Matching and Sorting in a Global Economy

Transcription

1 Matchng and Sortng n a Global Economy Gene M. Grossman Prnceton Unversty Elhanan Helpman Harvard Unversty and CIFAR Phlpp Krcher Unversty of Ednburgh September 203 Abstract We develop a neoclasscal trade model wth heterogeneous factors of producton. We consder a world wth two factors, labor and managers, each wth a dstrbuton of ablty levels. Producton combnes a manager of some type wth a group of workers. The output of a unt depends on the types of the two factors, wth complementarty between them, whle exhbtng dmnshng returns to the number of workers. We examne the sortng of factors to sectors and the matchng of factors wthn sectors, and we use the model to study the determnants of the trade pattern and the e ects of trade on the wage and salary dstrbutons. Fnally, we extend the model to nclude search frctons and consder the dstrbuton of employment rates. Keywords: heterogeneous labor, matchng, sortng, productvty, wage dstrbuton, nternatonal trade. JEL Class caton: F, F6 Part of ths research was completed whle Grossman was vstng STICERD at the London School of Economcs and CREI at the Unversty of Pompeu Fabra and whle Helpman was vstng Yonse Unversty as SK Chared Professor. They thank these nsttutons for ther hosptalty and support. The authors are grateful to Rohan Kekre, Ran Shorrer, and especally Kevn Lm for ther research assstance. Arnaud Costnot, Oleg Itskhok, Stephen Reddng, Dan Tre er, Jonathan Vogel and numerous semnar partcpants provded helpful comments and suggestons. Fnally, Grossman and Helpman thank the Natonal Scence Foundaton and Krcher thanks the European Research Councl for nancal support.

2 Introducton In ths paper, we study how nternatonal trade a ects the sortng of heterogenous workers and managers nto ndustres and the matchng of workers wth managers n producton unts. It s by now well known that rms n the same ndustry d er n sze, n the compostons of ther workforces, n the technologes and captal goods they use, and n the wages they pay to ther workers. Industres d er n factor ntenstes and n the margnal contrbutons of worker and manageral ablty to rm productvty. Workers d er n physcal attrbutes, n cogntve abltes, and n ther educaton, tranng, and experence. Although some studes of nternatonal trade have examned the assgnment of heterogeneous labor to d erent sectors and others have consdered the matchng of workers to heterogeneous teammates or technologes, relatvely lttle s known about the general problem of how factors sort and match n the open economy when several of these factors are d erentated, when xed quanttes of one mpart decreasng returns to the others, and when ndustres d er n ther factor ntenstes and n the usefulness of factor qualty. Our paper addresses these more general, allocatonal ssues and ther mplcatons for factor rewards. Because workers and managers are heterogeneous, our analyss sheds lght on the mpact of trade on the dstrbuton of wages and manageral salares, and thereby on the mpact of trade on earnngs nequalty. By allowng for worker, manager, and ndustry heterogenety, we can better understand a number of ssues concernng the pattern and consequences of nternatonal trade. Frst, we can study how countres dstrbutons of d erentated factors, n conjuncton wth ther aggregate endowments of these factors, determne ther comparatve advantage n the varous sectors. Bombardn et al. (202) provde evdence, for example, that countres skll dspersons have a quanttatvely smlar mpact on trade ows as do ther aggregate endowments of human captal. Second, we can nvestgate how trade n uences factor returns across the entre ncome dstrbuton, a ectng more than just the relatve compensaton pad to one factor versus another or to workers employed n one ndustry versus another. These addtonal dmensons of nequalty can be useful for understandng recent ndngs of substantal varaton n wages that s not easly explaned by observable worker characterstcs. Helpman et al. (202) show, for example, that wthn-ndustry wage varaton accounts for a majorty of wage nequalty n Brazl even after controllng for workers occupatons. Thrd, we can examne how globalzaton a ects the dstrbuton of employment rates across skll levels n a settng wth search-and-matchng frctons. The lterature on the sortng of workers to ndustres ncludes recent work by Costnot (2009), Costnot and Vogel (200), and Ohnsorge and Tre er (2007), as well as earler work by Mussa (982) and Ru n (988). All of these authors emphasze the comparatve advantage that the varous types of labor have when employed n d erent ndustres. They study the determnants of We use the term sortng to refer to the allocaton of heterogeneous factors to d erent ndustres and the term matchng to refer to the combnaton of d erentated factors wthn an ndustry.

3 the trade pattern n countres that d er n the compostons of ther labor forces and the mpact that trade has on ncome nequalty across the skll or ablty spectrum. But most assume a lnear relatonshp between labor nput (of a gven qualty) and output or, what amounts to the same, an absence of nteractons between quanttes of labor and quanttes of other factors of producton. Models wth one worker per rm or wth a lnear relatonshp between labor quantty and output cannot speak to the determnants of a rm s captal ntensty or ts manager s span of control. The matchng of workers to technologes wthn an ndustry s the focus of work by Yeaple (2005) and Sampson (202). These authors also assume a producton functon wth constant returns to labor and thus omt nteractons between labor and any other factors of producton. 2 Smlarly, Grossman and Magg (2000) study the parng of workers who perform d erent producton tasks, but n a context wth exactly two workers per rm and therefore no scope for varaton n factor ntensty or rm sze. The work of Antràs et al. (2006) does allow for endogenous span of control n a model wth matchng of workers and managers, but thers s a one-sector model wth nternatonal producton teams and they assume a partcular technology that tghtly lnks the qualty and the quantty of labor that a gven manager can oversee. Our analyss extends a famlar trade model wth two sectors, two factors, and perfectlycompettve product markets. Whle most of our analyss assumes frctonless factor markets, we also consder an economy wth search and matchng frctons. We call one factor labor and assume throughout that workers are d erentated along a sngle dmenson that we term ablty. Workers wth greater ablty are assumed to be more productve n both ndustres, but the contrbuton of ablty to output may d er across uses. We refer to the second nput as managers, although we mght alternatvely thnk of them as machnes. Smlar to workers, managers generally d er n ablty (or machnes d er n qualty) and more able managers contrbute more to output n both sectors, albet to an extent that may vary by ndustry. The modelng of an ndustry s technology resembles that n Lucas (978) and the extenson provded by Eeckhout and Krcher (202) to allow for heterogenety of both factors and decreasng returns to the quantty of one gven the quantty of the other. Wth ths formulaton, we can address how the economy matches a xed but heterogenous supply of one nput (managers or machnes) wth a xed but heterogeneous supply of another (labor) n a settng where the relatve number of workers per manager s a matter for rms to decde. In the next secton, we lay out our basc model of an open economy wth two countres, two compettve ndustres, and two heterogeneous factors of producton. Secton 3 consders trade between countres that have heterogeneous workers but homogeneous managers. Our analyss of ths smpler settng ads n understandng the more general case dscussed n Sectons 4, where managers also are assumed to vary n ablty. We show that, wth homogeneous managers, the sortng of workers s guded by a cross-ndustry comparson of the rato of the elastcty of output wth respect to labor qualty to the elastcty of output wth respect to labor quantty. Ths can 2 Both of these authors assume, however, that rms produce d erentated products n a world of monopolstc competton, so that nputs of addtonal labor by a rm do generate decreasng returns n terms of revenue. Thus, these models do share some features wth the ones that we study below. 2

4 generate a smple sortng pattern n whch all the best workers wth ablty above some threshold level are employed n one sector and the remanng workers are employed n the other. But t also can generate more complex patterns n whch, for example, the most able and least able workers sort to one sector whle workers wth ntermedate levels of ablty are allocated to the other. Trade between countres wth smlar dstrbutons of worker talent s determned by ther aggregate factor endowments as n the Heckscher-Ohln model, whereas trade between countres wth smlar relatve endowments reveals a comparatve advantage for a country wth a superor dstrbuton of labor qualty (as re ected n a proportonal rghtward shft of ts talent dstrbuton) n the good produced by the ndustry n whch worker ablty contrbutes more elastcally to productvty. Wth homogeneous managers, relatve prce movements do not a ect wthn-sector relatve wages and therefore have no e ect on wage nequalty wthn ndustres. Across ndustres, the mpact of trade on wages re ects a blend of Stolper-Samuelson and Rcardo-Vner forces, as n models wth mperfect factor moblty such as Mussa (982) and Grossman (983). In Secton 4 we turn to the more nterestng case that arses when both factors are heterogeneous and there are complementartes between the qualtes of the two factors. If the productvty of a producton unt s a strctly log supermodular functon of the ablty levels of the manager and the workers, there s postve assortatve matchng of factors n each sector. That s, among the sets of workers and managers that sort to a gven sector, the better workers are matched wth the better managers. We provde su cent condtons under whch all of the workers wth ablty above some threshold level and all the managers wth ablty above some (d erent) threshold level sort to the same sector. We also provde condtons under whch the hgh-ablty workers sort to the same sector as the low-ablty managers. More complex sortng patterns are possble as well. When countres share the same dstrbutons of abltes and the sortng patterns do nvolve a sngle threshold for each factor, then the country endowed wth more managers per worker must export the manager-ntensve good. When there are strong complementartes between the types of workers and managers, the e ects of trade or trade lberalzaton on the wage dstrbuton are subtle and nterestng. An ncrease n the relatve prce of some good mght worsen the matches for all workers and mprove the matches for all managers, or vce versa. Alternatvely, a change n relatve prce mght mprove the matches for workers n one ndustry whle worsenng those for workers n the other. We dentfy condtons for these varous shfts n the matchng functons and dscuss ther mplcatons for factor rewards. In partcular, we dentfy stuatons n whch trade causes wthn-ndustry ncome nequalty to rse or fall and we show that the mpact of trade on an nput s wthn-sector earnngs nequalty can d er from the changes that occur across sectors. In Secton 5, we extend the analyss to nclude economes wth labor-market frctons by assumng that workers engage n drected search. In ths settng, each potental worker seeks a job at a rm of hs choosng and manages to be hred by that rm wth a probablty that depends on the number of applcants per vacancy. We show that, wth these search frctons, wage and employment rates both vary wth ablty; more able workers not only earn hgher wages but also 3

5 enjoy better job prospects. Moreover, we nd that trade a ects the nequalty n expected wages and n employment rates smlarly. Secton 6 contans some concludng remarks. 2 The Economc Envronment We examne a world economy comprsng two countres, two ndustres, and two factors of producton. We call one of the factors labor and refer to ndvduals as workers. Each country s endowed wth a contnuum of workers of varous types. The exogenous supply of workers of type q L n country c s L c c L (q L) for c = fa; Bg, where L c s the aggregate endowment of labor and c L (q L) s the probablty densty functon (pdf) over worker types. For ease of exposton, we assume throughout that c L (q L) s contnuous and strctly postve on a nte support SL c = [qc L mn ; qc L max ]. We refer to the second factor as managers, although we could as well have used the term machnes. Country c has a contnuum of managers of measure H c. We begn n Secton 3 by assumng that all managers (or machnes) are alke, so that the set of manager types SH c has a sngle element. Subsequently, we ntroduce manager heterogenety and then denote the supply of managers of type q H by H c c H (q H), wth c H (q H) contnuous and strctly postve on a nte support SH c = [qc H mn ; qc H max ].3 Frms n the two countres have access to dentcal, constant-returns-to-scale technologes. The output generated n an ndustry by a producton unt comprsng one manager and a group of workers re ects the number of workers that s combned wth the manager and the types of the employed factors. We could begn by specfyng output as a functon of the type of the manager and a lst of the types of all workers used n the producton unt. However, n many models of a manager s span of control, such as Sattnger (975), Garcano (2000), and Eeckhout and Krcher (202), rms have no ncentve to combne a gven manager wth a group of workers of d erent types. 4 We buld on the latter and, to conserve on notaton, smply assume that each rm combnes a manager of some gven type q H wth a group of ` workers of a common type q L, thereby generatng output of x = (q H ; q L ) `, 0 < <. () Here, s a parameter that re ects the dmnshng returns to combnng more workers wth a gven manager and (q H ; q L ) s a strctly ncreasng, twce contnuously d erentable, log supermodular functon that captures the complementartes between the types of the two factors. 5 We assume that 3 We focus on an envronment where factor endowments are nvarant to trade. Ths makes our results comparable to most prevous studes. Future work mght consder adjustments n factor endowments - e.g., takng the termnology of workers and managers lterally one mght study long-run skll acquston that turns workers nto managers. Or, thnkng of the second factor as machnery, one mght ncorporate nvestment n captal of d erent qualtes. 4 For example, the manager may be endowed wth a unt of tme that she must allocate among the varous workers, such that each worker s productvty ncreases wth the tme devoted to hm by the manager, albet wth dmnshng returns. The key assumpton here s that there s no teamwork or synergy between workers n a rm; they nteract only nsofar as they compete for the manager s tme. See Eeckhout and Krcher (202) for further dscusson. They show, n such a settng, that t s optmal for every rm to combne a manager of some type wth a group of workers that share a common type. 5 We adopt a Cobb-Douglas spec caton for factor quanttes n order to smplfy the analyss. Some of our results 4

6 factor type contrbutes to productvty n qualtatvely the same way n both sectors and, wthout further loss of generalty, order the types so =@q F > 0 for = ; 2 and F = H; L. Wth ths labelng conventon, we can refer to q F as the ablty of factor F. Note that the ndustres generally d er n the strength of the complementartes between factors, n the contrbutons of factor abltes to productvty, and n ther factor ntenstes. The rest of the model s famlar from neoclasscal trade theory. Consumers worldwde share dentcal and homothetc preferences. Frms hre workers and managers on frctonless natonal factor markets and engage n perfect competton on ntegrated world product markets. Countres trade freely, wth balanced trade. Note that we neglect for now the search frctons that are a realstc and nterestng feature of many markets wth heterogeneous factors. We shall extend the analyss to ncorporate such frctons n Secton 5 below. 3 Homogeneous Managers We are ultmately nterested n the sortng and matchng of two heterogeneous factors of producton. However, before we get to that, we consder a smpler envronment n whch one of the factors (managers) has a unform type. By examnng a settng wth managers of smlar ablty, we can gan nsght nto the sortng of the heterogeneous workers to ndustres wthout needng to concern ourselves wth the matchng of managers and workers. We wll ntroduce manager heterogenety n Secton 4 below. Suppose that all managers are dentcal and assume wthout further loss of generalty that ther common ablty level s q H =. Let ~ (q L ) (q L ; ) be the productvty n sector of workers of ablty q L when combned wth any manager who mght be employed there. Output per manager n sector can now be wrtten as x = ~ (q L )`, consderng the dmnshng returns to the number of workers. A key varable n the analyss wll be the rato of two elastctes that descrbe a sector s producton technology. One elastcty s " ~ (q L ) q L ~ 0 (q L )= ~ (q L ), whch re ects the responsveness of output to worker ablty n sector, holdng constant the number of workers per manager. The other elastcty s, whch s the responsveness of output to labor quantty, holdng constant the ablty of the workers. Let s L (q L ) " ~ (q L ) " ~ 2 (q L ) 2 be the d erence across sectors n these ratos. We assume for now that s L (q L ) has a unform sgn for all q L n the doman of the ablty dstrbuton and label the ndustres so that s L (q L ) > 0. More formally, we adopt for the tme beng the followng assumpton: Assumpton S H = fg and s L (q L ) > 0 for all q L 2 S A L [ SB L. would reman the same wth an arbtrary constant-returns-to-scale producton technology provded that there are no factor ntensty reversals. 5

7 A rm n sector chooses the number and type of ts workers (per manager) to maxmze (q L ; `) = p ~ (q L ) ` w (q L ) ` r, where p s the prce of good, w (q L ) s the wage of a worker wth ablty q L, and r s the salary of the representatve manager. 6 The rm can solve ts pro t maxmzaton problem n two stages. Frst, t calculates the optmal demand (per manager) for workers of ablty q L when the wage of such workers s w (q L ), whch yelds ` (q L ) = " # p ~ (q L ) : (2) w (q L ) Substtutng ths labor demand nto the pro t functon gves an expresson for pro ts that depends only on the ablty of the workers n the producton unt, namely ~ (q L ) = p ~ (q L ) w (q L ) r, (3) where ( ). In the second stage, the rm chooses q L to maxmze ~ (q L ). To characterze ths optmal choce, let Q L be the set of abltes of workers that sort nto sector and let Q nt L be the nteror of ths set. Snce the equlbrum wage functon must be everywhere contnuous, strctly ncreasng, and d erentable at all ponts n Q nt L, = ; 2, the rst-order condton of the second-stage maxmzaton problem mples " ~ (q L ) = " w (q L ) for all q L 2 Q nt L, (4) where " w (q L ) s the elastcty of the wage schedule wth respect to ablty. 7 Evdently, rms n sector choose ther workers so that the elastcty of output wth respect to ablty dvded by the elastcty of output wth respect to quantty s just equal to the elastcty of the wage schedule. 8 If (4) were to hold at just one value of q L 2 Q L, then all rms n ndustry would demand workers wth the same ablty level. Of course, such an outcome would not be consstent wth full employment of all types of workers. Instead, (4) must hold for all q L 2 Q nt L. In such crcumstances, the rms n sector are nd erent among the varous types of workers that are employed n the sector. Ths nd erence ncorporates not only the heterogeneous productvtes of the d erent workers, but also the optmal adjustment n the number of workers that the rm would make were t to swtch from one type of worker to another. The accompanyng adjustment 6 We suppress for now the country superscrpt c, because we focus on rms decsons n a sngle country. 7 The strct monotoncty of the wage functon follows from the strct monotoncty of the productvty functons ~ (q L); f wages were declnng over some range of abltes, all rms would prefer to hre the most able workers n ths range. The contnuty of the wage functon follows from the contnuty of the productvty functon; f wages were to jump at some q 0 L, rms would strctly prefer workers wth ablty a shade below q 0 L to workers wth ablty a shade above q 0 L, because the former would be only slghtly less productve but would cost dscretely less. In the appendx, we prove that the wage functon must be d erentable n the nteror of Q L for the more general case n whch managers are heterogeneous; that proof apples as well to the case wth homogeneous managers that we consder n ths secton. 8 Note that Costnot and Vogel (200) derve a smlar wage schedule, except that = for all for ther economy wth lnear output. 6

8 w(q L ) Sector wages Sector 2 wages 0.2 Sector shadow wages Sector 2 shadow wages q L Fgure : Wages of workers: homogeneous managers n quantty explans why t s the rato of the two elastctes and not just the responsveness of output to ablty that rms take nto account when they contemplate a change n the ablty levels of ther employees. The requrement that the wage functon has an elastcty " ~ (q L )= for all worker types that are hred n sector s equvalent to the requrement that the wage functon takes the form w (q L ) = w ~ (q L ) = for q L 2 Q L, (5) for some (endogenous) wage anchor, w. Ths wage functon dctates the sortng pattern for labor. Consder any worker type, say ql, that s hred n equlbrum by both sectors and s thus pad the same wage n both. Under Assumpton, workers wth ablty greater than ql can earn more n sector than n sector 2, because the wage that makes rms nd erent between these more able workers and workers of ablty q L s hgher there. Smlarly, workers wth ablty less than q L face better prospects n sector 2, because rms there are more wllng to sacr ce ablty after takng account of the optmal adjustment n quantty. It follows that the equlbrum sortng pattern has a sngle cuto level ql such that workers wth ablty above q L are employed n sector and those wth ablty below ql are employed n sector 2. Fgure shows the qualtatve features of the equlbrum wage schedule. The sold curve depcts what workers of d erent abltes actually are pad n equlbrum, consderng that those wth ablty q L ql are employed n sector and those wth ablty q L ql are employed n sector 2. The broken curves show what the wages for d erent types of workers would have to be n order to make rms n an ndustry nd erent between hrng these types and hrng the types of workers that actually are employed n the ndustry. From now on, we wll refer to these wages that re ect what rms n the opposte sector would be wllng to pay to replace ther actual employees wth these alternatve hres as the shadow wages. Notce that the shadow wages are less than the equlbrum wages pad to workers n ther actual sector of employment, as of course they must be. 7

9 Notce too that rms n ether sector are wllng to hre the workers wth the margnal ablty q L. We record our observatons about the equlbrum sortng pattern n Proposton Suppose that Assumpton holds. Then, n any compettve equlbrum wth employment n both sectors, the more able workers wth q L ql are employed n sector and the less able workers wth q L ql are employed n sector 2, for some q L 2 S L. The ntuton for ths sortng pattern should be apparent by now. Sortng s determned by comparng across sectors the ratos " ~ =. On the one hand, when " ~ s large, there s a bg return to movng hgher ablty workers to sector nasmuch as margnal ablty contrbutes greatly to productvty there. On the other hand, when s large, output n sector expands rapdly wth the number of employed workers, rrespectve of ther ablty. In such crcumstances, t makes economc sense to deploy relatvely large numbers of workers n the ndustry, and ths can be accomplshed at lower cost by hrng the workers of lesser ablty. The equlbrum sortng pattern re ects a trade-o between the returns to ablty and the returns to quantty. We can now record the remanng equlbrum condtons by nvokng labor-market clearng for the varous types of workers, mposng contnuty of the wage functon at ql, and addng a requrement that all actve rms must break even. We focus henceforth on equlbra characterzed by ncomplete specalzaton, whch arse whenever the endowment ratos and skll dstrbuton are not too extreme. Consder rst the aggregate supply and demand for workers wth ablty greater than q L. De ne e (q L ) = ~ (q L ) = `(q L ) as the e ectve labor hred per manager by a rm that employs workers wth ablty q L. Such a rm produces [e (q L )] unts of good for every manager t employs. Usng the expresson for labor demand (2) and consderng the wage schedule (5), every rm operatng n sector combnes the same amount of e ectve labor wth any one of ts managers, namely e = ( p =w ) =( ). It follows that the rms operatng n sector collectvely demand H e = H ( p =w ) =( ) unts of e ectve labor, where H s the measure of managers employed n sector. The total supply of e ectve labor s smply the measure of e ectve unts of labor among those that sort to sector. Equatng demand and supply gves Z p H = L w ~ (q L ) = L (q L ) dq L, for = ; 2: q L 2Q L Proposton tells us whch workers are employed n whch sectors,.e., Q L = [q L ; q L max] and Q L2 = [q L mn ; ql ]. So we can wrte Z p ql max H = L ~ w (q L ) = L (q L ) dq L (6) ql and Z H H 2 p 2 2 q L = L ~ w 2 (q L ) = 2 L (q L ) dq L (7) 2 q L mn 8

10 where, n (7), we have used the market-clearng condton for managers, H + H 2 = H. The wage functon must be contnuous at ql, or else the rms that hre these workers n sector could save dscretely on labor costs by downgradng ther workforce slghtly, whle sacr cng only a neglgble quantty of output. Contnuty of the wage schedule at ql mples that w ~ (q L) = = w 2 ~ 2 (q L) = 2. (8) Fnally, pro ts must be equal to zero for rms operatng n both sectors, assumng that the economy s ncompletely specalzed (otherwse they are zero n the actve sector and potentally negatve n the other). These requrements together wth (3) pn down the equlbrum salary for managers, r = p ~ (q L ) w (q L ), and also ensure that p w = 2 p 2 2 w (9) Equatons (6)-(9) jontly determne the margnal worker q L, the wage anchors w and w 2, and the measure of managers H employed n sector for any economy that produces postve amounts of both goods. The equlbrum salary of managers s gven by r = p w ; = ; 2: (0) In what follows, we are nterested n the determnants of the trade pattern between countres that d er n ther relatve endowments of labor to managers and n ther dstrbutons of worker ablty, and especally n how trade between such countres a ects ther dstrbutons of ncome. 3. Determnants of the Trade Pattern Consder two countres that trade freely at common world prces but that d er n some way n ther factor supples. Snce consumers have dentcal and homothetc tastes worldwde, the trade pattern between them can be dent ed by examnng the countres relatve outputs of the two goods at common prces. Accordngly, we nvestgate how a change n parameters re ectng factor endowments a ects relatve outputs of the two goods at gven prces. In each country, a rm n ndustry employs e = ( p =w ) =( ) unts of e ectve labor per manager, thereby producng e unts of good. Thus, aggregate output n sector s p X = H ; = ; 2: () w We can substtute the equal-pro t condton (9) nto ths expresson to elmnate the wage anchors. 9

11 Takng the rato of the resultng expressons yelds 9 X X 2 = H H H ( 2)p 2 ( )p, whch mples that the relatve output of good s greater n whchever country allocates the greater share of ts managers to producng that good. 3.. Relatve Factor Endowments Frst, suppose the two countres have the same dstrbutons of worker ablty but d er n ther relatve aggregate endowments, H= L. To nd the pattern of trade, we totally d erentate the four-equaton system comprsng (6)-(9) wth respect to H= L and examne how a change n relatve endowments a ects the allocaton of managers to sector. The algebra n the appendx establshes the followng proposton. Proposton 2 Suppose that Assumpton holds and that A L (q L) = B L (q L) for q L 2 SL A = SB L. Then country A exports the manager-ntensve good f and only f H A = L A > H B = L B. Proposton 2 represents, of course, an extenson of the Heckscher-Ohln theorem. When worker talent s dstrbuted smlarly n the two countres, the sortng of workers to sectors generates no comparatve advantages and so has no ndependent bearng on the trade pattern. Comparatve advantage s governed nstead by relatve quanttes of the factors, just as n the case of homogeneous labor. Of course, the heterogeneous workers do d er n ther sutablty for employment n the two sectors, whch means that supply elastctes wll re ect the d erence across sectors n the elastcty rato " ~ (q L )= and so too wll the volume of trade Dstrbutons of Labor Ablty Now suppose that the relatve number of managers and workers s the same n the two countres, but that country A has relatvely better workers n the sense that the pdf for worker ablty n country A s a rghtward shft (RS) of the smlar functon n country B. That s, B L (q L =) = A L (q L ) for all q L 2 SL A ; for some >, (2) whch has the nterpretaton that every worker n country A s tmes as productve as hs counterpart n the talent dstrbuton n country B: Agan, we need to totally d erentate the system of equatons (6)-(9) n order to dentfy the mpact of a rghtward shft n the talent dstrbuton on employment of managers n sector. The algebra n the appendx supports the followng concluson. 9 Ths condton can alternatvely be derved from the observaton that n sector the fracton of revenue s pad to managers,.e., ( ) p X = rh. 0

12 Proposton 3 Suppose that Assumpton holds, that H A = L A = H B = L B, and that A L (q L) s a rghtward shft of B L (q L) for some >. If " ~ (q 0 L ) > " ~ (q 00 j L ) for all q0 L ; q00 L 2 SA L [ SB L, 6= j, ; j 2 f; 2g, then country A exports good. The proposton states that the country that has the superor labor force exports the good produced n the ndustry where worker ablty contrbutes more elastcally to productvty. Notce that ths need not be the good produced by the country s most able workers nasmuch as sortng re ects the rankng of " ~ (q L )= versus " ~ 2 (q L )= 2, whereas the trade pattern depends only on the rankng of " ~ (q L ) versus " ~ 2 (q L ). Ths result can be understood by thnkng about the sources of comparatve advantage n ths settng. Wth H A = L A = H B = L B, the cross-sectoral d erence n factor ntensty s not a source of comparatve advantage for ether country. Meanwhle, wth " ~ (q L ) d erent from " ~ 2 (q L ), worker ablty contrbutes d erently to productvty n the two sectors. Country A, whch s relatvely better endowed wth more able workers, enjoys a technologcal comparatve advantage n the ndustry n whch ablty matters more for output The E ects of Trade on Income Dstrbuton We are especally nterested n the relatonshp between trade and ncome dstrbuton n a world wth heterogeneous factors of producton. As n other neoclasscal trade models, commodty prces medate the lnk between trade and earnngs. The openng of trade (or subsequent trade lberalzaton) generates an ncrease n the relatve prce of a country s export good, whch n turn alters the demand for d erent factors and factor types. Accordngly, we examne the comparatve statc response of the wage schedule and manageral salares to a change n the relatve prce of the nal goods. Note rst that the wage functon (5) pns down the relatve wages of the varous workers who are employed n a gven sector. A change n relatve prce can alter the relatve pay only of workers who are ntally or ultmately employed n d erent ndustres. The calculatons n the appendx establsh the followng ndngs. Proposton 4 Suppose that Assumpton holds. Then when ^p > 0, () ^w > ^w 2 ; () f 2 ; then ^w > ^p > ^r > 0 > ^w 2 ; () f > 2 and s L (q L ) 0; then ^w ^w 2 > ^p > 0 > ^r; and (v) f < 2 and s L (q L ) 0; then ^r > ^p > 0 > ^w ^w 2. Part () of Proposton 4 says that any worker employed n ndustry gans relatve to any worker employed n ndustry 2 when the relatve prce of good rses. The remanng parts of the proposton capture the two dstnct n uences on factor returns n an economy wth heterogeneous 0 In the specal case n whch ~ (ql) s a power functon for = ; 2,.e., ~ (ql) = aq L for some a; > 0, " ~ (ql) 0 > " ~ j (ql) 00 for all ql 0 and ql 00 f and only f > j. Moreover, n ths case, s L (q L) > 0 for all q L f and only f = > 2= 2. Evdently, the condtons of Proposton 3 are easly sats ed. When ~ (ql) s not a power functon for = ; 2, the requrement that " ~ (ql) 0 > " ~ j (ql) 00 for all ql; 0 ql 00 2 SL A [ SL B, 6= j, ; j 2 f; 2g s not trval, but t can be weakened nto a comparson of the average elastctes of productvty wth respect to ablty n the two sectors. See the proof of Proposton 3 n the appendx. In what follows, we use a hat over a varable to ndcate an ncremental, proportonal change;.e., ^z = dz=z.

13 labor. The cross-sectoral d erence n factor ntenstes ntroduces a force akn to that n the standard Heckscher-Ohln model wth homogeneous labor, whereby real wages tend to rse and real manageral salares tend to fall f the sector experencng the ncrease n relatve prce s the more labor ntensve of the two. But the heterogenety of labor mples that d erent workers are not equally pro cent as potental employees n the two sectors, whch ntroduces a force akn to that n the Rcardo-Vner model (see, e.g., Jones, 97). Indeed, our result s remnscent of ndngs n a model wth mperfect factor moblty (Mussa, 982) or partally moble captal (Grossman, 983). That s, f the factor ntensty d erence across ndustres s large (.e., 6= 2 ) and the force for nter-ndustry sortng of the margnal worker types s muted (.e., s L (ql ) 0), then all types of the factor used ntensvely n sector must gan, whle all types of the factor used ntensvely n sector 2 must lose (parts () and (v) of the proposton). On the other hand, f the factor ntensty d erence s small (.e., 2 ) and the d erent margnal worker types are mperfect substtutes n the two sectors (s L (ql ) > 0), then all workers ntally employed n the expandng sector wll gan, all workers that contnue to be employed n the contractng sector wll lose, and the e ect on the well beng of managers wll depend on ther consumpton pattern (part () of the proposton). In the former case, manageral salares rse n terms of the mport good but fall n terms of the export good, as n the Rcardo-Vner model wth moble managers and sector-spec c labor. In the latter case, real manageral salares rse f the export sector s manager ntensve and fall f t s labor ntensve, as n the Heckscher-Ohln model. In summary, when managers are homogeneous and thus matchng s ndetermnate, trade has no e ect on wthn-ndustry wage nequalty; the relatve earnngs of any par of workers employed n the same sector s pnned down by (5). 2 Meanwhle, an ncrease n the prce of good rases the wage anchor n sector relatve to the wage anchor n sector 2 (see part () of the proposton). And snce the hgher-ablty, hgher-wage workers are employed n sector, ths mples that by rasng wages n sector relatve to wages n sector 2 an ncrease n the prce of good ncreases overall wage nequalty n country A, whle reducng wage nequalty n country B. These results provde a benchmark for comparng those n Secton 4, where market forces determne the matchng of workers wth heterogeneous managers and where trade can a ect ncome dstrbuton by alterng the pattern of matches. 3.3 Sortng Reversals So far, we have used Assumpton to characterze the sortng of heterogeneous workers and the resultng trade structure. In ths nal part of the secton on homogeneous managers, we clarfy what can happen when s L (q L ) swtches sgn. Frst note that f ~ (q L ) s a power functon for = ; 2, the functon s L (q L ) does not depend on q L nasmuch as the elastctes of productvty wth respect to ablty then are constants. In such 2 We should perhaps menton that, by alterng the composton of workers n each sector, trade wll a ect any measure of wthn-ndustry wage nequalty (such as, for example, the Gn coe cent) that does not hold the set of workers n the comparson xed. 2

14 w(q L ) Sector wages Sector 2 wages 0.5 Sector shadow wages Sector 2 shadow wages q L Fgure 2: Wages wth a reversal of sortng crcumstances, s L (q L ) s ether always postve or always negatve, and we can assume s L (q L ) > 0 wthout loss of generalty, because ths only amounts to a partcular labelng of the sectors. However, when ~ (q L ) s not a power functon for some, the assumpton that s L (q L ) has a unform sgn for all q L 2 S L mposes meanngful restrctons on the forms of the productvty functons and the support of the dstrbuton of worker talent. Wthout these restrctons, we cannot be sure that the most able workers sort nto one sector and the least able workers sort nto the other. To llustrate what can happen when s L (q L ) changes sgns, suppose that the productvty of a rm n sector that hres workers of ablty q L s gven by ~ (q L ) = q L + = ; > 0; < 0 for = ; 2: (3) Ths spec caton mples a constant elastcty of substtuton between the ablty of workers and the ablty of managers n generatng the productvty of the rm, and that worker and manageral ablty are, n fact, complements. Of course, wth homogeneous managers, rms have no possblty to adjust manager type n order to take advantage of ths complementarty. Nonetheless, the CES spec caton for productvty represents a legtmate and even a plausble functonal form. When productvty takes the form ndcated n (3), the elastcty of productvty wth respect to worker ablty n sector s gven by " ~ (q L ) = q L = q L +. If 6= 2 then " ~ (q L ) " ~ 2 (q L ) necessarly swtches sgns on q L 2 [0; +) and therefore s L (q L ) may swtch sgns on the support of the dstrbuton of worker ablty, dependng on the ndustry factor ntenstes and the range of the talent dstrbuton. Fgure 2 depcts an equlbrum wage schedule for an economy n whch s L (q L ) < 0 for low values of q L and s L (q L ) > 0 for hgh values of q L. 3 In ths economy, the most and least able workers sort to sector whle a mddle range of workers s hred nto sector 2. The thn sold curves n the gure depct the wages of workers employed n sector as a functon of ther ablty, whle 3 See Lm (203) for the functonal forms and parameter values that were used to generate ths gure. 3

15 the thck sold curve depcts the wages of workers employed n sector 2. The broken thn curve depcts the shadow wage for workers n sector 2,.e., the wage o ers they could garner were they to seek jobs n sector. Smlarly, the broken thck curve depcts the shadow wages avalable n sector 2 for workers actually employed n sector. Clearly, each worker sorts nto the ndustry that o ers hm the hghest wage. Fgure 2 represents an economy n whch = 2 = 0:5,.e., the ndustres have smlar factor ntenstes. However, 6= 2, whch generates the d erent elastctes of productvty at d erent levels of ablty. The comparatve statcs reveal an nterestng response of wages to relatve prce changes for these parameter values. Inasmuch as the factor ntenstes are common to the two ndustres, there are no Stolper-Samuelson forces at work. But the workers that sort to sector are better suted for employment there than ther counterparts workng n sector 2. The forces akn to those n a Rcardo-Vner model mply that when p rses, the real wages of all workers employed n sector also rse, whle the real wages of all workers employed n sector 2 declne. In short, an ncrease n the relatve prce of good generates ncome gans for workers wth hgh or low wages but ncome losses for those n the mddle of the wage dstrbuton. 4 When the two sectors d er n ther factor ntenstes, the Stolper-Samuelson forces wll agan play a role n determnng the e ects of trade on the wage dstrbuton. Take, for example, a case n whch = 0:9 and 2 = 0:, so that sector s much more labor ntensve than sector 2. We have solved ths example numercally for varous sets of the other parameter values. 5 In all such cases, we found that an ncrease n the prce of good rases both wage anchors more than n proporton to the prce change, so that all workers gan n real ncome. Meanwhle, the salary of managers falls. These results are famlar from the Stolper-Samuelson theorem, and they are smlar to what we found wth great dspartes n factor ntenstes for economes that satsfy Assumpton. We nd as well that an ncrease n p bene ts workers employed n sector more than those employed n sector 2, n keepng wth our observatons that workers are partally spec c to ther ndustry of employment due to comparatve productvty d erences. 6 Prce changes do not a ect relatve wages for workers employed n the same ndustry, even f those workers are at opposte tals of the talent dstrbuton as s the case for some pars of workers that sort to sector. 4 For ths example, we calculate that a 5% ncrease n p rases the wage anchor w by 5:7%, whle depressng the wage anchor w 2 by 4:2%. Managers salares rse by 4:3%, whch s proportonately less than the ncrease n prce. 5 As one example, we have solved the model for the case n whch world prces are (p ; p 2) = (; ) and the economy has an aggregate endowment of H; L = (; ). In ths example, we assumed that worker ablty s drawn from a truncated Pareto dstrbuton on the support S L = [0:8; :8] wth the shape parameter 3, and that the technologcal parameters are gven by ( ; ; ) = (0:9; 0:7; ) and ( 2 ; 2; 2 ) = (0:; 0:3; 20). In the computed equlbrum, sector 2 employs workers wth q L 2 [:034; :2] and 0:953 managers. The wage anchors are w = 0:78 and w 2 = 0:434 whle the managers earn a salary of r = 0: Usng the parameter values detaled n the prevous footnote, we nd that a 5% ncrease n the prce p generates a wage hke of 5:6% for workers employed n sector, a wage hke of 5:4% for workers employed n sector 2, and a slary reducton of 0:6% for all managers. 4

16 4 Heterogeneous Managers We now ntroduce manager heterogenety and model the dversty of manager types smlarly to that for workers. More spec cally, we post a mass H c of managers n country c and a probablty densty c H (q H) of managers wth ablty q H for q H 2 SH c = [qc H mn ; qc H max ]. We take the supply of managers and ther ablty dstrbuton as gven throughout the analyss. To capture complementartes between workers and managers wthn a producton unt, we take the productvty functons (q H ; q L ) for = ; 2 to be log supermodular;.e., worker ablty contrbutes relatvely more to productvty when the better workers are teamed wth a more able manager than when they are teamed wth a less able manager. For convenence, we also assume that (q H ; q L ) s strctly ncreasng and twce contnuously d erentable. In such crcumstances, log supermodularty of the productvty functons mples that H (q H ; q L ) = (q H ; q L ) s ncreasng n q L and L (q H ; q L ) = (q H ; q L ) s ncreasng n q H, where F (q H ; q L ) s the partal dervatve of (q H ; q L ) wth respect to q F, F = H; L. For the most part, we shall focus on the case where (q H ; q L ) s strctly log supermodular and then we wll nvoke Assumpton 2 () S H = [q H mn ; q H max ], 0 < q H mn < q H max < +; () (q H ; q L ) s strctly ncreasng, twce contnuously d erentable, and strctly log supermodular for = ; 2. However, we wll also make occasonal reference to the case of weak log supermodularty that arses when the productvty functons are multplcatvely separable n ther two arguments. 7 We can proceed as before by treatng the rm s pro t maxmzaton problem n stages. Frst, the rm takes as gven the ablty of ts workers and of ts manager and chooses the sze of ts producton team. Ths yelds the labor demand per manager as a functon of the employee types, namely p ` (q L ; q H ) = (q H ; q L ). w (q L ) Substtutng ths expresson for labor demand nto that for rm pro ts yelds ~ (q H ; q L ) = p (q H ; q L ) w (q L ) r (q H ) : (4) Next, the rm dent es the most sutable workers to combne wth the manager, takng the contnuous and strctly ncreasng wage schedule as gven. 8 Ths yelds a pro t functon, (q H ) = max q L 2S L ~ (q H ; q L ), (5) for q H 2 S H ; = ; 2. Fnally, the rm selects q H to maxmze (q H ), gven the contnuous and 7 In partcular, we shall refer to a case of Cobb-Douglas productvty, whch arses when the productvty functons are multplcatvely separable and have constant elastctes of productvty wth respect to the ablty of ether factor. In such crcumstances, we can wrte (q H; q L) = q H q L for some > 0 and > 0: 8 The wage schedule must be contnuous and strctly ncreasng for reasons analogous to those that apply wth homogeneous managers. 5

17 strctly ncreasng salary schedule, r (q H ). In an equlbrum, rms must be nd erent between the varous managers that are employed n an ndustry, or else all would hre a partcular type (or types) and some managers would be unemployed. We show n the appendx that the soluton to the rm s pro t-maxmzaton problem and the requrement that rms must be nd erent among the managers that sort to an ndustry together generate equlbrum allocaton sets Q L and Q H that are unons of closed ntervals (where Q F represents the set of types of factor F that sorts to ndustry, for F = H; L and = ; 2). Moreover, under Assumpton 2, there must be postve assortatve matchng (PAM) of managers and workers wthn each sector. In other words, among the managers and workers that sort to a sector, the better workers are teamed wth the better managers. 9 apply economy-wde. However, as we shall see, PAM need not Let m (q H ) denote the soluton to (5);.e., m (q H ) s the common ablty level of the workers who would be teamed wth a manager of ablty q H f that manager happens to be employed n sector. The equlbrum matchng functon for the economy, m (q H ), conssts of m (q H ) for q H 2 Q H and m 2 (q H ) for q H 2 Q H2. The matchng functon generates a par of closed graphs M = [fq H ; q L g j q L 2 m (q H ) for all q H 2 Q H ], = ; 2, where M represents the producton unts that form n sector n equlbrum. These graphs comprse a unon of connected sets M n such that m (q H ) s contnuous and strctly ncreasng n each set but may jump dscontnuously between them. Consder an equlbrum wth ncomplete specalzaton, so that a postve measure of managers sorts to each sector. All rms that are actve n sector earn zero pro ts, whch mples r (q H ) = p [q H ; m (q H )] w [m (q H )] for all q H 2 Q H ; = ; 2: (6) Contnuty of the wage and salary schedules mples that both functons are d erentable almost everywhere. Moreover, pro t maxmzaton and (6) mply that, at all nteror ponts n a connected subset M n of M, the salary functon r () and the wage functon w () are d erentable; see the appendx for proof. It follows that the soluton to (5) must satsfy the rst-order condton m (q H ) L [q H ; m (q H )] [q H ; m (q H )] = m (q H) w 0 [m (q H )] w [m (q H )] for all fq H ; m (q H )g 2 M n;nt ; n 2 N ; = ; 2; where M n;nt s the nteror of the set M n. Also, 9 Wth strct log supermodularty of the productvty functon (q H; q L), PAM wthn sector follows drectly from the arguments n Eeckhout and Krcher (202). If (q H; q L) s only weakly log supermodular, as n the case of Cobb-Douglas productvty, there always exsts an equlbrum wth PAM n each ndustry, although the equlbrum s not unque. Indeed, n ths case, the matchng of workers and managers n a sector s not well determned by the model. Such ndetermnacy re ects that the relatve productvty of a team of more able workers compared to a team of less able workers s ndependent of the type of the manager. However, as we show n the appendx, the ndetermnacy of matchng n the Cobb-Douglas case does not mply ndetermnacy of allocaton sets, output levels, or factor prces; these outcomes n fact are unque. (7) 6

18 (6) and (7) mply that q H H [q H ; m (q H )] ( ) [q H ; m (q H )] = q Hr 0 (q H ) r (q H ) for all fq H ; m (q H )g 2 M n;nt ; n 2 N ; = ; 2. Notce that the left-hand sde of (7) represents a rato of elastctes, namely the elastcty of productvty n sector wth respect to worker ablty dvded by the elastcty of output wth respect to labor quantty. Pro t maxmzaton requres that ths rato be equal to the elastcty of the wage schedule. Ths s qute analogous to (4) for the case of homogeneous managers, except that now the elastcty of productvty wth respect to worker ablty re ects the matches that take place n equlbrum and therefore the sortng of managers to sectors. Equaton (8) s an analogous condton that leaves rms nd erent among the managers hred n sector ; t equates the quotent of the elastcty of productvty wth respect to manager ablty and the elastcty of output wth respect to the number of managers to the elastcty of the salary schedule Sortng wth Heterogeneous Managers How do workers and managers sort to ndustres? Recall Fgures and 2 n Secton 3 that show equlbrum wage and shadow wage functons for the case of homogeneous managers. We argued that the slope of the wage functon must be greater just to the rght of any boundary pont between connected sets of workers that sort to d erent ndustres than the slope of the shadow wage functon showng what the other ndustry would be wllng to pay. An analogous condton apples when managers are heterogeneous. Let q y L be the boundary between some sets of workers that sort to d erent ndustres, so that workers wth ablty just above q y L sort to one sector and workers wth ablty just below q y L sort to the other. In equlbrum, the wage functon w (q L) must be at least as steep to the rght of q y L as t s to the left; otherwse the rms that employ workers wth abltes just above q y L could earn greater pro ts by slghtly downgradng ther workforce and those that hre workers wth abltes below q y L could earn earn greater pro ts by slghtly upgradng thers. A smlar argument apples for managers, whch mples that the salary functon r (q H ) must be (weakly) steeper just to the rght of any boundary pont q y H than just to the left of such a pont. Now that we have lad out the equlbrum condtons that gude matchng and sortng, we turn to the requrements for factor-market clearng. To ths end, consder some connected set of managers [q Ha ; q H ] that sorts to ndustry and the set of workers [m (q Ha ) ; m (q H )] wth whom these managers are matched n equlbrum. A pro t-maxmzng rm n sector that employs a manager wth ablty q H and workers of ablty q L demands ` (q H ; q L ) = [ r (q H )] = [( ) w (q L )] workers per manager. Snce the matchng functon s everywhere ncreasng, t follows that H Z qh q Ha r (q) ( ) w [m (q)] H (q) dq = L Z m(qh ) m(q Ha ) L [m (q)] dq ; 20 The technologes exhbt constant returns to scale n the quanttes of the two factors, so the elastcty of output wth respect to the number of managers n sector s. (8) 7