Nonlinear Gravity

Wyatt J. Brooks and Pau S. Pujolàs�

November 2014

Abstract

We extend the results of Arkolakis, Costinot and Rodriguez-Clare (2012) to envi-

ronments with non-homotheticities and show how to measure the welfare gains from

trade in a broad class of static, non-CES trade models. In these models, the elasticity

of import intensity with respect to trade costs (trade elasticity) is a function, not a

constant, implying a nonlinear version of the gravity relationship commonly studied

in the empirical trade literature. We have two main results. First, we provide an ex-

plicit formula for the trade elasticity function, which allows it to be computed in any

particular non-homothetic model. Second, we prove that, even in this environment,

the elasticity of welfare with respect to trade intensity (welfare elasticity) is equal to

the reciprocal of the trade elasticity at every point. We provide several examples of

models that are impossible to solve analytically, yet where the welfare elasticity can

be solved in closed form using our procedure allowing for one to get a closed form for

the gains from trade. We also provide su¢ cient conditions to compare the gains from

trade implied by non-homothetic models to those implied by CES models.

Keywords: Gains from Trade, Gravity, Trade Costs, Non-Homotheticities

JEL Codes: F10, F12, F13, F14

1 Introduction

Trade models are typically divided into two types: those that are analytically tractable, which

allow for easy computation of the gains from a reduction in trade costs, and non-homothetic

models that replicate detailed features of the trade data. The �rst group includes many of

�Brooks: University of Notre Dame (email: wbrooks@nd.edu); Pujolas: McMaster University (email: pu-jolasp@mcmaster.ca). We would like to thank Je¤ Bergstrand, Juan Carlos Conesa, Svetlana Demidova,Antoine Gervais, Joe Kaboski, Tim Kehoe, Vova Lugovskyy and Je¤ Thurk for useful comments. We aregrateful to Nezih Guner, Joan Llull and MOVE at the UAB for providing us space where a substantialportion of this work was completed. All errors are ours alone.

1

the workhorse models of the trade literature, such as Krugman (1980), Eaton and Kortum

(2002), and Melitz (2003). These models can typically be solved in closed form. They

accomplish this by utilizing some form of a constant elasticity of substitution aggregator,

and hereafter we will refer to them as CES models. They can match data on aggregate trade

�ows, but are not designed to capture many of the well-known patterns of international

trade, such as di¤erences in income elasticity, supply elasticity or elasticity of substitution

across sectors. The second class of models is able to match these rich patterns of trade by

introducing complexities to the preference or production structure of the economy. These

include Markusen (1986), Fieler (2011), Carol, Fally and Markusen (2014) and many others.

The ability to capture these patterns of trade comes at the cost of analytical tractability.

These models typically cannot be solved in closed form, making it di¢ cult to do simple

comparative statics on trade costs. Therefore, these models typically are not used to measure

gains from trade, even though we know that they incorporate features that are consistent

with observed patterns of trade.

The analytical tractability of the CES models comes in part from the simple relationship

that the models have between import demand and trade costs. These models exhibit a gravity

form, in which the traditional linear gravity model employed by the empirical trade literature

is correctly speci�ed, such as that used by Anderson (1979) and Bergstrand (1985). In trade

models in which trade costs have a constant marginal e¤ect on the intensity of imports,

the gains from reducing trade costs can be measured easily. As demonstrated in Arkolakis,

Costinot and Rodriguez-Clare (2012), which will hereafter be referred to as ACR, these

models have the feature that the marginal change in real income from a marginal decrease

in import intensity (which we call the welfare elasticity) is:

@ log (W )

@ log(�0)=1

"T

where W is real income at the observed level of trade and �0 is one minus the ratio of

imports to domestic gross output. Here "T is the elasticity of import intensity to trade costs

(hereafter we refer to this as the trade elasticity), which is commonly estimated using linear

gravity equations in the empirical trade literature. This implies that the total gains from

trade1 are equal to:

log

�W

WA

�=1

"Tlog(�0)

where WA is real income in autarky. This simple formula gives an exact solution for gains

1Unless otherwise indicated, throughout the paper by gains from trade we mean how much higher is realincome at the observed level of trade than it would be in autarky.

2

from trade in a large class of static trade models, including all the CES models previously

mentioned. Measuring the gains from trade here is simple, requiring only information on

aggregate imports, and the commonly measured constant trade elasticity.

In this paper, we generalize this result to a much broader class of trade models, including

those with non-homothetic preferences and non-CES production structures.2 We do this

by proving two new results. First we show how to solve for the trade elasticity in these

environments. The trade elasticity is no longer a constant, but now is a function. That

is, trade costs no longer have a constant marginal e¤ect on import intensity. We provide

a general formula for the trade elasticity, and show many examples of models that are too

complex to be solved in closed form, yet where there is a closed form solution for the trade

elasticity. The formula takes into account many margins that may, potentially, be present

in the model under consideration, as well as general equilibrium e¤ects from the reduction

in trade costs.

The trade elasticity itself is useful as a means of measuring the welfare, as we show in

our second main result. We prove that the welfare elasticity, as de�ned above, is equal

to the reciprocal of the trade elasticity.3 This is true in ACR, where both the trade and

welfare elasticities are constants. However, in our environment, both are functions and the

relationship is true point-by-point. That is:

8� ; @ log (W (�))@ log(�0(�))

=1

"T (�)

where the functional dependence of "T on � is meant to emphasize that the trade elasticity

changes as trade costs change. Then, given the formula for the trade elasticity, this provides

a means of measuring gains from trade in these non-homothetic environments.

Given these results, it is natural to compare the results from these non-homothetic en-

vironments to those from a CES model. We give su¢ cient conditions on the function "T to

measure the bias in gains from trade if one was to incorrectly assume that the trade elasticity

was a constant. To do that, notice that the gains to going from autarky to the observed

2Our assumptions are similar to those in ACR, but we dispense with the assumed CES import demandstructure. We require that the household in the domestic country has utility over goods that is strictlyconcave, twice continuously di¤erentiable and strictly increasing. There is a single factor of production ineach country that is in �xed supply. Aggregate pro�ts are proportional to factor payments, and the model isstatic. Final goods are composed of intermediates from many countries that are aggregated using a constantreturns to scale (though not necessarily CES) aggregator.

3One drawback of the approach we develop is that we must assume that there is a representative householdwhose welfare we are measuring. If preferences are non-homothetic, then we cannot rely on the familiaraggregation results, except to assume that all households in the country of analysis are identical. Whilethis is an important and unrealistic assumption, the issue of the distributional consequences of gains fromtrade is not our central point. Therefore, we save the case of within-country inequality for future work. SeeFajgelbaum and Khandelwal (2014) as an example of current work on this topic.

3

level of trade is:

log

�W

WA

�=

Z 1

log(�)

1

"T (�)

@ log(�0)

@ log(�)d log(�) =

1

"T (�)log(�00)+

Z 1

log(�)

� log(�00)"T (�)2

@"T (�)

@ log(�)d log(�)

where the second equality comes from integrating by parts. The �rst term would be the gains

from trade if the trade elasticity was incorrectly assumed to be constant. As �00 2 (0; 1),then the last term has the same sign as the derivative of the trade elasticity. Therefore, if

the trade elasticity is an increasing function, then the actual gains from trade is higher than

if the trade elasticity was constant. If it is decreasing, then the opposite. Given the results

that can be used to solve for the trade elasticity, then it is relatively easy to determine if it

is increasing or decreasing. Therefore, our results allow us to quickly compare the gains in

a non-homothetic environment to the CES case.

Before presenting a full explanation of our results, we provide two examples to demon-

strate how the methods developed in this paper can be used. We also relate our results more

fully to ACR and discuss their empirical content. We will then move on to developing the

results in steps: presenting the simplest case in Section 2, then adding more elements to the

model in each section. In Section 6 we give su¢ cient conditions to compare the gains from

trade with in the non-homothetic environment to those in a CES model, and in Section 7

we conclude.

1.1 Examples and Discussion

First, consider a simple economy where two goods are produced competitively. Good 0 is

produced in the domestic country and good 1 is produced in the foreign country, each using

inelastically supplied labor as the single factor of production. The problem of the household

in the domestic country is given by:

U = max!c��1�00 + (1� !)c

�1

01

s:t: : p0c00 + �p1c01 � w0L0

Because the two goods have di¤erent income elasticities, even this simple model is di¢ cult

to solve in closed form, especially taking into account general equilibrium e¤ects. However,

although the model cannot be solved in closed form, the results developed in later sections

allow for the trade elasticity to be solved for in closed form. It is equal to:

"T = ��( � 1)��00 + �01

4

where �0j is the fraction of the domestic country�s total expenditure that goes to purchase

good j. Now we can see that the trade elasticity itself changes as the economy becomes more

or less open. This is true for the welfare elasticity as well, as it is equal to the reciprocal

of the trade elasticity. Moreover if we want to know if the gains from trade in this model

are higher or lower than they would be with a constant trade elasticity, the fact that �00 is

increasing in � and that �00 = 1 � �01 implies that the trade elasticity is increasing if thegood produced in the home country has lower income elasticity than that from abroad, and

decreasing if the opposite. As above, this tells us immediately if the gains from trade are

higher or lower than in the constant trade elasticity case.

This shows in a simple framework how the characteristics of the goods that are imported

and consumed domestically a¤ect the gains from trade when there are non-homotheticities.

Moreover, our methods allow those gains to be measured and compared to a constant trade

elasticity case simply. All this is true even though the model above cannot be solved in

closed form.

Our methods can be applied to far more complex models. Consider the model in Carol,

Fally and Markusen (2014), referred to hereafter as CFM. This model features constant

relative income elasticity (CRIE) preferences and an Eaton and Kortum (2002) production

structure. Suppose the household in the domestic country has the following utility function:

U = maxfQkg

KXk=1

�kQ1�1=�kk

such that each sector has a di¤erent relative income elasticity. The domestic country trades

with the rest of the world subject to an iceberg transportation cost � . Then the Eaton-

Kortum production structure implies that the price of good k is:4

pk = �

��k + 1� �k

�k

� 11��k �

T0k + T1k(w1�)��k�� 1

�k

where Tjk is the level of productivity of country j in sector k, �k is the sector-speci�c com-

parative advantage parameter, �k is the elasticity of substitution between sector k varieties

and � is the gamma function.

As CFM shows, there is considerable heterogeneity in income elasticity across sectors.

Their estimates of sector-level income elasticities imply that they range by a factor of 10. This

implies that the composition of consumption and imports varies considerably as countries

change their real income or their trade costs. Therefore, there may be some interesting

4Here, the domestic country is denoted with subscript 0 and the rest of the world with subscript 1. Thedomestic wage is numeraire.

5

insights into gains from trade that can be gained from examining how di¤erent countries

produce and trade in di¤erent sectors.

Our results allow us to measure gains from trade exactly in this environment, even though

the model itself cannot be solved in closed form. Applying results from later sections, we

�nd that the trade elasticity is equal to:5

"T (�) = �KXk=1

(�k + 1� �k)X0kE0

X1kE1

X0k+X1kI

+KXk=1

(1� �k)X1k

E1

" PKj=1 �j

X0jE0PK

l=1 �lX0l+X1l

I

#

where Xik is the domestic country�s expenditure on sector k goods from country i, Ej is total

expenditure on goods from country j and I is the domestic country�s total expenditure. Using

the result above, we �nd that this trade elasticity is equal to the reciprocal of the welfare

elasticity.

As this formula shows, the trade and welfare elasticities depend not just on sector-level

income elasticities and comparative advantage parameters, but also on the composition of

trade and production. This allows for the possibility that di¤erent countries with di¤erent

sectoral composition realize di¤erent gains from trade, even if those countries have the same

intensity of trade measured in the aggregate. For example, this shows that countries that

produce more in higher comparative advantage sectors (that is, sectors with a higher �)

realize higher gains from trade.

This example also demonstrates an important di¤erence between our results and those

of ACR, both in their interpretation and empirical content. The emphasis of ACR is on how

many di¤erent models have equivalent formulae for welfare gains from trade. Our results

take the opposite approach. For any particular model there is a di¤erent formula for the

trade elasticity, even for many of the same observables. That is, in our approach you start

with a speci�c model and use our results to �nd gains from trade. From that perspective, we

would interpret ACR as a case in which a class of models all coincide on the same solution.

In terms of empirical content, our approach also di¤ers sharply from ACR. A strength

of ACR is that the trade elasticity is a single number that is commonly estimated. Instead,

in our model, the trade elasticity depends on other parameters. As an example, consider

the CFM model discussed above. That paper uses a panel of trade and production data to

identify the sector-level parameters of the model, rather than a single, aggregate constant

trade elasticity. What we provide above is a means to use the CFM estimates for the sector-

level �k and �k parameters, together with trade and production data, to measure the overall

gains from trade. This logic can be extended to show how the sector-level estimates found in

5Note that if 8k; �k = � and �k = �, and fraction of expenditure by country is constant across sectors,then the trade elasticity is ��, as in ACR.

6

many empirical studies (such as Bergstrand (1990), Feenstra (1994), Broad and Weinstein

(2006) and Soderbery (2014)) can be used to �nd an aggregate measure of gains from trade.

In each of these cases, these sector-level estimates are combined with production and trade

data to get aggregate trade and welfare elasticities. Under this interpretation, this shows

how to extend the ACR results to economies with sector heterogeneity.

When comparing CES and non-homothetic models, one may be concerned about the

fact that every non-homothetic model has a CES model that implies the same gains from a

marginal change in trade costs. As these examples demonstrate, that fact does not imply

that the gains from trade with non-homotheticities are the same as those in ACR, even at

the margin. This is true because of heterogeneity both across countries and over time. In the

�rst example above, if all countries had the same �00 and �01, and there was some vanishingly

small variation in trade costs, the trade elasticity estimated by a linear gravity model would

be equal (in the limit) to the trade elasticity that we �nd, and the marginal gains from

trade in ACR would be the same as ours no matter what sort of non-homotheticity was

present in the true model. However, this is no longer true with any non-trivial variation

within the sample. Linear gravity models provide appropriate estimates of trade elasticities

for CES models, but are incorrectly speci�ed when there are non-homotheticities. Instead,

the empirical strategy employed must match the theoretical model under consideration, as

the CFM example above illustrates.

This paper is closely related to the many papers that investigate gains from trade in

models with sector-level heterogeneity, including Broda and Weinstein (2006), Ardeland and

Lugovskyy (2010), Feenstra and Weinstein (2010) and Blonigen and Soderbery (2010). The

strength of this paper is to nest many di¤erent environments and demonstrate how to directly

relate measured trade elasticities to changes in welfare. It is also closely related to other

recent papers that have expanded the results of the original ACR paper, such as Arkolakis,

Costinot, Donaldson and Rodriguez-Clare (2013) and Allen, Arkolakis and Takahashi (2014).

These papers are focused on the relationship between expanded models and the traditional,

linear gravity model. In this paper, we consider models with non-homotheticities so that

the linear gravity regression is incorrectly speci�ed. In this sense, we view our paper as

complementary to those.

7

2 Model with Two Countries

We begin by considering a two country trade model. We take the perspective that country 0

is the country of analysis, and country 1 is the rest of the world6. Our motivating question

is: how much higher is the real income of country 0 with the observed level of trade than it

would be in autarky? In this section, we show how to answer this question in a very general

way.

2.1 Household�s problem

The household in country 0 has an additively separable utility function7 and solves the

following problem:

max1Xn=0

Zi2n

uni(cn(i))di

st :1Xn=0

Zi2n

� 0npn(i)cn(i)di = I0 � w0L0

8n;8i : cn(i) � 0

We choose to have the wage in country 0 be numeraire. Here � is the iceberg transportation

cost8 on imported goods, and L0 is the endowment of labor available in country 0. We set

� 00 = 1. The set of goods available to the household in the domestic country is partitioned

by country of origin, so that the set of goods from country n that the country 0 household

can consume is n, which may have any cardinality. Here we require only that uni be

strictly increasing, strictly concave9 and twice di¤erentiable. In general, these preferences

are non-homothetic. The non-negativity constraint may generate an extensive margin in

consumption of di¤erent types of goods.

6Instead of two countries, our results are unchanged if there are N + 1 countries, where country 0 isthe country of analysis and the other N countries are all identical to one another. In that case, the resultshere are the same when considering trade cost changes between country 0 and all of the N other countriessimultaneously. The case with heterogeneous trading partners is considered in Section 3.

7The general, non-seperable case is considered in Section 4.8All trade costs in the model take the form of iceberg trade costs. As in Arkolakis, et al (2012), our

results do not go through if it is instead a tari¤ that is rebated to the household.9Due to strict concavity and the fact that the sets of goods produced by each country are disjoint, this

model excludes goods that are perfectly substitutable across countries. We do allow perfect substitutabilityin more complex production environments in Section 5, such as Eaton and Kortum (2002) production.

8

2.2 Production

All production in the model is done by competitive �nal goods producers who use technology

linear in labor. A �rm in country n selling product i has productivity zn(i). In order to sell

ymn (i) units to the household in country m, the �rm must produce �mnymn (i) units.The �rm

solves:

maxXm

�mnpn(i)ymn (i)� wnln(i)

s:t: :Xm

�mnymn (i) = zn(i)ln(i)

The fact that these �rms are competitive implies that their price is equal to their marginal

cost, and their pro�ts are zero:

pn(i) =wnzn(i)

2.3 Market Clearing

In each country, labor is inelastically supplied and the equilibrium wage clears the labor

market:

8m;Zi2m

lm(i)di = Lm

2.4 Trade Elasticity

As has been shown in ACR, the trade elasticity (that is, the elasticity of imports as a fraction

of domestic consumption to trade costs) is the main determinant of gains from trade in a

CES model. We de�ne the import share from country n as:

�0n =

Ri2n � 0npn(i)cn(i)diP1

m=0

Ri2m � 0mpm(i)cm(i)di

Then the trade elasticity is de�ned as:10

"T (� 01) �@ log(�01=�00)@ log(�01)

1 + @ log(w1)@ log(�01)

Here we write the elasticity as a function of trade costs to emphasize that in general it is not a

constant as it is in CES models. This means that changes in trade costs have a non-constant

10Notice that this is a modi�ed de�nition from that in ACR. Here the elasticity of real exchange rates (theratio of domestic and foreign wages) with respect to trade costs is not constant, so we adjust by that term.

9

marginal e¤ect on imports, hence a �nonlinear gravity" relationship.

To economize on notation, let us de�ne:

X0n(i) = � 0npn(i)cn(i)

E0n =

Zi2n

X0n(i)

�n(i) =

(0 if cn(i) = 0

u0n(i)cn(i)u00n(i)

if cn(i) > 0

Now we derive a formula that can be used to solve for the trade elasticity. This result is

summarized in Theorem 1.

Theorem 1 Whenever �00 2 (0; 1):

"T (� 01) =

�1 +

Ri21 �1(i)

X01(i)E01

di� R

i20 �0(i)X00(i)E00

diP1m=0

Ri2m �m(i)

X0m(i)I0

di

The proof of Theorem 1 is left to the appendix. The formula is rather complicated, but

the terms in each part are the expenditure-weighted curvatures of each good�s sub-utility

function evaluated at the current level of consumption.

2.5 Examples

Here we solve out several examples to demonstrate the procedure for �nding the trade

elasticity, and for discussion later in the paper.

2.5.1 Example 1: Constant Elasticity of Substitution (CES)

Constant elasticity of substitution is a special case of the preferences given above, and, as in

Arkolakis, et al (2012), CES preferences should yield a constant trade elasticity. Therefore,

�rst we should demonstrate that this is true.

CES preferences can be written as:

U(c00; c01) =�

� � 1c1�1=�00 +

�

� � 1c1�1=�01

Then,

8n : �n(i) = ��

10

Then applying Theorem 1, we get:

"T (� 01) =�� (1� �)

�� = 1� �

which is a constant. Moreover, this provides the same solution as the equivalent example in

ACR.

2.5.2 Example 2: Heterogeneous Income Elasticity (CRIE)

Next we show how to get the result from the �rst example in the introduction by solving a

slightly generalized case. Following Fieler (2011), suppose utility is non-homothetic and has

goods with di¤ering income elasticities. Speci�cally, suppose goods imported from the rest

of the world have a di¤erent income elasticity than those goods produced in the domestic

country. Let utility be given by:

U =

Z0

!0(i)�

�� 1c0(i)1�1=�di+

Z1

!1(i)

� 1c1(i)1�1= di

With these preferences, the ratio of income elasticities of these two goods is given by the

ratio of to �. Taking ratios of �rst and second derivatives yields:

8i; �0(i) = ��; �1(i) = �

Applying Theorem 1, we get:

"T (� 01) = �� ( � 1)��00 + �01

Now we see that the trade elasticity is no longer constant. As trade costs decline, �00 shrinks

and �01 (which is equal to 1� �00) grows. If � > > 1, then the trade elasticity gets largeras trade costs get larger (that is, �00 grows), and the opposite if > � > 1. Notice also that

if � = , we recover the CES case, as expected.

2.5.3 Example 3: Consumption Requirement

Markusen (1986) introduced the idea of consumption requirements into utility functions

in trade models to better match important facts about the trade patterns of goods across

countries. We �rst consider a simple case of this, and generlize it in the next example.

Suppose there is a good produced in the home country that requires a minimum consumption

level, and a manufacturing good imported from abroad that does not. Preferences are given

11

by:

U(c00; c01) = (c00 � �c)1�1=� + c1�1=�01

Then, letting L0 = 1,

8i : �0(i) = ���1� �c

c0(i)

�; �1(i) = ��

And,

"T (� 01) = �

�1� p0�c

�00

�(� � 1)

1� p0�cIf � > 1 and the household is capable of satisfying its consumption requirement, then the

trade elasticity decreases as �00 increases (that is, when trade costs increase). Note that the

opposite is true if �c < 0, assuming that �00 > 0.

2.5.4 Example 4: Many Goods with Additive Constants

Consider a case similar to Simonovska (2014), but where we do not require the additive

constants to be positive nor that they be the same for all goods. Suppose L0 = 1 and utility

is given by:

U =

Zi20

�0(i) log(c0(i) + �c0(i)) +

Zi21

�1(i) log(c1(i) + �c1(i))

s:t: : 1 �Zi20

p0(i)c0(i) +

Zi21

� 01p1(i)c1(i)

WhereP

n

Rn�n(i)di = 1 and imposing that

R1�c1(i)=a1(i)di > 0.11 De�ne:

�n(i) =

(0 if cn(i) = 0

1 if cn(i) > 0

Then applying Theorem 1 yields:

"T =

�1� 1

�01

Ri21 � 01p1(i)(c1(i) + �c1(i))�n(i)di

�1�00

Ri20 �p0(i)(c0(i) + �c0(i))�n(i)di

�Ri20 p0(i)(c0(i) + �c0(i))�n(i)di�

Ri21 � 01p1(i)(c1(i) + �c1(i))�n(i)di

11This second condition is equivalent to the trade elasticity being negative. Let us emphasize that we donot require each value of �c1(i) be positive.

12

Let � be the Lagrange multiplier on the household�s budget constraint. Then taking �rst

order conditions and using the de�nition of �00 we can prove two useful equalities:

1

�=�00 +

R0p0(i)�c0(i)diR

0�0(i)

= 1 +Xn

Zn

� 0npn(i)�cn(i)

Then using the de�nition of �01 and rearranging:Ri21 � 01p1(i)�c1(i)di

�01= �1 +

�00 +R0p0(i)�c0(i)di

�01

R1�1(i)diR

0�0(i)di

Then we can rewrite the trade elasticity as a function of only country 0 prices (that are

independent of trade costs), preference parameters, and �00:

"T =

R0�0(i)�n(i)

�00

1�

�00 +R0p0(i)�c0(i)�n(i)di

1� �00

R1�1(i)�n(i)diR

0�0(i)�n(i)di

!

Interestingly, none of the values of c1(i) enter the trade elasticity. That information is already

encoded in the import penetration term �00. Not surprisingly, whether the trade elasticity

here is increasing or decreasing is ambiguous, and depends on the signs and magnitudes of

the �c terms.

This demonstrates that there are cases with many goods that di¤er in complex ways in

which the trade elasticity can be solved in closed form. However, it is not true that every

possible preference speci�cation has a closed form for the trade elasticity. Even this example

cannot be solved in closed form except in the logarithmic case. Our result is useful because

it does work in many cases, hence expanding the set of trade models that are analytically

tractable, and because the trade elasticity we derive is directly related to the welfare gains

from trade, as discussed in the next subsection.

2.6 Welfare Elasticity

The trade elasticity is useful because of its connection to the welfare gains from trade. ACR

showed, in models with constant trade elasticities, the welfare elasticity is equal to the

reciprocal of the trade elasticity, which makes computing the welfare gains from trade trivial

in that class of models. We show that this is also true in our environment, even though the

trade elasticity is no longer a constant. This result is summarized in Theorem 2 below. We

�rst de�ne some notation.

In this environment there is not a perfect price index so changes in real income are not

13

a simple function of changes in the price level. Instead we de�ne changes in real income as

the compensating variation needed to make the representative household indi¤erent between

the allocation they receive facing current trade costs and current income, and the allocation

they would receive with new trade costs. We will denote real income as W .

The dual of the household�s problem described above is:

I = min

1Xn=0

Zi2n

� 0npn(i)cn(i)di

s:t: �U =

1Xn=0

Zi2n

uni(cn(i))di

Note that I is not real income. For a given �U , I is increasing in � 01 since the household

would need more units of income to a¤ord the same utility level. Instead, changes in real

income are de�ned as the equivalent loss in income associated with an increase in trade costs.

Therefore,@ log(W )

@ log(� 01)= � @ log(I)

@ log(� 01)

Following Arkolakis, et al (2012), we de�ne the welfare elasticity as the elasticity of real

income with respect to changes in expenditure on domestic goods, �00. That is,

"W (� 01) �@ log(W )@ log(�01)

@ log(�00)@ log(�01)

= �@ log(I)@ log(�01)

@ log(�00)@ log(�01)

As with the trade elasticity, the welfare elasticity is explicitly written as a function of trade

costs to emphasize that it is not a constant. The relationship between the welfare and trade

elasticities is described in Theorem 2.

Theorem 2 For all � 01,

"W (� 01) =1

"T (� 01)

Proof. First note that �00 + �01 = 1, so that

@�00@ log(� 01)

= � @�01@ log(� 01)

=) �00@ log(�00)

@ log(� 01)= ��01

@ log(�01)

@ log(� 01)

Then notice that:

@ log(�01=�00)

@ log(� 01)=@ log(�01)

@ log(� 01)�@ log(�00)@ log(� 01)

= ��

�001� �00

+ 1

�@ log(�00)

@ log(� 01)= � 1

1� �00@ log(�00)

@ log(� 01)

14

Using the envelope theorem on the dual of the consumer�s problem written above, and noting

that 8i, p1(i) = w1=a1(i), yields:

@ log(I)

@ log(� 01)= (1� �00)

�1 +

@ log(w1)

@ log(� 01)

�Then,

1

"W (� 01)= �

@ log(�00)@ log(�01)

@ log(I)@ log(�01)

= � 1

1� �00

@ log(�00)@ log(�01)

1 + @ log(w1)@ log(�01)

= "T (� 01)

This theorem provides an easy means of solving for gains from trade, even in models

that are not analytically tractable. Example 2 above is a model that cannot (except in

particular parameter cases) be solved in closed form. However, using Theorem 1 we can

solve for the trade elasticity. Using Theorem 2 to get the welfare elasticity, we can then

solve for gains from trade in two ways. First, the gains from a marginal increase in import

intensity is immediate and given by the welfare elasticity. Second, we can measure gains from

trade from autarky to the observed level of trade by integrating over the marginal welfare

elasticities. This second exercise may be complicated depending on the particular form that

the welfare elasticity takes. A very simple case, for instance, is Example 2 above, where the

welfare elasticity is linear in �00 so that integration can be done by hand.

3 Extension to Model with N Trading Partners

We now analyze the e¤ects on country 0 of changing trade costs with its N other trading

partners. We assume that a parameter � governs trade costs between country 0 and all

its trading partners.12 Then we have two theorems analogous to Theorems 1 and 2 in this

environment. First, we de�ne the trade elasticity as:

"T (�) =

@ log�1��00�00

�@ log(�)PN

n=1�0n1��00

�1 + @ log(wn)

@ log(�)

�Notice that if all N trading partners are identical, then this trade elasticity is the same as

that given in Section 2.

12That is, �0j = ~�0j� and we will be considering changes in the common component � .

15

Theorem 3 Whenever �00 2 (0; 1),

"T (�) =

�1 +

PNn=1

(1+ @ log(wn)@ log(�) )PN

m=1 E0m(1+@ log(wm)@ log(�) )

Ri2n �n(i)X0n(i)di

�Ri20 �0(i)

X00(i)E00

diP1m=0

Ri2m �m(i)

X0m(i)I0

di

The proof is left to the appendix. The complication introduced with multiple trading

partners is that wages may have di¤ering responses in di¤erent countries, so that the e¤ective

change in the price of goods may di¤er across countries. This general equilibrium e¤ect would

require one to know how wages change as trade costs change in order to compute the trade

elasticity.13 With this de�nition of the trade elasticity, then the analogue of Theorem 2 is

essentially unchanged from before. The welfare elasticity is de�ned as before:

"W (�) =

@ log(W )@ log(�)

@ log(�00)@ log(�)

Theorem 4 8� ;"W (�) =

1

"T (�)

Proof. Applying the envelope theorem to the dual of the household�s problem,

@ log(W )

@ log(�)= �@ log(I)

@ log(�)= �

NXn=1

�0n

�1 +

@ log(wn)

@ log(�)

�

and

@ (1� �00)@ log(�)

= � @�00@ log(�)

=)@ log

�1��00�00

�@ log(�)

= ��1 +

�001� �00

�@ log (�00)

@ log(�)= � 1

1� �00@ log (�00)

@ log(�)

Then

"W (�) = �@ log(I)@ log(�)

@ log(�00)@ log(�)

=

11��00

PNn=1 �0n

�1 + @ log(wn)

@ log(�)

�� 11��00

@ log(�00)@ log(�)

=

PNn=1

�0n1��00

�1 + @ log(wn)

@ log(�)

�@ log

�1��00�00

�@ log(�)

=1

"T (�)

The usefulness of this result is limited by the presence of the general equilibrium terms.

In principle, one could ignore these terms and only measure the partial equilibrium gains

13Notice that if preferences are CES as in Example 1 above, we see that the terms for the changes in wagescancel out. This illustrates how CES models avoid this issue entirely.

16

from trade. Alternatively, one could use the labor market clearing conditions to solve for the

general equilibrium terms. In the appendix, we show how to solve for those terms in general.

4 Non-Seperable Preferences

For simplicity in previous sections we assumed that the utility function was additively seper-

able. In this section we dispense with that assumption and show that our results are un-

changed. That is, the household�s problem is now:

maxU�ffcn(i)gi2ngNn=0

�s:t: :

NXn=0

� 0npn(i)cn(i) � I0

This speci�cation allows for complementarity or substitutability of goods both within and

between countries.

Let H be the Hessian matrix of U . Because U is strictly concave and twice continuously

di¤erentiable, H is negative de�nite and invertible.14 The n(i) row of H contains all the

second order partial derivatives of good i in country n with all other goods. It is useful to

de�ne the following terms:

Am(j) =NXn=1

�1 +

@ log(wn)

@ log(�)

�Zn

H�1(n(i);m(j))

@U

@cn(i)�n(i)di

Bm(j) =NXn=0

Zn

H�1(n(i);m(j))

@U

@cn(i)�n(i)di

Then the general case of Theorem 1 is as follows:

Theorem 5 Whenever �00 2 (0; 1),

"T = �1�00

R0p0(j)A0(j)

L0PN

n=1 �0n

�1 + @ log(wn)

@ log(�)

�+ 1�00

R0p0(j)B0(j)PN

m=0

Rm� 0mpm(j)Bm(j)

241 + PNm=0

Rm� 0mpm(j)Am(j)

L0PN

n=1

�1 + @ log(wn)

@ log(�)

��0n

35The proof of this theorem is available in the appendix. Note that this generalizes the

previous results. If U was additively seperable, then H is a diagonal matrix, and so is H�1.

14If H is an in�nite matrix, additional regularly assumptions may be necessary. In that case, by inversewe mean the left-hand reciprocal of H, as described in Cooke (2014). A su¢ cient condition that we usein our example is that the set of goods can be partitioned into disjoint, �nite subsets fSjg such thati 2 Sn; j 2 Sm;m 6= n =) Uij = 0 so that H is block diagonal with �nitely sized blocks.

17

This implies that:

8j8n : Bn(j) =u0n(j)

u00n(j)In(j)

8j : A0(j) = 0

8j8n � 1 : An(j) =

�1 +

@ log(wn)

@ log(�)

�u0n(j)

u00n(j)In(j)

Substituting this into the formula in Theorem 6 returns the result from Theorem 4.

Due to the greater �exibility of this preference structure, solving for the trade elasticity

is necessarily more complicated. Therefore, we provide an example to show how this result

can be used to generate interesting results.

4.1 Example: Heterogeneous Elasticities of Substitution

Suppose there are two countries, and for each good produced in country 0, there is a corre-

sponding good in country 1 that is its imperfect substitute. Furthermore, we allow for the

elasticity of substitution to vary for each of these pairs, so that some goods are more sub-

stitutable across countries than others (e.g., grain is more substitutable between countries

than wine). Preferences are given by:

U =

Z

�(i)�c0(i)

(i)�1 (i) + c1(i)

(i)�1 (i)

� (i)�

��1 (i)�1

di

Here we require that 8i; (i) > 1 and � > 1. The elasticity of substitution between any goodand its corresponding good from the other country is � (i). Its elasticity of substitutionwith any other good is determined by �.

This preference structure is useful because it admits a block diagonal Hessian. That is,

consider the following matrix:

h(i) =

"@2U

(@c0(i))2

@2U(@c0(i)@c1(i))

@2U(@c1(i)@c0(i))

@2U(@c1(i))

2

#

Then the Hessian of U is composed of the block matrices h(i) along the diagonal, and zeros

everywhere else. Then the inverse of H is also block diagonal where each block is the inverse

of h(i). Because h(i) is 2x2, it is very easy to compute its inverse. We can write it easily

18

with some notation for product-level expenditure:

X0n(j) = � 0npn(j)cn(j)

X0n =

Z

� 0npn(j)cn(j)dj

We get:

h�1(i) =

"� c0(j)u00(j)

(j)X01(j)+�X00(j)X00(j)+X01(j)

c0(j)u01(j)

( (j)��)X01(j)X00(j)+X01(j)

c1(j)u00(j)

( (j)��)X00(j)X00(j)+X01(j)

� c1(j)u01(j)

(j)X00(j)+�X01(j)X00(j)+X01(j)

#Applying the de�nitions of A and B above:

A0(j) =

�1 +

@ log(w1)

@ log(� 01)

�c0(j)

( (j)� �)X01(j)

X00(j) +X01(j)

A1(j) = ��1 +

@ log(w1)

@ log(� 01)

�c1(j)

(j)X00(j) + �X01(j)

X00(j) +X01(j)

B0(j) = ��c0(j)B1(j) = ��c1(j)

Then applying the formula for the trade elasticity implies:

"T = 1� � �Z

( (j)� �)X00(j)X00

X01(j)X01

X00(j)+X01(j)X00+X01

dj

This result is interesting because it incorporates information about the patterns of trade and

consumption into the trade elasticity. The trade elasticity depends on each good�s domestic

expenditure relative to total domestic spending, the imports of each good relative to total

imports, and the total expenditure on that good relative to the total budget.

5 Extension to Other Types of Production

Up to this point we have only considered economies with simple production: each good can

only be produced by one country. We relax this assumption and show how to extend the

results we previously developed.

Suppose now that household�s problem15 can be written as follows:

15In this section, we once again assume additive separability of preferences to economize on notation asboth general preferences and general production is cumbersome.

19

max

Zi2ui(ci)di

st :

Zi2picidi = I0 � w0L0 +�0

8i : ci � 0

where we assume that all pro�ts from the production sector �0 are lump sum rebated to

the household. The di¤erence from before is that now goods are not partitioned by country.

We assume �nal goods are produced by competitive intermediaries. These �rms purchase

intermediate goods both from the domestic country and from abroad, and aggregate those

goods into the �nal good in each sector i. The price of the �nal good incorporates trade

costs from importing goods, as well as the prices of all intermediate goods. For now we do

not need to specify how these intermediate goods are aggregated, but we will discuss that

in detail in the examples that follow.

The problem of the competitive intermediary �rm is given by:

pi = minXn

Zj2�n(i)

� 0nqn(j)x0n(j)dj

st : 1 = Fi�ffx(j)gj2�n(i)gn

�8n; j : x0n(j) � 0

We make four assumptions about the intermediate good sector. First, Fi is a constant

returns to scale function. Second, the prices of intermediate goods qn(j) are linear in country

n wages and independent of trade costs. Third, we assume that �n(i); the set of available

intermediate goods from country n in industry i; is independent of wages and trade costs.16

Fourth, aggregate pro�ts from the production sector in country 0;�0;are proportional to the

wage in country 0.17 This last assumption is an important restriction in that it does not

allow us to consider environments that have pro-competitive e¤ects of trade liberalization,

which is an important ongoing area of research (see Arkolakis, Costinot, Donaldson and

Rodriguez-Clare (2014), Feenstra (2014), and Edmond, Midrigan and Xu (2014)).

Let X0n(i) be the total expenditure in country 0 on intermediate goods purchased in

16Let us emphasize that this does not imply that there is no extensive margin. We will show that Eaton-Kortum is a special case of this, for example.17Given the non-homothetic nature of this environment, zero pro�ts is the most relevant case of this

assumption. Perfect competition is su¢ cient for zero aggregate pro�ts, but an environment with a �xed costto produce and a zero pro�t condition on entry also generates zero aggregate pro�ts.

20

country n to produce the �nal good in sector i. Then de�ne:

X0n =

Z

X0n(i);

X0(i) =Xn

X0n(i)

�0n(i) =X0n(i)

X0(i)

Then �00(i) is the fraction of country 0 expenditure in sector i that is not imported. Then

for �00 de�ned as before, we can write:

�0n =

RX0n(i)

I0=

R�0n(i)X0(i)

I0

Lemma 6@ log(pi)

@ log(�)=

NXn=1

�1 +

@ log(wn)

@ log(�)

��0n(i)

The proof of this is immediate by applying the envelope theorem to the intermediate

�rm�s problem, and using the fact that intermediate good prices are linear in wages.

Lastly, we de�ne the change in the fraction of expenditure in sector i from the domestic

country as trade costs change as:

�i =

@ log(�00(i))@ log(�)PN

n=1

�1 + @ log(wn)

@ log(�)

��0n(i)

Now we can prove results analogous to those in previous sections:

Theorem 7 Whenever �00 2 (0; 1):

"T (�) = �Z

X00(i)

X00

24��i + 1 + u0iciu00i

�PNn=1

�1 + @ log(wn)

@ log(�)

��0n(i)PN

n=1

�1 + @ log(wn)

@ log(�)

��0n

35 di+

Z

X0(i)

I0

�1 +

u0iciu00i

�PNn=1

�1 + @ log(wn)

@ log(�)

��0n(i)PN

n=1

�1 + @ log(wn)

@ log(�)

��0n

di

24R X00(i)X00

u0iciu00idiR

X0(i)I0

u0iciu00idi

35

21

If country 0 has only one trading partner, then this simpli�es to:

"T (�) = �Z

��i + 1 +

u0iciu00i

� X01(i)X01

X00(i)X00

X0(i)I0

di

+

Z

�1 +

u0iciu00i

�X01(i)

X01

di

24R u0iciu00i

X00(i)X00

diR

u0iciu00i

X0(i)I0di

35Theorem 8 8� ;

"T (�) =1

"W (�)

Proof. Now the dual of the household�s problem implies:

@ log(I)

@ log(�)=

Z

@ log(pi)

@ log(�)

X0(i)

I=

NXn=1

�1 +

@ log(wn)

@ log(�)

�Z

X0n(i)

I=

NXn=1

�1 +

@ log(wn)

@ log(�)

��0n

substituting this into the de�nition of the welfare elasticity yields:

"W (�) = �@ log(I)@ log(�)

@ log(�00)@ log(�)

= �

PNn=1

�1 + @ log(wn)

@ log(�)

��0n

@ log(�00)@ log(�)

=

PNn=1

�0n1��00

�1 + @ log(wn)

@ log(�)

�@ log

�1��00�00

�@ log(�)

=1

"T (�)

Di¤erent production environments have di¤erent implications for the welfare elasticity

only in so far as they imply di¤erent values of �i. To illustrate this, we will provide two

examples of production environments commonly used in the trade literature, and one with

a non-CES production environment.

5.1 Armington Production

Suppose each country produces one intermediate good for each industry. Production of this

intermediate good is competitive, and has linear technology that transforms one unit of labor

into zn(i) units of the intermediate good. These intermediates are imperfect substitutes for

one another and every country consumes all of the varieties produced by each country. The

intermediary�s aggregator is:

Fi =

NXn=0

�1�ini x

�i�1�i0ni

! �i�i�1

22

Clearly Fi is constant returns to scale, and the set of intermediates is �xed. Because inter-

mediate goods are competitive there are no pro�ts and prices are given by:

qni =wnzni

Hence, this production environment satis�es all out assumptions above. Then solving the

intermediary�s problem, it can easily be shown that:

�00(i) =q0ix00iP

n � 0nqnix0ni=

�0iq1��i0iP

n �ni (� 0nqni)1��i

@ log(�00(i))

@ log(�)= (�i � 1)

PNn=1

�1 + @ log(wn)

@ log(�)

��ni (� 0nqni)

1��iPn �ni (� 0nqni)

1��i

= (�i � 1)NXn=1

�1 +

@ log(wn)

@ log(�)

��0n(i)

=) �i = �i � 1

In this case we have a closed form solution for how production enters the trade elasticity.

In this environment, it enters very simply as a constant. This is potentially useful for empir-

ical applications as a way to measure the implied gains from trade given supply elasticities

that vary by industry.

5.2 Eaton-Kortum Production

The Armington environment is very simple, but we get a very similar result when we consider

the production environment of Eaton-Kortum (2002). Their model has that each good is

composed of a unit measure of intermediate goods, and that each intermediate is produced in

each country. Intermediates of the same type from di¤erent countries are perfect substitutes,

and intermediates of di¤erent types of imperfect substitutes with one another. The industry-

level aggregator is:

Fi =

0@Z 1

0

NXn=0

xni(j)

!�i�1�i

dj

1A�i

�i�1

Clearly this aggregator is constant returns to scale, which satis�es the �rst assumption.

Also, the set of potential varieties is �xed, which satis�es the third assumption. We assume

there is only one factor of production, and that �rms in each country are competitive. We also

allow the elasticity of substitution across intermediates to vary by industry. Each country n

has e¢ ciency zni(j) for producing intermediate j in industry i. Therefore, the price of each

23

intermediate is:

qni(j) =wnzin(j)

Prices are linear in wages and there are zero pro�ts, which satisfy the second and fourth

assumptions. Because they are perfect substitutes, the competitive intermediaries purchase

each intermediate good from the lowest cost producer. By assuming that the country-

industry productivity level follows the Frechet distribution, they derive the following price

for the industry i good:

pi = �

��i + 1� �i

�i

� 11��i

NXk=0

Tki(wk� 0k)��i

!� 1�i

where �i and Tki are distributional parameters of the Frechet distribution: Tki controls the

absolute advantage of country k in industry i and �i controls comparative advantage in

industry i. Here � is the gamma function, which is well-de�ned when �i > �i � 1, which weassume holds for each industry.

The Eaton-Kortum model also implies that:

�0n(i) =Tni(wn� 0n)

��iPNk=0 Tki(wk� 0k)

��i

Now solving for the trade elasticity only requires to solve for the change in the intensity

of use of domestic goods within each industry as the trade costs change, which can be solved

for easily:

@ log(�00(i))

@ log(�)= �i

NXn=1

�1 +

@ log(wn)

@ log(�)

��0n(i)

=) �i = �i

The only di¤erence between the Armington model and the Eaton-Kortum model, there-

fore, is the interpretation of these production terms: �i � 1 in the Armington environment,and �i in the Eaton-Kortum environment. These have very di¤erent interpretations within

the context of the models that derive them, but as in ACR, they provide the same impli-

cation for welfare gains from trade. Like in ACR, this is driven by the underlying CES

structure inherent in both models. However, the fact that it has this simple form is not

implied by the assumptions made thus far, such as constant returns to scale aggregation.

This is demonstrated in the next example.

24

5.3 Case with Non-CES Production

As an example of a production structure that is not based on a CES aggregator, consider

the following constant returns to scale aggregator in a two country economy:

Fi =

�x�i�1�i1i + x

�i�1�i0i x

��i�1�i

��i�1�i

�1i

� �i�i�1

Each good is produced with a linear technology in each country by competitive �rms.

Then the prices of these intermediate goods are:

qni =wnzni

where zni is the productivity in sector i in country n. Solving the intermediary�s problem

above implies:

@ log(�00(i))

@ log(�)=

�1 +

@ log(w1)

@ log(�)

��01(i)

�i � �i + �i(�i � 1)�01(i)�i � �i�00(i)

=) �i =�i � �i + �i(�i � 1)�01(i)

�i � �i + �i�01(i)

Note that �i = �i yields the Armington case given above.

6 Comparison to CES Gains from Trade

The procedure described in section 2 is useful because it provides a means of measuring

gains from trade in a simple way even for complex models. However, this is only interesting

if it provides new insights into the determinants of gains from trade not already present in

the CES trade models. Therefore, an important question to ask is how the gains from trade

implied by the procedure above di¤er from those models.

To answer this question, we derive su¢ cient conditions on the trade elasticity, "T , for

the gains from trade implied by the Arkolakis, et al (2012) formula to be an upper or lower

bound on the gains implied by the procedure described in section 2. To do this, �rst consider

the gains from trade implied by a given trade elasticity, "T . Rearranging the de�nition of

the welfare elasticity and applying Theorem 2 implies:Z �AUT

�01

d log(W ) =

Z �AUT

�01

1

"T (�)d log(�00)

Recall that the trade elasticity is only de�ned when �00 and �01 are both positive. Therefore

25

here the upper limit of integration is the supremum of trade costs where that is true, which

may or may not be �nite18.

If the trade elasticity were a constant, the formula from Arkolakis, et al (2012) is imme-

diate. However, here it is a function. Therefore we proceed by integrating by parts:Z �AUT

�01

d log(W ) = log

�WAUT

W TRADE

�= � 1

"T (� 01)log(�00) +

Z �AUT

�01

log(�00)

"T (�)2@"T

@ log(�)d log(�)

or,

log

�W TRADE

WAUT

�=

1

"T (� 01)log(�00) +

Z �AUT

�01

� log(�00)"T (�)2

@"T@ log(�)

d log(�)

Again, the �rst term is exactly the formula from Arkolakis, et al (2012), while the second

term depends crucially on how the trade elasticity changes. Obviously, if the trade elasticity

is a constant the second term is zero. We can compare the gains from trade implied by this

formula directly to that implied by the ACR formula as follows:

Theorem 9 Let the �gains from trade at � 01" be de�ned as the increase in real income

associated with decreasing trade costs from �AUT to � 01. Consider two versions of the model

described in Section 2 with the same value of �00: one with a variable trade elasticity "T (�)

where "T is bounded away from zero, and one with a constant trade elasticity equal to " =

"T (� 01). Then the model with a variable trade elasticity has higher (lower) gains from trade

at � 01 than the model with a constant elasticity if "T is a monotone increasing (decreasing)

function of � :

Proof. Note that 8� ; �00 2 (0; 1) =) log(�00) < 0; and clearly "T (�)2 > 0: Therefore, for

all � ,

sign

�� log(�00)"T (�)2

@"T@ log(�)

�= sign

�@"T

@ log(�)

�Suppose that "T is increasing in � . Then the term within the integral is positive for all � ,

hence:

0 <

Z �AUT

�01

� log(�00)"T (�)2

@"T@ log(�)

d log(�) = log

�W TRADE

WAUT

�� 1

"T (� 01)log(�00)

The �rst term on the right hand side is the gains from trade in the model with a variable

trade elasticity, and the second term is the gains from trade in the constant elasticity model.

Therefore, the gains from trade are higher in the variable elasticity model than in the constant

elasticity model. The same argument applies for the case with a decreasing trade elasticity.

18For example, if c0 is produced in country 0 and c1 is produced in country 1, then if �c > 0 and U =

c1�1=�0 + (c1 + �c)

1�1=�, �AUT is �nite, whereas if �c = 0 it is in�nite.

26

The usefulness of this su¢ cient condition is apparent as an easy way to check how non-

homotheticities a¤ect gains from trade. In Example 3 above, we can see immediately that the

trade elasticity is decreasing in �00, and since �00 is increasing in trade costs, also decreasing

in � . Applying Theorem 3 above, we can then immediately say that the gains from trade

are therefore smaller in that model than they would be in a model with a constant trade

elasticity. This conclusion can be reached quickly without having to solve the full model.

Likewise, in Example 2, we can see that whether or not the ACR equation is an upper or

lower bound depends on if the income elasticity of domestic or foreign goods is larger. We

could intrepret this to say that countries that produce highly income elastic goods have

gains from trade higher than implied by the ACR equation, while countries that produce

low income elasticity goods have lower gains from trade. This points to a role for country

heterogeneity in determining the gains from trade, which is absent in CES models.

This result demonstrates that the ACR result is useful even in this environment with

general preferences as a bound on the possible gains from trade implied by classes of non-

homothetic models.

7 Conclusion

In this paper we provide a general methodology for solving for gains from trade in static trade

models, even for models that are themselves di¢ cult to solve. We provide several examples

of models that cannot be solved in closed form, yet which have closed form solutions for

their marginal gains from trade when applying our results. This greatly expands the set of

models that can be used to estimate gains from trade in a tractable way. Moreover, this

methodology implies that gains from trade can, in general, depend on the details of patterns

of trade.

An important extension that we plan to study in the future is unequal gains from trade in

a model where households have heterogeneous income within the same country. As discussed

in the introduction, assuming a representative household, as we have done in this paper, is

awkward, given the non-homothetic environment. Allowing for heterogeneous income means

people in the domestic country with di¤erent income levels may have di¤erent levels of gains

from trade.

Likewise, in our ongoing work, Brooks and Pujolas (2014), we apply the results developed

in this paper to see the implications for gains from trade. Assuming all countries have the

same underlying parameters (such as utility functions), we use di¤erences in the composition

of trade and production to see how the marginal trade elasticities di¤er across countries.

Moreover, for any particular country we can integrate over their marginal trade elasticities

27

to �nd their total gains from trade relative to autarky and compare it to that implied by the

ACR formula.

References

[1] Allen, T., C. Arkolakis, and Y. Takahashi (2014), "Universal Gravity", mimeo: Yale

University.

[2] Anderson, J. (1979), "A Theoretical Foundation for the Gravity Equation", American

Economic Review, 69(1), 106-116.

[3] Arkolakis, C., A. Costinot and A. Rodriguez-Clare (2012), "New Trade Models, Same

Old Gains?", American Economic Review, 102(1), 94-130.

[4] Arkolakis, C., A. Costinot, D. Donaldson, and A. Rodriguez-Clare (2014), "The Elusive

Pro-Competitive E¤ects of Trade", mimeo: Yale University.

[5] Ardelean, A, and V. Lugoskyy (2010), "Domestic Productivity and Variety Gains from

Trade", Journal of International Economics, 80(2), 280-291.

[6] Bergstrand, J. (1985), "The Gravity Equation in International Trade: Some Micro-

economic Foundations and Empirical Evidence", Review of Economics and Statistics,

67(3), 474-481.

[7] Bergstrand, J. (1990), "The Hecksher-Ohlin-Samuelson Model, the Linder Hypothesis,

and the Determinants of Bilateral Intra-Industry Trade", Economic Journal, 100(403),

1216-1229.

[8] Blonigen, B. and A. Soderbery (2010), "Measuring the Bene�ts of Foreign Product

Variety with an Accurate Variety Set", Journal of International Economics, 82(2), 168-

180.

[9] Broda, C. and D. Weinstein (2006), "Globalization and Gains from Variety", Quarterly

Journal of Economics, 121(2), 541-585.

[10] Brooks, W. and P. Pujolas (2014), "Gains from Trade: the Role of Composition", mimeo:

University of Notre Dame.

[11] Carol, J., T. Fally and J. Markusen (2014), "International Trade Puzzles: A Solution

Linking Production and Preferences", Quarterly Journal of Economics, 129(3), 1501-

1522.

28

[12] Cooke, R. (2014), In�nite Matrices and Sequence Spaces, Dover Publications.

[13] Eaton, J. and S. Kortum (2002), "Technology, Geography, and Trade", Econometrica,

70(5), 1741-1779.

[14] Edmond, C., V. Midrigan and D. Xu (2014), "Competition, Markups, and the Gains

from International Trade", NBER Working Paper No. 18041.

[15] Fajgelbaum, P. and A. Khandelwal (2014), "Measuring the Unequal Gains from Trade",

NBER Working Paper No. 20331.

[16] Feenstra, R. (1994), "New Product Varieties and the Measurement of International

Prices", American Economic Review, 84(1), 157-177.

[17] Feenstra, R. (2014), "Restoring the Product Variety and Pro-competitive Gains from

Trade with Heterogeneous Firms and Bounded Productivity", NBER Working Paper

No. 19833.

[18] Feenstra, R. and D. Weinstein (2010), "Globalization, Markups, and U.S. Welfare",

NBER Working Paper No. 15749.

[19] Fieler, A., (2011), "Non-Homotheticity and Bilateral Trade: Evidence and a Quantita-

tive Explanation", Econometrica, 79(4), 1069-1101.

[20] Krugman, P. (1980), "Scale Economies, Product Di¤erentiation and the Pattern of

Trade", American Economic Review, 70(5), 950-959.

[21] Markusen, J. (1986), "Explaining the Volume of Trade: An Eclectic Approach", Amer-

ican Economic Review, 76(5), 1002-1011.

[22] Melitz, M. (2003), "The Impact of Trade on Aggregate Industry Productivity and Intra-

Industry Reallocations", Econometrica, 71(6), 1695-1725.

[23] Simonovska, I, (2014), "Income Di¤erences and Prices of Tradables: Insights from an

Online Retailer", NBER Working Paper No. 16233.

[24] Soderbery, A. (2014), "Estimating Import Supply and Demand Elasticities: Analysis

and Implications", mimeo: Purdue University.

29

8 Appendix

8.1 Proof of Theorems 1 and 4

Theorem 4 generalizes Theorem 1, so we �rst prove Theorem 4 and show that it implies

Theorem 1. We begin by noting that the wage in country 0 is numeraire and rearranging

the de�niton of �00:

E0 = �00L0 =

Z0

p0(i)c0(i) =

Z0

c0(i)

a0(i)

Let the Lagrange multiplier on the household�s non-negativity constraint be �n(i). We write

the �rst order condition of the country 0 household�s problem as:

u0n(cn(i); i) = �� 0npn(i)� �n(i) =) cn(i) = (u0n(i))

�1(�� 0npn(i)� �n(i))

where � is the Lagrange multiplier on the household budget constraint. Because each u

function is strictly increasing, strictly concave and twice di¤erentiable, the inverse of the

�rst derivative exists and is itself di¤erentiable. Notice that �n(i) is equal to:

�n(i) =

(0 if cn(i) > 0

�� 0npn(i)� u0n(0; i) if cn(i) = 0

It is useful to note that it�s derivative is therefore:

8n � 1 :@�n(i)

@ log(� 0n)=

(0 if cn(i) > 0

�� 0npn(i)h1 + @ log(wn)

@ log(�0n)+ @ log(�)

@ log(�0n)

iif cn(i) = 0

n = 0 =) @�0(i)

@ log(� 0n)=

(0 if c0(i) > 0

�p0(i)@ log(�)@ log(�0n)

if c0(i) = 0

Then:

L0�00 =

Z0

(u00(i))�1 (�p0(i)� �0(i))a0(i)

@ log(�00)

@ log(� 0n)=

@ log(�)@ log(�0n)

R0p0(i)

�p0(i)u000 (i)

� (1� I0(i))p0(i)�p0(i)u000 (i)

L0�00=

@ log(�)@ log(�0n)

R0p0(i)

u00(i)u000 (i)

I0(i)

L0�00

30

Here we use the fact that, since u0n(k) is di¤erentiable and strictly decreasing (hence, always

strictly negative), then:

@ (u0n(i))�1 (�� 0npn(i)� �n(i))@ log(� 0n)

=�� 0npn(i)

h1 + @ log(wn)

@ log(�0n)+ @ log(�)

@ log(�0n)

i� @�n(i)

@ log(�0n)

u00n(u0n(i)

�1(�� 0npn(i)� �n(i)); i)

=

(0 if cn(i) = 0h

1 + @ log(wn)@ log(�0n)

+ @ log(�)@ log(�0n)

iu0n(cn(i);i)u00n(cn(i);i)

if cn(i) > 0

Next we have to solve for the change in the Lagrange multiplier on the budget constraint.

The budget constraint of the household in country 0 is:

L0 =1Xn=0

Zn

� 0npn(i)cn(i) =1Xn=0

Zn

� 0npn(i) (u0n(i))

�1(�� 0npn(i))

Then:

@ log(�)

@ log(�)= �

PNn=1

RnIn(i)

�� 0npn(i)cn(i) + � 0npn(i)

u0n(i)u00n(i)

��1 + @ log(wn)

@ log(�)

�PN

n=0

RnIn(i)� 0npn(i)

u0n(i)u00n(i)

= �

PNn=1

�1 + @ log(wn)

@ log(�)

��0n +

PNn=1

�1 + @ log(wn)

@ log(�)

� RnIn(i)� 0npn(i)

u0n(i)u00n(i)PN

n=0

RnIn(i)� 0npn(i)

u0n(i)u00n(i)

Combining this with the equation above and substituting into the de�nition of "T yields:

"T =�@ log(�00)

@ log(�)PNn=1 �0n

�1 + @ log(wn)

@ log(�)

� = R0 p0(i) u00(i)u000 (i)I0(i)

L0�00

1 +

PNn=1(1+

@ log(wn)@ log(�) )

Rn

In(i)�0npn(i)u0n(i)u00n(i)PN

n=1(1+@ log(wn)@ log(�) )�0nPN

n=0

RnIn(i)� 0npn(i)

u0n(i)u00n(i)

This is the term that appears in Theorem 4. To get the term in Theorem 1, assume that

N = 1:

"T =

R0p0(i)

u00(i)u000 (i)

I0(i)

L0�00

1 +

�1+

@ log(w1)@ log(�)

� R1I1(i)�01p1(i)

u01(i)u001 (i)�

1+@ log(w1)@ log(�)

��01P1

n=0

RnIn(i)� 0npn(i)

u0n(i)u00n(i)

=

R0p0(i)

u00(i)u000 (i)

I0(i)

L0�00

1 + 1�01

R1I1(i)� 01p1(i)

u01(i)u001 (i)P1

n=0

RnIn(i)� 0npn(i)

u0n(i)u00n(i)

which is equivalent to the statement of Theorem 1.

31

8.2 Solving for Changes in Wages with N Trading Partners

With N trading partners, there are terms that appear in the trade elasticity based on the

changes in wages in all countries besides country 0 (where the change is zero as w0 is nu-

meraire). To solve for these terms, we use information from the N labor market clearing

conditions of the trading partners. Di¤erentiating the labor market clearing conditions with

respect to changes in log-trade costs implies:

8n � 1 : 1 +@ log(wn)

@ log(�)=

PNm=1

Rn�mnpn(i)

um0nium00ni

�Rn� 0npn(i)

�c0n(i) +

u00niu000ni

�PN

m=1

Rn�mnpn(i)

um0nium00ni

�

�PN

m=0@ log(�m)@ log(�)

Rn�mnpn(i)

um0nium00niPN

m=1

Rn�mnpn(i)

um0nium00ni

where we use m superscripts to denote the consumption and Lagrange multipliers of other

countries. Di¤erentiating each of the N budget constraints of the trading partners implies:

8m � 1 :@ log(�m)

@ log(�)=

�1 + @ log(wm)

@ log(�)

�wmLm +

PNn=1

Rn�mnpn(i)

um0nium00ni

�R0�m0p0(i)

um00ium000iPN

n=0

Rn�mnpn(i)

um0nium00ni

�

�

PNn=1

�1 + @ log(wn)

@ log(�)

� Rn�mnpn(i)

�cn(i) +

um0nium00ni

�PN

n=0

Rn�mnpn(i)

um0nium00ni

and di¤erentiating country 0�s budget constraint yields:

@ log(�0)

@ log(�)= �

PNn=1

�1 + @ log(wn)

@ log(�)

� Rn� 0npn(i)

�cn(i) +

u00niu000ni

�PN

n=0

Rn� 0npn(i)

u00niu000ni

Notice that we now have 2N+1 equations and 2N+1 unknowns: the changes in Lagrange

multipliers, and the changes in wages. Moreover, they are all linear equations. Therefore,

we proceed by substituting all the terms derived from the budget constraints into the terms

derived from the labor market clearing conditions, which leaves N linear equations and N

32

unknowns. To write those terms we de�ne the following terms:

Gn = 1�

Rn� 0npn(i)

�c0n(i) +

u00niu000ni

��PN

k=1

Rn� knpn(i)

uk0niuk00ni

R0�k0p0(i)

uk00iuk000i

�PNl=1

Rl�klpl(i)

uk0liuk00liPN

l=0

Rl�klpl(i)

uk0li

uk00liPN

l=1

Rl�nlpl(i)

un0liun00li

Hkn =

wkLk

Rn

�knpn(i)uk0niuk00niPN

l=0

Rl�klpl(i)

uk0li

uk00li

�PN

l=0

Rn� lnpn(i)

ul0niul00ni

Rk� lkpk(i)

�clk(i)+

ul0kiul00ki

�PNm=0

Rm

� lmpm(i)ul0miul00miPN

l=1

Rl�nlpl(i)

un0liun00li

Wn = 1 +@ log(wn)

@ log(�)

Then let H be a matrix where the (k; n) element is Hkn, G be a column vector where the n

column is Gn, and W be a column vector where the n column is Wn. Then:

W = G+WH =) W = (I �H)�1G

Given W , these values can be substituted into the equation in Theorem 4 to give the trade

elasticity with multiple trading partners.

8.3 Proof of Theorem 6

If preferences are non-seperable, then the household�s �rst order condition is:

8n � 1 :@U

@cn(i)= �� 0npn(i)� �n(i)

n = 0 :@U

@c0(i)= �p0(i)� �0(i)

Each term on the left hand side can depend on any number of the other goods available to

the household. Next we di¤erentiate each such �rst order condition with respect to log-trade

costs:

cn(i) > 0 =) @

@ log(�)

�@U

@cn(i)

�=

NXm=0

Zm

Im(j)@cm(j)

@ log(�)

@2U

@cn(i)@cm(j)dj =

= �� 0npn(i)

�1 +

@ log(wn)

@ log(�)+@ log(�)

@ log(�)

�

33

Let H be the Hessian of U . Let H be a matrix constructed by deleting every row and

column whose good has zero consumption. In Section 6, there is some discussion of su¢ cient

conditions for H and its inverse to be well-de�ned, which we assume here. Let n(k) be the

index of good k in country n. Then:

@cm(j)

@ log(�)=@ log(�)

@ log(�)

NXn=0

Zn

H�1n(i);m(j)

@U

@cn(i)In(i)di+

NXn=1

�1 +

@ log(wn)

@ log(�)

�Zn

H�1n(i);m(j)

@U

@cn(i)In(i)di

Di¤erentiating the budget constraint of the household yields:

0 =

Z0

p0(j)@c0(j)

@ log(�)dj +

NXn=1

Zn

�pn(j)cn(j)

�1 +

@ log(wn)

@ log(�)

�+

NXn=1

Zn

� 0npn(j)@cn(j)

@ log(�)dj

0 =@ log(�)

@ log(�)

NXm=0

Zm

� 0mpm(j)NXn=0

Zn

H�1n(i);m(j)

@U

@cn(i)In(i)di+

NXn=1

�1 +

@ log(wn)

@ log(�)

��0n

+NXm=0

Zm

� 0mpm(j)NXn=1

�1 +

@ log(wn)

@ log(�)

�Zn

H�1n(i);m(j)

@U

@cn(i)In(i)didj

which implies:

@ log(�)

@ log(�)=

PNn=1

�1 + @ log(wn)

@ log(�)

� h�0n +

PNm=0

Rm� 0mpm(j)

RnH�1n(i);m(j)

@U@cn(i)

In(i)didji

�PN

m=0

Rm� 0mpm(j)

PNn=0

RnH�1n(i);m(j)

@U@cn(i)

In(i)didj

Recall the de�nition of �00:

L0�00 =

Z0

p0(j)c0(j) =)@ log(�00)

@ log(�)=

R0p0(j)

@c0(j)@ log(�)

L0�00

Using the de�nitions of the An(i) and Bn(i) terms from Section 5, we can simplify these

terms as:

@ log(�)

@ log(�)= �

PNn=1

�1 + @ log(wn)

@ log(�)

��0n +

PNm=0

Rm� 0mpm(j)Am(j)PN

m=0

Rm� 0mpm(j)Bm(j)

@c0(j)

@ log(�)= A0(j)�B0(j)

PNn=1

�1 + @ log(wn)

@ log(�)

��0n +

PNm=0

Rm� 0mpm(j)Am(j)PN

m=0

Rm� 0mpm(j)Bm(j)

34

This implies:

@ log(�00)

@ log(�)=

R0p0(j)A0(j)

L0�00�R0p0(j)B0(j)

L0�00

PNn=1

�1 + @ log(wn)

@ log(�)

��0n +

PNm=0

Rm� 0mpm(j)Am(j)PN

m=0

Rm� 0mpm(j)Bm(j)

Applying the de�nition of the trade elasticity yields:

"T = �1

L0�00

R0p0(j)A0(j)PN

n=1 �0n

�1 + @ log(wn)

@ log(�)

�+ 1�00

R0p0(j)B0(j)PN

m=0

Rm� 0mpm(j)Bm(j)

241 + PNm=0

Rm� 0mpm(j)Am(j)

L0PN

n=1

�1 + @ log(wn)

@ log(�)

��0n

35This completes the proof.

8.4 Proof of Theorem 8

Rewriting the de�nition of �0 yields:

�00I0 =

Z

�00(i)p0(i)c0(i)di

which implies:

@ log(�00)

@ log(�)=

R

h@ log(�00(i))@ log(�)

+ @ log(p0(i))@ log(�)

+�@ log(p0(i))@ log(�)

+ @ log(�0)@ log(�)

�u00(i)

c0(i)u000 (i)

i�00(i)p0(i)c0(i)di

�00I0

Then the budget constraint of the household implies:

@ log(�0)

@ log(�)= �

R@ log(p0(i))@ log(�)

p0(i)c0(i)�1 +

u00(i)c0(i)u000 (i)

�Rp0(i)c0(i)

u00(i)c0(i)u000 (i)

di

Therefore:

@ log(�00)

@ log(�)=

Z

�@ log(�00(i))

@ log(�)+@ log(p0(i))

@ log(�)

�1 +

u00(i)

c0(i)u000(i)

��X00(i)

X00

di

�Z

u00(i)

c0(i)u000(i)

X00(i)

X00

di

R@ log(p0(i))@ log(�)

p0(i)c0(i)�1 +

u00(i)c0(i)u000 (i)

�diR

p0(i)c0(i)

u00(i)c0(i)u000 (i)

di

35

The trade elasticity "T is equal to:

"T = �@ log(�00)@ log(�)PN

n=1

�1 + @ log(wn)

@ log(�)

��0n

Combining the previous two equations and using the lemma to plug in the changes in prices

yields the result.

36