DISCUSSION PAPER SERIES No. 9831 MARKET SIZE, ENTREPRENEURSHIP, AND INCOME INEQUALITY Kristian Behrens, Dmitry Pokrovsky and Evgeny Zhelobodko INTERNATIONAL TRADE AND REGIONAL ECONOMICS ABCD www.epr.org Available online at: www.epr.org/pubs/dps/dp9831.php www.ssrn.om/xxx/xxx/xxx
ISSN 0265-8003 MARKET SIZE, ENTREPRENEURSHIP, AND INCOME INEQUALITY Kristian Behrens, University of Quebe, Montreal, National Researh University, CIRPÉE and CEPR Dmitry Pokrovsky, Higher Shool of Eonomis, Mosow Evgeny Zhelobodko, Novosibirsk State University Disussion Paper No. 9831 February 2014 Centre for Eonomi Poliy Researh 77 Bastwik Street, London EC1V 3PZ, UK Tel: (44 20) 7183 8801, Fax: (44 20) 7183 8820 Email: epr@epr.org, Website: www.epr.org This Disussion Paper is issued under the auspies of the Centre s researh programme in INTERNATIONAL TRADE AND REGIONAL ECONOMICS. Any opinions expressed here are those of the author(s) and not those of the Centre for Eonomi Poliy Researh. Researh disseminated by CEPR may inlude views on poliy, but the Centre itself takes no institutional poliy positions. The Centre for Eonomi Poliy Researh was established in 1983 as an eduational harity, to promote independent analysis and publi disussion of open eonomies and the relations among them. It is pluralist and nonpartisan, bringing eonomi researh to bear on the analysis of medium- and long-run poliy questions. These Disussion Papers often represent preliminary or inomplete work, irulated to enourage disussion and omment. Citation and use of suh a paper should take aount of its provisional harater. Copyright: Kristian Behrens, Dmitry Pokrovsky and Evgeny Zhelobodko
CEPR Disussion Paper No. 9831 February 2014 ABSTRACT Market Size, Entrepreneurship, and Inome Inequality* We develop a monopolisti ompetition model with two setors and heterogeneous agents who self-selet into entrepreneurship, depending on entrepreneurial ability. The effet of market size on the equilibrium share of entrepreneurs ruially hinges on properties of the lower-tier utility funtion for differentiated varieties its elastiity of substitution and its Arrow-Pratt index of relative risk aversion. We show that the share of entrepreneurs, and the utoff for self-seletion into entrepreneurship, an inrease or derease with market size. The properties of the underlying ability distribution largely determine how inome inequality hanges with market size. JEL Classifiation: D31, D43, L11 and L26 Keywords: entrepreneurship, heterogeneous agents, inome inequality, market size and monopolisti ompetition Kristian Behrens University of Quebe, Montreal Department of Eonomis Case postale 8888 Suursale Centre-Ville Montréal Montréal, QC H3C 3P8 CANADA Email: behrens.kristian@uqam.a For further Disussion Papers by this author see: www.epr.org/pubs/new-dps/dplist.asp?authorid=159512 Dmitry Pokrovsky Higher Shool of Eonomis National Researh University Myasnitskaya ulitsa 20 Mosow 101000 RUSSIA Email: dm.pokrovsky@gmail.om For further Disussion Papers by this author see: www.epr.org/pubs/new-dps/dplist.asp?authorid=178018 *In memoriam Evgeny Zhelobodko (1973 2013). Novosibirsk State University, Russia; and National Researh University Higher Shool of Eonomis, Russia. Submitted 14 January 2014
1 Introdution Market size matters. It matters for produtivity, whih inreases by about 2 4% when the density of eonomi ativity doubles (Rosenthal and Strange, 2004). These produtivity gains an be asribed to any ombination of inreasing returns to sale, indivisibilities, and a variety of agglomeration externalities input-output linkages, labor market pooling, and knowledge spillovers that operate more strongly in bigger and denser markets (Duranton and Puga, 2004). Size also matters for the loation hoies of heterogeneous individuals. The size elastiity of the share of ollege graduates in us metropolitan statistial areas is, for example, 6.8% in the 2000 Census data. More effiient firms and more talented workers sort into larger ities (Baldwin and Okubo, 2008; Combes, Duranton, and Gobillon, 2008; Behrens, Duranton, and Robert-Nioud, 2014a), whih partly explains those ities produtivity advantage. The theoretial effets of market size on produtivity and sorting are fairly well understood and doumented by now. The same holds true for their empirial magnitudes. Little is still known, however, about the effets of size on firm seletion, self-seletion into entrepreneurship, and inome inequality. Do larger and denser markets provide a tougher ompetitive environment where less effiient firms and entrepreneurs fail more often? Do these markets provide greater inentives to set up firms and engage in entrepreneurship? And how do ompetition and self-seletion, driven by market size, translate into earnings inequality among heterogeneous individuals? These are the questions we theoretially address in this paper. The existing empirial evidene linking market size to firm seletion and entrepreneurship is inonlusive. Syverson (2004) douments that firm seletion is tougher in larger markets in the us ready-mix onrete industry. Combes, Duranton, Gobillon, Puga, and Roux (2012), however, find that firm seletion measured as the left trunation of the produtivity distribution is tiny and explains little to nothing of the produtivity advantage of large Frenh ities. Their results show that a right shift in the produtivity distribution, due to agglomeration effets that benefit all firms, is the key driver of higher produtivity in larger ities. This right shift operates more strongly for more produtive firms, thus dilating the produtivity distribution. Earnings inequality should, therefore, rise with ity size. That larger ities are indeed more unequal plaes has been doumented by Glaeser, Tobio, and Resseger (2010), Baum-Snow and Pavan (2014), and Behrens and Robert-Nioud (2014). The latter find that the size elastiity of the Gini oeffiient is 1.7% for the 507 largest us metropolitan areas in 2007. The ontribution of firm seletion to inequality remains, however, unlear for now. Turning to entrepreneurship, still less is known about the effets of market size and density on the ex ante deision to beome an entrepreneur, and on the subsequent ex post survival probability. Behrens et al. (2014a) doument that the share of self-employed a proxy for entrepreneurship is roughly onstant aross a sample of 276 us metropolitan statistial areas in the 2000 Census. Hene, there seems to be no link between market size and entrepreneurship. 1
Di Addario and Vuri (2010) find that the share of entrepreneurs grows when the population and employment density of Italian provines inreases. However, one individual harateristis and eduation are ontrolled for, the share of entrepreneurs dereases with market size. The probability of young Italian ollege graduates to be entrepreneurs three years after graduation dereases by 2 3 perentage points when the population density of a provine doubles. About one third of this seletion effet seems to be explained by inreased ompetition among entrepreneurs within industries. However, onditional on survival, suessful entrepreneurs in dense provines reap the benefits of agglomeration: their inome elastiity with respet to ity size is about 2 3%. Sato, Tabuhi, and Yamamoto (2010) find similar results for Japanese ities. Using survey data, they doument that the ex ante share of individuals who desire to beome entrepreneurs is higher in larger and denser ities: a 10% inrease in density inreases the share of prospetive entrepreneurs by about 1%. It however redues it ex post by more than that, so that the observed rate of entrepreneurship is lower in denser Japanese ities. To summarize, the empirial evidene suggests that larger markets have more prospetive entrepreneurs (more entrants), but ex post only a smaller share of those entrants survive (tougher seletion). Those who do survive in larger markets perform, however, signifiantly better. The objetive of our paper is to develop a monopolisti ompetition model of entrepreneurial hoie and seletion to investigate the effets of market size on entrepreneurship and inequality. To this end, we extend the model by Zhelobodko, Kokovin, Parenti, and Thisse (2012) to allow for heterogeneous agents à la Melitz (2003), who make an oupational hoie à la Luas (1978): they either beome entrepreneurs and produe differentiated varieties under monopolisti ompetition, or they beome workers that are hired by the entrepreneurs (as in Behrens et al., 2014a). To derive general results, we make only few assumptions on agents preferenes and on the underlying distribution of types in the population. Within that framework, we show that the effets of market size on the share of entrepreneurs, the output of their firms, the pries they harge, and their inomes, ruially hinges on two properties of the subutility funtion for differentiated varieties its elastiity of substitution and its Arrow-Pratt index of relative risk aversion. The ability utoff for self-seletion into entrepreneurship an either inrease or derease with market size, and so does the share of entrepreneurs and inome inequality in the eonomy. We also find that the underlying distribution of types ruially determines whether or not measured inequality rises or falls with market size. When taken together, our results suggest that the effets of market size on entrepreneurship and inequality are theoretially ambiguous and ruially hinge on modeling hoies. We thus annot expet to get lear-ut results, whih prompts us to be areful when interpreting the existing empirial evidene. Our findings also suggest that more empirial researh on this topi is required. In the end, sine theory an generate a large variety of responses of seletion and inequality to hanges in market size, the empirial razor is needed to sort out the theoretial speifiations onsistent with the stylized fats. 2
Our work builds on and extends several reent theoretial ontributions that aim to explain the links between market size, seletion, and inequality. Behrens and Robert-Nioud (2014) extend the model by Melitz and Ottaviano (2008) to an urban setting. They show that seletion is tougher in larger markets, and that this tougher seletion inreases inome inequality. In their model, inequality is driven by both those who fail the bottom of the inome distribution and by those who sueed well the top of the inome distribution. Our model an generate similar patterns. However, whereas market size benefits disproportionately the most able agents in Behrens and Robert-Nioud (2014), that effet an be reversed in our model, depending on the preferene struture. Behrens et al. (2014a) build on the onstant elastiity of substitution (heneforth, es) model and investigate the effets of market size on agglomeration, sorting, and seletion. One of their key findings is that seletion is independent of market size, so that the produtivity gains in larger ities are exlusively driven by agglomeration externalities and sorting along skills. Larger ities are also not more unequal in their framework. We show that the same results hold in our model for the es ase. Whereas little seletion in response to differenes in market size seems to align with empirial evidene, onstant inome inequality does not. Those two stylized fats annot be easily reoniled within simple monopolisti ompetition models. The remainder of this paper is organized as follows. Setion 2 presents the model and derives some omparative stati results for firm-level variables. Setion 3 proves existene and uniqueness of equilibrium and disusses the intuition underlying the effets of market size on the self-seletion utoff for entrepreneurship. Setion 4 ontains our key results linking the share of entrepreneurs to market size. Setion 5 then investigates and simulates the impats of market size on inequality, both among entrepreneurs and for the eonomy in general. Finally, Setion 6 onludes. All proofs and tehnial details as well as extensions of the model and supplementary proofs are relegated to an extensive set of appendies. 2 The Model Consider a losed eonomy with a single prodution fator, labor. There are L agents, eah endowed with one effiieny unit of labor. Hene, the maximum effetive labor supply equals L. There are two setors: a traditional setor and a modern one. Prodution in the traditional setor ours under onstant returns to sale and perfet ompetition. One unit of effetive labor produes one unit of a homogeneous output. Firms in the modern setor produe a differentiated good under monopolisti ompetition and firm-level inreasing returns to sale. Firms in that setor are run by entrepreneurs who differ by produtivity. Eah agent is free to hoose the setor he wants to work in, and labor is perfetly mobile aross setors. Although agents are homogeneous in their labor endowments, they differ in their innate entrepreneurial ability. The type of an agent is denoted by, whih we interpret 3
as the marginal ost of a potential firm launhed by an agent of this type. Hene, more able agents with a higher 1/, whih an be thought off as a measure of entrepreneurial ability an organize more effiient firms that operate with lower marginal ost. 1 We assume that is ontinuously distributed on [; ] with an at least twie ontinuously differentiable umulative distribution funtion Γ( ). We denote the assoiated density by γ( ). In what follows, we hoose the prie p A of the traditional good as the numeraire, and denote the wage in that setor by w A. Perfet ompetition then implies that w A = p A = 1. Perfet mobility of workers aross setors further implies that wages are equalized: w A = w M = w 1, where w M denotes the wage in the modern setor. Varieties in the modern setor an be ostlessly differentiated so that, in equilibrium, eah entrepreneur produes a distint variety. 2 Eah agent knows ex ante his entrepreneurial ability,, and takes it into onsideration when making his oupational hoie: (i) to beome an entrepreneur who operates a firm with marginal osts ; or (ii) to beome a salaried worker supplying his unit of labor to the fator market to earn the market wage. Oupational hoies are based on the highest inome an agent an seure. Sine a higher produtivity 1/ maps into higher profit, the most produtive agents in terms of entrepreneurial ability 1/ will operate as entrepreneurs, whereas the other agents will be workers. Formally, denote the optimal profit of an agent of type by π(). Let ĉ denote the type of agent who is indifferent between being an entrepreneur or being a worker. For that type, π(ĉ) = w, whereas π() > w for all < ĉ and vie versa. The self-seletion utoff for entrepreneurship, ĉ, is endogenous and depends on the harateristis of the eonomy suh as its size and its underlying ability distribution and on onsumers preferenes. Finally, the share of entrepreneurs in the eonomy is given by Γ(ĉ), with an ĉ assoiated mass of firms N L γ()d. 2.1 Preferenes and Demand We assume that all agents have idential quasi-linear preferenes. We model the utility derived from the onsumption of the differentiated good by a two-tier utility funtion, where the lower-tier is additively separable aross varieties. This is a standard assumption in models of monopolisti ompetition (e.g., Dixit and Stiglitz, 1977; Matsuyama, 1995; Zhelobodko et al., 2012). In our setting, there are Lγ() type- entrepreneurs and, therefore, Lγ() type- varieties. All type- varietes are produed at the same marginal ost and enter utility symmetrially. They will thus be onsumed in the same quantity and have the same prie. 1 The assumption that more able individuals beome entrepreneurs is the same as in, e.g., Luas (1978) or Behrens et al. (2014a). An alternative view is that entrepreneurs must be jaks-of-all-trades, i.e., have a balaned set of skills without neessarily being exellent anywhere (see Lazear, 2004, 2005). Poshke (2013) presents a model in whih both high-ability and low-ability agents beome entrepreneurs. The same an our in the model by Pokrovsky and Sharunova (2014), where agents are heterogeneous along two dimensions their entrepreneurial ability and their produtivity as a worker. 2 Agents with the same produtivity 1/ set up different firms that produe distint varieties of the good. 4
Let x x(), p p(x ), and γ γ() denote the quantity onsumed and prie harged for a variety of type with density γ. The lower-tier utility from onsuming all type- varieties then equals u(x )Lγ, and the expenditure for those varieties is p x Lγ. Given our assumptions, utility is expressed as follows: ( U V L u(x )γ d ) +A, (1) where A denotes the onsumption of the traditional good; and where u is an inreasing, onave, thrie ontinuously differentiable subutility funtion that satisfies u (x) < for all x > 0. We also impose the natural ondition that u(0) = 0 and assume that eah agent has an endowment A of the traditional good. This endowment is assumed to be large enough for the onsumption of the homogeneous good to be positive for all agents in equilibrium. Turning to the upper-tier funtion, V, we assume thatv( ) ln( ) in what follows (see, e.g., Oyama, Sato, Tabuhi, and Thisse, 2011; and Kukharskyy, 2012). In that ase, the existene of an interior equilibrium, where prodution of the differentiated good takes plae, is guaranteed. We explain this result in greater detail in Setion 3 and show in Supplemental Appendix D.2 that our key results extend to an arbitrary funtion V, provided an interior equilibrium exists. An agent with ability θ [;] has the following budget onstraint: L p x γ d+a = I(θ)+A, (2) where his inome I(θ) whih depends on his type θ is given by { 1 if θ (ĉ;] I(θ) = π(θ) if θ [;ĉ]. (3) The inverse demand for a type- variety is obtained from the onsumer s first-order onditions. That inverse demand is proportional to the marginal utility of the variety, and inversely proportional to some market aggregate, µ, that ats as a demand shifter: p(x ) = u (x ), for all [;ĉ], where µ L µ u(x )γ d. (4) Given the previous assumptions on u, it is lear that p(x) dereases from some stritly positive value to zero with the onsumption levelx. Observe that the higher the value ofµ, the lower the individual demand for eah variety, and the more the onsumers diversify their demand aross varieties for any given inome. This stronger demand for variety stimulates entrepreneurship by giving low-ability agents inentives to beome entrepreneurs. Observe thatµis the same for 5
all onsumers. It is, therefore, a universal harateristi a suffiient statisti of the market, whih is taken into onsideration by the produers when determining their optimal prie for the varieties they produe. Sine µ is a demand shifter, it is naturally linked to the degree of demand fragmentation, as in Zhelobodko et al. (2012). Yet, as shown later, a more fragmented demand does not neessarily imply that the market beomes more ompetitive in the lassial sense that the markups harged by firms are lower. Some omments are in order. First, even though agents differ by inomes, their onsumption of any type- variety will be the same. This is due to our assumption of idential quasilinear preferenes, whih rules out inome effets and makes the model tratable. Seond, a non-linear upper-tier utility funtion V( ) is required to guarantee that the substitution effet between the numeraire and the differentiated good is non-trivial. Observed that, in the general ase, the demand shifter µ in equation (4) is given by µ 1/V ( ) ĉ L u(x )γ d, whih is independent of the utoff ĉ and of market size L when V is linear. In that ase, market size has no bearing on the toughness of ompetition and on inequality, two key questions we want to investigate in the present paper. Last, let us stress again that the utoff type, ĉ, is endogenously determined and orresponds to the type of agent who is indifferent between working as an employee and starting a business as an entrepreneur. In what follows, the onavity properties of the lower-tier utility u( ) will prove important. We therefore introdue some notation and a few onepts. Define r f (x) xf (x) f (x), (5) whih is a measure of the onavity of f at x. When applied to the subutility funtion u, we obtain a measure of love-for-variety, given by the Arrow-Pratt index of relative risk aversion: r u (x) xu (x) u (x). (6) Note that the more onave is the lower-tier utility, the higher is the onsumer s relative lovefor-variety (heneforth, rlv). Let us further introdue the following notation for the elastiity of a funtion f with respet to its argument x: E x [f(x)]. Analoguously, we denote by e x [f(x, )] the partial elastiity of f with respet to x, holding all other variables onstant. Using that notation, the elastiity of demand for any variety of the differentiated good and, therefore, the markups that entrepreneurs an harge for that variety are haraterized by the Arrow- Pratt index r u of the lower-tier utility funtion. Using equation (4), it an be expressed as the opposite of the elastiity of the marginal utility as follows: r u (x) = u (x)x u (x) = E x [u (x)] = E x [p(x)] = p (x)x p(x). (7) 6
Expression (7) is the rlv alulated at the point of individual onsumption, x. It measures the urvature of the funtion u at that point. Under standard assumptions on u, the value r u (x) is between 0 and 1. Its preise behavior whether it is inreasing or dereasing depends on how different u is from a power funtion. For the latter, it is well known that the rlv is onstant. In the ase of an inreasing funtion r u, we are in the presene of inreasing elastiity of demand (heneforth, ied); whereas in the opposite ase, we are in the presene of dereasing elastiity of demand (heneforth, ded). The ase of a onstant r u for any value of x orresponds to the speial ase of es preferenes, whih play a partiular role as a borderline ase. 3 2.2 Prodution and Self-Seletion into Entrepreneurship Eah entrepreneur takes his inverse demand funtion, p( ), the demand shifter µ, and the wage w 1 as given. An entrepreneur of type- maximizes his profit π() = [p(x ) ]Lx with respet to output, x. 4 The first-order ondition p (x )x +p(x ) = yields the optimal prie and output, the latter being proportional to the individual onsumption of the variety. Rewriting the first-order ondition in terms of the markup, M() [p(x ) ]/p(x ), and realling the expression for the elastiity of inverse demand (7), we obtain a very simple ondition for the profit-maximizing pries in terms of the rlv: M() = r u (x ) p() = 1 r u (x ). (8) Using the first-order ondition (4) of the onsumer problem, and the first-order ondition (8) of the produer problem, the individual onsumption of a type- variety is a solution to the following equation: u (x )[1 r u (x )] = µ (9) for any given value of µ. The latter is, of ourse, determined from the general equilibrium of the model. A suffiient seond-order ondition for profit maximization is given by 2 π() x 2 = p (x )x + 2p (x ) = u (x )x µ + 2 u (x ) µ = u (x ) µ [ 2+ u (x )x u (x ) ] < 0, whih, realling the definition (5) of r f, an be rewritten as follows in terms of the rlv: r u (x ) < 2. (10) 3 Sine ied and ded are loal onepts that depend on the onsumption level x, a funtion may be ied over some range and ded over some other. We illustrate this point in the Supplemental Appendix D.3. 4 Choosing pries or outputs yields the same outome under monopolisti ompetition with a ontinuum of firms (Vives, 2001). 7
Condition (10) states the lassial ondition that the demand funtion (4) must not be too onvex for optimal pries to exist. The maximized operational profit an be rewritten in terms of the rlv as follows: π() = [p(x ) ]x L = r u(x ) 1 r u (x ) Lx. (11) As in Luas (1978), agents self-selet into entrepreneurship when that ativity is more profitable for them than being employees. A type- agent ompares his type with the utoff type to make a deision regarding his oupational hoie. Only agents whose marginal ost is lower than the utoff, < ĉ, set up firms. The mass of entrepreneurs in the eonomy thus equals N LΓ(ĉ). More formally, using (11), the indifferene ondition between entrepreneurship and salaried work an be expressed as: π(ĉ) = w 1 r u (xĉ) 1 r u (xĉ)ĉlx ĉ = 1. (12) Condition (12) pins down the equilibrium self-seletion utoff and, therefore, the share of entrepreneurs in the eonomy. The utoff ĉ whih an be viewed as the ability threshold of ative entrepreneurs is another market aggregate that is taken as given by agents. The higher the value ofĉ, the more entrepreneurs and thus produt diversity is supported by the market in equilibrium. Yet, the higher ĉ, the lower the average ability of entrepreneurs. The equilibrium trade-off between produt diversity a high value of ĉ and onsumption quantity a low value ofĉ will ruially determine how the share of entrepreneurs reats to hanges in market size. That trade-off fundamentally depends on onsumers preferenes. 2.3 Entrepreneurial Talent and Firm-Level Outomes Before investigating in depth the effets of market size on the self-seletion utoff and on inome inequality, we first look at firm-level equilibrium harateristis with respet to the ability of their entrepreneurs. Quite naturally, more able agents set up more effiient firms whih produe larger output, harge lower pries, have larger market shares, and earn larger profits. Note, however, that it is not possible to derive lear results on markups: depending on the underlying preferenes, more produtive firms may harge either higher or lower markups. 5 Let π π() for notational onveniene. Formally, we an prove the following results: Proposition 1 (Entrepreneurial Ability and Firm-Level Outomes). Consider two entrepreneurs with abilities 1 < 2. The more produtive entrepreneur, 1/ 1 > 1/ 2, runs a firm that: 5 In Melitz and Ottaviano (2008) and in Behrens, Mion, Murata, and Südekum (2014b), more produtive firms harge higher markups that are dereasing in market size. As shown in the seond paper, even if preditions on hanges in markups are unambiguous for eah market, it is diffiult to make statements about how markups hange in a multi-region trading world. For the ase of inreasing elastiity of substitution preferenes, where markups inrease with market size, see Bertoletti, Fumagalli, and Poletti (2009) and Zhelobodko et al. (2012). 8
(i) produes more output, x 1 > x 2 ; (ii) harges a lower prie, p(x 1 ) < p(x 2 ); and (iii) earns higher profits, π 1 > π 2, than the firm of the less produtive entrepreneur. Markups are dereasing with firm-level produtivity if r u is dereasing, and inreasing otherwise. Proof. See Appendix B.1. Agents with greater entrepreneurial ability organize firms with lower marginal ost, whih allows them to harge lower pries. Sine preferenes are symmetri aross varieties, demand is higher for heaper goods, i.e., more effiient firms have a larger market share. The elastiity of demand for eah variety being less than one, and given the properties of the subutility funtion u onavity, zero utility at zero onsumption larger sales volumes translate into higher revenue: E x [p(x)xl] = 1 r u (x) > 0. Markups, however, an be either inreasing or dereasing with the individual onsumption of a variety produed by a partiular firm, depending on the properties of r u. Put differently, depending on preferenes, it might turn out that more produtive firms harge higher markups or the other way round. Sine markups and output sold may a priori move in opposite diretions with produtivity, the total effet of produtivity on profits may be ambiguous. Nevertheless, we an show that even if r u is a dereasing funtion, the effet of an inrease in revenue exeeds the negative effet of the derease in the markup. From (7) and from Corollary A.2 in Appendix A, it follows that E x [π(x)] = E x [r u (x)p(x)x] = 2 r u (x) > 0, where the last inequality holds beause of the seond-order onditions (10) for profit maximization. Hene, irrespetive of whether more produtive firms harge higher or lower markups, they earn higher profits. 6 3 Equilibrium An equilibrium is haraterized by ĉ, µ, {x } [;ĉ], i.e., the two market statistis the selfseletion utoff, ĉ, and the demand shifter, µ and outputs suh that: (i) all onsumers maximize utility; (ii) produers maximize profits onditional on their inverse demand p( ) and the two market statistis; (iii) agents optimally hoose oupations, i.e., type-ĉ agents are indifferent between being workers or entrepreneurs, whereas all other agents pik the oupation that seures them the highest returns; and (iv) all markets lear. We an solve for equilibrium using the marginal utility of inome (4), the first-order onditions (9) for onsumers and produers, and the indifferene ondition (12). All the other variables espeially the pries and the onsumption of the numeraire good an then be retrieved from the first-order onditions of the onsumer problem and from the budget onstraint. We an prove the following result: 6 Reent empirial evidene strongly suggests that more produtive firms harge higher markups (see, e.g., De Loeker, Goldberg, Khandelwal, and Pavnik, 2012, for the ase of trade liberalization in India). This is also true in reent models of monopolisti ompetition with variable markups (Melitz and Ottaviano, 2008; Behrens et al., 2014b), where the pass-through of produtivity gains to onsumers is less than 100%. 9
µ µ ICC L 1 ICC L 15 SSC L 1 SSC L 15 ICC L 1 ICC L 15 SSC L 1 SSC L 15 2 1 ĉ 1 2 ĉ (a) Market size and tougher seletion (b) Market size and milder seletion Figure 1: Equilibrium Patterns and Changes in ĉ with Respet to L. Proposition 2 (Existene and Uniqueness). Conditions (4), (9), and (12) determine an equilibrium in terms of individual onsumptions, {x } [;ĉ], the demand shifter, µ, and the self-seletion utoff, ĉ. The equilibrium exists and is unique. Proof. See Appendix B.2 Proposition 2 establishes that the equilibrium exists and is unique when the upper-tier utility funtion is logarithmi. Note that with an arbitrary upper-tier utility funtion, V, an interior equilibrium need not exist (see the Supplemental Appendix D.1). The reason is that, depending on preferenes and on the underlying ability distribution, even the most produtive entrepreneurs may not be able to make profits that exeed the wage they ould seure as workers. In that ase, no agent wants to beome an entrepreneur and we have a orner equilibrium. We rule out this ase as it is obviously not of great interest to our analysis. Until now, we have only shown that the equilibrium exists and is unique. Clearly, in our monopolisti ompetition framework, its omparative stati properties depend on preferenes and on the distribution of entrepreneurial abilities among agents. To build intuition, we begin with a graphial analysis of the equilibrium. This analysis shows that the demand shifter µ, always inreases as the market expands. The hange in the self-seletion utoff ĉ and thus in the share of entrepreneurs an, however, go either way. 7 The two panels of Figure 1 depit the equilibrium in (ĉ;µ)-spae, at the intersetion of the urves µ 1 (ĉ L,) derived from (9), when ombined with the market aggregate (4) and µ 2 (ĉ L) derived from (9) when ombined with equation (12). We refer to the former urve as the intensity of ompetition ondition, i, and to the latter urve as the self-seletion ondition, ss. Both depend on the 7 Reall that the share of entrpreneurs equals Γ(ĉ). When Γ( ) and are given, hanges in ĉ thus diretly map into hanges in the share of entrepreneurs. Things are more ompliated when we onsider hanges in the distribution or in its support, two aspets that we will analyze in Setion 5. 10
parameter L. As illustrated in both panels of Figure 1, an inrease in market size L shifts both the i and the ss urves upwards, thus inreasing µ. We indeed show in the next setion that an inrease in market size, L, always inreases µ sine E L [µ] > 0. As already stated, the hange in the self-seletion utoff, ĉ, in response to a hange in L is more diffiult to analyze. A detailed analysis of those hanges is relegated to the next setion, but Figure 1 already shows that the utoff an either derease (panel (a)) or inrease (panel (b)). To understand why we annot generally expet unambiguous results, observe that the seond equilibrium ondition (9) depends on the marginal utility u( ) and the rlv. The third equilibrium ondition (12) an be rewritten as µ = u(xĉ)e u [xĉ]r u (xĉ)l, whih depends on the sale elastiity and the rlv. The properties of the first ondition (4) also depend on the properties of the subutility u( ). Thus, whether the share Γ(ĉ) of entrepreneurs in the eonomy inreases or dereases with market size L whih depends on the diretion of the shift in the self-seletion utoff ĉ ruially depends on preferenes via r u and E u. This is illustrated by panel (a) of Figure 1, whih depits the ase where the share of entrepreneurs dereases with market size (beause seletion beomes tougher ), whereas panel (b) depits the opposite ase where the share of entrepreneurs inreases (beause seletion beomes milder ). In the speial ase of es preferenes, both urves shift suh that the utoff remains unhanged. The intuition underlying the two ases is as follows. Observe that an inrease in population L orresponds to both a proportional inrease in the number of produers of eah type, as well as to a orresponding inrease in the number of onsumers. An inrease in the number of available varieties leads to a derease in their individual onsumption and, therefore, fores the least effiient entrepreneurs out of the market. Yet, the inrease in the number of onsumers gives entrepreneurs with lower skills inentives to enter the market, beause a larger market allows them to earn more than as a worker, even if they need to harge lower markups. We thus have two opposing fores due to market expansion, whih influene the employment struture of the eonomy. An inrease in market size shifts the i urve µ 1 (ĉ L) along the ss urve µ 2 (ĉ L,). Individual demands derease with L as the demand shifter µ inreases, and so does the seletion utoff ĉ. less effiient entrepreneurs. At the same time, entry ours from the bottom with The intuition is that when onsumers fragment their demand more strongly, there is a gap for entrepreneurs who an thus enter the market (Noke, 2006). Hene, ĉ inreases beause the ss urve µ 2 (ĉ L,) now shifts along the i urve µ1 (ĉ L) so that their rossing point moves to the right. However, sine the intensity of ompetition µ has hanged, optimal pries have hanged too. In partiular, pries fall as more entrepreneurs enter the market, thus reduing demand further and inreasing ompetition. Beause of that hange, survival beomes more diffiult, and the strength of this effet determines how muh the new self-seletion utoff will shift to the right. Note that the shift may be very small, so that the new equilibrium value ĉ an be smaller than the initial one (see pane (a)l of Figure 1). In other words, the share of entrepreneurs in the eonomy may fall as the market gets larger. 11
While the graphial analysis serves to build intuition, we show more formally in the next setion under what onditions eah ase arises. In partiular, we larify how market size influenes the share of entrepreneurs that operate in the eonomy, i.e., how it affets the selfseletion of agents. We then show, in Setion 5, how profits and inomes hange. This allows us to trae out hanges in inome inequality in response to hanges in the size of the eonomy. As usual, we an interpret an inrease in market size as opening the eonomy to free trade, whih makes our results on market size also useful for understanding the impat of international trade on inequality and seletion under monopolisti ompetition with more general preferene strutures than those usually used in the literature. 8 Alternatively, our results are useful in a ross-setional ontext, e.g., to understand why and how entrepreneurship and inome inequality may vary aross ities of different sizes. 4 Market Size and Entrepreneurship We now analyze more formally how hanges in market size as measured by population L affet the equilibrium of the model. We investigate, in partiular, the influene of market size on hanges in the self-seletion utoff, ĉ. For a fixed distribution of types, hanges in ĉ diretly map into hanges in the share of entrepreneurs and are, therefore, of partiular interest to us. The demand shifter, µ, hanges simultaneously with the self-seletion utoff in response to hanges in market size. We show that µ always inreases regardless of the preferenes and of the distribution of entrepreneurial abilities. The hange in the self-seletion utoff depends, however, on preferenes, and an be either positive or negative. Put differently, larger markets an either exbibit a larger or a smaller share of entrepreneurs, and a higher or lower ability threshold for self-seletion into entrepreneurship. We provide a full analytial haraterization of the onditions under whih eah ase ours. As we will see, the ase of es preferenes is bordeline in the sense that, given these preferenes, the share of entrepreneurs is independent of market size (see, e.g., Behrens et al., 2014a). As we have illustrated graphially in Figure 1, hanges in the two market statistis are ruial: the demand shifter, µ, and the self-seletion utoff, ĉ. We thus present all our results on the effet of market size in terms of elastiities of ĉ and µ with respet to population L. The impat on all other endogeneous variables pries, ouputs, profits an then be retrieved from the first-order onditions, given µ and ĉ, and we disuss this further in Setion 5 below when fousing on the impats of market size on inome inequality. In Setion 5, we also analyze in more detail how hanges in the underlying ability distribution γ( ) affet the share of entrepreneurs and inome inequality in the eonomy. 8 See Behrens and Murata (2012) and Behrens et al. (2014b) for an analysis using ara preferenes, and Kihko, Kokovin, and Zhelobodko (2013) for a two-fator two-setor model using quasi-linear preferenes. 12
4.1 CES Lower-Tier Utility Funtion To make the analysis more transparent, we first onsider the speial ase of es preferenes. These preferenes turn out to be neutral, in the sense that there is a onstant share of entrepreneurs in the eonomy. We do not provide formal proofs of that laim here beause it is a partiular ase of Proposition 3 that we prove below when we haraterize the possible hanges in the utoff ĉ depending on the properties of the lower-tier utility funtion. Assume hene that the lower-tier utility funtion is given by u(x) = x ρ, with 0 < ρ < 1. As shown by Zhelobodko et al. (2012), given these preferenes the seletion utoff in a Melitz-type model is onstant. However, firms in the Melitz model are not linked to individuals and their deisions with respet to their oupational hoie. This is done in, among others, Ohyama et al. (2011) and Kukharskyy (2012). In those two papers whih use es as the lower-tier utility funtion, and the logarithm as the upper-tier utility funtion market size does not influene the equilibrium share of entrepreneurs, the size of the firms, and pries. The same result holds true in Behrens et al. (2014a), who onsider a model in whih heterogeneous agents sort aross ities and self-selet into entrepreneurship based on a ombination of their talent and a random produtivity shok. In their model, preferenes are es but not quasi-linear, and there is no sunk ost of the Melitz-type. Instead, entrepreneurs self-selet into the oupation that seures them the highest returns, as they do in our paper. More generally, seletion is onstant in one-setor es models with inome effets (see, e.g., Pokrovsky and Skolkova, 2013), and this is also true in our paper. 9 It an readily be verified that when preferenes are given by the following utility funtion ( ) U = ĉ ln xρ Lγ d +A, the utoff ĉ is independent of market size. Neither the pries nor the output and as a onsequene the share of entrepreneurs or their (relative) inomes vary with market size. This implies that there is also no effet of market size on inome inequality, as we verify more formally in Setion 5 below. 4.2 General Lower-Tier Utility Funtion We now establish our key results onerning market size and entrepreneurship for the more general preferene struture given by (1), with V( ) ln( ). 10 Hene, instead of es prefer- 9 Behrens et al. (2014a) show that with imperfet sorting of types aross ities, the self-seletion utoff varies even under es preferenes. This result is entirely due to the fat that ities attrat more talented people, and it is the hange in skill omposition that an make seletion tougher in bigger markets. However, there is no pure effet of size on seletion, and even if the seletion utoff does vary aross ities, it does not vary substantially. We show in Setion 5 that these results need to be qualified in non-es models: differenes in seletion aross markets may be sizable. 10 In the Supplemental Appendix D.2, we extend our results to the ase of an arbitrary upper-tier utility funtion. Hene, our results arry through to this more general ase, provided that we have an interior equilibrium (see Appendix D.1 for a disussion). To simplify the exposition, we stik to log-preferenes in the main text and relegate more general results to the appendix. 13
enes, we use an unspeifi subutility u( ). As shown in Setion 3, the shift in the i and in the ss urves an lead to an inrease or a derease in the utoff ĉ. The shift in the urves depends on the harateristis of u( ), i.e., on the sale elastiity E x [u] and on the hange in the measure of onavity, r u ( ). Changes in the latter orrespond either to: (i) dereasing elastiity of demand (ded; r u < 0); (ii) inreasing elastiity of demand (ied; r u > 0); or onstant elastiity of demand (r u = 0, whih is the es ase we disussed before). It is important to emphasize from the start that, in general, all these properties are loal, i.e., they depend on the equilibrium onsumption level x. As shown in the Supplemental Appendix D.3, there exist well-behaved utility funtions that are ied over some range, and ded over some other range. The funtion u an even have onstant elastiity for a zero-measure set of points. However, onstant elastiity everywhere is equivalent to u( ) being a power funtion. Also, the sale elastiity an inrease over some range and derease over some other range. The impat of market size on the self-seletion utoff ĉ, and thus on the share of entrepreneurs Γ(ĉ), an be summarized as follows for any given equilibrium onsumption x: Proposition 3 (Market Size and the Share of Entrepreneurs). An inrease in market size, L, hanges the share of entrepreneurs in the following way: Proof. See Appendix B.4 (pointwise at x) ded ied r u < 0 r u = 0 r u > 0 E x[u] > 0 ĉ? ĉ = onst ĉ E x[u] = 0 ĉ = onst ĉ = onst ĉ = onst E x[u] < 0 ĉ ĉ = onst ĉ? To understand these results, observe that the onsumers willingness to pay for variety is haraterized by the rlv (Zhelobodko et al., 2012). The ied ase orresponds to an inreasing willingness of agents to pay for produt diversity as their inome and onsumption level expand. As should be lear, an inreasing willingness to pay for diversity provides strong market inentives for the provision of more variety, whih eteris paribus pushes towards a larger share of entrepreneurs in the eonomy. At the same time, a higher rlv raises markups and pries, whih pushes down individual demands and, eventually, profits. In the latter ase, it is tougher for the marginal entrepreneurs to survive. Given those two opposing hannels, it is not surprising that the diretion of hange in the share of entrepreneurs depends preisely on the behavior of both the rlv and the sale elastiity of the utility funtion. As mentioned before, under an inreasing measure of onavity of the subutility funtion (r u > 0), a fall in individual onsumption x due to an inrease in market sizeldereases eteris paribus r u and, therefore, pries. This requires that entrepreneurs must be able to sell a larger total quantity Lx to remain in the market. Sine pries of more produtive entrepreneurs fall 14
relatively less than those of less produtive entrepreneurs, the share of entrepreneurs dereases with L. The reason is that under ied relatively small inreases in onsumption map into relatively large hanges in utility. In that ase, agents have a strong inentive to inrease their spendings on lower-pried varieties, whih are preisely those provided by the more produtive entrepreneurs. In a nutshell, this leaves a smaller market size for the less produtive entrepreneurs, whih then go out of business. In the opposite ase, the outome is symmetri. Under ded and a dereasing measure of onavity of the utility funtion, a fall in individual onsumption x due to an inrease in market size L inreases r u and, therefore, pries. This allows less produtive entrepreneurs to profitably operate in the market so that the share of entrepreneurs inreases. A few additional omments are in order. First, as mentioned in the foregoing subsetion, es preferenes are the border ase between ied and ded. In that ase, the oupational struture of the eonomy is independent of the size of the market for any equilibrium value of x. For other preferenes, we may have loally absene of an effet of size on seletion, but this annot hold generally. Seond, there are two ases where an inrease in market size has a priori an ambiguous impat on the share of entrepreneurs. The first one, orresponding to a derease in the elastiity of the utility funtion a satiation effet and an inrease in the elastiity of demand, seems the more plausible one. In that ase, a larger L and an assoiated smaller x for eah variety, inrease the sale elastiity of utility, as well as derease markups and pries. A higher sale elastiity pushed towards more demand for the more produtive entrepreneurs, thus leaving less room for the less produtive entrepreneurs. As a result, the self-seletion utoff dereases and the share of entrepreneurs falls. 4.3 Extensions As shown in Appendix C.1 and Appendix C.2, the model an easily be extended to inorporate shoks to onsumers preferenes and to entrepreneurs produtivity. Without going into details, we an note the following. First, ommon preferene shoks that inrease the value onsumers attah to the differentiated good will, eteris paribus, inrease the share of entrepreneurs in the eonomy. At the same time, firms sizes and entrepreneurs profits also inrease. As we will show in the next setion, these hanges translate into larger inequality between entrepreneurs, and between workers and entrepreneurs. Thus, eonomies where onsumers attah more weight to the differentiated good produed by the entrepreneurs have a larger share of the latter and are also more unequal. 11 Seond, a ommon shok to the produtivity of all entrepreneurs in equilibrium is formally equivalent to a hange in the size of the market, L (see Appendix C.2 for the proof). Thus, 11 This finding also suggests that the reorientation of onsumer spending towards differentiated goods whih are more intensive in high-skilled labor lead to inreasing inome inequality. The relationship between trade, demand shifts, and inome inequality is an interesting avenue to explore in future researh. 15
onditional on an initial equilibrium and a distribution of abilities, ommon hanges to the ability of entrepreneurs and hanges in market size have similar effets on the eonomy. A poliy that would affet the produtivity of all entrepreneurs in the same way ould thus, e.g., be a perfet substitute for hanges in market size due to migration of agents aross ountries or ities. It is important to point out that, sine hanges in market size have an ambiguous effet on the self-seletion utoff and the share of entrepreneurs, the same holds true for produtivity shoks. Thus, a positive shok may very well lead to an inrease or a derease of the selfseletion utoff and the share of entrepreneurs, depending on the underlying preferenes. 5 Market Size, Ability Distribution, and Inome Inequality We now investigate how hanges in market size and in the ability distribution affet the distribution of inome aross heterogeneous agents. As shown in the previous setion, hanges in market size and the resulting hanges in the market aggregates indue hanges in the share of entrepreneurs. These hanges in employment struture lead, in turn, to hanges in the harateristis of ative firms their pries, outputs, and profits whih influene the distribution of inome in the eonomy as a whole and among entrepreneurs in partiular. Following Behrens and Robert-Nioud (2014), the effets of hanges in market size on entrepreneurs profits will be referred to as the intensive margin. Changes in the employment struture also affet the distribution of inomes via a omposition effet workers and entrepreneurs earn different inomes, and some entrepreneurs may swith to being workers or the other way round. We refer to this as the extensive margin. 12 We first investigate how entrepreneurs profits hange with market size. To this end, we look at how the whole profit distribution hanges as the size of the market inreases. The key finding is that larger markets an either inrease or derease the inequality in profits. A simple measure of inequality the profit ratio at different perentiles of the ability distribution, whih is aking to the interquartile range is analyzed later. Finally, we onlude by illustrating our model numerially for more omplex measures of inequality like the Gini index, whih subsume hanges at both the intensive and the extensive margins. 5.1 Changes in Profits Reall that the equilibrium profit of a type- entrepreneur is equal to π = (p )x L = r u(x ) 1 r u (x ) Lx. (13) 12 See Behrens and Robert-Nioud (2014) for the analysis of inome inequality in an urban model based on Melitz and Ottaviano (2008). They deompose hanges in inome inequality into an extensive and an intensive margin, and show that both margins ontribute to inrease inequality in that model. 16