Task Trade between Similar Countries

Gene M. Grossman
Princeton University

Esteban Rossi-Hansberg
Princeton University

January 2009

Abstract
We propose a theory of task trade between countries that have similar relative factor endowments but may di¤er in size. Firms produce di¤erentiated goods by performing a continuum oftasks, each of which generates local spillovers. Tasks can be performed at home or abroad, buto¤shoring entails costs that vary by task. In equilibrium, the tasks with the highest o¤shoring
costs may not be traded. Among the remainder, those with the relatively higher o¤shoring costs
are performed in the country that has the higher wage and higher aggregate output. We discuss
the relationship between equilibrium wages, equilibrium outputs, and relative country size andexamine how the pattern of specialization re‡ects the key parameters of the model.

We thank Aykut Ahlatç¬o¼
glu and Shlomi Kramer for research assistance, Sebastian Benz, Philipp Ehrl, Gordon
Hanson, Giovanni Maggi, Marc Melitz, Jens Wrona and Kei-Mu Yi for discussions and comments, and the National
Science Foundation (under grants SES 0211748 and SES 0451712) and Sloan Foundation for research support. Any
opinions, …ndings, and conclusions or recommendations expressed in this paper are those of the authors and do notnecessarily re‡ect the views of the National Science Foundation or any other organization.

1

Introduction

Modern production assigns a prominent role to international task trade. The delivery of a good orservice to a consumer typically requires the completion of a myriad of di¤erent tasks. Increasingly,
the performance of these tasks is spread across the globe, with an impressive share of o¤shore
production in the value of many …nal goods. As a result, international trade is less today a matterof countries’specialization in particular industries and more about their specialization in particularoccupations and tasks.
Much has been written about the growth of o¤shoring between countries that stand at di¤erent
levels of development, i.e., countries that have dissimilar factor endowments and disparate technological capabilities.1 Yet, as important as this sort of o¤shoring is becoming in world trade, it pales
in comparison to task trade between similar countries. Not only does most trade ‡ow between andamong the advanced industrialized economies, but these economies are engaging in an ever more
intricate web of production-sharing arrangements.
The Boeing 787 Dreamliner is a case in point.2 The production of this new midsize jet involves
43 suppliers spread over 135 sites around the world. Boeing relies heavily on local expertise when
making its sourcing decisions. The wings are produced in Japan, the engines in the United Kingdom
and the United States, the ‡aps and ailerons in Canada and Australia, the fuselage in Japan, Italy
and the United States, the horizontal stabilizers in Italy, the landing gear in France, and the doors in
Sweden and France. O¤shore production accounts for close to 70 per cent of the many thousands ofparts used to assemble the jet (Newhouse, 2007, p.29). Some parts are produced in foreign a¢ liatesof the Boeing Corporation while others are supplied under international outsourcing agreements.
The countries that perform the various tasks display no clear pattern of technological advantage.
Rather, experience and local knowledge play a central role. Apparently, expertise most often
derives from similar tasks being performed for other Boeing projects or for related industries, suchas military aviation and automobile production.3
Aggregate data on production sharing among the developed countries is di¢ cult to come by.4
Yet, hints of the substantial magnitude of such task trade abound. As one example, we point to thelocation of the stocks of U.S. foreign direct investment and of the employment of foreign a¢ liatesof U.S. …rms. Figure 1 shows the U.S. direct investment position in several regions of the world.
Not only is Europe the site of more than half of the accumulated foreign assets of U.S. …rms, but
the most recent FDI ‡ows are adding to its lead. Canada, another “similar” country in terms of
1

See, for example, Jones and Kierzkowski (1990, 2001), Feenstra and Hanson (1996), Deardor¤ (2001a, 2001b), Yi(2004), Egger and Falkinger (2003), Kohler (2004a, 2004b), Baldwin and Robert-Nicoud (2007), and Rodriguez-Clare(2007), as well as our own work in Grossman and Rossi-Hansberg (2008b).
2
The sourcing of Boeing’s parts for the 787 is detailed at http://www.boeing.com/commercial/787family/background.html. See Newhouse (2007) for further discussion.
3
See “A Cleverer Way to Build a Boeing,” The Financial Times, 07/08/2007.
4
Many researchers have documented production sharing for particular countries in their trade with the rest of theworld; see, for example, Campa and Goldberg (1997), Hummels, Rapoport and Yi (1998), Yeats (2001), Hummels,
Ishii and Yi (2001), Amiti and Wei (2005, 2006), and Hanson, Mataloni and Slaughter (2005). However, none ofthese authors identi…es the share of task trade that takes place between countries at similar stages of development.

1relative factor endowments and technological capabilities, accounts for an additional ten percent of
the accumulated foreign investment. All told, more than 60 percent of U.S. FDI resides in Europe
and Canada. Figure 2 shows the geographic spread of employment in foreign a¢ liates of U.S. …rms
since 1997, the …rst year for which such data are available. Clearly, employment in Europe dwarfs
that in the other locations. Not all of the activity in foreign a¢ liates of U.S. …rms representso¤shoring, nor does production sharing necessarily require an ownership relationship. Still, the…gures on FDI and employment in foreign a¢ liates suggest the importance of similar countries aspartners in U.S. task trade.

Figure 1: U.S. Direct Investment Position Abroad on a Historical-Cost Basis
700
Europe600

Latin America and Other Western Hemisphere
Asia and PacificCanadaBillions of 1983 Dollars

500

AfricaMiddle East

400

300

200

100

19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
20
06

0

Figure 2: Employment of all U.S. Foreign Affiliates
4500
4000

EuropeLatin America and Other Western Hemisphere

Thousands of Employees

3500

Asia and PacificCanada3000

AfricaMiddle East

2500
2000
1500
1000
500
0
1997

1998

1999

2000

2001

2

2002

2003

2004

2005

In this paper, we formulate a theory of task trade between similar countries. In our model,…rms incur entry costs to develop the know-how to produce particular goods. Production of any
good requires the completion of a continuum of tasks. The set of required tasks is the same forall goods, yet the resulting products are di¤erentiated in the eyes of consumers. Producers of
the …nal goods engage in monopolistic competition and sell their wares to consumers who holdconstant-elasticity-of-substitution (CES) preferences.
In keeping with the anecdotal evidence cited above, our treatment of production sharing emphasizes the role of local knowledge and specialized expertise. Our approach shares with the “newtrade theory” a focus on increasing returns to scale as a force that induces concentration of production. But whereas the most familiar models in that literature feature trade in …nal goods— forwhich scale economies internal to the …rm may be most pertinent— our focus on task trade dictatesa di¤erent approach. The expertise to produce a unique good may well reside in a single …rm,
but the expertise to perform a narrow task rarely does so. Rather, it is often embodied in a poolof specialized labor, be they engineers with speci…c training or workers with shared experience.
This suggests that localized knowledge at the task level may re‡ect external economies of scalerather than (or in addition to) internal economies. We take this notion to the extreme by assumingthat productivity in performing a task varies with the frequency with which it is performed in aparticular location, irrespective of the identify of the …rm or …rms performing the function.
The location of each task balances two competing forces. On the one hand, the external
economies of scale provide …rms with an incentive to locate each task in the country where others
are performing it. On the other hand, it is costly for …rms to organize and monitor the performanceof tasks in countries di¤erent from where their headquarters are located. Our model features heterogeneous o¤shoring costs to capture the reality that some tasks are easier to separate from …rms’
headquarters than others. For example, routine tasks can be performed remotely at relatively little
extra cost, because instructions can be expressed unambiguously and conveyed easily to workers,
with little need for interaction with central management. Other tasks may require greater adaptation to circumstances, so proximity to headquarters may be more important.5 Our analysis links
the pattern of specialization by task to the distribution of o¤shoring costs.
When small …rms operate in an environment with external economies of scale, they face anobvious coordination problem. If other …rms are performing an activity in some location, it may bemost pro…table to join them there, even if all other economic forces point to a di¤erent outcome.
As a result of the potential coordination failures, multiple equilibria can arise. Multiplicity ofequilibrium has plagued models with production externalities, where “history”and “expectations”
play a role in determining …nal outcomes. With a continuum of tasks and the possibility for
self-ful…lling expectations for each of them, it might appear that little could be said about theequilibrium allocation of tasks across countries. Yet the environment with many tasks suggests asolution to the coordination problem that narrows the set of equilibria dramatically. In particular,
5

Autor et al. (2003) have emphasized this distinction between routine and non-routine tasks and provided ameasure of this concept. Levy and Murnane (2004) and Grossman and Rossi-Hansberg (2008) have applied theconcept to explain variation in the costs of o¤shoring.

3

we recognize that …rms can perform tasks on behalf of others. The opportunity for a …rm to perform
a task for many producers means that it potentially can internalize the externalities of locational
choice. Such a supplier need not be large in relation to its industry, because even if it dominates
the performance of a particular task, there are many other tasks to be done. By introducing thepossibility for outsourcing tasks, we …nd that we can cut through the coordination issues and sayquite a lot about the pattern of specialization in equilibrium.
Our main proposition relates the pattern of specialization by task to equilibrium relative wages
and equilibrium aggregate outputs. It states that all …rms perform the tasks that are most costly
to o¤shore in the country of their headquarters. Among the remaining tasks, those that are easiest
to o¤shore concentrate in the country that has lower wages and lesser aggregate output, while thosethat are more di¢ cult to o¤shore concentrate in the country that has higher wages and greateraggregate output. Depending on the overall level of o¤shoring costs, the general equilibrium may beunique or not. With o¤shoring costs su¢ ciently high that identical-sized countries would engage inno production sharing, the unique equilibrium with unequal-sized countries has higher wages in thelarger country. When o¤shoring costs are not so high, there will be multiple equilibria in a world
with countries of nearly identical size, but a unique equilibrium when country sizes diverge. In
the former case, there is one equilibrium in which wages are higher in the (slightly) larger country,
another in which wages are higher in the (slightly) smaller country, and a third equilibrium withequal wages. In the latter case, the (much) larger country enjoys the higher wages.
Our …nding is best understood by thinking …rst about e¢ ciency considerations. The costs ofcommunication and coordination can outweigh the potential gains from agglomeration for tasks
that are quite costly to o¤shore. These tasks are e¢ ciently performed in both locations. The
other tasks should be concentrated in one location, which means they will generate o¤shoring costs
for one set of producers. Aggregate costs can be minimized if the tasks that are most costly too¤shore are performed in the country with the larger number of producers or the greater output
per …rm, because …rms headquartered in this country perform these tasks the greatest total numberof times. Since not all tasks can be performed in one country, it is e¢ cient to locate those withmodest o¤shoring costs in the country with fewer …rms or less output per …rm. We will …nd thatthe market allocation that results from potential internalization by large providers of tasks is notalways e¢ cient, because deviant suppliers can appeal to one set of national producers without
taking into account the harm they cause to others. Still, the competitive forces mimic the socially
optimal ones, at least qualitatively. Moreover, the equilibrium allocation of the di¢ cult-to-o¤shore
tasks to the country with greater aggregate output gives this country a cost advantage in producing…nal goods. This, in turn, justi…es its greater scale of output per …rm and also a higher equilibrium
wage.
Once we have proven our basic characterization of the equilibrium allocation, we rely on numerical computations to explore the relationship between equilibrium outcomes and the key parametersof the model. In particular, we study the connection between the pattern of specialization and theextent of increasing returns to scale, the extent of product di¤erentiation, and the size of o¤shoring4

costs. We …nd that larger size di¤erences between countries generate a broader range of tradedtasks and imply larger wage di¤erentials as long as some tasks are performed in both countries.
Stronger external economies of scale and higher elasticities of substitution have similar implications for the extent of production sharing and for relative wages. Not surprisingly, a reduction ino¤shoring costs induces more task trade and tends to improve welfare.
We are not aware of other e¤orts to explain the pattern of task trade between similar countries
in the nascent literature on o¤shoring. There have, of course, been many attempts in recent years
to understand the high volume of goods trade between countries at similar levels of development.
The early writings on product variety and trade (e.g., Dixit and Norman, 1980, Krugman, 1979, andHelpman, 1981) were designed exactly for this purpose, but had little to say about the pattern ofspecialization and trade. Trade patterns between similar countries have been the focus of researchon external economies of scale at the industry level, on trade in di¤erentiated products bearing
transport costs, and on comparative advantage that derives from di¤erences in the distribution offactor endowments for countries that share similar aggregate endowment ratios. This last approach(e.g., Grossman and Maggi, 2000, and Ohnsorge and Tre‡er, 2007) is quite di¤erent from the one
pursued here, so we do not discuss it any further.
Markusen and Melvin (1981) and Ethier (1982) were the …rst to explore the determinants ofthe trade pattern in general-equilibrium settings with Marshallian production externalities.6 Bothconsidered two-sector economies with constant returns to scale in one industry and increasing
returns to scale due to external economies in the other. Both established the existence of a stableequilibrium in which the larger country specializes in and exports the good produced with increasing
returns. Although their results are super…cially similar to ours, the underlying economics are quitedi¤erent. In their models, stability (and e¢ ciency) dictate concentration of the increasing-returns
industry in a single country. The smaller country may lack su¢ cient resources to satisfy world
demand for this good, even if it is completely specialized, whereas the larger country always cando so.7 In our context of task trade, the ability to accommodate world demand never is at issue,
because any small task can easily be concentrated in either location. The pattern of specialization
does not rest on country size per se, but rather on the interplay between the scale of aggregate
production of …nal goods and the o¤shoring costs.
Trade costs feature prominently in the literature on “home-market e¤ects,” which began with
Krugman (1980) and Helpman and Krugman (1985). The latter studied a world economy producing
di¤erentiated varieties subject to internal increasing returns to scale and a homogenous good withconstant returns to scale. When the di¤erentiated products are costly to trade and the homogeneous
product is not, the large country exports the former and imports the latter. The larger country has
a larger home market, and the shipping costs translate proximity to consumers into a pro…tability
6See, also, Melvin (1969) for an early contribution, Krugman (1991) and Matsuyama (1991) on dynamic stability
issues, and Helpman (1984) for a survey and further discussion.
7
In Grossman and Rossi-Hansberg (2008a), we study a trade model with Marshallian externalities in a continuumof …nal-goods industries. In such a setting, country size plays no role in determining the chain of comparativeadvantage. It does, however, e¤ect the cut-o¤ between goods that are produced in each country.

5

advantage for local …rms. Davis (1995) points out that the home-market result rests on the assumed
asymmetry in transport costs across sectors; if the homogenous good is as costly to trade as thedi¤erentiated products, sectoral trade between similar economies is balanced. Amity (1998) revisits
the issue in a model with two increasing-returns industries that di¤er in trade costs. She showsthat the larger country exports the goods that are more costly to ship, because its larger market
provides local …rms with a relatively greater advantage in that industry. Holmes and Stevens (2004)
and Hanson and Xiang (2004) extend her result to a world with many industries and heterogeneoustransport costs.
The results in the literature on market-size e¤ects and heterogeneous trade costs resemble ours.
In both cases, locational advantages give rise to factor-price di¤erentials, as the country that bears
the higher transport cost must o¤set this disadvantage in order that its factors be fully employed.
In the literature on the home-market e¤ect, the scale economies are internal to the …rm and thecost di¤erences stem solely from market size. In contrast, external economies of scale seem more
pertinent for modeling production sharing. The recognition of such externalities requires us toaddress coordination issues and the role of producers who potentially can internalize these bene…ts.
In our context, the scale economies drive each traded task to be performed in a single location,
whereas in the models of internal economies of scale, incomplete specialization is the more typicaloutcome. Also, for task trade, the scale of …nal-goods production and not the location of …nal
demand determines the pattern of specialization. Production of …nal goods is related to country
size, but the country that produces more …nal output need not be the one that is larger in size.
The remainder of this paper is organized as follows. In Section 2, we develop our model ofo¤shoring, discuss the equilibrium allocation of tasks given factor prices and aggregate outputs, andlay out the conditions for a general equilibrium. Section 3 begins with some illustrative examples
that facilitate our consideration of the uniqueness of equilibrium and some equilibrium properties.
The section proceeds to a discussion of the relationship between country size, aggregate output of…nal goods, and relative wages, and presents our main result on the pattern of specialization. In
Section 4, we use numerical methods to study the relationship between the pattern of task trade
and the key parameters of the model. Section 5 concludes.

2

The Model

Production requires many “tasks.” Each such task can be performed close to a …rm’s national
headquarters or at a foreign location. If a task is performed o¤shore, the …rm bears an extra costof coordinating production and communicating with distant workers. The cost of o¤shoring varies
by task. Some require more frequent and intense interaction between workers and managers, whileothers are easier to perform from a distance.
We study an environment with external economies of scale at the task level. A …rm’s productivity in performing a task in a particular location increases with the total scale of performance ofthe task by all …rms in that same location. As in the literature on increasing returns to scale at6the industry level, the external economies are meant to capture the presence of localized knowledge
spillovers.8
There are two countries, East and West. Each country is endowed with …xed supplies of two
primary factors, managers and workers. In East, the supplies of these factors are H and L, respectively; in West, they are H and L . The similarity of the two countries is re‡ected in theiridentical relative factor supplies; H=L = H =L . However, the sizes of the two countries need not
be the same.
A producer must perform (or procure) a unit measure of tasks to generate a unit of …nal
output. The set of tasks required of di¤erent producers is the same, but their resulting outputs aredi¤erentiated in the eyes of consumers. Let> 1 be the elasticity of substitution between any pairof …nal products. The world market for these goods is characterized by monopolistic competition,with (constant) mark-up pricing and zero pro…ts. We abstract from any cost of transporting …nal
goods in order to highlight the costliness of o¤shoring.
The tasks that comprise a …rm’s variable cost are performed by workers alone. A …rm canperform a task locally or o¤shore, and it can do so either in-house or by outsourcing the task toanother …rm. In addition to the production (or procurement) costs, each …rm must hire managers
to oversee production and coordinate the performance of the various tasks. A …rm requires f
managers in the country of its headquarters as a …xed cost of doing business. By paying this …xed
cost, it gains the capacity to perform the continuum of tasks in a set of locations of its choice.
In this paper, we do not address the choice between vertical integration and outsourcing. Instead, we assume that …rms use the same technology when performing tasks for themselves as when
performing them for others. Moreover, …rms must pay a small extra cost to acquire the capability
to serve as an external provider of a task. In equilibrium, no …rm has any incentive to pay this cost,so all tasks are performed in-house. Notwithstanding this outcome, the potential for outsourcing
plays a meaningful role in our analysis. A …rm that can perform a task for many others can (partially) internalize the externality associated with the choice of location. Although the equilibria
that we describe feature no outsourcing, the possibility that a single …rm might perform a task
for others eliminates many potential equilibria and allows us to characterize the allocation of tasks
across national boundaries.
We model the siting of each task as a multi-stage game. In the …rst stage post entry, …rms
choose— for each task i— whether to pay the cost that would prepare them to serve as a supplier.
The capability to outsource a given task requires a small numberof additional managers perunit of task. A …rm that bears this cost for task i can perform the task on behalf of any or allother producers. A …rm that does not incur the cost cannot supply the task to others. At thesame time, each …rm selects a “tentative” location for task i. This location can be changed at thenext stage, but only at a (small) cost. In the second stage, …rms that have the capacity to serve
as suppliers quote prices. Since the tasks are performed speci…cally for a …nal producer, we allow
8

On this point, see for example the discussions in Marshall (1920), Helpman (1984), Romer (1986), and Lucas(2002), among others.

7

for price discrimination; that is, a supplier may quote one price to perform the task on behalf of…rms headquartered in the East and a di¤erent price to perform the task for …rms headquarteredin the West. The prices include any o¤shoring costs (which are described further below). At thesame time that prices are quoted, …rms choose their “…nal” locations for each task. If a …rm’s…nal location di¤ers from the tentative location chosen at the prior stage, then its variable cost isa multiple(slightly in excess of one) times as great as what it otherwise would have been. In the…nal stage of the siting game, each …rm decides whether to perform task i for itself or to procure
the task from the supplier that has o¤ered it the lowest price.
There are external economies in the performance of every task that impart increasing returns
to scale at the national level. Suppose that task i is performed a total of Xij times in some country
j. Then a …rm that has its headquarters in country j and that chose country j as its tentative and…nal location for task i can perform the task in-house with 1=A(Xij ) workers per unit of output,
where A( ) is continuously di¤erentiable, increasing, and concave. The labor requirement is thesame for a …rm that has invested in outsourcing capability and that seeks to perform task i incountry j for other …rms headquartered there. If, instead, a …rm performs task i in country j
for a producer (itself or another) with headquarters in country j 0 , and if country j was also its
tentative location for task i, then it bears the (higher) per unit labor requirementt(i)=A(Xij ).

Here, t(i) > 1 re‡ects the cost of o¤shoring task i. All of the labor requirements are multiple byfor …rms that switch their location between the initial and …nal designation. These switching costsa¤ord potential suppliers the opportunity to make pro…ts in case the …rms tentatively coordinate
on the “wrong” location.
Our modeling of o¤shoring costs mirrors that in Grossman and Rossi-Hansberg (2008b). The
schedule t(i) captures the heterogeneity of these costs across tasks. We index tasks so that t0 (i) > 0.
Tasks with low indexes are those for which instructions can be communicated internationally withlittle loss of information. In contrast, remote performance of tasks with high indexes is problematic,
because these tasks must be monitored closely by headquarters and require intensive interaction
between managers and workers. The parameteris a technological parameter that we will use in ourcomparative statics to model improvements in communication technology and other technologicaladvances that facilitate coordination of activities at a distance.
For each task, we seek a sub-game perfect equilibrium in the location game. We shall …nd
that, for some tasks and some parameter values, certain con…gurations of market participants afterinvestments in outsourcing capability and choices of tentative locations imply the non-existenceof pure-strategy equilibrium in the subsequent stage game. We assume that participants dislike
such outcomes and so avoid these con…gurations along the equilibrium path.9 We also invoke
coalition-proofness as a re…nement of the set of sub-game perfect equilibria.10
9Alternatively, and perhaps more convincingly, we could allow for mixed strategies in prices and …nal locations for
con…gurations that do not admit a pure-strategy equilibrium. We …nd that the mixed strategy equilibria are di¢ cultto charactererize, but some special cases with a …nite number of producers imply that there exists a range of values
for = (
1) that support the task locations that we study here as the unique outcomes.
10
An equilibrium is coalition proof if no group of …rms can jointly change their actions at some stage in such a
way that all members of the group bene…t given the actions of non-members and that no member of the group has

8

The equilibrium of the location game determines the country in which each task is performedby each …rm, and the corresponding cost. Firms mark up their per-unit costs of producing goods
to maximize pro…ts. The general equilibrium determines the measure of producers in each country,
the outputs and prices of all varieties of the …nal good, and the factor prices in each country. In
the following sections, we lay out the equilibrium conditions, beginning with those that guide thesiting of a given task i.

2.1

Location of Tasks

Firms take the wage rates and aggregate output levels in the two countries as given. They decide
whether to invest in outsourcing capability and where to install their capacity to perform each task,…rst tentatively then permanently. Later, they perform a task for themselves or subcontract withanother …rm depending on whether the lowest quoted price for outsourcing (if any) exceeds or falls
short of the cost they would bear by performing the task in-house.
Along the equilibrium path, no …rm has any incentive to invest in outsourcing capability. Suchinvestments are costly and when …rms choose their equilibrium locations, none has any advantage
over its rivals that would allow it to recoup these costs. So we will begin by investigating …rms’
locational choices when they anticipate an absence of external suppliers.
Let’s hypothesize that all …rms tentatively and permanently locate a task i in East and ask
whether any …rm or coalition of …rms has reason to locate di¤erently without investing in outsourcing capability. If some …rms have an incentive to do so, it will be those headquartered inWest, because these …rms must bear an o¤shoring cost when performing the task abroad. They
face a trade-o¤, however, inasmuch as the savings in o¤shoring costs would come at the expenseof scale economies. By choosing East as its location for task i, a Western producer anticipates a
per-unit cost for the task oft(i)w=A(nx + n x ), where w is the wage of a production worker

in East, n and n are the measures of producers in East and West, respectively, and x and x
are the respective per-…rm outputs. A choice of West, if matched by other Western …rms, would
generate a per-unit cost of w =A(n x ). The deviation is pro…table for the Western …rms if andonly if t(i)w=A(nx + n x ) > w =A(n x ).11 The hypothesized location of task i in East therefore
requires iI, where I is de…ned by 12t(I) =w A (nx + n x ).
w
A(n x )

Equation (1) provides a limit on what tasks can be concentrated in East. For i(1)
I, thea further incentive to deviate from the altered action given the altered actions of all other group members besides
itself. In our context, the group deviations will involve all …rms with a common nationality swtiching their location
for a task from one country to the other.
11
Note that all Western …rms bene…t from the switch when any one does, and none has any incentive to maintainits capacity for task i in East if the others switch to West.
12
We have invoked coalition-proofness to make this argument. Note, however, that a tentative and permanent
concentration of task i in East would be susceptible to a deviant …rm that invests in outsourcing capability, locatesin West, and quotes a price just below t(i)w=A (nx + n x ). Therefore, the re…nement of coalition-proofness is not
required to eliminate the possibility of an equilibrium with task i concentrated in East when i > I.

9o¤shoring costs are not so high as to outweigh the bene…ts of the scale economies and the associatedbene…t or cost of any di¤erence in wages. Even if all Western …rms were to coordinate a move oftask i from East to West, the cost savings would not su¢ ce to o¤set the loss in productivity
from performing task i at smaller scale. But if i > I, such a coordinated move would be in theWestern …rms’individual and joint interest. By a similar argument, we can rule out, for iI, anequilibrium in which all …rms locate their capacity for task i in their native country: Such a location
would induce a deviation by the Western …rms, who would prefer to site the task in East to gain
the bene…ts of greater scale. Our discussion presumes the existence of a task with index between
zero and one such that the two alternatives of concentrated performance in East and dispersed
performance are equally costly for Western …rms. If no task can be concentrated in East without
threat that the Western …rms will switch their location to avoid the o¤shoring costs, then we assign
I = 0. If all tasks are immune to pro…table deviation of this sort, we assign I = 1.
An analogous condition applies to concentration of task i in West. Then, Eastern …rms might
bene…t from having the task performed closer to their national headquarters, which would conserveon their o¤shoring costs. A potential equilibrium with all …rms performing task i in West— inwhich Eastern …rms would face a per-unit cost oft(i)w =A(nx + n x )— might be underminedby a deviation by the Eastern …rms, who could perform the task near their headquarters at cost
w=A (nx) : Such a deviation would be pro…table ift(i)w =A(nx + n x ) > w=A(nx).13 This

determines another marginal task, I , de…ned byt(I ) =w A (nx + n x )
,
w
A(nx)

such that task i can be concentrated in West only if i(2)

I and can be performed locally by …rms inboth countries only if i > I . Again, we set I = 0 if the prescribed deviation always is pro…table
for Eastern …rms and I = 1 if it is never so.
Taking stock, we have shown thus far that for any task i > max[I; I ], the only candidate
for equilibrium is one with local production by all …rms and no task trade, as the high cost ofo¤shoring discourages the realization of scale economies for these tasks. For any task imin[I; I ],concentration in either country remains a possible outcome, insofar as no …rm or group of …rms has
an incentive to deviate without investing in outsourcing capability. Finally, for tasks with indexes
between min[I; I ] and max[I; I ], there is one candidate equilibrium, since these tasks cannot be
dispersed and cannot be concentrated in one of the two countries.
Let’s consider further the tasks with low o¤shoring costs; i.e., those for which imin[I; I ].

Might the potential for outsourcing discipline their location? The answer is a¢ rmative, provided
the cost of outsourcing capability is relatively small compared to the switching costs. Suppose thatall …rms but one tentatively locate task i in East and decline to invest in outsourcing capability.
Now let a deviant locate in West and pay the small extra cost that allows it to serve others. If
13
Again, the joint deviation is not necessary inasmuch as a single deviant could set up capacity in East and invest
in the capabilitiy to supply other Eastern …rms, so as to upset an allocation with all capacity for task i concentrated
in West.

10

the deviant anticipates that the other …rms will choose East as their …nal location despite its own
behavior, it can o¤er the Eastern …rms a price slightly less than w=A(nx + n x ) and the Western…rms a price slightly less than t(i)w=A(nx + n x ). At these prices, all …rms will …nd the option
to purchase task i from the deviant …rm to be more attractive than in-house production. Assuming
no response from the others, the deviant would earn by this strategy pro…ts equal to

d (i)

w
A (nx + n x )

t(i)w
t(i)w
nx +
A (nx + n x )A (nx + n x )w
n x ,A (nx + n x )where the …rst term is the pro…t (positive or negative) that the deviant would make by performingtask i for …rms headquartered in East and the second term is the pro…t that the deviant would makeby performing task i for …rms headquartered in West. If
of task i in East is immune to this deviation. But if

d (i) < 0, the hypothesized concentrationd (i) > 0, the …rms tentatively located in East

must formulate a best response. By retaining their capacity for task i in East, the Western …rms
face the price t(i)w=A(nx + n x ). A switch of location to West would promise a production cost
no greater than w =A(nx + n x ), considering that the Eastern …rms would surely buy from theprovider at the o¤ered price of w=A(nx + n x ). So, the Western …rms have incentive to switch
their locations for task i. Anticipating this, the deviant …rm would not set prices that presume thatits customers are located in East, but instead would quote the pricest(i)w =A(nx + n x ) andw =A(nx + n x ) for Eastern and Western …rms that presume a switch of location. The Western…rms buy from the deviant, because they can do no better producing for themselves. Given that,
the Eastern …rms would face a small scale of production were they to serve themselves from theEast. The fact that i

I implies that they too prefer to buy from the deviant despite the includedo¤shoring cost. The deviant’s pro…ts at these prices are (which exceed the cost of the investment if = (Note that if1) w [ t(i)nx + n x ] =A(nx + n x ),1) is su¢ ciently small.

d (i) > 0, task i is more e¢ ciently performed in West, while if

d (i) < 0, task

i is more e¢ ciently performed in East. So an analogous argument establishes the existence of apro…table deviation for task i, under similar conditions on = (locate in West and1) ; when all …rms tentatively

d (i) < 0. Then, the deviant locates in East with an eye toward the cost savings

it can achieve by performing the task for all producers there. The deviant’s action induces a switchof location for …rms headquartered in East, therefore also for those headquartered in West, andthe deviant’s pro…ts ultimately come from its (small) productivity advantage over those that haveswitched locations. Such a deviation is not pro…table whend (i) > 0:

We de…ne a task J as the solution tot(J) =

wnx w n x,
w nx wn x

if the solution falls between 0 and 1, and assign J = 0 if
for all i. If J 2 (0; 1), thend (J) = 0.

d (i) < 0 for all i and J = 1 if

(3)d (i) > 0

Now suppose that West has a lower wage than East11

in equilibrium. In the appendix, we show that one of two statements must be true.14 Either (i)d (i) > 0 or

d (i) < 0 for all i, or (ii) J > 0 andd (i) > 0 for i < J while

The potential for outsourcing pins down the location of tasks with irange, if i

d (i) < 0 for i > J.

min[I; I ]. In thisJ

min[I; I ] and w < w, then the unique sub-game perfect equilibrium has taski concentrated in West.15 If J < imin[I; I _] and w < w, then the unique equilibrium hastask i concentrated in East. These statements are reversed for w > w; for example, in this caseiJmin[I; I ] implies concentration of task i in East.

We return to the tasks with intermediate o¤shoring costs such that min[I; I ] < i < max[I; I ].
We have observed that such tasks cannot be dispersed and cannot be concentrated in one of the twocountries. Might a deviant that invests in outsourcing capability upset a potential equilibrium withtask i concentrated in the other country? To see what can happen, consider for example parametervalues that give rise to I < J and consider a task i 2 (I ; J). This task cannot be concentrated inWest in equilibrium, because Eastern …rms would deviate and relocate their capacity to East. The
candidate equilibrium has all …rms tentatively and …nally locating task i in East. Now, a deviant

that invests in outsourcing capability might perceive a pro…t opportunity, since i < J implies
d (i) > 0.

However, if the deviant quotes the prices w=A (nx + n x ) andt(i)w=A (nx + n x )for Eastern and Western …rms, respectively, the Western …rms will have an incentive to switch theirlocation to West. This cuts into the pro…ts that the deviant can make by selling to them, and thedeviation remains pro…table only if the deviant can also induce a switch of location by the Eastern…rms. However, with i > I , the Eastern …rms prefer to produce for themselves at smaller scalethan to bear the cost of o¤shoring to West. There are no prices that the deviant can quote thatgenerate positive pro…ts after the induced response by other …rms. In fact, the stage game thatensues after investment by a single potential supplier has no equilibrium in pure strategies. If thedeviant o¤ers a pair of prices assuming that all …rms will keep their capacity for task i in East, theWestern …rms will switch, but the Eastern …rms will not. Then the deviant prefers to set higher
prices which, however, leave the Western …rms with a preference for staying in East. We assume
that no deviation takes place in such circumstances.16
14

An analogous statement to what follows applies when East has the lower wage. We will discuss below theprospects for an equilibrium with equal wages in the two countries and the properties thereof.
15
When i J min[I; I ] and w < w, there exists an equilibrium in which all …rms locate in West. To see this,
suppose that a deviant were to invest in outsourcing capability and to locate tentatively in East. In the ensuing
sub-game, the remaining …rms would not switch their locations to East, because if they did so, the deviant also would
switch (to West) so that it could supply the market at lesser total cost. But, anticipating this, the other …rms would
not wish to switch their locations. Rather, the unique Nash equilibrium in the sub-game with a deviant tentatively
located in East has the deviant switching its location to West while the other …rms retain their locations there, sothat the deviant earns no pro…ts to cover the …xed cost of outsourcing. Similarly, when J < i < min[I; I ] andw < w, there exists an equilibrium with all …rms located in East that is immune to deviation by a …rm that invests
in outsourcing capability and tentatively locates in West. Analogous arguments apply to the cases in which w > w.
16
Alternatively, we might allow mixed strategies in sub-games that admit no pure-strategy equilibrium, as we noted
in footnote 9. We have investigated mixed strategies following an investment in outsourcing capability and found thegeneral case di¢ cult to solve. In several special cases with …nite numbers of …rms, the expected operating pro…ts forthe deviant are positive, but signi…cantly smaller than those available to a deviant when i < I < J. This means
that, for a range of values of = (1), the deviant can discipline the choice of equilibrium for tasks with i < I < J
and yet not upset an equilibrium with production concentrated in East for tasks with I < i < J.

12

We can summarize the …ndings in this section as follows. In equilibrium, …rms perform all tasks
in-house and none bears the cost of outsourcing capability or of switching locations. For a rangeof tasks with low o¤shoring costs, …rms in each country would rather pay the cost of o¤shoring toreap the bene…ts of scale economies than perform the task at home at smaller scale. These tasks
concentrate in whichever country o¤ers the lower aggregate (production plus o¤shoring) costs,
because the opposite location would invite deviation by a …rm that could pro…t by investing inoutsourcing capability. For a range of tasks with intermediate o¤shoring costs, …rms concentratein the one country in which the foreign …rms there have no incentive to relocate to their homecountry in order to conserve on o¤shoring costs at the expense of scale economies. Finally, fora range of tasks with high o¤shoring costs, …rms …nd these costs too high to bear. These tasks
are dispersed, as all …rms perform them in the country of their headquarters. Whereas the tasks
with low o¤shoring costs are performed in the e¢ cient location, those with higher o¤shoring costs
need not be. Firms make their location choices without regard to the externality it may impose
on others, and a potential supplier cannot internalize the externality when all the …rms of a given
nationality would rather perform the task locally than reap the scale bene…ts of concentrated
production abroad.

2.2

General Equilibrium

The remainder of the equilibrium conditions are more familiar. Firms price their products optimallyin the light of the demands they face, while free entry drives pro…ts to zero. Meanwhile, factor
markets clear in each country.
Let c and c denote the unit cost of a typical …nal good in East and West, respectively, whichre‡ect the equilibrium locations of the various tasks. Let E denote the set of tasks that are con-centrated in East, W the set of tasks that are concentrated in West, and B the set of tasks thatare performed locally by …rms in both countries. Of course tasks in E represent o¤shoring for …rms

headquartered in West, while tasks in W represent o¤shoring for …rms headquartered in East.
There is no o¤shoring of the tasks in B. In view of the costs of o¤shoring and the di¤erent scalesof output for tasks that are traded and non-traded, we havec=wM (E)w T (W)wM (B)++A(nx + n x ) A(nx + n x )
A(nx)

(4)

c =

wT (E)w M (W)w M (B)++,A(nx + n x ) A(nx + n x )A(n x )

(5)and

where M (Z) is the Lebesgue measure of Z for Z = fE; W; Bg. T (Z) is given by
T (Z) =

Zt(i)dii2Z

for Z = E and W. So T (Z) =A(nx + n x ) is the total amount of labor needed per unit of output to13

perform the tasks in Z for an o¤shore …rm when labor productivity is A(nx + n x ).17 In (4), thethree terms are the total cost to a …rm headquartered in East of the tasks that are concentrated in

East, the tasks that are concentrated in West, and the tasks that are dispersed, respectively. Theinterpretation of (5) is similar.
Given c and c , the …rms practice mark-up pricing. This yields, via the demand functions, arelationship between relative costs of Eastern and Western …rms and relative quantities produced(and consumed) of the di¤erent varieties, namely
x
c
=
x
c
where, as de…ned before,,(6)> 1 denotes the elasticity of substitution between any pair of varieties.

Free entry equates variable pro…ts to …xed costs, which are sf for a …rm headquartered in Eastand s f for a …rm headquartered in West, where s and s are the salaries of managers in East andWest, respectively. By familiar calculations, the zero-pro…t conditions imply
s=
and
s =

cx
f(

1)

c x
.
f(
1)

(7)

(8)

Finally, we have the factor-market clearing conditions. Managers are employed only in headquarters, where they perform activities that are independent of scale. In each country f managers
are needed per …rm, which implies
nf = H(9)n f =H .

(10)

and

Workers in each country are employed in tasks that are performed locally by national …rms andin a¢ liates of foreign …rms. Tasks in E do not use any Western labor and tasks in W do not
use any Eastern labor. Considering the demands by local and foreign …rms for the tasks that areconcentrated in one country and for those that are dispersed, we have

and

T (E)
M (B)
M (E)
nx +
n x +
nx = LA(nx + n x )A(nx + n x )
A(nx)

(11)

T (W)
M (W)
M (B)
nx +
n x +
n x =L .
A(nx + n x )A(nx + n x )A(n x )

(12)17
We assume E, W, and B are elements of the Borel
algebra and that t( ) is Lebesgue measurable. For the casein which w 6= w , we will …nd that the sets E, W, and B are connected intervals, so the integral that de…nes T ( ) is astandard Riemann integral. If w = w ; the theory imposes no structure on the sets E and W (B is still a connectedinterval). In this case, we restrict attention to sets E and W that are elements of the Borel -algebra and use theLebesgue integral. Of course, this restriction has no e¤ect on the general equilibrium properties of our economy.

14

The three terms on the right-hand side of (11) are, respectively, the labor employed in Eastern…rms to perform tasks that are concentrated in East, the labor employed in Eastern subsidiaries of…rms based in West, and the labor employed by Eastern …rms in tasks that are not traded. Theinterpretation of the terms in (12) is analogous.
We have not yet chosen a numeraire. Let w = 1. Then (9) and (10) determine n and n . Given
the allocation of tasks to the sets E, W and B and the equilibrium values of n and n , (4)-(6) and(11)-(12) determine c; c ; x; x and w.18 Finally, (7) and (8) determine s and s residually. Our

next task is to characterize the patterns of specialization that can emerge in equilibrium.

3

Patterns of Specialization

In this section, we explore the patterns of specialization that can emerge when there is task trade
between similar countries. We use a combination of numerical and analytical methods to describe
equilibrium con…gurations of task allocation. We begin by illustrating examples of patterns that
can arise when o¤shoring costs are, respectively, high and low. We then provide a general result thatlinks task allocation to relative wages and relative aggregate outputs and discuss the relationshipbetween these endogenous variables and relative country size.

3.1

Equilibrium Allocations with High O¤shoring Costs

Figure 3 depicts a typical outcome when o¤shoring costs are reasonably high. The …gure is drawn
for the case of a linear o¤shoring-cost schedule, with t(i) = 1+i andtake the form A(X) = X , with= 2. The external economies= 0:8. For other parameters, we take

= 2, f = 1, L + L = 2,

and H + H = 2.
The top panel of the …gure shows the threshold tasks I; I and J for di¤erent divisions of theworld’s labor supply between the two countries. In all cases, we take endowments of both factors tobe equal, so that H = L and H = L . Using the limits for pro…table deviations, we can describe
the equilibrium allocation of tasks across countries. The bottom panel shows the corresponding
relative wage; recall that the foreign wage serves as numeraire. For ease of interpretation, we
distinguish visually the outcomes with w > 1 from those with w < 1; the former are depicted withthick, dark curves, the latter with curves that are thinner and lighter in shade.
The …gure shows that when o¤shoring costs are high and the world’s resources are divided
almost evenly between the two countries, I = I = 0. For a range of values of L on either sideof unity, concentration of any task in some country would be undermined by a deviant o¤ering
to perform that task for …rms headquartered in the other country. The deviant could pro…t by
avoiding the high o¤shoring costs. In the unique equilibrium that arises for su¢ ciently highandL close to L , no o¤shoring takes place. As the bottom panel shows, the larger country enjoys
18
The solution to these …ve equations for given E, W and B is unique. However, the sets E, W and B are themselves
determined by the equilibrium values of the other variables. This suggests the possibility of multiple equilbria, which
we discuss further below.

15the relatively higher wage, as its superior scale gives it a productivity advantage in performing alltasks.
Figure 3: Equilibria and the Relative Size of Countries( β = 2, σ = 2, θ =.8, f = 1)
1
0.9

I (w > 1 Equilibrium)
I* (w > 1 Equilibrium)
J (w > 1 Equilibrium)
I (w < 1 Equilibrium)
I* (w < 1 Equilibrium)
J (w < 1 Equilibrium)

0.8

I, I* and J

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.5

0.6

0.7

0.8

0.9

1

1.1
*

1.2

1.3

1.4

1.5

1.2

1.3

1.4

1.5

*

L=H=2-L =2-H

3
w > 1 Equilibrium
w < 1 Equilibrium

2.5

w

2

1.5

1

0.5

0
0.5

0.6

0.7

0.8

0.9

1

1.1

L = H = 2 - L* = 2 - H*

For a given value of, the incentives to o¤shore grow as the countries become more unequalin size. When one country is substantially larger than the other, …rms in the smaller country havemuch to gain by performing some tasks abroad. The productivity gains associated with the scaleeconomies outweigh the relatively high costs of o¤shoring. Yet it may not be possible for task tradeto ‡ow only in one direction. If …rms headquartered in the small country o¤shore tasks to the large
country, their foreign subsidiaries use resources that otherwise would be employed by local …rms.
16

An incipient excess demand for the large country’s workers puts upward pressure on the wage there,
which in turn creates an incentive for …rms headquartered in the large country to perform tasks in
the lower-wage country.
Consider circumstances in which East is substantially larger than West.19 We see in the …gurethat when L is su¢ ciently greater than L , I > I > 0 and J > I . The tasks with iI

can be concentrated in either country without risk that national producers of the other country
will be drawn home by a deviant supplier. These tasks are more e¢ ciently concentrated in Westthan East, because the wage in West is lower and the burden of o¤shoring these tasks is not sogreat. The next range of tasks has I

i < I. Some or all of these tasks may be more cheaply

concentrated in West than in East.20

However, the siting of these tasks in West is undermined bypotential deviation by the Eastern …rms, who would prefer to locate and produce for themselves inEast. Therefore, the tasks in this range are performed only in East, where the large scale of output
makes up for the moderately high cost of o¤shoring borne by Western …rms. Finally, for tasks withi > I, even specialization by the high-wage country is not sustainable, inasmuch as a coalition of…rms from the low-wage country would like to produce the task back home. These tasks that are
most costly to o¤shore are performed locally in both countries in the general equilibrium.
Notice that, for all L and L depicted in Figure 3, if o¤shoring occurs at all, task trade ‡ows
in both directions. This is a consequence of the particular parameter values used in the …gure,speci…cally our choice of= 2.21 However, one feature of the equilibria depicted in the …gure ismore general. For all values of L and L , if a task i is concentrated in the low-wage country andanother task i0 is concentrated in the high-wage country, then i0 > i. We will see in Section 3.3 that
if wages in the two countries are not equal, the allocation of tasks always obeys this rule: tasks
that are performed only in the high-wage country bear higher o¤shoring costs than those that are
performed only in the low-wage country.
The …gure shows the larger country enjoying the higher wage. Its wage advantage derives fromtwo sources. First, as before, the larger country bene…ts from having greater scale in the tasks
that are performed locally in both countries. But now the larger country bene…ts as well from theallocation of the traded tasks. In this allocation, the smaller country performs locally the tasks
that are easiest to o¤shore, while the larger country performs locally tasks that are more costly
to o¤shore. This redounds to the bene…t of the larger country’s factors of production. For allparameter values that we have examined, there exists an equilibrium in which the larger country
has the higher wage. But, as we shall see in the next section, there can sometimes exist a second
equilibrium in which the smaller country enjoys the higher wage. In such circumstances, it remains
true that the low-wage country captures the tasks that are easiest to o¤shore and the high-wagecountry captures those that are more di¢ cult to o¤shore.
19

Notice the symmetry of the …gure. Everything that we say about equilibria with L > 1 applies as well to equilibria
with L < 1, with the names of the countries reversed.
20
The tasks below the thick dotted line can be performed at lower total cost in the low-wage West than in high-wage
East; those above the dotted line can be done at lower total cost in East.
21
When = 2 and A(X) = X , I = 0 if and only if I = 0. This property of Figure 3 does not apply for othervalues of or for other forms of the externality function when = 2.

17

3.2

Equilibrium Allocations with Low O¤shoring Costs

Figure 4 is drawn for similar parameter values as Figure 3, except that

= 1:1. The …gure depicts

three equilibria that exist when resources are almost evenly divided between the two countries. For
L signi…cantly greater than L (or vice versa) the equilibrium is unique.
Figure 4: Equilibria and the Relative Size of Countries( β = 1.1, σ = 2, θ =.8, f = 1)
1
I (w > 1 Equilibrium)
I* (w > 1 Equilibrium)
J (w > 1 Equilibrium)
I (w < 1 Equilibrium)
I* (w < 1 Equilibrium)
J (w < 1 Equilibrium)
I = I* (w = 1 Equilibrium)

0.9
0.8

I, I* and J

0.7
0.6

M ( E ) (w = 1 Equilibrium)
M ( W ) (w = 1 Equilibrium)

0.5
0.4
0.3
0.2
0.5

0.6

0.7

0.8

0.9

1

1.1
*

1.2

1.3

1.4

1.5

1.2

1.3

1.4

1.5

*

L=H=2-L =2-H

1.4
w > 1 Equilibrium
w < 1 Equilibrium
w = 1 Equilibrium

1.3

1.2

w

1.1

1

0.9

0.8

0.7
0.5

0.6

0.7

0.8

0.9

1

1.1

L = H = 2 - L* = 2 - H*

The various equilibria in Figure 4 are distinguished by the thickness and shading of the curves.
Consider the three curves in the top panel that are thickest and darkest and look …rst at the portionsof these curves that apply for L

L . The curves describe an equilibrium with I > I > J > 0.

Tasks with i less than J are most cheaply performed in West and no pro…table deviation prevents18

them from concentrating there. Tasks with i between J and I are most cheaply performed in
East and again nothing prevents them from locating there. Finally, tasks with i > I cannot
be concentrated, because a deviant supplier in either country could attract the business of local
producers. Notice that for L su¢ ciently large, I = 1, which means that no tasks are performed
in both countries. This re‡ects the modest cost of o¤shoring for the parameters used to draw the…gure. Of course, if t(i) were not linear but instead rose rapidly with i, then the equilibrium would
always feature some non-traded tasks. The bottom panel of Figure 4 again shows the associatedrelative wage. The thick curve corresponding to the equilibrium just described has w > 1 for allL

L . The larger country has the higher wage for the reasons discussed at the end of Section 3.1.
But notice that the thick curves do not begin at L = L = 1. In other words, we can havean equilibrium in which the smaller country has the higher wage and exports the tasks that haveintermediate o¤shoring costs. For L slightly smaller than L and w > 1, the ordering of theboundary values is I > I > J as before. Again, West alone performs the tasks with ialone performs the tasks with J < i

J, EastI, and both countries perform tasks with i > I. Here,

East generates greater aggregate output than West (i.e., nx > n x ) despite its smaller size andcorrespondingly smaller endowment of managers. The shortfall in the number of its …rms comparedto West (n < n due to H < H ) is more than made up by greater sales per …rm (x > x ). Because
the East has a greater scale of output, it enjoys a productivity advantage in the tasks that are
performed locally by all …rms. It also bene…ts by capturing the tasks that are more di¢ cult too¤shore among those that are traded. Its overall cost advantage (c < c ) underlies its superior sales
per variety, which in turn justi…es its higher wage and the pattern of specialization.
The requirement for the East to perform the tasks with intermediate o¤shoring costs for producers worldwide strains its small resource base. If L is su¢ ciently smaller than L , the East lacks
the workers it would need to perform a su¢ cient range of tasks that are relatively costly to o¤shore,
and then its costs would not be low enough to justify its larger scale and higher wage. In suchcircumstances, an equilibrium in which the smaller country has the higher wage and the higher
aggregate output does not exist. For the parameters used to draw Figure 4, an equilibrium withw > 1 exists for L > :96, but not otherwise.
The thick curves in Figure 4 are analogous to the thick curves in Figure 3. In both cases, thethickness refers to the fact that w > 1; i.e., the wage in East is greater than the wage in West.
Recall that, in Figure 3, the thick curves incorporate a range of values of L for which w > 1 andI = I = 0. As the o¤shoring costs fall, this range of values shrinks and eventually disappears.
Once these costs are low enough that o¤shoring takes place even when L = L , the outcome looks
qualitatively like that in Figure 4 instead of that in Figure 3.
The curves of medium shade and thickness depict a second equilibrium, analogous to the similarly shaded curves in Figure 3. Notice how they re‡ect the thicker curves across the vertical at
L = 1; i.e., an equilibrium with L < L corresponds to one with L > L , except for the reversalof country names. For L slightly greater than L , the curves of medium thickness represent anequilibrium in which the wage is higher in West despite its (slightly) smaller size and the tasks with19

intermediate trading costs are concentrated there. This equilibrium exists for L > L for much thesame reason as does the thick equilibrium with w > 1 when L < L . For L su¢ ciently larger thanL , an equilibrium with w < 1 cannot be sustained.
Finally, the …gure depicts a third equilibrium that exists for exactly the same range of L and Lthat admits the coexistence of a thick equilibrium with w > 1 and a medium-thick equilibrium withw < 1. The equilibrium represented by the thin curves features equal wages in the two countries.
If wages are the same in the two countries, all task can be performed at lesser aggregate costin the country that has the larger scale of aggregate production. No task could be concentrated in
the country with the smaller aggregate output, because such an outcome would be undermined bya price-discriminating supplier locating in the country with greater output. But with countries ofsimilar size, if all traded tasks were concentrated in the same country, the two labor markets could
not clear. It follows that an equilibrium with equal wages must have equal aggregate outputs aswell; that is, nx = n x .
With wages and aggregate outputs equalized, there is nothing to determine the siting of any task
for which specialization is viable. Nonetheless, the unit cost equations (4) and (5) and the labormarket clearing conditions (11) and (12) determine the measures of traded tasks that are performedin each country and the aggregate o¤shoring costs borne by producers of either nationality. Also,with w = w and nx = n x , the incentives for a deviant supplier to upset an equilibrium withconcentrated task performance are the same for both countries. Therefore, I = I and this common
value represents the boundary between traded and non-traded tasks. The …gure shows M (E) andM (W) for the equal-wage equilibrium, as well as I = I .
Notice that M (W) meets up with the J curve for the thick equilibrium at its left-most extreme,
while M (E) meets up with the J curve for the medium-thick equilibrium at its right-most extreme.
These are also the values of L and L at which the relative wages converge to one in the thick andmedium-thick equilibria, respectively. In other words, the equal-wage equilibrium and the thickequilibrium converge as the relative wage approaches one (from above) in the latter. Similarly, theequal-wage equilibrium and the medium-thick equilibrium converge as the relative wage approaches
one (from below) in the latter.
Although the equal-wage equilibrium has an indeterminate pattern of specialization, there are
two constraints on the allocation of the traded tasks. First, an equilibrium allocation must satisfy
T (E)

T ([0; M (E)]), because the total o¤shoring costs for tasks concentrated in East must be atleast the cost of o¤shoring the measure M (E) of tasks that are least costly to o¤shore. Second,
the allocation of tasks must obey T (E)T ([0; I])

T ([0; M (W)]), because the o¤shoring costs

for tasks concentrated in East can be at most the cost of o¤shoring the measure M (E) of tradedtasks that are most costly to o¤shore; i.e., it is maximized when the measure M (W) of tasks withthe lowest o¤shoring costs locate in West. An equilibrium with equal wages in which the measure
M (E) of tasks that are least costly to o¤shore is concentrated in East is identical to the limiting
equilibrium with w < 1 as w ! 1. And the equilibrium with equal wages in which the measure

M (E) of tasks that are most costly to o¤shore is concentrated in East is identical to the limiting20

equilibrium with w > 1 as w ! 1. This explains the convergence of the various thin and thicker
curves in Figure 4. When the gap between L and L grows too large, one of the constraints must

be violated, and so the equal-wage equilibrium ceases to exist.
We o¤er one further observation about the equal-wage equilibrium. Although our model lacks
explicit dynamics, the equal-wage equilibrium has a knife-edge property that suggests instability
under plausible adjustment mechanisms. Suppose we perturb such an equilibrium by misallocatinga few tasks in such a way that total production costs in the two countries remain unchanged. Thenthe labor markets will fail to clear, which will exert pressure on the relative wage. As soon as thewage equality is broken, the remaining traded tasks will relocate so that those with low o¤shoring
costs are concentrated in the low-wage country and those with intermediate o¤shoring costs areconcentrated in the high-wage country. In other words, a small perturbation creates incentives
for a large reallocation of resources and moves the economy into the neighborhood of one of thetwo equilibria with unequal wages. For this reason, we do not consider further the equal-wageequilibrium in the remainder of this paper.

3.3

Task Allocation: A General Result

Let us summarize the lessons from the two examples. First, if task trade is relatively costly, thegeneral equilibrium may be unique. In the extreme, all tasks are performed locally and the potential
bene…ts from specialization are foreclosed. But when countries di¤er greatly in size, specialization
can occur even with reasonably high costs of o¤shoring. Producers in a very small country will
pay a sizeable o¤shoring cost to reap the bene…ts of scale economies. As they do so, the decline
in relative demand for labor in the their country spells a reduction in the country’s relative wage.
This in turn induces …rms based in the large country to relocate tasks to pro…t from the cheaperlabor abroad. As the technology for communication and coordination improves, the endowment
gap necessary for o¤shoring to take place shrinks. When o¤shoring costs are su¢ ciently low, task
trade occurs between countries of similar size and multiple equilibria can exist. The country withthe higher wage produces more aggregate output of …nal goods and performs the traded tasks thatare more di¢ cult to o¤shore. This country thereby enjoys a cost advantage that validates its higher
wage and greater sales per …rm. There will be one equilibrium in which the larger country captures
the higher wage and, if the size disparity is not too great, another in which the smaller country
captures the higher wage. When these two equilibria are present there exists a third equilibriumwith equal wages, but this one has knife-edge properties that suggest instability.
The equilibria with unequal wages share a common pattern of specialization. In all of them, thetasks with the lowest o¤shoring costs are performed only in the country with the lower wage andlesser aggregate output, while the tasks with intermediate o¤shoring costs are performed only inthe country with the higher wage and greater aggregate output. Tasks with the highest o¤shoring
costs are performed locally by …rms in both countries. We now proceed to show that this patternof specialization arises quite generally.
Consider any equilibrium with unequal wages in the two countries. For concreteness, let w > 1,21

so that the wage in East exceeds that in West. Then as we have argued previously, tasks with i > I
cannot be concentrated in East and tasks with i > I cannot be concentrated in West. Among thosetasks with indexes below both I and I , those with i > J are performed only in the East and thosewith iJ are performed only in the West. These restrictions on where tasks can and cannot be

performed have several immediate implications. First, tasks with i < min[I ; J] are performed onlyin West, because no deviation by an outsourcing …rm or a coalition of Eastern …rms is pro…table forthese tasks. Second, tasks with i > max[I; I ] are dispersed, because concentration is underminedby a deviant that serves producers of a single nationality. Third, tasks with I

i < max[I; I ]are performed only in East, because concentration of these task is viable there, whereas the same
is not true of concentration in West.

Figure 5

J > I*
a.
0

I*

I

J´

W

1

J´´

E

B

b.
I

0

I*

1

J
B

W
J < I*
c.
0

I

I*

J
W

1
B

d.
0

J
W

I

I*
E

1
B

Panels a and b of Figure 5 depict patterns of specialization when J > I . The former has
II, while the latter has I > I. In either case, the tasks with iWest. In panel a, tasks with I < i

I are performed only in

I are performed only in East. No such range of tasks exists inpanel b. Finally, in each panel, the tasks with i > max[I; I ] are performed locally by …rms in both22

countries. Notice that the location of J plays no role in these circumstances; e.g., the allocation oftasks in panel a is the same for J = J 0 or J = J 00 .
Panels c and d of Figure 5 depict the possible outcomes when J < I . The former has I < J
while the latter has I > I . These are the only possibilities when J < I , because the following
lemma rules out any con…guration with w > 1 and J < I < I .
Lemma 1 If w > 1, then J < I implies I > I .
Proof. See the appendix.
When the ordering is as depicted in panel c, the tasks with i < I < J cannot be concentrated
in East, because such an allocation is undermined by an outsourcing …rm that locates in West andtarget all producers. But the tasks with i > I also cannot be concentrated in East, because suchan allocation is undermined by a coalition of Western …rms that prefers to produce the task inWest. It follows that no task is concentrated in East, much like in panel b. Tasks with i

I areperformed only in West, while tasks with i > I are dispersed.
When the ordering of panel d applies, tasks with i

J are concentrated in West, where theaggregate costs of performing these tasks is minimized. Tasks with J < i

I are concentrated in

East, where again the aggregate costs of performing these tasks is minimized. Finally, tasks withi > I are performed locally by all …rms, because concentration would be susceptible to the threatof a pro…table deviation by a …rm serving Western producers.
The patterns that appear in Figure 5 apply to any equilibrium with w > 1.22 Moreover, thefollowing lemma states that wages and scale go hand in hand.
Lemma 2 w > 1 if and only if nx > n x .
Proof. See the appendix.
By enumerating all of the possible orderings of I; I and J that can arise when w > 1 andinvestigating the pattern of specialization in each, we have established the following general result:
Proposition 1 The pattern of specialization is characterized by(i) concentrated performance of tasks with the lowest o¤ shoring costs in the country with low
wages and low aggregate output,(ii) concentrated performance of tasks with intermediate o¤ shoring costs in the country withhigh wages and high aggregate output, and(iii) dispersed performance of tasks with the highest o¤ shoring costs in both countries.
22

The patterns that arise when w < 1 are analgous. For example, when w < 1 and the ordering of threshold tasks
is I < J < I , then the tasks with i I are in E, the tasks with I < i I are in W, and the tasks with I > I are
in B. This pattern is analagous to that in Figure 5a; the other panels have similar analogs.

23

The proposition does not exclude the possibility that no tasks are concentrated in one of thecountries, or that no tasks are performed ubiquitously; i.e., one or more of the sets E; W and B

may be empty.

The pattern of specialization described by Proposition 1 holds intuitive appeal in the light ofour previous discussion. Tasks that are very costly to o¤shore are performed locally, for obvious
reasons. For the other tasks, …rms in the country with the smaller aggregate output have the mostto gain from moving tasks abroad, while those in the country with the larger aggregate output
have the most to lose from the communication and coordination costs.23 Market forces drive thetasks that are most di¢ cult to o¤shore (among those that are traded) to the country with thelarger aggregate output to reap the cost savings. In the process, the wage there is bid up, creating
incentives for …rms in the high-output country to o¤shore tasks that can readily be moved to
the low-wage location. Moreover, the pattern of specialization conforms with global e¢ ciency asconcerns the location of traded tasks; those with high o¤shoring costs are concentrated in the
country with greater aggregate output to conserve on transactions costs. Those with low o¤shoring
costs are concentrated in the country with lesser aggregate output to conserve on resource use.
But the extensive margin of o¤shoring is not e¢ cient, because a deviant can induce a group of
producers to source near their national headquarters without regard for the adverse e¤ect on the
productivity of …rms headquartered elsewhere.

4

Comparative Statics

In this section, we explore the relationship between the equilibrium outcomes and some of the
key parameters of the model. In particular, we focus on the connection between the pattern ofspecialization –as revealed by the threshold tasks I; I and J –and the extent of increasing returns
to scale, the extent of product di¤erentiation, and the size of o¤shoring costs. Our main purpose
is to gain a better understanding of how the pattern and volume of task trade are determined in
the model, along with the implications for relative wages.
Our analysis uses numerical methods. We present only a few examples which, however, are illustrative of our …ndings for an extensive exploration of the parameter space. We assume throughoutthat the spillover function takes the form A(X) = X , with

< 1 as required for concavity. Also,o¤shoring costs are given by (1 + i) for i 2 [0; 1]. For the most part, we present results for para-

meters that imply zero task trade between countries of similar size, so that the equilibrium for all
values of L and L is unique.

4.1

Extent of Increasing Returns to Scale

We begin with the strength of external economies, as captured by . When
constant and independent of the scale at which a task is performed. As
23= 0, productivity is

grows larger, returns toAggregate output need not correspond to country size, as the smaller country may produce more per brand if its
lower costs generate greater demand. The equilibrium in Figure 4 in which the smaller country has higher wages is
one in which its aggregate output of …nal goods exceeds that in the larger country.

24scale increase. We restrict attention to spillover functions with

< 1, because a function with anelasticity greater than one would allow unbounded output with …nite resources.
Figure 6: Relative Country Size and Economies of Scale( β = 1.8, σ = 2, f = 1, J is never an effective threshold)
1
0.9
0.8
0.7

I (θ = .9)
I (θ = .8)
I (θ = .7)
I* (θ = .9)
I* (θ = .8)
I* (θ = .7)

I and I*

0.6
0.5
0.4
0.3
0.2
0.1
0

1

1.05

1.1

1.15

1.2

1.25

1.3

*

*

1.35

1.4

1.45

1.5

L=H=2-L =2-H

2.2

2

w

1.8

θ = .9θ = .8
θ = .7

1.6

1.4

1.2

1

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

L = H = 2 - L* = 2 - H*

Figure 6 shows equilibria for three di¤erent values of . The computations that underlie this…gure use

= 1:8,= 2, f = 1 and L+L = H +H = 2. For these parameter values, the e¤ective

determinants of task allocation are the threshold values I and I , which identify the tasks that canbe concentrated in each country without being susceptible to deviation by a …rm that locates in theother and attracts local …rms there as customers. We suppress the J curves to minimize clutter,as the value of J has no bearing on the equilibria that are depicted. In all these cases, tasks with25

i

I (if any) are concentrated in the low-wage West, tasks with I < i

I are concentrated in

the high-wage East, and tasks with i > I (if any) are dispersed.24
The top panel in Figure 6 shows that I increases withfor all values of L as long as I < 1.

When some tasks are performed locally in both countries, stronger scale economies mean greater
incentives for concentration of task performance and so a smaller set of dispersed tasks. The set of
dispersed tasks also shrinks as the gap in resource endowments grows, for reasons that we discussed
in Section 3.1.
In the …gure, both I and I

I grow monotonically withfor all values of L andsuch thatI < 1. That is, as long as there are some tasks that are performed locally in both countries, astrengthening of scale economies expands the measure of tasks that is concentrated in each one.25
In such circumstances, an increase inalso increases the relative wage of the high-wage country,

as can be seen in the bottom panel of the …gure. As the set of tasks that is concentrated in eachcountry grows, the demand for Eastern labor grows by more than the demand for Western labor,
because the East is performing tasks that are more costly to o¤shore than those performed in
the West. Moreover, an increase incauses the set E to shift to the right and thus makes thetasks concentrated in East even more labor demanding relative to those concentrated in West than

before. The growth in the relative demand for Eastern labor exerts upward pressure on its relative
wage.
Once I = 1, however, this mechanism — of expansion in I that causes an increase in relative
demand for Eastern labor— can no longer operate. Then a further rise in

has no e¤ect on I .

The impact e¤ect of this further strengthening of scale economies is to enhance the incentives forconcentration in West. But any expansion of the set of tasks performed in West exerts upward
pressure on its relative wage, which dims the enthusiasm of the Eastern …rms for performing themarginal tasks abroad. In the end, aggregate outputs rise in both countries by the same percentage
as the increase in labor productivity, the pattern of specialization remains unchanged, and therelative wage of the East falls (see the bottom panel of Figure 6, where the fall in w asincreases

form 0.7 to 0.9 is slight, but visible).

4.2

Extent of Product Di¤erentiation

The extent of product di¤erentiation is captured by the elasticity of substitution, . We need atleast a unitary elasticity of substitution between goods for pro…t-maximizing prices to be …nite. As
the …nal goods become closer substitutes, any cost advantage that producers in one country enjoy
relative to producers in the other results in a larger ratio of sales per …rm; see equation (6). Recall
that producers based in the high-wage country have lower per unit costs as a result of their greater
24

For = 0:9, we …nd I > 0, I > 0, and w > 1 when the countries are identical in size (L = L = H = H ). The
existence of this asymmetric equilibrium with positive task trade and unequal wages when the countries are identicalin size implies that there exist two other equilibria when = 0:9 and L is su¢ ciently close to L , one with w = 1
and another with w < 1. We do not show these equilibria in Figure 6.
25
At given wages, an increase in dampens pro…tability for a deviant that would sell to only one set of national
producers and therefore both I and I rise at the initial relative wage.

26

productivity in tasks that are performed locally in both countries and their country’s specializationin tasks that have relatively greater o¤shoring costs. Thus, an increase inmagni…es the scaleadvantage of the high-wage country.
Figure 7: Relative Country Size and the Elasticity of Substitution( β = 1.8, θ = .8, f = 1, J is never an effective threshold)
1
0.9
0.8
I (σ = 2.4)
I (σ = 2)
I (σ = 1.6)
I* (σ = 2.4)
I* (σ = 2)
I* (σ = 1.6)

0.7

I and I*

0.6
0.5
0.4
0.3
0.2
0.1
0

1

1.05

1.1

1.15

1.2

1.25

1.3

*

*

1.35

1.4

1.45

1.5

L=H=2-L =2-H

2.2

2

w

1.8
σ = 2.4σ=2
σ = 1.6

1.6

1.4

1.2

1

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

L = H = 2 - L* = 2 - H*

Figure 7 uses the same parameter values as Figure 6, except that we …x
three values of

at 0.8 and consider

ranging from 1.6 to 2.4. Again, the values of I and I determine the equilibrium

allocation of tasks, so we suppress the J curves. Although it is di¢ cult to see in the …gure, we…nd that if> 2, there exists a range of country sizes for which only …rms in the West performtasks o¤shore (I > I = 0), while if< 2, there is a range of country sizes for which only …rms
27

in the East perform tasks o¤shore (I > I = 0): But, in either case, if the countries are su¢ cientlydi¤erent in size, task trade ‡ows in both directions. As we have noted previously, the case of=2is special, because it has tasks concentrated in both countries whenever any tasks are concentrated
in one.
Consider …rst a division of resources such as L = 1:15, for which 1 > I > I > 0; i.e., the sets
W, E and B are all non-empty. The relative cost advantage of Eastern …rms that derives from theallocation of traded tasks results in a larger ratio of sales per …rm as substitutability increases. As

a result of this and the fact that the numbers of …rms are …xed by the endowments of managerial
talent, nx grows while n x shrinks. The reduction in aggregate output by Western …rms enhances
their incentive to o¤shore tasks to East as compared to performing them at smaller scale at home.
Thus, I grows. The expansion of the set of tasks concentrated in the East spells an increase indemand for labor there, which bids up the relative wage w and causes more tasks with moderateo¤shoring costs to locate in West; that is, w and I increase with

as well.

Now consider a division of resources such as L = 1:45, for which I = 1. Once the di¤erencein country size and the elasticity of substitution are such that no tasks are performed in bothcountries, a further increase in product substitutability cannot expand the set E at the expense of
the set B. Nonetheless, we …nd that a rise inincreases the relative wage of the East due to themagni…ed scale advantage it derives from its lesser costs. In the bottom panel, w increases with ,albeit to a lesser extent than for more equal country sizes. The rise in w diminishes Western …rms’
incentives to concentrate the marginal tasks in East, while the fall in the scale of aggregate output
by Western …rms has the opposite e¤ect by reducing the productivity of dispersed production there.
For the parameters that underlie Figure 7, the former e¤ect (barely) dominates. Therefore, we seethat I falls slightly as

4.3

increases from 1.6 to 2.4.

Size of O¤shoring Costs

Finally, we examine variations in o¤shoring costs that might result, for example, from improvements
in communication technology. Figures 8 and 9 are drawn for a given division of the world’s resources
with L = H = 1:1 and L = H = 0:9: Both …gures assumeof the equilibrium for two di¤erent values of , namely
display outcomes for o¤shoring costs ranging between= 2 and f = 1 and both show features= 0:7 and= 1 and= 0:8. Finally, the …gures= 2.

The two panels of Figure 8 show I; I and J for the alternative values of the scale economy parameter . Consider …rst the uppermost panel, which depicts the case with greater scale economies.
When o¤shoring is not very costly, I = 1, which means that all tasks are concentrated in one country or the other. The tasks with i

J are performed in West, while those with i > J are performedin East. In this case, aggregate production plus o¤shoring costs are minimized for every task. Thisis possible, because the strong scale economies make concentration desirable and the modest o¤shoring costs make deviations to dispersed production relatively unappealing. For tasks i > J,concentrated production in West is undermined by a deviant to East who sets prices su¢ ciently
low to attract all producers as customers. The boundary at I plays no role in the equilibrium,
28

because the tasks just above and just below I are concentrated in East by dint of having indexes
greater than J, so the fact that a deviation to East would be pro…table were they instead to beconcentrated in West is irrelevant.
Figure 8: Offshoring Costs and Economies of Scale**( L = H = 1.1; L = H = 0.9, σ = 2, f = 1)
1
I (θ = .8)
I* (θ = .8)
J (θ = .8)

0.9
0.8

I, I* and J

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1

1.1

1.2

1.3

1.4

1.5
β

1.6

1.7

1.8

1.9

2

1
I (θ = .7)
I* (θ = .7)
J (θ = .7)

0.9
0.8

I, I* and J

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1

1.1

1.2

1.3

1.4

1.5
β

1.6

1.7

1.8

1.9

2

As o¤shoring costs increase, the thresholds I and I fall. The greater o¤shoring costs enhance
the pro…tability of a deviant who serves only local producers. For= 1:1, it is no longer possible

to concentrate the performance of all tasks. Rather, those with i

J are concentrated in West,

those with J < i

I are concentrated in East, and those with i > I are performed locally inboth countries. For= 1:2, the threshold J no longer plays a role in the allocation of tasks, as I
29instead determines the boundary between tasks that are concentrated in West and in East. Stillfurther increases in

reduce the sets of tasks concentrated in each country and expand the set thatis dispersed. Finally, forthe case of

greater than 1.92, o¤shoring does not take place. As noted previously,= 2 that is depicted here has I and I reaching zero for the same value of , whichmeans that each country’s o¤shore production ceases when its hosting of foreign producers does as
well.
Comparing the top and bottom panels of Figure 8, we see again that stronger scale economies
imply greater concentration of tasks. The set W, which comprises tasks with 0

is larger for every value ofimin[I ; J],in the top panel than in the bottom panel, as is the set E, which

comprises tasks with min[I ; J] < i

I. Otherwise, the bottom panel is qualitatively similar to

the top.
As the o¤shoring costs increase, the relative wage of the East …rst increases, then decreases
slightly, and ultimately becomes constant (see the top panel of Figure 9). The increase in therelative wage re‡ects the fact that the West su¤ers more from higher o¤shoring costs due to its
smaller scale for dispersed tasks. This e¤ect is more pronounced when scale economies are strong,which explains why the thick curve rises more steeply than the thinner curve. Once the o¤shoring
costs reach a level such that few tasks are traded, another e¤ect dominates. Workers in the East
bene…t relative to those in the West from the fact that their country specializes in tasks thatare more costly to o¤shore. But this relative bene…t disappears as

grows large. Finally, wheno¤shoring costs are so high as to choke o¤ task trade, further increases in these costs have no
bearing on the countries’wages.
The bottom panel of Figure 9 tells a related story about aggregate welfare. We have found amonotonic relationship between o¤shoring costs and welfare in each country for a wide range ofparameter values. The adverse e¤ects of higher o¤shoring costs are especially pronounced whenis large, because the potential gains from specialization are greater in such circumstances. The…gure shows that the West fares better when scale economies are strong than when they are weak(compare the thick and thin dark curves) when
when

is large. Whenis low, but the opposite is true (albeit barely so)

is small, the West gains from trading a wide range of tasks, the moreso the stronger are the increasing returns to scale. But when o¤shoring costs are high, little or notask trade takes place. Then a strengthening of scale economies reduces the relative cost of …nal
goods in the larger East, which in turn induces consumers to substitute toward their goods. The
scale of production in the small West can fall as

30

increases, in which case its welfare may decline.

Figure 9: Offshoring Costs and Economies of Scale**( L = H = 1.1; L = H = 0.9, σ = 2, f = 1)
1.55
1.5
θ = 0.8
θ = 0.7

1.45
1.4

w

1.35
1.3
1.25
1.2
1.15
1.1

1

1.1

1.2

1.3

1.4

1.5
β

1.6

1.7

1.8

1.9

2

10
9
East Welfare (θ = .8)
West Welfare (θ = .8)
East Welfare (θ = .7)
West Welfare (θ = .7)

8
7

Welfare

6
5
4
3
2
1
0

5

1

1.1

1.2

1.3

1.4

1.5
β

1.6

1.7

1.8

1.9

2

Concluding Remarks

We have developed a theory of task trade between similar countries. When o¤shoring costs are
not too high, …rms concentrate certain tasks in particular locations in order to realize external
economies of scale.

The potential for outsourcing allows them to overcome some aspects of thecoordination problem inherent in this. Our theory predicts the pattern of specialization by task forcountries that di¤er only in size. We …nd that there always exists an equilibrium in which the larger
country has higher wages and greater aggregate output of …nal goods. If o¤shoring costs are low
31enough and the countries are not too di¤erent in size, there may exist another equilibrium in which
the smaller country has the higher wages and greater aggregate output. In either case, the country
with the higher wages and output performs the tasks— among those that are concentrated— thatare more di¢ cult and costly to o¤shore.
Our main empirical prediction links the pattern of specialization in tasks to relative wages. To
test this prediction, we would need to identify the characteristics of tasks performed in di¤erent
countries, which is by no means an easy thing to do. However, Autor et al. (2003) have shownthat it is possible to distinguish the tasks performed in a country using data on the distributionof workers across occupations and information about the type of work performed by individuals ineach narrowly-de…ned occupational category. They have measured the specialization of the U.S.
economy across …ve task categories: routine and manual, routine and cognitive, non-routine andinteractive, non-routine and analytic, and non-routine and manual. Since the 1980’s, the UnitedStates has been specializing more in tasks that are non-routine and either interactive or analytic,
and less in the other three categories of tasks.
Spitz-Oener (2006) has conducted a similar exercise using German data. She …nds that thepattern of specialization across tasks has evolved similarly in Germany as in the United States,
except that Germany is performing more tasks that are non-routine and manual over time, unlike
the United States. The evidence supports the plausible conclusion that routine tasks are migrating
to low-income countries like China, India and Mexico, with the high-income countries specializing
increasingly in the set of non-routine tasks. But the evidence also suggests that Germany is specializing in a di¤erent set of non-routine tasks than the United States, namely, those that are moremanual in nature. Given that Germany is smaller than the United States in terms of aggregate
output and it has lower wages, our theory predicts that it should specialize in tasks that are relatively easier to o¤shore. Our prediction accords with the available evidence to the extent that
(non-routine) manual tasks can more readily be organized and coordinated from a distance thaninteractive or analytic tasks. This ranking of relative o¤shoring costs seems plausible to us, but wecould …nd no direct evidence to con…rm it.
Ideally, empirical research on task trade would begin by classifying tasks according to therelative ease of o¤shoring. More data on o¤shoring are becoming available as awareness of thisphenomenon grows, so it may soon be possible to measure the o¤shoring costs for di¤erent tasks.
Once that is possible, it will also be possible to study the pattern of specialization by task. We
hope that our theory can help guide such e¤orts.

32

6

Appendix

We …rst prove the claim we made in Section 2.1. We then proceed to the proofs of Lemmas 1 and
2.
Claim 1 Either (i)d (i) > 0 or

d (i) < 0 for all i, or (ii) J > 0 andd (i) > 0 for i < J whiled (i) < 0 for i > J.

Proof. Without loss of generality assume w > 1. The aggregate cost of performing task i in the
East minus the aggregate cost of performing it in the West is proportional (since it is divided by
A (nx + n x )) to(i; nx; n x ; w)d (i; nx; n x ; w) A (nx + n x )= wnx + t (i) wn x
= (wnxFirst assume that n x

nx: Then nx

n x )t (i) (nxwn x ) :wn x < 0 which implies mini

(0; nx; n x ; w) since t0 (i) > 0 for all i. Then, since(0; nx; n x ; w) > wnx= (w(i; nx; n x ; w) =t (0) > 1;n x

nx + wn x1) (nx + n x ) > 0:

So all tasks have higher aggregate cost in the East; i.e.
Now suppose instead that nx > n x : Then wnx
t (0) > 1 is close enough to one that(n x + t (i) nx)

d (i) > 0 for all i and J = 1.

n x > nx

wn x . Suppose …rst that(0; nx; n x ; w) > 0: Then tasks in the neighborhood oftask 0 yield lower costs in the West. Since t0 (i) > 0 for all i, either there exists J > 0 such that
(J; nx; n x ; w) = 0, in which case tasks with i > J have lower cost in the East ( d (i) < 0) andtasks with i < J have lower cost in West ( d (i) > 0), or (wnxn x ) > t (1) (nx

wn x ) in

which case(i; nx; n x ; w) > 0 for all i and all tasks have lower cost in the West ( d (i) > 0 and

J = 1). Ift (0) is such thatlower costs in the East, namely,(0; nx; n x ; w) < 0; then since t0 (i) > 0 for all i, all tasks haved (i) < 0 and J = 0.

Lemma 1 If w > 1, J < I implies I > I .
Proof. The proof of Claim 1 in the Appendix guarantees that if w > 1 then n x > nx implies
J = 1. So we can limit our attention to circumstances with nx > n x . Note thatt(I ) =wA (nx + n x )A(nx)t(I) =A (nx + n x ):wA(n x )and33

To establish a contradiction, we suppose that J < I and I > I: ThenwA (nx + n x )A (nx + n x )>A(nx)wA(n x )
and
w2 >

A(nx)
:
A(n x )

(13)

From the de…nition of J, we know thatt(J)t(I) =

wnx n xnx wn xA (nx + n x ):wA(n x )(14)

Since the denominators are both positive for J 2 (0; 1), the left-hand side has the same sign as(n x ; nx; w)
2w A(n x )nx

wA(n x )n x

A (nx + n x ) nx + wA (nx + n x ) n x

But then (13) implies that(n x ; nx; w)> A(nx)nx

wA(n x )n x

= wn x [A (nx + n x )> n x [A (nx + n x )A (nx + n x ) nx + wA (nx + n x ) n x

A(n x )] + nx [A (nx)A(n x )] + nx [A (nx)A (nx + n x )]A (nx + n x )] :De…ne the last term on the right-hand side as(n x ; nx)
and note that(n xnx) A (nx + n x )n x A(n x ) + nxA(nx);( ) is continuously di¤erentiable in both arguments and(nx; nx) = 0:

Calculate the partial derivative of(n x ; nx) with respect to the second argument,0
2 (n x ; nx) = A(nx) + nxA (nx)(nxn x ) A0 (n x + nx)

Then
2 (0; nx) = 0and02 (nx; nx) = A(nx) + nxA (nx)34

A(2nx)0;

A(n x + nx):

where the inequality follows from the concavity of A ( ). Note also that
12 (n x ; nx) =(nxn x )A00 (n x + nx)

by the concavity of A ( ). Then, since

2 ( ) is continuous,nx

n x . Since(nx; nx) = 0 and

2 (n x ; nx)

that(n x ; nx)

0 for all nx

0;

2 (n x ; nx)

0 for all nx

0 for all n x

0 andn x , it follows by continuity

n x . Hence, if w > 1, I > I, and nx > n x , we obtain that(n x ; nx; w) > 0; which implies by (14) that J > I. This establishes our contradiction.

Lemma 2 w > 1 if and only if nx > n x .
Proof. We consider three mutually exhaustive cases: (i) I
I < I and L

I , (ii) I < I and L > L , and (iii)L .

(i) From the de…nitions of I and I ,t(I)w
A(nx + n x )1
A(n x )t(I )A(nx + n x )

w
:
A(nx)

and

The …rst inequality can strict when I = 1, the second when I = 0. ThereforeA(nx + n x )wA(n x )t(I)wA(nx + n x )
;
A(nx)

t(I )

which implies thatA(nx)A(n x )

w2 > 1.

So nx > n x .
(ii) To establish a contradiction, suppose that nx
ThenL=n x . From Figure 5, I < I implies E = ;.

M (B)nx
M (B)n x>L >A(nx)A(n x )

which implies
A(nx)
A(n x )<:nx
n x
But A( ) concave, A(0)0; and nx

n x , imply that
A(nx)
nx

A(n x ):
n x

This contradicts the supposition that nx < n x .

35

(iii) To establish a contradiction, suppose that nx
L = (1andL > (1I )I )n x . Labor-market clearing impliesnx
A (nx)n x
nx + n x+I;A (n x )A (nx + n x )since T (I ) > I for all I :From manager-market clearing, and H = L and H = L ; this implies thatn x
nx+n xI ) A(nx ) + I A(nx+n x )

(1
L
n
=
>
n
L

nx
I ) A(nx)(1Multiplying both sides by nx=(n x ), we obtain
1 Ix
A(n x ) + I>
xNote that nx

nx+n x
n x
1 I
A(nx)

1
A(nx+n x ):n x and w > 1 imply thatw(1 I )T (I )c
A(nx) + A(nx+n x )
= 1 I
I
c
A(n x ) + A(nx+n x )1:

CES preferences and goods-market clearing yield
x
c
=
x
c
and since;> 1; this implies
x
x

c
:
c

Given that T (I ) > I and w > 1
x
x
or

1 I
I
c
A(nx) + A(nx+n x )
> 1 I
I
c
A(n x ) + A(nx+n x )1 I
Ix
A(n x ) + A(nx+n x )
< 1 I
:
I
x
A(nx) + A(nx+n x )Therefore, in order for an equilibrium to exhibit nx < n x it has to be the case that
1 I
Ix
A(n x ) + A(nx+n x )
>
1 I
I
x
+
A(nx)
A(nx+n x )>1 I
A(n x ) + I

36

nx+n x
n x
1 I
A(nx)

1
A(nx+n x ):But note that I =A (nx + n x ) > 0 and (nx + n x ) =n x > 1 so
1 I
I
1 I
A(n x ) + A(nx+n x )A(n x ) + I<
1 I
I
A(nx) + A(nx+n x )which contradicts the previous string of inequalities.

37

nx+n x
n x
1 I
A(nx)

1
A(nx+n x );7References

Amiti, Mary (1998), “Inter-industry Trade in Manufactures: Does Country Size Matter?,”Journalof International Economics, 44:2, 231-255.
Amiti, Mary and Shang-Jin Wei (2005), “Fear of Service Outsourcing: Is it Justi…ed?,”Economic
Policy, 20:42, 308-347.
Amiti, Mary and Shang-Jin Wei (2006), “Service O¤shoring, Productivity and Employment: Evidence from the US,” CEPR Discussion Paper No. 5475.
Autor David, Frank Levy and Richard Murnane (2003), “The Skill Content Of Recent Technological Change: An Empirical Exploration,” Quarterly Journal of Economics, 118:4, 1279-1333.
Baldwin, Richard and Frederic Robert-Nicoud (2007), “O¤shoring: General Equilibrium E¤ects
on Wages, Production and Trade,” NBER Working Paper No. 12991.
Campa, José M., and Linda S. Goldberg (1997), “The Evolving External Orientation of Manufacturing Industries: Evidence from four Countries,” Federal Reserve Bank of New York
Economic Policy Review, 3:2, 53-81.
Davis, Donald R. (1995), “Intra-industry Trade: A Heckscher-Ohlin-Ricardo Approach,” Journalof International Economics, 39:3-4, 201-226.
Deardor¤, Alan V. (2001a), “Fragmentation across Cones,” in S.W. Arndt and H. Kierzkowski,
eds., Fragmentation: New Production Patterns in the World Economy, Oxford: Oxford University Press.
Deardor¤, Alan V. (2001b), “Fragmentation in Simple Trade Models,” North American Journalof Economics and Finance, 12:2, 121-137.
Dixit, Avinash K. and Victor Norman (1980), Theory of International Trade, Cambridge: Cambridge University Press.
Egger, Hartmut and Falkinger, Josef (2003), “The Distributional E¤ects of International Outsourcing in a 2x2 Model,”North American Journal of Economics and Finance, 14:2, 189-206.
Ethier, Wilfred J. (1982), “Decreasing Costs in International Trade and Grank Graham’s Argument for Protection,” Econometrica, 50:5, 1243-1268.
Feenstra Robert C. and Gordon H. Hanson (1996), “Foreign Investment, Outsourcing and Relative
Wages,” in R.C. Feenstra, G.M. Grossman and D.A. Irwin, eds., The Political Economy ofTrade Policy: Papers in Honor of Jagdish Bhagwati, Cambridge, MA: The MIT Press.
Grossman, Gene M. and Giovanni Maggi (2000), “Diversity and Trade,” American Economic
Review, 90:5, 1255-1275.
38

Grossman, Gene M. and Esteban Rossi-Hansberg (2008a), “External Economies and International
Trade Redux,” manuscript, Princeton University.
Grossman, Gene M. and Esteban Rossi-Hansberg (2008b), “Trading Tasks: A Simple Theory ofO¤shoring,” American Economic Review, 98:5, forthcoming.
Hanson, Gordon H. and Chong Xiang (2004), “The Home-Market E¤ect and Bilateral Trade
Patterns,” American Economic Review, 94:4, 1108-1129.
Hanson, Gordon H., Raymond J. Mataloni and Matthew J. Slaughter (2005), “Vertical Production
Networks in Multinational Firms,” Review of Economics and Statistics, 87:4, 664-678.
Helpman, Elhanan (1981), “International Trade in the Presence of Product Di¤erentiation, Economies
of Scale and Monopolistic Competition : A Chamberlin-Heckscher-Ohlin Approach,”Journalof International Economics, 11:3, 305-340.
Helpman, Elhanan (1984), “Increasing Returns, Imperfect Markets, and Trade Theory,” in R.
Jones and P. Kenen, eds, Handbook of International Economics, New York: Elsevier.
Helpman, Elhanan and Paul R. Krugman (1985), Market Structure and Foreign Trade: Increasing
Returns, Imperfect Competition and the International Economy, Cambridge, MA: The MIT
Press.
Holmes, Thomas J. and John J. Stevens (2004), “Geographic Concentration and Establishment
Size: Analysis in an Alternative Economic Geography Model,”Journal of Economic Geography, 4:3, 227-250.
Hummels, David, Jun Ishii and Kei-Mu Yi (2001), “The Nature and Growth of Vertical Specialization in World Trade,” Journal of International Economics, 54:1, 75-96.
Hummels, David, Dana Rapoport, and Kei-Mu Yi (1998), “Vertical Specialization and the Changing Nature of World Trade,”Federal Reserve Bank of New York Economic Policy Review, 4:2,
79-99.
Jones, Ronald W. and Henryk Kierzkowski (1990), “The Role of Services in Production andInternational Trade: A Theoretical Framework,” in R. Jones and A. Kruegers, eds., The
Political Economy of International Trade: Essays in Honor of Robert E. Baldwin, Oxford:
Basil Blackwell.
Jones, Ronald W. and Henryk Kierzkowski (2001), “Globalization and the Consequences of International Fragmentation,” in R. Dornbusch, ed., Money, Capital Mobility and Trade: Essaysin Honor of Robert A. Mundell, Cambridge, MA: The MIT Press.
Kohler, Wilhelm (2004a), “Aspects of International Fragmentation,”Review of International Economics, 12:5, 793-816.
39

Kohler, Wilhelm (2004b), “International Outsourcing and Factor Prices with Multistage Production,” Economic Journal, 114:1, 166-168.
Krugman, Paul R. (1979), “Increasing Returns, Monopolistic Competition, and International
trade,” Journal of International Economics, 9:4, 469-479.
Krugman, Paul R. (1980), “Scale Economies, Product Di¤erentiation, and the Pattern of Trade,”
American Economic Review, 70:5, 950-959.
Krugman, Paul R. (1991), “Increasing Returns and Economic Geography,” Journal of Political
Economy, 99:3, 483-499.
Levy, Frank and Richard Murnane (2004), The New Division of Labor, Princeton: Princeton
University Press.
Lucas, Robert E. (2002), Lectures on Economic Growth, Cambridge, MA: Harvard University
Press.
Markusen James R. and James R. Melvin (1981), “Trade, Factor Prices, and the Gains from Tradewith Increasing Returns to Scale,” Canadian Journal of Economics, 14:3, 450-469.
Marshall, Alfred (1920), Principles of Economics, London: Macmillan.
Matsuyama, Kiminori (1991), “Increasing Returns, Industrialization, and Indeterminacy of Equilibrium,” Quarterly Journal of Economics, 106:2, 617-650.
Melvin, James R. (1969), “Increasing Returns to Scale as a Determinant of Trade,” Canadian
Journal of Economics, 2:3, 389-402.
Newhouse, John (2007), Boeing versus Airbus: The Inside Story of the Greatest International
Competition in Business, New York: A. A. Knopf.
Ohnsorge, Franziska and Daniel Tre‡er (2007), “Sorting It Out: International Trade with Heterogeneous Workers,” Journal of Political Economy, 115:5, 868-892.
Rodriguez-Clare, Andres (2007), “O¤ shoring in a Ricardian World,” NBER Working Paper No.
13203.
Romer, Paul M. (1986), “Increasing Returns and Long-run Growth,”Journal of Political Economy,
94:5, 1002-1037.
Spitz-Oener, Alexandra (2006), “Technical Change, Job Tasks, and Rising Educational Demands:
Looking outside the Wage Structure,” Journal of Labor Economics, 24:2, 235-270.
Yeats, Alexander J. (2001), “Just How Big Is Global Production Sharing?”in S.W. Arndt and H.
Kierzkowski, eds., Fragmentation: New Production Patterns in the World Economy, Oxford:
Oxford University Press.
40

Yi, Kei-Mu (2003), “Can Vertical Specialization Explain the Growth of World Trade?,” Journalof Political Economy, 111:1, 52-102.

41