Nonlinear Gravity Wyatt J. Brooks and Pau S. Pujolàs November 2014

Nonlinear Gravity
Wyatt J. Brooks and Pau S. Pujolàs
November 2014
Abstract
We extend the results of Arkolakis, Costinot and Rodriguez-Clare (2012) to environments 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 explicit 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: pujolasp@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 are
grateful to Nezih Guner, Joan Llull and MOVE at the UAB for providing us space where a substantial
portion 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:
1
@ log (W )
=
@ log( 0 )
"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:
1
W
=
log
log( 0 )
A
W
"T
where W A is real income in autarky. This simple formula gives an exact solution for gains
1
Unless otherwise indicated, throughout the paper by gains from trade we mean how much higher is real
income 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 ;
1
@ log (W ( ))
=
@ log( 0 ( ))
"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 environments 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
2
Our assumptions are similar to those in ACR, but we dispense with the assumed CES import demand
structure. We require that the household in the domestic country has utility over goods that is strictly
concave, twice continuously di¤erentiable and strictly increasing. There is a single factor of production in
each country that is in …xed supply. Aggregate pro…ts are proportional to factor payments, and the model is
static. Final goods are composed of intermediates from many countries that are aggregated using a constant
returns to scale (though not necessarily CES) aggregator.
3
One drawback of the approach we develop is that we must assume that there is a representative household
whose welfare we are measuring. If preferences are non-homothetic, then we cannot rely on the familiar
aggregation results, except to assume that all households in the country of analysis are identical. While
this is an important and unrealistic assumption, the issue of the distributional consequences of gains from
trade is not our central point. Therefore, we save the case of within-country inequality for future work. See
Fajgelbaum and Khandelwal (2014) as an example of current work on this topic.
3
level of trade is:
Z 1
1 @ log( 0 )
1
W
=
d log( ) =
log(
log
A
W
"T ( )
log( ) "T ( ) @ log( )
Z
00 )+
1
log( )
log( 00 ) @"T ( )
d log( )
"T ( )2 @ 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 demonstrate 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:
1
1
U = max !c00 + (1
s:t:
:
!)c01
p0 c00 + p1 c01
w0 L0
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 =
00
4
1)
+
01
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 the
good 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 = max
fQk g
K
X
1 1=
k Qk
k
k=1
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 EatonKortum production structure implies that the price of good k is:4
pk =
k
+1
1
k
1
k
T0k + T1k (w1 )
k
1
k
k
where Tjk is the level of productivity of country j in sector k, k is the sector-speci…c comparative 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
4
Here, the domestic country is denoted with subscript 0 and the rest of the world with subscript 1. The
domestic 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 ( ) =
K
X
k=1
(
k
+1
k)
X0k X1k
E0 E1
X0k +X1k
I
+
K
X
k=1
(1
X1k
k)
E1
" PK
X0j
j=1 j E0
PK
X0l +X1l
l=1 l
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 sectorlevel 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
5
Note 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:
max
1 Z
X
n=0
st :
1 Z
X
n=0
8n; 8i : cn (i)
i2
uni (cn (i))di
i2
n
0n pn (i)cn (i)di
= I0
w0 L0
n
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.
6
Instead of two countries, our results are unchanged if there are N + 1 countries, where country 0 is
the country of analysis and the other N countries are all identical to one another. In that case, the results
here are the same when considering trade cost changes between country 0 and all of the N other countries
simultaneously. The case with heterogeneous trading partners is considered in Section 3.
7
The general, non-seperable case is considered in Section 4.
8
All 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.
9
Due 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 substitutability
in 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
ynm (i) units to the household in country m, the …rm must produce mn ynm (i) units.The …rm
solves:
max
s:t: :
X
m
mn pn (i)yn (i)
wn ln (i)
m
X
m
mn yn (i)
= zn (i)ln (i)
m
The fact that these …rms are competitive implies that their price is equal to their marginal
cost, and their pro…ts are zero:
wn
pn (i) =
zn (i)
2.3
Market Clearing
In each country, labor is inelastically supplied and the equilibrium wage clears the labor
market:
Z
8m;
lm (i)di = Lm
i2
2.4
m
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
= P1
R
i2
m=0
Then the trade elasticity is de…ned as:10
"T (
Rn
01 )
i2
0n pn (i)cn (i)di
m
0m pm (i)cm (i)di
@ log( 01 = 00 )
@ log( 01 )
log(w1 )
1 + @@ 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
10
Notice that this is a modi…ed de…nition from that in ACR. Here the elasticity of real exchange rates (the
ratio 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) =
Z0n
E0n =
i2
n (i)
(
=
pn (i)cn (i)
X0n (i)
n
0
u0n (i)
cn (i)u00
n (i)
if cn (i) = 0
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):
1+
"T (
01 )
=
R
i2
1
P1
X01 (i)
1 (i) E01 di
m=0
R
i2
m
R
i2
m (i)
0
X00 (i)
0 (i) E00 di
X0m (i)
di
I0
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 ) =
1 1=
1
c00
Then,
8n :
n (i)
10
=
+
1 1=
1
c01
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=
Z
! 0 (i)
0
1 1=
1
c0 (i)
di +
Z
! 1 (i)
1
1
c1 (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)
=
01 )
=
;
1 (i)
=
Applying Theorem 1, we get:
"T (
(
00
1)
+
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
> > 1, then the trade elasticity gets larger
00 ) grows. If
as 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:
c)1
U (c00 ; c01 ) = (c00
1 1=
1=
+ c01
Then, letting L0 = 1,
8i :
0 (i)
=
1
c
c0 (i)
;
p0 c
(
1 (i)
=
And,
1
"T (
01 )
00
=
1
1)
p0 c
If > 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:
Z
U =
0 (i) log(c0 (i)
i2
s:t:
:
1
Z
0
i2
Where
P R
n
n (i)di
n
Z
0
+ c0 (i)) +
1 (i) log(c1 (i) + c1 (i))
i2 1
Z
p0 (i)c0 (i) +
01 p1 (i)c1 (i)
i2
= 1 and imposing that
n (i)
=
(
1
R
1
c1 (i)=a1 (i)di > 0.11 De…ne:
0 if cn (i) = 0
1 if cn (i) > 0
Then applying Theorem 1 yields:
1
"T =
11
1
01
R
R
i2
i2
0
1
01 p1 (i)(c1 (i)
+ c1 (i))
p0 (i)(c0 (i) + c0 (i))
n (i)di
n (i)di
R
i2
1
00
1
R
i2
0
p0 (i)(c0 (i) + c0 (i))
01 p1 (i)(c1 (i)
+ c1 (i))
n (i)di
n (i)di
This second condition is equivalent to the trade elasticity being negative. Let us emphasize that we do
not 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
=
+
Then using the de…nition of
R
i2
1
R
0
01
and rearranging:
R
p0 (i)c0 (i)di
0
01 p1 (i)c1 (i)di
0 (i)
=
=1+
XZ
n
1+
00
01
+
R
0
0n pn (i)cn (i)
n
p0 (i)c0 (i)di
01
R
R
1
0
1 (i)di
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 =
R
0
0 (i)
n (i)
00
1
00 +
R
p0 (i)c0 (i)
0
1
00
n (i)di
R
R
1
0
1 (i)
n (i)di
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
1 Z
X
n=0
s:t:U =
1 Z
X
i2
n=0
0n pn (i)cn (i)di
i2
n
uni (cn (i))di
n
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(I)
@ log(W )
=
@ log( 01 )
@ 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 (
@ log(W )
@ log( 01 )
@ log( 00 )
@ log( 01 )
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 (
Proof. First note that
00
@ 00
=
@ log( 01 )
+
01
01 )
1
"T ( 01 )
=
= 1, so that
@ 01
=)
@ log( 01 )
00
@ log(
@ log(
00 )
01 )
=
01
@ log(
@ log(
01 )
01 )
Then notice that:
@ log(
@ log( 01 = 00 )
=
@ log( 01 )
@ log(
01 )
@ log(
@ log(
01 )
00 )
01 )
=
00
1
14
+1
00
@ log(
@ log(
00 )
01 )
=
1
1
@ log(
00 @ log(
00 )
01 )
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)
= (1
@ log( 01 )
00 )
1+
@ log(w1 )
@ log( 01 )
Then,
1
=
"W ( 01 )
@ log( 00 )
@ log( 01 )
@ log(I)
@ log( 01 )
=
1
1
00
1
@ log( 00 )
@ log( 01 )
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:
@ log
"T ( ) = P
N
n=1 1
1
00
00
@ log( )
0n
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.
12
That is,
0j
= ~0j and we will be considering changes in the common component .
15
Theorem 3 Whenever
1+
"T ( ) =
00
PN
n=1
2 (0; 1),
n)
R
(1+ @@log(w
log( ) )
PN
@ log(wm )
i2
m=1 E0m (1+ @ log( ) )
P1 R
m=0
i2
m
R
n (i)X0n (i)di
n
m (i)
X0m (i)
di
I0
i2
0
X00 (i)
0 (i) E00 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 ( ) =
Theorem 4 8 ;
@ log(W )
@ log( )
@ log( 00 )
@ log( )
"W ( ) =
1
"T ( )
Proof. Applying the envelope theorem to the dual of the household’s problem,
@ log(W )
=
@ log( )
N
X
@ log(I)
=
@ log( )
1+
0n
n=1
@ log(wn )
@ log( )
and
@ (1
00 )
=
@ log( )
@ log 1 0000
@ 00
=)
@ log( )
@ log( )
=
1+
00
1
00
@ log ( 00 )
=
@ log( )
1
1
@ log ( 00 )
00 @ log( )
Then
"W ( ) =
@ log(I)
@ log( )
@ log( 00 )
@ log( )
1
=
1
00
PN
0n
n=1
1
1
00
1+
@ log(wn )
@ log( )
@ log( 00 )
@ log( )
=
PN
n=1 1
0n
00
@ log
1+
1
00
00
@ log(wn )
@ log( )
=
1
"T ( )
@ log( )
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
13
Notice that if preferences are CES as in Example 1 above, we see that the terms for the changes in wages
cancel 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 seperable. In this section we dispense with that assumption and show that our results are unchanged. That is, the household’s problem is now:
max U ffcn (i)gi2 n gN
n=0
N
X
s:t: :
0n pn (i)cn (i)
I0
n=0
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) =
Bm (j) =
N
X
n=1
N Z
X
n=0
Z
@ log(wn )
1+
@ log( )
1
H(n(i);m(j))
n
1
H(n(i);m(j))
n
@U
@cn (i)
@U
@cn (i)
n (i)di
n (i)di
Then the general case of Theorem 1 is as follows:
Theorem 5 Whenever
1
00
"T =
L0
PN
R
n=1
00
2 (0; 1),
p0 (j)A0 (j)
0
0n
1+
@ log(wn )
@ log( )
1
+ PN
00
m=0
R
R
p0 (j)B0 (j)
0
m
0m pm (j)Bm (j)
2
41 +
PN
m=0
L0
PN
R
n=1
m
0m pm (j)Am (j)
1+
@ log(wn )
@ log( )
The 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 .
14
If H is an in…nite matrix, additional regularly assumptions may be necessary. In that case, by inverse
we mean the left-hand reciprocal of H, as described in Cooke (2014). A su¢ cient condition that we use
in our example is that the set of goods can be partitioned into disjoint, …nite subsets fSj g such that
i 2 Sn ; j 2 Sm ; m 6= n =) Uij = 0 so that H is block diagonal with …nitely sized blocks.
17
0n
3
5
This implies that:
8j8n
:
8j
:
8j8n
u0n (j)
In (j)
u00n (j)
A0 (j) = 0
@ log(wn )
1 : An (j) = 1 +
@ log( )
Bn (j) =
u0n (j)
In (j)
u00n (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 corresponding 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 substitutable 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 good
and its corresponding good from the other country is
(i). Its elasticity of substitution
with 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
(@c1 (i)@c0 (i))
@2U
(@c0 (i)@c1 (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) =
X0n =
We get:
h 1 (i) =
"
c0 (j)
u00 (j)
c1 (j)
u00 (j)
Z0n
pn (j)cn (j)
0n pn (j)cn (j)dj
(j)X01 (j)+ X00 (j)
X00 (j)+X01 (j)
( (j) )X00 (j)
X00 (j)+X01 (j)
c0 (j)
u01 (j)
c1 (j)
u01 (j)
( (j) )X01 (j)
X00 (j)+X01 (j)
(j)X00 (j)+ X01 (j)
X00 (j)+X01 (j)
#
Applying the de…nitions of A and B above:
B0 (j) =
@ log(w1 )
( (j)
)X01 (j)
c0 (j)
@ log( 01 )
X00 (j) + X01 (j)
@ log(w1 )
(j)X00 (j) + X01 (j)
1+
c1 (j)
@ log( 01 )
X00 (j) + X01 (j)
c0 (j)
B1 (j) =
c1 (j)
A0 (j) =
A1 (j) =
1+
Then applying the formula for the trade elasticity implies:
"T = 1
Z
X00 (j) X01 (j)
( (j)
X00
X01
) X00
dj
(j)+X01 (j)
X00 +X01
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:
15
In this section, we once again assume additive separability of preferences to economize on notation as
both general preferences and general production is cumbersome.
19
st :
max
Z
Z
ui (ci )di
i2
pi ci di = I0
w0 L0 +
0
i2
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 = min
XZ
n
0n qn (j)x0n (j)dj
j2
n (i)
st
:
1 = Fi ffx(j)gj2
8n; j
:
x0n (j)
n (i)
gn
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
16
Let us emphasize that this does not imply that there is no extensive margin. We will show that EatonKortum is a special case of this, for example.
17
Given 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 cost
to 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:
Z
X0n (i);
X
X0 (i) =
X0n (i)
X0n =
n
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
Lemma 6
R
X0n (i)
=
I0
R
0n (i)X0 (i)
I0
@ log(wn )
@ log(pi ) X
=
1+
@ log( )
@ log( )
n=1
N
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
=P
N
@ log( 00 (i))
@ log( )
n=1
1+
@ log(wn )
@ log( )
0n (i)
Now we can prove results analogous to those in previous sections:
Theorem 7 Whenever
Z
"T ( ) =
+
Z
00
2 (0; 1):
2
X00 (i) 4
X00
X0 (i)
I0
1+
i
+1+
u0i
ci u00i
PN
u0i
1+
n=1
ci u00i
PN
n=1
PN
n=1
PN
n=1
21
1+
@ log(wn )
@ log( )
1+
@ log(wn )
@ log( )
1+
@ log(wn )
@ log( )
0n (i)
@ log(wn )
@ log( )
0n (i)
0n
0n
2R
di 4 R
3
5 di
X00 (i) u0i
di
X00 ci u00
i
X0 (i) u0i
di
I0 ci u00
i
3
5
If country 0 has only one trading partner, then this simpli…es to:
Z
"T ( ) =
+
i+1+
Z
1+
Theorem 8 8 ;
X01 (i) X00 (i)
X01
X00
di
X0 (i)
I0
2
R u0i X00 (i) 3
di
X01 (i)
ci u00
X00
i
u0i
ci u00i
u0i
ci u00i
X01
"T ( ) =
di 4 R
u0i X0 (i)
di
ci u00
I0
i
5
1
"W ( )
Proof. Now the dual of the household’s problem implies:
@ log(I)
=
@ log( )
Z
Z
@ log(pi ) X0 (i) X
@ log(wn )
=
1+
@ log( ) I
@ log( )
n=1
N
X0n (i) X
@ log(wn )
=
1+
I
@ log( )
n=1
N
0n
substituting this into the de…nition of the welfare elasticity yields:
"W ( ) =
@ log(I)
@ log( )
@ log( 00 )
@ log( )
=
PN
n=1
1+
@ log(wn )
@ log( )
@ log( 00 )
@ log( )
0n
=
PN
n=1 1
0n
00
@ log
1+
1
00
00
@ log(wn )
@ log( )
=
1
"T ( )
@ log( )
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:
! i1
N
i
1
i 1
X
i
i
Fi =
x
ni 0ni
n=0
22
Clearly Fi is constant returns to scale, and the set of intermediates is …xed. Because intermediate goods are competitive there are no pro…ts and prices are given by:
qni =
wn
zni
Hence, this production environment satis…es all out assumptions above. Then solving the
intermediary’s problem, it can easily be shown that:
00 (i)
@ log( 00 (i))
@ log( )
=
=
=
=)
1 i
q0i x00i
0i q0i
=P
1 i
n 0n qni x0ni
n ni ( 0n qni )
PN
1
@ log(wn )
ni ( 0n qni )
n=1 1 + @ log( )
( i 1)
P
1 i
n ni ( 0n qni )
N
X
@ log(wn )
( i 1)
1+
0n (i)
@ log( )
n=1
P
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 empirical 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 industrylevel aggregator is:
0
! i 1 1 i i1
Z 1 X
N
i
dj A
Fi = @
xni (j)
0
n=0
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) =
wn
zin (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 countryindustry productivity level follows the Frechet distribution, they derive the following price
for the industry i good:
pi =
i
1
+1
i
1
i
i
N
X
Tki (wk
0k )
k=0
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 we
assume holds for each industry.
The Eaton-Kortum model also implies that:
0n (i)
Tni (wn 0n ) i
= PN
k=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
N
X
1+
n=1
=)
i
=
@ log(wn )
@ log( )
0n (i)
i
The only di¤erence between the Armington model and the Eaton-Kortum model, therefore, 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 implication 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:
1
i
Fi =
x1i
i
1
i
+ x0i
1
i
x1i
i
1
i
i
i
i
1
i
Each good is produced with a linear technology in each country by competitive …rms.
Then the prices of these intermediate goods are:
qni =
wn
zni
where zni is the productivity in sector i in country n. Solving the intermediary’s problem
above implies:
@ log( 00 (i))
@ log( )
=
=)
Note that
6
i
=
i
@ log(w1 )
i
i+
01 (i)
@ log( )
i
1) 01 (i)
i
i + i( i
=
i
i + i 01 (i)
1+
i
i( i
i
1)
00 (i)
01 (i)
yields the Armington case given above.
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
AU T
d log(W ) =
01
Z
AU T
01
Recall that the trade elasticity is only de…ned when
25
1
d log(
"T ( )
00
and
01
00 )
are both positive. Therefore
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 immediate. However, here it is a function. Therefore we proceed by integrating by parts:
Z
AU T
d log(W ) = log
01
W AU T
W T RADE
=
1
log(
"T ( 01 )
or,
log
W T RADE
W AU T
1
=
log(
"T ( 01 )
00 ) +
Z
00 )
AU T
01
+
Z
AU T
01
log( 00 ) @"T
d log( )
"T ( )2 @ log( )
log( 00 ) @"T
d log( )
"T ( )2 @ 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 AU T 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 ,
@"T
log( 00 ) @"T
= sign
sign
2
"T ( ) @ log( )
@ log( )
Suppose that "T is increasing in . Then the term within the integral is positive for all ,
hence:
Z AU T
log( 00 ) @"T
W T RADE
1
0<
d
log(
)
=
log
log( 00 )
2
AU
T
"T ( ) @ log( )
W
"T ( 01 )
01
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.
18
For example, if c0 is produced in country 0 and c1 is produced in country 1, then if c > 0 and U =
+ (c1 + c)1 1= , AU T is …nite, whereas if c = 0 it is in…nite.
1 1=
c0
26
The usefulness of this su¢ cient condition is apparent as an easy way to check how nonhomotheticities 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 nonhomothetic 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
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University.
[2] Anderson, J. (1979), "A Theoretical Foundation for the Gravity Equation", American
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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.
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Trade", Journal of International Economics, 80(2), 280-291.
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University of Notre Dame.
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Linking Production and Preferences", Quarterly Journal of Economics, 129(3), 15011522.
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.
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from International Trade", NBER Working Paper No. 18041.
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Trade with Heterogeneous Firms and Bounded Productivity", NBER Working Paper
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NBER Working Paper No. 15749.
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Online Retailer", NBER Working Paper No. 16233.
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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 :
Z
Z
c0 (i)
E0 = 00 L0 =
p0 (i)c0 (i) =
a0 (i)
0
0
Let the Lagrange multiplier on the household’s non-negativity constraint be
the …rst order condition of the country 0 household’s problem as:
u0n (cn (i); i) =
0n pn (i)
n (i)
=) cn (i) = (u0n (i))
1
(
We write
n (i).
0n pn (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
u0n (0; i) if cn (i) = 0
0n pn (i)
It is useful to note that it’s derivative is therefore:
@ n (i)
=
1:
@ log( 0n )
8n
(
0n pn (i)
@ 0 (i)
=
n = 0 =)
@ log( 0n )
(
h
0
1+
@ log(wn )
@ log( 0n )
0
log( )
p0 (i) @@log(
0n )
i if cn (i) > 0
log( )
+ @@log(
if cn (i) = 0
0n )
if c0 (i) > 0
if c0 (i) = 0
Then:
L0
@ log(
@ log(
00
00 )
0n )
=
=
Z
1
(u00 (i))
0
@ log( )
@ log( 0n )
R
0
( p0 (i)
a0 (i)
(i)
p0 (i) up000(i)
0
L0
0 (i))
(1
(i)
I0 (i))p0 (i) up000(i)
0
00
30
=
@ log( )
@ log( 0n )
R
u0 (i)
0
p0 (i) u000 (i) I0 (i)
L0
0
00
Here we use the fact that, since u0n (k) is di¤erentiable and strictly decreasing (hence, always
strictly negative), then:
@
(u0n (i)) 1
( 0n pn (i)
@ log( 0n )
n (i))
h
p
(i)
1+
0n n
=
u00n (u0n (i) 1 (
(
=
@ log(wn )
@ log( 0n )
h
+
@ log( )
@ log( 0n )
0n pn (i)
0
@ log(wn )
@ log( 0n )
1+
+
@ log( )
@ log( 0n )
i
i
@ n (i)
@ log( 0n )
n (i)); i)
if cn (i) = 0
u0n (cn (i);i)
u00
n (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 =
1 Z
X
n=0
0n pn (i)cn (i)
1 Z
X
=
n
n=0
1
0
0n pn (i) (un (i))
(
0n pn (i))
n
Then:
PN R
@ log( )
=
@ log( )
n=1
PN
0n pn (i)cn (i)
PN R
n=0
1+
n=1
=
In (i)
n
@ log(wn )
@ log( )
0n
n
+
In (i)
PN
PN R
n=0
+
u0n (i)
0n pn (i) u00 (i)
u0n (i)
0n pn (i) u00 (i)
n=1
n
1+
n
n
1+
In (i)
@ log(wn )
@ log( )
u0n (i)
0n pn (i) u00 (i)
n
R
n
@ log(wn )
@ log( )
In (i)
u0n (i)
0n pn (i) u00 (i)
n
Combining this with the equation above and substituting into the de…nition of "T yields:
"T = P
N
n=1
@ log( 00 )
@ log( )
0n
1+
@ log(wn )
@ log( )
=
R
0
u0 (i)
p0 (i) u000 (i) I0 (i) 1
0
L0
00
R
PN
0
u (i)
n)
I (i) 0n pn (i) un
(1+ @@log(w
00 (i)
log( ) )
n n
n
+
PN
@ log(wn )
1+
n=1 (
@ log( ) ) 0n
PN R
0 (i)
In (i) 0n pn (i) uu00n (i)
n=0
n
n=1
n
This is the term that appears in Theorem 4. To get the term in Theorem 1, assume that
N = 1:
"T
R
1+
R
u01 (i)
I1 (i) 01 p1 (i) u00
1
1 (i)
@ log(w )
1+ @ log( 1)
01
@ log(w1 )
@ log( )
p0 (i) u000 (i) I0 (i) 1 +
0
=
P1 R
u0n (i)
L0 00
I
(i)
p
(i)
n
0n
n
n=0
u00
n
n (i)
R
R
0 (i)
u
u00 (i)
p0 (i) u00 (i) I0 (i) 1 + 101 1 I1 (i) 01 p1 (i) u001 (i)
0
0
1
=
P1 R
u0n (i)
L0 00
I
(i)
p
(i)
n
0n n
n=0
u00 (i)
n
u0 (i)
0
n
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 numeraire). 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
PN
@ log(wn )
=
1:1+
@ log( )
PN @ log( m ) R
m=0 @ log( )
PN
m=1
R
m=1
n
R
um0
ni
mn pn (i) um00
n
ni
PN
R
m=1
um0
ni
mn pn (i) um00
ni
n
R
n
0n pn (i)
c0n (i) +
u00
ni
u000
ni
um0
ni
mn pn (i) um00
ni
um0
ni
mn pn (i) um00
n
ni
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 )
1:
=
@ log( )
PN
n=1
1+
@ log(wn )
@ log( )
PN R
n=0
@ log(wm )
@ log( )
R
P
wm Lm + N
n=1
PN R
n=0
R
n
n
mn pn (i)
n
cn (i) +
n
R
um0
ni
mn pn (i) um00
ni
um0
ni
mn pn (i) um00
ni
0
um0
0i
m0 p0 (i) um00
0i
um0
ni
um00
ni
um0
ni
mn pn (i) um00
ni
and di¤erentiating country 0’s budget constraint yields:
@ log( 0 )
=
@ log( )
PN
n=1
1+
@ log(wn )
@ log( )
PN R
n=0
R
n
n
0n pn (i)
cn (i) +
u00
ni
u000
ni
u00
ni
0n pn (i) u000
ni
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
R
n
0n pn (i)
c0n (i) +
PN R
u00
ni
u000
ni
k=1
n
PN R
l=1
R
wk Lk P
n
N
l=0
Hkn =
R
uk0
ni
kn pn (i) k00
u
ni
uk0
li
kl pl (i) k00
u
l
li
PN R
l=0
PN R
l=1
Wn = 1 +
@ log(wn )
@ log( )
l
n
uk0
ni
kn pn (i) uk00
ni
l
nl pl (i)
R
k
ul0
ni
ln pn (i) ul00 P
N
ni
nl pl (i)
un0
li
un00
li
0
R
m
PN R
uk0
0i
k0 p0 (i) k00
u
0i
PN R
l=0
un0
li
un00
li
lk pk (i)
m=0
R
uk0
li
kl pl (i) k00
u
li
uk0
li
kl pl (i) k00
u
li
l=1
l
l
ul0
ki
ul00
ki
ul0
mi
lm pm (i) l00
u
mi
clk (i)+
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 + W H =) W = (I
H)
1
G
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:
@U
= 0n pn (i)
n (i)
@cn (i)
@U
n = 0:
= p0 (i)
0 (i)
@c0 (i)
8n
1:
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:
N Z
X
@
@U
cn (i) > 0 =)
=
@ log( ) @cn (i)
m=0
=
0n pn (i)
1+
Im (j)
m
@ log(wn ) @ log( )
+
@ log( )
@ log( )
33
@cm (j)
@2U
dj =
@ log( ) @cn (i)@cm (j)
^ be the Hessian of U . Let H be a matrix constructed by deleting every row and
Let H
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( ) X
=
@ log( )
@ log( ) n=0
N
Z
1
Hn(i);m(j)
n
N
X
@ log(wn )
@U
In (i)di+
1+
@cn (i)
@ log( )
n=1
Z
1
Hn(i);m(j)
n
@U
In (i)di
@cn (i)
Di¤erentiating the budget constraint of the household yields:
Z
N Z
N Z
X
X
@c0 (j)
@ log(wn )
@cn (j)
p0 (j)
0 =
dj +
pn (j)cn (j) 1 +
+
dj
0n pn (j)
@ log( )
@ log( )
@ log( )
n
n
0
n=1
n=1
N Z
N
N Z
X
X
@U
@ log(wn )
@ log( ) X
1
Hn(i);m(j)
1+
In (i)di +
0 =
0m pm (j)
0n
@ log( ) m=0 m
@cn (i)
@ log( )
n
n=0
n=1
Z
N
N Z
X
X
@ log(wn )
@U
1
1+
Hn(i);m(j)
+
In (i)didj
0m pm (j)
@ log( )
@cn (i)
m
n
n=1
m=0
which implies:
@ log( )
=
@ log( )
PN
n=1
1+
PN
m=0
Recall the de…nition of
L0
@ log(wn )
@ log( )
00
00 :
=
Z
R
h
0n
+
PN
m=0
0m pm (j)
m
R
m
0m pm (j)
PN R
n=0
n
R
n
1
@U
Hn(i);m(j)
I (i)didj
@cn (i) n
1
@U
Hn(i);m(j)
I (i)didj
@cn (i) n
@ log( 00 )
p0 (j)c0 (j) =)
=
@ log( )
0
R
0
i
0 (j)
p0 (j) @@clog(
)
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( )
PN
n=1
@c0 (j)
= A0 (j)
@ log( )
1+
@ log(wn )
@ log( )
PN
m=0
B0 (j)
PN
n=1
0n
R
m
1+
+
PN
m=0
R
0m pm (j)Am (j)
m
0m pm (j)Bm (j)
@ log(wn )
@ log( )
PN
m=0
34
0n
R
m
+
PN
m=0
R
m
0m pm (j)Am (j)
0m pm (j)Bm (j)
This implies:
R
@ log( 00 )
=
@ log( )
R
p0 (j)A0 (j)
0
L0
00
p0 (j)B0 (j)
0
L0
00
PN
n=1
@ log(wn )
@ log( )
1+
PN
m=0
Applying the de…nition of the trade elasticity yields:
1
"T =
L0
PN
00
n=1
R
0n
0
1
p0 (j)A0 (j)
1+
+ PN
@ log(wn )
@ log( )
m=0
This completes the proof.
8.4
00
R
R
0
m
p0 (j)B0 (j)
0m pm (j)Bm (j)
2
0n
R
41 +
+
PN
m=0
R
m
0m pm (j)Am (j)
0m pm (j)Bm (j)
m
PN
m=0
L0
PN
R
m
n=1
0m pm (j)Am (j)
1+
@ log(wn )
@ log( )
Proof of Theorem 8
Rewriting the de…nition of
0
yields:
00 I0
=
Z
00 (i)p0 (i)c0 (i)di
+
@ log(p0 (i))
@ log( )
which implies:
@ log( 00 )
=
@ log( )
R h @ log(
00 (i))
@ log( )
+
@ log(p0 (i))
@ log( )
+
@ log( 0 )
@ log( )
u00 (i)
c0 (i)u00
0 (i)
00 I0
i
00 (i)p0 (i)c0 (i)di
Then the budget constraint of the household implies:
@ log( 0 )
=
@ log( )
Therefore:
@ log( 00 )
=
@ log( )
Z
R
@ log(p0 (i))
p0 (i)c0 (i)
@ log( )
R
1+
u00 (i)
c0 (i)u00
0 (i)
u0 (i)
0
p0 (i)c0 (i) c0 (i)u
00 (i) di
0
@ log( 00 (i)) @ log(p0 (i))
u00 (i)
X00 (i)
+
1+
di
00
@ log( )
@ log( )
c0 (i)u0 (i)
X00
R @ log(p0 (i))
u00 (i)
Z
p
(i)c
(i)
1
+
di
0
00 (i)
0
0
@
log(
)
c
(i)u
u0 (i) X00 (i)
0
0
di
R
u00 (i)
c0 (i)u000 (i) X00
p0 (i)c0 (i) c0 (i)u
00 (i) di
0
35
0n
3
5
The trade elasticity "T is equal to:
"T =
PN
n=1
@ log( 00 )
@ log( )
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