NBER WORKING PAPER SERIES LEARNING AND (OR) DOING: HUMAN CAPITAL INVESTMENTS AND OPTIMAL TAXATION. Stefanie Stantcheva

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1 NBER WORKING PAPER SERIES LEARNING AND OR DOING: HUMAN CAPITAL INVESTMENTS AND OPTIMAL TAXATION Sefanie Sancheva Working Paper 238 hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 050 Massachuses Avenue Cambridge, MA 0238 July 205 I wan o hank James Poerba, Ivan Werning, and Emmanuel Saez for insighful commens. I also hank paricipans a he MIT Public Finance Lunch and seminar paricipans a Berkeley, Michigan, Princeon, Sanford, UPenn, Yale, Wharon, and Sana Barbara for heir useful feedback and commens. The views expressed herein are hose of he auhor and do no necessarily reflec he views of he Naional Bureau of Economic Research. NBER working papers are circulaed for discussion and commen purposes. They have no been peerreviewed or been subjec o he review by he NBER Board of Direcors ha accompanies official NBER publicaions. 205 by Sefanie Sancheva. All righs reserved. Shor secions of ex, no o exceed wo paragraphs, may be quoed wihou explici permission provided ha full credi, including noice, is given o he source.

2 Learning and or Doing: Human Capial Invesmens and Opimal Taxaion Sefanie Sancheva NBER Working Paper No. 238 July 205 JEL No. H2,H23,H24,H53,J24 ABSTRACT This paper considers a dynamic axaion problem when agens can allocae heir ime beween working and invesing in heir human capial. Time invesmen in human capial, or "raining," increases he wage and can inerac wih an agen's inrinsic, exogenous, and sochasic earnings abiliy. I also ineracs wih boh curren and fuure labor supply and here can be eiher "learning-and-doing" when labor and raining are complemens, like for on-he-job raining or "learning-or-doing" when labor and raining are subsiues, like for college. Agens' abiliies and labor supply are privae informaion o hem, which leads o a dynamic mechanism design problem wih incenive compaibiliy consrains. A he opimum, he subsidy on raining ime is se so as o balance he oal labor supply effec of he subsidy and is disribuional consequences. In a one-period version of he model, paricularly simple relaions arise a he opimum beween he labor wedge and he raining wedge ha can also be used o es for he Pareo efficiency of exising ax and subsidy sysems. In he limi case of learning-by-doing when raining is a direc by-produc of labor or in he case in which agens who are more able a work are also more able a raining, here are imporan modificaions o he labor wedge. Sefanie Sancheva Deparmen of Economics Liauer Cener 232 Harvard Universiy Cambridge, MA 0238 and NBER ssancheva@fas.harvard.edu

3 Inroducion Mos people spend a lo of ime acquiring human capial hroughou heir life. These ime invesmens can ake many differen forms: formal college, coninuing educaion, online degrees, on-he-job raining, or vocaional raining. Mirroring he variey of possible raining ypes, here is also a plehora of differen policies hroughou he world ha aim o encourage people o inves ime in improving heir skills. They include subsidies, grans or loans for ime spen in college, firm incenives for job raining, regulaions regarding on-he-job raining, or mixed arrangemens such as appreniceships. 2 A he same ime, oher imporan and far-reaching policies, such as income axaion or social insurance, could have an ambiguous effec on ime invesmens in human capial. Indeed, hey no only affec he reurns o ime invesmens, bu also he allocaion of ime beween raining and working. Sandard models of social insurance and axaion ypically focus only on ime spen working, absracing from he fac ha ime can also be allocaed o improving one s produciviy. 3 Therefore, his paper sudies he following issues: How should a comprehensive social insurance and ax sysem ake ino accoun ime spen acquiring human capial, called raining? Wha makes ime invesmens in human capial differen from oher invesmens, such as invesmens in physical capial or maerial invesmens in human capial? Wha parameers do we need o know o deermine he opimal policies regarding raining ime and he opimal ax sysem in he presence of raining? The firs goal of his paper is o provide a flexible life cycle framework ha can capure he unique characerisics of raining ime in order o answer hese quesions. The second goal is o solve for he opimal axaion and raining policies over he life cycle in his general model, and o poin ou how hey depend on some imporan parameers. The peculiariy of ime invesmens in human capial is ha hey are inrinsically linked o a given agen: heir reurns depend on an agen s ype and heir coss depend on he agen s labor supply. Accordingly, he wage of agens in he model depends on heir age, heir endogenous human capial sock, acquired hrough raining, and on an exogenous, sochasic, and persisen earnings abiliy. The sochasic abiliy causes earnings flucuaions agains which agens ry o insure hemselves by adjusing heir labor supply, raining choices, and risk-free savings. Abiliy and raining can inerac in any arbirary way in he wage funcion and he wage funcion can change wih age, allowing for Heckman 976a indirecly infers ha 30% of ime in early working years is spen raining. 40% of aduls paricipae in formal and/or non-formal educaion in a given year in OECD counries, and receive on average 988 hours of insrucion in non-formal educaion during life, ou of which 75 are job-relaed OECD, The mos noable example of which is probably he German appreniceship sysem ha combines formal raining wih hands-on experience. 3 See Mirrlees 97, Saez 200, Golosov e al. 2006, Albanesi and Slee 2006, Farhi and Werning 203. The lieraure review will presen some recen papers ha have ried o relax his assumpion.

4 differen impacs of raining a differen poins in life. In he model, agens can inves ime in raining in every year of heir working lives. They incur a disuiliy cos from working and raining, and he disuiliy specificaion allows for any age-dependen ineracion beween work and raining effor. To characerize his ineracion, I inroduce wo conceps Learning-or-doing and Learning-and-doing. The former designaes he case in which labor and raining ime are subsiues: ime spen working canno be spen raining. In he limi, his corresponds o he sandard opporuniy cos of ime model as in Ben-Porah 976. Learning-and-doing is he case in which labor and raining are complemens. For insance, raining could reduce he marginal disuiliy of working by making agens more aware of how o opimize heir work effor. The limi case here is he canonical learning-by-doing model in which raining is a direc by-produc of labor. The governmen wans o minimize he resource cos of providing any level of redisribuion and insurance, bu faces asymmeric informaion abou agens abiliies and labor effors. Because of his informaional asymmery, I solve a dynamic mechanism design problem wih incenive compaibiliy consrains. In order o build he inuiion for he resuls, I firs presen a simple one-period version of he dynamic model, which feaures a single realizaion of uncerainy and a one-sho invesmen in human capial. I inroduce he noion of a ne wedge, as a measure of he ne incenive ha is provided for raining, afer one has compensaed he agen for he exisence of oher disorions, such as he labor ax and, in he dynamic model, he savings ax. A posiive ne wedge means ha he governmen wans o encourage raining acquisiion on ne. The ne wedge has hree effecs: The firs is o increase he wage by simulaing invesmens in raining, which in urn increases labor supply. A he same ime, he presence of learning-or-doing or learning-and-doing direcly affecs labor supply posiively or negaively. The hird effec is he differenial impac of raining on people wih differen abiliies ha can increase or reduce inequaliy, depending on he wage funcion. The opimal subsidy balances hese hree effecs. A he opimum, wheher raining needs o be encouraged or discouraged on ne depends on wo crucial parameers: he Hicksian coefficien of complemenariy beween raining and abiliy in he wage funcion and he Hicksian coefficien of complemenariy beween raining and labor in he disuiliy funcion. The ne wedge decreases he more complemenary human capial and abiliy are in he wage funcion, decreases he more learning-or-doing here is, and increases he more learning-and-doing here is. I derive a simple relaion beween he labor wedge and he raining wedge a he opimum. In some special cases, his relaion has o remain consan over life and can be used o easily es for he Pareo efficiency of any ax and subsidy sysem. 2

5 In he dynamic model, he sign of he ne wedge also depends on he ineracion of conemporaneous raining wih fuure labor supply, and he sign of his effec is exacly he opposie of he one on he ineracion wih curren labor supply. The ampliude of he ne wedge is deermined by an insurance moive and he persisence of he abiliy shocks facors which usually affec he dynamic labor ax in models wihou human capial. 4 The labor wedge is no fundamenally affeced by he presence of raining, excep in wo cases. The firs is he limi case of learning-by-doing, in which raining is a direc by produc of unobservable labor supply. In his case, he labor wedge indirecly provides incenives for human capial acquisiion as well as for labor supply. The second case is when more able agens are no only more producive a working, bu also a raining. Relaed Lieraure: This paper conribues o he long-sanding opimal axaion lieraure as developed by Mirrlees 97 and Saez 200, and especially o he dynamic axaion lieraure, as in Kocherlakoa 2005, Albanesi and Slee 2006, Golosov e al. 2006, Kapicka 203b, Werning 2007a,b,c, Baaglini and Coae 2008a,b. Recen axaion papers seek o expand and improve he canonical opimal axaion framework see Rohschild and Scheuer 204a,b, Scheuer 203a,b, and Shourideh 202. The life cycle resuls direcly speak o age-dependen axaion as in Weinzierl 20. Paricularly relevan are he dynamic axaion models of Golosov, Tsyvinski and Troshkin 203, Shourideh and Troshkin 202, Golosov, Tsyvinski and Werquin 204, Werquin 204, and, mos imporanly, he life cycle axaion model in Farhi and Werning 203. This paper is also relaed o he large lieraure on human capial acquisiion, saring wih Becker 964, Ben-Porah 967, and Heckman 976a,b. The findings in Heckman, Todd and Lochner 2005 can jusify he assumpion of a disuiliy cos of human capial invesmen, above and beyond a pure opporuniy cos of ime. Some papers have considered axaion in he presence of human capial invesmens, bu have focused exclusively on moneary human capial expenses in saic models see Bovenberg and Jacobs 2005, Maldonado, 2007, Bovenberg and Jacobs, 20, DaCosa and Maesri, Gelber and Weinzierl 202 consider parenal invesmens in children s fuure abiliies. Dynamic papers ha examine he impac of axaion on human capial bear imporan differences o he curren paper. Bohacek and Kapicka 2008 or Kapicka 203 focus on heerogeneiy, wihou uncerainy in human capial reurns. Findeisen and Sachs 203 focus on a one-sho moneary invesmen during college, differen from he life cycle ime invesmens wih progressive realizaion of uncerainy in his paper. Complemenary papers are hose by Kapicka and Neira 204 wih 4 See for insance, Farhi and Werning

6 unobservable human capial, and Krueger and Ludwig 203 wih an overlapping generaions model and Ramsey axaion. The closes paper is Sancheva 204, which considers moneary invesmens in human capial. Time invesmens, as already emphasized earlier in he inroducion, are very differen under asymmeric informaion. 5 Their coss inerac wih he unobservable labor supply, which affecs he incenive provision mechanism in a way ha moneary expenses on human capial or physical capial invesmens do no. Accordingly, he resuls are subsanially differen and inroduce new conceps such as he learning-or-doing or learning-and-doing, which urn ou o be he major drivers of he opimal raining policies. The res of he paper is organized as follows. Secion 2 presens a model of work and raining over he life cycle. Secion 3 solves for he opimal policies in a simple one-period version of he model which helps build he inuiions. Secion 4 ses up he dynamic planning problem and explains he soluion mehod. Secion 5 presens he resuls on he opimal dynamic raining policies. Secion 6 digs deeper ino how he opimal labor wedge depends on he specificaion of he model wih raining. Secion 7 numerically illusraes some of he resuls. Secion 8 concludes. 2 A model of work and raining In his Secion, I presen a dynamic model wih work and raining. I hen solve for he firs-bes allocaion if here is no asymmeric informaion beween he Planner and agens. 2. Environmen The economy consiss of agens who live for T years, during which hey work and spend ime invesing in human capial. Agens who work l 0 hours in period a a wage rae w earn a gross income y = w l. Each period, agens can build heir sock of human capial by spending ime. These ime invesmens in human capial will be called raining. They could represen ime spen in college, in formal classes or raining courses, ime spen reading, or, imporanly, ime spen for on-he-job raining. Le z denoe he sock of raining ime, or he sock of human capial of an agen, a ime and i he incremenal raining ime acquired in period. Human capial z, evolves according o z = z + i. The disuiliy cos o an agen who provides l 0 unis of work and spends i 0 unis in raining 5 Noe ha under perfec informaion, in he firs bes, and wih no oher fricions, any ime cos could be convered ino an equivalen moneary cos and he disincion beween ime and money invesmens would be blurred. This is no longer rue under asymmeric informaion, unless raining does no inerac wih labor supply, a special case which will be explored in he paper. 4

7 is φ l, i. φ is sricly increasing and convex in each of is argumens φl,i i, φl,i l > 0, l, i > 0; 2 φl,i, 2 φl,i 0, l, i; however, no assumpion is ye made on he cross parial 2 φl,i l 2 i 2 l i. 6 The wage rae w is deermined by he raining acquired as of ime and by a sochasic abiliy : w = w, z w is sricly increasing and concave in each of is argumens w m > 0, 2 w m 2 0 for m =, z. Again, here are no resricions placed on he cross-parials ye. Noe ha because he wage depends on ime, which is he equivalen of age for he given cohor, human capial and abiliy are allowed o have differen wage impacs a differen imes in life. Agens sar heir life a ime = wih a heerogeneous ype, disribued according o a densiy funcion f. Earning abiliy in each period is privae informaion, and evolves according o a Markov process wih a ime-varying ransiion funcion f, over a fixed suppor Θ [, ]. The abiliy shock can be inerpreed in several differen ways. I could be a healh shock, an idiosyncraic labor marke shock, or a shock o he reurn o an agen s skills. To follow he axaion lieraure, will simply be called abiliy hroughou. The agen s per period uiliy is separable in consumpion and in he effor required for labor and raining: ũ c, y, z ;, z = u c φ u is increasing, wice coninuously differeniable, and concave. y w, z, z z Denoe by he hisory of abiliy shocks up o period, by Θ he se of possible hisories a, and by P he probabiliy of a hisory, P f...f 2 2 f. An allocaion {x } specifies consumpion, oupu, and he raining sock for each period, condiional on he hisory, i.e., x = { x } Θ discouned by a facor β, is given by: = { c, y, z } Θ. The expeced lifeime uiliy from an allocaion, U { c, y, z } = T = where, wih some abuse of noaion, d d...d. [ β u c y φ w, z, z z ] P d Le w m, denoe he parial of he wage funcion wih respec o argumen m m {, z}, and w mn, he second order parial wih respec o argumens m,n {, z} {, z}. There are wo imporan parameers ha drive he opimal policies. The firs is he Hicksian coefficien of complemenariy beween abiliy and raining in he wage funcion a ime, denoed by 6 The Appendix links his model o he sandard Ben-Porah model. 5

8 ρ z : 7 ρz w zw w z w 2 A posiive Hicksian complemenariy beween human capial z and abiliy means ha he effec of abiliy on he wage is amplified by human capial z, i.e, ha human capial and abiliy are complemens w z 0. Pu differenly, human capial compounds he exposure of he agen o sochasic abiliy and o risk. A Hicksian complemenariy greaer han means ha he wage elasiciy wih respec o abiliy is increasing in human capial, i.e., z w w 0. 8 Some simple cases can illusrae he complemenariy coefficien: A separable wage such as w = + h z, where h is an increasing, concave funcion, has ρ z, = 0. A muliplicaive wage w = h z has ρ z, =. A CES wage such as: has ρ z, = ρ. w = [ α ρ + α 2 z ρ] ρ Learning-and-doing and Learning-or-doing For raining ime, he mos imporan parameer will be he one ha characerizes he ineracion beween raining and labor. Define ρ φ lz o be he Hicksian complemenariy coefficien beween labor and raining in he disuiliy funcion φ: ρ φ lz φ lzφ φ l φ z 4 where he parials and second order parials of φ are denoed by: φ z, φ z, φ l, φ l, φ lz, 2 φ l z. How can we inerpre his disuiliy funcion? I is really jus a general way of capuring everyhing in beween wo polar cases we ofen consider in he labor lieraure, namely an opporuniy cos of ime specificaion, as in he canonical Ben-Porah 967 model, and he learning-by-doing model of Arrow 962. In he former, raining ime and work ime are perfec subsiues and ime spen raining canno be spen working. This can be capured by a disuiliy funcion linear in i and l, wih a oal ime endowmen consrain. In he learning-by-doing case, he disuiliy is a Leonieff funcion. 9 Bu in beween hese wo polar cases, here are many oher possibiliies. A less exreme case of opporuniy cos of ime is when raining and labor are subsiues, bu no perfec ones. Time spen raining akes away effor from working. Faigue effecs may se in, a a convex rae. In his case, he 7 On sudies of he Hicksian complemenariy as see Hicks 970, Samuelson 973, Bovenberg and Jacobs Equivalenly, he wage elasiciy wih respec o human capial is increasing in abiliy. 9 This equivalence beween raining and labor in he learning-by-doing case ignores he fac ha, in he planning problem, labor is unobservable while raining is no. Secion 6.2 will consider his limi case in more deail. 6

9 Hicksian complemenariy beween raining and labor in he disuiliy funcion is posiive ρ φ lz, > 0, a case I will label Learning-or-doing. Similarly, a less exreme case of learning-by-doing is when labor supply and raining are complemens over a leas some range ρ φ lz, < 0. The reasoning could be he same as he sandard learning-by-doing jusificaion: skills or experience acquired by working migh make raining less cosly a he margin. Or, less commonly hough, he guaranee of regular raining, which gives workers a prospecive for progress, migh boos moivaion and make labor effor seem less painful. I will call his case Learning-and-doing. In addiion, a differen levels of raining and labor supply, differen effecs could dominae: for insance, labor and raining could be complemenary a low levels, bu subsiuable a high levels of labor supply. The formulaion is general enough o accommodae all hese cases. Of course, he resuls on opimal raining policies in boh he saic and dynamic cases Secions 3 and 5 will srongly depend on his ineracion beween raining and labor. 2.2 Firs bes planning problem Before urning o he planning problem under asymmeric informaion, le us firs consider he firs bes soluion ha would arise if he Planner had he same informaion as he agen. In he firs bes, he Planner seeks o maximize he lifeime expeced uiliy in subjec o he ineremporal resource consrain, where R is he gross, exogenously given, borrowing rae: T = c y P d 0 R Θ Suppose ha β = R. There is perfec insurance agains earnings risk, as consumpion is equalized across all hisories: u c = λ, wih λ he muliplier on he resource consrain. Oupu and raining for each, given any level z, solve simulaneously: y φ l, w, z, z z = w, z λ φ z, φ,l l, w λ, z z, z z = T τ= R τ E w z,τ φ,l l,τ w τ λ, z τ z τ λ Hence, higher abiliy agens produce more oupu, for a given level of raining. The opimal raining equaes he marginal disuiliy of raining o is marginal benefi. The marginal expeced benefi of raining is increasing in abiliy, as long as raining and abiliy are complemens in he wage funcion 2 w z 0 and assumpion 2 i holds. The cos of raining, however, can also rise or fall wih abiliy depending on wheher here is learning-or-doing or learning-and-doing. If he laer is prevalen, or if here is no ineracion beween labor supply and raining φ is separable, hen unambiguously higher 7

10 ypes boh work more and rain more. However, if learning-or-doing is prevalen, higher ypes face boh a higher marginal benefi, and a higher marginal disuiliy from raining. The ne effec is ambiguous. When here is asymmeric informaion abou he ypes beween he Planner and he agens, his firs-bes allocaion is no longer achievable. The second-bes allocaion will have o ake ino accoun he agens incenive compaibiliy consrains, an issue o which I urn now. 3 Illusraing he Opimal Policies in a Simple One-Period Model Before solving he full-fledged dynamic planning problem, i is informaive o draw ou all he main effecs in a one-period model. The dynamic soluion in Secion 5 will exend hese resuls, and i is very helpful o build up he inuiion from he saic case. 3. Gross wedges versus ne wedges In he second bes, abiliy is privae informaion o he agen. Consumpion and oupu are observable. Because of he asymmeric informaion, he allocaions will be disored. The marginal disorions in agens choices can be described using wedges, which represen he implici local marginal ax and subsidy raes see Golosov e al for an inroducion o he wedges in dynamic public finance. The wedges capure he magniudes and signs of disorions in he allocaion relaive o he laissezfaire or, auarky allocaion. They are jus definiions of variables ha, evaluaed a he opimal second-bes allocaion, will characerize ha allocaion in a way analogous o axes. Expressing he properies of he opimal allocaion in erms of wedges allows us o appeal o inuiions relaed o a sandard ax sysem. In he one-period model, suppose ha ypes are disribued according o a disribuion f on Θ and ha he saring sock of raining is zero. Hence, z = i. For a given ype, define he labor wedge τ L and he raining wedge τ Z as follows: τ L φ ll, z w, zu c τ Z φ z u c τ Lw z l 6 The labor wedge τ L is defined as he gap beween he marginal rae of subsiuion and he marginal rae of ransformaion beween consumpion and labor. 5 I is akin o a locally linear marginal ax because, wihou any governmen inervenion, i would be zero. On he oher hand, if he governmen would se he marginal labor ax for agens of ype equal o τ L and le hese agens opimize heir labor supply in a small neighborhood around heir opimal allocaion, and condiional on he opimal raining choice, hey would choose heir labor supply so ha his relaion would hold. 8

11 The implici subsidy on raining τ Z can be hough of as he incremenal pay received by an agen for raining for one more uni of ime. If he implici subsidy is posiive, human capial is disored upwards. Like any subsidy, i is defined as he gap beween marginal cos and marginal benefi. The only subley is ha, since raining enails a disuiliy cos, he righ-hand side is scaled by marginal uiliy in order o conver i o a dollar amoun. Wihou he presence of raining, he wedges jus defined would be sufficien o characerize he allocaion. However, here are wo simulaneous disorions: saying ha he wegde on human capial, τ Z, is zero does no acually mean human capial is undisored if here is a posiive labor wegde τ L > 0, hen human capial is sill disored downwards, as par of is reurn is axed away. I would hus be useful o find a measure of he ne disorion on human capial, he one ha goes beyond jus compensaing for he presence of a labor disorion. We can ask: wha is he subsidy on human capial needed o exacly undo he labor disorion on human capial? To answer his, le s consider a hough-experimen. Suppose ha we se he raining subsidy o be τ Z = τ L φz u c. Then, locally, for a given labor supply l, he agen solves: max uwz, l τ L + τ Z z φz, l z The firs-order condiion of he agen wih respec o raining hen yields: w z lu c τ L + u cτ L u c φ zz, l = 0 where we have used he supposed relaion beween τ L and τ Z. This implies ha: τ L w z lu c φ z = 0, i.e., condiional on he labor choice, human capial is se opimally. This relaion beween τ L and τ Z is he equivalen of making raining coss fully ax deducible from he income ax base. More precisely, he agen is allowed o conver his disuiliy coss ino moneary equivalens, and hen deduc his moneary equivalen from his income ax base. 0 Inspired by his example, le us define he ne wedge on raining as he gross wedge minus he amoun needed o compensae he agen for he disorion on labor. φ z Definiion The saic ne wedge on raining ime z is defined as: z τ Z τ L φz u c φz u c τ Z τ L 7 As a maer of definiion, he ne wedge is scaled in he denominaor by he ne moneary cos of raining and he ne of ax rae. This will serve o make he opimal formulas simpler. 0 This idea of full deducibiliy holds perfecly locally and condiional on he labor choice. Of course, given ha he wedges are nonlinear and condiional on he ype, he analogy does no hold globally. 9

12 I will be useful o denoe by ε xy he elasiciy of any variable x wih respec o variable y, ε xy d log x /d log y. Le ε u and ε c be he uncompensaed and compensaed labor supply elasiciies o he ne wage, holding raining fixed. 3.2 The one-period planning problem I briefly se up he planning problem in he saic case: he full-fledged dynamic problem is rigorously explained in Secion 5 and ness his simple saic case. Hence, many echnical deails in paricular, abou how he firs-order approach works are omied here. Hence, he reader ineresed in he derivaions can refer o Secion 5 and o he proofs in he Appendix. Le ω denoe he uiliy of an agen of ype : ω = uc φl, z. I use a firs-order approach ha consiss in replacing he incenive compaibiliy consrains by envelope condiions of he agens. The envelope condiion ells us how he uiliy for each ype has o vary a he opimum and how he informaional ren received by each ype evolves. In his one-period case, he envelope condiion is simply: 2 ω := ω = w w lφ ll, z The one-period planning problem is hen o minimize resources: min c lwz, fd subjec o he envelope condiion and he uiliy consrain for each ype: ω ω. where he vecor of lower bounds for uiliy is arbirarily given. 3.3 The opimal raining wedge The nex proposiion a special case of he full fledged, dynamic opimum in Proposiion 2 shows how he ne wedge is se a he opimum. Proposiion A he opimum, he ne subsidy for raining and he labor wedge are se according o: z = τ L ε c τl + ε u ρ z ε φz ρ φ ε lz 8 wz ε Wih uiliy separable in consumpion and labor, c is he Frisch elasiciy of labor. +ε u ε c 2 This is jus a special case of he dynamic envelope condiion in 6. 0

13 where: ε φz d logφ/d logz is he elasiciy of disuiliy wih respec o raining ime. ρ φ lz is he Hicksian coefficien of complemenariy beween l and z in he disuiliy φ as defined in 4. In his one-period model, here are hree effecs from subsidizing raining ha are balanced a he opimum. Firs, a higher ne wedge increases invesmens in raining, and, hence, increases he wage and labor supply. This is he so-called labor supply effec. However, here is also a differenial effec on differen abiliy ypes. If ρ z > 0, raining benefis able agens more and furher spreads ou earnings inequaliy. On he oher hand, if ρ s < 0, earnings inequaliy is compressed by raining. This is he so-called inequaliy effec. Given ha abiliy is sochasic here, a complemenariy o abiliy also increases risk exposure. In mechanism design language, he inequaliy effec is jus he resul of he informaional ren ha needs o be forfeied o higher abiliy agens. When ρ z > 0, he informaional ren increases in raining. The hird effec is he direc ineracion wih labor supply hrough he disuiliy funcion, i.e., eiher learning-or-doing or learning-and-doing. In he formula, ε φz ε wz ρ φ lz, is he oal effec of he bonus on labor, i.e., he sum of he labor supply effec he increase in labor due o he higher wage, and eiher one of he learning-and-doing or learning-or-doing effecs, which can increase or decrease work. only if Rearranging his, we can see ha he oal effec of a raining subsidy on labor is posiive if and ε wz > ε φl z wih ε φl z log φ l / log z i.e., if and only if he wage is more sensiive o raining han he marginal disuiliy of work is. 3 The quesion hen is, wheher he increase in oal resources from he oal labor effec of raining more han makes up for he increased ren ransfers he inequaliy effec. The ne subsidy on raining will be posiive if and only if he answer o his quesion is yes. Corollary The saic ne subsidy on raining is posiive if and only if: ε φz ε wz ρ φ lz > ρ z 9 Wih learning-and-doing ρ φ lz < 0, as long as abiliy and human capial are no oo complemenary say, ρ z <, he ne subsidy on raining is posiive. Inuiively, raining does no disrac from labor effor, and so i s good o encourage i as long as high ypes do no disproporionaely benefi from i. However, if here is learning-or-doing, raining makes labor supply more cosly. In his case, even 3 Noe ha: ε φz, /ε wz, ρ φ lz, = ε φ l z,/ε wz,.

14 if he coefficien of complemenariy ρ z is small, i migh no be sufficienly small o compensae for he los work effor. Tesing for Pareo efficiency: Noe ha he simple relaion derived beween he labor wedge and he raining wedge in 8 can be used o es for he opimaliy of an allocaion. In he special cases highlighed in he nex wo subsecions, ha relaion is consan over ime and does no depend on he exac allocaion. In his case, we can use he raio of he labor wedge and he raining wedge o es for Pareo efficiency of any ax schedule. As a special case, noe ha if he wage is muliplicaively separable and φ is addiively separable, we have ρ z = and ρ φ lz = 0, and, accordingly, z = 0. This case urns ou o be of special ineres, as explained nex. 3.4 A special case: an applicaion of Akinson-Sigliz The special case in which he wage is muliplicaively separable and φ is addiively separable is an applicaion of he Akinson-Sigliz heorem Akinson and Sigliz, 976. Le he separable disuiliy funcion φl, z be given by: φl, z = φ l+φ 2 z. The separable uiliy hen saisfies all assumpions of he Akinson-Sigliz heorem, where he wo commodiies are c and z. We can direcly show ha, in his case, he ax schedule should be neural wih respec o human capial by using a simple variaional argumen. Recall ha he objecive is o maximize ne resources or minimize ne cos: c, ẑ l, ẑw, ẑ 0 where ẑ is a perurbaion from z and c, ẑ, l, ẑ are he soluions o he wo consrains below, as a funcion of ẑ. Taking he firs-order condiion wih respec o ẑ yields: c ẑ l w, ẑ l, ẑ w ẑ ẑ There are, as previously explained, wo consrains on his problem. condiion for an agen of ype, given here by: The case wih ρ z = is he case where w w given hese parameers, his consrain yields dl/dz = 0. gives: The firs one is he envelope ω = w w lφ l 2 does no change as z changes, so ha a he opimum The second consrain is ha uiliy for each ype is unchanged. Using he implici funcion heorem u cdc φ ldl φ 2 zdz = 0 3 2

15 Subsiuing his ino he firs-order condiion, yields: φ 2 z u c l w z = 0 4 Hence, condiional on l and c, z should no be disored. If we replace he definiions of τ Z and τ L from above, and rearrange o make z appear, his will be equivalen o z = Oher Special cases There are hree oher special cases ha are illusraive because hey yield paricularly simple relaions a he opimum beween he raining wedge and he labor wedge. Noe ha each of hese special relaions will also hold in he dynamic model, if human capial fully depreciaes beween periods. Muliplicaively separable wage and Cobb-Douglas disuiliy: Suppose ha he wage is muliplicaively separable and he disuiliy is Cobb-Douglas: w, z = z φl, z = γα lγ z α. In his case, ρ φ lz =, ε φz = α, and ε wz =, so ha he formula for he opimal wedge yields a very simple negaive relaion beween he opimal labor wedge and he opimal raining wedge a any poin in he skill disribuion: z = τ L α τl γ Based on he fac ha a every inerior ype τ L > 0, we can sign he wedges. 4 z < 0 a any inerior ype. In his case, CES wage and separable disuiliy: Suppose ha he wage akes a CES form w, z = ρ + z ρ ρ and he disuiliy is separable wih φl, z = φ l + φ 2 z. Then, we have ha ρ z = ρ and ρ φ lz = 0 so ha a he opimum: z = τ L ε c τl ρ + εu 4 I is shown ha he opimal labor wedge is posiive for all inerior ypes in he proof of proposiion 2. There, i is also shown ha τ L is opimally zero for he lowes and highes ypes. 3

16 If he disuiliy is in addiion isoelasic in labor wih φ l = γ lγ, hen we have: z = τ L ρ τl γ Hence, he ne wedge on raining is posiive if and only if ρ z <, i.e., if abiliy and raining are no oo complemenary in generaing earnings. Separable wage and Cobb-Douglas disuiliy: Suppose ha he wage is separable wih w, z = + z and he disuiliy is again Cobb-Douglas as above. Then he opimal raining wedge is se according o: z = τ L α τl γ In his case, he opimal wedge is again negaive since α >. See also Secion 5.3 for a version of his special case in he dynamic model. Equipped wih he inuiions and resuls from his simple one-period model, I now urn o solving he full dynamic life cycle model. 4 The dynamic planning problem In he dynamic problem, he planner canno see he abiliy realizaion in any period. Thus, he wage realizaion w, z, or labor supply l = y /w are also unknown. Oupu y, consumpion c, and raining z on he oher hand are sill observable in every period. The soluion mehod of such a dynamic mechanism design problem is based on Farhi and Werning 203 and Sancheva 204. I is hence presened only briefly here. The reader can refer o Sancheva 204 for a more deailed presenaion of his approach. To solve for he consrained efficien allocaions, I suppose ha he planner designs a direc revelaion mechanism, in which agens repor heir ypes each period and he planner assigns allocaions as a funcion of he hisory of repors made. Each agen has a hisory of repors as a funcion of his rue hisory of shocks, and his hisory of repors is denoed by r. For any reporing sraegy r, define he coninuaion value afer hisory o be: ω r = u cr y r φ w, zr, z r z r + β ω r + f + + d + 4

17 The coninuaion value under ruhful revelaion, ω, is he unique soluion o: ω = u c y φ w, z, z z + β ω + f + + d + Incenive compaibiliy requires ha ruh-elling yields a weakly higher coninuaion uiliy han any reporing sraegy r, afer all hisories : Assumpion i ũ c, y, z;, z and φl,i l iii f has full suppor on Θ. IC : ω ω r, r 5 w,z l w are bounded. ii f exiss and is bounded. The firs order approach consiss in replacing he incenive consrain by he envelope condiion of he agen. As in Secion 3, i consrains he informaional ren of each ype of agen. In his dynamic model, here is an addiional fuure erm ha capures he fac ha he agen has some advance informaion abou his fuure ype because of he Markov persisence. ω := ω = w, l φ l, l, z z + β w ω + f + + d + The envelope condiion is only a necessary condiion for incenive compaibiliy. 5 I use a numerical ex pos verificaion procedure o ensure ha he soluion found is indeed incenive compaible see Farhi and Werning 203 and Sancheva 204 for more deails. The nex sep is o rewrie he problem recursively. Denoe he gradien of he expeced fuure coninuaion uiliy wih respec o he ype by: The envelope condiion can hen be rewrien as: 6 ω + f + + d + 7 ω = w, w l φ l, l, z z + β 8 Le v be he expeced fuure coninuaion uiliy: v ω + f + + d + 9 Coninuaion uiliy ω can hence be rewrien as: ω = u c y φ w, z, z z + βv 20 5 An applicaion of Theorem 2 in Milgrom and Segal 2002, under assumpion. 5

18 Define he expeced coninuaion cos of he planner a ime, given v,,, and z : T Kv,,, z, = min[ R τ c τ τ y τ τ P τ d τ ] τ= where, wih some abuse of noaion, d τ = d τ d τ...d, and P τ = f τ τ τ...f. A recursive formulaion of he relaxed program is hen for 2: K v,,, z, = min c w, z l + R K v,,, z, + f d 2 subjec o: ω = u c φ l, z z + βv ω = w, l φ l, l, z z + β w v = ω f d = ω f d where he maximizaion is over c, l, z, ω, v,. For period =, suppose all agens have idenical iniial human capial level z 0. 6 for = is hen indexed by U Θ, he se of arge lifeime uiliies U for each ype : The problem K U Θ, = min c w, z l + R K v,,, z, 2f d s.. : ω = u c φ l, z z 0 + βv. ω = w, w l φ l, l, z + β ω U Technical assumpions: The following assumpions are sufficien condiions o deermine he sign of he opimal wedges and are used only for ha purpose in some proofs. 7 Assumpion 2 i f b d f s d,,, and b > s ; ii 0,, ; f f iii v > 0 for all, iv 2 v K 0 and K 0. v 2 They guaranee in his order ha ha here is firs-order sochasic dominance, ha a form of monoone likelihood raio propery is saisfied, ha he expeced coninuaion uiliy is increasing in he ype, and ha he resource cos is increasing and convex in promised uiliy. 6 Oherwise, given ha human capial is observable, we would need o condiion on he iniial levels. 7 In addiion, all heoreical resuls on he signs of he wedges are indeed saisfied in he simulaions secion 7. 6

19 5 Opimal dynamic raining policies This secion characerizes he allocaions, obained as soluions o he relaxed recursive program above. 5. The dynamic gross and ne wedges As already explained in he one-period model in Secion 3, marginal disorions in agens choices can be described using wedges, which represen he implici local marginal ax and subsidy raes. In he dynamic program, wedges are defined afer each hisory. They hence represen implici axes and subsidies, locally, and condiional on a given hisory. Define he inraemporal wedge on labor τ L, he ineremporal wedge on savings or capial τ K, and he human capial wedge τ Z as follows: τ L φ l, l, z z w, z u c τ K u c Rβ E u c + τ Z τ L w z, l + [ φz, u ] u c βe + c + φ z,+ u c u + c The inraemporal labor wedge was already explained in Secion 3. The savings or capial wedge τ K is defined as he difference beween he expeced marginal rae of ineremporal subsiuion and he reurn on savings and is sandard in he dynamic public finance lieraure see Golosov e al As in he saic case, he raining subsidy τ Z can be viewed as he incremenal pay received by an agen for one more uni of raining ime. I is again equal o he difference beween marginal cos and marginal benefi. Bu he marginal benefi now lass ino all fuure periods. To wrie his recursively, I express i as a funcion of he fuure marginal cos. Solved forward, 24 would yield he full fuure sream of marginal benefis. Recall ha he marginal disuiliy cos is scaled by marginal uiliy in order o be expressed as an equivalen moneary cos. The following definiions will be needed for he dynamic formulas below. For any variable x, define he insurance facor of x, ξ x,+ : ξ x,+ Cov β u + u, x + / E β u + u E x + wih ξ x,+ [, ]. If x is a flow o he agen, i is a good hedge if ξ < 0, and a bad hedge oherwise. Wih some abuse of noaion, define also: βu ξ x,+ Cov + u βu, x + / E + u E x + which, up o an addiive consan, capures he same risk properies as ξ x,+. 7

20 Denoe by ε xy, he elasiciy of x o y, ε xy d log x /d log y. Le ε u and ε c be he uncompensaed and compensaed labor supply elasiciies o he ne wage, holding boh savings and raining fixed. 8 As in he simple one-period model of Secion 3, i would be useful o have a measure of he ne disorion and real incenive effec on raining, ha akes ino accoun he need o filer ou whaever goes owards compensaing for oher disorions. By analogy o he one-period case, we can define a ne wedge on raining, z, ha looks more complicaed, bu follows he same logic. Definiion 2 The ne wedge or ne bonus on raining ime z is defined as: z τ Z τ L φz, u c d + PZ φz, u c d τz τ L 25 φz, u c d τ K R τ K φ z, u c φz,+ R τ K ξφ /u,+ E u + c + cos, convered ino moneary unis. P Z ξ φ /u,+ τ L E φ z,+ u + c + is he dynamic risk-adjused disuiliy is he risk-adjused savings disorion. Seing he ne wedge o zero can be hough of as making raining coss locally fully ax deducible from axable income in a dynamic risk-adjused way. Recall ha in he saic case, a zero ne wedge corresponded o conemporaneous full deducibiliy. In he dynamic case, we need o ake ino accoun hree more effecs. Firs, he cos is dynamic, due o he fac ha raining acquired oday persiss ino he fuure. Second, marginal uiliy is no smooh across saes and ime and, hence, he agen does no value one dollar of ransfer oday he same as one dollar of ransfer omorrow. Thus, we need o reweigh any moneary amoun by he raio of marginal uiliies. Finally, here is a savings wedge ha disors ineremporal ransfers and which also needs o be filered ou. The ne wedge is posiive if he governmen wans o encourage raining on ne, i.e., above and beyond simply compensaing for he disorive effecs of he oher wedges. 5.2 The opimal dynamic raining subsidy The following proposiion characerizes he ne raining subsidy a he opimum. 8 I.e., ε c and ε u are defined as in he saic framework e.g., Saez, 200, a consan savings and raining ime: ε u = φ l l, i /l + φ 2 ll,i u c u c 2 φ ll l, i φ ll 2 u c u c 2 ε c = φ l l, i /l φ ll l, i φ ll 2 u c 2 u c Again, wih per-period uiliy separable in consumpion and labor, ε c +ε u εc is he Frisch elasiciy of labor. 8

21 Proposiion 2 i A he opimum, he ne raining subsidy is given by: z = τ L ε c τl + ε u ρ z, ε φz, ρ φ lz, + τ ε wz, R E L+ + ε c + τl + ε u + w z,+ l + ε φz,+ ρ φ lz,+ w z, l ε wz,+ wih ε φz, = d log φ /d log z he elasiciy of disuiliy wih respec o raining ime. ρ φ lz, is he Hicksian coefficien of complemenariy beween l and z in he disuiliy φ. The labor wedges τl and τ L+ + are a heir opimal levels. ii The ne raining wedges are zero a he boom and he op of he abiliy disribuion: z, = z, = 0 As in he one-period case, we see he labor supply effec, he inequaliy effec, and he conemporaneous learning-or-doing or learning-and-doing effecs. However, in his dynamic model, he subsidy on raining ime has an addiional direc ineracion wih fuure labor supply hrough he disuiliy funcion. Even if raining divers ime away from conemporaneous labor supply, he effecs on fuure labor supply can moivae a posiive ne subsidy. Whaever he paern of complemenariy or subsiuabiliy beween conemporaneous labor supply and raining, he relaion beween curren raining and fuure labor supply is is mirror image. If he former are complemens, he laer are subsiues, and vice versa, because invesing in human capial oday means having o inves less omorrow o reach any given level. 9 If here is learning-or-doing, he ne wedge co-moves posiively wih he fuure income ax rae τ L+, bu negaively wih he curren ax rae τ L. Inuiively, if here is a higher conemporaneous wedge on labor, here already is an indirec simulus o raining because raining is a subsiue for labor; hence he need for an addiional simulus hrough a ne subsidy is reduced. However, a higher fuure labor wedge is a simulus for raining in he fuure, which is a subsiue for raining oday. Hence, o simulae raining oday, here is a need for a higher ne subsidy. Corollary 2 illusraes wo cases, among oher possible ones. Corollary 2 Under assumpion 2: i The ne subsidy is posiive if ε φz ε wz ρ φ lz, ρ z, and ρ φ lz, 0. ii The ne subsidy is negaive if ε φz ε wz ρ φ lz, ρ z, and ρ φ lz, 0. The sufficien condiions in i guaranee ha, alhough raining divers ime away from conemporaneous labor effor, he oal labor supply effec ε φz ε wz ρ φ lz, he direc labor supply effec plus he 9 This is because only he flow i = z z, and no he sock z eners he disuiliy. This reasoning is also only valid a inerior soluions for raining. 26 9

22 ineracion of labor and raining ouweighs he inequaliy effec ρ z. On he oher hand, he sufficien condiions in ii imply ha, despie simulaing conemporaneous labor supply hrough learning-anddoing, raining disproporionaely benefis high produciviy workers. One can rever back o he one-period special cases in Secion 3.5 if i is assumed ha he raining sock depreciaes enirely each period. In his case, all he simple relaions derived in ha secion beween he labor wedge and he raining wedge coninue o hold. Life cycle evoluion of he opimal ne raining wedge: I is naural o hink ha he parameers affecing he opimal ne wedge in 26 are age-dependen and would be evolving over he life cycle. In paricular, he ineracion beween labor and raining could be very differen a differen ages in life somehing ha is easily accommodaed by he general, age-dependen disuiliy funcion φ. I is an ineresing empirical quesion how labor supply and he acquisiion of raining evolve over life, and he answer urns ou o be imporan for opimal ax consideraions as well. 5.3 A special case: separabiliy beween raining and labor The peculiariy of ime spen invesing in human capial, unlike moneary human capial invesmens or physical capial, is ha i ineracs wih labor supply. Wih perfec informaion his would no maer, as any given disuiliy cos can be convered ino an equivalen moneary cos. This is no longer rue under asymmeric informaion, unless he disuiliy funcion is separable in labor and human capial 2 φ z l. = 0 Then, as corollary 3 shows, he raining subsidy is se only as a funcion of is complemenariy wih abiliy. Corollary 3 If 2 φ z l i The ne wedge saisfies: ii = 0 and assumpion 2 holds: τ L ε c z = τl + ε u ρ z, 27 z 0 ρ z, ii If he wage funcion is CES wih parameer ρ as in 3 and he disuiliy is separable and isoelasic φ l, i = γ lγ + η iη γ >, η >, 28 hen he ne wedge saisfies: τ L ρ z/ τl = γ 20

23 Firs, when here no longer is a direc ineracion wih labor supply, he ne simulus o raining is posiive if and only if raining is no oo complemenary o abiliy ρ z. In addiion, because he fuure ineracion wih labor supply no longer maers, he raining wedge and labor wedge are se according o a conemporaneous inverse elasiciy rule, in which he raining wedge is higher whenever is complemenariy wih abiliy is lower. Wih a CES wage funcion, his relaion is paricularly simple: in any period, he relaion beween he raining wedge and he labor wedge needs o say consan. 6 The opimal labor wedge in he presence of raining Wha happens o he labor wedge in he presence of raining? I is naural o hink ha he presence of raining, given is srong ineracion wih labor, would affec he opimal labor wedge. However, i urns ou ha in he case jus analyzed, he presence of learning-or-doing or learning-and-doing does no really change he labor wedge. The specificaion of he model maers a lo for his resul. I herefore consider wo modificaions o he model, which have imporan implicaions for he labor wedge. The firs is he limi case of learning-by-doing, in which all raining comes direcly as a byproduc of he unobservable labor supply. The second is he case in which higher abiliy agens are no only more producive a producing oupu, bu are also more able a raining. 6. Does learning-or-doing versus learning-and-doing maer for he opimal labor disorion? In he model analyzed up o here, he labor wage is only slighly affeced by he presence of human capial, bu no by eiher learning-or-doing or learning-and-doing. The full formula for he labor wedge is given in he nex proposiion. Proposiion 3 i A he opimum, he labor wedge is equal o: τl τl = µ u c ε w, f + ε u ε c 29 wih µ = η + κ, where η can be rewrien recursively as a funcion of he pas labor wedge, τ L, : and η = κ = τ L [ Rβ τl, u c u c ii τ L, = τ L, = 0,. u c 2 ε c + ε u ε w, f s d s u c m f m dm f ]

24 Hence, he only effec of raining is o inroduce he wage elasiciy wih respec o abiliy ε w,, which is usually consan and equal o one in a model wihou raining in which w =. Wih raining, a higher elasiciy amplifies he labor wedge as i increases he value of insurance and redisribuion. However, i does no direcly maer wheher here is learning-or-doing or learning-and-doing. The sandard zero disorion a he boom and he op resuls from he saic Mirrlees model coninue o apply. The muliplier µ on he envelope condiion capures wo effecs ha amplify or dampen he labor wedge over he lifecycle. Because he raining wedge in 26 was derived in relaion o he labor wedge, hey also influence he magniude of he ne raining wedge. The firs effec is an insurance moive, capured by κ, familiar from he saic axaion lieraure. I would be zero wih linear uiliy. The erm η capures he previous period s labor wegde, hence indirecly he previous period s insurance moive, weighed by a measure of abiliy persisence. In he limi, if is idenically and independenly disribued iid, only he conemporaneous insurance moive κ maers. 6.2 The limi case of Learning-by-Doing A limi case of learning-and-doing would be he learning-by-doing model inroduced by Arrow 962, which posis ha agens acquire human capial hrough work iself. Wih asymmeric informaion, his limi case acually leads o very differen implicaions for he labor wedge. This is because human capial wih learning-by-doing is a direc by-produc of labor, bu labor is unobserved by he governmen. There is no separae insrumen for raining and all he incenives have o be se indirecly hrough he labor wedge. Indeed, his case is much closer o a case wih unobservable raining. As soon as raining is a direc by-produc of labor, eiher he governmen will be able o perfecly infer labor effor from observing raining if we mainain he assumpion ha raining is observable or i has o be ha raining like labor is unobservable. Learning-by-doing can be modelled by leing he invesmen in raining be he direc resul of labor, no of a separae effor, so ha raining ime as previously defined, i is equal o i = i l where i is an increasing, concave funcion of l. The sock of human capial z is z = τ= i l τ. For simpliciy, I will consider a linear funcion wih i = l, bu he more general formulaion gives similar resuls. The nex proposiion highlighs how he opimal labor wedge needs o be modified in order o ake ino accoun he presence of learning-by-doing. Proposiion 4 Wih learning-by-doing, such ha i = l and z = τ= l τ : 22

25 i The opimal labor wedge is given by: τ L + w z, l τ L τ L w = µ f u c ε w + R E [ w+ w [ + ε u ε c w ] z l [ ρ z ] w τ L,+ τ L τ L,+ µ + τ L f + ε w,+ + + ε u + ε c + ] u c + ii Excep in he las period = T, he zero ax resuls a he boom and he op no longer hold: τ L, 0 and τ L, 0. Three elemens are noeworhy in his formula relaive o 29. Firs, he relevan efficiency cos of axaion is no longer jus he own-price elasiciy, as capured by +εu ε c. Insead, i also includes he incenive effec of raining ime, as capured by he Hicksian coefficien of complemenariy ρ z. All else equal, a lower Hicksian coefficien of complemenariy of raining and abiliy ends o decrease he labor wedge because i is aracive o encourage raining acquisiion which corresponds o supplying more labor when here is learning-by-doing. The reason for his was explained in Secion 3: when he Hicksian coefficien of complemenariy is below, raining has a redisribuive and insurance effec. Secondly, he opimal labor wedge, which is normally an inraemporal wedge, now becomes forward-looking and dynamic. Labor supply now has an ineremporal effec and lasing impacs because i indirecly builds human capial. There is a need o ake ino accoun he nex period s incenive effecs of having a higher human capial sock and he nex period s ax. Third, and relaed o he dynamic implicaions of he labor wedge, he zero ax resul a he boom and op of he abiliy disribuion no longer holds excep in he las period, which is akin o a saic model. The ineremporal effecs of he labor wedge lead o differen incenive implicaions. 6.3 Higher abiliy for working and raining There is anoher naural modificaion of he model which also has differen implicaions for he labor wedge. Suppose ha higher abiliy agens are no only more producive a producing oupu, bu are also more producive a raining, i.e., hey have a lower marginal cos for any raining achieved. In his case, he per-period uiliy of an agen of curren ype akes he form: y uc φ w, z, z z The envelope condiion needs o be modified accordingly see he Appendix and is now given by: ω = w lφ l + z z w 2 φ i + β where φ i is he derivaive of φ wih respec o is full second argumen