Monopolistic Competition and Optimum Product Diversity Under Firm Heterogeneity*

Transcription

1 Monopolistic Competition an Optimum Prouct Diversity Uner Firm Heterogeneity* Swati Dhingra CEP, Lonon School of Economics John Morrow CEP, Lonon School of Economics This Draft: May 1, 213 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 sources 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 George Alessanria, Costas Arkolakis, Roc Armenter, Any Bernar, Satyajit Chatterjee, Davin Chor, Steve Durlauf, Charles Engel, Thibault Fally, Rob Feenstra, Keith Hea, Wolfgang Keller, Jim Lin, Emanuel Ornelas, Gianmarco Ottaviano, Mathieu Parenti, Nina Pavcnik, Tom Sampson, Daniel Sturm, Jacques Thisse, John Van Reenen, Ariel Weinberger, Ben Zissimos an Mian Zhu for insightful comments, Katheryn Russ an Anres Roriguez-Clare for AEA iscussions an Tim Besley for avice. This paper has benefite from helpful comments of participants at AEA 211 an 213, Davis, DIME-ISGEP 21, ETSG 212, HSE St Petersburg, ISI, FIW, LSE, Louvain, Mannheim, Oxfor, Philaelphia Fe, Princeton, Wisconsin an Yale. Swati thanks the IES (Princeton) for their hospitality. A preliminary raft was a issertation chapter at Wisconsin in 21. *The first line is the title of Dixit an Stiglitz (1977). Contact: [email protected]. 1

2 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 sources 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 E.g., Bartelsman an Doms (2); Tybout (23); Feenstra (26); Bernar, Jensen, Reing an Schott (27). 2 E.g., Pavcnik (22); Asplun an Nocke (26); Foster et al. (21); Melitz an Reing (212). 3 E.g., Spence (1976); Venables (1985); Mankiw an Whinston (1986); Stiglitz (1986). 2

3 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. Firm heterogeneity oes not introuce any new istortions, but firms earn positive profits. This result seems surprising, base on the logic of average cost pricing which 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 forthcoming). 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 are perfectly aligne with social benefits only uner CES eman. More generally, market power inuces istortions relative to optimal allocations an eman-sie elasticities etermine these istortions. The pattern of istortions is etermine by two elasticities: the eman elasticity, which 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. 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

4 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) 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. 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 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. (forthcoming) emonstrate 4

5 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 markups matter for both welfare gains an allocational efficiency. 9 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. 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. 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. 5

6 Section 6 conclues. 2 Moel 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. We aopt the VES eman structure of Dixit an Stiglitz an the heterogeneous firm framework of Melitz, an refer to this setting as the Dixit-Stiglitz-Melitz framework. In this Section, we briefly recap the implications of asymmetric costs for consumers, firms an equilibrium outcomes. 2.1 Consumers A mass L of ientical consumers in an economy are each enowe with one unit of labor an face a wage rate w normalize to one. Preferences are ientical across all consumers. Let M e enote the mass of entering varieties an q(c) enote the quantity consume of variety c by each consumer. A consumer has preferences over ifferentiate goos U(M e,q) which take the general VES form: U(M e,q) M e u(q(c))g. (1) Here u enotes utility from an iniviual variety an u(q)g enotes utility from a unit bunle of ifferentiate varieties. Uner CES preferences, u(q) = q ρ as specifie in Dixit-Stiglitz an Krugman (198). 1 More generally, we assume preferences satisfy usual regularity conitions which guarantee well efine consumer an firm problems. Definition 1. (Regular Preferences) u satisfies the following: u() is normalize to zero, u is twice continuously ifferentiable, increasing an concave, (u (q) q) is strictly ecreasing in quantity, an the elasticity of marginal utility µ(q) qu (q)/u (q) is less than one. For each variety c, VES preferences inuce an inverse eman p(q(c)) = u (q(c))/δ where δ is a consumer s buget multiplier. As u is strictly increasing an concave, for any fixe price vector the consumer s maximization problem is concave. The necessary conition which etermines the inverse eman is sufficient, an has a solution provie inaa conitions on 1 The specific CES form in Melitz is U(M e,q) Me 1/ρ ( (q(c)) ρ G) 1/ρ but the normalization of the exponent 1/ρ in Equation (1) will not play a role in allocation ecisions. 6

7 u. 11 Multiplying both sies of the inverse eman by q(c) an aggregating over all c, the buget multiplier is δ = M e c u (q(c)) q(c)g. 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. Each firm prouces a single variety, so the mass of entering firms is the mass of entering varieties M e. Upon entry, each firm receives a unit cost c rawn from a istribution G with continuously ifferentiable pf g. 12 After entry, shoul a firm prouce, it incurs a fixe cost of prouction f. Each firm faces an inverse eman of p(q(c)) = u (q(c))/δ an acts as a monopolist of variety c. Post entry, the profit of firm c is π(c) where π(c) max q(c) [p(q(c)) c]q(c)l f. The regularity conitions guarantee the monopolist s FOC is optimal an the quantity choice is etermine by the equality of marginal revenue an marginal cost. Specifically, p + q u (q)/δ = c an the markup rate is (p(c) c)/p(c) = qu (q)/u (q). This shows that the elasticity of marginal utility summarizes the inverse eman elasticity as µ(q) qu (q)/u (q) = ln p(q)/ lnq = (p(c) c)/p(c). 2.3 Market Equilibrium Profit maximization implies firms prouce if they can earn non-negative profits. We enote the cutoff cost level of firms that are inifferent between proucing an exiting from the market as c. The cutoff cost c is fixe by the zero profit conition, π(c ) =. Since firms with cost raws higher than the cutoff level o not prouce, the mass of proucers is 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, π(c)g = f e. The next two Sections examine the efficiency properties of this Dixit-Stiglitz-Melitz framework. 11 Utility functions not satisfying inaa conitions are permissible but may require parametric restrictions to ensure existence. We will assume inaa conitions on utility an revenue, though they are not necessary for all results. 12 Some aitional regularity conitions on G are require for existence of a market equilibrium in Melitz. 7

8 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. Nonetheless, 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 all VES eman structures except CES. Following Dixit an Stiglitz, we start with an exposition of efficiency uner CES eman an then iscuss market inefficiency uner VES eman. 3.1 Welfare uner Isoelastic Deman A policymaker maximizes iniviual welfare U as given in Equation (1). 13 The policymaker is unconstraine an chooses the mass of entrants, quantities an types of firms that prouce. At the optimum, zero quantities will be chosen for varieties above a cost threshol c. Therefore, all optimal allocational ecisions can be summarize by quantity q(c), potential variety M e an prouctivity c. 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) 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. Proposition 1 shows the market provies the first-best quantity, variety an prouctivity. Proposition 1. Every market equilibrium of a CES economy is socially optimal. The proof of Proposition 1 iffers from stanar symmetric firm monopolistic competition results because optimal quantity varies non-trivially with unit cost, variety an cutoff prouctivity. We iscuss the rationale for optimality below. In symmetric firm moels with CES eman, firms charge positive markups which result in lower quantities than those implie by marginal cost pricing. However, the markup is constant so the market price (an hence marginal utility) is proportional to unit cost, ensuring proportionate reuction in quantity from the level that woul be observe uner marginal cost pricing 13 Free entry implies zero expecte profits, so the focus is on consumer welfare. 8

9 (Baumol an Brafor 197). Moreover, free entry ensures price equals average cost so profits exactly finance the fixe cost of prouction. The market therefore inuces firms to inirectly internalize the effects of higher variety on consumer surplus, resulting in an efficient market equilibrium (Grossman an Helpman 1993). With heterogeneous firms, markups continue to be constant, which implies profits are heterogeneous. One might imagine enforcing average cost pricing across ifferent firms woul inuce an efficient allocation but, average cost pricing is too low to compensate firms because it will not cover ex ante entry costs. Instea, the market ensures prices above average costs at a level that internalizes the losses face by exiting firms. Post entry, surviving firms charge prices higher than average costs (p(c) [cq(c) + f /L]/q(c)) which compensates them for the possibility of paying f e to enter an then being too unprouctive to survive. CES eman ensures that c an M e are at optimal levels that fix p(c ), thereby fixing absolute prices to optimal levels. The marginal entrant imposes a business stealing externality on other firms, but also oes not account for the variety gain an prouctivity loss from its entry. These effects exactly offset each other, an wages inuce by the market exactly reflect the shaow value of resources at the optimal allocation. The way in which CES preferences cause firms to optimally internalize aggregate economic conitions can be mae clear by efining the elasticity of utility ε(q) u (q) q/u(q) an the 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, there is a multiplier λ which encapsulates the shaow cost of labor. The social surplus is u(q) λcq an the optimal quantities ensure u (q(c)) = λc. Therefore, the social markup is 1 ε(q) =1 u (q) q/u(q) =(u(q) λcq)/u(q). (Social Markup) For any optimal allocation, a quantity that maximizes social benefit from variety c solves max q (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 9

10 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. A irect implication of Proposition 1 is that laissez faire inustrial policy is optimal uner constant elasticity eman. In the next subsection, we examine the role of variable elasticities on market efficiency Welfare beyon Isoelastic Deman Efficiency of the market equilibrium in a Dixit-Stiglitz-Melitz framework is tie to CES eman. To highlight this, we consier the general class of variable elasticity of substitution (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. 15 Proposition 2. socially optimal is that u is CES. Proof. Online Appenix. Uner VES eman, a necessary conition for the market equilibrium to be 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. 14 The CES efficiency result may seem surprising in the context of Dixit an Stiglitz (1977) who fin that market allocations are secon-best but not first-best. Dixit an Stiglitz consier two sectors (a ifferentiate goos sector an a homogeneous goos sector) an assume a general utility function to aggregate across these goos. This causes the markups charge in the homogeneous an ifferentiate goos to iffer, leaing to inefficient market allocations. In keeping with Melitz, we consier a single sector to evelop results for market efficiency in terms of markups. 15 CES eman is necessary but not sufficient for efficiency. To see this, exten the CES eman of Melitz to CES-Benassy preferences U(M e,c,q) ν(m e ) c q(c)ρ g(c)c. Here u is CES but varieties an the unit bunle are value ifferently through ν(m e ). Market allocations uner CES-Benassy are the same as CES. However, firms o not fully internalize consumers taste for variety, leaing to suboptimal allocations. 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 are optimal only if taste for variety exactly equals the markup to cost ratio, an Helpman an Krugman (1985) an Feenstra an Kee (28) erive a GDP function for this economy. Cole an Davies (forthcoming) further highlight variety istortions by introucing existence values for variety. 1

11 Proposition 3. Uner VES eman, the market maximizes aggregate real revenue. This result shows that the market resource allocation is generally not aligne with the social optimum uner VES eman. The market an efficient 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 Social For CES eman, u(q) = q ρ while u (q)q = ρq ρ implying revenue maximization is perfectly aligne with welfare maximization. Outsie of CES, quantities prouce by firms are too low 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. This leas us to an examination of the nature of misallocations inuce by the market. 4 Market Distortions an Variable Elasticities Although we have ientifie the conflict between private markups µ (q) capture by firms an social markups 1 ε (q) that woul maximize welfare as the source of istortions, we have not investigate the nature of these istortions. In this Section, we characterize how the market allocates resources relative to the social optimum in terms of markups. Specifically, the bias in market quantity an prouctivity is etermine by how private an social markups vary with quantity (µ (q) an (1 ε(q)) ). We start with a iscussion of markup an quantity patterns, an then show that ifferent markup patterns inuce very ifferent biases in market allocations. We summarize the pattern of istortions an iscuss empirical evience for ifferent eman characteristics. To highlight the importance of firm heterogeneity an variable markups, we finally compare our results with istortions uner symmetric firms. 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 11

12 have both high q an high markups. When µ (q) <, small boutique firms charge higher markups. For CES eman, markups are constant (µ = ). The sign of (1 ε(q)) etermines how social markups vary with quantity. When it is positive (1 ε(q)) >, social markups are higher at higher levels of quantity. As above, this implies a negative correlation between social markups 1 ε an unit costs c. Conversely, when (1 ε(q)) <, the boutique varieties which are consume in small quantities provie relatively higher social markups. Uner CES preferences, (1 ε(q)) is again zero. To bring out the istinction in istortions for ifferent markup patterns, Definition 2 below characterizes preferences as aligne when private an social markups move in the same irection an misaligne when they move in ifferent irections. Definition 2. 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. 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, 16 HARA an generalize CES (propose by Dixit an Stiglitz). 17 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 Quantity, Prouctivity an Entry Distortions We now characterize the misallocations by eman characteristics. The istortions in quantity, prouctivity an entry are iscusse in turn. 16 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. 17 The parameter restrictions are ρ (,1), α > q/(ρ 1) for HARA an α > q for Generalize CES. 12

13 4.2.1 Quantity Bias Quantity istortions across firms epen on whether private an social incentives are aligne or misaligne. We show that when private an social markups 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 quantities is: Private MB [ ( 1 µ q mkt )] MC = u ( q mkt) /δ c = u (q opt )/λ c = Social MB MC. When incentives are misaligne, market an optimal quantities are too high or too low across all varieties. In particular, when µ > > (1 ε), the market over-rewars firms proucing higher quantities an all firms over-prouce q mkt (c) > q opt (c). When µ < < (1 ε), market prouction is too low (q mkt (c) < q opt (c)). Therefore, firms are either over-reware (µ > ) for proucing q or uner-reware (µ < ), 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. 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 over-rewars high cost firms who impose an externality on low cost firms. When µ,(1 ε) <, the bias is reverse an quantities are biase towars low cost firms. Therefore, when private an social markups are aligne, the market uner or over prouces quantity, epening on a firm s costs. Proposition 4 summarizes the bias in market quantities. Proposition 4. When preferences are misaligne, q mkt (c) an q opt (c) never cross: 1. If µ > > (1 ε), market quantities are too high: q mkt (c) > q opt (c). 2. If µ < < (1 ε), 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). 1. If µ > an (1 ε) >, q mkt (c) < q opt (c) for c < c an q mkt (c) > q opt (c) for c > c. 2. If µ < an (1 ε) <, q mkt (c) > q opt (c) for c < c an q mkt (c) < q opt (c) for c > c. 13

14 This shows the misallocation in prouction iffers across firms, an variable eman elasticities characterize the pattern of misallocations Prouctivity Bias The istortion in firm selection is etermine by the relation between social markups an quantity. Proposition 5 shows that market prouctivity is either too low or high, epening on whether social markups are increasing or ecreasing. Revenue of the cutoff prouctivity firm is proportional to u (q)q while its contribution to utility is u(q). Therefore, the gap in prouctivity cutoffs is etermine by ε(q) an the market bias epens on ε (q). Increasing social markups (1 ε) > encourage higher optimal quantity at lower costs. In general equilibrium, this translates into a lower cost cutoff at the optimum, so market costs are too high. Proposition 5. 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 4 an 5 explain how the market misallocates resources across firms. Figure 1 illustrates the bias in firm-level prouction for aligne an misaligne preferences when private markups increase in quantity. Figure 1: Bias in Firm Prouction by Preferences (a) Misaligne: µ > > (1 ε) (b) Aligne: µ > an (1 ε) > A comparison of mass of entrants in the market an the optimum is generally har to make. Quantity an prouctivity istortions have opposing effects on potential entry so the bias in mass of entrants epens on the magnitues of exogenous parameters. Focusing on a linear 14

15 eman setting with Pareto cost raws, Nocco, Ottaviano an Salto (213) fin that the mass of firms cannot be unambiguously ranke. The bias in potential entry epens on the relative magnitues of eman an cost parameters. As the quantity an prouctivity istortions can be assesse by examining the eman parameters, the next sub-section iscusses the finings of empirical work on estimating markups Empirical Evience for Deman Characteristics This Section has shown that the unerlying eman structure can lea to very ifferent istortions. For ease of reference, Table 2 summarizes the misallocations by eman characteristics. 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 As the pattern of misallocation epens on how private an social markups vary with quantity, a natural question is whether empirical work can ientify which case in Table 2 is relevant. Systematic empirical evience on the relationship between markups an quantities is sparse (Weyl an Fabinger 212). However, existing stuies suggest that the relationship iffers across markets, an therefore we cannot restrict attention to a single case. For example, De Loecker, Golberg, Khanelwal an Pavcnik (212) irectly estimate the cross-sectional relationship for large Inian manufacturers an fin private markups are increasing in quantity µ (q) >. 18 With irect information on prices an costs, Cunningham (211) instea fins evience for ecreasing private markups among rugstore proucts in the US. Social markups are rarely observable, an there is lack of consensus on how they respon to quantity (Vives 18 The bulk of empirical work on pass-through rates an firm selection also suggests private markups increase with quantities. However, some stuies suggest markups ecrease with quantities as they fin a rise in markups after entry (Zheloboko et al. forthcoming). 15

16 21). Spence suggests social markups ecrease with quantity while Dixit an Stiglitz propose increasing social markups. Therefore, we cannot rule out specific cases without further empirical investigation of the market uner consieration Comparison with Symmetric Firms In the remainer of this Section, we compare the bias in market allocations uner symmetric an heterogeneous firms. Dixit an Stiglitz fin that only the elasticity of utility matters for quantity misallocation an the elasticity of eman is not relevant for etermining efficiency of prouction levels. We state their result below an iscuss how prouctivity ifferences affect istortions an efficiency analysis. Proposition 6. 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). In terms of etermining misallocations, the symmetric firm case simplifies the analysis as we nee only compare two ecisions, quantity an number of firms. In contrast, etermining misallocations across heterogeneous firms is less obvious because quantities vary by firm prouctivity. Further, the bias in quantities an prouctivity can have opposing implications for the bias in firm entry. For instance, when firms prouce too little quantity, there is ownwar pressure on wages an high cost firms are able to survive in the market. A higher cost cutoff in turn bis up wages, so firm quantities an the cost cutoff have opposite effects on the ex ante profitability of 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, when µ > an (1 ε) >, excess prouction by small firms imposes an externality on large firms. Large 19 Distinguishing 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]. 16

17 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 both the elasticity of utility an the inverse eman elasticity etermine resource misallocations. Uner symmetric firms, only the elasticity of utility etermines misallocations an the inverse eman elasticity oes not matter. Specifically, Proposition 6 oes not epen on µ (q). The presence of firm heterogeneity funamentally changes the qualitative analysis. When markups vary, firms with ifferent prouctivity levels charge ifferent markups. This affects their quantity ecisions as well as their incentives to enter. Therefore, firm heterogeneity an variable markups alter the stanar policy rules for correcting misallocation of resources by the market. 2 5 Efficiency an Market Size Increases in market size encourage competition, so we might expect that integrate markets 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 monopolistic 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 are able to show that welfare gains are relate to the eman-sie elasticities iscusse earlier. To unerstan the potential of market expansion in eliminating istortions, we examine efficiency in large markets. Large integrate markets can eliminate istortions, while preserving firm heterogeneity. 2 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). 17

18 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 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 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 from trae. 21 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. We summarize this result in Proposition 8. Proposition 8. Market expansion increases welfare when preferences are aligne. (Provie (µq) 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 private an social markups in the same irection. The market therefore incentivizes firms towars the right allocation an provies higher welfare. 21 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 ). 18

19 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. 22 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 22 How large markets nee to be to justify this approximation is an open quantitative question. 19

20 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 expo-power utility where u(q) = [1 exp ( αq 1 ρ) ]/α for ρ (,1). It nests the CES 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 11 in the Appenix. In wors, Proposition 11 shows that when markups are interior an the cost cutoff converges, 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 > ) but fixe costs still play a role in etermining which firms prouce ( lim c mkt < ). We therefore make the following assumption, which by Proposition 11 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. 2

21 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 = 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. 23 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. 23 From a technical stanpoint, this guarantees entry is well behave, avoiing pathological sequences of potential equilibria as market size grows large. 21

22 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. 24 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 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 CES Efficiency with Trae Frictions We have examine how opening to trae with small an large markets affects istortions. Conceptualizing integration as access to new markets enables us to provie a theoretical benchmark. A more realistic scenario however is one with partial trae liberalization where international trae entails aitional costs. In this sub-section, we introuce trae frictions as in Melitz an 24 The rate at which markups converge epens on c an is in any case enogenous (see Appenix). 22

23 show that the CES economy continues to be efficient. We then argue that trae frictions introuce istributional issues, which we o not aress in this paper. Let τ 1 enote the iceberg trae cost an f x enote the fixe cost of exporting goos abroa. When τ = 1 an f x =, the economy faces no trae frictions in integrating with worl markets. Proposition 1 shows that the autarkic an integrate market allocations are efficient uner CES eman. This implies that a worl planner woul never levy trae taxes even when it coul collect tax revenues by choosing τ > 1 or f x >. The CES efficiency result is therefore robust to enogenously chosen trae frictions. As Proposition 1 below shows, CES eman ensures the market picks the right allocations even in the presence of exogenous trae frictions. 25 Proposition 1. Every market equilibrium of ientical open Melitz economies with trae frictions is socially optimal. Proof. Online Appenix. Proposition 1 is striking in that the ifferences in firm costs o not generate inefficiencies espite heterogeneity of profits an the ifferent effects that trae frictions will have on firm behavior. Furthermore, selection of firms performs the function of allocating aitional resources optimally without any informational requirements. Uner CES eman, laissez faire inustrial policy is optimal for the worl economy. 26 The CES efficiency results of Propositions 1 an 1 imply that the higher prouctivity cutoff of an open Melitz economy is not optimal in autarky. This seems counter-intuitive, as Melitz shows that trae provies prouctivity an welfare gains by reallocating resources towars low cost firms. Why then is the lower cost cutoff of the open economy inefficient in autarky? Proposition 1 shows trae frictions make a new mix of prouctivity an variety efficient. The market minimizes losses from trae frictions by weeing out high cost firms. Conitional on trae costs, market selection of firms is optimal. In autarky, choosing a prouctivity cutoff that correspons to a higher level of frictions woul provie prouctivity gains at the expense of too little variety, an woul ecrease welfare Technically, we nee to be careful in specifying the policymaker s objective function in the presence of multiple countries. Formal etails are in the Online Appenix an we note here that the policymaker maximizes per capita worl welfare. 26 However, terms of trae externalities may exist an lea to a breakown of laissez faire policies (Demiova an Roriguez-Clare 29). Moreover, Chor (29) consiers policy intervention in the presence of multinationals an a homogeneous goos sector. 27 Another implication of market efficiency is that exogenous shocks (such as changes in trae frictions) affect worl welfare only through their irect effect on welfare. As market allocations maximize worl welfare, the inirect effects can be ignore when stuying the impact of exogenous shocks on welfare uner CES eman (for example, Atkeson an Burstein 21). 23

24 Moeling trae between equally size countries makes the role of trae frictions clear cut. When countries iffer in size, trae frictions introuce cross-country istributional issues which obscure the pure efficiency question. Specifically, consier two countries of ifferent sizes with CES eman. Market allocations are efficient when these countries trae with each other an face no trae frictions. These market allocations maximize social welfare with equal Pareto weights assigne to every iniviual in the two countries. Introucing trae frictions will continue to inuce efficient market allocations, but with unequal Pareto weights. Let ω mx enote the Pareto weight on welfare of country m from consuming goos of country x. Following Proposition 7, ω mx can be efine to ensure the market allocation is an interior solution to: max q,c,m e x L x M x e ω mx Me x m { m c mx c xm u (q mx (c)) q mx (c)l m G [τ xm cq xm (c)l m + f xm ]G + f e } where for each x. This shows the market is implicitly favoring certain consumers, so that resource allocation reflects istributional outcomes in aition to cost competitiveness. As our focus is on efficiency, we moel the stylize case of frictionless trae an consier more general eman structures which can explain a greater range of market outcomes. The cross-country istribution of welfare gains is important but beyon the focus of this stuy. We leave this avenue to future research an conclue in the next Section. 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 an trae frictions. This is because market outcomes maximize revenue, an uner CES eman, private an social incentives are perfectly aligne. 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 nature of market istortions epens 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 24

25 starting point to unerstan istortions across firms, enriching the moel with market-specific features can yiel better policy insights. 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), Arkolakis, C., A. Costinot, an A. Roriguez-Clare, New trae moels, same ol gains?, American Economic Review, 212, 12 (1), ,, D. Donalson, an A. Roriguez-Clare, The Elusive Pro-Competitive Effects of Trae, Working Paper, 212. Asplun, M. an V. Nocke, Firm turnover in imperfectly competitive markets, The Review of Economic Stuies, 26, 73 (2). Atkeson, A. an Burstein, Innovation, Firm Dynamics, an international Trae, Journal of Political Economy, 21, 118 (3), Balwin, R. E. an F. Robert-Nicou, Trae an growth with heterogeneous firms, Journal of International Economics, 28, 74 (1), Bartelsman, E. J. an M. Doms, Unerstaning prouctivity: Lessons from longituinal microata, Journal of Economic literature, 2, 38 (3). Baumol, W. J. an D. F. Brafor, Optimal Departures From Marginal Cost Pricing, The American Economic Review, 197, 6 (3), Behrens, Kristian an Yasusaa Murata, Trae, competition, an efficiency, Journal of International Economics, 212, 87 (1), Benassy, J. P., Taste for variety an optimum prouction patterns in monopolistic competition, Economics Letters, 1996, 52 (1),

26 Bernar, A. B., J. B. Jensen, S. J. Reing, an P. K. Schott, Firms in International Trae, The Journal of Economic Perspectives, 27, 21 (3), , J. Eaton, J. B. Jensen, an S. Kortum, Plants an Prouctivity in International Trae, American Economic Review, 23. Bilbiie, F. O., F. Ghironi, an M. J. Melitz, Monopoly power an enogenous variety in ynamic stochastic general equilibrium: istortions an remeies, manuscript, University of Oxfor, Boston College, an Princeton University, 26. Bulow, J. I. an P. Pfleierer, A note on the effect of cost changes on prices, The Journal of Political Economy, 1983, 91 (1), Campbell, J. R. an H. A. Hopenhayn, Market Size Matters, Journal of Inustrial Economics, 25, 53 (1), Chor, D., Subsiies for FDI: Implications from a moel with heterogeneous firms, Journal of International Economics, 29, 78 (1), Cole, Matthew T. an Ronal B. Davies, Royale with Cheese: The Effect of Globalization on the Variety of Goos, Review of Development Economics, forthcoming. Cunningham, T., Comparisons an Choice, Working Paper, 211. e Blas, B. an K. Russ, Unerstaning Markups in the Open Economy uner Bertran Competition, NBER Working Papers, 21. Demiova, S. an A. Roriguez-Clare, Trae policy uner firm-level heterogeneity in a small economy, Journal of International Economics, 29, 78 (1), Dixit, A. K. an J. E. Stiglitz, Monopolistic Competition an Optimum Prouct Diversity, The American Economic Review, 1977, 67 (3), an V. Norman, Theory of international trae, Cambrige Univ. Press, Eckel, Carsten, Globalization an specialization, Journal of International Economics, May 28, 75 (1), Epifani, P. an G. Gancia, Trae, markup heterogeneity an misallocations, Journal of International Economics, 211, 83 (1), Feenstra, R. an H. L. Kee, Export variety an country prouctivity: Estimating the monopolistic competition moel with enogenous prouctivity, Journal of International Economics, 28, 74 (2), Feenstra, R. C., A homothetic utility function for monopolistic competition moels, without constant price elasticity, Economics Letters, 23, 78 (1),

27 , New Evience on the Gains from Trae, Review of Worl Economics, 26, 142 (4), Foster, L., J. C. Haltiwanger, an C. J. Krizan, Aggregate prouctivity growth. Lessons from microeconomic evience, in New evelopments in prouctivity analysis, University of Chicago Press, 21., J. Haltiwanger, an C. Syverson, Reallocation, firm turnover, an efficiency: Selection on prouctivity or profitability?, American Economic Review, 28, 98 (1), Grossman, Gene M. an Elhanan Helpman, Innovation an Growth in the Global Economy, MIT Press, Hart, O. D., Monopolistic competition in the spirit of Chamberlin: A general moel, The Review of Economic Stuies, 1985, 52 (4), 529. Helpman, E. an P. R. Krugman, Market Structure an Foreign Trae: increasing returns, imperfect competition, an the international economy, MIT Press, 1985., O. Itskhoki, an S. J. Reing, Trae an Labor Market Outcomes, NBER Working Paper 16662, 211. Holt, C. A. an S. K. Laury, Risk aversion an incentive effects, American Economic Review, 22, 92 (5), Katayama, H., S. Lu, an J. R. Tybout, Firm-level prouctivity stuies: illusions an a solution, International Journal of Inustrial Organization, 29, 27 (3), Khan, M. A. an Y. Sun, Non-cooperative games with many players, Hanbook of Game Theory with Economic Applications, 22, 3, Krugman, P., Increasing Returns, Monopolistic Competition, an International Trae, Journal of International Economics, 1979, 9 (4), Krugman, P. R., Is free trae passé?, The Journal of Economic Perspectives, 1987, 1 (2). Krugman, Paul R., Scale Economies, Prouct Differentiation, an the Pattern of Trae, American Economic Review, 198, 7 (5), Kuhn, K. U. an X. Vives, Excess entry, vertical integration, an welfare, The Ran Journal of Economics, 1999, 3 (4), Loecker, Jan De, Pinelopi K. Golberg, Amit K. Khanelwal, an Nina Pavcnik, Prices, Markups an Trae Reform, Working Paper, March 212. Mankiw, N. G. an M. D. Whinston, Free entry an social inefficiency, The RAND Journal of Economics, 1986, pp

28 Matsuyama, Kiminori, Complementarities an Cumulative Processes in Moels of Monopolistic Competition, Journal of Economic Literature, June 1995, 33 (2), Melitz, M. J. an S. J. Reing, Heterogeneous Firms an Trae, Hanbook of International Trae (commissione), August 212. Melitz, Marc an Daniel Trefler, Gains from Trae when Firms Matter, Journal of Economic Perspectives, 212, 26. Melitz, Marc J., The Impact of Trae on Intra-Inustry Reallocations an Aggregate Inustry Prouctivity, Econometrica, 23, 71 (6), an Gianmarco I. P. Ottaviano, Market Size, Trae, an Prouctivity, Review of Economic Stuies, October 28, 75 (1), Melvin an R. D. Warne, Monopoly an the theory of international trae, Journal of International Economics, 1973, 3 (2), Nocco, Antonella, Gianmarco I. P. Ottaviano, an Matteo Salto, Monopolistic Competition an Optimum Prouct Selection: Why an How Heterogeneity Matters, CEP Discussion Paper, April 213, 126. Pavcnik, N., Trae Liberalization, Exit, an Prouctivity Improvements: Evience from Chilean Plants, The Review of Economic Stuies, 22, 69 (1), Perloff, Jeffrey M. an Steven C. Salop, Equilibrium with Prouct Differentiation, The Review of Economic Stuies, 1985, 52 (1). Post, T., M. J. Van en Assem, G. Baltussen, an R. H. Thaler, Deal or no eal? Decision making uner risk in a large-payoff game show, The American Economic Review, 28, 98 (1), Ruin, W., Principles of mathematical analysis, McGraw-Hill New York, Saha, A., Expo-power utility: A flexible form for absolute an relative risk aversion, American Journal of Agricultural Economics, 1993, pp Samuelson, P. A., The monopolistic competition revolution, Monopolistic competition theory: stuies in impact, 1967, pp Solow, R. M., Monopolistic competition an macroeconomic theory, Cambrige University Press, Spence, M., Prouct Selection, Fixe Costs, an Monopolistic Competition, The Review of Economic Stuies, 1976, 43 (2), Stiglitz, J. E., Towars a more general theory of monopolistic competition, Prices, competition an equilibrium, 1986, p

29 Syverson, C., Market Structure an Prouctivity: A Concrete Example, Journal of Political Economy, 24, 112 (6), , What Determines Prouctivity?, Journal of Economic Literature, 211, 49 (2). Troutman, J. L., Variational calculus an optimal control: Optimization with elementary convexity, New York: Springer-Verlag, Tybout, J. R., Plant-an firm-level evience on "new" trae theories, Hanbook of International Trae, 23, 1, Venables, A. J., Trae an trae policy with imperfect competition: The case of ientical proucts an free entry, Journal of International Economics, 1985, 19 (1-2), Vives, X., Oligopoly pricing: ol ieas an new tools, The MIT press, 21. Weyl, E. G. an M. Fabinger, Pass-through as an Economic Tool, University of Chicago, mimeo, September 212. Zheloboko, Evgeny, Sergey Kokovin, Mathieu Parenti, an Jacques-François Thisse, Monopolistic competition in general equilibrium: Beyon the CES, Econometrica, forthcoming. A Appenix: Proofs A.1 A Folk Theorem In this context, we nee to efine the policy space. Provie M e an q(c), an assuming without loss of generality that all of q(c) is consume, allocations are etermine. The only question remaining is what class of q(c) the policymaker is allowe to choose from. A sufficiently rich class for our purposes is q(c) which are positive an continuously ifferentiable on some close interval an zero otherwise. This follows from the basic principle that a policymaker will utilize low cost firms before higher cost firms. Formally, we restrict q to be in sets of the form Q [,c ] {q C 1,> on [,c ] an otherwise}. 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 = 29

30 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. 28 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). 29 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 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 28 Since h is linear in x, H is linear an since v is strictly concave in x (using ρ < 1) so is V. 29 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. 3

31 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. 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, 31

32 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 4. 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) An it follows from Equations (1) we have [ ( )] ( ) 1 µ q mkt (c) u q mkt (c) /u ( q opt (c) ) = δ/λ. (11) 32

33 Suppose µ > > (1 ε), an it is sufficient to show inf 1 µ ( q mkt (c) ) σ, since then c c mkt 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 5. 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 characterizing the solution to every optimum also imply β (α) = v α (q(c ))/(c q(c ) + f /L), 33

34 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. Average elasticity of utility changes ue to a ifferent cost cutoff an quantity allocations across firms. An increase in market size raises the marginal utility of income at the rate of average markups lnδ/ lnl = 34

35 µ pqg/ pqg µ. From lnδ/ lnl an ln ε/ lnl, the change in welfare is [ ] [ lnu lnl = u(q(c )) c g(c ) ug ε (1 µ ) (ε ε)( µ µ ) + µ µ ε 1 µ 1 µ + µ q/µ µ ] εu ε ug G. When preferences are aligne, the first 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 secon term in square brackets is also positive when preferences are aligne, given regularity conitions in Proposition 8. Proof of Proposition 8. Following the iscussion above, it is sufficient to show that for γ (c) (µ + µ q/(1 µ)) 1 (εu/ ε ug), µ ε 1 µ 1 µ + µ q/µ µ εu ε ug G = [1 ε + µ q/(1 µ) ] γg. (13) This clearly hols for µ, an for the other case where preferences are aligne, we have µ < < ε. Expaning Equation (13) shows that [1 ε + µ q/(1 µ) ] γg = [1 ε µ]γg [µ µ]γg. Since ε >, 1 ε µ > an [1 ε µ]γg + 1 >. Therefore, it is sufficient to show that [µ µ]γg >. This sufficient conition is equivalent to u µ G ug µη u ug G (14) where η (c) γ (c) ( ug/u)/ γ. Since η (u/ ug)g = 1 an µ/c >, it follows that if η/c <, then Equation (14) hols by stochastic ominance. As η/c < iff η/q >, we examine the sign of η/q below. { sign{η/q} = sign ln ( µ + µ q/(1 µ) ) ( 1 ε/ ε ) } γ / lnq = sign { ( µ q + 2µ ) q/(1 µ) + ( ε q/ε µ q/(1 µ) )( µ + µ q/(1 µ) )}. The aitional hypothesis that (µq) guarantees that each term above is positive, so η/q > an we conclue Equation (14) hols, giving the result. 35

36 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 11. Assume markups are interior. Then uner the market allocation: 1. lim c mkt = iff lim p ( c mkt ) = iff lim Lq ( c mkt ) =. 2. lim c mkt = iff lim p ( c mkt ) = iff lim Lq ( c mkt ) =. 3. lim c mkt (, ) iff lim p ( c mkt ) (, ) iff lim Lq ( c mkt ) (, ). Similarly, uner the optimal( allocation: ) ( ) ( ) 1. lim c opt = iff lim u q c opt /λq c opt = iff lim Lq c opt =. ( ) ( ) ( ) 2. lim c opt = iff lim u q c opt /λq c opt = iff lim Lq c opt =. ( ) ( ) ( ) 3. lim c opt (, ) iff lim u q c opt /λq c opt (, ) iff lim Lq c opt (, ). 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 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 )) 36

37 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 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) 37

38 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 ). 38

39 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 39