DP2010-11

Firm Heterogeneity and Location Choic e

Toshihiro OKUBO
April 1, 2010

Firm Heterogeneity and Location Choice
Toshihiro Okubo
Kobe University

April 2010
ABSTRACT

Heterogeneity in firm productivity affects the location patterns of firm and agglomeration.
Here we provide an economic geography model, involving forward and backward linkages
driven by the migration of a footloose entrepreneur (capital owner) with different productivity.
As a result we find a sorting equilibrium characterised by co-agglomeration of similar
productivity firms, however, in contrast to previous studies, unproductive firms are more likely
to agglomerate than their more productive counterparts. This is due to the increasingly severe
competition induced by productive firms. Productive firms prevent severe local competition
through their co-agglomeration. In terms of social welfare, although the sorting equilibrium
involves higher social welfare than a perfectly symmetric pattern of firm location, the market
outcome is sub-optimal and induces too much agglomeration.
JEL F15, F23.
Keywords: heterogeneous firms, footloose entrepreneurs, competition, productivity,
economic geography

1. INTRODUCTION
Heterogeneity in firm productivity has recently been one of the most important issues inspatial aspects of economics. Not only economic researchers but also national governments andpolicy makers have drawn attention to how firm heterogeneity affects firm location andgeographical concentration, and how high productivity firms can be attracted to a specific area
to induce agglomeration in order to boost national average productivity (World Bank, 2009).
There is some empirical evidence regarding firm heterogeneity and location patterns, forexample the observation of a core region characterised by severe competition, which pushes
lower productivity firms outwards towards the periphery (Syverson, 2004; Asplund and Nocke,2006). However the precise nature of the relationship between firm location and productivity is
still ill-defined. As Duranton and Overman (2005) observe empirically, location patterns arehighly heterogeneous across industries. In some sectors large scale firms are dispersed acrossregions while smaller firms are more concentrated geographically. They suggest that the spatial
pattern is influenced by firm or sector characteristics and the way in which sectors/regions areclassified. Furthermore, although dense areas are more productive than periphery in some cases,

2.1 Rokkoudai Nada Kobe Hyogo, Japan. okubo@rieb.kobe-u.ac.jp

2
this is less obvious, such as the example that downtown Detroit is more productive than suburbs
such as Silicon Valley (Glaeser and Kahn, 2004). Similarly Au and Henderson (2005), using
data on Chinese cities, found an inverted-U relationship between real wage per worker andurban size, which indicates that larger sized markets (cities) might not attract high productivity
firms or might attract more low productivity firms.
Based on the empirical evidence there might be another possible location pattern when firm
productivity is heterogeneous. It can be hypothesised that productive firms cause severe local
competition due to their lower prices and higher market share and thus they can deter coagglomeration in the large market and relocate from the large metropolitan area (core) tosuburbs (periphery) while maintaining better access to the core region. Urban area locations
sometimes cause firm to lower their productivity due to urban congestion effects, such as trafficjams, higher wage and higher land rents. These negative location factors might provide an
incentive for productive firms to escape to the suburbs. It is important to study these negative
urban congestion effects, but this paper explains firm location patterns by a “self-selection”
mechanism, in which severe market competition, which is self-induced by productive firms,
promotes their relocation to suburbs. Thus any urban congestion concerns and negative
externalities are out of our scope. Our interest is in market competition caused by thegeographical concentration of high productivity firms without any urban negative externalities.
In order to investigate the hypothesis described, this paper studies the impact of firm
heterogeneity on firm location in an economic geography model.

1.1. Literature Review and Our Model
This paper constructs a simple model on the basis of recent advancements of two strands ofliterature. One strand surrounds the footloose entrepreneur (FE) model in the economic
geography literature and the other is concerned with the heterogeneous-firm trade (HFT) modelof Melitz (2003) in the international trade literature.
The spatial aspects of economics have been discussed by the “new economic geography”
models, initiated by the core-periphery (CP) model of Krugman (1991). These models study the
relationship between trade costs and the location patterns of firms, agglomeration processes andthe driving forces behind agglomeration and dispersion. However, the CP model has two weak
aspects, first is its analytic intractability, the second is that it ignores heterogeneous firm
productivity, which is our central focus in this paper.
The first weakness has been resolved by models in subsequent studies. A set of models has
been proposed which provide more tractable frameworks such as Ottaviano, Tabuchi and Thisse3(2002). Forslid (1999) Forslid and Ottaviano (2002) and Baldwin et al. (2003, Ch.4) provide ananalytically solvable version of CP model called the footloose entrepreneur (FE) model. The FEmodel involves the migration of capital owners (or entrepreneurs), which causes demand-linked
circular causality driven by a migrant expenditure shift as well as cost-linked circular causality
driven by a decreasing cost of living due to more local producers in an areas. Later, Pfluger(2004), successfully obtained a simpler model with more analytical solutions, providing the FEmodel with the quasi-linear utility function.1 Since the quasi-linear utility function excludes
income effects and dampens demand-linked circular causality the agglomeration effect weakens
and catastrophic agglomeration never happens, resulting instead in a gradual agglomeration
process via trade cost reduction. This paper adopts the FE model of Pfluger (2004) due to its
tractability and simplicity as well as for the reasons described below.
A technical reason for using the quasi-linear utility function model is that we conduct anintensive analysis of firm heterogeneity rather than labour heterogeneity and migration of theentrepreneur is a key factor. When firms are heterogeneous, or when entrepreneurs have avarying level of talent, capital returns, via migrant entrepreneurs’ income, vary acrossentrepreneurs. This would incur a heterogeneous income effect across entrepreneurs and aheterogeneous demand shift through their migration, resulting in a mixture of heterogeneousfirms with heterogeneous labour migration. Thus the quasi-linear utility function reduces theheterogeneous demand/income effect in order to distinguish it from the previous demand/labour
heterogeneity literature (e.g. Tabuchi and Thisse, 2002) and instead highlight the impact of firm
heterogeneity on firm location and market competition.2 Thus in this paper we require that
entrepreneurs uniformly spend a unit of demand regardless of their income, although migration
still creates demand linkages. A single firm production shift corresponds to a constant uniform
shift of a unit of expenditure by a migrated entrepreneur. On the other hand, firms areheterogeneous and thus the impact of a production shift on competition and the cost of living is
influenced by the location pattern of firms with varying productivities. Productive firms cause
severe competition and improve the cost of living by lowering prices. For this reason, we adopt
the FE model with a quasi-linear utility function.
The second weakness, ignorance of firm heterogeneity, can be solved by incorporating the
HFT model of Melitz (2003) into an economic geography model. The HFT models focus on the1

Pfluger and Sudekum (2008), an extension of Pfluger (2004) involves welfare analysis with housing market.
Heterogeneity on the demand side is studied by Tabuchi and Thisse (2002) and Okubo and Picard (2008). The
sorting of skilled labour is modelled by Mori and Turrini (2005). Empirically Combes et al. (2003) found labour
sorting in France.
2

4
links between firm productivity and export behaviour on the one hand and the impact of tradeliberalisation on industry productivity on the other.3 Melitz (2003) allows for firm heterogeneity
and sunk market entry costs. In his model firms have heterogeneous marginal costs and the mostproductive firms, those with the lowest marginal costs, enjoy higher market shares and operating
profits. As a result the most productive firms will enjoy sales that are large enough to cover the
fixed domestic market entry costs and the most productive among these will export to foreign
markets and enjoy sales that are large enough to cover the fixed export costs. Tradeliberalisation raises only exporters’ profits while reducing local producers’ profits, this is known
as the profit share shifting effect, thus forcing the least efficient local producers to exit themarket, allowing the most productive local firms to enter the export market, the co-called
selection effect. The location of firms, however, is ignored as firms are assumed to locate in thenation in which they are ‘born’. In contrast to Melitz (2003) the aim of this paper is toinvestigate the way in which firm heterogeneity and firm location interact rather than focus on
trade patterns and export behaviour.
Economic geography models with firm heterogeneity are not entirely new, Nocke (2006)
first modelled spatial sorting, in which talented entrepreneurs enter a larger market and lesstalented ones choose to locate in a smaller market. Baldwin and Okubo (2006) extended his idea
to “new economic geography” and first developed the Footloose Capital (FC) model of Martin
and Rogers (1995) with firm heterogeneity a la Melitz (2003).4 The main results of Baldwin andOkubo (2006) are 1) firm heterogeneity works as dispersion force, however, firm heterogeneity
per se never affects the break and sustain points but just the moderate agglomeration process. 2)
The most productive firms are the most footloose because the productive firms are moresensitive to profit gap. Accordingly productive firms in smaller markets are more likely to
relocate to the larger market. This means that larger market has productive firms, while small
market has only unproductive firms (“spatial selection/sorting effect”).
The conclusions noted above does not represent all that is known regarding the interaction offirm heterogeneity and location. Baldwin and Okubo (2006) is based on the simplest economic
geography model, the FC model, in which capital returns are repatriated to the origin due to thefact there is no labour migration allowed, thus firm location is determined only by nominal
profits. By contrast, the inclusion of labour (entrepreneur) migration in the FE model alters thefirm location decision. A firm’s location is determined by real profits, which means migration
3

See also Helpman, Melitz and Yeaple (2004), Melitz and Ottaviano (2008) and Chaney (2008).
Later, Okubo (2009) added intermediate output linkages (the FC vertical linkage model) to Baldwin and Okubo(2006).
4

5
takes into account the gap in nominal profit as well as the cost of living. In particular firm
heterogeneity impacts upon competition as well as the cost of living. The agglomeration of highproductivity firms produce severe competition, resulting in reductions in market shares andprofitability, thus deterring these firms from co-agglomeration.
We have identified several results which contrast with existing heterogeneous firm economic
geography models. Firstly, the earliest movers are unproductive firms; since the migration of
productive firms causes more severe competition productive firms are deterred from coagglomeration, low productivity firms are more footloose in location choice. Secondly, spatialsorting occurs as productive firms will be diversified and unproductive firms create
agglomeration in one location. This is also in contrast with previous models in which productive
firms are likely to create agglomeration. Thirdly, firm heterogeneity works as an agglomeration
force, which is also in contrast with Baldwin and Okubo (2006). Firm heterogeneity is more
likely to cause an unstable symmetric initial equilibrium and encouraged full agglomeration,
seen through decreased break and sustain points. These results are in contrast with Nocke (2006)
and Baldwin and Okubo (2006). Finally, spatial sorting can improve global welfare whencompared to a perfectly symmetric equilibrium. However, spatial sorting in market outcomesinvolves too much agglomeration, compared with the social optimal spatial sorting equilibrium.
All of these results cannot be derived from HFT models (Melitz, 2003; Melitz and Ottaviano,2008), in these models the most productive firms are likely to be more footloose and engage in
FDI, while low productivity firms are more likely to exit market. Thus they cannot derivefootloose unproductive firms, hence they do not observe co-agglomeration of low productivity
firms and spatial sorting in terms of firm productivity.
The rest of the paper is organised in 6 sections, Section 2 presents the basic model. Section 3
explores the long-run equilibrium. Section 4 studies social welfare and optimal equilibrium path.
Finally concluding remarks are provided in Section 5.

2. THE BASIC MODEL AND SHORT-RUN EQUILIBRIUM
2.1. Footloose Entrepreneur Model
The model works across two regions, the North and the South, two sectors, differentiatedmanufacturing goods in the “M” sector, and a homogenous agriculture good “A” sector, and two
factors, labour, L, and capital, K. There are two categories of people, workers, who provides one
unit of labour (L) and are bound to the land and secondly entrepreneurs, who own one unit ofcapital (K) and is inter-regionally mobile together with their capital/firm. Thus labour, L is inter-6
regionally immobile, while capital, K, is inter-regionally mobile just as workers andentrepreneurs are immobile and mobile respectively. We define the total endowment of
resources in the world as Lw + K w where Lw and Kw are the worldwide endowments of labourand capital exogenously given as K w = 1 Lw = β . Thus the total global population, which issimply the sum of workers and entrepreneurs, is 1+β. The two regions are symmetric in all
aspects, specifically tastes, technology, openness to trade, and their relative factor endowmentsof labour and capital. The tastes of the representative consumer in each region are quasi-linear:
(1)

1-1/σ
U = μ ln C M + C A , C M ≡ ⎛⎜ ∫ ci
di ⎞⎟
⎝ i∈Θ
⎠

1 /( 1-1/σ ), 0 < μ <1<σwhere CM and CA are consumption of the composite of all differentiated varieties of M goods,
and consumption of the homogenous A good respectively. μ measures the share of expenditure
spent on M-sector varieties, Θ is the set of varieties available in a typical region, and σ is theconstant elasticity of substitution between any two M-sector varieties.
The A-sector is characterised by perfect competition, constant returns to scale, and zero trade
costs. The production of the A-sector good involves only labour and good A is taken to be thenumeraire. The M-sector is characterised by increasing returns, Dixit-Stiglitz monopolisticcompetition and iceberg trading costs. As usual, t ≥ 1units must be shipped in order to sell one
unit in the foreign market. Following the standard FE model, since one unit of capital, which isowned by an entrepreneur, creates one firm the migration of entrepreneurs/capital corresponds
to firm migration. Thus the total number of firms in the world is unity due to K w = 1 .
In contrast to standard FE model, firms in M-sector are heterogeneous in terms of labour
productivity. For simplicity we have two types of firms with different levels of productivity.
One type is unproductive firm and thus requires more units of labour, that is it has a higher
marginal cost, we label these as ‘H’ firms, while the other type is a more productive firm andthus requires less units of labour, corresponding to lower marginal cost firms, which we refer toas ‘L’ firms. Each firm is required to use one unit of capital, representing as fixed cost. The costfunction of a typical Northern firm of type j is:(2)

π j + a j wx j

1 ≥ φ ≡ t 1−σ ≥ 0 , j ∈ ( H , L)

where π j represents the reward to capital as fixed costs and the second term is variable costs. xj
is firm-level output, a j is that firm’s labour requirement, and w is the labour’s reward, or wage.
The parameter, φ, which plays a critical role in the analysis, is an indicator of the “freeness” of7trade in that φ ranges from zero, when trade is prohibitively expensive (t = ∞) to unity when
trade is perfectly free (t = 1). The firm-specific unit-input coefficient is a H for H firms and a L
for L firms, where a H > aL .
Each region is endowed with an equal share of firms at the initial equilibrium. We define theworldwide mass of varieties/capital to be equal to unity, hence each region’s mass offirms/capital is equal to ½ at the initial equilibrium. Each region begins with the same
proportion of H- and L-firms; α (1- α ) denotes the proportion of H-firms (L-firms) in thecomposition of the total number of firms across the regions. The share of firm types, α ,characterised by productivity distributions, is exogenously given as in Melitz (2003). Since eachfirm is associated with a particular unit of capital it is natural to assign the source of firm
heterogeneity to its capital, i.e. each unit of capital in each region is associated with a particularmarginal cost as measured by the firm-specific unit-input coefficients a H and a L .
The main focus of this paper is firm heterogeneity. As the difference between a H and a L
increases firms become more heterogeneous in terms of labour productivity. Firm share is also
relevant: for example as α approaches zero or one a single type of firm is dominant and thus
firms are almost homogeneous and the setting becomes almost identical to that in the standard
FE model. On the other hand, when α is close to 0.5, two types of firms coexist in almost equal
proportions. In order to highlight firm heterogeneity we exogenously create a substantial
difference between a H and a L with α ≈ 0.5 under the co-existence of two firm types.
Specifically we assume that
(3)

a
α
< H
1 − α aL

This means that the cost difference, which determines firm heterogeneity, is larger than therelative proportion of H and L firms.
Utility maximisation gives us the demand function for the ith variety of the M-goods as:
(4)

ci =

pi

∫

h∈Θ

-σ

ph dj + ∫
1-σ

k∈Θ*

φpk 1-σ dk

μE

where ci and pi are the consumption and price of variety i and E is total expenditure in the
North. Southern demand functions are isomorphic. Here we adopt the standard convention ofdenoting Southern variables with a “*” superscript.

8

2.2. The Short-run Equilibrium
We begin by examining a symmetric equilibrium where the worldwide mass of firms, which we
normalise to unity, is evenly split with ½ of all firms locating their production in each of the tworegions. We also fix the proportion of each region dedicated to the M and A sectors to be equalwith α / 2 and (1 − α ) / 2 being the type of each firm locating their production in each of theregions.
Due to constant returns, perfect competition, and zero trade costs in the A-sector, the price of the
A-sector good is identical in both markets and so equates marginal costs across the regions. The
equalisation of prices and marginal costs implies that equilibrium wages must also be identicalin the two regions. Choosing a number of units of A, without loss of generality, such that theunit labour-input requirement is unity, we have w = w* = 1 since the price of A is unity.
Utility maximisation implies the usual CES demand functions for each variety of M, taken
together with the assumed presence of Dixit-Stiglitz monopolistic competition, imply that ‘mill
pricing’ is optimal. That is, a typical M-sector firm based in the North charges a producer pricethat is a constant mark-up over marginal costs, with all trading costs passed on to consumers intheir entirety, specifically:
(5)

pH =

aH
1−1/ σ

pH* =

pL =

aL
1 −1/ σ

pL* =

ta H
1-1/ σta L
1-1/ σWe can combine the fact that we are in a symmetric equilibrium, so that each type of firm
producing in the North and the South is ½, with expression (5) to obtain the marginal cost
implied by a H and a L , so that price index in North, P, can be written as:
1−σ
1−σ
1−σ
1−σ
⎛
⎛ aH ⎞⎛ ta H ⎞⎛ aL ⎞⎛ ta L ⎞ ⎞⎟⎜P = αn H ⎜⎟ + α (1 − nH )⎜⎟ + (1 − α )nL ⎜⎟ + (1 − α )(1 − nL )⎜⎟⎜⎝ 1 −1/ σ ⎠⎝ 1−1/ σ ⎠⎝ 1−1/ σ ⎠⎝ 1 − 1 / σ ⎠ ⎟⎠⎝μ /(1−σ )1=() μ /(1−σ ) αnH a1H−σ + α (1 − nH )φa1H−σ + (1 − α )nL a1L−σ + (1 − α )(1 − nL )φa1L−σ
1−σ(1 − 1 / σ )where nH (1- nH ) denotes the Northern (Southern) share of H-firms and nL (1- nL ) denotes the()

Northern (Southern) share of L-firms, noting that nH = nL = 1 / 2 at the initial equilibrium.
Simplifying this and using an analogous approach to the Southern region yields(6)

Δ = αnH a1H−σ + α (1 − nH )φa1H−σ + (1 − α )nL a1L−σ + (1 − α )(1 − nL )φa1L−σ
Δ* = αnH φa1H−σ + α (1 − nH )a1H−σ + (1 − α )nLφa1L−σ + (1 − α )(1 − nL )a1L−σμ /(1−σ )9where, for notational convenience, we defineΔ ≡ (1 − 1/ σ )1−σ(P)(1−σ ) / μ

and Δ* ≡ (1 − 1/ σ )

1−σ

(P )

* (1−σ ) / μ. Note that Δ is the denominator of the

CES demand function.
A firm’s operating profit is critical to this analysis since, given the structure of the model, thereward to a unit of capital is the operating profit of the firm with which it is associated. To
calculate this we need to know the level of expenditure in each market. For example using thequasi-linear utility function (1) Northern and Southern expenditure on all M-goods, E and E*
respectively, can be written as:

μE * = μ (L* + K * )μE = μ (L + K )(7)

Endowments in each region are thus L = L* =β2

, K = αn H + (1 − α )n L ,and K* = α(1− nH ) + (1−α)(1− nL ) . Total initial endowments are K w = 1 Lw = β . We note that L
and L* are exogenously given and equally allocated between regions, while K and K*
correspond to firm location and thus are endogenously determined, initially defined as K = 0.5
and K * = 0.5 ( nH = nL = 1 / 2 ). Using this information, operating profits for a typical North
based firm with a unit-input coefficient of a H and a L can be written as:(8)

π [aH ] = γBaH 1−σ , π [aL ] = γBaL1−σ ;where B ≡ (EE*+φ * )
Δ
Δ

γ≡

μ
σ

a Southern firm is π *[aH ] = γB *aH

E=
1−σ

β
2

+ K , and E * =and π *[aL ] = γB*aL

1−σ

β
2

+ K * . The analogous formula forwhere B * ≡ (φE E*+
) . Here theΔ Δ*

B’s indicate the market potential in the North and South respectively. Using the indirect utility
function corresponding to (1), entrepreneurs are inter-regionally mobile in search of higher
utility, or real capital rewards, a fact which is expressed as(9) VH ≡ μ (ln μ − 1) + π [aH ] − ln P = μ (ln μ − 1) + γBaH

VL ≡ μ (ln μ − 1) + π [aL ] − ln P = μ (ln μ − 1) + γBaL

1−σ

1−σ

−

−

μ
ln Δ − μ ln(1 − 1 / σ )1−σμln Δ − μ ln(1 − 1 / σ )
1−σ

10

3. THE LONG-RUN EQUILIBRIUM
3.1. Relocation tendencies
Starting with a situation where firms are evenly divided between the regions we consider the gap
in real capital rewards between regions as an incentive for migration for each type ofentrepreneur. Since firms are heterogeneous an additional complication must be addressed;
whether H-firms or L-firms are the first to relocate. Using (8) and (9), the real reward gap thatan entrepreneur faces when locating in the North as opposed to the South is:μ(ln Δ − ln Δ* )
1−σ
(10)
μ
1−σ
VL − VL* = π [aL ] − ln P − π *[aL ] + ln P* = γ ( B − B* )aL −(ln Δ − ln Δ* )1−σ
VH − VH* = π [aH ] − ln P − π *[aH ] + ln P* = γ ( B − B* )aH

1−σ−

At the initial equilibrium the two regions are identical and thus the gap is zero, that is
VH − VH* = 0 and VL − VL* = 0 . The gap in real capital rewards is composed of two terms: the firstterm is the nominal capital reward gap, which varies between firm types. This term can itself be
divided into two parts, the first depending on the market potential gap, B-B*, and the second on
the firm-specific productivity term, a 1−σ . Entrepreneur migration will alter the B’s via impacts
on the Δ’s and E’s. As a H > a L we know that the gap between the L-firms is always larger thanthat between the H-firms. The second part of the gap in real capital rewards is the living costeffect, which comes from the price index gap and is independent of firm type.
In the symmetric outcome, B = B*, E=E*, and Δ=Δ*, so no firm (or entrepreneur) has anincentive to move. However, if the symmetry in firm location is lost all firms have an incentive
to move. At this point the standard economic geography question arises: will the relocation ofsome firms produce self-reinforcing agglomeration with all firms moving to the North, or will
the movement be self-correcting inducing firms to move to restore symmetry? If a slight
positive shock to nH and nL shifts VH − VH* and VL − VL* from zero to a positive number then thesymmetric case is unstable, if the shock turns VH − VH* and VL − VL* negative, symmetry is stable.
Importantly, the response of V − V * to the shock is idiosyncratic due to the heterogeneity of thefirst term in (10), indicating that there exists a certain trade cost level at which one type of firm
is stable and the other is unstable in the symmetric equilibrium.

3.2. Break point and Spatial Sorting
We are interested in evaluating the shocks starting from a position of symmetry, where no firms
have yet migrated. Following the standard economic geographic procedures we evaluate

11∂ (VH − VH* )∂ (VL − VL* )andin order to obtain the level of trade “freeness” where this derivative∂nH∂nL
is zero. Unlike in the standard model, the two types of firms have different migration motives.
Thus, the break point corresponds to when the first firm breaks the symmetric equilibrium andmoves to the other region. Technically this means that we differentiate the real profit gap ofeach firm in terms of firm share at the symmetric equilibrium, given the other type of firms is∂(VH − VH* )∂(VL − VL* )= 0 and= 0 in terms of φ, the smaller φ
fixed. Hence solving
∂n H
∂n L
n = 0.5
n = 0.5
L

H

is the break point.

(11)⎛ E E* ⎞μ
VH − VH* = γ (1 − φ )⎜⎜ − * ⎟⎟h −(ln Δ − ln Δ* )⎝ Δ Δ ⎠ 1−σ⎛ E E* ⎞μ
VL − VL* = γ (1 − φ )⎜⎜ − * ⎟⎟l −(ln Δ − ln Δ* )
Δ
Δ
σ
1
−
⎝
⎠

where h ≡ aH

1−σ

E* = L* + K * =β2l ≡ aL1−σ, E = L+K =β2+ αnH + (1 − α )nL ,+ α(1− nH ) + (1 − α)(1 − nL ) , Δ = αnH h + α (1 − nH )φh + (1 − α )nL l + (1 − α )(1 − nL )φland Δ* = αnH φh + α (1 − nH )h + (1 − α )nLφl + (1 − α )(1 − nL )l .

Solving∂ (VH − VH* )∂ (VL − VL* )= 0 and= 0 in terms of φ, we get φ HB and
∂n H
∂
n
L
n = 0.5
n = 0.5
L

H

φ LB respectively and consequently derive that φ HB < φ LB .5 This indicates that H firms are the firstmovers and break the symmetric equilibrium when trade costs fall. Thus, the break point shouldbe φ B = φ HB , specifically:
(12)

5

6

φB =

(σ − 1)(1 + β )h − (2σ − 1)(αh + (1 − α )l ) 6.
(σ − 1)(1 + β )h + (2σ − 1)(αh + (1 − α )l )See Appendix 1 for derivation.
In the standard quasi-linear FE model, the homogeneous firm model, h=l gives the break point

as φ B = (σ − 1)(1 + β ) − (2σ − 1) , which is identical to that in Pfluger (2004). To keep our analysis interesting, we(σ − 1)(1 + β ) + (2σ − 1)

12Result 1: The first firms to break the symmetric equilibrium are high cost, unproductive
firms. Unproductive firms are more footloose.

This result is contrasts with the standard results from economic geography models with firm
heterogeneity. In Baldwin and Okubo (2006), productive firms are first movers and are more
likely to relocate to the larger market and create agglomeration because they are more sensitive
to the nominal profit gap.
When firms are more heterogeneous, i.e. there is a larger h -l difference, the break point
decreases. This means that firm heterogeneity is more likely to break the symmetric equilibrium
and promote spatial sorting. Thus we can conclude that firm heterogeneity works as an
agglomeration force.
Result 2: Firm heterogeneity works as an agglomeration force. When firms are moreheterogeneous, characterised by a more substantial difference in costs, the break point islower and the symmetric equilibrium is more likely to be broken.

Once H firms deviate from the South to the North at the break point, the deviation causes thereal reward gap of L firms to become negative, i.e. VL − VL* < 0 , because(13)∂ (VL − VL* )4αμ (2σ − 1)(h − l )
=
<0.
2
3
∂nH
(
−
1
)
(
1
+
)
h
σ
β
B
n = 0.5, n = 0.5,φ =φH

L

Therefore, the deviation of H-firms to the North at the break point, when starting from thesymmetric equilibrium, induces L firms to move to the South. In particular, since L firms supply
low price products and thus competition is driven by L firm migration to the South competition
becomes more severe in the South. The increased number of L firms causes severe competitionin the South, which drives H firm relocation to the North, thus driving spatial sorting.

3.3. Equilibrium
We now study the long-run equilibrium when trade costs are lower than the break point. As
discussed above, H firms relocate to the North at the break point, pushing L firms out to theSouth. The mechanism for agglomeration involves the same agglomeration and dispersion
forces as in the standard FE model. Once the H firms migrate from South to North, the demand
shift through entrepreneur migration increases E and decreases E* in (11). As a result B
exclude the case of full agglomeration for any trade cost. As in the standard model, the black-hole condition is
given as (σ − 1)(1 + β ) h > α h + (1 − α )l .
( 2σ − 1)13increases, which is an example of so-called demand-linked circular causality, which attracts
more firms to the North, the demand linked circular causality works as an agglomeration force.
The migration involves firm relocation and thus the production shift raises Δ and falls Δ* ,which indicates a fall of the Northern price index and a rise of the Southern price index. The
improvement in the cost of living in the North attracts more entrepreneurs, which is a cost linked
circular causality, which works as an agglomeration force. On the other hand, cost-linked
circular causality causes more severe competition due to the lower price index, which decreases
B, the so-called congestion effect which acts as a dispersion force.
In the long-run equilibrium, real capital rewards to entrepreneurs should be equal across the tworegions when firms are dispersed, or when firms agglomerate in one location, real capital
rewards are not equal. In contrast to the standard model, we have two types of firms with adifferentiated profit gap due to h<l, as seen in (11). Thus both types of firm cannot equalise theirreal capital reward gaps simultaneously, i.e. VH − VH* = 0 and VL − VL* = 0 with 0 < n H < 1
and 0 < nL < 1 . Based on this, when H firms (L firms) move to the North (the South) there arethree possibilities of real reward gap and location patterns in the long-run equilibrium.
Case 1) VH − VH* = 0 and VL − VL* < 0 , thus 0 < n H < 1 and n L = 0 .
Case 2) VH − VH* > 0 and VL − VL* < 0 , thus nH = 1 and n L = 0 .
Case 3) VH − VH* > 0 and VL − VL* = 0 , thus n H = 1 and 0 < n L < 1 .
However, Cases 1 and 2 never happen and the long-run equilibrium can only be of the type inCase 3.7 At the break point, all H firms concentrate in the North, hence n H = 1 , while L firms
locate in both countries. The share of L firms, n L , is determined by the levelling of the real
reward:μ⎛ E E *⎞
(14) VL − VL* = γ (1 − φ )⎜ −(ln Δ − ln Δ* ) = 0
⎟l −
Δ
Δ
*
1
−
σ
⎝
⎠
where Δ = αh + (1 − α )nL l + (1 − α )(1 − nL )φl , Δ* = αφh + (1 − α )nLφl + (1 − α )(1 − nL )l ,
E=

7

β
2

+ α + (1 − α )n L and E * =See proof in Appendix 2.

β2+ (1 − α )(1 − n L ) .

14
The equilibrium location of the agglomeration of H firms in one region and dispersion of L
firms is attributed to the self-induced local competition. Since the co-agglomeration of L firms
triggers more severe local competition than that of H-firms, due to lower marginal costs andprices, the self-induced competition deters the co-agglomeration of L firms. At the break point
H firms can create full agglomeration immediately, but L-firms cannot. On the other hand, theagglomeration of H firms increases the size of Northern demand, which accommodates some of
the L firms.
1−σ

A simple manipulation using (3) yields α < aH < ⎛⎜⎜ aL ⎞⎟⎟1 − α aL ⎝ aH ⎠=l . Using this condition we alwayshkeep Δ > Δ* and B < B* , implying that the demand shift of E and E* involves a smaller impact
from migration than the production shift in Δ and Δ* . The cost of living is lower in the Northdue to the agglomeration of H-firms coupled with some the L firms, this also causes more
intense competition and reduces market potential and profitability in North. On the other hand,
the South is protected from Northern competition and thus has higher market potential,
regardless of the higher cost of living.
In this model, demand-linked circular causality as an agglomeration force is independent of firm
heterogeneity due to the quasi-linear utility function, thus, the causality is relatively weak. In
contrast, firm heterogeneity affects Δ > Δ* , because all H firms and some L firms locate in the
North. Thus the cost of living effect as an agglomeration force is substantially larger in the
North although the North is benefits from lower prices. Simultaneously a congestion effect acts
as dispersion force, as we observe higher Δ and decreases in B. A substantial congestion effectand a weak demand-linked effect lead to less market potential in North, i.e. B < B* .
In parallel to the standard FE model n L does not have an explicit form in the solution. Usingsome parameter values often adopted in the standard economic geography literature, Figure 1
plots Northern shares of H and L firms at the long-run equilibrium (h=1, l=2, σ=4, μ=0.5,α=0.5
and β=5)8. At the break point all H firms concentrate in the North, this case is known as
catastrophic agglomeration, while L firms locate in both regions. This creates two asymmetricregions: a bigger Northern market and a smaller market in the South. As trade costs fall, market
competition in the two regions is brought closer due to better market access, in addition the
difference in the cost of living is smaller. This attracts more firms to the bigger Northern market
8

aH = 1 and a L = 2 −1 / 3 ≈ 0.79 . We discuss the case of the first deviation of H firms to the North. The opposite could

happen if H firms first deviate from the symmetric equilibrium to the South at the break point and thus the figure isa mirror of Figure 1.

15
and gradual agglomeration happens among L firms. A fall of trade costs gradually raises thenumber of L firms and all L firms finally concentrate in the North at the sustain point.

Figure 1: Equilibrium in terms of trade costs

Result 3: When trade costs fall below the break point, spatial sorting occurs. All high cost
(unproductive) firms concentrate in the North at the break point, whereas low cost(productive) firms diversify their location across both countries. As trade costs fall, lowcost firms gradually relocate to North. Finally full agglomeration arises at the sustain
point.

This result is a sharp contrast to any previous heterogeneous-firm models specifically in that
productive (unproductive) firms are likely to agglomerate (diversify) in terms of their location.
It is also interesting is to see the catastrophic agglomeration of H firms as well as the gradualagglomeration of L firms at the break point.

3.4. Full Agglomeration and Sustain Point
To study the stability of the full agglomeration equilibrium, we evaluate the real reward gap at
n H = 1 and n L = 1 , the point where all firms have located in the North. To find the sustain point,

we solve for the φ where VL − VL* is just positive. Solving (8)-(10) with n H = 1 and n L = 1 , we
get the sustain point denoted as φ S , which is implicitly defined as9:
(15)

⎞
1
1−φ S ⎛ ββ ⎞⎛l⎟⎜⎜ S − − 1⎟⎟⎜⎜ln φ S =1−σσ ⎝ 2φ2 ⎠⎝ αh + (1 − α )l ⎟⎠More firm heterogeneity, represented by an increased gap in costs between L and H firms
decreases the sustain point,

dφ S
dφ S
< 0 and> 0 .10 This indicates that firm heterogeneitydldh

promotes the agglomeration process and works as an agglomeration force.
Result 4: Firm heterogeneity works as agglomeration force. More firm heterogeneity ismore likely to create full agglomeration and thus decrease the sustain point.
9

In the standard homogeneous-firm model, using (15) and substituting with h=l, the sustain point is derived from⎞1
1−φ S ⎛ ββ⎜ln φ S =− − 1 ⎟⎟ .
1−σσ ⎜⎝ 2φ S 2⎠

10

See Appendix 3 for the proof.

16

3.5. Symmetric Equilibrium and Sorting Equilibrium
Up until the last section we started from a setting with two symmetric regions with extremely
high trade costs and then trade costs fall. At the break point symmetry breaks and catastrophic
agglomeration of H firms and gradual agglomeration of L firms arises. In contrast this section
starts with free trade and full agglomeration in the North, under these conditions L firms start to
relocate to the South at the sustain point and more L firms relocate to the South as trade costs
are higher. This is the same argument as in the standard economic geography model, however,
the equilibrium with high trade costs is not the same (see Figure 1).
Even at the break point H firms never relocate to the South due to the fact that VH − VH* > 0 ,while L-firms maintain a degree of dispersion with 0 < n L < 1 to satisfy VL − VL* = 0 . Unless all
L-firms concentrate in the South, the H firms never break away from being fully agglomerated
in the Northern market because the real profit gap is strictly positive, VH − VH* > 0 . In addition,
it is impossible to simultaneously satisfy with both equations VL − VL* = 0 and VH − VH* = 0 as
mentioned above. We note that both VL − VL* = 0 and VH − VH* = 0 are satisfied only at thesymmetric equilibrium and that once location patterns become asymmetric between regions bothare not simultaneously satisfied. Thus as trade costs are increase n L falls, helping to maintainthe full agglomeration of the H-firms, i.e. n H = 1 . This sorting equilibrium is stable even whenφ B > φ , in other words even when trade costs are sufficiently high the location pattern cannot
replicate the symmetric equilibrium because the sorting equilibrium is always stable, i.e. n H = 1
and 0 < n L < 1 .
In contrast to standard economic geography models we have two stable equilibria with hightrade costs, as shown in Figure 1. One is the symmetric equilibrium, in which the two regions
are symmetric in terms of the firms located within them: each type of firm locates equally
locates across regions. While the symmetric equilibrium is stable with high trade costs it isunstable at the break point as lower trade costs induce agglomeration. The other equilibrium isthe sorting equilibrium, one region, which becomes the core, has all the unproductive firms andsome of the productive firms, while the other, which becomes the periphery, has someproductive firms only. Trade cost reduction reduces the intensity of Northern competition andreduces the gap in market competition, thus allowing more productive firms to locate in the coreregion and consequently all firms co-agglomerate in core.

17

4. WELFARE ANALYSISThis section studies social welfare. The central issue in this paper is firm heterogeneity and theimpact on firm location. Thus this section mainly discusses regional welfare gaps and profit
gaps between two types of firms and later studies socially optimal welfare levels and discusses
optimal firm location patterns.
The indirect quasi-linear utility function for a representative consumer can be specified as

μ (ln μ − 1) + y − ln P , where y denotes individual labour or an entrepreneur’s income (wages orcapital returns). Thus, social welfare in the North and South is given as the aggregation ofindividual’s welfare:W = μ (ln μ − 1)( L + K ) + Y − ( L + K ) ln P and W * = μ (ln μ − 1)( L* + K * ) + Y * − ( L* + K * ) ln P *where Y = β / 2 + αnH π H + (1 − α )nLπ L and Y * = β / 2 + α (1 − nH )π H* + (1 − α )(1 − nL )π L* .
The social welfare gap between two regions, W − W * , is plotted in Figure 2 (h=1, l=2,

σ=4,μ=0.5,α=0.5 and β=5). Below the break point, φ B > φ , social welfare in the symmetricequilibrium is invariant across the regions. Northern social welfare relatively increases once thesymmetric equilibrium breaks and asymmetric equilibrium arises because the North attracts
more firms resulting in an increasing population due to increased numbers of entrepreneurs anda better cost of living due to there being more L-firms. Once full agglomeration arises thewelfare gap is a function of φ , given as W − W * = γ (1 + β ) +μln φ , in which the second
1−σterm indicates the regional difference in the cost of living effect. As trade costs fall under full
agglomeration, the welfare gap declines due to the improvement of the cost of living in theSouth.

Figure 2: Regional social welfare gap.

The nominal profit gap between L and H firms π L − π H is plotted in Figure 3 (h=1, l=2, σ=4,
μ=0.5,α=0.5 and β=5). The profit gap in the sorting equilibrium initially decreases, thenincreases as trade costs fall. Above the sustain point, the gap remains constant due to the full⎛
β +1 ⎞⎟⎟(l − h) . Under gradualagglomeration of firms in the North, hence π L − π H = γ ⎜⎜⎝ αh + (1 − α )l ⎠agglomeration all H firms concentrate in the North and then more L firms relocate to the North18
with a rise of Δ. Meanwhile market potential in the North becomes lower due to the smaller
increase in E relative to the larger increase in Δ. The decreased market potential in the North
results in a lowering of nominal profits in the case of gradual agglomeration. Due to the type
difference, i.e. h and l, lowering profit is larger in L firms than in H firms. This is why the profit
gap between firm types in Figure 3 first declines in the sorting equilibrium, however as trade
costs fall, the gap between Δ and Δ* decreases due to better market access, which moderates therise of Δ caused by L-firm migration and this raises Northern market potential because of
keeping the rise in E. This causes the turn from a fall to a rise in the profit gap due to trade cost
reduction.

Figure 3: Profit gap between L and H firms.

We now evaluate whether our market equilibrium is optimal in terms of global welfare. To
discuss socially optimal welfare, we suppose a social planner can choose the location of eachtype of firm (i.e. they set nL and nH ) in order to maximise welfare across the two countries.11
Figures 4 provides a 3-D plot of the worldwide welfare (plotted on the vertical axes) in terms ofnL and nH (the two horizontal axes) with high trade costs (φ=0.3) and small trade costs (φ=0.8),

using parameter values, h=1, l=2, σ=4, μ=0.5,α=0.5 and β=3. The global welfare is given as thesum of two regional social welfares: i.e. W W ≡ W + W * . As seen in Figure 4-a, the global
welfare appears saddle-shaped with small trade costs and saddle points at nH = 1 and 0 < nL < 1
is higher than any symmetric equilibrium. The maximum welfare point is at nH = 1 andnL = 0.5282 (or nH = 0 and nL = 0.4717 in the case where H firms first deviate to theSouth).Hence we conclude that the sorting equilibrium dominates the symmetric equilibrium.
When firms experience higher trade costs we see that the saddle shape is retained in Figure 4-b,
this tells us that nH = 1 and nL = 1 (or nH = 0 and nL = 0 in the transposed case) yield the
maximum worldwide welfare, which is higher than that achieved at the symmetric equilibrium,on the other hand nH = 1 and nL = 0 (or nH = 0 and nL = 1 in the transposed case) yields theminimum total welfare. The symmetric equilibrium is thus proved to not represent maxima in
11

As discussed in Baldwin et al. (2003, Ch.11), the first-best outcome is to exclude all distortion. The planner
imposes prices equal to marginal costs and decides firm distributions with a lump-sum transfer from consumers tofirms. The second-best outcome is that firms are free to set prices to maximise their profits. Then the social plannerchooses firm distribution. In our model, the first and second social optimal equilibrium results in an identical firm
distribution, because quasi-linear utility function excludes an income effect, as discussed in Pfluger and Sudekum(2008).

19
terms of global welfare. The existence of the saddle shape in Figures 3 and 4 is mathematically2∂ 2W W ∂ 2W W ⎛ ∂ 2W W ⎞⎟⎟ . At theproved with the Hessian being given as Ω(nH , nL ) ≡− ⎜⎜∂nH2 ∂nL2⎝ ∂nH ∂nL ⎠symmetric equilibrium nL = nH = 0.5 ,

∂W W
∂W W
=0
= 0 yields∂nL∂nH

16α 2 (1 − α ) 2 μ 2 (l − h) 2 (1 − φ ) 2
Ω( n H , nL ) = −< 0 . This means that the worldwide function must(σ − 1) 2 (αh + (1 − α )l ) 2 (1 + φ ) 2

always be saddle-shaped and the symmetric equilibrium will not be associated with the
maximum level of worldwide welfare. Our saddle shape indicates that the socially optimal firmshares are always nH = 1 (or nH = 0 ).
Thus as shown in Figure 4, the sorting equilibrium, nH = 1 (or nH = 0 ) and 0 < nL < 1 with hightrade costs and full agglomeration, nH = 1 and nL = 1 (or nH = 0 and nL = 0 ) with small trade
costs can maximise global welfare. We thus conclude that the symmetric equilibrium is not
optimal in terms of maximising worldwide welfare in wither case.

Figure 4-a and Figure 4-b: Social welfare.

We now turn our attention to the sorting equilibrium and its associated socially optimal firmshares which maximise worldwide welfare, this is denoted by n~ and n~ , compared with theLHmarket equilibrium of nL and nH . Figure 5 plots n~L and n~H (h=1, l=2, σ=4, μ=0.5, α=0.5 and
β=3) in terms of φ. As discussed in previous sections nH =1 is always a market equilibrium(Section 3) and n~H =1 is socially optimal (see Figure 4). As a result of maximising worldwide
welfare the socially optimal n~L is always smaller than the market equilibrium and the sustain
point is higher than under the market outcome. This indicates that the market sorting equilibrium
involves more agglomeration than is socially optimal. The concentration of all H firms in the
North creates larger demand due to the concentration of entrepreneurs associated with H firms
but lower competition due to their low productivity and higher supply price, whichaccommodates L firms’ location in the North. On the other hand, workers are immobile and areequally distributed across the two regions. This causes there to be too much agglomeration interms of the socially optimal outcome. For this reason, the market outcome has too many firms
in the North, unless full agglomeration occurs. As total entrepreneurs/capital relatively decreases

20
and the total labour increases, this diminishes the discrepancy between the socially optimal andmarket driven outcomes.

Figure 5: Social optimal firm distribution.

Result 5: The sorting equilibrium would improve global social welfare compared to thesymmetric equilibrium. However, the sorting equilibrium is not socially optimal as thesocially optimal sorting equilibrium involves less agglomeration than in the market
outcome, or more simply the market outcome involves too much agglomeration.

5. CONCLUSIONSThis paper studies the impact of firm heterogeneity on location patterns and constructs an FE
model with firm heterogeneity. In contrast to other firm heterogeneity models, this paper takes
into account labour (entrepreneur) migration, which creates an expenditure shift from one region
to the other. We showed under spatial sorting that the severe competition in the core would be
more likely to lead productive firms to locate in the periphery, whereas all unproductive firms
concentrate their production in the core.
In this setting we encounter several results which contrast with other firm heterogeneity models.
First, spatial sorting occurs, in which low cost/high productivity firms locate in both regions,while high cost/unproductive firms concentrate in a single region. Productive firms are lesslikely to co-agglomerate due to their self-induced competition and are more likely to diversify
their location. Secondly, firm heterogeneity works as an agglomeration force, which decreases
the break and sustain points. Thirdly, the symmetric equilibrium is not optimal in terms ofworldwide welfare and the sorting equilibrium can improve welfare. However, the sorting
equilibrium involves too much agglomeration and is not socially optimal.
These results might provide some explanation of the current empirical evidence. For example,
small firms in some sectors are more likely to concentrate in one region (Duranton and Overman,
2005). Big Chinese cities witness lower real wage rates than intermediate sized cities, whichmight be due to low productivity firms’ location (Au and Henderson, 2005).
A possible extension to this work is to incorporate the heterogeneous labour literature (Tabuchi
and Thisse, 2002) or to add more urban issues (e.g. land rent) or some public policies such as21
corporate tax and subsidy to the model presented here. Further extension might include acontinuum of firm types.12

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12

See Appendix 4 for the model with three types of firm (intermediate productivity firms).

22
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APPENDIX 1. BREAK POINT
Solving∂ (VH − VH* )∂ (VL − VL* )= 0 and= 0 in terms of φ, we get
∂n H
∂
n
L
n = 0.5
n = 0.5LH(σ − 1)(1 + β )h − (2σ − 1)(αh + (1 − α )l )(σ − 1)(1 + β )l − (2σ − 1)(αh + (1 − α )l )and φLB =(σ − 1)(1 + β )h + (2σ − 1)(αh + (1 − α )l )(σ − 1)(1 + β )l + (2σ − 1)(αh + (1 − α )l )respectively. Taking differences between these yields,

φHB =2(2σ − 1)(σ − 1)(1 + β )(h − l )(αh + (1 − α )l )< 0.
{(σ − 1)(1 + β )h + (2σ − 1)(αh + (1 − α )l )}{(σ − 1)(1 + β )l + (2σ − 1)(αh + (1 − α )l )}φ HB is always smaller than φ LB . Thus we adopt φ HB as break point.

φHB − φLB =APPENDIX 2. LOCATION PATTERN IN THE LONG-RUN EQUILIBRIUM
Case 1) VH − VH* = 0 and VL − VL* < 0 then 0 < n H < 1 and n L = 0
Using n L = 0 , we can immediately derive Δ − Δ* = (1 − φ )(α (2nH − 1)h − (1 − α )l ) < 0 .
Hence Δ < Δ* . The price index effect, −μ(ln Δ − ln Δ* ) , is negative. To satisfy VH − VH* = 0 ,1−σ

the profit gaps γ ( B − B * )l and γ ( B − B* )h should be positive. Due to γ ( B − B * )l > γ ( B − B* )h , if
VH − VH* = 0 , then VL − VL* > 0 should also hold. This represents a contradiction, hence weconclude that this case will not occur.
Case 2) VH − VH* > 0 and VL − VL* < 0 then n H = 1 and n L = 0

23Using n H = 1 and n L = 0 , Δ − Δ* = (1 − φ )(αh − (1 − α )l ) < 0 . The price index effect,−μ(ln Δ − ln Δ* ) , is negative. Suppose that VH − VH* > 0 , the profit gaps γ ( B − B* )l and
1−σγ ( B − B * )h should be both positive. Due to γ ( B − B * )l > γ ( B − B* )h > 0 , if VH − VH* > 0 , thenVL − VL* > 0 should also hold. This is a contradiction, hence we conclude that this case does notoccur.
Case 3) VH − VH* > 0 and VL − VL* = 0 then n H = 1 and 0 < nL < 1 .
nL is a decreasing function in VL [nL ] − VL*[nL ] . Suppose that αh + (1 − α )(2nL − 1)l = 0 . Then nL
is given as n L =1αh2(1 − φ )(l − h)αμ. VL [n L ] − VL*[n L ] => 0 . This indicates that−
2 2(1 − α )l(1 + φ )(αh + (1 − α )l )σthe long-run equilibrium, to satisfy VL − VL* = 0 , should always be more than n L , i.e. nL > n L .
Thus αh + (1 − α )(2nL − 1)l > 0 holds in the equilibrium, resulting in Δ > Δ* . Using thisrelationship, since we discuss the case of a substantial cost gap, i.e. l is sufficiently larger than h,2(1−α)(1− nL )(Δ + Δ* ) + β(Δ − Δ* ) − 2Δ*
the gap between Δ and Δ is large and thus B − B = −< 0 and
2ΔΔ***

VH − VH* > VL − VL* = 0 . We conclude that this case could occur.

APPENDIX 3. FIRM HETEROGENEITY IMPACTl> 1 . Since we assume a substantial firm heterogeneity, χ should
αh + (1 − α )l
dχ
αh
=
> 0,
have a large value in our paper. Then we obtaindl (αh + (1 − α )l ) 2
dχ
αl
=−
< 0.
(αh + (1 − α )l ) 2
dhFirst we define χ ≡When firm heterogeneity is substantial, i.e. there is a larger χ, using (15), we can deriveββdφ S − 2σφ S − χ (σ − 1)((φ S ) 2 (2 + β ) − β )<=< 0 , because φ S <should be
S
S
S
S
dχ
φ (σ − 1)(1 − φ )( βφ + 2φ − β )
β +2
β +2

( ) < β β+ 2 ⇔ (φ ) (β + 2) − β < 0satisfied in order for the sustain point to exist, resulting in φ S

2

S 2

dφ S
dφ S
< 0 and> 0 , hence more firmdldhheterogeneity, i.e. increased an l and/or decreased h leads to a decrease the sustain point.

and we also obtain βφ S + 2φ S − β < 0 . Thus,

24

APPENDIX 4. THREE FIRM TYPE MODEL
Our model can be extended to include more types of firms. Here we add a third type of firm,intermediate cost/productivity firms (so-called M-firm). For simplicity, we assume the numberof each type of firms is equal. In this framework we maintain our key results. Solving∂ (VM − VM* )∂ (VL − VL* )∂ (VH − VH* )=0,= 0 and= 0 and choosing the smallest
∂nH
n
n
∂
∂
M
L
n = 0.5
n = 0.5
n = 0.5
LM

H

one is the break point. As a result, the first movers are H firms. The break point isφB =(σ − 1)(1 + β )h − 1 (2σ − 1)(h + l + m)
3
. Above the break point, spatial sorting occurs. The
1(σ − 1)(1 + β )h + (2σ − 1)(h + l + m)
3long-run equilibrium is derived either from Case 1) VH − VH* > 0 , VM − VM* > 0 and VL − VL* = 0
or from Case 2) VH − VH* > 0 , VM − VM* = 0 and VL − VL* < 0 . The first case is likely to occur
when h and m are close. Once the trade costs are at the break point, all H and M firms
immediately concentrate in the North, while L firms gradually relocate to the North due to trade
cost reduction. This is parallel to the two firm type case discussed in the main text. The secondcase is likely to occur when m is large and close to l, that is nH = 1 0 < nM < 1 and nL = 0 above
the break point. In this case nM increases as trade costs fall and all M firms concentrate in the
North under low trade costs. Then, as in the first case, L firms gradually move to the North as
trade costs fall.
In Case 2, since the North represents less severe competition due to the agglomeration of H
firms and the South has more competition due to the agglomeration of L firms, more M firms
locate in the North. As trade costs fall, the degree of competition becomes closer and thedemand size becomes much more important. Since the North has a higher population the North
attracts more M firms. Then M firms agglomerates in the North while, as in Case 1, L firms
gradually move to the North.

nH

nL



nH
Sorting Equilibrium

Symmetric Eq

nL



B

S

Figure 1: Equilibrium in terms of trade costsWelfare Gap, W-W*

Full Agglomeration

Sorting Equilibrium



0
s
B
Symmetric Eq

Figure 2: Regional social welfare gap

1

Nominal Profit Gap between types

Symmetric eq

Full Agglomeration

Sorting eq

0



B

s

Figure 3: Profit gap between L and H firms

1Figure 4-a Social welfare.

Figure 4-b: Social welfare

nH

nL



n~H nH

Market eq

nL

0.5282

n~L
Socially optimal eq




0.3

S

0.8

Figure 5: Social optimal firm distribution