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 theNational Bureau of Economic Research.

†Department of Economics, the Chinese University of Hong Kong.‡Sauder School of Business, the University of British Columbia.

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 appli-

cation, 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 differ-

entiated 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

1Tybout (2003) discusses empirical work the mark-up reducing or pro-competitive effects of trade. Pavc-

nik (2002) is an example of work on the impact of reallocation.

1

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 pro-

ductivity 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 = 1 where the markup distribution is degenerate at the simple monopoly level. Moving

to the Bertrand competition of = 2 clearly 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 = 2 to = 4 is less than from = 1 to = 2, and

doubling from = 4 to = 8 is 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 firm heterogeneity that we build into the model, the outcome

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

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 mark-

ups; there are simply twice as many monopoly firms 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-over-

productivity 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, say at the 20th percentile, where all firms below the cutoff exit and all firms above

the cutoff stay. In Melitz (2003), an expansion of trade raises the cutoff, say to the 30th

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

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 com-

petitors, 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

locations as different regions within the same country and allow factor mobility. For the

rest of the paper, we take the factors at each location as fixed. We can think of this as an

analysis in international trade where factors are immobile, or that part of a regional analysis

which conditions on location choice.

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 1 is 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

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 () and in general depends

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

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

the second stage, at each location and for each good , 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 , the lower

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 thatexpressions 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

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

when the size is very large. There are four parts. The first 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.

2The log of a Fréchet random variable has variance 262 that depends upon the shape parameter ,

but not the scaling parameter.3Others include Helpman, Melitz, and Yeaple (2004), Chaney (2008), and Eaton, Kortum and Kramarz

(2008).4The Pareto and Frechet distribution are tail equivalent in the sense that the tail probabilities are pro-

portional 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

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 (1 ≤ 1 2 ≤ 2|) (3)

= Pr [1 ≤ 2|] + Pr [2 1 ≤ 1 2 ≤ 2|]=

(2) +

£ (1)−

(2)¤ (2)

−1

= −−2 +

h−

−1 − −

−2

i−(−1)

−2 ,

for 1 ≥ 2. To see this, note that if the first 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 possible ways one of the 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

= , (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 entrepre-

neurs 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

12(1 2; ) = −−2 +

h−

−1 − −

−2i−(−

)−2 ,

Plugging =P∞

=0

!into the bracketed term and taking limits yields

8

lim→∞

12(1 2; ) = −−2 +

£−2 − −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 , as well as the technology component 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→0

Pr( = 1| ≥ 1 ) = 1.

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

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 1.

3.3 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 produc-

tivity from all sources and analogously let 2() be the second highest from all sources.

BEJK show the joint distribution of 1 and 2 equals

12(1 2) = −Φ−2 + Φ

£−2 − −1

¤−Φ

−2

for the index Φ defined by

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

be interpreted as the sum of entrepreneurial activity across all source locations, weighted by

the cost efficiency 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(). The price equals

() = min

Ã1

2()

1()

!. (8)

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

() = min

Ã2()

1()

!(9)

BEJK show that the distribution of the markup is a truncated Pareto,

Pr(() ≤ ) =

(1−− 1 ≤

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

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 share of revenue at each location must equal the share

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 equilibriumwith 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.

Helpman showed that (1− ) 1 is a sufficient condition for the symmetric (or dis-

persion) 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− ) 1 partly 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

5Stability is defined in the standard way of this literature. We formally define it in the proof.

12

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 . That is: (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

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,

1 = max

n1

o,

and 2 is the second highest across all sources.

Location is the source to if the most efficient entrepreneur at has higher cost-adjusted

productivity of serving , compared to the most efficient at all other locations. This happens

with probability

≡ Pr∙1 max

6=

n1

o¸=

P

=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 , and is

discounted with a higher wage or transportation cost. The denominator of

Φ ≡X

=1

, (13)

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 is the same as in BEJK, mark-ups are different. To determine mark-ups, we

first derive the joint density of the first and second highest productivities 1 and 2 from

all source locations. Conditioned upon the first highest being from source , a separate

appendix shows that the joint density is

∗12(1 2) = Φ (Φ − ) 2−−11 −−12 −

−1−(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

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

of draws goes to infinity, 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 , we obtain the

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

− Φ( − 1) , for , (15)

and equals () = 1, above the monopoly mark-up, ≥ . (ii) An increase in the number

of entrepreneurs , from any source location with nondegenerate ( 0) cost efficiency

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-up is shifts up with the cost efficiency of the source location, and shifts down with

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 , and from (15), we can see

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. cost efficiency

6In 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

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

and sold in destination . Demand for any differentiated good at can be written as

() = ◦−, where ◦ is scaling parameter that will depend upon the price index at

destination of the composite good, as well as the size of the location. Revenue at price

can then be written () ≡ () = ◦1− . Let be expected revenue for a good sold

to destination , conditioned on originating from source , and fixing 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 11. Multiplying this by the quantity of demand at

price results in a level of variable cost equal to =³11

´◦

−. 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.

For the variables we have just defined, we have the following result.

Proposition 3. (i) Expected revenues at location for a good originating at location can

be written as

= ◦

Z ∞

0

Z 1

0

(1 2)1−∗12(1 2)12 (16)

= ◦Γ

µ + 1−

¶× ,

16

where ◦ is a demand scaler, and is defined by

≡ −(−1)Φ

£ + (Φ − )

¤− +1−

+Φ (Φ − )

×h¡Φ − +

−¢− +1− −Φ

− +1−

i

and again = ( − 1) is the monopoly mark-up.(ii) Expected variable costs at location , for a good originating at , can be written

= ◦Γ

µ + 1−

¶×

for defined by

≡ −Φ

£ + (Φ − )

¤− +1−

+ Φ (Φ − )− +1−

µ + 1−

1 +

¶

and is defined by

=

µ1 +

1

1 + 2 −

2 +

1

−

Φ −

¶−−1−

µ1 +

1

1 + 2 −

2 +

1

−

−

Φ −

¶

where is the hypergeometric function.

(iii) For any nongenerate ( 0) source , an increase in strictly increases expected

revenue and strictly decreases the sales-weighted mark-up share, which can be written

as _ = 1 − . 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× . 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 , which are held fixed, it cancels out in what we do and can be ignored. We

17

focus on the last term , as this varies with the number of competitors at each location.

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 then drops out and the

formula reduces to

=

Ã

()1

!−(−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 − 1and that is the maximum amount cost heterogeneity permitted in the parameter space.

Otherwise, things are undefined. At the limit,

lim→(−1)

= 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, so Φ = , the unweighted number of firms. If ≥ 2, at the limitwhere is large we have

lim→∞

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

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 mark-

ups 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-to-

head 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 of variety. Here we ask, how do the

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.

We can drop the and location indicators and normalize draw quality to = 1, so that

Φ = is just the total number of firms. It is convenient here to write expected revenue ()

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

= Γ

µ + 1−

¶ −1−1Ã ∞X

=1

()

! −1−1

.

19

The welfare measure is the inverse of the price index, ignoring the multiplicative Γ term,

=

à ∞X=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 , in which

case the welfare measure reduces to

_( ) =³()

´ 1−1

= 1

−1 ()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 12)

_( )

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 theCompetition Variety Ratio at the lower bound for and at the other extreme of large ,

lim→−1

_ _(1 ) = 1

lim→∞

_ _(1 ) =

− 12− 1−1

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 first note that increasing , which makes the productivity

draws more similar, has two off-setting impacts on the ratio. On the one hand, greater sim-

ilarity 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-shape in Figure 2. In the limit as gets large, the productivity draws are identical, the

duopoly mark-up goes to zero, and the productivity gain from selection is zero. Thus for

20

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, the ratio is one,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 numberof 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 . Figure 4 plots how the average mark-

up 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. The impact of going from three to four is even smaller. It is well understood

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.

7For = 2, the ratio is minimized at = 25, where the ratio is .82. In the limit for large , the ratio is

maximized at = 33 where it equals 1.06.

21

5.2 The Impact of Market Size with Free Entry into Entrepre-

neurship

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.

5.2.1 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 entrepreneurshiplevel . The Poisson probability distribution over the , conditioned upon and that ≥ 1equals

1+ ( ) =−

(1− −)!, ≥ 1.

We use this to calculate the revenue share of goods with entrepreneur counts , given

entrepreneurship level . This equals

Rshare( ) =1+ ( )()P∞=1

1+ ( )()

.

The cumulative revenue share of goods with counts less than or equal to is

cumRshare( ) =

X=1

Rshare( ).

That this is strictly decreasing in (stochastic dominance), follows from the stochastic

dominance property of 1+ from an increase in and the fact that () increases in

from Proposition 3.8

The average markup given is the average across the different using revenue weights,

_() =

∞X=1

Rshare( )_().

Because of monotonicity of _() and the stochastic dominance property of

8It 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

Rshare( ), _() strictly decreases in , with limits of the monopoly and

BEJK shares at the extremes, i.e.,

lim→0

_() =1

lim→∞

_() =1

1 + ,

At the equilibrium level of entrepreneurship, the operating profit share of revenues equals

the entrepreneurship share of the population, i.e.

_() =

. (17)

Let ∗() be the unique solution to (17) given population . We have

Proposition 4. (i) The entrepreneurship level ∗() strictly increases in population while

the share ∗() strictly declines. (ii) The equilibrium markup distribution shifts down

in a first-order stochastic dominance way as population increases. (iii) The elasticity of

entrepreneurship with respect to changes in population,

() ≡ ∗()

is less than one but goes to one at the two extremes,

lim→0

() = 1

lim→∞

() = 1.

The elasticity of the mark-up share,

_() ≡ _()

_()

is strictly negative but is zero at the two limits,

lim→0

_() = 0

lim→∞

_() = 0.

Proof. See appendix.

Proposition 3 says that increases in market size shift mark-ups down and lower the share

of individuals choosing entrepreneurship. Yet these effects disappear in the Dixit-Stiglitz

23

and BEJK limits.

5.2.2 The Impact of Market Size in Pictures

We conclude this section by illustrating with graphs the impact of increasing market size on

welfare and on the three margins of welfare gain: greater variety, higher productivity from

more selection, and lower mark-ups.

We use = 2 and = 4, but the results for other parameter values follow the same

pattern. With these parameters, when the population is small and the model looks like

Dixit-Stigliz, the mark-up share is 1 = 5 which will equal the share of the population

entering entrepreneurship. Thus when population = 1, and half enter entrepreneurship,

≈ 05, virtually all entrepreneurs will be monopolists and product variety will be 05. At

the other extreme, when population is large and the model looks like BEJK, the mark-up

share is 1 (1 + ) = 2. Then when population = 100, and entrepreneurship ≈ 20,

virtually all of the unit measure of possible goods will be available in the market place, and

on average each good will have ≈ 20 head-to-head competitors.The top row of graphs in Figure 5 plot the equilibrium levels of variety, average mark-

ups, average productivity, and welfare as a function of the market size , in log scale. The

bottom row plots the elasticities of the four variables with respect to market size.

Note first that on the left when population is small, variety is proportional to population.

Virtually all entrepreneurs are monopolists, so increasing market size increases the number

of monopolists. There is no pro-competitive effect, as average mark-ups are flat. There

is no productivity effect from increased selection, so average productivity is flat. These are

all the things we find in the Dixit-Stiglitz model. The elasticity welfare and market size is

constant at 1 ( − 1) = 1, and is due to increases in variety.At the other extreme when population is large, all goods are offered, so variety is flat.

There is selection, so productivity increases with size. The effect of competition has leveled

off, and margins do not change with size. These are all the things we find in BEJK. The

elasticity of welfare is constant at 1 = 25, and is due to increases in productivity.

Now look between the extremes. There is a pro-competitive effect as average mark-ups

decrease sharply with market size, in contrast to what happens at the extremes where the

relationship is flat. There is a productivity effect of selection. Also variety increases.

These three margins combine to create welfare gains from market size increases that are

bigger than in the BEJK region, where the only source of gain is higher productivity.

24

References

Arkolakis, C., A. Costinot, and A. Rodriguez-Clare, “New trade models, same old gains?,”

American Economic Review, forthcoming.

Arnold, B. C., N. Balakrishnan, and H. N. Nagaraja (1992), A First Course in Order Sta-

tistics, New York: John Wiley & Sons.

Asplund, M. and V. Nocke, “Firm turnover in imperfectly competitive markets,” Review of

Economic Studies (2006) 73, 295—327.

Baldwin, R. E. and T. Okubo (2006), “Heterogeneous firms, agglomeration and economic

geography: spatial selection and sorting,” Journal of Economic Geography, 6(3): 323-346.

Behrens, K., G. Duranton, and F. Robert-Nicoud (2010), “Productive cities: Sorting, selec-

tion, and agglomeration,” working paper, University of Toronto.

Bernard, A. B., J. Eaton, J. B. Jensen and S. Kortum (2003), “Plants and productivity in

international trade,” American Economic Review, 93(4): 1268-1290.

Brander, James and Paul Krugman, A ‘reciprocal dumping’ model of international trade,

Journal of International Economics, Volume 15, Issues 3-4, November 1983, Pages 313-

321

Chaney, T. (2008), “Distorted Gravity: the Intensive and Extensive Margins of International

Trade,” American Economic Review, 98(4): 1707-1721.

Combes, P.-P., G. Duranton, L. Gobillon, D. Puga, and S. Roux (2009), “The productivity

advantages of large cities: Distinguishing agglomeration from firm selection,” working

paper, University of Toronto.

Dixit, A. K. and J. E. Stiglitz (1977), “Monopolistic competition and optimum product

diversity,” American Economic Review, 67(3): 297-308.

Duranton, G. and D. Puga (2001), “Nursery cities: Urban diversity, process innovation, and

the life cycle of products,” American Economic Review, Vol. 91, No. 5 (Dec., 2001), pp.

1454-1477.

Eaton, J., and S. Kortum (2002), “Technology, geography, and trade,” Econometrica, 70(5):

1741-1779.

Eaton, J., S. Kortum, and F. Kramarz (forthcoming), “An anatomy of international trade:

Evidence from French firms,” NBER Working Paper No. 14610.

Embrechts, P., C. Kluppelberg, and T. Mikosch (1997), Modelling Extremal Events for In-

surance and Finance, Berlin: Springer-Verlag.

25

Foster, L., J. Haltiwanger, and C. Syverson (2008), “Reallocation, firm turnover, and effi-

ciency: Selection on productivity and profitability?,” American Economic Review, 98(1):

394-425.

Fujita, M., P. Krugman and A. J. Venables (1999), The Spatial Economy: Cities, Regions,

and International Trade, The MIT Press.

Helpman, E. (1998), “The size of regions,” in D.Pines, E.Sadka and Y.Zilcha (eds.), Topics

in Public Economics, pp. 33-54, New York: Cambridge University Press.

Helpman, Elhanan, Oleg Itskhoki, and Steven Redding, Inequality and Unemployment in a

Global Economy", Econometrica, 78(4), 1239—1283, 2010.

Holmes, Thomas J. and James A. Schmitz, Jr. “Competition and Productivity: A Review

of the Evidence” Annual Review of Economics, Vol. 2, September 2010

Hopenhayn, Hugo A. Entry, Exit, and firm Dynamics in Long Run Equilibrium,” Economet-

rica Vol. 60, No. 5 (Sep., 1992), pp. 1127-1150

Horstmann, Ignatius J. and James R. Markusen, Up the average cost curve: Inefficient entry

and the new protectionism, Journal of International Economics, Volume 20, Issues 3-4,

May 1986, Pages 225-247

Krugman, P. (1991), “Increasing returns and economic geography,” Journal of Political Econ-

omy, 99(3): 483-499.

Markusen, James R. , Trade and the gains from trade with imperfect competition, Journal

of International Economics, Volume 11, Issue 4, November 1981, Pages 531-551,

Melitz, Marc J. (2003), “The impact of trade on intra-industry reallocations and aggregate

industry productivity,” Econometrica, 71(6): 1695-1725.

Melitz, Marc J. and G. Ottaviano (2008), “Market size, trade, and productivity,” Review of

Economic Studies, 75(1): 295-316.

Neary, J. Peter, “Presidential Address: Globalization and Market Structure,” Journal of

the European Economic Association Vol. 1, No. 2/3, Papers and Proceedings of the

Seventeenth Annual Congress of the European Economic Association (Apr. - May, 2003),

pp. 245-271.

Peter Neary, J. (2010), Two and a Half Theories of Trade. The World Economy, 33: 1—19.

Nocke, V. (2006), “A gap for me: Entrepreneurs and entry,” Journal of the European Eco-

nomic Association, Sept 2006, 4(5):929-959.

Ottaviano, G., T. Tabuchi and J.-F. Thisse (2002), “Agglomeration and trade revisited,”

International Economic Review, 43(2): 409-435.

26

Pavcnik N. 2002. Trade Liberalization, Exit, and Productivity Improvement: Evidence from

Chilean Plants. Review of Economic Studies. 69(1): 245-76.

Redding, Steven, “Theories of Heterogeneous Firms and Trade", Annual Review of Eco-

nomics, forthcoming, 2011,

Redding, S. and D. Sturm (2008), "The costs of remoteness: Evidence from German division

and reunification" American Economic Review, 98(5): 1766-1797.

Tybout JR. 2003. Plant- and Firm-Level Evidence on ‘New’ Trade Theories. In Handbook

of International Economics, ed. E Kwan Choi and J Harrigan, 13: 388-435. Oxford:

Basil-Blackwell. 435 pp.

Syverson, C. (2004), “Market structure and productivity: A concrete example,” Journal of

Political Economy, 112(6): 1181-1222.

Venables, Anthony J. , Trade and trade policy with imperfect competition: The case of

identical products and free entry, Journal of International Economics, Volume 19, Issues

1-2, August 1985, Pages 1-1

27

0.5

11

.5D

ensi

ty

-1 0 1 2 3lnx1

Figure 1 Comparison of Frechet and Pareto

(Same Mean and Variance)

Plot of ln(1-F) versus ln(x) for Frechet and Pareto

Density Plots (Curve is best-fitting log normal plot)

Frechet Pareto

01

23

Den

sity

0 .5 1 1.5 2 2.5lnx2

-10

-50

5

-1 0 1 2 3lnx

lnrank1 lnrank2

Figure 2 The Competition Variety Ratio: The Relative Gain of Doubling n and Halving Variety

As a Function of θ for various levels of σ and Starting Point Monopoly

Figure 3 The Competition Variety Ratio: The Relative Gain of Doubling n and Halving Variety

As a Function of θ, Fixing σ = 2 and Varying Starting n

Figure 4 The Impact on Mark-up Share of Increasing n as a Function of θ

Fixing σ = 2

Figure 5 How Equilibrium Variables Change with Market Size

Levels and Elasticities for σ = 2 and θ = 4.