AB Dscusson Papr 7/2011 Artcls on labour markt ssus ncom taxs, subsds to ducaton, and nvstmnts n human captal Conctta Mndolccho Dmtr Paoln Tto Ptra
ncom taxs, subsds to ducaton, and nvstmnts n human captal Conctta Mndolccho AB Dmtr Paoln DER and CRENoS, Unvrstà d Sassar Tto Ptra DSE, Unvrstà d Bologna Mt dr Rh AB-Dscusson Papr wll das Forschungsnsttut dr Bundsagntur für Arbt dn Dalog mt dr xtrnn Wssnschaft ntnsvrn. Durch d rasch Vrbrtung von Forschungsrgbnssn übr das ntrnt soll noch vor Drucklgung Krtk angrgt und Qualtät gschrt wrdn. Th AB Dscusson Papr s publshd by th rsarch nsttut of th Grman Fdral Employmnt Agncy n ordr to ntnsfy th dalogu wth th scntfc communty. Th prompt publcaton of th latst rsarch rsults va th ntrnt ntnds to stmulat crtcsm and to nsur rsarch qualty at an arly stag bfor prntng. AB-Dscusson Papr 7/2011 2
Contnts Abstract.......................................... 4 Zusammnfassung..................................... 4 1 ntroducton...................................... 5 2 Th Modl....................................... 8 3 Equlbrum...................................... 10 4 Effcncy proprts of qulbra........................... 14 4.1 Constrand optmal allocatons......................... 16 4.2 Wlfar mprovng tax polcs.......................... 17 5 Conclusons...................................... 19 6 Appndx........................................ 20 7 Rfrncs....................................... 28 AB-Dscusson Papr 7/2011 3
Abstract W study a two-sctor conomy wth nvstmnts n human and physcal captal and mprfct labor markts. Human and physcal captal ar htrognous. Workrs and frms ndognously slct th sctor thy ar actv n and choos th amount of thr sctor-spcfc nvstmnts. To ntr th hgh-skll sctor, workrs must pay a fxd cost that w ntrprt as a drct cost of ducaton. Gvn th dstrbuton of th agnts across sctors, n qulbrum, n ach sctor thr s undrnvstmnt n both human and physcal captal, du to non-contractblty of nvstmnts. A scond sourc of nffcncy s rlatd to th slf-slcton of th agnts nto th two sctors: typcally too many workrs nvst n ducaton. Undr sutabl rstrctons on th paramtrs, th jont ffct of th two dstortons s that qulbra ar charactrzd by too many popl nvstng too lttl ffort n th hgh skll sctor. W also analyz th wlfar proprts of qulbra and study th ffcts of svral tax polcs on th total xpctd surplus. n partcular, w consdr th qulbrum assocatd wth a flat labor ncom tax. Undr sutabl rstrctons on th paramtrs, a rvnu nutral progrssv chang n th margnal tax rats s wlfar mprovng. Zusammnfassung Wr untrsuchn n Zw-Sktorn-Ökonom mt nvsttonn n Human- und physschs Kaptal und unvollkommnn Arbtsmärktn. Human- und physschs Kaptal snd htrogn. Arbtr und Untrnhmn wähln ndogn sowohl dn Sktor, n dm s tätg wrdn, als auch d Mng dr sktorspzfschn nvsttonn. Für Arbtr falln Fxkostn an, falls s m Sktor tätg wrdn wolln, dr ausschlßlch hoch qualfzrt Arbtr bschäftgt. Für n ggbn Vrtlung dr Agntn übr d Sktorn st das Glchgwcht dsr Ökonom durch Untrnvstton n Human- und physschs Kaptal n bdn Sktorn gknnzchnt. Ursächlch dafür st d Annahm, dass nvsttonn ncht vrtragsfähg snd. En zwt Ursach von nffznz st d Slbstslkton von Agntn n d bdn Sktorn: typschrws wähln zu vl Arbtr dn Sktor für Hochqualfzrt. Zusammn bwrkn ds bdn Vrzrrungn, dass m Glchgwcht zu vl Arbtr m Hochqualfzrtn-Sktor tätg wrdn wolln, d dab abr nsgsamt zu wng Bldungsanstrngung n Humankaptal nvstrn. Wtr untrsuchn wr d glchgwchtgn Wohlfahrtswrkungn von Sturn. Es zgt sch, dass für ralstsch Paramtrrstrktonn n budgtnutral progrssv Ändrung dr margnaln Stursätz wohlfahrtsstgrnd wrkt. JEL classfcaton: J24; H2 Kywords: Human captal; Effcncy; Labour ncom tax Acknowldgmnts: Ths papr s part of th Ph.D. dssrtaton at RES, Unvrstè Catholqu d Louvan, of th frst author. W thank for hlpful commnts th mmbrs of hr dssrtatons commtt R. Boucchkn, V. Vandnbrgh, M. Blot, B. Dcrus, B. Van dr Lndn, P. Pstau, and partcpants of smnars at th Unvrsty of Luxmbourg, at th Unvrstà d Salrno, at ASSET 2009, at th 9 th Journs L.-A. Grard Vart, at ESWC 2010 and at EEA 2010. W acknowldg th fnancal support of MUR-PRN 2006 and Fondazon Banco d Sardgna. Th frst author acknowldgs th support of th Programma Vstng Profssor of th Unvrstà d Sassar taly. AB-Dscusson Papr 7/2011 4
1. ntroducton n th last fw dcads, causs and consquncs of nvstmnts n human captal hav bn a cntral fld of rsarch du to svral motvatons. Among thm, th rlvanc of human captal xtrnalts n growth thory, and th ssus rlatd to th dynamcs of th wag prmum and, mor gnrally, to th voluton of ncom dstrbuton. Stll, th analyss of human captal xtrnalts s far from sttld from both th mprcal and th thortcal vwponts. Emprcally, t s not obvous that thr ar sgnfcant, postv dffrncs btwn socal and prvat rturns, at last at th lvl of subsds prvalng n most Wstrn countrs. 1 From a thortcal vwpont, th mcroconomc mchansm gnratng th xtrnalty s not fully undrstood. A bttr undrstandng of ts natur has polcy rlvanc. Ths s tru vn f on s wllng to tak for grantd that thr ar no sgnfcant, unxplotd, postv xtrnalts, bcaus ths s typcally obtand wth hgh subsds to ducaton. 2 n ths papr, w xtnd th mcroconomc analyss of th dstortons rlatd to nvstmnts n human captal and drv som rsults on th wlfar ffcts of dffrnt polcs: fxd taxs/subsds on th drct cost of th acquston of hgh skll human captal, and taxs on labor ncom, or - quvalntly n our st-up - on th nvstmnt n human captal. W consdr conoms wth thr ky faturs: 1. Ex-ant, workrs ar htrognous, whl frms ar dntcal, 2. nvstmnts n human and physcal captal ar non-contractbl, 3. Thr ar two sparat sctors mployng dffrnt knds of human and physcal captal, so that an agnt must choos both th lvl of hs/hr nvstmnt and ts typ. Th conomy s bascally a two-sctor gnralzaton, wth sctor spcfc nputs, of th modl consdrd n Acmoglu 1996, whch ams to provd an xplct qulbrum foundaton for th xstnc of postv xtrnalts rlatd to human captal accumulaton. n hs framwork, frms and workrs choos th amount of thr nvstmnts. Thn, thy ar matchd randomly but prsrvng full mploymnt, and ncom dstrbuton s dtrmnd by a barganng procss. Whl a smlar analyss could b carrd out n svral framworks wth th proprts lstd abov, w focus th analyss on a Roy modl of nvstmnts n human captal whch s as clos as possbl to th on analyzd by Acmoglu. ndd, aftr agnts hav chosn th sctor thy ar gong to b actv n,.., th natur of thr nvstmnt, our modl rducs to a par of sparatd Acmoglu s conoms. n our st-up, ncom dstrbuton taks plac through barganng, too. Howvr, whn workrs ar htrognous, th drvng faturs of our rsults ar asymmtrc nformaton on th workrs typs and non-contractblty of th nvstmnts. Th barganng st-up s, of cours, mportant, but t dos not affct som ky aspcts of th wlfar rsults. 3 Our man dpartur from Acmoglu 1996 s that w adopt th noton of human captal put forth n Roy 1951: thr ar dstnct markts for hgh skll and low skll labor, and w assum that thy ar prfctly non-substtutabl. Howvr, contrary to what s oftn assumd n Roy modls, onc a workr has slctd th typ of human captal sh wants to acqur, sh stll has 1 For th U.S.A., a ngatv concluson s rachd, for nstanc, by Hckman, Layn-Farrar, and Todd 1996 and by Acmoglu and Angrst 2001. For E.U. countrs, th rsults n D la Funt 2003 ar also ngatv. S also Krugr and Lndhal 2000. 2 n 2005, n th OECD avrag, 85.5% of th drct cost of ducaton all lvls ncludd s fnancd by publc sourcs s OECD 2008, Tabl B3.1, p. 251. Th EU19 avrag s 90.5%. At th trtary lvl, ths prcntags ar, rspctvly, 73.1% and 82.5% Tabl B3.2b, p. 253. 3 ndd, on can dfn conoms wth prfctly compttv spot labor markts, asymmtrc nformaton and lack of contractblty, whr thr s stll a ngatv xtrnalty n human captal nvstmnts. AB-Dscusson Papr 7/2011 5
to dcd how much ffort to nvst. Thn, human captal translats on-to-on nto ffcncy unts of hgh skll low skll, rspctvly labor. 4 Hnc, ach workr maks two sparat chocs, at th ntnsv and th xtnsv margn. Most of th rcnt ltratur taks a dffrnt pont of vw, adoptng th ffcncy unts approach wth homognous human captal, thrfor rulng out, by assumpton, all th consquncs of slf-slcton of agnts nto dffrnt labor markts, whch ar, nstad, rlvant from both th thortcal and th mprcal vwponts. 5 Wth mprfct markts and slf-slcton of workrs nto dffrnt labor markts, two dstnct dstortons ar at work. Lack of contractblty of nvstmnts and th barganng st-up gnrat a hold-up problm, nducng an nffcntly low lvl of nvstmnts, n human and physcal captal of both typs hnc, n ach sctor. Ths s th ky mchansm at play n Acmoglu s papr. Scondly, gvn that workrs ar htrognous, whn a subst of thm swtchs from on sctor to th othr, thr s an mpact on th dstrbuton of rturns of th frms, hnc on thr optmal nvstmnts. n turn, ths affcts th optmal lvl of nvstmnts of workrs. Ths scond potntal sourc of dstorton s ndpndnt of th random matchng st-up, and s at work vn whn spot labour markts ar prfctly compttv but lack of contractblty and asymmtrc nformaton hold. 6 Ths mchansm has bn analyzd n th conomcs of ducaton ltratur at last snc Btts 1998. 7 Thrfor, n our st-up, publc polcs hav two dstnct ffcts on xpctd total surplus, our masur of wlfar. Th frst s thr mpact on th lvl of th optmal nvstmnts of th agnts acqurng a sctor-spcfc skll: w wll rfr to t as ncntv ffct. Th scond s thr mpact on th agnts dstrbuton across markts,.., th composton ffct. n pur Roy modls wth slf-slcton, but no choc of th nvstmnt ffort only th composton ffct s at play. n pur ffcncy-unts modls wthout slf-slcton only th ncntv ffct s at work. As usual, a hold-up problm on th rturns on th nvstmnt n human captal nducs undrnvstmnt n ducaton: lss workrs nvst n ducaton and ach workr nvsts lss ffort than n th cas of full appropraton of th margnal rturn of th nvstmnt. Th mpact on wlfar of th composton ffct s lss obvous. An mprovmnt of th condtonally xpctd lvl of human captal has always a postv ffct on qulbrum utlts of all th workrs and on th profts of th frms whch rman actv n th sam sctor. Th xpctd producr s surplus of th frms whch swtch sctor may actually dcras, but, undr sutabl rstrctons on th paramtrs, th total ffct s always postv. Bar n mnd that, n our conomy, thr s always full mploymnt and, thrfor, th classcal congston xtrnalty, charactrzd by th volaton of th Hosos condton, s absnt. W consdr two sparat sctors, usng sctor spcfc nputs hgh/low skll human and physcal captal. Th crucal proprty s that human and physcal captal ar htrognous. To dntfy on typ of captal wth on sctor somwhat smplfs th st-up and sharpns th wlfar rsults. Howvr, th two dstnct dstortons would b at work vn wth just on productv 4 As usual, w can also ntrprt ffort n th acquston of human captal as lastc supply of labor of a gvn skll. 5 A survy supportng ths clam s n Sattngr 1993. A mor rcnt dscussons of th dffrnt mprcal mplcatons of ffcncy unts vs. Roy modls s, for nstanc, n Carnro, Hckman, and Vytlacl 2001. nvstmnts n human captal n a two-sctor conomy wth frctons du to random matchng but wth prfctly nlastc supply of human and physcal captal hav bn studd n Sattngr 2003, Charlot and Dcrus 2005, and Mndolccho, Paoln, and Ptra 2010. 6 Wth prfctly compttv spot labor markt, th hold-up problm dsappars, and n ach sctor takng as gvn th dstrbuton af agnts nvstmnts ar at thr constrand ffcnt lvl. Howvr, du to asymmtrc nformaton and lack of contractblty, th composton ffct stll nducs constrand nffcncy of qulbra, whch ar always charactrzd by ovrnvstmnt n ducaton. 7 n th contxt of random matchng modls, t has bn frst studd n Charlot and Dcrus 2005. AB-Dscusson Papr 7/2011 6
sctor mployng both hgh and low skll labor, provdd that thr s a suffcnt dgr of substtutablty. Bar n mnd that whnvr n what follows w mnton th two sctor structur of th conomy, w mplctly man that th two sctors us dffrnt knds of human and physcal captal. Fnally, n our full mploymnt st-up, th lastcty of nvstmnts n physcal captal plays a ky rol. ndvdual ffort of workrs dpnds upon th composton of th pool of workrs n a sctor only ndrctly, bcaus of ts drct ffct on th optmal lvl of nvstmnts n physcal captal, whch, n turn, s ncrasng n th condtonal xpctaton of th ffort of th workrs actv n a markt. Hnc, our modl adopts, n a parsmonous way, th smplst structur of th conomy whch may dlvr th basc nsght. W provd two sts of rsults concrnng th ffcncy proprts of th qulbra. Frst, w show that an approprat polcy of subsds to th nvstmnts and taxs on th drct costs of ducaton can mplmnt th constrand ffcnt allocaton. Scondly, w consdr th qulbrum assocatd wth an arbtrary but not too hgh flat labor ncom tax and study th wlfar ffcts of changs n th tax structur. Ths allows us to gt som ntuton concrnng th rlatv magntuds of ncntv and composton ffcts. Som of th rsults n Acmoglu 1996 survv n our class of conoms. For nstanc, n both cass, th human captal xtrnalty s rlatd to ts sctor-spcfc avrag lvl. Thr ar, on th othr hand, sharp dffrncs wth rspct to th polcy prscrptons: n th on-sctor modl, subsds to nvstmnts n human captal or to labor supply ar unambguously bnfcal. Ths s bcaus only th ncntv ffct s at play: a subsdy to th nvstmnts n human captal or a rducton of th labor ncom tax rat of any subst of agnts ncrass thm and, thrfor, thr xpctd valu as a frst ordr ffct. Ths has a postv mpact on th frms nvstmnt dcsons and, n turn, furthr ncrass th optmal nvstmnt of all th workrs. Ths chan of postv fdbacks guarants that ths s wlfar mprovng. To rformulat th pont dffrntly: n on-sctor conoms, thr s a unqu dstorton nducd by th hold-up problm whch nducs undrnvstmnt for both frms and workrs. Any polcy ncrasng th nvstmnts of any subst of agnts s wlfar mprovng. Wth two sctors, th ncntv ffct of a polcy can b strngthnd, waknd, or ovrturnd, by ts composton ffct. Consdr, for nstanc, a rducton n th margnal tax rat on low labor ncom n our st-up: on th ncom of low-skll workrs. f total factor productvts ar suffcntly dvrs across sctors and workrs suffcntly htrognous, ths always ncrass total surplus, bcaus th postv ffct on ndvdual ffort n th low-skll sctor s strngthnd by th composton ffct,.., by th mprovmnt of th xpctd human captal of th pool of workrs n both markts. An ncras n taxs on th drct costs of ducaton also ncrass total surplus, just bcaus of ts composton ffct. On th othr hand, a dcras n th margnal ncom tax rat for hgh-skll workrs has a frst ordr postv ncntv ffct on thr nvstmnts, but a ngatv composton ffct. Hnc, t always has a ngatv mpact on th qulbrum utlty of low-skll workrs and on th qulbrum profts of th frms actv n that sctor. Th total ffct for th agnts actv n th hgh-skll sctor may b postv or ngatv, accordng to th magntuds of th postv ncntv ffct and th ngatv composton ffct. W provd a robust xampl whr th ffct of such a tax rat rducton on total surplus s ngatv. W conclud consdrng rvnu nutral tax changs: th most ntrstng rsult s that, undr our assumptons, a progrssv chang n th margnal labor ncom tax rats s wlfar mprovng. Thr s a larg ltratur on th ffcts of subsds to ducaton and of labor ncom taxs on accumulaton of human captal. Th usual argumnts favorng subsds hng thr on thr postv xtrnalty ffcts, or on th xstnc of lqudty constrants. Addtonally, subsds AB-Dscusson Papr 7/2011 7
to ducaton hav bn analyzd as on of th componnts of th optmal mx of rdstrbutv polcs s Bovnbrg and Jacobs 2005, Jacobs 2005, 2007, Jacobs and Bovnbrg 2008, Jacobs, Schndlr and Yang 2009, Schndlr and Wgrt 2008, 2009. Th last two aspcts may b both mprcally and thortcally mportant, but w abstract from thm, focussng th analyss on th pur ffcncy ssu rlatd to th prsnc of a hold-up problm and of slf-slcton. Th classcal analyss of th ffcts of labor ncom tax on nvstmnts n human captal startd wth th smnal paprs by Bn-Porath 1970, Boskn 1975 and Hckman 1976. 8 A flat labor ncom tax has a ngatv mpact on human captal accumulaton just bcaus of th non-dductblty of th drct costs of ducaton. On th othr hand, by dprssng th nt ntrst rat, n fully spcfd lf-cycl modls of consumr bhavor, a tax on total ncom may actually hav a postv ffct. Eaton and Rosn 1980 xtnd th analyss to unnsurabl multplcatv wag uncrtanty, pontng out that a flat arnng tax affcts nvstmnts n human captal through ts ffcts on thr rsknss and va an ncom ffct on th atttud toward rsk s, also, Andrbrg and Andrsson 2003, and Andrbrg 2009. Consdr now a progrssv ncom tax compard wth a rvnu-nutral flat on. Th canoncal concluson s that t dscourags nvstmnts at th hgh skll lvl, whl t may ncourag thm for th lss sklld. Whl ths ltratur provds us wth many nsghts, t mostly dals wth conoms whr thr s no slf-slcton nto dffrnt sklls, so that on of th ky mchansm at work n our conomy s absnt. Also, bar n mnd that, n our st-up, at th qulbrum, workrs fac no uncrtanty, so that th mchansm pontd out n Eaton and Rosn 1980 s absnt. A fnal rmark: w consdr nvstmnts n ducaton as a bnchmark cas whr htrognous agnts mak chocs nvolvng both th xtnsv and th ntnsv margns and whr th composton ffct mattrs. Thr ar many othr possbl applcatons of th sam basc framwork, such as chocs nvolvng mgraton. 2. Th Modl Th conomy s composd by two sparat producton sctors, dnotd by s {n, }. Workrs dnotd by a subscrpt whn w rfr to ndvduals, whn w rfr to thr st and frms dnotd by j and J, rspctvly can choos to ntr on of th two-sctor, payng a fxd cost. Workrs costs, c n, c, ar xognous, and can b ntrprtd as prvat, drct, fxd costs of ducaton tutons and th lk. W dnot frms costs d n J, d J. Thy ar ndognously dtrmnd, and wll b dscussd latr on. Thr ar two ntrvals of qual lngth of workrs and frms, Ω = Ω J [θ, θ] R ++, both ndowd wth th Lbsgu masur. Each ntrval s parttond nto two sts, {Ω n, Ω } ΩP and {Ω n J, Ω J } ΩP J, dtrmnd ndognously. Lt µωs µωs J dnot th masur of th st Ω s Ωs J, rspctvly. n sctor s, producton rqurs a frm j wth physcal captal ks j and a workr wth stock of human captal h s. Onc th parttons ΩP and ΩP ar gvn, ach sctor of th conomy rducs to th st-up studd n Acmoglu 1996. Frms ar dntcal, and choos thr nvstmnts n physcal captal to maxmz thr xpctd profts. Workrs choos thr nvstmnts n human captal to maxmz thr xpctd utlts. Th conomy lasts on prod, dvdd nto svral subprods. n subprod 0, frms and workrs ntr on of th two sctors and carry out thr nvstmnts. At 1, ach frm actv n sctor s 8 n our conomy, on obtans substantally dntcal rsults consdrng drct non-lnar subsds to ffort and subsds to th drct costs of ducaton. Prvous, rlatd work n ths ara ncluds Blanknau 2005, Blanknau and Camra 2006, 2009, Caucutt and Kumar 2003, Lloyd-Ells 2000, Sahn 2004, and Su 2004. AB-Dscusson Papr 7/2011 8
s matchd wth a workr actv n th sam sctor w wll b mor prcs on ths ssu latr on. n th fnal subprod, producton taks plac and th total output of ach match s splt accordng to th Nash barganng soluton wth xognous wghts β and 1 β. 9 Evdntly, gvn that nvstmnts ar carrd out bfor matchs tak plac, agnts cannot contract wth thr partnrs a gvn lvl of nvstmnt. Ths s on of th ky faturs of th conomy. For ach workr actv n sctor s, th utlty functon s U s C s, h s = C s 1 h s1+γ δ 1 + Γ, whr C s dnots consumpton, h s s th amount of human captal or th labor supply. Lt c s b th fxd cost of th nvstmnt n sctor s human captal. Thn, n th absnc of taxs and subsds, f workr s actv n sctor s and matchd wth frm j, C s s gvn by labor ncom mnus c s. Workrs ar htrognous bcaus of th paramtr δ, ndxng thr margnal dsutlty of ffort: ctrs parbus, largr valus of δ ar assocatd wth hghr valus of th optmal choc of human captal. Wthout any ssntal loss of gnralty, w assum that δ =, and that δ s unformly dstrbutd on [θ, θ], θ > 0. Mor gnral assumptons on th dstrbuton of δ would not chang any ssntal rsult. Tchnologs ar dscrbd by a par of Cobb-Douglas producton functons wth constant rturns to scal. Whn actv n sctor s, and matchd wth workr wth human captal h s, frm j has producton functon yj s = A s h sα k s1 α j, wth A > A n. Lt p b th unt prc of physcal captal n both sctors. Ths mpls som loss of gnralty, but smplfs notaton and computatons. Most mportant, smlar rsults hold for p p n. As w wll s, gvn any arbtrary partton of workrs and frms compatbl wth full mploymnt, xpctd producrs surpluss ar postv n both sctors and always largr n sctor. To avod addtonal complcatons not rally grman to our man ssu, and to mantan th smlarty wth Acmoglu s modl, w want to consdr an conomy wth full mploymnt at th qulbrum. Ths rqurs that, at th qulbrum, ach agnt s actually matchd wth a partnr. W assum, as mplct n Acmoglu 1996, that th matchng functon guarants wth probablty on a match to ach agnt, provdd that µ Ω s = µ Ωs J.10 Gvn th focus of th papr, th partton Ω P must b dtrmnd ndognously. Hnc, to guarant full mploymnt, w nd that, at ach qulbrum, µω s = µ Ωs J. Th asst way to obtan ths proprty s to ntroduc a fatur of th conomy such that qulbrum xpctd profts ar always qual n th two sctors. On way to obtan ths s to assum that th tchnology xplotd n sctor n s fr, whl th on adoptd n sctor s protctd by a patnt, ownd by som outsd agnt. 11 Th rght to us th patnt s auctond off to frms bfor th frm-workr-match s obtand. 12 Gvn that, at an qulbrum, xpctd profts n both sctors must b dntcal, th qulbrum royalts must b qual to th postv dffrnc btwn th xpctd producr s surpluss n th two sctors. Thn, at ach qulbrum, ach frm s ndffrnt among sctors, so that w can choos Ω P J wth µ Ωs = µ Ωs J, th proprty w 9 For a ratonalzaton of ths allocaton rul n ths contxt, s th Appndx n Acmoglu 1996. W assum that β s sctor-nvarant. Gvn that t s xognous, to lt t vary across sctors would just ntroduc mor notaton wthout provdng any substantv addtonal nsght. 10 A commonly usd functon whch dlvrs ths proprty s π s j = mn{µωs,µωs } J µω s J, whr π s j s th probablty of a match for a frm actv n sctor s. 11 Clarly, nothng would chang f ach tchnology wr subjct to a dstnct patnt. 12 An aucton dlvrng th rsult w nd s basd on closd nvlop, frst prc bds by th frms. Th royalty s allocatd to ach frm bddng th maxmum prc. AB-Dscusson Papr 7/2011 9
ar lookng for. Altrnatvly, on could adopt a structur basd on a contnuum of sparat slands, wth an dntcal numbr of frms and workrs on ach sland, no moblty across slands and asymmtrc nformaton on th workrs typs. 13 Wthout any loss of gnralty, th prcs of both knds of output ar st qual to 1 and, thrfor, omttd. Fnally, notc that thr ar always thr addtonal, trval, qulbra: th ons whr all th workrs and th frms ar n on of th two sctors, and th on whr non s actv n any sctor. As usual, w gnor thm. 3. Equlbrum Latr on, w wll show that, at th qulbrum, t s always Ω = [δf, θ] or Ω =, whr δf dnot th qulbrum valu of th thrshold n th conomy wth frctons. 14 Hnc, w can rstrct th analyss to parttons Ω P and Ω P J dfnd by an arbtrary lvl of th thrshold, dnotd δ, and wrt Ω s J δ and Ω s δ. For futur rfrnc, w dtrmn th optmal amount of nvstmnts assumng that thr s a publc ntrvnton dfnd by a par of vctors ξ s τ s, ζ s, c s, ξ ξ, ξ n, dscrbng possbly sctor spcfc subsds and taxs. W assum that thr ar stp-lnar taxs on labor ncom wth rats τ s, s = n,, and on th cost of th nvstmnts n physcal captal wth rats ζ s, s = n,, and fxd taxs, or subsds, on th drct costs of ducaton, c s w wll always st c n = 0. W wrt th tax rats as sctor spcfc just to smplfy th notaton: at qulbrum, ths systm of taxs s somorphc to a systm of stp-lnar taxs on labor ncom and on nvstmnts n physcal captal. 15 Pck an arbtrary thrshold δ. f actv n sctor s, frm j slcts th valu of kj s xpctd profts maxmzaton problm choos kj s arg max E kj s Ω s δ 1 β A s h sα k s1 α j p 1 + ζ s k s j d s J solvng th 1 β A s E Ω s δ hsα k s1 α j p 1 + ζ s kj s d s J, Π s whr, gvn any random varabl x s, wth x s : Ω s R, or ys, wth y s : Ω s J R, E Ω s δ xs j Ω s δ xs d µω s δ ovr Ω s J δ. or E Ω s J δ ys dnots th condtonal xpctaton of xs ovr th st Ωs δ or of y s j Th par of maps K s δ;ξ, s = n,, dfns th optmal nvstmnt n physcal captal for th frms actv n th two sctors. Thy ar j nvarant bcaus frms n ach sctor ar dntcal, and dpnd upon th xognous vctor ξ, th arbtrary thrshold δ, and th condtonal xpctatons E Ω s δ hsα. Lt Π s δ, δ;ξ b th surplus bcaus nclusv of d s J of th frm matchd wth workr n sctor s. 13 A thrd altrnatv would b to assum that frms cannot mov across sctors. A non-null masur of frms s xognously assgnd to ach sctor. W thn pck a matchng functon whch always guarants that ach frm s matchd wth a workr and convrsly for ach non-trval partton of th workrs. As long as thr s a contnuum of agnts n ach sctor, ths can b don. Of cours, ths approach would brak down f w had a fnt numbr of agnts and, anyhow, s basd on a vry ad hoc trck. 14 Obvously, th workr wth δ = δ s ndffrnt btwn th two sctors. For convnnc, w assum that h/sh ntrs sctor. 15 Evdntly, th sam closd form could b obtand by usng taxs or subsds basd on th ffort n ducaton, whch, howvr, could not b drctly obsrvabl. AB-Dscusson Papr 7/2011 10
Th optmzaton problm of workr f actv n s s choos h s arg max E h s Ω s J δ U s. U s 1 τ s βa s h sα E Ω s J δ ks1 α j 1 h s1+γ δ 1 + Γ cs + c s. Th par of maps H s δ, δ; ξ, s = n,, dscrbs th optmal nvstmnts n human captal of th agnts n ach sctor. Lt V s δ, δ; ξ b th assocatd lvl of utlty of agnt, f actv n sctor s. Workr ntrs sctor f and only f F δ, δ; ξ V δ, δ; ξ V n δ, δ; ξ 0, whr F δ, δ;ξ s agnt s utlty gan du to hs nvstmnt n ducaton. t s asy to chck that, for ach gvn δ, ξ, F. s strctly ncrasng n δ. Dfnton 1. Gvn ξ, an qulbrum of th conomy wth frctons s a thrshold valu δ F [θ, θ], and a royalty d F J 0, such that:. K s δ F ;ξ solvs Π s, s = n for ach j = such that δ < δ F, s = for ach j = such that δ δ F ;. H s δ, δ F ;ξ solvs U s, s = n for δ < δ F, and s = for δ δ F ;. E Ω δ F Π δ, δ F ;ξ E Ω n Π n δ δf, δ F ;ξ = d F J > 0; v. F δ, δ F ;ξ 0 f and only f δ δ F. Frst, obsrv that th condtonal xpctatons E Ω s J δ ks1 α j, E Ω s δ hsα, s = n,, ar computd makng rfrnc to th actual valus {H s., K s.}, s = n,, so that w ar mposng ratonal xpctatons. Condtons mpos ndvdual optmalty n th choc of th nvstmnt. Condtons v mpos ndvdual optmalty n th choc of th sctor whr an agnt s actv. By, ach frm s ndffrnt btwn } bng actv n any of th two sctors, so that w can mpos Ω P J = ΩP {[θ, = δ F, [δ F, θ] by v. Th man rsults concrnng xstnc of qulbra and thr proprts ar summarzd n Proposton 1. Th proof s n th appndx. Hr w just provd an outln of th argumnt. Frst, gvn an arbtrary δ, w comput th valus of H s δ, δ; ξ, K s δ; ξ, s = n,, th dmand functons for nvstmnt n human and physcal captal obtand mposng that condtonal on δ xpctatons ar fulflld s qs. A3 and A4 n th appndx. Occasonally, w wll rfr to H s δ, δ;ξ, K s δ;ξ and th drvd maps Ṽ s δ, δ;ξ and Π s δ, δ;ξ as th qulbrum maps condtonal on δ. Lt F δ, δ;ξ b th analogous of map F., obtand usng H s δ, δ; ξ, K s δ; ξ. Gvn that F δ, δ;ξ s strctly ncrasng n δ, F δ, δ;ξ = 0 at δ = δ gvs us th qulbrum valu of th thrshold,.., δ F ξ. Hnc, δ F ξ s th soluton to th quaton F δ = δ, δ; ξ f δ; ξ c + c = 0, whr, by drct computaton and usng A3 and A4, f δ; ξ δ δ α 1+Γ α α 1+Γ α A E Ω δ δ α 1+Γ α 1 α 1+Γ χ ξ 1 A n E Ω n δ δ α 1+Γ α 1 α 1+Γ χ n ξ. AB-Dscusson Papr 7/2011 11
Th varabls χ s ξ, χ s ξ 1+Γ α 1+Γ 1 τ s 1+Γ Γ dpndng upon th xognous paramtrs. α 1 Γ β 1+Γ Γ 1 α1 β p1+ζ s 1+Γ1 α, ar scalars Proposton 1. Fx α, β and lt ξ = τ, τ, 0, 0, 0. Gvn Γ, A, A n ; ξ, thr ar { C, C } >> 0 such that, for almost vry c C, C, thr s an qulbrum wth thrshold valu δ F ξ θ, θ. Morovr, gvn Γ, A n, thr s A such that, for ach A > A, th qulbrum s unqu and f. δ=δ F > 0. Also, δf. τ > 0, δf. τ < 0, δf. n c > 0, δf. A < 0 and δf. A > 0, n whr δ F A, A n ;ξ s th functon assocatng wth th vctor ξ th unqu qulbrum thrshold. Th sam rsults hold, gvn A, A n, for Γ suffcntly small. Proof. S th appndx. Gvn th focus of th papr, t s convnnt to consdr a vctor ξ wth th statd proprts, just to smplfy computatons. Nothng rlvant dpnds upon ths rstrcton. n th followng, w wll mostly consdr th ladng cas whr f. δ=δ F > 0 at ach qulbrum thrshold. 16 Ths rstrcton dlvrs two dffrnt proprts for qulbra. Frst, f. > 0 at ach qulbrum thrshold mpls ts unqunss. Scondly, by th mplct functon thorm, th comparatv statcs proprts dpnd upon th drvatvs of th qulbrum condtons wth rspct to th xognous paramtrs A, A n, ξ and δ. Th sgns of ths drvatvs wth rspct to A, A n, ξ ar always unquly dfnd. Hnc, th comparatv statcs of th qulbrum thrshold just dpnds upon th sgn of f. δ=δ F, and to rstrct th analyss to conoms wth f. δ=δ F > 0 at ach δ F allows us to obtan wll-dfnd rsults. Dffrnt sts of rstrctons on th paramtrs would guarant that f. δ=δ F > 0. Th ons proposd abov sm farly wak and natural. That som addtonal rstrctons ar ncssary to obtan f. δ=δ F > 0 at ach qulbrum s shown n Exampl A1 n th appndx. Thr w construct an conomy wth f. > 0 for δ suffcntly clos to θ and ngatv for δ larg nough. Gvn that f. s contnuous on [θ, θ], for ths conomy f. has at last on local maxmum, δ. Hnc, ach conomy wth c such that c < fδ, and clos nough to fδ, has at last two qulbra. Th prcs magntud of th rstrcton on th rato A /A n obvously dpnd upon th prcs valus of α, β, Γ. Numrcal smulaton suggst that thy ar not ovrly rstrctv. Lt s compar th qulbrum allocaton of ths conomy to th on of th assocatd Walrasan conomy th on wth prfct contractblty and compttv wags. Thr ar thr man rsults. Fx ξ = 0. Parto nffcncy of qulbra s obvous, bcaus, n th conomy wth frctons, a frm s nvstmnt dos not dpnd upon th valu of δ of th workr t s matchd wth, whl t dos at any Parto ffcnt allocaton. Scond, th Walrasan qulbrum of th sam conomy domnats th qulbrum of ths conomy n trms of total xpctd surplus, but t s not ncssarly Parto supror. ndd, at ξ = 0, and usng A3 and A4 n th appndx, th physcal/human captal rato at th two allocatons satsfy K s δ F H s δ, δ F = 1 β 1 α E Ω s δ F δ 1 1+Γ α δ a 1+Γ α 1 α KW s δ H W s δ, whr th suprscrpt W dnots th qulbrum valus at th Walrasan allocaton. f δ F s larg nough, compard to θ, and for suffcntly small δ, n sctor n th trm n brackts s K always gratr than on, so that n δ F H > KW n δ n δ,δ F H W n δ. Ths mmdatly mpls that agnts wth a suffcntly low δ ar bttr off at th qulbrum of th frctonal conomy. A thrd 16 To avod msundrstandngs: F δ, δ, ξ s always strctly ncrasng n δ. Th functon f δ, ξ s obtand sttng δ = δ and t dos not ncssarly hav ths proprty. AB-Dscusson Papr 7/2011 12
obsrvaton s that th thrshold valu δ F can b thr lowr or hghr than ts valu n th Walrasan conomy. For nstanc, lt ξ = 0, st [θ, θ] = [1, 2], A = 2, A n = 1, α = β = 1/2, and Γ = 2. By drct computaton, on can vrfy that, for c < 0.7, δf < δ W, whl th oppost occurs for c > 0.71. Hnc, lack of contractblty always nducs Parto nffcncy bcaus of lowr than optmal nvstmnts, whl t has an ambguous ffct on th sz of th st of popl nvstng n ducaton. From ths vwpont, thrfor, t dos not nduc unambguously ovrducaton or undrducaton. Fnally, consdr th asymptotc bhavor of th qulbrum allocaton along any squnc {A v } v= v=1 wth A v A n. Lt f δ, A, A n b th functon obtand from f δ;ξ sttng ξ = 0 and makng xplct ts dpndnc on A, A n smlarly for δ F A, A n. t s asy to chck that lm δ θ f δ, A, A n > 0, for ach A A n. Hnc, thr s an ntrval of valus of c such that th assocatd qulbrum thrshold s strctly smallr than θ vn f A = A n. 17 Hnc, th qulbrum nvstmnts n hgh skll human captal s postv vn whn ths skll s compltly uslss, from th tchnologcal vwpont. Whn A = A n, th two sctors ar ssntally dntcal, whl to oprat n sctor, rqurs th us of costlr sklls. Thrfor, Parto ffcncy rqurs us to shut down ths sctor. Ths s smlar to what happns n sgnallng modls. Th man purpos of th papr s to analyz th polcy mplcatons of workrs slf-slcton nto dstnct labor markts. Howvr, t s ntrstng to consdr th comparatv statcs of qulbra, also bcaus th wlfar ffcts of dffrnt polcs coms through thr mpact on th qulbrum valus of th ndognous varabls. Lt φ ξ, A, A n. Lt w s δ, δ F, φ b workr s wag n sctor s. Th standard dvaton, σ Ω s δ F δf, φ, masurs th varablty of wags wthn sctor s. W P Ω δ F δf, φ s th wag prmum. 18 Proposton 2. Fx Γ, c f., α, β. Assum that rstrctons ar satsfd: 19 δ=δ F ξ dτ dτ n c da da n E Ω δ F ξ H.? +? + E Ω n ξ H n. + + + δf K.? +? + K n. + + + E Ω δ F ξ w.? +? + E Ω n δf ξ w n. + + + σ Ω n δf ξ. + + + W P Ω δ F ξ. + + > 0. At ξ = 0, th followng sgn. 17 Dpndng upon th valus of th othr paramtrs, w may hav at last two qulbra wth dffrnt thrsholds, or a unqu qulbrum. What s rlvant s that thr s always som lvl of nvstmnt n hgh sklls. 18 n gnral, thr ar thr dffrnt notons of wag prmum: w. w n. w. w n., E Ω δf α 1+Γ α w. w n. E Ω n δf. Du to lnarty of th wag functon wth rspct to δ n ach sctor, hr thy concd. 19 Each cll rports th sgn of th drvatv of th functon on th row wth rspct to th varabl on ts column. W omt th standard dvaton of th wags of sklld workrs. For ths varabl, t s mpossbl to rach any wll-dfnd, gnral rsult. For rasonabl valus of th paramtrs, α = 2 3 and Γ > 1 2, som numrcal smulatons show that th composton ffct has th sgn oppost to th on of δf φ. Thrfor, σ Ω δ F. φ s postv for φ {τ, A }, ngatv for φ { τ n, c, An}. and AB-Dscusson Papr 7/2011 13
All th rsults follow by tdous, but straghtforward, computatons. Th ntuton bhnd thm s basd on th ntracton of th ncntv and th composton ffct. For nstanc, consdr an ncras n τ n,.., n th margnal tax rat on th labor ncom of th n workrs. As a pur ncntv ffct, dτ n > 0 rducs thr ffort n ducaton, and pushs down th thrshold δ F. Hnc, bcaus of th composton ffct, t rducs th condtonal xpctd human captal of both low and hgh sklld workrs. Ths, n turn, rducs nvstmnts n physcal captal n both sctors. Ths ngatv fd-back strngthns th ntal mpacts. Hnc, th ffcts on xpctd human and physcal captals and on wags ar ngatv n both sctors. For th wag prmum, by drct computaton, t turns out that both drct and composton ffcts ar postv. Th standard dvaton of wags of unsklld workrs dcrass bcaus both ffcts ar ngatv. On th othr hand, dτ > 0 has unambguously a postv ffct on th lvl of human and, consquntly, of physcal captal and on th wags n th n sctor, bcaus t ncrass th valu of δ F ξ composton ffct. ndd, gvn that th xpctd human captal of th pool of n workrs ncrass bcaus of th ncras n τ, physcal captal also ncrass, stmulatng ths workrs optmal nvstmnts. Th mpact n th sctor s ambguous bcaus th ncntv ffct rducs th optmal nvstmnt n human captal. Howvr, th composton ffct acts n th oppost drcton, bcaus t nducs workrs wth a rlatvly low valu of δ to swtch to th low skll sctor. Ths has a postv mpact on th xpctd lvl of human captal n th sctor and, thrfor, on th nvstmnts n physcal captal, nducng a postv fd-back. Th ffct of xognous changs n tchnology, da, da n, can b xpland bascally n th sam way. n partcular, n ths st-up, skll basd tchncal chang da > 0, da n = 0 has a ngatv mpact on th xpctd human, and physcal, captal and on th wags n th low skll sctor, an ambguous mpact n th hgh skll on, and a postv ffct on th wag prmum. 4. Effcncy proprts of qulbra W hav alrady argud that th qulbra of th conomy wth frctons ar Parto nffcnt. W wll now show that thy do not satsfy thr a wakr crtron of constrand optmalty CO n th squl whch taks nto account th mprfctons whch charactrz th conomy. Most ntrstng s th analyss of thr nffcncy n trms of th amount, and typ, of nvstmnts. n th squl, w wll manly rfr to nvstmnts n human captal. Smlar consdratons hold for th ons n physcal captal. n our st-up, nffcncs can b of two dffrnt typs. Frst, an ndvdual can choos an amount of nvstmnt dffrnt from th CO on, gvn th partton Ω P assocatd wth th CO allocaton. W wll rfr to ths possbl sourc of nffcncy as undrnvstmnt or ovrnvstmnt n ducatonal ffort. Scondly, an agnt can choos to nvst n a typ of ducaton dffrnt from th on assgnd to hr at th CO allocaton. W wll say that thr s undrnvstmnt n ducatonal lvl whn agnt nvsts n ducaton n, whl, at th CO allocaton, sh should nvst n ducaton lvl. n th on-sctor modl, qulbra ar unambguously charactrzd by undrnvstmnt n ducatonal ffort. Hr, th sam ffct s at work: n ach sctor, gvn any arbtrary δ, an ncras n th nvstmnts of frms and workrs lads to a Parto mprovmnt. Onc w consdr an arbtrarly fxd thrshold δ, th argumnt s dntcal to th on n Acmoglu 1996. St ξ = 0 and omt t, for notatonal convnnc. Fx δ, so that ach sctor s dntcal to th conomy analyzd thr, and consdr a small chang n th nvstmnt of ach agnt. Th changs n utlts and producrs surplus valuatd at th qulbrum condtonal on δ par H s δ, δ, K s δ and takng nto account that nvstmnts n physcal captal ar j nvarant AB-Dscusson Papr 7/2011 14
ar gvn by αβa s [ ] 1 α Ks δ H s δ, δ 1 δ Hs δ, δ Γ dh + 1 α βa s Hs δ, δ K s δ α dk > 0 2 and 1 α 1 β A s E Ω s δ H s δ, δ α K s δ α p dk+ α 1 β A s Ks δ 1 α E Ω s δ H s δ, δ dh > 0, 1 α 3 rspctvly. Th nqualts hold bcaus th frst trms n parnthss n 2 and 3 ar zro, at th optmal solutons of Π s and U s, whl th scond trms ar postv. Hnc, gvn any δ, thr s undrnvstmnt n ducatonal ffort and physcal captal, n ach sctor. Ths stablshs, n a mor drct way, th Parto nffcncy of th qulbra of our conomy. n th two-sctor cas, thr s a scond potntal sourc of nffcncy, bcaus changs n th valu of δ may also ntal Parto mprovmnts. An ncras n th thrshold valu δ ncrass th condtonal xpctd amount of human captal n both sctors at th sam tm and, consquntly, nducs an ncras n th amount of physcal nvstmnts of frms n both α 1+Γ α sctors. ndd, gvn that δ s a strctly monotoncally ncrasng functon, E Ω s δ δ α 1+Γ α and, consquntly, usng A3 and A4, H s δ, δ > 0, for ach s and δ, 4 > 0 and K s δ rlvant, from A5, A6 and 4, for ach and δ, Ṽ s δ, δ > 0, for ach s and δ. Mor > 0 and E Ω s δ Π s δ, δ Ths proprts do not suffc to stablsh our clam, bcaus a chang n th thrshold nducs a jump n th producr s surplus for th frms shftng from on sctor to th othr. W wll gt back to ths ssu latr on. To analyz th wlfar proprts of qulbra, t s convnnt to ntroduc an xplct noton of constrand ffcncy. As usual n conoms wth mprfct markts, w consdr th mtaphor of a bnvolnt plannr choosng an allocaton whl facng constrants amng to captur th ons th agnts fac n th dcntralzd conomy. W provd two rsults. Frst, w show that thr ar constrand optmal allocatons Proposton 3, and that thy can b attand wth an approprat systm of taxs and subsds Corollary 1. Th amount of subsds and taxs s ntrly dctatd by th faturs of th CO allocaton, and thy can b qut larg. That s why, n Prop. 4 and 5, w study th ffcts of small changs n taxs and subsds on total surplus valuatd at th markt qulbrum, takng as gvn th actual dmand and supply functons of th agnts. Proposton 4 consdrs gnrc changs n taxs and subsds. n Proposton 5, w consdr rvnu nutral changs. Bar n mnd that, n th followng, w always consdr changs n total surplus. W ar not concrnd wth actual Parto mprovmnts. Howvr, gvn that utlty functons ar quas-lnar, an ncras n total surplus mmdatly translats modulo an approprat - and contngnt - systm of lump-sum taxs and transfrs nto a Parto mprovmnt. > 0. AB-Dscusson Papr 7/2011 15
4.1. Constrand optmal allocatons Th objctv functon of th plannr s P h s, ks j, Ωs, J Ωs, th sum of th xpctd utlts and producrs surpluss of th agnts. Hs polcy nstrumnts ar th parttons Ω P and ΩP J and a par of maps H COs δ, δ, K COs δ. W rstrct th parttons to hav th structur Ω δ = Ω J δ = [ δ, θ]. Gvn that frms ar x-ant dntcal, th nformatonal constrants mbddd nto th dfnton of P., and th proprts of th mplct matchng functon, to mpos ths structur on Ω P and ΩP J dos not ntal any loss of gnralty. Also, obsrv that, gvn that frms ar dntcal, xpctd total surplus and ralzd total surplus concd. W dfn an allocaton Constrand Optmal or CO f and only f t solvs th plannr s optmzaton problm. Lt δ CO b th lvl of th thrshold assocatd wth th CO allocaton. Proposton 3. Undr th mantand assumptons, thr s a CO allocaton. Equlbrum allocatons ar nvr CO, and ar charactrzd by undrnvstmnt n th amount of physcal captal and n ducatonal ffort. δ F > δ CO and δ F < δ CO can both occur. Proof. S th appndx. Th sourc of nffcncy consdrd by Acmoglu 1996 rappars n our st-up, bcaus, gvn any thrshold lvl δ, H COs δ, δ > H s δ, δ, for ach δ, and K COs δ > K s δ. On th othr hand, th rlaton btwn δ CO and δ F s not unvocal. n th proof n th appndx, w provd an xampl of an conomy such that δ F < δ CO f th drct costs of ducaton ar suffcntly low, whl δ F > δ CO for suffcntly hgh valus of c. n ntrprtng ths rsult, bar n mnd that, n computng δ F and δ CO, w us dffrnt nvstmnt functons: H s., K s. and H COs., K COs., rspctvly. On th othr hand, n Corollary 1, w show that, onc th optmal subsds τ, ζ ar ntroducd, to mplmnt th CO allocaton w always nd c > 0. Thus, gvn th optmal taxs, CO always rqurs us to shrnk th st of agnts nvstng n th hgh skll sctor. t s asy to s that th CO dstrbuton of nvstmnts n human and physcal captal can b attand wth an approprat systm of taxs and subsds. Gvn that prfrncs ar quas-lnar, th systm of tax and subsds can b balancd usng unform lump-sum taxs on workrs n th absnc of postv ndowmnts of consumpton goods, ths could ntal ngatv consumpton for som subst of agnts. Corollary 1. Thr s a systm of taxs and subsds ξ, wth c > 0, such that th assocatd qulbrum allocaton s CO. Proof. S th appndx. n our st-up as wll as n Acmoglu 1996, qulbra of th conomy wth frctons ar constrand nffcnt for ach valu of β, bcaus, at ξ = 0, vn f δ CO = δ F, for ach β, and H s δ, δ CO H COs δ, δ CO K s δ CO K COs δ CO = 1 β 1 α β 1 Γ 1, for ach s and, = 1 β 1+Γ α β 1 Γ 1, for ach s. n th usual random matchng modl, constrand ffcncy s obtand whn th Hososcondton s satsfd,.., whn β s qual to th absolut valu of th lastcty of th matchng AB-Dscusson Papr 7/2011 16
functon. On th contrary, n our conomy, gvn any thrshold δ, as obsrvd n Acmoglu 1996, p. 789, th xtrnalts ar rlatd to th valu of th futur matchs and ar always postv. Morovr, th dstrbuton of workrs across sctors may fal to b optmal, but ths nducs a markt falur dffrnt from th on du to th congston xtrnalty charactrzng th usual conoms wth random matchng. Ths s why hr th Hosos-condton has no conncton wth ffcncy. 4.2. Wlfar mprovng tax polcs W conclud consdrng th wlfar ffcts of altrnatv tax schms. Tax changs hav two dstnct ffcts. Frst, thy may chang th margnal rturn of th nvstmnt n ffort, gvn th typ of skll an ndvdual acqurs. Ths s a drct ncntv ffct. Scond, thy affct th dstrbuton of workrs n th two sctors. Ths s th composton ffct, whch, n turn, changs th margnal rturn of th nvstmnt n ffort bcaus of ts ffct on th optmal lvl of nvstmnt n physcal captal. Th rol of th composton ffct n our conomy has pcular faturs. A margnal chang n th thrshold δ ξ has no drct ffct on total workrs surplus, bcaus, by dfnton of qulbrum, V n δ = δ F ξ, δ F ξ = V δ = δ F ξ, δ F ξ. Smlarly, gvn that a frm s xpctd profts ar qual across sctors, th drct composton ffct on total xpctd profts s zro. Unfortunatly, n our st-up, du to th strctly postv royalts, typcally, at th margn, th producr s surplus n th hgh skll sctor s strctly largr than n th othr sctor. Ths follows from th partcular structur of our conomy, that w hav justfd abov. n th wlfar analyss, ths maks t hardr for our man rsults to hold, bcaus th ndrct mpact of th composton ffct has to b suffcntly larg to compnsat ts ngatv drct mpact on wlfar. Ths ndrct ffct gos frst through th postv mpact of th ncras n th condtonal xpctaton of th lvl of human captal n ach sctor on th frms nvstmnts n physcal captal. Gnrally spakng, th mchansm at work hr holds tru n a strongr form n any conomy wth slf-slcton of agnts n dstnct sctors and whr thr s som postv fd-back btwn th varabls of ntrst and th condtonal xpctaton of som fatur of th pool of agnts slf-slctng n on markt. 20 Thrfor, n th proof of th two fnal Propostons, w nd addtonal rstrctons on th xognous paramtrs, suffcnt to guarant that th composton ffct on wlfar of an ncras n th thrshold s postv. Thy ar formulatd mplctly, as an uppr bound on th valu of th qulbrum thrshold.., on th drct cost of ducaton, gvn th othr paramtrs. Thy do not appar unrasonabl. For nstanc, fx, as usual, α = 2 3, β = 1 10 and ξ = 0. St Ω = [1, 4]. For Γ 1,.., gvn an lastc ffort supply, and An A = 0.9, th composton ffct s always postv for δ F 1.7. Th ntrval of valus of δ F such that t s postv s dcrasng n Γ. Consdr as a startng pont an conomy wth a flat labor ncom tax. An ncras of taxs on th drct cost of ducaton c has a pur composton ffct, du to quas-lnarty of th utlty functons. Changs n th margnal tax rats hav both ncntv and composton ffcts: As obvous, an ncras n th margnal rat on hgh ncom ndvduals.., n our st up, on th hgh skll workrs has a ngatv drct ncntv ffct, but a postv composton ffct. Changs n th margnal tax rat on th low ncom workrs hav ngatv ncntv 20 Charlot and Dcrus 2005 consdr a two-sctors, dynamc random sarch modl. Thr, an ncras n th valu of th thrshold mprovs th condtonal xpctaton of th productvts of th workrs n both sctors. Ths maks t proftabl for frms to crat nw vacancs and, thrfor, lads to a dcras n unmploymnt n ach sctor. Ths may ntal a wlfar mprovmnt. n a way, n thr modl craton of vacancs has a rol smlar to th on playd hr by th ncras n physcal nvstmnts. A smlar mchansm s also at play n th modl wth slands and prfctly compttv spot labor markts outlnd at th nd of Scton 2. AB-Dscusson Papr 7/2011 17
and composton ffcts. Hnc, thy hav, unambguously, a ngatv mpact on wlfar. Th ntuton s farly smpl, also gvn Prop. 2 abov. For nstanc, dτ n > 0 has a drct, ngatv ncntv ffct on ffort n ths sctor. t also maks convnnt for som subst of workrs to mov to sctor, so that t movs down th valu of th thrshold. Hnc, t has a ngatv composton ffct on ffort and, n turn, on nvstmnts n physcal captal, n both sctors. To th contrary, n th cas of dτ > 0, th mpacts on total surplus of ncntv and composton ffcts hav oppost sgns and, undr sutabl condtons, th scond can actually domnat, so that w can obtan a wlfar mprovmnt by movng from a flat ncom tax to a progrssv on. Lt s mak formal th hurstc argumnt abov. Gvn ξ, workrs and frms choos thr ndvdually optmal bhavor. Lt Sξ b th xpctd total surplus at th qulbrum assocatd wth th vctor ξ of polcy nstrumnts. Lt R ξ b th total tax rvnu. Thn, Sδ F ξ; ξ s E Ω s Ω s δ F ξ Π s.dj + J δf ξ Ω s δf ξ Ṽ s.d Th frst st of rsults concrns th ffcts of a chang of on of th tax rats. + Rδ F ξ; ξ. Proposton 4. Consdr an qulbrum assocatd wth an arbtrary ξ = τ, τ, c, τ > 0 and suffcntly small, and satsfyng f. δ=δ F ξ > 0. Thn, for [ θ, θ ] larg nough and δ F ξ suffcntly clos to θ,. d c > 0, and suffcntly small, ncrass total surplus,. dτ n < 0, and suffcntly small, ncrass total surplus,. dτ < 0, and suffcntly small, may dcras total surplus. Th proofs of, ar n th appndx, whr w also stablsh that th wlfar ffct of a chang of τ s, n gnral, ndtrmnat. Th thrd statmnt s shown n Exampl A3, also n th appndx. Th two assumptons on th support [ θ, θ ] and th valu of δ F ξ guarant that th composton ffct SδF ξ;ξ δ F ξ s postv. Changs n xpctd surplus ar our masur of wlfar gans and losss. Howvr, dffrnt polcy nstrumnts hav dffrnt mplcatons also n trms of ndvdual wlfar. Undr th mantand assumptons, a dcras n th valu of τ n or an ncras of c has a postv mpact on th utlty lvl of all th workrs and on th xpctd profts of ach frm th ffct on th qulbrum lvl of th royalts s, howvr, ndtrmnat, n gnral. On th contrary, a dcras n τ has always a ngatv mpact on th utlty of all th workrs n sctor n and on th xpctd surplus of all th frms actv n ths sctor. t may hav a postv or ngatv mpact on utlty and surplus of agnts actv n sctor, and on th qulbrum royalts. Consdr now polcs whr rductons n th ncom taxs ar fnancd through taxs on th drct costs of ducaton, or by rvnu nutral changs dτ, dτ n. Proposton 5. Lt ξ = τ, τ, c, wth c 0. Consdr balancd budgt polcs dτ, dτ n and dτ s, d c. Undr th assumptons of Prop. 1, dτ, dτ n >> 0 and dτ n, d c >> 0, 0 ncras xpctd total surplus, dτ, d c >> 0, 0 may dcras t. Th proof s n th appndx. Th frst rsult mpls that som small dgr of progrssvnss n th labor ncom taxaton s wlfar mprovng. Undr th assumptons of Prop. 5, an ncras n th valu of th qulbrum thrshold s wlfar mprovng. n th proof, w show that a rvnu nutral polcy dτ, dτ n >> 0 always has a postv ffct on δ F ξ. Hnc, th composton ffct ncrass wlfar. t also turns out that, for τ suffcntly small w nd to AB-Dscusson Papr 7/2011 18
b on th ncrasng part of th Laffr curv, th drct ncntv ffct of a rvnu nutral tax chang s also postv. Thrfor, ths polcy chang s unambguously wlfar mprovng. Th scond rsult can b xpland along th sam lns. On th othr hand, a dcras n τ, balancd by an ncras n c has an ambguous ffct on wlfar. Th pur ncntv ffct of th polcy s wlfar mprovng. Th dffrnc wth rspct to th prvous cas s that now th total ffct of th polcy on th qulbrum thrshold dpnds n a non-trval way upon th paramtrs, bcaus dτ < 0 maks nvstmnt n ducaton mor appalng, whl d c > 0 acts n th oppost drcton. Th total ffct on wlfar s, thrfor, ndtrmnat. Howvr, n gnral, th rvnu nutral polcy dτ, c >> 0 has a largr postv or a smallr ngatv ffct on wlfar than a pur rducton of th margnal labor ncom tax rat. Fnally, w hav bn consdrng a sctor-contngnt vctor of subsdy rats τ, τ n. Ths s crtanly an unusual fatur of th polcy. Howvr, lt w s δ, δ F b agnt s labor ncom n sctor s. t s asy to chck that max w n δ, δ F w n δ F, δ F < w δ F, δ F mn w δ, δ F. Ω n δf Ω δf Hnc, gvn th proprts of th utlty functons, th sam rsults can b obtand wth a standard systm of stp-lnar taxs or subsds. 5. Conclusons Th papr consdrs a class of conoms whr w modl both xtnsv and ntnsv margns of nvstmnt chocs. Th man concluson s that th rsults typcally obtand n an ffcncy unt st-up whch consdrs only th ntnsv margn can fal to b robust to ts natural xtnson to a Roy s modl wth optmal choc of nvstmnts n human and physcal captal. Th ffcncy unt framwork ruls out, by assumpton, all th phnomna nducd by th slfslcton of th agnts nto dffrnt labor markts and, thrfor, all th wlfar consquncs rlatd to th composton ffct. Our analyss s carrd out for a smpl, paramtrc class of conoms. Ths allows us to comput xplctly th qulbra and th wlfar ffcts of dffrnt polcs, and to compar drctly our rsults wth th ons of Acmoglu 1996. Evdntly, to consdr quas-lnar utlty functon s rstrctv, n partcular n th analyss of th wlfar mpact of th varous polcs. Howvr, frst, an xtnson of th analyss to a rchr nvronmnt s possbl, but at a hgh cost n trms of analytcal tractablty. Scondly, all th rsults ar opn, so that thy crtanly survv n nvronmnts whr ncom ffcts ar suffcntly small. What mattrs most, th basc ntuton bhnd th wlfar rsults s strong, and thy should b robust to many possbl xtnsons of th basc st-up. Thr ar two man mssags of th papr: n nvronmnts charactrzd by lack of contractblty, rrvrsblty of th nvstmnts n human captal gnrats a hold-up problm. Ths tnds to dprss nvstmnts blow thr optmal lvl, so that a pcunary xtrnalty n human captal s gnratd. Howvr, f workrs slf-slct nto dstnct labor markts by nvstng n dffrnt typs of human captal, a scond dstorton arss whnvr wags ar an ncrasng functon of th condtonal xpctaton of th lvl of human captal of workrs actv n a markt. n our modl ths s nducd by th postv ffct of ths xpctaton on th lvl of th nvstmnts n physcal captal. Ths scond xtrnalty may nduc ovrnvstmnt n ducaton at th xtnsv margn. Whl both phnomna hav bn prvously dscussd n AB-Dscusson Papr 7/2011 19