arXiv:2404.09796v2  [econ.TH]  20 Mar 2025
Heterogeneity, trade integration and spatial inequality*
José M. Gaspar†
Abstract
We study the impact of economic integration on agglomeration in a model where all con-
sumers are inter-regionally mobile and have heterogeneous preferences regarding their residen-
tial location choices. This heterogeneity is the unique dispersion force in the model. We show
that, under reasonable values for the elasticity of substitution among varieties of consumption
goods, a higher trade integration always promotes more symmetric spatial patterns, reducing
the spatial inequality between regions, irrespective of the functional form of the dispersion
force. We also show that an increase in the degree of heterogeneity in preferences for location
leads to a more even spatial distribution of economic activities and thus also reduces the spatial
inequality between regions.
Keywords: heterogeneous location preferences; economic integration; economic geography; spa-
tial development.
JEL codes: R10, R12, R23
1
Introduction
In this paper, we study the possible spatial distributions in the 2-region Core-Periphery (CP) model
by Murata (2003), where all labour is free to migrate between regions and workers are heteroge-
neous regarding their preferences for location. Our aim is to investigate how spatial patterns evolve
*This work is dedicated to Carlos Hervés-Beloso. I am indebted to Soﬁa B. S. D. Castro and João Correia da Silva
for their invaluable comments and suggestions and also for their help in revising previous and related versions of this
work. I am also very thankful to the audience at the "Porto Workshop on Economic Theory and Applications - in
honor of Carlos Hervés-Beloso" for the fruitful discussions. This research has been ﬁnanced by Portuguese public
funds through FCT - Fundação para a Ciência e a Tecnologia, I.P., in the framework of the projects with references
UIDB/00731/2020 and PTDC/EGE-ECO/30080/2017 and by Centro de Economia e Finanças (CEF.UP), which is
ﬁnanced through FCT. Part of this research was developed while José M. Gaspar was a researcher at the Research
Centre in Management and Economics, Católica Porto Business School, Universidade Católica Portuguesa, through
the grant CEECIND/02741/2017.
†School of Economics and Management and CEF.UP, University of Porto. Email: jgaspar@fep.up.pt.
1

as regions become more integrated and whether the degree of heterogeneity has qualitative impact
on this relationship. Heterogeneity in preferences for location generates a local dispersion force
(akin to congestion), which we model as a utility penalty, following the recent work of Castro et
al. (2022). This framework can been shown to encompass various functional forms used in dis-
crete choice models of probabilistic migration, including the well-known Logit model (used e.g.
by Murata, 2003; Tabuchi and Thisse, 2002).
We show that, irrespective of the degree of heterogeneity, a higher trade integration always
promotes a more symmetric spatial distribution of economic activities, thus reducing the spatial
inequality in terms of industry size between regions. Since heterogeneity does not co-vary with
transportation costs, the latter only affect the strength of agglomeration forces due to increasing
returns. This is a reversion of the usual prediction that lower transport costs lead to agglomeration
(Krugman, 1991; Fujita et al., 1999, Chap. 5), and contrasts the ﬁndings that consumer heterogene-
ity leads to a bell-shaped relationship between decreasing transport costs and the spatial distribution
of economic activities (Murata, 2003; Tabuchi and Thisse, 2002). Since there is no immobile work-
force (no ﬁxed regional internal demand), there is a lower incentive for ﬁrms to relocate to smaller
markets in order to capture a higher share of local demand and avoid competition when transport
costs are higher. Krugman and Elizondo (1996), Helpman (1998) and Murata and Thisse (2005),
have also obtained similar results concerning the relationship between transportation costs and the
spatial distribution of economic activities. The ﬁrst includes a congestion cost in the core, the sec-
ond considers a non-tradable housing sector and the treatment is only numerical, while the latter’s
prediction cannot be disassociated from the interplay between inter-regional transportation costs
and intra-regional commuting costs. Recently, Tabuchi et al. (2018) have reached similar con-
clusions to ours arguing that falling transport costs increase the incentives for ﬁrms in peripheral
regions to increase production since they have a better access to the core, thus contributing to the
dispersion of economic activities. Allen and Arkolakis (2014) estimated that the removal of the
US Interstate Highway System, which, by limiting accesses, can be interpreted as an increase in
transportation costs, would cause a redistribution of the population from more economically remote
regions to less remote regions in the US. This adds empirical validity to our ﬁndings.
The rest of the paper is organized as follows. In section 2 we describe the model and the short-
run equilibrium. In section 3 we study the qualitative properties of the long-run equilibria. Finally,
section 4 is left for some concluding remarks.
2

2
The model
There are two regions L and R, symmetric in all respects, with a total population of mass 1. Con-
sumers are assumed to have heterogeneous preferences with respect to the region in which they
reside. We follow Castro et al. (2022) and assume that these preferences are described by a param-
eter x, uniformly distributed along the interval [0, 1]. The consumer with preference described by
x = 0 (resp. x = 1) has the highest preference for residing in region L (resp. region R). An agent
whose preference corresponds to x = 1/2 is indifferent between either region. We can thus refer
to each consumer with respect to his preference towards living in region L as x ∈[0, 1].
For a consumer with preference x, the utilities from living in region L and R are given, respec-
tively, by:
UL(x) =U (u (CL) , t(x))
UR(x) =U (u (CR) , t(1 −x)) ,
(1)
where Ci denotes the level of consumption of a consumer living in region i ∈{L, R}, and t :
[0, 1] 7→R is the utility penalty of consumer x associated with living in L, while t(1−x) is the utility
penalty the same consumer faces from living in region R. We assume that t(x) is differentiable,
such that t′(x) > 0, ∀x ∈[0, 1].
The number of agents in region L is given by h ∈[0, 1] and fully describes the spatial distribu-
tion of economic activities. In the short-run, the spatial distribution of workers and ﬁrms is ﬁxed
and an equilibrium is reached when all prices, wages, and output adjust to clear markets given h.
In what follows, the short-run equilibrium is equivalent to Murata (2003) and Tabuchi et al. (2018),
which we brieﬂy describe here. The consumption aggregate is a CES composite given by:
C =
ˆ
s∈S
c(s)
σ−1
σ ds

σ
σ−1
,
where s stands for the variety produced by each monopolistically competitive ﬁrm and σ is the
constant elasticity of substitution between any two varieties. The consumer is subject to the budget
constraint PiCi = wi, where Pi is the price index and wi is the nominal wage in region i. Utility
maximization by a consumer in region i yields the following demand for each manufactured variety
produced in j and consumed in i:
cij = p−σ
ij
P 1−σ
i
wi,
(2)
3

where:
Pi =
ˆ
s∈S
pi(s)1−σd(s)

1
1−σ
,
(3)
is the manufacturing price index for region i. A manufacturing ﬁrm faces the following cost func-
tion:
TC(q) = w (βq + α) ,
where q corresponds to a ﬁrm’s total production of manufacturing goods, β is the input requirement
(per unit of output) and α is the ﬁxed input requirement. The manufacturing good is subject to trade
barriers in the form of iceberg costs, τ ∈(1, +∞): a ﬁrm ships τ units of a good to a foreign region
for each unit that arrives there. Assuming free entry in the manufacturing sector, at equilibrium
ﬁrms will earn zero proﬁts, which translates into the following condition:
π ≡(p −βw) q −αw = 0,
which gives the ﬁrm’s total equilibrium output, symmetric across regions:
qL = qR = α(σ −1)
β
,
(4)
and the following proﬁt maximizing prices:
pLL =
βσ
σ −1wL and pRL = βστ
σ −1wL,
(5)
where pij is the price of a good produced in region i and consumed in region j. Labour-market
clearing implies that the number of agents in region L, equals labour employed by a ﬁrm times the
number of varieties (and hence ﬁrms) nL produced in region L:
h = nL (α + βqL) ,
from where, using (4), we get the number of ﬁrms in each region i:
nL = h
σα and nR = 1 −h
σα .
(6)
Choosing labour in region R as the numeraire, we can normalize wR to 1. The price indices in L
4

and R using (3) are given, after (6), by:
PL =
"
h
σα
 βσ
σ −1w
1−σ
+ 1 −h
σα
 βστ
σ −1
1−σ#
1
1−σ
,
PR =
"
1 −h
σα
 βσ
σ −1
1−σ
+ h
σα
 βστ
σ −1w
1−σ#
1
1−σ
.
(7)
Rewriting a ﬁrm’s proﬁt in region L as:
πL = (pLL −βw) [hcLL + (1 −h)τcRL] −αw,
the zero proﬁt condition yields, after using (2), (5) and (7), the following wage equation:
h =
wσ −φ
wσ −φ + w(w−σ −φ),
(8)
where φ ≡τ 1−σ ∈(0, 1) is the freeness of trade.
The nominal wage w can be implicitly deﬁned as a function of the spatial distribution in region
L, h ∈[0, 1].1 We have w(0) = φ1/σ < 1, w(1/2) = 1 and w(1) = φ−1/σ > 1. We say that L is
larger than R when h > 1/2, and smaller otherwise.
Figure 1: Short-run equilibrium relative wage as a function of the consumers in region L. We set
σ = 2 and φ ∈{0.1, 0.5, 0.7}.
Figure 1 illustrates w(h) for h ∈[0, 1] with parameter values σ = 2 and φ ∈{0.1, 0.5, 0.7}. The
following result corroborates this illustration and discusses the impact of trade barriers on wages.
1The conditions for application of the Implicit Function Theorem are shown to be satisﬁed in Appendix A.
5

Lemma 1. The relative nominal wage w is strictly increasing in h. When region L is larger
(smaller) than region R, the wage w is decreasing (increasing) in φ.
Proof. See Appendix A.
We conclude that higher trade barriers increase the wage divergence between the regions. The
intuition is as follows. When all workers are mobile, they can move to the region that offers them a
relatively better access to varieties. This advantage of the larger region becomes higher as transport
costs increase because markets become more focused on local demand. This increases expenditures
in the larger region relative to the smaller, which in turn pushes the nominal relative wage upwards.
We consider a general isoelastic sub-utility for consumption goods:





ui = C1−θ
i
−1
1 −θ
,
if θ ∈[0, 1) ∪(1, +∞)
ui = ln(Ci),
if θ = 1,
,
(9)
where for θ = 1 we take the limit value of the upper expression of ui. The parameter θ is a positive
agglomeration externality. It inﬂuences how a change in the regional consumption differential,
CL −CR, impacts the regional utility differential, uL −uR, i.e., it inﬂuences the strength of the
self-reinforcing agglomeration mechanism when one region is more populated than the other. The
speciﬁcation in (9) is quite general and encompasses the Murata (2003) model as a particular case
when θ = 0, which shall be of great interest for comparison purposes in Section 4.2
The utility differential from consumption goods, ∆u ≡uL −uR, is given by:
∆u =





η
1 −θ

w1−θ [(1 −h)φ + hw1−σ]
1−θ
σ−1 −[1 −h + hφw1−σ]
1−θ
σ−1

,
if θ ̸= 1
ln w +
1
σ −1 ln
h
hw1−σ+(1−h)φ
hφw1−σ+(1−h)
i
,
if θ = 1.
(10)
We adopt the normalizations by Fujita et al. (1999), i.e., ασ = 1 so that the number of consumers
in a region equals its number of ﬁrms. Moreover, we assume that (σ −1)/(σβ) = 1 so that the
price of each manufactured variety in a region equals its workers’ nominal wage. These imply that
η = 1.3
2We note in advance that the qualitative results of the model obtained in subsequent sections do not depend on θ
(c.f., Section 3.2).
3This choice is made mainly for convenience, as it allows us to abstract from changes in the variable input require-
ment β.
6

Figure 2: Utility differential ∆u = uL −uR for different levels of θ. Parameter values are σ = 2
and φ = 0.5.
Figure 2 depicts ∆u in (10) for different levels of θ. We observe that a higher θ increases
the utility differential ∆u for any spatial distribution h > 1/2. Therefore, if L is relatively more
industrialized, a higher θ increases the attractiveness of L relative to R for consumers. Thus, it
strengthens the agglomeration forces towards region L.
3
Long-run equilibria
In a long-run spatial equilibrium, each worker chooses to live in the region that provides a higher
utility. We follow Castro et al. (2022) and assume that t(x) enters additively in overall utility,
Ui(x), so that the utility penalty, and hence heterogeneity in preferences for location, are modelled
à la Hotelling (1929).4 In a long-run equilibrium, workers with x ∈[0, h) live in region L, and
workers with x ∈(h, 1] live in region R. Knowing from utility maximization that CL = w/PL and
CR = 1/PR, we rewrite the indirect utilities of the consumer with x = h as:
VL(h) = C1−θ
L
−1
1 −θ
−t(h)
VR(h) = C1−θ
R
−1
1 −θ
−t(1 −h),
(11)
where h satisﬁes the short-run equilibrium in (8).
4It could be interesting to consider the case where t(x) is multiplicative (e.g. as a percentage of Ui(x)). However,
the non-linear impact of t(x) would likely make analytical results harder to obtain.
7

3.1
Interior equilibria
Due to symmetry, we focus only on the case where L is larger or the same size as R, i.e., h ≥1/2.
We deﬁne an interior equilibrium h∗∈[1/2, 1) as a spatial distribution h that satisﬁes both (8)
and VL = VR. Such an equilibrium is said to be stable if d (VL −VR) /dh < 0. This leads to the
following Lemma.
Lemma 2. An interior equilibrium h∗∈[1/2, 1) is stable if:
ζ




ϕwσ 1−θ
σ−1 + ψ
w
 (
 1 −φ2
w
w2σ −[ϕ(w + 1)wσ] + w
) 1−θ
σ−1 

< t′ (h∗) −t′ (1 −h∗) ,
where h∗satisﬁes the short-run equilibrium condition in (8), and:
ζ =
w2σ −[φ(w + 1)wσ] + w
(σ −1) [(σ −1)φ + (σ −1)φw2σ + (−2σ + φ2 + 1) wσ];
ϕ = w−θ−σ {φ [σ + (σ −1)w] wσ −2σw + w} ;
ψ = (σ −1)φ + (1 −2σ)wσ + σφw.
Proof. See Appendix B.
An interior equilibrium is called symmetric dispersion if h∗= 1/2 and partial agglomeration
otherwise. While the former always exists, the latter depends on the form of t(x). Knowing that
w = 1 for h = 1/2 the stability condition in Lemma 2 for symmetric dispersion simpliﬁes to:
d∆u
dh
 1
2

≡2(2σ −1)(1 −φ)
 1+φ
2
 1−θ
σ−1
(σ −1)(2σ + φ −1)
< t′   1
2

.
(12)
As expected, symmetric dispersion is stable if, after a small increase in h, the relative utility gain
from consumption is lower than the increase in the utility penalty for the agent of type x = h.
A careful inspection of (12) allows us to conclude that the LHS is decreasing in φ if θ ≥1. For
θ < 1, it is decreasing in φ if σ ≥1 +
√
2/2 ≈1.71. Symmetric dispersion then becomes easier
to sustain under lower transportation costs if σ > 1 +
√
2/2, which, according to recent empirical
estimations for σ, is more than reasonable.5 As φ →1, symmetric dispersion is always stable
because t′   1
2

> 0. This warrants the following result.
Proposition 1. As φ increases from a low level, symmetric dispersion either remains stable or
becomes stable, if either conditions hold: (i). θ ∈[0, 1) and σ > 1 +
√
2/2; or (ii). θ ≥1.
5Estimations evidence that σ should be signiﬁcantly larger than unity (Crozet, 2004; Head and Mayer, 2004;
Niebuhr, 2006; Bosker et al., 2010). Anderson and Wincoop (2004), for instance, ﬁnd that it is likely to range from 5
to 10.
8

Thus, an increase in φ can never turn symmetric dispersion from stable to unstable, suggesting
that lower trade barriers always promote an even distribution of economic activities. Since, at
symmetric dispersion, consumers in each region have access to the same amount of manufactures,
an exogenous migration will induce a lower (higher) beneﬁt from local consumption goods in the
larger market if transport costs are lower (higher). This is captured by the fact that the relative
decrease in prices and increase in wages (at the symmetric equilibrium) is more pronounced when
transport costs are higher.
Let φ = φb ∈(0, 1) such that d∆u
dh
  1
2; φb

= t′   1
2; φb

. If φb exists, then symmetric dispersion
is stable for φ > φb and unstable for φ < φb. We have the following result.
Lemma 3. A curve of partial agglomeration equilibria h∗∈(1/2, 1) branches from φ = φb. If it
branches in the direction of φ < φb, partial agglomeration is stable.
Proof. See Appendix B.
The result above establishes existence of partial agglomeration in a neighbourhood of φ = φb.6
Let us now assume that a partial agglomeration h∗∈(1/2, 1) equilibrium exists and is stable for
some value of φ ∈(0, 1). Regarding the inﬂuence of trade integration on partial agglomeration
equilibria, we establish the following result.
Proposition 2. If a stable partial agglomeration h∗∈(1/2, 1) exists, it becomes more symmetric
as trade barriers decrease.
Proof. See Appendix B.
This result states that if most consumers reside in region L, increasing the freeness of trade
will lead to a smooth exodus from region L to region R, irrespective of the value of t(h∗). We
conclude from Propositions 1 and 2 that more integration leads agents to distribute more equally
among the two regions, as in Helpman (1998), who also considers that all labour is inter-regionally
mobile. This qualitative effect of φ does not depend on consumer heterogeneity. It thus contrasts
the ﬁndings that consumer heterogeneity leads to a bell-shaped relationship between decreasing
transport costs and agglomeration (Murata, 2003; Tabuchi and Thisse, 2002).
6Uniqueness is not ensured for the entire range of φ ∈(0, φb) as the model may undergo secondary bifurcations
along the primary branch of partial agglomeration equilbria, such as a saddle-node bifurcation (see e.g. Gaspar et al.
(2018, 2021) and Castro et al. (2021) for applications in economic geography).
9

3.2
Logit preferences for location
We begin this Section by noting that, for any value of θ, there exists a non-trivial subset of a suitably
deﬁned parameter space over which the qualitative results of the model remain invariant. Therefore,
for the remainder of this Section, we will assume that utility is logarithmic in consumption. This
greatly simpliﬁes the analysis and entails no loss of generality.
Assumption 1. Utility ui is logarithmic in Ci: θ = 1 =⇒ui = ln(Ci).
For the two region case, and following Tabuchi and Thisse (2002) and Murata (2003), we
assume that the probability that a consumer will choose to reside in region L is given by the Logit
model, written as:
ΦL(h) =
euL(h)/µ
euL(h)/µ + euR(h)/µ,
(13)
where µ ≥0 is a scale parameter which measures the dispersion of consumer preferences. If µ = 0,
consumers do not care about their location preferences but rather solely about relative wages. The
number of agents h is the same as the consumer x who is indifferent between living in region L or
in region R. Therefore, the probability ΦL(h) is tantamount to the indifferent consumer x = h (as
in Castro et al., 2022). Thus, using (13), we can write:
h =
euL(h)/µ
euL(h)/µ + euR(h)/µ.
(14)
Manipulating (14) yields:
uL + µ ln(1 −h) = uR + µ ln(h),
(15)
which rearranged yields the long-run equilibrium condition, ∆u(h) = ∆t(h), with t(x) = −µ ln(1−
x). This yields ∆t(h) = µ [ln h −ln(1 −h)], dubbed by Castro et al. (2022) as the “home-sweet-
home” effect, which comprises the model’s unique dispersion force generated by consumer hetero-
geneity in preferences for location.
Notice from (15) that t(h) = −µ ln(1 −h) > 0 for h ∈[0, 1]. The overall utility of a consumer
in region i thus lies on the interval (−∞, ui]. For a strictly positive µ, the consumer who likes
region R the most (at x = 1) will never want to live in L, because, at h = 1, the overall utility
loss from living in region L is unbounded. This means that the Logit model penalizes (beneﬁts) the
consumers who are less (more) willing to leave a region very strongly. Therefore, agglomeration in
any region (h ∈{0, 1}) is not possible. Now, suppose condition (12) does not hold and symmetric
dispersion is unstable. Then, we have d∆u
dh
  1
2

> t′   1
2

which means that ∆u(h) > t(h) −t(h) for
h = 1
2 +ǫ, with ǫ > 0 small enough. Since ∆u (1)−∆t(1) →−∞, then, by the Intermediate Value
10

Theorem, we must have ∆V = 0 for some h ∈(1/2, 1). Accordingly, a partial agglomeration
equilibrium h∗∈(1/2, 1) must always exist when symmetric dispersion is unstable.
The discussion above allows us to formalize our next result, which summarizes the possible
spatial outcomes under Logit type preferences, depending on the degree of heterogeneity µ.
Proposition 3. Under Logit type preferences and Assumption 1, the spatial distribution depends
on the level of consumer heterogeneity as follows:
• Symmetric dispersion is the only stable equilibrium if:
µ > µd ≡
(2σ −1)(1 −φ)
(σ −1)(2σ + φ −1);
(16)
• Partial agglomeration h∗∈
  1
2, 1

is the only stable equilibrium and is unique if µ < µd.
Proof. See Appendix C.
When consumer heterogeneity is low there is a single stable partial agglomeration equilibrium
that is very asymmetric (close to agglomeration). As consumer heterogeneity increases, partial
agglomeration corresponds to a more even distribution. Finally, if consumer heterogeneity is high
enough (µ > µd in (16)), consumers disperse symmetrically across the two regions.
Figure 3: Utility differential ∆u (dashed lines) and penalty differential t(h) −t(1 −h) (thick line) for
increasing levels of φ. For φ = 0.3 (upper dashed line), dispersion is unstable and partial agglomeration is
stable. For φ = 0.5 (medium dashed line) dispersion is unstable, partial agglomeration is stable and is less
asymmetric. For φ = 0.9 (lower dashed line) dispersion is stable and is the only equilibrium. The values for
the parameters are σ = 2.5, θ = 0 and µ = 0.2.
11

In Figure 3 we show how the spatial distribution of industry changes as regions become more
integrated. We set σ = 2.5 and µ = 0.2. For low levels of the freeness of trade, a unique stable
partial agglomeration equilibrium exists where most consumers reside in region L. This means that
there is just a small fraction of consumers in R that are not willing to forego their preferred region
because the gain in consumption goods from doing so is not high enough. As regions become
more integrated, the home-market effect becomes weaker and partial agglomeration becomes more
symmetric. Finally, for a high level of inter-regional integration, agglomeration forces are so weak
that no consumer is willing to leave his most preferred region and only symmetric dispersion is
stable.
Figure 4: Bifurcation diagrams. To the left, the bifurcation parameter is µ ∈[0, 1] with φ = 0.4. To the
right, we have φ ∈(0, 1) as the bifurcation parameter and set µ = 0.2. For both scenarios we use σ = 2 and
θ = 0.
The preceding analysis is summarized by the two bifurcation diagrams in Figure 4, along a
smooth parameter path where µ (left picture) and φ (right picture) increase. We uncover two pitch-
fork bifurcations in µ and φ. We conﬁrm analytically that they are supercritical for the case of logit
preferences and under Assumption 1.7 This implies that a curve of stable partial agglomeration
equilibria branches in the direction of µ < µd (left picture) and φ < φb (right picture), showing that
lower trade barriers or a higher heterogeneity promote more even spatial distributions. The picture
to the right differs from other supercritical pitchforks found in the literature (e.g., Pﬂüger, 2004;
7Employing φ as the bifurcation parameter, we have ∂3∆V
∂h3 ( 1
2; φb) < 0, with φb =
(2σ−1)[µ(σ−1)−1]
−(µ+2)σ+µ+1
and σ ∈

2, µ+1
µ

. Then by the proof of Lemma 3 (Appendix B) the pitchfork is supercritical. For the case of µ as the
bifurcation parameter, see the proof of Proposition 3 (Appendix C).
12

Ikeda et al., 2022) crucially in the sense that the direction of change in stability as φ increases is re-
versed, i.e., lower trade barriers leads to more symmetric spatial distributions, as expected from our
analysis. Regarding the effect of µ, it is quite intuitive that a higher scale of heterogeneity increases
local dispersion forces (i.e., dispersion forces that do not depend on φ) and thus also promotes a
more even distribution of economic activities.
4
Discussion and concluding remarks
We have seen that heterogeneity in preferences for location alone bears no impact on the rela-
tionship between trade integration and agglomeration, which is a monotonic decreasing one. This
contrasts the ﬁndings in other works with heterogeneity in consumer preferences, such as Tabuchi
and Thisse (2002) and Murata (2003), who show evidence of a bell-shaped relationship between
trade integration and the spatial distribution of industry, whereby agglomeration but is followed by
a re-dispersion phase as trade integration increases. The former’s setting differs from ours because
the authors consider an inter-regionally immobile workforce whose role as a dispersive force is
enhanced by higher transportation costs.
However, it is particularly worthwhile to discuss the results of Murata (2003), who uses a
model that is a particular case of ours if we set θ = 0 and consider logit type heterogeneity.
Murata (2003) found that the relationship between trade integration and agglomeration need not
be monotonic and depends on the degree of consumer heterogeneity, which is at odds with our
ﬁndings. For instance, for an intermediate degree of consumer heterogeneity (µ), Murata ﬁnds
that increasing trade integration initially fosters agglomeration and later leads to re-dispersion of
industry. However, these conclusions can be shown to stem from the author’s particular choice
of the value for the elasticity of substitution, σ = 1.25. As we have argued in Section 3, such a
low value is empirically implausible. For exceedingly low values (σ < 1.71), increasing returns at
the ﬁrm level are too strong. Strong enough that the utility gain at dispersion becomes increasing
in φ, instead of decreasing. This would justify an initial concentration of industry as a result
of an increase in φ. However, we have seen that for a plausible range of σ the utility gain from
consumption at symmetric dispersion always decreases with φ. Moreover, if θ > 0, this holds
even for lower values of σ.8 Therefore, a higher φ always promotes more equitable distributions as
opposed to asymmetric ones.
Hence, when workers are completely mobile, more trade integration ubiquitously leads to a
more even dispersion of spatial distributions among the two regions, irrespective of the degree of
8If θ ≥1, the result holds for σ > 1.
13

heterogeneity in location preferences. Therefore, a de facto lower inter-regional labour mobility
induced by consumer heterogeneity alone cannot account for the predictions that a higher inter-
regional integration will lead to more spatial inequality or an otherwise bell-shaped relationship
between the two.
By considering that all consumers are allowed to migrate if they so desire, we have shown that
a higher inter-regional trade integration always leads to more dispersed spatial distributions. This
result is independent of the level and impact of consumer heterogeneity. This conclusion may be of
potential use for policy makers. Namely, the predictions that globalization is likely to lead to higher
spatial inequality in the future (World Bank, 2009) may be reversed if policies are undertaken to
promote inter-regional mobility.
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Appendix A - Wages and freeness of trade
Proof of Lemma 1:
Consider h = f(w) in (8). Let us now deﬁne F(h, w) = f(w) −h = 0. Differentiation of F(h, w)
yields:
∂F(h, w)
∂w
=
wσG(wσ)
[w2σ −(w + 1)φwσ + w]2,
(17)
where:
G(wσ) = −

φ(σ −1) + (σ −1)φw2σ −
 2σ −φ2 −1

wσ
.
The derivative in (17) is zero if G(wσ) = 0. One can observe that G(wσ) has either two (real)
zeros given by some {w−, w+} , or none. Moreover, G(wσ) is concave in wσ. Since G (wσ = φ) =
σ [φ(1 −φ2)] > 0 and G (wσ = φ−1) = −[σφ (1 −φ−2)] > 0, it must be that G(wσ) > 0 for
wσ ∈[φ, φ−1] ⊂(w−, w+).
We now proceed to show that w ∈
 0, φ1/σ
and w ∈
 φ−1/σ, +∞

are not deﬁned in h ∈[0, 1].
Using (9), we have the following:
lim
w→0 h(w) = 0;
lim
w→+∞h(w) = 1;
h(φ1/σ) = 0;
h(φ−1/σ) = 1;
Since dh/dw = ∂F/∂w, we have that h(w) is increasing for wσ ∈(w−, w+). The limits above,
together with the knowledge that the zeros of dh/dw lie to the left and right of [φ, φ−1], ensure that
h(w) < 0 for w ∈
 0, φ1/σ
and h(w) > 1 for w ∈
 φ−1/σ, +∞

.
Knowing that ∂F/∂h = −1 and ∂F/∂w are continuous, by the IFT we can write w : h ∈
[0, 1] ⊂R 7→R such that dw/dh exists and F(h, w(h)) = 0.
16

Using implicit differentiation on (8), we get:
dw
dh = [w2σ −(w + 1)φwσ + w]2
wσG(wσ)
.
(18)
This derivative is positive for w ∈[φ1/σ, φ−1/σ], which implies that w(h) is increasing in [0, 1].
This concludes the proof for the ﬁrst statement.
From F(h, w) = 0 deﬁned above, differentiating with respect to φ, we get:
∂F
∂φ + ∂F
∂w
dw
dφ = 0 ⇐⇒
dw
dφ = −∂F
∂φ
 ∂F
∂w .
Using (8) we get:
∂F
∂φ =
wσ+1 (w2σ −1)
[w2σ −(w + 1)φwσ + w]2,
which is positive if h > 1/2, because the latter implies w > 1 (because w is increasing in h).
As a result, we have dw/dφ < 0. Therefore, the nominal wage w is decreasing in the freeness of
trade when L is the largest region (h > 1/2). Conversely, we have dw/dφ > 0 if h > 1/2. This
concludes the proof of the second statement.
□
Appendix B - Interior Equilibria
Proof of Lemma 2:
Taking the derivative of (10) with respect to h, substituting for w′ using (18) from Appendix A, and
evaluating at h∗using the short-run equilibrium condition given by (8), we reach:
d∆u
dh

h=h∗= ζ




ϕwσ 1−θ
σ−1 + ψ
w
 (
 1 −φ2
w
w2σ −[ϕ(w + 1)wσ] + w
) 1−θ
σ−1 

,
(19)
with:
ζ =
w2σ −[φ(w + 1)wσ] + w
(σ −1) [(σ −1)φ + (σ −1)φw2σ + (−2σ + φ2 + 1) wσ];
ϕ = w−θ−σ {φ [σ + (σ −1)w] wσ −2σw + w} ;
ψ = (σ −1)φ + (1 −2σ)wσ + σφw.
The fact that an interior equilibrium is stable if d∆u/dh < d [t(h) −t(1 −h)] /dh at h = h∗
17

concludes the proof.
□
Proof of Lemma 3:
Suppose there exists a value of φ = φb ∈(0, 1) such that d∆u
dh
 1
2; φb

= t′   1
2; φb

. Let ∆V =
VL −VR. Then it is possible to show the following:
∂∆V
∂h
 1
2; φb

= 0; ∂2∆V
∂h2
  1
2; φb

= 0; ∂∆V
∂φ
  1
2; φb

= 0;
∂2∆V
∂h∂φ
 1
2; φb

= −4η
  1+φb
2
 1−θ
σ−1 [θ(1 −φb) + 2σ + φb −3]
(σ −1)2(φb + 1)2
;
The last expression is negative if we assume that σ > 1 +
√
2/2. Next, we have:
∂3∆V
∂h3
  1
2; φb

= 16(σ −2)(2σ −3)(1 −φb)3   1+φb
2

1
σ−1
(σ −1)3(φb + 1)3
−2t′′′
1
2

.
If ∂3∆V
∂h3 ̸= 0, then, according to Guckenheimer and Holmes (2002), the model undergoes a pitchfork
bifurcation at symmetric dispersion and a curve of partial agglomeration equilibria branches at
φ = φb. If the pitchfork is supercritical (∂3∆V
∂h3
< 0), the curve branches in the direction of φ < φb
and partial agglomeration is stable. If it is subcritical (∂3∆V
∂h3
> 0), it branches in the direction of
φ > φb and partial agglomeration is unstable. The criticality of the bifurcation depends on the
speciﬁc functional form of t(x). This concludes the proof.
□
Proof of Proposition 2:
Assume that partial agglomeration h∗= h ∈
  1
2, 1

exists and is stable. Then, according to Castro
et al. (2022, Prop. 9, pp. 197), it becomes more symmetric if d∆u
dφ < 0. We have:
d∆u
dφ = ∂∆u
∂φ + ∂∆u
∂w
dw
dφ ,
which, by using (8) and (10), becomes:
−w
a3
(
a1

(1 −φ2) wσ+1
w2σ −(w + 1)φwσ + w
−θ+σ−2
σ−1
+ a2

w(1 −φ2)
w2σ −(w + 1)φwσ + w
−θ+σ−2
σ−1 )
(20)
18

where:
a1 = w1−θ (φwσ −1)

−2σ + 2(σ −1)φwσ + φ2 + 1

a2 = w−σ (wσ −φ)

2(σ −1)φ +
 −2σ + φ2 + 1

wσ
a3 = (σ −1)

w1−σ + wσ −(w + 1)φ
 
(σ −1)φ + (σ −1)φw2σ +
 −2σ + φ2 + 1

wσ
.
Notice that (20) is equivalent to
−
w
h
w(1−φ2)
w2σ−(w+1)φwσ+w
i−θ+σ−2
σ−1 
a1w−σ θ+σ−2
σ−1 + a2

a3
.
Thus, the sign of (20) is given by the sign of:
Ψ ≡−a1w−σ θ+σ−2
σ−1 + a2
a3
.
Since w > 1 for h ∈( 1
2, 1], we have a3 < 0 because (w1−σ + wσ −(w + 1)φ) > 0 and

(σ −1)φ + (σ −1)φw2σ +
 −2σ + φ2 + 1

wσ
< 0.
Next, taking the limit of Ψ as θ →+∞, we get:
lim
θ→+∞Ψ = w−σ (wσ −φ)

2(σ −1)φ +
 −2σ + φ2 + 1

wσ
< 0.
Differentiating Ψ with respect to θ yields:
ln(w)w−σ(θ+σ−2)
σ−1
−θ+1 (φwσ −1) [−2σ + 2(σ −1)φwσ + φ2 + 1]
σ −1
,
which is positive because 1 < wσ < 1/φ and [−2σ + 2(σ −1)φwσ + φ2 + 1] < 0. Thus, Ψ
remains negative for all θ ≥0. We conclude that (20) is negative and thus d∆u
dφ
< 0 for any
θ ∈[0, 1) ∪(1, +∞). For θ = 1, take ui = ln(Ci) to verify that the derivative is negative. We
conclude that d∆u
dφ < 0, ∀θ ≥0. This ﬁnishes the proof.
□
19

Appendix C - Logit heterogeneity
Proof of Proposition 3:
When t(x) = −µ ln(1 −x), we have:
t′ (h∗) −t′ (1 −h∗) = µ

1
h(1 −h)

.
Using θ = 1 in d∆u
dh given by (19), and rearranging, partial agglomeration is stable if:
−
w−σ−1 
w2σ −(w + 1)φwσ + w
2 Γ
(σ −1) (wσ −φ) (1 −φwσ) [(σ −1)φ + (σ −1)φw2σ + (−2σ + φ2 + 1) wσ] < 0,
(21)
where
Γ = φ

µ(σ −1)2 −2σ + 1

+ φ

µ(σ −1)2 −2σ + 1

w2σ−
−wσ 
φ2 [−(µ + 2)σ + µ + 1] + (2σ −1) [µ(σ −1) −1]
	
.
The numerator of (21) except Γ is positive, and so is (wσ −φ) (1 −φwσ). We have:
(σ −1)φ + (σ −1)φw2σ +
 −2σ + φ2 + 1

wσ < 0.
Therefore, (19) holds if and only if Γ < 0, which gives:
µ > µp ≡
(2σ −1)(wσ −φ)(1 −wσφ)
(σ −1) [(2σ −φ2 −1) wσ −(σ −1)φw2σ −(σ −1)φ].
The condition for partial agglomeration in (21) holds for any interior equilibrium including sym-
metric dispersion h∗= 1/2. At symmetric dispersion we have w = 1 and the condition above
simpliﬁes to:
µ > µd ≡
(2σ −1)(1 −φ)
(σ −1)(2σ + φ −1).
The derivative of µp with respect to X ≡wσ equals:
∂µp
∂X =
σ(2σ −1) (X2 −1) φ (φ2 −1)
(σ −1) [(σ −1) (X2 + 1) φ −2σX + Xφ2 + X]2 < 0.
We can also see that µp approaches zero as wσ approaches φ−1. Therefore, we conclude that
0 < µp < µd. We can show that symmetric dispersion undergoes a pitchfork bifurcation at µ = µd,
20

following a procedure similar to that of the proof of Lemma 3 (see Appendix B) by replacing φ with
µ as the bifurcation parameter. The corresponding derivatives have the same signs when evaluated
at µ = µd. Furthermore, we have:
∂3∆V
∂h3 ( 1
2; µd) = −64(1 −φ)φ {σ [φ(φ + 2) + 5] −φ2 −3}
(σ −1)(φ + 1)3(2σ + φ −1)
< 0,
which ensures that the pitchfork is supercritical and proves that a branch of stable partial equilibria
emerges in the direction of µ < µd. Since the state space is one-dimensional, multiple partial
agglomeration equilibria require the interchange of stability between consecutive equilibria, which
cannot happen when µ ∈(µp, µd).9 We thus have two possibilities: (i) if µ ∈(µp, ud), partial
agglomeration is the only stable equilibrium and is unique; and (ii) if µ > µd, symmetric dispersion
is the only stable equilibrium.
Next, we know from the text that a partial agglomeration equilibrium always exists if µ <
µd. Assume now, by way of contradiction, that µ ∈(0, µp). Then both dispersion and partial
agglomeration are unstable. Since the state space is one-dimensional, there can be only one unstable
partial agglomeration, which would require agglomeration to be stable. However, we know that
agglomeration is always unstable, which implies that partial agglomeration is always stable. Hence,
µ /∈(0, µp). Therefore, symmetric dispersion is the only stable equilibrium if µ > µd; otherwise,
partial agglomeration is the only stable equilibrium.
□
9This holds even if equilibria are irregular.
21
