Plants, Productivity, and Market Size, with Head-to-Head Competition

Transcription

1 Plants, Productivity, and Market Size, with Head-to-Head Competition Thomas J. Holmes Wen-Tai Hsu Sanghoon Lee June 23, 2011 Abstract. Note: The paper is for a conference presentation and is preliminary. The appendix it refers to is incomplete. Department of Economics, the University of Minnesota, Federal Reserve Bank of Minneapolis, and the National Bureau of Economic Research. Department of Economics, the Chinese University of Hong Kong. Sauder School of Business, the University of British Columbia.

2 1 Introduction The role that increased competition can play in driving down the mark-up between price and marginal cost has long been understood. A more recent literature has emphasized the role that increased competition plays in selection when firms are heterogeneous in productivity. When trade barriers fall and firms compete with a larger pool of potential rivals, the selection process is tougher and only the most efficient survive. Reallocating production from low to high productivity firms results in welfare gains. See Melitz (2003), Eaton and Kortum (2002), Bernard, Eaton, Jensen, and Kortum (2003), hereafter BEJK, and the recent survey of Redding (2011). Models that simultaneously capture both channels of increased trade the pro-competitive effect of lower mark-ups, and the higher-productivity effect of increased selection are useful for at least two reasons. First, both channels may be at work in any given empirical application, so it is useful to have one unified model that can incorporate both effects at the same time. 1 Second, while lumping the two channels together and evaluating a combined impact may suffice for some policy analysis, other contexts necessitate disentangling the two distinct impacts. This is particularly true when looking at the broader impacts of trade, as there may be significant differences across the two channels. For example, a promising recent literature has begun integrating models of labor markets with trade; see Helpman, Itskhoki, and Redding (2010). If increased trade reallocates production across firms, labor needs to be reallocated as well, and this necessitates a process of job search. In contrast, to the extent trade just lowers mark-ups, holding reallocation fixed, workers won t have to search for new jobs, and the broader impact is different. Another example is a literature, recently surveyed by Holmes and Schmitz (2010), that links increased competitive pressure, like profit margins getting squeezed, to within-plant productivity increases, which are distinctly different from productivity gains from reallocation across plants. This paper develops a model where there are firms capable of producing a given differentiated good, where is a finite number that may vary across goods. In cases where =1, the single firm capable of producing the good is a monopoly and the problem faced by the firm is the same as in the standard Dixit-Stigliz setup. For 2, there is oligopoly with head-to-head competitors that compete in a Bertrand fashion. If the competing firms were identical in cost, then Bertrand competition would drive price down to the common cost and the mark-up would be zero. However, with heterogeneity in productivity, as we assume here, the mark-up is positive even with Bertrand competition. A key assumption of the model is 1 Tybout (2003) discusses empirical work the mark-up reducing or pro-competitive effects of trade. Pavcnik (2002) is an example of work on the impact of reallocation. 1

3 that the set of possible differentiated goods is bounded. As trade barriers fall, and previously separated markets become integrated, the bound on the set of possible goods implies that there will be increasing overlap of firms across the given set of goods. For example, while before integration there might be only one firm competing in the market for a particular good at a particular location, after integration there might be two or more head-to-head competitors. It is actually quite an old idea that the elimination of trade barriers can lead to an increase in a finite number of head-to-head competitors for a particular good. (See Markusen (1981), Brander and Krugman (1983), Venables (1985), Horstmann and Markusen (1986) and more recently Neary (2003).) The contribution of this paper is to take the idea and put it to work in a model of selection with heterogeneity in productivity draws, and to show that it is possible to calculate the rich equilibrium structure in a tractable way. Our model builds on the approach of BEJK, which uses a Fréchet distribution for productivity draws. The significant difference is that we use a finite version of this distribution, while BEJK use a continuous limit. We show that as trade barriers decline and the size of the economy gets arbitrarily large, the number of head-to-head competitors for any particular good also gets large, and the economy goes to BEJK in the limit. A key result of the paper is that while our model is the same as BEJK for large, our model is very different from BEJK when is not large. In particular, in BEJK the distribution of the markup is fixed and does not change when trade barriers are reduced and economies are integrated. In such an event, prices indeed fall, but with increased selection, costs also fall. In an elegant result, the two forces exactly counterbalance. In contrast, with the finite of our model, markups strictly decrease with market size. For intuition, just consider the case of =1where the markup distribution is degenerate at the simple monopoly level. Moving to the Bertrand competition of =2clearly shifts mark-ups down. We find there is a kind of diminishing returns to increasing to lower mark-ups. The impact on margins of doubling from =2to =4is less than from =1to =2,and doubling from =4to =8is lesser still. In the limit, the impact of these doublings on mark-ups goes to zero. For intuition, consider the case where there is no heterogeneity and all firms have the same cost. Then when we add the second firm, price falls from the monopoly level all the way to marginal cost, and is constant for higher, an extreme form of diminishing returns. With the firmheterogeneitythatwebuildintothemodel,theoutcome is never extreme as this, but the example is suggestive of the basic forces at work. Neary (2010) has argued that the trade literature should work with small models of industry, and in this model small really makes a difference. Interestingly, at the other limit where the economy is small, the model converges to a Dixit-Stiglitz monopolistic competition model of trade (Krugman (1979)). When the 2

4 economy is small, the variety of goods is very small and the chance that any two entrepreneurs will draw the same product to become head-to-head competitors is very small; i.e. virtually all firms are monopolists. Near this limit, a doubling of market size has no impact on markups;therearesimplytwice asmanymonopolyfirms each charging the monopoly mark-up, which is invariant to market size. Thus at the extremes of market size, large where things are like BEJK, or small where things are like Dixit-Stiglitz, doublings of market size have no impact on mark-ups. But in our area of focus between the extremes, there is an impact. Here expansions of market size deliver welfare gains through three channels: lower mark-ups, more selection over productivity, and more variety. In contrast, in the Dixit-Stiglitz limit, there is only the variety effect, while in the BEJK limit there is only the selection-overproductivity effect. Melitz and Ottaviano (2008) is the first paper to develop a model that simultaneously captures the selection and reallocation effects of trade, highlighted in Melitz (2003), with the pro-competitive impact of increased trade on markups, highlighted in the earlier trade literature, beginning with Markusen (1981). Melitz and Ottaviano is a tractable framework that allows for rich detail, such as asymmetric countries. One difference to highlight is that the market structure in Melitz and Ottaviano is monopolistic competition, while here firms compete in a finite firm oligopoly (but are still small compared to the economy as a whole as in Neary (2003)). Another difference is preference structure. Melitz and Ottaviano employ the quasilinear-quadratic utility framework developed in Ottaviano, Tabuchi, and Thisse (2002), an approach which gives rise to a tractable linear demand system. Our paper employs the CES preference structure that has served as a workhorse model in trade for many years. In contrast to the zero income effects of a quasilinear structure, CES has unit income elasticity, a natural baseline case for many empirical applications. Also, the homothetic structure of CES, a property not shared by quadratic utility, allows for aggregation of demand to a representative consumer. This property is particularly useful as trade models begin to be applied to labor issues, such as how trade impacts wage inequality. We note a modeling difference in the way selection operates here compared to much of the literature. In the standard selection model of Hopenhayn (1992), there is a productivity cutoff, sayatthe20thpercentile,whereallfirms below the cutoff exit and all firms above the cutoff stay. In Melitz (2003), an expansion of trade raises the cutoff, saytothe30th percentile. That is, in larger markets there is more left-truncation of the productivity distribution. Syverson (2004) and Combes et al (2010) are empirical papers that test for increased left truncation in large markets. In our model of selection, there is no cutoff. Rather, with draws, the best survives and the remaining 1 do not. Formally, the productivity distribution of the surviving firm is that of a maximum order statistic, rather 3

5 than a left-truncation. In other words, if someone went looking for increased left-truncation in large markets in data generated by our model, they wouldn t find it, even though there is indeed more selection in large markets. We view these two approaches as abstractions that are probably best judged on their technical merits for the application at hand, as opposed to which is a better approximation of reality. The rest of the paper proceeds as follows. Section 2 describes the model. Section 3 proves the outcome is equivalent to BEJK when market size is large and equivalent to Dixit-Stigitz when the market size is small. As an application of these results, we determine the conditions under which there is agglomeration in a two-location regional version of the model. We show the model exhibits a diminishing returns property of agglomeration that is absent when the exercise is conducting in either limiting case. Section 4 characterizes the distribution of mark-ups and expected revenues, for multiple locations and general transportation costs. In addition to depending on the number of competitors, mark-ups depend upon characteristics of source locations, with goods originating from high cost locations tending to have low mark-ups. Section 5 uses the tools developed in Section 4 to examine the welfare impact of increasing market size. 2 Model There are locations indexed by 1, 2,... In part of Section 3, we will think of the locationsasdifferent regions within the same country and allow factor mobility. For the restofthepaper,wetakethefactorsateachlocationasfixed. We can think of this as an analysis in international trade where factors are immobile, or that part of a regional analysis whichconditionsonlocationchoice. The consumption composite is an aggregation of differentiated goods indexed by on the unit interval. It follows the standard CES form, µz 1 = 0 (()) 1 1 where 1is the elasticity of substitution. As we will explain, some of the goods [0 1] will be available for purchase by consumers and others will not. For any that is unavailable, of course () =0. In that part of the paper where we allow for factor mobility, we introduce land as a force of dispersion, following Helpman (1998). Preferences over the composite good and land 4

6 are represented by the utility function ( ) = 1 The supply of land at location is. Analogous to what Redding and Sturm (2008) do, we assume land rents at a particular location are distributed equally in a lump sum fashion among the population at the particular location. There are a measure individuals in the economy, each endowed with one unit of labor. In the regional version of the model with factor mobility, these individuals choose where to live. Let be the measure of individuals choosing location, = P =1. In the international trade version of the model, we take as fixed. There is an entrepreneurship task that requires a fixed unit of labor that we normalize to one. The alternative use of labor is for production work. Individuals choose to become an entrepreneur or a worker, and let and be the measure of individuals in each of these occupations at location. Each individual choosing entrepreneurship randomly draws a good and productivity. An individual drawing ( ) can produce units of output of good per unit of labor employed. The process of drawing ( ) is independent and identically distributed across all entrepreneurs. Assume the good is drawn uniformly across the unit interval of possible goods. The distribution of productivity draws is denoted () andingeneraldepends upon the location of the entrepreneur. We defer details about this distribution until the end of the section. As a result of this process, there will be an integer number of entrepreneurs capable of producing a particular good at a given location. Let () be this integer number for good at location. The distribution of counts across goods will depend upon the total measure of entrepreneurs at the location. For ease of exposition, leave the location implicit and write the measure of entrepreneurs at a particular location and time as. The process of entrepreneurs randomly drawing goods results in a Poisson distribution of entrepreneur counts across goods, ( ) =. (1)! Thus a fraction of goods (0)= will have zero entrepreneurs at the location capable of production, a fraction (1)= will have one capable entrepreneur, and so on. There is an iceberg transportation cost. Formally, to deliver one unit to destination from source location, 1 units of the good need to be shipped. Assume =1,so there is no transportation cost to ship locally. 5

7 We can think of there being three stages in the model. In the first stage, in the regional version of the model with labor mobility, individuals choose an occupation and where to live. In the international version, location is fixed, and individuals choose only their job. In thesecondstage,ateachlocationandforeachgood, there is Bertrand price competition between all entrepreneurs capable of producing good. Finally, in third stage, individuals make consumption decisions. In equilibrium, the following conditions must be satisfied. First, location decisions must be optimal (if factors are mobile) and job choice decisions must be optimal. Second, the price choices made in the price subgame for good at location is a Nash equilibrium; i.e. entrepreneurs maximize profits taking prices of head-to-head competitors of the same good as given. Note this set includes any entrepreneur capable of producing good at any location. Third, output and labor markets clear. We now turn to the details of the distribution function from which entrepreneurs draw productivity. The distribution is Fréchet, with c.d.f. () Pr [ ] =, where 0 is a scaling parameter and determines curvature. The parameter governs location-level efficiency. The parameter is a measure of similarity; the bigger, thelower the variance in productivity draws. As in Eaton and Kortum (2002), we assume 1. The role of this assumption is analogous to that of 1 in a standard Dixit-Stigliz model. If 1, heterogeneity in costs is so large relative to curvature of preferences that expressions involving welfare and price indices become undefined. Suppose we take independent draws at location from this distribution and let 1 denote the random realization of the highest value. The distribution is 1 ( 1 ; ) Pr( 1 1 ) = 1 ( 1 ) = 1 (2) Thus the distribution of the highest value is also Fréchet with scaling parameter and the original curvature parameter. The Fréchet is an extreme value distribution which means that the highest from a set of independent draws remains within the same distribution family. In the introduction we noted a idea in the literature [Syverson (2004) and Combes et al (2010)] of looking for increased left truncation for evidence of increased selection, a change in shape of the survivor s productivity distribution to become more compressed. Here as increases and selection is tougher, the shape of the distribution of the highest value remains the same, as it shifts to the right. In fact, looking at log productivity, as is typical in the 6

8 empirical literature, the variance is constant, independent of. 2 There is no sense that the distribution gets more compressed as increases. We motivate why we use the Fréchet instead of the Pareto, as Melitz and Ottaviano and other recent papers have done. 3 It is very useful for us that the Fréchet s is an extreme value distribution, as is in Eaton and Kortum (2001) and BEJK. This property is irrelevant in the other papers, which are models of monopolistic competition in which there is no need to take the maximum of a set of draws. While technical advantages dictate our choice, we note that the two distributions are closely related, in any case. Both have fat right tails; i.e., the chance of a very good draw decays at slower than an exponential rate. With this property, in the limit when markets get large, we will see below that the elasticity of welfare to market size remains positive with the Fréchet, just like in the other papers with the Pareto. In fact, the Fréchet looks very similar to the Pareto on the right side of the distribution, which is the relevant side for a model of selection. Top panel of Figure 1 plots ln(1 ) for c.d.f against ln() for the Fréchet with =4 along with the same plot for Pareto with the same mean and variance. This is a common way to plot fat-failed distributions. The distributions are very similar, two straight lines close to being on top of each other, except on the far left side where the Fréchet curves down while the Pareto remains linear. 4 The difference on the left is that the Fréchet density is bell-shaped while the Pareto density is downward-sloping throughout. The density plots at the bottom of Figure illustrate this and also display the best-fitting log normal curves. Empirical distributions of productivity are typically bell-shaped (e.g. Syverson (2004)), which is an argument in favor of the Fréchet. 3 Dixit-Stiglitz and BEJK as Extreme Cases This section determines the limiting outcomes when the size of the economy is very small or whenthesizeisverylarge. Therearefourparts. Thefirst solves for the joint distribution of first and second highest productivities and takes limits. The second and third parts report price and mark-up distributions in the two limiting cases. The fourth part puts these preliminary results to work to determine in the limiting cases the conditions for agglomeration in a two-location version of the model. 2 The log of a Fréchet random variable has variance that depends upon the shape parameter, but not the scaling parameter. 3 Others include Helpman, Melitz, and Yeaple (2004), Chaney (2008), and Eaton, Kortum and Kramarz (2008). 4 The Pareto and Frechet distribution are tail equivalent in the sense that the tail probabilities are proportional to each other in the limit. If the "tail indices" of the two distributions, of the Frechet and the power exponent of the Pareto, are chosen to be the same, then the slopes at the right tail in Figure 1 will be almost identical. 7

9 3.1 The Distribution of First and Second Highest Productivity We begin by deriving the joint distribution of the first and second highest productivity draws, 1 and 2 at location. The second highest matters, because with Bertrand competition, it can impact pricing. Given draws, the joint distribution is 12 ( 1 2 ; ) Pr ( ) (3) = Pr[ 1 2 ]+Pr[ ] = = 2 + ( 2 ) + ( 1 ) ( 2 ) ( 2 ) 1 i h 1 2 ( 1) for 1 2. Toseethis,notethatifthefirst highest is below 2 then the second highest is too, accounting for the first term in each of the last three lines. To understand the second term in these lines, observe there are possiblewaysoneofthe draws can be between 2 and 1 while the remaining 1 draws are below 2. We now look at the limit where there is a large number of entrepreneurs. Suppose that at location the measure of entrepreneurs can be written as 2, =, (4) for a constant and a parameter that scales up entrepreneurial activity at each location proportionately. As we scale up, suppose we also adjust the parameter of the underlying draw distribution, to keep the expected value of the maximum constant. =, (5) for a constant. Finally, we evaluate the joint distribution of the first and second highest productivities, conditioned upon the good having the expected number of capable entrepreneurs at source location, = =. (6) As we will discuss further below, as goes off to infinity, realizations of entrepreneur counts for a particular good will tend to be close to the expected value, a law of large numbers result. Plugging (4), (5), and (6) into the formula (3), yields h 12 ( 1 2 ; ) = i 2 ( ) 2, Plugging = P =0 into the bracketed term and taking limits yields! 8

10 lim 12( 1 2 ; ) = (7) This limit distribution where the number of draws goes off to infinity for each good is what BEJK use for the distribution of the first and second highest productivity firms at each location. The product corresponds to the location-level technology parameter in BEJK ( in BEJK notation). Here effective technology includes a component due to the entry level,aswellasthetechnologycomponent determining individual productivity draws. To examine limits in the opposite direction where is small, we first use the Poisson formula to write the probability that a good has one entrepreneur, conditioned on having at least one, Pr( =1 1 )= 1 with limit lim Pr( =1 1 )=1. 0 Thus, when the measure of entrepreneurs at is small, the probability of overlap in products goes to zero. If the at all locations are small, there will be virtually no overlap across locations either, so virtually each entrepreneur is a monopolist. That is, the environment approximates Dixit-Stitgliz, with heterogeneity in productivity like in Melitz (2003). Since there is only one draw per good, the distribution of productivity of the highest draw for each good is simply the underlying distribution of draws (). Having established what the productivity distributions look like in the limit economies, it is useful to review results in the literature about pricing and trade flows in the limit economies. For this review, we take as given the entrepreneurship level at each location and the wage. 3.2 At the Limit Where Entrepreneurs are Monopolists Let be the measure of entrepreneurs at each location, P =1 1. Suppose each entrepreneur has a monopoly over a particular differentiated good. Let be the wage at location. A firm at source location, drawing productivity, has a marginal cost of to deliver one unit. We invert marginal cost and call it the entrepreneur s cost-adjusted productivity of selling in market, =. 9

11 Cost-adjusted productivity is the amount of good the entrepreneur can deliver to location, per unit numeraire expenditure on input. As a monopolist with constant elasticity of demand, the price set by the entrepreneur at source to destination location is () = where is the monopoly mark-up over cost, = 1. The price index for the composite consumption good at location equals = " X =1 ( ) ( 1) # 1 1 where depends upon and. We define the mark-up share of revenue to equal revenues less variable costs, as a share of revenues. With monopoly, the mark-up share equals At the Limit Where Head-to-Head Competition is Large Next consider the limit where we take the scaling parameter off to infinity, according to (4), (5), (6), so for any good there is a large number of head-to-head competitors. The limit distribution is (7), which matches the assumption of BEJK. Thus, the results of BEJK apply in the limiting case. For a particular good sold at destination, let 1 () be highest cost-adjusted productivity from all sources and analogously let 2 () be the second highest from all sources. BEJK show the joint distribution of 1 and 2 equals for the index Φ defined by 12 ( 1 2 )= Φ 2 + Φ 2 1 Φ 2 Φ X ( ). =1 Note this is the same functional form as the limit joint distribution (7) of the first and second most productive within any location, with the index Φ in place of. The index Φ can 10

12 be interpreted as the sum of entrepreneurial activity across all source locations, weighted by thecostefficiency of each source in selling to. In particular, the entrepreneurial activity level from source is weighted by the source s technology level, and is discounted by its wage and the transportation cost to get to, with the degree of the discount determined by the Fréchet shape parameter. For a given destination and a given good, in the equilibrium of Bertrand competition, the entrepreneur with the highest cost-adjusted productivity delivering to gets the sale, i.e., the one with productivity 1 (). Thepriceequals () =min à 1 2 ()!. (8) 1 () That is, the most efficient entrepreneur sets price to match the cost of the second most efficient, unless that cost is above the most efficient entrepreneur s monopoly markup. The mark-up is Ã! 2 () () =min 1 () (9) BEJK show that the distribution of the markup is a truncated Pareto, ( 1 1 Pr( () ) = 1 (10) This distribution is the same, regardless of the destination, and regardless of the source location. BEJK show the price index takes the form = Φ 1 = " X =1 # 1 ( ) for a constant that is a function of and. They show that the mark-up share of revenue at each location equals 1(1 + ). In comparing the formula for the price index and the mark-up share, we see that the outcome for BEJK, given and, is the same as the outcome at the monopoly limit for raised to 0 =1+, aside from multiplicative constants on the price index terms. That is, in terms of aggregate variables like the price index, the BEJK economy looks like the limit monopoly economy, only with less curvature on demand (yielding lower markups than the original limit monopoly economy). This follows the point made by Arkolakis, Costinot, and Rodriguez-Clare (2009) that there is an equivalence between the Dixit-Stiglitz model and the BEJK model in terms of aggregate implications. 11

13 3.4 Equilibrium Agglomeration Near the Limits We put our limit results to work in determining when there is agglomeration. We use a familiar setup in the New Economic Geography literature. (See Krugman (1991) and Fujita, Krugman and A. J. Venables (1999) for a textbook treatment.) There are two locations and symmetric land endowments, 1 = 2, draw qualities 1 = 2, and stransportation costs, 21 = 12 = 1. The Dixit-Stiglitz version that we use is due to Helpman (1998). We will get a result in our model that is qualitatively different from what happens in a pure Dixit-Stigliz model or a regional version of a pure BEJK model. We note two standard equilibrium conditions. First, free entry into entrepreneurship implies the expected operating profit shareofrevenue ateachlocationmustequaltheshare of labor in fixed cost (i.e. the entrepreneur share). Second, utility is equated at the two locations, so individuals are indifferent to where they live. An equilibrium with agglomeration is one where more than half the population concentrates in one location. We start with what happens when population is small. Lemma 1. Take the limiting case of Dixit-Stiglitz. Define ( ) by ( ) 1 [1 + 2 ( 1)] + 1 2, 1 [1 + 2 ( 1)] 1 (i) If (1 ) ( ), the unique equilibrium is the symmetric outcome where half of the population locates in each place. (ii) If (1 ) ( ), the symmetric outcome is not stable and there is a stable agglomeration equilibrium that is unique up to the identity of which place gets more than half the people. 5 Proof. See the appendix. Helpmanshowedthat(1 ) 1is a sufficient condition for the symmetric (or dispersion) outcome to be the unique equilibrium outcome. Since 1 ( ), Helpman s result is contained in part (i) of the above. It is intuitive that dispersion is more likely, the higher the utility weight (1 ) on land and the greater the substitutability of products, which diminishes the value of picking a congested location to enjoy wide variety. Helpman determined what happens when (1 ) 1partly through simulation. Lemma 1 restates Helpman s result with an analytic expression. Next we consider the opposite extreme, where population is large. This limit is a version of the BEJK model in which population is mobile and in which the joint distribution of the first and second best productivities at each location is endogenous. (BEJK itself is a model of international trade in which population at each location is fixed and the distribution of 5 Stability is defined in the standard way of this literature. We formally define it in the proof. 12

14 productivities at each location is exogenous.) As shown above, the BEJK limit works just like the Dixit-Stigliz limit, with raised to 0 =1+, in terms of the price index and operating profit shares, which are the variables determining location choice and entrepreneurship entry. This implies the following Corollary to Lemma 1: Corollary. Begin with the version of the model as parameterized by the scaling parameter in (4), (5), and (6), and take the limit as goes to infinity. Then Lemma 1 holds with 1+ in place of. Thatis: (i)if(1 )(1+) (1 + ), the unique equilibrium is dispersion. (ii) If (1 )(1+) (1 + ), the unique stable outcome is agglomeration. So far in this subsection we have looked at the limits. Doing so enables us to exploit the very tractable properties of the limiting models. Our model is between the limits. In the next section, we will crunch through and explicitly take into account that the entrepreneur count for a particular good will be a finite integer, in general different than one. But before we do that, we use continuity to make a quick point that our model can yield qualitatively different results than either Dixit-Stiglitz or BEJK. Proposition 1. Vary the total population holding the rest of the parameters fixed. (i) If (1 ) ( ), then for small and for large, the unique equilibrium outcome is dispersion where half the population locates in each place. (ii) If (1 )(1+) (1+ ), then for small and for large, the unique stable equilibrium is agglomeration with an unequal distribution of population across locations. (iii) If (1 + ) ( ), but (1 + )(1+) (1 + ), then for small, agglomeration is the unique stable outcome while for large dispersion is the unique outcome. Proof. See appendix. In the Helpman limit where entrepreneurs are monopolists, changes in the size of the economy, as measured by population, have no impact on agglomeration. This is true about the BEJK limit as well. But as we can see above in part (iii), population does matter in our model. Agglomeration found at low population levels can disappear at high population levels. There is no equivalence theorem between our model and Dixit-Stigitz, or our model and BEJK. 4 Mark-Ups, Revenues, and Costs in the General Model This section determines the distribution of mark-ups and expected revenues and costs for a given good. For the analysis of this section, the entrepreneur count () at each location 13

15 is taken as given as well as the wage. As we will be holding fixed the particular good, for this rest of this section we leave the index implicit. As above, = ( ) is the cost-adjusted productivity for a particular entrepreneur at source for selling to destinaton, given the entrepreneur draws raw productivity. Define 1 to be the maximum cost-adjusted productivity for selling to destination across all entrepreneurs at source capable of producing the good, unless =0, in which case set 1 =0. As before, 1 is the highest to destination across all sources, n o 1 =max 1, and 2 is the second highest across all sources. Location isthesourceto ifthemostefficient entrepreneur at has higher cost-adjusted productivity of serving,comparedtothemostefficient at all other locations. This happens with probability n o Pr 1 max 1 = P 6= =1, (11) where = ( ), (12) is a summary measure of the quality of a productivity draw at source location for selling to destination. The quality adjustment is higher with a better technology,andis discounted with a higher wage or transportation cost. The denominator of X Φ, (13) =1 can be interpreted as the sum across locations of all the quality-adjusted draws for serving market Formula (11) for the probability that sells to is the same as in BEJK, with equal to the location technology parameter in that model. While isthesameasinbejk,mark-upsaredifferent. To determine mark-ups, we firstderivethejointdensityofthefirst and second highest productivities 1 and 2 from all source locations. Conditioned upon the first highest being from source, aseparate appendix shows that the joint density is 12( 1 2 )=Φ (Φ ) (Φ ) 2. (14) We use to indicate we are conditioning on the first highest 1 being from source location. Note the second highest can potentially be from any location. Also note the presence of 14

16 the term Φ in the expression, which takes the sum Φ over all the (quality-adjusted) draws in the economy, and subtracts out a single one from the source. (Again can be interpreted as the quality level of one draw.) In the limit of BEJK, where the number ofdrawsgoestoinfinity, subtracting out a single draw doesn t make any difference. With finite draws, it does make a difference. Using formula (9) for how the mark-up depends on 1 and 2, and integrating over the density (14) conditional on source, weobtainthe following result. Proposition 2. (i) Take as given the entrepreneur counts at each location for a particular good and use these to define, through (13), the sum Φ of quality adjusted draws for serving destination. Conditional on sales originating from source location, the distribution of the mark-up at destination is 1 () =1,for, (15) Φ ( 1) and equals () =1, above the monopoly mark-up,. (ii) An increase in the number of entrepreneurs,fromanysourcelocation with nondegenerate ( 0) costefficiency shifts down the distribution of mark-ups in a first-order stochastic dominance sense. 6 (iii) If there is at least one entrepreneur located outside the source, the distribution of the mark-upisshiftsupwiththecostefficiency ofthesourcelocation,andshiftsdownwith of other locations 6=, having an entrepreneur capable of producing the good, 0. Proof. See the appendix for the proof of (i). The proofs of (ii) and (iii) follow from definition (11) and equation (15). To see the intuition in this formula, note that the term Φ can be interpreted as one over the number of entrepreneurs obtaining draws, again with adjustments for draw quality. In the limit where the number of draws goes to infinity, Φ goes to zero. Hence, the limiting distribution goes to 1 1, the distribution (10) in BEJK. In the limit, increasing competition by doubling the number of draws has no impact on the mark-up distribution. But with finite draws, changing the number of competitors does impact the mark-up distribution. Mark-ups are highest in the case of monopoly. If there is a single entrepreneur at and none anywhere else, then Φ reduces to,andfrom(15),wecansee the the mark-up distribution is degenerate at the monopoly level ( 1). An interesting finding in Proposition 2 is that the distribution of mark-ups depends upon the source, a property that disappears in the limiting distribution 1 1. If the good originates from a source that is low cost in delivering to, i.e. costefficiency 6 In this paper, when we refer to shifting up (or down) a distribution, we will always mean in a first-order stochastic dominance sense, i.e., that the value of the c.d.f. decreases (or increases), 15

17 is high, then the mark-up tends to be large. If the source is high cost, the mark-up tends to be low. This is a intuitive result. Consider in particular the case where locations are symmetric, and wages and the technology parameters are the same everywhere. In this case, differences in cost efficiency across sources are due entirely to transportation cost. In particular, it implies that the distribution of mark-ups on imported goods to a location will be strictly lower than the mark-ups on locally-produced goods. The next step is to calculate expected sales revenues for a good originating in source andsoldindestination. Demand for any differentiated good at can be written as () =,where is scaling parameter that will depend upon the price index at destination ofthecompositegood,aswellasthesizeofthelocation. Revenueatprice can then be written () () = 1. Let be expected revenue for a good sold to destination, conditioned on originating from source, andfixing entrepreneur counts at each location, where the expectation is taken with respect to the random price realization. We can define an analogous measure for variable costs. The most efficient firm at location has cost-adjusted productivity 1. By the definition of the adjustment, variable cost per unit good delivered to equals 1 1. Multiplying this by the quantity of demand at price results in a level of variable cost equal to = ³1 1. Let be the expected level of variable costs, for a good sold to destination, conditioned on originating from source, where the expectation is taken over random price and productivity draws. Finally, define the mark-up share to be _ =1. This is one minus the variable cost share of revenue. It is the weighted average of the mark-up, using revenues as the weights. The revenue-weighted measure is interesting in its own right and we will need next section to determine equilibrium entry. Forthevariables wehavejustdefined, we have the following result. Proposition 3. (i) Expected revenues at location for a good originating at location can be written as = = Γ Z Z µ +1 ( 1 2 ) 1 12 ( 1 2 ) 1 2 (16) ˆ, 16

18 where is a demand scaler, and ˆ is defined by ˆ ( 1) Φ +(Φ ) +1 + Φ (Φ ) h Φ + +1 i Φ +1 and again = ( 1) is the monopoly mark-up. (ii) Expected variable costs at location, for a good originating at, canbewritten µ +1 = Γ ˆ for ˆ defined by ˆ Φ +(Φ ) +1 + Φ (Φ ) +1 µ and is defined by µ = Φ µ Φ where is the hypergeometric function. (iii) For any nongenerate ( 0) source, anincreasein strictly increases expected revenue and strictly decreases the sales-weighted mark-up share, which can be written as _ =1 ˆ ˆ. In the limit, lim _ =1 (1 + ), the BEJK level. If the firm is a monopolist, then _ =. Proof. See the separate appendix. The result provides analytic expressions that we use in the next section to evaluate the impact of expansions in market size. The result states that if the number of competitors is increased in any market, expected revenue increases. Since increases in competition lower prices and demand is elastic, this is straightforward. Finally, an increase in competition lowers the sales-weighted expected mark-up. The proposition states that expected revenue can be written as = Γ ˆ. The demand scaling can be ignored in what we look at here. The Gamma function term Γ subsumes the random effects on productivity coming from the Frechet. As this term depends only on and, whichareheldfixed, it cancels out in what we do and can be ignored. We 17

19 focus on the last term ˆ,asthisvarieswiththenumberofcompetitorsateachlocation. While the formula is complicated, we can gain intuition from looking at limiting cases. Suppose first there is a monopoly. Then the variable summing total draws reduces to Φ =, the one draw from location. The second term of ˆ then drops out and the formula reduces to Ã! ( 1) ˆ = ( ) 1 This equals the revenue when price is the monopoly mark-up over marginal cost ( ) 1. Again, that aspect of marginal cost related to the uncertainty of the draw has been factored out through Γ. Next consider what happens at extreme values of. The lower bound of is 1 and that is the maximum amount cost heterogeneity permitted in the parameter space. Otherwise, things are undefined. At the limit, lim ( 1) ˆ = Φ ( 1). Interestingly, expected revenue is the monopoly level, scaled up proportionately by quality adjusted count of draws Φ. At the lower bound of, cost heterogeneity is great; the most efficient entrepreneur will tend to have a substantial cost advantage over the next most efficient, so is effectively a monopolist. And the greater the number of draws, the greater the expected cost efficiency of the winner. Finally, to look at the case of high, assume there is a single location. Also normalize draw quality to =1,soΦ =, the unweighted number of firms. If 2, at the limit where is large we have lim ˆ single_location =1, 2. When is large, differences in productivity are negligible, so firms virtually have the same marginal cost. As is standard with Bertrand competition with common costs, if there are two or more more firms, price is driven down to this common cost, here normalized to one, and revenue is approximately 1 ( 1) =1. 5 Results with Frictionless Trade This section works out the model with frictionless trade between all locations, i.e. =1 for all destinations and sources. With frictionless trade, we can group all locations together and treat them as one aggregated location with population. We think of a 18

20 trade liberalization as expanding the size of the the economy by bringing more within the frictionless trade area. The first part of this section puts the formulas derived in the previous section to work on a central issue of the paper, analyzing the welfare benefit of increasing the number of competitors. The analysis takes into account both the pro-competitive effect of lower markups and the selection benefit of higher productivity. These benefits are evaluated against the benchmark of the benefit of variety. The second part imposes the free-entry condition on entrepreneurship. It determines how changes in market size impact welfare, breaking down the effects into impacts on margins, productivity, and variety. 5.1 Welfare and Increased Competition We evaluate the welfare benefits of increasing competition, i.e., the number of head-tohead competitors in a market. As highlighted in the Introduction, there are two benefits at work here. First, there is the pro-competitive effect of the increased competition in driving down mark-ups. Second, there is the higher productivity effect of increased selection. As a benchmark for measuring these gains, we relate them to the gains from variety. Specifically, we evaluate the relative welfare gain of doubling the number of competitors in each market, in exchange for halving the total variety of goods. This is a compelling benchmark to consider because it holds constant the fixed cost of entrepreneurial resources. There is much focus in the literature on the benefit ofvariety. Hereweask,howdothe benefits of increased competition, through the pro-competitive and productivity effects, stack up against this? The formulas in Proposition 3 simplify because there now is only one aggregate location. Wecandropthe and location indicators and normalize draw quality to =1,sothat Φ = is just the total number of firms. It is convenient here to write expected revenue ˆ() as a function of. To make the welfare comparison, we first need to write down the price index. Letting be the measure (or variety) of goods with entrepreneur count, the price index for the composite consumption good equals µ +1 = Γ 1 Ã 1 X ˆ() =1!

21 The welfare measure is the inverse of the price index, ignoring the multiplicative Γ term, Ã X = ˆ() =1! 1 1 (We can ignore land here since population is fixed throughout this subsection.) In this subsection, we focus on cases where all goods are symmetric in having the same, inwhich case the welfare measure reduces to _ ( ) = ³ 1 1 ˆ(). = 1 1 ˆ() 1 1, for variety level. As is standard with CES, the elasticity of welfare with respect to variety is 1 ( 1). Starting from a point with variety and each good having entrepreneur count, we define the Competition Variety Ratio as ( ) _ (2 1 2 ) _ ( ) the gain of doubling the number of competitors and halving variety, relative to the starting point. As we will emphasize, it depends on,, and, and we write it as such. Starting variety cancels out in the ratio. We begin our analysis setting the starting point level of competition to monopoly, =1. Recall that the lower bound for is 1. It is straightforward to evaluate the limits of the Competition Variety Ratio at the lower bound for and at the other extreme of large, lim (1) = 1 1 lim (1) = 1 2 In between these extremes, there is a U-shaped relationship between the ratio and. Figure 2 illustrates this relationship for several example values of. To interpret these results, we firstnotethatincreasing, which makes the productivity draws more similar, has two off-setting impacts on the ratio. On the one hand, greater similarity strengthens the pro-competitive effect of going from monopoly to duopoly in reducing mark-ups, and this tends to increase the ratio. On the other hand, the gain from selection is reduced, and this tends to decrease the ratio. These two off-setting forces account for the U-shapeinFigure2. Inthelimitas gets large, the productivity draws are identical, the duopoly mark-up goes to zero, and the productivity gain from selection is zero. Thus for

22 large, the entire gain from adding arises entirely through the pro-competitive effect. As we can see in the formula above, whether or not the ratio is above or below one for large depends upon whether or not is greater than two. Suppose we set 2, corresponding to cases where the monopoly mark-up does not exceed 50 percent. In this case, the ratio is no lower than.82 and no higher than 1.06 over the range of and. 7 We find it interesting how close to one this is. The welfare of duopoly over monopoly, combining the pro-competitive and productivity-selection gains, is roughly the same as the gain from variety, and potentially can even be slightly higher. This discussion so far has taken monopoly as the starting point. Next we consider trading off variety to get more competitors, when starting off with already more than one competitor. Figure 3 plots the ratio for various such starting points, fixing =2. Note first that in the limit when goes to its lower bound 1, theratioisone, regardless of initial. Here there is extreme cost heterogeneity. The most productive firm is so much more efficient than the second most, that effectively it is setting the monopoly mark-up. That is, Bertrand competition itself is not providing a disciplining impact on prices. Regardless of the starting, there is no pro-competitive effect of increasing. Here, the gains from increasing work entirely through the productivity effect of increased selection. In the limit where goes to 1, the productivity effect of doubling the number of draws exactly counterbalances the variety effect of halving the number of goods. Next turn to what is happening away from the lower bound of. Here the ratio shifts down, as the starting number increases. That is, the willingness to trade off variety to get more competitors is lower the more competitors you start with. The reason is diminishing returns in the pro-competitive effect of adding. Figure4plotshowtheaveragemarkup varies with and. The reduction in mark-up, from adding a third firm when there are already two, is small compared to how mark-ups fall when moving from monopoly to duopoly. Theimpactofgoingfromthreetofourisevensmaller. Itiswellunderstood that if firms have identical cost (corresponding to the limit of high here), adding a second firm to monopoly lowers the mark-up all the way to zero. The interesting result in Figure 4 is that second firm is key even when firms are heterogeneous to such a degree that average mark-ups are large. 7 For =2, the ratio is minimized at =25, where the ratio is.82. In the limit for large, theratiois maximized at =33 where it equals

23 5.2 The Impact of Market Size with Free Entry into Entrepreneurship We begin by deriving equilibrium entry into entrepreneurship as well as some analytical results about the impact of market size. We then illustrate the impact of market size with pictures Free Entry and Market Size So far in this section, like we did in the previous section, we have taken the entrepreneur counts as given for a particular good. Here, we make entrepreneur counts endogenous, depending on the equilibrium level of entrepreneurship. We will have to integrate over the different 1, conditioned on the entrepreneurship level. The Poisson probability distribution over the, conditioned upon and that 1 equals 1+ ( ) = (1 )!, 1. We use this to calculate the revenue share of goods with entrepreneur counts, given entrepreneurship level. This equals Rshare( ) = 1+ ( ) ˆ() P =1 1+ ( ) ˆ(). The cumulative revenue share of goods with counts less than or equal to is cumrshare( ) = X Rshare( ). =1 That this is strictly decreasing in (stochastic dominance), follows from the stochastic dominance property of 1+ from an increase in and the fact that ˆ() increases in from Proposition 3. 8 The average markup given is the average across the different using revenue weights, _() = X Rshare( )_(). =1 Because of monotonicity of _() and the stochastic dominance property of 8 It is straightforward to verify that the monotone likelihood ratio property on 1+ holds for increases in, which implies first-order stochastic dominance, i.e., 1+ ( ) strictly decreases in. 22