Taste for variety and optimum produt diversity in an open eonomy Javier Coto-Martínez City University Paul Levine University of Surrey Otober 0, 005 María D.C. Garía-Alonso University of Kent Abstrat We extend the Benassy (996) taste for variety model to an open eonomy setting. With the Benassy effet, the market equilibrium is ineffiient, openness redues the varieties provided in the unonstrained optimum and there are potential gains from international oordination. Keywords: Taste for variety; monopolisti ompetition; Benassy effet; open eonomy JEL lassifiation: D43, F Corresponding author: María D. C. Garía-Alonso, Department of Eonomis, The University of Kent, Canterbury, Kent CT 7NP, UK, e-mail: m..garia-alonso@kent.a.uk. Phone: (+44)07 87488. Fax: (+44)07 87850.
. Introdution In a reent paper, Benassy (996) presented an alternative to the representation of taste for variety traditionally used in the monopolisti ompetition literature (see e.g., Dixit and Stiglitz, 977 and Spene, 976). The Benassy (996) speifiation disentangles taste for variety from market power and substitutability measures. As a result, the monopolisti ompetition market equilibrium may no longer be effiient. This is done in a losed eonomy ontext. The purpose of this paper is to study the impliations of the Benassy speifiation for the effiieny of the market equilibrium in an open eonomy and the potential gains from international oordination. More speifially, we onsider a two way model of trade where firms have market power owing to the onsumers love for variety. We analyze the unonstrained open eonomy optimum and ompare it with the market equilibrium and the global soial planner outome. In addition, we study the impat of hanges in openness on firm numbers. We will argue that the Benassy speifiation may provideanexplanationforreenthangesinonentrationinindustriessuhas the defene industry. The paper is organized as follows, setion presents the model and finds the solution for the unonstrained small open eonomy equilibrium. Setion 3 ompares the different equilibria. Finally, setion 4 onludes the paper.. The model We onsider a two ountry model. There are n and n firms in ountries and respetively, eah produing a single variety of a differentiated good. Consumers preferenes in ountry are haraterized by the following utility funtion: U = w n ν+! X (d i ) Ã n i= +( w) n ν+! X (m j ) Ã n j= ; [0, ), [, ), ν>0, w [, ], (.) Flam and Helpman (985) made the first ontribution to the analysis of an government poliy under monopolisti ompetition in a small open eonomy. We use a partial equilibrium version of their model whih will learly illustrate the impat of the Benassy speifiation.
where, d j represents domesti onsumption of domesti variety j and m j represents domesti onsumption of foreign variety j, similarly for ountry. As in Benassy (996), the utility funtion inludes separate parameters to measure the elastiity of substitution aross varieties, σ = [, ), whih will later determine the mark up, and the taste of variety, determined by ν, whihmeasuresthe impat on utility of an inrease in the total number of varieties N, U N N U = v +. This speifiation allows for a more (ν > ) or less (ν < )tasteforvariety than in the Dixit-Stiglitz ase (ν = ). In (.) the parameter w [, ] represents the degree of home bias for domestially produed goods. 3 For w = we have autarky whilst the lower bound w = gives us the ase of the omplete integration of the two eonomies. Thus we an assoiate a lower home bias w with an inrease in openness... Demand funtions and equilibrium produer pries In this setion, we obtain the demand funtions using standard two stage maximization proedure. We define the following prie index assoiated to the domesti varieties in ountry : P d = n ν+ σ Ã n X i= (p i) σ! σ, (.) where p i is the prie onsumers pay for domesti variety i (superindex is introdued so as to distinguish between onsumer and produer pries). Straight- At the symmetri equilibrium aross varieties, the utility funtion is equal to U = µ w n ν+ d +( w) n ν+ m. We then onsider an expansion in the total number of varieties, keeping the relation between domesti and foreign varieties n n fixed. Then, putting n = kn U = Ã w ³(kN) ν+ d +( w) ³(( k) N) ν+ m!, thus, U N N U = v +. 3 For the ase where =, w is the share of domestially produed varieties in total onsumption. 3
forward utility maximization subjet to the standard onsumer budget onstraint results in the following demand funtion for eah domesti variety i µ p σ d i = w i n (σ )v P d P d P Y, (.3) where P =(wp d +( w)pm ), Pm is the prie index assoiated to the foreign varieties (similar to P d )andy is net domesti inome, whih is defined as Y = y +π +T, where y is an endowment inome, idential for both ountries, π is the total profits of the domesti firms and T is the domesti government lump sum transfer to the onsumers. 4 Similarly, demand for imported variety j is µ p σ j (σ )ν m j =( w) n P m P m P Y. (.4) We assume that all firms in both ountries have idential onstant marginal osts and fixed prodution osts F. At the symmetri firm equilibrium, the monopolisti ompetition assumption generates the mark-up equation p =,where p is the produer prie or monopoly prie any variety (this is also typial of the Dixit-Stiglitz-Spene monopolisti ompetition framework). This prie will not be affeted by domesti or foreign government poliy. Therefore, the domesti government an only affet domesti onsumer pries... Small open eonomy unonstrained optimum In this setion, we ompute the small open eonomy unonstrained non-ooperative equilibrium. In the unonstrained optimum, governments have three poliy instruments: a subsidy to domesti onsumption, an imports tariff and a fixed ost subsidy to the domesti firms. Governments will then be able to hoose the number of domesti firms, their objetive will be to maximize welfare subjet to the government s budget onstraint, for ountry this an be expressed as follows (similarly for ountry ): Xn Xn T = (p i p)d i + p j p m j n F. (.5) i= j= 4 The assumption that governments an use lump-sum taxation to finane their industrial poliy implies that their optimal poliy will be determined by the welfare effet, not by the government revenue needs. 4
We represent the subsidies as a wedge between produer and onsumer pries. Thus, (p i p) is the subsidy to variety i and p j p is the tariff on the imports of variety j. Finally, the government pays the fixed ost to domesti produers through lump-sum transfers n F. At the symmetri equilibrium, the domesti and foreign prie indexes are P d = n ν p d and P m = n ν p m respetively. Here, p d and p m represent the onsumer pries for domesti and imported varieties at the symmetri equilibrium (in what follows, we drop subindexes i and j to represent symmetri firm values). Weanthenwritethewelfaremaximizationproblemforountryas that of maximizing its indiret utility funtion V (n,n,p d,p m,y )= hw n ν p d +( w) n ν p m i Y, (.6) where Y = y + n (p )(m + d )+T, subjet to the budget onstraint T = n (p d p)d + n p m p m n F. Substituting T in Y, we an rewrite inome as: Y = y + n (p )m + n p d d + n p m p m n F. (.7) Our small open eonomy assumption means that the government does not take into aount the effet of n on m. The first order onditions are: V + V µ n p d + n (p d d Y ) d =0, (.8) p d V + V µ n p m + n (p m m Y p) m =0, (.9) p m V + V Y =0. (.0) n Y n Note that, applying Roy s identity 5 to (.8) and (.9), we get 5 From Roy s identity (x i (p, Y ) being a demand funtion) we know that V (p, Y ) p i + V (p, Y ) x i (p, Y )=0. Y 5
n (p d ) d p d =0, (.) n (p m p) m =0. (.) p m The above implies that the government will implement a onsumption subsidy to make onsumer prie for domesti varieties equal to the marginal ost of prodution p d =, and the government will not implement a tariff, p m = p. The reasons are that, under monopolisti ompetition, the government annot use a tariff to redue the prie of foreign firms and sine we have lump-sum taxation, we do not need to levy a tariff to generate revenue. This is also a feature of the Dixit-Stiglitz-Spene monopolisti ompetition framework. We now turn to equation (.0), there, V n, represents the taste for variety effet and V Y Y n represents an inome effet. To alulate the impat of a variation in the number of firms on domesti inome Y n, firstnotethatsubstituting p d = and p m = p in equation (.7), we obtain where Y = y + n (p )m n F, (.3) (p m ) ν +ν m =( w)n Y w(n ν p d ) +( w)(n ν p m ). (.4) Note that, given exports, an inrease in number of varieties inreases total profits, but, as an be seen from equation (.4), hanges in the number of the varieties would also affet the demand to eah firm. For simpliity, we assume that individual ountries are small in the sense that individual governments do nottakeintoaounttheeffet of a variation in n on m. 6 In this ase, 6 The small ountry assumption an be justified as the limit of an eonomy with N ountries. Variable export profits for ountry would then be π = n (p )D (p) Y (N ), N where D (p) is a general demand funtion and p is vetor of pries. Then, the derivative of variable profits with respet to n π =(p )D (p) Y n N (N ) + n (p ) D (p) p Y (N ). p n N 6
Y =(p )m F, (.5) n that is, an inrease in the number of varieties inreases the variable profits made by all the domesti firms through exports. Finally, introduing (.5) in (.0), substituting p = and using all the first order onditions, we obtain the number of firms per ountry in the small open eonomy unonstrained equilibrium (see Appendix for details): 3. Comparison of equilibria n u = y [ vw +( )( w)] F [ ( + v) w +( w)]. (.6) Table presents the number of varieties in the open eonomy unonstrained optimum, the market eonomy equilibrium n m, the losed eonomy unonstrained equilibrium n (obtained by setting w =in n u )andtheworldfirst best n s (with and without the Benassy effet). Table. Small eonomy equilibrium number of firms. Benassy ase: ν 6= Market eonomy equilibrium n m = y( ) F Closed eonomy unonstrained optimum n = yν (+ν)f World first best n s = yν (+ν)f Open eonomy unonstrained optimum n u = y[ vw+( )( w)] F [ (+v)w+( w)] Dixit-Stiglitz ase: ν = n m = y( ) F n = y( ) F n s = y( ) F n u = y( ) F The world first best an be obtained using standard maximization proedure (see Appendix for a details). The Dixit-Stiglitz ase for eah equilibrium type is D (p) π Now, if lim N p =0,then lim N n =(p )D (p)y. Therefore, the strategi effet of the variation in pries disappears. To obtain this result, it is important that we repliate the eonomy keeping the demand onstant (see Chari and Kehoe, 990). 7
obtained by setting ν =. It then beomes apparent that in the Dixit-Stiglitz ase, firm numbers oinide at all the equilibrium types. 7 Note that dnu > 0, therefore, the Benassy effet will inrease the unonstrained dv optimum number of firms (relative to the Dixit-Stiglitz ase) if ν > and derease it if ν <. This is onsistent with the idea that these two ases represent more and less love for variety that in the Dixit-Stiglitz ase. The following results illustrate the impat of the Benassy effet on the effiieny of the market equilibrium, the impat of openness on firm numbers and the potential gains from international poliy oordination. Proposition. If ν> ν<, the unonstrained small open eonomy optimum number of firms is higher (lower) than the number of varieties generated by the market equilibrium n u >n m (n u <n m ). Therefore, the Benassy effet (ν 6= ) makes the small open eonomy market equilibrium ineffiient. The intuition behind this result beomes apparent if we note that measures the magnitude of profits, whereas ν measures the soial benefit of introduing new varieties. Proposition. If ν> ν<,aninreaseinopenness(alowerw) redues (inreases) the optimal number of firms n >n u (n <n u ) in a small open eonomy. The above result implies that, with Benassy (ν 6= ), hanges in openness affet firm numbers. That is, the Benassy effet makes home bias relevant to the open eonomy unonstrained optimum. It is interesting to note that the inrease in the size of the market ould lead to a redution in the number of firms. This result is onsistent with observed hanges in onentration in industries where government subsidies are still prevalent. One suh ase is the military industry, where reent inreases in openness have led to inreases in onentration. This would suggest that a Benassy setting with ν > would be a suitable representation of preferenes in suh industry (see Dunne et al., 005). Proposition 3. If ν> ν<, the unonstrained small open eonomy optimum number of firms is lower (higher) than the world first best n u <n s (n u >n s ). 7 Note that the reason why n u and n m oinide is that (unlike the original Dixit-Stiglitz (977)) we do not inlude a numeraire. The purpose of this simplifiation is to illustrate the impat of the Benassy on the equilibrium number of firms in the different equilibria in a neat way. 8
The above result shows that the unonstrained optimum is generally ineffiient and there are then potential gains to be made from international oordination. The reason is that governments only take into aount the effet of an inrease in variety on the domesti onsumer wν. This effet redues the number of varieties below the global soial optimum. At the same time, firms obtain monopolisti profits by exporting to the other ountry,the government tries to inrease this profit, this is the reason by the number of varieties n u, depends both on the love for variety parameter and on the mark-up and may be both above or bellow the first best. 4. Conlusions In this paper, we have extended the Benassy (996) framework to an open eonomy. Using a simple utility funtion, we provide a suint illustration of the impat of the Benassy effet in an open eonomy. We prove that with Benassy, the small open eonomy market equilibrium beomes ineffiient, openness redues the varieties provided in the unonstrained optimum and there is potential gains from international oordination. If we relax the small eonomy assumption, the expression for the unonstrained optimum beomes more omplex but, still, the Benassy effet, a higher ν, inreases the number of firms. The market equilibrium, the losed eonomy optimum and the world first best remain (for any ν) as before. However, the unonstrained optimum for ν = hanges and hene, it is different from the other equilibria. Therefore, although the market equilibrium will still generate the world first best, there will be potential gains from poliy oordination if ountries try to maximize welfare individually, even if the Benassy effet is not present. 5. Referenes Benassy, J. P. (996). Taste for variety and optimum prodution patterns in monopolisti ompetition, Eonomi Letters 5, 4-47. Dunne, J. P., Garía-Alonso, M. C., Levine, P. and Smith, R. P. (005) Military Prourement, Industry Struture and Regional Conflit. University of Kent, Disussion Paper No. 05/0. 9
Chari, V. V. and Kehoe, P. J. (990) International oordination of fisal poliy in limiting eonomies, Journal of Politial Eonomy 98, 3, 67-636. Dixit, A. K. and Stiglitz, J. E. (977). Monopolisti ompetition and optimum produt diversity, Amerian Eonomi Review 76, 3, 97-308. Flam, H. and Helpman, E. (987) Industrial poliy under monopolisti ompetition, Journal of International Eonomis, 79-0. Spene, M. (976) Produt seletion, fixed osts and monopolisti ompetition, Review of Eonomi Studies 43, 7-35. 0
6. Appendix 6.. Small eonomy unonstrained optimum First, we obtain the derivatives orresponding to the first order ondition on the number of firms: V + V Y =0. n Y n Partially differentiating (.6) we have V p n v( ) h d w n v p d +( w) i n ν p m, ³ = vwy n V Y = Y n h w n v p d +( w) i n ν p Y m. n Given that p d = and p m = p, total inome is Y = y + n (p )m n F. When we differentiate Y, we assume that the ountry is small, therefore, the government does not take into aount the effet of n on m. In this ase, Note that m =( w) µ p m P m Y n =(p )m F. σ n (σ )ν P m P Substituting for all the prie indexes we get ν +ν p m =( w)n m w n ν p d +( w) n ν p Y. m Next, we introdue Y into the first order ondition for n to obtain n vwy n v v () w n v +( w)(n ν p) +(p )m F =0. Y.
In the symmetri equilibrium, we get vw n w () +( w)(p) Y +(p )m F =0. where the imports demand and budget onstraint equations beome m = ( w)(p) n w () +( w)(p) Y. and Y = y + n(p )m nf. Substituting m into Y,weget Y = [y nf ] w () +( w)(p) w () + ( w)(p) and substituting the above into m, we get [y nf ]( w)p m = n w () + ( w)(p). We now substitute the above two equations for Y and m in the first order ondition for n to get vw +(p )( w)p [y nf ] n w () + ( w)(p) F =0, whih gives n = vw +(p )( w)(p) y ( + v) w () +( w)(p) F. The final expression for n u, (.6), is obtained by substituting p = above. 6.. World soial planner optimum We drop subindexes to indiate a world soial planner who, sine ountries are idential, will hoose the same variables for both ountries from the outset to maximize the utility of the representative ountry U U = n w ν+ (d) +( w) (m) ;
subjet to: y = n(m + d)+nf. The Lagrangian is: L = n w ν+ (d) +( w) (m) + λ [y n(m + d) nf ]. and the fos are: L d = nν+ L m = nν+ L n =(ν +)nν w (d) +( w) (m) w (d) +( w) (m) w (d) λn =0, ( w) (m) λn =0, w (d) +( w) (m) λ [(m + d)+f ]=0, L =0:y = n(m + d)+nf. λ Dividing the first and seond fos above and simplifying we get ( w) m = w d, whih we introdue in the budget onstraint to find (y nf ) d = w, n (y nf ) m =( w). n Now, we divide the seond and third equations to get ( w) (m) (ν +) w (d) +( w) (m) = [(m + d)+f ] and substitute d and m to find the optimal number of firms per ountry n s yν = (ν +)F. as in Table. 3