QUANTIFYING THE GAP BETWEEN EQUILIBRIUM AND OPTIMUM UNDER MONOPOLISTIC COMPETITION

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1 Kristian Behrens, Giordano Mion, Yasusada Murata, Jens Suedekum QUANTIFYING THE GAP BETWEEN EQUILIBRIUM AND OPTIMUM UNDER MONOPOLISTIC COMPETITION BASIC RESEARCH PROGRAM WORKING PAPERS SERIES: ECONOMICS WP BRP 185/EC/218 This Working Paper is an output of a research proect impemented at the Nationa Research University Higher Schoo of Economics (HSE). Any opinions or caims contained in this Working Paper do not necessariy refect the views of HSE

2 Quantifying the gap between equiibrium and optimum under monopoistic competition * Kristian Behrens Giordano Mion Yasusada Murata Jens Suedekum January 1, 218 Abstract Equiibria and optima generay differ in imperfecty competitive markets. Whie this is we understood theoreticay, it is uncear how arge the wefare distortions are in the aggregate economy. Do they matter quantitativey? To answer this question, we deveop a muti-sector monopoistic competition mode with endogenous firm entry and seection, productivity, and markups. Using French and British data, we quantify the gap between the equiibrium and optima aocations. In our preferred specification, inefficiencies in the abor aocation and entry between sectors, as we as inefficient seection and output per firm within sectors, generate wefare osses of about 6 1% of gdp. Keywords: monopoistic competition; wefare distortions; equiibrium versus optimum; inefficient entry and seection; inter- and intra-sectora aocations JEL Cassification: D43; D5; L13. *This work was previousy circuated under the tite Distorted monopoistic competition. It contains statistica data from ONS which is Crown copyright and reproduced with the permission of the controer of HMSO and Queen s Printer for Scotand. The use of the ONS statistica data does not impy the endorsement of the ONS in reation to the interpretation or anaysis of the data. This work uses research datasets which may not exacty reproduce Nationa Statistics aggregates. Copyright of the statistica resuts may not be assigned, and pubishers of this data must have or obtain a icense from HMSO. The ONS data in these resuts are covered by the terms of the standard HMSO icense. Université du Québec à Montréa (esg-uqàm), Canada; Nationa Research University Higher Schoo of Economics, Russian Federation; and cepr. E-mai: behrens.kristian@uqam.ca. Department of Economics, University of Sussex; cep; cepr; and cesifo. E-mai: g.mion@sussex.ac.uk Nihon University Popuation Research Institute (nupri); University Research Center, Nihon University, Japan; and Nationa Research University Higher Schoo of Economics, Russian Federation. E-mai: murata.yasusada@nihon-u.ac.p Düssedorf Institute for Competition Economics (dice), Heinrich-Heine-Universität Düssedorf, Germany; cepr; and cesifo. E-mai: suedekum@dice.hhu.de 1

3 1 Introduction In imperfecty competitive markets, equiibria and optima generay differ in many respects such as the number of firms and firm-eve outputs. Whie this is we understood theoreticay, it is uncear how arge the wefare osses from these distortions are in the aggregate economy. Do they matter quantitativey? To answer this question, we deveop a muti-sector mode of monopoistic competition with endogenous firm entry and seection, productivity, and markups. Using data from France and the United Kingdom (UK), we quantify the gap between the equiibrium and optima aocations, and document patterns of inter- and intrasectora distortions that transate into wefare osses of about 6 1% of gdp. The wefare costs of monopoistic competition are hence sizabe. The theoretica iterature on equiibrium versus optimum aocations under monopoistic competition dates back at east to Dixit and Stigitz (1977). They anayze the tradeoff between product diversity and output per firm as a source of inefficiencies in genera equiibrium modes with unspecified utiity functions. More recenty, Zheobodko, Kokovin, Parenti, and Thisse (212) introduce heterogeneous firms into those modes, and Dhingra and Morrow (217) show that markets generay deiver a sociay inefficient seection of firms. Whie these insights are vauabe, they are derived from modes with a singe monopoisticay competitive sector. Extant studies thus abstract from a first-order feature of the data: sectors are highy heterogeneous. In France in 28, for exampe, there are 4,889 textie and footwear producers, which compete for an expenditure share of 2% by French consumers. Those firms operate, arguaby, in a different market and face different demands than the 4,67 manufacturers of wood products or the 124,22 heath and persona service providers, on which French consumers spend ess than.1% and amost 2% of aggregate income, respectivey. 1 Therefore, to answer the basic so-what question how great the overa wefare osses from imperfecty competitive markets are we need to enrich existing genera equiibrium modes to have both between-sector and within-sector heterogeneity. For instance, the textie industry may have some firms that produce too itte and others that produce too much, and at the same time, it may aso attract too many (or too few) firms and workers in equiibrium. This, in turn, means that some other industries may have fewer (or more) firms and workers than are sociay optima. Quantifying the magnitude of the aggregate wefare distortion in such a genera equiib- 1 See Tabe 1 in Section 4 for more detais about the data. The arge number of firms in each sector suggests that monopoistic competition is a reasonabe approximation of the market structure. 2

4 rium framework is important for at east two reasons. First, since the semina contribution by Dixit and Stigitz (1977), the existence of a gap between the equiibrium and optima aocations has been one of the most infuentia theoretica resuts in various appied fieds of economics. Yet, despite its importance, we are not aware of any attempt to put numbers on it whie taking into account heterogeneity both between and within sectors. Second, the rationae for government interventions in a particuar sector typicay reies on a partia equiibrium anaysis, thus ignoring the interdependencies between heterogeneous sectors. For exampe, the question of excess or insufficient entry into industries shoud be viewed from a genera equiibrium perspective: imited resources impy that excessive entry in some industries is ikey to go hand-in-hand with insufficient entry in others. Assessing the wefare costs of imperfect competition in a genera equiibrium setting is aso difficut for at east for two reasons. First, to capture misaocations within and between sectors, we need a mode with heterogeneous firms and sectors. Deveoping such a mode is chaenging, especiay with genera utiity functions and productivity distributions that can accommodate various specifications used in the iterature. Second, we need to compare the equiibrium and optima aocations. Whie the former is observabe from the data, the atter is not. It is thus not obvious how we can measure the gap between the equiibrium and the optimum quantifying something unobservabe is not an easy task. We address the first probem by buiding on Zheobodko et a. (212) and Dhingra and Morrow (217), who study the positive and normative aspects of a singe monopoisticay competitive industry. We extend their approach to incorporate mutipe sectors and aow the sectors to differ in many respects such as utiity functions and productivity distributions. Imposing standard assumptions on the upper-tier utiity function, we estabish existence and uniqueness of the equiibrium and optima aocations. Comparing those two aocations enabes us to characterize various distortions, which incude inefficiencies in the abor aocation and the masses of entrants between sectors, as we as inefficient firm seection and output per firm within sectors. We cope with the second probem by using a nove way to quantify the gap between the equiibrium and the optimum: the Aais surpus (Aais, 1943, 1977). Roughy speaking, we consider a panner who minimizes the resource cost whie achieving the equiibrium utiity eve. Since by definition the panner can do better than the market economy, it requires ess resources, thus generating a surpus. The advantage of the Aais surpus is that it can be used for comparing the equiibrium with the first-best aocation, i.e., even in contexts where equivaent or compensating variations or other reated criteria to compare different equiibria cannot be readiy appied due to a ack of prices. 3

5 Previewing our theoretica findings, we show that the distortion in the abor aocation depends on the interaction between two types of easticities: the easticities of the upper-tier utiity that govern the inter-sectora aocation; and the easticities of the subutiities that shape the intra-sectora aocation. These easticities are reated to the equiibrium and optima entry conditions. In equiibrium, entry in each sector occurs unti the expected revenue equas the expected cost of abor aocated to that sector. Since each firm takes consumers demands as given, its expected revenue depends on both types of easticities that govern consumers expenditure aocations between and within sectors. In contrast, the panner equates the margina socia benefit of entry with the margina socia cost of abor required for entry. Since the former equas the expected sectora utiity, it depends ony on the uppertier easticities and does not require information on the easticities of the subutiities. In fact, the panner does not face consumers demand functions when determining entry in each sector, whereas firms do. This difference creates distortions in the sectora abor aocation. One key message of muti-sector genera equiibrium modes is that, contrary to the conventiona approach that has studied singe industries in partia equiibrium, distortions in one sector depend on the characteristics of a sectors in the economy. Indeed, sectors are interdependent, so that an excessive abor aocation to some sectors, for exampe, impies an insufficient abor aocation to others. The inefficient abor aocation across sectors, in turn, causes distortions in entry patterns. In particuar, sectors with an excessive abor share tend to feature an excessive number of entrants. Thus, too many entrants in some sectors are accompanied by too few entrants in others, though this inter-sectora prediction on entry needs to be adusted by standard intra-sectora business steaing and imited appropriabiity effects as in Mankiw and Whinston (1986). Our genera framework nests many specifications in terms of utiity functions and productivity distributions that are used in the iterature. We take two of those specifications to data. The first one buids on Cobb-Dougas upper-tier utiity functions and constant easticity of substitution (ces) subutiity functions. Dhingra and Morrow (217) show that seection and firm-eve outputs are efficient in a singe-sector economy if and ony if the subutiity function is of the ces form. This resut is shown to hod in our muti-sector setting, thus impying that there are no intra-sectora distortions. However, with mutipe sectors, distortions in the abor aocation and firm entry sti arise in genera. Both disappear if and ony if the easticities of the ces subutiity functions are identica across a sectors. Otherwise, the abor aocation and firm entry are efficient within but not between sectors. In particuar, sectors with a higher easticity of the subutiity attract too many workers and firms, regardess of productivity distributions. 4

6 Our second exampe is a tractabe mode with variabe easticity of substitution (ves), where demands exhibit smaer price easticities at higher consumption eves. Unike the ces mode, this ves mode can account for variabe markups and incompete pass-through (e.g., Wey and Fabinger, 213; Mrázová and Neary, 217). It features a the kinds of distortions that we highight in the genera framework. We show that high-productivity firms aways produce too itte and ow-productivity firms too much, and that the market deivers too itte seection compared to the socia optimum. Entry and the abor aocation are aso inefficient, and with Pareto distributions the market aocates too many firms and workers to sectors where a arger mass of the productivity distribution is concentrated on ow-productivity firms. Previewing our empirica findings, we estabish four key resuts using data from France and the UK. First, there are substantia aggregate wefare distortions. In the muti-sector ves mode, they equa 6 1% of the tota abor input in either country. Second, inter-sectora misaocations are crucia for these aggregate distortions. When we constrain the economy to consist of a singe sector, thereby shutting down inefficiencies in entry and the abor aocation, the aggregate distortion can be 3% ower than the one predicted in the mutisector case. Put differenty, a singe-sector mode yieds downward-biased predictions for the tota wefare oss. Third, the muti-sector ces mode predicts an aggregate distortion of.3 2.5%, which is much smaer than the ves mode. The intuition is that this mode dispays by construction efficient seection and firm-eve outputs, thereby missing distortions within sectors. Last, we find simiar patterns of inefficient entry and seection between France and the UK. Insufficient entry arises amost excusivey for services, whie manufacturing sectors tend to exhibit excessive entry. Equiibrium firm seection is generay coser to optima one in manufacturing sectors. These resuts are robust to using different measures of firm size, e.g., empoyment or revenue, and different strategies to dea with fixed costs. Our paper is cosey reated to the recent iterature on the equiibrium and optima aocations in modes with a singe monopoisticay competitive sector, most notaby Zheobodko et a. (212), Nocco, Ottaviano, and Sato (214), Dhingra and Morrow (217), and Parenti, Ushchev, and Thisse (217). Reative to this recent strand of iterature, we make two contributions. First, we characterize both the equiibrium and optima aocations in a muti-sector monopoistic competition mode. Second, whie those papers focus excusivey on theory, we take our mode with heterogeneous sectors and firms to data to assess the quantitative importance of the distortions under monopoistic competition a question that remains unanswered since Spence (1976) and Dixit and Stigitz (1977). Our work is further reated to the cassic iterature in industria organization that studies wefare impications of 5

7 market power and inefficient entry for singe industries in partia equiibrium. Harberger (1954) is a semina reference for the former, and Mankiw and Whinston (1986) for the atter. Our monopoistic competition mode is compementary to this ine of research, and recognizes genera equiibrium interdependencies between sectors. The rest of the paper is organized as foows. Section 2 presents our genera mode, whie Section 3 turns to the specific sovabe exampes. The quantification procedure and resuts are discussed in Section 4. Section 5 concudes. 2 Genera mode Consider an economy with a mass L of agents. Each agent is both a consumer and a worker, and suppies ineasticay one unit of abor, which is the ony factor of production. There are = 1, 2,...,J sectors producing fina consumption goods. Each good is suppied as a continuum of differentiated varieties, and each variety is produced by a singe firm under monopoistic competition. Firms can differ by productivity, both within and between sectors. We denote by G the continuousy differentiabe cumuative distribution function, from which firms draw their margina abor requirement, m, after entering sector. An entrant need not operate and ony firms with high productivity 1/m survive. Let N E and m d be the mass of entrants and the margina abor requirement of the east productive firm in sector, respectivey. Given N E, a mass NE G (m d ) of varieties are then suppied by firms with m m d. 2.1 Equiibrium aocation The utiity maximization probem of a representative consumer is given by: max {q (m),,m} s.t. U U ( U 1,U 2,...,U J ) U N E J N E =1 u ( q (m) ) dg (m) p (m)q (m)dg (m) = w, (1) where U is a stricty increasing and stricty concave upper-tier utiity function that is twice continuousy differentiabe in a its arguments; u is a stricty increasing, stricty concave, and thrice continuousy differentiabe sector-specific subutiity function satisfying u () = 6

8 ; p (m) and q (m) are the price and consumption of a sector- variety produced with margina abor requirement m; and w denotes a consumer s income. im U ( U/ U ) = for a sectors to be active in equiibrium. We assume that Let λ denote the Lagrange mutipier associated with (1). The utiity-maximizing consumptions satisfy the foowing first-order conditions: u ( q (m) ) = λ p (m), where λ λ U/ U. (2) To aeviate notation, et p d p (m d ) and qd q (m d ) denote the price set and quantity sod by the east productive firm operating in sector, respectivey. From the first-order conditions (2), which hod for any sector and any firm with m m d, we then have u u (qd ( ) q (m) ) = pd p (m) and u (qd ) u (qd ) = λ p d λ p d, (3) which determine the equiibrium intra- and intersectora consumption patterns, respectivey. We assume that the abor market is competitive, and that workers are mobie across sectors. A firms hence take the common wage w as given. Turning to technoogy, entry into each sector requires to hire a sunk amount F of abor paid at the market wage. After paying the sunk cost, F w >, each firm draws its margina abor requirement m from G, which is known to a firms. Conditiona on surviva, production takes pace with constant margina cost, mw, and sector-specific fixed cost, f w. Let π (m) denote the operating profit of a firm with productivity 1/m, divided by the wage rate w. Making use of condition (2), and of the equivaence between price and quantity as the firm s choice variabe under monopoistic competition with a continuum of firms (Vives, 1999), the firm maximizes π (m) = L with respect to quantity q (m). [ u ( q (m) ) λ w m ] q (m) f (4) Athough λ w contains the information of a the other sectors by (2), each firm takes this market aggregate as given because there is a continuum of firms. From (4), the profit-maximizing price satisfies p (m) = mw 1 r u ( q (m) ), (5) 7

9 where r u (x) xu (x)/u (x) denotes the reative risk aversion or the reative ove for variety (Behrens and Murata, 27; Zheobodko et a., 212). 2 In what foows, we refer to 1/[1 r u (q (m))] as the private markup charged by a firm that produces output q (m). To estabish the existence and uniqueness of an equiibrium cutoff, (m d )eqm, and equiibrium quantities, q eqm (m) for a m [,m d ], we consider the zero cutoff profit (zcp) condition, given by π (m d ) =, and the zero expected profit (zep) condition, defined as π (m)dg (m) = F. Using (2), (4), and (5), the zcp and zep conditions can be expressed respectivey as foows: [ L 1 1 r u (q (m)) 1 [ ] 1 ( ) 1 m d 1 r u q d qd = f L, (6) ] mq (m)dg (m) = f G (m d )+F, (7) which even in our muti-sector economy aow us to prove the existence and uniqueness of the sectora cutoff and quantities. Formay, we have the foowing resut. Proposition 1 (Equiibrium cutoff and quantities) Assume that the fixed costs, f, and sunk costs, F, are not too arge. Then, the equiibrium cutoff and quantities {(m d )eqm, q eqm (m), m [,(m d )eqm ]} in each sector are uniquey determined. Proof See Appendix A.1. Turning to the abor aocation, L = N E[L mq (m)dg (m)+f G (m d )+F ], and the mass of entrants, N E, in each sector, we first provide two important expressions that must hod in equiibrium. 3 We then estabish the existence and uniqueness of the equiibrium abor aocation and entry. To this end, we introduce the foowing notation. Let E U,U U U U U and E u,q (m) u (q (m))q (m) u (q (m)) (8) denote the easticities of the upper-tier utiity and of the subutiity, respectivey. Let further ζ (q (m)) u (q (m)) u (q (m))dg (m) and ν (q (m)) 2 We assume that the second-order conditions for profit maximization, r u (x) xu u (q (m))q (m) u (q (m))q (m)dg (m) (x)/u (x) < 2 for a = 1, 2,...,J, hod (Zheobodko et a., 212, p.2771). 3 To aeviate notation, we henceforth suppress the eqm superscript when there is no possibe confusion. 8

10 denote the shares that a variety produced with margina abor requirement m in sector contributes to the ower-tier utiity U and to sectora expenditure, respectivey. Using these expressions, we obtain the foowing resut. Lemma 1 (Labor aocation and firm entry) Any equiibrium abor aocation in sector = 1, 2,...,J satisfies L = E U,U E u,q (m) J =1 E L, (9) U,U E u,q (m) where E u,q (m) E u,q (m)ζ (q (m))dg (m) is a weighted average of the easticities of the subutiity functions, where the weights are given by the contribution of each variety to the sectora utiity. Furthermore, any equiibrium mass of entrants satisfies N E 1 m d [1 r u (q (m))]ν (q (m))dg (m) = L f G (m d )+F. (1) Proof See Appendix B.1. Lemma 1 shows that, in any equiibrium, the abor aocation L can be expressed by the easticities E U,U of the upper-tier utiity function and the weighted average E u,q (m) of the easticities of the subutiity functions. We wi discuss the intuition for those terms in Section 2.3. The mass of entrants is affected not ony by L, but aso by effective entry cost f G (m d )+F, the distribution of the markup terms 1 r u (q (m)), and the expenditure shares ν (q (m)). It is worth emphasizing that we have not specified functiona forms for either utiity or productivity distributions to derive those resuts. Note that Lemma 1 does not yet impy existence and uniqueness of the equiibrium abor aocation and the equiibrium mass of entrants. The reason is that, whie the expression in the braces in (1) is uniquey determined by Proposition 1, the abor aocation L can depend on {N E } =1,2,...,J via E U,U. Thus, to estabish those properties, we impose some separabiity on the upper-tier utiity function. More specificay, assume that the derivative of the upper-tier utiity function with respect to the ower-tier utiity in each sector can be divided into an own-sector and an economy-wide component as foows: U U = γ U ξ Uξ, (11) whereγ >, ξ <, andξ are parameters. 4 Specification (11) incudes, for exampe, the 4 The crucia points are that, under condition (11), the ratio of the derivatives in (2) with respect to and 9

11 cases where the upper-tier utiity function is of the Cobb-Dougas or the ces form. When condition (11) hods, we can prove the foowing resut: Proposition 2 (Equiibrium abor aocation and firm entry) Assume that (11) hods. Then, the equiibrium abor aocation and masses of entrants {L eqm,(n E)eqm } =1,2,...,J are uniquey determined by (9) and (1). Proof See Appendix A Optima aocation Having anayzed the equiibrium aocation, we now turn to the optima aocation. 5 Assume that the panner chooses the quantities, cutoffs, and masses of entrants to maximize wefare subect to the resource constraint of the economy as foows: max L U ( ) U 1,U 2,...,U J {q (m),m d,ne,,m} s.t. U N E u (q (m))dg (m) { J } m d N E [Lmq (m)+f ] dg (m)+f = L. (12) =1 The panner has no contro over the uncertainty of the draws ofm, but knows the underying distributions G. Let δ denote the Lagrange mutipier associated with (12). The first-order conditions with respect to quantities, cutoffs, and the masses of entrants are given by: L u (q (m)) = δ m, δ δ U/ U (13) L u (q d) = Lm d δ qd +f (14) u (q (m)) δ dg (m) = [Lmq (m)+f ] dg (m)+f. (15) is independent of Nk E for k =,, and that it satisfies some monotonicity properties. Otherwise, the resuting system of equations becomes generay intractabe. 5 In the main text, we consider the prima first-best probem where the panner maximizes utiity subect to the economy s resource constraint. When quantifying the gap between the equiibrium and the optimum in Section 4, we wi anayze a dua probem where the panner minimizes the resource cost subect to a utiity eve. The atter aows us to derive the Aais surpus (Aais, 1943, 1977) that can be used for comparing the equiibrium with the first-best aocation, i.e., even in contexts where equivaent or compensating variations or other reated criteria to compare different equiibria cannot be readiy appied due to a ack of prices. More detais are reegated to Appendices D and F. 1

12 From the first-order conditions (13), which hod for any sector and any firm with m m d, we then have u u (qd ( ) q (m) ) = md m and u (qd ) u (qd ) = δ m d δ m d, (16) which determine the optima intra- and intersectora consumption patterns, respectivey. We start again with the cutoff and quantities. Noting that δ = u (q (m))/m for any vaue of m from (13), we can rewrite condition (15) as foows: L [ 1 E u,q (m) 1 ] mq (m)dg (m) = f G (m d )+F, (17) where E u,q (m) is defined in (8). We refer to 1/E u,q (m) as the socia markup that a firm with margina abor requirement m shoud optimay charge, and to m/e u,q (m) as the shadow price of a variety produced by a firm with m in sector. 6 Condition (17) which equates the margina socia benefit of entry in sector with its margina socia cost may then be understood as the zero expected socia profit (zesp) condition, which is anaogous to the zep condition (7). Furthermore, evauating (13) at m d and pugging the resuting expression into (14), we obtain an expression simiar to the zcp condition (6) as foows: ( 1 E u,q d 1 ) m d q d = f L, (18) which we ca the zero cutoff socia profit (zcsp) condition. Using (17) and (18), we can estabish the existence and uniqueness of the sectora cutoff and quantities. Proposition 3 (Optima cutoff and quantities) Assume that the fixed costs, f, and the sunk costs, F, are not too arge. Then, the optima cutoff and quantities {(m d )opt,q opt (m), m [,(m d )opt ]} in each sector are uniquey determined. Proof See Appendix A.3. Turning next to the optima abor aocation, L, and the optima masses of entrants, N E, we proceed in the same way as for the equiibrium case, and provide the foowing two expressions. 6 Dhingra and Morrow (217) refer to 1 E u,q (m) = [u (q (m)) δ mq (m)]/u (q (m)) as the socia markup, which captures the utiity from consumption of a variety net of its resource costs. Moreover, they abe [p (m) mw]/p (m) = r u (q (m)) as the private markup. We adopt their terminoogy but redefine the two markups in a sighty different way. 11

13 Lemma 2 (Labor aocation and firm entry) Any optima abor aocation in sector = 1, 2,...,J satisfies Furthermore, any optima mass of entrants satisfies L = E U,U J =1 E U,U L. (19) 1 m d N E E u,q = (m)ζ (q (m))dg (m) L f G (m d )+F. (2) Proof See Appendix B.2. Lemma 2 shows that, in any optimum, the abor aocation L can be expressed by the easticities E U,U of the upper-tier utiity. The mass of entrants is affected not ony by L, but aso by effective entry costs f G (m d )+F, the distribution of the socia markup terms E u,q (m), and the shares ζ (q (m)) that capture the reative contribution of a variety produced with margina abor requirement m to utiity in sector. Finay, simiary to the equiibrium anaysis, Lemma 2 does not yet impy the existence and uniqueness of the optima abor aocation and the optima masses of entrants. We thus impose again the separabiity condition (11) to estabish those properties as foows: Proposition 4 (Optima abor aocation and firm entry) Assume that (11) hods. Then, the optima abor aocation and masses of entrants {L opt,(n E)opt } =1,2,...,J are uniquey determined by (19) and (2). Proof See Appendix A Equiibrium versus optimum Having estabished existence and uniqueness of the equiibrium and optima aocations in Propositions 1 4, we now investigate the difference between these two aocations. The nove feature of our mode ies in abor and entry distortions between sectors. It is important to notice that characterizing abor and entry distortions for one sector requires information on a sectors. Put differenty, the abor aocation and, thus, entry are interdependent when there are mutipe sectors. Hence, entry distortions in our muti-sector mode generay differ from those in modes with a singe imperfecty competitive sector such as Mankiw and Whinston (1986) and Dhingra and Morrow (217). 12

14 To characterize the abor distortions, we compare expressions (9) from Lemma 1 with (19) from Lemma 2. We then obtain the foowing proposition. Proposition 5 (Distortions in the abor aocation) The equiibrium and optima abor aocations satisfy L eqm L opt if and ony if Υ E eqm U,U E eqm u,q (m) E opt U,U J =1 E eqm U,U E eqm u,q (m) J =1 E opt. (21) U,U Assume, without oss of generaity, that sectors are ordered such that Υ is non-decreasing in. If there are at east two different Υ s, then there exists a unique threshod {1, 2,...,J 1} such that the equiibrium abor aocation is not excessive for sectors, whereas it is excessive for sectors >. The equiibrium abor aocation is optima if and ony if a Υ terms are the same. Proof See Appendix A.5. As can be seen from (21), the interdependence of heterogeneous sectors is important for distortions in the abor aocation. Which sectors have an excessive abor aocation depends on two types of statistics: the easticities E U,U of the upper-tier utiity function, evauated at the equiibrium and the optimum; and the weighted averages E u,q (m) of the easticities of the subutiity functions, evauated at the equiibrium. To buid intuition for these statistics we focus on expressions (9) and (19) from Lemmas 1 and 2. The former comes from the equiibrium free entry condition (7) and the atter from the optima entry condition (15). Mutipying (7) and (15) by N E and rearranging, we have L eqm L opt = L(NE )eqm λ w = L(NE )opt δ u (q (m))q (m)dg (m) (22) u (q (m))dg (m). (23) The key difference is that the former refects zero expected profit by each firm, whereas the atter equates the margina socia benefit and the margina socia cost of entry for the panner to maximize socia wefare. Since firms and the panner have different obectives, the two aocations differ in genera. Using (22) and (23) we discuss them in terms of E U,U for equiibrium and optimum and of E u,q (m) for equiibrium. Easticities of the upper-tier utiity function. Expressions (22) and (23), together with the definition of λ in (2) and the definition of δ in (13), revea why L eqm and L opt invove the 13

15 easticities of the upper-tier utiity function, E eqm U,U and E opt U,U. Intuitivey, at the free entry equiibrium, the private cost wl eqm /(N E)eqm of an entrant in equation (22) is ust offset by the private benefit, i.e., firms expected revenue (L/λ ) u (q (m))q (m)dg (m). Since the atter depends on the inverse demand functions (2), the equiibrium abor aocation L eqm is affected by the easticities of the upper-tier utiity via λ. By contrast, the socia cost of an additiona entrant is proportiona to L opt /(N E)opt, which by equation (23) must be equa to (L/δ ) u (q (m))dg (m) at optimum. Note that the atter refects the (expected) margina socia benefit generated by the additiona entrant. Thus, the optima abor aocation L opt depends on the easticities of the upper-tier utiity via δ. It is worth emphasizing that even when the equiibrium and optima easticities of uppertier utiity in each sector are the same, i.e., E eqm U,U = E opt U,U, their sectora heterogeneity pays a crucia roe in the abor distortions as ong as there is sectora heterogeneity in E eqm u,q (m). Indeed, athough E eqm U,U and E opt U,U in the eft-hand side of (21) cance out when they are identica, the easticities in the right-hand side remain. We wi eaborate on this point in the next section where we iustrate some exampes. Weighted average of the easticities of the subutiity functions. Expressions (22) and (23) revea why L eqm depends on the weighted averages of the easticities of the subutiity functions, E eqm u,q (m), whereas Lopt does not. To understand this difference, reca that the private benefit of an entrant is given by (L/λ ) u (q (m))q (m)dg (m). Since this expected revenue for the entrant invoves the consumers inverse demand functions, the equiibrium abor aocation L eqm depends not ony on E eqm U,U via λ but aso on E eqm u,q (m) via u (q (m)). 7 Whie the equiibrium abor aocation is determined by the firms that care about zero expected profit conditiona on the consumers demand, the optima abor aocation is determined by the panner who maximizes socia wefare with respect to the mass of entrants. Since the atter does not invove the inverse demand functions, it is independent of E eqm u,q (m). Other things equa, the higher E eqm u,q (m) the more abor is aocated to sector by (9) because consumers aocate a arge share of their budget to that sector by (22). Furthermore, a sector with higher E eqm u,q (m) reative to the other sectors tends to dispay an excessive abor 7 The weighted average satisfies E eqm u,q (m) < 1 because E u,q (m) < 1 for a m [,md ] by concavity of u, and because E u,q (m) ζ (q (m))dg (m) < ζ (q (m))dg (m) = 1 by the definition of ζ (q (m)). 14

16 aocation by (21). To see why a sector with higher E eqm u,q (m) assume that E eqm U,U = E opt utiity. If E eqm u,q (m) = E eqm u 1,q 1 (m) tends to dispay an excessive abor aocation, U,U for a, which is the case with the Cobb-Dougas upper-tier hods for a = 1, then the equiibrium abor aocation is optima by (9) and (19). However, if E eqm u J,q J (m) > E eqm u,q (m) = E eqm u 1,q 1 (m) for = 2, 3,...,J 1, then expenditure on and thus the abor aocation to sector J gets arger at the expense of the other sectors, whereas the optima abor aocation does not change. Thus, sector J dispays an excessive abor aocation, whereas the other sectors = 1, 2,...,J 1 exhibit an insufficient abor aocation. The atter is a genera equiibrium effect: an excessive aocation to one sector must go hand in hand with an insufficient aocation to the other sectors. Note that what matters is the reative magnitude of E eqm u,q (m). Indeed, it is easy to see that a proportionate increase in E eqm u,q (m) for a does not affect the equiibrium aocation by (9) and, hence, excess or insufficient abor aocation by (21). Turning to entry distortions, we compare expression (1) from Lemma 1 with (2) from Lemma 2 to obtain the foowing proposition. Proposition 6 (Distortions in firm entry) The equiibrium and optimum masses of entrants satisfy (N E )eqm /(N E )opt 1, if and ony if L eqm L opt Proof fg ((m d )opt )+F f G ((m d 1 (m d ) eqm )eqm )+F [1 r u (q eqm (m))]ν (q eqm (m))dg (m) 1 (m d )opt E u,q opt (m) ζopt (q (m))dg (m) Expression (24) directy foows from (1) and (2). 1. (24) Expression (24) shows that (N E)eqm /(N E)opt depends on three terms. The first term L eqm /L opt vanishes in a singe-sector mode, because L eqm = L opt = L. In a muti-sector mode, however, the gap between L eqm pays a crucia roe, as mentioned above. The second and third terms in (24) capture two additiona margins, namey effective and L opt fixed costs and private and socia markups, which depend on the cutoffs and quantities both at equiibrium and optimum. Reca that by the proofs of Propositions 1 and 3, λ w and δ are uniquey determined without any information on the other sectors. Hence, even in our muti-sector framework, the anaysis of cutoff and quantity distortions in each sector turns out to work as in the singe-sector mode by Dhingra and Morrow (217). We sha not repeat their theoretica anaysis here, but we first briefy discuss them, and then provide specific exampes in the next section. Those exampes wi be taken to the data in Section 4. 15

17 Effective fixed costs. The second term in (24) shows that if the market deivers too itte seection, (m d )eqm > (m d )opt, entry tends to be insufficient. The reason is that the higher surviva probabiity in equiibrium, as compared to the optimum, increases the expected fixed costs that entrants have to pay. This reduces expected profitabiity and discourages entry more in equiibrium than in optimum. In contrast, other things equa, too much equiibrium seection, (m d )eqm < (m d )opt, eads to excessive entry. Private and socia markups. The ast term in (24) shows that the gap between equiibrium and optima entry depends on the private and socia markup terms, which may exacerbate or attenuate excess entry (Mankiw and Whinston, 1986; Dhingra and Morrow, 217). The numerator can be reated to the business steaing effect: the higher the private markups 1/[1 r u (q (m))], the more excessive the entry. The denominator, in turn, captures the imited appropriabiity effect: the greater the socia markups 1/E u,q (m), the more insufficient the entry. Thus, the ast term in (24) depends on the reative strength of these two effects, as we as on the weighting schemes ν (q (m)) and ζ (q (m)) that are determined by the properties the subutiity function u and the productivity distribution function G. To sum up, the difference between market equiibrium and socia optimum in terms of the abor aocation and firm entry across heterogeneous sectors depends, in genera, on four key ingredients: the easticities of the upper-tier utiity; the weighted averages of the easticities of the subutiities; effective fixed costs; and private and socia markups. Whie distortions in a singe-sector mode are characterized soey by u and G for that sector (Dhingra and Morrow, 217), in a muti-sector setting characterizing distortions for one sector requires additiona information on the easticities of the upper-tier utiity, E U,U, and the weighted averages of the easticities of the subutiities, E u,q (m), for a sectors. Hence, when assessing distortions we need to take into account the interdependence between heterogeneous sectors. 3 Exampes Our resuts in the Propositions and Lemmas presented so far are genera enough to encompass various specifications of utiity functions and productivity distributions used in the iterature. We now consider specific upper-tier utiity and subutiity functions that enabe us to express the two types of easticities, E U,U and E u,q (m), in simpe parametric forms. We then take the parametric modes to data in Section 4. 16

18 Concerning the subutiity function u, we first anayze in Section 3.1 the ubiquitous ces case that has dominated much of the iterature on monopoistic competition. We then turn to a tractabe variabe easticity of substitution (ves) mode in Section 3.2 In doing so, notice that the ower-tier utiity U in specification (1) does not nest the standard homothetic ces aggregator. To nest it, we consider a simpe monotonic transformation of the ower-tier utiity in (1) as Ũ (U ). In Section 3.1 we assume that Ũ (U ) = U 1/ρ [N E m d q (m) ρ dg (m)] 1/ρ, whereas we retain Ũ (U ) = U in Section 3.2. Even with the transformation Ũ of the ower-tier utiity, we can re-estabish the genera = resuts shown in Section 2, as ong as we et Ũ () =, Ũ >, and im U Ũ(U ) =, whie repacing the condition in (11) with where γ >, ξ <, and ξ are parameters. 8 U Ũ Ũ U = γ Ũ ξ Uξ, (25) Turning to the upper-tier utiity function, we consider in the remainder of this paper the standard ces form: U = { J =1 β [Ũ (U )] (σ 1)/σ } σ/(σ 1), where σ 1, β > for a, and J =1 β = 1. Thus, the easticity of the upper-tier utiity function is given by E U,U ( U/ Ũ )( Ũ / U )(U /U) = β ( Ũ / U )(U /Ũ )(Ũ /U) (σ 1)/σ. When σ 1, the upper-tier utiity reduces to the Cobb-Dougas form, U = J =1 [Ũ (U )] β, so that ( ) Ũ U E U,U = β. (26) U Ũ The Cobb-Dougas upper-tier utiity function aways satisfies condition (25) that guarantees the existence and uniqueness of the equiibrium and optima aocations. When the uppertier utiity function is of the ces form, whereas the ower-tier utiity is of the homothetic ces form with Ũ (U ) = U 1/ρ, we have ( U/ Ũ )( Ũ / U ) = (β /ρ )Ũ σ 1 σ ρ U 1/σ. Hence, in that case, it is required that (σ 1)/σ < ρ for condition (25) to hod with ξ <. 9 Retaining σ 1 for now, we consider two specific forms for the subutiity functions for which the weighted averages of the easticities of the subutiity functions dispay a simpe 8 The proofs are virtuay identica to the ones in Appendices A and B, except that U/ U needs to be repaced with ( U/ Ũ )( Ũ / U ). Observe that in a singe-sector mode, the choice of Ũ does not affect distortions because it is a monotonic transformation of the overa utiity in that case. In a muti-sector mode, however, sectora aocations and thus aggregate distortions are affected by Ũ. 9 Shoud (σ 1)/σ > ρ hod, goods are Hicks-Aen compements (see, e.g., Matsuyama, 1995) so that mutipe equiibria with some inactive sectors may arise. Since ξ < is not satisfied in that case, we excude it from our anaysis. 17

19 behavior. We wi return to the case with σ > 1 in Section 4 where we quantify the mode. 3.1 ces subutiity We first discuss the case of the ces subutiity that has been widey used in the iterature. Assume that u (q (m)) = q (m) ρ and Ũ (U ) = U 1/ρ, where ρ (, 1) for a sectors. The abor distortion in Proposition 5, and thus the first term of (24) that characterizes entry distortion in Proposition 6, depend on E eqm U,U, E opt U,U, and E eqm u,q (m). Using (26), the easticity of the upper-tier utiity function can be rewritten as E eqm U,U = E opt U,U = β /ρ. Furthermore, when the subutiity function is of the ces form, we know that E u,q (m) = ρ for a m, so that E eqm U,U = ρ by the definition in Lemma 1. The entry distortion in Proposition 6 depends aso on the cutoffs and quantities. Since we have shown that the cutoff and quantity distortions can be studied on a sector-by-sector basis even in our muti-sector mode, we can appy the singe-sector resut by Dhingra and Morrow (217), i.e., in the ces case (m d )eqm = (m d )opt and q eqm (m) = q opt (m) for a m irrespective of the underying productivity distribution G. Furthermore, since E u,q (m) = 1 r (q (m)) and ν (q (m)) = ζ (q (m)) hod for a m, the second and the third terms in (24) vanish, so that (N E)eqm /(N E)opt = L eqm /L opt. Hence, we can restate Propositions 5 and 6 for this specific exampe as foows: Coroary 1 (Distortions in the abor aocation and firm entry with ces subutiity) Assume that the subutiity function in each sector is of the ces form, u (q (m)) = q (m) ρ. Then, the abor aocation and the masses of entrants satisfy L eqm L opt and (N E)eqm (N E)opt, respectivey, if and ony if 1 ρ J =1 β. (27) /ρ Assume, without oss of generaity, that sectors are ordered such that ρ is non-decreasing in. If there are at east two different ρ s, there exists a unique threshod {1, 2,...,J 1} such that the equiibrium abor aocation and firm entry are not excessive for sectors, whereas they are excessive for sectors >. The equiibrium abor aocation and firm entry in the ces case are optima if and ony if a ρ s are the same. Proof See above. Severa comments are in order. First, since there are no cutoff and quantity distortions in the case of ces subutiity functions, the market equiibrium is fuy efficient if and ony if 18

20 the ρ s are the same across a sectors. However, there are distortions in the abor aocation and in the masses of entrants when the ρ s vary across sectors. 1 Second, ρ in the ces mode can be reated not ony to the inverse of the markup, but aso to E U,U and to E u,q (m). It is the atter two easticities that matter for the abor and entry distortions. The reason is that the difference between the equiibrium and optima abor aocations comes from E U,U = β /ρ and E u,q (m) = ρ, which are determined by the first derivatives of Ũ and u as seen from (26) and the definition of E u,q (m). In contrast, the markup depends on r u, which invoves the second derivative of u. Thus, in the case of the Cobb-Dougas upper-tier utiity and ces subutiity functions, markup heterogeneity is not a determinant of abor and entry distortions. Third, Coroary 1 hods irrespective of the functiona form for G. Hence, productivity distributions pay no roe in the optimaity of the market outcome for the standard case with the Cobb-Dougas upper-tier utiity and ces subutiity functions. Last, since Coroary 1 ony pertains to the cass of ces subutiity functions, it must not be read as a genera if and ony if resut for any subutiity function. Indeed, as we show in the next subsection, the abor aocation and entry can be efficient even when the subutiity function is not of the ces form. 3.2 ves subutiity We have so far examined the case of ces subutiity functions without cutoff and quantity distortions. We now turn to our ves exampe where a types of distortions cutoff, quantity, abor, and entry distortions can operate. Specificay, we consider the constant absoute risk aversion (cara) subutiity as in Behrens and Murata (27), u (q (m)) = 1 e α q (m), where α is a stricty positive parameter. This specification can be viewed as an exampe of the ves subutiity anayzed in the semina paper by Krugman (1979). It is anayticay tractabe, and generates demand functions exhibiting smaer price easticities at higher consumption eves. Unike the ces mode, this ves case can therefore account for the empiricay we-documented facts of incompete pass-through and higher markups charged by more productive firms within each sector. In what foows, we assume that Ũ (U ) = U, so that E eqm U,U = E opt U,U = β by (26). To express E eqm u,q (m) in a parametric form, we aso assume that G foows a Pareto distribution G (m) = ( m/m max ) k, where both the upper bounds m max > and the shape parameters 1 Hsieh and Kenow (29) consider a heterogeneous firms mode where the mass of firms is either fixed or invariant, and where the ρ s are the same across a sectors. In contrast, Epifani and Gancia (211) aow for heterogeneity in the ρ s across sectors, yet consider homogeneous firms within sectors. 19

21 k 1 may differ across sectors. We reegate most anaytica detais for the case with cara subutiities and Pareto productivity distributions to Appendix E. We show there that the equiibrium and optima cutoffs are given as foows: (m d ) eqm = [ ] α F (m max ) k 1 k +1 κ L and (m d ) opt = [ ] α F (m max ) k (k + 1) 2 1 k +1, (28) L where κ k e (k +1) 1 (1+z)( z 1 +z 2 ) (ze z ) k e z dz > is a function of the shape parameter k ony. Using expressions (28), we can estabish the foowing resut: Proposition 7 (Distortions in the cutoff and quantities with cara subutiity) Assume that the subutiity function in each sector is of the cara form u ( q (m) ) = 1 e α q (m), and that the productivity distribution foows a Pareto distribution, G (m) = (m/m max ) k. Then, the equiibrium cutoff exceeds the optima cutoff in each sector, i.e., (m d )eqm > (m d )opt. Furthermore, there exists a unique threshod m (,(md )opt ) such that q eqm (m) < q opt (m) for a m [,m ) and q eqm (m) > q opt (m) for a m (m,(md )eqm ). Proof See Appendix A.6. Three comments are in order. First, in this mode, more productive firms with m < m underproduce, whereas ess productive firms with m > m overproduce in equiibrium as compared to the optimum in each sector. Notice that both types of firms coexist in equiibrium since the threshod m satisfies the inequaities < m < (md )opt < (m d )eqm. 11 Second, using (28), the gap between the equiibrium and optima seection can be expressed as a simpe function of the sectora shape parameter ony: (m d )opt /(m d )eqm = [κ (k + 1) 2 ] 1/(k +1) < 1. Since this expression increases with k, the arger the vaue of k (i.e., a arger mass of the productivity distribution is concentrated on ow-productivity firms) the smaer is the magnitude of insufficient seection in sector. Finay, Proposition 7 hods on a sector-by-sector basis, regardess of the abor aocation and the masses of entrants. Thus, our resuts on cutoff and quantity distortions woud aso appy to a singe-sector version of the cara mode. Turning to the abor and entry distortions, the combination of cara subutiity functions and Pareto productivity distributions yieds the equiibrium and optima masses of entrants 11 This need not aways be the case, however. For exampe, Dhingra and Morrow (217) derive genera conditions for cutoff and quantity distortions in a singe-sector framework. In their mode with an arbitrary subutiity function and an arbitrary productivity distribution, it is possibe that m exceeds (md )eqm. In that case, a firms (even the east productive ones) woud underproduce, whereas in our mode some firms (the east productive ones) aways overproduce from a socia wefare point of view. 2

22 as foows (see expressions (E-15) (E-16) and (E-3) (E-31) in Appendix E): (N E ) eqm = Leqm and (N E ) opt = (k + 1)F L opt (k + 1)F. (29) Thus, as in the ces case, we have (N E)eqm /(N E)opt = L eqm /L opt we know that distortions in the abor aocation are determined by E eqm u,q (m). From Proposition 5, together with E eqm U,U = E opt U,U = β. When the subutiity function is of the cara form and the productivity distribution foows a Pareto distribution, we can show that E u,q (m) depends soey on the sectora shape parameter k as foows: Lemma 3 (Weighted average of the easticities of the cara subutiity functions) Assume that the subutiity function in each sector is of the cara form, u ( q (m) ) = 1 e α q (m), and that the productivity distribution foows a Pareto distribution, G (m) = (m/m max ) k. Then, the weighted average E u,q (m) of the easticities of the subutiity functions in each sector can be rewritten as θ 1 (1 z)ez 1 (1+z)e z 1( ze z 1) k 1 dz 1 (1 ez 1 )(1+z)e z 1 (ze z 1 ) k 1 dz. (3) Proof See Appendix B.3. To characterize the abor and entry distortions, we rank sectors such that θ 1 θ 2... θ J. Since θ is increasing in k, ranking sectors by θ is equivaent to ranking them by k. Pugging (3) into (21), using E U,U = β from the upper-tier Cobb-Dougas specification, and noting that (N E)eqm /(N E)opt = L eqm /L opt by (29), we can restate Propositions 5 and 6 for this exampe as foows: Coroary 2 (Distortions in the abor aocation and firm entry with cara subutiity) Assume that the subutiity function in each sector is of the cara form, u ( q (m) ) = 1 e α q (m), and that the productivity distribution foows a Pareto distribution, G (m) = (m/m max ) k. Then, the abor aocation and the masses of entrants satisfy L eqm L opt and (N E)eqm (N E)opt, respectivey, if and ony if θ J β θ. (31) =1 Assume, without oss of generaity, that sectors are ordered such that θ is non-decreasing in. If there are at east two different θ s, there exists a unique threshod {1, 2,...,J 1} such that the equiibrium abor aocation and firm entry are not excessive for sectors, whereas they are 21