GAINS FROM TRADE UNDER MONOPOLISTIC COMPETITION

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1 bs_bs_banner Pacific Economic Review, 2: (206) pp oi: 0./ GAINS FROM TRADE UNDER MONOPOLISTIC COMPETITION ROBERT C. FEENSTRA* University of California, Davis an National Bureau of Economic Research Abstract. The monopolistic competition moel in international trae preicts three sources of gains from trae that are not present in traitional moels: consumer gains from having access to new import varieties of ifferentiate proucts; gains from a reuction in firm markups ue to import competition; an gains from the self-selection of more efficient firms into export markets, provie that firms are heterogeneous in their prouctivities. With the ae assumption that the istribution of firms prouctivity is Pareto, an with a support that is unboune above, we argue that only the thir source of gains from trae, ue to the self-selection of firms, operates. This result helps to explain the simple formula for the gains from trae foun by Arkolakis et al. (202). If the Pareto istribution is boune above, however, then all three sources of gains from trae operate once again.. INTRODUCTION The monopolistic competition moel in international trae preicts three sources of gains from trae that are not present in traitional moels. First, there are the consumer gains from having access to new import varieties of ifferentiate proucts. That source of gains from grae has been emonstrate for the Unite States by Broa an Weinstein (2006), for example, using the formula for the exact constant elasticity of substitution (CES) price inex from Feenstra (994). These gains epen on one minus the share of expeniture on new imports, raise to a negative power that epens on the elasticity of substitution in consumption. Secon, the monopolistic competition moel also allows for gains from a reuction in firm markups ue to import competition. This secon source of gains was stresse in Krugman (979), but has been absent from much of the later literature ue to the assumption of CES preferences, leaing to constant markups. Research in Feenstra an Weinstein (200), using the translog expeniture function allows those gains to be measure for the Unite States. They fin that the welfare gains from reuce markups for the Unite States are of roughly the same orer of magnitue as the gains from prouct variety. The extension of the monopolistic competition moel to allow for heterogeneous firms, ue to Melitz (2003) an iscusse in Section 2, leas to a thir source of gains from the self-selection of more efficient firms into export markets. With the ae assumption that the istribution of firms prouctivity is Pareto, as in Chaney (2008), then that source of gains has been shown to have a very simple formula by Arkolakis et al. (202): the gains from trae epen on one minus the share of expeniture on imports (equal to the share of expeniture on omestic *Aress for Corresponence: Robert C. Feenstra, University of California, Davis, an National Bureau of Economic Research. rcfeenstra@ucavis.eu. The material in this article will appear in Chapter 6 of Robert C. Feenstra, Avance International Trae, 2n eition, to be release in 205 by the Princeton University Press.

2 36 R. C. FEENSTRA goos), raise to a negative power that epens on the shape parameter of the Pareto istribution. Surprisingly absent from this formula is the elasticity of substitution in consumption. How can this elasticity (neee to measure the gains from import variety) be absent? We argue that this result occurs because, with the Pareto assumption, the first two gains from trae o not occur. Specifically, the consumer gains from new import varieties o not occur because they cancel out with isappearing omestic varieties. This fining, emonstrate in Section 3, helps to explain the simple theoretical formula foun by Arkolakis et al. (202), where the gains from trae epen on the import share but are otherwise inepenent of the elasticity of substitution in consumption. Rather, their formula epens instea on the Pareto shape parameter, reflecting the self-selection of more efficient firms into exporting. We conclue the paper by iscussing current research that will allow all three sources of gains from trae to operate. To achieve this, we must abanon the assumption of a Pareto istribution of prouctivities with a support for prouctivities that is unboune above (as in Feenstra, 204). By putting a boun on the highest prouctivity, or by consiering istributions beyon the Pareto, then we can fin that prouct variety, pro-competitive effects an selection effects all contribute towars the gains from trae uner monopolistic competition. 2. MONOPOLISTIC COMPETITION WITH HETEROGENEOUS FIRMS We allow for a continuum of ifferentiate proucts in the CES utility function, which is written as: h i σ= ðσ Þ; U Ω cðωþ ðσ Þ=σ ω where we use ω Ω to inex the set of available proucts. With each consumer earning the wage of w, the eman for each prouct is: cðωþ pðωþ σ w P P ; () where h P Ω pðωþ σ ω i = σ ð Þ (2) enotes the theoretical price inex for the CES eman system. The elasticity of eman for each prouct is ln c(ω)/ ln p(ω)=σ. 2.. Autarky equilibrium As in Melitz (2003), a mass of entering firms, M e, pay a sunk cost to obtain a raw of prouctivity, ϕ. Initially, we examine the autarky equilibrium, where all firms have the fixe cost of f for selling in the omestic market. It will often

3 GAINS FROM TRADE UNDER MONOPOLISTIC COMPETITION 37 be convenient to inex variables by prouctivity ϕ rather than the generic inex ω. Because marginal cost is (w/ϕ), then with elasticity σ the markup of price over marginal cost is p(ϕ)/(w/ϕ) =[σ/(σ )]. For convenience, we will normalize the wage at unity, w =. It follows that profits from the omestic market are: π a ðϕþ pðϕþyðϕþ f þ yðϕþ yðϕþ ϕ ϕσ ð Þ f ; where f +[y(ϕ)/ϕ] is the labour neee to prouce output of y(ϕ), an, therefore, total costs with a wage of unity. From equation, total eman from the home population of L consumers is LcðϕÞ LpðϕÞ σ =Pa σ,wherep a is the autarky price inex. Market clearing requires that y(ϕ)=lc(ϕ) uner autarky. Using the prices p(ϕ)=σ/[ϕ (σ )], it follows that output is yðϕþ LcðϕÞ L σ σ Pa σ : ϕσ ð Þ Substituting this into the expression for profits, we obtain " # π a ðϕþ Lσ σ Pa σ ðσ Þ ϕ σ f : To summarize, using the MR = MC conition to obtain the optimal prices for firms, an using market clearing in autarky, we obtain the above expression for profits. Because σ >, it is clear that this expression is increasing in the prouctivity of the firm, ϕ. Inee, if we use the transformation of prouctivity ϕ σ,thenprofits are linearly increasing in ϕ σ. By setting profits equal to zero, we solve for ϕa σ from the zero-cutoff-profit (ZCP) conition: ZCP a : π a ðϕ a Þ 0 ϕa σ f σ σ LPa σ ðσ Þ σ : (3) We refer to ϕ a as the ZCP level of prouctivity. Firms with higher prouctivity earn strictly positive profits, an firms with lower prouctivity woul earn negative profits so they o not prouce an exit the market Trae equilibrium We allow for international trae between two countries. For simplicity, we shall assume that these countries are ientical, which is the same simplification that we use in the Krugman (979) moel. We continue to normalize the wage in

4 38 R. C. FEENSTRA both countries at unity. Unlike the Krugman moel, however, we cannot treat the opening of trae as ientical to an increase in country size. The reason this metho oes not work is that we assume firms have another fixe cost of f x that must be pai in orer to export. This extra cost can reflect the ifficulty of learning about foreign customers an setting up a istribution network abroa. In aition, we assume that firms face trae barriers when exporting, which we moel as iceberg trae costs introuce by Samuelson (952): in orer for one unit to arrive abroa, there must be τ > units prouce an sent, where the remainer (τ ) melts along the way. In this framework, there will still be a zero-profit-cutoff ϕ for firms serving the omestic market, where firms with lower prouctivity woul earn negative profits an exit, whereas firms with slightly higher prouctivity sell at home but o not fin it profitable to export. There will be a secon zero-profit-cutoff ϕ x for firms that also fin it profitable to export. Uner plausible conitions we will fin that ϕ x > ϕ, so that only the more prouctive firms export, which is consistent with a large amount of empirical evience. The omestic zero-profit conition is given by: ZCP : π ðϕ Þ 0 ϕ σ f σ σ LP σ ðσ Þ σ ; (4 ) where the cutoff prouctivity ϕ σ will iffer from autarky in equation 3 ue to a new value of the CES price inex P obtaine in the presence of trae. We will be more explicit about this price inex in the next section. For now, we will presume that there are gains from trae, which means that the cost of living is reuce, so that P < P a an ϕ > ϕ a. In the export market, the marginal cost of selling one unit abroa is (τw/ϕ), so the firm charges an optimal price of p x (ϕ)=[σ/(σ )] (τ/ϕ). It follows that profits from the export market are: π x ðϕþ p x ðϕþy x ðϕþ f x þ τy xðϕþ ϕ τy xðϕþ ϕσ ð Þ f x; where f x +[τy x (ϕ)/ϕ] is the labour neee to eliver export of y x (ϕ). Export eman from the foreign population of L consumers is y x ðϕþ Lp x ðϕþ σ =P σ, where P is the same CES price inex as at home ue to ientical countries. Using the solution for export price, foreign eman for exports is y x ðϕþ Lc x ðϕþ L τσ σ P σ : ϕσ ð Þ See, for example, the evience presente in Bernar et al. (2007).

5 GAINS FROM TRADE UNDER MONOPOLISTIC COMPETITION 39 Profits in the export market are then " # π x ðφþ Lτ σ σ σ P σ ðσ Þ fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} B x ϕ σ f x : It follows that the ZCP conition in the export market is ZCP x : π x ðϕ x Þ 0 ϕx σ f x B x f x τ σ σ σ LP σ ðσ Þ σ : (5) Diviing equation 5 by equation 4, we immeiately obtain ϕ x ϕ σ f xτ σ f : It follows that a sufficient conition to have ϕ x > ϕ is that f x τ σ > f,aswe shall assume. In this case, only a subset of the proucing omestic firms (that receive a prouctivity raw above ϕ ) will also have prouctivity above ϕ x an will, therefore, export. Note that the mass of surviving firms M is relate to the mass of entering firms M e by M = M e [ G(ϕ )], an the mass of exporting firms M x is relate to the mass of entering firms by M x = M e [ G(ϕ x )]. We nee to appeal to the full-employment conitions for the economies in orer to pin own these masses. The probabilities of survival [ G(ϕ )] an [ G(ϕ x )] are particularly simple when we follow Chaney (2008) an assume that the istribution of prouctivities across firms follows a Pareto istribution: GðφÞ ϕ θ for ϕ ; an θ > σ > 0 (6) With this Pareto assumption it turns out that the mass of entering firms M e is equal in autarky an with trae, an we shall make use of this result below. 3. GAINS FROM TRADE The CES price inex facing consumers is written in general terms in equation 2. Because firms earn zero expecte profits, there are no profits reistribute to consumers. So the income of each consumer equals the wage w, an their real income equal w/p. Normalizing the wage at unity, any reuction in price inex P therefore correspons to a rise in utility.

6 40 R. C. FEENSTRA By making the change in variables from the generic inex ω to the prouctivity ϕ, the CES price inex in equation 2 becomes: P M pðϕþ σ gðϕþ ϕ ½ Gðϕ ÞŠ ϕ þ M x p x ðϕþ σ ϕ x gðϕþ ½ Gðϕ x ÞŠ ϕ = σ ð Þ ; (7) where the first integral reflects the prices of the mass M of omestic firms with prouctivity ϕ ϕ, an the secon integral reflects the export prices of the mass M x of foreign firms with prouctivity ϕ ϕ x. Recall that we are assuming trae between two ientical countries, so these foreign export prices are written in the same manner as home import prices, an the mass of exporting firms is M x = M e [ G(ϕ x )] in both countries. We nee to show that the price inex falls as compare to autarky, so that there are overall gains from trae. In aition, we want to ientify the sources of these gains using the various terms in this price inex. There are three potential effects of trae. First, the presence of the secon integral in the price inex will raise the overall term in square brackets as compare to its autarky value, an, because it is raise to a negative power, therefore lower the price inex an raise welfare. This is the positive effect of import variety on welfare. However, the mass of surviving omestic firms equals M = M e [ G(ϕ )], an we expect that the prouctivity cutoff for omestic firms, ϕ, excees its autarky value, ϕ a, which means that some firms have exite. With fixe entry M e this leas to a fall in the mass of proucts, M, which lowers the overall term in square brackets an, therefore, raises the price inex. This is the negative effect of reuce omestic variety on welfare. The reason this offsetting effect occurs is that import competition lowers eman for omestic firms an forces the least prouctive of these to exit. However, that exit correspons to a positive selection effect of trae, which lowers the average omestic price through raising ϕ above ϕ a in the first integral an, therefore, raises welfare. We will argue that with the Pareto istribution, this thir effect is the only explanation for the gains from trae. In orer to solve for the average omestic price, we follow Melitz (2003) in specifying ϕ as average prouctivity: ϕ ϕ σ ϕ ð Þ gðϕþ ½ Gðϕ ÞŠ ϕ = σ ð Þ : (8) We note that a property of the Pareto istribution is that an integral like equation 8 is always a constant multiple of the lower boun of integration. That is, ϕ θ = ðσ Þ ϕ θ σ þ ; (9) as obtaine by evaluating the integral in equation 8. Continuing to normalize

7 GAINS FROM TRADE UNDER MONOPOLISTIC COMPETITION 4 w =, the average prices of omestic goos appearing in equation 7 are then: M pðϕþ σ gðϕþ ϕ ½ Gðϕ ÞŠ φ σ σ σ ϕ M σ ; (0) which uses the prices p(ϕ)=[σ/(σ )] (/ϕ) together with the efinition of average prouctivity in equation 8. When comparing autarky with free trae, we nee to take into account the changing price of omestic goos an their changing variety, as in equation 0, along with the fact the all importe goos are new. The ratio of CES price inexes, P /P a, is what was measure using an exact price inex in the theorem ue to Feenstra (994). We now apply that theorem using equation 0 to measure the effective price of omestic goos. We treat all importe proucts as new, an let λ enote the share of expeniture on omestic goos in the trae equilibrium. Then the theorem ue to Feenstra (994) leas to the following exact price inex comparing trae with autarky: P P a M σ =ϕ M σ a =ϕ a! λ σ : () The first term appearing on the right of equation is just the change in the average price of omestic goos, using equation 0. The secon term on the right is the spening on omestic goos relative to total spening in the trae equilibrium, or one minus the share of spening on new importe varieties. This term reflects the potential gains ue to new import varieties. We now show that those gains from new import varieties are offset ue to a reuce number of omestic varieties. Those offsetting forces are seen more easily be rewriting equation as: P P a =ϕ σ λ : (2 ) =ϕ a M =M a This formulation neatly captures the three potential sources of gains from trae in the Melitz moel: the selection effect, which is a rise in average prouctivity an is shown by the first term on the right; the gains ue to import variety, which is shown by the numerator of the secon term; an the losses ue to reuce omestic variety, M < M a, which is shown by the enominator of the secon term on the right. We now show that M /M a = λ in equation 2, so the welfare loss ue to the reuction in the number of omestic varieties just cancels with the welfare gain ue to new import varieties. This result is obtaine from iviing the cutoff prouctivities for omestic firms from equation 4, by the expression in autarky

8 42 from equation 3, to obtain: R. C. FEENSTRA ϕ =P : (3) ϕ a =P a We see that the change in the cutoff prouctivities closely matches the fall in the prices inexes. Provie that the cutoff prouctivity rises when trae is opene, ϕ > ϕ a, it follows that the price inex falls, P < P a. Therefore, opening trae leas to a rise in welfare. Our goal here is to ientify the source(s) of those gains from trae. Once we a the Pareto assumption, then the ratio of the cutoff prouctivities equals the ratio of the average prouctivities ϕ =ϕ a, as shown in equation 9. Then comparing equation 2 with equation 3 we immeiately see that that M /M a = λ, as we intene to show. It follows that: λ M M e G ð ϕ Þ ϕ θ ϕ M a M e Gðϕ a Þ ϕ a ϕ a λ =θ : The secon equality above follows from using the formulas for the mass of surviving firms, an then the thir equality follows because the mass of entering firms is equal in the autarky an trae equilibria, an also using the Pareto istribution in equation 6. We fin, therefore, that we obtain a particularly simple formula for the ratio of cutoff prouctivities by using λ (the share of expeniture going to omestic proucts) raise to negative, inverse of the Pareto parameter. However, because the cutoff prouctivities etermine the CES price inexes from equation 3, we have obtaine a very simple formula for the gains from trae that is also foun by Arkolakis et al. (202): 2 THEOREM. Normalizing the wage at unity an comparing autarky with free trae, the rise in welfare for the representative consumer in the Melitz Chaney moel equals =P λ =θ ; =P a where λ is the share of expeniture going to omestic proucts in the trae equilibrium. As note above, the fact that M /M a = λ in equation 2 means that the welfare loss ue to the reuction in the number of omestic varieties just cancels with the welfare gain ue to new import varieties. So while the change in the mass of consume varieties can either increase or ecrease from M a to 2 A longer erivation of this theorem, making use of the implie prouction-possibilities frontier in the Melitz (2003) moel, is provie by Feenstra (200).

9 GAINS FROM TRADE UNDER MONOPOLISTIC COMPETITION 43 M + M x, 3 this change in variety has zero welfare impact. Rather, we have shown that the only source of gains from trae in the Melitz Chaney moel becomes the selection effect, whereby exiting omestic firms lea to a rise in average prouctivity. Notice that these effects from the Melitz moel are very ifferent from the Krugman (980) moel with CES preferences, where neither the scale nor the number of omestic firms changes at all when trae is opene: the only source of gains from trae in Krugman (980) is the variety gain from new import varieties. Melitz an Reing (205) show that the heterogeneous firm moel allows for greater gains from trae ue to increase specialization: the mostprouctive firms become major exporters while the least-prouctive firms exit the market, with imports corresponingly higher than with homogeneous firms. It reassures us that the gains from trae from the Krugman (980) moel woul be even higher with heterogeneous firms. 4. CONCLUSIONS Arkolakis et al. (202) have shown that with CES preferences, the monopolistic competition moel gives rise to a remarkably simple formula for the gains from opening trae: those gains are equal to one minus the import share of the economy, raise to an exponent that epens on the specific etails of the moel. In the Krugman (980) moel with homogeneous firms, that exponent is the inverse of one minus the elasticity of substitution. In the Melitz Chaney moel with heterogeneous firms an a Pareto istribution for prouctivities, the exponent equals minus the inverse of the Pareto parameter. As we have argue in this article, that ifference in results is obtaine because import competition rives out a mass of omestic varieties that just cancels in welfare terms with increase import variety, so that the only remaining source of gains from trae is from the positive selection of firms (governe by the Pareto shape parameter). Arkolakis et al. (202) argue that a similar result hols in a broaer class of moels that allows for a family of preferences beyon the CES, incluing the aitively separable an the quaratic preferences. They fin that a similarly simple formula for the gains from trae (equal to one minus the import share raise to an exponent) continues to hol. That result hinges very strongly, however, on the assumption of a Pareto istribution for the prouctivities of firms that is unboune above, like in equation 6. Feenstra (204) shows that, even with preferences much more general that the CES case, the assumption of unboune Pareto eliminates any gains from trae ue to increase variety or ue to a changing istribution of markups. All that is left are the gains ue to the positive selection of firms. In contrast, once the Pareto istribution is boune or truncate above, then the gains from trae ue to increase prouct variety an reuce markups of firms are restore, although, interestingly, the formula in Arkolakis et al. (202) continues to hol as an upper boun to the total gains from trae. We conclue from this 3 Feenstra (205, problem 6.3) shows that M a > (<)M + M x as f x > (<)f.

10 44 R. C. FEENSTRA iscussion that along with examining preferences other than the CES, it equally important to examine istributions for firm prouctivity other than the unboune Pareto, an this will be an important irection for further research. REFERENCES Arkolakis C., A. Costinot an A. Roríguez-Clare (202) New Trae Moels, Same Ol Gains?, American Economic Review 02, Bernar A. B., J. B. Jensen, S. J. Reing an P. K. Schott (2007) Firms in International Trae, Journal of Economic Perspectives 2, Broa C. an D. E. Weinstein (2006) Globalization an the Gains from Variety, Quarterly Journal of Economics 2, Chaney T. (2008) Distorte Gravity: The Intensive an Extensive Margins of International Trae, American Economic Review 98, Feenstra R. C. (994) New Prouct Varieties an the Measurement of International Prices, American Economic Review 84, Feenstra R. C. (200) Measuring the Gains from Trae uner Monopolistic Competition, Canaian Journal of Economics 43, 28. Feenstra, R. C. (204) Restoring the Prouct Variety an Pro-competitive Gains from Trae with Heterogeneous Firms an Boune Prouctivity, NBER Working Paper No Feenstra R. C. (205) Avance International Trae, 2n en. Princeton: University Press. Feenstra, R. C. an D. Weinstein (200) Globalization, Competition, an U.S. Welfare, NBER Working Paper No Krugman P. R. (979) Increasing Returns, Monopolistic Competition an International Trae, Journal of International Economics 9, Krugman P. R. (980) Scale Economies, Prouct Differentiation, an the Pattern of Trae, American Economic Review 70, Melitz M. (2003) The Impact of Trae on Intra-Inustry Reallocations an Aggregate Inustry Prouctivity, Econometrica 7, Melitz M. an S. Reing (205) New Trae Moels, New Welfare Implications, American Economic Review 05, Samuelson P. A. (952) The Transfer Problem an Transport Costs: The Terms of Trae when Impeiments are Absent, Economic Journal 62,