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