The Impact of Integration on Productivity and Welfare Distortions Under Monopolistic Competition

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1 The Impact of Integration on Prouctivity an Welfare Distortions Uner Monopolistic Competition Swati Dhingra CEP, LSE an Princeton University John Morrow Centre for Economic Performance, LSE This Draft: November 27, 211 Abstract A funamental question in monopolistic competition theory is whether the market allocates resources optimally. This paper generalizes the Spence-Dixit-Stiglitz framework to heterogeneous firms, aressing when the market provies optimal quantities, variety an prouctivity. Uner constant elasticity eman, each firm prices above its average cost, yet we show market allocations are first-best. When eman elasticities vary, market allocations are not optimal an reflect the istortions of imperfect competition. After etermining the nature of market istortions, we investigate how integration may serve as a remey to imperfect competition. Both market istortions an the impact of integration epen on two eman sie elasticities, an we suggest richer eman structures to pin own these elasticities. We also show that integration eliminates istortions, provie the post-integration market is sufficiently large. JEL Coes: F1, L1, D6. Keywors: Selection, Monopolistic competition, Efficiency, Prouctivity, Social welfare, Deman elasticity. Acknowlegments. We thank Bob Staiger for continue guiance, Katheryn Russ for etaile iscussion an George Alessanria, Costas Arkolakis, Roc Armenter, Any Bernar, Satyajit Chatterjee, Davin Chor, Steve Durlauf, Charles Engel, Thibault Fally, Rob Feenstra, Keith Hea, Wolfgang Keller, Jim Lin, Mathieu Parenti, Nina Pavcnik, Steve Reing, Anres Roriguez-Clare, Jacques Thisse, John Van Reenen an Mian Zhu for insightful comments. This paper has benefite from helpful comments of participants at AEA, DIME-ISGEP, the Philaelphia Fe, Princeton University an Wisconsin-Maison. Preliminary raft circulate as When is Selection on Firm Prouctivity a Gain from Trae? in 26. Swati thanks the IES Princeton for their hospitality uring work on this paper. Contact: s.hingra@lse.ac.uk an j.morrow1@lse.ac.uk. 1

2 1 Introuction Empirical work has rawn attention to the high egree of heterogeneity in firm prouctivity an the impact of market integration on firm survival an markups (Bernar et al. 27, Feenstra 26). The introuction of firm heterogeneity in monopolistic competition moels has provie new insights into how inustries continually reallocate resources. A funamental question within this setting is whether the market allocates resources optimally. Symmetric firm moels explain when market allocations are optimal by explaining the traeoff between quantity an prouct variety. When firms are heterogeneous in prouctivity, we must also ask which types of firms shoul prouce an which shoul be shut own. This paper answers this question for a general eman structure, which allows rich interrelationships between markups, prouctivity an efficiency. We focus on three key aspects of market istortions. First, we characterize which eman systems among the general class consiere by Dixit an Stiglitz are efficient in open economies, an iscuss the nature of istortions inuce by imperfect competition. Secon, we stuy how economic integration affects the ability of markets to reallocate resources optimally. For instance, the istortions of imperfect competition may be mitigate with entry of foreign firms, implying that trae liberalization provies opportunities to correct market failure. The impact of integration epens crucially on eman system characteristics, which we iscuss in etail. Thir, we examine whether large markets will push outcomes towars what we efine as the monopolistically competitive limit, which eliminates istortions. This enables us to unerstan ifferences in the ability of small an large markets to allocate resources efficiently. Differences in prouctivity across firms alter optimal allocation ecisions in a funamental way. In symmetric firm moels, marginal cost pricing an average cost pricing serve as heuristics for first-best an secon-best resource allocations. In heterogeneous firm moels, inucing each firm to price at its marginal cost or average cost will not maximize welfare because neither scheme takes into account sunk entry costs or the effect of heterogeneity on input costs. Thus, ifferent levels of prouction maximize welfare than what marginal or average cost pricing imply. It coul be optimal to allocate resources towars firms with lower costs (to conserve resources) or towars firms with higher costs (to preserve variety). The relative position of a firm in the cost istribution matters. A better unerstaning of optimal resource allocations can inform policy, especially in regar to international integration. Starting with constant elasticity of substitution (CES) eman, we show that the close market equilibrium is first-best espite the existence of positive profits. This optimality result seems surprising, base on the logic of marginal or average cost pricing which is esigne to return proucer surplus back to consumers. With prouctivity ifferences, the market requires prices above average cost to inuce firms to enter an potentially take a loss. Free entry ensures the wege between 2

3 prices an average costs exactly finances sunk entry costs. Therefore, the market implements the first-best allocation an laissez-faire inustrial policy is optimal uner CES eman. How broaly oes this efficiency result hol? We generalize the eman structure to the variable elasticity of substitution (VES) form of Dixit an Stiglitz which provies a rich setting for a wie range of market outcomes (Spence 1976; Vives 21; Zheloboko et al. 211). 1 Since optimality is unique to CES eman, any other VES eman generates istortions. Due to imperfect competition, the market maximizes aggregate real revenue instea of maximizing welfare. The nature of istortions can be etermine by two eman sie elasticities, the inverse eman elasticity an the elasticity of utility. The misalignment of these elasticities pins own the bias in market allocations. As a heterogeneous cost environment presents a potentially severe information problem for policy, one potential tool for efficiency improvements is to increase competition through international integration. Melitz (23a) shows that uner CES eman, integration increases average prouctivity, resulting in welfare gains from trae. We show that this outcome is efficient. In the presence of fixe export costs, the firms a planner woul close own in the open economy are exactly those that woul not survive in the market. However, a planner woul not close own firms in the absence of export costs. Thus, the rise in prouctivity following trae provies welfare gains by optimally internalizing trae frictions. 2 Uner VES eman, firms respon to import competition by changing their markups, an prouctivity respons to market size (even in the absence of trae frictions). For this eman class, the impact of integration can be summarize by eman-sie elasticities. For instance, changes in market prouctivity epen on the inverse eman elasticity, while changes in optimal prouctivity epen on the elasticity of utility. A comparison of these elasticities etermines how integration affects the gap between market prouctivity an optimal prouctivity. When these elasticities are aligne, istortions will eventually issipate. However, when the elasticities are misaligne, integration can exacerbate prouctivity istortions. To capture the role of integration as a policy tool, we efine the monopolistically competitive limit in which firm heterogeneity persists, but market size is so large that quantity sol from a firm to each worker is negligible. In this limit, VES eman operates much like CES eman, an is therefore socially optimal. Thus, constant markups implie by CES eman might approximate richer eman structures in large economies. However, the monopolistically competitive limit may 1 CES eman provies a useful benchmark by forcing constant markups that ensure market size plays no role in prouctivity changes. However, recent stuies fin market size has an economically significant impact on prouctivity istributions across markets (Campbell an Hopenhayn 25, Syverson 24a, Syverson 24b). Variable elasticity eman enables us to capture these rich interrelationships. 2 Melitz assumes equally size countries traing with each other. Market allocations are efficient even with asymmetric countries. But in the presence of trae frictions, asymmetry of country sizes introuces istributional concerns which we o not aress in this paper. 3

4 require a market size which is unattainable even in fully integrate worl markets. Then estimation of richer eman structures becomes imperative in unerstaning how integration impacts market istortions. Consequently, our last results provie suggestions about how to assess istortions empirically. The paper is organize as follows. Section 2 relates this paper to previous work an Section 3 recaps trae moels with firm heterogeneity. Section 4 presents efficiency results in a close economy. Section 5 introuces international trae an contrasts the efficiency of CES eman with inefficiency of VES eman, also eriving a monopolistically competitive limit which shows how integration can eliminate istortions. Section 6 further analyzes the impact of integration on istortions. Section 7 gathers together some theoretical implications useful for esigning empirical strategies an Section 8 conclues. 2 Relate Work Our paper is relate to work on welfare gains in inustrial organization an international economics. The traeoff between variety an quantity occupies a prominent place in the inustrial organization literature (e.g., Economies 1989, Mankiw an Whinston 1986). We contribute to this literature by stuying the effects of firm heterogeneity an international trae. The analysis is motivate by efficiency properties which have been stuie at length in symmetric firm moels of monopolistic competition. 3 Recently, Bilbiie et al. (26) show the market equilibrium with symmetric firms is socially optimal if an only if preferences are CES. We generalize the result to heterogeneous firms an show that efficiency is unrelate to the prouctivity istribution of firms. To the best of our knowlege, this is the first paper to show market outcomes in Melitz are first best. 4 To highlight the potential scope of market imperfections, we generalize the well known CES eman structure to VES eman. In contemporaneous work, Zheloboko et al. (211) evelop complementary results for market outcomes uner VES eman an emonstrate its richness an tractability uner various assumptions such as multiple sectors an vertical ifferentiation. Unlike Zheloboko et al., our focus is on market efficiency. We also stuy the limiting behavior of a VES economy. A large literature examines whether 3 Spence (1976); Dixit an Stiglitz (1977); Bilbiie et al. (26); Behrens an Murata (29). 4 We consier this to be the proof of a folk theorem. The iea of efficiency in Melitz has been in the air. Within the heterogeneous firm literature, Balwin an Robert-Nicou (28) an Feenstra an Kee (28) iscuss certain efficiency properties of the Melitz economy. In their working paper, Atkeson an Burstein (21) consier a first orer approximation an numerical exercises to show that prouctivity increases are offset by reuctions in variety. We provie an analytical treatment to show the market equilibrium implements the unconstraine social optimum. Helpman et al. (211) consier the constraine social optimum in the presence of a homogeneous goo. Their approach iffers because the homogeneous goo fixes the marginal utility of income. 4

5 monopolistic competition arises as a limit to oligopolistic pricing an when monopolistic competition converges to perfect competition in symmetric firm moels (Vives 21, Chapter 6). This literature consiers the limiting behavior as the number of firms tens to infinity. Instea, we examine a monopolistically competitive moel with a continuum of firms so there are infinitely many firms even in an economy with finite market size. After establishing the equivalence of increase international trae an increase market size, we stuy the limiting behavior in terms of the moel primitive of market size becoming large. The finings of our paper are relate to an emerging literature on welfare gains in new trae moels. Generalizing Krugman (198) to heterogeneous firms, Melitz (23b) shows that opening to trae raises welfare through reallocation of resources towars high prouctivity firms. Consiering 48 countries exporting to the US in 198-2, Feenstra an Kee (28) estimate that rise in export variety accounts for an average 3.3 per cent rise in prouctivity an GDP for the exporting country. In recent influential work, Arkolakis et al. (forthcoming) show that the mapping between trae ata an welfare is the same across several ol an new trae moels with ifferent prouction structures. This equivalence hols for moels which permit welfare to be summarize by import shares an trae elasticities (that can be erive from gravity equations). Unlike Arkolakis et al., we vary the eman sie of new trae moels an focus on the optimality of market outcomes. Once the Spence-Dixit-Stiglitz eman framework is consiere, welfare inferences from import shares require aitional information about eman an become more structural in nature. A large boy of empirical stuies use firm level prouction ata to examine whether trae liberalization inuces exit of low prouctivity firms an increases sales of high prouctivity firms. 5 Our results characterize when observe prouctivity gains reflect a narrowing of the istortionary gap between market an optimal prouctivity. Therefore, our work is more in line with Tybout (23) an Katayama et al. (29) who point to the limitations of the empirical literature in mapping observe prouctivity gains to welfare an optimal policies. Our results speak irectly to the mixe finings about trae liberalization an prouctivity in the empirical literature. Following trae liberalization, some countries show a reallocation towars high prouctivity firms while others showing a reallocation towars low prouctivity firms. 6 Tybout (23) proposes that these mixe finings coul mean that the selection effects emphasize by Melitz are not robust, or that firm size is a poor proxy for prouctivity. We aress the first issue by examining the robustness of selection effects to general eman specifications. We show that ifferences in inverse eman elasticities inuce ifferent patterns of firm selection, reconciling the 5 For a etaile survey of the literature, the reaer is referre to Tybout (23). While prouctivity estimation is fraught with ifficulties in measuring technical efficiency, we focus on the relationship between prouctivity an welfare as exposite in the heterogeneous firm literature. 6 Interpreting ifferences in firm size as prouctivities, Tybout (23) notes that it was the high prouctivity firms that lost market share in Chile an Colombia while it was the low prouctivity firms that suffere a ecline in Morocco. 5

6 mixe evience for prouctivity changes across heterogeneous firms. The secon issue of prouctivity measurement has been aresse in several stuies. Instea of aressing measurement, we focus on how VES eman can better explain observe patterns. We show how observe markups an physical prouctivity vary with market size uner general eman specifications. Our finings reiterate the importance of isentangling changes in markups an prouctivity to unerstan the sources of welfare gains from trae. 3 Trae Moels with Heterogeneous Firms Trae moels with heterogeneous firms iffer from earlier trae moels with prouct ifferentiation in two significant ways. First, costs of prouction are unknown to firms before sunk costs of entry are incurre. Secon, firms are asymmetric in their costs of prouction, leaing to firm selection base on prouctivity. In this section we briefly recap the implications of asymmetric costs for consumers, firms an equilibrium outcomes. 3.1 Consumers A mass L of ientical consumers in an economy are each enowe with one unit of labor an face a wage rate w normalize to one. Preferences are ientical in the home an foreign countries. Let M e enote the mass of entering varieties an q(c) enote quantity consume of variety c by each consumer. A consumer has preferences over ifferentiate goos U(M e,q) which take the general VES form: U(M e,q) M e u(q(c))g. (VES) (1) Here u enotes utility from an iniviual variety an u(q)g enotes utility from a unit bunle of ifferentiate varieties. In a Melitz economy, preferences take the special CES form with u(q) = q ρ. 7 More generally, we assume preferences satisfy usual regularity conitions which guarantee well efine consumer an firm problems. Definition 1. (Regular Preferences) u satisfies the following conitions: 1. u() is normalize to zero. 2. u is twice continuously ifferentiable with u > an u <. 7 The specific CES form in Melitz is U(M e,q) M 1/ρ e ( (q(c)) ρ G) 1/ρ but the normalization of the exponent 1/ρ in Equation (1) will not play a role in allocation ecisions. 6

7 3. u guarantees each monopolist s FOC uniquely etermines their optimal quantity The elasticity of marginal utility µ(q) qu (q)/u (q) is less than one. For each goo inexe by c, VES preferences inuce an inverse eman p(q(c)) = u (q(c))/δ where δ is a consumer s buget multiplier. As u is strictly increasing an concave, for any fixe price vector the consumer s maximization problem is concave. The necessary conition which etermines the inverse eman is sufficient, an has a solution provie inaa conitions on u. 9 Multiplying both sies of the inverse eman by q(c) an aggregating over all c, the buget multiplier is δ = M e ca u (q(c)) q(c)g. 3.2 Firms There is a continuum of firms which may enter the market for ifferentiate goos, by paying a sunk entry cost of f e. Each firm prouces a single variety so the mass of entering firms is the mass of entering varieties M e. Upon entry, each firm receives a unit cost c rawn from a istribution G with continuously ifferentiable pf g. 1 After entry, shoul a firm prouce for the omestic market it faces a cost function TC(q(c)) cq(c) + f where f enotes the fixe cost of prouction. Each firm faces an inverse eman of p(q(c)) = u (q(c))/δ an acts as a monopolist of variety c. Post entry profit of the firm from omestic sales is π(c) where π(c) max q(c) [p(q(c)) c]q(c)l f. The regularity conitions guarantee the monopolist s FOC is optimal an the quantity choice is given by p + q u (q)/δ = c. (MR=MC) MR = MC ensures that the markup rate is (p(c) c)/p(c) = qu (q)/u (q) = µ(q(c)). Therefore, the elasticity of marginal utility summarizes the inverse eman elasticity as µ(q) qu (q)/u (q) = ln p(q)/ lnq. When the economy opens to trae, firms incur an iceberg transport cost τ 1 an a fixe cost f x in orer to export to other countries. As a result, firms face a cost function TC(q x (c)) τcq x (c) + f x an a eman function p(q x (c)) for sales to the export market. Profit from foreign sales is π x (c) max qx (c)[p(q x (c)) τc]q x (c)l f x an the optimal q x choice is given by a similar MR = MC conition. 8 Sufficient conitions for this are 2u + u q < or that u is the integral of a strictly ecreasing an concave function. 9 Utility functions not satisfying inaa conitions are permissible but may require parametric restrictions to ensure existence. 1 Some aitional regularity conitions on G are require for existence of a market equilibrium in Melitz. 7

8 3.3 Market equilibrium Profit maximization implies that firms prouce for the omestic an/or export markets if they can earn non-negative profits from sales in the omestic an/or export markets, respectively. We enote the cutoff cost level of firms that are inifferent between proucing an exiting from the omestic market as c a in autarky an c in the open economy. The cutoff cost level for firms inifferent between exporting an not proucing for the export market is enote by c x. Formally, let ι = a,,x enote autarky an the omestic an export markets of the open home economy respectively. Each c ι is fixe by the Zero Profit Conition (ZPC), π ι (c ι ) = for ι = a,,x. (ZPC) Since firms with cost raws higher than the cutoff level o not prouce, the mass of omestic proucers (M ι ) supplying to market ι is M ι = M e G(c ι ). In summary, each firm faces a two stage problem: in the secon stage it maximizes profits from omestic an export sales given a known cost raw, an in the first stage it ecies whether to enter given the expecte profits in the secon stage. We maintain the stanar free entry conition impose in monopolistic competition moels. Specifically, let Π(c) enote the total expecte profit from sales in all markets for a firm with cost raw c, then ex ante average Π net of sunk entry costs must be zero, Π(c)G = f e. (FE) The next two Sections examine the efficiency properties of this framework for close an open economies. 4 Efficiency in the Close Economy Having escribe an economy consisting of heterogeneous imperfectly competitive firms, we now examine the optimality of market outcomes in the close economy. Outsie of cases in which imperfect competition leas to competitive outcomes with zero profits, one woul generally expect the coexistence of positive markups an positive profits to inicate inefficiency through loss of consumer surplus. Nonetheless, this Section shows that CES eman combine with the Melitz prouction framework exhibits positive markups an profits for surviving firms, yet it is allocationally efficient. However, we also show that the usual relationship between imperfect competition an welfare, that private incentives are not aligne with optimal prouction patterns, is true for all VES eman structures except CES. 8

9 4.1 Welfare uner isoelastic eman In a close economy, a social planner maximizes iniviual welfare U as given in Equation (1). 11 The social planner is unconstraine an chooses the mass of entrants, quantities an which firms of various prouctivities prouce. At the optimum, zero quantities will be chosen for varieties above a cost threshol c a. Therefore, all optimal allocative ecisions can be summarize by quantity q(c), potential variety M e an prouctivity c a. Our approach for arriving at the optimal allocation is to think of optimal quantities ˆq(c) as being etermine implicitly by c a an M e so that per capita welfare can be written as ca U = M e u( ˆq(c))G. (2) After solving for each ˆq conitional on c a an M e, Equation (2) can be maximize in c a an M e. Proposition 1 shows the market provies the first-best quantity, variety an prouctivity. Proposition 1. Every market equilibrium of a close Melitz economy is socially optimal. Proof. See Appenix. The proof of Proposition 1 iffers from stanar symmetric firm monopolistic competition results because optimal quantity is a nontrivial function of unit cost, variety an cutoff prouctivity. As the proof is involve, we relegate etails to the Appenix an iscuss the rationale for optimality below. In homogeneous firm moels, we know that firms charge positive markups which result in lower quantities than those implie by marginal cost pricing. However, the markup is constant so the market price (an hence marginal utility) is proportional to unit cost, ensuring proportionate reuction in quantity from the level that woul be observe uner marginal cost pricing (Baumol an Brafor 197). Moreover, homogeneous firms choose price equal to average cost so the profit exactly finances the fixe cost of prouction. Each firm therefore internalizes the effect of higher variety on consumer surplus, resulting in an efficient market equilibrium (Grossman an Helpman 1993, Bilbiie et al. 26, Balwin an Robert-Nicou 28). With heterogeneous firms, markups continue to be constant, ensuring that market prices across firms are proportionate to unit costs. But, average cost pricing is too low to compensate firms for an efficient allocation, because it will not cover ex ante entry costs. The market ensures that surviving firms internalize the losses face by exiting firms, losses which are etermine by aggregate economic eman that epens on q(c), c a an M e. Post entry, surviving firms charge prices higher than average costs (p(c) [cq(c) + f /L]/q(c)) which compensates them for the possibility of paying f e to enter an then being too unprouctive to survive. CES eman ensures that c a an M e are at optimal levels that fix p(c a ), thereby fixing absolute prices to optimal levels. 11 Free entry implies zero expecte profits so the planner s focus is on consumer surplus. 9

10 The way in which CES preferences cause firms to optimally internalize aggregate economic conitions can be mae clear by efining the elasticity of utility ε(q) qu (q)/u(q) an the social markup 1 ε(q). We term 1 ε(q) the social markup because at the optimal allocation, it enotes the utility from consumption of a variety net of its resource cost. At the optimal allocation, there is a multiplier λ which encapsulates the shaow cost of utility an ensures u (q(c)) = λc. Therefore, the social markup is 1 ε(q) =1 u (q)/u(q) =(u(q) λcq)/u(q). (Social Markup) For any optimal allocation, a quantity that maximizes social benefit from variety c solves max q(c) L(1 ε(q(c)))u(q)/λ f = max q In contrast, the incentives that firms face in the market are L 1 ε(q) cq f. ε(q) µ(q) maxlµ(q(c))pq f = max q(c) 1 µ(q) cq f. Since ε an µ epen only on the primitive u(q), we can examine which preferences woul make firms choose optimal quantities. Clearly, if µ(q)/(1 µ(q)) is proportional to (1 ε(q))/ε(q), firms will choose optimal quantities q when they prouce, but the set of proucers might be smaller or larger than optimal, epening on which firms can make enough profits to clear the fixe cost f. For the market to also select the optimal range of prouctivity, µ(q)/(1 µ(q)) must not only be proportional to (1 ε(q))/ε(q), but in fact be the same. Examining CES eman, we see precisely that µ(q)/(1 µ(q)) = (1 ε(q))/ε(q) for all q. Thus, CES eman incentivizes exactly the right firms to prouce, in aition to proucing optimal quantities. A irect implication of Proposition 1 is that laissez faire inustrial policy is optimal uner CES eman. This efficiency result may seem surprising in the context of Dixit an Stiglitz (1977) who fin that market allocations are secon-best but not first-best uner CES eman for ifferentiate goos. Dixit an Stiglitz consier two sectors (a homogeneous goos sector an a ifferentiate goos sector) an assume a general utility function to aggregate across these goos. With a general utility function, the elasticity of substitution between the homogeneous an ifferentiate goos is not constant, leaing to inefficient market allocations. In the next subsection, we examine the role of elasticities in greater etail. In keeping with Melitz, we consier a single sector to evelop results for market efficiency in terms of elasticities. 1

11 4.2 Welfare beyon isoelastic eman Efficiency of the market equilibrium in a Melitz economy is tie to CES eman. To highlight the role of CES eman, we consier the general class of variable elasticity of substitution (VES) eman stuie by Dixit an Stiglitz (1977) as specifie in Equation (1). With regra to efficiency, comparison of FOCs for the market an optimal allocation shows constant markups are necessary for efficiency. Therefore, within the VES class, optimality of market allocations is unique to CES preferences. 12 Proposition 2. Uner VES eman, a necessary conition for the market equilibrium to be socially optimal is that u is CES. 13 Proof. Proof available upon request. Uner general VES eman, market allocations are not socially optimal. Market allocations o not maximize iniviual welfare. Proposition 3 shows that the market instea maximizes aggregate real revenue (M e u (q(c)) q(c) LG) generate in the economy. Proposition 3. economy. Uner VES eman, the market maximizes aggregate real revenue in the close Proof. See Appenix. Proposition 3 shows that market resource allocation is generally not aligne with the social optimum uner VES eman. The allocations of market an the social optima are solutions to: ca maxm e u (q(c)) q(c)g ca maxm e u(q(c))g { ca } where L M e [cq(c)l + f ]G + f e { ca } where L M e [cq(c)l + f ]G + f e Market Social For CES eman, u(q) = q ρ while u (q)q = ρq ρ implying revenue maximization is perfectly aligne with welfare maximization. Outsie of CES, quantities prouce by firms are too low 12 VES utility is aitively separable an therefore oes not inclue the quaratic utility of Melitz an Ottaviano (28) an the translog utility of Feenstra (23). However, Zheloboko et al. (211) show VES eman captures the qualitative features of market outcomes obtaine uner these forms of non-aitive utility. 13 For completeness, we note that constant elasticities of eman are necessary but not sufficient for optimality of market allocations. We exten the CES eman of Melitz to CES-Benassy preferences U(M e,c a,q) ν(m e ) c q(c)ρ g(c)c. In this example, u is CES but varieties an the unit bunle are value ifferently through ν(m e ). Market allocations uner CES-Benassy preferences are the same as with CES preferences of Melitz. However, firms o not fully internalize consumers taste for variety, leaing to suboptimal levels of quantity, variety an prouctivity. Following Benassy (1996), Bilbiie et al. (26) an Alessanria an Choi (27), when ν(m e ) = M ρ(ν B+1) e, these preferences isentangle taste for variety ν B from the markup to cost ratio (1 ρ)/ρ. Market allocations are optimal only if taste for variety exactly equals the markup to cost ratio (ν B = (1 ρ)/ρ). 11

12 or too high an in general equilibrium, this implies the average prouctivity of operating firms is also too low or too high. Market quantity, variety an prouctivity reflect istortions of imperfect competition, an therefore, increase competition through opening markets to trae might improve allocations. This leas us to an examination of the impact of trae on market istortions. 5 Efficiency in an Open Economy Motivate by empirical stuies of firm heterogeneity, Melitz (23b) shows that reallocation of resources towars high prouctivity firms provies a new source of gains from trae. In this Section, we examine how international trae affects market an optimal allocations in a Melitz economy. We start by showing that CES eman continues to inuce socially optimal allocations in an open economy. Uner VES eman, market allocations are suboptimal so we examine when market expansion from trae eventually mitigates the istortions of imperfect competition while preserving firm heterogeneity. 5.1 Welfare uner isoelastic eman Trae provies prouctivity gains by reallocating resources towars low cost firms. One might therefore expect artificially selecting low cost firms to prouce woul improve welfare in autarky. In fact, this is not the case. Proposition 1 shows that the autarkic market equilibrium is efficient. This implies that the open economy prouctivity level is unesirable in autarky. It reuces entry an generates too little variety. However, as Proposition 4 below shows, the prouctivity level selecte in an open economy is efficient. Thus trae itself makes reuce entry an reuce variety of home firms efficient. Proposition 4. Every market equilibrium of ientical open Melitz economies is socially optimal. Proof. See Appenix. Why is the higher prouctivity level of the open economy inefficient in autarky? Proposition 4 implies that market selection of firms is optimal if an increase in size can only be attaine at a cost of exogenous frictions (τ, f x ). Compare to a frictionless worl, trae frictions reuce the potential welfare gains from trae. The market minimizes the losses from frictions by weeing out the right firms. It bis up resource prices an eliminates low prouctivity firms. Conitional on trae costs, market selection of firms is optimal an provies a net welfare gain from trae. Proposition 4 is striking in that the ifferences in firm costs o not generate inefficiencies espite heterogeneity of profits an the ifferent effects that trae frictions will have on firm behavior. Furthermore, selection of firms performs the function of allocating aitional resources optimally 12

13 without any informational requirements. Uner CES eman, laissez faire inustrial policy is optimal for the worl economy. Market allocations maximize social welfare uner equal Pareto weights across the equally size countries. 14 Moeling trae between equally size countries makes the role of trae frictions extremely clear cut. When countries iffer in size, trae frictions introuce cross-country istributional issues which obscure the pure efficiency question. Specifically, consier two countries of ifferent sizes with cost istribution G(c) = (c/cmax) k an CES eman. Market allocations are efficient when these countries trae with each other an face no trae frictions. These market allocations maximize social welfare with equal Pareto weights assigne to every iniviual in the two countries. Introucing trae frictions will continue to inuce efficient market allocations, but with unequal Pareto weights. This shows the market is implicitly favoring certain consumers, so that firm selection patterns reflect istributional outcomes in aition to cost competitiveness. The cross-country istribution of welfare gains is important but beyon the focus of this stuy. In what follows, we wish to stuy efficiency rather than istribution so we moel the stylize case of frictionless trae an consier more general eman structures which can explain a greater range of trae effects. 5.2 Welfare beyon isoelastic eman We examine the impact of opening to international trae on a VES economy with no trae costs. As iscusse above, the absence of trae costs allows us to abstract from istributional issues. In particular, the market equilibrium between freely traing countries of sizes L 1,...,L n is ientical to the market equilibrium of a single autarkic country of size L = L L n. Thus, opening to trae is equivalent to an increase in market size, echoing Krugman (1979). This result is summarize as Proposition 5. Proposition 5. In the absence of trae costs, trae between countries of sizes L 1,...,L n has the same market outcome as a unifie market of size L = L L n. Proof. Available upon request, see also Krugman (1979). Proposition 5 allows us to think about increase trae as an increase in market size L of a close economy. An increase in market size has the ientical effect of increase competition (except now instea of new foreign competition, there is new omestic competition) which will impact efficiency through altering market istortions. We turn to efficiency properties of the open 14 However, terms of trae externalities may exist an lea to a breakown of laissez faire policies. Demiova an Roriguez-Clare (29) incorporate terms of trae consierations an provie omestic policies to obtain the first-best allocation in an open Melitz economy with Pareto cost raws. Chor (29) also consiers when policy intervention is appropriate in a heterogeneous firm moel with multinationals an a homogeneous goos sector. 13

14 VES economy, an investigate how far increase competition from trae can go towars improving market outcomes Market Efficiency uner VES Deman Having establishe that opening to trae is equivalent to an increase in the size of a VES economy, we can follow the same reasoning as in the close VES economy to infer that market allocations in an open economy are suboptimal. Marginal revenues o not correspon to marginal utilities so relative allocations in the market are not aligne with efficient allocations. When market allocations are suboptimal, opening to trae may take the economy further from the social optimum. For example, market expansion from trae may inuce exit of low prouctivity firms from the market when it is optimal to keep more low prouctivity firms with the purpose of preserving variety. This raises the question of when integration mitigates or exacerbates istortions. As acknowlege by Spence, perfectly general propositions are har to come by an the nature of istortions can be highly epenent on parameter magnitues. To make progress, we follow Stiglitz (1986) an first stuy market an optimal outcomes as market size becomes arbitrarily large. This allows us to examine when international trae enables markets to eventually mitigate istortions. We emphasize prouctivity istortions as they are new to moels with monopolistic competition an later we also examine how small increases in market size affect prouctivity istortions Market Efficiency in Large Markets We consier the effects of trae when worl markets are so large that the quantity supplie from each firm to each worker becomes negligible. For example, consier a small economy which integrates with the rest of the worl. Looking at efficiency in large markets explains when integrating with worl markets enables a small economy to overcome its market istortions. From a theoretical perspective we will term a large market the limit of the economy as the mass of workers L approaches infinity, an in practice we might expect that sufficiently large markets approximate this limiting case. 15 The large economy concept is similar in spirit to the iea of a competitive limit, although with heterogeneous firms at least three salient outcomes can occur. One outcome is that competitive pressures might wee out all firms but the most prouctive. This occurs for instance when marginal revenue is boune, as when u is quaratic or CARA (constant absolute risk aversion) as in Behrens an Murata (29). It may also happen that access to large markets allows even the least prouctive firms to amortize fixe costs an prouce. To retain the funamental properties of monopolistic competition with heterogeneous firms, we chart out a thir possibility between these two extremes: 15 How large markets nee to be to justify this approximation is an open quantitative question. 14

15 some, but not all, firms prouce. Accoringly, we consier large economies which satisfy the following assumption. Assumption. In the large economy, both the market an the social planner select a non-egenerate cost istribution in which some but not all entrants prouce. In consiering general VES eman, we maintain the previous regularity conitions for a market equilibrium an a a few more conitions about the behavior of u(q) when q is close to zero in orer to ai the analysis. Assumption. VES preferences exhibit the following properties for the large economy: 1. Quantity ratios istinguish price ratios for small q: If κ κ then lim p(κq)/p(q) lim p( κq)/p(q). q q 2. The inverse eman elasticity an elasticity of utility are boune away from an 1 for small quantities. Formally, lim q µ(q) an lim q ε(q) (,1). 3. For small quantities, the inverse eman elasticity an elasticity of utility are monotone. Formally, for all sufficiently small q, µ (q) an ε (q). Assumption 1 is a regularity conition which guarantees prouction levels across firms can be istinguishe if the firms charge istinct prices. In particular, this implies lim q u (q) =. Following Dixit an Stiglitz, Assumptions 2 an 3 imply that market prices converge to non-zero constants as market size grows an lim q 1 ε(q) = lim q µ(q) so the ae utility provie per labor unit in the social optimum converges to a non-zero constant as in Solow (1998). An example of a utility function satisfying Assumptions 2 an 3 is u(q) = q ρ bq for ρ (,1) an b >. 16 The main economic content of these assumptions is that they etermine when the economy converges to a monopolistically competitive limit, rather than a competitive limit. As the economy grows, each worker consumes a negligible quantity of each variety. At these low levels of quantity, the inverse eman elasticity oes not vanish an firms can still extract a positive markup µ. Uner the competitive limit, firms are left with no market power an µ rops to zero. Similarly, the social markup (1 ε) oes not rop to zero in the monopolistically competitive limit an each variety contributes at a positive rate to utility even at low levels of quantity. Combining these assumptions ensures the large economy goes to the monopolistically competitive limit, summarize as Proposition Kuhn an Vives (1999) fin that lim q 1 ε(q) = lim q µ(q) uner relatively mil assumptions. Let ν = V (q)q/v (q) where V (q) = u(q)/q in our notation. Then their assumption is that there exist positive constants ρ an κ such that lim q [ν(q) ν()]/q ρ = κ so ν(q) ν() has an asymptotic expansion at q = with a leaing term of the constant elasticity type. Examples of such utility functions inclue u(q) = q ρ e κq for ρ (,1) an κ < (1 ρ)/(1 + ρ)ρ, u(q) = q κ (1 q) for κ (,1) an u(q) = q(1 q κ ) for κ >. 15

16 Proposition 6. Uner the large economy assumptions, as market size L approaches infinity the market approaches the monopolistically competitive limit. This limit has the following characteristics: 1. Prices, markups an expecte profits converge to positive constants. 2. Per capita quantities q(c) go to zero, while aggregate quantities Lq(c) converge. 3. Relative quantities q(c)/q(c ) converge to (c/c ) 1/α with α = lim q µ(q). 4. The entrant per worker ratio M e /L converges. 5. The market an socially optimal allocations coincie. Proof. See Appenix. Proposition 6 shows that integration with large markets can push economies base on VES eman to the monopolistically competitive limit. In this limit, the inverse eman elasticity an the elasticity of utility become constant, ensuring the market outcome is socially optimal. Firms charge constant markups which exactly cross-subsiize entry of low prouctivity firms to preserve variety. This wipes out the istortions of imperfect competition as the economy becomes large. Intuitively, we can explain Proposition 6 in terms of our previous result that CES preferences inuce efficiency. In large markets, the quantity q(c) sol to any iniviual consumer goes to zero, so markups µ(q(c)) converge to the same constant inepenent of c. This convergence to constant markups aligns perfectly with those generate by CES preferences with an exponent equal to 1 lim q µ(q). Thus, large markets reuce market istortions until they are aligne with socially optimal objectives an utility provie by the market approaches the first-best level. It is somewhat remarkable that the large market outcome is socially optimal. Firms charge positive markups but they exactly recover both average costs an ex ante entry costs. Therefore, market allocations are first-best espite positive markups. Such persistence of imperfection in competition is consistent with the observation of Samuelson (1967) that the limit may be at an irreucible positive egree of imperfection (Khan an Sun 22). 17 While the monopolistically competitive limit is optimal espite imperfect competition, it is an open empirical question whether markets are sufficiently large for this to be a reasonable approximation to use in lieu of richer VES eman. When markets are small, variable markups are crucial in unerstaning the prouctivity gap between the market an the social optimum. In the next Section, we examine the impact of trae on the prouctivity gap in a small variable elasticity economy. 17 Stiglitz (1986) notes that the CES moel violates the assumptions of the competitive limit of the monopolistically competitive economy erive by Hart (1985) who assumes markups are completely wipe out in the limit. 16

17 6 Distortions an the Impact of Integration With variable markups, opening to trae can have positive or negative effects on prouctivity. Trae expans market size an has ifferent effects on profitability across firms. With constant markups, firm profitability is unaffecte by market expansion so trae affects prouctivity only through trae frictions an not market expansion. 18 Here we show that this is specific to CES eman. With VES eman, trae affects prouctivity even in the absence of trae frictions. We first show that the inverse eman elasticity etermines when the market selects more (or less) prouctive firms after trae. The optimal prouctivity change epens on the elasticity of utility. As iscusse above, market allocations in a VES economy are not optimal, so it is unclear if the irection of prouctivity changes is optimal. Accoringly, we compare the market an optimal outcomes to examine the istortions inuce by imperfect competition. 6.1 Prouctivity changes in the market The markup rate in a VES economy is µ(q) qu (q)/u (q). When µ (q) >, markups are positively correlate with quantity an when µ (q) <, markups are negatively correlate with quantity. Here we show that µ > implies that prouctivity rises following market expansion, an µ < implies prouctivity falls after market expansion. For CES eman, markups are constant (µ = ) an so are uncorrelate with any variable. This istinction between µ > an µ < is brought out by the richer VES eman structure an, as we now spell out, the sign of µ etermines the impact of trae on prouctivity. In a VES economy, inverse eman is p(q(c)) = u (q(c))/δ where δ is a consumer s buget multiplier. The multiplier δ is an aggregate eman shifter that increases with market size. This can be seen from the free entry conition (FE), c a [(p(c) c)q(c)l f ]G = f e. Differentiating FE with respect to L an applying the envelope theorem (an noting π(c a ) = f ), we have ca [(p(c) c)q(c) + L p(c)/ L q(c)]g =. The first term on the LHS above is the rise in profits from higher sales in a bigger market while the secon term reflects the shift in the resiual eman curve ue to market expansion. We can solve for the change in aggregate eman conitions as ln p(c)/ lnl = c a (p(c) c)q(c)g/ c a p(c)q(c)g. 18 Unlike prouctivity changes from trae frictions, prouctivity changes from market size effects of trae reflect an expansion in prouction possibilities of the economy. However, the interaction between trae frictions, selection an pro-competitive effects of trae can introuce new questions that we o not aress here. 17

18 Using the fact that p(c) c = µ(c) p(c), we have ca ca ln p(c)/ lnl = µ(c)p(c)q(c)g/ p(c)q(c)g <. As market size expans, more firms enter so resiual eman of each firm falls. The percentage fall is the average markup in the economy, an we now examine how this rise affects the ability of the cutoff firm to survive. From the cutoff cost conition (ZPC), (p(c a ) c a )q(c a )L = f. Differentiating with respect to to L an applying the envelope theorem, we have (p(c a ) c a )q(c a ) + L( p(c a )/ L c a /L)q(c a ) =. The first terms on the LHS is the rise in profit from higher sales in a bigger market an the secon term is the rop in resiual eman from market expansion. Re-writing for lnc a / lnl an ln p/ lnl, we have ln p(c a )/ lnl + (p(c a ) c a )/p(c a ) = ( lnc a / lnl) c a /p(c a ). (3) Substituting for ln p(c)/ lnl an noting µ = (p c)/p an 1 µ = c/p, we see that ca ca ( lnc a / lnl)(1 µ(c a )) = µ(c a ) µ(c) p(c)q(c)g/ p(c)q(c)g. (4) Since µ(c a ) (,1), lnc a / lnl is the same sign as the RHS of Equation (4) which epens on the markup of the cutoff firm relative to the average markup in the economy. Our regularity conitions guarantee high cost firms prouce lower quantities so q(c) is ecreasing in c an sign µ(c)/c = sign (µ (q)q (c)) = sign µ (q). Therefore, the change in the inverse eman elasticity µ (q) etermines how market expansion from trae changes the cutoff cost level. When µ (q) >, lnc a / lnl < an when µ (q) <, lnc a / lnl >. The intuitive explanation is as follows. When µ (q) >, high prouctivity firms sell more q an charge higher markups. This is the case stuie by Krugman (1979) an we refer to it as aictive preferences because firms are able to charge higher markups when they sell higher quantities. This means markups are negatively correlate with unit costs so less prouctive firms have both low q an low markups. With market expansion, the rise in competition (δ) squeezes prices an the less prouctive firms are the least able to cushion this price rop through markups. They cannot survive so the cutoff cost level rops. Conversely, when µ (q) <, firms selling higher quantities charge lower markups, so we term these preferences congestive. Following market expansion, rise in competition forces a lower 18

19 supply of per capita quantity q, but µ (q) < implies a compensating increase in markups to cushion this competition. This increase in markups is strong enough to make even higher cost firms profitable an the cutoff cost level rises with market expansion. Low prouctivity firms sell the lowest q(c) an have the highest markups to cushion against import competition. These low prouctivity boutique firms fare much better following increase competition. This is consistent with Holmes an Stevens (21) who fin small US plants were less impacte than large plants uring the import surge from China. Uner CES preferences, the inverse eman elasticity is µ(q) = 1 ρ implying µ (q) =. Constant inverse eman elasticity implies there shoul be no correlation between markups an unit costs. Therefore, the cutoff cost level is not affecte by market size. The rop in resiual eman from higher competition exactly counterbalances the higher sales to a bigger market so firm ecisions are unaffecte. The cost cutoff oes not change with market size an only trae frictions inuce low prouctivity firms to exit the market. Uner VES eman, this is no longer true. Firms ajust their markups in response to market expansion an the ensuing firm entry, so profitability of the least prouctive firm is affecte by market size. Following market expansion from trae, prouctivity changes epen on whether the markup of low prouctivity firms provies enough cushion to absorb the ownwar shift in eman. We summarize the implications of aictive an congestive VES preferences in Proposition 7. We show that the market selects high prouctivity firms when preferences are aictive an inclues more low prouctivity firms when preferences are congestive. Proposition 7. Increases in market size (L) change the market cost cutoff (c a ) as follows: 1. When preferences are aictive, efine as µ (q) >, the cutoff ecreases with size. 2. When preferences are congestive, efine as µ (q) <, the cutoff increases with size. We have seen the correlation of markups with quantities characterizes prouctivity changes. It is unknown if these changes are optimal. For CES eman, this correlation is zero an prouctivity changes are optimal. VES eman allows non-zero correlation between markups an quantities so we have goo reason to expect that changes in market size funamentally alter the optimality of market allocations. We now aress this issue by consiering optimal prouctivity changes uner VES eman Optimal prouctivity changes In a parallel fashion to the market prouctivity changes, we consier two types of preferences base on the sign of ε (q) an show that they have ifferent implications for how trae affects 19