Modern Spatial Economics: A Primer∗
Treb Allen
Dartmouth and NBER
Costas Arkolakis
Yale and NBER
First Version: May 2017 This Version: February 2017
Abstract
We present a primer of the basic framework and solution techniques for models of
the spatial economy with mobility of trade and workers flows. We document a number
of facts about the flow of people and goods both within and across countries. We then
show how these empirical patterns are consistent with a simple version of one of the
most successful theories in modern economics: the gravity model. In this framework we
highlight the importance of mobility and trade frictions by considering the impact of
the Interstate Highway System under different scenarios. Our approach is designed for
teaching this material at an advanced level but is still accessible to audiences without
expertise on the topic.
∗We thanks Xiangliang Li and Jan Rouwendal for helpful comments. All errors are our own.
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1 Introduction
Space – Economic Science’s final frontier. Over the past twenty years a revolution in the
exploration of space in Economics led by a combined effort from geographers, trade, and
urban economists has finally brought the required technology to analyze space in all its
glory: An armada of tools from mathematics, physics, cartography, and computer science
has gradually formed the necessary equipment to explore space and its consequence for
growth and allocation of economic activity.
Spearheading this exploration is spatial theory’s dreadnought: The gravity model. This
mathematical design strikes a careful balance between two key ingredients behind any suc-
cessful theory. It is intuitive and pedagogical while at the same time rich enough to provide
an incredibly good fit to the empirical observations. In other words, it is beautiful and it
works like a charm!
The key force that spatial models with mobility of goods and people need to harness is
space itself. In an environment with N locations, mobility of goods and people implies that
N2 number of trade interactions and N2 migration flows have to be modeled. These many
complex interactions could potentially obscure the main forces that determine economic
activity across space. The resolution that the gravity model provides cuts right through
these complexities: It allows for as many (exongeous) frictions as relationships, but assumes
that the elasticity of the flows to (endogenous) push and pull factors are governed by only
a single parameter. This lets the theory capture the first-order impacts of the effects of
geography on economic outcomes while ignoring the (potentially) less important impact of
varying bilateral elasticities.
To understand the model and the impact of space on economic activity we present a
generalized, yet simple, version of the gravity model that allows for mobility of goods and
people across space limited by frictions specific to these flows.1 In particular, we employ
an extension of the Allen and Arkolakis (2014) framework with an exogenous population in
each location and mobility frictions across locations. Variations of this extension have been
more formally modeled by Tombe, Zhu, et al. (2015), Bryan and Morten (2015), Caliendo,
Dvorkin, and Parro (2015), Desmet, Nagy, and Rossi-Hansberg (2016), Faber and Gaubert
(2016), Allen, Morten, and Dobbin (2017), and Allen and Donaldson (2017). Within this
1Examples of gravity trade models included in our framework are perfect competition models such as Anderson (1979), Anderson and Van Wincoop (2003), Eaton and Kortum (2002), Caliendo and Parro (2015) monopolistic competition models such as Krugman (1980), Melitz (2003) as specified by Chaney (2008), Arkolakis, Demidova, Klenow, and Rodŕıguez-Clare (2008), Di Giovanni and Levchenko (2008), Dekle, Eaton, and Kortum (2008), and the Bertrand competition model of Bernard, Eaton, Jensen, and Kortum (2003). Economic geography models incorporated in our framework include Allen and Arkolakis (2014) and Redding (2016) and the geography-trade framework of Allen, Arkolakis, and Takahashi (2014).
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environment we can answer a number of important questions: What is the allocation of
economic activity across space and how is it determined by location fundamentals or bilateral
frictions of mobility. When does a solution of the model exists and when is it unique? Do
different assumptions on the frictions of moving goods and people imply that policies have
different implications?
Finally, we should note that the brief literature review of gravity models above is by no
means complete and refer the interested reader to the excellent review articles by Baldwin
and Taglioni (2006), Head and Mayer (2013), Costinot and Rodriguez-Clare (2013) and
Redding and Rossi-Hansberg (2017), where the latter two focus especially on quantitative
spatial models.
2 Gravity: The Evidence
We begin by explaining and documenting the “gravity” relationship. Empirically, the notion
of gravity introduced by Tinbergen (1962) postulates that flows decline with distance. We
illustrate that both the flow of goods (trade) and people (migration) exhibit gravity. More-
over, this gravity relationship is robust to different scales of distance: it is prevalent for the
flow of people and goods both across countries and within countries. Finally, gravity has
been present in the data for (at least) the past fifty years, and shows no signs of attenuating
over time.
2.1 Gravity in the flow of goods
We first examine the flow of goods (trade). We use two different data sets: the first, from
Head, Mayer, and Ries (2010), comprise the value of trade between between countries from
1948 to 2006; the second, from (CFS, 2007, 2012), are the Commodity Flow Surveys, comprise
the value of trade between U.S. states for 2007 and 2012. To reduce concerns of selection
bias, we constrain each sample to be balanced by only including origin-destination pairs that
reported positive trade flows for each year in the sample; moreover, to avoid having to take a
stand on what constitutes the “distance-to-self”, in each sample we exclude own trade flows.
Let Xijt be the value of trade flows from location i to location j in time t. The gravity re-
lationship postulates that the (log) of the value of trade flows declines in the distance between
locations, conditional on (endogenous) origin-specific push factors γTit and destination-specific
pull factors δTjt:
lnXijt = f (ln distij) + γ T it + δ
T jt, (1)
where ∂f ∂ ln distij
< 0. Figure 1 overlays a non-parametric function f (·) on top of a scatter
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plot of the relationship between log trade flows and log distance after partitioning out the
origin-year and destination-year fixed effects for international trade in both 1950 and 2000;
as is evident, there is a strong negative (and approximately log-linear) relationship in both
years, with the negative effect being especially pronounced in the year 2000. In Figure 2, we
impose a linear function f (ln distij) = γt ln distij and estimate the coefficient γt separately
for each year; we find a precisely estimated negative relationship that appears to be getting
stronger over time, with a coefficient γt of between -0.5 and -1.5.
This strong gravity relationship in the flow of goods is also prevalent within countries.
Figure 3 is the analog of Figure 1 for trade between U.S. states. In both 2007 and 2012 there
is a strong negative (and approximately log-linear) relationship between log trade flows and
log distance. As with the international trade flows, the trade coefficient is about -1 , showing
that the effect of distance is similar within and across countries.
We can also examine how the origin push factor γit and destination pull factor δjt are
correlated with various observables. Figure 4 shows that both the push and pull factors in
international trade are strongly positively correlated with GDP – even after partitioning out
time-invariant country effects and country-invariant year effects. Put another way, changes
in GDP within a country over time (relative to total world GDP) are strongly positively
correlated with both a country’s imports and exports. This strong positive correlation is
also present in within country trade flows, as is evident in Figure 5. Moreover, Figures 6
and 7 show a strong positive correlation between the push and pull factors both across and
within countries. As we will see below, this positive correlation will be predicted by a gravity
spatial model with balanced trade and symmetric trade costs.
2.2 Gravity in the flow of people
The flow of labor (migration) exhibits similar – but not identical – patterns as the flow
of goods. We analyze the flow of labor both across and within countries. For international
migration, we turn to the WBG (2011) dataset, which provides bilateral flows of people across
countries every ten years from 1960 to 2010. For intranational migration, we construct flows
of people across U.S. states using their state of birth and current location for each decennial
census from 1850 to 2000 from Ruggles, Fitch, Kelly Hall, and Sobek (2000). As with the
trade data, we consider a balanced sample of location pairs for which there is a positive flow
of people in all years and exclude own-flows of people (i.e. those that do not migrate).
As with the flow of goods, we can construct a simple empirical gravity specification for
the flow of people from location i to location j at time t, Lijt:
lnLijt = g (ln distij) + γ L it + δ
L jt, (2)
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where gravity implies ∂g ∂ ln distij
< 0. Figure 8 overlays a non-parametric function g (·) on top of a scatter plot of the relationship between log migration flows and log distance after
partitioning out the origin-year and destination-year fixed effects for international migra-
tion in both 1960 and 2010; like with the flow of goods, there is a strong negative (and
approximately log-linear) relationship in both years. In Figure 9, we impose a linear func-
tion g (ln distij) = γt ln distij and estimate the coefficient γt separately for each year of data;
again, as with the flow of goods, we find a precisely estimated negative relationship with a
coefficient γt of between -1 and -2.
Figure 10 shows that the gravity relationship also exists for within country migration and
is remarkably stable over the 150 years of data; indeed, as Figure 11 illustrates, the effect
of distance on the flow of people is nearly identical within the United States as it is across
countries, with a coefficient hovering of about -1.5. (It is interesting to note that unlike for
the flow of goods, the gravity coefficient of migration shows no evidence of getting more
negative over time).
While the gravity relationship with distance is quite similar for trade and migration, the
“push” and “pull” factors (γLit and δ L jt, respectively) are substantially different for migration.
Figure 12 shows that there is no systematic relationship between population and either the
push or pull factor across countries; within the United States, however, Figure 13 shows
the lagged population is strongly correlated with the push factor and the contemporaneous
population is strongly correlated with the pull factors.Unlike the flow of goods, there is no
systematic correlation between the push and pull factors across countries (Figure 14), but
there is a negative correlation between push and pull factors across U.S. states (Figure 15).
As we will see below, this is consistent with a theoretical model of migration when the
population is not in a steady state and/or migration costs are not symmetric.
3 Gravity: A Simple Framework
We now introduce a simple model that can generate the prevalence of gravity in the data.
Suppose there are N locations, where in what follows we define the set S ≡ {1, …, N}, each producing a differentiated good. The only factor is labor, and we denote the allocation of
labor in location i ∈ S as Li and assume the total world labor endowment is ∑
i∈S Li = L̄.
Given the evidence from the previous section that gravity holds both within and across coun-
tries, locations can be interpreted as either regions within a country or countries themselves.
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3.1 Demand for Goods: Gravity on Trade Flows
We assume that workers have identical Constant Elasticity of Substitution (CES) preferences
over the differentiated varieties produced in each different location. The total welfare in
location i ∈ S, Wi, can be written as:
Wi =
(∑ j
q σ−1 σ
ji
) σ σ−1
ui, (3)
where qsi is the per-capita quantity of the variety produced in location s and consumed in
location i, σ ∈ (1,∞) is the elasticity of substitution between goods ω, and ui is the local amenity, discussed below.2
Each worker in location i earns a wage wi and thus the budget constraint is∑ j
pjiqji = wi (4)
where psi is the price of good from location s in i. Optimization of the worker utility, equation
(3), subject to the budget constraint, equation (4), yields the total expenditure in location
j on the differentiated variety from location i:
Xij = (pij) 1−σ P σ−1j wjLj for all j (5)
where Lj is the total number of workers residing in location j (determined endogenously
below) and Pj ≡ (∑
i (pij) 1−σ) 11−σ is the Dixit-Stiglitz price index.
The production function of each variety is linear in labor and the productivity in location
i is denoted by Ai. Thus, the cost of producing variety i is pi = wi/Ai. Shipping the good
from i to final destination j incurs an “iceberg” trade friction, where τij ≥ 1 units must be shipped in order for one unit to arrive. Thus, the price faced by location j for a factor from
i can be written as:
pij = wi Ai τij, (6)
where τij are bilateral trade frictions. Substituting this solution to equation (5) and rear-
ranging we obtain
Xij = (τij) 1−σ ( wi Ai
)1−σ P σ−1j wjLj. (7)
This equation is the modern version of a gravity equation initially derived by Anderson (1979)
2While the model attains a non-trivial solution even for σ ∈ (0, 1), we focus on the case where σ > 1 so that the elasticity of trade flows to trade costs is negative.
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and is ubiquitous in modern work in international trade. It is characterized by a bilateral
term that is a combination of model parameters, trade costs and the trade elasticity, and
origin and destination specific terms which are combinations of endogenous variables and
parameters.
More recent work provides a wealth of microfoundations for this structural equation based
on comparative advantage, increasing returns, or firm heterogeneity (see for example, Eaton
and Kortum (2002); Chaney (2008); Eaton, Kortum, and Kramarz (2011); Arkolakis (2010);
Arkolakis, Costinot, and Rodŕıguez-Clare (2012); Allen and Arkolakis (2014); Redding (2016)
among others). Taking log of this equation generates the empirical gravity trade equation
(1) presented in Section 2.
To incorporate agglomeration forces in production – which Allen and Arkolakis (2014)
show creates an isomorphism with the monopolistic competition models with free entry as
in Krugman (1980) – we assume that the productivity of a location is subject to spillovers:
Ai = ĀiL α i where Āi is the exogenous productivity.
3 We focus on the empirically relevant
cases of α ≥ 0, capturing agglomeration externalities due to endogenous entry, scale effects etc.
3.2 Demand for Labor: Gravity on Worker Flows
We next determine the allocation of labor across location. We assume that there is an
initial (exogenous) distribution of workers across all locations i denoted by L0i , from which
all workers choose where to live subject to migration frictions. In particular, the indirect
utility function of an owner of one unit of aggregate factor originating from location i and
moving to location j is equal to the product of the utility realized in the destination and a
bilateral migration disutility νij :
Wij = wj Pj uj × νij,
where νij = (Lij/L0i )
−β
µij depends both on an (exogenous) iceberg migration friction µij ≥ 1
and on the (endogenous) number of migrating workers Lij. The parameter β ≥ 0 governs the extent to which migration flows create congestion externalities.
In equilibrium, labor mobility implies that the utility of all agents originating from i is
equalized:
Wi = wj Pj
ūj µij
( Lij/L
0 i
)−β . (8)
3See Allen and Arkolakis (2014) for a precise discussion of the various isomorphisms to this formulation.
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Inverting this expression, we obtain the number of workers that migrate from i to j:
Lij =
( wj Pj
ūj µij
) 1 β
W 1 β
i
L0i . (9)
Equation (9) is a gravity equation on worker flows as it determines the share of workers in
location i as a function of the real wage in location i. By taking logs, it provides a theoretical
justification of the empirical gravity specification for the flow of people in equation (2) in
Section 2.
We should at this point note that when modeling bilateral migration flows many authors
choose alternative, possibly more intuitive, formulations. These formulations yield the same
functional form as (8) but capture a variety of microfoundations such as competition for an
immobile factor (e.g. land or housing markets) or heterogeneous location preferences across
workers. In this latter approach, an agent’s utility in location j is the product of the local real
wage times a heterogeneous component. This heterogeneity results to different decisions for
otherwise identical agents. Assuming a Frechet distribution for this heterogeneity following
Eaton and Kortum (2002); Ahlfeldt, Redding, Sturm, and Wolf (2015); Redding (2016) leads
to a similar formulation to equation (9), as discussed in Allen and Arkolakis (2014).
3.3 Closing the Model
To close the model we need to satisfy four equilibrium conditions. The first two are associated
with the flow of goods. First, the total amount of labor used for the production of goods for
all countries equals the labor available in each country i. Written in terms of labor payments,
this implies that the total payments accrued to labor in location i must equal to the sales of
this location to all the locations in the world, including i,
wiLi = ∑ j
Xij. (10)
The second equilibrium condition is that total expenditure equals total labor payments
and in turn this equals total payments for goods produced for location i,
Ej = wiLi = ∑ i
Xij. (11)
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Using equation (7) this expression can be written,
wjLj = ∑ i
(τij) 1−σ ( wi Ai
)1−σ P σ−1i wjLj =⇒
P 1−σi = ∑ i
(τij) 1−σ ( wi Ai
)1−σ , (12)
which is the expression for the Dixit-Stiglitz price index defined above.
The third and fourth equilibrium conditions are associated with the flow of labor. The
third condition is that the initial population in location i is equal to the total flows of persons
from location i, i.e.:
L0i = ∑ j
Lij (13)
Combined with the migration gravity equation (9) above allows us to write equilibrium
welfare of migrants from location i Wi as the CES aggregate of their bilateral utility:
L0i = ∑ j
( wj Pj
ūj µij
) 1 β
W 1 β
i
L0i ⇐⇒ Wi =
(∑ j
( wj Pj
ūj µij
) 1 β
)β . (14)
Substituting this expression for welfare back into the migration gravity equation then allows
us to write migration shares of people analogously to expenditure shares on goods:
Lij/L 0 i =
( wj Pj
ūj µij
) 1 β
∑ j
( wj Pj
ūj µij
) 1 β
. (15)
Finally, the fourth equilibrium condition requires that the in-flow of migrants to location
i is equal to its total population:
Lj = ∑ i∈S
Lij. (16)
Define the geography of the economy as the set of trade costs {τij}, migration frictions {µij}, productivities
{ Āi }
, amenities {ūi}, and initial population {L0i }. For any set of elasticities {σ, α, β} and any geography, an equilibrium is defined as a set of wages, labor allocations,
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price index, and welfare, that satisfy the following four equations:
wiLi = ∑ j
(τij) 1−σ (
wi ĀiLαi
)1−σ P σ−1i wjLj (17)
Pi =
∑ j
( τjiwj ĀjLαj
)1−σ 11−σ (18) Li =
∑ j∈S
(µji) − 1 β
( wi Pi ūi
) 1 β
(Wj) − 1 β L0j (19)
Wi =
(∑ j
( wj Pj
ūj µij
) 1 β
)β (20)
This yields a system of 4 × N equations and 4 × N unknowns (with one equation being redundant from Walras Law and one price being pinned down by a normalization). Note the
symmetry of the labor and goods market clearing conditions: each consists of one market
clearing condition and one composite “price index”: Pi for goods and Wi for labor.
Precise restrictions that guarantee existence and uniqueness are provided by Allen and
Donaldson (2017). Product differentiation implies that there are gains to moving to locations
with low population in order to provide labor for the global demand of the local good. In
addition agglomeration and dispersion forces act upon this basic mechanism. Intuitively, the
agglomeration forces, governed by parameter α, imply increased concentration of economic
activity. Dispersion forces, governed by parameter β, imply dispersion of economic activities
away from locations with large population. When agglomeration forces are stronger than
dispersion forces the possibility of multiple equilibria arises. In these cases agglomeration
may act as a self-sustaining force and equilibria where different locations are the ones with
the largest population can arise, similar to the spatial models of Krugman (1991); Helpman
(1998); Fujita, Krugman, and Venables (1999). In fact, in the version of the model where
migration costs are infinite, this happens exactly at α > β, as discussed in Allen and Arko-
lakis (2014). Existence of equilibria with positive population is always guaranteed. However,
when the agglomeration forces are very strong black hole equilibria with all the activity con-
centrated in one point may be the only ones that satisfy some refined notion of equilibrium
related to stability.4
4The system of equations has the form of the multi-equation multi-location gravity system analyzed by Allen, Arkolakis, and Li (2014). Using their approach equilibrium existence and uniqueness can be characterized in generalized gravity systems. In addition, their approach provides algorithms to compute the equilibrium of these multi-equation systems efficiently. Refined notions of stability are discussed in Allen and Arkolakis (2014).
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Finally, it may be apparent from the above discussion that particular microfoundations of
the two key gravity equations do not play a key role in determining many of the properties of
the model. A long tradition on modeling gravity in international trade flows is summarized
in Arkolakis, Costinot, and Rodŕıguez-Clare (2012) (see discussion in Proposition 2) by a
class of models governed by a single parameter, the elasticity of trade, hereby captured by
1− σ. In addition, more recently, microfoundations for the migration gravity equation have been provided as well. It can be shown that, as far as it concerns the positive properties and
the counterfactuals of this expanded class of models, with respect to wages and labor, two
composite parameters (in this case functions of the three parameters α, β and σ) determine
all its predictions. This class of models that includes geography models with labor mobil-
ity, intermediate inputs, non-traded goods and other economic forces is discussed in Allen,
Arkolakis, and Takahashi (2014).
4 Model Characterization
We proceed next to characterize this setup by imposing assumptions on the geography of
trade costs, τij, and the geography of fundamentals, i.e. productivities and amenities, Āi, ūi.
4.1 Geography and the distribution of economic activity
We first provide intuition about how geography shapes the spatial distribution of economic
activity; these results build upon Allen and Arkolakis (2014), Allen, Arkolakis, and Takahashi
(2014) and Allen and Donaldson (2017).
Suppose that trade costs are bilaterally symmetric, i.e. τij = τji for all i and j. Then
thegoods gravity equation (7), and two equilibrium conditions, equations (10) and (11),
together imply that the origin and destination fixed effects of the trade gravity equation are
equal up to scale, i.e.: ( wi ĀiLαi
)1−σ ∝ P σ−1i wiLi ∀i ∈ S (21)
This accords well with the finding in Section 2.1 that the origin and destination fixed effects
of the trade gravity equation as strongly correlated.
What about on the migration side? Suppose that migration costs are bilaterally symmet-
ric, i.e. µij = µji for all i and j. One might wonder if this symmetry, along with the labor
gravity equation (9), and the two labor adding up conditions, equations (14) and (16) corre-
spondingly imply that the origin and destination specific terms of the labor gravity equation
are equal up to scale. It turns out that the answer in general is no, unless the population
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distribution is in a steady state where L0i = Li. In that case (and only in that case) we have:( wi Pi ūi
) 1 β
∝ W − 1 β
i Li ∀i ∈ S (22)
Recall from Section 2.2 that we empirically find virtually no correlation in the origin and
destination fixed effects in the migration gravity equation both across countries and across
states within the U.S., suggesting that we are either far from a steady state or migration
costs are asymmetric (or both). Combining equations (21) and (22), we can express the
equilibrium steady-state population and wage of each location solely as a function of that
location’s productivity, amenity, and geographic location (with the effect of trade costs being
summarized by Pi and the effect of migration costs being summarized by Wi):
γ lnLi = 1
β (σ − 1) ln Āi +
σ
β ln ūi +
σ
β lnWi −
2σ − 1 β
lnPi + C1 (23)
σγ lnwi = σ (σ − 1) ln Āi + (α (σ − 1)− 1) σ
β ln ūi + (α (σ − 1)− 1)
σ
β lnWi − σ
( (σ − 1)
( 1− α
β
) − 1 β
) lnPi + C2,
(24)
where γ ≡ ( σ + 1
β (1− α (σ − 1))
) , C1 is a constant determined by the size of the aggregate
labor market and C2 is a constant determined by the choice of numeraire. Focusing on
the equilibrium distribution of population and assuming α < 1 σ−1 , we can see that more
productive places (higher Āi), higher amenity places (higher ūi), places with lower migration
costs (higher Wi) and places with lower trade costs (lower Pi) all have higher populations,
with the responsiveness of the population to the geography governed by the strength of
spillovers through the composite term γ.5
4.2 Special cases
We now illustrate a number of interesting special cases of this general framework. First,
consider the case where workers cannot move from their original location, i.e. where µij =∞ for all i 6= j. This is an assumption maintained in gravity trade models such as Anderson (1979); Eaton and Kortum (2002); Chaney (2008); Arkolakis (2010); Arkolakis, Costinot,
and Rodŕıguez-Clare (2012); Costinot and Rodriguez-Clare (2013); Allen, Arkolakis, and
Takahashi (2014). Since labor is fixed, we can determine the wage and the price index, wi
and Pi by using equations (17) and (18), equation (19) implies Li = L 0 i , and equation (20)
5When α ≥ 1σ−1 , the only type of stable equilibria possible is a black-hole equilibrium where all the population is concentrated in a single location; see Allen and Arkolakis (2014) for an in depth discussion and formal definition of “stability” in this context.
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simplifies to Wi = wi Pi ūi.
Second, consider the case where there are no migration frictions, i.e. µij = 1 for all i 6= j. This is an assumption maintained in economic geography models such as Krugman (1991);
Helpman (1998); Redding and Sturm (2008) and in the gravity new economic geography
models of Allen and Arkolakis (2014); Redding (2016). In this case, equation (20) implies
welfare is equalized across locations,
W = Wi = wi/Pi,
Moreover, equation (19) implies that the population residing in location i is
Li L̄
=
( wi Pi ūi
)1/β ∑
i′
( wi′ Pi′ ūi
)1/(β) . (25) A third special case is where there is both free trade and free migration, i.e. µij, τij = 1
for all i 6= j. This setup is the celebrated Rosen-Roback (1982) model put to use in a number of urban applications (see Glaeser and Gottlieb (2008) and Moretti (2011) for review
of applications of this model). With free labor mobility, equation (20) again implies that
Wi = W while with free goods mobility, equation (18) implies that the price index equalizes
across locations Pi = P . 6 Notice that in this special case equations (23) and (24) give an
explicit solution of wages and labor in terms of geography of each location. These solutions
are intuitive (e.g. under reasonable restrictions labor is increasing in productivities and
amenities) and have been heavily exploited by the urban literature.
4.3 Analytical Solutions
In this section, we provide analytical solutions for the equilibrium above for two simple
geographies. (Note that the characterization of the equilibrium from equations (23) and (24)
provide only partial characterization in the case with positive trade or migration frictions
as they are functions of two equilibrium objects, namely the price index and the expected
welfare of a location).
6In fact, the baseline Rosen-Roback (1982) framework also imposes that elasticity of substitution is infinite across goods, i.e. the goods are perfect substitutes. The assumption is not essential for what is obtained below.
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Two countries
In the special case of two countries we assume that the population is fully mobile across
locations so that utility equalization holds. Here we do not impose any restriction on the
geography of trade costs but we do impose the assumption of symmetry in the geography
of productivities and amenities so that Āi = ūi = 1 for all i. Using equation (10) and the
gravity equation for trade flowsimplies
wiLi = w 2−σ i P
σ−1 i Li + τ
1−σ ij w
1−σ i wjLjP
σ−1 j ⇐⇒
Li = w 1−σ i P
σ−1 i Li + τ
1−σ ij w
−σ i wjLjP
σ−1 j (26)
We next impose utility equalization resulting from the zero migration costs: wi Pi
= wj Pj
= W̄ .
This implies that: P σ−1j
P σ−1i = wσ−1j
wσ−1i .
Using the definition of the price index we have that
w1−σi τ 1−σ ij + w
1−σ j
w1−σj τ 1−σ ji + w
1−σ i
= wσ−1j
wσ−1i =⇒
( wi wj
) =
( τji τij
) 1 2
. (27)
Combining the two results yields:
Li = W̄ 1−σ ( Li + τ
1−σ ij
( wj wi
)σ Lj
) .
Now take the ratio of the two region’s populations:
Li Lj
= Li + τ
1−σ ij
( τij τji
)σ 2 Lj
τ 1−σji
( τji τij
)σ 2 Li + Lj
=
( Li Lj
) + τ 1−σij
( τij τji
)σ 2
τ 1−σji
( τji τij
)σ 2 Li Lj
+ 1
=⇒
Li Lj
=
( τij τji
) 1 2
(28)
In other words, if country j is more open that country i, τij < τji, then the population in
country i is smaller but wages are higher.
14
The Line
We now assume that the topography of the world is described by a line, largerly drawing
from the analysis of Allen and Arkolakis (2014). Let space S be the [−π, π] interval and suppose that α = β = 0 and Āi = ūi = 1, i.e. there are no spillovers and all locations
have homogeneous exogenous productivities and amenities. Suppose that trade costs are
instantaneous and apart from a border b in the middle of the line at location 0; that is, trade
costs between locations on the same side of the line are τis = e τ̃ |i−s| and those on different
sides are τis = e b+τ̃ |i−s|.7
Taking logs of equation (23) and differentiating yields the following differential equation:
∂ lnLi ∂i
= (1− 2σ) ∂ lnPi ∂i
. (29)
It is easy to show that ∂ lnP−π ∂i
= −τ̃ and ∂ lnPπ ∂i
= τ̃ in the two edges of the line and ∂ lnP0 ∂i
= τ̃ ( 1− e(1−σ)b
) / ( 1 + e(1−σ)b
) in the location of the border which gives us boundary
conditions for the value of the differential equation at locations i = −π,0, π. Intuitively, moving rightward while on the far left of the line reduces the distance to all other locations
by τ , thereby reducing the (log) price index by τ . To obtain a closed form solution to
equation (29), we differentiate equation (23) twice to show that the equilibrium satisfies the
following second order differential equation:
∂2
∂i2 Lσ̃i = k1L
σ̃ i for i ∈ (−π, 0) ∪ (0, π), (30)
where ˙˜ ≡ (σ − 1) / (2σ − 1)σ and k1 ≡ (1− σ)2 τ 2 + 2 (1− σ) τW 1−σ can be shown to be negative. Given the boundary conditions above, the equilibrium distribution of labor in
both intervals is characterized by the weighted sum of the cosine and sine functions (see
example 2.1.2.1 in Polyanin and Zaitsev (2002)):
Lσ̃i = k2 cos ( i √ −k1
) + k3
∣∣∣sin(i√−k1)∣∣∣ . The values of k1 and the ratio of k2 to k3 can be determined using the boundary conditions.
Given this ratio, the aggregate labor clearing condition determines their levels.8 Notice that
7This border cost is reminiscent of the one considered in Rossi-Hansberg (2005). As in that model, our model predicts that increases in the border cost will increase trade between locations that are not separated by border and decrease trade between locations separated by the border. Unlike Rossi-Hansberg (2005), however, in our model the border does not affect what good is produced (since each location produces a distinct differentiated variety) nor is there an amplification effect through spillovers (since spillovers are assumed to be local).
8More general formulations of the exogenous productivity or amenity functions result to more general
15
in the case of no border or an infinite border, the solution is the simple cosine function or
two cosine functions one in each side of the border, respectively, and k3 = 0, so that the
aggregate labor clearing condition directly solves for k2. 9
Figure 16 depicts the equilibrium labor allocation in this simple case for different values
of the instantaneous trade cost but no border. As the instantaneous trade cost increases, the
population concentrates in the middle of the interval where the locations are less economically
remote. The lower the trade costs, the less concentrated the population; in the extreme where
τ = 0, labor is equally allocated across space. With symmetric exogenous productivities and
amenities, wages are lower in the middle of the line to compensate for the lower price index.
Figure 17 shows how a border affects the equilibrium population distribution with a positive
instantaneous trade cost. As is evident, the larger the border, the more economic activity
moves toward the middle of each side in the line; in the limit where crossing the border is
infinitely costly, it is as if the two line segments existed in isolation.
Differences in exogenous productivities, amenities
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