offshoring in a knowledge economy* p ol a ...tion versus knowledge jobs), and (as matches are...

OFFSHORING IN A KNOWLEDGE ECONOMY

POL ANTRAS

LUIS GARICANO

ESTEBAN ROSSI-HANSBERG

How does the formation of cross-country teams affect the organization of workand the structure of wages To study this question we propose a theory of theassignment of heterogeneous agents into hierarchical teams where less skilledagents specialize in production and more skilled agents specialize in problemsolving We first analyze the properties of the competitive equilibrium of themodel in a closed economy and show that the model has a unique and efficientsolution We then study the equilibrium of a two-country model (North andSouth) where countries differ in their distributions of ability and in which agentsin different countries can join together in teams We refer to this type of integra-tion as globalization Globalization leads to better matches for all southern work-ers but only for the best northern workers As a result we show that globalizationincreases wage inequality among nonmanagers in the South but not necessarilyin the North We also study how globalization affects the size distribution of firmsand the patterns of consumption and trade in the global economy

I INTRODUCTION

A number of recent technological1 and institutional2 devel-opments have blurred the borders between national labor mar-kets and have allowed for the formation of international teamsThese developments have altered what teams of agents can do ata distance Some tasks such as data entry in consumer bankingsoftware upgrades and maintenance low-level customer handlingin call centers or standardized manufacturing processes are nowfrequently done offshore Other more knowledge-intensive tasks(such as data manipulation software development higher-endsales and service and RampD and product design in manufacturingindustries) continue to be undertaken domestically Broadly rou-tine tasks are offshored while more complex tasks are donedomestically Thus the traditional vertical division of laborwithin a team whereby some low skill agents (workers) under-

We thank several colleagues and seminar participants at various institu-tions as well as the Editor and two anonymous referees for very useful comments

1 Improvements in information technology have reduced the cost of inter-national data and voice transfer from prohibitively expensive to levels that arevirtually identical to within-country communication costs

2 Recent political and economic reforms in China India and Eastern Europehave substantially liberalized economic activity Meanwhile the worldwide de-regulation and competition in the telecommunications industry has contributedsubstantially to the drop in communication costs

copy 2006 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics February 2006

31

Pantras
Text Box
Reprinted from the QUARTERLY JOURNAL OF ECONOMICS published by the MIT Press copyright copy 2006 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology All rights reserved

take routine tasks and the highest skill agents (managers) spe-cialize in knowledge-intensive tasks can now take place acrosscountries

In our view what is important about this new division oflabor is that it alters the feasible matches between agentsrsquo skilltypes High skill agents in more developed countries can leveragetheir knowledge at lower cost by working with cheaper labor onroutine tasks and the better workers in less developed countriesare able to become part of international high value added teams

In this paper we present a simple framework that puts agentskill heterogeneity and matching at the center of the analysis Byallowing us to analyze changes in matching and in the supportingearnings functions our framework allows us to examine theimpact of offshoring on wages on occupational choices (produc-tion versus knowledge jobs) and (as matches are ldquomany-to-onerdquo)on the distribution of firm sizes

We model an economy in which production requires physicalinputs and knowledge and where a continuum of agents withheterogenous abilities sorts into teams competitively Agents ofdifferent skill levels form teams Less skilled agents (workers)specialize in production work and deal with routine tasks whilethe most skilled agents specialize in knowledge-intensive tasks(managers) Relative to less skilled managers better managersare able to increase more the productivity of all the workers intheir team as they are able to solve a wider range of the problemstheir team confronts in production Better production workersallow individuals to manage larger teams as workers can solvemore problems by themselves and require less help This resultsin a complementarity between manager and worker ability thatdetermines the identity of agents working and managing differ-ent teams It also determines through comparative advantagethe occupational choice of agents More able agents althoughmore productive as production workers want to set up their ownfirms and manage their own teams of workers instead of workingfor other managers

To study the impact of the formation of international teamsin this economy we study a simple one-sector two-country modelin which countries differ only in their skill distributions In par-ticular one country the North has a distribution of skills with arelatively high mean while the other country the South has adistribution of skills with a relatively low mean In our model theldquoskill overlaprdquo implied by these skill differences is captured by a

32 QUARTERLY JOURNAL OF ECONOMICS

single parameter which plays a crucial role in the analysis Theother key parameter in our model is the cost of communicatingknowledge within teams (ie the state of communication tech-nology) which determines the extent to which managers canleverage their knowledge via larger teams

We initially study the case in which cross-country teams areprohibitively expensive so that the equilibria in the North and inthe South correspond to those of two closed economies We thencompare these equilibria to that of a perfectly integrated inter-national economy where cross-border teams are as expensive aslocal ones We refer to this type of integration as ldquoglobalizationrdquo

We first show that globalization leads to the formation ofinternational teams in which northern managers supervise teamsof southern workers offshoring Offshoring thus allows for thegeographic separation of production and problem solving and thedelocation of physical production toward the South It leads to thecreation of routine jobs and an increase in production in theSouth and to the creation of knowledge-intensive jobs or firmsand a decrease in production in the North This implies that thepattern of trade is such that the South is a net exporter ofphysical goods while the North is a net exporter of knowledgeservices

Globalization also affects the level and structure of earningsof individuals both in the North and in the South We first showthat our model is consistent with the empirical regularity thatsouthern workers employed in multinational firms receive wagesthat are on average higher than those received by workers em-ployed in domestic firms (see Aitken Hanson and Harrison[1996] for empirical evidence) We next analyze how globalizationaffects income inequality within each of the two countries Weshow that globalization leads to an increase in within-workerwage inequality that is wage inequality among nonmanagers inthe South This prediction is consistent with the findings of severalempirical studies (eg Feenstra and Hanson [1997] Goldberg andPavcnik [2004] and references therein) These findings have re-ceived considerable attention in the international trade literaturesince they cannot be easily rationalized with standard factor propor-tions trade frameworks Our theory predicts an increase in within-worker inequality in the South as a result of changes in matchingglobalization improves the quality of the managers with whomsouthern workers are matched thus raising the productivity ofthese workers and thereby leading to an increase in their marginal

33OFFSHORING IN A KNOWLEDGE ECONOMY

return to skill This effect is reinforced by an occupational choiceeffect more agents become workers hence increasing the range ofabilities in the worker skill distribution

The effect on within-worker wage inequality in the North ismore complicated On the one hand low skill workers in theNorth face increased competition from southern workers and thistends to reduce their marginal return to skill On the other handour model highlights a new force leading to an increase in within-worker wage inequality in the North When more low skill agentsare available the time of high skill managers becomes morescarce and workers who are better able to economize on this timebecome relatively more valuable As a result the value of moreskilled workers relative to less skilled ones increases as does thedifference between the ability of the managers they are matchedwith When either the skill overlap or communication costs aresufficiently low so that high skill managers are particularlyvaluable and scarce this last effect dominates and globalizationincreases wage inequality not only in the South but also in theNorth Conversely when communication costs and the skill over-lap are sufficiently large the former effect dominates and off-shoring is associated with lower wage inequality in the NorthThis may help rationalize the findings of Feenstra and Hanson[1996b 1999] that offshoring raised wage inequality in theUnited States in the 1980s but not in the 1970s Our theorysuggests that these findings can be explained by lower commu-nication costs and deeper trade integration with less developedcountries in the 1980s than in the 1970s3

Which firms engage in offshoring When the skill overlap islarge and communication costs are high only the most productiveand larger firms will engage in offshoring while when the skilloverlap is small and communication costs are low the firms thatengage in offshoring will actually be the least productive firmsthose controlled by the lowest skill managers More generally weshow that the ldquoqualityrdquo of offshoring as measured by the averageskill level of the workers that form international teams relative tothe skill level of all southern workers is weakly increasing inboth the skill overlap and communication costs At the same timewe show that the ldquoquantityrdquo of offshoring as measured by the

3 The ability of our model to deliver the level of income for all agents in theeconomy also allows us to identify the winners and losers from globalization Inparticular in subsection IVB we show that there is always a subset of workerswho are hurt by globalization


proportion of southern workers who work for northern managersis instead weakly decreasing in both communication costs and theskill overlap and converges to zero as the skill distributionscompletely overlap We also study how occupational choices thesize distribution of firms and wage inequality are affected bythese same parameters

One of the advantages of our approach is that offshoring isnot only the result of the relative aggregate supply of skills butrather follows from the competitive sorting of agents with differ-ent skill levels into teams Paraphrasing Sattinger [1993] wagesin the economy play an allocative role rather than simply beingrewards for the possession of particular characteristics This al-lows us to derive conclusions on the characteristics of offshoringfirms as well as on the distribution of wages Most other efforts tounderstand offshoring do not have this feature Feenstra andHanson [1996a 1997 2003] for example assume factor endow-ments of skilled and unskilled workers in the North and Southand a production function that uses these inputs to produce eitherintermediate or final goods4 In these models offshoring is theresult of foreign direct investment and leads to changes in wageinequality as a result of changes in the sectoral composition ofproduction Their work is important in that it determines thechanges in wages due to these sectoral (in inputs or output)compositional changes5 In general however it is silent aboutchanges in wage inequality within narrowly defined sectors aswell as on the cross-sectional characteristics of offshoring firms6

Our paper is closely related to the work of Grossman andMaggi [2000] and Kremer and Maskin [2003] in that they alsostudy the relationship between patterns of trade and patterns ofmatching between the skill of different workers7 Grossman and

4 Feenstra and Hanson [1997] assume that the supply of skill and unskilledlabor does respond to relative wages but they do not model the occupationalchoice decision or the sorting of agents into production teams

5 Feenstra and Hanson [2003] stress that these changes in the sectoralcomposition of production may occur within industries and may therefore not bepicked up by industry-level price indices (cf Lawrence and Slaughter [1993])

6 Other papers have developed frameworks with similar characteristics thatalso abstract from the dimensions that we focus on in particular Acemoglu[2003] Bernard Robertson and Schott [2004] Zhu and Trefler [2005] and Ver-hoogen [2004]

7 Nocke and Yeaple [2004] present an assignment model of FDI but focus onthe matching between brands of different quality and entrepreneurs of heteroge-neous ability Our paper is also related to a branch in the literature that hasstressed the importance of heterogeneity in understanding the differential impactthat globalization may have on different types of firms or workers (eg Manesse


Maggi consider the consequences of different types of productionfunctions involving substitutability or complementarity in skillsfor the patterns of specialization and trade A maintained as-sumption in their analysis is that international teams are notallowed to form

Our work is more closely related to Kremer and Maskin[2003] who study the patterns of trade and wages that resultfrom production functions that are characterized by complemen-tarity between inputs and imperfect substitutability betweenthem Consistent with any production function that may hope toaddress within-worker wage inequality the production functionwe study involves skill complementarity imperfect substitutabil-ity between workersrsquo skill and differential sensitivity to the skillof different workers (see Kremer and Maskin [1997]) Our modelhowever is novel in four key dimensions First following Gari-cano and Rossi-Hansberg [2005 2004] it is the only one to involvehierarchical one-to-many matching (rather than one-to-onematching) where a manager is endogenously matched with apotentially large number of workers and can potentially raise theoutput of all of them Second the identity of managers andworkers is endogenous and is the result of an occupational choicedecision Third the actual team production function results nat-urally and endogenously from a production process that does notassume skill complementarities but rather derives them from thespecialization of agents in different aspects of the processmdashpro-duction and knowledge Fourth the relation between the skill ofthe manager and that of the worker is mediated by communica-tion technology that is the state of communication technologydetermines the extent to which a manager can leverage hisknowledge by communicating it to many or few production work-ers As a result of these differences we are able to move beyondprevious contributions in formally analyzing how the process ofglobalization interacts with the state of communication technol-ogies in determining the worldwide organization of productionand the structure of rewards that support it8

and Turrini [2001] Melitz [2003] Helpman Melitz and Yeaple [2004] Antrasand Helpman [2004] and Yeaple [2005])

8 For previous equilibrium models of the allocation of heterogeneous agentsto hierarchical teams but which do not involve matching between workers andmanagers see Lucas [1978] Rosen [1982] and Waldman [1984]


Our paper differs from Garicano and Rossi-Hansberg [2005]in that we simplify the analysis by taking the skill level of agentsas exogenous and we limit team sizes to two layers The payofffor this simplification is that we provide a closed-form solution tothe model and most importantly we are able to study the rela-tionship between matching and wage inequality across countries

The rest of the paper is organized as follows Section IIpresents the basic framework for a closed economy and showsexistence uniqueness and optimality Section III constructs anequilibrium in the integrated economy and discusses its basicproperties Section IV discusses the effects of economic integra-tion or globalization Section V presents comparative statics withrespect to communication costs and the skill overlap and SectionVI concludes All the proofs in the paper are relegated to theAppendix

II THE MODEL

Agents are endowed with one unit of time and a skill level zThe distribution of skills in the population is given by a cumula-tive distribution function G( z) with density g( z) that for themoment we will assume has support in [0z ] with z 1 Agentsrank consumption according to a linear utility function so theyare income maximizers given that we normalize the price of theonly good in the economy to one

Our theory of the organization of production follows Garicano[2000] Production is done by teams with one manager and pro-duction workers Workers spend their time producing Whileproducing they face a problem that has to be solved for produc-tion to happen If a worker knows the solution to her problem shesolves it and produces one unit of output If she does not know thesolution she can ask her manager If the manager knows thesolution to the problem the manager solves it immediately com-municates the solution to the worker and she produces one unitof output The manager spends 0 h 1 units of time commu-nicating what she knows to the worker no matter if she knows thesolution to the problem or not The skill level of an agent deter-mines the set of problems she can solve An agent with skill z cansolve all problems that require knowledge between 0 and z Wenormalize the set of problems so that the skill level z is also the


proportion of problems an agent can solve9 Hence a manager ina team with n workers of skill zp faces the following timeconstraint

h1 zpn 1

and so can deal with n( zp) 1[h(1 zp)] workers10 Productionin a team formed by a manager with skill level zm and workerswith skill zp is therefore given by zmn( zp) Given wages manag-ers choose the ability of their workers to maximize rents11

(1) R zm maxzp

zm wzpnzp maxzp

zm wzp

h1 zp

subject to

h1 zpn 1

The first-order condition of this problem is given by

(2) w zp zm w zp

1 zp

Agents choose whether to become managers or workers so asto maximize their utility that is their income Hence given theirability z they solve max R( z) w( z) This implies that theearnings function the envelope of the wage and rent functionsgiven occupational choices will be continuous

In equilibrium labor markets clear Namely at the equilib-rium wages and earnings the supply and demand of productionworkers equalize at all skill levels Let w be an equilibriumwage function and let the equilibrium occupational choice deci-sion be such that agents with skill levels in [0z] become work-ers and agents in [ zz ] become managers Agents with knowl-

9 The upper bound z thus represents the fraction of problems that themost-skilled agent in the economy can solve

10 In principle the interpretation of our technology given in the text re-quires us to address the stochastic element in the arrival of problems which couldresult in congestion and queuing Doing so would not we believe add to theeconomics of the question at hand An alternative interpretation that circum-vents the need to address these issues is that each worker draws a continuum ofproblems of measure one Workers then solve the problems that they can giventheir skill level and ask managers for help on the measure of problems that theydo not know how to solve Then h would be interpreted as the time cost for amanager of helping on a measure one of problems

11 Note also that we have assumed that a manager with ability zm hiresworkers of homogeneous ability zp In Antras Garicano and Rossi-Hansberg[2005] we generalize the technology and show that this assumption is without lossof generality


edge z are indifferent This restriction on equilibrium occupa-tions turns out to be without loss of generality as Theorem 1below shows Let m( z) be the skill level of the manager of aworker with ability z We prove in Theorem 1 that an equilibriumallocation of this economy has to satisfy positive sorting andtherefore that m is invertible Then labor market clearing im-plies that

0

zp

g zdz m0

m zp

nm1 z g zdz for all zp z

where m1( z) is the ability of the workers hired by a manager ofability z The left-hand side of this equation is the supply ofworkers between 0 and zp The right-hand side is the demand forworkers by managers between m(0) z and m( zp) Marketclearing is guaranteed when supply equals demand for every skilllevel of workers zp z Substituting for n and deriving withrespect to zp we obtain that as long as z z and m( z) isincreasing (positive sorting)

(3) m z h1 zg z

gm z

Notice that in this economy positive sorting is always guaranteedbecause of the complementarity between workersrsquo and managersrsquotalent (see equation (1)) Hence better workers always work forbetter managers a property we will exploit intensively belowThis differential equation together with the two boundary con-ditions m(0) z and m( z) z determines the equilibriumassignment function Notice that the equilibrium assignment ofworkers to managers can be determined without knowing wagesand rents once positive sorting is established12 Note also thatsince managers lead teams with many workers matching ismany-to-one This implies that m although single valued has aslope smaller than g( z)g(m( z)) and so a given mass of workersis matched with a smaller mass of managers

12 Of course equilibrium wages and rents sustain this assignment as anequilibrium allocation We can compute assignments independently of wagessince the span of control of managers is a technological restriction of the problemManagers add agents to their teams until they do not have any time left If agentscould acquire skill or could work by themselves this helpful property of oureconomy would be lost and the analysis would be much more complicatedGaricano and Rossi-Hansberg [2005 2004] present closed economy frameworksthat incorporate these dimensions


A competitive equilibrium in our economy is therefore givenby a wage function w a rent function R an assignment functionm and occupational choice decisions (summarized by z) suchthat managers maximize rents ((2) is satisfied and w( z) R( z))13 agents maximize utility (w( z) R( z)) and labormarkets clear ((3) is satisfied together with m(0) z andm( z) z ) The following theorem shows that an equilibrium ofthis economy exists as long as h is lower than a threshold h14 Italso shows that if an equilibrium exists it is unique efficientexhibits positive sorting and can be characterized by a thresholdz as we have done so far On top of this we can show in generalthat the earnings function max R( z) w( z) is strictly convexin z

THEOREM 1 There exists a threshold h 0 such that if h [0h] there exists a unique competitive equilibrium of thiseconomy In equilibrium the set of managers and the set ofworkers are connected the equilibrium exhibits positive sort-ing and the earnings function is strictly convex Further-more the equilibrium allocation is efficient

IIA Equilibrium in the Closed Economy

Consider a world formed by two independent economieswhere agents can only form teams with other agents in the sameeconomy The first one that we call the North is exactly asdescribed before but with a uniform distribution of skills in thepopulation GN( z) z for z [01] with density gN( z) 1 Inthe North the best agents of the economy can therefore solve allthe problems that arise in production The second economy thatwe call the South also has a uniform distribution of skills but thesupport of the distribution is the interval [0] for 1 withGS( z) z for z [0] and density gS( z) 1 The bestagents in the South can thus solve only a fraction of theproblems that they face while producing The North is therefore

13 The second condition is needed to guarantee that managers at z do notprofit from hiring agents with abilities slightly above z The condition is neces-sary given that (2) only holds for z (0z) but not for z Garicano andRossi-Hansberg [2005] show that this condition would always be satisfied if wewere to allow agents to produce individually as well as in teams

14 The reason that we need to restrict h for an equilibrium to exist is that wedo not allow agents to be self-employed If we were to allow them to work on theirown we could guarantee existence for all 0 h 1 In the rest of the paper wewill analyze the case in which we specify the distribution of abilities to be apiecewise uniform density and in this case we can show that h 085


relatively better endowed with skilled agents but both countriesare identical in all other respects including population size Wewill often refer to the parameter as the skill overlap The choiceof a uniform assumption has the virtue of allowing us to solve thewhole model analytically15 It also implies that h 085 and sobelow we will focus only on h [0085]

The northern economy is just a special case of the southerneconomy when is equal to 1 Hence we start by describing anequilibrium in the South All the expressions are identical for theNorth if we substitute 1 Using (3) and the boundary condi-tion mS(0) zS we obtain that

(4) mS z zS hz1 1frasl2 z

and using mS( zS) we can solve for the threshold ability zS

(5) zS 1 h 1 h2 2h1

h

That is all agents with skill between 0 and zS become workersand all agents with skill between zS and become managers Itis easy to show that zS increases as communication technologyimproves that is as h declines Intuitively now managers canhave larger teams so in equilibrium there are fewer managersand more workers In an economy with more skilled agentslarger zS is higher There are two forces that determine thiseffect First as increases and therefore the density 1 de-creases given the size of teams agents with higher skill decide tobecome workers Second the best agents manage larger teamswhich reduces the set of managers and increases the set of work-ers Thus an economy with higher or lower h is an economy inwhich the skill levels of the agents that become workers is more

15 The distribution of skills across countries depends on both the distribu-tion of innate ability and the technology available to transform endowed innateability into skills used in production Start with the same innate distribution ofability in both countries but let economic institutional or cultural factors resultin different technologies to transform ability into skill Suppose that an agent withability z that goes through say the compulsory education system obtains eiz unitsof marketable skills i S N where eN eS if the North has a better educationsystem Then a uniform distribution of innate abilities in both countries withnormalized support in [01] leads to our assumption on the distributions of skillsafter normalizing eN 1 and letting eSeN 1 Generally any technologythat results in skills that are proportional to the level of ability will implydistributions of skills with supports that overlap only for low skill levels Some ofour results will depend on this property of the distribution of skills See Garicanoand Rossi-Hansberg [2005] for a framework in which agents can choose theamount of skills given their innate ability


dispersed This higher skill dispersion will lead to higher mea-sured within-worker wage inequality We call this effect the oc-cupational choice effect

A characteristic of this equilibrium is that because of posi-tive sorting more skilled managers lead teams with more skilledworkers Since the size of a firm is uniquely determined by theskill levels of its workers and by an economywide parameter hhigher skilled agents work in larger firms Because managers ofthese firms have more skill they solve a larger proportion of theproblems they face and so these firms are more productive theaverage product of labor is higher As we will now see this willresult in both managers and workers in these teams earningmore per unit of skill the wage and rent functions will be convexin the level of skill (see also Theorem 1 above)

Equation (2) together with wS( zS) RS( zS) implies thatthe equilibrium wage function is given by

(6) wS z zS S1 z 1frasl2 hz2

where

(7) S hzS1 1frasl2hzS

1 h hzS

The slope of the wage function the marginal return to skill forworkers is thus given by

wS z S hz

Hence the wage function is convex the marginal return to skillincreases with the skill level This force is captured by the qua-dratic term 1frasl2hz2 and it reflects the imperfect substitutabilitybetween workers of different skillmdashthe skill price per unit of skillvaries with the skill level Throughout the paper we refer to thisforce as the complementarity effect There is a second determi-nant of the marginal return to skill the one given by S which isdetermined by the supply and demand of worker skill in equilib-rium the competition effect

The marginal return to skill can be shown to be an increasingfunction of h As communication costs decrease given the thresh-old zS team size increases Since the difference between the skilllevels of the managers of two different workers will be smaller thelarger are team sizes complementarity between worker and man-ager skills implies a decrease in the marginal return to skill a


decrease in the complementarity effect This is reinforced by asecond effect resulting from lower communication costs reducingthe demand for workersrsquo skills which reduces their baselineprice a decrease in the competition effect

The marginal return to skill is also increasing in since Sis an increasing function of zS which in turn increases with Inthis case the complementarity effect is unchanged In contrastthe competition effect increases since agents are more skilledthere are too few workers per manager at the old threshold whichrequires raising workersrsquo return to skill in equilibrium Againworkers are matched with better managers and this increasesthe returns to their own skill

After solving for the distribution of wages we turn next tothe analysis of managerial rents From equation (1) managerialrents are given by

RS z z wSmS

1 z

h1 mS1 z

Using the envelope condition the marginal return to skill formanagers is given by

RS z 1h1 mS1 z

Given that the assignment function is increasing (positive sort-ing) the rent function is convex the marginal return to skill formanagers increases with their skill level (see Theorem 1 above)Note also that the marginal return to skill for managers is equalto the number of workers in their team Hence every time wederive conclusions about firm size the same applies for the mar-ginal return to skill of managers

A worker of ability z works for a manager with ability m( z)This means that the total output produced by this worker is givenby m( z) Total production in the South is therefore given by16

(8) YS 0

zS

mS z gS zdz 1

6zS26 3h hzS

It is easy to verify that YS decreases with h and increases with The reasoning is simple the larger is h the higher are commu-

16 Equivalently output may be calculated as the integral over managerialskill of the production function n(m1( z)) z Both expressions yield the sameresult as one results from a change of the variable of integration in the other


nication costs the less managers can leverage their knowledgeand the lower is the implied average productivity As increasesthe average skill level in the economy increases which also leadsto larger output

III EQUILIBRIUM IN THE WORLD ECONOMY

Consider a world economy formed by the two countries de-scribed above North and South In the world equilibrium agentscan form production teams with agents in their own country orwith agents in the other country We assume that the cost ofcommunicating the solution to a problem h is the same whethercommunication happens between agents in the same or in differ-ent countries17

The equilibrium in the world economy is similar to an equi-librium in the individual countries once we adjust the distribu-tion of talent in the population The distribution of skills in theworld population is given by the sum of the distribution of skillsin the South and in the North (and so it is not a probabilitydistribution since it integrates to 2) namely

gW z (1 ) if 0 z 1 if z 1

The construction of an equilibrium in this economy parallels theone for a closed economy with one caveat Since the density ofskills in the world is not continuous the derivative of the assign-ment function is not continuous However an equilibrium alloca-tion must be such that the earnings function is continuous anddifferentiable for all z except at the threshold that divides work-ers and managers at which it is not differentiable just as in theclosed economy If this condition was not satisfied some manag-ers and workers would have incentives to form new teams Thisimplies that the assignment function is continuous and Theorem1 applies unchanged for the world economy

17 We could add an extra cost of communicating with agents in anothercountry However this extra cost would then influence the formation of interna-tional teams directly and would open a wedge between wages in different coun-tries thereby greatly complicating the analysis of the economic forces in theequilibrium of our setup Furthermore this added complexity would be associatedwith relatively small gains in terms of new results or economic insights unless weallowed for multiple layers of management within a firm We develop some ofthese extensions in Antras Garicano and Rossi-Hansberg [2006]


Depending on the value of h and we can show that thereare two types of equilibria in the world The first one is anequilibrium in which all agents in the South are workers Sincethere are no managers in the South all of them work for northernmanagers That is they all work in international teams Positivesorting implies that because they are the lowest quality workersin the world (there is an identical set of workers in the North plussome more skilled ones) they work for the worst managers in theNorth Hence international teams are associated with the worstworkers and managers in the world We call this the Low QualityOffshoring Equilibrium (LQE)

The second type of equilibrium is one in which some of theagents in the South are managers This equilibrium features theless skilled workers in the South working for southern managersand the more skilled ones working for the best managers in theNorth We call this the High Quality Offshoring Equilibrium(HQE)18 All our results are derived under the assumption thatinternational teams are formed only if managers strictly prefer tohire foreign workers than domestic ones19

In general the set of parameter values that determines theboundary between these two equilibria is a nonlinear function ofh and that we will determine below and which we plot inFigure I We analyze each equilibrium in turn

IIIA Low Quality Offshoring Equilibrium

Denote by zWL the threshold that separates the ability of theagents who choose to be workers or managers in a LQE In orderfor the world equilibrium to be a LQE it must be the case that zWL (ie all agents in the South are workers) For anassignment to satisfy the world labor market equilibrium condi-tion it has to satisfy (3) or in this case

m z ((1 ))h(1 z) if 0 z h1 z if z zWL

Equilibrium in the labor market also implies that m(0) zWLand m( zWL) 1 In order for the wage function to be differen-tiable (see (2)) the assignment function has to be continuous at

18 In Section V we will define a precise measure of offshoring quality and wewill show that in a LQE the quality of offshoring is always lower than in a HQEthus justifying the names chosen for the two types of world equilibria in our setup

19 We are effectively selecting the equilibrium with the least amount ofoffshoring This is analogous to the approach in Helpman [1984]


all z zWL and in particular at This characteristic of theequilibrium allocation provides another boundary condition of theproblem Using the two differential equations and the threeboundary conditions we can solve for the assignment function

(9) mWL z zWL ((1 ))hz(1 1frasl2 z) if 0 z

zWL h(1 1frasl2 ) hz(1 1frasl2 z) if z zWL

as well as for the threshold

(10) zWL 1 h 1 h23

h

Again simple differentiation verifies that zWL the set of workersin the world decreases with h and increases with where theintuition is similar to the one for the closed economy Note thatthe assignment function is continuous but not differentiable at

In order for the world to be in a LQE we need to guaranteethat zWL or

(11) h 21 2 2

The right-hand side of the inequality is decreasing in and isequal to zero for 1 and equal to one for 0 This condition

FIGURE ITypes of Equilibria


with equality is the curve that separates the parameter setwhere we obtain each equilibrium and that was plotted in Figure I

Maximization of rents by managers implies that wages haveto satisfy (2) Furthermore in order for agents not to have incen-tives to join other firms in the economy which would be willing tohire them we also know that the earnings function has to becontinuous In particular the wage function has to be continuousat and wages and rents have to be equal at zWL The lattercondition is given by wWL( zWL) RWL( zWL) Combining allthese conditions we obtain

(12) wWL z

zWL 1L(1 z) 1frasl2 ((1 ))hz2 if 0 z

zWL h(1 1frasl2 ) 2L(1 z) 1frasl2 hz2 if z zWL

where

(13) 1L hzWL1 h 1frasl2 hzWL 1frasl2 h2

1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL

for h h (see the proof of Theorem 1) Hence the earningsfunction has a kink a nondifferentiability at zWL This impliesthat given that the wage and rent functions are convex themarginal return to skill is larger for managers than for workers

Figure II summarizes what we have discussed about a LQEAgents with skill in [0] in the South and North work for north-ern managers with skill in [ zWLmWL()] Agents in the Northwith skill in [zWL] work for managers in the North with skill in[mWL()1] The wage function of all workers in the world andthe rent function of northern managers is a continuous anddifferentiable function of skill The marginal return to skill ofmanagers is larger than that of workers

IIIB High Quality Offshoring Equilibrium

A HQE is such that the highest skilled agents in the Southdecide to become managers If we denote by zWH the threshold


that divides occupations in a HQE it must be the case that zWH Positive sorting implies that since managers in the South aresome of the lowest skill managers in the world they are matchedwith the lowest skilled agents In particular they are matchedwith agents in the set [0z] where the threshold z is defined bythe worker type that works for the best agent in the Southnamely m( z) Agents with skill lower than z work formanagers in their own country (since we focus on the equilibriumwith the least amount of offshoring) and workers with skillgreater than z work in international teams Then labor marketclearing implies that

mWH z h(1 z) if 0 z z

((1 ))h(1 z) if z z zWH

which restates condition (3) for this case together with the sameboundary conditions as in the LQE m(0) zWH and m( zWH) 1 On top of this we have to guarantee again that the equilibriumassignment function is continuous in particular at z in order forthe wage function to be differentiable These conditions thenresult in an equilibrium assignment function given by

FIGURE IILow Quality Offshoring Equilibrium


(14) mWH z zWH hz(1 1frasl2 z) if 0 z z

zWH (1)hz(1 1frasl2 z) ((1 ))hz(1 1frasl2 z) if z z zWH

and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h

One can verify again that zWH is decreasing in h and increasingin Using the definition of z we also obtain that

(16) z 1 1 2 zWH

h

It is straightforward to show that the condition that ensures thatthis world equilibrium is a HQE (ie zW ) is the reciprocal ofcondition (11)

Again maximization of rents implies that condition (2) has tobe satisfied together with wWH( zWH) RWH( zWH) and continu-ity of wages at z Solving the two differential equations weobtain that

(17) wWH z

zWH 1H(1 z) 1frasl2 hz2 if 0 z zzWH (1)hz(1 1frasl2 z) 2H(1 z) 1frasl2 1 hz2 if z z zWH

where

(18)

1H hzWH1 1hz 1 2hzWH 1 2hz

2

1 h hzWH

and

2H 1H h z

As in the LQE it is easy to show that the marginal return to skillis increasing in the level of skill and is larger for managers thanfor workers

Let us summarize what we have shown for the HQE usingFigure III Agents with skill in [0z] work in national firms formanagers with skill in [ zWH] Agents with skill between z


m1() and zWH work for northern managers with skill in [1]This set of managers includes the ones that manage internationalteams As before the earnings function is continuous and differ-entiable everywhere except for zWH in which its slope increasesdiscreetly

IV EFFECTS OF GLOBALIZATION

We study here the impact of an exogenous policy or techno-logical change that we call globalization and that allows for theformation of international teams We analyze its effects on thecomposition of teams occupational choices and the rewardsstructure of the economy To do so we compare the world equi-librium of Section III with the autarkic equilibria in the Northand South that we described in Section II

IVA Matching Occupational Choice and Firm Characteristics

To compare the open and autarkic equilibria we use FigureIV The figure presents the matching functions in autarky and

FIGURE IIIHigh Quality Offshoring Equilibrium


FIGURE IVaMatching before (mN( z) mS( z)) and after (mW( z)) globalization in a LQE

FIGURE IVbMatching before (mN( z) mS( z)) and after (mW( z)) globalization in a HQE


the two types of world equilibria The blending of the two skilldistributions produces a rearrangement of the matches for bothnorthern and southern workers Independently of the equilibriumwe are studying all workers in the South strictly improve theirmatch This is the case even for southern workers who do notmatch with international managers since some southern man-agers become workers in international teams and the absence ofthese managers increases the quality of the match of everyworker Agents who were managers before globalization mayeither become workers (as there is a supply of higher qualitymanagers who can do their problem solving job better) or remainas managers In the latter case they are matched with lower skillworkers precisely because some of the southern managers whowere previously managing low quality workers have becomeworkers and the remaining managers are left to hire lower qual-ity agents In other words while workers always benefit from thehigher quality managers available for matching managersrsquomatches suffer from the increasing competition of better interna-tional managers

The picture is considerably different for workers in theNorth The key change is in the opportunities of the middle-skilled agents in the North Previously they were not ldquogoodenoughrdquo to be team managers After globalization there is a set oflow-skilled agents who need managing As a result some of thesemarginal workers become managers of low skilled agents Thisimplies that matches of northern agents with sufficiently low skillnecessarily become worse However the highly skilled workers inthe North now have less competition since some of their highlyskilled competitors particularly the ones who were previouslymatched with the best northern managers have become manag-ers Hence as we show formally in Proposition 1 and illustrate inFigure IV there is a skill level below which workers have worsematches while above it northern workers improve their matchesThe following proposition formalizes these results20

PROPOSITION 1 Globalization has the following effects on teamformation

20 When the distinction between LQE and HQE is not relevant we denotevariables in the world economy with a subscript W We follow this notation for allvariables and functions


(i) The mass of southern workers and the mass of northernmanagers both increase ie zS zW zN

(ii) (a) Southern workers who were already workers arematched with a better manager

(b) Southern managers who remain managers arematched with worse workers

(c) Southern managers who become workers arematched with a northern manager

(iii) (a) There exists a unique threshold such that allnorthern workers who remain workers with z are matched with a worse manager while thosewith z are matched with a better manager

(b) All northern managers who were already managerswith z mW() are matched with a better workerwhile those with z mW() are matched with aworse worker

Part (i) of Proposition 1 implies that globalization leads to thecreation of routine (worker) jobs in the South and to their de-struction in the North Similarly if firms are identified by themanagers who run them we can conclude that globalization leadsto firm destruction in the South and to firm creation in the NorthParts (ii) and (iii) in turn imply that the size of the largest firmsin the North decreases while some other firms will becomelarger since some managers in the North improve their match Incontrast all businesses based in the South that remain aliveshrink This leads to the following corollary

COROLLARY 1 Globalization leads to routine job creation and firmdestruction in the South and to routine job destruction andfirm creation in the North Furthermore it compresses thesupport of the size distribution of firms in both countries andreduces the size of all surviving southern firms

Proposition 1 also implies that the best workers in the Southare in international teams and thus work for the most productiveand larger firms doing business in the South This sorting mayprovide a rationale for the often-found evidence that ldquosouthernrdquoworkers employed in multinational firms receive wages that areon average higher than those received by workers employed indomestic firms (see for instance Aitken Harrison and Lipsey[1996] and Lipsey and Sjoholm [2004]) More specifically a ratio-nale for the regression result is simply that those who hold


offshored jobs are unobservably more skilled than those who donot and so they are matched with better managers21 In sum

COROLLARY 2 The best workers in the South work for northernmanagers and receive higher wages than southern workerswho are employed by southern managers

IVB Wage Inequality Returns to Skill and Wage Levels

The previous subsection focused on the implications of ourtheory for quantities Corresponding to those quantities there areequilibrium effects of globalization on prices That is workersrsquowages and managerial earnings must be such that matches arerearranged in the way we have described

We first need to propose a set of measures that will help uscharacterize the effect of globalization on the distribution ofwages and in particular wage inequality One potential measureof wage inequality is the ratio of the wage of the highest skilledagent and the wage of the lowest skilled agent (eg w( zW)w(0))The problem with this measure is that it combines the level andslope effects on the wage distribution in a way that is not alwaysstraightforward to disentangle To avoid this problem we focuson changes in the absolute difference between the wage of thehighest skill workers and that of the lowest skilled ones That iswW( zW) wW(0) We will use this measure consistently everytime we talk about wage inequality

An alternative measure of changes in wage inequality in thecontext of our model is the change in the nonlinear (quadratic)term in the wage equation This term which we refer to as thecomplementarity effect measures the premium that a workerreceives for possessing a particular skill level in excess of whatseveral separate workers would receive for possessing the sameaggregate amount of skill In other words the term reflects theextent to which workers with different skill levels are imperfectsubstitutes in production

These two measures directly relate changes in wages tochanges in matching Our preferred interpretation of the empiri-

21 Controlling linearly for the skill of workers is unlikely to solve thisproblem as earnings are the result as we showed above of the interactionbetween the skill of the worker and that of the (higher quality) internationalmanager In particular Aitken Hanson and Harrison [1996] only distinguishbetween skilled and unskilled workers and Lipsey and Sjoholm [2004] controllinearly for educational attainment of workers None of these controls eliminatesthe relationship between wages and multinationals generated by our framework


cal evidence concerning wage inequality in the South (eg Feen-stra and Hanson [1997] and Hanson and Harrison [1999]) mapsthis evidence to changes in within-worker wage inequality in ourmodel since all blue-collar (unskilled) workers but also somewhite-collar (skilled) workers should be considered workers andnot managers in our theory22 Of course some white-collar work-ers would probably be considered managers Still given the em-pirical definitions of occupations we believe that the best way tointerpret a high-skillndashlow-skill wage gap in our model is as thegap between the wages of the most and least skilled workerswhich is our measure of wage inequality We discuss for com-pleteness the impact of globalization on some overall measures ofinequality at the end of this section23

The analysis of the impact of globalization on inequalityfollows quite directly from the changes in matching First in-equality within southern workers unambiguously increases Themarginal value of workersrsquo skill is driven by the skill of themanager with whom they are matched which increases for allsouthern workers Thus the sum of the complementarity and thecompetition effects unambiguously leads to higher returns to skillin this case Moreover measured within-worker inequality willincrease even more since the mass of workers in the Southunambiguously increases (occupational choice effect)

PROPOSITION 2 Globalization increases within-worker wage in-equality in the South Furthermore it increases the marginalreturn to skill for southern workers at all skill levels

Consider next the effects of globalization on northern wageinequality Globalization decreases the quality of the match ofthose northern workers who are relatively unskilled and in-creases it for the more skilled among them As we could expectgiven that the marginal return to skill of all workers is a functionof the quality of the match the returns to skill for relatively low

22 Feenstra and Hanson [1997] and Hanson and Harrison [1999] defineunskilled workers as blue-collar workers and skilled workers as white-collarworkers White-collar workers are in charge of tasks such as management prod-uct development administration and general office tasks The latter three taskscan be interpreted as worker tasks in our model The occupational distinction inour theory is not between blue- or white-collar workers but between low levelfront-line routine tasks (including white-collar tasks such as handling the calls ata call center or processing the back office paper work for a bank) and specializedproblem solving dealing with exceptions namely managerial tasks

23 To our knowledge the effect of offshoring on these measures of overallincome inequality has not been studied in the empirical literature


skilled northern workers go down and the returns to skill for themore skilled ones go up The equilibrium effect on the marginalreturn to skill can again be decomposed in two First becausenow there is more competition from workers in the South thebaseline return per unit of skill always goes down (1W N 0) as we can verify numerically for all parameter values24

Second since there are relatively more workers with low skillin the world than in the North an increase in the skill level ofworkers increases the quality of their managers more after glob-alization Thus the complementarity effect tends to increase themarginal return to skill This second effect is larger the higherthe ability of the workers since they are part of larger teams aslong as southern and northern workers compete for the samemanager In fact for workers without skill z 0 this effect isnot present so the first effect has to dominate and the marginalreturn to skill decreases Numerically we can show that thesecond effect dominates for workers with skill above a certainthreshold and so the marginal return to skill increases for themThe threshold ability at which both effects are identical is afunction of the parameters h and The lower the moresouthern agents are being added at each skill level where workersin both countries compete and so the larger the set of abilities inwhich the complementarity effect dominates The lower h thesmaller the competition effect and so again the threshold ofabilities decreases

In order to understand the effects on wage inequality weneed to combine this reasoning with the occupational choice ef-fect In particular the fact that after globalization fewer agents inthe North become workers which reduces wage disparity Nu-merically we can conclude that wage inequality in the Northincreases when h and are small but decreases when theseparameters are large As mentioned in the introduction thisprediction is consistent with the findings of Feenstra and Hanson[1996b 1999] who reported a significant positive effect of offshor-ing on U S wage inequality in the 1980s but not in the 1970sWe summarize these results below

24 In our two-parameter model it is straightforward to analyze numericallydifferent equilibrium values for a tight grid of the whole parameter space In theworking paper version of the paper [Antras Garicano and Rossi-Hansberg 2005]we provide graphical illustrations of all the numerical results discussed below


SUMMARY 1 Globalization increases within-worker wage inequal-ity in the North if h and are sufficiently low but it de-creases it if h and are sufficiently high Furthermoreglobalization decreases the marginal return to skill of allnorthern workers with knowledge z below a threshold butincreases the marginal return to skill of all northern workersabove this threshold

We are also interested in studying the effect of globalizationon the level of wages which is the result of two effects on onehand southern workers face more competition from low skillnorthern workers on the other hand globalization improves theirmatch In contrast in the North workers also face more compe-tition but they do not always improve their match (as describedin Proposition 1) If we focus attention on the effect of globaliza-tion on the lowest skilled agents (for which the match deterio-rates in the North) we can show numerically the followingresults

SUMMARY 2 Globalization has the following effects on wages

(i) Increases the wages of low skilled southern workers forlow h and but decreases them for high h and

(ii) Decreases the wages of low skilled northern workers forlow h and but increases them for high h and

(iii) It decreases the wage of at least some low skilledagents

Our model also allows us to derive some conclusions on wageinequality among managers in both countries In particular re-member that the marginal return to skill of managers is given bythe size of their team From Proposition 1 we know that allmanagers in the South will have smaller teams and so the mar-ginal return to skill for them decreases Since there are also fewerof them within-manager income inequality in the South de-creases In the North there are two opposing forces First fromProposition 1 we know that the lowest skill managers who werein managerial positions before globalization will have largerteams but the best managers will have smaller teams Thisimplies that the return to skill of low ability managers increasesand that of high ability managers decreases Second there aremore managers in the North so the occupational choice effectleads to more income inequality among managers This reasoningleads to the following corollary of Proposition 1


COROLLARY 3 Globalization has the following effects on within-manager income inequality and on the marginal return toskill of managers

(i) Globalization decreases within-manager income in-equality and the marginal return to skill of all southernmanagers

(ii) Globalization increases the marginal return to skill fornorthern managers with knowledge z below a thresholdbut decreases it for the rest

We now turn to analyze the predictions of the model for othermeasures of inequality The changes in within-worker inequalityin the North together with compositional changes suggest thatinequality is more likely to increase at the top of the northernskill distribution than at the bottom Intuitively some northernworkers improve their matches but some others suffer fromsouthern competition and get worse matchesmdashthus the increasein within-worker inequality is moderate However globalizationincreases the share of northern agents who are managers andinequality within managers is always large since managers areable to leverage their skill over an entire team Indeed we findnumerically that inequality in the North as given by the gapbetween the earnings of the ninetieth percentile agent and thoseof the fiftieth percentile agent (the 9050 gap) increases for all hand for all In contrast the 5010 gap may or not increase Wealso find numerically that the expected increase in the 9050 gapwhere the expectation taken over all h and (assuming uniformdistributions for both parameters) is larger than the expectedincrease in the 5010 gap Both of these findings suggest thatoffshoring may provide an explanation for the empirical finding(see Murphy and Welch [2001] or Autor Katz and Kearney[2004]) that wage inequality in the United States has stabilizedat the bottom of the distribution but continues to grow at thetop25

The fact that the model allows us to characterize the wholeearnings distribution suggests that we can compute any measureof inequality and for instance we can study the effect of offshor-

25 Autor Levy and Murnane [2003] develop a model that differentiatesbetween routine and nonroutine tasks where only the aggregate units of factorsenter the production function Such a model could also potentially be applied tostudy offshoring but would miss the effect of offshoring on matching and there-fore within-group wage inequality that is central in our analysis


ing on overall inequality Finding robust results for these mea-sures is unlikely since wage levels will be affected by the relativemasses of agents at each point of the distribution and we haveshown analytically that offshoring leads to contradictory effectson inequality within managers and within workers We computethe expected changes in overall inequality that globalizationbrings about when the South is relatively unskilled (expectationsare taken over 05 and h) In the North globalization leads toan increase in the standard deviation of income (wages andrents) in the 9010 gap and in the managerworker gap (asmeasured by the difference between the mean managerial rentand the mean worker wage) In the South globalization impliesan increase in the standard deviation of income and in the man-ager worker gap but a drop in the 9010 gap The reason thatthese measures differ in the South is that for a given wagefunction the first one takes into account the relative mass ofworkers to managers while the last one does not26

IVC Production Consumption and Trade

As argued above Theorem 1 applies also to the equilibrium ofthe world economy and therefore the equilibrium is unique andefficient As a result since in the world economy we could alwaysreplicate the equilibrium in the closed economies of the North andthe South we know that in our framework there are alwayswelfare gains from international offshoring The following corol-lary summarizes this conclusion

COROLLARY 4 Globalization increases total production in theworld economy That is there are gains from trade

How are these gains distributed between the countries De-fine a countryrsquos physical output as the quantity of goods that areproduced by its workers since they are the ones who combinelabor and knowledge to produce In the South Proposition 1implies that there are more and better matched workers and sototal physical output increases In the North there are fewerworkers and some of them have worse matches We can shownumerically that for all combinations of h and physical pro-duction decreases in the North Hence in terms of physical value

26 We find that when the South is relatively skilled ( 05) the samechanges take place in the North and the South except that the managerworkergap decreases in the South


added the ldquowinnerrdquo of globalization is the South The reason whythe North produces less after globalization is that physical pro-duction does not take into account that managersrsquo rents have tobe repatriated Managers consume in their own country and theyreceivemdashas compensation for the time spent helping and commu-nicating with workers abroadmdashpart of the production of theseworkers These rents can be substantial and in fact imply thatconsumption in both countries increases and since utility is lin-ear so does welfare

This difference in consumption and production outcomes isreflected in the trade balance of these countries In particular theSouth features net exports of physical goods while the Northfeatures net exports of knowledge services27 Furthermore ifknowledge transactions are not registered as imports for theSouth and exports for the North the trade balance of the north-ern country will be in deficit and that of the southern country insurplus This deficit and surplus is however not evidence of animbalance but just the result of the potential misrecording ofknowledge transactions This reasoning suggests that some ofthese forces may be at play when we look at the trade balance ofthe United States with some of its Asian trade partners likeChina28 We summarize these conclusions below

SUMMARY 3 Globalization has the following effects on physicalproduction consumption and the trade balance

(i) It increases physical production in the South and de-creases physical production in the North

(ii) It increases consumption (and thus welfare) in bothcountries

(iii) The pattern of trade is such that the South exportsmanufactures and the North exports knowledgeservices

(iv) If knowledge transactions are not reported globaliza-tion generates a trade surplus in the South and a tradedeficit in the North

27 Indeed using mW( z) w( z) it is straightforward to show that YWS CWS 0 which of course implies that YWN CWN 0

28 Note that this misrecording will not be evident in the capital account aslong as some of the managerrsquos rents are kept offshore


IVD Generalizations

Our setup has assumed particular functional forms for thedistributions of ability We now briefly outline the robustness ofsome key results to more general distributions First we canshow that for globalization to lead to the creation of routine jobsin the South and to their destruction in the North (ie zS zW zN) all that is required is that the distribution of ability in theNorth first-order stochastically dominates that in the South (ieGN( z) GS( z) for all z)

Second we note that the complementarity effect (which isnovel in our setup) is generally given by the expression

w z hg z

gm z

m z

1 z

which highlights the close relationship between matching andwage inequality in our setup29 Furthermore if in addition tofirst-order stochastic dominance we assume that the distributionsof ability in the South and the world satisfy

(19)gS z

gS z

gW z

gW zfor z z z

then we can show that mW( z) mS( z) and w W( z) w S( z) forall z zS These conditions ensure that from the point of view ofthe South globalization improves the match of all southern work-ers (with the implications for the size distribution of firms notedin subsection IVA) and leads to an increase in wage inequality onaccount of the complementarity effect

Note that these conditions are only sufficient not necessaryand that they guarantee improved matches and a larger comple-mentarity effect for all ability levels in the South If any of theseassumptions on the distributions does not hold it may be the casethat the complementarity effect decreases with integration in theSouth To illustrate this consider the case where the distribu-tions in the North and South are both uniform but the range of

29 This characterization of the complementarity effect allows us to deriveconclusions on the wage effects of changes in population size Consider our setupwith overlapping uniform distributions of ability Suppose for example thatpopulation size in the South increases keeping the support of the distribution ofabilities constant but increasing the density by the same amount for all abilitylevels Then it is straightforward to see that the ratio gW( z)gW(mW( z)) willweakly increase while gi( z)gi(mi( z)) will be unaffected for i N S Thus thelarger the population size in the South the larger the increase that globalizationwill cause through the complementarity effect on wage inequality


abilities in the South is given by [01] and in the North by [1]so condition (19) is not satisfied In this case there exists a suchthat for some z [0zS]

w W z hgW z

gWmW z h

11

h w S z

That is integration may lead to a decrease in the complementa-rity effect Note however that in the above example it is still thecase that GN( z) GS( z) for all z and thus the patterns ofspecialization in the global economy are analogous to those dis-cussed in our benchmark case

V COMPARATIVE STATICS IN THE WORLD ECONOMY

In this section we analyze the effect of changes in communi-cation costs (h) and the skill overlap () on the equilibriumoutcome of the integrated economy For brevity we only report theimpact of these changes on occupational choices characteristics(quantity and quality) of international offshoring and matchingWe also discuss briefly some implications of these changes for thestructure of earnings in the world economy More details on thesecomparative statics are provided in Antras Garicano and Rossi-Hansberg [2005]

Occupational Choice As h decreases managers have accessto a communication technology that allows them to deal withlarger teams As a result the number of workers in the Southweakly decreases with h and the number of workers in the Northdecreases with h The number of managers in the South in turnweakly increases with h and the number of managers in theNorth increases with h As increases agents in the Southbecome relatively more skilled This implies that the number ofworkers in the South weakly decreases with whereas thenumber of workers in the North increases In contrast the num-ber of managers in the South weakly increases with whereasthe number of managers in the North decreases with Theseconclusions are implied by the following proposition

PROPOSITION 3 The skill of the worldrsquos most-skilled worker ( zW) isdecreasing in communication costs (h) and increasing in theskill overlap ()


The Quantity and Quality of Offshoring To analyze thequantity of offshoring we need a measure that captures theextent to which firms in the economy are formed by nationalversus international teams Therefore we define the quantity ofoffshoring as the proportion of southern workers that work forinternational teams In a LQE all agents in the South are work-ers in international teams so it follows that our measure ofoffshoring equals one In contrast in a HQE the quantity ofoffshoring is given by

OW zWH zzWH

The proportion of workers in international teams is always lessthan one in a HQE but converges to one as we change parametersto approach the boundary between both type of equilibria Hencein a LQE there is always more offshoring than in a HQE

We also want to derive conclusions on the characteristics ofthe firms that engage in offshoring For this purpose we definethe quality of offshoring as the average skill level of the workersthat form international teams relative to the skill level of allsouthern workers

QW zWH zzWH

In a HQE the quality of offshoring is always larger than one andagain converges to one as we change parameter values in a waythat approaches the boundary between equilibria Clearly OW 2 QW and so the same forces that cause an increase inoffshoring will simultaneously reduce its quality With thesemeasures at hand we derive the next result

PROPOSITION 4 The quantity of international offshoring is weaklydecreasing in communication costs (h) and the skill overlap() Furthermore the quality of offshoring is weakly increas-ing in h and

Matching and Firm Characteristics A fall in h improves theskill level of the worst manager and so improves the match of theworst workers z 0 Conversely the best worker before the fallin h is now matched with a worse manager Moreover we canshow that the direct effect of the technological improvement onfirm size dominates these effects thereby leading to an increasein the size of all firms in the economy In contrast changes in have distinct effects on small versus large firms As increasesworkers with low skill levels will be matched with better manag-


ers and will thus work for more productive firms Workers withhigh skill levels will be matched with worse managers and so theproductivity of the firms for which they work will decrease Weformalize these conclusions in the next proposition

PROPOSITION 5 A decrease in communication costs (h) has thefollowing effects on matching and firm size

(i) It improves the match for workers below a thresholdskill level 13 while it worsens the match for workers(who were already workers) above 13

(ii) It increases the size of all firmsAn increase in the skill overlap () has the following effectson matching and firm size

(i) It improves the match for workers below a thresholdskill level while it worsens the match for workers withskill above this threshold

(ii) It increases the size of the largest firms and decreasesthe size of the smallest firms

(iii) It increases the size of the largest offshorers and de-creases the size of the smallest offshorers

(iv) It increases the size of all nonoffshorers in a LQE butdecreases the size of all nonoffshorers in a HQE

Managerial Rents and Wages Concerning managerial rentsa decrease in h increases the marginal return to skill for manag-ers since all firms grow and the marginal return to skill ofmanagers is equal to firm size The effect of on firm size is notthe same for all firms We can however conclude that the mar-ginal return to skill of the worst managers decreases and themarginal return to skill of the best managers increases

Now consider the effect of changes in h and on the wagestructure First the complementarity effect (the quadratic termin both (12) and (17)) increases with h and weakly decreases with The reason is that as h increases team sizes decrease and sohaving more skill will imply matching with a much better man-ager In contrast an increase in leads to a decrease in densityfor all skill levels in the South (and thus for the levels in whichnorthern and southern workers compete) Hence a slightly betterworker will now match with only a slightly better managerSecond h has an ambiguous effect and increases the competi-tion effect as measured by the baseline unit of skill (1L and1H) The latter effect follows from increasing workersrsquo skills in


a LQE and making them more scarce in a HQE Third as shownin Proposition 3 the occupational choice effect decreases with hand increases with since the set of workers increases andshrinks respectively30

VI CONCLUSIONS

We have developed a theory of offshoring in which agentswith heterogeneous abilities sort into teams competitively Wehave analyzed the effects of globalization on the organization ofwork the size distribution of firms and the structure of earningsof individuals and we have illustrated how these outcomes inturn determine the patterns of production consumption andinternational trade in the global economy

We have shown that the effects of globalization interact innontrivial ways with the state of communication technologiesFor example in our model globalization always increases within-worker wage inequality in the South but it increases within-worker inequality in the North only if the costs of communicatingknowledge are relatively low Similarly we have shown that thecharacteristics of international offshoring also depend very muchon the state of communication technologies the lower are com-munication costs the higher is the amount of international off-shoring but the lower is its quality

In order to highlight the main forces in the model our theo-retical framework has abstracted from certain aspects that arecentral in shaping the international organization of productionFirst we have imposed that production is undertaken by two-layer teams consisting of a manager and a set of workers It wouldbe interesting to incorporate the possibility of both self-employ-ment and multiple layers in our model This would open the doorfor a study of how globalization affects the incentives to offshoreor not to different countries as well as the way it affects thehierarchical structure of firms Second we have presented apurely technological theory of the international organization ofproduction A caveat of this approach is that we can explain why

30 The overall effect of h and on wage inequality results from the combi-nation of these three effects In Antras Garicano and Rossi-Hansberg [2005] weshow for instance that our model is consistent with an increase in the relativeendowment of skilled agents in the South (a higher ) leading to increasedwithin-worker inequality in that country This is consistent with the findings ofZhu and Trefler [2005] and is not easily rationalizable with standard generalequilibrium models


a northern manager might have an incentive to form a team witha group of southern workers but we have less to say about whythis international exchange of knowledge will occur within firmboundaries (ie within multinationals) rather than througharmrsquos length subcontracting or licensing It would be interestingto incorporate contractual frictions in our setup in order to obtaina more well-defined trade-off between in-house versus armrsquos-length offshoring31

APPENDIX

Proof of Theorem 1 We first show that an equilibrium of thiseconomy exhibits positive sorting Let ( zmzp) denote the rentsof a manager of ability zm and hires workers with ability zp Fromour definitions above we know that (m( z)z) R(m( z)) if mis the equilibrium assignment function In equilibrium we knowthat managers choose the ability of their workers optimally so

zmzp

zp 0

Totally differentiating this expression we obtain that

zm

zp

2zmzpzp2

2zmzpzpzm

The numerator has to be negative since managers are maximiz-ing rents in equilibrium To show that the denominator is posi-tive notice that

2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0

Since the argument is valid for all workers we conclude that inan equilibrium allocation m( z) 0 for all workers with ability z

31 In a similar vein Antras [2003 2005] and Antras and Helpman [2004]embed the property-rights approach of Grossman and Hart [1986] in standardtrade models Grossman and Helpman [2004] in turn develop a model of theinternational organization of production that shares certain features with themultitask approach of Holmstrom and Milgrom [1994]


Let w be an equilibrium wage function and let W(w) andM(w) be the equilibrium set of agents who become workers andmanagers respectively Let m( z) be the skill level of the managerof a worker with ability z then

h W0zp

1 z g zdz Mm0m zp

g zdz all zp W

Deriving with respect to zp we obtain that as long as zp is in theinterior of W and m( z) is increasing as we showed above

m zp h1 zpg zp

gm zp

We want to prove that an equilibrium in this economy im-plies that W(w) is a connected interval Suppose that it is not Inparticular suppose that W [a1a2] [a3a4] and M [a2a3] [a4a5] Then given a1 and a3 we know that m(a1) a2 andm(a2) a3 Combining these conditions with the differentialequation for wages (and continuity of wages at a2) and the ex-pression abovemdashthat have to hold in the interior of Wmdashwe cansolve for a wage function w13 a rent function R13 and a thresholda2 Similarly given a3 and a5 we can solve for a wage functionw35 a rent function R35 and a threshold a4 that satisfy all theequilibrium conditions in the interval [a3a5] In order for W andM to represent equilibrium occupational choices we have to guar-antee that agents in the interval [a3a5] do not want to formteams with agents in the interval [a1a3] The first necessarycondition is that

R13a3 w35a3

Since if R13(a3) w35(a3) agents with skill above but arbi-trarily close to a3 would like to become managers If R13(a3) w35(a3) agents with skill marginally below a3 would like tobecome workers and agents at a4 would like to hire them at awage marginally larger than R13(a3) The next condition is that

(20) limz1a3

R13z

z limz2a3

w35z

z

To show that this is the case notice that

R13a3

z 1

h1 a2 1


by the envelope theorem We will prove that the inequality abovehas to hold in equilibrium in two distinct cases for the case whenw13(a2) a2 and for the case when w13(a2) a2 Suppose thatw13(a2) a2 then since R13( z) z 1 for all z [a2a3] weknow that R13(a3) a3 and since R13(a3) w35(a3) w35(a3) a3 Then since h 1 and a2 a3 a4 z 1 we can concludethat

R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z

which proves condition (20) if w13(a2) a2 Now suppose thatw13(a2) a2 then since w35(a3) R13(a3) we can rewrite theright-hand side of the inequality as

w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2

Proving that condition (20) holds then amounts to prove that

a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2

But this is trivially satisfied given that a4 z 1 and w13(a2) a2

We have established that condition (20) has to hold in equi-librium but then a4 would like to hire a3 ε at a better wagethan what he makes as a manager and a3 ε would accept theoffer To show this consider the rents that a4 would get fromhiring a3 ε at wage R13(a3 ε)

a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0

where the inequality comes from the result above Hence anallocation where W is not connected implies that there are incen-tives for agents to form different teams This implies that an


equilibrium requires that W be a connected interval of the form[0z] Hence in equilibrium m(0) z and m( z) z Anallocation that (i) satisfies the two differential equations above(ii) satisfies the previous boundary conditions for assignmentfunction m and (iii) yields a continuous earnings function existsand is unique (see Garicano and Rossi-Hansberg [2005])

The final step is to prove that such an allocation is in fact anequilibrium For this we need to prove that there exists an h 0 such that the allocation guaranteed to exist by the above rea-soning is such that R( z) w( z) To show this we use asimilar argument to the one above Consider the incentives of amanager with ability z to hire a worker with ability z ε Herprofits are given by

z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z

since R( z) w( z) Hence in equilibrium it has to be the casethat R( z) w( z) in order for this term to be negative Butnotice that since R( z) w( z)

w z z w z

1 z hz z w0

1 zh

which is smaller than 1h if hz w(0) 1 Hence since w(0) 1 (if not rents of all managers would be negative) this impliesthat there exists an h 0 such that for all h h

w z 1h R z

Hence for all h h there exists a unique competitive equilib-rium in this economy Given that markets are complete andcompetitive this implies that the equilibrium allocation in theeconomy is Pareto optimal

To show that the earnings function is convex first notice thatfrom equation (3)

w zp hg zp

gm zp 0


while the rent function is such that

R zm m1 zm

h1 m1 zm2 0

where the last inequality follows from positive sorting

Proof of Proposition 1 (i) We first show that the mass ofworkers increases in the South This is obviously the case in aLQE because zS zWL32 On the other hand for the case ofa HQE it suffices to show that zS zWH but this follows fromsimple inspection of the formulas for these thresholds That themass of workers decreases in the North follows from zWH zNand zWL zN which are both clearly true from the expressionsfor these thresholds

(ii) For the first statement we want to show that bothmS( z) mWL( z) and mS( z) mWH( z) for all z zS The firstinequality follows directly from zWL zS in a LQE SimilarlyzWH zS immediately implies that mS( z) mWH( z) for z zFor the interval z z zWH it is useful to rewrite mWH( z) asmWH( z) hz(1 1frasl2 z) zWH (1)h[ z(1 1frasl2 z) z(1 1frasl2 z)] The inequality then follows from zWH zS and the factthat x(1 1frasl2 x) is nondecreasing in x for x [01] The secondstatement is an immediate corollary of this first result For thethird statement it is sufficient to show that mWH( zS) for allzS z zWH But notice that with a couple of substitutionsmWH( zS) ((1 ))( zWH zS) and the resultfollows from the monotonicity of mWH33

(iii) To prove the first part we simply write mW( z) mN( z)for each of the two equilibria For the LQE one this equals

mWLz mNz zWL zN (h)z(1 1frasl2 z) if 0 z

zWL zN h(1 1frasl2 ) if z zWL

which is nondecreasing in z is negative for low enough z and ispositive for high enough z (notice that zWL zN h(1 1frasl2 ) 0 is implied by mN( zWL) 1) For the HQE case this equals

32 There are parameter values for which zS zWL but these are inconsis-tent with the existence of a LQE

33 Notice that mWH( zS) hzS(1 1frasl2 zS) zWH (1)h[ zS(1 1frasl2 zS) z(1 1frasl2 z)] The two substitutions are hzS(1 1frasl2 zS) zS andhz(1 1frasl2 z) zWH


mWH z mN z

zWH zN if 0 z z

zWH zN (1)hzp(1 1frasl2 zp) (1)hz(1 1frasl2 z) if z z zWH

which is again nondecreasing in z is negative for low enough zand is positive for high enough z (the latter is implied bymN( zWH) 1) The second part follows immediately since thematching functions are monotonic and thus invertible That is atthe same point at which workers are matched with better man-agers managers are matched with worse workers See Figure IVfor an illustration

Proof of Proposition 2 Let us start with the last claim Thedifference in the marginal return to skill in the South with andwithout globalization is given by

1L S hz if 0 z and h 21 2 21L S h if z zS and h 21 2 21H S if 0 z z and h 21 2 21H S hz z if z z zS and h 21 2 2

It is thus sufficient to show that 1L S and 1H S That 1L S follows directly from the expressions after realizing that zWL zSand zWL 1frasl2 in a LQE For 1H S rewrite (18) as

1H hzWH1 1frasl2 hzWH

1 h hzWH

h2 2 zWH z2

1 h hzWH S

where the inequality follows since the first term is increasing inzWH and zWH zS and the second term is positive This result com-bined with zWH zS implies that wage inequality in the South in-creases with globalization

Proof of Proposition 3 Simple differentiation of (10) yields

zWL

h 1 h23 1

h21 h23 0

and differentiation of (15) results in

zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0


where the sign follows from

1 h h2 1 21 h2 1

1 2h 4h2

The conclusions on the set of workers and managers follow di-rectly from this result and the definitions of a LQE and HQE

Differentiating zW with respect to results in

zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0

The last two statements follow from this result as well as from thefact that zWH is decreasing in (ie ( zWH) zWH) asshown in the proof of Proposition 4

Proof of Proposition 4 First consider the effect of h on thequantity of offshoring The measure of the quantity of offshoringis given by

OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)

That is the quantity of offshoring is the proportion of southernworkers in international teams The quantity of workers engagedin offshoring is in turn given by in a LQE and by zWH z ina HQE We first prove the first statement of the propositionnamely that zWH z is a decreasing function of h Toward acontradiction suppose that zWH z is a weakly increasingfunction of h Then the number of workers hired by northernmanagers in [1] weakly increases with h But as we show belowin the proof of Proposition 5 firm size (given by 1h(1 mWH

1 ( z)))is decreasing in h for any skill level zm of the manager Hencesince the number of managers in [1] has not changed thenumber of workers in their firms must have gone down a con-tradiction Hence zWH z decreases with h or

zWH

h z

h 0


Moving to the first statement of the proposition notice thatsince z zW the above inequality implies that

zWH

h z z

h zWH

and thus zzW is increasing in h This in turn implies that thequantity of offshoring is weakly decreasing in h

We now turn to the effect of on the quantity of offshoringSimple differentiation and equation (14) imply that

zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH

and so ( zzWH) 0 if zWH ( zWH) But simplealgebra delivers

zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1

Now note that v1(h) v2(h) if and only if

v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0

which is clearly true Hence

zzWH

0


which implies from the definition of OW that the quantity ofoffshoring is strictly decreasing in

Finally we want to show that the measure of workers en-gaged in offshoring also decreases with that is

zWH z

0

To see this notice that

zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0

where we have used twice that zWH (zWH) as well asmWH(z)

Proof of Proposition 5 From Proposition 3 a decrease in hincreases zW say from zW0 to zW1 From the boundary conditionmW(0) zW it follows that the worst agent is matched with abetter manager Similarly the boundary condition mW( zW) 1implies that the match for workers with zp zW0 worsens Itremains to show that the change in the match is a monotonicfunction of the skill of the worker But simple inspection of theformulas for mWL( zp) and mWH( zp) reveals that 2mWL( zp)h zp 0 because zp(1 1frasl2 zp) is increasing in zp To prove thesecond claim we need to show that h(1 mW

1( zm)) increases inh for all zm in [ zW1] This amounts to computing these partialderivatives for each segment of each equilibrium and showingthat they are positive Simple but tedious derivation then con-firms the first result

To understand the effect of changes in note that makinguse of the boundary conditions as well zWL 0 it is straight-forward to see that the least-skilled worker is matched with abetter manager while the ex ante most-skilled worker is matchedwith a worse manager For claim (i) it remains to show that thechange in the match is a monotonic function of the skill of the


worker ie 2mW( zp) z 0 Again this is clear from inspec-tion of the formulas for mWL( z) and mWH( z) because z(1 1frasl2 z)is increasing in zp Hence for each equilibrium there exist athresholds 13i zWj j H L such that all workers with z 13jare matched with a better manager while all workers with z 13jare matched with a worse manager

A corollary of this result is that all managers with skill belowmWj(13j) are matched with lower-skilled workers while all man-agers with skill above mWj(13j) are matched with higher-skilledworkers This immediately delivers claim (ii) because rememberthat firm size is the inverse of h(1 mW

1( zm))For claims (iii) and (iv) notice that 2mWL( z) z 0 for

z zWL which implies (given the effect on the boundaryagents) that 0 13L Similarly 2mWH( z) z 0 for 0 zp z implies that z 13H zWH By the monotonicity ofmW( z) these inequalities in turn imply zWL mWL(13L) mWL() and mWH(13H) 1 To see that this is sufficient forclaims (iii) and (iv) simply remember that the interval of man-agers that offshore in each equilibrium is ( zWLmWL()) and (1) respectively while the interval of nonoffshorers is (mWL()1)and ( zWH) respectively

HARVARD UNIVERSITY

UNIVERSITY OF CHICAGO

PRINCETON UNIVERSITY

REFERENCES

Acemoglu Daron ldquoPatterns of Skill Premiardquo Review of Economic Studies LXX(2003) 199ndash230

Aitken Brian Gordon H Hanson and Ann E Harrison ldquoWages and ForeignOwnership A Comparative Study of Mexico Venezuela and the UnitedStatesrdquo Journal of International Economics XL (1996) 345ndash371

Antras Pol ldquoFirms Contracts and Trade Structurerdquo Quarterly Journal of Eco-nomics CXVIII (2003) 1375ndash1418

mdashmdash ldquoIncomplete Contracts and the Product Cyclerdquo American Economic ReviewXCV (2005) 1054ndash1073

Antras Pol and Elhanan Helpman ldquoGlobal Sourcingrdquo Journal of Political Econ-omy CXII (2004) 552ndash580

Antras Pol Luis Garicano and Esteban Rossi-Hansberg ldquoOffshoring in a Knowl-edge Economyrdquo National Bureau of Economic Research Working Paper No11094 January 2005

Antras Pol Luis Garicano and Esteban Rossi-Hansberg ldquoExtensive OffshoringThe Role of Middle Managementrdquo mimeo Harvard University January 2006

Autor David H Lawrence F Katz and Melissa S Kearney ldquoTrends in U SWage Inequality Re-Assessing the Revisionistsrdquo mimeo Massachusetts In-stitute of Technology 2004

Autor David H Frank Levy and Richard J Murname ldquoThe Skill Content ofRecent Technological Change An Empirical Explorationrdquo Quarterly Journalof Economics CXVIII (2003) 1279ndash1334


Bernard Andrew Raymond Robertson and Peter Schott ldquoIs Mexico a LumpyCountryrdquo National Bureau of Economic Research Working Paper No 10898November 2004

Feenstra Robert C and Gordon H Hanson ldquoForeign Investment Outsourcingand Relative Wagesrdquo in Robert C Feenstra Gene M Grossman and DouglasA Irwin eds Political Economy of Trade Policy Essays in Honor of JagdishBhagwati (Cambridge MA MIT Press 1996a) pp 89ndash127

Feenstra Robert C and Gordon H Hanson ldquoGlobalization Outsourcing andWage Inequalityrdquo American Economic Review LXXXVI (1996b) 240ndash245

Feenstra Robert C and Gordon H Hanson ldquoForeign Direct Investment andRelative Wages Evidence from Mexicorsquos Maquiladorasrdquo Journal of Interna-tional Economics XLII (1997) 371ndash393

Feenstra Robert C and Gordon H Hanson ldquoThe Impact of Outsourcing andHigh-Technology Capital on Wages Estimates for the United States 1972ndash1990rdquo Quarterly Journal of Economics CXIV (1999) 907ndash940

Feenstra Robert C and Gordon H Hanson ldquoGlobal Production Sharing andRising Inequality A Survey of Trade and Wagesrdquo in E Kwan Choi and JamesHarrigan eds Handbook of International Trade (Oxford UK Basil-Black-well 2003) pp 146ndash185

Garicano Luis ldquoHierarchies and the Organization of Knowledge in ProductionrdquoJournal of Political Economy CVIII (2000) 874ndash904

Garicano Luis and Esteban Rossi-Hansberg ldquoOrganization and Inequality in aKnowledge Economyrdquo National Bureau of Economic Research Working PaperNo 11458 June 2005

Garicano Luis and Esteban Rossi-Hansberg ldquoInequality and the Organization ofKnowledgerdquo American Economic Review Papers and Proceedings XCIV(2004) 197ndash202

Goldberg Pinelopi Koujianou and Nina Pavcnik ldquoTrade Inequality and PovertyWhat Do We Know Evidence from Recent Trade Liberalization Episodes inDeveloping Countriesrdquo in Susan M Collins and Dani Rodrik eds BrookingsTrade Forum 2004 Globalization Poverty and Inequality (Washington DCBrookings Institution Press 2004) pp 223ndash269

Grossman Sanford J and Oliver D Hart ldquoThe Costs and Benefits of OwnershipA Theory of Vertical and Lateral Integrationrdquo Journal of Political EconomyXCIV (1986) 691ndash719

Grossman Gene M and Elhanan Helpman ldquoManagerial Incentives and theInternational Organization of Productionrdquo Journal of International Econom-ics LXIII (2004) 237ndash262

Grossman Gene M and Giovanni Maggi ldquoDiversity and Traderdquo American Eco-nomic Review XC (2000) 1255ndash1275

Hanson Gordon H and Ann E Harrison ldquoTrade Liberalization and Wage In-equality in Mexicordquo Industrial and Labor Relations Review LII (1999)271ndash288

Helpman Elhanan ldquoA Simple Theory of International Trade with MultinationalCorporationsrdquo Journal of Political Economy XCII (1984) 451ndash471

Helpman Elhanan Marc J Melitz and Stephen R Yeaple ldquoExports versus FDIwith Heterogeneous Firmsrdquo American Economic Review XCIII (2004)300ndash316

Holmstrom Bengt and Paul R Milgrom ldquoThe Firm as an Incentive SystemrdquoAmerican Economic Review LXXXIV (1994) 972ndash991

Kremer Michael and Eric Maskin ldquoWage Inequality and Segregation by Skillrdquomimeo Harvard University 1997

Kremer Michael and Eric Maskin ldquoGlobalization and Inequalityrdquo mimeo Har-vard University 2003

Lawrence Robert Z and Matthew J Slaughter ldquoInternational Trade and Ameri-can Wages Great Sucking Sound or Small Hiccuprdquo Brookings Papers onEconomic Activity 2 (1993) 161ndash210

Lipsey Robert E and Frederik Sjoholm ldquoForeign Direct Investment Educationand Wages in Indonesian Manufacturingrdquo Journal of Development Econom-ics LXXIII (2004) 415ndash422

Lucas Jr Robert ldquoOn the Size Distribution of Business Firmsrdquo Bell Journal ofEconomics IX (1978) 508ndash523


Manasse Paolo and Alessandro Turrini ldquoTrade Wages and lsquoSuperstarsrsquordquo Jour-nal of International Economics LIV (2001) 97ndash117

Melitz Marc J ldquoThe Impact of Trade on Intra-Industry Reallocations and Ag-gregate Industry Productivityrdquo Econometrica LXXI (2003) 1695ndash1725

Murphy Kevin M and Finis Welch ldquoWage Differentials in the 1990s Is the GlassHalf-Full or Half-Emptyrdquo in Finis Welch ed The Causes and Consequencesof Increasing Inequality (Chicago IL University of Chicago Press 2001) pp341ndash364

Nocke Volker and Stephen R Yeaple ldquoAn Assignment Theory of Foreign DirectInvestmentrdquo mimeo University of Pennsylvania 2004

Rosen Sherwin ldquoAuthority Control and the Distribution of Earningsrdquo BellJournal of Economics XIII (1982) 311ndash323

Sattinger Michael J ldquoAssignment Models of the Distribution of Earningsrdquo Jour-nal of Economic Literature XXXI (1993) 831ndash880

Verhoogen Eric A ldquoTrade Quality Upgrading and Wage Inequality in the Mexi-can Manufacturing Sector Theory and Evidence from an Exchange RateShockrdquo mimeo University of California at Berkeley 2004

Waldman Michael ldquoWorker Allocation Hierarchies and the Wage DistributionrdquoReview of Economic Studies LI (1984) 95ndash109

Yeaple Stephen R ldquoA Simple Model of Firm Heterogeneity International Tradeand Wagesrdquo Journal of International Economics LXV (2005) 1ndash20

Zhu Susan C and Daniel Trefler ldquoTrade and Inequality in Developing Coun-tries A General Equilibrium Analysisrdquo Journal of International EconomicsLXV (2005) 21ndash48


take routine tasks and the highest skill agents (managers) spe-cialize in knowledge-intensive tasks can now take place acrosscountries

In our view what is important about this new division oflabor is that it alters the feasible matches between agentsrsquo skilltypes High skill agents in more developed countries can leveragetheir knowledge at lower cost by working with cheaper labor onroutine tasks and the better workers in less developed countriesare able to become part of international high value added teams

In this paper we present a simple framework that puts agentskill heterogeneity and matching at the center of the analysis Byallowing us to analyze changes in matching and in the supportingearnings functions our framework allows us to examine theimpact of offshoring on wages on occupational choices (produc-tion versus knowledge jobs) and (as matches are ldquomany-to-onerdquo)on the distribution of firm sizes

We model an economy in which production requires physicalinputs and knowledge and where a continuum of agents withheterogenous abilities sorts into teams competitively Agents ofdifferent skill levels form teams Less skilled agents (workers)specialize in production work and deal with routine tasks whilethe most skilled agents specialize in knowledge-intensive tasks(managers) Relative to less skilled managers better managersare able to increase more the productivity of all the workers intheir team as they are able to solve a wider range of the problemstheir team confronts in production Better production workersallow individuals to manage larger teams as workers can solvemore problems by themselves and require less help This resultsin a complementarity between manager and worker ability thatdetermines the identity of agents working and managing differ-ent teams It also determines through comparative advantagethe occupational choice of agents More able agents althoughmore productive as production workers want to set up their ownfirms and manage their own teams of workers instead of workingfor other managers

To study the impact of the formation of international teamsin this economy we study a simple one-sector two-country modelin which countries differ only in their skill distributions In par-ticular one country the North has a distribution of skills with arelatively high mean while the other country the South has adistribution of skills with a relatively low mean In our model theldquoskill overlaprdquo implied by these skill differences is captured by a



























II THE MODEL





h1 zpn 1


(1) R zm maxzp

zm wzpnzp maxzp

zm wzp

h1 zp

subject to

h1 zpn 1


(2) w zp zm w zp

1 zp








0

zp

g zdz m0

m zp



(3) m z h1 zg z

gm z















(5) zS 1 h 1 h2 2h1

h








where


1 h hzS


wS z S hz







RS z z wSmS

1 z

h1 mS1 z


RS z 1h1 mS1 z



(8) YS 0

zS

mS z gS zdz 1

6zS26 3h hzS








gW z (1 ) if 0 z 1 if z 1


















(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES






































































II THE MODEL





h1 zpn 1


(1) R zm maxzp

zm wzpnzp maxzp

zm wzp

h1 zp

subject to

h1 zpn 1


(2) w zp zm w zp

1 zp








0

zp

g zdz m0

m zp



(3) m z h1 zg z

gm z















(5) zS 1 h 1 h2 2h1

h








where


1 h hzS


wS z S hz







RS z z wSmS

1 z

h1 mS1 z


RS z 1h1 mS1 z



(8) YS 0

zS

mS z gS zdz 1

6zS26 3h hzS








gW z (1 ) if 0 z 1 if z 1


















(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES














































h1 zpn 1


(1) R zm maxzp

zm wzpnzp maxzp

zm wzp

h1 zp

subject to

h1 zpn 1


(2) w zp zm w zp

1 zp








0

zp

g zdz m0

m zp



(3) m z h1 zg z

gm z















(5) zS 1 h 1 h2 2h1

h








where


1 h hzS


wS z S hz







RS z z wSmS

1 z

h1 mS1 z


RS z 1h1 mS1 z



(8) YS 0

zS

mS z gS zdz 1

6zS26 3h hzS








gW z (1 ) if 0 z 1 if z 1


















(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES














































0

zp

g zdz m0

m zp



(3) m z h1 zg z

gm z















(5) zS 1 h 1 h2 2h1

h








where


1 h hzS


wS z S hz







RS z z wSmS

1 z

h1 mS1 z


RS z 1h1 mS1 z



(8) YS 0

zS

mS z gS zdz 1

6zS26 3h hzS








gW z (1 ) if 0 z 1 if z 1


















(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES
























































(5) zS 1 h 1 h2 2h1

h








where


1 h hzS


wS z S hz







RS z z wSmS

1 z

h1 mS1 z


RS z 1h1 mS1 z



(8) YS 0

zS

mS z gS zdz 1

6zS26 3h hzS








gW z (1 ) if 0 z 1 if z 1


















(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES

















































where


1 h hzS


wS z S hz







RS z z wSmS

1 z

h1 mS1 z


RS z 1h1 mS1 z



(8) YS 0

zS

mS z gS zdz 1

6zS26 3h hzS








gW z (1 ) if 0 z 1 if z 1


















(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES
















































RS z z wSmS

1 z

h1 mS1 z


RS z 1h1 mS1 z



(8) YS 0

zS

mS z gS zdz 1

6zS26 3h hzS








gW z (1 ) if 0 z 1 if z 1


















(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES

















































gW z (1 ) if 0 z 1 if z 1


















(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES



























































(10) zWL 1 h 1 h23

h



(11) h 21 2 2






(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES















































(12) wWL z



where


1 h hzWL

and

2L 1L h

Note that at zWL

wWL zWL 1 wWL zWL

1 zWL

1h RWL zWL








((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES















































((1 ))h(1 z) if z z zWH






and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES















































and a threshold

(15) zWH 1 h 1 h2 1 1 2h

h


(16) z 1 1 2 zWH

h



(17) wWH z


where

(18)


2

1 h hzWH

and

2H 1H h z


















































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES



























































































































IVD Generalizations



w z hg z

gm z

m z

1 z


(19)gS z

gS z

gW z

gW zfor z z z






w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES














































w W z hgW z

gWmW z h

11

h w S z








OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










HARVARD UNIVERSITY



REFERENCES














































OW zWH zzWH



QW zWH zzWH

















VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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VI CONCLUSIONS







APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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APPENDIX


zmzp

zp 0


zm

zp

2zmzpzp2

2zmzpzpzm


2 zmzp

zp zm

1h1 zp

zp

1h1 zp

2 0

Hence

zm

zp 0





h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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h W0zp

1 z g zdz Mm0m zp

g zdz all zp W


m zp h1 zpg zp

gm zp


R13a3 w35a3


(20) limz1a3

R13z

z limz2a3

w35z

z


R13a3

z 1

h1 a2 1



R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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R13a3

z 1

h1 a2 1

a4 w35a3

1 a3

w35a3

z


w35a3

z a4 w35a3

1 a3

a4h1 a2 a3 w13a2

1 a3h1 a2


a4h1 a2 w13a2 1

or

a4 1 w13a2

h1 a2



a4 a3 ε a4 R13a3 ε

h1 a3 ε

and note that

limε30

a4 a3 ε

ε R13a3 w35a3

h1 a3 0





z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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z z ε z R z ε

h1 z ε

so

limε30

z z ε

ε wz Rz

h1 z


w z z w z

1 z hz z w0

1 zh


w z 1h R z



w zp hg zp

gm zp 0



R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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R zm m1 zm

h1 m1 zm2 0











mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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mWH z mN z

zWH zN if 0 z z







1 h hzWH

h2 2 zWH z2

1 h hzWH S



zWL

h 1 h23 1

h21 h23 0


zWH

h

1 h h 1 1 h2 1 1 2h

1h21 h2 1 1 2h 0



1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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1 h h2 1 21 h2 1

1 2h 4h2



zWL

h

21 h23 0

and

zWH

2

1 21 1 h2 2h1

1

0



OW 1 if h 2(1 )(2 2)1 (zzWH) if h 2(1 )(2 2)



zWH

h z

h 0



zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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zWH

h z z

h zWH



zzWH

11 zhzWH

2 zWH1 zWH

hz1 z zWH

11 zhzWH

2 zWH zWH

1frasl2 hz

2

11 zhzWH

2 zWH zWH


zWH zWH

v1h v2h

h 121 12h 2h h2 h2 1

where

v1h 12h 1

1 1 (2h 2h h2 h2 1)

v2h 2h 2 2h h2 2 2h2 2h2 h22 1


v1h2 v2h2

42hh 2 h2 2 2h2 2h2 h22 1 0


zzWH

0




zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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zWH z

0


zWH z

12 zWH

zWH

h1 z1

zWH

z

12h1 z

1 zWH

hz1 z

12h1 z

1 zWH

hz1 1frasl2 z

12h1 z

1 zWH

zWH 0










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