leader Many artcles have quoted ths proposton For a recent example, Magg (1999, p575) stated as follows: The optmal unlateral subsdy s the one that shfts the domestc frm s reacton functon n such a way that t ntersects the foregn reacton functon R(q ) at the Stackelberg pont However, there was lttle contrbuton that the orgnal quantty competton be n the way of the Stackelberg competton n the context of the strategc trade polcy In ths paper, the optmal subsdy polcy s reexamned under the Stackelberg leader-follower competton 4
The man obectve s to nvestgate the effects of the sequental-move between two exportng frms under the Stackelberg model and also the effects of the sequental decson-makng between ther subsdzng governments on the szes of subsdy, frm s proft, and natonal welfare We pay attenton not only to the smultaneous decson on subsdy by governments, whch has usually been analyzed by the exstng lterature, but also to the sequental decson For the sequental-move game on strategc trade polcy, there are several artcles that we should refer to In the two-country model, Syropoulos (1994) showed that the governments may choose tarffs sequentally under perfect competton Colle (1994) showed that the domestc government sets tarff at frst and then the foregn government sets export subsdy under Cournot quantty competton In the thrd-country model, Arvan (1991) concluded that demand uncertanty may cause the sequental-move of the polcy choce by governments Shvakumar (1993) ntroduced the export quota and showed that the restrcted quantty competton and demand uncertanty cause the sequental decson of trade polcy by governments Although the exstng lterature analyzed the endogenous tmng of polcy-makng by governments, n ths paper n whch the tmng of polcy-makng s exogenous, we focus on examnng the effects of the dfferent tmng on decson-makng on the effectveness of trade polcy Recently, Ohkawa, Okamura, and Tawada (2002) endogenzed the tmng of government nterventon under nternatonal olgopoly Ther paper s closely related wth our paper n the sense that the sequental-move game by governments s analyzed n the thrd-country model Dfferent from our concern, however, they focused on the relatonshp between the number of frms and the endogenous tmng of the polcy decson by governments and not dealng wth Stackelberg competton between frms In our work, we ntroduce the sequental-move game by 5
frms, that s, the Stackelberg quantty competton 1 In other related papers, Neary and Leahy (2000) examned optmal trade and ndustral polcy n dynamc olgopolstc markets Applyng a generalzed model, they analyzed the strategc nteracton between frms and between governments n a 2-stage game However, they focused only on the stuaton n whch the economc agents act smultaneously n each stage Lkewse, Balboa, Daughety, and Renganum (2004) dealt wth a 2-stage game whch ncludes both Cournot and Stackelberg compettons at the 2nd stage Although they obtaned ndependently some of the results that are shown n our paper, they dd not deal wth the sequental decsonmakng by the ntervenng governments By comparng the smultaneous and the sequental moves made by frms and governments, we obtan some nterestng results Two man results are as follows: Frst, under the Stackelberg quantty competton, when the governments decde the export subsdes smultaneously and n advance, the orgnal Stackelberg leader frm produces as f t was the follower Dfferent from the Cournot model, under the Stackelberg model, the subsdy polcy by the government, whch can subsdze the leader frm, s almost nullfed Second, under the sequental-move game n whch the government, whch can subsdze the leader frm, decdes the subsdy level n advance, the leader s proft s less than the follower s proft, although the frst-mover advantage of the government s mantaned and the leader produces more than the follower The paper presents one of the theoretcal foundatons on the mportance of that the tmng of polcy decson to the effectveness of trade polcy The remander of the artcle s organzed as follows Secton 2 descrbes the model Secton 1 In the context of ndustral organzaton, there are many artcles that argue the endogenous tmng under duopolstc competton For a representatve paper, for example, see Hamlton and Slutsky (1990) 6
3 derves subsdy, output, proft, and domestc welfare n the equlbrum and analyzes the relatonshp between the dfferent structures In Secton 4, the calculaton results about the varables are summarzed under each case and the comparatve statcs are made wth regard to the dfferent structures on tmng Some results on the dfferent tmng of decson-makng are also presented Concludng remarks are presented n Secton 5 2 The model Two dentcal frms, one from country and one from country, produce homogeneous goods and sell n a thrd country We consder the mperfect quantty competton model n the thrd country àlabrander and Spencer (1985) It s assumed that snce both frms produce only for the thrd market, there s no consumpton effects for the exportng countres 2 The frm n country () are denoted by the ndex (respectvely ) Because the frms are dentcal, they can be nterpreted as nterchangeable Frm (frm ) produces quantty q (resp q ) The total quantty s Q q + q The argument s lmted to the lnear demand and lnear cost for smplfcaton of analyss The nverse demand functon s denoted by P (Q) a bq and the constant margnal cost by c It s assumed that a>c and b>0 Government, that les n country, can mplement the per unt export subsdy, s 0, as a means of trade polcy It s defned that e c s 3 The proft that frm maxmzes s denoted by π (q,q ; s,s ) (P (Q) c + s )q = 2 Ths knd of assumpton s usual n the context of the strategc trade polcy as t smplfes the analyss 3 It s shown that the sgn of e s ndetermnate n the followng analyss The compensaton from the government may be hgher than the margnal cost 7
(P (Q) e )q The soluton concept s the subgame perfect equlbrum The welfare of country s denoted by G (s,s ), whch conssts of the proft from the exportng frm mnus the cost of the export subsdy: G (s,s ) π (q,q ; s,s ) s q Government maxmzes ths welfare The tmng of the game s as follows: 1st stage: Governments choose subsdy levels smultaneously or sequentally 2nd stage: Frms choose output levels smultaneously or sequentally Subsdy polces can be commtted by both governments and can be observed by both frms n advance of the competton stage In the next secton, the subsdy, the output, the proft, and the welfare n the equlbrum are derved by backward nducton Both the Cournot and the Stackelberg leader-follower duopolstc competton are analyzed 3 Analyss In ths secton, the subsdy, the output, the proft, and the domestc welfare n the equlbrum are derved for all classfed cases In the frst step, the subgame at the second stage s solved In the begnnng, the smultaneous output choce, that s, Cournot quantty competton at ths subgame s examned Then we proceed to examne the sequental output choce, that s, the Stackelberg duopoly 8
31 Subgame at the second stage 311 Cournot competton Gven the subsdes (s,s ), both frms maxmze ther profts The frst-order condton for frm to maxmze ts proft s as follows: π =(a b(q + q ) e ) bq =0 4 The reacton functon of frm s q = R (q )= a bq e In order to obtan the output levels under the Cournot duopolstc competton, the ntersecton of the reacton functons s solved as follows: (q C (s,s ),q C (s,s ))=( a 2e + e 3b, a 2e + e ) 5 (1) 3b If there s no subsdy, the Cournot outcome s as follows: (q C(0, 0),qC (0, 0)) = ( a 2c +c 3b, a 2c +c 3b ) The total quantty s Q C = 2a e e 3b, the prce s P (Q C )= a+e +e 3, and the proft margn s P (Q C ) e = a 2e +e 3 = bq C as follows: The proft levels under the Cournot competton are calculated (π C (s,s ),π C (s,s ))=(b(q C ) 2,b(q C ) 2 )=( (a 2e + e ) 2 9b, (a 2e + e ) 2 ) (2) 9b 312 Stackelberg competton In the sequental-move game, under the Stackelberg competton, suppose that frm s the Stackelberg leader and frm s the follower wthout loss of generalty Frm, antcpatng the reacton of frm to ts own output choce q, that s, q = R (q ), maxmzes the proft functon π (q,q ) That s, the followng maxmzaton problem s solved: max q π (q,r (q )) Note that R (q )= 1 2 4 The subscrpt of the proft denotes the partal dervatve by q, that s, π π q The second-order condton s satsfed because π = <0 5 For the output to be postve, t must be assumed that a 2e + e > 0 When there s no subsdy, t s assumed that a 2c + c > 0 9
The foc s π + π R (q )=((a b(q + R (q )) e ) bq ) bq ( 1 2 )=06 The Stackelberg output pars are derved as follows: (q S (s,s ),q S (s,s ))=( a 2e + e, a 3e +2e ) 7 (3) If there s no subsdy, the Stackelberg outcome s as follows: (q S(0, 0),qS (0, 0)) = ( a 2c +c, a 3c +2c ) As a well-known fact n the olgopoly theory, f the subsdy pars (s,s ) are dentcal, q C (s,s ) <q S (s,s ) and q C (s,s ) >q S (s,s ) are satsfed 8 The total quantty s Q S = 3a 2e e and the prce s P (Q S )= a+2e +e 4 Q S >Q C and P (Q C ) >P(Q S ) are satsfed The proft margn s P (Q S ) e = a 2e +e 4 = b 2 qs and P (Q S ) e = a 3e +2e 4 = bq S The proft levels under the Stackelberg competton are calculated as follows: (π S (s,s ),π S (s,s ))=( b 2 (qs ) 2,b(q S ) 2 )=( (a 2e + e ) 2 It s satsfed that π C (s,s ) <π S (s,s ) and π C (s,s ) >π S (s,s ), (s,s ), (a 3e +2e ) 2 ) (4) For the followng analyss, the results of the comparatve statcs are presented as follows: q C (s,s ) s = 2 3b > 0, qc (s,s ) s = 1 3b < 0, qs (s,s ) s = 1 b > 0, qs (s,s ) s = 1 < 0, qs (s,s ) s = 3 > 0, and qs (s,s ) s = 1 < 0 6 The soc s satsfed because π +π R (q )+(π +π R (q ))R (q )+π R (q )= +b/2+b/2 = b <0 7 For the output to be postve, t must be assumed that a 3e +2e > 0 In the case of no subsdy, t s assumed that a 3c +2c > 0 8 And also t s well-known that q S q C = a 2e +e >q C 6b q S = a 2e +e, (s 1,s ) That s, the total quantty expands under the Stackelberg competton 10
32 Subsdy decson at the frst stage At the frst stage, government maxmzes the welfare n country as follows: max s 0 G (s,s ) π (q,q ; s,s ) s q The foc for government to maxmze ts welfare s as follows: G (s,s ) = π (q,q ; s,s ) q q s =0, (5) s s s f s 0 (nteror soluton) If G (s,s ) s < 0, the soluton s s = 0 (corner soluton) 9 10 Fnally, the dfferent tmng of decson-makng among frms and governments are classfed nto fve cases In Case A and Case B, the unlateral and the blateral nterventon by government(s) under the Cournot competton are examned In Case C and Case D, the unlateral and the blateral nterventon by government(s) under the Stackelberg competton are examned In Case E, the stuaton n whch all players sequentally decde s analyzed In the followng subsecton, we nvestgate all cases n sequence See Fgure 1 The superscrpts, C and S stand for Cournot and Stackelberg equlbrum respectvely for notatonal convenence < Fgure 1 around here > 9 Ths s derved from the Kuhn-Tucker slackness condton 10 Under the followng analyss, the soc s satsfed and the soluton s nteror, unque, and stable These are confrmed by tedous calculaton snce the demand and cost are lnear 11
Cournot competton A unlateral nterventon government 1st stage Stackelberg competton C unlateral nterventon C-1 Government ntervenes government 1st stage 2nd stage frm frm 2nd stage frm B blateral nterventon B-1 smultaneous decson-makng government government 1st stage 2nd stage 2nd stage frm frm B-2 sequental decson-makng government government 1st stage frm frm E wholly sequental decson-makng 1st stage 2nd stage 3rd stage 4th stage government frm government frm frm C-2 Government ntervenes 1st stage 2nd stage government frm frm D blateral nterventon D-1 smultaneous decson-makng 1st stage 2nd stage government frm government frm D-2 sequental decson-makng (Government moves frst) government 1st stage government 2nd stage frm frm D-3 sequental decson-makng (Government moves frst) 1st stage government government 2nd stage frm frm Fgure 1: Nodes of decson-makng 12
A Unlateral nterventon under Cournot competton Frst, we examne the unlateral nterventon case n whch only government subsdzes under the Cournot competton As s = 0, that s, e = c, government maxmzes the followng obectve: max s 0 G C (s, 0) = π (q C (s, 0),q C (s, 0); s, 0) s q C (s, 0) The foc for government s as follows: GC (s,0) s = π q C s + π q C s + π s q C s q C s =0 11 It s calculated that π = bq and π s = q, and π =0 s satsfed by the foc of the Cournot equlbrum By substtutng them nto the foc, t s obtaned that ( bq )( 1 3b )+q q s 2 3b = 0, that s, s = b 2 qc Arrangng ths equaton, the optmal subsdy level, s uc s derved as follows: s uc = a 2c + c (6) 4 In ths case, the Cournot output s calculated as follows: (q C (s uc, 0),q C (s uc, 0)) = ( a 2c + c Ths result s summarzed n the followng proposton, a 3c +2c )[=(q S (0, 0),q S (0, 0))] (7) Proposton 1 Under the Cournot competton, the unlateral nterventon by government changes the market structure from the Cournot duopoly to the Stackelberg one n whch frm s the leader Ths proposton s ust a corollary of Proposton 3 n Brander and Spencer (1985, p89) The optmal subsdy has the proft-shftng effect and moves the Cournot competton to the Stackelberg leader-follower poston Ths result s well-known n the context of strategc subsdy polcy 12 As a result of the unlateral subsdy, the proft of frm, whch s subsdzed by the 11 The soc s satsfed because 2 G C (s,s ) s 2 = π q C s + π q C s qc s = ( )( 2 3b )+( b)( 1 3b ) 2 3b = (1 + 2 3b ) < 0 12 It s shown that q C (s uc, 0) >q C (0, 0) and q C (s uc, 0) <q C (0, 0) 13
government, rses That s, π C (s uc, 0)(= π S (0, 0)) >π C (0, 0) and π C (s uc, 0)(= π S (0, 0)) < π C (0, 0) Also, t s evdent that the subsdy of government expands the welfare n country () (resp contracts), that s, G C (s uc, 0)(= max s G C (s, 0)) >G C (0, 0) and G C (s uc, 0)(= π C (s uc, 0)) <G C (0, 0)(= π C (0, 0)) 13 B Blateral nterventon under Cournot competton Next, we analyze the blateral nterventon case n whch both governments subsdze under the Cournot competton The smultaneous and the sequental decson of subsdy are examned n sequence B-1 Smultaneous decson of subsdy Consder that the smultaneous decson of subsdes (s,s ) by both governments has the smlar tmng of decson as the Cournot quantty competton Gven s, government maxmzes the welfare wth regard to ts own subsdy as follows: max s 0 G C (s,s ) = π (q C(s,s ),q C(s,s ); s,s ) s q C The foc s as follows: G C (s,s ) s = π q C s + π q C s + π s q s q C s =0 (8) Lke Case A, the foc s arranged as s = b 2 qc = a 2e +e 6 The reacton functon s derved as s = R (s )= s +a 2c +c 4 In order to examne the smultaneous decson on the subsdy levels, the ntersecton of the reacton functons of both governments s solved The subsdy level n the equlbrum s obtaned as follows: 13 As s uc b(q C ) 2 = (a 2c +c ) 2 9b s bcc = a 3c +2c (9) 5 = b 2 qc, the welfare of country s G C (s uc, 0) = b(q C ) 2 s uc q C = b 2 (qc ) 2 = (a 2c +c ) 2 (> G C (0, 0) = 14
By substtutng the subsdy level, s bcc, the Cournot output levels n the equlbrum are obtaned under the smultaneous decson of subsdy (q C (sbcc,s bcc ),q C (sbcc,s bcc ))=( 2(a 3c +2c ) 5b, 2(a 3c +2c ) ) (10) 5b Frst, comparng the subsdy levels under the unlateral and the blateral cases, the followng lemma can be stated Lemma 1 (comparson of the subsdy levels under Case A and Case B-1) The subsdy under the unlateral nterventon s larger than under the blateral nterventon That s, s uc >s bcc 14 Ths lemma mples that under the blateral nterventon, there s a strategc nteracton between governments about the subsdy settng and, as a result, the mpact of the subsdy on the output of the frm s smaller than the mpact under the unlateral nterventon Then we proceed to compare the output levels under the unlateral and blateral nterventon Proposton 2 (comparson of the output levels under Case A and Case B-1) Under the Cournot competton, (a) the output level of frm () under the blateral nterventon s smaller (resp larger) than under the unlateral nterventon by government That s, q C (suc, 0)(= q S (0, 0)) >qc (sbcc,s bcc ) and q C (suc, 0)(= q S(0, 0)) <qc (sbcc,s bcc ) (b) When the margnal costs of the frms are almost dentcal, the output level under the blateral nterventon s larger than under nonnterventon That s, f a 8c +7c > 0, 15 then q C(0, 0) <qc (sbcc,s bcc ) 14 s uc s bcc = a 2c +c a 3c +2c = a 3c +2c > 0 4 5 20 15 If the frms have dentcal margnal cost, c c = c, ths condton s satsfed, because a 8c +7c = a 3c +2c +5(c c ) > 0 15
Due to strategc substtutes on output, subsdzng by rval government results n the output reducton of the own frm As the reacton functons of both frms shft outwards by subsdzng blaterally, as a result, the output competton under the blateral nterventon becomes more severe than wthout nterventon Ths s an example of the prsoner s dlemma As for the frm s proft, π C = b(q C )2, the relaton about the proft sze s obtaned from the prevous relaton about the output sze From Proposton 2, t s obtaned that π C (s uc, 0) > π C (s bcc π C (s bcc,s bcc ), and π C (s uc, 0) <π C (s bcc,s bcc ) When frms are almost dentcal, π C (0, 0) <,s bcc ) Ths mples that the subsdy from government () makes the proft of frm larger (resp smaller) and the blateral nterventon makes the profts of both frms larger than wthout nterventon Fnally, the effect of the subsdy on the welfare s examned By drect calculaton, t s obtaned that G C (0, 0) = (a 2c +c ) 2 9b, G C (s uc (a 3c +2c ) 2 G C (s bcc G C (s bcc, and G C (s bcc,s bcc )= 2(a 3c +2c ) 2 25b, 0) = (a 2c +c ) 2, G C (s uc, 0) = π C (s uc, 0) = 16 Hence G C (s uc, 0) >G C (0, 0) and,s bcc ) >G C (s uc, 0) 17 If the costs are almost dentcal, then G C (0, 0) >,s bcc ) 18 That s, the blateral nterventon falls nto the prsoner s dlemma for both governments Ths s ust a corollary of Proposton 5 n Brander and Spencer (1985, p95) Ths result s restated n the followng proposton See also Table 1 Proposton 3 (comparson of the welfare under nonnterventon and Case B-1) Under the Cournot competton, when frms have almost dentcal margnal costs, the welfare under the blateral nterventon s smaller than under nonnterventon That s, the blateral 16 It s shown that G C (s uc, 0) >G C (s bcc when the costs are dentcal, c = c, because G C (s uc,s bcc ), f 9a +13c 22c > 0 Ths condton s suffcently satsfed, 0) G C (s bcc,s bcc )= (a 2c +c ) 2 2(a 3c +2c ) 2 = 25b (a 3c +2c )(9a+13c 22c ) 200b > 0, f 9a +13c 22c =9(a 3c +2c ) + 40(c c ) > 0 17 G C (s bcc,s bcc ) G C (s uc, 0) = 2(a 3c +2c ) 2 (a 3c +2c ) 2 = 7 (a 3c +2c ) 2 > 0 25b 400 b 18 G C (s bcc,s bcc ) G C (0, 0) = 450(c c ) 2 (7a 11c +4c ) 2 < 0, f c 7 225b = c 16
nterventon falls nto the prsoner s dlemma for both governments government nonnterventon nterventon government nonnterventon G C (0, 0), G C (0, 0) G C (0,s uc ), G C (0,s uc ) nterventon G C (s uc, 0), G C (s uc, 0) G C (s bcc,s bcc ), G C (s bcc,s bcc ) If the costs are almost dentcal, G C (0,s uc ) <G C (s bcc,s bcc ) <G C (0, 0) <G C (s uc, 0) government nonnterventon nterventon government nonnterventon (a 2c +c ) 2 9b, (a 2c +c ) 2 9b (a 3c +2c ) 2, (a 2c +c ) 2 nterventon (a 2c +c ) 2, (a 3c +2c ) 2 2(a 3c +2c ) 2 25b, 2(a 3c +2c ) 2 25b Table 1: Welfare under nonnterventon, Case A, and Case B-1 B-2 Sequental decson of subsdy Next, we consder the sequental decson of subsdy (s,s ) as a sequental-move game between both governments Frst, government decdes the subsdy level, s, and then, after observng s, government decdes s That s, government () acts as the Stackelberg leader (resp follower ) wth regard to the subsdy choce Snce s s gven, the follower government decdes the subsdy s = R (s )= s +a 2c +c 4 Note that R (s )= 1 4 < 0 Ths reacton functon s nduced by the leader government who solves the followng maxmzaton problem: max s 0 G C (s,r (s )) = π (q C (s,r (s )),q C (s,r (s )); s,r (s )) s q C (s,r (s )) The foc s as follows: G C (s,r (s )) = π (q C(s,R (s )),q C(s,R (s )); s,r (s )) q (s,r (s )) s ( q + q R s s s s (s )) = π( q + q R s s (s )) + π( q + q R s s (s ))+( π + π R s s (s )) q s ( q + q R s s (s )) = 0 (11) 17
Lke Case A, t s satsfed that π = 0,π = bq, π s = q, and π s = 0 Arrangng the foc, π ( q s a 2(c s )+(c R (s )) 3 3b + q s R (s )) s ( q s + q s R (s )) = 0 That s, s = 3 q (s,r (s )) = s derved The subsdy levels are obtaned as follows: s bsc = a 3c +2c 3 = R (s bsc )= a 4c +3c (12) 6 By substtutng (s bsc ) nto the Cournot output levels, the followng equatons are derved: (q C (sbsc ),q C (sbsc ))=( a 3c +2c, a 4c +3c ) 19 (13) 3b Frst, comparng the subsdy levels under the unlateral and the blateral cases, the followng lemma can be stated Lemma 2 (comparson of the subsdy levels under Case A and Case B-2) When frms have almost dentcal margnal costs, the subsdy under the unlateral nterventon s smaller than that of the leader government under the blateral nterventon The subsdy under the unlateral nterventon s larger than that of the follower government under the blateral nterventon That s, f a 6c +5c > 0, then s uc <s bsc and s uc >s bsc 20 Dfferent from Case B-1, when the sequental decson of subsdy s made by governments under the blateral nterventon, the subsdy of the frst-mover government s larger and the subsdy of the follower government s smaller compared to the unlateral case Then we proceed to compare the output levels under the unlateral and the blateral nterventon 19 For the output and the subsdy to be postve, t s assumed that a 4c +3c > 0 throughout the followng analyss 20 s uc s bsc = a 6c +5c < 0, f a 6c 12 +5c = a c 5(c c ) > 0 s uc s bsc = a 3c +2c > 0 12 18
Proposton 4 (comparson of the output levels under nonnterventon, Case A, and B-2) Under the Cournot competton, (a) whether the output level of frm under the blateral sequental nterventon s smaller than that under the unlateral nterventon by government, depends on the relatve szes of the margnal costs of the frms That s, q C (suc, 0) q C(sbSC ) c c 21 When the margnal costs are almost dentcal, the output level of frm under the blateral nterventon s smaller than that under the unlateral nterventon ether by government or That s, f a 7c +6c > 0, then q C (suc q C (0,suC ) >q C(sbSC ) 22, 0) <q C (sbsc ) or f a 6c +5c > 0, then (b) When the margnal costs are almost dentcal, the output level of frm under the blateral nterventon s larger than that under nonnterventon That s, f a 5c +4c > 0, then q C(0, 0) <qc (sbsc ) 23 Whether the output level of frm under the blateral nterventon s smaller than that under nonnterventon, depends on the relatve szes of the margnal costs of the frms That s, q C(0, 0) qc (sbsc ) c c 24 As a corollary of ths proposton, f the costs are dentcal, that s, c = c, then q C (suc, 0) = q C (sbsc ) and q C (0, 0) = qc (sbsc ) When the costs are dentcal, the output level of frm under the blateral nterventon by the leader government s equal to that under the unlateral nterventon by government And also the output level of frm s the same, whether under the blateral nterventon by the follower government or under nonnterventon The subsdy polcy by the government has two effects on output The frst s to shft the reacton 21 q C (s uc 22 q C (s bsc q C (0,s uc 23 q C (s bsc, 0) q C (s bsc ) q C (s bsc )= c c T 0 c T c ) q C (s uc, 0) = a 7c +6c > 0, f a 7c 1 +6c > 0 )= a 6c +5c > 0, f a 6c 6b +5c > 0 ) q C (0, 0) = a 5c +4c > 0, f a 5c 6b +4c > 0 24 q C (0, 0) q C (s bsc )= c c T 0 c b S c 19
functon outwards gvng the home frm an advantage on output competton under strategc substtutes The second s to adust to compettve dstorton orgnated from costs dfferences If the costs are dentcal, the second effect does not appear and, through the subsdy, the output s adusted at the level of the Stackelberg leader As for the frm s proft, π C = b(q C )2, the relaton about the proft sze s obtaned from the prevous relaton about the output sze From Proposton 4, t s obtaned that π C (s uc, 0) π C (s bsc π C (s bsc )fc c, π C (s uc, 0) <π C (s bsc ), f a 7c +6c > 0, and π C (0,s uc ) > ), f a 6c +5c > 0 Moreover, t s satsfed that f a 5c +4c > 0, then π C (0, 0) <π C (s bsc ), and f c c, then π C (0, 0) π C (s bsc ) When the frm s cost s hgher (lower) than that of the rval, the frm that s subsdzed by the leader government prefers the unlateral (resp blateral) nterventon to the blateral (resp unlateral) one When frms are almost dentcal, the frm that s subsdzed by the follower government always prefers the unlateral nterventon to the blateral one Comparng nonnterventon wth the blateral one, the leader government always prefers the blateral nterventon, whle the follower government wll prefer the blateral nterventon to nonnterventon only f the margnal cost s lower Fnally, the effect of the subsdy on the welfare s examned By drect calculaton, t s obtaned that G C (s bsc ) = (a 3c +2c ) 2 1 and G C (s bsc ) = (a 4c +3c ) 2 1 25 Hence G C (s uc, 0) >G C (0, 0) and G C (s bsc )= (a 3c +2c ) 2 1 >G C (0,s uc )= (a 3c +2c ) 2 If the costs are almost dentcal, G C (s bsc G C (0, 0) >G C (s bsc ) and G C (0, 0) >G C (s bsc ) <G C (s uc, 0) 26 If the costs are dentcal, then ) 27 Dfferent from Case B-1, when 25 Because G C (s uc, 0) G C (s bcc,s bcc )= (a c ) 2 6(c c ) 2 f a c > ± 6(c c ) It s satsfed when the costs are almost dentcal 26 G C (s bsc 2, t s shown that G C (s uc, 0) >G C (s bsc ), ) G C (s uc, 0) = (a 6c +5c ) 2 72(c c ) 2 < 0, f (a 6c 14 +5c ) 2 72(c c ) 2 > 0 27 G C (s bsc ) G C (0, 0) = (a 2c +c ) 2 12(c c ) 2 < 0, f (a 2c 36b + c ) 2 12(c c ) 2 > 0 20
the costs are almost dentcal, the leader government chooses to ntervene and the follower government chooses not to ntervene The frst-mover advantage of government on the subsdy choce can deter the rval follower government from exercsng the subsdy Ths result s summarzed n the followng proposton See also Table 2 Proposton 5 (comparson of the welfare under nonnterventon and Case B-2) Under the Cournot competton, when frms have almost dentcal margnal costs, the welfare under the blateral nterventon s smaller than that under nonnterventon In the equlbrum, the result s that only the leader government ntervenes In ths case, the prsoner s dlemma of the blateral nterventon for both governments s avoded government nonnterventon nterventon government nonnterventon G C (0, 0), G C (0, 0) G C (0,s uc ), G C (0,s uc ) nterventon G C (s uc, 0), G C (s uc, 0) G C (s bsc ), G C (s bsc ) If the costs are almost dentcal, G C (0, 0) <G C (s uc G C (s bsc ) <G C (s uc, 0), G C (0, 0) >G C (s bsc, 0), G C (0,s uc ) <G C (s bsc ), ), and G C (0, 0) >G C (s bsc ) government nonnterventon nterventon government nonnterventon (a 2c +c ) 2 9b, (a 2c +c ) 2 9b (a 3c +2c ) 2, (a 2c +c ) 2 nterventon (a 2c +c ) 2, (a 3c +2c ) 2 (a 3c +2c ) 2 1, (a 4c +3c ) 2 1 Table 2: Welfare under nonnterventon, Case A, and Case B-2 C Unlateral nterventon under Stackelberg model Now, we proceed to examne the unlateral nterventon under the Stackelberg model G C (s bsc ) G C (0, 0) = (a c ) 2 8(c c ) 2 < 0, f (a c 1 ) 2 8(c c ) 2 > 0 21
C-1 Unlateral nterventon by government We examne the case n whch government, whose frm s the Stackelberg leader, ntervenes As s = 0, that s, e = c, government maxmzes the followng obectves: max s 0 G S (s, 0) = π (q S (s, 0),q S (s, 0); s, 0) s q S The foc s as follows: G S (s,0) s = dπ (q S (s,0),q S (s,0);s,0) ds q s q s = π q S s + π q S s + π s q s q S s 0 28 By the foc of the Stackelberg leader, t s satsfed that π = π R (q )= b 2 q By substtutng π = bq and π s = q nto the foc, ( b 2 q )( 1 b )+( bq )( 1 )+q 1 q s b = s 1 b 0 That s, the subsdy level s zero, s us =0 In ths case, the Stackelberg output levels n the equlbrum are as follows: (q S (s us, 0),q S (s us, 0))[= (q S (0, 0),q S (0, 0))] = ( a 2c + c, a 3c +2c ) (14) The optmal subsdy moves to the Stackelberg leader-follower poston Ths result s summarzed n the followng proposton Proposton 6 Under the Stackelberg competton n whch frm s the leader, the unlateral nterventon by the government of the leader frm has no effect That s, there s no subsdy Ths proposton s ust another corollary of Proposton 3 n Brander and Spencer (1985, p89) The optmal subsdy has the proft-shftng effect and moves the Cournot competton to the Stackelberg leader-follower poston Under the Stackelberg competton, the government of the leader frm has nothng to do Wthout subsdy, the proft of frm s π S (s us, 0) = π S (0, 0) = b 2 (qs )2 and π S (s us, 0) = π S (0, 0) = b(q S)2 The welfare s G S (s us, 0) = G S (0, 0) = π S (0, 0) and G S (s us, 0) = π S (0, 0) C-2 Unlateral nterventon by government 28 The soc s satsfed because 2 G S (s,s ) s 2 = π q S s +π q S s qs s =( ) 1 +( b)( 1 ) 1 = ( 3 + 1 ) < 0 b b 2 b 22
We examne the case n whch government, whose frm s the Stackelberg follower, ntervenes As s =0,e = c, government maxmzes the followng obectves: max s 0 G S (0,s )= π (q S (0,s ),q S (0,s ); 0,s ) s q S The foc of government s as follows: G S (0,s ) s = dπ (q S (0,s ),q S (0,s );0,s ) ds q s q s = π q S s + π q S s + π s q s q S s =0 29 By the foc of the Stackelberg follower, π = 0 s satsfed By substtutng π = bq and π s foc, ( bq )( 1 )+q q s 3 =0 s = 3 q = a 3(c s )+2c 6 = q nto the s us = a 3c +2c (15) 3 In ths case, the Stackelberg output levels n the equlbrum are as follows: (q S (0,s us ),q S (0,s us ))=( a 4c +3c 3b, a 3c +2c ) (16) Ths output level s equvalent to that n Case B-2 The followng proposton s obtaned Proposton 7 (comparson of the output levels under Case B-2 and Case C-2) Under the Stackelberg competton n whch frm s the leader, the unlateral nterventon by the government of the follower frm yelds the same result on output as the blateral sequental nterventon under Cournot competton n whch government s the frst-mover That s, (q S (0,suS ),q S (0,suS ))=(q C (sbsc ),q C(sbSC )) Ths proposton mples that the subsdy of the government works as f t changes the competton mode from Stackelberg to Cournot The optmal subsdy mproves the Stackelbergfollower poston compared to the Cournot one Even f frm s the Stackelberg follower under quantty competton, the optmal subsdy by government makes the dsadvantage of the follower reduce untl t dsappears By substtutng (16) nto π S (s,s )= b 2 (qs )2 and π S (s,s )=b(q S)2, the profts of the 29 The socs satsfed because 2 G S (s,s ) s 2 = π q S s +π q S s qs s =( ) 3 +( b)( 1 ) 3 = (1+ 3 ) < 0 23
frms are obtaned: π S (0,s us )= (a 4c +3c ) 2 1 and π S (0,s us )= (a 3c +2c ) 2 It s shown that π S (s us, 0) = (a 2c +c ) 2 >π S (0,s us ) and π S (s us, 0) = (a 3c +2c ) 2 <π S (0,s us ) 30 Fnally, the welfare s examned It s obtaned that G S (0,s us )=π S (0,s us )= (a 4c +3c ) 2 1 < G S (s us, 0) = π S (s us, 0) = (a 2c +c ) 2 Also t s obtaned that G S (0,s us )=b(q S)2 s us q S = (a 3c +2c ) 2 1 >G S (s us, 0) = π S (s us, 0) = (a 3c +2c ) 2 For smplfcaton of analyss, t s assumed that c c = c throughout the followng analyss Under ths dentcal assumpton, the followng proposton s obtaned: Proposton 8 (comparson of the proft and the welfare levels under Case C-1 and Case C-2) Consder the Stackelberg competton n whch frm s the leader Suppose that the costs are dentcal In the case n whch the government of the follower frm ntervenes unlaterally, the proft of the follower frm (the leader frm ) s larger (resp smaller) than the proft of the leader frm (resp the follower frm ) n the case n whch the government of the leader frm ntervenes unlaterally That s, π S (s us, 0) = (a c)2 π S (s us, 0) = (a c)2, π S (0,s us, π S (0,s us )= (a c)2 1, )= (a c)2 When government ntervenes unlaterally, the welfare of government () s smaller than that of government (resp ) when government ntervenes unlaterally That s, G S (s us, 0) = (a c)2 G S (s us, 0) = (a c)2, G S (0,s us, G S (0,s us )= (a c)2 1, )= (a c)2 1 Note that ths proposton also holds when frms have almost dentcal cost When the costs are almost dentcal, although t looks at frst glance that the subsdzed leader frm may enoy hgher proft than that of the subsdzed follower, the prevous proposton shows that such 30 Because π S (0,s us satsfed that π S (0,s us q S (0,s us )= a 3c +2c > 0 6b )=(G S (0,s us ) <π S (s us )=) b 2 (qs (0,s us )) 2 and π S (s us, 0), f and only f q S (0,s us ) <q S (s us, 0) = (G S (s us, 0) =) b 2 (qs (s us, 0)) 2,ts, 0) It s shown that q S (s us, 0) 24
a preconceved dea s ncorrect Ths mplcaton s derved from the fact that government does not subsdze at all, because the advantage of the Stackelberg leader had already been acqured by frm n Case C-1 On the other hand, ths ntuton s correct when the welfare s consdered Even f the government makes the follower frm recover the frst-mover advantage through nterventon, t takes an extra cost to subsdze ths frm D Blateral nterventon under Stackelberg model We analyze the blateral nterventon under Stackelberg model D-1 Smultaneous decson of subsdy We consder the smultaneous decson of subsdy (s,s ), smlar to the smultaneous quantty choce n the Cournot model Government, whose frm s leader, maxmzes the followng obectve: gven s, max s 0 G S (s,s )=π (q S(s,s ),q S(s,s ); s,s ) s q S The foc s as follows: GS (s,s ) s = dπ (q S(s,s ),q S(s,s );s,s ) ds q s q s = π q S s +π q S s + π q s q s S s =0 By the foc of the Stackelberg leader, t s satsfed that π = π R (q )= b 2 q By substtutng π = bq and π s = q nto the foc, ( b 2 q ) 1 b +( bq )( 1 )+q q s 1 b = s 1 b 0 s bcs = 0 The reacton functon s s bcs = R (s bcs )=0 Government, whose frm s the follower, maxmzes the followng obectve: gven s, max s 0 G S (s,s )=π (q S (s,s ),q S (s,s ); s,s ) s q S The foc s as follows: GS (s,s ) s = dπ (q S (s,s ),q S (s,s );s,s ) ds q s q s = π q S s + π q S s + π s q s q S s = 0 From the foc of the Stackelberg follower, t s satsfed that π = 0 By substtutng π = bq and π s nto the foc, ( bq )( 1 )+q q s 3 =0 s = 3 q = a 3(c s )+2(c s ) 6 The reacton functon s s bcs = R (s bcs )= 2s +a 3c +2c 3 In order to work out the smultaneous decson of the subsdy levels by both governments, = q 25
the ntersecton of the reacton functons s reduced as follows: Note that s bcs = s us s bcs = 0 and s bcs =0,s bcs = s us = a 3c +2c (17) 3 = a 3c +2c 3 By substtutng s nto the outputs, the optmal Stackelberg output levels are obtaned as follows: Hence q S (sbcs (q S (sbcs,s bcs,s bcs )=q S (0,suS ),q S (sbcs,s bcs ) and q S (sbcs ))=( a 4c +3c 3b,s bcs )=q S(0,suS ), a 3c +2c ) (18) In ths case, as a result of the smultaneous decson of subsdy levels, dfferently from the Cournot model, under the Stackelberg model, the subsdy polcy of the government to the leader frm s nullfed In Case D-1, the Stackelberg leader and the follower have the same behavor as n Case C-2 Proposton 9 (equvalence between Case D-1 and Case C-2) Consder the Stackelberg competton n whch frm s the leader The result n the equlbrum s the same under Case D-1 and under Case C-2 The government whose frm s the leader does not subsdze ts frm at all Also the proft and the welfare are the same as those under Case C-2 That s, the profts are π S (s bcs,s bcs )(= π S (0,s us )= (a 4c +3c ) 2 1 The welfares are G S (s bcs,s bcs = G S (0,s us )= (a 3c +2c ) 2 1 and π S (s bcs )=π S (s bcs,s bcs )= (a 4c +3c ) 2 1,s bcs )(= π S (0,s us )) = (a 3c +2c ) 2 and G S (s bcs,s bcs ) D-2 Sequental decson of subsdy (s s ) Then, we examne the sequental decson of subsdy (s s ) In ths case, the government of the Stackelberg leader moves frst and then the government of the follower, after observng s, decdes the subsdy level, s = R (s ) 26
Government maxmzes the followng obectve: gven s, max s 0 G S (s,s )= π (q S (s,s ),q S (s,s ); s,s ) s q S The foc s as follows: GS (s,s ) s = dπ (q S (s,s ),q S (s,s );s,s ) ds q s q s = π q S s + π q S s + π s q s q S s = 0 From the foc of the Stackelberg follower, t s satsfed that π = 0 By substtutng π = bq and π s = q nto the foc, ( bq )( 1 )+q q s 3 =0 s = 3 q = a 3(c s )+2(c s ) 6 The reacton functon s s bss = R (s bss )= 2s +a 3c +2c 3 The slope s R (sbss )= 2 3 Ths maxmzaton problem s the same procedure taken by the follower government n Case D-1 The leader government nduces ths reacton functon, R, and solves the followng maxmzaton problem: max s 0 G S (s,r (s )) = π (q S (s,r (s )),q S (s,r (s )); s,r (s )) s q S (s,r (s )) The foc s as follows: dg S (s,r (s )) ds = dπ (q S(s,R (s )),q S(s,R (s ));s,r (s )) ds q S(s,R (s )) s ( qs s + qs s R (s )) = π ( q s + q s R (s ))+π ( q s + q s R (s ))+( π s + π s R (s )) q s ( q s + q s R (s )) = 0 Smlar to Case C, t s satsfed that π = π R (q )= b 2 q By substtutng π = bq, π s = q, and π s = 0 nto the foc, b 2 q ( q s + q s R (s )) bq ( q s + q s R (s )) s ( q s + q s R (s ))=0 s = b 4 q = a 2(c s )+(c R (s )) 8 Under the sequental decson, the optmal subsdy levels are as follows: s bss = a 4c +3c, s bss 8 = R (s bss )= a 5c +4c (19) 4 Note that s bss >s bss f the costs are almost dentcal 31 In ths case, the subsdy to the follower s larger than that to the leader By substtutng s nto the output, the optmal Stackelberg output levels are obtaned as follows: (q S (sbss,s bss ), q S (sbss,s bss ))=( a 4c +3c, 3(a 5c +4c ) ) (20) 31 It s satsfed that s bss >s bss,fa 13c +12c = a 3c +2c + 10(c c ) > 0, because s bss s bss = a 13c +12c 8 27
Note that q S (sbss,s bss ) >q S (sbss,s bss ), f the costs are almost dentcal 32 In ths case, the output of the leader s larger than that of the follower In partcular, when costs are dentcal, q S (sbss,s bss )= a c >q S (sbss,s bss )= 3(a c) From (π S,π S )=( b 2 (qs )2,b(q S)2 ), the profts are calculated as follows: π S (s bss,s bss )= (a 4c +3c ) 2 and π S (s bss,s bss )= 9(a 5c +4c ) 2 When the costs are dentcal, t s worth notng 6 that π S (s bss,s bss )= (a c)2 <π S (s bss follower s larger than that of the leader,s bss )= 9(a c)2 6 In other words, the proft of the From G S = π S s q, the welfares are calculated as follows: G S (s bss,s bss )= (a 4c +3c ) 2 and G S (s bss,s bss )= 3(a 5c +4c ) 2 6 When the costs are dentcal, then G S (s bss,s bss )= (a c) 2 > G S (s bss,s bss ) = 3(a c)2 6 Wth regard to the welfare, the welfare of the leader government s larger than that of the follower Ths result s summarzed n the followng proposton Proposton 10 (proft and welfare levels under Case D-2) Consder the Stackelberg competton n whch frm s the leader In the symmetrc equlbrum under Case D-2, the proft of the leader s less than that of the follower The welfare of the leader government s larger than that of the follower government Although the leader produces more than the follower, the government of the leader subsdzes less than that of the follower As a result, the proft of the leader s less than that of the follower and the welfare of the frst-mover government s larger than that of the second-mover government Ths result provdes a new vewpont consderng the subsdy polcy determned sequentally Furthermore, by comparng the welfare under nonnterventon wth that under Case D-2, t s obtaned that G S (0, 0) = (a 2c +c ) 2 >G S (s bss,s bss )= (a 4c +3c ) 2 and G S (0, 0) = 32 q S (s bss,s bss ) q S (s bss,s bss )= a c 27(c c ) > 0, f a 28c +27c > 0 28
(a 3c +2c ) 2 >G S (s bss,s bss )= 3(a 5c +4c ) 2 6, f the costs are almost dentcal 33 Thus, the followng proposton s derved Proposton 11 (comparson of the welfare under nonnterventon and Case D-2) Under the Stackelberg competton n whch frm s the leader, when the costs are almost dentcal, the welfares of the governments under the blateral nterventon n Case D-2 are smaller than under nonnterventon That s, the blateral nterventon falls nto the prsoner s dlemma D-3 Sequental decson of subsdy (s s ) Fnally, we examne the sequental decson of subsdy (s s ) Ths case s the adverse case of Case D-2 wth regard to the tmng of decson-makng by governments In ths case, the government of the Stackelberg follower frm moves frst and then the government of the leader frm, after observng s, decdes the subsdy s = R (s ) The follower government maxmzes the followng obectve: gven s, max s 0 G S (s,s )= π (q S (s,s ),q S (s,s ); s,s ) s q S The foc s as follows: GS (s,s ) s = dπ (q S(s,s ),q S(s,s );s,s ) ds q s q s = π q S s + π q S s + π s q s q S s = 0 From the foc of Stackelberg leader, t s satsfed that π = π R (q )= b 2 q By substtutng π = bq and π s = q nto the foc, ( b 2 q ) 1 b +( bq )( 1 )+q 1 q s b = s b 0 s = 0 The reacton functon s s bss = R (s bss ) = 0 Ths s the same procedure taken by the leader government n Case D-1 The leader government nduces ths reacton and solves the followng maxmzaton problem substtutng s bss s q S (0,s ) The foc s as follows: = R (s bss ) = 0: max s 0 G S (0,s )=π (q S(0,s ),q S(0,s ); 0,s ) dg S (0,s ) ds = dπ (q S(0,s ),q S(0,s );0,s ) ds q S(0,s q ) s S s = π q s + π q s + π s q s q s = 0 Smlar to Case C-2, t s satsfed that π =0 By 33 G S (0, 0) G S (s bss,s bss )= (a c ) 2 8(c c ) 2 and G S (0, 0) G S (s bss,s bss )= (a 4c +3c ) 2 48(c c ) 2 6 29
substtutng π = bq and π s = q nto the foc, bq q s s q s =0 s = 3 q = a 3(c s )+2c 6 s = a 3c +2c 3 Under the sequental decson, the optmal subsdy levels are as follows: s bss =0,s bss = a 3c +2c (21) 3 By substtutng s nto the output, the optmal Stackelberg output levels are as follows: (q S (sbss,s bss ), q S (sbss,s bss ))=( a 4c +3c 3b, a 3c +2c ) (22) The equlbrum n ths case s the same as that n Case D-1 (and also Case C-2) Proposton 12 (equvalence between Case D-3 and Case D-1 (Case C-2)) Consder the Stackelberg competton n whch frm s the leader The result n the equlbrum under Case D-3 s the same as that under Case D-1 (and Case C-2) The government whose frm s the leader does not subsdze ts frm at all In ths case, the follower government of the country where there s the Stackelberg leader frm has nothng to do, by the same reason as Case D-1 and Case C-2 Also the proft and the welfare are the same as those under Case D-1 and Case C-2 That s, the profts are π S (s bss,s bss )= (a 4c +3c ) 2 1 and π S (s bss,s bss )= (a 3c +2c ) 2 The welfares are G S (s bss,s bss )= (a 4c +3c ) 2 1 and G S (s bss,s bss )= (a 3c +2c ) 2 1 We compare the proft and the welfare n Case D-2 wth those n Case D-3 When the costs are dentcal, the followng nequalty s satsfed: π S (s bss π S (s bss G S (s bss,s bss,s bss advantage stll exsts ) <π S (s bss ) <G S (s bss,s bss,s bss,s bss ) <π S (s bss,s bss ) < ) 34 And t s also satsfed that G S (s bss,s bss ) < ) <G S (s bss,s bss ) 35 For the government, the frst-mover 34 By drect calculaton, t s obtaned that (a c)2 1 35 By drect calculaton, t s obtaned that (a c)2 1 < (a c)2 < 3(a c)2 6 < 9(a c)2 6 < (a c)2 < (a c)2 < (a c)2 1 30
E Wholly sequental decson As the remanng possble combnaton, we examne the wholly sequental decson (s q s q ) Frst, we consder the blateral nterventon of sequental decson-makng: s q s q 36 The equlbrum can be solved by backward nducton In the subgame at the fourth stage, the foc for the proft maxmzaton of frm, gven (s,q,s ), s as follows: π =(a b(q + q ) e ) bq = 0 The reacton functon s q = R s (q,s )= a bq e ; Rs (q,s ) s = 1 Note that ths reacton functon does not depend on s In the subgame at the thrd stage, the subsdy decson by government s determned by maxmzng G (s,q,s,q ) π (q,q ; s,s ) s q Government maxmzes the followng obectve: gven (s,q ), max s 0 G (s,q,s,q ), st q = R s(q,s )= a bq e, that s, max s 0 G (s,q,s,r s(q,s )) The foc for government s dg (s,q,s,r s (q,s )) ds = dπ (R s (q,s ),q ;s,s ) ds q s q s = 0, f the soluton s nteror (s 0) If dg (s,q,s,r s (q,s )) ds < 0, the soluton s corner, s = 0 The foc s rewrtten as follows: dπ (R s (q,s ),q ;s,s ) ds q s q s = π R s (q,s ) s G (s,q,s,r s (q,s )) s = + π s q s R s (q,s ) s = 0 By substtutng π = 0, the foc of frm, and π s = q nto the foc of government, we obtan 0 1 + q 1 q s = s 1 0 sbs = 0 As a result, the subsdy level s zero, regardless of any q The reacton functon s s bs = R s (q ) = 0 Thus q = R s (q, 0) = a bq c In the subgame at the second stage, the output choce by frm s solved as follows: gven s, nducng s bs = 0 and q = R s(q, 0) = a bq c, frm maxmzes the proft functon π (q,r s (q, 0); s, 0) That s, the maxmzng problem s max q π (q,r s (q, 0); s, 0), notng that R s (q )= 1 2 The foc s π +π Rs (q )=((a b(q +R s(q, 0)) e ) bq ) bq ( 1 2 ) = 0 Ths result s the same as for the Stackelberg equlbrum wth no subsdy: q = R s (s )= a 2e +c = 36 Note that f s = 0, the unlateral nterventon of sequental decson-makng s analyzed: q s q If s = 0, Ths s Case C-1 31