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Diversity and Trade

By GENE M. GROSSMAN AND GIOVANNI MAGGI*

We develop a competitive model of trade between countries with similar aggregatefactor endowments. The trade pattern reflects differences in the distribution of talentacross the labor forces of the two countries. The country with a relatively homo-geneous population exports the good produced by a technology with complemen-tarities between tasks. The country with a more diverse workforce exports the goodfor which individual success is more important. Imperfect observability of talentstrengthens the forces of comparative advantage. Finally, we examine the effects oftrade on income distribution and the composition of firms in each industry.(JELF11, D51)

A large fraction of world trade takes placebetween countries with similar aggregate factorendowments and similar technological capabil-ities. Moreover, much of this trade involvesgoods with similar factor intensities. Expla-nations for these now-familiar observationsabound. Increasing returns to scale can accountfor trade in goods that are technologically alikebut differentiated in the eyes of consumers (e.g.,Paul R. Krugman, 1979). Oligopolistic compe-tition can account for two-way trade in identicalproducts under certain assumptions about oli-gopolistic conduct (e.g., James A. Brander,1991). Small technological differences acrosscountries can account for high volumes of tradein goods with similar factor intensities whenfactor endowment ratios are similar and thereare more goods than factors (Donald Davis,1995).

But all of these explanations share the impli-cation that the pattern of trade is essentiallyarbitrary. If trade is motivated by scale econo-mies, theory predicts the number of varietiesexported by each country, but not which ones. Iftrade reflects oligopolistic competition and

countries have similar cost structures, only thenumbers of competitors in the countries affecttheir market shares, and these numbers are notexplained by the models. If trade is generatedby small technological differences, the patternof such differences inevitably will be difficult topredict.

Yet it seems that there are systematic differ-ences in the trading patterns of, say, the UnitedStates and Japan, or Germany and Italy. Japantends to excel in industries requiring care andprecision in a long sequence of productionstages. Its exports include many sophisticatedconsumer goods, such as automobiles and high-end consumer electronics. Whereas the UnitedStates exports many goods and services whosevalue reflects disproportionately the input of afew very talented individuals. Its highly suc-cessful software industry is an example of this.So too are some of the financial services ema-nating from Wall Street. Similar to Japan, Ger-many’s comparative advantage resides in itsmanufacturing sector; it exports passenger cars,industrial equipment, and chemicals. Again,quality control plays a large part in explainingits success in these sectors. For Italy, like theUnited States, international competitiveness of-ten reflects the outstanding performance of oneor a few persons. Italy is known, for example,for its innovative furniture styles, its fashiondesigns, and its movies.

In this paper we argue that not only aggregatefactor endowments but also the distribution ofthose factors (especially human capital) across anation’s population matters for the pattern of

* Grossman: Woodrow Wilson School, Princeton Uni-versity, Princeton, NJ 08544; Maggi: Department of Eco-nomics, Princeton University, Princeton, NJ 08544. We aregrateful to Dilip Abreu, Susan Athey, Avinash Dixit, Gio-vanni Forni, Elhanan Helpman, Marciano Siniscalchi, MartiSubrahmanyam, Tim van Zandt, two anonymous referees,and especially Sam Kortum for helpful discussions andremarks. Grossman also thanks the National Science Foun-dation for financial support and IIES in Stockholm, EPRUin Copenhagen, and IDEI in Toulouse for their hospitality.

1255

specialization and trade. Countries with thesame aggregate endowment ratios may special-ize in producing distinct types of goods, evenwhen the goods use the factors with the sameaverage intensities. For example, the ratio ofhuman capital to other factors may be similar infirms producing computer software and digitaltape players, and the United States and Japanmay have similar aggregate ratios of human tophysical capital. Still, our model would predicta determinate pattern of trade, with the UnitedStates exporting the goods like software andJapan exporting those like tape players. We willargue that the trade pattern reflects the greaterdiversityof talent in the United States comparedto Japan.

There is ample evidence that countries differsignificantly in the relative homogeneity of theirlabor forces. For example, Table 1 providesinformation on the distributions of test scoresfrom the International Adult Literacy Survey(IALS). The collaborators in this study admin-istered a common test of work-related literacyskills to a large sample of adults in Europe andNorth America (see OECD and Human Re-sources Development Canada, 1997). The tableshows the standard deviation, fifth percentileand ninety-fifth percentile scores—each as afraction of the national mean—for six countrieson three different literacy scales. The findingsindicate a more widely spread distribution ofskills in the United States, the United Kingdom,and Canada than in Sweden, the Netherlands,and Germany. Similar variation in skill distri-butions is reported in the Third International

Mathematics and Science Study (TIMSS)(Maryellen Harmon et al., 1997), a study thatmeasured science and mathematics skillsamong students in their final year of secondaryschool in 21 countries. In this study, SouthAfrica, the Czech Republic, and the UnitedStates show the greatest diversity in test results,whereas scores vary much less in the Nether-lands, France, and Iceland.1

Our predictions about the trade pattern willbe based on an assumption about the nature ofdifferent technologies, in particular the way inwhich workers of different skill levels interact.As Paul Milgrom and John Roberts (1990) andMichael Kremer (1993) have emphasized, thereare some activities—complicated manufactur-ing processes being a prime example—whereworkers performing different tasks are highlycomplementary. Kremer describes the “O-Ringproduction function” in which failure at anytask destroys the entire value of the project.This is a particular form of complementarityin which a production unit is only as strongas its weakest link. Milgrom and Roberts ex-plore the implications of a more general notionof complementarity: two tasks are complemen-tary if performing one better raises the marginal

1 Japan did not participate in either of these studies.However, Japan did participate in the TIMSS survey of themathematics skills among seventh and eighth graders. Thisstudy finds, for example, a standard deviation for eighth-grade math achievement in Japan that is 16.9 percent of themean, as compared to 18.2 percent in the United States; seeAlbert E. Beaton et al. (1996).

TABLE 1—DIVERSITY IN LITERACY SKILLS

PERSONSAGED 16–65, 1994–1995

Prose Document Quantitative

s

m

F0.05

m

F0.95

m

s

m

F0.05

m

F0.95

m

s

m

F0.05

m

F0.95

m

United States 0.25 0.50 1.34 0.26 0.47 1.37 0.25 0.50 1.37United Kingdom 0.24 0.57 1.32 0.26 0.48 1.35 0.25 0.53 1.35Canada 0.23 0.52 1.30 0.26 0.54 1.35 0.23 0.55 1.34Sweden 0.17 0.71 1.27 0.18 0.72 1.27 0.18 0.71 1.28Netherlands 0.16 0.72 1.23 0.17 0.71 1.24 0.17 0.70 1.25Germany 0.16 0.72 1.27 0.16 0.73 1.27 0.15 0.74 1.25

Notes:Based on scales with a range of 0–500 points.s/m is the standard deviation of scale score divided by the mean.F0.05/mis the fifth percentile score divided by the mean.F0.95/m is the ninety-fifth percentile score divided by the mean.Source:OECD and Human Resources Development Canada (1997 Tables 1.1 and 1.11).

1256 THE AMERICAN ECONOMIC REVIEW DECEMBER 2000

product of better performance in the other. Thisnotion of complementarity corresponds to amathematical property of the production func-tion known assupermodularity.

The opposite of supermodularity issubmodu-larity. When a production function is submodular,superior performance of one task mitigates theneed for superior performance in the others. Weargue that such a property may characterize someproduction processes, especially those requiringcreativity or problem solving. These processesshare the feature that value reflects disproportion-ately the best idea or the luckiest draw. If severalinventors seek a similar invention with differentapproaches, the most successful one will generatemost of the revenues. If two software engineersattempt to solve a specific programming problem,it only takes one brilliant idea to solve the prob-lem. In these situations—as we argue in moredetail below—output becomes a submodularfunction of the human inputs; that is, the marginalvalue of having a more talented inventor (or in-vestment adviser, etc.) increases when the abilitiesof others working with him on the same projectare less.

Supermodularity and submodularity carryvery different implications for the optimal or-ganization of production. If a technology issupermodular and tasks are symmetric, effi-ciency requiresself-matching,as Kremer (1993)has shown. Workers should be sorted so thatthose of similar ability work together. In con-trast, when a technology is submodular,cross-matching is indicated. The most talentedindividuals should be matched with less tal-ented counterparts, so that talent is not “wast-ed.” Competitive forces will induce such sortingin a perfectly competitive equilibrium.

Another important difference between su-permodular and submodular technologies hasto do with returns to an individual’s talentwithin his or her production unit. Supposethere are overall constant returns to skill in afirm; that is, if the human capital of all em-ployees is doubled, the value of the outputgoes up by a factor of two. Then there will bedecreasing returns to individual talent for aworker contributing to a supermodular pro-duction process, but increasing returns to tal-ent for one engaged in a submodular process.This means that the individuals with thegreatest talent are most productive when they

pursue activities requiring mostly individualsuccess.

How, then, will a trading equilibrium lookif sectors differ in the technological relation-ship between talents in a firm? Section IIdeals with the case where technologies aresupermodular in each of two sectors, but thedegree of complementarity between tasks dif-fers in the two. We show that, if individuals’abilities are perfectly observable and there areconstant returns to talent, then countries withaccess to similar technologies will not trade,even if the distributions of talent in the twoare different. But the distribution of talentbegins to matter once one of the sectors ex-hibits submodularity. In Section III we exam-ine the pattern of trade between countrieswith identical technologies but different dis-tributions of talent, when one sector is super-modular and the other is submodular. Weidentify sufficient conditions under which thecountry with a more diverse talent pool ex-hibits a comparative advantage in producingthe good or service with a submodular pro-duction technology.

In Section IV we study the consequences ofthe trade that results from differences in thedistribution of talent. First we show that, in thecountry with the greater diversity of talent, anopening of trade induces exit of firms at bothends of the skill distribution. As resources movefrom the supermodular to the submodular in-dustry, the firms that close in the former sectorare those that initially employed the highest-skilled and lowest-skilled workers in the indus-try. Next we consider the effect of trade on thewage distribution. Again focusing on the coun-try with greater diversity, we find that tradeincreases the real wages of those at the bottomof the skill distribution, while reducing the realwages of a group of higher-skilled workers.Those at the very top of the distribution maygain or lose. These results reflect a Stolper-Samuelson like tendency for trade to benefitworkers in the export sector, but also the factthat movements in relative prices induce achange in the distribution of surpluswithin eachfirm in the submodular sector. The latter effectfavors low-skill workers in the country with amore diverse labor force and high-skill workersin the country with a more homogeneous laborforce.

1257VOL. 90 NO. 5 GROSSMAN AND MAGGI: DIVERSITY AND TRADE

Section V extends the model to allow forimperfect observability of talent. We show thatimperfect observability gives the country witha more homogeneous population an absoluteadvantage in producing goods where comple-mentarities exist. It therefore strengthens ourpredictions about the pattern of trade. SectionVI contains some concluding remarks.

Before launching into the analysis, we beginin the next section with a discussion of super-modular and submodular technologies. We givesome examples of each type of production func-tion, and establish what each implies for theefficient matching of workers. We also discusswhy submodularity may be an appropriate as-sumption for sectors where creativity matters.

I. Supermodularity and Submodularity

Consider a production process involving twotasks. The tasks are indivisible; each must beperformed by exactly one worker. LetF(tA, tB)be the output from the process when the firsttask is performed by a worker with talenttA andthe second by a worker with talenttB. Output ismeasured in quality-adjusted units. For simplic-ity, we assume that the tasks are symmetric; i.e.,F(tA, tB) 5 F(tB, tA).2 We also assume thatF[ is homogeneous of degree one in the talentlevels of the two workers.3

A. Only As Strong As the Weakest Link

The functionF[ is supermodular if

(1) F~tA , tB! 1 F~t9A , t9B!

# F@min$tA , t9A%, min$tB , t9B%#

1 F@max$tA , t9A%, max$tB , t9B%#.

Donald M. Topkis (1978) and Milgrom andRoberts (1990) have shown that, under rather

weak conditions, this definition is equivalent toF12 $ 0; i.e., the marginal product of talent ineach task is increasing in the talent used toperform the other.4 This is the sense in whichthe two tasks are complementary in producingoutput or value.

When F is homogeneous of degree one,F12 $ 0 implies thatF is quasi-concave. There-fore, its isoquants are convex. Figure 1 depicts

2 Asymmetry between tasks greatly complicates theproblem of optimal sorting; see Kremer and Eric Maskin(1996).

3 In the concluding section we discuss how differingreturns to talent in different sectors could itself generatetrade between countries with different distributions of talentbut similar aggregate endowments and technological capa-bilities.

4 Topkis assumed thatF is twice continuously differen-tiable. Milgrom and Roberts weakened this to the assump-tion thatF can be written as an indefinite double integral ofa functionFij .

FIGURE 1. COMPLEMENTARY TALENTS


two examples. In panel a, the extent of comple-mentarity is moderate; if the talent of the workerperforming one task is reduced, this can becompensated by an increase in the talent of theworker performing the second. In panel b,complementarity is extreme. Here, output isfully determined by the skill of the less ableworker in the pair. Panel a applies, for example,when the production function isF(tA, tB) 5[ tA

u 1 tBu ] 1/u, with u , 1. Notice that this

function is symmetric and has constant returnsto talent, as we have assumed. Panel b is thelimiting case asu3 2`, or equivalently,F(tA,tB) 5 min{ tA, tB}.

Milgrom and Roberts (1990) and Kremer(1993) have argued that many manufacturingprocesses exhibit important complementaritiesof the sort reflected in Figure 1. Kremer de-scribes a production technology in which failurein any one of a number of tasks causes value todrop to zero. If there were two such tasks, andif the probability of success in each were pro-portional to the square root of the talent levelsof the person performing it, then we would havea symmetric, constant-returns-to-talent and su-permodular production function such as the oneused here. More generally, as Kremer notes,small mistakes at one or a few production stagessometimes cause large reductions in the value ofa good, as with “irregular” clothing or automo-biles with “minor” design flaws. If the numberor probability of mistakes at any stage is relatedto the talent of the worker undertaking thatstage, then a supermodular production functionis the likely result.

The optimality of self-sorting follows readilyfrom the definition of supermodularity. Supposethere were two firms each with a worker oftalent t1 matched with a worker of talentt2,t1 Þ t2. By the assumed symmetry of theproduction function, output of these firms is thesame no matter how workers within each firmare assigned to the two tasks. Suppose, then,that in each firm the worker with talent levelt1performs taskA and the one with talent levelt2performs taskB. Then, by inequality (1), totaloutput would rise (or at least, not fall) if thefirms were reorganized so that the twot1 work-ers were matched together and similarly the twot2 workers. More generally, symmetry and su-permodularity imply that, with any four work-ers, total output is greatest when the two with

the most talent are paired together and the twowith the least talent are paired together.

A final observation concerns the returns to anindividual worker’s ability. If returns to talentare constant for the firm as a whole, then themarginal products are homogeneous of degreezero. This implies thattAF11 1 tBF12 5 0, orthat F11 has the opposite sign fromF12. SinceF12 . 0, there are decreasing returns to indi-vidual talent within a firm.

B. It Only Takes One

Submodularity is defined to be the oppositeof supermodularity. That is,F[ is submodular,if

(2) F~tA , tB! 1 F~t9A , t9B!

$ F@min$tA , t9A%, min$tB , t9B%#

1 F@max$tA , t9A%, max$tB , t9B%#.

This is equivalent toF12 # 0 in most circum-stances; increasing the talent of one workerreducesthe marginal value of increasing thetalent of the other.

Submodularity applies when workers aresubstitutable in creating value. Although it isbest of course for all tasks to be performed to ashigh a standard as possible, better in submodu-lar processes that some be done superbly andothers only tolerably well than that all be com-pleted at a medium level of competence. Suchappears to be the case, for example, in theperforming arts; an Oscar-winning performancecombined with a decent supporting cast is likelyto generate more revenue than one that is goodbut not great across the board. It is also true ofmany research activities, where an outstandingidea and some silly ones are worth more than aset of reasonable but not sterling suggestions.Other examples might include legal and con-sulting services, and architectural and fashiondesigns.

An example helps to explain why creativeprocesses may exhibit submodularity. Supposethere are two workers (or teams) engaged inparallel creative endeavors. This may be anattempt to improve the quality of a word pro-cessor, to design a new sport coat, or to developa winning legal argument. Suppose further that


the firm employing these two will find it prof-itable to pursue only the better of the two cre-ations, since both are attempts to solve the sameperceived problem. Letmi be the quality of thecreation by workeri , who has talentti. Sincethe creative process is random, we may think ofmi as being drawn from some cumulative dis-tribution function V(m; ti) with nonnegativesupport. More talented workers are more likelyto succeed, in the sense thatV(m; tA) first-orderstochastically dominatesV(m; tB) if tA . tB.This implies V2(m, t) # 0 for all t. Finally,suppose the two employees work independentlyand that their draws are independent of oneanother.

We refer to “output” here as the expectedvalue of the superior draw. Clearly, this is afunction of the talent levels of the two employ-ees; call the functionF(tA, tB). Now it isstraightforward to show that, for any distribu-tion V, the functionF is submodular.5 Intu-itively, when the talent level of one softwarewriter (designer or lawyer) is raised, this in-creases the likelihood of a good draw from him.This means, in turn, that it is more likely that thesecond software writer’s efforts will be redun-dant.6 Accordingly, there is less incentive toinvest in talent for this second draw.

The spirit of the example survives when thetwo creations are not perfect substitutes. Thenthe firm might choose to pursue both originalideas. Still, its revenues will be a submodularfunction of the talents of the two employees if

greater success of one dramatically shrinks thepotential market for the other.7

The above example can alternatively be in-terpreted as describing a pure problem-solvingactivity. Suppose two engineers work indepen-dently at solving a problem, which is settled themoment one of them comes up with a solution.The value of the project depends on the speedwith which the team resolves the issue: ifTiis the time it takes engineeri to come up witha solution, the value of the project isF5 E min{ T1, T2}. If a higher talent level shiftsthe distribution ofTi in the sense of first-order-stochastic dominance, thenF is submodular inthe underlying talents.

More generally, workers’ talents often aresubstitutable when they contribute in parallel toachieving a target outcome or a target level ofoutput. Such targets come about, for example,when the cost of a shortfall or “failure” exceedsthe value of an overrun. Parallel contributions tofixed targets would describe, for example, theinputs by members of a sports team, the flightattendants on an airline crew, the workers on arepair crew, or the professionals on a designteam. In each case, the marginal product of acontributor’s ability is positive, but the presenceof a very talented individual mitigates the needfor others.8

When a submodular function is also homo-geneous of degree one, its isoquants are con-cave. Figure 2 illustrates a moderate and anextreme example. In panel a, the ability of thelesser-talented employee still matters for the over-all value of the project. A constant-elasticity-of-substitution (CES) production functionF(tA, tB) 5[tA

u 1 tBu]1/u has the shape depicted there foru . 1.

Figure 2b depicts the limiting case asu 3 `,whenF(tA, tB) 5 max(tA, tB). Then the talent ofthe superior worker fully determines the effec-tive output.

Submodularity makes cross-matching opti-mal. For any four workers, output is greatest

5 Let z 5 max{mA, mB}. Then z is a random variablewith cumulative distribution functionV( z, tA)V( z, tB). Itsexpectation can be written as

F~tA , tB! 5 E0

`

@1 2 V~z, tA!V~z, tB!# dz.

Then F12 5 2*0` V2( z, tA)V2( z, tB) dz # 0.

6 In the Microsoft Corporation, for example, the prevail-ing mode of production involves separate efforts by severalsmall teams of software engineers who work in parallel ona new program (or new version of an existing program).Each team develops one or more ideas for features toinclude in the program. The proposals are then ranked andonly those receiving the highest rankings are incorporated inthe final version of the product (see Michael A. Cusumanoand Richard W. Selby, 1995). This procedure pits the soft-ware engineers in competition with one another and renderstheir talents substitutable in Microsoft’s production process.

7 Sherwin Rosen (1981) describes properties of demandthat give rise to such a pattern of revenues, in his seminalarticle on “superstars.”

8 Basketball aficionados should consider the sequence ofho-hum centers that the Chicago Bulls employed to supporttheir stars, Michael Jordan and Scottie Pippen. Televisionstations seem to follow a similar strategy in pairing talk-show hosts with “sidekicks.”


when the two with the greatest and least talentwork together and those with intermediate tal-ents work together, as compared to any otherpossible pairings. Moreover, cross-matched pairswill be observed in a competitive equilibrium,because firms employing superstars will be lesswilling to pay for a supporting cast than firmsstocked with workers of more modest talents.Note that the assumed symmetry of the tasks isnot needed for these arguments. In fact, whentasks are asymmetric, the forces leading tocross-matching in a submodular sector are even

stronger. Not only are there increasing returnsto individual talent within a pair (F12 , 0 andF homogeneous of degree one impliesF11 .0), but a firm can benefit by assigning its moretalented worker to the more important task andits less talented worker to the less importanttask. We shall assume symmetry of tasks in allsectors for ease of exposition, but actually theassumption is restrictive only as it applies tosupermodular activities.9

II. Complementarities and Trade

Now consider an economy with two sectors,C and S. Production in each sector requiresexactly two workers, each performing a differ-ent task, as described in the previous section.Output by a pair of workers in sectori is F i(tA,tB), where tj denotes the skill level of theworker performing taskj . Both Fc and Fs aresupermodular functions, although the degree ofsupermodularity may differ across sectors, asfor example whenF i 5 [ tA

u i 1 tBu i] 1/u i for i 5 c,

s, anduc , us # 1. Also,F i is strictly increas-ing in both arguments, symmetric, and has con-stant returns to talent.

There are two countries, home and foreign,each with a continuum of workers. LetL andL* be the measures of their labor forces. Theskill distributions in the two countries arerepresented by cumulative distribution func-tions F(t) and F*( t), so that, for example,F(t) L is the measure of workers in the homecountry with talent less than or equal tot.These distributions are taken to be exogenous,reflecting perhaps the educational philoso-phies of the two countries as well as variationin achievement due to differences in socio-economic background. We take the distribu-tions of talent to be atomless, which isequivalent to requiring thatF and F* beeverywhere continuous. Lettmin and tmax de-note the minimum and maximum talent levels

9 More formally, letF(tA, tB) be any globally submodu-lar, but not necessarily symmetric, technology. A firm canassign workers to tasks optimally. Therefore, a firm withtwo workers of talenttA and tB can produceF(tA, tB) 5max{F(tA, tB), F(tB, tA)} units of output.F[ is symmet-ric by construction; it is also globally submodular. There-fore, the technology facing a firm is symmetric even whenthe underlying tasks are not.

FIGURE 2. SUBSTITUTABLE TALENTS


in the home population and similarly,t*min andt*max denote the minimum and maximum lev-els in the foreign country. Every worker’stalent t is perfectly observable, both to him-self and to all potential employers. Thustmight represent a worker’s years of schoolingor his score on a readily administered aptitudetest.

Preferences in the home and foreign countriesare identical and homothetic. We seek to char-acterize a competitive, free-trade equilibrium.Such an equilibrium exists, because the tech-nologies have constant returns to scale at theindustry level; if an industry doubles its em-ployment of workers at every talent level, thevolume of output doubles.

A competitive equilibrium maximizes thevalue of national output at given prices. Thus,production in each sector is technically effi-cient. The following lemma formalizes our ob-servation that efficiency requires self-matchingin any sector characterized by a supermodulartechnology.

LEMMA 1: If F i is supermodular, then for anyallocation of resources to sector i, industry outputis maximized when tA 5 tB in all firms in i.

PROOF:See Appendix A.

Armed with the knowledge that competi-tive pressures induce self-matching in bothsectors, we are ready to describe the produc-tion possibility frontiers (PPFs) of the twocountries. A typical home firm in sectoriemploys two workers of some talent level, sayt. Its output isF i(t, t) 5 l i t, where l i [F i(1, 1). This follows from the assumption ofconstant returns to talent. Accordingly, aggre-gate output in sectori is Yi 5 l iTi / 2, whereTi measures the aggregate talent (or humancapital) allocated to sectori . The PPF is gen-erated by varying the split of talent betweensectors, subject to the constraint thatTc 1Ts 5 T [ L * tdF, the country’s aggregateendowment of talent. Evidently, the PPF islinear, with a slope of2lc/ls.

In the foreign country, the PPF has the sameslope. With proportional production possibili-ties and identical, homothetic preferences, thetwo countries do not trade.

PROPOSITION 1:If F i(tA, tB) is symmetric,supermodular, and homogeneous of degree onefor i 5 c, s, then the free-trade equilibrium hasno trade.

With perfect information and constant returnsto talent, differences in the degree of workercomplementarity in the two sectors are notenough to generate a comparative advantage foreither country in producing either of the twogoods. Although the countries may differ intheir distributions of talent, in equilibriumworkers in each country match with partners ofsimilar ability. Then the output of a pair isproportional to their shared talent level, and anindustry’s output is proportional to the aggre-gate amount of talent allocated to the industry.With no differences in technology, there is nobasis for trade.

III. Complementarity, Substitutability,and Trade

Now let the two sectors differ in terms of thenature of the relationship between the individ-uals working together in a firm. In one sector, apair of workers performs complementary tasks.Then the marginal product of an individual’stalent is greater the more able is his co-worker.This is theC sector,C for “complementarity.”In the other sector the workers toil on substi-tutable tasks. What is most important in thissector is that at least one of the tasks be per-formed especially well. Although output in-creases with the skill of any worker, themarginal product of talent is higher the lesscapable is ones partner. We useS for this sectorto denote “substitutability.”

Again, a competitive equilibrium exists(there are constant returns to scale in each in-dustry) and maximizes the value of output atdomestic prices. To characterize the generalequilibrium, we look first at the organization offirms in each sector, then at the allocation ofworkers to the two sectors.

In the sector with complementary tasks,Lemma 1 dictates the matching of workers withsimilar abilities. Now consider the other sector.Let Fs(t) be a cumulative distribution functionsuch thatFs(t) L represents the allocation oftalent to sectorS. Lemma 2 describes how theseworkers will be paired.


LEMMA 2: If Fs is submodular, then for anyallocation of talentFs(t) L to sector S, indus-try output is maximized by maximal cross-matching; i.e., each worker of talent t is pairedwith a worker of talent ms(t), where ms(t) isdefined implicitly byFs[ms(t)] 5 1 2 Fs(t).


In Lemma 2, maximal cross-matching meansthe following. For any set of individuals employedin the S industry, it is efficient for the most-talented worker to be paired with the least-talentedworker, the second-most-talented worker to bepaired with the second-least-talented worker, andso on. The demands for the various levels of skillby profit-maximizing firms ensure that the effi-cient pairings are realized in a competitiveequilibrium.

The next question is how the aggregate poolof talentF(t) L will be divided between the twosectors. Of course, the allocation of resourcesdepends upon the relative price. But we canmake the following qualitative statement thatholds true at any price.

LEMMA 3: To produce any given output Ys ofgood S, it is efficient to allocate to that sectorall workers with talents t# t and all workerswith talents t $ m( t ), where m(t) is definedimplicitly by F[m(t)] 5 1 2 F(t) and t solvesYs 5 L *tmin

t Fs[ t, m(t)] dF(t).


Lemma 3 says that theS sector attracts themost-talented and least-talented workers. Thisallocation reflects the fact that there are increas-ing returns to a single worker’s talent in thesector with substitutable tasks;F11

s , F22s . 0,

whereasF11c , F22

c , 0. And once a highlytalented worker has been assigned to theS sec-tor, there are diminishing marginal costs of re-ducing the talent of his partner.

We are now ready to describe the pattern oftrade between two countries with different dis-tributions of talent. We consider several possi-ble ways in which these distributions mightdiffer. In each case, we assume that the densityfunctions f [ dF/dt and f* [ dF*/ dt aresymmetric about their means.

PROPOSITION 2:If F(t) 5 F*( t), then thefree-trade equilibrium has no trade.

PROOF:To show that there is no trade, it suffices to

show thatMRT5 MRT* implies Ys/Yc 5 Y*s/Y*c,whereMRT 5 2dYs/dYc is the marginal rate oftransformation in the home country, andMRT* 52dY*s/dY*c is that in the foreign country. This isbecause each country produces in a competitiveequilibrium where the relative price is equal to themarginal rate of transformation. In a free-tradeequilibrium, the marginal rates of transformationare equalized, and this implies no trade if theoutput ratios are same and demands are identicaland homothetic.

Let t be the least-talented worker in the super-modular sector in the home country. By Lemma 3and the symmetry off, the most-talented workerin that sector has talentm(t) 5 2t# 2 t, wheret# 5T/L is the average talent level in the home country.Total output of goodC is

(3) Yc 5 L Et

2t# 2 t

Fc~t, t!f~t! dt

5lct#

2@F~2t# 2 t! 2 F~ t!#,

in view of Lemma 1. That of goodS is

(4) Ys 5 L Etmin

t

Fs@t, m~t!#f~t! dt

5 L Etmin

t

Fs~t, 2t# 2 t!f~t! dt,

in view of Lemma 2 and the symmetry off,which implies m(t) 5 2t# 2 t. The homemarginal rate of transformation can be calcu-lated as

(5) MRT5 2dYs/dt

dYc /dt5

Fs~ t, 2t# 2 t!

lct#.

Similarly, that abroad is


(6) MRT* 5Fs~ t*, 2 t#* 2 t* !

lct#*,

wheret#* is the average talent level in the for-eign country andt* is the least-talented individ-ual working in theC sector there.

Whenf 5 f*, the mean talent levelst# andt#*are the same. ThereforeMRT5 MRT* if and onlyif t 5 t*; i.e., if the marginal worker in the super-modular sector has the same talent in both coun-tries. But then it follows immediately from (3) and(4) and the analogs for the foreign country thatwhen t 5 t* and f 5 f*, Ys/Yc 5 Y*s/Y*c.

It is not surprising that, if the two countries arescalar replicas of one another, they do not trade.The potential matches in each country are thesame. The larger country simply can have more ofeach type of match. Since there are constant re-turns to scale at the industry level, more matchesmeans proportionally more output. In short, coun-tries with the same technologies and same distri-butions of talent face the same trade-off inproducing one good relative to the other. Theirproduction possibility curves have the same shape.

The next proposition indicates that no tradetakes place also when every worker in onecountry has a similar fraction (or multiple) ofthe skills of his counterpart in the talent distri-bution of the other.

PROPOSITION 3:If F(t) 5 F*( bt) for allt [ [ tmin, tmax], then the free-trade equilibriumhas no trade.

PROOF:Note that the aggregate endowment of talent in

the foreign country isbL*/L times that in thehome country; i.e.,T* 5 bTL*/L. Again, we focuson the implications ofMRT5 MRT*. SinceFs ishomogeneous of degree one and increasing inboth arguments, (5) and (6) implyMRT5 MRT*if and only if t* 5 bt. With t* 5 bt andF(t) 5F*(bt), Y*s 5 bL*Ys/L and Y*c 5 bL*Yc/L.10

Therefore,MRT5 MRT* implies Ys/Yc 5 Y*s/Y*c.

This proposition reflects our assumption thatthere are constant returns to talent in each sec-tor. The potential matches in the foreign countryare similar to those in the home country, exceptthat every member of a pair in the foreigncountry hasb times as much talent as his coun-terpart in a pairing in the home country. Withconstant returns to talent, this means that eachpotential team in the foreign country can pro-duceb times as much output as the analogousteam in the home country. It follows that, onceagain, the production possibility curves have thesame shape.

Finally, we come to the case of distributionsthat differ in thediversityof talent. We employthe following definition of diversity.

Definition 1: The distribution of talentF ismore diverse thanF* if F(t) . F*( t) for t ,t9 andF(t) , F*( t) for t . t9, for somet9 .tmin.

With this definition, if F is more diverse thanF* and the mean talent level is the same in eachcountry (t# 5 t#*), then F must be a mean-preserving spread ofF*. However, the combi-nation of equal mean and greater diversity is abit more restrictive than a mean-preservingspread, because we allow the two cumulativedistribution functions to cross only once.11 Thesingle-crossing property seems reasonable fordescribing a distribution with uniformly “fattertails”; in any event, we use it only to provide acondition which is sufficient (not necessary) fora particular pattern of trade.

Intuitively, the country with a more diverse

10 Note that

Y*c 5lcL*

2 Eb t

2b t# 2 b t

t* f* ~t* ! dt*,

sincet*min 5 btmin, and t* 5 b t. Now make the change ofvariablev 5 t*/ b. Then we have

Y*c 5lcL*

2 Et

2t# 2 t

bvf~v! dv 5bL* Yc

L,

becausef*( t*) 5 f*( bv) 5 f(v)/b anddt* 5 bdv. TheequalityY*s 5 bL* Ys/L is established similarly.

11 Andreu Mas-Colell et al. (1995 p. 199) provide anexample of a mean-preserving spread for which the c.d.f.’scross several times. Our notion of diversity corresponds towhat in decision theory is known as a simple increase inrisk; see, for example, Jack Meyer and Michael B. Ormiston(1989).


distribution of talent has the greater possibilitiesfor pairing very talented individuals with onesat the opposite end of the talent distribution.The pairs with one extremely talented memberare especially productive in the sector with sub-stitutable tasks, as we have seen. Accordingly,the country with the more diverse distributionenjoys an advantage in producing this good.Meanwhile, productivity in the sector with com-plementary tasks depends only on the averagequality of the pairs, which is the same in twocountries with similar mean endowments of hu-man capital. It follows that comparative advan-tage in theS industry resides in the country withthe greater proportion of workers whose talentsare extreme. In Proposition 4, we make thisclaim more precise.

PROPOSITION 4:Suppose t# 5 t#* and Fmore diverse thanF*. Then the home countryexports good S and imports good C in a free-trade equilibrium.

PROOF:Let t9 , t#* be a point in the support ofF*

and let t0 , t9. Define a single, symmetricmean-preserving spread (SSMPS) ofF* as anarbitrary decrease in densitydf* on [ t9, t9 1dt] and on [m(t9), m(t9 2 dt)], accompaniedby an equal increase in density on [t0, t0 1 dt]and [m(t0), m(t0 2 dt)]. Then it is straightfor-ward to show, using the same methods as in theproof of Theorem 1(b) in Michael Rothschildand Joseph E. Stiglitz (1970), that ifF(t) andF*( t) are symmetric,t# 5 t#*, and F is morediverse thanF*, F can be generated fromF*by a sequence of SSMPS’s.

Sincet# 5 t#*, (5) and (6) imply thatMRT 5MRT* if and only if t 5 t*. Consider the ratioof outputs in the two countries whenMRT 5MRT*. By Proposition 2, these ratios would bethe same if the countries had identical distribu-tions of talent. ButF is more diverse, whichmeans that it can be generated fromF* by asequence of SSMPS’s. There are three possibleeffects of any SSMPS. Ift* , t0 , t9, thenoutputs of both goods are unaffected by thechange in distribution. Ift0 , t* , t9, then theoutput of goodS must rise, while the output ofgoodC falls. Finally, if t0 , t9 , t*, the outputof good S rises, while the output of goodCremains the same. The increase in the output of

good S reflects the fact thatFs[ t, m(t)] is aconvex function oft, sinceFs is submodularand homogeneous of degree one. It follows thateach SSMPS increases or does not change theratio of output of goodS relative to the output ofgood C, at a givenMRT. Therefore,Ys/Yc .Y*s/Y*c for t 5 t*. This pattern of relativeoutputs plus identical and homothetic prefer-ences implies that the home country exportsgood S and imports goodC at any commonrelative pricep.

Figure 3 depicts the production possibilityfrontiers for two countries of similar size (L 5L*) and the same mean talent level (t# 5 t#*)when the home country has a more diversedistribution of talent than the foreign country.The maximum potential output of goodC is thesame in both countries, inasmuch as output inthe C industry depends only on the total talentallocated to the industry, and two countries ofsimilar size and similar mean talent levels havethe same aggregate human capital. However,the home country is better equipped to producea given amount of goodS, because this countryhas more of the most talented people, and firmswith such superstars are especially productive inproducing goodS. The proof of Proposition 4

FIGURE 3. COMPARATIVE ADVANTAGE

WITH SIMILAR MEAN TALENT LEVELS

AND DIFFERENT DIVERSITY


establishes that, where the two PPF’s have thesame slope, the relative output of goodS ishigher in the country with greater diversity. Thetrade pattern follows immediately from this.

Proposition 4 predicts the pattern of trade onlyfor countries with the same mean level of talent.But it is straightforward to extend the propositionto include countries that differ in mean talentlevels, using Proposition 3. What we need to do isto look at the diversity of talent in a hypotheticalhome economy that has the same mean talentlevel as that abroad. In particular, let us define thedistributionFma[ such thatFma(rt) 5 F(t) for allt [ [tmin, tmax], where r [ t#*/ t#. The superscriptdesignates “mean adjusted,” inasmuch as themean of a random variable drawn from the distri-bution Fma is r times the mean of a randomvariable drawn fromF, and thus is equal tot#* byconstruction. By Proposition 3, the productionpossibility frontier for the hypothetical economyhas the same shape as that for the home economy;the one is a radial expansion of the other. Now ifthe hypothetical economy has a more diverse dis-tribution of talent than the foreign country, theproof of Proposition 4 tells us that it must producerelatively more of goodS when the two face thesame price. It follows that the home country alsoproduces relatively more of goodSwhenMRT5MRT*. The extended proposition follows.

PROPOSITION 49: Let F(t) and F*( t) besymmetric distributions. DefineFma[ suchthat Fma(rt ) 5 F(t) for all t , where r[ t#*/ t#.If Fma is more diverse thanF*, the homecountry exports good S and imports good C ina free-trade equilibrium.

IV. The Effects of Trade

In this section, we derive some consequencesof the trade that results from differences in thedistribution of talent. For concreteness, we fo-cus on the home country with its greater diver-sity of skills. This country exports goodS in thefree-trade equilibrium. Since trade reduces therelative price of a country’s import good, weconsider the effects of a decline in the relativeprice p of goodC.

A novel prediction of our model followsstraightforwardly from Lemma 3. It concernsthe effects of trade on the composition of firmsin the import-competing sector. In any equilib-

rium, p 5 MRT, so (5) impliesdt/dp , 0.Therefore, a fall in the price of goodC increasest and reducesm( t ); i.e., there is exit fromthe industry that producesC. It followsimmediately that the firms that exit the import-competing sector—which are those that pairedworkers of talentt and those that paired workersof talentm( t )—are the ones that were the mostand least productive among those operating inthe industry before the opening of trade.12

The model has interesting implications aswell for the change in the distribution of in-come. To explore these, we need the equationsthat determine the equilibrium wages of work-ers of different talent levels. Workers with tal-ents betweent and m( t ) toil in the C sector.Each member of a matched pair contributesequally to output. Competition ensures that themembers of a team are paid the same and thattheir joint earnings exhaust the value of whatthey produce. Therefore,

(7) w~t! 5plc t

2for t [ @ t, m~ t !#.

Firms in theS sector also earn zero profits.Thus,

(8) w~t! 1 w@m~t!# 5 Fs@t, m~t!#

for t [ @tmin, t #.

Each firm maximizes profits, which means thatit must (weakly) prefer to hire its actual work-force than to hire a different individual for itsteam. Consider the firm that hires a worker oftalent m(s) 5 2t# 2 s to perform task 2. Inequilibrium, this firm also hires the worker oftalents to perform task 1. If instead it hired aworker of slightly greater ability,s 1 ds, itsoutput would be greater byFs[s 1 ds, m(s)] 2Fs[s, m(s)], and the wage bill would be higherby w(s 1 ds) 2 w(s). Thus, it must be thatFs[s 1 ds, m(s)] 2 Fs[s, m(s)] # w(s 1ds) 2 w(s). Now consider a firm that in equi-

12 In the foreign country, the entrants in theC sector asa result of trade will be more and less productive per workerthan the firms that operate under autarky.


librium hires workers of talents 1 ds andm(s 1 ds). This firm must not profit by down-grading the talent of the worker it hires to per-form task 1; this impliesFs[s 1 ds, m(s 1ds)] 2 Fs[s, m(s 1 ds)] $ w(s 1 ds) 2w(s). Dividing each of these two inequalities byds and taking the limit asds3 0, we find thatprofit maximization impliesF1

s[s, m(s)] 5w9(s). This is true for alls , t. Moreover,equation (7) gives us the wage for the marginalworker in theSsector with talentt. For workerswith less talent thant, we have

(9) w~t! 5lcpt

22 E

t

t

F1s@s, m~s!# ds

for t [ @tmin, t !.

The wages for workers with talents at the upperend of the distribution can be found by combin-ing (8) and (9).

Figure 4 depicts the wage distribution asso-ciated with a given relative pricep. We havedrawn the schedule as linear in the range [t,m( t )] and convex outside this range. Linearityfor t [ [ t, m( t )] follows directly from (7). Fort , t, (9) implies w0(t) 5 F11

s [ t, m(t)] 2F12

s [ t, m(t)], sincem9(t) 5 21. But F11s . 0

andF12s , 0 by the supermodularity ofFs and

the assumption of constant returns to talent.Therefore,w0 . 0. Convexity in the ranget [[m( t ), tmax] can be established similarly.

Next we consider the effects of a decline in therelative price of goodC. We first derive the effectsof this price change on nominal wages (in terms ofgood S), then we draw the implications for realincome levels. We know that a fall inp increasest and reducesm(t)—say, to the points labeledt9andm(t9) in the figure. For workers in the import-competing sector before and afterp falls, nominalwages fall in proportion to the decline inp. Theconvexity of w(t) outside [t9, m(t9)] ensures aless-than-proportional nominal wage reduction (ora wage increase) for workers who begin in theimport-competing sector and move to the exportsector; i.e., those witht [ [t, t9] and those witht [[m(t9), m(t)].

We now will prove that the nominal wage ofa worker with talentt rises, while that of aworker with talentm( t ) falls. Notice that these

workers are paired together in the export sectorboth before and after the decline in the price ofgood C. They share the income from sales ofFs[( t, m( t )] bothex anteandex post;therefore,the movements in their nominal wages reflect achange in the division of surplus.

To establish our claim, we differentiate (9)with respect tow, t andp at a talent levelt ,t. This gives

(10) dw~t!

5lcp

2dt 2 F1

s@ t, m~ t !#dt 1lct

2dp.

The equality of price and the marginal rate oftransformation impliesplct# 5 Fs[t, m(t)] by (5).Also, constant returns to talent impliesFs[t,m(t)] 5 tF1

s[t, m(t)] 1 m(t)F2s[t, m(t)]. And the

symmetry of the talent distribution impliest 1m(t) 5 2t#. Using all of these facts, we can rewrite(10) as

dw~t! 5m~ t !

2t#$F2

s@ t, m~ t !# 2 F1s@ t, m~ t !#%dt

1lct

2dp.

Finally, we differentiate the condition forp 5MRT with respect top and t, which gives

lct#dp 5 $F1s@ t, m~ t!# 2 F2

s@ t, m~ t !#%dt.

FIGURE 4. EFFECTS OFTRADE ON WAGES


Substituting, we find

dw~t! 5lc

2@ t 2 m~ t !#dp.

Sincet , m( t ), dw(t)/dp , 0 for all t , t.This means that all workers with talents lessthan t see their nominal wage rise when theprice of goodC falls. By continuity, a workerwith talent equal tot also experiences a wagehike. It follows from (8) that the nominal wageof all workers with talent greater than or equalto m( t ) must fall. The figure has been drawn toreflect these findings.

It is now easy to derive qualitative conclu-sions about the implications of trade for thewell-being of the various workers in the countrythat exports goodS. Clearly, trade benefits thelow-skill individuals who work in the ex-portables sector both before and after an open-ing of trade (t # t ), as their real wages rise interms of either good. We also know that theindividuals who work in the importables sectorbefore and after the opening of trade (t [ [ t9,m( t9)]) are worse off, because their purchasingpower in terms of goodS declines, and that interms of goodC does not change. Whether theremaining individuals (i.e., those who switchsectors and those at the high end of the distri-bution) gain or lose from trade depends onconsumption shares. Figure 5 identifies the ben-eficiaries of trade as a function of the consump-tion share of goodC.

The key features of Figure 5 are summarizedin the following proposition.

PROPOSITION 5:In a country that exportsgood S, trade (a) benefits all workers with tal-ent less than or equal to some critical level(thatincreases with the consumption share of goodC); (b) harms all workers with talent in a rangeimmediately above the critical level; and(c)benefits a range of workers with the highestskills if the consumption share of good C islarge, but not if the consumption share of goodC is small.

Statement (a) follows from the fact that nom-inal wage changes are monotonic int, and thatreal incomes rise for the least-skilled workers.The welfare effects described in the proposition

reflect the two distinct influences of trade onreal incomes. First, there is a direct effect of theincrease in the relative price of the exportablegood, which harms the workers employed in theimportables sector and benefits those employedin the exportables sector. Second, there is anindirect effect resulting from the reallocation ofproceeds within firms producing goodS, whichfavors the lesser-skilled workers.13

V. Imperfect Matching

We have seen that comparative advantage incertain manufacturing activities resides in thecountry with the more homogeneous laborforce. These are the activities that require thecompletion of several complementary tasks,where poor performance in any one greatlydamages the value of the product.

However, the model does not predict anyabsolute advantage in these sectors for a country

13 We have found no simple intuition to explain thereallocation that takes place within firms in sectorS. Beforethe opening of trade, the worker with talentt earns just hisopportunity value in theC sector. With trade, this opportu-nity value falls, but the worker becomes inframarginal andso earns a rent. The outside option falls in proportion to theworker’s ability, t. But the size of the rent is proportional tothe ability of the partner,m( t ). Sincem( t ) . t, the lower-skilled member of the team gains, while the higher-skilledmember loses.

FIGURE 5. WINNERS AND LOSERS FROMTRADE


like Japan that seems to have a relatively tightdistribution of talent levels. Indeed, with perfectself-matching, productivity in the supermodularsector depends only on the average level oftalent, which tends to vary little between coun-tries with similar per capita incomes. Yet casualobservation suggests that Japan excels in activ-ities like automobile production exactly becauseits firms are characterized by a more consistentlevel of competence. This consistency makesthe Japanese firm better able than some of itsrivals to control quality and costs, because thereare fewer slip-ups in the production process.

A simple modification of our model serves tobring its predictions more into line with thisapparent reality. What is needed is an environ-ment where the realized mix of talents in firmsseeking uniformity differs from place to place.Such an environment arises naturally whenworkers’ abilities are only imperfectly ob-served. Our goal in this section is to extend ormodel to allow for (partially) hidden talents.

Consider then an economy like the one de-scribed above, except that now an individual’stalent is the product of two factors. One factor,qi , we assume to be perfectly observable, whilethe other,ei , is known neither to the worker norto his potential employers.14 We specify

(11) t i 5 qi ei ,

whereqi can be thought of as a measure of theindividual’s years of schooling andei as anindication of the (personal) “effectiveness” ofthat schooling.

The distributions of observable and unob-servable ability are exogenous and independent.We useF(q) and F*( q*) to describe the cu-mulative distribution functions for educationalattainment in the home and foreign countries,and C(e) and C*( e) for the respective distri-butions of the unobservable components of tal-ent. Differences in the latter can arise from

differences in educational policy, as for exam-ple when one country devotes more resources toenrichment classes for the top-end students,while the other spends more to ensure thatstruggling students are brought up to speed.Without further loss of generality, we normalizethe noise in the educational processes so that themean ofe in each country is equal to one.

How will workers be paired in firms, nowthat their talents are partially hidden? If idio-syncratic risks can be diversified, as we shallassume, then the pairings will be done to max-imize aggregate industry output, given theresources allocated to each sector.15 In sectori ,total output is given by

Yi 5 Eqmin

qmax Eqmin

qmax

F i~qA , qB! dGi~qA , qB!,

where Gi(qA, qB) is the fraction of firms insectori employing a worker of observable talentless than or equal toqA performing task A andone of observable talent less than or equal toqBperforming task B; andF i(qA, qB) 5 ** Fi(qAeA,qBeB) dC(eA) dC(eB) is the expected output of afirm employing a pair of workers with observabletalent levelsqA andqB. But note that

2F i

qA qB

5EE eAeBF12i ~qAeA , qBeB! dC~eA! dC~eB!,

which has the same sign asF12i . Therefore

F c(qA, qB) is supermodular andFs(qA, qB) issubmodular. By reasoning similar to that usedin proving Lemmas 1 and 2, we conclude thatfirms in the C sector will match workers ofsimilar observable talent, while those in theSsector will practice maximal cross-matching.

Consider two countries of similar size. Sup-14 In a more complete model we would want to makeei

specific to certain tasks and processes, in which case itwould be reasonable to suppose that the workers do notknow perfectly their own aptitudes. Alternatively, we couldallow the workers to know their ownei as long as simpleemployment contracts are used that do not allow firms tosort workers by ability.

15 Although the output of each firm is random, totaloutput is determinate, because there is a large number ofinfinitesimal firms. We assume that owners of these firms(who earn zero expected profits) hold a diversified portfolioof shares in the different firms in the industry.


pose the distribution ofq is the same or morediverse in the home country than in the foreigncountry, and that the educational process athome is noisier than that abroad (in the sensethat C is a mean-preserving spread ofC*).Figure 6 depicts the production possibilities forthese countries. If all labor in each country wereallocated to theC sector, output in the foreigncountry would exceed that at home. The reasonis simple. In each country, there will be mis-matches in the pairing of workers producinggood C, because firms hiring these workerscannot perfectly observe their talents. The mis-matches cause per capita output to fall short oflct#/ 2. But the average mismatch in the homecountry will be larger than that in the foreigncountry, because the unobserved component oftalent has a noisier distribution there. It followsthat the country with a more homogeneousworkforce has a competitive advantage in pro-ducing goodC.16

By a similar token, the noise in the educationprocess adds to the home country’s productivityadvantage in producing goodS. Not only do thematches in this country involve workers of moredisparate observable talent, on average, but ineach match the noisiness ofeA and eB raisesexpected output.17 It can readily be shown thatthe production possibility frontiers for twocountries of equal size cross only once—as il-lustrated in the figure.

Imperfect matching reinforces the pattern ofcomparative advantage identified in Section III.On the one hand, the unobserved noise in thetalent distribution causes problems for all firmsoperating in the sector with production comple-mentarities, but the more so in the country withthe noisier distribution ofe. On the other hand,the greater randomness in the schooling processof the home country generates relatively moreextremely talented individuals, who are espe-cially productive in the sector with substitutabletasks. Therefore, the home country imports

16 A typical firm in the C sector matches two workerswith the same observable talent, sayq. The expected outputfor two workers with observable talentq is F c(q, q) 5** qFc(eA, eB)c(eA)c(eB)deAdeB in the home country andF c* (q, q) 5 ** qFc(eA, eB)c*( eA)c*( eB)deAdeB in theforeign country. LetFc(eA, q) [ * qFc(eA, eB)c(eB)deB

and Fc*( eA,q) [ * qFc(eA, eB)c*( eB)deB. Since Fc isconcave in each argument andc is a mean-preservingspread ofc*, Fc(eA, q) , Fc*( eA,q) for all eA andq. Also,

integration preserves concavity, soFc and Fc* are bothconcave ineA.

We haveF c(q, q) 5 * Fc(eA, q)c(eA)deA andF c*( q,q) 5 * Fc*( eA, q)c*( eA)deA, hence

F c~q,q!2F c* ~q,q!

5E @~Fc~eA,q!2Fc* ~eA , q!!c~eA!#deA

1E @Fc* ~eA,q!~c~eA!2c* ~eA!!#deA.

We have already established thatFc(eA, q) , Fc* (eA, q)for all eA andq, so the first term is negative. SinceFc* is aconcave function ofeA andc is a mean-preserving spread ofc*, the second term is negative as well. We conclude thatthe expected output of two workers of observable talentq isless in the country with the noisier education process.

17 The argument is analogous to that used in the preced-ing footnote. In particular, we can evaluate the expectedoutput of a pair of workers with observable talentsq andm(q), when the noise in the educational process has densityc at home andc* in the foreign country. To do so, wedefine Fs(eA, q) [ * Fs[eAq, eBm(q)]c(eB)deB andFs*( eA, q) [ * Fs[eAq, eBm(q)]c*( eB) deB so thatF s[q,m(q)] 5 * Fs( z )c(eA)deA and F s*[ q, m(q)] 5* Fs*( z )c*( eA)deA. By familiar reasoning,Fs(eA, q) .Fs*( eA, q) for all eA andq (becauseFs is convex in botharguments) and convex ineA. We can then argue thatF s[q,m(q)] . F s*[ q, m(q)], which is the result we are after.

FIGURE 6. COMPARATIVE ADVANTAGE WITH

IMPERFECTMATCHING


goodC and exports goodS, as recorded in thefollowing proposition.

PROPOSITION 6:Let F(q) andF*(q) be sym-metric distributions of observable talent, with t5t#* and F equally or more diverse thanF*. LetC(e) and C*(e) be symmetric distributions ofunobservable talent, withC noisier than C*.Then the home country exports good S and im-ports good C in a free-trade equilibrium.

PROOF:Let q and q* be the workers with the least

(observable) talent in theC sector of the homeand foreign countries, respectively. The mar-ginal rates of transformation in the two coun-tries are given by

MRT~q! 5F s~q, 2q# 2q!

12F c~q, q!11

2F c~2q# 2q, 2q# 2q!

and

MRT* ~q* !

5F s* ~q*, 2q# 2q* !

12F c* ~q*, q* !11

2F c*(2q# 2q*, 2q# 2q*)

respectively, where use has been made of thefact that there is self-matching in theC sectorand maximal cross-matching in theS sector inboth countries, that the workers with extremetalents produce goodS in both countries, andthat the countries share the same mean level ofobservable talent,q# . By the arguments of foot-notes 16 and 17,MRT(q) , MRT*( q), be-cause the greater noise in the home distributionof e raisesF s and lowersF c at any commonvalue ofq. Also, dMRT(q)/dq , 0. It followsthat, in a free-trade equilibrium withMRT 5MRT*, q . q*.

Arguments analogous to those used in theproof of Proposition 4, combined with thoseused in footnotes 16 and 17, can be used toestablish that, whenF is equally or more di-verse thanF*, relative output of goodS isgreater at home, for any common value ofq. Afortiori, Ys/Yc . Y*s/Y*c when q . q*. It fol -lows that the home country produces relativelymore output of goodS when the countries

face the same international price. Given identi-cal and homothetic preferences, the home coun-try must export this good in a free-tradeequilibrium.

We note in passing that, with imperfectmatching, trade would take place between twocountries with different educational processeseven if tasks were complementary in all produc-tion activities. Suppose, for example, that allworkers in each country had the same observ-able talentq. Then matching in each countrywould be at random, and both PPF’s would belinear. The marginal rate of transformation inthe home country would beF s(1, 1)/F c(1, 1),which need not be the same as the marginal rateof transformationF s*(1, 1)/F c*(1, 1). Onemight expect the country with the more consis-tent educational process to enjoy a comparativeadvantage in the sector where complementari-ties are more important; but we have yet to finda definition of the degree of supermodularity forwhich a general result of this sort holds.

VI. Concluding Remarks

We have developed a model in which thedistribution of talent matters for comparativeadvantage and trade. There can be trade be-tween countries with similar aggregate factorendowments, provided human capital is morewidely dispersed in one country than the other.The country with a relatively homogeneouspopulation exports the good with a productiontechnology characterized by complementaritiesbetween workers. In this sector, effective outputis greater when all tasks are done reasonablywell than when some are performed superblyand others miserably. Efficient organization ofproduction requires the matching of workerswith similar talent, and this is more readilyaccomplished in the country with a homoge-neous workforce.

Meanwhile, the country with a diverse pop-ulation exports the good whose technology ischaracterized by substitutability between em-ployees. In this sector, the firms with a superstarengaged in some task and lesser talented indi-viduals completing the others outperform thosewith a uniform caliber of worker of mediumtalent. A country with a more diverse popula-tion has a greater proportion of superstars, and


so it has a productivity advantage in these star-sensitive activities.

Ours is not the only model that would gen-erate trade between countries with differingdistributions of human capital. A model withone worker per firm and with increasing re-turns to talent in some sectors and decreasingor constant returns to talent in others woulddeliver similar predictions to the ones wehave provided here. We have chosen to em-phasize the forces guiding the internal orga-nization of the firm, because these seemimportant in explaining, for example, Japan’ssuccess in automobile production. Japanesefirms are reputed to have uniformly compe-tent personnel, which contributes to a highstandard of quality control. At the same time,the United States is a more individualisticsociety, and this seems relevant to its successin certain types of activities. However, we donot deny that differing returns to talent indifferent sectors could be a part of the story aswell. Ultimately, which is more important forthe trade pattern (if either) is an empiricalquestion, which we hope to address in futureresearch.

We have employed a number of simplifyingassumptions, which perhaps limit the generalityof our conclusions. The two that trouble us themost are the requirement that both tasks becompleted by every producer and that, in thesector with complementarities, the tasks con-tribute symmetrically to output. By making thetasks essential, we render it difficult to interpretsome as parallel creative activities. After all,parallel activities are undertaken at the discre-tion of the firm. In Appendix B, we show thatour results extend to situations where firms canchoose to employ a single worker, if certainfurther conditions hold. Suppose a firm in sectori with only one employee (of talentt) producesoutput F i(t, 0). In the C sector, where thetechnology is supermodular, firms alwayschoose to hire two workers. However, in theSsector, team production arises only if there aresome fixed set-up costs. We show that for arange of parametersK representing the fixedcost of operating a firm in theS sector, thegeneral equilibrium has pairs of workers in allfirms in both sectors. Moreover, the trade pat-tern conforms to our Proposition 4, with themore diverse country exporting goodS.

We leave the extension to asymmetric tasksfor future research.

APPENDIX A: PROOFS OF THELEMMAS

PROOF OF LEMMA 1:Let Fi(t) L be an arbitrary allocation of talent

to sectori , with t [ [ tmin, tmax]. We representthe allocation of these workers to firms by abivariate cumulative distribution functionG(tA,tB) with support [tmin, tmax] 3 [ tmin, tmax]. Thisfunction gives the fraction of firms with aworker of talent less than or equal totA per-forming taskA and one of talent less than orequal to tB performing taskB. Without lossof generality, we can restrict attention to func-tions G[ that are symmetric about the diago-nal, in view of the symmetry of the two tasks.The resource constraint impliesGA(t) L #F i(t) L/ 2, where GA(t) [ G(t, tmax) is themarginal cumulative distribution function fortask A. This constraint binds, because everyworker has a strictly positive marginal product.

Total output in sectori is Yi 5 L ** F i(tA,tB) dG(tA, tB). We now show thatYi is maxi-mized by an allocation functionG that puts allweight on the diagonal. Take any other functionG such thatGA(t) L 5 Fi(t) L/ 2. Now consideran arbitrary off-diagonal point (t9A, t9B), t9A Þt9B. We have G(t9A, t9B) L # GA(t9A) L 5F i(t9A) L/ 2 5 GA(t9A) L 5 G(t9A, t9B) L. (Thelast equality follows from the fact thatG hasweight only on the diagonal.) Therefore, anyGfirst-order stochastically dominatesG. But then,sinceF12

i $ 0, Corollary 5 in Haim Levy andJacob Paroush (1974) implies that outputYimust be at least as great underG as underG.

PROOF OF LEMMA 2:Again, we represent the allocation of workers

to firms by a bivariate cumulative distributionfunction G(tA, tB) that is symmetric about thediagonal. The resource constraint implies2GA(t) L # Fs(t) L. Again, the constraintbinds.

Total output in sectorS is Ys 5 L ** Fs(tA,tB) dG(tA, tB). We want to show thatYs ismaximized by an allocation functionG thatputs all weight on the curvetB 5 ms(tA).Take any other functionG such thatGA(t)L 5Fs(t)L and consider an arbitrary off-diagonalpoint (t9A, t9B), t9A Þ t9B. If (t9A, t9B) lies to the left of the


curvetB 5 ms(tA), then clearlyG(t9A, t9B)L $ G(t9A,t9B)L. Suppose that (t9A, t9B) lies to the right of thecurve. Note thatGB(t9A, t9B) 5 GA(t9A) 1 GB(t9B) 21 1 M(t9A, t9B), whereM(t9A, t9B) [ Pr(tA $ t9A, tB $t9B) under distributionG. Similarly GB(t9A, t9B) 5GA(t9A) 1 GB(t9B) 2 1 1 M(t9A, t9B), whereM(t9A,t9B) [ Pr(tA $ t9A, tB $ t9B) under distributionG.SinceG(t9A) 5 G(t9A) 5 Fs(t9A) L/ 2, G(t9B) 5G(t9B) 5 Fs(t9B) L/ 2, andM(t9A, t9B) $ M(t9A,t9B) 5 0, it follows that G(t9A, t9B) L $ G(t9A,t9B) L in this case as well. Therefore,G first-order stochastically dominatesG.

Since maximizingYs is equivalent to mini-mizing 2 L ** Fs( tA, tB) dG( tA, tB), and2F12

s $ 0, the Levy-Paroush result impliesthat output in sectorS must be at least asgreat underG as underG.

PROOF OF LEMMA 3:

Suppose that all workers are in theC sectorand consider the minimum cost (in terms ofYc) method to produce some small amountYsof good S. If we reallocate to sectorS equalmeasures of workers of typetA and tB, theloss of output in sectorC will be proportionalto lc(tA 1 tB), while the maximal output ofgoodS (in view of Lemma 2) will be propor-tional to Fs(tA, tB). The least-cost method ofproducing goodS is the choice oftA and tBthat maximizesFs(tA, tB)/lc(tA 1 tB).

If this problem were to have a solution (t9A,t9B) Þ (tmin, tmax), either t9A 5 t9B or it wouldneed to satisfy at least one of the followingfirst-order conditions:

(A1)Fj

s~t9A , t9B!~t9A 1 t9B! 2 Fs~t9A , t9B!

lc~t9A 1 t9B!2 5 0

for j 5 1, 2.

Constant returns to talent impliesFs 5 t9AF1s 1

t9BF2s. This and the symmetry ofFs[ imply that

each of the first-order conditions is satisfied ifand only ift9A 5 t9B. However, all points witht9A5 t9B are minimum points, not maximum points,because the second-order condition for a maxi-mum is violated whenevertA 5 tB.18 We con-

clude that the ratioFs(tA, tB)/lc(tA 1 tB)reaches a maximum at an extreme point, where(tA, tB) 5 (tmin, tmax).

The same argument can be applied recur-sively, for larger amounts of goodS. That is, theleast-cost method to produce any incrementalamount of goodS uses the workers, amongthose remaining in sectorC, with the mostextreme (minimum and maximum) talent levels.

APPENDIX B: INESSENTIAL TASKS

In this Appendix, we relax the assumptionthat firms must perform both tasks in order togenerate output. That is, we allow firms to staffonly one of the two possible positions. Surely, ifthe tasks represent parallel creative activities, itis not necessary that both be performed foroutput to be positive. Now, a firm that employsa single worker of talentt producesF i(t, 0) insectori , whereas one that deploys no talent toeither task has output of zero.

Notice first that, in the absence of fixed set-upcosts, all firms in theS sector will hire a singleemployee. This conclusion follows immediatelyfrom the assumed properties of the technology,since Fs submodular impliesFs(tA, tB) ,Fs(tA, 0) 1 Fs(0, tB); i.e., total output ishigher when workers labor separately thanwhen they toil together. On the other hand, thesupermodularity ofFc implies Fc(tA, tB) .Fc(tA, 0) 1 Fc(0, tB), so firms in theC sectorprefer to hire teams.

Teams may emerge in theSsector if there arefixed costs of operating firms there. To explorethis possibility, we suppose that each firm pro-ducing goodS must pay a fixed cost ofK inunits of this good. The net output of a pair ofworkers of talenttA and tB is Fs(tA, tB) 2 K,whereas an individual with talentt producesFs(t, 0) 2 K. All other assumptions are thesame as in Section III.

Despite the fixed costs at the firm level, theSindustry has constant returns to scale. An indi-vidual team is too small to affect prices, andteams cannot expand by adding extra workers.Thus, a competitive equilibrium can exist in

18 Differentiating (A1) with respect tot9A and evaluatingat t9A 5 t9B 5 t9 givesF11

s (t9, t9)/(2lct9)2. This is positive,

becauseF11s (t9, t9) 5 2F12

s (t9, t9) by the zero-degreehomogeneity ofF1

s and2F12s (t9, t9) . 0 by the submodu-

larity of Fs.


which the owners of firms earn zero profits. Asbefore, the competitive equilibrium maximizesthe value of national output given prices.

Consider the production equilibrium for agiven relative pricep. We will show that, forsome values ofp andK, all firms in both sectorsemploy pairs of workers. Then, if preferencesand fixed costs are such thatp andK fall in theindicated range, the equilibrium is like the onewe described before.

We use the efficiency of the competitiveequilibrium to rule out firms with single work-ers. Such firms are never efficient in theCsector, so we concentrate on single-workerfirms in theS sector. No such firm can exist inequilibrium if even the most productive workerwould generate more value by toiling as part ofa matched pair in sectorC. Therefore, there willbe no single-worker firms if

(B1) p@Fs~tmax, 0! 2 K# ,lc

2tmax.

This condition is satisfied by combinations ofpandK that fall below the increasing and convexcurve CC in Figure A1. Also, single-workerfirms will not form in sectorS if even the least-productive pair could produce more net outputas a team than its members could produce sep-arately. If firms in theS sector do indeed em-

ploy pairs of workers, then efficiency dictatesmaximal cross-matching. With such pairings,there can be no team in the sector that producesless output than one with two workers of talentt#. These workers could produceFs(t#, t#) 2 K asa team, whereas they could each produceFs(t#,0) 2 K by working alone. For an absence ofsingle-worker firms, it is sufficient thatF s(t#, t#) 2K . 2[Fs(t#, 0) 2 K], or

(B2) K . 2Fs~t#, 0! 2 Fs~t#, t#!.

These are the points to the right ofSS in Fig-ure A1.

Finally, positive production of both goodsrequires that the most-productive teams in sec-tor S produce more value there than their mem-bers could produce in sectorC, while the least-productive pairs (with median talent) do theopposite. Thus, diversified production requires

(B3) p@Fs~tmin, tmax! 2 K#

.lc

2tmin 1

lc

2tmax 5 lct#

and

(B4) p@Fs~t#, t#! 2 K# , lct#.

The first of these conditions is satisfied bypoints in the figure to the right of curveAA, thesecond by points to the left of curveBB. Adiversified equilibrium with no single-workerfirms obtains for values ofp andK that satisfy(B3), (B4), and either (B1) or (B2), a nonemptyregion in (p, K) space.19 Thus, for certain val-ues of K and certain preferences, the equilib-rium has only team production in both sectors.

Suppose now that parameters are such thatboth countries are incompletely specialized andthat all firms employ two workers. In such anequilibrium, the country with the more diverseworkforce must export goodS and import goodC. The proof of this statement follows the same

19 Both AA andBB are aboveCC whereK 5 0. Bothcurves are increasing and convex. CurveBB intersectsCCexactly once, whereasAA may or may not intersectCC.

FIGURE A1. ENDOGENOUSTEAMS WITH INESSENTIAL TASKS


lines as that of Proposition 4, except that netoutput Fs(tA, tB) 2 K replacesFs(tA, tB) inthat proof.

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