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General Equilibrium Analysis of the
Eaton-Kortum Model of International Trade
Fernando Alvarez
University of Chicago and NBER
Robert E. Lucas, Jr.
University of Chicago and Federal Reserve Bank of Minneapolis
April, 2004
1. Introduction
Eaton and Kortum (2002) have proposed a new theory of international trade,
an economical and versatile parameterization of the models with a continuum of
tradeable goods that Dornbusch, Fischer, and Samuelson (1977) and Wilson (1980)
introduced many years ago. In the theory, constant-returns producers in di¤erent
countries are subject to idiosyncratic productivity shocks. Buyers of any good search
over producers in di¤erent countries for the lowest price, and trade assigns production
of any good to the most e¢ cient producers, subject to costs of transportation and
We thank Jonathan Eaton, Tim Kehoe, Sam Kortum, Ellen McGrattan, Sergey Mityakov, Kim
Ruhl, Ed Prescott, Phil Reny, Nancy Stokey, Chad Syverson and participants in the December, 2003,
conference at Torcuato di Tella for helpful discussions. Constantino Hevia, Natalia Kovrijnykh, and
Oleksiy Kryvtsov provided many suggestions and able assistance.
1
other impediments. The gains from trade are larger the larger is the variance of
individual productivities, which is the key parameter in the model.
The model shares with those of the new trade theory1 the important ability to
deal sensibly with intra-industry trade: trade in similar categories of goods between
similarly endowed countries. But unlike the earlier theory, the Eaton-Kortum model
is competitive, involving no xed costs and no monopoly rents. Of course, xed costs
and monopoly rents are present in reality, too, but theories based solely on these
motives have proved di¢ cult to calibrate and have seriously underpredicted postwar
increases in trade volumes following tari¤ reductions.2
One aim of this paper is to restate the economic logic of the Eaton-Kortum
model of trade in a particular general equilibrium context. In the next section, we
will introduce the basic ideas using a closed economy with a production technology of
the Eaton-Kortum type. In Section 3, we dene an equilibriumwith trade in a world of
many countries, closely related but not identical to the Eaton-Kortum theory. Section
4 gives su¢ cient conditions for this equilibrium to exist. The problem of determining
whether the equilibrium is unique and of nding an algorithm to construct it is
addressed in Section 5.
A second goal of the paper is to nd out whether a model of this general type
can be made consistent with some of the main features of world trade in the postwar
period. In Section 6, we calibrate some of the main parameters of the theory using
mostly U.S. data, imputing these parameters to the rest of the world. Section 7
discusses some instructive special cases that are simple enough to work out by hand.
Using estimates from Section 6, we examine the implications of these special cases
of the theory for the volume of trade, and the way that trade volume behaves as
1See, for example, Ethier (1979, 1982), Krugman (1979), Helpman (1981), and the Helpman and
Krugman (1985) monograph.2See Kehoe (2002) and Bergoeing and Kehoe (2003).
2
a function of size, and compare these implications to data on total GDP and trade
volumes for the 60 largest economies. Section 8 goes over the same ground numerically
with more realistic assumptions. We apply the algorithm described in Section 5,
calibrating the model to observed GDPs, and introducing measured tari¤ rates.
Our nal goal is to use the quantitative theory to estimate the welfare gains
from hypothetical trade liberalizations. Comparisons between free trade and autarchy
are carried out in Sections 7. Section 9 studies the optimal tari¤ policy of a small
economy. We also calculate the e¤ects of a world-wide liberalization in which every
countrys tari¤s are set to zero. Section 10 relates the theory to growth accounting:
the partitioning of cross-section di¤erences in incomes into their ultimate sources.
Conclusions are contained in Section 11.
2. Preferences, Technology, and Closed Economy Equilibrium
The Eaton-Kortum model is Ricardian, with a continuum of goods produced
under a constant-returns technology. The new idea is a two-parameter probabilistic
model that generates the input requirements for producing each good. It will be
useful to introduce this model of a technology in the simpler context of a single,
closed economy before turning to the study of a model of a world of n countries in
Section 3.
We develop a purely static model in which labor is the only primary (non-
produced) factor of production, and production requires only labor and produced,
intermediate goods as inputs. In the model, there are L consumers, each of whom
supplies one unit of labor, to which no disutility is attached, and produces and con-
sumes a single good in quantity c.3 This nal good is produced with labor services
3We call L population, LPc total GDP, and c real GDP per capita. This usage will be ne
through the development of the theory in Sections 2-5. When we calibrate and apply versions of
the model, we will need to interpret these variables more carefully so as to accomodate physical and
3
and a symmetric Spence-Dixit-Stiglitz aggregate q,
q =
Zi
q(i)11=di
=(1)(2.1)
of a continuum of produced goods. We call these produced goods tradeables,with
an eye toward the role they will play in later sections.
Individual tradeable goods are in turn produced with labor and the tradeables
aggregate (2.1). Thus, consider a given tradeable q(i): Let s(i) be the labor used to
produce this good and let qm(i) be the level of the materials aggregate q, dened in
(2.1), used to produce q(i): The production technology relating these inputs and the
output they imply is assumed to be
q(i) = x(i)s(i)qm(i)1:
As in Eaton and Kortum (2002), the individual productivities x(i) are stochastic,
drawn from an exponential density, and then amplied in percentage terms by the
parameter :4 Note that a low x-value means a high productivity level (and a low
unit cost).
To build a theory on this basis, we would need to put enough structure on
these functions of i so that the integral in (2.1) has meaning. Instead of doing this
directly, we re-label the goods as follows. The only parameter that varies across these
goods i is this productivity level x(i), and all goods q(i) enter symmetrically in the
aggregate (2.1). It will be convenient, then, simply to re-name each material good by
its productivity x, to re-write the aggregate (2.1) in the form
q =
Z 1
0
exq(x)11=dx
=(1); (2.2)
human capital di¤erences.4We are using for the parameter that Eaton and Kortum call 1=, so that in this paper a larger
means a larger variance in individual productivities.
4
where is the parameter of the exponential distribution from which productivities
are drawn, and to re-state the production functions of the individual materials as
q(x) = xs(x)qm(x)1: (2.3)
We speak of good x; " and so on.
The production of the non-tradeable nal good is given by a Cobb-Douglas func-
tion of the tradeable aggregate qf and the labor input sf :
c = sf q1f : (2.4)
The labor and tradeables inputs allocated to each production process must sum to
the totals available. In per capita terms, this means that
sf +
Z 1
0
exs(x)dx = 1; (2.5)
q = qm + qf ; (2.6)
and
qm =
Z 1
0
exqm(x)dx: (2.7)
To sum up, feasible per capita allocations are numbers y; sf ; q; qm; and qf and func-
tions s(x); q(x); and qm(x) on R+ that satisfy (2.2)-(2.7).
Let the prices of individual tradeables be p(x). Producers of all kinds will choose
purchases of the individual goods so as to obtain the tradeables aggregate at minimum
unit cost pm, say. That is, they will solve
pmq = minq(x)
Z 1
0
exp(x)q(x)dx
subject to
Z 1
0
exq(x)11=dx
=(1) q:
5
This problem is solved by the function
q(x) =
Z 1
0
exp(u)1du
=1p(x)q:
It follows that
pm =
Z 1
0
exp(x)1dx
1=1: (2.8)
The individual production levels can be restated as
q(x) = pmp(x)q: (2.9)
Similarly, given the price w of the labor input and the aggregate materials price
pm; a nal goods producer will choose labor and goods inputs so as to minimize the
unit cost p of the nal good. That is,
pc = mins;q[ws+ pmq]
subject to
sq1 c:
This problem is solved by the values
sf =
1
1 pmw
1c (2.10)
and
qf =
1
w
pm
c: (2.11)
It follows that
p = (1 )1+wp1m : (2.12)
Finally, given a price w of the labor input and an aggregate tradeable goods price
pm; any particular tradeable goods producer x will choose labor and goods inputs so
as to minimize the unit cost p(x) of his production, q(x). That is, he will solve
p(x)q(x) = min`;q[w`+ pmq] subject to x`q1 q(x):
6
This problem is solved by the values
s(x) = x
1
1 pmw
1q(x) (2.13)
qm(x) = x1
w
pm
q(x) (2.14)
It follows that
p(x) = Bxwp1m ; (2.15)
where
B = (1 )1+:
In this Ricardian model, we can rst solve for the equilibrium prices p; pm, and
p(x) in terms of the wage w. Then we can use these prices to calculate equilibrium
quantities. Combining (2.8) and (2.15), we have
pm =
Z 1
0
ex(Bxwp1m )1dx
1=(1)= Bwp1m
Z 1
0
ezz(1)dz
1=(1); (2.16)
using the change of variable z = x. We write A(; ); or sometimes just A, for
A(; ) =
Z 1
0
ezz(1)dz
1=(1):
The integral in brackets is the Gamma function (), evaluated at the argument
= 1 + (1 ): Convergence of the integral requires
1 + (1 ) > 0; (2.17)
which we assume to hold throughout this paper.5 In terms of A, (2.16) can be written
5If were too large to satisfy (2.17), the integral in (2.16) would not converge. Economically, this
would mean unbounded production of the tradeable aggregate, as labor is concentrated on goods
where x is near zero (where x is very high). Changes in the parameter will a¤ect the units in
which tradeables are measured, and hence relative prices that depend on these units. The allocation
of labor and materials between the two sectors is not a¤ected. See note 8.
7
as
pm = ABwp1m :
and solving for pm yields
pm = (AB)1==w: (2.18)
Substituting from (2.18) back into (2.15) then yields the prices of individual
tradeables:
p(x) = A(1)=B1=x(1)=w: (2.19)
The price of the nal good is, from (2.12) and (2.18),
p = (1 )1+(AB)(1)=(1)=w: (2.20)
Notice that all these prices, p; pm; and p(x) are di¤erent multiples of the price of the
wage rate w. This is a labor theory of value: Everything is priced according to its
labor content.
The shares of labor and materials inputs in the output value of each tradeable
good x are and 1 respectively. Then the same equality must obtain for the
aggregates:
=w(1 sf )
pmqand 1 =
qmq: (2.21)
Using (2.6) we have qf = q and then the relative price formula (2.18) gives
1 sf = (AB)1==qf : (2.22)
A second equation involving sf and qf is obtained from (2.10) and (2.11):
sfqf=
1
pmw
:
Using (2.18) again,sfqf=
1
(AB)1==: (2.23)
8
The two equations (2.22) and (2.23) can be solved for sf and qf :
sf = (2.24)
and
qf = (1 )(AB)1==: (2.25)
From these equations, all equilibrium quantities can be calculated, just as equi-
librium prices can be calculated from (2.18)-(2.20). National income per capita, in
dollars, is w, and this must equal per capita nominal GDP, pc. Multiplying the gures
by L gives the economy totals. Real GDP per capita, which equals utility in the units
we are using, is
c =w
p= (1 )1(AB)(1)=(1)=; (2.26)
using (2.20).
3. General Equilibrium
The technology proposed by Eaton and Kortum for the production of tradeables,
described in the last section, involves a continuum of goods, produced under constant
returns with labor requirements that vary in a smooth, exogenously given way, dened
by the parameter pair (; ). This is a special case of the technologies proposed by
Dornbusch, Fischer, Samuelson (1977) and Wilson (1980). An international trade
theory based on this technology can thus be developed along the lines of these papers.
Specically, we consider an equilibrium in a world of n countries, all with the
structure described in Sections 2, in which trade is balanced. Let total labor en-
dowments be L = (L1; :::; Ln), where Li is the total units of labor in i. The ex-
ponential distributions that dene each countrys technology have the parameters
= (1; :::; n). Labor is not mobile. We use w = (w1; :::; wn) for the vector of
9
wages in the individual countries. Preferences and the technology parameters ; ;
and are common to all countries. The structure of production in each country is
exactly as described in Section 2, except that now tradeables are traded, subject to
transportation costs and tari¤s.
Transportation costs are dened in physical, iceberg terms: we assume that
one unit of any tradeable good shipped from j to i results in ij units arriving in i.
Interpreting the terms ij as representing costs that are proportional to distance, it is
natural to assume that ij > 0; ij 1; with equality if i = j; ij = ji for all i; j;
and
ij ikkj for all i; j; k: (3.1)
We also want to consider tari¤s that distort relative prices but do not entail a
physical loss of resources. In practice, trade barriers take many forms, but here we
consider only at rate tari¤s levied by country i on goods imported from j, and where
the proceeds are rebated as lump sum payments to the people living in i. Dene !ij
to be the fraction of each dollar spent in i on goods made in j that arrives as payment
to a seller in j.
In the closed economy analysis of Section 2 we exploited the assumptions of
competition and constant returns to solve for all equilibrium prices as multiples of
the wage w, with coe¢ cients depending only on the technology. With this done, we
then calculated equilibrium quantities. This same two-stage procedure can be applied
to the case of many countries, though of course each stage is more complicated.
A new notation for the commodity space is needed. Let x = (x1; :::; xn) be the
vector of technology draws for any given tradeable good for the n countries. We refer
to good x;as before, but now x 2 Rn+: Assume that these draws are independent
across countries, so that the joint density of x is
(x) = (Qni=1 i) expf
Pni=1 ixig:
10
Use qi(x) for the consumption of tradeable good x in country i, and qi for consumption
in i of the aggregate,
qi =
Zqi(x)
11=(x)dx
=(1):
(HereRdenotes integration over Rn
+:) Let pi(x) be the prices paid for tradeable good
x by producers in i. Let
pmi =
Zpi(x)
1(x)dx
1=(1)(3.2)
be the price in i for a unit of the aggregate. Analogous to (2.9), we have
qi(x) = pmipi(x)qi; i = 1; :::; n: (3.3)
The tradeable good x = (x1; :::; xn) is available in i at the unit prices
Bx1w1p1m1
1
i1!i1; :::; Bxnw
np1mn
1
in!in;
which reect both production (labor and intermediate inputs) costs and transporta-
tion and tari¤ costs. All producers in i buy at the same, lowest price:
pi(x) = Bminj
"wj p
1mj
ij!ijxj
#: (3.4)
Note that without assumption (3.1), the right side of (3.4) would not necessarily be
the least cost way of obtaining good x in country i.
Dene the sets Bij Rn+ by
Bij = fx 2 Rn+ : pi(x) = B
wj p1mj
ij!ijxjg:
That is, Bij is the set of goods that people in i want to buy from producers in j. For
each i, [jBij = Rn+. Note that the sets Bij are all cones, dened equivalently by
Bij = fx 2 Rn+ :
wj p
1mj
ij!ij
!1=xj
wkp
1mk
ik!ij
!1=xk for all kg: (3.5)
11
The price index pmi of tradeables in i must be calculated country by country.
Using (3.2) and (3.4), we have
p1mi = B1nXj=1
wj p
1mj
ij!ij
!1 ZBij
x(1)j (x)dx: (3.6)
Using (3.5), we haveZBij
x(1)j (x)dx =
j
(P
k aijkk)1+(1)A
1 (3.7)
where A = A(; ) is the constant dened in Section 2 and where
aijk =
wj p
1mj ik!ik
wkp1mk ij!ij
!1=:
Combining (3.6) and (3.7) and simplifying yields
pmi(w) = AB
0@ nXj=1
wj pmj(w)
1
ij!ij
!1=j
1A
; i = 1; :::; n: (3.8)
We view (3.8) as n equations in the prices pm = (pm1; :::; pmn); to be solved for pm as
a function of the wage vector w: It is the same formula as (7) and (9) in Eaton and
Kortum (2002). The solution to (3.8), which will be studied in detail in Section 4, is
the analogue to (2.19) in Section 2. Notice that the country identier i appears on
the right side of (3.8) only via the parameters ij and !ij:
Next we dene functions Dij(w) that describe the way each producer in i dis-
tributes his total spending pmiqi on tradeables over purchases from each country j:
That is, a producer in i spends Dijpmiqi on goods produced in j. These expendi-
tures will be integrals of spending over the sets Bij dened above. For any x 2 Bij;
pi(x) = Bxjwj p1mj =(ij!ij) and the quantity purchased is, from (3.3),
qi(x) = pmi
Bxjw
j p1mj
ij!ij
!qi:
12
Then expenditures on x 2 Bij are
pi(x)qi(x) = pmi
Bxjw
j p1mj
ij!ij
!1qi:
It follows that
Dijpmiqi =
ZBij
pi(x)qi(x)(x)dx
= (AB)1j
(P
k aijkk)1+(1)p
miqi
xjw
j p1mj
ij!ij
!1;
using (3.7). Continuing, using (3.8),
Xk
aijkk =
wj p
1mj
ij!ij
!1=Xk
wkp
1mk
ik!ik
!1=k
=
wj p
1mj
ij!ij
!1=p1=mi (AB)1=
so that
Dij(w) = (AB)1= pmi(w)
1=
wj pmj(w)
1
ij!ij
!1=j: (3.9)
Note thatP
j Dij(w) = 1:
We now impose trade balance. Under this assumption, the dollar payments for
tradeables owing into i from the rest of the world must equal the payments owing
out of i to the rest of the world. Firms in i spend a total of Lipmiqi dollars on
tradeables, including both transportation costs and tari¤ payments. Of this amount,
Lipmiqi
nXj=1
Dij!ij
reaches sellers in all countries. (The rest is collected in taxes, and rebated as a lump
sum to consumers in i.)
Buyers in j spend a total of LjpmjqjDji dollars for tradeables from i, but of this
total only
LjpmjqjDji!ji
13
reaches sellers in i. The rest remains in j, as rebated tax receipts. Trade balance
then requires that the condition
Lipmiqi
nXj=1
Dij!ij =nXj=1
LjpmjqjDji!ji (3.10)
must hold. Notice that the term LipmiqiDii!ii country is spending on home-
produced tradeables appears on both sides of (3.10). Cancelling thus yields the
usual denition of trade balance: payments to foreigners equal receipts from foreign-
ers.
Our strategy for constructing the equilibrium in this world economy draws on
the analysis of a single, closed economy in Section 2. As in that section, we rst note
that all prices in all countries can be expressed in terms of wages. In the present case,
wages are a vector w = (w1; :::; wn) and we use the n equations (3.8) to solve for the
prices pm = (pm1; :::; pmn) of the tradeables aggregate in all countries. This problem
is the subject of Theorem 1 in the next section. With tradeables prices expressed as
functions of wages (and of the tax rates and other parameters involved in (3.8)), (3.9)
expresses the expenditure shares Dij in (3.9) as functions of wages and of tax rates,
too. Then (3.10) can be viewed an equation in wages w and the vector q of tradeables
consumption.
The impact of the rest of the world on the behavior of individual producers in
i is entirely determined by pmi: In the absence of taxes if !ij = 1 for all i; j the
trade balance condition (3.10) reduces to
Lipmiqi =
nXj=1
LjpmjqjDji:
Also in the absence of taxes, the equilibrium quantities can be calculated from the
relative price pmi=wi, exactly as we did in Section 2. In this case, (2.24) implies that
sfi = ; and then the share formula (2.21) implies
Lipmiqi = (1 )Liwi: (3.11)
14
Applying this fact to both sides of the trade balance condition and cancelling, we
obtain
Liwi =nXj=1
LjwjDji(w); i = 1; :::; n: (3.12)
We do not need to restrict the transportation cost parameters ij to reduce (3.10) to
(3.12) because the e¤ects of these costs are entirely captured in (3.8).
Theorem 2 in the next section provides conditions that ensure that (3.12) has
a solution w, but since our objective is to be able to analyze the e¤ects of changes
in tari¤ policies, we cannot stop with this special case. Nor can we make use of the
share formulas (2.21) to simplify (3.10) in the general case: The presence of indirect
business taxes implies that (3.11) will not hold. Taxes appear in (3.8) too, but they
appear separately in (3.10) because taxes receipts are recycled back to consumers as
lump sum transfers. To deal with the general case, it will be useful to review the
national income and product accounts for a country, i.
TABLE 1
Value-added in services Labor income in services
Lipici Lipmiqfi Liwisi
Value-added in tradeables Labor income in tradeablesPj LjpmjqjDji!ji Lipmiqmi Liwi(1 si)
Value-added in importing Indirect business taxes
Lipmiqi LipmiqiP
j Dij!ij LipmiqiP
j 6=iDij(1 !ij)
GDP Total labor income plus indirect taxes
Lipici Liwi + LipmiqiP
j Dij(1 !ij)
15
Table 1 provides the accounts for country i, viewed as a three sector economy. All
entries are in dollars. The left side gives value-added in each sector; the right side
gives factor payments (labor payments plus indirect business taxes). Two of these
sectors are services (nal goods) and tradeables. The third is an importing sector, in
which rms buy tradeable goods from both home and foreign producers, pay import
duties to their own government, and resell the goods to home producers. This is a
constant-returns, free-entry activity, so of course selling prices must be marked up
exactly to cover the taxes. If !ij = 1; the entries for this sector would be zero in both
columns. To verify that the three sector value-added terms sum to GDP, one needs
to use the trade balance condition (3.10).
We now use these accounts as an aid in calculating the fraction si of country is
labor used in nal goods production. To do this, let
Fi(w) =nXj=1
Dij(w)!ij (3.13)
denote the fraction of country is spending on tradeables that reaches producers (as
opposed to the home government). We will verify
si(w) =[1 (1 )Fi(w)]
(1 )Fi(w) + [1 (1 )Fi(w)]: (3.14)
Evidently, without taxes Fi = 1 and (3.14) implies that si = :
To verify (3.14), we use the share formulas from both producing sectors, the
resource constraint (2.6) on tradeables, and the trade balance condition (3.10). The
share formulas in nal goods production are
wisi = pici and pmiqfi = (1 )pici;
implying that
wisi =
1 pmiqfi: (3.15)
16
The share formulas for tradeables production are
Liwi(1 si) = nXj=1
LjpmjqjDji!ji = LipmiqiFi (3.16)
and
Lipmiqmi = (1 )
nXj=1
LjpmjqjDji!ji = (1 )LipmiqiFi: (3.17)
where the second equality is each line follows from trade balance (3.10) and the
denition of Fi: From (3.17) and the fact that qi = qfi + qmi we have that
qfi = qi[1 (1 )Fi]: (3.18)
Then (3.15) and (3.18) imply
wisi =
1 pmiqi[1 (1 )Fi] ; (3.19)
and (3.19) and (3.16) imply
wi(1 si) = pmiqiFi: (3.20)
Finally, eliminating pmiqi between (3.19) and (3.20) and simplifying yields (3.14).
We next use the formulas (3.13), (3.14) and (3.20) to reduce the trade balance
equation (3.10) to a system of n equations in the n wage rates w; just as we derived
equations (3.12) for the no-tax special case. From (3.20),
pmiqi =wi(1 si)
Fi:
Inserting this expression into (3.10), we have
Liwi(1 si) =
nXj=1
Ljwj(1 sj)
FjDji!ji: (3.21)
17
We view solving these equations as nding the zeros of an excess demand system
Z(w):6,
Zi(w) =1
wi
"nXj=1
Ljwj(1 sj(w))
Fj(w)Dji(w)!ji Liwi(1 si(w))
#: (3.22)
We sum up this section in the
Denition. An equilibrium is a wage vector w 2 Rn++ such that Zi(w) = 0 for
i = 1; :::; n, where the functions pmi(w) satisfy (3.8), the functions Dij(w) satisfy
(3.9), the functions Fi(w) satisfy (3.13), and the functions si(w) satisfy (3.14).
As in the closed economy analysis of Section 2, the full set of equilibrium prices
and quantities are readily determined once equilibrium wages are known. The unique
solution to (3.8), analyzed in Theorem 1 in the next section, gives tradeable goods
prices, as we will describe in Theorem 1 in the next section. Then (3.9) describes the
allocation of every countrys spending on tradeables, and (3.13) and (3.14) give the
equilibrium allocation si of labor in country i. Then (3.15) and (3.18) determine qi
and qfi. Final goods production ci is determined by (2.4).
4. Existence of Equilibrium
The economy we analyze is specied by the technology parameters ; ; and
; common to all countries, the country-specic populations and technology levels
L = (L1; :::; Ln) and = (1; :::; n); the transportation parameters [ij]; and the tax
parameters [!ij]: All these numbers are strictly positive. Moreover, we impose
Assumptions (A):
6Calling equations (3.22) an excess demand system could mislead, since goods prices have
been solved for (in terms of wages) and trade balance has been used in its derivation. The equations
describe excess demands for each countrys labor only, as functions of wages, just as in Wilson
(1980). But whatever terminology one prefers, (3.22) has the mathematical properties of excess
demand systems that will let us apply standard results from general equilibrium theory.
18
(A1) ; < 1,
(A2) 1 + (1 ) > 0;
and for some numbers and !,
(A3) 0 < ij 1 and 0 < ! !ij 1:
Under these assumptions, we study the existence and (in the next section) the
uniqueness of solutions to the excess demand system (3.22). Before turning directly to
these issues, Theorem 1 characterizes the function pm() : Rn++ ! Rn
+ relating trade-
able goods prices to wage rates, dened implicitly by equations (3.8). Then Theorem
2 shows that the excess demand system (3.22) satises the su¢ cient conditions for a
theorem on the existence of equilibrium in an n-good exchange economy. Theorem 3,
in Section 5, gives one set of assumptions that imply that this solution is unique.
To study (3.8), it is convenient to restate these equations in terms of the logsepmi = log(pmi) and ewi = log(wi) of prices and wages:epmi = log(AB) log
nXj=1
(ij!ij)1= expf1
[(1 )epmj + ewj]gj! ;
i = 1; :::; n: Dene the function f : Rn Rn ! Rn so that these n equations are
epm = f(epm; ew): (4.1)
Let S = [ij] be the n n matrix with elements
ij =(ij!ij)
1= (p1mj wj )1=jPn
k=1 (ik!ik)1= (p1mk w
k )1=k
; (4.2)
so that@fi(epm; ew)@epmj = (1 )ij:
The Jacobian of the system epm f(epm; ew) with respect to epm is then I (1 )S:
We note that S is a stochastic matrix (ij > 0 for all i; j andP
j ij = 1 for all i) and
that 2 (0; 1), so that the inverse of this Jacobean is the strictly positive matrix
[I (1 )S]1 =
1Xi=0
(1 )iSi: (4.3)
19
If (3.8) has a di¤erentiable solution epm( ew), its derivatives are given by the formulas@epm@ ewk = [I (1 )S]1k; (4.4)
where k = (1k; :::; nk) denotes the k-th column of S.
Theorem 1. Under the assumptions (A), for any w 2 Rn++ there is a unique pm(w)
that satises (3.8). For each i, the function pmi (w) is
(i) continuously di¤erentiable on Rn++;
(ii) homogenous of degree one,
(iii) strictly increasing in w,
(iv) strictly decreasing in the parameters ij and !ij, and
(v) satises the bounds
pm(w) pmi (w) pm(w);
for all w 2 Rn++, where
pm(w) =
AB
!
1= nXj=1
w=j j
!=(4.5)
and
pm(w) = (AB)1=
nXj=1
w=j j
!=; (4.6)
(vi) the derivatives of epm( ew) satisfyk @epmi( ew)
@ ewk k; (4.7)
where for each k, k = mini ik and k = maxi ik:
Proof. That the homogeneity and monotonicity properties (ii), (iii) and (iv) must
hold for any solution is evident from the properties of the functions fi(epm; ew) in20
w; ij; !ij, and pmi. To verify the bounds (v), note rst that if ij!ij = a for all i; j
for any constant a then (3.8) is solved by
pmi(w) =
AB
a
1= nXj=1
w=j j
!=for all i. This fact together with properties (iii) and (iv) implies that any solution to
(3.8) must satisfy the bounds (v).
In view of the bounds (v), we can nd a compact, convex set C Rn such that
f(; ew) : C! C. We use the norm
kzk =Xi
j zi j
on C: For any z; z 2 C; the mean value theorem applied to f implies that
(f(z))i (f(z))i = (1 )ij(zj zj)
where the functions ij are evaluated at some convex combination of z and z: It follows
from the fact thatP
j ij = 1 that
kf(z) f(z)k (1 )kz zk;
or that f is a contraction on S. Then the contraction mapping theorem implies the
existence of a unique xed point epm( ew) for f and a unique solution pm(w) to (3.8).The Jacobean of the system (4.1) has the inverse (4.3), so the implicit function
theorem implies that epm( ew) is continuously di¤erentiable everywhere. To verify thebounds (vi), we use the fact that k is the largest coordinate in k and write k k,
where is a vector of ones. Thus (4.4) implies
@epm@ ewk [I (1 )S]1k: (4.8)
Now S = , since S is a stochastic matrix, implying that
[I (1 )S] =
21
or that
[I (1 )S]1 = :
Thus (4.8) is equivalent to@epm@ ewk k:
This veries the upper bound in (4.7). An analogous argument shows that
@epm@ ewk
k:
This completes the proof of Theorem 1. To prove that an equilibrium exists, we will apply an existence theorem for an
exchange economy with n goods to the demand system Z(w) dened in (3.14).
Theorem 2: Under assumptions (A) there is a w 2 Rn++ such that
Z(w) = 0:
Proof. We verify that Z(w) has the properties
(i) Z(w) is continuous,
(ii) Z(w) is homogeneous of degree zero,
(iii) w Z(w) = 0 for all w 2 Rn++ (Walrass Law),
(iv) for k = maxj Lj > 0; Zi(w) > k for all i = 1; :::; n and w 2 Rn++;
and
(v) if wm ! w0, where w0 6= 0 and w0i = 0 for some i, then
maxjfZj(wm)g ! 1: (4.9)
Then the result will follow from Proposition 17.C.1 of Mas-Colell, Whinston, and
Green (1995), p. 585.
(i) The continuity of pmi is part (i) of Theorem 1. The continuity of the functions
Dij is then evident from (3.9). The functions Fi dened in (3.13) are continuous,
22
and are uniformly bounded from below by !: The functions si dened in (3.14) are
continuous. The continuity of Z then follows from (3.22).
(ii) From Theorem 1, pmi is homogeneous of degree one. Then (3.9) and (3.12)
imply that Dij are homogeneous of degree zero, and it is immediate that Fi, si, and
Zi all have this property.
(iii) To verify Walrass Law, restate (3.22) as
wiZi =
nXj=1
Ljwj(1 sj)1
FjDji!ji Liwi(1 si)
and sum over i to get:
nXi=1
wiZi =nXi=1
nXj=1
Ljwj(1 sj)1
FjDji!ji
nXi=1
Liwi(1 si)
=nXj=1
Ljwj(1 sj)nXi=1
1
FjDji!ji
nXi=1
Liwi(1 si)
= 0
using (3.13).
The proofs of parts (iv) and (v) are in Appendix A.
5. Uniqueness and Computation of Equilibrium
In this section we establish a su¢ cient condition for the equilibrium of Section 4
to be unique. To do so, we add to Assumption (A) the assumption that the import
duties !ij levied by country i are uniform over all source countries j; so that we write
!ij = !i for i 6= j and !ii = 1: The main result of this section is
Theorem 3. If assumptions (A) hold, if !ij = !i for all i 6= j; and if
(!)2= 1 ; (5.4)
; (5.5)
23
and
1 !
; (5.6)
there is exactly one solution to Z(w) = 0 that satisesPn
i=1wi = 1:
Proof. In Appendix B, we use the results from Theorem 1, (iii) and (v), to establish
that Z has the gross substitute property:
@Zi(w)
@wk> 0 for all i; k; i 6= k; for all w 2 R++:: (5.7)
(Since Z is homogeneous of degree zero, (5.7) will imply that
@Zi(w)
@wi> 0 for all i; for all w 2 R++:)
Then the result will follow from Proposition 17.F.3 of Mas-Colell, Whinston, and
Green (1995), p. 613.
Direct inspection of the su¢ cient conditions (5.4) and (5.6) shows that they
are satised if the tari¤ and transportation costs are small enough, that is if ! and
are close enough to one. For the parameter values for ; and proposed in
Section 6; conditions (5.4) and (5.6) are only satised for small tari¤s. For instance, if
= 0:75, = 0:5 and = 0:15; condition (5.4) is satised if tari¤s and transportation
cost are no higher than 2.5% each (i.e. ! = 0:95 ). For the same parameters,
condition (5.6) is satised if tari¤s are no higher than 40% (i.e. != 0:6). Condition
(5.5), requiring > , is easily satised for the benchmark calibration presented later
on.
Of course, the conditions (5.4)-(5.6) are su¢ cient, not necessary, conditions for
the gross substitute property to obtain. For the case of two countries, it can be shown
that (5.6) alone is su¢ cient and (5.4) and (5.5) are not required at all. Our numerical
experience also conrms that the gross substitutes property holds under much wider
conditions than (5.4)-(5.6), including quite high tari¤ and transportation costs.
24
We nish this section by discussing the algorithm that we use to compute equi-
librium. The gross substitutes property established in Theorem 3 suggests the use
of a discrete time analogue of the continuous time tatonnement process. Let w be
dened as
w =
(w 2 Rn
+ :nXi=1
wiLi = 1
):
Then we dene the function T , mapping w into itself as follows:
T (w)i = wi (1 + Zi(w)=Li) ; i = 1; :::; n; (5.8)
where is an arbitrary constant satisfying 2 (0; 1]: To interpret (5.8), notice that
Zi (w) =Li is country i0s labor excess demand per unit of labor. Thus, T prescribes
that the percentage increase in country is wage be in proportion to a scaled version
of country is excess demand. To see that T : w ! w; note rst that T (w)i 0 if
1 + Zi (w) =Li 0; since Zi is bounded below by Li by part (iv) of Theorem 2.
Note second that for any w 2 w
nXi=1
T (w)i Li =Xi
wi
1 +
ZiLi
Li =
nXi=1
wiLi + nXi=1
wiZi (w) = 1
where the last equality uses WalrasLaw. To calculate T (w) numerically, one rst
needs to calculate pm (w), the solution to (3.8): We used an algorithm based on the
contraction property of the function f , dened in (4.1), and used to prove Theorem
1.
This function T is closely related to a continuous time version of the tatonnement
process. To see this, interpret T dynamically as giving the value T (w) to w(t + v)
whenever w(t) takes the value w. Then (5.8) becomes
wi(t+ v) = wi(t) (1 + Zi(w(t))=Li) ;
orwi(t+ v) wi(t)
wi(t)v=Zi(w(t))
Li:
25
Letting ! 0; we obtaind logwi(t)
dt=Zi(w(t))
Li: (5.9)
Although (5.9) di¤ers from the standard tatonnement process, given by dwi=dt =
ci Z (wi (t)) for some constant ci, it has the same stability properties : If Z satises
the gross substitute property, the di¤erential equation (5.9) converges globally to the
unique equilibrium wage. A proof that (5.9) converges to the unique equilibrium can
be constructed by showing that L (w) = maxi fZi (w) =Lig is Lyapounov function for
this system. In our computational experiments, we found that setting the parame-
ter of (5.8) equal to one always produced monotone convergence, in the sense of
sequences with decreasing Lyapounov function L (w).
6. Calibration
The general structure of the theory is now in place. In the rest of the paper,
we will use a series of algebraic examples and numerical simulations to get an under-
standing of the properties of the model, of its ability to account for some of the main
features of world trade, and of its implications for the e¤ects of some simulated policy
changes. We intend these inquiries to be quantitative, so we will need estimated val-
ues for the parameters ; ; and that are assumed to be constant across economies,
and for the endowments L = (L1; :::; Ln); the technology parameters = (1; :::; n),
and the matrices [ij] and [!ij] that describe transportation costs and tari¤ policies.
At this stage of our research, the preliminary character of these calibrations cannot be
overstressed: There is a vast amount of evidence bearing on these parameter values
that we do not use at all, and the comparisons with evidence that we do carry out
will be frankly impressionistic.
For the substitution parameter used in forming the tradeables aggregate, we
used a conventional value of 2: The results we report are not at all sensitive to this
26
choice.7 For and ; we use the estimates 0.75 and 0.5, based on U.S. Bureau
of Economic Analysis data and related data from other countries from the United
Nations and the World Bank. Conceptually, in our theory, is the share of labor
in the total value of tradeables produced, and is closely related to the fraction
si of employment that is in the non-tradeables (nal goods) sector. We discuss the
relationship of these theoretical magnitudes to observation, and then review recent
evidence.
The theory divides production into two categories: tradeables and non-tradeables.
Provisionally, we used value-added, employment, and capital in agriculture, mining,
and manufacturing in the U.S. to estimate value-added, employment, and capital in
tradeablesproduction. Using the BEA input-output tables, the value-added share
of these sectors was about 0.2 for the U.S. for the years 1996-99, consistent with an
value of 0.8. Using employment shares would yield = 0:82; and xed capital shares
would imply = 0:73:
In fact, according toWorld Development Indicators (WDI: http://www.worldbank.org)
data for the U.S. for the same years, trade in goods was only about 77% of total trade
(exports plus imports over two) in goods and services. The average of this gure for
the countries listed in Table 2 (below) is 0.8. These gures led us to augment the
tradeables share to 0.25 = .2/.8. We use = 0:75 in all the simulations reported
below.
The United Nations Common Database (UNCDB: http://unstats.un.org) for
1993 reports value-added in agriculture, mining, and manufacturing averaging around
0.3 for the OECD countries, and levels ranging to 0.5 and higher for poorer coun-
tries. The OECD input-output tables(http://www.oecd.org) for 1990 imply an
7See note 6. The parameter does not a¤ect the expenditure shares Dij (see (3.9)), and so does
not a¤ect the variables Fi and si (see (3.13) and (3.14)), and so does not a¤ect equilibrium wages
(see (3.22)).
27
value of .72 for the OECD countries. In short, 0.75 seems a reasonable value for
in the industrialized world and a serious overstatement for economies that are still
substantially pre-industrial.
To calibrate the parameter , we need to think of the primary factor Li as labor-
plus-capitalor perhaps equipped laborand to identify wiLi with total value added,
not just compensation of employees. Based on the BEA input-output tables for 1996-
99, the ratio of value added in manufacturing to the total value of production in this
sector was about 0.38. This gure can be compared to the U.N. (UNIDO Industrial
Statistics database) estimate of a world average value of 0.38 in manufacturing for
1998. The OECD input-output table for 1990 gives an average of 0.38 in agriculture,
mining, and manufacturing. Since labors share in most services is higher, including
tradeable services in total tradeables would require a higher value of . For instance,
in the 1997 U.S. input-output table, the average of the ratio of value added to gross
product across sectors, weighted by the share of each sector in U.S. exports, is 0.5.
Based on these considerations, we use = 0:5 throughout this paper.
The parameter describing the variability of the idiosyncratic component to
productivity will prove to be central in quantitative applications of the theory. It
cannot be estimated from accounting data in any simple way. Eaton and Kortum
report estimates based on international price and wage data that range from 0.08 to
0.28.8 In our numerical experiments we let vary from 0.10 to 0.25.
There is a wealth of evidence that could be used to calibrate the matrix [ij] and
[!ij]: See Eaton and Kortum (2002), and the references therein to the vast literature
8Eaton and Kortum (2002), Table VIII, p. 1767. There is enormous, persistent variability in
value added per worker in individual U.S. plants (see Baily et al. (1992), Bernard et al. (2003), and
Syverson (2003)). It is hard to infer the across-economy variance we want from these data alone,
but Bernard et al. report the estimate = 0:25 based on productivity comparisons between U.S.
plants that export and those that do not. See also Melitz (2003).
28
on gravity modelsthat deals with the e¤ects of distance on trade. In this paper, we
will not incorporate spatial e¤ects in realistic ways, treating ij; i 6= j, as a constant
that functions as an intercept term to match average, observed trade volumes. The
tari¤ data we used to calibrate [!ij] is included in Table 2, described below.
Neither the endowments L = (L1; :::; Ln) nor the technology parameters =
(1; :::; n) can be observed directly, and the problem of inferring their values from
characteristics we can observe will be the focus of Sections 8-10. Here we simply
describe the limited, aggregative data set we use for this and other purposes.
We use the 2002 WDI cdRom to assemble a cross-section of the 59 economies
with the largest total GDPs. These countries and the variables measured for each are
listed in Table 2. We also include a residual, rest-of-world category (with 5 percent
of world GDP), treated as one of the 60 economies. Column (1) of the table is the
Penn World Table measure of per capita real GDP. We used this variable to order the
countries. For each country, ve variables are recorded, along with the utility gains
from a simulated tari¤ reform described in Section 9..
Column (1) is total GDP, denoted Y = (Y1; :::; Yn): These are IMF-based nominal
values, converted to U.S. dollars at market exchange rates (where available). They
are not on a purchasing power parity basis. In the table they are expressed as frac-
tions of total world GDP. These ows, and all the import and export ows that we
used were averaged over the years 1994-2000 in order to reduce the importance of
trade imbalances and year to year uctuations, about which the theory evidently has
nothing to say.
Column (2) is trade volume, denoted V = (V1; :::; Vn), dened as the average of
the values of imports and exports, also from the 2002 WDI cdRom, divided by GDP.
Both imports and exports are dened to include services as well as goods.
Column (3) reports the ratio of the consumption goods deator for each country
to an index of the prices of machinery and equipment, from the 1996 benchmark year
29
in the Penn World Table. We will use them as observations P = (P1; :::; Pn) on the
prices (p1=pm1; :::; pn=pmn) in the theory.
Column (4) of Table 2 lists estimates of average 1996-2000 import tari¤ rates for
each country. These are unweighted averages of ad valorem tari¤s applied to di¤erent
commodities. They are available in the World Bank database Data on Trade and
Import Barriers, and are described in Dollar and Kraay (2004). (For the three
countries for which we do not have tari¤ data, indicated by asterisks, we substituted
tari¤s from a second source: ratios of import duties to imports, from WDI 2002. For
the residual ROW, we used the average of the rest of the column.)
Column (5) contains simulation results that are discussed in Section 9. Column
(6) is a 1994-2000 average of per capita income, on a purchasing power basis, from
the Penn World Table. This last series is used only in Figure 10.1.
INSERT TABLE 2
7. Examples
The algorithm proposed in Section 5 makes it easy to compute equilibria with
many countries, di¤ering arbitrarily, but it will be instructive to work through some
examples rst that are simple enough to solve by hand. We derive the predictions
of special cases of the theory for the behavior of trade volumes and the gains from
trade, as measured by the e¤ects of changes in trade on real consumption.
For future reference, we start with the derivation of some useful formulas for trade
volumes and gains, under the assumption used also in Theorem 3 that tari¤s are
uniform: !ij = !i for all i 6= j: We rst derive expressions for the value of imports
Ii and the volume of trade vi; dened as the ratio of the value of imports to GDP.
The value of imports Ii is the fraction of tradeable expenditures bought abroad,
Ii = LipmiqiXj 6=i
Dij:
30
From the share formula (3.20), using (3.14) to eliminate si; and collecting terms,
Ii = Liwi(1 )
+ ( )Fi(1Dii) : (7.1)
Now GDP equals wages plus indirect business taxes,
Lipici = Liwi + Ii (1 !i) ;
so using the expression (7.1) for imports implies
Lipici = Liwi
1 +
(1 ) (1 !i)
+ ( )Fi(1Dii)
: (7.2)
Then dividing and using the denition (3.13) of Fi and the uniform tari¤ assumption,
we have
vi =Ii
Lipici=
(1 )
(Dii= (1Dii) + !i) + 1 !i: (7.3)
Notice that (1 )= is an upper bound for vi:
Using these formulas, we consider rst the case of costless trade: ij = !ij = 1,
all i; j:We solve for each countrys wages wi, the prices of non-tradeable goods relative
to tradeables pi=pmi, the shares in world GDP Liwi=P
j Ljwj, and the volume of trade
vi; all as functions of the parameters Li and i. With costless trade every country
buys the intermediate inputs from the same lowest cost producer, so the pmi are the
same for all countries and given by
pm = (AB)1=
nXj=1
w=j j
!=:
Inserting this information into (3.9) yields
Dij(w) =
nXk=1
w=k k
!1w=j j: (7.4)
Notice that the expenditure sharesDij do not depend on the identity i of the importer.
With ! = 1; expression (3.14) gives si = , and the excess demand functions (3.22)
31
can now be written
Zi(w) =
1
nXj=1
Dij(w)
wiLjwj Li
!:
Equating Zi(w) to 0 and applying (7.4) , we solve for equilibrium wages
wi =
iLi
=(+); (7.5)
where the parameter , which does not depend on i, will be set by whatever normal-
ization we choose for w.
Setting = 1; total GDP for country i is
Liwi = L=(+)i
=(+)i ; (7.6)
a geometric mean of productivity in tradeables i and labor in e¢ ciency units Li.
Notice that if = 0; so that there is no variation of productivities, then country
is GDP Liwi is simply Li: Using (2.12) and (7.5), the price of the nal non-tradeable
goods relative tradeables goods in country i is given by
pipm
= (1 )1+iLi
=(+)pm : (7.7)
so that countries with high productivity i in tradeables have a high relative price
of non-tradeables.9 From (7.6) and (7.7), we can solve for Li and i in terms of
observables:
Li =1
kYiP
1=i (7.8)
and
i = k=YiP=()i : (7.9)
9The idea that countries with a more advanced technology will have a high relative price of
non-tradeables is known as the Balassa-Samuelson e¤ect (Balassa (1964), Samuelson (1964)). Its
time-series analogue is a key implication of the Greenwood, Hercowitz, Krusell (1997) model of
growth, where technological change occurs in investment goods production only.
32
In this world of costless trade, then, data on GDPs Yi and on relative prices Pi can
be used to infer each countrys labor endowment Li and its tradeables productivity
parameter i: We will see in Section 10 that the idea of using relative price data Pi
to separate the e¤ects of Li and i on production can also be applied in the general
case where tari¤ and transportation costs are positive.
The volume of trade vi in the costless trade case is given by
vi =1
(1Dii) =
1
1 LiwiP
j Ljwj
!(7.10)
so that the volume of trade decreases with the share Liwi=P
j Ljwj of country i in
world GDP:
Our second example explores a di¤erent special case. We study a symmetric
equilibrium with equal sized countries Li = L = 1; identical technologies, i = , and
uniform transportation costs and tari¤s, described by
ij = and !ij = ! if i 6= j
and ii = !ii = 1: In these circumstances, we conjecture a common equilibrium wage
wi = w; all i, and normalize it to w = 1: Everyone will face the same tradeables price
pm, and the formula (3.8) can be solved for
pm =(AB)1=
(1 + (n 1)(!)1=)= =: (7.11)
The price of nal goods is given by (2.12), which with w = 1 and pm given by (7.11)
yields
p =(1 )1+ (AB)(1)=
[1 + (n 1)(!)1=](1)=(1)=: (7.12)
With w = 1; (3.9) implies
Dij =(!)1=
1 + (n 1)(!)1=
33
for i 6= j, so thatDii
1Dii
=1
(n 1)(!)1= :
Then applying (7.3), the imports/GDP ratio v equals
v =1
(n 1)(!)1=1 + (1 + ! !)(n 1)(!)1== : (7.13)
(With ! = 1, this formula becomes
v =1
(n 1)1=1 + (n 1)1= :
With = 1; it reduces to (7.10) with GDP shares equal to 1=n.)
Nominal GDP per capita in this example is
pc = 1 + I (1 !) = 1 + vpc(1 !);
with v given by (7.13), and consumption, or utility per unit of labor, is given by
c =1
1 v(1 !)
1
p: (7.14)
We calculate the utility gain from eliminating a tari¤!: Denote by c; I and p the
levels of consumption, imports, and the consumption price corresponding to a tari¤
! and by c0; I0 and p0 the values corresponding to the case of no tari¤: ! = 1: Then
using (7.12).
c0c= [1 v(1 !)]
[1 + (n 1)1=][1 + (n 1)(!)1=]
(1)=: (7.15)
We use equations (7.13) and (7.15) to derive an expression for the gain
log (c0=c) of going from pure autarchy, ! = 0, to costless trade, = ! = 1: Specializ-
ing (7.14) to the autarchy-costless-trade comparison, we get
log (c0=c) = (1 )
log(n) : (7.16)
34
In the last section we argued that the values = :75 and = :5 are empirically
reasonable, at least for the high income countries. We can think of n in (7.16) as
the ratio of world GDP to the home countrys, so that taking values from Table 2,
n = 3:6 = 1=:28 for the United States, n = 6:2 for Japan, and n = 170 for Denmark.
Using the value 0.15 for ; (7.16) then implies the gain estimate
= (:075) log(n):
In percentage terms, this formula implies benets of 10 percent of consumption for
the U.S., 14 percent for Japan, and 38 percent for Denmark. These are fantasy
calculations even ideally free trade is not costless trade but they give useful upper
bounds for the magnitude of gains we will be discussing in the rest of the paper.
Next we return to the case of positive transportation costs and tari¤s, retaining
symmetry. The properties of the volume and welfare gain functions dened in (7.13),
and (7.14)-(7.15) are illustrated in Figures 7.1 through 7.5. In all cases, we used the
values = 0:75; = 0:5; and = :75. In the gures, the values of and the tari¤
parameter ! are varied, as shown. Figure 7.1 shows the properties of the gains of
going from autarchy to free (but not costless: ! = 1, = :75) trade plotted against
the size of a typical economy in this world of identical economies. Three values
were used. These gains are a decreasing function of size and of .
Figure 7.2 shows the gains of eliminating a tari¤ corresponding to ! = :9 for the
same three values of . Of course, the gains from eliminating a 10 percent tari¤are far
smaller than the gains from moving from autarchy: Compare the scale of the vertical
axis to that in Figure 7.1. Notice too that the gains in Figure 7.2 are not always
decreasing in the the size of the country: This is due to the e¤ect of the revenue
from the tari¤. Figure 7.3 plots the gains from eliminating a tari¤ ! for di¤erent
values of ! holding xed at 0.15. Note that the di¤erence in the welfare gains
from eliminating a 30 percent versus a 20 percent tari¤ is smaller than the di¤erence
35
between eliminating a 40 percent versus a 30 percent tari¤.
Figures 7.4 and 7.5 are plots of the relation between the volume of trade, (7.13),
measured as the ratio of import value to GDP, and the size of the economy. The
volume of trade is decreasing in size, and is bounded above by the ratio of (1 )
to (1 (1 )!), which equals 0.45 for our benchmark parameter values and ! =
0:9. Figure 7.5 shows the same curve for di¤erent values. Small economies have
trade volumes that nearly attain the bound for even very small values. For large
economies, increases in have large e¤ects for less than .15, but by = :25 or
larger, trade volume approaches the upper bound. In Figure 7.5, is held at 0.15 and
! is varied between 0.7 and 1. The e¤ects of variation in ! are large for economies of
all sizes.
Figures 7.1 - 7.5 refer to symmetric world economies, where all economies have the
same size and technology levels. These are not cross-sections. The scatter of points
in Figure 7.6 are GDPs, Yi; and Import to GDP ratios, Vi; for 60 large countries:
columns (1) and (2) in Table 2. These data are a cross-section. But the continuous
curve on the picture is calculated for a symmetric world, just as in Figures 7.4 and
7.5, except that the x -axis is transformed to logs so that one can see the observed
pairs for small countries. The curve is positioned so as to correspond to an average
tari¤ of 10% (! = 0:9):
In a world with very di¤erent national policies toward trade one would not ex-
pect equal trade volumes at each GDP level, even if the assumptions underlying the
construction of Figure 7.6 were correct: The points should not lie on the theoret-
ical curve. If the theory were accurate, the rich economies with similar and more
or less free trade roughly, the OECD should near the curve, and the protectionist
economies should fall below it by larger amounts.
There are also four striking outliers in the gure: Hong Kong, Singapore, Malaysia,
and Belgium, with trade volumes much higher than others, and much higher than
36
our theoretical upper bound. It is a characteristic of port cities that a high vol-
ume of goods passes through, counted as imports when they enter and exports when
they leave. Countries in which such ports are important would appear as high "
countries in our parameterization, so it is possible that relaxing the assumption that
is uniform across economies would yield a better t of the volume-size curves in
Figures 7.6 and 8.2, below. Since can be calibrated category-by-category (as we
did for manufactures and services in Section 6) it should be possible to pursue this
conjecture, but we have not done so.
INSERT FIGURES 7.1 - 7.6
8. Volume of Trade
The algorithm described in Section 5 lets us replace the theoretical curve in Fig-
ure 7.6, based on an assumption of symmetry, with the volume predictions of the gen-
eral theory, calibrated to t the actual distribution of economies by size. In addition,
the general theory lets us incorporate other kinds of international di¤erences for
example, di¤erences in tari¤ policies into the trade volume predictions. We do this
in the section, in two ways.
In constructing the theoretical curve on Figure 7.6, we treated observations
on each countrys GDP as proportional to that countrys labor endowment, Li.
Evidently, this assumption requires a view of labor input as corrected for skill
di¤erences e¤ective labor and as we remarked in Section 6, it requires a correc-
tion for di¤erences in physical capital per worker too. In addition, we assumed in
constructing the gure that each countrys technology parameter was proportional
to its GDP so that i = kLi for some constant k > 0. In e¤ect, then, the GDP
observations Y are taken as estimates of both L and : Equation (7.13), with the
indicated parameter values, was used to get the value of the trade volume function
37
at each GDP level.
Once the assumption of identical countries is dropped, as we do in this section,
there is no reason for equilibrium wages to be equal, and if they are not, observed
GDPs Y cannot be taken as direct observations on labor-capital endowments L: Even
without tari¤ distortions, Yi will be the product wiLi; and neither w nor L can be
directly inferred from observations on Y . What can be done about this depends on
what other data are used. We discuss several possibilities in the next three sections.
The simplest calibration method uses the theory to infer w and L from the data
on Y only. To do this, it seems a natural starting point to think of the parameter in
any country to be proportional to that countrys e¤ective labor endowment L: That
is, we assume that if country 1 has twice the labor endowment of country 2, that
country will also have twice as many drawsfrom the distribution of productivities.
With exponentially distributed productivities, this means 1 = 22, and in general,
that the vector is assumed to be proportional to the endowment vector L. This
is assumption of uniform ratios i=Li surely has more appeal that assuming uniform
levels i: In the latter case, there would be enormous diseconomies of size: Denmark
would be the low cost producer of as many goods as the United States is, but with
its much smaller workforce, Danish wage rates would be bid up to much higher levels
the wages in the U.S.10
Of course, these are not the only possibilities. The assumption = kL is at best
a kind of steady state or very long run hypothesis, in the spirit of Kremers (1993)
idea that the stock of useful ideas should be proportional to the number of people.
10See the costless trade formula (7.5), for example. Note that although the hypothesis = kL
avoids an unrealistic diseconomy of scale, it leaves in place an unrealistic scale economy. In the
theory, transportation costs within an economy, no matter how large, are taken to be zero. Insofar
as the parameters ij measure the resources used in moving goods over space, this is a deciency
that can only be xed by introducing some actual geography.
38
We take it as a convenient point of departure. Under this assumption, the equilibrium
condition
Z(w;L; ) = 0; (8.1)
written so as to emphasize the dependence of the excess demand system Z on L and
, is specialized to Z(w;L; kL) = 0: The choice of the constant k is just a matter of
the units chosen for tradeables and labor input. We set it equal to one:
Z(w;L; L) = 0: (8.2)
A second set of equations in the variables w and L is given by the GDP-equals-national
income conditions
L "(w;L) = Y; (8.3)
where "i(w; ) is wi adjusted for indirect taxes using the function of the equilibrium
wage vector dened in equation (7.2):
"i(w; ) = wi
1 +
(1 ) (1 !i)
+ ( )Fi(w; )(1Dii(w; ))
: (8.4)
We view (8.2) and (8.3) as 2n equations in the pair (w;L), given the data Y .
We describe the algorithm used to solve (8.2)-(8.3). Dene w(; L) to be the
function that solves (8.1). Its values can be calculated using the algorithm described
in Section 6. Dene ' by
'i(L) =Yi
"i(w(L;L); L)=
nXj=1
Yj"j(w(L;L); L)
: (8.5)
Then clearly ' maps the n-dimensional simplex n into itself, and if L is a xed point
of ', the pair (w(L;L); L) satises (8.2)-(8.3). We located a xed point by iterating
using (8.5), applying the algorithm from Section 5 at each iteration, from an initial
guess for L. In practice, this algorithm always converged to a strictly positive xed
point.
39
Figure 8.1 displays the equilibrium wages calculated in this way as a function
of size (GDP share). The benchmark parameters used in Figures 7.1-7.5 were used,
and the same range of values. The equilibrium wages are increasing with size,
reecting the scale economy in transportation enjoyed by larger economies. Since the
technology level is assumed proportional to size L in the construction of the gure,
it cannot provide a second source of wage variation.
Given equilibrium wages and endowments computed in this way, the analogues to
Figures 7.1-7.5 are readily constructed. We were surprised to nd that the new gures
constructed in this way were very similar to the gures based on the assumption of
equal size countries and wage equality, even though these two sets of assumptions
seem very di¤erent.
This nding is illustrated in Figure 8.2. This gure is the exact analogue to
Figure 7.6, except that the very high volume countries have been left o¤ so as to get
higher resolution on the others. The volume and GDP data used in both gures
are the same. The theoretical curve plotted in Figure 7.6, based on a symmetric
model with identical countries and uniform wages, is reproduced on Figure 8.2, as
is new second curve, constructed by solving the general equilibrium system with
endowments L calibrated in the way we have just described. Despite the completely
di¤erent computational methods used to construct them, the two curves are very
similar, except for the largest economies, Japan and the U.S., where the symmetric
model predicts a smaller volume than more realistic one does. This is due to the e¤ects
of size on wages in the calibrated economy, shown on Figure 8.1. The implication
we draw from the similarity of the two curves is that even though an economys size
relative to the world economy matters for the determination of trade volume, the way
the rest of the world is congured matters very little.
INSERT FIGURES 8.1 - 8.2
40
Neither of these curves is a particularly good t: They pick up the e¤ects of
size on trade volume, and nothing else. Some other factors were remarked on in our
discussion of Figure 7.6, and other possible inuences will occur to anyone. Here
we examine the possible e¤ects of tari¤ policies, under the assumption also used
in Sections 5 and 7 that each country i imposes a uniform tari¤ factor !i on all
countries j.
We introduce tari¤s simply by repeating the simulation (8.2)-(8.3) with a uniform
tari¤ factor of ! = 0:9 replaced by the vector = (1; :::;n) of observed tari¤
factors, obtained from the tari¤ rates from column (4) of Table 2, interpreting i as
the uniform tari¤ factor that country i imposes on all imports. Results are shown on
Figure 8.3, based on the same GDP and volume data as Figures 7.6 and 8.2. The
os are the data, the continuous curve is obtained from the calibrated model with a
uniform tari¤ ! = 0:9: The asterisks are predictions from the calibrated model using
the tari¤ factors implied by Table 2, column (4).
INSERT FIGURE 8.3
As a basis for comparison, we ran a regression of volume on GDP and tari¤s
levels for the 60 countries. The results were
log(Vi) = a (0:23) log(Yi) (0:029)(100)(1 i): (8.4)
The associated R2 was .34. The same statistic but with the estimates Vi of Vi calcu-
lated from the theory (the xs on Figure 8.3) was also R2 = (:58)2 = :34. With the
uniform tari¤ imposed, the comparable statistic was R2 = (:45)2 = :20:
The slope parameters in (8.4) are freely chosen to t the data. The theoretical
parameter = :15 was selected, among other considerations, to t the volume and
GDP data. But the e¤ect of tari¤s derived from the calibrated model was not selected
in any way to improve the t, and no actual tari¤ data (beyond average levels) was
41
used in the calibration. Yet the tari¤s have exactly the same (considerable) ability to
improve the t when constrained to work through our theory as with a freely chosen
regression coe¢ cient.
9. Gains from Trade
In Section 7 we studied the gains from trade using autarchy versus costless trade
examples and hypothetical tari¤ reductions in the context of a symmetric world econ-
omy. These exercises served to illustrate the properties of the theory, but the numbers
they yield are not closely related to realistic changes in trade policy from actual cur-
rent levels. In this section, we move a little closer to this objective, using the general
version of the theory and the measured tari¤ factors used in Section 8.
Results of a specic, world-wide tari¤ reduction are described below and dis-
played in gure 9.2. But before turning to these results, it will be helpful to study
the e¤ects of unilateral tari¤ changes in a small economy, or to calculate the best-
response functionfor a small economy, taking the tari¤ policies of rest of the world
as given. Studying this problem will help us to interpret the results of a uniform,
multilateral tari¤ changes.
We focus on the case of economy 1, say.We use the notation L1 = (L2; L3; :::Ln) and
1 = (2; 3; :::; n) to denote the parameters corresponding to countries other than
1, and similarly with w1; c1; and pm1: Assume that country 1 applies a uniform
tari¤ ! to all its imports, and assume that all other countries apply a common tari¤b! to country 1s exports. Our interest is in analyzing the behavior of country 1swelfare (nal goods consumption) c1 as a function of the pair (!; b!):
To make precise the idea that country 1 is small, consider a sequence of world
economies fLr; rg with (Lr; r) ! (L; ); and with r1=Lr1 = k > 0 along the
sequence: Let fwr; cr; prmg denote the corresponding sequence of equilibrium values,
and let (w; c; pm) be the corresponding equilibrium values of the limiting economy.
42
We study the behavior of equilibrium as
Lr1 ! 0: (9.1)
In Appendix C we show that in the limit, the behavior of economy 1 does not
matter at all for the other n1 economies, in the sense that (w1; c1; pm1) is equal
to the equilibrium of a world economy with n1 countries and endowments (L1; 1).
We solve for the limiting behavior of economy 1, (w1; c1; pm1): In particular, we show
that welfare in economy 1 is described by the formula
c1(!) = K1!(1)=(+) [1 + ( 1)!] [+ ( )!]()=(+) ; (9.2)
where the constant K1 does not depend on ! (though it does depend on the tari¤
level b! of the rest of the world).The expression (9.2) can be used to calculate the optimal tari¤: the level of
! that maximizes utility c1 for country 1: One can show, provided that < ;
that there is a unique ! that maximizes c1 and that the optimal tari¤ is strictly
positive (that ! 2 (0; 1)) and increasing in : This result is quite intuitive. For
small decreases, there are small di¤erences across countries, and hence a given
increase in tari¤s produces a large decrease in the demands for the products of country
1: Consequently, the optimal tari¤ is decreasing in : Figure 9.1, based on (9.2),
illustrates the way utility c1 varies with ! for di¤erent values. The vertical axis in
the gure is log(c1(!)) log(c1(1)):
Should we be surprised at this persistence of market power as the economy be-
comes vanishingly small? Not under the Eaton-Kortum technology: Any economy,
no matter how small, has some goods which it is extremely e¢ cient at producing.
Subject to transportation costs, a small country can serve a large part of the world
market for these particular goods. In our case, this market power cannot be exploited
by individual sellers, because others in the same economy have free access to the ef-
cient, constant returns technology. But as Figure 9.1 illustrates, it can be exploited
43
by the government.11
INSERT FIGURE 9.1
It follows from these observations that a Nash equilibrium of a world-wide tari¤
game involving many small countries would involve strictly positive tari¤ levels for
every country. We do not go further into computing such an equilibrium. Instead, we
calculated the analogue to the equilibrium shown in Figure 8.3 that results when the
observed tari¤ factors are replaced with the free-trade factors (1; 1; :::; 1): We then
calculated the percentage increase i in consumption the country 1 would receive
if the actual tari¤s were reduced from the levels reected in to zero everywhere.
These gains are reported, as percents, in Column (5) of Table 2. They are shown in
Figure 9.2, plotted against each countrys initial tari¤ rate, i:
INSERT FIGURE 9.2
One can see the optimal tari¤ structure in Figure 9.1 reected in the U-shaped
pattern of gains from trade shown in Figure 9.2. The gure shows the e¤ect of a tari¤
reform beginning from a situation in which tari¤s vary realistically cross-sectionally
and ending with all tari¤s at zero. In the post-reform situation, every country would
like to have its tari¤ at the best response to a world of zero tari¤s. Countries with
initial tari¤s near this level lose the most from moving their own tari¤s to zero, though
they still gain from otherstari¤ reductions. Countries with very high initial tari¤s
gain from a reduction to the optimal tari¤, but then lose some of these gains back as
they continue toward zero. Countries with very low initial tari¤s were never at their
optimal tari¤, so they only gain from othersreductions. Finally, we can see that for
small countries with tari¤s near 10%, the welfare gains are similar to those shown on
Figure 7.2, computed under the symmetric world assumption.
11A similar point is made in Helpman and Krugman (1989) and Gros (1987), in a context of
imperfect competition.
44
10. Sources of Income Di¤erences
Our nal simulation involves dropping the proportionality assumption = kL
and instead using the relative price data P given in column (3) of Table 2 to estimate
and L separately. Equations (7.8) and (7.9) illustrated the way that evidence on
relative prices can be used to identify i and Li separately in the context of a costless
trade example. In the general case, equation (2.12) implies that equilibrium wages
and prices satisfypipmi
= (1 )1+
wipmi(w; )
for all i, where the notation emphasizes that the right side can be computed as a
function of w and : Write (w; ) for the n-vector of right side values, and view
the relative prices on the left as the theoretical counterparts to the observed relative
prices Pi in column (3) of Table 2. Then we can obtain estimates of w;L; and by
solving
Z(w;L; ) = 0; (10.1)
L "(w; ) = Y; (10.2)
and
(w; ) = P; (10.3)
where " is dened implicitly in (7.2). The system (10.1)-(10.3) consists of 3n equations
to be solved for the 3n unknowns w;L; and :
To solve (10.1)-(10.3) we use an algorithm that parallels the one described in
Section 8. We construct a function : nn ! nn whose xed point (; L)
calibrates the model. As in Section 8, we use w(; L) for the solution to (10.1).
The rst n coordinates of exactly parallel (8.5) but without the requirement that
= L :
i(; L) =Yi
"i(w(; L); )=
nXj=1
Yj"j(w(; L); )
; i = 1; :::; n: (10.4)
45
As in Section 8, if i(; L) = Li then (10.2) is satised up to a normalization.
To derive the remaining n coordinates we proceed in three steps. First note that
Zi(w; ; L) = 0 is equivalent to
wi(; L) = i(; L)=(+)
iLi
=(+)(10.5)
where i(; L) is dened by
i(; L) =nXj=1
Ljwj (1 sj)
(1 si )Fj
pmj
p(1)mi
ij!ijAB
!1=!ji:
(We use the superscript (*) to indicate functions evaluated at w(; L):) Second, we
insert wi(; L) from (10.5) into (10.3) to dene
bn+i(; L) = " Pi(1 )1+
1=pmi(w
(; L); )
#(+)=Li
i(; L);
for i = 1; :::; n. By construction, if bn+i(; L) = i for all i then (10.3) is satised.
The third and nal step is to normalize:
n+i(; L) = bn+i(; L)= nXj=1
bn+j(; L):While a vector solving n+i(; L) = i does not satisfy (10.3) it matches the relative
prices Pi=Pj: Using the homogeneity of w(; L) in and the homogeneity of pm(w; )
in (w; ); we can always rescale so that a solution to n+i(; L) = i does satisfy
(10.3). As in Section 8, we calculate the xed points of by iterating on an initial
pair (; L):
In the 60 country data set, the observed relative prices Pi are correlated with per
capita GDPs, yi, and so are the equilibrium wage rates wi obtained by solving (10.1)-
(10.3). Figure 10.1 plots Pi and wi against yi. The behavior of i=Li implied by the
wage behavior shown on the gure contrasts with the simulations of Sections 8 and 9,
which require constancy of i=Li: In the calculations of Section 8 and 9, wage rates per
46
unit of e¤ective labor vary only due to variations in the economys size (Figure 8.1), so
the remaining variation in earnings per capita (that is, most of it) must be attributed
to variation in human and physical capital per person. The calculation in this section
permits variation in i=Li to be an independent source of wage variation. Figure 10.2
illustrates this di¤erence, comparing simulated wages from the two simulations. One
can see that variation in country size, the only variable a¤ecting equilibrium wages
in Sections 8 and 9, is but a minor source of variation compared to variation as
estimated using the relative price data.
INSERT FIGURES 10.1 AND 10.2
These calculations illustrate the fact that, as suggested by the costless trade
example in Section 7, relative price information can in principle be used to iden-
tify individual countriestechnology parameters i in the manner dened by (10.1)-
(10.3). Whether this procedure, applied to the particular price data and sample of
countries that we have used in fact provides good estimates of parameters that can
be interpreted as cross-section di¤erences in the average e¢ ciency of manufacturing
technologies is a harder question, and one we do not claim to have resolved.
As Figure 10.1 shows, the i variation implied by (10.1)-(10.3) in our application
is very much the result of the attempt to stretch the theory to account for the behavior
of very poor and very rich economies alike. We use one set of parameters to describe
the production technology in economies like the United States and in economies like
India, so we are obliged to view millions of impoverished people in rural Asia and
elsewhere as though they were engaged in the same mix of activities that American
workers are engaged in, only doing them very badly.12
This same issue was present in the background of Sections 8 and 9 as well, where
we used the same 60 country data set. There it was hidden from view by our focus12The empirical results in Eaton and Kortum (2002), our only independent source of information
in the parameter , are also based on data from the OEDC countries only.
47
on each countrys total GDP, ignoring the breakdown into population and GDP per
person. By constraining i=Li to equal one, as we did in those simulations, we simply
precluded ourselves from learning anything new about the relative importance of hu-
man and physical capital di¤erences and technology di¤erences in the determination
of per capita incomes. But it does not follow that by relaxing this constraint we have
learned something about this issue, and we remain doubtful that we have.
Of course, the focus of Sections 8 and 9 was on the determination of trade ows
and the consequences of policy changes that a¤ect these ows, and we think we have
made some progress on these questions. It is reassuring that the simulations of this
section, with i=Li left free, give very similar answers on the volume of, and gains
from, trade as the constrained simulations do.
11. Conclusion
We think of this paper as a kind of trial run of a particular version of the Eaton
and Kortum trade theory. As we formulated the theory, the problem of solving
for equilibrium prices and quantities can be reduced to solving for the vector of
equilibrium wages in the n countries that comprise the world economy, very much
along the lines of Wilsons (1980) analysis. We have shown that such an equilibrium
exists under reasonable conditions and that under somewhat tighter assumptions
it will be unique. We have proposed and tested an algorithm that is essentially a
tatonnement process for calculating equilibria. We have discovered that toy versions
of the theory can provide surprisingly accurate approximations to predictions about
wages, trade volumes and gains from trade, so pencil-and-paper calculations can be
used to provide inexpensive checks on quantitative conjectures and to help interpret
simulation results.
For the most part, objects in the theory match up naturally to counterparts
in the national income and product accounts, input-output accounts, and standard
48
trade statistics. This makes much of the calibration easy to carry out, lets us focus
attention sharply on small regions of the theorys very large parameter space, and fa-
cilitates interpretation of simulation results. These features are essential to successful
quantitative economics.
The calibrated model accounts fairly well for the overall volume of world trade in
the 2000, and for the way volume varies cross-sectionally with an economys size and
tari¤ levels. With its assumption of continuous trade balance, the theory is obviously
not designed to interpret short term uctuations. We have not tested the theorys
ability to account for trends in trade volumes (as studied, for example, in Yi (2003))
nor have we responded to Kehoes (2002) challenge to provide a satisfactory account
of the e¤ects of NAFTA or other important trade agreements. These issues are high
on our agenda, as they are on every quantitative international economists.
Fitting the model to aggregate size and volume data gives some information on
the size of the variance parameter values smaller than 0.1 or larger than .25 do
not yield good versions of Figure 7.6 but this is a very wide range for so crucial a
parameter. The best available estimates are those we cited from Eaton and Kortum
(2002), but these authors do not claim much precision, either. There is so much good
productivity data available now, at very ne levels of detail, that it should be possible
to get reasonably accurate, direct information on i for di¤erent economies and for
the parameter , which we assume to be common.13
There is also an uncomfortable amount of latitude in the interpretation and
calibration of the parameters ij that describe transportation costs. Values like 0.8
or .75 are far too low to be attributed to physical shipping costs. (See, for example,
Hummels (1999).) They should also include time costs of dealing with regulations,
inspections, and other resource-using activities associated with international trade
13As in Baxter (1992), this can be done without disturbing the Ricardian character of long run
behavior.
49
any costs that cannot be recycled back to consumers as we assume tari¤s costs are.
There is much evidence that might be brought to bear on this issue, too.
We have kept the analysis and empirical work in this paper on a strictly static
basis, in order to keep complications within bounds and to understand better the con-
nections with other trade theories. The cost of this decision was to leave the models
many connections to growth theory unexplored. One natural direction, already ex-
amined by Eaton and Kortum (1999), will be to introduce technology di¤usion by
introducing a law of motion for the parameters . Another is to introduce investment
goods and physical and human capital accumulation. Perhaps in some combination
such extensions can help us to discover the long-sought theoretical link between trade
and growth. The static gains from trade that we have calculated here, like the gains
implied by earlier models, seem too small to contribute much to this goal.
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53
Appendix A: Proofs of Theorem 2, (iv) and (v).
Proof of part (iv).The lower bound on Zi(w) is implied by
Zi(w) =1
wi
"nXj=1
Ljwj(1 sj)1
FjDji!ji Liwi(1 si)
#
Li(1 si)
Li:
Proof of part (v). Suppose that fwmg is a sequence in Rn++, that w
m ! w0 6= 0,
and that w0i = 0 for some i. We need to verify that (4.9) holds for this sequence. For
any w 2 Rn++, we have
maxkZk(w) = max
k
"nXj=1
Ljwjwk(1 sj)
1
FjDjk!jk Lk(1 sk)
#
maxk
nXj=1
Ljwjwk(1 sj)Djk!jk max
kLk
maxk;j
Ljwjwk(1 sj)Djk!jk max
kLk:
By Assumption (A), !jk !; implying in turn that the functions Fi take values in
[!; 1]. Then (3.14) implies that the shares 1 sj are uniformly bounded away from
zero. Thus (4.9) will be proved if it can be shown that
maxk;j
wmjwmk
Djk(wm)!1 (A.1)
for the wage sequence fwmg:
From (3.9) we have, for any w;
Djk(w) = (AB)1= p
1=mj
wkp
1mk
jk!jk
!1=k
(AB)1= (!)1= kw=k p1=mj p
(1)=mk
54
or that
Djk(w) (AB)1= (!)1= kpmkwk
= pmjpmk
1=: (A.2)
Using (3.8) directly,pmjpmk
1==
"nXr=1
wr p
1mr
jr!jr
1=r
#1 nXr=1
wr p
1mr
kr!ir
1=r (!)1= : (A.3)
Using the lower bound established in Theorem 1, (v), we havepmkwk
= w
=k (AB)1=
nXr=1
w=r r
!1: (A.4)
It follows from (A.2)-(A.4) that
wjwkDjk(w) (!)2= k
nXr=1
wkwr
=r
!1wjwk
and therefore that for all w;
maxk;j
wjwkDjk(w) (!)2=
minkk
nXr=1
mink wkwr
=r
!1maxj wjmink wk
(!)2=minkk
nXr=1
r
!1maxj wjmink wk
:
Since the wm ! w0 6= 0 with w0i = 0 for some i
maxjwmj ! max
jw0j > 0
and
minkwmk ! min
kw0k = 0:
This veries (A.1) and hence (4.9) and completes the proof of Theorem 2.
55
Appendix B: Proof of Theorem 3.
We show that Z has the gross substitute property (5.7):
@Zi(w)
@wk> 0 for all i; k; i 6= k; for all w 2 R++::
Before calculating the derivatives, note that Zi can be written as
Zi(w) =(1 )
wi
"nXj=1
LjwjDji!j+ ( )Fj
LiwiFi+ ( )Fi
#;
using (3.14) to substitute for sj in (3.22) and using the fact that
(1 ) Fj + [1 (1 )Fj] = + ( )Fj:
Thus we can write the derivatives in (5.7) as
@Zi(w)
@wk=
(1 )
wi
nXj=1;j 6=i;j 6=k
Ljwj!j@
@wk
Dji
+ ( )Fj
+(1 )
wiLk!k
@
@wk
wkDki
+ ( )Fk
+(1 ) Li
@
@wk
Dii Fi
+ ( )Fi
:
In the following three steps we sign each of the three terms of this derivative.
- Step (ia): If (5.4) holds, then
@
@wkDji > 0
for all j 6= i: To see this, notice that direct computation gives
@Dji
@wk=Dji
wk
1
@~pmj@ ~wk
(1 )@~pmi@ ~wk
:
Let
k
wkp
1mk
1=kPn
r=1
wr p
1mr
1=r
56
so that the denitions of ik; k; andk imply
(!)1= k k ik k
1
(!)1=k
for all i: Then Theorem 1 (vi) implies
@Dji
@wk Dji
wk
1
hk (1 ) k
i Dji
wk
1
k
"(!)1= (1 )
(!)1=
#=
Dji
wk
1
(!)1=k
h(!)2= (1 )
iwhich is positive if (5.4) holds.
- Step (ib): If then for j 6= k
@
@wk(+ ( )Fj) 0:
To see this, notice that under the assumption of uniform tari¤s
Fj = !j + (1 !j)Djj = !j + (1 !j) (AB)1=
pmjwj
=j: (B.1)
Hence@
@wk(+ ( )Fj) = ( ) (1 !j)
@Djj
@wk
and for j 6= i@Djj
@wk= (=)
Djj
pmj
@pmj@wk
> 0;
by (iii) in Theorem 1. Thus if (5.5) holds (if ) then
@
@wk(+ ( )Fj) = ( ) (1 !j) (=)
Djj
pmj
@pmj@wk
0:
Clearly, (ia) and (iib) imply that
nXj=1;j 6=i;j 6=k
Ljwj!j@
@wk
Dji
+ ( )Fj
> 0:
57
- Step (ii): If (5.5) and (5.6) hold for k 6= i;
@
@wk
wkDki
+ ( )Fk
> 0:
To see this notice that direct computation gives
@
@wk
wkDki
+ ( )Fk
=
(Dki + wk@Dki=@wk) (+ ( )Fk) wkDki ( ) @Fk=@wk
[+ ( )Fk]2
In step (ia) we verify that (5.4 ) implies that @Dki=@wk 0 so that it will su¢ ce to
show thatDki (+ ( )Fk) wkDki ( ) @Fk=@wk
[+ ( )Fk]2 0:
Using (B.1),
@Fk@wk
= (1 !k)@Dkk
@wk= (1 !k)
Dkk
wk
@~pmk@ ~wk
1
(1 !k)1
wk(=)
since Dkk 1 and @~pmk=@ ~wk > 0: Thus
@
@wk
wkDki
+ ( )Fk
Dki
(+ ( )Fk) + ( ) (1 !k) (=)
[+ ( )Fk]2
=Dki
[+ ( )Fk]
1 ( ) (1 !k) (=)
[+ ( )Fk]
and since Fk 2 [!; 1] and
@
@wk
wkDki
+ ( )Fk
Dki
[+ ( )Fk]
1 ( ) (1 !k) (=)
=
Dki
[+ ( )Fk]
1 ( ) (1 !k)
and thus if if condition (5.6) holds the inequality in step ii) is veried.
- Step (iii) For k 6= i;
@
@wk
Dii Fi
+ ( )Fi
> 0 :
58
To see this, use (B.1) so that
Dii Fi+ ( )Fi
=!i (1Dii)
+ ( ) (!i + (1 !i)Dii):
Hence
@
@wk
Dii Fi
+ ( )Fi
=
@
@Dii
!i (1Dii)
+ ( ) (!i + (1 !i)Dii)
@Dii
@wk
with@Dii
@wk= (=)
Dii
pmi
@pmi@wk
> 0
and
@
@Dii
!i (1Dii)
+ ( ) (!i + (1 !i)Dii)
=
!i (+ ( ) (!i + (1 !i)Dii)) + !i (1Dii) ( ) (1 !i)
[+ ( ) (!i + (1 !i)Dii)]2
=!i
[+ ( ) (!i + (1 !i)Dii)]2 > 0
which establishes the inequality in step (iii).
This shows that Z satises the gross substitute property, and hence that the
equilibrium is unique.
Appendix C: Behavior of the Limiting Economies
We verify that as Lr1 ! 0, the limiting behavior (w1; c1; pm1) of the other
n 1 economies is equal to the equilibrium of a world economy with n 1 countries
and endowments (L1; 1). We show that for the limiting behavior of economy 1,
(w1; c1; pm1); is given by
w1 =
+ ( )!
!1(1)=k
=(+) b!(1+)=(+) w1; (C.1)
c1 = !(1)=(+) [1 + ( 1)!]
[+ ( )!]()=(+) b!(1)(1+)=(+) k (1)+ c1; (C.2)
59
and
pm1 = pm1 = !1; (C.3)
where the numbers pm1; w1 and c1 do not depend on !; b! and k:We proceed under the hypothesis that w1 2 (0;1) ; which we verify later on.
Then when 1 = kL1 = 0; (3.8) implies
pmi = AB
0@ nXj=2
wj p
1mj
ij!ij
!1=j
1A
;
for all i and, using the assumption country 1 imposes a uniform tari¤ !,
pm1 = !1AB
0@ nXj=2
wj p
1mj
1j
!1=j
1A
= !1 pm1:
The second equality denes pm1 and veries (8.4).
The fraction of country js expenditures on tradeables produced by country 1 is
Dj1(w) = (AB)1=
pmjj1!j
p1m1 w1
1=1 = 0:
As Lr1 ! 0, this fraction goes to zero for each j, limr!1Dj1 (wr) = 0: Inspection of
the expression for Zi (w) then conrms that when 1 = kL1 = 0 the excess demand
system Z1 (w) (Z2 (w) ; Z3 (w) ; :::Zn (w)) = 0 does not depend on w1; b!; !: Thecontinuity of Z implies that w1 solving Z1 (w1; (w1)) = 0 is the desired limit.
The next step is to derive an expression for the limit for economy 1. We take
w1 as given in the previous step. For L1 > 0; we have that Z1 (w1; w1) = 0 is
equivalent to Z1 (w1; w1) =L1 = 0, so we analyze the latter expression. It can be
shown, using the formulas (3.13) and (3.14) that for L1 > 0
Z1(w1; w1)
L1= 0
60
is equivalent to
nXj=2
Ljwj(1 sj)
Fj(AB)1= (pmjj1)
1=k
1
p1m1 w1
1=!1+1=
= w1+=1
(1 )F1+ ( )F1
"1 1
F1(AB)1=
pm1w1
=L1
#
As 1 = kL1 ! 0; then D11 ! 0; and hence F1 = D11 + (1D11)! ! !: Thus,
taking the limit as L1 ! 0 yields
nXj=2
Ljwj(1 sj)
Fj(AB)1= (pmjj1)
1=k !(1+)=
= w1+=1
(1 )!
+ ( )!(pm1)
(1)= :
Solving for w1 yields (C.1), where it can be seen that the factor w1 does not depend
on !; b! and k:Finally we turn to the calculation of c1: From (7.2) we have that with uniform
tari¤s
p1c1L1 = L1w1
1 +
(1 ) (1 !1)
+ ( )F1(1D11)
:
and since D11 ! 0 and F1 ! !1 as L1 ! 0 then
limL1!0
p1c1 = w1
1 +
(1 ) (1 !1)
+ ( )!1
= w1
!1 + (1 !1)
+ ( )!1
:
Using the expression for p1 in (2.12),
c1 =1
(1 )1+
w1pm1
1 ! + (1 !)
+ ( )!
:
Using the expression for pm1 in (C.3) and for w1 in (C.1),
c1 =1
(1 )1+
!
pm1
1 ! + (1 !)
+ ( )!
+ ( )!
(1 )!
!
pm1
(1)=G (w1) k b!(1+)=!
(1)+
;
61
where
G (w1) =
"nXj=2
Ljwj (1 sj)
Fj(AB)1=(pmjj1)
1=
#=(+):
Collecting terms involving !, b! and k :c1 = k
(1)+ !
1+ [1 + ( 1)!] [+ ( )!]
++ b! (1)(1+)
+ c1
where it can be seen that the factor c1 does not depend on !, b! or k:c1 =
(pm1)(1)[ 1++ ]
(1 )1+
G (w1)
(1 )
(1)+
This completes the proof.
62
TABLE 2: DATA AND SIMULATIONS RESULTS
Size Trade Volume Relative Price Tariff (in %) Welfare Gain Per Capita GDPCountry Name GDP as % of Imports /GDP Consumption / Mean across of eliminating PPP Adjusted
World GDP Mach. & Equipt goods tariffs, in percent US =1 Yi [1] Vi [2] Pi [3] Ωi [4] [5] yi [6]
United States 27.99 0.10 1.37 5.40 0.15 1.00Japan 15.69 0.08 1.65 5.48 0.25 0.83Germany 7.35 0.27 1.48 5.86 0.41 0.76Rest of the World 5.25 0.32 0.70 11.49 * 0.20 0.10France 4.86 0.24 1.55 5.86 0.52 0.73United Kingdom 4.30 0.28 1.28 5.86 0.56 0.70Italy 3.85 0.25 1.27 5.86 0.59 0.71China 2.86 0.22 0.70 (*) 18.58 0.47 0.10Brazil 2.29 0.09 1.00 13.73 0.42 0.23Canada 2.06 0.33 1.30 5.10 0.84 0.82Spain 1.93 0.26 1.35 5.86 0.79 0.56Mexico 1.40 0.29 0.72 14.26 0.56 0.26India 1.34 0.13 0.70 (*) 33.44 2.79 0.07Australia 1.32 0.17 1.13 5.30 0.94 0.76Netherlands 1.31 0.48 1.51 5.86 0.89 0.75Russian Federation 1.09 0.28 0.48 12.47 0.63 0.24Argentina 0.94 0.11 1.02 12.40 0.66 0.39Switzerland 0.91 0.31 1.70 0.68 * 1.56 0.88Belgium 0.85 0.73 1.61 5.86 0.99 0.80Sweden 0.80 0.39 1.61 5.86 1.00 0.71Austria 0.72 0.35 1.57 5.86 1.02 0.79Turkey 0.61 0.25 0.62 12.20 0.75 0.21Indonesia 0.59 0.32 0.60 9.88 0.81 0.10Denmark 0.58 0.35 1.50 5.86 1.05 0.82Hong Kong, China 0.52 1.39 1.70 0.00 1.74 0.76Norway 0.50 0.36 1.67 4.04 1.24 0.90Thailand 0.49 0.49 0.64 18.00 0.91 0.21Poland 0.47 0.27 0.65 15.55 0.81 0.25Saudi Arabia 0.47 0.36 0.70 (*) 12.48 0.78 0.37South Africa 0.47 0.24 0.70 (*) 8.30 0.92 0.29Finland 0.41 0.34 1.39 5.86 1.10 0.70Greece 0.40 0.19 1.08 5.86 1.10 0.48Portugal 0.36 0.34 0.97 5.86 1.11 0.50Israel 0.32 0.39 1.65 7.55 1.00 0.60Iran, Islamic Rep. 0.32 0.21 0.48 4.90 1.21 0.18Colombia 0.31 0.19 0.70 (*) 11.70 0.85 0.20Venezuela, RB 0.29 0.24 0.71 12.28 0.84 0.20Malaysia 0.29 1.00 0.70 (*) 9.18 0.93 0.26Singapore 0.29 1.62 2.02 0.16 1.78 0.67Ireland 0.26 0.63 1.29 5.86 1.14 0.72Egypt, Arab Rep. 0.25 0.22 0.22 27.60 2.05 0.11Philippines 0.25 0.48 0.67 11.22 0.88 0.12
Chile 0.23 0.29 0.82 10.25 0.91 0.28Pakistan 0.20 0.18 0.76 39.90 5.16 0.06New Zealand 0.19 0.26 1.29 4.84 1.25 0.61Peru 0.18 0.16 0.72 13.30 0.88 0.15Czech Republic 0.18 0.60 0.52 7.04 1.09 0.43Algeria 0.16 0.27 0.70 (*) 24.60 1.63 0.16Hungary 0.15 0.46 0.52 13.42 0.89 0.35Ukraine 0.15 0.47 0.24 10.20 0.94 0.12Bangladesh 0.14 0.15 0.68 21.30 1.27 0.05Romania 0.12 0.31 0.38 16.03 0.95 0.21Morocco 0.11 0.30 0.50 30.77 2.68 0.11Nigeria 0.11 0.41 0.61 24.06 1.58 0.03Vietnam 0.08 0.25 0.28 15.15 0.94 0.06Belarus 0.08 0.62 0.30 12.63 0.92 0.20Kazakhstan 0.07 0.40 0.37 1.15 * 1.71 0.16Slovak Republic 0.06 0.64 0.43 6.88 1.14 0.32Tunisia 0.06 0.45 0.40 31.17 2.77 0.18Sri Lanka 0.05 0.41 0.57 7.67 * 1.09 0.10
Sources: [1]: Share in world gdp. GDP in current dollars. From WDI 2002 cdRom, average 1994-2000 [2]: 0.5*(Exports+Imports)/GDP, all in current dollars. From WDI 2002 cdRom, average 1994-2000 [3]: Price of machinery and equipment relative to consumption, from benchmark PWT year 1996. Average of other countries for ROW. The countries marked with (*) are not in the 1996 PWT benchmark, so we use the average for the other countries. [3] displays the reciprocal of this number. [4]: Average 1996-2000 ad-valorem tariff rate, simple average across products, from Dollar and Kraay, using worldbank database "Data on Trade and Imports Barriers", when available. When it is not available, indicated as *, import duties/imports from WDI 2002. Average of other 59 countries for ROW [5]: Calculations described in section 9, Figure 9.2 [6]: Pen World Table average 1994-2000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
10
20
30
40
50
60
FIGURE 7.1: WELFARE GAINS FROM GOING FROM AUTARCHY TO FREE TRADEThree values of dispersion θ
WE
LFA
RE
GA
IN IN
PE
RC
EN
T: lo
g di
ffere
nce
in c
onsu
mpt
ion
x 10
0
COUNTRY SIZE: fraction of world GDP
κ = 0.75
β = 0.5
α = 0.75
Parameters
θ = 0.1
θ = 0.15
θ = 0.25
0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FIGURE 7.2: WELFARE GAINS FROM ELIMINATING A 10% TARIFFThree values of dispersion θ
WE
LFA
RE
GA
IN IN
PE
RC
EN
T: lo
g di
ffere
nce
in c
onsu
mpt
ion
x 10
0
COUNTRY SIZE: fraction of world GDP
ω = 0.9
κ = 0.75
β = 0.5
α = 0.75
Parameters θ = 0.1
θ = 0.15
θ = 0.25
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
2
4
6
8
10
12
FIGURE 7.3: WELFARE GAINS FROM ELIMINATING A TARIFF FACTOR ωThree values of ω
WE
LFA
RE
GA
IN IN
PE
RC
EN
T: lo
g di
ffere
nce
in c
onsu
mpt
ion
x 10
0
COUNTRY SIZE: fraction of world GDP
θ = 0.15
κ = 0.75
β = 0.5
α = 0.75
Parameters
ω = 0.8
ω = 0.7
ω = 0.6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
FIGURE 7.4: VOLUME OF TRADEThree values of dispersion parameter θ
IMPO
RT
TO G
DP
RAT
IO: v
COUNTRY SIZE: fraction of world GDP
ω = 0.9
κ = 0.75
β = 0.5
α = 0.75
Parameters
θ = 0.1
θ = 0.15 θ = 0.25
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
FIGURE 7.5: VOLUME OF TRADEThree values of tariff factor ω
IMPO
RT
TO G
DP
RAT
IO: v
COUNTRY SIZE: fraction of world GDP
θ = 0.15
κ = 0.75
β = 0.5
α = 0.75
Parameters
ω = 0.7 ω = 0.8ω = 0.9
-8 -7 -6 -5 -4 -3 -2 -10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8FIGURE 7.6: VOLUME OF TRADE
IMPO
RT
TO G
DP
RAT
IO
COUNTRY SIZE: log of fraction of world GDP
θ = 0.15
ω = 0.9
κ = 0.75
β = 0.5
α = 0.75
Parameters Singapore
Hong Kong
Malaysia
USAJapanArgentina
Belgium
Ukraine
Bangladesh
Germany
-7 -6 -5 -4 -3 -2-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
FIGURE 8.1: WAGES VERSUS SIZEThree values of θ
COUNTRY SIZE: log share of GDP
WA
GE
S in
logs
Parameters
α = 0.75
β = 0.5
κ = 0.75
ω = 0.9
θ = 0.1
θ = 0.15
θ = 0.25
-6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
FIGURE 8.2: VOLUME OF TRADE
IMPO
RT
TO G
DP
RAT
IO
COUNTRY SIZE: log of share of GDP
θ = 0.15
ω = 0.9
κ = 0.75
β = 0.5
α = 0.75
Parameters
CALIBRATED
SYMMETRICUSA
Japan
Argentina
Sweden
Belgium
Germany
Netherlands
Peru
Czech Republic
Bangladesh
Brazil
Canada
-6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
FIGURE 8.3 VOLUME VERSUS SIZEData (o), Model w/uniform tariffs (solid), Model w/tariffs from Table 2 (*)
COUNTRY SIZE: log of share of GDP
IMP
OR
TS T
O G
DP
RA
TIO Parameters
α = 0.75
β = 0.5
κ = 0.75
ωuniform = 0.9
θ = 0.15
ρ (log Vdata , log vmodel ωi ) = 0.58
ρ (log Vdata , log vmodel uniform ω ) = 0.45
0 5 10 15 20 25 30-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
FIGURE 9.1: WELFARE EFFECTS OF TARIFF FACTOR ω FOR A SMALL OPEN ECONOMYThree values of θ
TARIFF IN PERCENT: (1 - ω) x 100
WEL
FAR
E G
AIN
REL
ATIV
E TO
ZER
O T
ARIF
F IN
PER
CEN
T: lo
g di
ffere
nce
x 10
0
Parameters
α =0.75
β =0.5
θ =0.1
θ =0.15
θ =0.25
0 5 10 15 20 25 30 35 400
1
2
3
4
5
6FIGURE 9.2: WELFARE GAINS FROM ELIMINATING TARIFFS
TARIFFS BEFORE REDUCTION IN PERCENT: (1 - Ωi) x 100
WE
LFA
RE
GA
IN IN
PE
RC
EN
T: lo
g di
ffere
nce
x 10
0
Parameters
α = 0.75
β = 0.5
κ = 0.75
θ = 0.15
USAJapan
ArgentinaPeru
Czech RepublicBangladesh
Brazil
Nigeria
Pakistan
Hong Kong
India
France
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
FIGURE 10.1: MEASURED RELATIVE PRICES P AND SIMULATED WAGESWages: (*), and Relative Prices P: (o)
Real per capita GDP PPP adjusted, normalized to US=1
WA
GE
S a
nd R
ELA
TIV
E P
RIC
ES
P, n
orm
aliz
ed to
US
=1
Parameters for simulated wage
α = 0.75
β = 0.5
κ = 0.75
θ = 0.15
-8 -7 -6 -5 -4 -3 -2 -10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2FIGURE 10.2: WAGES IN THE TWO CALIBRATIONS
COUNTRY SIZE: log of share of GDP
WAG
ES
Parameters
α = 0.75
β = 0.5
κ = 0.75
θ = 0.15
σ (log w ) = 0.08 for constant λ / L
σ (log w ) = 0.63 for variable λ / L