Purchase and rental subsidies in durable-good oligopolies* 1

Transcription

1 Hacienda Pública Epañola / Review of Public Economic, 3-(/05): , Intituto de Etudio Ficale DOI: /HPE-RPE.5.. Puchae and ental ubidie in duable-good oligopolie* AMAGOIA SAGASTA JOSÉ M. USATEGUI Univeity of the Baque County Abtact Received: July, 03 Accepted: Febuay, 05 We fully chaacteize the effect of pe unit ubidie fo the puchae and ental of duable good, conideing ubidie in the peent and ubidie in the futue, unde impefect competition. We how how welfae i affected by the imultaneou conideation of ubidie on enting and elling and how the effect of a change in one of thoe ubidie depend on it inteaction with othe ubidie. Among othe eult we explain why a ubidy in the futue mitigate the commitment poblem of poduce when they ell and ent the duable good in the peent, we find the egion in the pace of ubidie whee a ubidy on ale in the peent imultaneouly inceae the conume, poduce and total uplue, and we how that the cot of ubidie may change in the oppoite diection to the diection of change in any one ubidy. Keywod: ubidie, duable good, oligopoly, inteaction between ubidie, cot of ubidie. JEL Claification: H3, L3, H, H4.. Intoduction Given an oligopoly of fim that may ell and alo ent a duable good (enting-elling fim), we analyze and compae the effect of pe unit ubidie on puchae and ental of the good in the peent and in the futue. We povide a full chaacteization of the effect of uch ubidie on pice, quantitie, conume and poduce uplue, total uplu, and total ubidy cot. We note that peent and futue ubidie have intetempoal effect: Each ubidy affect the level of ale and ental and thei pice in the peent and in the futue. * The autho thank two anonymou efeee fo helpful comment and uggetion on ealie daft of the pape. The uual diclaime applie. Financial uppot fom Miniteio de Ciencia e Innovación and Miniteio de Educación y Ciencia (ECO0-3580) and fom Depatamento de Educación, Univeidade e Invetigación del Gobieno Vaco (IT and IT-783-3) i gatefully acknowledged. Coepondence: Depatamento de Fundamento del Análii Económico II and BRiDGE; Facultad de CC. Económica y Empeaiale; Univeidad del Paí Vaco UPV/EHU; Avda. Lehendakai Agie, 83; 4805 Bilbao; Spain. joemaia.uategui@ehu. e, amagoia.agata@ehu.e.

2 amagoia agata and joé m. uategui We tudy how welfae vaiable ae affected by the imultaneou conideation of ubidie on enting and elling. The effect of a change in one of thoe ubidie depend on how it inteact with othe ubidie. We how that when thee ae no othe ubidie on puchae o ental a po-competitive effect can be expected of an inceae in the ubidy on puchaing in the peent and of a eduction in the ubidy on enting in the peent o in the ubidy in the futue (and an anti-competitive effect can be expected of change in oppoite diection in thoe ubidie). Nevethele, when thee ae othe ubidie on enting o puchaing the diection of ome of the effect of a change in a ubidy on welfae vaiable may diffe, ince they depend on the level of the othe ubidie. We pove that in thi latte cae fim pofit may change in the ame diection a a ubidy on puchaing in the peent and in the oppoite diection to a ubidy in the futue, that the cot of ubidie may change in the oppoite diection to the diection of change in any of the ubidie and that total uplu (when the ocial cot of a unit of public fund i ) may change in the ame diection a a ubidy on enting in the peent. Hence, inteaction between ubidie concen not only the extent to which change in a ubidy affect welfae vai able but alo the diection of ome of thoe effect. The effect of change in ubidie ae elated to the tategic and commitment mecha nim undelying fim behavio. We how that change in ubidie affect the maket powe of oligopolit (thei ability to often the effect of the time inconitency poblem). An inceae in the ubidy on ental in the peent o in a ubidy in the futue mitigate the poduce commitment poblem with buye in the peent. Sale in the peent become le attactive when the latte ubidy i inceaed a the unit of the good ented in the peent will alo be ented in the futue and will eceive the inceaed ubidy. By contat, an inceae in the ubidy on puchaing in the peent povide incentive fo poduce to inceae thei ale now, which futhe inceae thei commitment poblem. Uing a ubidy on puchae but not on ental in the peent educe the maket powe that oligopolit exet by alo enting the duable good. A an inceae in a ubidy alo inceae the willing ne to pay of conume eligible fo that ubidy, an inceae in the ubidy on puchae in the peent ha two oppoite effect on fim pofit. Howeve, in the cae of non-duable good ubidie diectly benefit fim without impoing any cot on them, ince thee ae no commitment poblem with buye. An inceae in the ubidy on puchaing in the peent unde an oligopoly inceae the incentive of each poduce to teal maket hae fom it ival by inceaing ale in the peent. Thoe incentive inceae the poduce commitment poblem unde an oligopoly. To iolate the effect of that tategic behavio of fim we alo tudy the cae of a monopo ly, whee uch incentive ae abent. We how that an inceae in the ubidy on puchaing in the peent alway inceae conume uplu, poduce uplu, total uplu and the cot of ubidie unde a monopoly. Unde an oligopoly the ame occu, but only fo ome itu ation whee thee ae alo ubidie on ental. We find the egion in the pace of ubidie whee the poduce uplu change in the ame diection a a ubidy on puchaing in the peent fo the oligopoly cae. Nevethele, it cannot be concluded that the egion of the pace of ubidie whee an inceae in the ubidy on puchaing in the peent inceae conume uplu, poduce uplu and total uplu i lage unde a monopoly than unde

3 Puchae and ental ubidie in duable-good oligopolie 3 an oligopoly, becaue the egion of the pace of ubidie whee the analyi applie (pice and quantitie ae non-negative) change with the numbe of fim. We detemine what change in ubidie ae bette fo avoiding pice inceae and what change in ubidie favo total output, hot-un output and output in the futue. We alo note that thee ae imultaneou change in ubidie that imply patial compenation, o einfocement, of the effect on mot pice and quantitie 3. Ou eult ae baed on the inteaction between ubidie, the intetempoal effect of ubidie, the diffeent implication of ubidie on enting and on puchaing in the peent, impefect competition, the tategic behavio of each fim in tealing ale fom it ival in the peent and in the futue, and the poduce commitment poblem. Poduce of duable good may have incentive to ell ome unit of the good and to ent the et of thei output even if they face the time inconitency poblem. Calton and Getne (989) pove that tategic inteaction between ival povide an oligopolit with a eaon to chooe to ell ome of it output athe than ent it, in contat to the behavio of a monopolit, which will chooe to ent all it output 4. Although the pofit of each oligopolit would be geate if they all only ented the good (leaing would olve the time inconitency poblem), enting become le attactive fo a fim that face competition fom othe fim. Sale allow a fim to captue pat of the maket, wheea enting may eult in futue maket hae being lot to ival ince all fim can compete again fo conume that leaed in the pat. Nevethele, poduce do not ell all thei output o a to mitigate the poduce commitment poblem with buye in the peent (to etain ome maket powe by educing competition between futue and peent ale) 5. The famewok fo ou analyi i thi oligopoly of fim that ent and ell the duable good in the peent, a tudied in Calton and Getne (989). Hence, coe point in ou analyi include not only the poduce commitment poblem but alo the tategic inteaction among competito that lead each oligopolit to chooe both to ent and to ell. Ou wok i elated to the liteatue on the effect of ubidie and taxe when thee i impefect competition in the poduction of the duable good. Some eult apply to ituation whee fim ell thei output: Chen et al. (00) invetigate the effectivene of a futue ale tax eduction in timulating ale and pofit among duable good manufactue (in the U.S. automobile induty). Goeing and Boyce (996) pove that an excie tax on output may inceae a duable good monopolit maket powe and pofit though it effect on the poduce commitment poblem when the good i old. In a context whee poduction by a duable good monopolit caue pollution, Dikill and Hoowitz (007) how that an effluent tax offet by a lump-um ubidy may inceae both the fim pofit and ocial welfae a the effluent tax povide a cedible commitment that etict futue upply by a monopolit elle. The effect of taxe on ale and ental when thee i imultaneou enting and elling have been analyzed befoe. Goeing (0) how that a duable good monopolit that face a cuent ad-valoem tax wihe to concuently ell and ent in the peent. Kim et al. (04)

4 4 amagoia agata and joé m. uategui analyze how an ad-valoem tax, with a contant ate in the peent and in the futue, affect the leae-ale tategy of a duable-good monopolit, and dicu it implication on monopolitic behavio. Ou tudy diffe fom thee analye a we conide an oligopolitic maket tuctue and pe unit ubidie that may be diffeent fo puchae and fo ental and that may diffe in the peent and in the futue 6. Non-duable good ae old, but duable good can uually be ented a well a bought and fo ome uch good the ental maket i becoming inceaingly impotant 7. Govenment ubidize puchae and ental in ome duable good indutie, e.g. ca puchae in eveal countie and puchae and ental of agicultual machiney 8. Moeove, conume pending on duable good tend to be moe cloely elated to the economic cycle than pending on non duable good and evice, a the fome can be moe eadily potponed in time of economic hadhip 9. Given the weakne in demand fo duable good duing lowdown, govenment have alo intoduced ubidie in eceion tageted at pending on conume duable 0. The et of the pape i oganized a follow: Section intoduce the model. The deciion of fim unde puchae and ental ubidie ae invetigated in Section 3. The effect of ubidie on pice and quantitie ae tudied in Section 4. A welfae analyi of ubidie i peented in Section 5. Section 6 conclude. The Appendix contain the poof of the eult and define the feaible egion in the pace of ubidie whee the analyi applie.. Model We conide an oligopolitic induty with n ³ identical fim that poduce a homogeneou duable good. Enty into the induty i aumed to be unpofitable o unfeaible. Thee ae two dicete peiod of time: peent ( t ) and futue ( t ). Fim may both ell and ent thei output (enting-elling fim) in the fit peiod. Given that the econd peiod i the lat one, enting i identical to elling in that peiod. Denote by q i and q i, epectively, the quantity old and the quantity ented by fim i in the fit peiod. Hence, q + q i i i the quantity poduced by fim i in the fit peiod and q i denote the quantity old (ented) by fim i in the econd peiod. Theefoe, q i -q i and q i + q i ae, epectively the quantity poduced by fim i in the econd peiod and the total (fit peiod plu econd peiod) poduced by fim i. Fo the coeponding quantitie at induty level wite Q, intead of q, and eliminate the i ubcipt. We conide ituation whee thee i enting and elling in t ( q i >0 and q i >0) and whee all unit ented in t ae old (o ented) in t and new unit of the good ae poduced in the econd peiod ( q > q i i ). The quantity of the duable good ued in t in the maket i thu Q +Q.

5 Puchae and ental ubidie in duable-good oligopolie 5 The invee demand fo evice of the duable good i aumed to be contant ove time. Thi invee ental demand function fo the evice of the duable good in each peiod i pq ( ) a- bq, whee Q epeent the quantity ued by conume in the peiod (o the tock of the duable good available in the peiod) and ab, > 0. All agent paticipating in the maket have pefect, complete infomation and potential ue of the good have pefect foeight. We conide that thee i a pefect econd-hand maket fo the duable good. Ou analyi take place unde thee implifying aumption that do not affect the eult. We conide fit that the dicount facto i fo all agent paticipating in the maket. Secondly, we aume that the unit of the duable good poduced in the fit peiod do not depeciate ove time. Thu, evey unit ued in the fit peiod can be ued in the econd peiod without depeciation. Finally, the maginal cot of poduction of each fim in each peiod i c 0. The analyi poceed in two tage. In the fit tage the egulato et ubidie fo the two peiod. We conide that the egulato can commit to ubidie and announce them at the beginning of the fit peiod. In the econd tage fim engage in quantity competition. Fim choice in each peiod ae imultaneou. Each fim chooe it level of poduction in each peiod and the diviion of output between enting and elling in the fit peiod, taking a given the deciion on poduction, enting and elling of it competito. The objective of each fim i to maximize the um of it pofit in the two peiod. The competition game between fim i theefoe non-coopeative. The olution concept ued i that of a ubgame pefect Nah equilibium in pue tategie. In each peiod each fim maximize the peent dicounted value of pofit tating fom that peiod. Theefoe, the olution ae deived by backwad induction fom the lat peiod of the econd tage. We conide that each buye of a unit of the duable good in t eceive a pe unit ubidy of, each ente of a unit of the duable good in t eceive a pe unit ubidy of, and each conume who puchae o ent a unit of the duable good in t eceive a pe unit ubidy of. Let p, p and p denote, epectively, the maket pice of a unit of the duable good old in the fit peiod, of a unit of the duable good ented in the fit peiod, and of a unit of the duable good old o ented in the econd peiod. The net pice paid by conume ae, epectively, p -, p - and p -. The poibility of abitage by conume implie: p p +( p ) The net ale pice in the fit peiod equal the dicounted team of the expected net ental pice in the two peiod. Thi condition i incopoated into the analyi. A conume

6 6 amagoia agata and joé m. uategui with high willingne to pay fo the good in the fit peiod and with vey low willingne to pay fo the good in the econd peiod would ent the good only in the fit peiod. Fom the definition of the invee ental demand function we have: p - a-bq -bq and p a b - - Q -bq The net ental pice paid by conume in each peiod i linealy elated to the tock of the duable good available in the peiod. The econd peiod tock conit of the total (fit peiod plu econd peiod) output by fim. A a conequence of the abence of abitage we alo have: p a bq bq + (a bq bq ) We conide that pe unit ubidie and paamete value ae uch that p - >0, p - >0 and p - >0. The egion in the thee-dimenional (,, ) pace of ubidie whee the analyi applie i woked out in Section B of the Appendix. The limit of that egion depend on the paamete of the model and ae defined by the equiement of poitive net ental and ale pice, poitive ental and ale in the peent, and ental in t geate than ental in t. At the end of Section B of the Appendix we pecify the inteval of value fo each pe unit ubidy whee the analyi applie when thee ae no othe ubidie. Pe unit taxe ae negative pe unit ubidie. Ou analyi can be applied to the analyi of pe unit taxe on puchae o ental of duable good, povided that the containt obtained in Section B of the Appendix hold. The effect of an inceae in a pe unit tax would be the ame a the effect of a eduction in the coeponding pe unit ubidy. 3. Fim deciion with pe unit ubidie on puchaing and enting Thi ection examine how ubidie affect poduction, elling, and enting in each peiod. Let p i i and p denote, epectively, the pofit in peiod and in peiod of fim i, with i i i i,..., n. The peent value of fim i pofit i then witten a p p + p. At the beginning of peiod t each fim chooe it econd peiod ale (ental) to maximize it pofit. Fim have no incentive to take into account in peiod the capital lo bone by buye of unit of the duable good in peiod one. Hence, in peiod t each active fim i, with i,..., n, olve the following poblem: i max p (a bq b Q + )q. q i i

7 Puchae and ental ubidie in duable-good oligopolie 7 The fit ode condition of thi poblem i : a bq bq bq i + 0, () that i, maginal evenue fo fim i in the econd peiod equal maginal cot (which i 0 ). Adding up the n fit ode condition ove i give: a bq + q. () i bn ( +) Thi behavio of fim in peiod t tun out a fim face the time inconitency poblem fit noted by Coae (97). A Coae (97) point out, a duable good poduce that ell it output and ha no commitment ability lack any incentive to take into account the decline in the value of unit peviouly old that ae in the hand of conume when it decide it futue output. A conume have pefect foeight, they ealize that each fim will chooe it econd peiod poduction to atify (). Hence, () become an expectation con taint on each enting-elling fim which implie that econd peiod poduction i implic itly detemined by fit peiod output. In peiod t, each fim chooe the level of ale and ental, q and i q, that i maximize the peent value of it pofit ubject to (). Thu, each fim i, with i,...,n, olve the following poblem: max p i + p i [(a bq bq + a bq bq + q ) + i { q,q i i } + ( a bq bq + )q i + (a bq bq + ) q ] i ubject to (). p i A backwad induction implie that 0, the fit ode condition of thi poblem ae: q i (p i + p i ) a bq bq + bq bq i i 0 q i (p i + pi ) a b q i Q bq + a bq bq + q b i (3) Q Q i bq i bq i bq i b q b i q i 0 q q i i whee Q -i Q -q i. Thee equation equate maginal cot to maginal evenue fom ental in the fit peiod and maginal evenue fom ale in that peiod, epectively.

8 8 amagoia agata and joé m. uategui The expeion of maginal evenue in (3) incopoate all the elevant apect of the maximization poblem when good ae duable and thee i impefect competition. Fo in tance, the maginal evenue fom ale in the fit peiod may be decompoed in the follow ing tem 3 : a bq bq + a bq bq + + ( bq ) q q i ale pice impefect competition effect + ( bq ) + ( bq i bq ) q i q i concuent enting and elling effect duability effect Q Q + ( b q ) + ( b i i q i ) q q q i q i intetempoal inconitency effect tategic effect The concuent enting and elling effect take into account the effect of ale in the fit peiod on the pice of ental in that peiod. The duability effect i a conequence of the duability of the good. The intetempoal inconitency effect captue the effect of the im pact of fit peiod ale on ale (ental) in the econd peiod in a context in which poduc e in peiod do not take into account the capital lo bone by buye of the good in the fit peiod. The tategic effect cove the implication of n >. The impefect competition, du ability and concuent enting and elling effect ae negative. A () implie that thee i a Q ymmetic olution fo fim deciion in peiod it can be concluded fom () that Q q i i and q ae negative and, hence, that the intetempoal inconitency and tategic effect i ae poitive. Taking into account () and the ymmety of the olution fo fim deciion in () and (3) the fit ode condition of the poblem of maximization of p i + p i may be witten a: (p i + p i ) a bq bq + bq i bq i 0 q i (p i + p i ) a bq n a bq bq + + q i n + bq i a+ bq i + n n a bq + bq + b q + b i n + n + i n + bn ( +) (4) a bq a bq bq + n (a+ bq ( i n + n + ) + ) + bq i 0 n + (n +)

9 Puchae and ental ubidie in duable-good oligopolie 9 A paamete value that imply inteio olution ae aumed, fim deciion ae ob tained fom () and (4). 4. Effect of ubidie on pice and quantitie Table A and B how the ign of the fit patial deivative of quantitie and pice with epect to each ubidy. Hence, in thee table a + ign mean a vaiation of the co eponding vaiable in the ame diection a the ubidy and a - ign mean a vaiation in the oppoite diection to the ubidy. We wite 0 when the vaiable i independent of the level of the ubidy. All thee ign ae obtained in the poof of Popoition below. Table A EFFECTS OF SUBSIDIES: QUANTITIES q i q i q i q i + q i Table B EFFECTS OF SUBSIDIES: PRICES p p - p p - p p The following effect of ubidie on quantitie and pice ae emphaized: Popoition. If fim poducing a duable good can ent and ell thei output, then in equilibium: i) fit peiod ale, futue output and total output change in the ame diection a and in the oppoite diection to and ; ii) ental in the peent and in the futue change in the oppoite diection to and in the ame diection a and ; and iii) maket and conume ale pice in the peent and maket and conume pice in the futue change in the oppoite diection to and in the ame diection a and.

10 0 amagoia agata and joé m. uategui Poof: See Appendix. The poof of Popoition alo how that fit peiod output ( Q +Q ) doe not depend on o on but inceae with, and that poduction in the futue ( Q -Q ) inceae with and deceae with and with. Hence, in ou model an inceae in favo hot-un poduction while an inceae in (o a deceae in o in ) i bette fo induc ing inceae in futue poduction and in total output. When i inceaed output in the fit peiod doe not change but lack of commitment by poduce and the eduction in ental in the peent imply an inceae in futue poduction and, a a conequence, a deceae in the maket ale pice in the peent. Fom Table A and B it can be infeed that thee i a po-competitive effect of an inceae in (o of a deceae in o in ) and thee i an anti-competitive effect of an inceae in o in (o of a deceae in ). That po-competitive effect occu though an induced inceae in ale in peiod t which i not offet by a eduction in ental in the econd peiod. Thee po-competitive and anti-competitive effect of ubidie ae di cued futhe in the next ection. If imultaneou change in eveal ubidie ae conideed it hould be noted fom Table A and B that a change in the ame diection in and (o in and ) implie patial offetting of the effect on mot pice and quantitie. By contat, mot of thoe ef fect would be einfoced if and (o and ) change in oppoite diection o if and change in the ame diection. The poof of Popoition can be ued to tudy the effect of a change in n. We find that an inceae in n inceae the poitive patial deivative Q Q Q,,, Q and Q Q Q, and deceae the negative patial deivative Q,, and Q. Thi implie, fo intance, ome offetting in total output, and in fact we find that a n inceae (Q +Q ) become le poitive and (Q +Q ) (Q and +Q ) become le nega tive Welfae analyi Thi ection analyze how ubidie on puchaing and ental in the peent and the ubidy in the futue affect the conume uplu, the pofit of fim, the cot of ubidie and the total uplu. The effect of ubidie obtained in thi ection deive fom the effect

11 Puchae and ental ubidie in duable-good oligopolie of ubidie on quantitie and pice dicued in the peviou ection. The following defini- tion ae ued: Conume uplu: CS Q + Q (a bxd ) x ( p ) Q (p )Q 0 + Q +Q 0 (a bxb ) dx (p )Q Poduce uplu: b ((Q +Q ) +(Q +Q ) ) Cot of ubidie: Π p Q + p Q + pq SUB Q + Q + Q We meaue total uplu ( TS ) a the um of conume uplu and poduce uplu net of the cot of ubidie. Conide that the ocial cot of public fund i g, with g ³. Thi give ie to the following: TS( g ) CS + Π g SUB a(q + Q ) + aq ( +Q ) b (( Q +Q ) +(Q +Q ) ) (g )( Q + Q + Q When the ubidy inceaed (o deceaed) i the only ubidy (the othe two ubidie ae zeo) the following i obtained: ) Popoition. When thee ae no othe ubidie: i) CS and TS ( g ) change in the ame diection a and in the oppoite diec tion to and ; ii) P change in the oppoite diection to and in the ame diection a and ; and iii) the cot of ubidie change in the ame diection a each ubidy.

12 amagoia agata and joé m. uategui Poof: See Appendix. The eult in Popoition ae independent of n, povided that n ³ (the cae of a monopoly i conideed below). Table peent the ign of the fit patial deivative of CS, P, SUB and TS ( g ) with epect to each ubidy obtained in the poof of that Popoition: Table MARGINAL EFFECTS OF EACH SUBSIDY WHEN IT IS THE ONLY SUBSIDY g ) CS P SUB TS ( When the ubidy modified i the only ubidy poduce uppot a deceae in o inceae in o in. A change in a ubidy affect the maket powe of poduce. An inceae in (o a deceae in ) inceae the poduce commitment poblem with buy e in the peent a ental in the peent deceae. It theefoe educe the maket powe that poduce exet by alo enting the duable good (buye will be willing to pay le fo the duable good in the fit peiod a they know that poduce will inceae poduction in the futue). Moeove, unde an oligopoly an inceae in (o a deceae in ) einfoce the incentive of fim to teal maket hae fom thei ival in the peent and in the futue by inceaing in ale in the peent and thi futhe inceae the poduce commitment poblem. When i inceaed thee effect ae found to dominate the poitive diect effect of the inceae in on the willingne to pay of buye in the peent. It may eem that an inceae in (an inceae in the pe unit ubidy in the futue) will alo inceae the poduce commitment poblem with buye in the peent but the eult in the peviou ection eveal that in peiod t it implie an inceae in ental and a educ tion in ale by the ame amount. Sale in the peent become le attactive with an inceae in becaue the unit of the good ented in the peent will alo be ented in the futue and will eceive the inceaed ubidy. Hence, with fim that ent and ell thei output an inceae in a pe unit ubidy in the futue educe the poduce commitment poblem and inceae the poduce uplu, a it alo inceae the willingne of conume to pay fo the good in the econd peiod. An inceae in educe the incentive of fim to teal maket hae fom thei ival in the peent and in the futue by inceaing ale in the pe ent. Ealie pape have hown that the expectation of a futue tax mitigate the poduce commitment poblem with buye in the peent in context whee poduce only ell the good (ee Goeing and Boyce (996), Dikill and Hoowitz (007) and Chen, Eteban and

13 Puchae and ental ubidie in duable-good oligopolie 3 Shum (00)). A pe unit ubidy i a pe unit negative tax, but we find hee that a pe unit ubidy in the futue mitigate the poduce commitment poblem. The explanation fo thi divegence of eult concening the effect of a futue ubidy (o tax) on the poduce commitment poblem i that we conide that poduce both ent and ell the duable good. Fom Popoition it emege that conume may not be favoed by an inceae in a ubidy. It i not only that buye, fo intance, may not be favoed by a ubidy on ental: Table B how that ente of the good in the econd peiod pay a highe pice when inceae. In thi cae poduce pefe to educe poduction in the lat peiod in a way that implie a bigge inceae in p than in. A CS deceae with and, an inceae in any of thee ubidie would have to be ooted in equity citeia (fo intance, ente of the duable good in the fit peiod pay le when i inceaed) o in ome othe eaon. When thee ae no othe ubidie pat i) and ii) in Popoition ae a conequence of the po-competitive effect of an inceae in (o of a deceae in o in ) and of the anti-competitive effect of an inceae in o in (o of a deceae in ) pointed out in the peviou ection. The deceae in total uplu, even if g, with a deceae in o with an inceae in o in i elated to the deceae in total poduction caued by the change in ubidie. A an inceae in g educe total uplu, an inceae in will alo educe total uplu if g i big enough. The ign of ome of the maginal effect of a change in a ubidy may be diffeent fom thoe obtained in Popoition when thee ae poitive level of one o two of the othe ubidie. We have: Popoition 3. When thee ae poitive level of one o two of the othe ubidie all the eult in Popoition hold except: i) P may change in the ame diection a o in the oppoite diection to ; ii) TS ( g ) may change in the ame diection a ; and iii) the cot of ubidie may change in the oppoite diection to the diection of change in any of the ubidie. Poof: See Appendix. Table 3 peent the ign of the fit patial deivative of CS, P, SUB and TS ( g ) with epect to each ubidy obtained in the poof of Popoition 3 ( + / mean that the coeponding vaiable may change in the ame diection a the ubidy o in the oppoite diection to it):

14 4 amagoia agata and joé m. uategui Table 3 MARGINAL EFFECTS OF EACH SUBSIDY WHEN THERE ARE OTHER SUBSIDIES CS P SUB TS ( g ) + +/ +/ + + +/ +/ +/ +/ When thee ae poitive level of one o two of the othe ubidie the inteaction between ubidie may modify not only the ize but alo the diection of the effect of a change in a ubidy on ome vaiable. When i high with epect to and the diect effect on P of an inceae in may offet the negative effect of the inceae in the poduce commitment poblem. Moeove, a ale in the peent diminih with a high level of may offet the poitive effect on P of the mitigation in the poduce commitment poblem when i aied. Hence, when thee ae othe ubidie on ale o ental and i high with epect to and, poduce may uppot an inceae in o a deceae in. Table A help explain why the cot of ubidie may change in the oppoite diection to the diection of change in a ubidy when thee ae othe ubidie. The cot of ubidie may inceae afte a eduction in a that eduction induce inceae in ental in the pe ent and in the futue, and hence inceae the amount pent on ubidie on ental. Analo gouly, the cot of ubidie may alo inceae afte a eduction in, o in, a that change induce an inceae in ale in the peent, and hence inceae the amount pent on ubidie on ale in that peiod. The poibility of a eduction in the cot of ubidie with an inceae in, togethe with the mitigation of the poduce commitment poblem with that change in, i alo behind the feaibility of an inceae in TS( g ) with. Table 3 how that thee ae value of the paamete uch that CS, P and TS ( g ) inceae when i aied and thee ae poitive level of one o two of the othe ubidie. In the following Popoition we delimit the egion in the pace of ubidie whee thoe inceae occu: Popoition 4. An inceae in inceae CS, P and TS ( g ) imultaneouly when: n 4 + n 3 + n + nn ( )a (n ) a ( n +) max + +, + < n 4 + n 3 + n + n 4 + n 3 + n + n + (n +) n n + +, + +. (n +) (n +) 3 n + nn ( +) < min ( + + ) a +(n +) a

15 Puchae and ental ubidie in duable-good oligopolie 5 Poof: See Appendix. Fom the poof of Popoition and 3 it emege that when + pofit de ceae with. Hence, the egion of the pace of ubidie defined in Popoition 4 i within the egion > +. Fo intance, CS, P and TS ( g ) inceae imultane ouly with an inceae in if a 50, n 6, b, 8., 7 and 0. (the poof of Popoition and 3 include anothe example fo n ). Moeove, when n the condition in Popoition 4 imply > 5 a : ( )a 9 a ( + + ) a + ( + ) 3 8 a < > 3 + ) ( + ) ( The poof of Popoition 4, howeve, how that in the egion of the pace of ubidie defined in that Popoition the cot of ubidie paid by taxpaye inceae with. When and 0 it can be hown fom the poof of Popoition and 3 that CS, P and TS ( g ) inceae with (that i, with a imultaneou inceae in and ). Poceeding a in the poof of Popoition 4 it i poible to find the egion in the pace of ubidie whee Π TS <0, whee >0 and whee thee i a negative effect of each ubidy on the cot of ubidie when thee ae othe ubidie on puchae o ental. Each of thoe egion in the pace of ubidie i detemined by the coeponding condition obtained. Fom the poof of Popoition and 3 ome imple infomation can alo be ob tained on the aea whee thoe egion ae located in the pace of ubidie. When + pofit inceae with and the cot of ubidie inceae with, when total uplu deceae with, the cot of ubidie inceae with if + and ³ and, finally, the cot of ubidie inceae with if +. To detemine the effect of the incentive of fim to teal maket hae fom thei ival in the peent and in the futue by inceaing ale in the peent when i aied, we con ide below that thee i a monopoly intead of an oligopoly of poduce. The incentive to teal inceae the poduce commitment poblem and educe poduce uplu unde an oligopoly, but it i abent unde a monopoly. The following Popoition examine the ign of the patial deivative of CS, P, cot of ubidie and TS ( g ) with epect to each ubidy when a monopoly i conideed:

16 6 amagoia agata and joé m. uategui Popoition 5. Unde a monopoly: i) an inceae in inceae CS, P and TS ( g ) imultaneouly a it alway hold that Π >0; and ii) the et of the eult in Popoition and 3 hold except that it alway hold that SUB >0. Poof: See Appendix. Popoition 5 eveal that when thee ae no othe ubidie on puchae and ental the pofit of the monopolit inceae with, contay to the eult obtained in Popoition fo n ³. Moeove, when thee ae othe ubidie on puchae o ental thee i no am Π SUB biguity in the ign of and unde a monopoly. Poceeding a in Section B of the Appendix it i poible to obtain the feaible egion in the pace of ubidie whee the analyi of a monopoly applie (it can be hown, fo intance, that Q >0 only if > + ). Fom the compaion of Popoition 4 and 5-i) it cannot be concluded that the egion of the pace of ubidie whee a aie in inceae CS, P and TS ( g ) i lage unde a monopoly than unde an oligopoly. Conide that a, 0.4 and 0.. When n the condition in Popoition 4 i 0.58 < < 0.6. Fom ection B in the Ap pendix it can be hown that the inteval fo whee the analyi of monopoly applie i 0.65 < < Thee inteval do not inteect in that example. Nevethele, the et A of pai (, ) uch that thee exit a value of whee the analyi of monopoly applie i lage than the et B of pai (, ) uch that thee exit a value of whee the condition in Popoition 4 i fulfilled when n. Thi give: and B Ì A. A {(, ) /(, ) ( < a ) ( < a ) ( < a ) } B a a (, )/(, ) ( < ) ( < ) ( < ) ( 3 < a ) 8 6. Concluion We analyze the effect of pe unit ubidie on the puchaing and ental of duable good unde an oligopoly. We conide ubidie on ental and ubidie on puchae in the peent, and ubidie in the futue. Thee ubidie ae uch that pe unit ubidie may be diffeent fo puchae and fo ental and may alo diffe in the peent and in the futue.

17 Puchae and ental ubidie in duable-good oligopolie 7 Ou analyi eveal that thee i a po-competitive effect of an inceae in the ubidy on puchae in the peent and of a eduction in the ubidy on ental in the peent o in the ubidy in the futue (and thee i an anti-competitive effect of change in oppoite diection in thoe ubidie). Ou analyi may have the following policy implication: Fit of all, inteaction between ubidie mut be taken into account. To educe the maket powe of poduce unde an oligopoly the ubidy on puchae in the peent hould in geneal be inceaed o the ubidy on ental in the peent o the ubidy in the futue hould be deceaed. Howeve, if thee ae othe ubidie on puchae o ental poduce uplu may inceae with a ubidy on puchae in the peent (povided that the ubidy on ale in the peent i geate than the um of the ubidie on ental in the peent and in the futue). Hence, an inceae in that ubidy unde an oligopoly may imultaneouly inceae total uplu and conume and poduce uplue. Thi ame eult i alway obtained unde a monopoly in ou analyi. Nevethele, uch inceae in uplue ae accompanied by an inceae in the cot of ubidie. Ou eult eveal that an inceae in the ubidy on ental in the peent i the bet choice fo inceaing hot un poduction while inceaing the ubidy on puchae in the peent (o deceaing the ubidy on ental in the peent o the ubidy in the futue) inceae futue poduction and total output. If imultaneou change in ubidie ae conideed, it mut be taken into account that thoe change may imply a patial offetting o a einfocement of the effect on pice and quantitie. The analyi in Section 4 and 5 how, fo intance, that the diection of change in quantitie, in pice and in mot welfae vaiable ae oppoite fo a eduction in the ubidy on puchae in the peent and fo a eduction in the ubidy in the futue. Govenment often conide altenative eduction in ubidie a a conequence of budget limitation. We pove that a eduction in one ubidy may inceae the cot of ubidie when thee ae othe ubidie on puchae o ental of the duable good. Moeove, we find that when thee ae othe ubidie on puchae o ental, poduce may alo uppot eduction in the ubidy in the futue and that total uplu (when the ocial cot of a unit of public fund i ) may inceae with a eduction in the ubidy on ental in the peent. The eult fo the effect of ubidie on quantitie, pice and welfae vaiable mean that it i adviable to conduct a caeful empiical analyi of the paticula duable good maket to etimate the value of the elevant paamete befoe etting ubidie on puchae and ental in the peent and in the futue. Ou model alo enable the effect of imultaneou change in eveal ubidie to be analyzed once thoe paamete have been etimated. Some extenion of ou analyi ae immediately appaent. Conide that the demand fo the duable good i non-tationay, and in paticula that the invee demand function fo the evice of the duable good in the futue i p a + k bq, whee k may be poitive o

18 8 amagoia agata and joé m. uategui negative. In thi cae, if all agent in the maket have coect expectation of the change in demand ove time the effect of k will be the ame a the effect of a pe unit ubidy equal to k in the futue. A compaion between the effect of pe unit ubidie and thoe of ad-valoem ubi die on welfae vaiable would be ueful. A full compaion of the implication of thee two type of ubidie i left fo futue eeach. Nevethele, ome diffeence and imilaitie can be pointed out. On the one hand, Goeing (0) how that ad-valoem taxe in the peent lead a monopolit to ell ome of it unit in that peiod in a context whee thoe taxe ae chaged on the full value of the unit poduced in the peent, egadle of wheth e they ae old o ented. The eaon fo thi eult i that thee i a commitment poblem (alo) fo a enting fim a a ente wihe to commit to poducing moe in the futue (and le today) to leen the tax buden. A pe unit ubidie do not depend on ale o ental pice a ente doe not face thi commitment poblem in ou analyi. When and 0 we obtain fom Section A of the Appendix that a monopolit will ent all it output (wheea if n ³ then q >0 and i q >0 fo all i when i ³ 0 ). On the othe hand, Kim et al. (04), alo conideing a monopolit that may ent and ell a duable good, tudy the effect of a contant ad-valoem tax in the peent and in the futue and how that uch a tax may inceae ocial welfae. If and in ou model we obtain that total uplu, conume uplu, poduce uplu, ale and all ental pice inceae with 6. Note. We do not conide a paticula choice of ubidie baed on budget o equity conideation o a choice of uch ubidie that eek to maximize total uplu (the egulato i not well infomed on the paamete of the demand and cot function).. In duable good indutie fim face the time inconitency poblem fit noted by Coae (97). When a duable good poduce ell it output buye know that the fim will have an incentive to inceae poduction in the futue, educing the value of the exiting tock of unit held by buye. Hence, buye willingne to pay in the peent i educed. If conume ae ational, thi expected behavio of fim become an expectation containt on each enting-elling fim. A lage numbe of tudie have analyzed vaiou condition fo mitigat ing o avoiding the commitment poblem [ee Bulow (98), Bond and Samuelon (984), Kahn (986), Butz (990), Kap and Peloff (996), Saggi and Vetta (000) and Moita and Waldman (004)]. 3. Ou analyi alo applie to pe unit taxe on ale and ental, a hown below. 4. Bucovetky and Chilton (986) and Bulow (986) pove that a monopolit facing the theat of enty chooe to ell pat of the unit upplied, intead of enting them all. Deai and Puohit (998) how that a combination of leaing and elling may be optimal fo a monopolit if ented and old poduct depeciate at diffeent ate. 5. Bulow (986) alo how that in an oligopoly fim ue a mixtue of ale and ental in the peent. 6. A indicated by Saggi and Vetta (000), duable good maket ae pimaily oligopolitic athe than monopolitic. 7. Example of duable good that ae both ented and old include automobile, houehold appliance, comput e, copying machine and mechanical equipment (See Saggi and Veta, 000). 8. Fo intance, fom 950 to 980 many developing countie etablihed tate-un, ubidized tacto hie cheme to extend tacto-baed mechanization to mall fame. Since the late 990 the govenment of Ghana and othe Wet Afican countie have povided ubidie to fame who puchae agicultual ma chine, and ince 007 the govenment of Ghana ha been poviding ubidized agicultual machiney to individual fame and pivate entepie to offe tacto-hie evice to mallholde fame (ee Houou et al 03).

19 Puchae and ental ubidie in duable-good oligopolie 9 9. The pecentage of GDP accounted fo by duable conume good tend to fall duing eceion and ie du-ing boom. See Black and Cubet (00) fo an analyi of cycle in pending on duable good in Autalia ove the pat 50 yea and duing the cuent economic lowdown, and fo a compaion with the United State and othe economie. 0. Moto vehicle ubidie fo conume have been implemented in a numbe of countie, including China, Japan, the United State and ome Euopean countie, with highe ca ale and output epoted in many cae [ee Chen et al. (00)].. Altenatively, that conume could buy the good in the fit peiod paying p - and ell it back in the econd peiod fo a pice p -.. Thoughout the pape we aume that paamete and ubidie ae uch that inteio olution exit (ee Section A and B of the Appendix). 3. The decompoition fo the maginal evenue fom ental in the fit peiod i analogou, although much imple. (Q +Q 4. Nevethele, ) (Q +Q and ) have the ame value fo n 3 a fo n It can alo be hown that > when n 3 but we do not find that the diffeence between and inceae with n To obtain thee eult, note that etiction (B.) in thi cae only imply that a> n. Refeence Black, S. and T. Cubet, (00), Duable Good and the Buine Cycle, Bulletin Reeve Bank of Autalia Septembe: -8. Bond, E. W. and L. Samuelon, (984), Duable good monopolie with ational expectation and eplacement ale, Rand Jounal of Economic, 5: Bucovetky, S. and J. Chilton, (986), Concuent enting and elling in a duable-good monopoly unde theat of enty, Rand Jounal of Economic 7: Bulow, J., (98), Duable-Good Monopolit, Jounal of Political Economy 90: Bulow, J., (986), An Economic Theoy of Planned Obolecence, Quately Jounal of Economic 0: Butz, D., (990), Duable-Good monopoly and bet-pice poviion, Ameican Economic Review 80: Calton, D. and R. Getne, (989), Maket Powe and Mege in Duable-Good Indutie, Jounal of Law and Economic 3: S03-S6. Chen, J., S. Eteban and M. Shum, (00), Do ale tax cedit timulate the automobile maket?, Intenational Jounal of Indutial Oganization 8: Coae, R., (97), Duability and Monopoly, Jounal of Law and Economic 5: Deai, P. and D. Puohit, (998), Leaing and elling: Optimal maketing tategie fo a duable good fim, Management Science, 44: Dikill, R. and A.. W. Hoowitz, (007), The pollution haven paadox: Can an effluent tax impove both pofit and welfae?, The B.E. Jounal of Economic Analyi and Policy 30: -6. Goeing, G. and J. Boyce, (996), Taxation and maket powe when poduct ae duable, Jounal of Regulatoy Economic 9:

20 30 amagoia agata and joé m. uategui Goeing, G., (0), Taxation and duable good monopoly: Doe a cuent tax influence fim behavio?, Review of Economic & Finance (3): 0-8. Houou, N., Diao, X., Coa, F., Kolavalli, S., Jimah, K., Aboagye, P., (03), Agicultual Mechanization in Ghana: I Specialization in Agicultual Mechanization a Viable Buine Model?, IFPRI Dicuion Pape 055. Kahn, C., (986), The duable good monopolit and conitency with inceaing cot, Econometica, 54: Kap, L. and J. Peloff. (996), The Optimal Suppeion of a Low-Cot Technology by a Duable- Good Monopoly, Rand Jounal of Economic 7(): Kim, J., M. Kim and S. Chun, (04), Popety tax and it effect on tategic behavio of leaing and elling fo a duable-good monopolit, Intenational Review of Economic and Finance 9: Moita, H. and M. Waldman, (004), Duable good, monopoly maintenance, and time inconitency, Jounal of Economic and Management Stategy 3(): Saggi, K. and N. Vetta, (000), Leaing veu elling and fim efficiency in oligopoly, Economic Lette 66: Reumen En ete tabajo caacteizamo completamente lo efecto de lo ubidio en cuantía fija a la compa y al alquile de biene duadeo, conideando ubidio en el peente y ubidio en el futuo, en un contexto de competencia impefecta. Motamo cómo la conideación imultánea de ubidio a la compa y al alquile afecta al bieneta ocial y cómo lo efecto de cualquie cambio en uno de eo ubidio dependen de la inteacción ente lo ubidio exitente. Ente oto eultado deteminamo la egión del epacio de ubidio en la que un ubidio a la compa en el peente aumenta imultáneamente lo excedente de conumidoe y poductoe y el excedente total y motamo que el cote de lo ubidio puede cambia en diección opueta a la diección de vaiación en uno cualquiea de lo ubidio. Palaba clave: ubidio, biene duadeo, oligopolio, inteacción ente lo ubidio, cote de lo ubidio. Claificación JEL: H3, L3, H, H4.

21 Puchae and ental ubidie in duable-good oligopolie 3 Appendix A. Poof of Popoition Adding up the n fit ode condition in (4) ove i and olving, we obtain: an ( ) + ( n + ( ) q ) (n + n + ) i (n +)b a ( n +) n q ( + 4n + 3n + ) + ( n + n + 3n + ) i (n +)(n +)b (A.) a+ q + q i i (n +)b By eplacing q in equation () we get: i a n(+ n)( ) + n ( + n + ) q i (A.) (n +)b Theefoe, we have: ( n )a + (n + ) ( n + n + ) (n +) q i q i (n +)(n +)b na + ( n + ( ) ) q i + q i (n +)b (A.3) We alo obtain: an ( + n + )+ n + n ( n +) n ( n +) p (n +)(n +) a n p (A.4) n + a n( n + )( ) + n p n +

22 3 amagoia agata and joé m. uategui an ( + n + )+ n p + n( n +) n ( n +)( n +) ( )(n +) a + p (A.5) n + a n( n + )( ) + (n + n + ) a+ ( n + n)( p ) n + n + The effect on pice and quantitie of change in the level of any ubidy ae immediate fom equation (A.) to (A.5). In the next ection we obtain the egion in the pace of ubidie whee the analyi applie. The limit of that egion depend on the paamete of the model. B. Feaible egion in the pace of ubidie A Q > Q p > p > p we equie that a-bq -bq >0 to guaantee poitive net ental and ale pice. We have: (n +) +(n + n + ) a ( n ) q i >0 >, (n +) 3 + (n + 4n + 3n + ) + (n 3 + n + 3n + ), 3 a q i >0 < (n +) ( n + n + ) + ( n +) a(n ) q i > q i > (n +) and a+ n ( n +) + n a bq bq >0 <. nn ( +)

23 Puchae and ental ubidie in duable-good oligopolie 33 Hence, we equie, and uch that: ( n +) + (n + n + ) (n )a (n + n + ) + ( n + ) ( n )a max, < (n + ) (n +) a + (n 3 + 4n + 3n + ) + n 3 n ( + + 3n + ) a+ n( n +) + n < min, 3 (n +) n( n +) i.e., (n + n + ) (n )a (n )a (n +) max +, + < (n +) (n +) n + (n +) (n + n + ) a +(n +) a < min + +, + +. (n +) ( n +) 3 n + nn ( + ) (B.) Condition (B.) i atified if all ubidie ae equal to 0. Thi condition i alo atified if demand, a paametized by a, i high enough (the highe i n the geate i the value of a equied to fulfill (B.). An example of ituation whee (B.) i atified: a 00, n, 0 and < < Note alo that: ( n +) +(n + n + ) (n )a (n + n + ) + (n + ) (n )a > ( n +) (n +) >. When 0 we have fom (B.): When 0 we have fom (B.): (n )a a < (n +) <. 3 (n +) a (n )a 3 < <. n + n + 3n + n + n +

24 34 amagoia agata and joé m. uategui When 0 we have fom (B.): a (n )a 3 < <. n + 4n + 3n + (n + n + ) C. Poof of Popoition and 3 In thi poof we obtain the geneal expeion of the deivative of CS, P, SUB and TS( g ) with epect to each ubidy. Fom them we can tudy the ign of thoe deivative when the ubidy modified i the only one a thee ae no othe ubidie. To obtain the ign of the maginal effect of each ubidy on CS we note that: b (a + )n na +( )( n +) CS n + ( ) ( bn+ n + ) b CS (an + ( )( n +))( n +) n bn ( +) CS n a+ an + ( )( n +) + ( n ) b (n +) ( n +) ( an +( ) ( n +)) n CS bn ( +) Fom (B.) we obtain: an ( ) n + ( +) < n + (n +) an ( ) ( n +) ( )(n + ) + n + >0 an +( )( n + ) > 0,

25 Puchae and ental ubidie in duable-good oligopolie 35 an ( ) ( n +) a dcs dcs an >. Hence, >0 and <0. n + d d an ( + n ) an ( ) ( n Note alo that, a > +), we have fom (B.): n + n + an ( + n ) ( )(n +) + n + >0 a+ < ( + n)( an CS <0. + (n + ) n ) A p pq + p q + pq we ubtitute the expeion fo quantitie and pice i i i i obtained in Section A and we get that: ( ) ( 3 4 )( ) ( 3 4 ) p n n a + n + n + n + + n + n + n + i bn ( +) p i ( an ( n+ n 3 + n 4 +) (n +) (n +) b + (3n + 5n + n 3 + 4n 4 +3n 5 + n 6 + ) ( n n +)(n +) 4 + ( n + n 3 + n 4 +)(n +) ) ( an + ( n + n 3 + n 4 +)+ ( n + n 3 + n 4 + )( i p (n +) b )) Fom (B.) we have: n ( + n + ) a+ (n +) < (n +) (n +)

26 36 amagoia agata and joé m. uategui Hence, we get: p ( n )( n + an+ + n + n + a) i > ( n +)( n +) b Fom (B.) we alo know that. a + an ( ( +) a + > + <. n + nn ( +) n + ( n +) n ) n a an + + n A < + n + a, and n n an + + n + n + a < n + an + + n + n + a > 0, n we conclude that p i >0. If a 00, b, n 0, 3, and then p i pi 3.959, 4.76 and (B.) i atified a: max{ 3.587, 3.587} < < min{4.53,3.09}. Intead, if a 3, b, n, 8.3, p 6 and i 0. then.88, p i 0.57 and (B.) i atified a: max{ 3.587, 3.587} < < min{4.53,3.09}. When 0 and 0 we clealy get, fom the expeion above, >0. pi When 0and 0 we have fom (B.): a a a 0< a a < min,, a <. (n +) 3 nn ( +) ( n +) 3 ( n + ) 3 nn ( +) We alo have: p n( n )a (n + n 3 + n 4 +) i bn ( +)

27 Puchae and ental ubidie in duable-good oligopolie 37 When n ³ we get pi a nn )a ( <. 3 (n +) n+ n 3 + n 4 + <0 a nn ( )a ( n + n 3 + n 4 +) >0 becaue To obtain the ign of the maginal effect of each ubidy on TS( g ) we note that when g we have: TS an an ( ) + (n +) (n + ) (n + n + ) bn ( +) a ( n + ) 3 + ( n n + 3n + ) + ( n 3 + n + 3n + ) +an ( n +)( n +) b a n(+ n ) + n( n +) + ( n + n +) +an bn ( +) b ( a+ ) n na + ( n+) (n + ) + n bn ( +) ( n + ) b na ( a ( (n ) + +) bn ( +) n + n + +( )( n +) + + n ) n + n a+ na + ( n + ( ) ) + b n + n + na n + n + n + (a + ( )( n+) + + n ) bn ( +) n + n + n a+ ( ) na + n + ( ) + b n + n +

28 38 amagoia agata and joé m. uategui TS ( a+ n + n ( )(n + ))(n + ) n (n +) b TS ( ( n + ) 4 a(3 + n + n n 3 ) ( n +) 3 (n + 4n n n 4 +))n (n +) (n +) b TS ( a n + n ( )(n +))n ( n +) b Fom (B.) we have: a < TS TS n + nn ( +) Hence, >0 and <0. a n + n( )( n + > ) 0. If a, b, n 0,.3, and then TS and (B.) i atified a: max{.89,.89} < < min{4.035,.373}. Intead, if a 00, b, n 0, 3, TS and then and (B.) i atified a: max{ 3.587, 3.587} < < min{4.53,3.09}. Nevethele when 0 and 0 we have, fom the expeion above, TS <0. To obtain the ign of the maginal effect of each ubidy on SUB we note that: ( a+ )(n ) + (n + ) ( ) SUB n + bn ( +) (a+ ) ( n +) 3 ( ) +(n +)( ) + n ( n +)( n +)b a+ (n + n)( ) + n bn ( +)