The Welfare Effects of Transportation Infrastructure
Improvements∗
Treb Allen

Costas Arkolakis

Dartmouth and NBER

Yale and NBER

First draft: August 2016
This version: September 2016
Preliminary and Incomplete

Abstract
In this paper, we develop a framework to characterize the impact of infrastructure
investment on welfare. We suppose infrastructure investment affects the cost of shipping goods between directly connected locations and the total bilateral cost between
any two locations is determined by traders optimally traveling across the complete
transportation network. Our approach comprises two distinct but complementary characterizations: First, we characterize how infrastructure investment between any twoconnected locations decreases the total trade costs between all pairs of locations. Second, we characterize how the cost reduction between any two locations changes affects
welfare. We apply these results to shipment level data between U.S. cities to calculate
the welfare effects of improving each portion of the U.S. Interstate Highway System(IHS). We find very heterogeneous welfare effects of improvements to different sectionsof the IHS – reducing the travel time by 30 minutes on I-95 South from New Yorkto Philadelphia would increase aggregate U.S. welfare by 0.02%, whereas reducing thetravel time by 30 minutes from Seattle to Salt Lake City along I-84 East would only
increase aggregate U.S. welfare by one-hundredth of that.

∗
Comments are welcome. We thank David Atkin, Dave Donaldson, Pablo Fajgelbaum, Xiangliang Li,
and Sam Kortum for very helpful discussions.

11IntroductionMore than a trillion dollars is spent on investment in transportation across the world eachyear (Lefevre, Leipziger, and Raifman, 2014). These investments on the transportation networks are designed to enhance economic activity by reducing the economic costs of moving
goods and people across space. However, there is limited understanding both how investment
in transportation networks affect economic costs and how changes in economic costs affecteconomic outcomes. Better understanding of both questions is necessary in order to determine how to optimally allocate transportation investment to optimize economic outcomes.
In this paper, we develop a new spatial framework incorporating infrastructure investmentthat admits an analytical solution for how infrastructure investment of any two connected locations decreases the cost of travel between all bilateral pairs. We then embed this framework
in a standard spatial economic framework in order to characterize the general equilibrium
impact on welfare of any infrastructure investment. Finally, apply the theory to detailed
shipment level data on trade between U.S. cities in order to estimate the welfare effects ofimproving each portion of the U.S. Interstate Highway System.
To incorporate infrastructure investment, we suppose the world is composed of many
locations arranged on a weighted graph. Trade between locations requires traveling acrossthe network of the graph, incurring (iceberg) trade costs along the way. Infrastructure
investment can be undertaken to reduce the cost of any connection on the graph, which
reduces the bilateral trade costs both for the locations immediately connected and for all
trade flows using that route. As a result, characterizing how infrastructure investment affects
bilateral trade costs is quite difficult, as investment in infrastructure may potentially affecttrade costs between all other locations and traders may endogenously alter the route they
travel.
Previous applications of spatial models to understand the impact of transportation infrastructure have relied on computational algorithms such as Djikstra (see e.g. Donaldson(2012); Donaldson and Hornbeck (2012); Redding (2012); Alder et al. (2014)) to solve forthe optimal routes in graphs. Our approach, in contrast, is analytical. To make the problem
tractable, we suppose that trade is undertaken by a continuum of heterogeneous traders, whochoose the least cost route over the network but vary in their idiosyncratic cost of travelingover different paths. Using a combination of graph theory and properties of extreme value
distributions (in the spirit of Eaton and Kortum (2002)), we show that this assumption
admits a straightforward analytical relationship between the weighted graph of the networkand the resulting endogenous expected bilateral iceberg trade costs. Moreover, the framework admits an intuitive expression for the elasticity of a bilateral trade cost between any

2locations to an improvement in infrastructure between any two other locations.
We then show how this framework can be readily combined with standard general equilibrium spatial frameworks, including the economic geography framework of Allen and Arkolakis(2014) and the trade framework of Anderson (1979); Anderson and Van Wincoop (2003).
For both these frameworks, the elasticity of equilibrium welfare to a change in bilateral trade
costs between any two locations is equal to the fraction of world trade between those locations.Combining this elasticity with the elasticity of bilateral trade costs to infrastructure
investments yields a straightforward expression of the welfare effect of any transportation
infrastructure investment, which can be written as a “sufficient statistic” depending solely on
observed trade flows and the bilateral trade cost where the infrastructure investment occurs.
We then apply our framework to examine the welfare impact of improving each segmentof the U.S. Interstate Highway System (IHS). Using recently released shipment level micro
data from the Commodity Flow Survey, we show that our framework closely matches observed trade flows, and both the average and variability of distance traveled across shipments
between a given origin and destination. Finally, we estimate that the welfare impact of improvements to different segments of the IHS are very heterogeneous. Along the I-95 corridor
on the eastern seaboard and along I-70 in Indiana and Ohio, the effects of infrastructure
improvements are large; for example, reducing the travel time by 30 minutes on I-95 South
from New York to Philadelphia (which we estimate is the most important highway in theU.S.) would increase aggregate U.S. welfare by 0.02%. However, infrastructure improvements
elsewhere on the IHS have effects that are orders of magnitude smaller; for example, we estimate that reducing the travel time by 30 minutes from Seattle to Salt Lake City along I-84
East would only increase aggregate U.S. welfare by 0.0002%. Hence, the results highlight theneed to appropriately target infrastructure improvements in order to maximize their benefit.
There is a recent expanding literature on evaluating the impact of infrastructure investment in general equilibrium, such as Donaldson (2012), Allen and Arkolakis (2014),
Donaldson and Hornbeck (2012) in an inter-city context and the work of Ahlfeldt, Redding,Sturm, and Wolf (2012) in an intra-city context; see Redding and Turner (2014) for an
excellent review of the literature. The application of these models to the study of spatial
policy, however, has preceded more slowly, as the combination of many locations, generalequilibrium forces, and flexible spatial linkages have made it difficult to characterize optimal policy. For example, Ossa (2014) and Ossa (2015) follow a computational approach tocharacterize optimal tariff and tax policies in spatial frameworks, Alder et al. (2014) uses a
sequential algorithm to construct an approximation of the optimal transportation network,
while Allen, Arkolakis, and Takahashi (2014) and Allen, Arkolakis, and Li (2015) derive firstorder necessary conditions for the characterizations of the optimal investment in infrastruc3

ture. However, despite the huge investment in transportation infrastructure, very little is
known about the the impact of changes in the transportation network on welfare in a generalequilibrium setup. Two notable exceptions are Felbermayr and Tarasov (2015), who consider
infrastructure investment when the world is a line, and Fajgelbaum and Schaal (2016), who
use tools from the transportation economics literature to derive the optimal transport network in the presence of congestion. Relative to Felbermayr and Tarasov (2015), our paper
considers an arbitrary geography; whereas relative to Fajgelbaum and Schaal (2016), oursetup falls within the class of general equilibrium spatial gravity models that have provensuccessful in matching observed empirical patterns and performing counterfactuals using realworld data, see e.g. Redding and Rossi-Hansberg (2016).
The remainder of the paper proceeds as follows. The next section presents the framework
for the endogenous transportation costs and derives a number of key properties. Section 3
embeds the transportation cost framework into a the spatial economic model and derives theelasticity of aggregate welfare
derives the elasticity of welfare to a change in any bilateral trade costs in both economic
geography and trade frameworks. Section 4 then combines the two results to derive theelasticity of welfare to any infrastructure project. Section 5 concludes.

2

Endogenous transportation costs

In this section, we describe how to calculate the transportation costs between any two locations, accounting for the fact that agents endogenously choose the least cost route between
locations.

2.1

Setup

Consider a world composed of a finite number of locations i ∈ {1, ..., N }. Locations areorganized on a weighted graph with an associated infrastructure matrix T = [tij ≥ 1], wheretij indicates the iceberg trade cost incurred from moving directly from i to j (if i and j are
not directly connected in the graph, then tij = ∞). The top left panel of Figure 1 provides
an example of such a network, where N = 25 and locations are arrayed in a two-dimensionalgrid, with locations that are directly connected (i.e. tij is finite) if they are adjacent and notconnected (i.e. tij is infinite) otherwise.
Trade between i and j is undertaken by a continuum of heterogeneous traders ν ∈ [0, 1]
who travel along (endogenously chosen) paths to get from i to j. A path p between iand j is a sequence of locations beginning with location i and ending with location j4

{i = p0 , p1 , ..., pK = j}, where K is the length of path p.1 The aggregate trade cost fromi to j on a path p of length K, τ̃ij (p) is the product of the instantaneous trade costs along
the path:KYτ̃ij (p) =tpk−1 ,pk(1)
k=1

Each trader also incurs a path-specific idiosyncratic trade cost shock εij (p, ν), so the total
cost to trader ν of traveling along path p between i and j is τ̃ij (p) εij (p, ν). Let τij (ν)
indicate the cost trader ν incurs optimally choosing the path between i and j to minimize
the iceberg trade costs incurred:τij (ν) =minp∈PK ,K≥0τ̃ij (p) εij (p, ν) .

In what follows, we assume that εij (p, ν) is Frechet distributed with shape parameter θ > 0,
and we let Pij,K denote the set of all paths of length K that go from i to j.
Notice in our formulation that we allow traders to choose any possible path to ship a goodfrom i to j – including the most meandering of routes. The extent to which traders differ inwhich path they choose is determined by the shape parameter θ. In a sense, the parameterθ can be considered as capturing the possibility of mistakes and randomness in the choiceof routes, with higher values indicating greater agreement across traders. In the limit caseof no heterogeneity, θ → ∞, all traders choose the route with the minimum aggregate tradecost. The previous literature has focused exclusively on this limited case and relied upon
computational methods to calculate this least cost route (i.e. using the Djikstra algorithm as
in Donaldson (2012) and Donaldson and Hornbeck (2012) or the Fast Marching Method as inAllen and Arkolakis (2014). As will become evident below, the advantage of the framework
proposed here is that the solution takes a convenient analytical form allowing for explicit
characterizations for how changes in the infrastructure affect bilateral trade costs. For this,we view our formulation as relaxing the previous literature’s focus on the least cost route inorder to to better understand the impact of changes in the investment matrix on realized
trade costs. Finally, we should note that the concept of heterogeneous traders formulation
has been utilized previously by Allen and Atkin (2015). However, their formulation does
not consider the optimal route taken, as the the random trader shocks are realized on the
expected cost matrix instead of the infrastructure matrix.
1

Following the literature on graph theory, we assume that tii = ∞ to exclude paths that stay in the same
location; however, we allow traders shipping goods from i to i to choose the “null” path where they travel
nowhere and incur no trade costs (which is the only admissible path of length 0).

5

2.2

Optimal routes

Using the familiar derivations pioneered in Eaton and Kortum (2002) we can express the
expected trade cost τij from i to j across all traders as:τij ≡ Eν [τij (ν)] = c

∞ X
X

!− θ1
τ̃ij (p)−θ

,(2)

K=0 p∈PK

where c ≡ Γ

θ−1
θ


. Substituting equation (1) into equation (2) yields:
τij−θ = c−θ

∞ X Y
K
X

t−θ
pk−1 ,pk .

K=0 p∈PK k=1

In what follows, it proves useful to characterize the weighted adjacency matrix A =
aij ≡ t−θ
ij . Note that aij ∈ [0, 1], where 0 indicates there is no connection between i and j,
aij = 1 indicates a cost-less connection, and aij ∈ (0, 1) indicates a costly connection. We
can then write the expected trade cost as:τij−θ = c−θ

∞ X Y
K
X

apk−1 ,pk .

K=0 p∈PK k=1

The summing over all paths of length K can be written in a more convenient form by
explicitly summing across all locations that are traveled to first, second, etc. as follows:τij−θ = c−θ

∞
X



N
N X
X


K=0

k1 =1 k2 =1

..

N
X



ai,k1 × ak1 ,k2 × ... × akK−2 ,kK−1 × akK−1 ,j  ,kK−1 =1

where kn is the sub-index for the nth location arrived at on a particular path. This portion of
the expression in the parentheses, however, is equivalent to the (i, j) element of the weighted
adjacency matrix to the power K, i.e.:τij−θ = c−θ

∞
X

AK
ij ,

K=0

 
K
K
where AK = AK
ij , i.e. Aij is the (i, j) element of the matrix A , i.e. A to the matrix
power K. Furthermore, as long as the spectral radius of A is less than one,2 the geometricP
A sufficient condition for the spectral radius being less than one is if j t−θ
ij < 1 for all i. This will
necessarily be the case if either trade costs between connected locations are sufficiently large, the adjacency
matrix is sufficiently sparse, or the heterogeneity across traders is sufficiently small (i.e. θ is sufficiently
2

6

sum can be expressed as:∞X

AK = (I − A)−1 ≡ B,

K=0

where we call B = [bij ] the route cost matrix. Finally, the expected trade cost from i to j
can be written as a simple function of the route cost matrix:−1

τij = cbij θ .

(3)Equation (3) provides an analytical relationship between any given infrastructure matrix andthe resulting expected trade cost, accounting for traders choosing the least cost route.

2.3

Properties of the endogenous trade costs

In this subsection, we derive four important properties of this endogenous trade cost framework. The first two properties will prove helpful in providing an analytical characterizationof the welfare impact of transportation infrastructure improvements, whereas the latter two
properties will prove helpful in estimation the model parameters from the data.
2.3.1

The probability of using a particular link along the optimal path

We first characterize the probability that trade going from location i to location j travels
through the link tkl , which we refer to as πijkl . Given the extreme value distribution, theprobability of taking any particular path p of length K can be written as:
τij (p)−θ
⇐⇒πij (p) = P∞ P
0 −θ
K=0
p0 ∈Pij,K τij (p )K

1 Y
πij (p) =ap ,p ,
bij k=1 k−1 k(4)where the second line used the definition of τij (p) from equation (1) and the derivation forequation (3). As a result, we can calculate the probability of using link tkl when travelingfrom i to j by summing across all paths from i to j that use the link tkl :πijkl =∞K
1 X X Y
apk−1 ,pk .
bij K=0kl
k=1p∈Pij,K

kl
where Pij,Kis the set of all paths from k to l of length K that use link tij .

large).

7kl
Note that for any p ∈ Pij,K, there must exist some length B ∈ [1, 2, ..., K − 1] at which
the path arrives at link tkl so that this can be written as:

πijkl =

∞ K−1
1 XX



bij K=0 B=0



B
X Y




X

K−B−1
Y

q∈Pkj,K−B−1

k=1

apk−1 ,pk  × akl × p∈Pik,B k=1


aqk−1 ,qk 

As above, we can then explicitly enumerate all possible paths from i to k of length B andall possible paths from l to j of length K − B − 1:
πijkl =

∞ K−1
X
X



N
X

N
X

1

B−1 =11
···
bij K=0 B=0 k =1 kai,k1 × . . . × akB−1, k ×akl ×

N
X
k1 =1

N
X

···


al,k1 × . . . × akK−B−1, j  ,kK−B−1 =1

which again can be expressed more simply as elements of matrix powers of A :∞ K−1

πijkl =

1 XX BA × akl × AK−B−1
lj
bij K=0 B=0 ik

This is where it gets a little more difficult. Recall from matrix calculus that the derivativeof the power of a matrix can be written as:
DSK (A) C =K−1
XAB CAK−B−1 ,

(5)

B=0

where SK (A) = AK and C is an arbitrary matrix. Furthermore, recall from above that thegeometric series of a power of matrices can be written as:∞
X

AK = (I − A)−1 .

K=0Hence, we can right multiply both sides of equation (5) by C and differentiate to yield:∞
X

DSK (A) C = DT (A) C,

K=0

where T (A) ≡ (I − A)−1 . Recall from matrix calculus (see e.g. Weber and Arfken (2003))
that DT (A) C = (I − A)−1 C (I − A)−1 so that we have:
∞ K−1
X
X

AB CAK−B−1 = (I − A)−1 C (I − A)−1 .

K=0 B=08(6)

Define C to be an N × N matrix that takes the value of akl at row k and column l and zeros
everywhere else. Equation (6) then implies that:bik akl blj
⇐⇒
bij

θ
1 τij
kl
πij =
.
c τik tkl τlj

πijkl =

(7)

Equation (7) is intuitive: the numerator on the right hand side is the expected trade cost on
the least cost route from i to j, whereas the denominator is the expected trade cost on theleast cost route from i to j through the transportation link tkl . Hence, the more costly it isto travel through the link tkl relative to the unconstrained least cost route, the less likely atrader is to use the link tkl .
As an example, consider the network of locations presented in Figure 1. If we assume
that all connected locations have the same direct cost tij , then the top right panel depicts
the probability that an agent going from i = 1 (bottom left) to j = 25 (top left) travels
over each connection. As is evident, connections along the direct diagonal route are chosenmuch more often than other routes, and connections near the diagonal route are chosen more
often than those further away. As an additional example, the bottom left panel depicts theprobability for agents going from i = 1 (bottom left) to j = 15 (top center). In this case,there are four different paths that share the same least deterministic costs. As is evident,
traders are much more likely to use links on these routes than other links. However, there isalso heterogeneity in the probability traveled across these different links – for example, theconnection from 1 to 7 is traveled more often than the connection from 2 to 8. This occurs
because there are many more possible paths going from 1 to 15 that use the 1 to 7 link thanthat use the 2 to 8 link, so each trader is more likely to get an idiosyncratic draw that causes
her to choose a path using the former link.
It turns out that this same intuition is helpful in interpreting how changes in an infrastructure link affect bilateral trade costs, which we turn to next.
2.3.2

How changes to infrastructure affect endogenous trade costs

Consider now how a change in the transportation link tkl affects the bilateral trade costs∂ ln τ
from i to j, i.e. ∂ ln tklij for all i, j, k, l ∈ {1, ..., N }. First note that this elasticity is equal
to the elasticity of an element of the route cost matrix to a change in the element of the9weighted adjacency matrix A:−1
cbij θ

∂ ln∂ ln bij∂ ln τij 1 ==.
−∂ ln tkl∂ ln akl∂ ln a θkl

To calculate this elasticity, we parameterize the weighted adjacency matrix as a function ofa variable t follows:aif k 6= i or l 6= jij,
Akl (t) =t
if k = i and l = ji.e. changing t just increases akl . By defining Ckl (t) ≡ I − Akl (t), we have:"#∂ ln bijakl
dCkl (t)−1=×∂ ln akl
dt
bij

(8)

ij

Using the familiar expression for the derivative of an inverse of a parameterized matrix (seee.g. Weber and Arfken (2003)), we have:dCkl (t)−1dCkl (t)= −Ckl (t)−1
Ckl (t)−1 .
dtdtkl (t)
Note that dCdt=

0

if k 6= i or l 6= j

−1 if k = i and l = j

and Ckl (t)−1 = B so the derivative becomes:

dCkl (t)−1= BEkl B,
dt
where Ekl is an N × N matrix equal to one at (k, l) and zeros everywhere else. The (i, j)
component of this inverse is hence:"dCkl (t)−1dt"#= [BEkl B]ij =

#
XX
n

ij

m

kl
bim Emnbnj= bik blj ,

(9)

ij
−1

θ
so combining equations (8) and (9) and using the relationships akl = t−θkl and τij = cbij yields
a simple expression for how a change in the infrastructure matrix changes the endogenous
trade costs:θ∂ ln τijbik akl blj
τij
=
= c
.
(10)
∂ ln tklbijτik tkl τlj

10Note that the elasticity of trade costs from i to j to changes in the link tkl from (10) is exactly
equal to the probability that a route from i to j uses the tkl infrastructure from (7). Hence,
the more “out of the way” the transportation link tij is from the optimal path between k andl, the less frequently that path is used, and the smaller the effect an improvement in tij hason the reduction of trade costs between k and l.
2.3.3

The expected distance between two locations

We now characterize the expected distance traveled between two locations on the optimalroute. While traders do not directly care about distance (distance only matters inasmuch
as it affects trade costs), the following expressions will prove useful in estimating the model
parameters from the data.
Let dij be the distance between locations i and j, where dij is finite if i and j are directly
connected and infinite otherwise. Let d˜ij (p) be the distance between i and j on path p:d˜ij (p) =KX

dpk−1 ,pk

k=1

The expected distance from i to j is equal to the sum across all possible paths of theprobability of traveling over that path times its distance:
∞ X
K
 X

X
πij (p)dpk−1 ,pk .
E d˜ij (p) =K=0 p∈PK

k=1

Applying expression (4) for the probability of a particular path yields:
∞
K
K

X
1 X XY˜
ap ,p
dp ,p ,
E dij (p) =bij K=0 p∈P k=1 k−1 k k=1 k−1 kKwhich as above we can write by explicitly accounting for all possible paths:


∞
N X
N
N

X
X
X


1

E d˜ij (p) =..
ai,k1 × ... × akK−1 ,j di,k1 + ... + dkK−1 ,j  .
bij K=0k =1 k =1 k
=1


1

2

K−1

As in Section 2.3.1, this expression can be rearranged to note that distance dkl is incurredon any path that travels through link akl , allowing us to consider all paths that go through11

link akl on the B th stop along the way from i to j:
N X
N
∞ K−1
 c−θ XX
X
E d˜ij (p) = −θdkl
τij k=1 l=1
K=0 B=0



PN

PkN1 =1

PN

kB+1 =1

!

a
×
...
×
a
×
a
×
i,k1
kB−1 ,kkl
kB−1 =1

⇐⇒
PN
..
a
×
...
×
a
l,k
k
,k
B+1
K−1
kB+2 =1
kK−1 =1!

k2 =1 ..
P
N

PN

∞ K−1
N X
N
 c−θ X
X X
K−B−1
˜
E dij (p) = −θABik × dkl akl × Alj
τij K=0 B=0 k=1 l=1



(11)



P
PN
B
Define the matrix (D ◦ A) ≡ [dij aij ]ij . Note that AB (D ◦ A) AK−B−1 ij = N
k=1
l=1 Aik ×
dkl akl × AK−B−1, so that we can rewrite expected distance from expression (11) in matrix
lj
notation as follows:" ∞ K−1#XX
1
E d˜ij (p) =AB (D ◦ A) AK−B−1 .
bij K=0 B=0ij

We can then apply equation (6) to write:


1 
˜
E dij (p) =(I − A)−1 (D ◦ A) (I − A)−1 ij ⇐⇒bij [B (A ◦ D) B]ij
E d˜ij (p) =
,
[B]ij


where the operation “◦” is the Hadamard (element-wise) product. This can be written
equivalently in “trade notation” as:

θ
N
N X
 X
1 τij
˜
dkl
E dij (p) =.
c
τ
ik tkl τlk
k=1 l=1(12)Equation (12) is intuitive: from equation (7), the probability of a trader going from i toθ
τijj traveling through link tkl is 1c τik tkl; as a result, the expected distance that a traderτlk
travels is simply the sum across all links multiplied by the probability each link is used.
2.3.4

The variation across traders in distance traveled between two locations

Finally, we derive a measure of the variation across traders in the distance traveled between
two locations. This expression will prove helpful when comparing the model predictions to
the data below. A natural measure of the variation would be the standard deviation of
distance traveled across all traders going from i to j. We instead employ an alternative
measure of the variation which admits a simple analytical formulation: the “LogMeanExp

12

minus mean”, which for a random variable x is defined as:”LogMeanExp minus mean of x” ≡ log E [exp (x)] − E (x) .
Because log E [exp (x)] is convex, the “LogMeanExp minus mean” of x is increasing in thevariance of x.3
Using an analogous derivation as in Section 2.2 and applying equation (12) for the expected distance traveled, the LogMeanExp minus mean can be calculated as:
log (E (exp dij (p))) − E d˜ij (p) = log !
(I − (A ◦ exp (D)))−1 ij
[B]ij

−

[B (A ◦ D) B]ij,
[B]ij

where exp (D) ≡ [exp dij ] is a matrix of the exponential of distances.

3

The welfare effects of changing bilateral trade costs

We now introduce a spatial economic setup which we incorporate with the endogenous transportation network setup presented in Section 2 in order to characterize the welfare effect ofchanging the transportation infrastructure. We consider two variants of the setup – one inwhich agents are perfectly immobile and one in which agents are perfectly mobile – andpresent new derivations of how a change in any bilateral trade cost affects the endogenous
aggregate welfare.

3.1

A Spatial Economic Framework

Setup Suppose that the N locations are inhabited by an exogenous measure L̄ of agents.
On the production side, an agent residing in location i supplies her endowed unit of labor
inelastically in order to produce Ai units of that locations differentiated variety, for which
she is compensated with a wage wi . Production occurs in a perfectly competitive frameworkwith labor as the only factor of production. On the consumption side, agents use their wage
to purchase a constant elasticity of substitution (CES) bundle of these differentiated varieties
with elasticity of substitution σ. To purchase goods, workers are randomly matched withtraders and hence face trade costs equal to the expected trade costs τij above.
Pn
Indeed, one can show that for any set of realizations of random variables, log N1 i=1 exp (xn ) ∈[max xn − log n, max xn ], so that the “LogMeanExp minus mean” can be interpreted as measuring the difference between the maximum of a random value and its mean. We show below that the “LogMeanExp minus
the mean” of distances traveled across traders is highly correlated (correlation of 0.89) with the standard
deviation of distance traveled.
313Equilibrium Given the perfect competition assumption, the price for a unit of a goodwifrom i for a consumer in location j is pij = τij A. Given the CES preferences of consumers,ithe value of goods shipped from location i to location j can be written as:
Xij = τij1−σ Aσ−1
wi1−σ Pjσ−1 Ej ,i(13)where Pj is the ideal Dixit-Stiglitz price index and Ej is the total expenditure of agents inlocation j.
In equilibrium, the following three conditions will hold:1. Total income Yi is equal to total sales:
Yi =

N
X

Xij

(14)

j=1

2. Total expenditure Ei is equal to total purchases:
Ei =

N
X

Xji

(15)

j=1

3. Total income is equal to total expenditure which is each to total payments to workers:
Yi = E i = w i L i ,(16)where Li is the measure of agents residing in location i.
The equilibrium welfare of a worker is given by Wi ≡ wPii ui , where ui is the amenity value ofliving in location i. Using this expression to substitute out for the price index and the third
equilibrium to substitute for Ej , we can rewrite equation (13) for the value of bilateral tradeas follows:

1−σ
τij
Xij =
wi1−σ Wj1−σ wjσ Lj .
(17)
Ai uj
Substituting the gravity equation (17) and the equilibrium condition (16) into equilibrium
conditions (14) and (15) yields a system of equilibrium conditions:wi Li =
wi Li =X  τij 1−σ
i
N
X
j=1

A i uj
τji
Aj ui

1−σ

14

wi1−σ Wj1−σ wjσ Lj(18)

wj1−σ Wi1−σ wiσ Li .

(19)

We do so under two different assumptions regarding labor mobility:
Assumption 1. Trade framework (Anderson, 1979; Anderson and Van Wincoop, 2003).
Workers are perfectly immobile, so the measure of workers in each location {Li } is exogenous.

Assumption 2. Economic geography framework (Allen and Arkolakis, 2014). Workers are
perfectly mobile, so the measures of workers in each location ensures that welfare is equalized
across all locations, i.e. Wi = W for all i ∈ {1, ..., N }.
In the trade framework, equations (18) and (19) can be solved directly to determine theequilibrium wage and welfare in all locations. In the economic geography framework, theimposition of welfare equalization results in the following system of equilibrium equations:wi Li = W

1−σX  τij 1−σiwi Li = W 1−σ

Ai uj

1−σ
N 
X
τji
j=1

Aj ui

wi1−σ wjσ Lj(20)

wj1−σ wiσ Li ,(21)

which can be solved for the equilibrium wage and population in all locations along with theaggregate welfare W .4
In what follows, we use the system of equilibrium conditions above to derive how changes
in transportation infrastructure affects the equilibrium welfare of agents. As an intermediate
step, we first derive the welfare effects of a changes in a particular (endogenous) trade cost.

3.2

Elasticity of welfare to changes in trade costs

We first examine how a change in a (endogenous) bilateral trade costs affects the aggregate
welfare. To do so, we construct a planner problem which maximizes aggregate welfare subject
to resource constraints and then show that the first order conditions of this problem are
identical to the equilibrium conditions (18) and (19) in the trade framework or equilibrium
conditions (20) and (21) in the economic geography framework. This alternative planner
formulation of the equilibrium then allows us to directly apply the envelope theorem tocalculate the elasticity of welfare to a change in trade costs.
4

See Allen, Arkolakis, and Takahashi (2014) for a detailed description of the properties of both types of
equilibria.

15

3.2.1

Trade framework

Consider the following “planner” problem which chooses wages and welfare to maximize aweighted average of (log) welfare subject to the constraint that aggregate labor income isequal to aggregate trade flows:max{wi ,Wi }NXωi ln Wi subject to:

i=1

1−σ
N
N
N 
1 X
1 X X τij
wi Li =
wi1−σ Wj1−σ wjσ Lj
1 − σ i=1
1 − σ i=1 j=1 Ai uj(22)

where ωi ≡ wi Li is the Pareto weight placed on the welfare of location i and the constraint
requires that total world labor income is equal to total world sales.
The associated Lagrangian is:NX

X X  τij 1−σ
X1λ
wi1−σ Wj1−σ wjσ Lj −wi LiL:ωi ln Wi −
1−σ
Ai uj
i
j
i
i=1

!

The first order conditions of this maximization problem can be written as:
1−σ
N 
X
τji
∂L
= 0 : ωi = λwj1−σ Wi1−σ wiσ Li∂ ln Wi
A
u
j i
j=1
1−σ
N 
X
τij
∂L
σ
= 0 : wi Li = (1 − σ)
wi1−σ Wj1−σ wjσ Lj + ωi
∂ ln wiAi uj
λj=1
Substituting in the weight ωi = wi Li and noting in equilibrium the Lagrange multiplier λ = 1,
we see immediately that the solution to the maximization problem (22) is the equilibrium
system of equations (18) and (19). As a result, we can apply the envelope theorem to derive
the elasticity of world welfare to a change in bilateral trade frictions τij :
N X∂ ln WiYi
i=1

YW∂ ln τij

=−

Xij
.
YW

(23)

Equation (23) – which has been derived using alternative methods by Atkeson and Burstein(2010) and Fan, Lai, and Qi (2013) – shows that the effect of a change in bilateral trade
friction between i and j on a weighted average of world welfare in a trade framework is
simply proportional to the fraction of world income comprised by trade flows from i to j.

16

3.2.2

Economic geography framework

Now consider the slight variant of the “planner” problem (22) which chooses wages, population, and welfare to maximizes the (equalized) log welfare across locations subject to thesame constraint that aggregate labor income is equal to aggregate trade flows:max{{wi },{Li },W }ln W subject to:

1−σ
N
N
N 
1 X X τij
1 X
wi Li =
wi1−σ W 1−σ wjσ Lj
1 − σ i=1
1 − σ i=1 j=1 Ai uj(24)

The associated Lagrangian is:X
X X  τij 1−σ
1
wi1−σ W 1−σ wjσ Lj −wi Li
λL : ln W −
1−σ
A
u
i
j
i
i
j

!

The first order conditions of this maximization problem can be written as:
XX
∂L
1
=0:
=λ
∂ ln W
W
i
j
λ=



τij
Ai uj

1−σ

wi1−σ W −σ wjσ Lj ⇐⇒

1
YW

1−σ
N 
X
X  τji 1−σ
τij
∂L
1−σ
1−σ σ
= 0 : wi Li = (1 − σ)wi Wj wj Lj + σ
wj1−σ W 1−σ wiσ Li∂ ln wi
A
u
A
u
i j
j i
j=1
j
(25)
∂L
= 0 :wi Li = W 1−σ∂ ln Li

X
j

τji
Aj ui

1−σ

wj1−σ wiσ Li(26)

Substituting equation (26) into equation (25) yields:
wi Li = W

1−σX  τij 1−σ
j

Ai uj

wi1−σ wjσ Lj(27)Hence, the first order conditions (26) and (27) correspond to the equilibrium conditions (20)and (21). As a result, we can apply the envelope theorem to determine the elasticity of
welfare to a change in the (endogenous) trade cost τij , which yields:∂ ln W∂L=× τij = λ∂ ln τij
∂τij



τij
Ai uj

1−σ

17

wi1−σ W 1−σ wjσ Lj =

Xij
.
YW

(28)

Hence, in both the case where labor is perfectly immobile and perfectly mobile, the elasticity
ln W
of aggregate welfare to a change a bilateral trade cost, ∂∂ ln, is equal to the fraction of world
τij
Xincome that is traded from that origin to that destination, Y Wij .

3.3

Elasticity of welfare to changes in infrastructure

A simple application of the product rule allows us to combine equation (10) – which provides the elasticity of bilateral trade costs between any pair of locations to a change in any
infrastructure – with equations (23) and (28) – which give us the elasticity of welfare withrespect to a change in bilateral trade costs in economic geography and trade frameworks,
respectively – to determine the welfare effect of any change in infrastructure:
N X
NX
d ln Wd ln τkl
d ln W=×.
d ln tijd ln τkl
d ln tij
k=1 l=1(29)

Note that because traders are optimally choosing their routes, any change in infrastructure
will affect all bilateral trade flows. Substituting in equations (10) and either (28) or (23) into
equation (29) yields:
N X
NX
d ln W
Xkl bki aij bjl
=
,
(30)
−
d ln tij
Y W bkl
k=1 l=1
which can be written in “trade notation” as:

θ
N
N X
X
Xkl
d ln W
τkl
θ
−
=c
.
W
d ln tijYτki tij τjl
k=1 l=1(31)

As an example, return to the two dimensional grid network considered in Figure 1 andsuppose all locations have identical productivities and amenities and labor is mobile. The
bottom right panel of Figure 1 depicts the elasticity of welfare to a change in each directconnection using equation (31). As is evident, reducing the cost of traveling over links inthe center of the grid have larger impacts on welfare than those in the periphery. Intuitively,
this is for two reasons: first, the connections in the center are more likely to be traveled thanon the periphery (see the bottom left panel of Figure 1) and hence will have larger effects onbilateral trade costs; second, because they are more centrally located, locations in the center
will have greater populations and economic output, so that the trade flows flowing throughthe central links will be larger on average. As a result, improvements to those links will havelarger effects on welfare.
We can further express this equation solely in terms of data and (i, j) variables by noting

18

from gravity equation (17) that we can write:

Xkl Xji
Xki Xjl=τkl τjiτki τjl1−σ,so that expression (30) becomes:
d ln W−=d ln tij

tij τji
c

−θ

θ

θ
1−σ

Xji



N X
N 
X
Xkl
Xki Xjl σ−1
k=1 l=1

YW

Xkl

.

(32)

Equation (32) provides a “sufficient statistic” of the welfare effect of any infrastructure
improvement that depends only observed bilateral trade flows and knowledge of the expectedtrade cost between j and i, τji , and the instantaneous trade cost, tij . Moreover, the result is
Xki Xjlis indicative of low trade costs from k to i and low trade
intuitive: a larger value of X
klcosts of j to l relative to trade costs between k and l, which implies that more trade from k
to l travels through i and j. As a result, an improvement in the infrastructure between i andj will have a larger effect on trade flows between k and l. Finally, the larger the value of θ,
the more responsive traders are to changes in the relative cost of different paths; conversely,
the larger the value of σ, the more responsive trade flows are to changes in trade costs (soa given ratio of trade flows is indicative of smaller differences in trade costs). As a result,
the relevant measure of the effect of an infrastructure improvement between i and j on trade θXki Xjl σ−1. To calculate the effect of suchflows between k and l in terms of observables is
Xkl
an infrastructure improvement on total world welfare, we then simply construct a weighted
average of this ratio, where the weights are the fraction of world trade between any two
kllocations YXW.

4

The welfare effects of improving the U.S. Interstate
Highway System

In this section, we apply the theoretical framework developed above to estimate the aggregate
welfare effect of the U.S. of improving each segment of the Interstate Highway System (IHS).
The IHS – depicted in Panel A of Figure 2 – is one of the largest infrastructure investments
in history (Kaszynski, 2000). It took more than thirty five years to construct at an estimatedcost $560 billion (in 2007 dollars), and total annual maintenance costs of the highway system
is approximately $130 billion (CBO, 1982; FHA, 2008; NSTIFC, 2009). However, little
is known about the relative importance of different segments of the IHS in terms of howeach affects the welfare of the U.S. population. Such knowledge is crucial for appropriately
19

targeting future investments in the improvement of the IHS.

4.1

Data

To assess the relative importance of each segment of the IHS, we use shipment level data
between 67 U.S. metropolitan statistical areas (which we refer to as “cities”) from the public
use micro data of the 2012 Commodity Flow Survey (CFS, 2012). Shipments between thesecities constitute 60% of the value of commodity flows within the United States. For eachof the 696,021 shipments on for-hire trucks between these cities, we observe the origin city,
the destination city, the value of the shipment, and the distance the truck traveled on its
chosen route.5 Consistent with the theoretical framework developed above, the CFS microdata demonstrate that there exists substantial heterogeneity across shipments in the distancethey travel between a particular origin city - destination city pair. Panel A of Figure 3 depicts
the distribution of distance traveled for the twenty origin city - destination city pairs withthe greatest observed trade flows. For example, the interquartile range of shipments fromNew York to Boston is 91 miles. While certainly part of this variation is due to differencesin the particular origin and destination within the two cities, it seems likely that differencesin the path chosen also are responsible for part of this variation.6
We proceed by constructing the graph representation of the IHS. We say that a pair ofcities in the CFS is directly connected by the IHS if the shortest route between the twocities on the IHS does not pass through another city.7 For each connected city pair, we thencalculate both the distance traveled and the time it takes to traverse the direct route using
Google Maps. The resulting graph of the IHS network is depicted in Panel B of Figure 2.
Finally, for all origin city - destination city pairs, we use the CFS micro data to construct an aggregate measure of the value of bilateral trade flows, the mean distance traveled
between the pair, the standard deviation of distance traveled between the pair, and the difference between the log of the mean of the exponential and the mean distance traveled (the“LogMeanExp minus Mean”).8 As discussed above, the “LogMeanExp minus Mean” is analternative measure of the variability of a random variable; consistent with this, Panel B of5

We constrain our analysis to for-hire trucks because the distance traveled for private trucks is top-coded
at 500 miles.
6
We examine the extent to which the model can capture the variation across cities in route distancevariability despite treating each city as a single point below.
7
We allow a small about of flexibility in defining the shortest route. For example, without traffic, travelingfrom Denver, CO to Seattle, WA takes 18 hours and 16 minutes via Salt Lake City on I-80 W and I-84 W
takes 18 hours and 31 minutes on I-25 N and I-90 W without passing through another CFS city. Because
the direct route is nearly as fast, we classify Denver and Seattle as directly connected on the IHS.
8
To control for outliers, we winsorize the distance traveled in the micro CFS data at the 10%/90% level
before calculating the variability measures. Both variability measures are measured in hundreds of miles.

20Figure 3 shows that the correlation between the standard deviation and the “LogMeanExp
minus Mean” is very high (0.89).

4.2

Estimation

In order to assess the elasticity of aggregate welfare to improvements to different segments of
the IHS, we need (1) a measure of the infrastructure matrix T = [tij ≥ 1] and (2) estimates ofthe shape parameter governing heterogeneity across routes θ and the elasticity of substitution
σ. To proceed, we assume that the trade cost incurred along the direct connection from iand j is a function of the time it takes to travel from i to j (in hours) timeij :tij = exp (κtimeij ) ,(33)

where κ > 0. The exponential functional form has been used extensively in the economic
geography literature, and has a number of attractive properties including (1) infrastructure
costs are always greater or equal to one, i.e. tij ≥ 1; (2) conditional on travel time, thenumber of locations through which a trader passes does not affect the trade costs that
trader incurs; and (3) the effect of a reduction in travel time on the percentage change inwelfare is proportional to the welfare elasticity to a change in the infrastructure cost, i.e.
ln W
∂ ln W= κ ∂∂ ln.
∂timeijtij
Given parametric assumption in equation (33), we just need to estimate the parameter
triplet {κ, θ, σ}. To proceed, we use a method of moments estimation strategy using threesets of moments: (1) the mean distance traveled between each origin city - destination city
pair; and (2) the observed value of bilateral trade flows between each origin city - destinationcity pair; and (3) an average trade cost 20% ad valorem tariff equivalent, consistent with theevidence from the literature on domestic distribution costs in the U.S., see e.g. Anderson
and Van Wincoop (2004). This provides us with 2 × N 2 + 1 = 8979 moments to estimate
the three parameters.
The three sets of moments match intuitively to each of the parameters. Given the parametric assumption in equation (33), aij = exp (− (θκ) distij ), so that the matrices A and B
depend only on the product θκ. From equation (12), this in turn implies that the expecteddistance between any origin city - destination city pair depends only on the product of θ andκ as well, so the first set of moments (the mean in distance traveled) are sufficient to identify
θκ. The second set of moments – the bilateral trade flows – can then be used to identify theratio of the elasticity of substitution and the trader heterogeneity σ−1. To see this, note thatθ
substituting equation (3) into the gravity equation (17) and taking logs yields the following

21

expression for bilateral trade flows:
ln Xij =σ−1θ
ln bij + ln γi + ln δj ,where ln γi and ln δj are origin and destination fixed effects, respectively. For any set matrix
B = [bij ], this implies that we can identify the ratio of parameters σ−1
by regressing log
θtrade flows on ln bij with a set of origin and destination fixed effects. Finally, from equation(3), we see that the level of trade costs are differentially affected by κ and θ, as bij depends−1on the product κθ but τij = cbij θ , allowing us to find the level of the θ that ensures the mean
trade cost matches our assumed value.

4.3

Estimation Results

Table 1 presents the results of the estimation. We estimate an effect of travel time (inhours)κ = 0.0108, indicating a five hour trip incurs an ad valorem tariff equivalent trade
cost of 5.5% . We estimate an elasticity of substitution σ = 7.92, implying a trade elasticity
of about 7, consistent with a large literature (e.g. Eaton and Kortum (2002)). Finally, we
estimate a trader heterogeneity parameter θ = 136.13, indicating reasonably small heterogeneity across different paths, which suggests most trade occurs on paths close to the least
cost route. The implied trade costs from these parameters are calibrated to have a mean of
20% ad valorem tariff equivalent, with a median of 16.9% and moderate variation (a 25th
percentile of 10.4% and a 75 percentile of 27.1%).
Figure 4 shows that the model is able to closely match the observed average distancetraveled in the data (with a correlation of 0.997) and the observed bilateral trade flows (witha correlation of 0.737). Moreover, we can also test the extent to which the model is able
to capture the observed heterogeneity in distance traveled between a particular origin city destination city pair. Reassuringly, the “LogMeanExp minus Mean” predicted by the modelis positively and statistically significantly correlated with the “LogMeanExp minus Mean”
observed in the data, albeit with a correlation of only 0.03. However, is able to closely capture
the observed variation across origin and destination cities in variability of routes. To see
this, we regress variability in traveled distance for a given origin-destination city pair acrossshipments on a set of origin and destination city fixed effects. Figure 5 shows the correlation
between those fixed effects in the CFS data to those estimated in the model is positiveand reasonably large (0.318 for origin fixed effects and 0.168 for destination fixed effects),
although the model predicts substantially more variation in route distance than exists in22the data.9 Note that this positive correlation exists despite the model not accounting for
across-city variation in the size of the Metropolitan Statistical Area, which is likely driving
part of variation in the data due to within-city differences in where the particular origin anddestination was for a given shipment.

4.4

The welfare effects of improving the U.S. Interstate Highway
SystemGiven our estimated parameters, we can estimate the probability of using any particularconnection for any origin city - destination city pair using equation (7). As an example,
Figure 6 depicts the probability a trader going from Los Angeles to New York uses eachpart of the Interstate Highway System. As is evident, there is reasonable variation in thepaths chosen, with traders employing both northern connections (e.g. Salt Lake City toOmaha) and southern connections (e.g. Phoenix to Oklahoma City) along the way. As
traders continue eastward, however, the geographic dispersion of connections narrow, as
routes converge through Indiana, Ohio, and Pennsylvania.
Finally, we estimate the welfare effects of improving the U.S. Interstate Highway System.
To do so, we apply equation (30) to all connected origin city - destination city pairs to calln W. Figure 7 presents the results. The portions of the IHS for which improvements
culate ∂∂ ln
tij
would have the largest impact on aggregate welfare are the I-95 corridor which connects the
major eastern seaboard cities (Washington, Baltimore, Philadelphia, and New York) andthe interstates in the states of Ohio and Indiana (“the crossroads of America”), throughwhich much of the U.S. trade flows. Table 2 reports the top 20 IHS segments for which
improvements would have the greatest welfare impacts. As is evident, the welfare impactof transportation infrastructure improvements is substantial: our parameter estimates imply
that a 30 minute reduction in the travel time from New York to Philadelphia – which given
our estimate of κ implies a fall in trade costs of about 0.5% – would increase aggregate U.S.
welfare by 0.02%. However, there is substantial heterogeneity across various parts of the IHS– for example, a 30 minute reduction in travel time from Seattle to Salt Lake City along I-84
East would only increase aggregate U.S. welfare by 0.0002%. Hence, these results highlightthe need to appropriately target funds for infrastructure improvements in order to maximize
the benefit.
9

Indeed, if we also include the variability of distance traveled as a moment, the estimated trader heterogeneity θ increases to 228, suggesting we may be potentially overstating the heterogeneity across traders.

235Conclusion

TBD

24

References
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on economic development,” Working Paper, University of North Carolina, Chapel Hill.
Allen, T., and C. Arkolakis (2014): “Trade and the Topography of the Spatial Economy,” The Quarterly Journal of Economics.
Allen, T., C. Arkolakis, and X. Li (2015): “Optimal City Structure,” Yale University,
mimeograph.
Allen, T., C. Arkolakis, and Y. Takahashi (2014): “Universal Gravity,” NBER Working Paper 20787.
Allen, T., and D. Atkin (2015): “Volatility, insurance, and the gains from trade,” Discussion paper, Working Paper.
Anderson, J. E. (1979): “A Theoretical Foundation for the Gravity Equation,” American
Economic Review, 69(1), 106–116.
Anderson, J. E., and E. Van Wincoop (2003): “Gravity with Gravitas: A Solution to
the Border Puzzle,” American Economic Review, 93(1), 170–192.
(2004): “Trade Costs,” Journal of Economic Literature, 42(3), 691–751.
Atkeson, A., and A. Burstein (2010): “Innovation, Firm Dynamics, and International
Trade,” Journal of Political Economy, 118(3), 433–489.
CBO (1982): “The interstate highway system: Issues and options,” Congressional Budget
Office of the United States.
CFS (2012): “2012 Commodity Flow Survey Public Use Microdata,” Bureau of Transportation Statistics.
Donaldson, D. (2012): “Railroads of the Raj: Estimating the Economic Impact of Transportation Infrastructure,” Manuscript, MIT.
Donaldson, D., and R. Hornbeck (2012): “Railroads and American Economic Growth:
New Data and Theory,” Discussion paper.

25

Eaton, J., and S. Kortum (2002): “Technology, Geography and Trade,” Econometrica,
70(5), 1741–1779.
Fajgelbaum, P., and E. Schaal (2016): “Optimal Transport Networks,” Discussion paper.
Fan, H., E. Lai, and H. Qi (2013): “Global Gains from Reduction of Trade Costs,”
Unpublished manuscript.
Felbermayr, G. J., and A. Tarasov (2015): “Trade and the Spatial Distribution ofTransport Infrastructure,” .
FHA (2008): “2008 status of the nation’s highways, bridges, and transit: Conditions andperfomance,” Federal Highway Administration and Federal Transit Administration.
Kaszynski, W. (2000): The American Highway: The History and Culture of Roads in the
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Surface Transportation Infrastructure Financing Commission.
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Academic Press.
26

Tables and Figures
Figure 1: An Example Geography
Example Graph

Probability of traveling across each link from i=1 to j=25
0.6

5

10

15

20

25

5

10

15

20

25

0.5
4

9

14

19

24

4

9

14

19

24

0.4

3

8

13

18

23

3

8

13

18

23

2

7

12

17

22

2

7

12

17

22

1

6

11

16

21

1

6

11

16

21

0.3

0.2

0.1

Probability of traveling across each link from i=1 to j=15

Welfare effect of improving a link
0.55

5

10

15

20

25

0.026
5

0.5

10

15

20

25

0.024
0.45
0.022
4

9

14

19

24

0.4

4

9

14

19

24

0.02
0.35
3

8

13

18

23

0.018

0.3

3

8

13

18

23

0.016

0.25

0.014

0.2
2

7

12

17

22

2

7

12

17

22

1

6

11

16

21

0.15
0.1
1

6

11

16

21

0.012
0.01

0.05

0.008

Notes: This figure provides an example geography. The top left frame depicts the graph ofthe connections between each of the 25 locations, where all connections are assumed to incur
an equal cost. The top right depicts the probability of traveling along each connection when
beginning at location 1 (bottom left) and traveling to location 25 (top right). The bottom
left frame also depicts the probability of traveling along each connection when beginning at
location 1 (bottom left) but now traveling to location 15 (top center). Finally, the bottom
right figure depicts the elasticity of aggregate welfare to an improvement in each of theconnections.

27

Figure 2: The U.S. Interstate Highway Network
Panel A: The Observed U.S. Interstate Highway Network
Seattle_^Portland

Minneapolis

_
^

_
^

MilwaukeeGrand Rapids_^

Chicago

Omaha

Salt Lake City

_
^

_
^

_
^

Denver

_^_^Wichita

San Jose

Indianapolis DaytonColumbus

Las Vegas

Tulsa_
Oklahoma City^_^

Phoenix_^

San Diego

_
^

_
^

Tucson_^_^Memphis, TN_^

Birmingham

DallasAtlanta

_
^

_
^

_
^

Austin_^San Antonio_^
Laredo, TX_^

Beaumont
Houston__LA^
Lake Charles,_ ^^Boston

_
^

_
^
_
^

Washington

Richmond, VACharleston_
Savannah^_^

El Paso

New York_^_^
Virginia Beach_^
Greensboro
Knoxville_ Raleigh_^Charlotte^_^
Greenville_^_^

Nashville_^

Los Angeles

Hartford_^
Philadelphia_
Baltimore ^_^Louisville/Jefferson County

_
^

_
^

Pittsburgh__ Cincinnati
^
_ ^
^
_
^
_
^

St. Louis

_
^

_
^

Fort Wayne

Rochester_^_ Cleveland
^
_
^

_
^

Kansas City_^

Sacramento

Buffalo

Detroit_^_^

MobileJacksonville_
New Orleans ^
_
^
_
^

_
^

OrlandoTampa ^_

Corpus Christi

_
^

_
^

Miami

_
^

Panel B: A Graphical Representation of the U.S. Interstate Highway Network
Seattle_^Portland

Minneapolis

_
^

_
^

MilwaukeeGrand Rapids_^

Chicago

Omaha

Salt Lake City

_
^

_
^

_
^

Denver

_^Wichita

San Jose_^

Indianapolis DaytonColumbus__ Cincinnati^_ ^^St. Louis

Las Vegas

Tulsa_
Oklahoma City^_^

Los Angeles_^

San Diego

_
^

_
^

Phoenix_^

Tucson_^Knoxville

Birmingham

Atlanta_^

Memphis, TN_^

Dallas_^Greensboro_^

Greenville

_
^

_
^

Austin_^San Antonio_^
Laredo, TX_^

Beaumont
Houston__LA^
Lake Charles,_ ^^Corpus Christi_^

Mobile_New Orleans ^_^_^Boston_^_^Raleigh

Richmond, VA_^
Virginia Beach
_
^

_
^

Charleston_
Savannah^_^

El Paso_Charlotte^

_
^

_
^

_
^
_
^

Washington_^

Nashville

Hartford_^
Philadelphia_
Baltimore ^

_
^

_
^

_
^

_
^

New York

Pittsburgh

Louisville/Jefferson County

_
^

_
^

Fort Wayne

Rochester_^_ Cleveland
^
_
^

_
^

Kansas City_^

Sacramento

Buffalo

Detroit_^_^Jacksonville_^

OrlandoTampa ^
_

_
^

Miami

_
^

Notes: This figure depicts the U.S. Interstate highway network. Panel A shows the observed
interstate highway network along with the cities included in the Commodity Flow Survey.
Panel B depicts the graphical representation of the direct connections between cities along
the interstate highway system. Two cities are considered connected if the fastest route
between the locations along the interstate highway system does not pass through anothercity. The color of the links indicates the driving distance between the connected cities, withblue indicating a longer distance and red indicating shorter distances.

28

200

400

600

.015
.01
.005
0

0

400

600

.015
.01

400

600

0

500 1000 1500 2000

0

.005

.01

New York to Jacksonville

0

500

0

500

1000

St. Louis to Columbus

.005

.005

.01

.01

San Jose to Los Angeles.015

200 400 600 800

200

0
200

New York to Boston000Memphis, TN to Los Angeles

.005

.01

.015

600

.005
0

Philadelphia to New York0 .002 .004 .006 .008

.005
0

400

0

.01
1000

0
100 200 300 400

1000

.005
500

.01

.01
.005
0
0

500

0
0

Philadelphia to Columbus

200

Memphis, TN to Chicago

0

600

0

Memphis, TN to Pittsburgh

Las Vegas to Salt Lake City

0

100

200

300

0

.015

.005
400

0

.002 .004 .006

600

0
200

300

.01
400

.01

0 .005 .01 .015 .02
0

200

.005
200

.015

0

Memphis, TN to Oklahoma City Memphis, TN to Philadelphia

New York to Philadelphia

100

Memphis, TN to Baltimore

00
50 100 150 200

0

.015

1000

.005

.005
0
0

.005

.005
0

0

500

.01

.015

0

Memphis, TN to Atlanta

Cincinnati to Philadelphia

.01

.01

.015

Chicago to Indianapolis

.005

.005
0

100 200 300 400

Los Angeles to San Diego

.01

.015

0

Density

Chicago to Dallas

.01

.01

Boston to New York

.015

Figure 3: The Variability of Route Distances Across Shipments
Panel A: Variation in Distance Traveled between Major City Pairs

0

200

400

600

0

200

400

600

0

Log Mean Exponential Distance - Mean Distance
.5
1
1.5

Panel B: Correlation between Different Measures of Variability

0

1
2
3
Std.dev. of distance along route by truck

4

Notes: This figure shows the variability in distances traveled from a given origin to a given
destination across shipments in the CFS micro data. Panel A presents the histogram of
distances traveled for the twenty origin-destination city pairs with the largest observed valueof shipments. Panel B presents the relationship between the standard deviation in distancetraveled between a particular city pair to the log of the mean of the exponential distancetraveled, where distances are measured in hundreds of miles. Both measures of variability
are calculated across all for-hire truck shipments in the micro CFS data and winsorized at
the 10%/90% level. The correlation between the two measures is 0.89. The two outlying
origin-destination pairs are shipments from Jacksonville to Memphis and from Salt Lake City
to El Paso.

29

Figure 4: Model Fit

Mean distance35

30

Observed in CFS

25

20

15

10

5

Correlation:0.997

0
0

5

10

15

20

25

30

35

Predicted by model

Log Trade Flows

Observed in CFS

-5

-10

-15

Correlation:0.737

-20
-15

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

Predicted by model

Notes: This figure depicts the ability of the model to match the moments in the data. The
top panel reports the relationship between the model predicted and observed mean distance(in hundreds of miles) across all shipments within each city pairs. The middle panel reports
the relationship between the model predicted and observed difference between the log ofthe mean of the exponential of the distance and the mean distance, which measures theheterogeneity in which routes are chosen between city pairs. The bottom panel reports therelationship between the observed trade flows and those predicted from a gravity regression
with origin and destination fixed effects using the model predicted average bilateral trade
costs.

30

Figure 5: Predicted and Observed Variability of Route Distances

Dest nat on C ty

.04

Origin City
04

New York

03.03

New York

02

.02

Miami

Jacksonville

Da a

San Jose

Detroit.01

BostonAtlanta

Memphis, TN

01

Data

Salt Lake City

Knoxville
Washington

Los Angeles
TucsonSan JoseGreenville
Birmingham

Los Angeles

Miami

St. Louis

BostonPhiladelphia

Indianapolis
Cleveland
ChicagoPhiladelphia Fort Wayne

ChicagoGrand Columbus
RapidsDaytonDetroitNashville
Charlotte
Virginia
Beach
RochesterKansas City
Milwaukee
TucsonNew
Orleans
Pittsburgh
Minneapolis
DallasLas Vegas
Greensboro
RaleighOrlandoCharleston
Tulsa
Sacramento
AustinCincinnati
Baton Rouge, MobileLA
HoustonBaltimore
Phoenix
Omaha
BuffaloMemphis,TN
Richmond, Hartford
VA
Tampa
Wichita
Savannah
SeattleOklahomaCityLouisville/Jefferson County
San Diego
DenverBeaumont
San Antonio
El Paso
Laredo, TX

PortlandCorpus Christi
Lake Charles, LA

-4

-2

0
Model

2

4

Indianapolis
Pittsburgh
Fort W
Wayne
Milwaukee
Birmingham
Nashville
Kansas City Minneapolis
New Orleans
CharlotteCharlottKnoxville
Greensboro
Tulsa
Greenville
Washington
W
Louis
Orlando St.RaleighVirginia BeachGrand Rapids
Salt Lake City
Cincinnati
Tampa
Richmond,
VA Rochester
BaltimoreJacksonvilleCharleston
Louisville/Jefferson
Buffalo
Dayton County
MobileSavannah
Las VegasDenver
Laredo, TX
San Diego
Hartford
Oklahoma
CityRouge, LA
Baton
Corpus Christi
WichitaW
Omaha
Beaumont
Lake Charles, LA
Seattle0

El Paso

AustinHoustonSan Antonio
Phoenix

4

Correlation: .318

Columbus
Cleveland

Atlanta

DallasSacramento

01

-.01

0

Portland

2

0
Mode

2

4

Co e a on 168

Notes: This figure compares the observed and model predicted variability of distance traveledby origin and destination cities. To do so, we regress variability in traveled distance for agiven origin-destination city pair across shipments (measured as the difference between thelog of the mean of the exponential distance traveled ) on a set of origin and destination city
fixed effects. The figure depicts the relationship between those fixed effects in the CFS data
to those estimated in the model.

31

32

_
^San Diego_^

Los Angeles_^

Las Vegas_Tucson^_^Phoenix_^

Salt Lake City_^El Paso_^Denver

ChicagoMilwaukee

BuffaloRochester

_
^

Miami

0.058818 - 0.977932

Boston__ ^^_ Detroit
Hartford_^
Omaha__Cleveland ^^Fort Wayne ^
New
York
_
^
_
^
_ Pittsburgh^_^
Philadelphia_DaytonColumbus^
Indianapolis_^_ Baltimore^Kansas City
_ ^^St. Louis_ ^^_County^Louisville/Jefferson
__
^
_
_
^
_
^
Wichita
Richmond, VA_^
Cincinnati
Washington_^_ Tulsa^_Virginia Beach^
Nashville Knoxville Greensboro
Raleigh_ LA to NY probabilities^
Oklahoma City
Charlotte
Memphis, TN
_
_
^
^
_^
^
Greenville
_
^
_
_
^
0.000000_^_ BirminghamAtlanta ^^_0.000001 - 0.000000
Charleston
Dallas_^0.000001 - 0.000000_^
Savannah_^_^
0.000001 - 0.000008_^
Mobile0.000009 - 0.000032
Jacksonville
AustinBeaumont Baton Rouge, LA
0.000033 - 0.000210_^
San Antonio
_
_
^
Lake^
Charles, LA_ Houston
^
_
^
_
^
_
0.000211 - 0.001606_ New Orleans^^
Orlando_Corpus Christi^
Tampa
0.001607 - 0.007088
Laredo, TX_^
0.007089 - 0.058817
_
^
_
_ ^
^

_
^

Minneapolis

Notes: This figure depicts the probability of a trader going from Los Angeles to New York using each particular connection of
the Interstate Highway System. This is equivalent to the elasticity of trade costs from LA to NY to a change in the cost of
traveling over each connection.

San Jose_^_^

Sacramento

Portland_^_^

SeattleFigure 6: Probability of Using Different Highways from LA to NY

33_San Diego^_^

Los Angeles_^

Las Vegas_Tucson^_^Phoenix_^

Salt Lake City_^El Paso_^Denver

ChicagoMilwaukee

BuffaloRochester

Boston__ ^^_ Detroit
Hartford_^
Omaha__Cleveland ^^Fort Wayne ^
New
York
_
^
_
^
_ Pittsburgh^_^
Philadelphia_DaytonColumbus^
Indianapolis_^_ Baltimore^Kansas City
_ ^^St. Louis_ ^^_County^Louisville/Jefferson
__
^
_
_
^
_
^
Wichita
Richmond, VA_^
Cincinnati
Washington_^_^ Tulsa_Virginia Beach^
Nashville Knoxville Greensboro
Raleigh_ Welfare elasticity^
Oklahoma City
Charlotte
Memphis,TN__ Greenville
^
_^
^
_
^
_
_^
^
0.000367 - 0.001347_^_ BirminghamAtlanta ^^_0.001348 - 0.002593
Charleston
Dallas
_^0.002594 - 0.003803_^
Savannah_^_^
0.003804 - 0.004815_^
Mobile0.004816 - 0.006455
Jacksonville
AustinBeaumont Baton Rouge, LA
Houston0.006456 - 0.008267_^
San Antonio
_
^
_
^
Lake
Charles,
LA
_ ^
^
_
^
_
_
^
0.008268 - 0.010332_^
Orlando_Corpus Christi^
New Orleans
Tampa
0.010333 - 0.014548
Laredo, TX_^
0.014549 - 0.021815
_
^
_
_ ^
^
0.021816 - 0.044808
Miami
_
^

_
^

Minneapolis

Notes: This figure depicts the elasticity of total welfare in the U.S. to a change in the direct trade cost between any two
locations, accounting for both how that change affects bilateral trade costs between all locations and the general equilibrium
response to those changes in trade costs. The color of each link indicates its relative elasticity, with blue indicating a smaller
welfare effect and red indicating a larger welfare effect.

San Jose_^_^

Sacramento

Portland_^_^

SeattleFigure 7: The Welfare Effects of Transportation Infrastructure Improvements

Table 1: Estimation
Model parameters
Parameter
Effect of distance on direct
trade cost
Trader heterogeneity
Elasticity of substitution

Symbolκ

MomentMean path distance

Estimated Correlation: predictedvalue
vs. observed
0.01080.997

θσTrade flows
Trade costs

136.137.92370.737
1Implied trade costs
Mean
25% perc. Median
75% perc.
1.2 (calibrated)
1.104
1.169
1.271
Notes: This table reports the estimated parameter values and summary statistics for the
implied bilateral trade costs. The three parameters were estimated to most closely matchthe three reported moments; note that the estimation procedure simultaneously estimatedall parameters by minimizing all moments, so the assignment of parameter to moment aboveis heuristic.

34

Table 2: Top 20 Highways
Origin City
Destination City
Interstate Welfare Elasticity
1 New York
Philadelphia
95 (South) 0.04481
2 Baltimore
Washington
95 (South) 0.04477
3 Columbus
Dayton70 (West) 0.0438
4 Dayton
Columbus
70 (East)
0.04094
5 Dayton
Cincinnati
75 (South) 0.04007
6 Philadelphia
Baltimore
95 (South) 0.03961
7 Cincinnati
Dayton75 (North) 0.03691
8 Philadelphia
New York
95 (South) 0.03292
9 Washington
Baltimore
95 (North) 0.03069
10 Washington
Richmond, VA
95 (South) 0.03025
11 Beaumont
Houston10 (West) 0.02874
12 Tulsa
Oklahoma City
44 (West) 0.02874
13 Pittsburgh
Columbus
70(West)0.02868
14 Lake Charles, LA
Beaumont
10 (West) 0.02858
15 Baltimore
Philadelphia
95 (North) 0.02793
16 Columbus
Cincinnati
71 (South) 0.02742
17 Cincinnati
Louisville/Jefferson County 71 (South) 0.02662
18 Louisville/Jefferson County Nashville
65 (South) 0.02659
19 Columbus
Cleveland
71 (North) 0.02629
20 Nashville
Memphis, TN
40 (West) 0.02617
Notes: This table reports the twenty (one-way) interstate links that have the greatest improvement on U.S. welfare for a given percentage reduction in trade costs on that connection.

35