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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Cambidge: MIT Press.

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versity of Chicago working paper.

[19] Kehoe, Timothy J. 2002. An Evaluation of the Performance of Applied General Equi-

librium Models of the Impact of NAFTA.Federal Reserve Bank of Minneapolis

Research Department Sta¤ Report.

[20] Michael Kremer. 1993. Population Growth and Technological Change: One Million

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tional Trade.Journal of International Economics. 9: 469-480.

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ory. 1995. New York: Oxford University Press.

[23] Mark J. Melitz. 2003. The Impact of Trade on Intra-Industry Reallocations and

Aggregate Industry Productivity.Econometrica. 71: 1695-1726.

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nomics and Statistics, 46: 145-154.

52

[25] Spence, A. Michael. 1976. Product Selection, Fixed Costs, and Monopolistic Com-

petition.Review of Economic Studies. 43: 217-235.

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University of Chicago working paper.

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Journal of Political Economy. 111: 52-102.

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