reading discussion

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:

ln Xijt = 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:

ln Lijt = 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

Pσ−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 Pσ−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 Pσ−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 ĀiL

α i

)1−σ Pσ−1i wjLj (17)

Pi =

 ∑

j

( τjiwj ĀjL

α j

)1−σ 1

1−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 ĀiL

α i

)1−σ ∝ Pσ−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 ln Li = 1

β (σ − 1) ln Āi +

σ

β ln ūi +

σ

β ln Wi −

2σ − 1 β

ln Pi + C1 (23)

σγ ln wi = σ (σ − 1) ln Āi + (α (σ − 1) − 1) σ

β ln ūi + (α (σ − 1) − 1)

σ

β ln Wi −σ

( (σ − 1)

( 1 −

α

β

) −

1

β

) ln Pi + 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 ≥ 1 σ−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: Pσ−1j

Pσ−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

τ1−σji

( τji τij

)σ 2 Li + Lj

=

( Li Lj

) + τ1−σij

( τij τji

)σ 2

τ1−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:

∂ ln Li ∂i

= (1 − 2σ) ∂ ln Pi ∂i

. (29)

It is easy to show that ∂ ln P−π

∂i = −τ̃ and ∂ ln Pπ

∂i = τ̃ in the two edges of the line and

∂ ln P0 ∂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 τ2 + 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 and the spillovers also play a key role

in determining the equilibrium allocation of labor and wages. The numerical solutions for

these cases can be found in Allen and Arkolakis (2014).

5 Taking the model to the data

In this section, we discuss how one can estimate the parameters of the structural model from

the previous section and use the model to perform counterfactuals. We conclude the section

by illustrating how different assumptions regarding the mobility of goods and labor affects

such counterfactual results.

5.1 Estimation

We first discuss how to estimate all the model parameters using readily available data. One

particular highlight of the methodology that follows is that it does not require observing all

bilateral flows of goods and labor. This is helpful, as oftentimes (especially in sub-national

data) such bilateral data is unavailable.

specifications of the second order differential equation illustrated above (see Polyanin and Zaitsev (2002) section 2.1.2 for a number of tractable examples).

9 Mossay and Picard (2011) obtain a characterization of the population based on the cosine function in a model where there is no trade but agglomeration of population arises due to social interactions that decline linearly with distance. In their case, population density may be zero in some locations while in our case the CES Armington assumption generates a strong dispersion force that guarantees that the equilibrium is regular when agglomeration forces are not too strong, as discussed in Theorem 2.

16

Estimating Bilateral Frictions

Assume that the log of the trade and migration bilateral frictions (to the power of their

respective elasticities, i.e. i.e. τ1−σij and µ 1 β

ij) are functions of observables (e.g. distance),

i.e. (1 −σ) ln τij = f ( XTij; γ

T )

and 1 β

ln µij = g ( XLij; γ

L ) , where f (·) and g (·) are functions

known up to a vector of parameters γT and γL, respectively. Then taking logs of the goods

and labor gravity equations (7) and (9) yield:

ln Xij = f ( XTij; γ

T )

+ γTi + δ T j + ε

T ij (31)

ln Lij = g ( XLij; γ

L )

+ γLi + δ L j + ε

L ij, (32)

where γTi and γ L i are the origin fixed effects of the trade and labor gravity equations, δ

T j and

δLj are the respective destination fixed effects, and we interpret ε T ij and ε

L ij as measurement

error in the observed bilateral flows. Note that equations (31) and (32) are virtually identical

to the empirical gravity equations (1) and (2)presented in Section 2; the key difference is we

have now provided a theoretical justification for them.

By far the most common assumption in the literature is that bilateral frictions are in-

creasing in bilateral distance distij, i.e. f (·) = γT ln distij and g (·) = γL ln distij. However, several recent innovations have allowed researchers to go beyond simply using straight-line

distances between locations. Donaldson and Hornbeck (2012); Donaldson (forthcoming) uses

Dijkstra’s algorithm to calculate the least cost route between locations on a graph of the

transportation network. Allen and Arkolakis (2014) apply the Fast Marching Method to

calculate the least cost route between locations over a continuous geography. Both these

methods rely on algorithms originally developed in computer science, leaving f (·) and g (·) implicit functions of the underlying geography. More recently Allen and Arkolakis (2017)

derive a closed form solution for f (·) and g (·) as a function of the underlying transportation network assuming that many heterogeneous traders each choose the least cost route over the

network.

Regardless of the choice of f (·) and g (·), the fixed effects regression only recovers esti- mates of the combination of the bilateral frictions and their respective elasticities, i.e. τ1−σij

and µ −1 β

ij . However, as we will see, this is all that is needed to recover the remainder of

the parameters and conduct counterfactuals. To make this clear, define Tij ≡ τ1−σij and

Mij ≡ µ −1 β

ij for what follows. Moreover, note that estimation can be accomplished even if

bilateral flow data is only available for a subset of location pairs, as long as the observables

are available for all bilateral pairs. This is helpful, for example, if trade flow data is only

available for a subset of countries, but one wishes to construct a model for the entire world.

17

Recovering Location fundamentals and estimating model elasticities

Given estimates of Tij ≡ τ1−σij and Mij ≡ µ −1 β

ij , we can use the equilibrium structure of

the model to recover information about endogenous outcomes in each location, namely the

productivities Ai, the amenities ui, the price index Pi, and the welfare Wi.

To see this, we re-write the equilibrium equations (17)-(20) as follows:

pσ−1i = ∑ j

Tij

( Yj Yi

) Pσ−1i (33)

( Pσ−1i

)−1 = ∑ j

Tji ( pσ−1j

)−1 (34)

( ω

1 β

i

)−1 = ∑ j

Mji

( L0j Li

)( W

1 β

j

)−1 (35)

W 1 β

i = ∑ j

Mijω 1 β

j (36)

where pi ≡ wiĀiLαi is the price of a good produced in location i and ωi ≡ wi Pi ūi is the welfare of

residents residing in location i. Since {Tij} and {Mij} were estimated in the previous step and assuming the income Yi, initial population L

0 i and current population Li are all observed

in the data, it can be shown (see Allen and Donaldson (2017)) that there exists a unique

(to-scale) set of

{ pσ−1i ,P

σ−1 i ,ω

1 β

i ,W 1 β

i

} that are consistent with equations (34)-(36). That

is, the equilibrium structure of the model allows one to uniquely invert the model to recover

these parameters for each location.

There are three important things to note about this inversion procedure. First, the

inversion itself does not require knowledge of the any model elasticities, i.e. conditional on

the bilateral frictions {Tij} and {Mij}, the choice of σ, β, or α does not affect the equilibrium{ pσ−1i ,P

σ−1 i ,ω

1 β

i ,W 1 β

i

} . Second, if one does know the values of σ, β, and α, one can identify

the exogenous productivity and amenity of each location, since:

ln ( pσ−1i

) = (σ − 1) ln wi −α (σ − 1) ln Li − (σ − 1) ln Āi (37)

ln

( ω

1 β

i

) =

1

β ln wi Pi

+ 1

β ln ūi, (38)

where the left hand side of both equations (ln ( pσ−1i

) and ln

( ω

1 β

i

) ) are values recovered from

the inversion and the right hand side are either observed in the data (the ln wi and ln Li), are

recovered from the inversion (the ln Pσ−1i ) or are the exogenous productivities and amenities

18

(the ln Āi and ln ūi).

Third, and perhaps most importantly, note that equations (37) and (38) are estimating

equations that allow one to recover the model elasticities σ, α, and β. Specifically, one can

regress the ln ( pσ−1i

) recovered from the model inversion on the observed wages ln wi and

population ln Li to recover estimates of σ and α. Given the estimate of σ, one can construct

Pi from the model inversion of P σ−1 i and then regress ln

( ω

1 β

i

) on the observed real wage to

recover β. Crucially, because the exogenous productivity and amenity of each location are

the residuals of these two equations – and the model tells us that the wage and population

will be correlated with the exogenous productivities and amenities (see equations (23) and

(24)), ordinary least squares estimation will yield biased estimates. Instead, estimation of

the structural parameters requires valid instruments for the wage (both real and nominal)

and population that are both correlated with the observed wage and population but uncor-

related with the fundamental productivity and amenity of a location. Two plausible sets of

instruments are as follows. First, Allen, Arkolakis, and Takahashi (2014) suggest that the

model structure could be used to generate instruments. In particular, the authors calculate

the equilibrium wages, price indices, and populations of a hypothetical world where bilateral

frictions are a solely a function of observed bilateral distance and local characteristics (e.g.

productivity and amenities) are solely a function of known geographic variables (e.g. dis-

tance to coast, ruggedness, soil quality, etc.). One could then use these equilibrium variables

as instruments for the actual wages, price indices, and populations. This procedure is valid

as long as the geographic variables used are also controlled for directly in the regression;

indeed, Adao, Arkolakis, and Esposito (2017) show that a two-stage procedure using such

a model implied instrument is optimal in the sense that it minimizes the variance of the

estimates. An alternative approach would be to use a shock to either trade or migration

costs; for example Allen, Morten, and Dobbin (2017) use the construction of wall segments

along the border between the U.S. and Mexico as a shock to migration costs. This can be

implemented either in a reduced form way (e.g. instrumenting for wages, populations, and

the price indices using distance to the border wall, as in Ahlfeldt, Redding, Sturm, and Wolf

(2015)) or by using the structure of the model to predict how the trade cost shock changes

each of the endogenous variables and using these model-predicted changes in wages, pop-

ulations, and price indices to estimate the elasticities using the first-differenced versions of

equations (37) and (38).

19

5.2 Conducting Counterfactuals

Given estimates of Tij ≡ τ1−σij and Mij ≡ µ −1 β

ij and the structural elasticities σ, α, and β from

the previous section, it is straightforward to use the equilibrium structure of the model to

conduct counterfactuals. Consider any counterfactual change in the bilateral frictions that

change {Tij,Mij} to { T ′ij,M

′ ij

} . Following Dekle, Eaton, and Kortum (2008), we can express

the equilibrium system using the “exact hat algebra” approach where the notation x̂ ≡ x ′

x

is the ratio of the counterfactual value of a variable its current value. We start by writing

equilibrium equations (17)-(20) for the future period and using the the identity x′ = x̂×x as follows:

Ŷi = ∑ j

πijT̂ijp̂i 1−σP̂σ−1i (39)

P̂ 1−σi = ∑ j

χjiT̂jip̂j 1−σ (40)

L̂i = ∑ j∈S

ρjiM̂jiω̂ 1 β

i Ŵ −1 β

j (41)

Ŵ 1 β

i = ∑ j

υijM̂ijω̂ 1 β

j , (42)

where πij ≡ Tijp

1−σ i P

σ−1 j Yj

Yi =

Xij Yi

is the trade export share, χji ≡ Tjip

1−σ j P

σ−1 i Yi

Yi =

Xji Yi

as

the trade import share, ρji ≡ MjiW

− 1 β

j L 0 jω

1 β i

Li =

Lji Li

is the in-migration share, and υij ≡ MijW

− 1 β

i L 0 iω

1 β j

L0i =

Lij L0i

is the out migration share. There are several important things to note

about the system of equations (39)-(42). First, the kernel of each equation – i.e. πij, χji, ρji,

and υij – are solely a function of observed variables (income Yi, initial and final populations

Li and L 0 i ), estimated variables (the bilateral frictions Tij and Mij) and variables that are re-

covered from the model inversion (i.e.

{ pσ−1i ,P

σ−1 i ,ω

1 β

i ,W 1 β

i

} ). Moreover, no element of the

kernel requires any knowledge of the model elasticities to recover. Hence, even if the bilateral

shares are not directly observed, all necessary components of the shares are observed so that

the kernels of each equation can be treated as observable. This shows that the “exact hat

algebra” approach of Dekle, Eaton, and Kortum (2008) does not require observing bilateral

trade or migration flows to implement. Second, the change in bilateral frictions M̂ij and

T̂ij depend on the counterfactual of interest, so they are known as well. Hence, equations

(39)-(42) comprise 4N equations for 6N unknowns, namely

{ Ŷi, L̂i, p̂i

1−σ, P̂σ−1i , ω̂ 1 β

i ,Ŵ −1 β

j

} .

Since the 4N equations in the system (39)-(42) do not depend on any of the model elas-

20

ticities {σ,α,β}, one might reasonably ask how these elasticities affect the counterfactuals. The answer is that the choice of model elasticities affects how one can express the changes in

incomes Ŷi and populations L̂i as a function of the changes in the other endogenous variables{ p̂i

1−σ, P̂σ−1i , ω̂ 1 β

i ,Ŵ −1 β

j

} . Taking first differences of equations (37) and (37) – using the fact

that ˆ̄Ai = ˆ̄ui = 1 and ln Ŷi = ln L̂i + ln ŵi – and inverting yields:

ln Ŷi = β (1 + α)

α ln ω̂

1 β

i − 1 + α

α (σ − 1) ln P̂σ−1i −

1

α (σ − 1) ln p̂i

1−σ

ln L̂i = β

α ln ω̂

1 β

i − 1

α (σ − 1) ln P̂σ−1i −

1 −α α (1 + α) (σ − 1)

ln p̂i 1−σ,

which allows us to re-write equations the 4N equations in the system (39)-(42) as a function

of only 4N unknowns:

( ω̂

1 β

i

)β(1+α) α (

P̂σ−1i

)− 1+α α(σ−1) (

p̂i 1−σ)− 1α(σ−1) = ∑

j

πijT̂ijp̂i 1−σP̂σ−1i (43)(

P̂σ−1i

)−1 = ∑ j

χjiT̂jip̂j 1−σ (44)

( ω̂

1 β

i

)β α (

P̂σ−1i

)− 1 α(σ−1) (

p̂i 1−σ)− 1−αα(1+α)(σ−1) = ∑

j∈S

ρjiM̂jiω̂ 1 β

i Ŵ −1 β

j (45)

( Ŵ −1 β

j

)−1 = ∑ j

υijM̂ijω̂ 1 β

j , (46)

and a normalization on prices. These four equations can then be solved to determine the

effect of the counterfactual on all endogenous variables.

5.3 An Example: The Interstate Highway System

Finally, we illustrate the procedure above with a counterfactual based on Allen and Arkolakis

(2014): the effect of the construction of the Interstate Highway System (IHS). In particular,

we consider five different versions of that counterfactual with different assumptions regarding

the mobility of goods and labor: first, we consider a “trade” model where labor is fixed in each

location and the movement of goods is costly; second, we consider the reverse where each

location is in autarky (trade costs are infinite) but migration is possible (but costly); third,

we consider an “economic geography” model where trade is costly but the movement of labor

is costless; fourth, we consider a “labor” model where trade is costless but the movement of

labor is costly; and finally we consider the full model where both labor and trade are mobile

21

but subject to bilateral frictions.

For each version of the model, we use data on the wage and population from the 2000

Census as in Allen and Arkolakis (2014) and invert the model using the methodology dis-

cussed in Section 5.1 to recover the unique set of variables for each location so that the model

exactly matches the data given assumptions on labor and goods mobility.10 (For models with

labor mobility, we assume that the population in the 2000 Census is the steady state popu-

lation). We use the estimates of the removal of the IHS from Allen and Arkolakis (2014) to

construct T̂ij and M̂ij (assuming migration costs and trade costs are affected equally) and

apply the counterfactual procedure from Section 5.2 to calculate the change in populations

L̂i and real wages ŵi P̂i

in each location. For all simulations, we set σ = 4, α = 0.1, and

β = 0.25, which are (roughly) in line with estimates from the existing literature, see e.g.

Simonovska and Waugh (2014) and Rosenthal and Strange (2004).

Figures 18 through 22 present the results; each figure shows the spatial distribution in the

change in population and real wage across all U.S. counties by decile, with blue indicating

the greatest decline and red indicating the greatest increase. The most important thing to

see from the simulations is that the assumptions regarding the mobility of goods and people

play a hugely important role in dictating the effects of the removal of the IHS. For example,

in the first counterfactual, there is zero correlation between the change in population and

the change in real wages – for the simple reason that the population is assumed to be

immobile (see Figure 18). Contrast this with the second counterfactual, where there is a

perfect correlation between the (log) change in population and the (log) change in real wages

– for the simple reason that the change in real wages is determined solely by the change in

the location population when each location is in autarky (see Figure 19). This is also true

when trade is costly but migration is costless (as is assumed in Allen and Arkolakis (2014)),

as welfare will be equalized across all locations (see Figure 20). However, when migration

is costly (and trade is possible), the correlation between the change in population and the

change in real wages is far from perfect, as Figures 21 and 22 highlight.

Table 1 depicts the pair-wise correlations in predicted changes in real wages and popu-

lations across the different model variants. It is somewhat surprising to see how varied the

predictions for population changes are: for example, the predictions of model variant #2

(no trade, costly migration) is strongly negatively correlated with the other model variants.

Even across the other model variants with labor mobility, the correlation in predicted labor

changes is as low as 0.74 (comparing costly trade, free migration to costly trade, costly mi-

10To ease computation, we approximate the model variants with “infinitely” costly trade or migration with very large (but finite) trade or migration costs, respectively. Similarly, we approximate model variants with “costless” trade or migration with very small (but positive) trade or migration costs, respectively.

22

gration). In contrast, the model variants (with the exception of variant #2) largely agree on

the change in real wages, with all pair-wise correlations exceeding 0.95.

6 Conclusion

In this primer, we present a review of the simple gravity framework that nests many leading

spatial economic models, succinctly incorporates a rich real world geography and matches a

number of important empirical patterns in the flow of people and goods. Using a set of newly

developed solution techniques and mathematical methods the model can be solved efficiently

and be used to conduct relevant policy experiments. Within this environment, the analysis

of the economic impact of the interstate highway network highlights the importance of the

assumptions on the mobility of labor and goods for evaluating infrastructure policies. We

expect that these new tools will further advance the way economists conduct the exploration

of space - Economic Science’s final frontier.

23

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Economics, 101, 148–167.

Redding, S. J., and E. A. Rossi-Hansberg (2017): “Quantitative Spatial Economics,”

Annual Review of Economics, 9(1).

Roback, J. (1982): “Wages, rents, and the quality of life,” The Journal of Political Economy,

pp. 1257–1278.

Rosenthal, S. S., and W. C. Strange (2004): “Evidence on the nature and sources of

agglomeration economies,” Handbook of regional and urban economics, 4, 2119–2171.

Rossi-Hansberg, E. (2005): “A Spatial Theory of Trade,” American Economic Review,

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pp. 259–284.

Simonovska, I., and M. E. Waugh (2014): “The elasticity of trade: Estimates and

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27

Figures and Tables

Figure 1: Across country gravity: Trade flows between countries over time

-1 0

-5 0

5 Lo

g Tr

ad e

flo w

s

-3 -2 -1 0 1 2 Log Distance

1950 2000

Notes : Data are from Head, Mayer, and Ries (2010). Only bilateral pairs with observed trade flows in both 1950 and 2000 are included. The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5 after partitioning out the origin- year and destination-year fixed effects.

28

Figure 2: Across country trade gravity: The gravity coefficient over time

-2 -1

.5 -1

-.5 0

G ra

vi ty

c oe

ffi ci

en t

1950 1960 1970 1980 1990 2000 2010 Year

Notes : Data are from Head, Mayer, and Ries (2010). This figure plots the estimated coef- ficient of distance in a trade gravity regression with origin-year and destination-year fixed effects over time. The bars indicate the 95% confidence interval, where the standard errors are two-way clustered by country of origin and country of destination.

29

Figure 3: Within country gravity: Trade flows between U.S. states

-5 0

5 Lo

g tra

de fl

ow s

-3 -2 -1 0 1 2 Log distance

2007 2012

Notes : Data are from 2007 and 2012 Commodity flow surveys (CFS, 2007, 2012). The figure excludes trade flows within each state. The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5 after partitioning out the origin-year and destination-year fixed effects.

30

Figure 4: Location size and gravity fixed effects: Trade flows between countries

-2 -1

0 1

2 Lo

g G

D P

-4 -2 0 2 4 Origin fixed effect

Origin fixed effects and GDP

-2 -1

0 1

2 Lo

g G

D P

-4 -2 0 2 Destination fixed effect

Destination fixed effects and GDP

Notes : Data are from Head, Mayer, and Ries (2010). The country-year fixed effects are from a series gravity regression of trade flows on distanceXyear and origin-year and destination- year fixed effects. The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5 after partitioning out both a country and year fixed effect.

31

Figure 5: Location size and gravity fixed effects: Trade flows between U.S. states

6 8

10 12

14 Lo

g to

ta l s

al es

-10 -5 0 5 Origin fixed effect

2007 2012

Origin fixed effects and total sales

9 10

11 12

13 14

Lo g

to ta

l p ur

ch as

es

-4 -2 0 2 4 Destination fixed effect

2007 2012

Destination fixed effects and total purchases

Notes : Data are from 2007 and 2012 Commodity flow surveys (CFS, 2007, 2012). Fixed effects are from a gravity regression of trade flows on distance and origin and destination fixed effects within year. Total sales (purchases) are caculated by summing trade flows across all destinations (origins). The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5.

32

Figure 6: Origin and destination fixed effects: Trade flows between countries

-4 -2

0 2

D es

tin at

io n

fix ed

e ffe

ct

-4 -2 0 2 4 Origin fixed effect

Notes : Data are from Head, Mayer, and Ries (2010). The country-year fixed effects are from a gravity regression of trade flows on distanceXyear and origin-year and destination- year fixed effects for each year. The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5 after partitioning out both a country and year fixed effect.

33

Figure 7: Origin and destination fixed effects: Trade flows between U.S. states

-4 -2

0 2

4 D

es tin

at io

n fix

ed e

ffe ct

-4 -2 0 2 4 Origin fixed effect

2007 2012

Notes : Data are from 2007 and 2012 Commodity flow surveys (CFS, 2007, 2012). Fixed effects are from a gravity regression of trade flows on distance and origin and destination fixed effects within year. The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5.

34

Figure 8: Across country gravity: Migration flows between countries over time

-1 0

-5 0

5 10

Lo g

po pu

la tio

n flo

w s

-4 -2 0 2 Log distance

1960 2010

Notes : Data are from Yeats (1998). Excludes own country population shares (i.e. non- migrants). The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5 after partitioning out the origin-year and destination-year fixed effects.

35

Figure 9: Across country gravity: The migration gravity coefficient over time

-2 -1

.5 -1

-.5 0

G ra

vi ty

c oe

ffi ci

en t

1960 1970 1980 1990 2000 2010 Year

Notes : Data are from Yeats (1998). This figure plots the estimated coefficient of distance in a gravity migration regression with origin-year and destination-year fixed effects over time. The bars indicate the 95% confidence interval, where the standard errors are two-way clustered by country of origin and country of destination.

36

Figure 10: Within country gravity: Migration flows between U.S. states

-4 -2

0 2

4 Lo

g po

pu la

tio n

flo w

s

-1 0 1 2 Log distance

1850 2000

Notes : Data are from the 1850 and 2000 U.S. Censuses Ruggles, Fitch, Kelly Hall, and Sobek (2000), where migration flows are comparing current state of residence of 25-34 year olds to their state of birth. The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5 after partitioning out the origin-year and destination-year fixed effects.

37

Figure 11: Within country gravity: The migration gravity coefficient over time

-2 -1

.5 -1

-.5 0

G ra

vi ty

c oe

ffi ci

en t

1850 1900 1950 2000 Year

Notes : Data are from the U.S. Censuses from 1850 to 2000 Ruggles, Fitch, Kelly Hall, and Sobek (2000), where migration flows are comparing current state of residence of 25-34 year olds to their state of birth. This figure plots the estimated coefficient of distance in a gravity migration regression with origin-year and destination-year fixed effects over time. The bars indicate the 95% confidence interval, where the standard errors are two-way clustered by state of origin and state of destination.

38

Figure 12: Location population and gravity fixed effects: Migration flows between countries

-1 -.5

0 .5

1 Lo

g P

op ul

at io

n

-4 -2 0 2 4 Origin fixed effect

Origin fixed effects and Population

-1 -.5

0 .5

1 1.

5 Lo

g P

op ul

at io

n

-4 -2 0 2 4 Destination fixed effect

Destination fixed effects and Population

Notes : Data are from Yeats (1998). The country-year fixed effects are from a gravity regres- sion of migration flows on distanceXyear and origin-year and destination-year fixed effects. The thick lines are from a non parametric regression with Epanechnikov kernel and band- width of 0.5 after partitioning out both a country and year fixed effect.

39

Figure 13: Location size and gravity fixed effects: Migration flows between U.S. states

-3 -2

-1 0

1 2

Lo g

P op

ul at

io n

(p re

vi ou

s de

ca de

)

-3 -2 -1 0 1 2 Origin fixed effect

Origin fixed effects and Population

-2 -1

0 1

2 Lo

g P

op ul

at io

n

-2 -1 0 1 2 Destination fixed effect

Destination fixed effects and Population

Notes : Data are from the U.S. Censuses from 1850 to 2000 Ruggles, Fitch, Kelly Hall, and Sobek (2000), where migration flows are comparing current state of residence of 25-34 year olds to their state of birth. The state-year fixed effects are from a series gravity regression of migration flows on distanceXyear and origin-year and destination-year fixed effects . The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5 after partitioning out both a state and year fixed effect.

40

Figure 14: Origin and destination fixed effects: Migration flows between countries

-4 -2

0 2

4 D

es tin

at io

n fix

ed e

ffe ct

-4 -2 0 2 4 Origin fixed effect

Notes : Data are from Yeats (1998). The country-year fixed effects are from a gravity regres- sion of migration flows on distanceXyear and origin-year and destination-year fixed effects. The thick lines are from a non parametric regression with Epanechnikov kernel and band- width of 0.5 after partitioning out both a country and year fixed effect.

41

Figure 15: Origin and destination fixed effects: Migration flows between U.S. states

-2 -1

0 1

2 D

es tin

at io

n fix

ed e

ffe ct

-3 -2 -1 0 1 2 Origin fixed effect

Notes : Data are from the U.S. Censuses from 1850 to 2000 Ruggles, Fitch, Kelly Hall, and Sobek (2000), where migration flows are comparing current state of residence of 25-34 year olds to their state of birth. Fixed effects are from a gravity regression of migration flows on distanceXyear and origin-year and destination-year fixed effects. The thick lines are from a non parametric regression with Epanechnikov kernel and bandwidth of 0.5 after partitioning out both a state and year fixed effect.

42

Figure 16: Economic activity on a line: Trade costs

Notes : This figure shows how the equilibrium distribution of population along a line is affected by changes in the trade cost. When trade is costless, the population is equal along the entire line. As trade becomes more costly, the population becomes increasingly concentrated in the center of the line where the consumption bundle is cheapest. Source: Allen and Arkolakis (2014).

43

Figure 17: Economic activity on a line: Border costs

Notes : This figure shows how the equilibrium distribution of population along a line is affected by the presence of a border in the center of the line. As crossing the border becomes increasingly costly, the equilibrium distribution of population moves toward the center of each half of the line. Source: Allen and Arkolakis (2014).

44

Figure 18: Effect of removing the Interstate Highway System: No migration, costly trade

Change in Population

Change in Real Wage Notes : These maps show the predicted change in population and real wage in every county from removing the Interstate Highway System under the assumption that migration is in- finitely costly and trade is costly. The model is calibrated to match the observed population and income in each county from the 2000 Census. The effect of the IHS on trade costs are taken from Allen and Arkolakis (2014). The model parameters assumed are an elasticity of substitution (σ)of 5, a productivity spillover (α) of 0.1 and a disamenity spillover (β) of 0.25, yielding trade and migration elasticities of 4. The color of a county indicates its decile, with blue indicating the greatest decline and red indicating the greatest increase.

45

Figure 19: Effect of removing the Interstate Highway System: No trade, costly migration

Change in Population

Change in Real Wage Notes : These maps show the predicted change in population and real wage in every county from removing the Interstate Highway System under the assumption that migration is costly and trade is infinitely costly. The model is calibrated to match the observed population and income in each county from the 2000 Census. The effect of the IHS on migration costs are set equal to the change in trade costs estimated in Allen and Arkolakis (2014). The model parameters assumed are an elasticity of substitution (σ)of 5, a productivity spillover (α) of 0.1 and a disamenity spillover (β) of 0.25, yielding trade and migration elasticities of 4. The color of a county indicates its decile, with blue indicating the greatest decline and red indicating the greatest increase.

46

Figure 20: Effect of removing the Interstate Highway System: Costly trade, free migration

Change in Population

Change in Real Wage Notes : These maps show the predicted change in population and real wage in every county from removing the Interstate Highway System under the assumption that migration is costless and trade is costly. The model is calibrated to match the observed population and income in each county from the 2000 Census. The effect of the IHS on trade costs are equal to those estimated in Allen and Arkolakis (2014). The model parameters assumed are an elasticity of substitution (σ) of 5, a productivity spillover (α) of 0.1 and a disamenity spillover (β) of 0.25, yielding trade and migration elasticities of 4. The color of a county indicates its decile, with blue indicating the greatest decline and red indicating the greatest increase.

47

Figure 21: Effect of removing the Interstate Highway System: Free trade, costly migration

Change in Population

Change in Real Wage Notes : These maps show the predicted change in population and real wage in every county from removing the Interstate Highway System under the assumption that migration is costly and trade is costless. The model is calibrated to match the observed population and income in each county from the 2000 Census. The effect of the IHS on migration costs are equal to those estimated for the change in trade costs in Allen and Arkolakis (2014). The model parameters assumed are an elasticity of substitution (σ)of 5, a productivity spillover (α) of 0.1 and a disamenity spillover (β) of 0.25, yielding trade and migration elasticities of 4. The color of a county indicates its decile, with blue indicating the greatest decline and red indicating the greatest increase.

48

Figure 22: Effect of removing the Interstate Highway System: Costly trade, costly migration

Change in Population

Change in Real Wage Notes : These maps show the predicted change in population and real wage in every county from removing the Interstate Highway System under the assumption that both migration and trade are costly. The model is calibrated to match the observed population and income in each county from the 2000 Census. The effect of the IHS on both migration and trade costs are equal to those estimated for the change in trade costs in Allen and Arkolakis (2014). The model parameters assumed are an elasticity of substitution (σ)of 5, a productivity spillover (α) of 0.1 and a disamenity spillover (β) of 0.25, yielding trade and migration elasticities of 4. The color of a county indicates its decile, with blue indicating the greatest decline and red indicating the greatest increase.

49

T a b le

1 :

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n o f

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c t u a l

p r e d ic

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a c r o s s

d if

f e r e n t

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s p e c if

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P o p u la

ti o n

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co st

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sh o w

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50