Monopolistic Competition and Optimum Product Diversity Under Firm Heterogeneity*

Monopolistic Competition an Optimum Prouct Diversity Uner Firm Heterogeneity* Swati Dhingra CEP, Lonon School of Economics & CEPR John Morrow CEP, Lonon School of Economics This Draft: June 3, 214 Abstract Empirical work has rawn attention to the high egree of prouctivity ifferences within inustries, an its role in resource allocation. This paper examines the allocational efficiency of such markets. Prouctivity ifferences introuce two new margins of potential inefficiency: selection of the right istribution of firms an allocation of the right quantities across firms. We show that these consierations impact welfare an policy analysis. Market power across firms leas to istortions in resource allocation. Deman-sie elasticities etermine how resources are misallocate an when increase competition from market expansion provies welfare gains. JEL Coes: F1, L1, D6. Keywors: Efficiency, Prouctivity, Social welfare, Deman elasticity, Markups. Acknowlegments. We thank Bob Staiger for continue guiance an Steve Reing for encouragement. We are grateful to G Alessanria, C Arkolakis, R Armenter, A Bernar, S Chatterjee, D Chor, S Durlauf, C Engel, T Fally, R Feenstra, K Hea, W Keller, J Lin, E Ornelas, G Ottaviano, M Parenti, N Pavcnik, T Sampson, D Sturm, J Thisse, J Van Reenen, A Weinberger, B Zissimos an M Zhu for insightful comments, K Russ an A Roriguez-Clare for AEA iscussions an T Besley for avice. This paper has benefite from helpful comments of participants at AEA 211 an 213, CEPR-ERWIT, CEPR-IO, Columbia, Davis, DIME-ISGEP 21, ETSG 212, Georgetown, Harvar KSG, HSE St Petersburg, ISI, FIW, LSE, Louvain, Mannheim, Marylan, NBER, Oxfor, Philaelphia Fe, Princeton, Toronto, Virginia Daaren, Wisconsin an Yale. Swati thanks the IES (Princeton) for their hospitality. We acknowlege the financial support from Portuguese national funs by FCT (Funacao para a Ciencia e a Tecnologia) project PTDC/EGE-ECO/122115/21. A preliminary raft was a issertation chapter at Wisconsin in 21. *The first line is the title of Dixit an Stiglitz (1977). Contact: s.hingra@lse.ac.uk; j.morrow1@lse.ac.uk. 1

1 Introuction Empirical work has rawn attention to the high egree of heterogeneity in firm prouctivity, an the constant reallocation of resources across ifferent firms. 1 The focus on prouctivity ifferences has provie new insights into market outcomes such as inustrial prouctivity, firm pricing an welfare gains from policy changes. 2 When firms iffer in prouctivity, the istribution of resources across firms also affects the allocational efficiency of markets. In a recent survey, Syverson (211) notes the gap between social benefits an costs across firms has not been aequately examine, an this limite unerstaning has mae it ifficult to implement policies to reuce istortions (pp. 359). This paper examines allocational efficiency in markets where firms iffer in prouctivity. We focus on three key questions. First, oes the market allocate resources efficiently? Secon, what is the nature of istortions, if any? Thir, can economic integration reuce istortions through increase competition? Symmetric firm moels explain when resource allocation is efficient by examining the traeoff between quantity an prouct variety in imperfectly competitive markets. 3 When firms iffer in prouctivity, we must also ask which types of firms shoul prouce an which shoul be shut own. Firm ifferences in prouctivity introuce two new margins of potential inefficiency: selection of the right istribution of firms an allocation of the right quantities across firms. For example, it coul be welfare-improving to skew resources towars firms with lower costs (to conserve resources) or towars firms with higher costs (to preserve variety). Furthermore, ifferences in market power across firms lea to new trae-offs between variety an quantity. These consierations impact optimal policy rules in a funamental way, istinct from markets with symmetric costs. One contribution of the paper is to unerstan how these consierations affect welfare an policy analysis. A secon contribution of the paper is to show when increase competition improves welfare an efficiency. When market allocations are inefficient, increase competition (from trae or growth) may exacerbate istortions an lea to welfare losses (Helpman an Krugman 1985). A secon-best worl offers no guarantee of welfare gains from trae. But, by creating larger, more competitive markets, trae may reuce the istortions associate with imperfect competition an provie welfare gains (Krugman 1987). This insight is even more relevant in a heterogeneous cost environment because of new sources of potential inefficiency. We explain when integration provies welfare gains by aligning private an social incentives. As a benchmark, 1 Example, Bartelsman an Doms (2); Tybout (23); Feenstra (26); Bernar, Jensen, Reing an Schott (27). 2 Example, Pavcnik (22); Asplun an Nocke (26); Foster et al. (21); Melitz an Reing (212). 3 Example, Spence (1976); Venables (1985); Mankiw an Whinston (1986); Stiglitz (1986). 2

we show integration with large worl markets provies a policy option to correct istortions. 4 To unerstan efficiency in general equilibrium, we examine resource allocation in the stanar setting of a monopolistically competitive inustry with heterogeneous firm prouctivity an free entry (e.g. Melitz 23). We begin our analysis by consiering constant elasticity of substitution (CES) eman. In this setting, we show market allocations are efficient, espite ifferences in firm prouctivity. This is striking, as it requires the market to inuce optimal resource allocations across aggregate variety, quantity an prouctivity. As in symmetric firm moels, there are two sources of potential inefficiency: the inability of firms to appropriate the full consumer surplus an to account for business stealing from other firms. CES eman uniquely ensures these two externalities exactly offset each other. Firm heterogeneity oes not introuce any new istortions because the magnitue of these externalities oes not vary across firms. Firms earn positive profits which seems surprising base on the logic of average cost pricing that is esigne to return proucer surplus to consumers. When prouctivity iffers, the market requires prices above average costs to inuce firms to enter an potentially take a loss. Free entry ensures the wege between prices an average costs exactly finances sunk entry costs, an positive profits are efficient. Therefore, the market implements the first-best allocation an laissez faire inustrial policy is optimal. 5 What inuces market efficiency an how broaly oes this result hol? We generalize the eman structure to the variable elasticity of substitution form of Dixit an Stiglitz (1977), which provies a rich setting for a wie range of market outcomes (Vives 21; Zheloboko, Kokovin, Parenti an Thisse 212). When eman elasticity varies with quantity an firms vary in prouctivity, markups vary within a market. This accounts for the stylize facts that firms are rarely equally prouctive an markups are unlikely to be constant. 6 Introucing this empirically relevant feature of variable elasticities turns out to be crucial in unerstaning istortions. When elasticities vary, firms iffer in market power an market allocations reflect the istortions of imperfect competition. Nonetheless, we show the market maximizes real revenues. This is similar to perfect competition moels, but now market power implies private benefits to firms 4 International integration is equivalent to an expansion in market size (e.g., Krugman 1979). As our focus is on efficiency, we abstract from trae frictions which introuce cross-country istributional issues. 5 Melitz (23) consiers both variable an fixe costs of exporting. In a separate note, we show that the open Melitz economy is efficient, even with trae frictions. In the presence of fixe export costs, the firms a policymaker woul close own in the open economy are exactly those that woul not survive in the market. However, a policymaker 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. 6 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 matters for firm size (Campbell an Hopenhayn 25) an prouctivity ispersion (Syverson 24). Foster, Haltiwanger an Syverson (28) show that profitability rather than prouctivity is more important for firm selection, suggesting a role for richer eman specifications. For further iscussion, see Melitz an Trefler (212). 3

are perfectly aligne with social benefits only uner CES eman. More generally, the appropriability an business stealing effects nee not exactly offset each other, an firm heterogeneity introuces a new source of potential inefficiency. When firm iffer in prouctivity, entry of an aitional variety shifts business across the entire istribution of firms an inuces istortions relative to optimal allocations. The pattern of istortions is etermine by two elasticities: the eman elasticity, which measures market incentives through markups, an the elasticity of utility, which measures social incentives through a firm s contribution to welfare. We show that the way in which these incentives iffer characterizes the precise nature of misallocations. This also yiels two new insights relating prouctivity ifferences to misallocations. First, ifferences in market power across firms imply misallocations are not uniform: some firms over-prouce while others unerprouce within the same market. For instance, the market may favor excess entry of low prouctivity firms, thereby imposing an externality on high prouctivity firms who en up proucing too little. Secon, ifferences in market power impact economy-wie outcomes. The istribution of markups affects ex ante profitability, an therefore the economy-wie trae-off between aggregate quantity an variety. This is in sharp contrast to symmetric firm markets, where markups (or eman elasticities) o not matter for misallocations, as emphasize by Dixit an Stiglitz (1977) an Vives (21). Differences in prouctivity unerline the importance of eman elasticity for allocational efficiency, an complement the message of Weyl an Fabinger (212) an Parenti et al. (214) that richer eman systems enable a better unerstaning of market outcomes. As misallocations vary by firm prouctivity, one potential policy option that oes not require firm-level information is international integration. The iea of introucing foreign competition to improve efficiency goes back at least to Melvin an Warne (1973). We show that market integration always provies welfare gains when private an social incentives are aligne, which again is characterize by the eman elasticity an the elasticity of utility. This result ties the Helpman-Krugman characterization of gains from trae to the welfare approach of Spence- Dixit-Stiglitz. Symmetric firm moels with CES eman provie a lower boun for the welfare gains from integration. Gains from trae uner aligne preferences are higher ue to selection of the right istribution of firms an an allocation of the right quantities across firms. As a benchmark for unerstaning efficiency gains, we follow the literature on imperfect competition in large markets an examine whether integration with large global markets leas to allocative efficiency (Vives 21, Chapter 6). Integration with large markets will push outcomes towars a new concept, the CES limit, where firms converge to charging constant markups. Unlike a perfectly competitive limit (Hart 1985), prouctivity ispersion an market power persist in the 4

CES limit. Yet the market is efficient an integration with large global markets is therefore a first-best policy to eliminate the istortions of imperfect competition. However, as the limit may require a market size which is unattainable even in fully integrate worl markets, integration may be an incomplete tool to reuce istortions. Relate Work. Our paper is relate to work on firm behavior an welfare in inustrial organization an international economics. As mentione earlier, the trae-off between quantity an variety occupies a prominent place in the stuy of imperfect competition. We contribute to this literature by stuying these issues in markets where prouctivity ifferences are important. To highlight the potential scope of market imperfections, we consier variable elasticity of substitution (VES) eman. In contemporaneous work, Zheloboko et al. (212)emonstrate the richness an tractability of VES market outcomes uner various assumptions such as multiple sectors an vertical ifferentiation. 7 The focus on richer eman systems is similar to Weyl an Fabinger (212) who characterize several inustrial organization results in terms of pass-through rates. Unlike these papers, we examine the efficiency of market allocations, so our finings epen on both the elasticity of utility an the eman elasticity. To the best of our knowlege, this is the first paper to show market outcomes with heterogeneous firms are first-best uner CES eman. 8 The finings of our paper are also relate to a traition of work on welfare gains from trae. Helpman an Krugman (1985) an Dixit an Norman (1988) examine when trae is beneficial uner imperfect competition. We generalize their fining an link it to moel primitives of eman elasticities, proviing new results even in the symmetric firm literature. In recent influential work, Arkolakis et al. (212a,b) show richer moels of firm heterogeneity an variable markups are neee for these microfounations to affect welfare gains from trae. In line with this insight, we generalize the eman structure an show that firm heterogeneity an variable 7 While VES utility oes not inclue the quaratic utility of Melitz an Ottaviano (28) an the translog utility of Feenstra (23), Zheloboko et al. show it captures the qualitative features of market outcomes uner these forms of non-aitive utility. 8 We consier this to be the proof of a folk theorem which has been in the air. Matsuyama (1995) an Bilbiie, Ghironi an Melitz (26) fin the market equilibrium with symmetric firms is socially optimal only when preferences are CES. Epifani an Gancia (211) generalize this to multiple sectors while Eckel (28) examines efficiency when firms affect the price inex. 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 prouctivity increases are offset by reuctions in variety. We provie an analytical treatment to show the market equilibrium implements the unconstraine social optimum. Helpman, Itskhoki an Reing (211) consier the constraine social optimum. Their approach iffers because the homogeneous goo fixes the marginal utility of income. Our work is closest to Feenstra an Kee who focus on the CES case. Consiering 48 countries exporting to the US in 198-2, they also estimate that rise in export variety accounts for an average 3.3 per cent rise in prouctivity an GDP for the exporting country. 5

markups matter for both welfare gains an allocational efficiency. 9 As in Melitz an Reing (213), we fin that the cost istribution matters for the magnitue of welfare gains from integration. Builing on Bernar, Eaton, Jensen an Kortum (23), e Blas an Russ (21) also examine the role of variable markups in welfare gains but o not consier efficiency. We follow the irection of Tybout (23) an Katayama, Lu an Tybout (29) who suggest the nee to map prouctivity gains to welfare an optimal policies. The paper is organize as follows. Section 2 recaps the stanar monopolistic competition framework with firm heterogeneity. Section 3 contrasts efficiency of CES eman with inefficiency of VES eman an Section 4 characterizes the istortions in resource allocation. Section 5 examines welfare gains from integration, eriving a limit result for large markets. Section 6 conclues. 2 Moel We aopt the VES eman structure of Dixit an Stiglitz within the heterogeneous firm framework of Melitz. Monopolistic competition moels with heterogeneous firms iffer from earlier 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. This Section lays out the moel an recaps the implications of asymmetric costs for consumers, firms an equilibrium outcomes. 2.1 Consumers We explain the VES eman structure an then iscuss consumer eman. The exposition for consumer eman closely follows Zheloboko et al. (212) which works with a similar setting an buils on work by Vives (21). An economy consists of a mass L of ientical workers, each enowe with one unit of labor an facing a wage rate w normalize to one. Workers have ientical preferences for a ifferentiate goo. The ifferentiate goo is mae available as a continuum N of horizontally ifferentiate varieties inexe by i [,N]. Given prices p i for the varieties, every worker chooses quantity q i for each of the varieties to maximize her utility subject to her buget constraint. 9 For instance, linear VES eman an Pareto cost raws fit the gravity moel, but firm heterogeneity still matters for market efficiency. More generally, VES eman is not neste in the Arkolakis et al. moels an oes not satisfy a log-linear relation between import shares an welfare gains, as illustrate in the Online Appenix. 6

Preferences over ifferentiate goos take the general VES form: U(q) N u(q i )i (1) where u( ) is thrice continuously ifferentiable, strictly increasing an strictly concave on (, ), an u() is normalize to zero. The concavity of u ensures consumers love variety an prefer to sprea their consumption over all available varieties. Here u(q i ) enotes utility from an iniviual variety i. Uner CES preferences, u(q i ) = q ρ i an Krugman (198). 1 as specifie in Dixit-Stiglitz For each variety i, VES preferences inuce an inverse eman p(q i ) = u (q i )/δ where δ is the 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. 11 Multiplying both sies of the inverse eman by q i an aggregating over all i, the buget multiplier is δ = N u (q i ) q i i. The consumer buget multiplier δ will act as a eman shifter an the inverse eman will inherit the properties of the marginal utility u (q i ). particular, the inverse eman elasticity ln p i / lnq i equals the elasticity of marginal utility µ(q i ) q i u (q i )/u (q i ), which enables us to characterize market allocations in terms of eman primitives. Uner CES preferences, the elasticity of marginal utility is constant an the inverse eman elasticity oes not respon to consumption ( ln p i / lnq i = µ(q i ) = 1 ρ). When µ (q i ) >, the inverse eman of a variety becomes more elastic as its consumption increases. The opposite hols for µ (q i ) <, where the eman for a variety becomes less elastic as its price rises. The inverse eman elasticity summarizes market eman, an will enable a characterization of market outcomes. A policymaker maximizes utility, an is not concerne with market prices. Therefore, we efine the elasticity of utility ε(q i ) u (q i )q i /u(q i ), which will enable a characterization of optimal allocations. The elasticity of utility can be unerstoo as follows. The real expeniture on variety i is u (q i )q i an the contribution of variety i to welfare is u(q i ). Therefore, 1 ε(q i ) = (u(q i ) u (q i )q i )/u(q i ) enotes the proportion of social benefits not capture by real expeniture when introucing variety i. Uner CES preferences, the elasticity of utility is constant an 1 ε(q i ) = 1 ρ. For (1 ε(q i )) <, the welfare contribution of a 1 The specific CES form in Melitz is U(q) ( (q ρ i i) 1/ρ but the normalization of the exponent 1/ρ in Equation (1) will not play a role in allocation ecisions. 11 Aitional assumptions to guarantee existence an uniqueness of the market equilibrium are in a separate note available online. Utility functions not satisfying inaa conitions are permissible but may require parametric restrictions to ensure existence. In 7

variety relative to expeniture is more elastic when its consumption is low. For (1 ε(q i )) >, the welfare contribution of a variety is more sensitive when more of it is consume. We iscuss the interpretation of these elasticities in more etail. 2.1.1 Interpretation of Elasticities Zheloboko et al. (212) show that the elasticity of marginal utility µ(q i ) can also be interprete in terms of substitution across varieties. For symmetric consumption levels (q i = q), this elasticity equals the inverse of the elasticity of substitution between any two varieties. For µ (q) >, higher consumption per variety or fewer varieties for a given total quantity, inuces a lower elasticity of substitution between varieties. Consumers perceive varieties as being less ifferentiate when they consume more, but this relationship oes not carry over to heterogeneous consumption levels. For symmetric consumption levels, Vives (21) points out that 1 ε(q) is the egree of preference for variety as it measures the proportion of the utility gain from aing a variety, holing quantity per firm fixe. Extrapolating to heterogeneous varieties, 1 ε(q) measures the relative contribution of variety to total utility from aing another variety, holing the average quantity level q an the ispersion of quantities across varieties fixe. If 1 ε(q) =, there is no preference for variety, an the composition of consumption is irrelevant for welfare. If 1 ε(q) = 1, utility epens only on variety, not quantity per variety. For (1 ε(q)) >, consumers have a higher preference for variety when they consume more per variety. This can be explaine in a framework following Kuhn an Vives (1999). Utility can be re-written to explicitly account for taste for variety, U NqV (q) for q such that q i V (q i )i = qv (q) (Q/N)V (Q/N) where Q is total quantity. Holing average quantity q fixe, aing a variety increases utility by U/N = qv (q). This gain consists of a pure variety effect on welfare, holing total quantity fixe: U/N = QV (q) ( Q/N 2). Utility also rises ue to an increase in total quantity, holing variety fixe: U/Q = [V (Q/N) + QV (Q/N)/N](Q/N). Since the total quantity increase is Q/N = q, the output effect is given by U/Q =V(q)q[1 +V (q)q/v (q)]. The two effects a up to give the total effect of aing a variety at constant quantity per firm. The ratio of the variety effect to the total utility gain from aing a variety equals 1 ε(q) = V (q)q/v (q) at the average quantity q. 2.2 Firms There is a continuum of firms which may enter the market for ifferentiate goos, by paying a sunk entry cost of f e >. The mass of entering firms is enote by M e. Firms are monop- 8

olistically competitive an each firm prouces a single unique variety. A firm faces an inverse eman of p(q i ) = u (q i )/δ for variety i. It acts as a monopolist of its unique variety but takes aggregate eman conitions δ as given. Upon entry, each firm receives a unit cost c rawn from a istribution G with continuously ifferentiable pf g. Each variety can therefore be inexe by the unit cost c of its proucer. After entry, shoul a firm prouce, it incurs a fixe cost of prouction f >. Profit maximization implies firms prouce if they can earn non-negative profits net of the fixe costs of prouction. A firm with cost raw c chooses its quantity q(c) to max q(c) [p(q(c)) c]q(c)l an q(c) > if π(c) = max q(c) [p(q(c)) c]q(c)l f >. To ensure the firm s quantity FOC is optimal, we assume marginal revenue is strictly ecreasing in quantity an the elasticity of marginal utility µ(q) = qu (q)/u (q) is less than one. A firm chooses its quantity to equate marginal revenue an marginal cost (p + q u (q)/δ = c), an concavity of the firm problem ensures low cost firms supply higher quantities an charge lower prices. The markup charge by a firm with cost raw c is (p(c) c)/p(c) = q(c)u (q(c))/u (q(c)). This shows that the elasticity of marginal utility µ(q) summarizes the markup: µ(q(c)) = q(c)u (q(c))/u (q(c)) = (p(c) c)/p(c). When µ (q) >, low cost firms supply higher quantities at higher markups. 2.3 Market Equilibrium Profits fall with unit cost c, an the cutoff cost level of firms that are inifferent between proucing an exiting from the market is enote by c. The cutoff cost c is fixe by the zero profit conition, π(c ) =. Firms with cost raws higher than the cutoff level earn negative profits an o not prouce. The mass of proucing firms in equilibrium is therefore M = M e G(c ). In summary, each firm faces a two stage problem: in the secon stage it maximizes profits given a known cost raw, an in the first stage it ecies whether to enter given the expecte profits in the secon stage. To stuy the Chamberlinian traeoff between quantity an variety, we maintain the stanar free entry conition impose in monopolistic competition moels. Specifically, ex ante average profit net of sunk entry costs must be zero, c π(c)g = f e. This free entry conition along with the consumer s buget constraint ensures that the resources use by firms equal the total resources in the economy, L = M e [ c (cq(c)l + f )G + f e]. 9

2.4 Social Optimum To assess the efficiency of resource allocation in the market equilibrium, we now escribe the policymaker s optimal allocation. A policymaker maximizes iniviual welfare U as given in Equation (1) by choosing the mass of entrants, quantities an types of firms that prouce. 12 The policymaker can choose any allocation of resources that oes not excee the total resources in the economy. However, she faces the same entry process as for the market: a sunk entry cost f e must be pai to get a unit cost raw from G(c). Fixe costs of prouction imply that the policymaker chooses zero quantities for varieties above a cost threshol. Therefore, all optimal allocation ecisions can be summarize by quantity q(c), potential variety M e an a prouctivity cutoff c. The policymaker chooses q(c), c an M e to c { c max M e u(q(c))g where L M e [cq(c)l + f ]G + f e }. Our approach for arriving at the optimal allocation is to think of optimal quantities q opt (c) as being etermine implicitly by c an M e so that per capita welfare can be written as c U = M e u(q opt (c))g. (2) Optimal quantities ensure marginal utility equals social marginal cost of a variety, u (q opt (c)) = λc where λ is the resource multiplier for fixe c an M e. Note that q(c) is a function of λc that maximizes U an epens on both the istribution of costs an aggregate entry ecisions. Fixing the optimal λ an showing sufficiency of such caniate quantity functions is hanle using variational calculus techniques in the Appenix. After solving for each q opt conitional on c an M e, Equation (2) can be maximize in c an M e. Of course, substantial work is involve in showing sufficiency, but we relegate this to the Appenix. The next two Sections compare the market an optimal allocations in this framework. 3 Market Efficiency Having escribe an economy consisting of heterogeneous, imperfectly competitive firms, we now examine efficiency of market allocations. Outsie of cases in which imperfect competition leas to competitive outcomes with zero profits, one woul expect the coexistence of positive markups an positive profits to inicate inefficiency through loss of consumer surplus. Nonethe- 12 Free entry implies zero expecte profits, so the focus is on consumer welfare. 1

less, this Section shows that CES eman uner firm heterogeneity exhibits positive markups an profits for surviving firms, yet it is allocationally efficient. However, this is a special case. Private incentives are not aligne with optimal prouction patterns for any VES eman structure except CES. Following Dixit an Stiglitz, we start with efficiency uner CES eman an then explain market inefficiency uner VES eman. We then iscuss the externalities arising in the market an the reasons for efficiency uner CES eman. 3.1 Market an Optimal Allocations Proposition 1 shows the market provies the first-best quantity, variety an prouctivity. The proof of Proposition 1 iffers from symmetric firm monopolistic competition results because optimal quantity varies non-trivially with unit cost, variety an cutoff prouctivity. The main fining is that laissez faire inustrial policy is optimal uner CES eman. Proposition 1. Every market equilibrium of a CES economy is socially optimal. Proposition 1 shows that the market allocation is optimal uner CES eman an we now contrast the market allocation across symmetric an heterogeneous firms. When firms are symmetric, resource allocation reflects average cost pricing. Firms charge positive markups which result in lower quantities than those implie by marginal cost pricing. Even though firms o not charge marginal costs, their market price (an hence marginal utility) is proportional to marginal cost because markups are constant. This ensures proportionate reuctions in quantity from the level that woul be observe uner marginal cost pricing (Baumol an Brafor 197). These reuce quantity levels are efficient because the marginal utility of income ajusts to ensure that the ratio of marginal utility to marginal cost of a variety coincies with the social value of labor (u (q)/c = δ/(1 µ) = λ). Free entry equates price to average cost, an the markup exactly finances the fixe cost of an aitional variety. The market therefore inuces an efficient allocation. With heterogeneous firms, markups continue to be constant an marginal utility is proportional to marginal cost. One might infer enforcing average cost pricing across ifferent firms woul inuce an efficient allocation, as in symmetric firm moels. But average cost pricing is too low to compensate firms because it will not cover ex ante entry costs. The market ensures prices above average costs at a level that internalizes the losses face by exiting firms. Entry is at optimal levels that fix p(c ), thereby fixing absolute prices to optimal levels. Post entry, surviving firms charge prices higher than average costs (p(c) [cq(c) + f /L]/q(c)) an the markups exactly compensate them for the possibility of paying f e to enter an then being too unprouctive to survive. 11

The way in which CES preferences cause firms to optimally internalize aggregate economic conitions can be mae clear through a variety-specific explanation. The elasticity of utility ε(q) u (q) q/u(q) can be use to efine a social markup 1 ε(q). We term 1 ε(q) the social markup because it enotes the utility from consumption of a variety net of its resource cost. At the optimal allocation, the multiplier λ encapsulates the social value of labor an the social surplus from a variety is u(q) λcq. At the optimal quantity, u (q(c)) = λc an the social markup is 1 ε(q) =1 u (q) q/u(q) =(u(q) λcq)/u(q). (Social Markup) For any optimal allocation, the quantity that maximizes social benefit from variety c solves max q (u(q)/λ cq)l f = 1 ε(qopt (c)) ε(q opt cq opt (c)l f. (c)) In contrast, the incentives that firms face in the market are base on the private markup µ(q) = (p(q) c)/p(q), an firms solve: max q (p(q)q cq)l f = µ(qmkt (c)) 1 µ(q mkt (c)) cqmkt (c)l f. Since ε an µ epen only on the primitive u(q), we can examine what eman structures woul make the economy optimally select firms. Clearly, if private markups µ(q) coincie with social markups 1 ε(q), profits will be the same at every unit cost. Examining CES eman, we see precisely that µ(q) = 1 ε(q) for all q. Thus, CES eman incentivizes exactly the right firms to prouce. Since the optimal set of firms prouce uner CES eman, an private an social profits are the same, market entry will also be optimal. As entry M e an the cost cutoff c are optimal, the competition between firms aligns the buget multiplier δ to ensure optimal quantities. Efficiency of the market equilibrium in our framework is tie to CES eman. To highlight this, we consier the general class of VES eman specifie in Equation (1). Direct 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. Proposition 2. socially optimal is that u is CES. Proof. Online Appenix. Uner VES eman, a necessary conition for the market equilibrium to be 12

Uner general VES eman, market allocations are not efficient an o not maximize iniviual welfare. Proposition 3 shows that the market instea maximizes aggregate real revenue (M e u (q(c)) q(c)g) generate in the economy. Proposition 3. Uner VES eman, the market maximizes aggregate real revenue. Proposition 3 shows ecentralize profit maximization coincies with centralize revenue maximization. While firms have no iniviual influence over entry M e or consumers marginal utility of income δ, they o have ecentralize control over quantities q(c) an the ecision whether to prouce at all. A shaow value of labor ˆδ from a policymaker who wishes to maximize real revenue acts[ exactly like δ, since firms solve max q L[u (q)/δ c]q while the policymaker solves max q L u (q) ˆδc ] q an clearly this results in the same (iniviual) quantity an prouction ecisions at δ = ˆδ. Therefore ecentralize profit maximization coincies with centralize revenue maximization if the marginal utility of income an shaow value of labor happen to coincie, conitional on equivalent entry. That δ = ˆδ happens in the marketplace comes not from firms (who take δ as exogenous), but from consumers who internalize aggregate firm ecisions an ientify their marginal utility of income with the real value of their labor. That entry in the market matches entry chosen by a revenue maximizing policymaker comes from the ex ante ecisions of firms which aggregates market outcomes through rational expectations. This result shows that the market an optimal allocations are generally not aligne uner VES eman. The market an optimal allocations are solutions to: c { c } max M e u (q(c)) q(c)g where L M e [cq(c)l + f ]G + f e c max M e u(q(c))g { c } where L M e [cq(c)l + f ]G + f e Market Optimum For CES eman, u(q) = q ρ while u (q)q = ρq ρ implying revenue maximization is perfectly aligne with welfare maximization. The CES result is therefore a limiting case of allocations uner VES eman. Outsie of CES, quantities prouce by firms are too low or too high an in general equilibrium, this implies prouctivity of operating firms is also too low or too high. Market quantity, variety an prouctivity reflect istortions of imperfect competition. To unerstan these istortions, the next sub-section explains the externalities arising in the market an the subsequent Section examines the nature of misallocations. 13

3.2 Unerstaning Externalities Although straightforwar, the variety-level explanation of comparing private an social markups obscures the externalities at play in firm ecisions. The market results in revenue-maximizing allocations that reflect externalities arising from private incentives. This sub-section iscusses market externalities an the reasons for CES efficiency when firms iffer in prouctivity. Uner symmetric firms, Mankiw an Whinston (1986) show that there are two market externalities. First, firms cannot capture the entire surplus generate by their prouction, an this lack of appropriability iscourages firm entry. This is summarize by the elasticity of utility which measures the proportion of utility from a variety not capture by the real revenues (1 ε(q) = 1 u (q)q/u(q)). Secon, firms o not internalize the ownwar pressure impose by their prouction on prices of other firms, an this business stealing effect tens to encourage too much entry. This externality is summarize by the inverse eman elasticity µ(q). Uner CES eman, the appropriability externality exactly counteracts the business stealing externality an there is no incentive to eviate from optimal entry (Grossman an Helpman 1993). Our setting iffers from stanar symmetric firm moels in two respects. First, firms are heterogeneous so the market must ensure an optimal selection of firms for prouction an the optimal istribution of quantities across these firms. Secon, wages are etermine enogenously an the marginal utility of income is not fixe by an outsie goo (as is typical in symmetric firm moels). We therefore generalize the efficiency analysis from Vives (21) to heterogeneous firms an enogenous marginal utility of income. To unerstan the potential sources of inefficiency, we now examine how a ecline in firm entry affects the real expeniture neee to maintain welfare. We are intereste in the trae-off between variety N = M e G(c ) an quantities q(c), formulate as a uniform scaling of quantities s(n) that maintains consumer welfare when variety changes for a given istribution of proucers. To monetize this trae-off, we efine an expeniture function e ( p(c,n),n,u mkt) at the market level of welfare, U mkt, an prices p(c,n) that support a uniform scaling of quantities s(n) as above. As real incomes are δ = e, this necessitates p(c,n) = u (s(n)q(c))/δ (N) 14

an consequently at market prices (where s(n) = 1), the change in real expeniture is c lne/ lnn = 1 + ln = 1 + s (N)N u (s(n)q(c))s(n)q(c)g(c)/ lnn. c u (q(c))q(c)[1 µ (q(c))]g(c)/(δ/n) which consists of the irect effect of entry on expeniture through a change in variety an the inirect effects through quantity an price per firm. In particular, s (N) = 1/Nε where ε c u (q)qg/ c u(q)g.13 Letting µ c u (q)qµ (q)g/ c u (q)qg, the change in real expeniture is therefore lne/ lnn = [1 ε µ]/ε. When firm are symmetric, lne/ lnn = [1 ε µ]/ε for ε an µ evaluate at the market quantity. This highlights two externalities arising in the market. First, firms are unable to appropriate the full consumer surplus through revenues as measure by (1 ε). Lower entry requires higher real expeniture to maintain welfare because consumers have a taste for variety. Secon, firms o not account for the effect of their sales on the eman for other firms proucts. This business stealing externality is measure by µ. Lower entry reuces business stealing an requires less real expeniture to maintain welfare. Uner symmetric firms an CES eman, the market allocation is efficient because the appropriability externality balances the business stealing externality (1 ε µ = ), leaing to optimal entry an prouction. When firms iffer in prouctivity, the change in real expeniture neee to maintain welfare upon entry is lne lnn = 1 ε (1 ε) } {{ } Appropriability + µ }{{} Business Stealing u (µ(q) µ) (q)q u (q)qg G } {{ }. Business Shifting c + for µ c u(q) µ (q)g/ c u(q)g. As earlier, the first an secon terms measure the appropriability externality an the business stealing externality. With heterogeneous firms, these two externalities are represente by the average across all varieties. The thir term represents the business shifting effect of entry. It consists of the revenue-weighte average of the eviation in business stealing across firms (µ µ) an summarizes whose business suffers upon entry. 13 This is because the change in welfare ( = 1 + ln c u(s(n)q(c))g(c)/ lnn) gives = 1 + s (N)N c u (q(c))q(c)g(c)/ c u(q(c))g(c). 15

Uner CES eman or symmetric firms, all firms charge the same markup an business shifting oes not arise. More generally, business shifting arises when firms iffer in prouctivity. This leas us to an examination of the istribution of misallocations inuce by the market. 4 Market Distortions an Variable Elasticities Having ientifie externalities, we characterize how the market allocates resources relative to the social optimum. In their symmetric firm setting, Dixit an Stiglitz (1977) examine when the market uner-prouces an over-prouces. They fin that the bias in market allocation is etermine by how the elasticity of utility varies with quantity (1 ε(q)). When firms iffer in prouctivity, we show that the variation in the inverse eman elasticity µ (q) also matters for the bias in market allocations. We start with a iscussion of markup an quantity patterns an then iscuss how these eman patterns etermine misallocations in symmetric firm moels. Uner firm heterogeneity, ifferent eman patterns inuce ifferent misallocations. We first summarize the misallocations by eman patterns an then iscuss empirical evience for ifferent eman elasticities. Finally, we consier extensions of the basic framework to unerstan the robustness of the misallocations. 4.1 Markup an Quantity Patterns We will show that the relationship between markups an quantity characterizes istortions. It is therefore useful to efine preferences by the signs of µ (q) an (1 ε(q)). When µ (q) >, private markups are positively correlate with quantity. This is the case stuie by Krugman (1979): firms are able to charge higher markups when they sell higher quantities. Our regularity conitions guarantee low cost firms prouce higher quantities (Section 3.1), so low cost firms have both high q an high markups. When µ (q) <, small boutique firms charge higher markups. Similarly, the sign of (1 ε(q)) etermines how social markups vary with quantity. For CES eman, private an social markups are constant (µ =, (1 ε) = ). To bring out the istinction in istortions for ifferent markup patterns, Definition 1 below characterizes preferences as aligne when private an social markups move in the same irection an misaligne when they move in ifferent irections. Definition 1. Private an social incentives are aligne when µ an (1 ε) have the same sign. Conversely, incentives are misaligne when µ an (1 ε) have ifferent signs. 16

To fix ieas, Table 1 summarizes µ an (1 ε) for commonly use utility functions. Among the forms of u(q) consiere are expo-power, 14 HARA an generalize CES (propose by Dixit an Stiglitz). 15 Table 1: Private an Social Markups for Common Utility Forms µ > (1 ε) < (1 ε) > Generalize CES (α > ): (q + α) ρ CARA, Quaratic HARA (α > ): (q/(1 ρ)+α)ρ α ρ ρ/(1 ρ) Expo-power (α > ): 1 exp( αq1 ρ ) α µ < HARA (α < ): (q/(1 ρ)+α)ρ α ρ ρ/(1 ρ) Expo-power (α < ): 1 exp( αq1 ρ ) α Generalize CES (α < ): (q + α) ρ 4.2 Misallocations uner Symmetric Firms Dixit an Stiglitz examine how the market allocation eviates from the optimal allocation. They fin that the elasticity of utility etermines the bias in prouction an entry. We state their result below an iscuss how prouctivity ifferences affect istortions subsequently. Proposition 4. Uner symmetric firms, the pattern of misallocation is as follows: 1. If (1 ε) <, market quantities are too high an market entry is too low. 2. If (1 ε) >, market quantities are too low an market entry is too high. Proof. Dixit an Stiglitz (1977). Variation in the elasticity of utility summarizes the ifference between the lack of appropriability an business stealing because ε q/ε = 1 ε µ. When (1 ε) >, the business stealing externality outweighs the appropriability externality. Firms ignore the negative effect of entry on prices an the market provies too much variety. When (1 ε) <, the business stealing externality is smaller an the market provies too little variety. Uner symmetric firms, the business shifting effect is irrelevant an the variation in firm markups µ (q) oes not affect the bias in market allocations. The symmetric firm case simplifies the analysis of misallocations as the traeoff is between two ecisions: quantity an entry. In contrast, etermining misallocations across heterogeneous 14 The expo-power utility was propose by Saha (1993) an recently use by Holt an Laury (22) an Post, Van en Assem, Baltussen an Thaler (28) to moel risk aversion empirically. 15 The parameter restrictions are ρ (,1), α > q/(ρ 1) for HARA an α > q for Generalize CES. 17

firms is less obvious because quantities vary by firm prouctivity, an this variation epens on entry an selection. Further, the business shifting effect epens on the istribution of markups an can have ifferent signs epening on the variation in private an social markups. The next sub-section explains these misallocations for heterogeneous firms. Examining misallocations across the entire istribution of firms reveals two substantive results. First, as we might expect, the misallocation of resources across firms iffers by prouctivity. An interesting fining is that this heterogeneity in misallocation can be severe enough that some firms over-prouce while others uner-prouce. For example, as we will show below, when µ > an (1 ε) >, excess prouction by small firms imposes an externality on large firms. Large firms prouce below their optimal scale an too many small firms enter the market. In this case, the market iverts resources away from large firms towars small firms. Secon, accounting for firm heterogeneity shows that both the elasticity of utility an the inverse eman elasticity etermine resource misallocations. When firms are symmetric, only the elasticity of utility etermines misallocations an the inverse eman elasticity oes not matter (Proposition 4). The presence of firm heterogeneity funamentally changes the qualitative analysis. When markups vary, firms with ifferent prouctivity levels charge ifferent markups. This creates a new externality an affects the quantity an entry ecisions. Therefore, firm heterogeneity an variable markups alter the stanar policy rules for correcting misallocation of resources. 4.3 Quantity, Prouctivity an Entry Distortions We now characterize the misallocations by eman characteristics. The istortions in quantity, prouctivity an entry are iscusse in turn. The sign of the bias in market outcomes epens on both µ an (1 ε). 4.3.1 Quantity Bias Quantity istortions across firms epen on whether private an social incentives are aligne or misaligne. We show that when private an social incentives are misaligne, market quantities q mkt (c) are uniformly too high or low relative to optimal quantities q opt (c). In contrast, when private an social markups are aligne, whether firms over-prouce or uner-prouce epens on their prouctivity. The relationship between market an optimal quantities is fixe by FOCs for revenue maximization an welfare maximization. The market chooses [1 µ(q mkt )]u (q mkt ) = δc, while the optimal quantity is given by u (q opt ) = λc. Therefore, the relationship of market an optimal 18

quantities is [ ( Firm MB 1 µ q mkt )] Social MB = u ( q mkt) u (q opt = δc Firm MC = ) λc Social MC. The ratio of real revenue to welfare δ/λ epens on entry, prouctivity an the istribution of quantities. It summarizes the inustry-wie istortions through the lack of appropriability an business stealing across all varieties. The variety-specific externality arises from business shifting which is capture by µ(q mkt (c)). When incentives are misaligne, market an optimal quantities are too high or too low across all varieties an the irection of this bias is similar to the symmetric firm case. In particular, when (1 ε) < < µ, the market over-rewars firms proucing higher quantities an all firms over-prouce q mkt (c) > q opt (c). When (1 ε) > > µ, market prouction is too low (q mkt (c) < q opt (c)). Therefore, firms are either over-reware for proucing q or unerreware, an quantities are istorte in the same irection for all firms. When incentives are aligne, the gap between the market an social cost of resources (δ an λ) is small enough that quantities are not uniformly istorte across all firms. The business shifting effect can ominate the average appropriability an business stealing effects, leaing to ifferences in prouction bias across firms. Quantities are equal for some c where 1 µ ( q mkt (c ) ) = δ/λ. For all other varieties, quantities are still istorte. When µ,(1 ε) >, market prouction is biase towars high cost firms (q mkt < q opt for low c an q mkt > q opt for high c). The market shifts business away from low cost firms an over-rewars high cost firms. When µ,(1 ε) <, the bias is reverse an low cost firms over-prouce. Therefore, when private an social markups are aligne, whether the market uner or over prouces epens on a firm s costs. Proposition 5 summarizes the bias in market quantities. Proposition 5. When preferences are misaligne, q mkt (c) an q opt (c) never cross: 1. If (1 ε) < < µ, market quantities are too high: q mkt (c) > q opt (c). 2. If (1 ε) > > µ, market quantities are too low: q mkt (c) < q opt (c). In contrast, when preferences are aligne an inf q ε (q) >, q mkt (c) an q opt (c) have a unique crossing c (perhaps beyon market an optimal cost cutoffs). 3. If (1 ε) > an µ >, q mkt (c) < q opt (c) for c < c an q mkt (c) > q opt (c) for c > c. 4. If (1 ε) < an µ <, q mkt (c) > q opt (c) for c < c an q mkt (c) < q opt (c) for c > c. 19

4.3.2 Prouctivity Bias The istortion in firm selection is etermine by the relation between the elasticity of utility an quantity. Proposition 6 shows that market prouctivity is either too low or high, epening on whether social markups are increasing or ecreasing. We use this result now to epict the pattern of misallocation graphically, an iscuss the result further below. Proposition 6. Market prouctivity is too low or high, as follows: 1. If (1 ε) >, market prouctivity is too low: c mkt > c opt 2. If (1 ε) <, market prouctivity is too high: c mkt. < c opt. Propositions 5 an 6 show the market misallocates resources across firms, an variable eman elasticities characterize the pattern of these misallocations. Figure 1 illustrates the bias in firm-level prouction for aligne an misaligne preferences when private markups increase in quantity. For ease of reference, Table 2 summarizes the misallocations by eman characteristics. 16 A iscussion of the externalities at play in the results follow in the next sub-section. Figure 1: Bias in Firm Prouction by Preferences (a) Misaligne: µ > > (1 ε) (b) Aligne: µ > an (1 ε) > 16 Table 2 characterizes the qualitative role of eman elasticities in misallocations. Using a quantitative measure of istortions reiterates their importance. The loss from misallocations can be summarize by the ifference between social an market profits, evaluate at optimal allocations. This measure consists of the ifference between average social markup an average private markup (1 ε µ), an the covariance between social an private markups Cov(1 ε, µ). The covariance component shows that the istribution of markups matters for quantifying istortions, except when firms are symmetric or markups are constant (leaing to zero covariance). 2

Table 2: Distortions by Deman Characteristics µ > (1 ε) < (1 ε) > Quantities Too High: Quantities High-Cost Skewe: q mkt (c) > q opt (c) q mkt (c) < q opt (c) for c < c q mkt (c) > q opt (c) for c > c Prouctivity Too High: c mkt < c opt Prouctivity Too Low: c mkt > c opt µ < Quantities Low-Cost Skewe: q mkt (c) > q opt (c) for c < c q mkt (c) < q opt (c) for c > c Quantities Too Low: q mkt (c) < q opt (c) Prouctivity Too High: c mkt < c opt Prouctivity Too Low: c mkt > c opt 4.3.3 Unerstaning Externalities an Prouctivity While Proposition 6 follows from a general equilibrium analysis, the ecision to introuce a marginal variety can be intuitively explaine as follows. Uner increasing social markups (1 ε) >, the lack of appropriability of a marginal variety is lower than its business stealing effect. This encourages prouction of the marginal variety an the cost cutoff in the market is too high. Although the marginal variety steals business an shifts business across varieties, its impact is small an the bias in the cost cutoff is etermine by the elasticity of utility. We now illustrate this reasoning in a similar fashion as for entry in Section 3.1. We are intereste in the trae-off between prouctivity c an quantities q(c) for a uniform scaling of quantities that maintains consumer welfare when c changes, holing M e fixe. At market prices, evaluating the change in real expeniture to maintain consumer welfare upon a rise in the cost cutoff yiels c lne/ lnc = ln u (s(n)q(c))s(n)q(c)g(c)/ lnc = c g(c ) [ M e s (N)[1 µ] + u(q(c ))ε(q(c )) ] / ε c u(q(c))g(c). 21

Defining x x(q(c )), the change in real expeniture is 17 lne = u c g(c ) lnc ε u(q)g } (1 {{ ε ) } Appropriability + µ }{{} Business Stealing u (q)q (µ(q) µ ) u (q)qg G } {{ } Business Shifting As earlier, the change in real expeniture highlights the lack of appropriability, business stealing an business shifting. The marginal firm is unable to appropriate the full surplus it generates, an this appropriability externality is measure by (1 ε(q(c )). The marginal variety steals business from other firms (µ(q(c ))) an shifts business across them (µ(q) µ(q(c ))). Uner CES eman, the business stealing externality exactly outweighs the appropriability externality, an there is no business shifting. More generally, the externalities iffer an their net effect on the change in real expeniture can be signe. The change in real expeniture neee to maintain welfare upon a rise in the cost cutoff is lne/ lnc = ( u c g(c )/ + ) ug [ (1 ε µ) + (ε ε)]/ ε. The sign of the first term in square brackets is the sign of (1 ε). The secon term also takes the sign of (1 ε) because the marginal firm makes the lowest quantity. Although the marginal firm shifts business, this impact is smaller an the change in real expeniture neee to maintain welfare is etermine by the elasticity of utility. This analysis also highlights that a comparison of the mass of entrants in the market an the optimum is generally har to make. The change in real income neee to maintain welfare upon a fall in entry is lne/ lnn = (1 ε µ) + u (µ(q(c)) µ) (q(c)))q(c) u (q(c)))q(c)gg. Unlike the business shifting effect of the marginal variety, business shifting from entry nee not be ominate by the net effect from the appropriability externality an the business stealing externality. The first term in ln e/ ln N summarizes the traeoff between the appropriability externality an the business stealing effect an takes the sign of (1 ε). The secon term summarizes the business shifting effect an epens on the sign of µ. Consier the case with aligne preferences an increasing markups. Then the first term is positive an the secon term is negative. The business shifting effect ampens the other two externalities an lower real expeniture is neee to maintain welfare upon a fall in entry. 18 As the externalities move in opposite irec- 17 At the market allocation, M e s (N) = u(q(c ))/ε c u(q(c))g(c) because the change in welfare is = 1 + s (N)N c u (q(c))q(c)g(c)/ c u(q(c))g(c). 18 This is consistent with early insights from Vickers (1995) an Vives (21) arguing that an increase in entry hurts high prouctivity firms more than low prouctivity firms, an cost asymmetries lea to an improvement in 22

tions, the bias in potential entry M e an available variety M e G(c ) cannot be etermine without further information on eman an cost parameters. The net effect of the three externalities an hence the bias in potential entry epens on the relative magnitues of eman an cost parameters incluing the cost istribution G(c). 19 While firm heterogeneity makes entry istortions epenent on the cost istribution, the bias in quantity an prouctivity can be unambiguously inferre from the eman-sie elasticities. In the remainer of this Section, we first examine the robustness of these finings uner alternative moeling assumptions an then iscuss empirical work on estimating the eman-sie elasticities. 4.4 Extensions of the Basic Framework As many ifferent fiels of economics (such as macroeconomics an urban economics) use monopolistically competitive moels, we exten our basic framework to ifferent moelling assumptions use in these fiels to iscuss the robustness of CES efficiency an misallocations uner VES eman. Details are in an online Appenix an a summary of four key extensions is provie here. First, suppose the costs of prouction of a firm vary with its scale of prouction. To account for non-constant marginal costs, let the variable cost of prouction be ω(q) cq an assume 2ω + ω q > for all feasible quantities to ensure strict concavity of the firm problem. The market maximizes aggregate revenue uner non-constant marginal costs. As firms account for the interepenence between their unit costs an quantity, CES eman ensures the same traeoff between ifferent externalities an leas to efficient allocations (as shown in an online Appenix). Uner VES eman, the bias in quantity an prouctivity are the same as Propositions 5 an 6. Secon, let firms choose their avertising technology as in Arkolakis (21). A firm can reach a fraction n(c) of consumers by spening [ 1 (1 n(c)) 1 θ ] L ϑ f /(1 θ) units of labor for θ,ϑ [,1]. The prouction cost f therefore varies with the fraction of consumers that a firm chooses to reach. The market allocates resources efficiently uner CES eman when the costs of commencing prouction are akin to avertising costs. The market maximizes aggregate revenue an the quantity an prouctivity istortions are the same as earlier. A new insight is gaine from this richer moel of fixe costs. The market oes not choose the optimal levels of avertising. When (1 ε) >, low cost firms uner-avertise an reach too few consumers (n(c) is too low). High cost firms over-avertise an their n(c) is too high. For (1 ε) <, low the entry externality. 19 Focusing on a linear eman setting with Pareto cost raws, Nocco, Ottaviano an Salto (213) fin that the mass of firms cannot be unambiguously ranke. 23

cost firms in the market over-avertise while high cost firms uner-avertise. Thir, the efficiency an misallocation results are robust to introucing multiple sectors, conitional on the resource allocation for the sector. Following Zheloboko et al. (212), let the multi-sector utility function be U (q,q) where q is a homogeneous numeraire goo an Q M e u(q)g is the sub-utility from ifferentiate goos. Conitional on a resource allocation of (1 q ) towars ifferentiate goos, the bias in quantity an prouctivity is the same as earlier. The market allocation within the ifferentiate goos sector is efficient uner CES eman. This however oes not imply that the market an the optimum have the same level of (1 q ). For instance, in the Cobb-Douglass specification of Dixit an Stiglitz (1977), U (q,q) = q 1 γ Q γ, the optimal allocation for the homogeneous goo is q opt = 1 γ while the market allocation is q mkt = (1 γ)/(1 γ + γ ε). The markups charge in the homogeneous an the ifferentiate goos sectors iffer, leaing to inefficient market allocations. The markup for the homogeneous goo is one an the marginal utility of income is fixe by the homogeneous goo. Marginal cost pricing (p = c) therefore aligns the markups across the two sectors. Thus, Dixit an Stiglitz suggest marginal cost pricing an lumpsum entry subsiies to inuce optimal allocations across sectors. In keeping with Melitz, we consier a single sector an fin resources are optimally allocate in the market. In a single sector economy, prices are proportional to marginal costs (p = δc) but the marginal utility of income δ is no longer fixe by the homogeneous goo. Market allocations are first best as the marginal utility of income reflects the social cost of resources. 2 For completeness, we finally note that the CES eman of Melitz is also necessary for efficiency uner the CES-Benassy class of preferences. Benassy (1996) points out that the taste for variety uner Dixit-Stiglitz preferences is closely linke to the egree of market power of competitors. Taste for variety can be isentangle from market power through Benassy preferences U(M e,c,q) ν(m e ) c q(c)ρ g(c)c which value quantity an variety ifferently through ν(m e ). Following Benassy (1996) 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 uner CES-Benassy are the same as CES. However, firms o not fully internalize consumers taste for variety, leaing to suboptimal allocations. Market allocations are optimal only if taste for variety exactly equals the markup to cost ratio. 21 As the unerlying eman structure can lea to very ifferent istortions, the remainer of 2 In relate work, Behrens et al. (forthcoming) examine efficiency in a multi-sector moel with constant absolute risk aversion (CARA) preferences. 21 Helpman an Krugman (1985) an Feenstra an Kee (28) erive a GDP function for this economy, an Cole an Davies (forthcoming) highlight variety istortions by introucing existence values for variety. 24

this Section iscusses empirical evience for ifferent eman parameters. 4.5 Empirical Evience for Deman Characteristics The pattern of misallocation epens on eman-sie elasticities. A natural question is whether empirical work can ientify which case in Table 2 is relevant. Although the elasticity of utility is typically unobservable, the inverse eman elasticity (or firm markups) has been a subject of research in inustrial organization. A large empirical literature in inustrial organization shows a high level of markup ispersion across plants, an fins much larger markup ispersion within inustries rather than across inustries (example Klette 1999; Nishimura et al. 1999). The empirical relationship between markups an quantities is largely in line with increasing markups though there are inustries which show ecreasing markups. The empirical literature can be broaly classifie into papers that use price-cost margins to measure markups an those that use variants of the Hall methoology to estimate markups. 22 In a series of influential papers, Roberts an Supina (1996, 21) use physical output, revenue, an input expenitures to measure price-cost margins for a number of U.S. manufacture proucts an show the majority of proucts exhibit increasing markups. Focusing on proucts with little scope for vertical ifferentiation, they ocument a high an persistent level of price ispersion across plants for most proucts. They fin markups increase with plant size an often monotonically across quartiles of plant size for six of the thirteen proucts (polyester blen fabrics, brea, coffee, oak flooring, softwoo plywoo, newsprint). Two proucts (cotton sheeting, gasoline) show no significant change in markups with plant size. For the remaining four proucts (harwoo plywoo, vans, corrugate boxes an concrete), markups ecrease significantly with increases in plant size across the whole size istribution. One concern with the latter fining is that ecreasing markups might be riven by the ecision of large plants to operate in larger, more competitive markets, as shown by Syverson (24) for reay-mixe concrete. Stuies base on the Hall methoology largely fin a positive relationship between markups an quantity. In a careful stuy using ata on physical quantities, De Loecker et al. (212) fin markups are positively correlate with firm prouctivity of large Inian manufacturers uring 1989-23. De Loecker an Warzynski (212) estimate a positive correlation between markups an prouctivity for Slovenian manufacturing firms uring 1994-2 an Dhyne et al. (211) also fin markups are positively relate to firm prouctivity for Belgian brea manufacturers uring 1995-29. On the other han, a highly-cite stuy by Klette (1999) shows Norwegian 22 The Hall methoology estimates the price-cost markup as the slope coefficient from a regression of output growth on the share-weighte rate of input growth. A iscussion of this approach is provie in Tybout (23) an De Loecker an Golberg (213). 25

firms with higher markups ten to have lower prouctivity. 23 While the empirical literature largely fins increasing firm markups, social markups are rarely observable an early papers on monopolistic competition express a lack of consensus on how they respon to quantity. Spence (1976) suggests social markups increase with quantity while Dixit an Stiglitz propose ecreasing social markups. Vives (21) iscusses three reasons for consiering increasing private an social markups as the normal case (Chapter 6). First, for symmetric consumption, this woul imply that consumers have an increasing preference for variety an a higher inverse eman elasticity at a higher output per variety. Secon, aligne preferences are theoretically appealing because the elasticity of 1 ε equals the elasticity of µ in the limit as q approaches zero uner a relatively mil assumption. Finally, commonly-use preferences exhibit aligne preferences with increasing markups. For instance, (1 ε) > whenever µ > in the HARA class (as shown in Table 1). Moreover, the generalize CES example of Dixit an Stiglitz for ecreasing markups is not continuous at zero when it is appropriately normalize to ensure u() =. While we cannot rule out specific cases without further empirical investigation, the assumption of increasing private an social markups has appealing properties for theoretical work. 24 5 Efficiency an Market Size Having iscusse misallocations, this Section examines welfare an efficiency from integration with worl markets. The existence of gains from international trae is one of the most funamental results in economics (Costinot an Roriguez-Clare (213)). Increases in market size encourage competition, so we might expect that integration woul reuce market power an improve welfare. However, the following insight of Helpman an Krugman (1985) (pp. 179) is relevant: Unfortunately imperfect competition, even if takes as sanitize a form as monop- 23 A separate literature provies evience for increasing markups by estimating the price response to exchange rate fluctuations. The typical estimate for exchange rate pass through is less than one, which suggests increasing markups (because the pass-through rate correspons to (1 µ)/(1 µ + µ q/µ)). A iscussion of this literature is provie in Golberg an Knetter (1997) an more recently in Klenow an Malin (21). 24 While private markups can be estimate using pricing an prouction ata, istinguishing increasing an ecreasing social markups is more challenging as they are unlikely to be irectly observable. Consequently, for stanar firm level ata sets, policy inferences require more structure on eman. One approach is to use flexible eman systems that leave etermination of the four cases up to the ata. For example, the VES form u(q) = aq ρ + bq γ allows all sign combinations of ε (q) an µ (q) (Online Appenix). This form overlaps with the ajustable pass-through eman system (Bulow an Pfleierer 1983; Weyl an Fabinger 212). If sufficient ata is available, another approach is to recover ε(q) from price an quantity ata using ε(q) = p(q)q/ p(q)q or from markup an quantity ata using lnε(q)/q = q (µ(t)/t)t ln[ q exp{ s (µ(t)/t)t}s]. 26

olistic competition, oes not lea the economy to an optimum. As a result there is no guarantee that expaning the economy s opportunities, through trae or anything else, necessarily leas to a gain. We cannot prove in general that countries gain from trae in the ifferentiate proucts moel. Builing on this insight, we aress two relate questions. First, we examine when market expansion provies welfare gains. Having characterize istortions, we first show that welfare gains are relate to the eman-sie elasticities mentione earlier. Next, we examine efficiency in large markets to unerstan the potential of market expansion in eliminating istortions. We show large integrate markets can eliminate istortions, while preserving firm heterogeneity. Finally, we iscuss the role of firm heterogeneity an variable elasticities for quantitative work measuring the welfare gains from international trae. 5.1 Integration, Market Size an Efficiency We begin with the equivalence between market expansion an trae. Proposition 7 shows an economy can increase its market size by opening to trae with foreign markets. 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 1 +... + L n, echoing Krugman (1979). This result is summarize as Proposition 7. Proposition 7. Free trae between countries of sizes L 1,...,L n has the same market outcome as a unifie market of size L = L 1 +... + L n. Proof. Online Appenix an Krugman (1979). Proposition 7 implies that the market istortions etaile in Section 5 persist in integrate markets. Resource allocation in an integrate market is suboptimal, except uner CES eman. When markups vary, marginal revenues o not correspon to marginal utilities so market allocations are not aligne with efficient allocations. This is particularly important when consiering trae as a policy option, as it implies that 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. Helpman an Krugman (1985) provie sufficient conitions for welfare gains from trae. They show when prouctivity an variety o not ecline after integration, then there are gains 27

from trae. 25 In terms of primitives, we fin integration is always beneficial when preferences are aligne. This is true for any cost istribution, but requires a regularity conition for ecreasing private markups (2 + µ q/µ (1 µ) ). We summarize this in Proposition 8. Proposition 8. Market expansion increases welfare when preferences are aligne. (Provie 2 + µ q/µ (1 µ) whenever µ < ). The economic reasoning for Proposition 8 follows from similar responses of the two emansie elasticities to changes in quantity. An increase in market size increases competition an reuces per capita eman for each variety. When preferences are aligne, eman shifts alter the private an social markups in the same irection. The market therefore incentivizes firms towars the right allocation an provies higher welfare. Builing on this result, Bykaorov et al. (214) show that aligne preferences are necessary an sufficient for welfare gains from trae uner symmetric firms an variable marginal costs. The role of aligne markups in firm survival highlights how trae increases welfare. When aligne markups increase with quantity, a rise in market size forces out the least prouctive firms. Since social markups are positively correlate with quantity, the least prouctive firms also contribute relatively little to welfare an their exit is beneficial. When markups ecrease with quantity, small boutique firms contribute at a higher rate to welfare an are also able to survive after integration by charging higher markups. Integration enables the market to aapt their prouction in line with social incentives, leaing to welfare gains from trae. While integration can increase welfare, a more ambitious question is: can we ever expect trae to eliminate the istortions of imperfect competition? Following Stiglitz (1986), we stuy market an optimal outcomes as market size becomes arbitrarily large. Since small markets have insufficient competition, looking at large markets allows us to unerstan where market expansion is heae an when international trae enables markets to eventually mitigate istortions. 5.2 Efficiency in Large Markets We examine when integrating with large global markets enables a small economy to overcome its market istortions. From a theoretical perspective, we 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. 26 25 Specifically, let w enote the wage an C(w,q) = w(c + f /q) enote the average unit cost function for proucing q units of variety c. When firms are symmetric in c, trae is beneficial as long as variety oes not fall (M e M aut e ) an average unit cost of the autarky bunle is lower (C(w,q) q aut C(w,q aut ) q aut ). 26 How large markets nee to be to justify this approximation is an open quantitative question. 28

Large markets enable us to unerstan whether competition can eliminate istortions. For instance, when firms are symmetric, large markets eliminate istortions as per capita fixe costs fall to zero. This is because free entry leas to average cost pricing (p = c + f /ql), so the per capita fixe costs summarize market power. As market size grows arbitrarily large an per capita fixe costs fall to zero, markups isappear leaing to perfect competition an efficient allocations in large markets. Builing on this reasoning, we evelop the large market concept in two irections to unerstan the sources of inefficiency. First, we tie the conitions for efficiency to eman primitives, taking into account enogeneity of allocations. In the simple example above, this amounts to etermining how f /ql changes with market size uner ifferent moel primitives. Secon, we examine whether prouctivity ifferences are compatible with large markets. When firms are heterogeneous, simply knowing per capita fixe costs oes not explain the istribution of prouctivity, prices an quantity. 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 (e.g. Behrens an Murata 212). 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 uner prouctivity ifferences, we chart out a thir possibility between these two extremes: some, but not all, firms prouce. To o so, we maintain the previous regularity conitions for a market equilibrium. In orer to ai the analysis, we make three assumptions on eman at small quantities. The first assumption enables a clear istinction between the three salient outcomes in large markets. Assumption (Interior Markups). 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). The assumption of interior markups guarantees that as the quantity sol from a firm to a consumer becomes small (as happens for all positive unit cost firms), markups remain positive (µ > ) an prices remain boune (µ < 1). It also guarantees that the ae utility provie per labor unit at the optimum converges to a non-zero constant (e.g., Solow 1998, Kuhn an Vives 1999). An example of a class of utility functions satisfying interior markups is the expopower utility where u(q) = [1 exp ( αq 1 ρ) ]/α for ρ (,1). It nests CES preferences for α =. When markups are interior, there is a sharp taxonomy of what may happen to the istribution of costs, prices an total quantities (Lq(c)), as shown in Proposition 1 in the Appenix. In wors, Proposition 1 shows that when markups are interior an the cost cutoff converges, 29

one of three things must happen. 1) Only the lowest cost firms remain an prices go to zero (akin to perfect competition), while the lowest cost firms prouce infinite total quantities. 2) Post-entry, all firms prouce inepenent of cost while prices become unboune an the total quantities prouce become negligible, akin to a rentier case where firms prouce little after fixe costs are incurre. 3) The cost cutoff converges to a positive finite level, an a non-egenerate istribution of prices an total quantities persists. Although each of these possibilities might be of interest, we focus on the case when the limiting cost raw istribution exhibits heterogeneity ( lim c mkt L > ) but fixe costs still play a role in etermining which firms prouce ( lim c mkt L < ). We therefore make the following assumption, which by Proposition 1 will guarantee non-egenerate prices an total quantities: Assumption (Interior Convergence). In the large economy, the market an optimal allocations have a non-egenerate cost istribution in which some but not all entrants prouce. Uner interior markups an convergence, the economy converges to a monopolistically competitive limit istinct from the extremes of a perfectly competitive limit or a rentier 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. This is in sharp contrast to a competitive limit, in which 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, so each variety contributes at a positive rate to utility even at low levels of quantity. The monopolistically competitive limit is therefore consistent with positive markups which become more uniform with increase market size. In fact, this monopolistically competitive limit has a sharper characterization very close to the conitions which characterize a finite size market uner CES eman (incluing efficiency). We therefore refer to it as a CES limit an introuce one last regularity conition to obtain this result. Assumption (Market Ientification). Quantity ratios istinguish price ratios for small q: If κ κ then lim p(κq)/p(q) lim p( κq)/p(q). q q Market ientification guarantees prouction levels across firms can be istinguishe if the firms charge istinct prices as quantities sol become negligible. Combining these three assumptions of interior markups, convergence an ientification ensures the large economy goes to the CES limit, summarize as Proposition 9. The intuition for the role of these assumptions follows. As market size grows large, q so uner Interior Markups, (p c)/p = 3

µ (q) µ () an, finite but non-zero markups can persist in the large economy. Since profits are µ (q)/(1 µ (q)) Lcq, whether a particular firm survives in the large economy epens on how variable costs Lcq evolve with market size. Clearly, if variable costs iverge to zero for a firm with cost c, that firm must eventually exit, while if variable costs iverge to infinity, the firm must eventually enter. To arrive at the CES limit, necessarily variable costs must converge to a positive level, which requires convergence of the total quantity sol, Lq. However, since firms are embee in a heterogeneous environment where aggregate conitions impact firm behavior, the pointwise convergence of markups {µ (q(c))} is not sufficient to guarantee that total quantities {Lq(c)} are well behave in aggregate. What is sufficient is that prices {p(c)} can istinguish firms as market size grows large, thus the Market Ientification conition. 27 Proposition 9. Uner the above assumptions, as market size approaches infinity, outcomes approach the CES 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 Lq(c)/Lq(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. Proposition 9 shows that integration with large markets can push economies base on variable elasticity eman to the CES 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. While ealing with the assumptions of the market equilibrium is somewhat elicate (see Appenix), we can explain Proposition 9 intuitively 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. 28 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 istortions until market allocations are perfectly aligne with socially optimal objectives. It is somewhat remarkable that the large market outcome, which exhibits cost ifferences an remains imperfectly competitive, is socially optimal. Such persistence of imperfect competition is consistent with the observation of Samuelson (1967) that the limit may be at an 27 From a technical stanpoint, this guarantees entry is well behave, avoiing pathological sequences of potential equilibria as market size grows large. 28 The rate at which markups converge epens on c an is in any case enogenous (see Appenix). 31

irreucible positive egree of imperfection (Khan an Sun 22). Perloff an Salop (1985) also note that the markup isappears if the utility from a variety is boune, but unboune entry may not eliminate the markup when this conition is not met. We show that is precisely what happens at the CES limit. While the CES 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 variable elasticity eman. When integrate markets are small, variable markups are crucial in unerstaning istortions an aitional gains can be reape by using omestic policy in conjunction with trae policy. 5.3 Quantitative Literature on Welfare Gains from Trae A growing boy of work seeks to quantify the gains from international trae. New quantitative trae moels typically estimate welfare gains from trae uner CES eman. In an influential paper, Arkolakis et al. (212a) show that welfare in a moel with heterogeneous firms can be summarize by two sufficient statistics: the share of expeniture on omestically prouce goos an the elasticity of trae with respect to trae costs. As these sufficient statistics are common to heterogeneous an representative firm moels, welfare gains estimate from import shares an constant trae elasticities using trae ata are the same across heterogeneous an representative firm moels. However, the two moels only eliver the same estimates for welfare gains when the unerlying structural parameters for preferences an technology iffer across the moels. We use this insight of Melitz an Reing (213) to explain the relevance of our optimality results for the quantitative literature on gains from trae. Melitz an Reing fin that the heterogeneous firm moel of Melitz provies quantitatively higher gains from trae than an equivalent representative firm moel when the structural parameters are the same across these moels. As they mention, this can be unerstoo by appealing to the social optimality results for CES eman (Proposition 1). Consier initial equilibria in the heterogeneous an homogeneous firm moels that feature ientical aggregate statistics an welfare. In the homogeneous firm moel, unit cost is exogenously fixe, an hence remains unchange when the economy opens to trae. In the heterogeneous firm moel, the cost istribution changes when the economy opens to trae. In a companion note (Dhingra an Morrow 214), we show that the open economy equilibrium with trae frictions is efficient uner CES eman. Since the policymaker chooses to change the cost cutoff in an open economy, the open economy market allocation must yiel higher welfare than any other feasible allocation (where the unit cost is unchange). The allocation where the unit cost oes not change is ientical to the open economy equilibrium in the homogeneous firm moel. Therefore the open economy equilibrium in the heterogeneous firm moel must yiel higher welfare than the open econ- 32

omy equilibrium in the homogeneous firm moel. This shows that a quantitative trae moel with the same structural parameters across moels will provie higher welfare gains in a setting with firm heterogeneity. The optimality of market allocations ensures that firm heterogeneity increases the magnitue of welfare gains from trae. Departing from CES preferences, market allocations are no longer optimal. This raises the question of the role playe by firm heterogeneity in altering the magnitue of welfare gains from trae. While we o not moel trae costs, Proposition 8 shows market expansion through trae provies higher welfare gains when firms iffer in prouctivity. Uner aligne preferences an the regularity conition (2 + µ q/µ (1 µ) ), we iscuss when moels with firm heterogeneity an variable elasticities provie higher welfare gains from trae than representative firm moels. For a given change in real income, the welfare gains from trae epen on the ifferent assumptions on eman an firm costs. Welfare is U = M e u(q)g = δ/ ε where the average elasticity of utility is ε εug/ ug. An increase in market size increases real income at the rate of the average markup ( lnδ/ lnl = µ pqg/ pqg µ). The change in average elasticity can be ecompose into the change in ε(q) given u/ ug, an the change in the weights u/ ug. Let x x(q(c )), then the change in the average elasticity of utility is ln ε lnl = ε u ε lnq u ug lnl G + ε u ε ε lnq ug lnl G + ε ug (ε ε)c g(c ) lnc. lnl } {{ } Reallocation across heterogeneous firms The first term enotes the change ue to a fall in quantity per firm, holing fixe the share of each variety in the average elasticity. The secon an thir terms enote the change in the average elasticity of utility ue to a reallocation of resources across heterogeneous firms. Reallocation of resources across firms changes the share of each variety in the average elasticity of utility through ( u(q)/ c u(q)g). Using this ecomposition, we can explain the role of variable elasticities an firm heterogeneity in welfare gains from trae. For a given change in real income ( lnδ/ lnl = µ), we ecompose the gains from trae into gains for a representative firm an gains ue to ifferences in firm prouctivity. Defining the market outcome of a representative firm as the revenue-weighte average of heterogeneous firms, the gains from trae for a given change in real income are: u 33

( lnu 1 ε lnl = µ εu 1 ε + µ µ ε ug G + µ q/(1 µ) µ + µ } {{ } 1 ε ) εu G q/(1 µ) µ εug } {{ } CES VES & Representative Firm ε ε εu + µ µ + µ q/(1 µ) ε ug G + u c g(c ) ug ε(1 µ ) (ε ε)( µ µ ) } {{ } Quantity Reallocation } {{ } Firm Selection The first line contains the gains from trae for a representative firm. The first component is the welfare gain when firm markups are constant an the secon component shows how welfare gains change when markups vary with quantity. Uner CES eman, the welfare gain is the revenue-weighte average of 1 ε. VES eman as the secon component which is positive when markups are increasing an negative when markups are ecreasing with quantity. The secon line consists of the gains from trae arising ue to ifferences in firm prouctivity. The first component of the secon line is the welfare gain from changes in relative quantities across firms. When firms iffer in prouctivity, market size affects their output levels ifferently an resources are reallocate across firms. For aligne preferences, quantity reallocation increases the welfare gains from trae uner the regularity conition. The secon component shows the welfare gains from firm selection. Aligne preferences ensure the market selects the right firms as it expans an leas to higher welfare gains. Uner aligne preferences, reallocation of resources across heterogeneous firms increases the welfare gains from trae beyon those arising in a representative firm moel. As most empirical stuies are consistent with increasing markups (µ > ), structural estimates base on CES eman therefore provie a lower boun (1 ε) for the potential gains from trae. For a given change in real income, accounting for firm heterogeneity an increasing markups woul reveal higher welfare gains from trae. The magnitue of these aitional gains epens on the markup variation (through ε(q(c)) ε(q(c )) an µ (q(c))) an on the prouctivity istribution (through g(c )). 6 Conclusion This paper examines the efficiency of market allocations when firms vary in prouctivity an markups. Consiering the Spence-Dixit-Stiglitz framework, the efficiency of CES eman is vali even with prouctivity ifferences across firms. This is because market outcomes maximize revenue, an uner CES eman, private an social incentives are perfectly aligne. 34

Generalizing to variable elasticities of substitution, firms iffer in market power which affects the trae-off between quantity, variety an prouctivity. Unlike symmetric firm moels, the market istortions epen on the elasticity of eman an the elasticity of utility. Uner CES eman, these two elasticities are constant an miss out on meaningful trae-offs. When these elasticities vary, the pattern of misallocations epens on how eman elasticities change with quantities, so policy analysis shoul ascertain these elasticities an take this information into account. While the moeling framework we consier provies a theoretical starting point to unerstan istortions across firms, enriching the moel with market-specific features can yiel better policy insights. Neary an Mrazova (213) an Parenti et al. (214) provie further generalizations of eman an costs, an Bilbiie et al. (26) an more recently Opp et al. (213) consier ynamic misallocations. Future work can also provie guiance on the esign of implementable policies to realize further welfare gains. We focus on international integration as a key policy tool to realize potential gains. Market expansion oes not guarantee welfare gains uner imperfect competition. As Dixit an Norman (1988) put it, this may seem like a sa note on which to en. But we fin that integration provies welfare gains when the two eman-sie elasticities ensure private an social incentives are aligne. Integrating with large markets also hols out the possibility of approaching the CES limit, which inuces constant markups an therefore an efficient outcome. Even though integration can cause market an social objectives to perfectly align, How Large is Large? is an open question. Further work might quantify these relationships an thereby exhibit the scope of integration as a tool to improve the performance of imperfectly competitive markets. References Alessanria, G. an H. Choi, Do Sunk Costs of Exporting Matter for Net Export Dynamics?, The Quarterly Journal of Economics, 27, 122 (1), 289 336. Arkolakis, C., A. Costinot, an A. Roriguez-Clare, New trae moels, same ol gains?, American Economic Review, 212, 12 (1), 94 13.,, D. Donalson, an A. Roriguez-Clare, The Elusive Pro-Competitive Effects of Trae, Working Paper, 212. Arkolakis, Costas, Market Penetration Costs an the New Consumers Margin in International Trae, Journal of Political Economy, 21, 118 (6), 1151 1199. Asplun, M. an V. Nocke, Firm turnover in imperfectly competitive markets, The Review of Economic Stuies, 26, 73 (2). 35

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We maintain Melitz s assumptions which imply a unique market equilibrium, an use the following shorthan throughout the proofs: G(x) x g(c)c, R(x) x c ρ/(ρ 1) g(c)c. Proof of Proposition 1. Assume a market equilibrium exists, which guarantees that R(c) is finite for amissible c. First note that at both the market equilibrium an the social optimum, L/M e = f e + f G(c ) implies utility of zero so in both cases L/M e > f e + f G(c ). The policymaker s problem is c c max M e L q(c) ρ g(c)c subject to f e + f G(c ) + L cq(c)g(c)c = L/M e where the maximum is taken over choices of M e, c, q Q [,c ]. We will exhibit a globally optimal q (c) for each fixe (M e,c ) pair, reucing the policymaker s problem to a choice of M e an c. We then solve for M e as a function of c an finally solve for c. Fining q (c) for M e,c fixe. For convenience, efine the functionals V (q),h(q) by c V (q) L v(c, q(c))c, c H(q) L h(c, q(c))c where h(c,x) xcg(c) an v(c,x) x ρ g(c). One may show that V (q) λh(q) is strictly concave λ. 29 Now for fixe (M e,c ), consier the problem of fining q given by max V (q) subject to H(q) = L/M e f e f G(c ). (3) q Q [,c ] Following Troutman (1996), if some q maximizes V (q) λh(q) on Q [,c ] for some λ an satisfies the constraint then it is a solution to Equation (3). For any λ, a sufficient conition for some q to be a global maximum on Q [,c ] is D 2 v(c,q (c)) = λd 2 h(c,q (c)). (4) This follows because (4) implies for any such q, ξ s.t. q + ξ Q [,c ] we have δv (q ;ξ ) = λδh(q ;ξ ) (where δ enotes the Gateaux erivative in the irection of ξ ) an q is a global max since V (q) λh(q) is strictly concave. Conition (4) is ρq (c) ρ 1 g(c) = λcg(c) which implies q (c) = (λc/ρ) 1/(ρ 1). 3 From above, this q serves as a solution to maxv (q) provie that H(q ) = L/M e f e f G(c ). This will be satisfie by an appropriate λ since for fixe λ 29 Since h is linear in x, H is linear an since v is strictly concave in x (using ρ < 1) so is V. 3 By abuse of notation we allow q to be at c = since reformulation of the problem omitting this single point makes no ifference to allocations or utility which are all eventually integrate. 41

we have c H(q ) = L (λc/ρ) 1/(ρ 1) cg(c)c = L(λ/ρ) 1/(ρ 1) R(c ) so choosing λ as λ ρ (L/M e f e f G(c )) ρ 1 /L ρ 1 R(c ) ρ 1 makes q a solution. In summary, for each (M e,c ) a globally optimal q satisfying the resource constraint is q (c) = c 1/(ρ 1) (L/M e f e f G(c ))/LR(c ) (5) which must be > since L/M e f e f G(c ) must be > as iscusse at the beginning. Fining M e for c fixe. We may therefore consier maximizing W(M e,c ) where c W(M e,c ) M e L q (c) ρ g(c)c = M e L 1 ρ [L/M e f e f G(c )] ρ R(c ) 1 ρ. (6) Direct investigation yiels a unique solution to the FOC of M e (c ) = (1 ρ)l/( f e + f G(c )) an 2 W/ 2 M e < so this solution maximizes W. Fining c. Finally, we have maximal welfare for each fixe c from Equation (6), explicitly W(c ) W(M e (c ),c ). We may rule out c = as an optimum since this yiels zero utility. Solving this expression an taking logs shows that ln W(c ) = lnρ ρ (1 ρ) 1 ρ L 2 ρ + (1 ρ)[lnr(c ) ln( f e + f G(c ))]. Defining B(c ) lnr(c ) ln( f e + f G(c )) we see that to maximize ln W(c ) we nee maximize only B(c ). In orer to evaluate critical points of B, note that ifferentiating B an rearranging using R (c ) = c ρ/(ρ 1) g(c ) yiels { } B (c ) = c ρ/(ρ 1) R(c ) f /[ f e + f G(c )] /g(c )R(c ). (7) Since lim c c ρ/(ρ 1) = an lim c c ρ/(ρ 1) = while R(c ) an G(c ) are boune, there is a positive interval [a,b] outsie of which B (x) > for x a an B (x) < for x b. Clearly sup x (,a] B(x),sup x [b, ) B(x) < sup x [a,b] B(x) an therefore any global maximum of B occurs in (a,b). Since B is continuously ifferentiable, a maximum exists in [a,b] an all maxima occur at critical points of B. From Equation (7), B (c ) = iff R(c )/c ρ/(ρ 1) G(c ) = f e / f. For c that satisfy B (c ) =, Me an q are etermine an inspection shows the entire system correspons to the market allocation. Therefore B has a unique critical point, which is a global maximum that maximizes welfare. 42

A.2 VES Market Allocation Proof of Proposition 3. Consier a policymaker who faces a utility function v(q) u (q)q. Provie v(q) satisfies the regularity conitions use in the proof of optimality, it follows that the conitions below characterize the unique constraine maximum of LM e c u (q(c))q(c)g, where δ enotes the Lagrange multiplier: c u (q(c))q(c) + u (q(c)) = δc, u (q(c ))q(c )/(c q(c ) + f /L) = δ, ( c ) u (q(c))q(c)g/ [cq(c) + f /L]G + f e /L = δ, ( c ) M e Lcq(c) + f G + f e = L. Comparing these conitions, we see that if δ is the same as uner the market allocation, the first three equations respectively etermine each firm s optimal quantity choice, the ex post cost cutoff, an the zero profit conition while the fourth is the resource constraint an must hol uner the market allocation. Therefore if this system has a unique solution, the market allocation maximizes LM e c u (q(c))q(c)g. Since these conitions completely characterize every market equilibrium, the assume uniqueness of the market equilibrium guarantees such a unique solution. A.3 Static Distortion Results Proof of Proposition 5. The result relies on the following relationship we first prove: σ sup To see this recall δ = M mkt e δ/σ = M mkt e c mkt c c mkt ( ) ε q mkt (c) > δ/λ> inf c c opt ε ( q opt (c) ) σ. (8) c mkt u ( q mkt (c) ) q mkt (c)g so σ > δ/λ because ( ( ) ) ( ) ε q mkt (c) /σ u q mkt (c) c mkt G < Me mkt ( ) u q mkt (c) G (9) an λ is the maximum welfare per capita so λ > Me mkt c mkt u ( q mkt (c) ) G > δ/σ. A similar argument shows λσ < δ, giving Equation (8). Now note that [ ( ) ( )] u q mkt (c) q mkt (c) + u q mkt (c) /δ = c, u ( q opt (c) ) /λ = c. (1) 43

An it follows from Equations (1) we have [ ( )] ( ) 1 µ q mkt (c) u q mkt (c) /u ( q opt (c) ) = δ/λ. (11) Suppose µ > > (1 ε), an it is sufficient to show inf c c mkt 1 µ ( q mkt (c) ) σ, since then Equations (8) an (11) show that u ( q mkt (c) ) < u (q opt (c)) which implies q mkt (c) > q opt (c). Since µ > > (1 ε) an by assumption lim c q mkt (c) = = lim c q opt (c), inf c c mkt ( ) 1 µ q mkt (c) = lim 1 µ (q) = lim ε (q) + ε (q)q/ε (q) lim ε (q) = σ. q q q Similarly, if µ < < (1 ε) one may show that Equations (8) an (11) that q mkt (c) < q opt (c). sup c c mkt 1 µ ( q mkt (c) ) σ, implying from Now consier the cases when µ an ε have ifferent signs, an since inf q ε (q) >, from above in both cases it hols that inf q> 1 µ (q) = inf q> ε (q) an sup q> 1 µ (q) = sup q> ε (q). The arguments above have shown that sup q> ε (q) > δ/λ > inf q> ε (q) an therefore sup q> 1 µ (q) > δ/λ > inf q> 1 µ (q). It follows from Equation (11) that for some c, 1 µ ( q mkt (c ) ) = δ/λ an therefore u ( q mkt (c ) ) = u (q opt (c )) so q mkt (c ) = q opt (c ). Furthermore, q mkt (c) is strictly ecreasing in c so with µ, c is unique. Returning to Equation (11), using the fact that q mkt (c) is strictly ecreasing in c also shows the relative magnitues of q mkt (c) an q opt (c) for c c. Proof of Proposition 6. For α [,1], efine v α (q) αu (q)q + (1 α)u(q) an also efine w(q) u (q)q u(q) so v α (q) = u(q) + αw(q). Consier the continuum of maximization problems (inexe by α) efine as: c ( c max LM e v α (q(c))g subject to L M e Lcq(c) + f G + f e ). (12) M e,c,q(c) Let the Lagrange multiplier associate with each α in Equation (12) be written as β (α). By ap- pealing to the envelope theorem an ifferentiating (12) in M e we have β (α) = M c e v α (q(c))g an that β/α = M c e w(q(c))g = M c e u(q(c))[ε(q) 1]G <. The conitions char- 44

acterizing the solution to every optimum also imply β (α) = v α (q(c ))/(c q(c ) + f /L), whereby we arrive at v α (q(c ))/α = (β/α)(v α (q(c ))/β) + β ((c /α)q(c ) + c (q(c )/α)) = w(q(c )) + v α (q(c ))(q(c )/α) = w(q(c )) + βc (q(c )/α) so cancellation an rearrangement, using the expressions for β, β/α above shows βq(c )(c /α) = w(q(c )) (v α (q(c ))/β)(β/α) ( c ) c = w(q(c )) v α (q(c ))/M e v α (q(c))g M e w(q(c))g. We conclue that c /α when w(q(c )) c v α (q(c))g v α (q(c )) c w(q(c))g. Expaning this inequality we have (suppressing q(c) terms in integrans): c c c c w(q(c )) ug + αw(q(c )) wg u(q(c )) wg + αw(q(c )) wg. Cancellation an expansion again show this is equivalent to c c u (q(c ))q(c ) ug u(q(c )) u q(c)g. Finally, this expression can be rewritten ε (q(c )) c ε (q(c))u(q(c))g/ c u(q(c))g an since q(c) is strictly ecreasing in c, we see c /α when ε. Note that Equation (12) shows α = correspons to the social optimum while α = 1 correspons to the market equilibrium. It follows that when ε < that c /α > so we have c mkt > c opt an vice versa for ε >. A.4 Welfare Gains from Trae The sufficient conition for gains from trae follows from ifferentiating U = M e u(q)g = δ/ ε where the average elasticity of utility is ε εug/ ug. An increase in market size 45

raises the marginal utility of income at the rate of average markups lnδ/ lnl = µ pqg/ pqg µ. From lnδ/ lnl an ln ε/ lnl, the change in welfare is [ lnu lnl = µ 1 + ] [ ] 1 µ ε εu µ + µ q/(1 µ) ε ug G u c + g(c ) ug ε(1 µ ) (ε ε)( µ µ ). When preferences are aligne, the secon term in square brackets is positive because µ an (1 ε) move in the same irection. Change in the cost cutoff therefore has a positive effect on welfare, irrespective of the cost istribution G(c). The first term in square brackets is also positive when preferences are aligne, given the regularity conition (2 + µ q/µ (1 µ) ). Proof of Proposition 8. Following the iscussion above, it is sufficient to show that for γ (c) ε (µ + µ q/(1 µ)) 1, 1 + 1 µ ε µ + µ q/(1 µ) εu [1 ε ug G = ε + µ q/(1 µ) ] γu ε G. (13) ug This clearly hols for µ, an for the other case where preferences are aligne, we have µ < < ε. Expaning Equation (13) for γ γ (u/ ug)g shows that [1 ε + µ q/(1 µ) ] γu ε ug G =[1 ε µ] γ/ ε + 1 + [µ µ] γu ε ug G. Since ε >, 1 ε µ > an [1 ε µ] γ/ ε + 1 >. Therefore, it is sufficient to show that [µ µ] γu ε ugg >. This sufficient conition is equivalent to u µ G µ ug γu γ G (14) ug Since γ(c) (u/ γ ug)g = 1 an µ/c >, it follows that if γ/c <, then Equation (14) hols by stochastic ominance. As γ/c < iff γ/q >, we examine the sign of γ/q below. sign{γ/q} = sign { lnε ( µ + µ q/(1 µ) ) } 1 / lnq = sign { ( 2 + µ q/µ (1 µ) ) µ q + ( ε q/ε µ q/(1 µ) )( µ + µ q/(1 µ) )}. The aitional hypothesis that 2 + µ q/µ (1 µ) guarantees each term above is positive, so γ/q > an we conclue Equation (14) hols, giving the result. 46

A.5 Results Regaring the Impact of Large Markets To arrive at the large market result, we first state Lemmas characterizing convergence in the large market an then show market allocations coincie with optimal allocations. Detaile proofs of the Lemmas are in the Online Appenix. Lemma. As market size becomes large: 1. Market revenue is increasing in market size an goes to infinity. 2. At the optimum, utility per capita is increasing in market size an goes to infinity. 3. Market entry goes to infinity. Proof. Online Appenix. Lemma. For all market sizes an all positive marginal cost (c > ) firms: 1. Profits (π(c)) an social profits (ϖ(c) (1 ε(c))/ε(c) cq(c)l f ) are boune. 2. Total quantities (Lq(c)) in the market an optimal allocation are boune. Proof. Online Appenix. Proposition 1. Assume markups are interior. Then uner the market allocation: 1. lim c mkt L = iff lim p ( c mkt ) L = iff lim Lq ( c mkt ) L =. 2. lim c mkt L = iff lim p ( c mkt ) L = iff lim Lq ( c mkt ) L =. 3. lim c mkt L (, ) iff lim p ( c mkt ) L (, ) iff lim Lq ( c mkt ) L (, ). Similarly, uner the optimal( allocation: ) ( ) ( ) 1. lim c opt L = iff lim u q c opt L /λq c opt = iff lim Lq c opt L =. ( ) ( ) ( ) 2. lim c opt L = iff lim u q c opt L /λq c opt = iff lim Lq c opt L =. ( ) ( ) ( ) 3. lim c opt L (, ) iff lim u q c opt L /λq c opt (, ) iff lim Lq c opt L (, ). Proof. Note the following zero profit relationships that hol at the cost cutoff c a, suppressing the market superscripts throughout we have: u (q(c ))/δ f /[Lq(c ) µ q(c )/(1 µ q(c ))] = c, (15) Lc q(c ) µ q(c )/(1 µ q(c )) = f. (16) First, if lim Lq(c ) =, Equation (16) implies c µ q(c )/(1 µ q(c )). Clearly L q(c ) an since limµ (q) (,1), µ q(c )/(1 µ q(c )) is boune, an therefore q c. Now suppose c an since c u (q(c ))/δ, u (q(c ))/δ. Finally, if u (q(c ))/δ, since δ, necessarily q(c ) so we fin µ q(c )/(1 µ q(c )) 47

is boune. It follows from Equation (16) that Lc q(c ) is boune, so from Equation (15), Lq(c ) u (q(c ))/δ is boune so Lq(c ). If lim Lq(c ) =, q(c ) so from limµ (q) (,1), µ q(c )/(1 µ q(c )) is L q boune. Therefore from Equation (16), c. Now assume c so from (16), Lq(c ) µ q(c )/(1 µ q(c )) which implies with Equation (15) that u (q(c ))/δ. Finally, if u (q(c ))/δ, (15) shows c. The secon set of equivalences follows from examining the conitions for a firm at the limiting cost cutoff c (, ). The argument for the optimal allocation is similar. Lemma. Assume interior convergence. Then as market size grows large: 1. In the market, p(c) converges in (, ) for c > an Lq(c ) converges in (, ). 2. In the optimum, u q(c)/λq(c) an Lq(c ) converge in (, ) for c >. Proof. Online Appenix. Lemma. Assume interior convergence an large market ientification. Then for the market an social optimum, Lq(c) converges for c >. Proof. Online Appenix. Lemma. At extreme quantities, social an private markups align as follows: 1. If lim1 ε(q) < 1 then lim1 ε(q) = limµ(q). q q q 2. If lim 1 ε(q) < 1 then lim 1 ε(q) = lim µ(q). q q q Proof. Online Appenix. Lemma. Assume interior convergence an large market ientification. As market size grows large: 1. q(c)/q(c ) (c/c ) 1/α with α = lim q µ (q). 2. The cost cutoffs for the social optimum an market converge to the same value. 3. The entrant per worker ratios M e /L converge to the same value. Proof. Define ϒ(c/c ) by (the above results show this limit is well efine) ϒ(c/c ) lim q u (ϒ(c/c )q)/u (q) = c/c. We will show in fact that ϒ(c/c ) = (c/c ) α. It follows from the efinition that ϒ is weakly ecreasing, an the results above show ϒ is one to one, so it is strictly ecreasing. Define f q (z) u (zq)/u (q) so lim q f q (z) = ϒ 1 (z) for all ϒ 1 (z) (,1). Note f q(z) = u (zq)q/u (q) = µ(zq) u (zq)/zu (q) 48

so since limµ(zq) = µ (,1) an limu (zq)/zu (q) = ϒ 1 (z)/z, we know that lim f q q q q(z) = µ ϒ 1 (z)/z. On any strictly positive close interval I, µ an u (zq)/zu (q) are monotone in z so f q(z) converges uniformly on I as q. Ruin (1964) (Thm 7.17) shows lim f q(z) = lim f q(z)/z = µ ϒ 1 (z)/z = ϒ 1 (z)/z. (17) q q We conclue that ϒ 1 (z) is ifferentiable an thus continuous. Given the form euce in (17), ϒ 1 (z) is continuously ifferentiable. Since ϒ 1 (z)/z = 1/ϒ ϒ 1 (z), composing both sies with ϒ(z) an using (17) we have ϒ (z) = ϒ(z)/µ z. Therefore ϒ is CES, in particular ϒ(z) = z 1/µ. Finally, let c opt an c mkt be the limiting cost cutoffs as L for at the social optimum an market, respectively. Letting q opt (c), q mkt (c) enote the socially optimal an market quantities, we know from above that for all c > : ( ) q opt (c)/q opt c opt ( c opt /c ) 1/α, ( ) ( 1/α q mkt (c)/q mkt c mkt c mkt /c). (18) Now consier the conitions involving f e, c mkt π(c)g = f e = c opt ϖ(c)g. Expaning, c mkt L µ q mkt (c) 1 µ q mkt (c) cqmkt (c)g f G(c mkt It necessarily follows that cmkt lim L L c lim L L opt opt c ) = L 1 ε q opt (c) ε q opt cq opt (c)g f G(c opt (c) ). ( ) µ q mkt (c)/ 1 µ q mkt (c) cq mkt (c)g f G(c mkt ) = ( 1 ε q opt (c) ) /ε q opt (c) cq opt (c)g f G(c opt ). (19) Using Equation (18), we see that Lq opt (c) an Lq mkt (c) converge uniformly on any strictly positive close interval. Combine with the fact that limµ(q) = lim1 ε(q), we see from q q Equation (19) the limits of the µ/(1 µ) an (1 ε)/ε terms are equal an factor out of Equation (19), leaving lim L Lcmkt q mkt (c mkt lim L Lcopt q opt (c opt ) c mkt ) c opt (c/c mkt )(c/c mkt ) 1/α G f G(c mkt ) = (c/c opt )(c/c opt ) 1/α G f G(c opt ). 49

Noting f (1 µ )/µ = Lc mkt q mkt (c mkt ) = Lc opt q opt (c opt ), we therefore have lim L lim L c mkt opt c (c/c mkt ) 1 1/α (c mkt (c/c opt ) 1 1/α (c opt so that finally evaluating the limits, we have c mkt /c mkt /c opt [ ] (c/c mkt ) 1 1/α 1 G = c opt ) 1/α G G(c mkt ) = ) 1/α G G(c opt ) [ ] (c/c opt ) 1 1/α 1 G. (2) Letting h(w) ] w [(c/w) 1 1/α 1 G, we see that h (w) = w (1/α 1)c 1 1/α w 1/α 2 G an since α = µ (,1), h >. Since h is strictly increasing, there is a unique c opt, namely = c mkt between the market an social optimum as well. c opt such that Equation (2) hols. Checking the conitions for L/M e show they coincie 5