Trade and the Topography of the Spatial Economy∗
Treb Allen

Costas Arkolakis

Northwestern

Yale and NBER

First Version: November 2012
This Version: February 2014

Abstract
We develop a general equilibrium framework to determine the spatial distributionof economic activity on any surface with (nearly) any geography. Combining the gravity structure of trade with labor mobility, we provide conditions for the existence,uniqueness, and stability of a spatial economic equilibrium and derive a simple set of
equations which govern the relationship between economic activity and the geographyof the surface. We then use the framework to estimate the topography of trade costs,
productivities and amenities in the United States. We find that geographic locationaccounts for at least twenty percent of the spatial variation in U.S. income. Finally,
we calculate that the construction of the interstate highway system increased welfare
by 1.1 to 1.4 percent, which is substantially larger than its cost.

∗

We thank George Alessandria, Lorenzo Caliendo, Arnaud Costinot, Dave Donaldson, Gilles Duranton,
Michal Fabinger, Pablo Fajgelbaum, Gene Grossman, Gordon Hanson, Johannes Horner, Sam Kortum, Kiminori Matsuyama, Stephen Redding, Andres Rodriguez-Clare, Esteban Rossi-Hansberg, Antonios Stampoulis,
and Jonathan Vogel for their helpful suggestions and comments. Xiangliang Li provided excellent research
assistance. This paper started while Allen and Arkolakis were visitors at the International Economics Section
at Princeton University; the hospitality of whom we gratefully acknowledge. All errors are our own.

The Topography of the Spatial Economy

11

Introduction

There exists an enormous disparity in economic activity across space. For example, in theyear 2000, the population density in McLeod County, MN was 26 persons/km2 and the
payroll per capita was $13,543, while in Mercer County, NJ the population density was 369
persons/km2 and the payroll per capita was $20,795 (MPC, 2011b). Many explanations forthis disparity focus on the characteristics of a location that affect either the productivity or
the amenity value of living there (e.g. climate, natural resources, institutions, etc).1 These
explanations ignore the role of geographical location: if the local characteristics of McLeod
County were identical to those of Mercer County, such explanations would imply that the
two locations should have the same economic activity. In contrast, the theoretical literaturein spatial economics developed over the past few decades emphasizes that, because tradeover space is costly, geographical location plays an important role by affecting how remote a
location is from economic activity elsewhere.
How much of the observed spatial disparity in economic activity is due to geographic
location? Unfortunately, the simplicity of the spatial structure postulated in most spatial
economic models has restricted their direct applicability to a set of stylized examples. In
this paper, we resolve this tension between theory and data by developing a new frameworkthat allows us to determine the equilibrium spatial distribution of economic activity on
any surface with (nearly) any geography. With this framework, we perform a quantitative
empirical analysis to estimate the fraction of spatial inequality in incomes in the UnitedStates that is due to variation in trade costs arising from differences in geographic location.
Our theoretical framework relies on an economic and geographic component, which are
distinct but mutually compatible. The economic component combines the gravity structureof international trade with labor mobility to determine the equilibrium distribution of economic activity on a space with any continuous topography of exogenous productivity andamenity differences and any continuous bilateral iceberg trade costs.2 To incorporate thepossibility of productivity or congestion externalities, we allow for the overall productivity
and amenity in a location to endogenously depend on its population (“spillovers”). Given thissetup, we show that the equilibrium conditions can be formulated as a set of integral equations, which allows us to apply a set of conventional mathematical theorems to characterize
1

The literature examining the factors contributing to the productivity of a location is immense, see e.g.
Sachs (2001), Acemoglu, Johnson, and Robinson (2002).
2
The idea of analyzing economic activity on a surface has a long tradition, see e.g. Beckmann (1952);
Beckmann and Puu (1985, 1990).

The Topography of the Spatial Economy

2

conditions for the existence, uniqueness, and stability of a spatial economic equilibrium. In
turn, this equilibrium system yields simple relationships between the endogenous economic
outcomes and the underlying geography of the surface and highlights the role that spillovers
play in determining the equilibrium spatial distribution of economic activity.
The geographic component provides a micro-foundation for the bilateral trade costs.
We suppose that there exists a topography of instantaneous trade costs over a surface.
The bilateral trade costs are then equal to the accumulation of these instantaneous trade
costs over the least-cost route. We use methods from differential geometry to characterize
the bilateral trade costs between any two points in space. Combining the economic andgeographic frameworks, we provide several stylized examples of the mechanisms of the modeland derive closed-form solutions to the equilibrium distribution of population for some simple
geographies (e.g. the line).3
Combining the economic and geographic components, we analyze the real world distribution of economic activity throughout the continental United States. We begin by estimating
the underlying geography – the bilateral trade costs, productivities and amenities – of theUnited States. To estimate the bilateral trade costs, we combine detailed geographic information on the rail, road, and water networks with mode-specific bilateral trade shares toinfer the relative cost of travel using different modes of transportation. The procedure isgreatly facilitated by the “fast marching method” algorithm borrowed from computational
physics, which allows us to efficiently compute the trade cost along the least-cost route fromall locations to all other locations. Given the trade costs, we then identify the unique topography of composite productivities and amenities that exactly match the observed spatial
distribution of wages and population given the structure of the model.
We then perform two exercises using the estimated geography of the United States. First,
we estimate what fraction of the observed variation in income can be explained by geographic
location. Because the model yields a log-linear relationship between the income of a locationand its productivity, amenity, and its price index (which is a sufficient statistic for geographic
location), we can apply a standard decomposition technique to determine how much of theobserved variation in income the price index can explain. The decomposition implies thatat least twenty percent of the spatial variation in income across the United States in theyear 2000 can be explained by geographic location alone. Second, we examine the effect ofremoving the Interstate Highway System. We estimate that without the Interstate Highway
3

There is a (small) literature which provides analytical characterizations of the equilibrium distributionof economic activity across space, e.g. Matsuyama (1999) and Fabinger (2011).

The Topography of the Spatial Economy

3

System, welfare would decline by 1.1 to 1.4 percent, suggesting that the benefits of theInterstate Highway System substantially outweigh the costs.
Our framework departs from the seminal economic geography model of Krugman (1991)(which is extensively analyzed in Fujita, Krugman, and Venables (1999)) in two importantways. First, we dispense with the assumption of a homogeneous freely traded good, thereby
allowing nominal wages to vary across space. Second, we depart from the tradition of amonopolistic competition structure, instead using a perfect competition Armington setup
with differentiated varieties as in Anderson (1979) and Anderson and Van Wincoop (2003).
Unlike much of this literature, rather than taking a stand on the source of production orcongestion externalities we incorporate such spillovers by simply assuming that productivity
and amenities may depend in part on the local population.4 While ad-hoc, this assumption
allows us to show that for particular strengths of spillovers, our model becomes isomorphic tomany other spatial economic models, including the free entry monopolistic competition setup
similar to the one considered by Krugman (1980) and Krugman (1991) and the fixed amenity
framework of Helpman (1998) and Redding and Sturm (2008). By showing how spillovers
affect the existence, uniqueness, and stability of the equilibrium as well as the relationshipbetween equilibrium economic activity and the underlying geography, our framework provides
a link between these (previously distinct) spatial theories.
Our model is also related to a large literature on urban development based on the framework of Roback (1982), as in Kline and Moretti (2014) and Diamond (2012). These papers
assume free labor and capital mobility and costless trade of a homogeneous commodity.
While our model relies on differentiated goods to provide a dispersion force, it turns out thatwhen trade is costless, the equilibrium conditions for our model is equivalent to versions of
the Roback (1982) model; hence, our framework can be interpreted as an extension of the
Roback (1982) framework to a world with costly trade.
The primary goal of the paper is to provide an empirically implementable framework to
study the role of economic geography. While there has been much empirical work examining
the implication of space for the allocation of agents (Davis and Weinstein 2002, 2008) andwages (Hanson 2005; Breinlich 2006; Head and Mayer 2006; Amiti and Cameron 2007), there
has been little empirical application of the extensive body of theoretical research on economic
geography. In recent exceptions to this rule, Redding and Sturm (2008), Redding (2012),
4

Unlike Rossi-Hansberg (2005), we restrict such spillovers to be local. For the examination of microfoundations of spillovers, see for example Lucas and Rossi-Hansberg (2003), Duranton and Puga (2004), andRossi-Hansberg and Wright (2007)

The Topography of the Spatial Economy

4and Ahlfeldt, Redding, Sturm, and Wolf (2012) use a quantitative framework to analyze thespatial distribution of economic activity. This paper follows in their tradition, and develops
a number tools to facilitate future quantitative analysis of economic geography.
Finally, our empirical work is related to the recent literature estimating the impact of
the transportation network on economic output. Donaldson (2012) and Cervantes (2012)
consider the impact of railroads in India and the US, respectively, when labor is immobile,
while Donaldson and Hornbeck (2012) consider the impact of the railroad network in the US
when labor is mobile. While we show how such transportation networks can be incorporated
in our framework, we can also incorporate geographical characteristics (e.g. mountains) that
do not have obvious network representations.
The remainder of the paper is organized as follows. The next section presents the theoretical framework and the third section presents the empirical analysis. The last section
concludes.

2

Theoretical framework

This section describes our theoretical framework. It comprises three subsections. We firstpresent the economic component of the framework, where we describe the equilibrium distribution of economic activity in a space with arbitrary trade costs. Second, we present thegeographic component of the framework, where we define and characterize geographic trade
costs that arise from moving goods across a surface. Finally, we combine the economic andgeographic components to characterize the equilibrium distribution of economic activity ona surface for several simple examples of the model.

2.1

Economic component

In this subsection, we present the economic component of our framework and characterize
the existence, uniqueness, and stability of a spatial equilibrium.
2.1.1

Setup

The world consists of a continuum of locations i ∈ S, where S is a closed bounded set ofa finite dimensional Euclidean space with the Euclidean norm as its metric.5 Each location
5

The continuum of locations is not important for much of what follows. In particular, as we discuss below,
Theorems 1 and 2 generalize for the case of discrete number of locations; however, Proposition 1 is only true

The Topography of the Spatial Economy

5i ∈ S produces a unique differentiated variety of a good. Trade is costly: trade costs areof the iceberg form and are described by the function T : S × S → [1, ∞), where T (i, j) is
the quantity of a good needed to be shipped from location i in order for a unit of a good toarrive in location j. We normalize T (i, i) = 1 for all locations.
The world is inhabited by a measure L̄ of workers who are freely mobile across locations
and derive utility from the consumption of differentiated varieties and the local amenity.
In particular, we assume workers have identical Constant Elasticity of Substitution (CES)
preferences over the continuum of differentiated varieties, so that the total welfare in locationi ∈ S, W (i), can be written as:ˆq (s, i)W (i) =

σ−1
σ

σ
 σ−1

ds

u (i) ,

s∈S

where q (s, i) 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 u (i) is the
local amenity.
Labor is the only factor of production. Each worker provides a unit of labor inelasticallyin the location where she lives, for which she is compensated with a wage. A worker inlocation i produces A (i) units of a good, where A (i) is the local productivity. Production isassumed to be perfectly competitive. We define the functions L : S → R+ and w : S → R++
to be the density of workers and their wage, respectively.
In order to allow for the possibility of productivity or congestion externalities, bothproductivity and amenities may depend on the density of workers. In particular, we assume
that overall (or composite) productivity in location i can be written as:
A (i) = Ā (i) L (i)α ,(1)where Ā (i) is the exogenous component of productivity inherent to location i and α ∈ Rdetermines the extent to which productivity is affected by the population density. Similarly,
we assume that the overall amenity in location i can be written as:
u (i) = ū (i) L (i)β ,(2)where ū (i) is the exogenous utility derived from living in location i inherent to the location
with a continuum of locations since it relies on the fact that a change in the population in one location does
not affect the price index.

The Topography of the Spatial Economy

6and β ∈ R determines the extent to which amenities are affected by the population density.
In what follows, we refer to α and β as governing the strength of productivity and amenity
spillovers, respectively. While we make no theoretical restrictions regarding α or β, in what
follows we focus on the empirically relevant cases of α ≥ 0 and β ≤ 0. It is important
to note that these spillovers are assumed to be local in nature (i.e. they do not affect the
productivity or amenities in nearby regions).
In Appendix A.2, we show how particular productivity and amenity spillovers make1
, our
our framework isomorphic to other spatial economic models. In particular, if α = σ−1model is isomorphic to a monopolistically competitive framework with differentiated varieties
and free entry, where the number of varieties produced in a location is proportional to its
population. The productivity spillover can be interpreted in this sense as an agglomeration
externality caused by more entry in markets with a larger size, as in the standard geography
setup of Krugman (1991).
1
Similarly, if α = σ−1and β = − 1−δ
, our model is isomorphic to the Helpman (1998)δ
Redding (2012) framework with 1 − δ being the budget share spent on an immobile factor,
e.g. land or housing. In this case, the value of β is negative capturing the inelastic supplyof land or housing and the resulting congestion externality through their increased prices.
In this case, the amenity spillover can be interpreted as also capturing the disutility ofhigher housing prices. The model is also isomorphic to a model where land is a factor of
production if α = 1 − δ, where δ is the share of labor in the Cobb-Douglass production
function, in which case the productivity spillover can be interpreted as also capturing thediminishing returns to labor in the production function. Finally, if β = − 1θ , our model isisomorphic to one where workers have heterogeneous preferences (drawn from an extreme
value distribution) for living in different locations, so that the amenity spillover can beinterpreted as the extent to which workers differ in their locational preferences. Notice thatthe isomorphisms we discuss above regards trade flows, wages, population and welfare, but
not necessarily other aspects of these models. Independently of their interpretation, thedegree of this agglomeration and dispersion externalities are crucial to guarantee uniquenessand existence of a spatial equilibrium, as we will discuss in detail below.
We define the geography of S to be the set of functions Ā, ū, and T , where Ā and ū
comprise the local characteristics and T comprises the geographic location. S is said to havea regular geography if Ā, ū, and T are continuous and bounded above and below by strictly
positive numbers. We define the distribution of economic activity to be the set of functions´
w and L, where we normalize S w (s) ds = 1.

The Topography of the Spatial Economy
2.1.2

7

Gravity

We first determine bilateral trade flows as a function of the geography of the surface, thewages, and the labor supply. The function X (i, j) expresses the value of bilateral trade flows
from location i to location j, where X : S × S → R+ . Using the CES assumption, and thefact that with perfect competition the final price of the good produced in location i and soldw(i)T (i, j), the value ofin location j is equal to the marginal production and shipping cost, A(i)location j’s imports from location i can be expressed as:X (i, j) =T (i, j) w (i)A (i) P (j)1−σw (j) L (j) ,(3)where P (j) is the CES price index with
ˆ
1−σ

P (j)

T (s, j)1−σ A (s)σ−1 w (s)1−σ ds.

=

(4)

S

2.1.3

Equilibrium

The CES assumption implies that the welfare of living in a particular location can be written
as an indirect function of the real wage and the overall amenity value:
W (i) =w (i)u (i) .
P (i)(5)

Welfare is said to be equalized if for all i ∈ S there exists a W > 0 such that W (i) ≤ W , withequality if L (i) > 0. That is, welfare is equalized if the welfare of living in every inhabited
location is the same and the welfare of living in every uninhabited location is no greater than
the welfare of the inhabited locations.
Markets are said to clear if the income is equal to the value of goods sold in all locations,i.e. for all i ∈ S :ˆ
w (i) L (i) =X (i, s) ds.

(6)

S

Given a regular geography with parameters σ, α, and β, we define a spatial equilibrium
as a distribution of economic activity such that (i) markets clear; (ii) welfare is equalized;and (iii) the aggregate labor market clears:

The Topography of the Spatial Economy8ˆ
L (s) ds = L̄.

(7)

S

In what follows, we pay particular attention to spatial equilibria with the following features.
A spatial equilibrium is said to be regular if w and L are continuous and every location is
inhabited, i.e. for all i ∈ S, L (i) > 0. A spatial equilibrium is said to be point-wise locally(i)< 0 for all i ∈ S. Intuitively, a point-wise locally stable equilibrium is one
stable if dWdL(i)where no small number of workers can increase their welfare by moving to another location.6
2.1.4

Existence, uniqueness, and stability

We now discuss sufficient conditions for the existence and uniqueness of regular spatial
equilibria. Using equations (3) to substitute out for trade flows and the indirect utility
function (5), we can write the market clearing condition (6) for all i ∈ S as:ˆ
L (i) w (i)σW (s)1−σ T (i, s)1−σ A (i)σ−1 u (s)σ−1 L (s) w (s)σ ds.

=

(8)

S

Combining the indirect utility function (5) with the price index (4) yields:
ˆ
1−σ

w (i)

W (i)1−σ T (s, i)1−σ A (s)σ−1 u (i)σ−1 w (s)1−σ ds.

=

(9)

S

When there are no productivity or amenity spillovers (i.e. α = β = 0 so that A (i) = Ā (i)and u (i) = ū (i)) and welfare is equalized so that W (i) = W for all i ∈ S equations (8) and(9) are linear operators whose eigenfunctions are L (i) w (i)σ and w (i)1−σ , respectively. Notethat the kernels of the two equations are transposes of each other. As a result, we have thefollowing theorem:
Theorem 1 Consider a regular geography with exogenous productivity and amenities. Then:i) there exists a unique spatial equilibrium and this equilibrium is regular; andii) this equilibrium can be computed as the uniform limit of a simple iterative procedure.
Proof. See Appendix A.1.1.
Equations (8) and (9) can be viewed as a linear system of equations for which extensionsof standard results in linear algebra guarantee the existence and uniqueness of a positive
6

This concept of stability is an adaptation of the one first introduced by Krugman (1991) to a continuumof locations.

9

The Topography of the Spatial Economy

solution. Part (ii) of the theorem guarantees that the equilibrium wages and population canbe calculated quickly without the need of a good prior guess.
When there are productivity or amenity spillovers and welfare is equalized, substituting
equations (1) and (2) into equations (8) and (9) yields the following two equations:ˆ
1−α(σ−1)L (i)σw (i) = W

1−σT (i, s)1−σ Ā (i)σ−1 ū (s) σ−1 L (s)1+β(σ−1) w (s)σ ds,(10)

T (s, i)1−σ Ā (s) σ−1 ū (i) σ−1 w (s)1−σ L (s)α(σ−1) ds.

(11)

S

ˆ
1−σ

w (i)

L (i)

β(1−σ)=W

1−σ
S

Equations (10) and (11) are a system of two non-linear integral equations, which have only
recently begun to be studied in the mathematics literature (see e.g. Yang and O’Regan(2005)). However, when bilateral trade costs are symmetric, i.e. T (i, s) = T (s, i) for alli, s ∈ S, it turns out that the system can be written as a single non-linear integral equation,
which will allow us to provide a simple characterization of the equilibrium system.7 To see
this, suppose that the left hand sides of equations (8) and (9) are equal up to scale:
L (i) w (i)σ A (i)1−σ = φw (i)1−σ u (i)1−σ ,(12)where φ > 0 is some scalar. Given equations (1) and (2) governing the strength of spillovers,
it is straightforward to show that if equation (12) holds, then any functions w (i) and L (i)satisfying equation (10) will also satisfy (11) (and vice versa). We prove in the subsequent
theorem that for any regular equilibrium, equation (12) is the unique relationship between
L (i) and w (i) such that equations (10) and (11) hold.
Substituting equations (12), (1) and (2) into either equation (10) or (11) yields (aftersome algebra):
ˆ
σ̃γ1

L (i)

= ū (i)(1−σ̃)(σ−1)σ̃(σ−1)Ā (i)

W

1−σ

1−σ

T (s, i)(1−σ̃)(σ−1)

Ā (s)ū (s)σ̃(σ−1)



L (s)

σ̃γ1

 γγ2
1

ds,

S

(13)
7

It is interesting to note that this method of reducing a system of non-linear equations into a single nonlinear equation when trade costs are symmetric can also be applied more generally to prove the existence
and uniqueness of the equilibrium of trade models where welfare does not necessarily equalize, see Allen and
Arkolakis (2013).

The Topography of the Spatial Economy

10

where
γ1 ≡ 1 − α (σ − 1) − βσ,γ2 ≡ 1 + ασ + (σ − 1) β,and σ̃ ≡ (σ − 1) / (2σ − 1).
Note that equation (13) characterizes the equilibrium distribution of labor as a function
only of the underlying geography of the surface; wages, in particular, do not enter. Equation(13) is a non-linear integral equation known as a homogeneous Hammerstein equation of
the second kind (see, e.g. p.807 of Polyanin and Manzhirov, 2008). If equation (13) has asolution for L (i) and W 1−σ then equilibrium wages can be determined from equation (12)using the aggregate labor clearing condition to determine the scalar φ. The next theorem
discusses the conditions for existence and uniqueness of spatial equilibria for γ1 6= 0.
Theorem 2 Consider a regular geography with overall productivity and amenity functions
specified in equations (1) and (2), respectively, and assume that iceberg trade costs are symmetric and parameters are such that γ1 6= 0. Then:i) there exists a regular spatial equilibrium;ii) if γ1 > 0, all equilibria are regular;iii) if γγ12 ∈ [−1, 1], the spatial equilibrium is unique; and if γγ12 ∈ (−1, 1], it can be computed
as the uniform limit of a simple iterative procedure.
Proof. See Appendix A.1.2.
Note that γγ21 ∈ [−1, 1] implies γ1 > 0, so that part (iii) holds only if part (ii) holds as well.
It is straightforward to show that if γ1 = 0 there is (generically) no regular spatial equilibrium
satisfying equations (10) and (11). Finally, the following proposition characterizes when aspatial equilibria is point-wise locally stable.
Proposition 1 Consider a regular geography with overall productivity and amenity functions specified in equations (1) and (2), respectively, and assume that iceberg trade costs are
symmetric and parameters are such that γ1 6= 0. Then if γ1 < 0, no regular equilibria is
point-wise locally stable, and if γ1 > 0, all equilibria are point-wise locally stable.
Proof. See Appendix A.1.4.

The Topography of the Spatial Economy

11

To get intuition for this result notice that when markets clear, the welfare of living in alocation can be written as:´
W (i) =

 σ1
1−σ
σ−1
γ1
σ−1
T
(i,
s)
P
(s)
w
(s)
L
(s)
ds
S
Ā (i) σ ū (i) L (i)− σ .
P (i)

(14)

The parameter γ1 is the partial elasticity of welfare with respect to the population in alocation. Expression 14 shows that if a small number of workers moves to a location, thewelfare in that location will decrease if and only if γ1 > 0.8
Figure 1 depicts the ranges of α ≥ 0 and β ≤ 0 and the different cases of equilibrium
uniqueness and stability with σ = 9 (a complete characterization for α, β ∈ R is presented
in Appendix A). The graph is divided in four regions with sufficient conditions on α andβ for uniqueness and stability. Focusing on the range where α ∈ [0, 1] and β ∈ [−1, 0],
we see that γγ21 ∈ [−1, 1] if and only if α + β ≤ 0, so there is a unique stable equilibrium
regardless of the economic geography as long as dispersion forces are at least as strong as
agglomeration forces. When α + β > 0 but is small, there exists an equilibrium that is stable(since γ1 > 0) but it need not be unique (since γ2 /γ1 > 1). We provide specific examples ofthe possible multiple equilibria below. However if α + β increases enough so that γ1 ≤ 0, theagglomeration forces are sufficiently strong that they can induce complete concentration ina single location, i.e. a black-hole. Black holes are the only possible equilibria when γ1 = 0;
however, if γ1 < 0, regular equilibria also exist (although they are not point-wise locallystable).9The existence and uniqueness results of the Theorem 1 and 2 generalize for a discrete
number of locations, as we discuss in the Appendix, in which case the set S is finite or
countable.10 However, with a discrete number of locations, stability has to be analyzed
in a case-by-case basis as in Fujita, Krugman, and Venables (1999) since a change in the
population in one location will affect the price index.
8

The fact that the competitive equilibrium exists and is stable means that the spatial impossibility resultof Starrett (1978) does not apply in our case. The difference arises from the fact that in our model, theproduction set of firms differs across locations because of the Armington assumption.
9
Notice that if α ≥ 0 and β ≤ 0 and γ1 < 0, the condition for uniqueness is not satisfied. However, there
do exist alternative configurations of α < 0 and β > 0 in which there is a unique point-wise locally unstableequilibrium; see Appendix A.
10Continuity extends to the discrete topology in a trivial way since any function´ in a discrete topology iscontinuous.
If S is finite or countable a Lebesgue integral can be considered and “ S ” is formally equivalentP
to “ S ”.

The Topography of the Spatial Economy
2.1.5

12

Equilibrium economic activity and the underlying geography

When trade costs are symmetric, equations (5) and (12) (along with equations (1) and (2)governing the strength of spillovers) imply that both wages w (i) and population L (i) canbe written as log linear functions of the local characteristics and the price index:
γ1 ln w (i) = Cw − β (σ − 1) ln Ā (i) − (1 − α (σ − 1)) ln ū (i) + (1 + (σ − 1) (β − α)) ln P (i) ,(15)
γ1 ln L (i) = CL + (σ − 1) ln Ā (i) + σ ln ū (i) + (1 − 2σ) ln P (i) ,(16)

where the scalars Cw and CL are determined by the wage normalization and the labormarket clearing, respectively. Equations (15) and (16) provide three important implications
regarding the relationship between the equilibrium distribution of economic activity and thegeography of the space. First, because bilateral trade costs only appear in the price index,
the price index is a sufficient statistic for geographic location. Second, as long as γ1 > 0, thepopulation will be higher in locations with high exogenous productivities and amenities, andlower in locations with higher price indices. In contrast, the equilibrium wages will decreaseas the underlying productivity increases and may increase or decrease as the exogenous
amenity of a location or price index increases, depending on the signs of 1 − α (σ − 1) and1 + (σ − 1) (β − α), respectively. Third, conditional on the price index, productivity andamenity spillovers only change the elasticity of the equilibrium distribution of economic
activity to the underlying geography. If γ1 > 0, stronger spillovers (i.e. larger α or β)
result in the equilibrium distribution of population becoming more sensitive to underlying
geographic differences.11

2.2

Geographic component

In this subsection, we present a micro-foundation for the bilateral trade cost function by
assuming that bilateral trade costs are the total trade costs incurred traveling from an origin
to a destination along the least-cost route.
Suppose now that S is a compact manifold in RN .12 In what follows, we focus on the onedimensional cases where S is a finite line or a finite circle and the two-dimensional case where
11

Unless trade costs are zero, the strength of productivity and amenity spillovers will also affect theequilibrium distribution of population through general equilibrium effects on the price index.
12
A manifold is a topological space that is locally Euclidean, or intuitively, a space that can be ‘charted’
in the Euclidean space.

The Topography of the Spatial Economy

13

S is a finite plane, although the following results hold for any finite-dimensional manifold.
Let τ : S → R+ be a continuous function where τ (i) gives the “instantaneous” trade cost
incurred by crossing point i ∈ S. Define t (i, j) be the solution to the following least-cost
path minimization problem:ˆ 1
t (i, j) = infg∈Γ(i,j)τ (κ (t))
0dg (t)
dt,
dt

(17)

where g : [0, 1] → S is a path and Γ (i, j) ≡ {g ∈ C 1 |g (0) = i, g (1) = j} is the set of all
possible continuous and once-differentiable paths that lead from location i to location j.
The notation k·k stands for the Euclidean norm. If the bilateral trade cost function T issuch that for all i, j ∈ S, T (i, j) = f (t (i, j)), for some monotonically increasing function
with f (0) = 1, we say that the bilateral trade costs are geographic. Note that when bilateral
trade costs are geographic, there exists a unique mapping from the instantaneous trade costfunction τ (which has a domain of S) to the bilateral trade costs T (which has a domain
of S × S), so that assuming trade costs are geographic reduces the dimensionality of the
problem by its square root.
Geographic trade costs provide a flexible means of approximating the true costs associated with moving goods across space. Transportation networks such as roads and railroad
can be incorporated by assuming that the instantaneous trade costs are lower where roads
or railroads exist. Borders can be incorporated by constructing (positive measure) “walls”
between regions where the instantaneous trade costs are large; such “walls” can also be placedalongside roads or railroads to so that they are accessible at only a finite number of entrance
ramps or stations. The instantaneous trade costs can also reflect differences in natural geography, such as ruggedness, water, etc. Two properties of geographic trade costs deserve
special mention. First, because traveling over a particular point i ∈ S incurs the same cost
regardless of the direction of travel, geographic trade costs are symmetric, i.e. for all i, j ∈ S,
T (i, j) = T (j, i) . Second, because the topography of the surface is smooth, nearby locations
will face similar trade costs to all other destinations. Formally, for all s, i, j ∈ S, we havelims→i T (s, j) = T (i, j). While we believe these are attractive properties for trade costs
arising from transportation costs, they abstract from alternative sources of trade costs, e.g.
origin-specific tariffs or information frictions (see e.g. Allen (2012)). We will allow for suchnon-geographic trade costs when we estimate the total bilateral trade costs for the UnitedStates in Section 3.
Equation (17) is a well studied problem that arises in a number of fields. For any origin

The Topography of the Spatial Economy

14i ∈ S and destination j ∈ S, its solution is characterized by the following eikonal partial
differential equation (see e.g. Mantegazza and Mennucci (2003)):||∇t (i, j) || = τ (j) ,(18)

where the gradient is taken with respect to the destination j.
Because we care only about the total bilateral trade costs (rather than the actual leastcost route), for our purposes it suffices to focus on the set of iso-cost contours, i.e. the setof curves defined by the set of destinations {j|t (i, j) = C} for all C. Equation (18) implies
that as C increases, the iso-cost contour expands outward at a rate inversely proportional tothe instantaneous trade cost in a direction that is orthogonal to the contour curve. Hence,the evolution of the contour of the bilateral trade costs is equivalent to the propagation of awave front outward from the origin along the surface at a speed inversely proportional to theinstantaneous trade cost. Intuitively, when instantaneous trade costs are large, the iso-cost
contour expands more slowly, reflecting the fact that a given increase in distance results ina larger increase in the total geographic trade costs.
For any initial point i ∈ S, it is possible to determine the bilateral trade costs to allother destinations j ∈ S using a simple iterative procedure based on the eikonal equation(18). Given any contour set {j|t (i, j) = C}, we can construct for each j ∈ {j|t (i, j) = C}εand normal to the iso-cost contour. By connecting the ends
a vector from j of length τ (j)of these vectors, we arrive at a new contour set {j 0 |t (i, j 0 ) = C + ε}. Figure 2 illustrates
the propagation process. By starting from an arbitrarily small contour around i, we canapply this process iteratively to determine the complete set of iso-cost contours and hence
the bilateral trade cost from i to all destinations j ∈ S. This algorithm is known as the Fast
Marching Method (FMM) (see Sethian (1996, 1999)).
The FMM relies on the fact that because the instantaneous trade costs are positive
everywhere, the bilateral trade costs will only increase as you move away from the origin.
As a result, subsequent contours can be constructed using only the immediately previous
contour. This has a number of implications. First, the FMM is extremely efficient, with arun time of O (n log n), where n is the number pixels approximating the instantaneous trade
cost function τ . Practically speaking, even with high resolution images of τ , the FMM takes
less than a second to determine the distance from any i to all j ∈ S (however, because FMM
has to be run separately for every origin, determining trade costs from all locations to allother locations can take a couple hours at high resolutions).

The Topography of the Spatial Economy

15

Second, the FMM bears a close resemblance to the Dijkstra algorithm used to calculate
shortest paths over graphs, which also relies on an outward expansion from the origin. Indeed,
the FMM can be interpreted as a generalization of Dijkstra to continuous spaces: bilateralcosts can determined by approximating a surface with a grid (i.e. a network) and taking
the appropriate weighted average over different paths along the grid (see Tsitsiklis (1995)).
However, it is important to note that applying the Dijkstra algorithm directly using a grid
to approximate the space will not result in accurate bilateral distances because of so-called“digitization bias.” Digitization bias arises because any chosen grid necessarily restricts
the possible directions of travel, biasing estimated distances upwards, where the bias issystematically correlated with how different the optimal path is from the allowed directionsof travel over the grid (see e.g. Mitchell and Keirsey (1984)).
Third, the FMM can be easily generalized to allow for the direction of travel to affecttrade costs, allowing it incorporate such physical realities like elevation changes or one-way
roads, for example. This is because only two pieces of information are required to determine
the vector at a point j ∈ {j|t (i, j) = C} used to construct the subsequent iso-cost contour:
1) the slope of the current iso-cost contour (which determines the direction of the vector);and 2) the instantaneous trade cost (which determines the length of the vector). Because
the direction of the vector does not depend on the instantaneous trade cost, we can simply~~
allow the instantaneous trade to depend on the direction of travel d, i.e. τ i, d . We provide
a simple example of the direction of travel mattering in Section 2.3 below. Note, however,that if instantaneous trade costs are affected by the direction of travel, total bilateral trade
costs will no longer be symmetric as Theorem 2 requires.
For the rest of the analysis we will use a specific formulation for the geographic costs:
T (i, j) = et(i,j) . This exponential form has the interpretation that the instantaneous trade
costs are of iceberg form, as it is the limit of the product of many incremental iceberg costs as
the distance between the increments tends to zero.13 That is, the exponential form provides
a micro-foundation for why the total bilateral trade costs are of an iceberg form. However, it
can be shown (see the online appendix) that any log sub-additive monotonically increasing
function f such that f (0) = 1 will generate bilateral iceberg trade costs that are weakly
greater than one and satisfy the triangular inequality, i.e. T (i, j) ≤ T (i, k) T (k, j) for alli, k, j.
13

´b

In other words, e a τ (x)dx =Qba (1 + τ (x) dx), whereQb

a denotes a type II product integral.

The Topography of the Spatial Economy

2.3

16

ExamplesIn this subsection, we present solutions for two simple manifolds when trade costs are geographic: the line and the circle. These two cases help us to illustrate the different types of
equilibria that may arise and discuss their stability properties.
The line
Let S be the [−π, π] interval and suppose that α = β = 0 and Ā (i) = ū (i) = 1, i.e. there
are no spillovers and all locations have homogeneous exogenous productivities and amenities.
Suppose that instantaneous trade costs are constant, i.e. τ (i) = τ for all i ∈ S apart from aborder b in the middle of the line; that is, trade costs between locations on the same side ofthe line are T (i, s) = eτ |i−s| and those on different sides are T (i, s) = eb+τ |i−s| .14 While the
T function in this case is discontinuous, so that the sufficient conditions of Theorems 1 and
2 are not satisfied, we can still obtain a unique explicit solution.
Taking logs of equation (16) and differentiating yields the following differential equation:∂ ln P (i)∂ ln L (i)= (1 − 2σ)
.
∂i
∂i

(19)

It is easy to show that ∂ ln P∂i(−π) = −τ and ∂ ln∂iP (π) = τ in the two edges of the line and∂ ln P (0)= τ 1 − e(1−σ)b / 1 + e(1−σ)b in the location of the border which gives us boundary∂i
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 locationsby τ , thereby reducing the (log) price index by τ . To obtain a closed form solution toequation (19), we differentiate equation (13) twice to show that the equilibrium satisfies thefollowing second order differential equation:∂2
L (i)σ̃ = k1 L (i)σ̃ for i ∈ (−π, 0) ∪ (0, π),
2
∂i

(20)

where k1 ≡ (1 − σ)2 τ 2 + 2 (1 − σ) τ W 1−σ . Given the boundary conditions above, the equilibrium distribution of labor for each line segment [−π, 0] and [0, π] is characterized by theweighted sum of the cosine and sine functions (see example 8.8.16 in Polyanin and Manzhirov
14

This border cost is reminiscent of the one considered in Rossi-Hansberg (2005). As in that model, ourmodel 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 adistinct differentiated variety) nor is there an amplification effect through spillovers (since spillovers are
assumed to be local).

The Topography of the Spatial Economy

17(2008)):p 
 p  σ̃1
L (i) = 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.15 Notice thatin 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 theaggregate labor clearing condition directly solves for k2 .16
Figure 3 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, thepopulation 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 whereτ = 0, labor is equally allocated across space. With symmetric exogenous productivities andamenities, wages are lower in the middle of the line to compensate for the lower price index.
Figure 4 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 isinfinitely costly, it is as if the two line segments existed in isolation.
Differences in exogenous productivities, amenities and the spillovers also play a key rolein determining the equilibrium allocation of labor and wages. We use numerical methods tocompute these more general cases. Assume, for example, that there are no spillovers, butAĀ (i) = e σ−1 i . Then the differential equation becomes:∂ ln L (i)∂ ln P (i) ,= Ai + (1 − 2σ)∂i∂iso that the equilibrium distribution of population is shifted rightward when A > 0. Figure
5 depicts this reallocation of labor toward locations with higher productivity. In this case,it can be shown that an analytical solution of L (i) exists in terms of Bessel functions of thefirst and the second kind.
15More general formulations of the exogenous productivity or amenity functions result to more general
specifications of the second order differential equation illustrated above (see Polyanin and Zaitsev (2002)section 8.1 for a number of tractable examples).
16
Mossay and Picard (2011) obtain a characterization of the population based on the cosine function in amodel 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 isregular when agglomeration forces are not too strong, as discussed in Theorem 2.

The Topography of the Spatial Economy

18

A different result is obtained if we increase the parameter α that regulates productivity
spillovers, but leave exogenous productivities homogeneous. As mentioned in the previous
subsection, as long as γ1 > 0, this change increases the elasticity of the labor supply tochanges in the geography, which increases the concentration of population in the alreadyhighly populated locations. Figure 6 depicts the population for higher values of α, andthe resulting increase in the concentration. Notice that further increases in α, to the pointthat γ1 < 0, results in a completely different regular spatial equilibrium where most of the
population is concentrated at the two edges of the line. This equilibrium, however, is not
locally point-wise stable, as a small number of workers could move from the edges to thecenter and become better off.
Finally, we can consider what would happen if the instantaneous trade costs depended
on the direction of travel. Suppose that the cost of traveling to the right on the line is τrwhile the cost of traveling to the left on the line is τl , where τr ≥ τl . Figure 7 illustrates
that is it becomes increasingly costly to travel to the right relative to travel to the left, theequilibrium distribution of the population shifts leftward, where the price index is lowest.
The circle
The example of the circle illustrates the possibility of multiplicity of spatial equilibria.
Figure 8 shows the cases α + β = 0 (left panel) and α + β > 0 (right panel). When α + β = 0
there is a unique equilibrium with symmetric population across all locations. This remains anequilibrium when α + β > 0, but there are also (a continuum of) additional equilibria, where
any location on the circle could be the one where economic activity is more concentrated.
Thus, γ1 = 1, which corresponds to α + β = 0, is a bifurcation point that moves us froma parameter space with a unique spatial equilibrium to one with a continuum of equilibria.
When α + β > 0 higher trade costs may act as an additional agglomeration force, favoring
differentially regions with already concentrated economic activity.17
It is possible to obtain a characterization of the equilibrium in a circle when two borders
are located into symmetrically opposite points on the circle. Using the methodology of
Fabinger (2011) we can obtain an approximation of the solution for the population function
using Fourier series for small values of the border. As expected, this approximation implies
that as the cost of the border increases population moves away from the border and its
details are provided in the online appendix.
17

If we further increase α + β to the point that the sign of γ1 turns negative we can only find numericallyone regular spatial equilibrium, which is again the symmetric one. This equilibrium is not point-wise locallystable, as increasing the population of any point in the circle increases the welfare workers living there.

The Topography of the Spatial Economy

19

In the line and circle examples above, trade costs act as an agglomeration force. However,in economic geography models such as those of Krugman (1991); Fujita, Krugman, andVenables (1999), trade costs also generate a dispersion effect by creating a home market
for the manufacturing sector. This effect arises from the presence of an additional sector
with zero trade costs. In the online appendix, we incorporate a second sector in our modeland show that in the case of a line, increasing trade costs in one sector will reduce theagglomeration of economic activity only if the trade costs in the other sector are sufficiently
small.

3

The topography of the real world spatial economy

In this section, we use the model developed in Section 2 to analyze the actual topography of
economic activity in the continental United States. The section is composed of three parts.
In the first part, we estimate the underlying geography of the United States. In the second
part, we determine the fraction of the observed spatial variation in income due to geographic
location. In the third part, we examine the welfare impact and the resulting redistribution of
economic activity arising from the construction of the Interstate Highway System. In what
follows, we assume the elasticity of substitution σ = 9, which, consistent with Eaton andKortum (2002), yields a trade elasticity of eight.18

3.1

Determining the real world geography

The goal of this subsection is to recover the underlying geography of the continental UnitedStates, namely the bilateral trade cost function T and the topography of exogenous productivities Ā and amenities ū. To do so, we proceed in two steps. We first estimate trade
costs using the observed transportation networks in order to best match the observed bilateral trade flows between locations. We then find the unique overall productivities A andamenities u that generate the observed distribution of wages and population given the trade
costs. Given particular values of α and β, we can then back out the underlying exogenous
productivities Ā and amenities ū.
To estimate the underlying geography of the U.S., we rely on different types of data which
we summarize here; see Appendix B for details. The first type of data is the complete high18

While this is toward the high end of the accepted range of international trade elasticities, because theelasticity of substitution is between products produced in different locations within a country, it seems
reasonable to assume it is higher than the elasticity of substitution of products across countries.

The Topography of the Spatial Economy

20

way, rail, and navigable water networks in the United States, which we collect from several
sources (NDC, 1999; CTA, 2003; NHPN, 2005). Figure 9 depicts the networks; the networks
are quite detailed and include the entire U.S. highway system (400,000 miles of interstates,
other highways and arterial roads), all railroads in the U.S. (approximately 140,000 miles),and all navigable waterways (approximately 190,000 miles). Using GIS software, we project
the transportation networks onto a 1032 × 760 pixel image of the United States, which we
use to construct the mode-specific instantaneous trade cost function.
The second type of data is bilateral trade flow data, which we take from 2007 Commodity
Flow Survey (CFS, 2007). The CFS is the primary source of within-U.S. domestic freightshipments and the only public source of commodity flow data by U.S. highways. It is collected
every five years as a part of the Economic Census and reports the value of trade flows between
each CFS area and every other CFS area by each mode of travel.19 We treat each CFS area
as a single location, and assign its location on the image of the United States using thelatitude and longitude of its centroid. In what follows, we focus on four modes of travel:road, rail, water, and air. The left panel of Figure 10 depicts how the share of each modeof travel varies with straight-line distance in the data. The vast majority of trade (in value
terms) in the United States is shipped via road; however, this fraction declines as distanceincreases.
The third type of data is county-level income and demographic characteristics, which
we take from the 2000 U.S. Census (MPC, 2011b). Figure 12 depicts the observed spatial
distribution of relative labor and wages. We treat each of the 3,109 counties in the contiguousUnited States as a distinct location and assign each a location on the image of the UnitedStates using the latitude and longitude of their centroid.
A few words are necessary regarding the assumption that each CFS area (in the estimationof trade costs) and each county (in the estimation of overall productivities and amenities) are
distinct locations. To calculate an equilibrium, it is necessary to approximate the continuous
space with a discrete number of locations. However, there is a trade-off in determining the
optimal size of each discrete location. The major advantage of a finer discretization (i.e.
more locations) is that the approximation of the continuous space solution improves. There
are two disadvantages of a finer discretization. The first is practical: the greater the numberof locations, the more computationally intense the problem; the second is conceptual: thesmaller each discrete location, the more egregious the assumptions regarding no commuting
19

The CFS micro-data, which is not publicly available, reports establishment level shipment data at thezip-code level; see Hillberry and Hummels (2008).

The Topography of the Spatial Economy

21

and no spatial productivity and amenity spillovers become.20 We feel that treating eachcounty as a distinct location provides a reasonable balance of the two trade-offs.
3.1.1

Step #1: Estimating trade costs

We first estimate the bilateral trade cost function T . The basic procedure is as follows: forany origin-destination pair, we apply the Fast Marching Method (FMM) algorithm to theobserved transportation network to get a (normalized) distance between the two locations
for each mode of travel (road, rail, water, and air). We then compare these mode-specificdistances to the observed mode-specific bilateral trade shares using a discrete choice framework to infer the relative geographic trade cost of each mode of travel. Given the structureof the discrete choice framework, we can combine these estimates to determine the total
geographic trade cost up to scale. Finally, we estimate the scale using the observed bilateral
trade levels and the gravity equation implied by the model. The last step has the advantage
of allowing us to incorporate proxies for non-geographic trade costs.
We begin by determining the normalized mode-specific distance between all locationsin the United States. Using the detailed transportation networks data detailed above, wecreate an instantaneous cost function τm : S → R++ , where locations i on the network areassigned a low value of τm and locations off the network are assigned a high value τm (seeAppendix B.3 for details). For any origin i ∈ S and destination j ∈ S and mode m ∈ M , wecan apply the FMM algorithm using τm to determine the normalized mode-specific distancedm (i, j). We normalize the scale of distance so that the cost of traveling the width of theUnited States would be one if there existed a straight-line route via a particular network.
We estimate the relative costs of trade across different modes of transport below.
Before proceeding, it is informative to note that simple reduced form regressions showthat the normalized mode-specific distances dm (i, j) do indeed appear to be capturing thecost of traveling via different modes of travel. Table 1 reports the results of regressions of themode-specific value of bilateral trade flows on the normalized mode-specific distances, conditional on origin and destination fixed effects. The log value of road shipments is strongly
negatively correlated with the log road distance (column 1), and remains so even conditionalon straight-line distance (column 2). Conditional on road distance, there is no statisticallysignificant relationship between road shipments and rail distance, while, increases in water
distance are actually associated with greater shipments via road (column 3), suggesting that
20
In the Online Appendix, we extend the model to allow for commuting and find similar estimated amenities
and productivities.

The Topography of the Spatial Economy

22

traders substitute across modes of transport. Similar patterns are present for shipments via
rail (columns 4-6) and water (columns 7-9), although the results are not as statistically significant, possibly because there are fewer observations and the different measures of distance
are highly correlated.21
We next determine the relative cost of trade across different modes of transport using a
discrete choice framework.While it would be possible to estimate travel cost parameters using
variation in bilateral trade levels across origins and destinations without using a discrete
choice framework, such a procedure would be subject to concerns about the endogeneityof the location of transportation networks (e.g. there exists a highway between Chicago
and New York because the two cities trade a large amount with each other). In contrast,
the discrete choice framework provides a method of estimating travel cost parameters using
mode-specific trade shares between a given origin and destination (e.g. what fraction of trade
between Chicago and New York occurs via rail rather than road). This procedure effectively
controls for the overall level of bilateral trade flows, mitigating endogeneity concerns.
Suppose for every origin i ∈ S and destination j ∈ S there exists a mass of identicaltraders who choose a particular mode of transport in order to minimize the trade costs incurred from shipping a unit amount from i to j. Suppose there are m ∈ {1, ..., M } modesof transport and the iceberg cost of trader t shipping goods from i to j using mode m is
exp (τm dm (i, j) + fm + νtm ), where τm is the mode-specific variable cost, fm is the modespecific that is fixed with respect to distance, and νtm is a trader-mode specific idiosyncratic
cost.22 Finally, suppose that νtm is distributed i.i.d. across traders and modes of trans−θxportation with a Gumbel distribution with shape parameter θ, i.e. Pr{ν ≤ x} = e−e .23−θ(Note that this implies Pr{eν ≤ x} = e−x , i.e. eν is distributed according to a Fréchet
distribution with shape parameter θ.)
Let πm (i, j) denote the fraction of trade shipped from i to j using mode of transportation
m. Given the distribution assumption of νtm , it is straightforward to show that:exp (−am dm (i, j) − bm ),πm (i, j) = P
k (exp (−ak dk (i, j) − bk ))
21

(21)

The results are similar if we constrain our analysis to only trade between metropolitan statistical areas
rather than all CFS areas; results available upon request.
22
While the introduction of a fixed cost violates the continuity assumption of Section 2, this is not apractical concern here because we consider only a discrete number of locations.
23
Our discrete choice framework bears a resemblance to the one presented in Lux (2011); in that framework,
there were a continuum of goods, where each goods had an idiosyncratic mode-specific transportation costs;
here, there is a single good but a continuum of traders and each trader is assumed to have an idiosyncratic
mode-specific transportation cost.

The Topography of the Spatial Economy

23

where am ≡ θτm and bm ≡ θfm . Given mode specific distances {dm } : M × S × S → R+ ,we can estimate {am } and {bm } using equation (21) by choosing {am } and {bm } such thatthe predicted mode-specific share of bilateral trade most closely match the observed modespecific trade shares. As is standard in discrete choice estimation, mode-specific trade sharesare invariant to a multiplicative shifter on the trade costs. To pin down the relative scale,
we assume that traders do not incur a fixed cost of traveling via road. We then estimate{am } and {bm } from equation (21) using a non-linear least squares routine.
Given our estimates of {am } and {bm }, we can estimate total bilateral trade costs using
the observed level of bilateral trade flows. From the discrete choice framework, the averagegeographic trade cost incurred in trading from i to j, Tg (i, j), is:
!− θ1
  X
1
1
(exp (−am dm (i, j) − bm )).
Tg (i, j) = Γ
θ
θ
m

(22)

Suppose that total trade costs T are a composite function of geographic trade costs Tg andnon-geographic trade costs Tng , where the latter can be approximated by a vector of nongeographic bilateral observables C (i, j), e.g. similarity in language and ethnicities. Taking
logs of equation (3) and substituting in the functional forms of Tg and Tng yields the following
gravity equation:ln Xij =σ − 1 X
exp −âm dmij − b̂m + (1 − σ) β 0 ln Cij + δi + δj + εij ,
ln
θ
m

(23)

where the notation indicates that that we observe a finite number of bilateral trade flows.
Hence, given an elasticity of trade σ, we can estimate θ (which thereby determines {τm }and {fm }) and β (which determines the non-geographic trade costs). Note that varying theelasticity of trade will simply scale the estimates accordingly; this is the well known result(see e.g. Anderson and Van Wincoop (2003)) that observed trade flows are matched equallywell with a high trade elasticity and a low level of trade costs or vice versa.
Table 2 reports estimated values of the mode-specific variable and fixed trade costs {τm }and {fm }, the estimated shape parameter θ, and the effect of each non-geographic observable
on trade costs. Because the estimation procedure is a multiple-stage process, we calculatebootstrapped standard errors derived from re-doing the entire estimation procedure 1,000
times. We do the estimation for trade flows between all CFS areas as well as trade flows
only between metropolitan statistical areas (MSAs) (or subsets thereof). While the former

The Topography of the Spatial Economy

24

sample has more observations, the latter sample corresponds more closely with our theoreticalconception of a location; reassuringly, the results in the two samples are very similar.
The right panel of Figure 10 depicts how the estimated mode-specific costs vary with
distance. Given that the vast majority of trade occurs over roads, it is not surprising thattravel via roads is always estimated to have the lowest cost, regardless of distance. As
distance between origin and destination increases, however, the cost of travel via air, water,and rail decline relative to travel via road, which is consistent with the declining shareof trade occurring via road with distance. Overall, the magnitude of the trade costs isroughly consistent with estimates of domestic trade costs in the literature (e.g. Anderson and
Van Wincoop (2004) estimate an iceberg trade cost of 55 percent for domestic distribution
costs in a representative rich country). The estimated non-geographic trade costs also appear
reasonable. Trade costs are estimated to be approximately 30 percentage points lower when
the origin and destination are in the same state, and a 10% increase in the ethnic similarity
between an origin and destination is associated with a 9% decline in trade costs. Somewhat
surprisingly, trade costs are estimated to increase with the similarity in languages between
the origin and destination, although this effect is not statistically significant when the sample
only includes MSAs.
How well do the estimated trade costs predict trade flows? The top panel of Figure 11
compares the bilateral trade flows predicted by the estimated trade costs to those observedin the CFS. Overall, the predicted trade flows can explain 65% of the observed variationin trade flows, and there does not appear to be any systematic bias in the estimates withthe observed volume of trade shares. The bottom panel of Figure 11 shows that there is
not any systematic relationship between the predicted and actual trade flows and distance,suggesting that the assumed exponential relationship between trade costs and distance is areasonable approximation.
3.1.2

Step #2: Identifying productivities and amenities

Suppose we observe trade costs and the equilibrium distribution of economic activity. Can
we identify the underlying topography of overall productivities and amenities? The following
theorem guarantees that for any observed distribution of economic activity, there exists a
unique topography of overall productivities and amenities that generate that equilibrium.
Theorem 3 For any continuous functions w and L and continuous symmetric function T ,all bounded above and below by strictly positive numbers, there exists unique (to-scale) positive

The Topography of the Spatial Economy

25

and continuous functions A and u such that w and L comprise the regular spatial equilibrium
for the geography defined by T , Ā = AL−α and ū = uL−β .
Proof. See Appendix A.1.5.
It is important to note that Theorem 3 does not rely upon the assumed relationships
governing spillovers in equations (1) and (2); hence, the theorem applies for any strength orsource of spillovers, including for example spillovers that occur across space. In general, if therelationship between the strength of spillovers and the population distribution is known, thenbecause the distribution of labor is observed, the underlying productivities and amenities canbe determined by inverting the relationship given A and u. Given our assumed functional
form of spillovers, Ā and ū can be identified given L, α and β using equations (1) and (2).
The converse of this is that the strength of spillovers (in our case, α and β) cannot be
identified from the observed cross-sectional distribution of wages and population: for any
α and β, unique functions Ā and ū can be chosen to generate the composite productivities
necessary to generate the observed equilibrium distribution of economic activity.24
Intuitively, the identification of composite productivities and amenities from the observed
distribution of population and welfare works as follows. Consider two points i and j ∈ Swith the same geographic locations, i.e. T (i, s) = T (j, s) for all s ∈ S. Because the twopoints have the same geographic location, they share the same price index, which implies
the (observed) ratio of their nominal wages is equal to the ratio of their real wages. Because
welfare is the same in both locations, it must be the case that any difference in relative real
wages must be fully compensated by differences in amenities; hence the relative amenities are
simply the inverse of the relative wages. Similarly, because the two locations have the samegeographic location, differences in demand for their produce arises only because of differencesin their marginal costs of production, which depends only on wages and productivity. Frommarket clearing, income is equal to the total quantity sold, so the relative productivity of thetwo locations can be inferred by comparing the total income and wages in each location.25
Equations (1) and (2) simply extend this intuition to allow for differences in trade costs
across locations.
Using equations (1) and (2) and the bilateral trade costs estimated in the previous section,
we identify the unique composite amenities and productivities of each U.S. county in the24

Ellison and Glaeser (1997) make a similar point about the inability to disentangle the natural advantage
of a location from spillovers using cross-sectional data alone.
25
In particular, it is straightforward to show that market clearing implies A (i) /A (j) =1σσ((L (i) w (i) ) / (L (j) w (j) )) σ−1 .

The Topography of the Spatial Economy

26

year 2000 that are consistent with the observed distribution of labor and wages from the2000 U.S. Census. Figure 13 depicts the unique distribution of composite amenities andproductivities that are consistent with the estimated trade costs and the observed distributionof labor and population. Composite amenities are much lower in more populated counties,
while composite productivities are much higher. Figure 14 depicts the resulting exogenous
amenities and productivities when α = 0.1 and β = −0.3 (values which are roughly consistent
with the estimates of productivity spillovers from Rosenthal and Strange (2004) and the shareof income spent on housing BLS (2000)).26 The topography of exogenous productivities andamenities seem reasonable; amenities in southern Florida, southern California, and Arizona
are high, while amenities in the central of the United States are low; productivities arehighest along the eastern seaboard and in the upper Midwest and low in places like Montana,
Nebraska, and West Texas. Note that there is only a weak positive correlation (0.12) between
exogenous amenities and productivities.

3.2

Importance of geographic location

Given the estimated geography of the United States, we can determine the fraction of theobserved variation in incomes Y (i) ≡ w (i) L (i) that is due to the geographic location ofi ∈ S. To do so, note that combining equations (15) and (16) yields the following expression:
γ1
ln Y (i) = Cw + CL + (1 − β) ln Ā (i) + (1 + α) ln ū (i) − (2 + α − β) ln P (i) .
σ−1

(24)

Equation (24) provides a log linear relationship between the observed income in location i,
the exogenous productivities and amenities, and the price index. As reduced form evidence
that geographic location matters for the distribution of income in the United States, Figure
15 depicts the geographic variation in the estimated price index. There exists substantial heterogeneity in the price index; both within and across states, counties with better geographic
location (i.e. lower price indices) are wealthier.
26

Rosenthal and Strange (2004) summarize estimates for the increase of productivity when population
doubles of around 3-8%. We chose a roughly higher spillover term of α = 0.1 since our model is a perfect competition model that ignores the effects of entry on overall output, but as already discussed, these
additional spillovers map directly to a higher parameter α. Depending on whether one includes “housefurnishings and equipment” and “household operations” with “shelter”, The BLS (2000) reports that 18.7% to24.7% of household expenditure is spent on housing (18.7% when “shelter” and half of “housefurnishings andequipment” is included, 24.7% when all three categories are included). Given our isomorphism that implies
β = δ/ (1 − δ), where δ is the share of expenditure spent on housing, the range of relevant parameters for β
is 0.23 to 0.325; we choose a parameter of β = −0.3 for our baseline experiment.

The Topography of the Spatial Economy

27

To determine the relative contribution of the effect of local characteristics (i.e. Ā (i) andū (i)) and geographic location (i.e. P (i)) to the spatial dispersion of income, we apply a
Shapley decomposition (see Shorrocks (2013)) to equation (24). The Shapley decomposition
determines the expected marginal contribution of the local characteristics and the geographic
location to the total variation in observed incomes; intuitively, it provides a way of assigning
what fraction of the R2 of a regression is due to each set of explanatory variables. Because we
do not observe the strength of spillovers (i.e. α and β), but they are necessary to determine
amenities and productivities, we report the results of the decomposition for all combinationsof α ∈ [0, 1] and β ∈ [−1, 0].
It should be noted that if the trade cost function is mis-specified (and hence the priceindex ln P (i) is measured with error), the model would erroneously rely on amenity and productivity differences to explain observed differences in incomes, thereby biasing downwards
the estimated contribution of the price index. As a result, the contribution of the price index
should be considered a lower bound for the importance of the geography of trade costs in
explaining the differences in income across space.
Figure 16 reports the fraction of the spatial variation in income in the United States inthe year 2000 that can be attributed to geographic location rather than local characteristics.
While the exact value of the decomposition depends on the strength of spillovers, the decomposition suggests that at least 20% of the observed spatial variation in income is due togeographic location, and geographic location may be responsible for upwards of 70% of theobserved variation in income (if the spillovers are such that α = 0.23 and β = −0.14). Hence,
the results suggest that a substantial fraction of the spatial variation in incomes across theUnited States can be explained by variation in trade costs due to geographic location.

3.3

The effects of the Interstate Highway System

Given the estimated geography of the United States, we can also examine how changes to
the geography affect the equilibrium spatial distribution of population and wages and overall
welfare. This section provides an illustrative example of such counterfactual analysis byexamining what would happen if the Interstate Highway System (IHS) were removed.
The counterfactual procedure is straightforward. We first re-calculate the bilateral trade
cost function T using the estimates from Section 3.1.1 supposing that there were no interstate
highways, but keeping all other modes of transportation (including other national highways
and arterial roads) unchanged. For a given strength of spillovers α and β, we hold fixed

The Topography of the Spatial Economy

28

the exogenous productivities Ā and amenities ū at the values estimated in Section 3.1.2
and recalculate the equilibrium distribution of labor, wages and the overall welfare level
under these alternative trade costs using equations (12) and (13). Because the effect ofremoving the IHS will depend on the strength of spillovers, we do the counterfactual formany combinations of α ∈ [0, 1] and β ∈ [−1, 0] such that α + β ≤ 0, a restriction whichfrom Theorem 2 guarantees the uniqueness of equilibrium.
To illustrate the effect of the removal of the IHS on trade costs, Figure 17 presents therelative change in the price index (holding wages, population, and productivities fixed atobserved levels). As is evident, the price index rose the most in the Rocky Mountains,
indicating that locations there saw the greatest increase in economic remoteness, whereas
the price index in California and the Eastern seaboard increased by less. There are two
reasons for these differences: first, locations in California and the Eastern seaboard had better
alternative modes of transportation (see Figure 9); additionally, locations in California andthe Eastern seaboard purchased more goods from nearby locations (since a greater amountof production was concentrated nearby), so they relied less on the IHS. The importance ofthe latter effect, however, depends on how the spatial distribution of population (and hence
production) will endogenously change in response to changes in the trade costs. Figure 4
shows how removing the IHS changes the spatial distribution of the population. Consistent
with the fact that California and the Eastern Seaboard incur relatively small increase ineconomic remoteness, there is a redistribution of the population toward those locations andaway from the Rocky Mountains. However, the redistribution of population across spacedepends importantly on the strength of spillovers: when spillovers are absent (top map ofFigure 4), there is substantially less local variation of population changes than when α = 0.1
and β = −0.3.
Finally, Figure 19 presents the effect of the removal of the IHS on welfare for a large
number of different spillover strengths. Depending on the strength of spillovers, we estimatethat removing the IHS would result in a decline in welfare of between 1.1−1.4%.27 Given thisestimate of the welfare losses of removing the IHS, a simple back-of-the-envelope calculationsuggests that the benefits of the IHS substantially outweigh its costs. According to theCongressional Budget Office, the total cost of constructing the IHS was $560 billion (in 2007
dollars); assuming a 5% annual cost of capital, this amounts to roughly $28 billion a year
27

This estimated welfare loss arises only from the additional cost of trading goods. To the extent that the
IHS had other benefits (e.g. facilitating passenger travel), the welfare loss of removing the IHS would beeven greater.

The Topography of the Spatial Economy

29(CBO, 1982). The total cost of maintaining the entire highway system is approximately$130 billion a year (FHA, 2008; NSTIFC, 2009). If we assume that half of that expenseis spent on the IHS,28 this suggests the total annual cost of building and maintaining the
IHS is approximately $100 billion. In comparison, the U.S. GDP in 2007 was $14.25 trillion;
since preferences are assumed to be homothetic, if removing the IHS would decrease (static)
welfare by 1.1 − 1.4%, the model implies the monetary value of the IHS is between $150 and$200 billion 2007 dollars, suggesting an overall return on investment of at least 50%, or an) a year.
annualized return of at least 9% ( 150−100560

4Conclusion

We view this paper as taking a number of steps toward the rigorous quantification of spatial
theory. First, we develop a unified general equilibrium framework combining labor mobility, gravity, and productivity and amenity spillovers. Within this framework, we establish
conditions for the existence and uniqueness of a spatial equilibrium and derive relationships
between the equilibrium distribution of economic activity and the underlying geography.
Given the isomorphisms of our framework to multiple existing frameworks in the literature,
we see this as helping to, in the words of Duranton (2008), “provide a unified general equilibrium approach to spatial economics and end the often uneasy coexistence between urban
systems and the new economic geography.” Second, we provide a micro-foundation of trade
costs as the accumulation of instantaneous trade costs over the least-cost route on a surface.
We then develop tools to apply our framework to the analysis of detailed real world data on
spatial economic activity.
This framework could be extended to address a number of other questions, including:What is the optimal spatial taxation scheme in both the short-run and long-run? Whattransportation system maximizes social welfare? How would removing restrictions on crosscountry migration affect the equilibrium distribution of economic activity?
28

The IHS accounts for about one quarter of all passenger miles on the system, but the maintenance costs
are likely higher per passenger miles than other highways (Duranton and Turner, 2012).

The Topography of the Spatial Economy

30

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Tables and figures

36

Table 1: Commodity flows and mode-specific shipping distances
Dependent variable:
Road Shipments
Rail Shipments
Water Shipments
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Log road distance-1.369*** -0.945*** -1.362***-0.0610.371(0.015)(0.160)(0.141)(0.506)(3.028)
Log rail distance-0.083-0.457*** -0.382 -0.296-0.622(0.147)(0.056)(0.421) (0.515)(3.067)
Log water distance
0.083***-0.109 -0.730* -0.444 -0.498(0.032)(0.100) (0.349) (0.754) (0.911)
Log straightline distance-0.407***-0.069-0.313(0.154)(0.388)(0.718)
Origin Fixed Effect
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Destination Fixed Effect Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
R-squared
0.488
0.489
0.489
0.051
0.051
0.052
0.384
0.403
0.397
Observations
9177
9177
9177
1434
1434
1434
58
58
58
Notes: Ordinary least squares. Each observation is the observed (log) value traded froma CFS region to another CFS region in 2007 by a particular mode of transport. Standard
errors are reported in parentheses. Stars indicate statistical significance: * p<.10 ** p<.05*** p<.01.

The Topography of the Spatial Economy
37

Table 2: Estimated mode-specific relative cost of travel
Only MSAs
All CFS Areas
Geographic trade costs
Road
Rail
Water
Air
Road
Rail
Water
Variable cost
0.5636*** 0.1434*** 0.0779*** 0.0026
0.4542*** 0.1156*** 0.0628***(0.0233)(0.0210)(0.0265)(0.0120)(0.0063)(0.0199)(0.0085)
Fixed cost
00.4219*** 0.5407*** 0.5734*** 00.34***
0.4358***N/A(0.0235)(0.0375)
N/A(0.0097)(0.0236)(0.0129)
Estimated shape parame14.225***17.6509***ter (θ)(0.3375)(1.4194)
Non-geographic
trade
costs
Similar ethnicity-0.0888***-0.0803***(0.0275)(0.0153)
0.0286
Similar language
0.063***(0.0223)(0.0359)-0.0135
Similar migrants-0.0191(0.0119)(0.0203)-0.3104***
Same state-0.2984***(0.0101)(0.0176)
R-squared (within)
0.44870.4113R-squared (overall)
0.6456
0.5995Observations with posi- 9601
9601
9601
9601
3266
3266
3266
tive bilateral flows
Observations with posi- 9311
1499
78
1016
3144
340
26
tive mode-specific bilateral flows
Notes: This table shows the estimated cost of traveling via different modes of travel. Relative
costs are estimated using the mode-specific shares of total bilateral trade values from the 2007
Commodity Flow Survey. The mode specific bilateral cost is exp(variablecost ∗ distance +f ixedcost), where the distance of the width of the country is normalized to one. The shapeparameter (θ) is estimated from a gravity regression and pins down the scale of the variable
and fixed costs (see the text for details). Similarity in ethnicity, language, and migrants ismeasured by the correlation across respective census categories between the counties nearest
to the origin and destination CFS area. Boostrapped standard errors (from 1000 repetitions)
reported in parentheses.
471

3266

Air
0.0021(0.0176)
0.4621***(0.0264)

The Topography of the Spatial Economy
38

The Topography of the Spatial Economy

39

Figure 1: Equilibria with amenity and productivity spillovers

Notes: This figure shows the regions of values for the productivity spillover α and the amenity
spillover β for which there exists an equilibrium, for which there exists a point-wise locallystable equilibrium, and whether that equilibrium is unique. The elasticity of substitutions σ
is chosen to equal 9.

The Topography of the Spatial Economy

40

Figure 2: Propagation of geographic trade costs

Notes: This figure shows how the geographic trade costs evolve across a surface. Given a
contour of points on a surface such that the geographic trade cost to location i is equal to a
constant C (the solid line), for an arbitrarily small ε > 0, we can construct the contour line
for bilateral trade costs C + ε (the dashed line) by propagating the initial contour outwards
at a rate inversely proportional to the instantaneous trade cost.

The Topography of the Spatial Economy

41

Figure 3: Economic activity on a line: Trade costs

Notes: This figure shows how the equilibrium distribution of population along a line isaffected by changes in the trade cost. When trade is costless, the population is equal along theentire line. As trade becomes more costly, the population becomes increasingly concentrated
in the center of the line where the consumption bundle is cheapest.

The Topography of the Spatial Economy

42

Figure 4: Economic activity on a line: Border costs

Notes: This figure shows how the equilibrium distribution of population along a line isaffected by the presence of a border in the center of the line. As crossing the border becomesincreasingly costly, the equilibrium distribution of population moves toward the center of
each half of the line.

The Topography of the Spatial Economy

43

Figure 5: Economic activity on a line: Exogenous productivity differences

Notes: This figure depicts how the equilibrium distribution of population along a line isaffected by exogenous differences in productivity across space. With homogeneous productivities, and positive trade costs, the population is concentrated at the center of the line.
When productivity is higher toward the right, the population concentrates in regions to theright of the center of the line.

The Topography of the Spatial Economy

44

Figure 6: Economic activity on a line: Productivity spillovers

Notes: This figure shows how the equilibrium distribution of population along a line isaffected by varying degrees of productivity spillovers. As the productivity spillovers increase,the population becomes increasingly concentrated in the center of the line. A non-degenerate
equilibrium can be maintained as long as γ1 = 1 − α (σ − 1) − σβ > 0.

The Topography of the Spatial Economy

45

Figure 7: Economic activity on a line: Direction of travel

Notes: This figure shows how the equilibrium distribution of population along a line isaffected by instantaneous trade costs that depend on the direction of travel. As the costof traveling to the right becomes increasingly more expensive than traveling to the left, theequilibrium distribution of population shifts toward the left.

The Topography of the Spatial Economy

46

Figure 8: Economic activity on a circle: Multiple equilibria

Notes: This figure provides an example of multiple equilibria when the surface is a one
dimensional circle. The left panel shows the unique homogeneous distribution of population
along the circle when α + β = 0. When α + β > 0 (here α = 0.01 and β = 0), uniqueness is
no longer guaranteed. In the case of the circle, there are uncountably many equilibria, eachof which has an increased concentration of population around a different point of the circle.
The right panel depicts two such equilibria.

The Topography of the Spatial Economy

47

Figure 9: U.S. transportation networks

Notes: This figure shows each of the observed transportation cost networks. Interstate
highways are black, other U.S. highways are dark gray, and arterial roads are light gray.
Class A railroads are dark red, class B railroads are light red, and other railroads are pink.
Navigable waterways are blue.

48

The Topography of the Spatial Economy

Figure 10: Mode-specific bilateral trade shares by distance

Estimated trade costs

1

0

.2

1.2

Mode-specific share ofbilateral trade value
.6
.4

.8

Estimated iceberg trade cost
1.4
1.6
1.8

1

Observed trade shares

0

.2
.4
.6
.8
Normalized straight-line distance

0

.2
.4
.6
.8
Normalized straight-line distance

Road

Air

Road

Air

Rail

Water

Rail

Water

Notes: This figure shows the relationship between mode specific trade flows and distance.
The left panel shows how the share of bilateral trade (measured in value) by each mode oftransport varies with the straight-line distance between the origin and destination. Eachline is a non-parametric local mean smoothed regression using an Epanechnikov kernel witha bandwidth of 0.1. 99% confidence intervals are reported in grey. The right panel shows
how the estimated trade costs for each mode of transportation vary with distance. In bothpanels, distance is normalized so that the width of the United States has distance of one.

The Topography of the Spatial Economy

49

Figure 11: Assessing the predicted trade costs

Notes: This figure assesses the quality of the estimated trade costs. The top panel compares
the bilateral trade flows implied by the estimated trade costs with the bilateral trade flows
observed in the 2007 Commodity Flow Survey. The bottom panel shows that the difference
between the trade flows implied by the estimated trade costs and the observed bilateral tradeflows (i.e. the residuals) are uncorrelated with straight-line distance.

The Topography of the Spatial Economy

50

Figure 12: United States population density and wages in 2000

Population density

Wages
Notes: This figure shows the relative population density (top) and wages (bottom) within
the United States in the year 2000 by decile. The data are reported at the county level; red(blue) indicate higher (lower) deciles. (Source: MPC (2011a)).

The Topography of the Spatial Economy

51

Figure 13: Estimated composite productivity and amenity

Composite productivity

Composite amenity
Notes: This figure shows the estimated composite productivity (top) and amenity (bottom)
by decile. The data are reported at the county level; red (blue) indicate higher (lower)
deciles.

The Topography of the Spatial Economy

52

Figure 14: Estimated exogenous productivity and amenity

Exogenous productivity

Exogenous amenity
Notes: This figure shows the estimated exogenous productivity Ā (top) and amenity ū(bottom) by decile assuming α = 0.1 and β = −0.3. The data are reported at the county
level; red (blue) indicate higher (lower) deciles.

The Topography of the Spatial Economy

53

Figure 15: Estimated price index

Notes: This figure shows the estimated price index by decile. The data are reported at thecounty level; red (blue) indicate higher (lower) deciles.

The Topography of the Spatial Economy

54

Figure 16: Fraction of spatial inequality of income due to geographic location in the UnitedStates

Notes: This figure shows the fraction of the observed variation in income across space in the
U.S. in the year 2000 that is due to geographic location. The decomposition is calculated
for all constellations of productivity spillover strength α ∈ [0, 1] and β ∈ [−1, 0].

The Topography of the Spatial Economy

55

Figure 17: Estimated increase in the price index from removing the Interstate Highway
SystemNotes: This figure depicts the estimated increase in the price index (by decile) across spacefrom removing the Interstate Highway System (IHS), holding wages and productivities constant at the 2000 U.S. levels. Red (blue) indicate higher (lower) deciles (e.g. the removal of
the IHS disproportionately increased the economic remoteness in red regions).

The Topography of the Spatial Economy

56

Figure 18: Estimated change in the population from removing the Interstate Highway System

α = 0, β = 0

α = 0.1, β = −0.3
Notes: This figure shows the estimated change in population (in deciles) from the removalof the Interstate Highway System (IHS). The top map reports the estimated population
changes when there are no spillovers (i.e. α = β = 0), while the bottom map reports theestimated population changes when spillovers are chosen to approximately match those fromthe literature (i.e. α = 0.1 and β = −0.3). Red (blue) indicates higher (lower) deciles (e.g.
the removal of the IHS increased the relative population in red areas).

The Topography of the Spatial Economy

57

Figure 19: Estimated decline in welfare from removing the Interstate Highway System

Notes: This figure shows the estimated decline in welfare (in percentage terms) from the removal of the Interstate Highway System (IHS) for each combination of productivity spilloverstrength α ∈ [0, 1] and β ∈ [−1, 0] such that α + β ≤ 0.

The Topography of the Spatial Economy

A

58

Theory Appendix

This Appendix is composed of two subsections. In the first, we prove Theorems 1 and 2 andProposition 1 regarding the existence, uniqueness, and point-wise local stability of a spatial
equilibrium, as well as Theorem 3, regarding the identification of exogenous productivities
and amenities. In the second, we discuss the isomorphisms existing between our frameworkand other spatial economic models.

A.1

Proofs of Theorems

In this section, we prove the theorems in the main text. The proofs rely heavily on resultsfrom the study of integral equations, for which Zabreyko, Koshelev, Krasnosel’skii, Mikhlin,
Rakovshchik, and Stetsenko (1975) and Polyanin and Manzhirov (2008) are handy references.
The proofs of the theorems apply to compact intervals S ⊂ RN but for convenience we provide
references with results for connected and compact subsets of RN .
A.1.1

Proof of Theorem 1

Note that when α = β = 0, equation (10) can be written as:ˆg (i) = λK (s, i) g (s) ds,

(25)

S

where g (i) ≡ L (i) w (i)σ in unknown, K (s, i) ≡ T (i, s)1−σ A (i)σ−1 u (s)σ−1 is known, andλ ≡ W 1−σ is unknown. We can also re-write equation (11) in an identical form:ˆf (i) = λK (i, s) f (s) ds,

(26)

S

where f (i) ≡ w (i)1−σ is unknown, K (i, s) ≡ T (s, i)1−σ u (i)σ−1 A (s)σ−1 is the transpose ofK (s, i) , and λ ≡ W 1−σ is unknown.
Part (i) As mentioned in the text, equations (25) and (26) are eigenfunctions. Furthermore, because in both cases all components of the Kernel are continuous and bounded aboveand below by a positive number, each kernel K (s, i) is also continuous and bounded aboveand below by a positive number. As a result, by a generalization of Jentzsch’s theorem(see e.g. Theorem 3 of Birkhoff (1957) where S is a Banach lattice or p.648 of Polyanin

The Topography of the Spatial Economy

59

and Manzhirov (2008) where S is a connected interval of R.29 ), there exists a unique (toscale) strictly positive function g (i) and constant λ1 that solves equation (25) and a unique(to-scale) strictly positive function f (i) and constant λ2 that solves equation (26).
It remains to show that λ1 = λ2 . From one of the Fredholm Theorems (see Theorem
1.3 on p. 31 of Zabreyko, Koshelev, Krasnosel’skii, Mikhlin, Rakovshchik, and Stetsenko(1975)30 ), because the kernel of equation (26) is the transpose of the kernel equation (25),
λ1 is a characteristic value of equation (26) and λ2 is a characteristic value of equation (25).
In addition, notice that the constants λ1 and λ2 correspond to the smallest characteristic
values of equations (25) and (26), respectively.31 Suppose that λ1 > λ2 , we will arrive at acontradiction. In that case equation (26) has a characteristic value smaller than λ2 , which isa contradiction of Jentzsch’s theorem. Similarly, we get a contradiction if we assume λ2 > λ1 .
Therefore, λ1 = λ2 , so that there exists unique (to-scale), strictly positive functions g (i) andf (i) that solve equations (25) and (26). Because g (i) ≡ L (i) w (i)σ and f (i) ≡ w (i)1−σ ,wages and the labor supply can be determined (up to scale) immediately from g (i) and f (i).
To prove that the equilibrium is regular we need to argue that L (i) , w (i) are strictly
positive and continuous functions for all i. The proof that all regions are populated, and thus
L (i) , w (i) > 0, is given in the proof of Theorem 2 for any γ1 > 0 . The proof of continuity
is given in Part (ii) below.
Part (ii) The solution of the wages and the labor, up to scale, is the uniform limit of
the successive approximation´
K (i, s) fn (s) ds,
fn+1 (i) = ´ ´SK (i, s) fn (s) dsdi
S S

(27)

as shown by Birkhoff (1957), starting from an arbitrary guess of the function f0 (i). In
practice, we find that the convergence of equation (27) is rapid for both f (i) and g (i).
1Solving for equilibrium wages and the labor supply is then straightforward, as w (i) = g (i) 1−σσ
σ−1

and L (i) = L̄ ´ f (i)g(i) σ−1. Note that once convergence occurs, the normalization identifiesσ
S f (s)g(s)29dsNote that the compactness of S and the continuity of K (s, i) are sufficient but not necessary conditions
to apply Theorem 3 of Birkhoff (1957). Related to that, the boundeness of K (s, i) above and below by apositive number is a stronger requirement than the linear transformation of K (s, i) is uniformly bounded.
30
Note that because λ1 is real, the complex conjugate of λ1 , λ̄1 , is equal to λ1 (and likewise for λ2 andλ̄2 ).
31
See, for example, Krasnosel’Skii and Boron (1964) p.232. This statement derives from the results ofTheorems 2.11 and 2.13, p.78 and 81, with the required conditions on the kernel stated in Theorem 2.10,
page 76. The conditions require that the kernel is bounded above and below by a positive number, which
we have already assumed (see Theorem 2.2).

60

The Topography of the Spatial Economy´σ
W 1−σ , i.e. W 1−σ = L̄/ S f (s) g (s) σ−1 ds.
Notice that if we start with a continuous guess f0 (i) the operator (27) is continuous andthus {fn (i)}n∈N is a sequence of continuous functions. By the uniform convergence theorem
and the uniform limit result above the limit of this sequence is also continuous and thus
f (i) is also continuous. Since we proved that the equilibrium solutions are positive andcontinuous, we proved that the equilibrium is regular, completing the proof of the theorem.
A.1.2

Proof of Theorem 2

We first show that if there exists a regular spatial equilibrium, then equation (12) is theunique relationship between w (i) and L (i) that satisfies equations (10) and (11). Suppose
there exists a regular spatial equilibrium, i.e. there exists continuous functions w (i) andL (i) bounded above and below by positive numbers that satisfy equations (10) and (11).
Define the function φ : S → R+ as follows:
φ (i) ≡

L (i)1−α(σ−1) w (i)σ Ā (i)1−σw (i)1−σ ū (i)1−σ L (i)β(1−σ).

Note that φ (i) is positive, continuous, and bounded above and below by strictly positive
numbers. Suppose too that T (i, s) = T (s, i). Then from equations (10) and (11) we have:´
φ (i) = ´ S

T (i, s)1−σ ū (s)σ−1 L (s)1+β(σ−1) w (s)σ ds

T (s, i)1−σ Ā (s) σ−1 w (s)1−σ L (s)α(σ−1) ds
S´F (s, i)φ (s)β ds
S
φ (i) = ´,F (s, i)φ (s)β−1 ds
S

⇐⇒
(28)

where F (s, i) ≡ T (i, s)1−σ ū (s)(1−β)(σ−1) Ā (s)β(σ−1) L (s)1+β(σ−1)+β((α−β)(σ−1)−1) w (s)σ+β(1−2σ)
and we use the assumed symmetry of trade costs, i.e. T (i, s) = T (s, i). Note that F (s, i)
is positive, continuous, and bounded above and below by strictly positive numbers. We canthen write (28) as´

φ (i)βF (s, i)φ (s)β dsS=´φ (i)β−1F (s, i)φ (s)β−1 ds
S

.

(29)

β

φ(i)
Define λ (i) ≡ ´ F (s,i)φ(s). Note that because φ (i) is positive, continuous, and boundedβdsS

above and below by strictly positive numbers, so too is γ (i). Define functions g1 (i) ≡ φ (i)βThe Topography of the Spatial Economy

61

and g2 (i) ≡ φ (i)β−1 . Then we can rewrite equation (29) as the following set of equations:ˆ
g1 (i) =λ(i)F (s, i)g1 (s)ds(30)

λ(i)F (s, i)g2 (s)ds

(31)

ˆS
g2 (i) =S

Because λ (i) F (s, i) is positive, continuous, and bounded above and below by strictly positive numbers, the generalized Jentzsch theorem implies that there exists a unique (to-scale)strictly positive function that satisfies both equations (30) and (31), i.e. g1 (i) = Cg2 (i),where C is a constant. As a result, φ (i)β = Cφ (i)β−1 , or equivalently, φ (i) = C. Substituting in the definition of φ (i) into φ (i) = C immediately yields equation (12). Hence equation(12) is the unique relationship between w (i) and L (i) that satisfies equations (10) and (11)for a regular spatial equilibrium.
Since equation (12) holds for any regular equilibrium, it is sufficient to consider it alongwith equation (13) to determine existence and uniqueness of a regular equilibrium rather
than equations (10) and (11) directly. Note that we can rewrite equation (13) as a nonlinear
integral equationˆγ2

K (s, i) f (s) γ1 ds,

f (i) = λ(32)

Swhere f (i) ≡ L (i)γ1 , λ = W 1−σ , andK (s, i) ≡ ū (i)(1−eσ)(σ−1) Ā (i)σe(σ−1) T (s, i)1−σ Ā (s)(1−eσ)(σ−1) ū (s)σe(σ−1) .
In fact, instead of characterizing (32) it suffices to find the solution for the combined variableγ

12fe(i) = f (i) λ γ1 −1 . To see this, notice thatˆ
fe(i) =
Sf (i) λ

1
γ2
γ1 −1

h
i γγ2
˜
K (s, i) f (s) 1 ds ⇐⇒
ˆ

=

(33)


 γγ2
1
1
γ2
−1
K (s, i) f (s) λ γ1ds,S

which is equivalent to (32). In our case f (i) ≡ L (i)γ1 and the labor market clearing constraint implies
1− γ −γL̄ = λ2ˆfe(s)1/γ1 ds,

1

S

(34)

The Topography of the Spatial Economy

62

and thus, for each solution for fe(s), a unique solution for λ.32 Therefore, finding a solution
for fe(i) gives us the solution for f (i) and the eigenvalue of the system λ, which in our case isinversely related to welfare. Given the above preliminaries we proceed to prove the different
parts of Theorem 2.
Part (i) We first prove existence of a regular spatial equilibrium. To do so we candirectly use Theorem 2 of Karlin and Nirenberg (1967) to establish existence for Equation(32). Their Theorem 2 shows the existence of a continuous solution f (i) for a Hammersteinequation of the second kindˆK (s, i) φ (s, f (s)) ds,

f (i) = λS

where φ is a continuous function and where the bounds of integration are given by the min
and the max of K (s, i) /F (K (., i)) with F (K (., i)) an arbitrary linear functional such that
F (f ) = 1 and F (K (·, s)) > 0 for all s ∈ S. Furthermore, the solution f (i) is bounded> 0 and bounded above by b ≡ maxi,s∈S F K(i,s)> 0. For our
below by a ≡ mini,s∈S F K(i,s)(K(·,s))(K(·,s))γ2´
1
purposes, φ (s, f (s)) = f (s) γ1 and F (f ) ≡ L̄1 S f (s) γ1 ds. Note both that F (f ) = 1 from´
1labor market clearing and F (K (·, s)) = L̄1 K (i, s) γ1 ds > 0 for all s ∈ S since K (s, i) isbounded above and below by a positive number, so the theory applies. Note too that for
Karlin and Nirenberg (1967) S = [0, 1]. However, as they point out, and as is easily verified
from the steps of the proof of their Theorem 2, their result applies for any domain in RN
and, thus, for a compact interval, which completes the proof of existence.33
Part (ii) To prove that for γ1 > 0 all equilibria are regular notice that we need to provethat all locations are inhabited in equilibrium and that the equilibrium wages and labor
functions are continuous. For the first part, note that substituting the gravity equation(equation (3)) and the market clearing condition (equation (6)) into the indirect utility
function (equation (5)) yields:´
W (i) =
32

 σ1
1−σ
σ−1
γ1
σ−1
T
(i,
s)
P
(s)
w
(s)
L
(s)
ds
S
Ā (i) σ ū (i) L (i)− σ .
P (i)

(35)

Using the latest formulation it is easy to show that increasing L̄ does not affect the distribution of laboracross locations. In particular, an increase in L̄ does not affect fe(i) , given equation (33), and thus translates
only to a change in overall welfare, λ = W 1−σ , but not to a change in the distribution of labor acrosslocations.
33
The proof involves constructing a compact operator that maps the convex set of all continuous functions
f (s) into itself and consequently applying Schauder’s fixed point theorem. These steps do not depend on
the domain of the integration.

63

The Topography of the Spatial Economy

Notice that if γ1 > 0 expression (35) guarantees that every location is populated: the utilityof moving to an uninhabited location is infinite. To show that every equilibrium is continuous,
we need to prove that for any  > 0 there exists a δ > 0 such that ks1 − s2 k < δ implies
that that f˜ (s1 ) − f˜ (s2 ) < , i.e. f˜ is continuous, and thus f is continuous. Assume that
we have an equilibrium with every location population and a resulting eigenvalue λ, we will
establish continuity. Note first that K (s, i) is assumed to be continuous and S is compact,
so that by the Heine-Cantor theorem, K (s, i) is uniformly continuous on S. Then for any
 > 0 there exists a δ > 0 so that ks1 − s2 k < δ implies |K (s, s1 ) − K (s, s2 )| <  with
s1 , s2 ∈ S. Suppose ks1 − s2 k < δ. Then we have:ˆγ

2(K (s, s1 ) − K (s, s2 )) f˜ (s) γ1 ds

f˜ (s1 ) − f˜ (s2 ) =ˆSγ2≤|K (s, s1 ) − K (s, s2 )| f˜ (s) γ1 ds
Sˆ
γ2≤  f˜ (s) γ1 ds
γ2
Sˆ
1
˜
γ1
|S|1−γ2≤f (s) dsSγ2

≤ L̄ λ γ2 −γ1 |S|1−γ2γ2

where the second to last line used Holder’s inequality, the last line used equation (34), and´|S| ≡ S ds. Hence for any  > 0, we can choose a δ > 0 such that ks1 − s2 k < δ implies that
and thus f˜ (s1 ) − f˜ (s2 ) < , establishing continuity.
|K (s, s1 ) − K (s, s2 )| <γ2
L̄γ2 λ γ2 −γ1 |S|1−γ2Part (iii) We now prove uniqueness of a regular equilibrium when | γγ12 | ≤ 1.
We already discussed the case γ2 = γ1 (which would occur if α + β = 0).
Next suppose instead that | γγ12 | < 1. In this case, we can apply Theorem 2.19 from
Zabreyko, Koshelev, Krasnosel’skii, Mikhlin, Rakovshchik, and Stetsenko (1975) (p.401),
which states that if i) K (i, s) is positive and continuous and ii) fe(s) is strictly positive andit is non-decreasing and fe(s) /sc is non-increasing for c ∈ (0, 1) or it is non-increasing andfe(s) sc is increasing for c ∈ (0, 1) then there there exists a unique positive solution to equation(33). Furthermore, that solution is the uniform limit of the successive approximations:ˆ
fen+1 (i) =Sγ2K (s, i) fen (s) γ1 ds.

(36)

The Topography of the Spatial Economy

64

for any arbitrary non-zero, non-negative fe0 (i) .34 Notice that i) is satisfied given the restricγ2
tions on K (i, s) and fe(s) γ1 satisfies ii) and in particular with γγ12 ∈ [0, 1), the first condition
or with γγ21 ∈ (−1, 0) the second condition. Thus, there exists a unique positive function fe(i)that solves (33) and uniqueness is proved when | γγ21 | < 1
Finally, assume that γγ12 = −1. For this case Remark 1 of Karlin and Nirenberg (1967)
implies that as long as K (s, i) continuous, non-negative and K (i, i) > 0, there exists a
unique continuous and positive function f (i) that satisfies (32) for γγ12 = −1, in the case of
S = [0, 1]. Their argument for uniqueness trivially extends to any compact subset of S ofRN .
It remains to characterize the parameter space when | γγ21 | ≤ 1 in terms of α, β and σ.
; it can then be shown that:
Recall that γγ21 = 1+ασ+(σ−1)β1−α(σ−1)−βσ
γ2| | ≤ 1 ⇐⇒
γ1



1+α
β+α<
and β + α ≤ min{2 (1 + α) , 0} if γ1 > 0or
σ


1+α
and β + α ≥ max{2 (1 + α) , 0} if γ1 < 0.
β+α>σNote that if γ1 > 0 (which implies 1+α> β + α) and α ≥ 0 then a sufficient condition forσuniqueness is that β + α ≤ 0, as we discuss in the main text.
A.1.3

Proofs of Theorems 1 and 2 for a discrete number of locations

Theorem 1 This theorem extends in straightforward manner to the case of a discrete numberof locations. The analogous result to Jentzsch’s theorem for matrices – related to the caseof a discrete number of locations – is the celebrated Perron-Frobenius theorem (in fact,
Jentzsch’s theorem is a generalization of Perron-Frobenius theorem, which regards matrices
and eigenvectors, for continuous kernels and eigenfunctions). The analogous result to theFredholm theorem used comes from the fact that for a square matrix its eigenvalue is thesame as the eigenvalue of its transpose. Finally, the matrix operator in this case is ergodic
in the sense that an iterative approach as the one in (27) converges to the true solution. The
algorithm in this case is the same as for the case of continuous variables.
Theorem 2, part (i) We first prove existence of a regular spatial equilibrium. To do so
we can directly use Theorem 1.1 of Krasnosel’skii, Armstrong, and Burlak (1964) to establish
34

Notice that there is a typo in the statement of the second condition of Theorem 2.19 in Zabreyko,
Koshelev, Krasnosel’skii, Mikhlin, Rakovshchik, and Stetsenko (1975). A statement of the Theorem for aconnected compact interval in R is given by Polyanin and Manzhirov (2008) p. 831.

The Topography of the Spatial Economy

65

existence of a regular equilibrium for Equation (32). The Theorem states that for Γ to be theboundary of an open set defined on a Banach space of positive functions, define a positive
continuous and compact operator T (i.e. one that maps positive, continuous and compact
functions to positive continuous and compact functions) for which
inf kTxk > 0x∈ΓThen the operator has at least one positive eigenvector x0 such that
Tx0 = λx0
to which it corresponds a positive eigenvalue λ.
Our assumptions guarantee that the Kernel of the integral equation is bounded, continuous and positive and thus the operator is positive, continuous and compact. In addition,
given that the operator is bounded below and transforms positive functions to functions thatare strictly bounded above zero satisfying the requirements of the Theorem. Thus, there
exists at least one positive eigenvector (eigenfunction) and a corresponding eigenvalue thatsolve Equation (32).
Theorem 2, parts (ii) and (iii) First notice that for γ1 > 0 the same argument as in
the continuum case establishes that no location will be inhabited in a spatial equilibrium.
The uniqueness proof for Theorem 2 also applies to the case of a discrete number of locationsfor γ2 /γ1 ∈ (0, 1]. In particular, Fujimoto and Krause (1985) show that any operator T thatis strictly increasing and satisfies T (λx) = f (λ) T (x) with f : R+ → R+ such that f (λ) /λis non-increasing and f (0) = 0, has a unique positive solution and is strongly ergodic.
X
γ2
Our operator in the discrete case is T (f ) =K (s, i) f (s) γ1 and γ2 /γ1 ∈ (0, 1] all thesesrestrictions on the operator are satisfied, proving the result. Finally for γ2 /γ1 ∈ [−1, 0) thetheorem of Karlin and Nirenberg (1967) directly applies for the discrete case as in Part iii ofTheorem 2, which completes the uniqueness proof.

66

The Topography of the Spatial Economy
A.1.4

Proof of Proposition 1

Consider a regular equilibrium satisfying equations (10) and (11). Taking the derivative of
welfare in location i with respect to the population in location i from equation (35) yields: ´dW (i)γ1=− dL (i)σS

1−σ

T (i, s)w (s) L (s) ds
P (i)

 σ1


γ1

σ−1

Ā (i) σ ū (i) L (i)− σ −1  ,´since changes in the population in location i do not affectlocation i has zero measure. As a result:dW (i)= −sign (γ1 ) .
sign
dL (i)

1( S T (i,s)1−σ w(s)L(s)ds) σP (i)becauseFrom the definition of point-wise local stability, it immediately follows that if γ1 < 0, theequilibrium is point-wise locally stable and if γ1 > 0 the equilibrium is point-wise locallystable, thereby proving the Proposition.
A.1.5

Proof of Theorem 3

Combining the price index (equation (4)) and the indirect utility function (equation (5)) and
imposing utility equalization yields:
ˆ
1−σ

u (i)

=W

T (s, i)1−σ w (i)σ−1 w (s)1−σ A (s)σ−1 ds.

1−σ

(37)

S

In addition, equation (12) can be rearranged as follows:
A (s)σ−1 =1L (s) w (s)2σ−1 u (s)σ−1 .
φ(38)

Substituting equation (38) into equation (37) yields:
u (i)

1−σ

W 1−σ
=
φ

ˆ
T (s, i)1−σ w (i)σ−1 w (s)σ L (s) u (s)σ−1 ds.
S(39)The Topography of the Spatial Economy

67

1−σ

Define the functions f (i) ≡ u (i)1−σ and K (s, i) ≡ W φ T (s, i)1−σ w (i)σ−1 w (s)σ L (s).
Then equation (39) can be rewritten as:ˆ
f (i) =K (s, i)
S

1
ds.
f (s)

(40)

Equation (40) arises in models of signal theory and was first studied by Nowosad (1966) inthe case that K (s, i) is symmetric and S = [0, 1]. Since this equation is essentially the sameas equation (33) when γ2 /γ1 = −1 the argument for uniqueness in Theorem 2 directly applies
here. In particular, note that because w, L, and T are continuous and bounded above and1−σbelow by strictly positive numbers and W φ is strictly positive, the kernel K is continuous
and K (i, i) > 0 for all i ∈ S. As a result, we can apply Theorem 2 and Remark 1 of Karlinand Nirenberg (1967) to equation (40), which imply that there exists a unique continuous
positive function u (i) satisfying equation (39) for any connected compact subspace of RN .
The continuous and positive function A can then be determined using equation (40). Notethat A and u are only identified up to scale, as the constants φ and W offset any changes in
the normalizations of A and u, respectively.

A.2

Isomorphisms

We study two separate types of isomorphisms of our elementary gravity model with labormobility to richer gravity trade models. First, we show that our main setup can be shown
to be isomorphic to the class of gravity trade models considered by Arkolakis, Costinot, andRodrı́guez-Clare (2012) if an equilibrium with labor mobility is considered in that setup.
Second, we show that our setup is isomorphic to a new economic geography model as in
Krugman (1991), but when an inelastic supply of housing (amenity) is introduced, as in
Helpman (1998). Finally, we show an isomorphism to a gravity model where workers haveidiosyncratic utility shocks for each location. In all these exercises we consider a surface S
with a continuum of locations.
Gravity models Arkolakis, Costinot, and Rodrı́guez-Clare (2012) consider gravity trademodels with exogenous entry and free entry. Models with exogenous entry include Eaton andKortum (2002), Chaney (2008)-Melitz (2003), and of course, the Armington (1969) setup.
The gravity trade relationships and the labor market clearing conditions of these models arevery similar. As long as the models are set to have the same bilateral trade costs, population, and also the elasticity of trade is set to the same value, their technology parameters

The Topography of the Spatial Economy

68

can be adjusted so that they are formally isomorphic. This elasticity of trade parameter isthe Frechet curvature parameter in Eaton and Kortum (2002), the Pareto curvature parameter in Chaney (2008)-Melitz (2003), and the CES demand elasticity in Armington (1969).
Given this formal isomorphism, introducing labor mobility simply extends the isomorphism
to a labor mobility equilibrium, as we have introduced in the main text with exogenous
productivities and amenities.
The isomorphism carries on in the case of models with free entry, but allowances haveto be made in order for a non-degenerate equilibrium to emerge. Models of free entry analyzed by Arkolakis, Costinot, and Rodrı́guez-Clare (2012) include Krugman (1980), and theMelitz (2003) model with Pareto distributed productivities considered by Arkolakis, Demidova, Klenow, and Rodrı́guez-Clare (2008). The assumption of free entry implies that firms
need to hire f e units of local labor to produce a unique differentiated variety in a location, and in the resulting equilibrium the number of entrants is proportional to population,
Ni ∝ Li . When labor is allowed to move, it is straightforward to show that our setup withproduction spillovers and α = 1/ (σ − 1) is isomorphic to the free entry models discussed
above. Unfortunately, α = 1/ (σ − 1) and β = 0 =⇒ γ1 = 0, and the only equilibrium
is a “black-hole” equilibrium where all production concentrates in one location. For a nondegenerate equilibrium to arise, the production externality needs to be less strong, whichcorresponds to allowing a negative production externality in the Arkolakis, Costinot, andRodrı́guez-Clare (2012) setup with free entry, and respectively setting α < 1/ (σ − 1) in ourmodel.
Non-tradable sector (e.g. housing) A formal isomorphism can be derived with theHelpman (1998)-Redding (2012) setup. That setup assumes that workers spend a constantshare δ of their income on differentiated goods and a share 1 − γ to local non-tradable goods(often referred to as “housing”). A preference structure that gives rise to this is a monotonic
transformation of a Cobb-Douglas aggregator with a coefficient one on the differentiatedgoods and (1 − δ) /δ on housing. The differentiated sector is as in Krugman (1980) andKrugman (1991) while earnings from land are equally divided by workers residing in thatlocation. In equilibrium with labor mobility, a constant share of income is earned from wages
and rents. To formally map this model to our setup we need to set α = 1/ (σ − 1) and also

The Topography of the Spatial Economy69

β = − (1 − δ) /δ. For the equilibrium to be unique we require thatα + β ≥ 0 ⇐⇒

1 + (δ − 1) (σ)≥0δ (σ − 1)This condition is discussed as a sufficient condition for the existence mobility equilibrium in
an N -location world, in Redding (2012). In fact, Theorem 2 implies that this condition is
sufficient for a unique labor mobility equilibrium in our setup, while the sufficient conditions
for existence are weaker. It is easy to check the rest of the parts of the isomorphism, i.e.
that the gravity relationship of trade is exactly the same and the trade balance conditionis the same. Notice that similar isomorphisms can be derived if other free entry setups are
considered instead of the Krugman (1980) one.
Land as a factor of production Suppose that land M (i) and labor L (i) are combinedin a Cobb-Douglass production function to produce output: Q (i) = L (i)γ M (i)1−γ , where
M (i) is fixed. Suppose too that workers living in location i ∈ S each own an equal share of(i)r (i) . From the first order conditions of thethe land, so that their total income is w (i) + M
L(i)production function, the total income of each worker is γ1 w (i). Since production in our modelis Q (i) = A (i) L (i)1+α , a formal isomorphism exists where productivity A (i) = M (i)1−γ ,α = γ − 1, and wages are scaled by γ1 .
Worker Heterogeneity We now build a formal isomorphism to a model where workers
have idiosyncratic utility shocks in each location. Notice that this isomorphism holds for any
finite number of locations. Suppose that a worker ω receives welfare U (i, ω) from living inlocation i ∈ S, where:ˆU (i, ω) =

q (s)

σ−1
σ

σ
 σ−1

ds

u (i) υ (i, ω) ,s∈Sand υ (i, ω) is distributed i.i.d. Frechet across people and locations with shape parameter−θθ, i.e. P r [υ ≤ u] = e−u . If workers choose to live in the location with their highest
idiosyncratic utility, then for any two locations i, s ∈ S, the ratio of the population densities
can be written as a function of the non-idiosyncratic welfare:


θ

w(i)
u (i)
P (i)

L (i)
=
L (s)
w(s)

P (s)

u (s)

θ ⇐⇒

1
1
w (i)
w (s)
u (i) L (i)− θ =u (s) L (s)− θ .
P (i)
P (s)

The Topography of the Spatial Economy

70

Since u (i) = ū (i) L (i)β , the above condition is isomorphic to the utility equalization condition presented in the main text with the alternative β̃ = β − 1θ . Hence, adding heterogeneous
worker preferences simply creates an additional dispersion force.
Capital and fixed factors of production, labor and capital mobility, no trade
costs (Roback, 1982) Consider the following model based on Roback (1982). Supposethat there are three factors of production: labor L (i), a fixed factor F (i), and capital
K (i) which are combined together in a constant returns to scale Cobb-Douglass production
function to produce a homogeneous tradable good Q (i). It is assumed that labor and capitalare perfectly mobile and the tradable good is traded costlessly. Finally, suppose that workers
receive utility from consuming the tradable good, a local non-tradable good (“housing”) H (i)and a local amenity u (i), where the utility function is also assumed to be Cobb-Douglass.
Note that this framework nests, among others, Kline and Moretti (2014) and Diamond (2012)(with a single type of worker). Let the price of capital be κ (i), the price of the tradable
good be P (i), the price of the non-tradable good be R (i), and the wage be w (i).
First we derive the welfare equalization condition. Because workers are perfectly mobile,
welfare is equalized across locations. Given the Cobb-Douglass utility function, we canrepresent the utility equalization condition using the indirect utility function:ln W = ln w (i) − (1 − ζ) ln P (i) − ζ ln R (i) + ln u (i) ,where ζ is the fraction consumers spend on the non-tradable good. Suppose that rent canbe written as:
ln R (i) = ln C (i) + γ ln L (i) ,where γ > 0 is the elasticity of rent to the population and C (i) are (exogenous) local
production costs. Since it is assumed that goods are perfectly traded, we have P (i) = P .
Then we can rewrite utility equalization as:
ln W̃ = ln w (i) − ζγ ln L (i) + ln u (i) − ζ ln C (i) ,(41)

where ln W̃ ≡ ln W + (1 − ζ).
We now derive the equilibrium wage condition, which arises from the first order conditionof profit maximizing firms. Let the production function be:
Q (i) = A (i) L (i)δ K (i)λ F (i)1−δ−λ ,71

The Topography of the Spatial Economy

where A (i) is the local productivity. Taking the first order conditions of capital and substituting it into the first order condition for wages yields:1λln λ +
ln A (i) +ln w (i) = ln δ +
1−λ
1−λ



δ−1+λ
1−λ


ln L (i) −λ1−δ−λln κ (i) +ln F (i) .
1−λ1−λSince it is assumed capital is perfectly mobile, we have κ (i) = κ so that we can write:1
ln w (i) = C2 +ln A (i) +
1−λ



δ−1+λ
1−λ


ln L (i) +1−σ−λln F (i) ,
1−λ

(42)

λ
λ
where C2 ≡ ln δ + 1−λln λ − 1−λln κ.
Let us now compare this to this model. Welfare equalization (equation (5)) along withthe amenity spillovers (equation (2)) yields:

ln W = ln w (i) + ln ū (i) + β ln L (i) − ln P (i) .
Note from equation (4) that if trade is costless, then P (i) is constant. Hence we have:ln W̃ = ln w (i) + ln ū (i) + β ln L (i) − ln P (i) .

(43)

Comparing equation (43) to equation (41), the two are equivalent if β = −ζγ and ln ū (i) =ln u (i) − ζ ln C (i) .
Recall that combining market clearing, welfare equalization, and imposing the symmetryof trade costs yields equation (12), which can be written as:ln w (i) =σ−1(σ − 1) (α − β) − 1
1−σ1ln φ +
ln Ā (i) +ln L (i) +ln ū (i) (44)2σ − 12σ − 12σ − 12σ − 1σ−1Comparing equation (44) to equation (42), the two are equivalent (to scale) if 2σ−1=
1
1−σ, (σ−1)(α−β)−1= δ−1+λ, and 2σ−1ln ū (i) = 1−δ−λln F (i) . Because there are three de1−λ
2σ−1
1−λ
1−λ
σ−1
grees of freedom {σ, α, β}, we can ensure that all three equations {β = −ζγ, 2σ−1=(σ−1)(α−β)−11
1−σ,= δ−1+λ} are satisfied. However, in general, 2σ−1ln ū (i) = 1−δ−λln F (i)
1−λ
2σ−1
1−λ
1−λ
and ln ū (i) = ln u (i) − ζ ln C (i) will not be both satisfied simultaneously. Hence, an isomorphism exists if either there are arbitrary differences in amenities or arbitrary differences inthe supply of the fixed factor of production, but not both.

The Topography of the Spatial Economy

B

72

Data appendix

This appendix describes the data used in Section 3.

B.1

Population, wage, and demographic data

The population, wage, and demographic data comes from the 2000 U.S. Census. We use
county level data retrieved from the National Historic Geographic Information System MPC(2011a), of which there are 3, 109 in the continental United States. To construct the distribution of labor L : S → R+ , we take the total population of the county in 2000, divide it by
the area of the county (which is calculated in GIS), and normalize the population so it has amean of one. To construct the distribution of wages w : S → R++ , we divide the aggregate
wage or salary income in 1999 variable by the total population of the county and normalize
the wage so it has a mean of one. We use demographic data from the 2000 U.S. Census
to construct non-geographic measures of bilateral trade costs. Between all bilateral county
pairs, we calculate the correlation in ethnicity (using the number of people in each county ineach of the 14 U.S. Census race categories), the correlation in language (using the numberof people aged 18 to 64 in each county who speak English, Spanish, other Indo-European
languages, Asian and Pacific Island languages, and other languages primarily at home), andthe correlation in migrants (using the number of people in each county who are born in Europe, Asia, Africa, Oceania, Americas, at sea, and everywhere else). Finally, we use GIS tocalculate the latitude and longitude of the centroid of every county, which we use to matchthe county data to the transportation network data.

B.2

Bilateral trade flow data

The bilateral trade flow data comes from the 2007 U.S. Commodity Flow Survey (CFS, 2007).
For every pair of the 122 CFS areas and for each mode of travel, the CFS reports the quantityand value of goods shipped by mode of travel. We classify the modes of travel into fivecategories: road (corresponding to CFS categories truck, for-hire truck, and private truck),water (corresponding to CFS categories deep draft, Great Lakes, shallow draft, and water),rail (corresponding to the CFS category rail), air (corresponding to the CFS category), andother (which includes all other CFS categories, notably those reporting multiple modes oftravel). We use the Google Maps API to determine the latitude and longitude of the centroidof each CFS area. This allows us to place each CFS area on the transportation network andThe Topography of the Spatial Economy

73

allows us to determine the demographic data of each CFS area (which we equate the thedemographic data of the nearest county). We also distinguish between CFS areas thatare either parts of or entire metropolitan statistical areas (MSAs) (e.g. the IL part of the
Chicago-Naperville-Michigan City MSA) versus those that are not (e.g. the remainder ofIL), as the former correspond more closely to our theoretical concept of a location.

B.3

Transportation network data

The transportation network data are collected from three sources. Geographic data on
the road network is taken from the National Highway Planning Network (NHPN, 2005).
According to its meta data, the NHPN consists of over 400,000 miles of the nation’s highways comprised of rural arterials, urban principal arterials and all National Highway Systemroutes. In our analysis, we distinguish between interstate highways, national non-Interstate
highways, and arterial roads. Geographic data on the railroad network is from the Center forTransportation Analysis Railroad Network dataset (CTA, 2003). It contains every railroad
route in the U.S., Canada, and Mexico that has been active since 1993. For each railroad
line, it reports a “subjective rating of line importance, and implicitly quality.” In our analysis, we distinguish between class-A railroad lines, class-B railroad lines, and all other classes.
Geographic data on the water network is from the U.S. Army Corps of Engineers Navigation
Data Center (NDC, 1999). It includes all navigable inland waterways, off-shore water routes,
and routes via the Great Lakes and the Saint Lawrence Seaway.
To determine the mode-specific normalized transportation cost between any two locationsin the U.S., we construct a cost raster for each of the transportation networks. To do so, we
first convert each element of each mode network into a raster file covering the continental
United States using the 1983 North American GCS projection. Each raster file has resolution
1032 × 760, so that each pixel corresponds roughly to a 5km by 5km square. To construct
the normalized road network cost raster, we assign all pixels with interstate highways a cost
, all pixels with arterial roads aof one, all pixels with non-interstate highways a cost of 70
55
70
70
cost of 35 , and all other pixels a cost of 20 , where the costs are chosen to (roughly) reflect
differences in relative travel speeds. To construct the normalized railroad cost raster, weassign all pixels with class A railroads a cost of one, all pixels with class B railroads a cost of
1.25, all pixels with other railroads a cost of 1.5, and all other pixels a cost of 3. To construct
the normalized water cost raster, we assign all pixels with navigable waterways a cost of one
and all other pixels a cost of 10. To construct the normalized air cost raster, we assign all

The Topography of the Spatial Economy

74

pixels a cost of one (reflecting the fact that the cost of traveling through the air is (roughly)
uniformly costly regardless of the location.