The Gravity Equation in International Trade:
An Explanation∗Thomas Chaney†
University of Chicago, NBER and CEPR
September, 2011PRELIMINARY AND INCOMPLETE

Abstract
The gravity equation in international trade is one of the most robust empirical findingin economics: bilateral trade between two countries is proportional to their respective sizes,
measured by their GDP, and inversely proportional to the geographic distance between them.
While the role of economic size is well understood, the role played by distance remains a mystery. In this paper, I propose the first explanation for the gravity equation in international
trade. This explanation is based on the emergence of a stable international network of importers and exporters. Firms can only export into markets in which they have a contact. Theyacquire contacts by gradually meeting the contacts of their contacts. I show that if, as observed
empirically, (i) the distribution of the number of foreign countries accessed by exporters is fat
tailed, (ii) there is a large turnover in exports, with firms often going in and out of individual foreign markets, and (iii) geographic distance hinders the initial acquisition of contacts
in an arbitrary way, then trade is proportional to country size, and inversely proportional todistance. Data on firm level, sectoral, and aggregate trade support further predictions of themodel.

∗

I want to thank Fernando Alvarez, Xavier Gabaix, Sam Kortum and Bob Lucas for helpful discussions. I am
indebted to Jong Hyun Chung, Stefano Mosso and Adriaan Ten Kate for their research assistance.
†
Contact: Department of Economics, The University of Chicago, Chicago, IL 60637. Tel: 773-702-5403. Email:tchaney@uchicago.edu.

1Introduction
Fifty years ago, Jan Tinbergen (1962) used an analogy with Newton’s universal law of gravitation
to describe the patterns of bilateral aggregate trade flows between two countries A and B as
“proportional to the gross national products of those countries and inversely proportional to the
distance between them,”

TA,B ∝(GDPA )α (GDPB )β(DistAB )ζ
with α, β, ζ ≈ 1. The so called “gravity equation” in international trade has proven surprisingly
stable over time and across diﬀerent samples of countries and methodologies. It stands among themost stable and robust empirical regularities in economics.
While the role of economic size (α, β ≈ 1) is well understood in a variety of theoretical settings,

to this day no explanation for the role of distance (ζ ≈ 1) has been found. This paper oﬀers suchan explanation for the first time.

The empirical evidence for the gravity equation in international trade is strong. Both therole of distance and economic size are remarkably stable over time, across diﬀerent countries,
and using various econometric methods. Disdier and Head (2008) use a meta-analysis of 1,467
estimates of the distance coeﬃcient ζ in gravity type equations in 103 papers. There is someamount of dispersion in the estimated distance coeﬃcient, with a weighted mean eﬀect of 1.07
(the unweighted mean is 0.9), and 90% of the estimates lying between 0.28 and 1.55. Despite thisdispersion, the distance coeﬃcient ζ has been remarkably stable, hovering around 1 over morethan a century of data. If anything, Disdier and Head (2008) find a slight increase in the distancecoeﬃcient since 1950. The size coeﬃcients α and β are also stable and close to 1. Anderson andvan Wincoop (2003) show how to estimate gravity equations in a manner that is consistent witha simple Armington model, and how to deal especially with diﬀerences in country sizes.1 Silva
Santos and Tenreyro (2006), Helpman, Melitz and Rubinstein (2008) and Eaton, Kortum andSotelo (2011) show how to accommodate zeros in the matrix of bilateral trade flows to estimate
gravity equations.
Existing theoretical models can easily explain the role of economic size in shaping trade flows,
but none explains the role of distance. Krugman’s (1980) seminal contribution was motivated1
McCallum (1995) measures a very large negative eﬀect of the US-Canada border. Anderson and van Wincoop(2003) show that the large diﬀerence in the size of the US and Canada explains this seemingly implausible bordereﬀect.

2

in part by the empirical regularity of the gravity equation. His model explains how in the aggregate, trade flows are proportional to country size, and inversely related to trade barriers. To
the extent that distance proxies for trade barriers, his model can also explain why distance has anegative impact on trade flows in general, but it has nothing else to say about the precise role ofdistance. Several others have shown that the same type of predictions as Krugman can be derived
in various other settings. Anderson (1979) derives a similar gravity equation under the Armington
assumption that goods are diﬀerentiated by country of origin. Eaton and Kortum (2002) derive asimilar gravity equation in a modern version of trade driven by Ricardian comparative advantages.
Chaney (2008) extends the Melitz (2003) model to derive a similar gravity equation in a modelwith heterogeneous firms. Arkolakis, Costinot and Rodriguez-Clare (forthcoming) show that thesame gravity equation can be derived in many settings with or without heterogeneous firms.
None of these models however can explain the precise role played by distance. The fact thatthe distance elasticity of trade has remained stable around -1 over such a long time and oversuch diverse countries is almost a direct rejection of these models. In all of these models, granted
that trade costs increase with distance in a log-linear way, the distance elasticity of trade is the
product of some deep parameters of the model2 with the distance elasticity of trade barriers. To
explain why the distance coeﬃcient is close to -1, those models need some mysterious alignmentof those deep parameters. Even if that magical alignment were to happen in a particular year, fora particular sector and a particular country, it is hard to understand how it could survive beyond
that point for more than a century. The technology of transportation, the political impediments
to trade, the nature of the goods traded, as well as the relative importance of the countries tradingthese goods all have undergone some tremendous change over the course of the last century. In
other words, all the deep parameters identified by the various existing trade theories have been
evolving over time, while the empirical distance coeﬃcient in the gravity equation has remained
essentially constant.
This paper oﬀers the first explanation that is immune to this critique. I explain not only therole of economic size, which is straightforward, but also the role of distance. This explanation isbased on the emergence of a stable network of importers and exporters. I assume that there are
two ways for firms to circumvent the barriers associated with international trade. The first one isto pay a direct cost for creating a foreign contact. This cost is in essence similar to the trade cost
2

The demand elasticity in the Krugman and Armington models, the dispersion of productivities across firms in
the Eaton and Kortum model, and a combination of both in the Melitz-Chaney model.

3assumed in all existing trade models. The second one is to “talk” with one’s existing contacts, andlearn about the contacts of one’s contacts. This second channel requires direct interaction. Whileadvances in the technology for transportation or communication will surely aﬀect the first type ofcost, and may even aﬀect the frequency of the second type of interaction, it does not change theneed for direct interaction. In my model, the geographic distribution of any one firm’s exports
does depend on how distance aﬀects the direct cost of creating contacts. But in the aggregate,
the details of this distance function vanish, and the gravity equation emerges. This is the main
contribution of this paper: even if technological, political or economic changes aﬀect the particularshape of firm level exports, in the aggregate, the gravity equation remains essentially unaﬀected.
The remainder of the paper is organized as follows. In section 1, I present a theoretical modelof firm level and aggregate trade. In section 1.1, I spell out an economic model of trade subject
to matching frictions. In section 1.2, I characterize the patterns of firm level trade. In section 1.3,
I show that aggregate trade obeys the gravity equation. In section 2, I test empirically the main
theoretical predictions of the model. I relegate to Appendix A all mathematical proofs, and toAppendix B the description of the data and robustness checks.

1

TheoryIn this section, I develop a simple model of the formation of a stable network of importers andexporters. The model is a extension of the Krugman (1980) model of international trade in
diﬀerentiated goods subject to matching frictions similar to the Chaney (2011) model of trade
networks.

1.1

A model of trade subject to matching frictions

This model is purposefully simple, and is meant to illustrate how the proposed dynamic model offirm trade can be derived in a classical trade setting. The hasty reader may skip this section so as
to focus her attention on the formation of a stable network of exporters in the following section
1.2.
There are two types of goods: final goods and intermediate inputs. Final goods are producedby combining diﬀerentiated intermediate inputs and labor. Intermediate inputs are themselves
produced by combining diﬀerentiated inputs and labor, so that the economy features roundabout4

production. Final goods are sold locally to consumers on a perfectly competitive market. Intermediate inputs are produced and distributed worldwide by monopolistically competitive firms. I
will focus most of my attention on the production and trade of these intermediate goods. Due tomatching frictions, intermediate input firms source their inputs from, and sell their output to a
subset of producers only.
The static problem of the firm: Consider what happens within period t. For the moment,
I drop the time t index. Firm i buys intermediate inputs from a continuum of suppliers k ∈ Ki

and sells its output to a continuum of customers j ∈ Ji . Both Ki and Ji will be endogenously de-termined dynamically below. Firm i combines labor with qi (k) units of diﬀerentiated intermediate
inputs from each supplier k to produce Qi units of output,
Qi =

�ˆ

qi (k)

σ−1
σ

k∈Ki

dk

�α σ

σ−1

L1−α
i

(1)

with 0 < α < 1 the share of intermediate inputs in production and σ > 1 the elasticity of
substitution between any two intermediate inputs. Firm i faces the same iso-elastic demand fromany customer j ∈ Ji ,pj (i) qj (i) = ´pj (i)1−σk∈Kj pj (k)

1−σ

dk

(2)

αXj

with pj (i) the price charged by i to customer j, and Xj the total spending on intermediate inputs
and labor by j. Given these iso-elastic demands, firm i charges all its customers the same constantσmark-up, σ−1, over its marginal cost,σpj (i) = pi =
w1−α
σ−1

�ˆ

pi (k)

1−σ

dk

k∈Ki

� α

1−σ

(3)

with w the competitive wage rate. For simplicity, I will consider a symmetric equilibrium where
all firms within a cohort have the same number of suppliers and customers3 and therefore charge
the same price: �Kj � = �K� and pj (k) = p for any j, k �= i. Given this symmetry assumption,

the demand equation (2) and the pricing equation (3), the total sales of firm i only depend on thenumber of suppliers and of customers,�ˆpi Q i =pi qj (i) dj = �Ki �α × �Ji � × α
j∈Ji

3

X
�K�1+α�(4)In such a symmetric equilibrium, all the complexity of the input-output structure of the economy is assumed
away. See Carvalho (2010), Acemoglu, Ozdaglar and Tahbaz-Saleh (2010) or Atalay, Chaney, Hortacsu and Syverson(2011) for models with such a complex structure.

5

To save on notations, I will make from now on a slight abuse of language and use K (resp. J) todenote the mass of suppliers (resp. customers), instead of �K� (resp. �J�).

It is clear from the previous Equation (4) that the number of suppliers and customers increases

output, sales, and ultimately profits. The mass of suppliers, Ki , plays a role equivalent to capital,
or to a productivity shifter. I will use the “capital” analogy, and denote by Ii the “investment” inacquiring new contacts. The notion that the diversity of intermediate inputs plays a role similar
to capital has been presented since at least Romer (1990).4 The mass of customers, Ji , plays arole equivalent to a proportional demand shifter. A firm is willing to pay for information about
new upstream and downstream contacts, as well as sell the information it has about its existing
contacts. I now turn to this dynamic decision.
The dynamic problem of the firm: A firm is born with a mass K0 = J0 of suppliersand customers. Those contacts are randomly distributed along the real line R according to thesymmetric p.d.f. g0 . One can think that a potential entrant pays a fixed entry cost to create a
new firm. This fixed cost buys both the blueprint for a new variety and this initial set of upstream
and downstream contacts. This fixed cost will be reimbursed by a discounted stream of per period
profit over the lifespan of the firm. Imposing a free entry condition would pin down the numberof entrants each period.
Once born, a firm expands its set of suppliers and customers. While a priori, firm i may
buy and sell information about both suppliers and customers, I assume instead that i actively
buys information about new suppliers from its existing suppliers, actively sells information about
suppliers to its existing customers, but passively waits to be contacted by downstream firms. In
a symmetric equilibrium, this simplification is innocuous since firms have as many suppliers as
customers on average: if i is a supplier to j then j is a customer of i. The sequence of contact
formation is depicted in Figure 1.
I assume that a firm always has the option of directly searching for suppliers on its own. Thisoutside option technology oﬀers new names at a given constant marginal cost. Facing the threatof this outside option, firm i sets a constant price pI to reveal the name of one of its suppliers
to its existing customers. The price pI is set just low enough to prevent firm j ∈ Ji to look forcontact directly instead. Just as i sells information about its suppliers, it also buys at the same
price pI information about new suppliers from its existing suppliers.
4

See among many examples the theoretical model of di Giovanni and Levchenko (2010) or the empirical evidenceof Halpern, Koren and Szeidl (2011) for two recent applications of this notion in trade.

6Figure 1: Firms buy and sell information about suppliers and customers.
Notes: The straight solid arrows represent input-output linkages: e.g. firm k supplies intermediate inputs to firm i.
The curvy dotted lines represent information linkages: e.g. firm k sells to firm i information about a new supplierk� . After these information exchanges take place, firm i has a new supplier, k’, and a new customer, j’.

In addition to the direct cost pI of buying information, firm i faces a convex adjustment cost
G (Ii , Ki ). The adjustment cost function is assumed increasing and convex in I, GI , GII > 0,
decreasing in K, GK < 0, and homogenous of degree one in I and K. This convex adjustment
cost function is analogous to the adjustment cost assumed in the classical investment literature,
as in Lucas (1967) or Hayashi (1982). As in the investment literature, I assume that the more
suppliers a firm already has (Ki ), the more eﬃcient it is at acquiring new suppliers (Ii ), in sucha way that the adjustment cost G is proportional to Ki for a given investment share Ii /Ki . As
in Lucas (1967), this homogeneity assumption will warrantee that Gibrat’s law holds, in the sensethat the growth rate of a firm is independent of its size.
Note that as in classical investment models, firm i has two reason for accumulating more
suppliers, i.e. “investing” in Ki : first, a higher Ki increases its productivity and profits,;but it also
lowers the future cost of “investment” in acquiring new suppliers, Ii . However, while firm i sells
information about its suppliers to its customers, having more suppliers does not change firm i’s
future prospect for selling more information: the price firm i sets for selling information about
suppliers, pI , is set by an arbitrage condition, and the number of requests for names firm i receives
depends on the decisions of its customers, j ∈ Ji . At each point in time t, firm i receives I (t)requests for names, where I (t) depends on the “investment” decision of downstream firms, which7

is beyond firm i’s control.
Finally, I assume that firm i’s existing contacts are lost at an exogenous rate δ.
Firm i solves the following dynamic optimization problem,
�
�
�
�
ˆ +∞
α αJi (t) X (t)−ρtmax V (0) =e
Ki (t)− pI (t) Ii (t) + pI I (t) − G (Ii (t) , Ki (t)) dt
Ii (t)σK (t)1+α
0(5)s.t. K̇i = Ii − δKi
Firm i maximizes a discounted stream of profits, with an exogenous discount rate ρ. The firstterm represents per period profits, net of spending and receipts on information acquisition. It
is a fraction 1/σ of the aggregate sales derived in Equation (4). In addition, firm i purchases
information about Ii new suppliers at a price pI each, and it sells information about I suppliers
at a price pI each. Finally, firm i pays to convex adjustment cost G to acquire new suppliers.
The solution to this classical problem is such that the “investment” rate is independent of thestock of “capital” (Gibrat’s law). In other words, firm i increases its number of suppliers Ki at a
rate that is independent of Ki ,Ii = β

�p

I

w

�

, ρ, δ × Ki

where the function β sumarizes the contributions of the production function and the adjustment
cost function that are relevant for the optimal investment decision. In general equilibrium, I could
solve for the sequence of prices pI (t) /w (t). In this model as in any model where growth is driven
by the accumulation of one factor of production (here K) combined with labor under constantreturns to scale, growth ultimately is driven by population growth, as in Solow (1956). While I do
not explicitly solve for such a steady state growth path, I will assume that the economy is alongsuch a path, so that I/K = β is constant.
Because all firms are charging the same price pI per contact information, firm i has no reason
to direct its search for new suppliers to any particular k ∈ Ki . To break this indeterminacy, I

assume that the Ii new names come uniformly from all existing suppliers Ki . This means thatany one of the existing suppliers k ∈ Ki reveals one of the names of its suppliers, k � ∈ Kk , with aprobability βdt over a small time interval dt. To break the indeterminacy of which name k � ∈ Kk

gets revealed by firm k, I simply assume that k draws k � at random among its existing Kk contacts.
Here is a recap of the conclusions of the model: A firm is born with an initial mass K0of suppliers, distributed geographically according to the p.d.f. g0 . Subsequently, contacts are8

randomly created at a rate β and lost at a rate δ, with each new contact coming from the suppliersof the firm’s existing suppliers. The next section characterizes explicitly the dynamic evolution offirm level trade flows, i.e. trade between the suppliers and customers of this model.

1.2

Firm level trade flows

In this section, I spell out a dynamic model of firm level trade flows that incorporates the key results
derived from the economic model in the previous section. With the exception of the population
growth rate γ, all the parameters introduced in this section (K0 , g0 .β, δ) are the same as the onesin the economic model above. I treat the arrival rate of new contacts, β, as a parameter, knowing
from the model above that it is the solution to the dynamic optimization problem of the firm.
Heuristically, the model is as follows.
New firms are continuously born. When a firm is born, it randomly contacts a geographicallybiased mass of firms over the entire world. After this initial period, contacts are randomly lostand created. Old contacts are lost to i.i.d. shocks. New contacts are created in the following way:each period, with some probability, a firm receives names from the contact lists of its existing
contacts. In other words, a firm gradually meets the contacts of its contacts, who themselvesacquire contacts in a similar way, etc.
Formally, the model is as follows.
Space: Firms are uniformly distributed over an infinite one-dimensional continuous space
represented by R. Each coordinate along that line can be thought of as representing a city, andcountries can be thought of as an arbitrary partition of that space, where a country is then acollection of cities, or an interval of the real line.
Time: Time is continuous. In every location, new firms are born continuously, with the
population of firms in each location growing at a constant rate γ, where γ stands for “growth”. At
time t, there is the same density of firms eγt in every location, where I normalize the population
at t = 0 to 1. As the model is perfectly symmetric, I will focus my attention on a firm located at
the origin.
Birth of a firm: When a firm is born, it samples a mass K0 of contacts, distributed geographically according to the p.d.f. g0 (·). So the mass of contacts it acquires in the interval [a, b] is´b
K0 a g0 (x) dx. I assume that g0 is symmetric and has a finite variance, but can take any arbitrary
9

shape otherwise. For simplicity, I assume that when a firm is born, it samples contacts only among
other newly born firms: firms within each cohort gradually get connected to each other.5
Death of a firm: I assume that firms are infinitely lived. This assumption is inocuous, andall results would carry through if firms are hit by random Poisson death shocks. A positive death
rate for firms would simply be added to the death rate of contacts below.
Birth of contacts: New contacts are continuously created as follows. At any point in time,each existing contact may reveal one of its own contacts according to a Poisson process with arrival
rate β, where β stands for “birth”.
Death of contacts: Existing contacts are continuously lost according to a Poisson processwith arrival rate δ, where δ stands for “death”.
I assume γ > β − δ > 0. While the second assumption β − δ > 0 is not required to derive myresults, it would generate counter-factual predictions, such as an infinitly long tail of infinitesimallysmall firms and firm sizes shrinking on average.
I will now define two concepts: the function ft describes the geographic distribution of thecontacts of a firm of age t, and the variable Kt describes the total mass of contacts of this firm,ˆ+ft : R → R and Kt ≡ft (x) dx(6)Rft (x) is the density of contacts of a firm of age t in location x. In other words, the mass of contacts´ba firm of age t in the interval [a, b] is a ft (x) dx. The total mass of contacts a firm of age t has

worldwide is then Kt . Note that as ft does not sum up to 1, it is not a probability density. The
normalized ft /Kt on the other hand is a well defined p.d.f.
The distribution of contacts evolves recursively according to the following Partial Diﬀerential
Equation,∂ft (x)=β∂tft (x − y)× ft (y) dy − δft (x)
Kt
R

ˆ

(7)

with the initial condition f0 (x) = K0 g0 (x).
I multiply both sides of the equation by dx for a rigorous interpretation. The first termwith the integral sign on the right hand side of Equation (7) corresponds to the creation of new
contacts. It can be decomposed into four components, β, ft (x−y)
Kt dx, ft (y) dy and the integral
5

While this simplifying assumption is strong, relaxing it would force me to keep track of the entire system offirms simultaneously, and would render the model analytically intractable. Numerical simulations suggest that themain results of the paper are robust to relaxing this assumption.

10

sign

´

y∈R .

The first component, β, correspond to the Poisson arrival of new information from afirm’s contacts. With a probability βdt over a small time interval dt, any one of a firm’s contact
in location y will reveal the name of one of her own contacts. The second component, ft (x−y)
Kt dx,
corresponds to the probability that a contact in location y reveals the name of one of her contacts
in a neighborhood dx of x.6 Note here that I impose the simplifying assumption that a firm of
age t only meets other firms in the same cohort, who themselves have the same distribution ft .
The third component, ft (y) dy, corresponds to the fact that a firm of age t has potentially several
contacts in a neighborhood dy of y (exactly ft (y) dy of them), each of whom can potentially release´
the name of one of its contacts in x. The fourth component, y∈R , corresponds to the fact that the
information about new contacts in x can potentially be intermediated via contacts in any locationy ∈ R. The second term with the minus sign on the right hand side of Equation (7) corresponds

to the destruction of old contacts. Any one of the existing ft (x) dx contacts of a firm of age t in aneighborhood dx of x may be destroyed with the same probability δdt over a small time interval
dt.
The Partial Diﬀerential Equation (7) admits an explicit analytical solution, which I relegate toAppendix A in the interest of conciseness. While the mathematically less inclined reader may skip
the derivation of this solution, it contains a number of analytical tools that may prove useful ina variety of economic settings. The analytical solution to the geographic distribution of contacts
ft allows me to derive closed-form solutions for the number of contacts of an individual firm, its
distribution within the population, and the geographic location of these contacts. Formal proofsof all results are provided in Appendix A.
First, the model predicts that as a firm ages, the number (mass) of its contacts increases,
Kt = K0 e(β−δ)t(8)The total number of a firm’s contacts grows at a constant rate equal to the net birth rate ofcontacts (birth rate β minus death date δ).
Second, as both the number of a firm’s contacts and the number of firms grow exponentially,
the model predicts that the distribution of the number (mass) of contacts within the population
6

ft
Since the distribution ft sums up to Kt , the normalized Kt
is a well defined p.d.f. that sums up to one.
Moreover, the distribution of contacts for a firm located in y is the same as for a firm located in the origin (y = 0),
where all coordinates are simply shifted by the constant −y: f0,t (x) = fy,t (x − y).

11

is Pareto distributed. The fraction F (K) of firms with K or fewer contacts is given by,
F (K) = 1 −

�

K
K0

�− γ

β−δ

for K ≥ K0

(9)

From Equation (8), young firms have fewer contacts than old ones. The larger is the growth rateof the population as a whole, γ, the more young firms relative to old ones, the fewer firms with alarge number of contacts, and the thinner the upper tail of the Pareto distribution of the numberof contacts. From Equation (8) also, the higher is the growth rate of a firm’s contacts, the larger
the mass of contacts of old firms relative to young ones. The larger is the net birth rate of new
contacts, β − δ, the more firms with many contacts, and the fatter the upper tail of the Pareto
distribution of the number of contacts.

If, as is approximately verified in the data, the cross-sectional distribution of the sizes ofexporters is close to a Zipf’s law, then we should expect the Pareto shape parameter to be closeγγ
to 1, β−δ≈ 1+ .7 Beyond this empirical evidence, the assumption that β−δ≈ 1+ seems to bea good candidate for a stationary system, where the number of contacts of existing firms grows

approximately at the same rate as the population as a whole. While deviations from this stationary
benchmark are to be expected in the data, these deviations ought not to be large.
Third, the model predicts that as a firm ages, not only does it acquire more contacts, but
those contacts become increasingly dispersed over space. Let me denote by fK the geographic
distribution of contacts of a firm with K contacts.8 The average (squared) distance from thecontact of a firm with K contacts, ∆ (K), increases with its number of contacts,∆ (K) ≡
where ∆0 ≡

´

ˆ

R

x

2 fK (x)Kdx = ∆0

�

K
K0

� β

β−δ

(10)

2
R x g0 (x) dx is the average (squared) distance from a firm’s initial contacts. Whilea firm’s initial contacts are some distance away, each wave of new contacts come from firms who
are themselves further away. As a consequence, each wave of new contacts is geographically moredispersed than the previous one. This eﬀect is compounded by the fact that a firm’s contacts arealso acquiring more and more remote contacts. Since a firm acquires more contacts as it ages, themore contacts a firm has, the more dispersed these contacts are.
7
Note that this simple model with a constant growth rate of the population and of the number of contacts
corresponds to the Steindl (1965) model of firm growth. More elaborate stochastic models such as Gabaix (1999)
or Luttmer (2007) deliver an invariant size distribution that is close to Zipf’s law in a more general set-up. I choose
to use the simpler Steindl model while adding substantial complexity on the geographic dimension of the model. I
conjecture that including the stochastic elements of Gabaix (1999) or Luttmer (2007) would not change my results.
8
fK = ft(K) with t (K) s.t. Kt(K) = K.

12

Note that the particular moment ∆ (K) only depends on two parameters of the distribution
g0 : ∆0 and K0 . For any two distributions g0 and g0� that share the same ∆0 and K0 , the average(squared) distance from a firm’s contact will evolve in the exact same way as a firm acquires
more contacts, no matter how diﬀerent g0 and g0� are otherwise. This result arises for the same
reason that the n-th derivative of the composition of several functions only depends on their firstn derivatives: ∆ (K) is the second moment of the p.d.f. gK = fK /K, which is given by the second
derivative of the moment generating function of gK ; this second derivative does not depend on
any derivative of the moment generating function of g0 above the second one.
Note the following interesting special case, and asymptotic result: if the initial distribution ofcontacts g0 is a Laplace distribution (a symmetric two-sided exponential distribution), then eachsubsequent distribution gt = ft /Kt is also Laplace, with the variance increasing at a constant rate
β. Moreover, for any initial distribution g0 , the distribution gt converges to a Laplace distribution
as t grows large.
Having characterized the distribution of contacts for all firms, I analyize next the aggregate
distribution of contacts, and the structure of aggregate trade flows in this economy.

1.3

Aggregate trade flows

Each firm trades one unit with each of its contacts. I have shown in the previous section thatolder firms have more numerous and dispersed contacts. Knowing the distribution of contacts ofeach firm, I can characterize the patterns of aggregate trade flows between firms in any set of
locations. The following lemma and proposition show that aggregate trade flows in this model
obey the gravity equation in international trade.

Lemma 1 For any distribution g0 of initial contacts that is symmetric and admits a finite variance, aggregate trade flows between two countries A and B are approximately proportional to theirrespective sizes (GDPA and GDPB ), and inversely related to the distance between them (DistA,B ),TA,B ∝
with � ≡ 2 min�rate of contacts.

�γ−(β−δ),1
βGDPA × GDPB(DistA,B )1+�

, γ the population growth rate and β (resp. δ) the birth (resp. death)13

Proposition 1 If the distribution of export sizes among individual firms is close to Zipf ’s law,then aggregate trade flows between two countries are approximately proportional to their respective
sizes and inversely proportional to the distance between them. The canonical gravity equation holds,
TA,B ∝

GDPA × GDPBDistA,BThe role of economic size in this model is relatively straightforward, and in essence similar to
the role of size in existing trade models. If an exporting country doubles in size, it has twice as
many firms (each with its own foreign contacts) and aggregate exports double. Symmetrically, if
an importing country doubles in size, its aggregate imports double. Note also that as in traditional
trade models, this argument is exact only for the case of a small economy, i.e. one that has anegligible size relative to the rest of the world. If a country becomes a non-negligible fractionof the world, part of world trade will now take place within its borders, so that the elasticity ofaggregate trade with respect to size eventually decreases below one for large countries.
The role of distance on the other hand is novel compared to existing trade models.
While the exact intuition behind the precise functional forms in Lemma 1 is mathematicallyarduous, a simplified explanation would be as follows. Each cohort has a diﬀerent distributionof contacts. From Equation (9), the distribution of the number of contacts in the population
is a power law. From Equation (10), the variance of the distributions of contacts of each firm
(the average squared distance from the firm’s contacts) is a power function of the number ofcontacts of this firm. So the variances of the various distributions of contacts are themselves
power law distributed. It turns out that the aggregation of a family of distributions with powerlaw distributed variances is approximately a power law. This result is powerful and holds nomatter what the exact shape of each distribution is. In particular, I do not need to impose any
restriction on how distance aﬀects the formation of contacts.9
To understand why the aggregation of a family of distributions with power distributed variances is approximately a power law itself, consider the following simplified set up: assume thateach of these distributions can be approximated by a uniform distribution. A firm with K�contacts with a variance ∆ (K) has therefore a constant density K/4 ∆ (K) over the interval
9While I assume that distance aﬀects the creation of initial contacts, I only impose that new contacts are
symmetric (they are equally likely to be formed “eastward” or “westward”), and they occur on average at a finite(squared) distance. Beyond these two minimal regularity conditions, the relationship can take any arbitrary shape.

14

�

�
�
�
−2 ∆ (K), +2 ∆ (K) . Only those firms that have contacts distributed with a standard devi-

ation higher than x/2 will export at a distance x. The aggregate amount exported at a distancex is then the sum (integral) of the number (density) of contacts of each of those firms. Since the�K’s are power law distributed, and the ∆ (K) are a power function of K, the amount exported
is a power function of x (the integral of a power function is a power function). Formally, using

Equation (9) for the distribution of K’s and Equation (10) for ∆ (K), the fraction (density) offirms that export at a distance x, which I denote ϕ (x), is given by the following expression,�� ˆ +∞
�k1
�
ϕ x = 2 ∆ (K) ∝dF (k) ∝γ−(β−δ)1+2
4 ∆ (k)Kβx

The intuition for why a higher γ, lower β or higher δ increase the exponent on distance inthe gravity equation is more straightforward. The contacts of younger firms are geographicallyless dispersed than those of older firms. The faster the population growth rate, i.e. the higherγ, the more younger firms there are relative to older ones: aggregate trade declines faster with
distance. The less frequently firms acquire new contacts, i.e. the lower β, the fewer chances firms
have to expand their network of contacts towards longer distances: firm level and aggregate tradedeclines faster with distance. δ plays the opposite role to β: the higher δ, the faster aggregate
trade declines with distance.
Proposition 1 shows that the -1 distance elasticity of aggregate trade is related to Zipf’s law for
the distribution of the size of firm level exports. Formally, it is the same assumption that generates
��γ
Zipf’s law for the distribution of firm level exports β−δ≈ 1+ that also makes aggregate trade��+ . In this model, firms thatapproximately inversely proportional to distance 1 + 2 γ−(β−δ)≈
1βexport a lot, i.e. firms with many contacts, are also firms that export far away. The same
parameter condition that gives the highest share of total exports to large firms, Zipf’s law, also
gives the highest share in aggregate exports to firms that export far away. With exports a powerfunction of distance, this corresponds to the gravity equation with a -1 distance elasticity of trade.
This result however is not tautological. Zipf’s law describes the distribution of total exports
of individual firms within the population. It says nothing about where those exports go. While
Zipf’s law is a statement about how much diﬀerent firms export, the gravity equation is a statement
about where firms export.10
10

The mathematical properties that generate Zipf’s law and the gravity equation are also diﬀerent. Zipf’s law isderived as the solution to a diﬀerential equation, while the gravity equation is derived from the regularly-varying
property of a sequence of functions. The only direct connection between both results is that the same stationarity
condition is required to get a -1 coeﬃcient for the power law distribution of firm exports and for the distanceelasticity of trade.

15

On a more conceptual level, this model departs from existing traditional models in its treatment of distance and trade barriers. In existing models, distance captures or proxies physical tradebarriers. In this model, distance captures informational barriers and the network that transmits
information. As in the Krugman (1980) model, the premise of this model is that if left unhindered,
all firms would export to all countries. In the Krugman (1980) model, trade barriers are the only
impediment to trade; they can be circumvented to the extent that firms can cover those trade
costs. In this model on the other hand, while informational barriers can also be circumvented
by paying some direct cost (the g0 function is a very general reduced form for the direct cost of
information acquisition), more importantly, they can be circumvented by the direct interaction
between people. This feature implies that information about potential foreign contacts is transmitted along individual connections. Advances in transportation or communication technologies
aﬀect physical trade barriers as well as the direct cost of information (the function g0 ), but not
the need for direct interactions. A model that only features direct costs will fail to explain why
distance plays the same role today as it did a century ago. In this model on the other hand,the shape of aggregate trade flows is immune to changes in the g0 function. If direct interactions
between people play a role today as it did a century ago, this model predicts that the role for
distance will remain unchanged.

2

Empirics

THIS SECTION IS CURRENTLY VERY PRELIMINARY AND INCOMPLETE...
The theoretical model above predicts that if the distribution of firm level total exports is close
to Zipf’s law, and if the average (squared) distance of a firm’s exports is a power function of thisfirm’s number of contacts, then aggregate exports follow the gravity equation.
To confirm this prediction, using firm level data on export for France in 1992, I show in Figure
2 the relation between the log of the rank of a firm, versus the log of its size. The relationship isvery close to Zipf’s law for large exporters. In Figure 3, I show the relation between the log of theaverage (squared) distance from a firm’s exports, ∆ (K) versus the log of the number of foreigncountries it exports to. The relationship between ∆ (K) and K is well approximated by a powerfunction. The estimated slopes of these two relationships would predict according to my modelthat the number of firms that export to country c at a distance Distc should be proportional to
1.16
1/Dist1.17
c . In the data, it is proportional to 1/Distc .

16

Figure 2: The distribution of firm level total exports is Zipf.
Notes: This graph shows on a log-log scale the fraction (in percentiles of the population) of firms that export morethan x as a function of the value x of a firm’s total exports (in million French Francs). This distribution is wellapproximated by Zipf’s law for the largest firms, as shown by the straight fitted line in this log-log scale. The
estimated slope is -1.0386 (s.e. 0.00185, adj-R2 =99.24%). Data: all French exporters, 1992.

Conclusion
This paper oﬀers the first theoretical explanation for the gravity equation in international tradein the sense that it explains not only why trade is proportional to size, but also the mysterious-1 distance elasticity of trade. This explanation is immune to the critique that the impact of
distance on trade ought to change with changes in the technology for trading goods, in the typesof goods traded, in the political barriers to trade, in the set of countries involved in trade, etc.
As long as the individuals that make up firms engage in direct communication with their clientsand suppliers, and as long as information permeates through these direct interactions, one ought
to expect that aggregate trade is close to proportional to country size and inversely proportional
to distance.

17

Figure 3: Average (squared) distance of exports versus number of export destinations.
Notes: This graph shows on a log-log scale the average (squared) distance of exports, ∆ (K), among firms thatexport to K foreign countries, as a function of K. The relationship is close to a straight line in this log-log scale,suggesting that ∆ (K) is well approximated by a power function of K. Distances are measured in thousand km’s.
Data: all French exporters, 1992.

References
Acemoglu, Daron, Asuman Ozdaglar and Alireza Tahbaz-Saleh. 2010. “Cascades in Networks and Aggregate Volatility,” MIT, mimeo.
Ahn, JaeBin, Amit K. Khandelwal and Shang-Jin Wei. Forthcoming. “The Role of Intermediaries in Facilitating Trade,” Journal of International Economics.
Anderson, James E. 1979. “A Theoretical Foundation for the Gravity Equation.” American
Economic Review, 69(1): 106–16.
Anderson, James E., and Eric van Wincoop. 2003. “Gravity with Gravitas: A Solution to theBorder Puzzle,” American Economic Review, 93(1): 170–92.
Antràs, Pol and Arnaud Costinot. Forthcoming. “Intermediated Trade,” Quarterly Journal of
Economics.
18

Arkolakis, Costas, Arnaud Costinot and Andres RodrÌguez-Clare. Forthcoming. “New
Trade Model, Same Old Gains?” American Economic Review.
Atalay, Enghin, Ali Hortaçsu, James W. Roberts, and Chad Syverson. 2010. “On the
Network Structure of Production,” University of Chicago, mimeo.
Atalay, Enghin. 2011. “An Exact Expression for the In-degree Distribution of the Jackson andRogers (2007) Model,” University of Chicago, mimeo.
Barabási, Albert-László and Réka Albert. 1999. “Emergence of Scaling in Random Networks,”
Science, 286: 509-12.
Bramoullé, Yann and Brian W. Rogers. 2009. “Diversity and Popularity in Social Networks,”
CIRPEE Working Paper No. 09-03.
Burchardi, Konrad B. and Tarek A. Hassan. 2010. “The Economic Impact of Social Ties:
Evidence from German Reunification.” Chicago Booth, mimeo.
Carvalho, Vasco M. 2010. “Aggregate Fluctuations and the Network Structure of Intersectoral
Trade,” CREI mimeo.
Chaney, Thomas. 2008. “Distorted Gravity: The Intensive and Extensive Margins of International Trade,” American Economic Review, 98(4): 1707-21.
Chaney, Thomas. 2011. “The Network Structure of International Trade,” NBER WP 16753.
Combes, Pierre-Philippe, Miren Lafourcade and Thierry Mayer. 2005. “The Trade-Creating
Eﬀects of Business and Social Networks: Evidence from France,” Journal of International Economics, 66(1):1-29.
de Haan, Laurens. 1976. “An Abel-Tauber Theorem for Laplace Transforms,” Journal of the
London Mathematical Society, s2-13(3): 537-542.
Disdier, Anne-Célia and Keith Head. 2008. “The Puzzling Persistence of the Distance Eﬀect on
Bilateral Trade,” Review of Economics and Statistics, 90(1): 37-48.
di Giovanni, Julian and Andrei A. Levchenko. 2010. “Firm Entry, Trade, and Welfare in Zipf’s
World,” University of Michigan mimeo.
Eaton, Jonathan, Marcela Eslava, C.J. Krizan, Maurice Kugler, and James Tybout. 2010.
“A Search and Learning Model of Export Dynamics,” Penn State University, mimeo.
Eaton, Jonathan and Samuel Kortum. 2002. “Technology, Geography, and Trade,” Econometrica, 70(5): 1741–79.
Eaton, Jonathan, Samuel Kortum, and Francis Kramarz. Forthcoming. “An Anatomy ofInternational Trade: Evidence from French Firms,” Econometrica.
19

Eaton, Jonathan, Samuel Kortum, and Sebastian Sotelo. 2010. “International Trade: Linking
Micro and Macro,” University of Chicago, mimeo.
Erdös, Paul and Alfréd Rényi. 1959. “On Random Graphs,” Publicationes Mathematicae, 6:2907.
Feenstra, Robert C., Robert E. Lipsey, Haiyan Deng, Alyson C. Ma, and Hengyong Mo.
2004. “World Trade Flows: 1962-2000,” NBER WP 11040.
Gabaix, Xavier. 1999. “Zipf Law for Cities: an Explanation,” Quarterly Journal of Economics,
114(3): 739-67.
Gabaix, Xavier. 2008. “Power Laws,” The New Palgrave Dictionary of Economics. Second
Edition. Eds. Steven N. Durlauf and Lawrence E. Blume. Palgrave Macmillan, 2008.
Halpern, Lázló, Miklós Koren and Adam Szeidl. 2009. “Imported Inputs and Productivity,”
Central European University mimeo.
Hayashi, Fumio. 1982. “Tobin’s Marginal q and Average q: A Neoclassical Interpretation,”
Econometrica, 50:213-224.
Helpman, Elhanan, Marc J. Melitz, and Yona Rubinstein. 2008. “Estimating Trade Flows:
Trading Partners and Trading Volumes,” Quarterly Journal of Economics, 123: 441-487.
Hidalgo, César A., B. Klinger, Albert-László Barabási, and Ricardo Hausmann. 2007. “The
Product Space Conditions the Development of Nations” Science, 317(5837): 482-487.
Jackson, Matthew O. 2010. “An Overview of Social Networks and Economic Applications,” The
Handbook of Social Economics, edited by J. Benhabib, A. Bisin, and M.O. Jackson, North Holland
Press.
Jackson, Matthew O., and Brian W. Rogers. 2007. “Meeting Strangers and Friends of Friends:
How Random Are Social Networks?” American Economic Review, 97(3): 890-915.
Krugman, Paul. 1980. “Scale Economies, Product Diﬀerentiation, and the Patterns of Trade,”
American Economic Review, 70(5): 950-59.
αβ2Kummer, Ernst Eduard. 1836. “Über die hypergeometrische Reihe 1 + 1.γ
x + α(α+1)β(β+1)
1.2.γ(γ+1) x +α(α+1)(α+2)β(β+1)(β+2) 3
x + etc.”
1.2.3.γ(γ+1)(γ+2)(in German). Journal für die reine und angewandte Mathematik,

15: 39–83.
Lucas, Robert E. Jr. 1967. “Adjustment Costs and the Theory of Supply,” Journal of Political
Economy, 75:321-334.
Luttmer, Erzo G. J. 2007. “Selection, Growth, and the Size Distribution of Firms,” Quarterly
Journal of Economics, 122(3): 1103-44.
20

Mayer, Thierry, and Soledad Zignago. 2006. “Notes on CEPII’s Distances Measures,” mimeo.
McCallum, John. 1995. “National Borders Matter: Canada-U.S. Regional Trade Patterns,”
American Economic Review, 85(3): 615-623.
McPherson, Miller, Lynn Smith-Lovin, and James M. Cook. 2001. “Birds of a Feather:
Homophily in Social Networks,” Annual Review of Sociology, 27: 414-44
Melitz, Marc J. 2003. “The Impact of Trade on Intra-Industry Reallocation and Aggregate
Industry Productivity,” Econometrica, 71(6): 1695-1725.
Rauch, James E. and Vitor Trindade. 2002. “Ethnic Chinese Networks in International Trade,”
Journal of International Economics, 84(1): 116-30.
Romer, Paul M. 1990. “Endogenous Technological Change,” Journal of Political Economy, 98(5):S71-S102.
Santos Silva, J.M.C. and Silvana Tenreyro. 2006. “The Log of Gravity,” Review of Economics
and Statistics, 88:641-658.
Slater, Lucy Joan. 1966. Generalized Hypergeometric Functions. Cambridge, UK: Cambridge
University Press.
Solow, Robert M. 1956. “A Contribution to the Theory of Economic Growth,” Quarterly Journalof Economics. 70 (1): 65–94.
Steindl, Josef. 1965. “Random Processes and the Growth of Firms,” Charles Griﬃn, London.
Tinbergen, Jan. 1962. “An Analysis of World Trade Flows,” in Shaping the World Economy,edited by Jan Tinbergen. New York, NY: Twentieth Century Fund.

21

APPENDIXA

Mathematical proofs

Proposition 2 The geographic distribution of the contacts of a firm of age t is given by,��(β−δ)t B [g (x)]Ke00ft (x) = B −1
(1 − eβt ) B [g0 (x)] + eβtwhere B is the two-sided bilateral Laplace transform,11 B −1 its inverse, K0 is the mass of initial

contacts of a newly born firm, g0 is the p.d.f. of these initial contacts, β (resp. δ) is the Poisson
birth (resp. death) rate of new (resp. old) contacts.
Proof. Recognizing a convolution product12 in Equation (7), I can rewrite it in a compact
form,
∂ft
ft ∗ ft
=β
− δft
∂t
Kt

(11)

with initial condition f0 = K0 g0 . I will first solve for Kt , and then solve for ft . Integrating
Equation (11) over R, and using the fact that the integral of the convolution of two functions is
the product of their integrals, I derive an ordinary diﬀerential equation for Kt ,∂Kt
Kt × K t=β− δKt = (β − δ) Kt∂t
Kt
with initial condition K0 . This ODE admits the simple solution,
Kt = K0 e(β−δ)tPlugging this result into Equation (11), taking the two-sided Laplace transform of this equation (Idenote by fˆ the transform of f ), and using the convolution theorem which states that the Laplacetransform of the convolution of two function is the product of their Laplace transforms, I get thefollowing ordinary diﬀerential equation,
∂ fˆt
fˆt2
=β
− δ fˆt∂t
K0 e(β−δ)t11The two-sided Laplace transform is closely related to the moment-generating
For a random variable� function.
�
X with a p.d.f. f , the moment generating function µX is defined as µX (s) = E esX , while the Laplace transform��´B [f ] is defined as B [f ] (s) = E e−sX = R e−sx f (x) dx, so that µX (−s) = B [f ] (s). This definition extends topositive functions which are not probability densities.
12
Remember that the p.d.f. of the sum of two random variables is the convolution of their p.d.f.’s.

22with initial condition fˆ0 = K0 ĝ0 . This ODE admits the solution,fˆt =K0 e(β−δ)t ĝ0(1 − eβt ) ĝ0 + eβt

Taking the inverse Laplace transform, I recover the proposed solution for ft .

Corollary Equations (8), (9) and (10) are satisfied,
Kt = K0 e(β−δ)t�� γK − β−δ
F (K) = 1 −for K ≥ K0
K0
�
� β
K β−δ
∆ (K) = ∆0
for K ≥ K0
K0
Proof. Equation (8). Using the property of the Laplace transform, the total mass of contactsof a firm of age t, Kt , is the Laplace transform fˆt (s) evaluated at zero,
Kt = fˆt (0) = K0 e(β−δ)twhere I used the fact that since g0 is a well defined p.d.f. that sums up to 1, ĝ0 (0) = 1.
Equation (9). The formula for Kt provides the following relation between a firm’s numberof contacts and its age,
t

e =

�

Kt
K0

� 1

β−δ

The population grows at an exponential rate γ so that the fraction of firms younger than t is��
1 − e−γt . Since a firm of age t has a total number of contacts Kt , using the above expression
for et , I get the proposed formula for the fraction of firms with fewer than K contacts,
F (K) = 1 −

�

K
K0

�− γ

β−δ

Equation (10). The average (squared) distance between a firm of age t and its contacts,∆t , is the variance of the p.d.f. ft /Kt of the distribution of this firm’s contacts. Again using
the property of the Laplace transform, this variance is simply the second derivative of f�
t /Kt (s)evaluated at zero. Simple algebra gives this second derivative,
�
��
�
�
�
��
eβt ĝ0 ” (s) eβt − 1 ĝ0 − eβt − 2ĝ0� (s)2 eβt − 1f�
t /Kt ” (s) =3((eβt − 1) ĝ0 (s) − eβt )23

Since g0 is a well defined symmetric p.d.f. with finite variance, I can use the following propertiesof its Laplace transform: ĝ0 (0) = 1 (a p.d.f. sums up to 1), ĝ0� (0) = 0 (g0 is symmetric) andĝ0 ” (0) = ∆0 (g0 has a finite variance ∆0 ). The previous expression evaluated at zero simplifies
into the proposed formula,βt∆t = f�
t /Kt ” (0) = ∆0 e
1Plugging the expression et = (Kt /K0 ) β−δ into the above formula for ∆t , I derive the proposed
relationship between a firm’s total number of contacts K and the average (squared) distance fromits contacts, ∆ (K),
∆ (K) = ∆0

�

K
K0

� β

β−δ

� �
�
Proposition 3 If the initial distribution of contacts is Laplace, g0 ∼ Laplace 0, ∆0 /2 , then���
the distribution of contacts remains Laplace at all subsequent period, gt = ft /Mt ∼ Laplace 0, eβt/2 ∆0 /2 .
Moreover, for any initial distribution g0 , asymptotically as t → ∞, the p.d.f. of contacts,��
gt = ft /Kt , converges to a Laplace 0, eβt/2 distribution,
1
ĝt (s)∝2
t→∞ 1+(eβt/2 s)��1gt (x)∝ 2eβt/2
exp − |x| /eβt/2t→∞Proof. For simplicity, consider a normalized Laplace distribution g0 ∼ Laplace (0, 1). The�1
proof can trivially be extended to ∆0 /2 �= 1. The Laplace transform of g0 is ĝ0 (s) = 1+s2.
From Proposition 2, the Laplace transform of g is then,ĝt (s) =fˆt (s)ĝ0 (s)
1
=
=
�
�2
βt
βt
Kt
(1 − e ) ĝ0 (s) + e
1 + eβt/2 s��
where one recognizes the Laplace transform of a Laplace 0, eβt/2 distribution.
From Equation (11), I derive an ordinary diﬀerential equation for ĝt ,��∂ĝt = β ĝt2 − ĝt∂t��Now postulate that ĝt is of the form ĝt (s) = h eβt/2 s, t . Then from the previous equation I
derive a partial diﬀerential equation for h (y, t),
�
�
∂
1 ∂
2
h=β h −h+ y h
∂t
2 ∂y
24

∂
Accepting without proof that limt→∞ ∂t
h = 0, h must asymptotically satisfy the following ordinary

diﬀerential equation,
1 ∂y h = h − h22 ∂ywhich admits the solution,h (y) =

1
1 + y2This completes the proof.
Lemma 1 (reminded) For any distribution g0 of initial contacts that is symmetric and admits afinite variance, aggregate trade flows between two countries A and B are approximately proportional
to their respective sizes (GDPA and GDPB ), and inversely related to the distance between them(DistA,B ),with � ≡ 2 min�rate of contacts.

�γ−(β−δ),1βTA,B ∝

GDPA × GDPB(DistA,B )1+�

, γ the population growth rate and β (resp. δ) the birth (resp. death)

Proof. I will prove first that aggregate trade is proportional to economic size, and second that
it is inversely proportional to distance raised to the power (1 + �).
Size: In any location x, all firms of the same age t have the same volume of exports towards
and the same volume of import from any other location. For any λ > 0, if a location, or any setof locations (any country) produces λ times as much in the aggregate, it will export and importλ times as much in the aggregate. Aggregate trade flows between any arbitrary set of locations(countries) are therefore proportional to the size of the importing and exporting countries.
Distance: Denote by ϕ (x) the p.d.f. of aggregate exports from the origin towards any locationx ∈ R. It is the weighted average of the exports of firms in the origin of all ages towards locationx, normalized to sum up to 1,

ϕ (x) ≡

γ−β+δ
M0

ˆ ∞

e−γt ft (x) dt

0

I will prove that ϕ (x) is equal to 1/x1+� for x → +∞, up to a slowly varying function L,13
ϕ (x) = L (x) ×
13

1
x1+�

A function L is said to be slowly varying around +∞ i.i.f.
lim

x→+∞

L (λx)= 1, ∀λ > 0
L (x)

25

Step 1: By virtue of Karamata’s abelian and tauberian theorem, the p.d.f. ϕ (x) is equal to
1/x1+� for x → +∞, up to a slowly varying function i.i.f. its Laplace transform ϕ̂ is such that1 − ϕ̂ (s) is equal to s� for s → 0, up to a slowly varying function. See for instance de Haan (1976)
for an application of Karamata’s theorem to p.d.f.’s. Formally, this means that I need to prove,
1 − ϕ̂ (λs)= λ� , ∀λ > 0s↓0 1 − ϕ̂ (s)lim

Step 2: Taking the two-sided Laplace transform of ϕ which I denote by ϕ̂, and using theproperties of the Laplace transform, the formula for ft in Proposition 2 and simple algebra, I get,ˆγ − β + δ ∞ −γt ˆϕ̂ =e ft dtK00ˆγ − β + δ ∞ −γt K0 e(β−δ)t ĝ0
=
e
dt
K0
(1 − eβt ) ĝ0 + eβt
0
ˆ ∞
ĝ0
= (γ − β + δ)e−(γ−π+δ)t βt
dt
e
(1
−
ĝ0 ) + ĝ0
0
�
�n
ĝ0
∞
ĝ0 −1γ−β+δ �=−βn + (γ − β + δ) /β
n=1

where I iteratively integrate by part to get the last expression.
Step 3: To save on notations, I introduce α = γ−(β−δ)
so that � = 2 min [α, 1]. Manipulatingβ
the previous expression ϕ̂, recognizing Gauss’ hypergeometric function 2 F1 , and invoking one
among the hundreds of useful properties of the hypergeometric function,14 I get,1 − ϕ̂ (s) = 1 + α∞��ĝ0 (s)ĝ0 (s)−1

�n

n+α
�
�n
∞
�
(1)n (α)nĝ0 (s)=(1 + α)n n! ĝ0 (s) − 1
n=0��ĝ0 (s)= 2 F1 1, α, 1 + α,ĝ0 (s) − 1
�
�−1 �
�
�−k
∞
Γ(α − 1)Γ(1 + α)ĝ0 (s)(1)k (1 − α)kĝ0 (s)=−Γ(α)Γ(α)
ĝ0 (s) − 1
k!(2 − α)kĝ0 (s) − 1
k=0
�
�−α �
�
�−k
∞
Γ(1 − α)Γ(1 + α)ĝ0 (s)(α)k (0)kĝ0 (s)+−Γ(1)Γ(1)ĝ0 (s) − 1
k!(α)kĝ0 (s) − 1
k=0
��� ĝ0 (s) �� > 1 and a non-integer α
for a suﬃciently small s such that ��ĝ0 (s) − 1 �
�
�
�−1
�
�−1 �
�
�−α
α
ĝ0 (s)ĝ0 (s)ĝ0 (s)=−+ Γ(1 − α)Γ(1 + α) −2 F1 1, 1 − α, 2 − α,α−1ĝ0 (s) − 1
ĝ0 (s) − 1
ĝ0 (s) − 1
n=1

14
See http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/02/02/ or a modified
version of Kummer’s Theorem (1836), as presented by Slater (1966) in Equation 1.7.1.3 on page 31.

26

Step 4: The following lemma will prove useful. If g is the p.d.f. of a random variable X
symmetric around the origin and with a finite variance 0 < V ar (X) < +∞, then its Laplacetransform ĝ is such that for any λ > 0,1−ĝ(λs)ĝ(λs)
lim 1−ĝ(s) = λ2
s↓0
ĝ(s)

To prove this lemma, note that g being a well defined p.d.f., ĝ (0) = 1. Using l’Hôpital’s rule,1−ĝ(λs)∂ĝ (λs)1 − ĝ (λs) ĝ (s)ĝ(λs)
lim 1−ĝ(s) = lim= λ lim ∂s∂s↓0s↓0 1 − ĝ (s) ĝ (λs)
s↓0∂s ĝ (s)ĝ(s)

I use the known result that ĝ (k) (0) = (−1)k µk where µk is X’s k-th moment. Since X is symmetric,
its first moment is zero, and ĝ � (0) = µ1 = 0. The limit is again indeterminate. Applying l’Hôpital’srule a second time, and by the assumption 0 < V ar (X) = µ2 − µ21 = µ2 < +∞, I prove theproposed lemma,∂

λ lim ∂s∂s↓0ĝ (λs)∂s ĝ (s)∂22

= λ2 lim ∂s∂ 2
s↓0ĝ (λs)∂s2ĝ (s)

= λ2

µ2
= λ2
µ2

Note that the assumption of finite variance is a suﬃcient but not a necessary condition. For
example, Student’s t-distribution with 2 degrees of freedom satisfies the desired property although
its variance is infinite. �
�−1
0 (s)
0 (s)
Step 5: Let h(s) = − ĝ0ĝ(s)−1= ĝ0ĝ(s)−1and note that h(0) = 0 and 1 − ϕ̂ (0) = 0. Using
l’Hôpital’s rule and the above lemma for the penultimate equality, I can now characterize the limitof interest,∂ϕ̂(λs)
1 − ϕ̂(λs)= lim ∂s∂s↓0 1 − ϕ̂ (s)s↓0∂s ϕ̂(s)
�
�
∂
α
α
α−1
h(λs)
F
(1,
1
−
α,
2
−
α,
−h(λs))
+
h(λs)
F
(2,
2
−
α,
3
−
α,
−h(λs))
+
Γ(1
−
α)Γ(1
+
α)α(h(λs))
2
1
2
1
∂s
α−1
2−α
�
�
= lim
∂
α
α
s↓0
α−1
h(s)
F
(1,
1
−
α,
2
−
α,
−h(s))
+
h(s)
F
(2,
2
−
α,
3
−
α,
−h(s))
+
Γ(1
−
α)Γ(1
+
α)α(h(s))
2 1∂sα−1 2 12−α�
λ2when α > 1 so that the second and third terms vanish=
λ2α when α < 1 so that the first and second terms vanish

lim

= λ�

This completes the proof.
Proposition 1 (reminded) If the distribution of export sizes among individual firms is close
to Zipf ’s law, then aggregate trade flows between two countries are approximately proportional totheir respective sizes and inversely proportional to the distance between them. The canonical gravity
27equation holds,
TA,B ∝

GDPA × GDPBDistA,B

Proof. From Equation (9), the distribution of export volumes among individual firms is closeγ
to Zipf’s law if β−δ
≈ 1+ . Plugging this condition into Lemma 1, one gets � ≈ 0+ , so that the

canonical gravity equation holds for aggregate trade flows.

B

Data

28