Abstract We compare the effect of a trade quota on products quality selection under Cournot and Bertrand competition. In contrast with the Cournot

Transcription

1 Astrat We ompare the effet of a trae uota on prouts uality seletion uner Cournot an Bertran ompetition. In ontrast with the Cournot ase, a uota never inues uality owngraing uner Bertran ompetition, moreover the omain of effetive uotas is larger uner Bertran than uner Cournot.

2 Trae Quotas, Quality Seletion an the Moe of Competition Niolas Boar Xavier Y. Wauthy Septemer 16, Introution Despite of a marke tren towars freer trae, uotas an VERs are still present in many inustries. The Market Aess Setoral an Trae Barriers Dataase maintaine y the European Commission provies ample eviene on this point. 1 As reently argue y Maggi an Roriguez-Clare (000), the politial inuene of omesti prouers an/or importers may explain why governments are likely to prefer uotas to tariffs. As a matterof fat a vast theoretial literature alreay helps us to etter assess the likely inuene of suh trae restraints. Still, some key issues nee to e larie, in partiular regaring the relationship etween uota effets an the more of ompetition. A pathreaking paper in this respet is Krishna (199). She shows that eause of the uantitative restritions they impose at the market ompetition stage, uotas have ualitatively ifferent impat epening on whether rms ompete in uantity or prie. On the other han, the reent Inustrial Organization literature shows that market stage effets ten to spillover to earlier stages of the game where long term eision (suh as the seletion of prouts attriutes) are mae. In a trae ontext, we may for instane expet that uotas affet prouts uality seletion ut also that their impat on euilirium uality hoie epens on the moe of oligopolisti ompetition. Herguera, Kujal an Petrakis (000) arefuly stuy the impat of uotas on uality seletion y the rms uner Cournot ompetition. However, in view of Krishna (199)s nings, it is not ovious that the onlusions they reah to exten to a Bertran ompetition framework. The present note preisely aims at The researh of the seon author enete from the support of Interuniversity Attration Pole Program-Belgian State- Feeral Offie for Sienti, Tehnial an Cultural Affairs uner ontrat PAI 5/6. University of Girona Faultés universitaires Saint-Louis, Brussels & CORE See the wesite 1

3 testing the roustness of Herguera, Kujal an Petrakis (000) onlusions to the moe of ompetition. To this en, we evelop a very simple example. Suppose a omesti government wishes to offer protetion to an emerging inustry faing international ompetition. For simpliity, we assume that the foreign prouer is alreay ommitte to its uality efore the uota is enfore. This allows us to onentrate on the uality seletion of the omesti rm only. In orer to analyze the role of the ompetition moe, we onsier in turn the optimal uality seletion uner Cournot an Bertran ompetition in the last stage. In our framework, the preition of Herguera, Kujal an Petrakis (000) is that, as ompare to the free trae euilirium, the omesti rm is likely to selet a higher uality in the presene of a restritive uota. However, if the uota is set just aove the foreign sales uner free trae euilirium it is effetive in the sense that it will alter the uality hoie of the omesti rm. More preisely, the omesti rm selets a lower uality, as ompare to free trae. Comparing uality seletion uner Bertran an Cournot ompetition, we show that for a large omain of uota values, the impat of the uota on uality seletion oes not epen on the moe of ompetition. However, we also show that, as ompare to Cournot, the range where the uota is effetive is signiantly larger uner Bertran ompetition. Moreover, uner Bertran, the uota never inues uality owngraing y the omesti rm, as ompare to the free trae enhmark.. The Moel Consumers preferenes are erive from Mussa an Rosen (197). Consumer i exhiits a taste for uality i an erives an iniret utility isp when onsuming a prout of uality s, ought at prie p. Not onsuming yiels a utility of 0. Consumers types are uniformly istriute in the [0, 1] interval. The ensity is 1, an is taken as a measure of the omesti market size. s In orer to proue a uality level s, a rm has to inur a sunk ost () s. Marginal ost is zero for simpliity. The seuene of eisions is the following. Before the game starts, a foreign prouer sells as a monopolist in the omesti market a prout of uality s f 1. We assume that the monopolist is ommitte to this uality level. 3 Then the government hooses a level for the uota from an interval [ min, 1] with min > Note that sf 1is inee the optimal hoie of a monopolist uner the ost assumption s s () 3 Thus, we assume that swithing to another uality is too ostly for the monopolist. This assumption may reet the fat that the monopolist is also ative in other foreign markets an oes not n it protale to alter its uality solely for the omesti market sujet to trae restritions. 3

4 0. Given the uota, the omesti rm selets its uality s, efore oth rms ompete in the last stage of the game. We onsier in turn Cournot an Bertran ompetition. Notie that given the speiation of the sunk ost, we may restriit the analysis of omesti uality seletion to s < 1, i.e. uality leapfrogging y the omesti rm is not an issue in our setting. 3. Cournot Competition We rst solve the last stage of the game, an then we go akwar to stuy uality seletion.the analysis of the Cournot game is straightforwar. Given ualities, inverse emans are given y p 1x x s p (1 x x ) s x ( ) 1 s s f f f Uner Free Trae, the Cournot euilirium is iretly otaine as ; ( ) s xf s s. (1) () (3) s s ( s) Domesti prots at the Cournot euilirium are given y. Maximizing this funtion with respet the s, we otain the optimal uality seletion y the omesti prouer. The only relevant real root to the rst orer onition is s This enes the optimal uality seletion uner Free Trae. Deman aresse to the foreign rm at the SPE is ompute as x f( s ) Assume now that the government has set a uota at level. If the uota is ining at the Cournot stage, we have xf. Using (), the market learing prie for the omesti prout is ene as p (1 x ) s () It is then iret to show that at the uota-onstraine Cournot euilirium 1 (1 ) s s we have x. By maximizing with respet to s, we immeiately otain s (1 ). Thus, the optimal uality seletion is ereasing in the uota. Now, we aress the following uestion: in whih range is the uota ining? One ualities have een selete, it is well known that the uota is effetive if only it is stritly elow the level of foreign eman at the Cournot euilirium. However, sine the uota is ommitte to efore omesti uality is selete, a The introution of a lower oun for the uota is mae to avoi tehnial prolems in the erivation of priing games. The importane of this assumption is isusse elow. 5 Note that the optimal value an e ompute analytially. However, to avoi umersome expressions, we retain here, as well as for the results the ome, numerial approximations.

5 uota set aove the Free Trae enhmark x effet on uality seletion. s s s s ( s ) (1 ) ( ) s s Proposition 3.1. If <. 50 (the Free Trae enhmark) the omesti rm selets a uality aove the Free Trae Euilirium level; if [. 50,. 55], the omesti rm selets a uality elow the Free Trae Euilirium level; If >. 55, the Free Trae Euilirium uality level otains. Notie that (1 ). 961 < In other wors, when we reah the level where the uota starts to affet the optimal uality hoie, the omesti uality jumps own. At this step it is useful to istinguish two notions of effetivenessfor a uota. In general, we laim that a uota is effetive if it alters euilirium outomes. Given this enition, we ene a short-run effetive uotas as altering Cournot outomes given ualities, an long-run effetive uotas as altering euilirium outomes through their effet on uality seletion. Using these enitions, we notie that uner Cournot ompetition, uotas aove the foreign sales, onitional on ualities, are not short-run effetive. However, uotas slightly aove the Free Trae sugame perfet euilirium are long-run effetive.. Bertran Competition f ( s ) may e effetive through its Notie rst, using (3), that > whenever >. 60. > xf ( s) Thus, there exist uota levels, aove the Free Trae enhmark for whih the foreifn rm is uota-onstraine in a Cournot euilirium. To enfore this euilirium, the omesti rm must selet a uality level elow the free trae one. Reall inee that s is ereasing in the uota. The intuition is straightforwar. Suppose the uota is slightly aove the Free Trae enhmark. The omesti rm may enet from the protetion of the uota if only it inates foreign sales at the Cournot euilirium. This is ahieve through uality owngraing. In orer to ientify the range in whih this strategy is optimal for the omesti prouer, we ompare an. Solving for, we ( ) otain as the uniue relevant real root. 55 <. 60 whih therefore enes the ritial value elow whih the uota is effetive when ommitte to efore omesti uality is selete. The next proposition haraterizes the impat of a uota uner Cournot ompetition. Uner Bertran ompetition, emans aresse to the rms are given y x f x pf p 1 1 s ps f p s (1 s ) (5) (6) 5

6 Let us now ene rms est reply in the Free Trae priing game: ( ) 1 s f p + p f ps f f( pf) (7) () Straightforwar omputations yiel the uniue Nash euilirium: p s (1 s ) ; (1 s p ) f. s s (9) 1 s p x( p) 1 s. s pfp 1s s(1 s) s ( s) Using these values, we note that x, xf an. Solving the rst orer onition on for s, we otain the optimal uality seletion of the omesti rm as s. 193 an the assoiate eman for the foreign rm is x f ( s ). 55. Let us then onsier the presene of a uota. As shown y Krishna (199), the rst key ifferene etween Cournot an Bertran ompetition is that, uner prie ompetition, uotas aove Free Trae are (almost) always short-run effetive. This is so eause a uota eeply alters the struture of the priing game itself. Therefore, we rst have to stuy in etails the impliation of the uota on prie ompetiton itself efore we turn to the analysis of uality seletion. The keypoint is the following: Uner prie ompetition, onsumers may e ratione y the foreign prouer if its eman exees the uota level. Ratione onsumers may then report their purhase on the omesti rm, whose effetive sales inrease. These rationing spillovers estroy the gloal onavity of the omesti rms payoff an therey make the existene of a pure strategy euilirium prolemati. In other wors, the presene of the uota inue Bertran-Egeworth ompetition at the market stage. The analysis of this prolem has een arefuly stuie in Krishna (199) an we apply her methoology to our partiular framework. Notie rst that in the Mussa an Rosen moel, an if the omesti rm sells the low uality, any onsumer that is ratione y the high uality foreign rm prefers to report his purhase on the omesti prout rather than to refrain from uying. Aoringly, whenever xf( p,p f) >, the numer of onsumers who report their purhase on rm is given y xf( p,pf). It is is then iret to show that the resiual eman aresse to the omesti rm is Using (5) we now ene p as the prie of the omesti rm suh that the uota is exatly ining. Formally, we solve 1 to otain (10) f f. p ( p ) (1 )(1 s ) + p (11) 6

7 Whenever p p, the free trae analysis applies an rms eman are given y (5) an (6). Whenever p p, the uota is ining an rms eman are x( p) an respetively. Comparing the erivatives of the omesti eman funtion using (5) an (10), we oserve that the omesti rms eman exhiits an outwar kink at p p. This kink estroys the onavity of its prots. Thus, the existene of a pure strategy euilirium is not guarantee. We now haraterize the nature of euilirium in the priing game. To this en we ene rm fs est reply in the uota game, whih we enote ϕf. Whenever xf( p,pf), i.e. whenever p p, the uota is not ining. Aoringly the foreign rm is etter off playing ff( p) as ene y (7). Whenever p p, the pfp foreign rm is etter off selling the uota at the highest prie. Solving 1 1s we otain p ( p ) (1 )(1 s ) + p (1). Thus, we have: { ϕ ( p ) 1 (1 ) with prots eual to f f ff( p) iff p p p ( p ) iff p p f Notie that the est reply is kinke an ontinuous, reeting the fat that the foreigners prot in onave in own prie. Using (10) we now ene y ( p) px ( p) the prot of the omesti rm when it enets from spillovers. Notie that ( p) exhiits a loal maximum for ps s s. In the omain where the uota is not p f s (1 s) ining, the est reply is f( pf) as ene y (), yieling payoff ( p f). p f s (1 ) Solving (1 s ) s for p f, we ene the ritial value pf (1 ) 1 s for whih the omesti rm is inifferent etween the two strategies. The est reply orresponene is therefore: { ps p p ϕ( pf) iff f f (1) f( pf) iff pf p f Comining (13) an (1), we oserve that there is only one aniate for a pure strategy euilirium: the Free Trae Euilirium as ene y (9). 6 A neessary an suffiient onition for this aniate to e an euilirium is that p f > p f. Diret omputations show that this onition is satise if an only 1 > s if ( s ) 1 s. When this onition is not satise, there exist no pure strategy euilirium. The natural aniate for a mixe strategy euilirium is then the following one: The omesti rm ranomizes etween ps an f( p f) while hoosing the weight to put on eah pure strategy to ensure that p f is inee 6 s It is inee immeiate to hek using (1) that p f < pf ( p ), whih is suffiient to rule out any pure strategy euilirium aniate in the uota ining omain of pries. Therefore, the only remaining aniates must lie in free trae region. There is only one suh aniate: the free trae euilirium. (13) 7

8 aest reply for the foreign rm against the mixture. We all this euilirium the Krishna euilirium. 7 Diret omputations show that ( s ) > s, for all s. Aoringly, whatever the omesti uality level, there always exist uota values aove the free trae euilirium level suh that the uota is short-run effetive. In partiular, evaluating at the Free Trae sugame perfet euilirium, we have ( s ). 579 > x f( s ). 55. Notie that this is never the ase uner Cournot ompetition. This ontrast in the short-run effetivness of a uota epening on the moe of ompetition is the essene of Krishna (199)s nings. Let us assume for the moment that the Krishna euilirium always exist when the Free Trae one oes not an stuy the issue of uality seletion. It is not neessary to ompute the mixe strategy expliitely for our present purpose. Inee, the keypoint here is to note that in this euilirium, the omesti rm (1 ) s earns exatly s in the Krishna euilirium. Notie that this is exatly euivalent to the Cournot euilirium payoffs uner a ining uota. Therefore, if the Krishna euilirium is playe at the prie ompetition stage, optimal uality seletion y the omesti rm is iential to the uality seletion mae uner (1 ) Cournot, i.e. s (1 ), yieling a payoff when we take the sunk ost into aount. In orer to ene the omain in whih the uota affets uality seletion, onsier the optimal uality seletion uner Free Trae: s As note aove, we nee a large enough uota >( s ). 579 for the Free Trae euilirium to exist in the priing game. On the other han, it is always possile for the omesti rm to enfore the Krishna euilirium y hoosing a high enough uality. In orer to ientify the optimal strategy, we ompare the orresponing payoffs. Diret omputations iniate that s (1 s) s (1 ) ( s ) < whenever <. 5301, where enes the ritial value aove whih the uota is neither long-run nor short-run effetive uner Bertran ompetition. Our nings are summarize in the following proposition. Proposition.1. If r <. 55 (the Free Trae enhmark), the uota inues the seletion of a higher omesti uality; any [. 55,. 5301], is effetive an inue a higher omesti uality; If >. 5301, the uota is totally ineffetive. Notie that (1 ). 0 > In other wors, when we reah the ritial level at whih the uota starts affeting uality hoie, omesti uality jumps up. A seon omment pertains to the existene of the Krishna euilirium. Inee, this euilirium oes not always exist. For this euilirium to exist, the 7 We refer the intereste reaer to Krishna (199), Theorem, for the etaile onstrution of this euilirium. In a mixe strategy, the euilirium payoff an e ompute at any of the rms atom. Sine s the foreign rm faes a pure strategy, its euilirium payoff, ompute at p must e s. (1 )

9 foreign eman must satisfy the non-negativity onstraint xf( f( p f),p f) > 0). Diret omputations show that this is the ase if only > 1 1 s. In other wors, there exists a lower oun on the uota value, whih epens positivelytively on s elow whih the Krishna euilirium oes not exist. This onition imposes restritions on the amissile values of the uota only if s > 3/. hols Notie then that (1 ) > 3/ if only < As a onseuene, our analysis is fully ompelling if we assume min > Hene our initial assumption of a lower oun on the amissile values for the uota Comparing Cournot an Bertran As shown y Krishna (199), the impat of a uota at the market stage epens on the moe of ompetition. This result is est oserve in our framework y noting that a uota set slitghly aove Free Trae is never short-run effetive uner Cournot while it is always short-run effetive uner Bertran. However, a uota also have long-run impliations. By omparing uality seletion y the omesti prouer, we may now assess the long-run effetivness of uotas. In orer to assess the epenene of uality seletion to the moe of ompetition, it is est to refer Figure 1. 9 insert gure 1 aout here The ol line epits the uality seletion as a funtion of the uota value uner Cournot. The ashe one epits the uality seletion uner Bertran. Notie rst that >. Thus the range for an effetive uota is larger uner Bertran than uner Cournot. Seon, note that for very restritive uotas the uality seletion is invariant to the moe of ompetition. Last, as ompare to their respetive Free Trae values, the optimal uality annot erease eause of the uota uner Bertran ompetition while there exists a omain of uota values, aove the Free Trae enhmark for whih the uota inues uality owngraing. Notie nally that when the uota eomes tighter the egree of prout ifferentiation ereases. These results have een otaine uner uite a restritive framework. Assuming that the foreign prouer oes not alter its uality seletion as a response to the uota onsieraly eases the analysis. However, it is our elief that they apture some asi impliations of uotas on uality seletion. In Boar an Wauthy (000), we onsier a more general game where the omesti an the foreign rms are free to hoose any uality level, an thus inue any uality When the Krishna euilirium oes not exist, there exists fully mixe strategy euiliria involving a nite numer of atoms. We have not een ale to fully haraterize them. However, Levitan an Shuik (197) provie a haraterization for the partiular ase where s 1. For the relevant omain of uota values, the payoff of the omesti rm at s 1ominates the Free Trae enhmark. Hene, in this omain, it must e the ase that the euilirium uality is larger than the Free Trae one. 9

10 ranking etween the omesti an the foreign prout after the uota is implemente. We reah ualitatively similar onlusions: inentives to uality seletion uner Bertran ompetition are in line with those prevailing uner Cournot whenever the uota is effetive. The egree of prout ifferentiation ereases when the uota eomes tighter. When it eomes long-run effetive, a uota inues a marke uality upgraing uner Bertran. Referenes [1] Boar N. an X. Wauthy (000), Quantity restritions an enogenous uality hoie uner prie ompetition, mimeo [] Herguera I., P. Kujal an E. Petrakis (000), Quantity restritions an enogenous uality hoie, International Journal of Inustrial Organization, 1, [3] Krishna K (199), Trae restritions as failitating praties, Journal of International Eonomis, 6, [] Levitan R. an M. Shuik (197), Prie uopoly an apaity onstraints, International Eonomi Review, 13, [5] Maggi G. an A. Roriguez-Clare (000), Import penetration an the politis of trae protetion, Journal of International Eonomis, 51, 7-30 [6] Mussa M; an S. Rosen (197), Monopoly an prout uality, Journal of Eonomi Theory, 1,