University of Zurich Deprtment of Economics Center for Institutions, Policy nd Culture in the Development Process Working Pper Series Working Pper No. 421 Eduction nd Optiml Dynmic Txtion: The Role of Income-Contingent Student Lons Sebstin Findeisen April 2013
Eduction nd Optiml Dynmic Txtion: The Role of Income-Contingent Student Lons Sebstin Findeisen University of Zurich Dominik Schs University of Konstnz First version: October 21, 2011 This version: April 19, 2013 Abstrct We study Preto optiml tx nd eduction policies when humn cpitl upon lbor mrket entry is endogenous nd individuls fce wge uncertinty. Though optiml lbor distortions re history-dependent, i.e. depend on income nd eduction, simple policy instruments cn yield the desired distortions: single nonliner lbor income tx schedule combined with income-contingent lons. To tke the model to the (US) dt, we simplify the model to binry eduction decision (grduting from college or not). We find tht for low nd intermedite incomes the lbor supply decision of college grdutes should be distorted more hevily thn for individuls without college degree. As consequence, the optiml student lon repyment schedule increses in income for this rnge. This result holds long the Preto frontier. We compre the second best to sitution where lon repyment is restricted to be independent from income nd find significnt welfre gins. JEL-clssifiction: H21, H23, I21 Keywords: Optiml dynmic txtion, Eduction, Implementtion Contct: sebstin.findeisen@uzh.ch, dominik.schs@uni-konstnz.de. We owe specil thnks to our dvisors Friedrich Breyer nd Fbrizio Zilibotti for ongoing support nd vluble comments. We lso thnk Mnuel Amdor, Dn Anderberg, Crlos d Cost, Emmnuel Frhi, Mike Golosov, Bs Jcobs, Normnn Lorenz, Elen Mttn, Florin Scheuer, Dirk Schindler, Kjetil Storesletten, Aleh Tsyvinski, Iván Werning nd Christoph Winter for encourgement nd helpful discussions. We lso like to thnk udiences t IIPF (August 2012, Dresden), Munich (July 2012), NORMAC (August 2012, Stroemstdt) PET (June 2011, Bloomington), SED (June 2012, Limssol), SMYE (April 2012, Mnnheim), Stnford (October 2011), Yle (Mrch 2011) nd VFS (September 2012, Goettingen) for their comments. We thnk Stefn Voigt for vluble reserch ssistnce. Prts of this reserch were undertken while Sebstin Findeisen ws visiting Yle nd Dominik Schs ws visiting Stnford. We re grteful for their hospitlity. Sebstin Findeisen cknowledges finncil support from the ERC Advnced Grnt IPCDP-229883 nd the University of Zurich (Forschungskredit of the University of Zurich, grnt no. 53210603).
1 Introduction The deepening wge nd income inequlity observed in mny countries over the lst severl decdes is often linked to chnges in the reltive returns to eduction. In recent survey, Acemoglu nd Autor (2011) note tht the college wge premium hs been rising stedily in the US for the lst 30 yers. The sme holds true for mny other dvnced economies, s n incresing college wge premium hs contributed to n increse in the dispersion of the distribution of wges nd income (Krueger, Perri, Pistferri nd Violnte 2010). Motivted by this tight connection between economic inequlity nd differences in eductionl ttinment, we tckle the optiml design of integrted income tx nd eduction policy systems in this pper. We nlyze dynmic heterogeneous gents economy, which consists of n eduction nd working period. Unlike former ppers in the literture, we ssume both, tht individuls re ex-nte heterogeneous nd the eductionl investment is risky. 1 The resulting Preto optiml lbor distortions re history-dependent, implying tht implicit tx rtes should condition on eduction in ddition to income. History-dependent lloctions nd lbor wedges re stndrd outcome in dynmic tx problems (Golosov, Tsyvinski nd Werning 2007, Kocherlkot 2010). We show tht in our setting Preto optiml lloctions cn be implemented by simple policy instruments: income-contingent student lons in ddition to stndrd txes on lbor ernings nd svings. We pply our model to the cse of binry eduction decision: going to college or entering the lbor mrket directly fter high-school. We use income dt to clibrte our model. The min focus of this numericl exercise is on the following question: given tht we know tht income-contingent student lons implement the optimum, how should they vry with income? We find tht the optiml repyment schemes re long the Preto frontier incresing for low nd intermedite income t n lmost liner slope. A second question we ddress is: cn we guge the welfre gins of such policies reltive to simpler policies? We compre the optiml policies to scenrio where repyment cnnot be income-contingent nd find significnt welfre gins. The ide of income-contingent student lons hs been proposed s erly s 1955 1 With the exception of Bénbou (2002), no pper hs looked t optiml policies in frmework with heterogeneity nd uncertinty yet. His pproch differs from ours in tht he does not solve for the second-best but compres the use of progressive income txes with progressive eduction subsidies. 2
by Milton Friedmn (1955), who envisioned repyment mounts to be proportionl to income, i.e. linerly incresing repyment schedule. Severl countries like the United Kingdom, Austrli or New Zelnd currently hve similr systems in plce, where repyment is proportionl to income within certin income rnge. In this sense, our model develops second-best rgument for the use of such incomecontingent repyment rtes. Importntly, to rrive t this conclusion, we do not impose ny restrictions on policy instruments in the spirit of the dynmic pproch to public finnce. It is noteworthy tht the implementtion result itself is not necessrily cse for Friedmn s proposl. In theory, repyment could lso be decresing in income if college students re lredy over-insured (s compred to the second-best optiml insurnce) by the income tx; in such cse it would be optiml to lower their effective mrginl tx rtes by hving decresing repyment schedule. Looking t rel world income distributions, however, our Mirrleesin nlysis yields the opposite normtive conclusion: robust numericl result is tht for low nd intermedite incomes, repyment should indeed be incresing in income. In our nlysis, we first concentrte on generl model, where the focus lies on theory. We study n environment chrcterized by privte informtion, which evolves stochsticlly over time. Individuls re born with different innte bilities nd decide on their humn cpitl investment. After obtining eduction, gents lern their skill level tht cn be interpreted s wge, they work nd generte income. The plnner cnnot observe innte bility, work effort nd the skill level (wge). However, we do ssume tht the plnner cn observe eductionl investment. Lbor mrket risk is tken into ccount in our model by the fct tht individuls fce distribution of wges before entering the lbor mrket. This distribution function depends on cquired humn cpitl nd innte bility. When deriving nlyticl results, we try to impose miniml structure on this stochstic process nd work with generl distribution function. These modelling ssumptions re quite flexible nd in line with empiricl evidence s we rgue in more detil when introducing the model. As hinted t bove, the optiml lloction fetures history-dependent lbor wedges, which is stndrd in dynmic models with informtionl frictions. We derive simple formul for the optiml lbor wedge tht cn be decomposed into two terms: one tht is equivlent to the seminl formuls of Mirrlees (1971) nd Sez (2001) nd one tht cptures the optiml provision of dynmic incentives. We show tht the dynmic 3
incentive term hs very simple intuition, relted to how responsive the distribution of skills is to innte bilities. Our theory lso presents novel results on how individul eduction decisions re distorted by government intervention in constrined Preto optiml lloctions. This depends on two forces. First, implicit eduction subsidies re used to offset the distortionry impct of lbor txes on eductionl decisions. Intuitively, the gins from investing into eduction re prtly reped by the government vi income txes. Therefore the government uses n eduction subsidy to internlize this fiscl externlity. Second, the plnner wnts to tx eduction in cse he wnts to redistribute from high to low innte bility types t the mrgin. This tx effect is incresing in the complementrity between innte bility nd eduction. Next, we move on to suggesting wys how Preto optim cn be decentrlized. We show tht the following policy instruments cn implement the desired lloctions: student lons with income-contingent repyment, lbor income tx schedule tht solely depends on income nd nonliner svings tx. The lbor wedge consists of the mrginl tx rte on lbor income nd the mrginl lon repyment. The eduction wedge is implemented by the chnge in lon size s function of eduction nd how the repyment schedule chnges with the lon size. In ddition, we provide n lterntive implementtion with eduction-dependent income txes similr to the ide of grdute tx which hs been proposed in the populr debte in the United Kingdom. Finlly, we move on from theory nd present simple ppliction of the model, feturing binry eduction decision nd only two different ex-nte types. Importntly, this enbles us to use estimtes of fctul s well s counterfctul income distributions from the lbor economics literture (Cunh nd Heckmn 2007, 2008), both of which re needed s n input to simulte optiml policies. We follow Sez (2001) nd ppend Preto tils to the distributions of income before we clibrte the skill distribution. Under the we believe resonble ssumption tht the college income distribution hs thicker tils, the repyment schedule is incresing in income for Utilitrin plnner. Importntly, this result is stronger thn tht, s it holds long the Preto frontier. Under the more conservtive ssumption tht top incomes re distributed ccording to the sme Preto prmeter, the repyment schedule is still incresing for low to intermedite incomes nd roughly flt fterwrds. 4
We guge the welfre gins reltive to sitution where lbor wedges re constrined to be function of income only but the plnner cn otherwise fully optimize. Concerning the policy instruments mentioned bove, this is equivlent to restricting lon repyment to be independent of income. The key simplifying ssumption to mke this comprison trctble is the bsence of income effects. We show tht in this cse, constrining lbor wedges to be function of income only is equivlent to restricting income to be function of the wge only. Compring the second best to this solution, we find welfre gins rnging from bout 0.2% to 0.6% of lifetime consumption nd we show how these gins chnge with risk-version nd societl preferences for redistribution. Our welfre comprison is lso methodologicl contribution becuse we re the first to use first-order pproch for the cse where lbor txes re constrined to depend on current income only nd cn therefore use very fine type spce. Relted Literture. Severl previous ppers hve studied problems of optiml income txtion nd their reltion to eduction decisions. One strnd of inquiry hs worked under the ssumption of ex-nte homogeneity nd risky humn cpitl. D Cost nd Mestri (2007) show tht humn cpitl should lwys be encourged in the second-best optimum. Anderberg (2009) emphsizes tht the risk properties of humn cpitl re crucil for the question whether nd how eduction should be distorted reltive to first-best rule. 2 In sttic setting with heterogeneity but without uncertinty, Bovenberg nd Jcobs (2005) nlyze how endogenous eduction lters the result of the Mirrleesin tx problem nd conclude tht distortions on the eduction mrgin through income txtion should be offset by eduction subsidies. Reltedly, Bohcek nd Kpick (2008) study dynmic model with certinty nd obtin similr results. 3 Our pper is connected to the New Dynmic Public Finnce literture. Concerning the chrcteriztion of Preto optiml lloctions, it is relted to two recent contribu- 2 Focusing on liner policy instruments Anderberg nd Andersson (2003) s well s Jcobs, Schindler nd Yng (2012) lso discuss the importnce of the risk properties of humn cpitl. 3 Kpick (2006) introduces non-observble endogenous humn cpitl into dynmic, non-stochstic Mirrlees model where txes cn only be conditioned on current income. He shows tht mrginl tx rtes re lowered due to the eduction mrgin. Bovenberg nd Jcobs (2005) consider observble nd unobservble investment nd provide n interesting discussion bout different implictions. Grochulski nd Piskorski (2010) focus on the implictions of unobservble humn cpitl investment for cpitl txtion in n ex-nte homogeneous gent setting with uncertinty. Reltedly, Shourideh (2012) studies optiml txtion of cpitl income with heterogeneous gents nd cpitl income shocks. 5
tions tht hve nlyzed optiml lbor wedges in dynmic Mirrlees economies without endogenous eduction but with productivity shocks in every period, which we bstrct from. Frhi nd Werning (2012) chrcterize the evolution of the lbor wedge over time nlyticlly nd numericlly in such n environment. Golosov, Troshkin nd Tsyvinski (2011) derive nd illustrte optiml lbor wedges in the cross section, connecting their generlized formuls to the sttic Mirrlees-Sez expressions. Concerning the implementtion of history-dependent lloctions, our pper is relted to Golosov nd Tsyvinski (2006) who consider n environment with bsorbing disbility shocks nd present n implementtion in which disbility insurnce conditions on sset testing. 4 Similr s Golosov nd Tsyvinski (2006), we nlyze model with reltively simple stochstic environment, which enbles us to present decentrliztions of Preto optim with simple policy instruments. Further, our implementtion builds on recent work of Werning (2011), who shows tht decentrlized implementtions of incentive comptible lloctions exist, which re chrcterized by non-liner cpitl txes tht do not condition on current income nd cn even be history-independent. The dependency of the tx schedules on chrcteristics other thn income is relted to the ide of tgging, s proposed by Akerlof (1978). 5 Wheres our nlysis shres the feture of this method in tht the socil plnner uses dditionl informtion to tilor mrginl tx rtes for ech eduction level to the respective skill distribution, it differs substntilly in the sense tht eduction is not n immutble tg, but rther n endogenous vrible. 6 Finlly, the pper is relted to Luttmer nd Zeckhuser (2008), who consider setting in which gents hve some form of privte informtion bout distribution of outcomes. They pply this ide numericlly to the cse of income risk nd two exnte types who self-select into two different tx schedules bsed on privte signl bout their bility. For this purpose the uthors interpret the college entry decision s proxy for this privte signl. In contrst, in our model the distribution of lbor 4 Kocherlkot (2005) provides n implementtion bsed on income-contingent welth txes in very generl stochstic environment. Albnesi nd Sleet (2006) derive n implementtion with welth-contingent income txes for the cse of iid shocks. 5 More recently tgging is investigted by Cremer, Ghvri nd Lozchmeur (2010), Mnkiw nd Weinzierl (2010) s well s Weinzierl (2011b). 6 A similr logic rises in the recent pper of Scheuer (2012) in model with endogenous occuptionl choices. 6
mrket skills depends on innte bility nd eduction jointly nd importntly, the eduction decision of individuls is endogenous. This pper is orgnized s follows. Section 2 contins the bsics of the model. In Section 3, we investigte dynmic incentive comptibility nd describe the mjor properties of constrined efficient lloctions. Decentrlized implementtions of constrined efficient lloctions re provided in Section 4. We pply our model to the cse of binry eduction decision in Section 5. Section 6 concludes. 2 The Model 2.1 Technology nd Preferences Individuls, whose mss is normlized to one, live for two periods. In the first period they cquire humn cpitl nd in the second period they work. Individul lbor mrket bility in period two is stochstic nd the distribution depends on eductionl investment nd initil type. We now formlize these ides. Individuls differ in innte bility θ, which cn be interpreted s one dimensionl ggregte of (non-)cognitive skills, I.Q. nd fmily bckground, nd is distributed in the intervl [θ, θ] ccording to the cumultive density function (cdf) F (θ). After individuls lern their type θ, which is privte informtion, they mke n eductionl investment z. Flow utility in period 1 is ssumed to be u(c 1 ), so we bstrct from ny effort costs of eduction. In period two individuls drw their lbor mrket bility from continuous conditionl cdf G( z, θ), which depends on innte bility θ nd eduction z nd hs bounded support [, ]. When individuls lern their type, they mke lbor-leisure decision. We ssume tht preferences in period 2 re given by: 7 u(c 2 ) Ψ ( y ). With this specifiction of the model, we cpture mny empiricl regulrities. First, ssuming G( z, θ) to be non-degenerte, our model cptures the importnt fct of uncertinty in the lbor mrket nd risky eductionl investment. See e.g. Cunh nd Heckmn (2008) or Chen (2008) for recent contributions. 7 We stick to seprble preferences in the nlytic prt becuse the formul for the optiml lbor wedge will be very esy to interpret. For the numericl simultions in Section 5, we sometimes deprt from the seprbility ssumption. 7
Second, we llow this cdf to be function of innte bility θ nd thereby cpture the fct tht inequlity in ernings is to certin extent lso determined by innte bility. Tber (2001) presents findings suggesting tht much of the rise in the college premium my be ttributed to rise in the demnd for unobserved skills, which re predetermined nd independent of eduction. Indirect evidence for the importnce of unobserved skills comes from the strong persistence of within eduction group inequlity (Acemoglu nd Autor 2011). Third, the cdf G being function of z cptures the returns to eduction. Importntly, for most of our results, we do not impose certin ssumption on the pttern of these returns. Thus, our frmework cn either cpture risk-incresing or riskdecresing humn cpitl investment. Fourth, s long s 2 G( z,θ) θ z 0, returns to eductionl investment differ in innte bility θ. E.g., Crneiro nd Heckmn (2005) document tht the returns cn differ by s much s 19% points cross individuls for one yer of college. See lso Lemieux (2006) for evidence on heterogeneity in returns. To shrpen few nlyticl results, it turns out helpful to plce some structure on the behvior of G( z, θ): Assumption 1: G( z, θ) F OSD G( z, θ) G( z, θ) G( z, θ), for ll z < z nd for ll (θ, ). Assumption 2: G( z, θ ) F OSD G( z, θ) G( z, θ ) G( z, θ), for ll θ < θ nd for ll (z, ). Assumption 3: 2 G( z,θ) θ z 0 for ll θ, z. These ssumption will not be needed to derive our min results, but help to illustrte importnt spects of the model. Whenever n ssumption is needed for result, we refer to it. The first nd the second one cpture the notion tht eduction nd innte bility should both hve direct effect on lbor mrket outcomes represented by first-order stochstic dominnce shift; rther nturl wy of ordering distributions. The third one cptures their interction nd respects the compelling evidence of complementrity between erly bility nd eductionl investment. When simulting optiml policies, we tke into ccount tht the eduction period is shorter thn the working period nd then consider T e identicl eduction periods 8
nd T w identicl working periods. Hence, we bsiclly just multiply period one by T e nd period two by T w. This lso implies tht we bstrct from further shocks to idiosyncrtic lbor productivity. This simplifies nd helps to focus the nlysis on the eduction-txtion link. In the empiricl literture, there is no ultimte consensus on the reltive importnce of heterogeneity before lbor mrket entry for lifetime inequlity, but different pproches hve ttributed mjor role to it. In recent work, Huggett, Ventur nd Yron (2011) estimte structurl life-cycle model nd find tht differences relized t the ge of 23 cn ccount for more of the vrition in lifetime outcomes thn do shocks received over the working lifetime. A stndrd reference is Ken nd Wolpin (1997), who ttribute n stonishing 90% to heterogeneity relized before lbor mrket entry, while Storesletten, Telmer nd Yron (2004) estimte number of bout 50%. 2.2 The Lissez Fire Equilibrium To ly out the bsic properties of the model, we strt with the chrcteriztion of the lissez-fire equilibrium without government intervention. In the second period, fter gents hve lerned their lbor mrket skill, they choose lbor supply, tking svings or privte debt s given. This gives rise to the indirect utility function: ( y ) v 2 (, s(θ)) = mx u (c 2 ) Ψ y,c 2 s.t. c 2 = y + Rs(θ). Individuls utility functions re well-behved u(.) is ssumed to be incresing, t lest twice continuously differentible nd strictly concve, nd Ψ(.) to be incresing, t lest twice continuously differentible nd strictly convex. The prmeter is n individul s lbor mrket skill, mening tht individuls with higher need to provide less lbor effort to ern given income y. In the first period, gents decide how much to invest in eduction, nd mke consumption-sving decision. Agents hve ccess to risk-free one period bond mrket; we impose no short-sle or enforcement constrints nd n exogenous gross return R. This defines the indirect utility function: V (θ) = mx s,z,c 1 u(c 1 ) + β v 2 (, s)g( z, θ)d s.t. c 1 + z = s, 9
where s re svings. As lredy stted in the lst section, we model the conditionl distribution of skills g( z, θ) s being determined by n gent s eduction level z nd her innte bility θ. Moreover, we focus on eductionl investment s direct monetry cost. This is consistent with the ide tht tuition fees nd other monetry expenses re the most importnt fctors on the cost side driving eductionl decisions. It is lso in line with foregone ernings interprettion, where more eduction delys lbor mrket entry. z cn be sum of both fctors. We now present the min properties of the equilibrium without government intervention: Proposition 2.1. The lissez-fire lloction hs the following properties: (i) The Euler Eqution holds: u (c 1 (θ)) = βr u (c 2 (θ, ))dg( z(θ), θ) ( ) (ii) Lbor supply is undistorted: Ψ y(θ,) 1 = u (c 2 (θ)). (iii) The mrginl cost of eduction is equlized to mrginl benefits: u (c 1 (θ)) = β v 2(θ, ) g( z(θ),θ) z d (iv) If Assumptions 1-3 hold, eductionl investment is incresing nd svings re decresing in innte bility, i.e. z (θ) > 0 nd s (θ) < 0. Proof. See Appendix A.1 Prts (i)-(iii) follow directly from the first-order conditions. They re unsurprising properties, stting tht privte mrginl rtes of substitution re equted to technicl mrginl rtes of trnsformtion on the lbor, cpitl, nd eduction mrket. Prt (iv) sttes tht without government policies, eduction nd svings re monotone in innte bility θ if Assumptions 1-3 re fulfilled. The proof provides instructive intuition for this result. It is sufficient to show tht the objective is supermodulr in ll choice vribles nd type θ (see Milgrom nd Shnnon (1994)). Inserting the budget constrint gives the problem reduced to two choices s nd z: mx s,z U(s, z; θ,, β) = 10
u( s z) + β v 2(, s)g( z, θ)d. This objective is supermodulr in credit tken s, eduction z nd type if nd only if: 2 U(s, z; θ,.) s θ 2 U(s, z; θ,.) s z 2 U(s, z; θ,.) z θ < 0 (1) < 0 (2) > 0. (3) In Appendix A.1 we show tht ll inequlities hold. Equtions (1) nd (2) imply tht the return to svings is lower for higher θ types nd with higher eduction, since expected lbor skills re lso higher. Eqution (3) holds, since innte bilities nd eduction re complementry to ech other. Tken together the direct effects of being of higher type on credit nd eduction re being reinforced by the reltionship between the endogenous vribles. So fr, we hve ssumed no limits on the bility of gents to borrow ginst future lbor income. Imposing n d hoc constrint of the form s φ, where φ is some negtive number, leves most of the results from Proposition 2.1 unffected. Notbly, constrined gents will not be ble to smooth consumption intertemporlly s much s desired. Still eduction levels will be incresing in type: Corollry 2.2. Suppose Assumptions 1,2 nd 3 hold. If gents fce borrowing constrints s φ, eduction is monotone in type θ, i.e. z (θ) > 0 in the lissez-fire equilibrium. Proof. Above some threshold type, gents rech the borrowing limit nd set s equl to φ. Of those gents higher types still fce the greter returns to eduction becuse of the complementrity nd therefore choose higher level of z. The empiricl literture hs documented sorting into eduction, bsed on heterogeneous expected returns (Cunh nd Heckmn 2007). The monotonicity of eduction in the lissez-fire equilibrium is consistent with tht fct. 2.3 Wedges For lter purposes when we nlyze optiml lloctions nd the respective tx nd eduction finnce systems tht cn implement such lloctions, it is useful to define 11
three wedges. They re equl to implicit mrginl tx rtes on svings, lbor income nd eduction, respectively: Svings wedge: τ s (θ) = 1 u (c 1 (θ)) βr u (c 2 (θ, ))g( z, θ)d where R is the gross return on svings. τ s (θ) > (<)0 implies downwrd (upwrd) distortion of svings. Lbor wedge: The lbor wedge is positive (negtive) if n individul works less (more) thn it would t the intervention-free mrket price (which is her productivity level ). Formlly the lbor wedge reds s: τ y (θ, ) = 1 ( ) Ψ y(θ,) 1 u (c 2 (θ, )). Eduction wedge: Here positive (negtive) wedge corresponds to n upwrd (downwrd) distortion of the eduction decision. Formlly the eduction wedge reds s τ z (θ) = 1 β v 2(θ, ) g( z(θ),θ) d z(θ) u. (c 1 (θ)) 3 Constrined Preto Optiml Alloctions In this section, we chrcterize constrined Preto efficient lloctions, where constrined refers to the government being unble to observe gents type θ in period one nd in period two. In Subsection 3.1, we show tht the problem is trctble using first-order pproch. In ddition, we provide necessry s well s sufficient conditions for this pproch to be vlid. In Subsection 3.2, we nlyze optimlity conditions nd their consequences for optiml policies. In Subsection 3.3, we explore the model using numericl simultions. 3.1 Incentive Comptibility We cst the problem s sequentil, dynmic mechnism gents report n initil type θ in the first period nd, fter uncertinty hs mterilized, report their productivity in the second period. The plnner ssigns initil consumption levels c 1 (θ) nd 12
eduction levels z(θ) to individuls with innte bility θ. Moreover, with ech report there comes sequence of utility promises for the next period {v 2 (θ, )} [,]. In the second period, the screening tkes plce over consumption levels c 2 (θ, ) nd lbor supply y(θ, ). All these quntities define n lloction in the economy. Dynmic incentive comptibility is ensured bckwrds, so we strt nlyzing the problem from the second period. 3.1.1 Second Period Incentive Comptibility By the reveltion principle, we cn restrict ttention to direct mechnisms. Suppose tht in the first period gents hve mde truthful reports r θ (θ) = θ, lthough this is not necessry nd just simplifies the exposition. 8 Conditions for this to be true re given in the next subsection. Conditionl on this report, the second period incentive constrint must be met for ny history of types (θ, ) nd reporting strtegy r (): [ ] [ ] y(θ, ) y(θ, r ()) u (c 2 (θ, )) Ψ u (c 2 (θ, r ())) Ψ, r (), θ. Define the ssocited indirect utility function of the gents s: [ ] y(θ, v 2 (θ, ) = mx u (c r ()) 2 (θ, r ())) Ψ. r () Like in stndrd Mirrleesin problem preferences stisfy single-crossing for given first-period reports. For globl incentive comptibility it is, hence, necessry nd sufficient tht ll locl envelope conditions hold: v 2 (θ, ) ( ) y(θ, ) y(θ, ) = Ψ 2, (4) nd the usul monotonicity condition, stting tht y(θ, ) is non-decresing in bility levels, is stisfied: y(θ, ) 0. (5) 8 The reson is tht in the second period the utility is function of, r () nd r θ (θ) but not of θ. 13
3.1.2 First Period Incentive Comptibility In the first period, n gent tkes into ccount the effect of her report bout θ on future utility. First period incentive comptibility is ensured if nd only if the double continuum of wek inequlities holds: U(θ, θ) = u (c 1 (θ)) + β u (c 1 (r θ (θ))) + β v 2 (θ, )dg( z(θ), θ) v 2 (r θ (θ), )dg( z(r θ (θ)), θ) = U(θ, r θ (θ)), θ, r θ (θ), where U(θ, r θ (θ)) is the expected utility of n individul of type θ reporting r θ (θ). The ssocited vlue function is: V (θ) = mx r θ (θ) u (c 1 (r θ (θ))) + β v 2 (r θ (θ), )dg( z(r θ (θ)), θ). (6) By using the FOC of (6) one cn esily derive the following envelope condition dv (θ) dθ = β v 2 (θ, ) g( z(θ), θ) d, (7) θ which cn esily be incorported into Lgrngin. As often done in screening problems, our strtegy for solving the second-best problem is to work with relxed problem with only restrictions (4) nd (7) being imposed nd then check expost whether incentive comptibility is fulfilled. In the numericl explortions in Section 3.3 we find tht incentive comptibility is lwys stisfied nd therefore the first-order pproch is vlid for the primitives we consider. 9 Next, we present set of sufficient conditions. Lemm 3.1. Suppose Assumptions 2 nd 3 hold, conditions (4), (5), (7) re stisfied nd we hve: (i) y(θ,) θ > 0, 9 Our results of this section on dynmic incentive comptibility re relted to previous work in the optiml non-liner pricing literture by Courty nd Li (2000). They study optiml pricing schemes of monopolist fcing consumers with stochstic tstes. In our cse the distribution of types tomorrow is endogenous since eduction is choice. In recent contributions, Kpick (2011) s well s Pvn, Segl nd Toikk (2011) investigte the robustness nd vlidity of the Mirrleesin first-order pproch in lrge clss of generl dynmic environments. 14
(ii) z(θ) θ > 0, then the considered lloction is incentive comptible. Proof. See Appendix A.2. This lemm implies tht insted of directly ex-post verifying whether period one incentive comptibility is stisfied in n lloction, one cn lterntively check these two simple monotonicity conditions; if they re fulfilled, then the lloction is incentive comptible. Wheres condition (ii) is lwys fulfilled in our numericl exmples, condition (i) ws often violted for very low ; we will comment on the resons in Section 3.3 when we present numericl illustrtions of the model. 3.2 Properties of Constrined Preto Optiml Alloctions The plnner mximizes θ θ u(c 1 (θ))d F (θ) + β θ subject to (4), (7) nd the resource constrint: θ θ [ c 1 (θ) z(θ) + 1 R θ v 2 (θ, )dg( z(θ), θ)d F (θ) (c 2 (θ, ) y(θ, ))dg( z(θ), θ) ] df (θ) = 0. We let the plnner ssign Preto weights F (θ) to individuls, depending (solely) on their initil skill level. Any distribution of these weights, which we normlize to stisfy θ f(θ)dθ = 1, corresponds to one point on the Preto frontier. λ θ R denotes the multipliers on the resource constrint nd η(θ) the multiplier function of the first-period envelope conditions. In Appendix A.3 the Lgrngin nd the first-order conditions of the problem re stted. 10 We now chrcterize the wedges of second-best Preto optiml lloctions. 10 As with the locliztion of the incentive constrints, we verify numericlly tht our solution fulfilling the first-order conditions indeed ttins the mximum. 15
3.2.1 Lbor Distortions The following proposition chrcterizes the optiml lbor wedge. 11 Proposition 3.2. At ny constrined Preto optimum, lbor wedges stisfy: where τ y (θ, ) 1 τ y (θ, ) = 1 + εu (θ, ) u (c 2 (θ, )) ε c [A(θ, ) + B(θ, )], (θ, ) g( z(θ), θ) A(θ, ) =G( z(θ), θ) 1 G( z(θ) G( z(θ), θ) [ 1 u (c 2 (θ, )) dg( z(θ), θ) 1 u (c 2 (θ, )) dg( z(θ), θ) ] B(θ, ) = 1 [1 G( z(θ), θ)] Rβ η(θ), f(θ)λ R θ where ε u (θ, ) (ε c (θ, )) is the uncompensted (compensted) lbor supply elsticity of type (θ, ) nd Proof. See Appendix A.4.2. η(θ) = F (θ) θ θ θ θ 1 u (c 1(θ)) f(θ)dθ. 1 u (c f(θ)dθ 1(θ)) Elsticities ply double role for the optiml lbor wedge. On the one hnd, higher compensted elsticity increses the excess burden of lbor distortions nd is therefore inversely relted to optiml lbor wedges; on the other hnd, higher uncompensted elsticity trnsltes into higher income inequlity for given skill distribution, mking insurnce more vluble nd therefore tends to increse optiml lbor wedges. Moreover, the weighted mss g( z(θ), θ) of gents whose lbor supply is distorted by the tx is negtively relted to the mrginl tx reflecting dedweight loss rgument. The term u (c 2 (θ, )) cn be interpreted s cpturing income effects 11 In recent pper Golosov, Troshkin nd Tsyvinski (2011) provide formuls for dynmic optiml lbor wedges with exogenous humn cpitl, connecting them to empiricl observbles in the spirit of the contributions of Dimond (1998) nd Sez (2001) for the sttic Mirrlees model. 16
for individuls with low consumption income effects re stronger nd therefore the disincentive effect of mrginl tx rtes is wekened. Conceptully, the lbor wedge consists of two prts in our dynmic economy. The first one, A(θ, ), is vrition of the optiml tx formul in the sttic Mirrlees cse, with the difference tht for ech initil type θ there is one seprte function A(θ, ). A(θ, ) disppers if gents re risk neutrl nd therefore second period insurnce is not concern. With risk-version, however, optiml policies provide insurnce ginst the lbor mrket risk gents fce. Eduction enters through its effect on the conditionl distribution of skills. The term is equivlent to the tx formul from the stndrd sttic Mirrlees problem with utilitrin welfre weights nd, s shown in Appendix A.4.2, it cn be rewritten s in Sez (2001). The second term B(θ, ) is novel nd shows how lbor tx rtes re used to optimlly supply dynmic incentives. In contrst to A(θ, ), it is independent of riskpreferences, but vnishes with ex-nte homogeneous gents. Fixing, the implicit tx rte is proportionl to [1 G( z(θ),θ)] θ, which mesures the chnge in the probbility of becoming higher type thn. Higher initil types hve, eduction constnt, higher probbility of reching skill level bove. For two neighboring θ the plnner djusts the lbor wedges of the lower type to deter devition in the first period. The increse in the implicit mrginl tx rte increses verge txes for ll skills nd mkes mimicking unttrctive for the higher type. The effect is bigger, the more importnt the effect of innte bilities on lbor mrket outcomes. The term lso increses in η(θ) which cptures the redistributive preferences of the plnner t the mrgin. Note tht this term cn lso be negtive, especilly if Preto weights re in fvor of high θ-types. Finlly, no-distortion t the top nd bottom result goes through since B(θ, ) = B(θ, ) = A(θ, ) = A(θ, ) = 0. 3.2.2 Eduction Distortions The following proposition chrcterizes optiml eduction policies. 17
Proposition 3.3. At ny constrined Preto optimum, the eduction wedge is given by: τ z (θ) = 1 R Proof. See Appendix A.4.3. + βη(θ) λ R f(θ) (y(θ, ) c 2 (θ, )) v 2 (θ, ) g( z(θ), θ) d z(θ) 2 G( z(θ), θ) d. z(θ) θ The first term cptures the expected mrginl fiscl gin of n increse in eduction. One cn show tht it is lwys positive under Assumption 1 (FOSD shift of eduction) nd positive lbor wedges. Investing dollr more into eduction increses the expected obligtion of n gent. The first prt of the eduction wedge exctly offsets this effect from the lbor wedge. Bovenberg nd Jcobs (2005) hve discovered this effect for the sttic Mirrlees model, wheres we show this fiscl externlity prt of the wedge extends to the setting with uncertinty, holding in expecttion. We now turn to the second term. Under Assumption 3 the cross-derivtive 2 G( z(θ),θ) z(θ) θ is negtive nd v2(θ,) is positive everywhere by second-period incentive comptibility. Further, for redistributive preferences η(θ) is typiclly positive. Then the second prt of the eduction wedge is negtive nd cts s n implicit tx on eduction. By distorting eduction downwrd, the plnner relxes binding incentive constrints nd cn redistribute more effectively in line with her preferences. This is consequence of the complementrity ssumption, stting tht gents endowed with higher innte skills gin more from eduction t the mrgin. The bundle of lower type, hence, becomes less ttrctive from the perspective of n gent if eduction is downwrd distorted. Such n intuition is fmilir from the stndrd sttic Mirrlees model concerning positive mrginl income tx rtes on the interior of the skill set. Reltedly, for this incentive term zero t the top nd t the bottom (θ, θ) result holds. 12 12 Jcobs nd Bovenberg (2011) discuss devitions from first-best rule for the eduction subsidy for generl ernings function in the cse without uncertinty. Our result is similr to their first result tht complementrity in eduction nd bility leds to tx on eduction. They lso consider the degree of complementrity between lbor supply nd eduction which might cll for n eduction subsidy in contrst. This second effect is not cptured in our environment since the returns to lbor supply once uncertinty hs mterilized re independent from the eduction choice. 18
3.2.3 Svings Distortions It turns out tht the presence of eduction, which endogenously ffects the probbility distribution of tomorrow s skills, does not chnge the prescription of positive intertemporl wedge, stemming from the optimlity of the Inverse Euler eqution in dynmic Mirrleesin models. 13 Some mnipultions of the first-order conditions yield the following proposition: Proposition 3.4. In ny constrined Preto optimum, the inverse Euler eqution holds: 1 u (c 1 (θ)) = 1 βr Proof. See Appendix A.4.1. [ 1 1 u g( z(θ), θ)d = (c 2 (θ, )) βr E θ 1 u (c 2 (θ, )) Jensen s inequlity then implies βe [u (c 2 (θ, ))] > u (c 1 (θ)) the optiml lloction dicttes wedge between the intertemporl rte of substitution nd trnsformtion; svings re discourged. ]. 3.2.4 Multiple Eduction nd Working Periods We hve worked with two-period life-cycle model so fr. We now briefly show how our results for optiml lloctions cn esily be extended to the cse of multiple eduction nd working life periods. For our numericl simultions, this enbles us to ccount for the fct tht the time spent in (tertiry) eduction reltive to the working life is smll. Let T e be the number of yers eduction tkes plce nd T w the number of yers n individul works, such tht T w + T e = T. During eduction, so for the first T e yers in the life-cycle, consumption is constnt nd given by c e (θ), consumption during the work period, c w (θ, ) nd income erned y w (θ, ) re lso constnt, s cn be shown under our ssumption of seprble preferences. However, wheres there is no intertemporl wedge within the yers of eduction nd within one s working life, the inverse Euler eqution still governs the reltionship between c e (θ) nd c w (θ, ) generting the fmilir svings distortion. Assuming 13 Dimond nd Mirrlees (1978) nd Rogerson (1985) were the first to derive it. In n importnt pper reviving the interest in the result, Golosov, Kocherlkot nd Tsyvinski (2003) generlized it to lrge clss of dynmic environments, most importntly llowing for rbitrry skill processes. 19
βr = 1, the lbor wedges re given by the nlogous expressions s in Proposition 3.4. Also the eduction wedge remins unffected, except tht the expression is djusted for the time-horizon since the mrginl benefits nd incentive costs of eduction depend on the length of the working life period: with βr = 1. τ z (θ) = T t=t e+1 + β t 1 (y w (θ, ) c w θ, )) T t=t e+1 βt 1 η(θ) λ R f(θ) v 2 (θ, ) g( z(θ), θ) d z(θ) 2 G( z(θ), θ) d, z(θ) θ 3.3 Numericl Illustrtion In this section we numericlly explore our model in n illustrtive mnner. We consider two skill distributions s our primitives G( z, θ) tht led to very similr equilibrium wge distributions nd eductionl expenses for ctul given policies. In one of the cses, the distribution function is chrcterized by strong complementrity between innte skills nd eduction. In the other, there is less complementrity nd the direct effects of eduction nd innte skills dominte. We solve for the Utilitrin optimum, so f(θ) = f(θ) θ. The utility function is: U(c, l) = c1 ρ 1 ρ (y/)σ, σ where we set σ = 3, implying Frisch elsticity of 0.5 nd the CRRA coefficient to ρ = 2. The yerly risk-free rte is ssumed to be 3% (R=1.03) nd β = 1/R. We ssume tht lbor mrket bilities re distributed log-normlly following common prctice nd impose the loction prmeter µ to be function of θ nd z. Concerning θ, we ssume uniform distribution within [0.1, 1]. Cse () - Strong Complementrity: The functionl form of the loction prmeter is: µ(θ, z) = 1.7 + 1.5θ 0.5 z 0.15. In this cse, individuls re the sme if they do not cquire ny eduction t ll. However, the more eduction they cquire the stronger re the differences in the 20
loction prmeters. This inequlity in µ will be reinforced by the fct tht gents hve incentives to self-select into different eduction levels becuse of heterogeneous returns. Cse (b) - Strong Direct Effect: In the second cse we ssume, µ(θ, z) = 1.5 + θ + 0.75z 0.25. In this cse, individuls re lredy very different from the outset, i.e. if nobody cquires ny eduction. The difference in the loction prmeter then stys constnt for n uniform increse in z cross gents. Although 2 µ(θ,z) θ z = 0 in this cse, Assumption 3 is fulfilled for the relevnt rnge. However, innte skills nd eduction re weker complements s compred to Cse (). The respective prmeters for the two cses s well s the respective constnt mrginl costs of eduction were chosen such tht with n pproximtion of the current tx nd college subsidy system in the US, the model roughly replictes per-cpit expenditures on college eduction nd the centers of the intervl of the loction prmeters of the log-norml distributions is equl to the empiricl vlue of the wge distribution. 14 Figure 1 illustrtes optiml eduction wedges for the two cses. In both cses, the optiml lloction fetures positive implicit eduction subsidies round 40%, which re reltively flt cross innte types. The min difference between the two cses lies in the incentive effect. When innte skills nd eduction re complements, the plnner finds it optiml to tx eduction reltive to first best in line with Proposition 3.3. In Cse () this incentive effect becomes s lrge s 17% wheres in Cse (b) it hovers round zero. Figure 2 illustrtes the optiml lbor wedges from Proposition 3.2. Pnel () displys the optiml lbor wedge s function of income. 15 Drker regions refer to 14 Following Gllipoli, Meghir nd Violnte (2011) we set the lbor income tx to flt rte of 27% nd lump sum trnsfer of one sixth of lbor income per cpit. We introduce yerly eduction subsidy of 24%. In both cses, for these given policy instruments, verge college eduction expenditures per yer re roughly 30% of yerly medin income; long run verge for the US (Gllipoli, Meghir nd Violnte 2011). The relized vlues of µ(θ, z) re within the rnge [2.02, 3.34], centered round 2.76, the vlue of the lognorml fit for the US wge distribution found by Mnkiw, Weinzierl nd Ygn (2009); s them, we set the scle prmeter equl to 0.565. 15 To economize on spce we only show the figures for Cse (b) here. The grphs for Cse () turn out to look nerly identicl. 21
0.5 0.5 0.4 0.4 Wedge 0.3 0.2 0.1 Fiscl Externlity Eduction Wedge Incentive Effect Wedge 0.3 0.2 0.1 Fiscl Externlity Eduction Wedge Incentive Effect 0 0 0.1 0.1 0.2 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Thet () Strong complementrity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Thet (b) Strong direct effect Figure 1: Optiml Eduction Wedges 1 1 12 1 0.9 10 0.8 Lbor Wedge 0.7 0.6 0.5 0.4 Lbor Wedge Decomposition 8 6 4 Incentive Effects Insurnce Effects 0.3 2 0.2 0.1 0 20 40 60 80 100 120 140 160 Wge 0.1 0 0 20 40 60 80 100 120 140 160 Wge 0.1 () Optiml Lbor Wedges (b) Lbor Wedge Decomposition Figure 2: Optiml Lbor Wedges innte low types nd lighter regions to innte high types. The picture shows tht higher innte types fce high lbor wedges, wheres the shpe of the wedges does not vry with θ. 16 In the next pnel (b), we illustrte the decomposition from Proposition 3.2 into the insurnce term nd the incentive term by plotting A(θ, ) nd B(θ, ). The set of insurnce effects A(θ, ) lies bove the set of incentive effects B(θ, ). Still, especilly t the beginning of the income distribution incentive effects contribute to higher implicit tx rtes. The grphs lso revels tht these incentive effects re of more importnce for higher innte types on verge. 16 Since low incomes the distortions re strongly incresing in θ, condition (i) of Lemm 3.1 is typiclly not fulfilled for low. 22
4 Implementtion So fr we only considered direct mechnism, in which individuls mke reports bout their relized type nd the plnner ssigns bundles of consumption, lbor supply nd eduction s functions of the reports. The focus in the chrcteriztion of the optiml lloction ws on wedges or implicit price distortions of the lloction. In this section, we explore two decentrlized implementtions of constrined Preto optim. 4.1 Implementtion One: Student Grnts nd Income Txes Conditioning on Eduction 4.1.1 Two Periods The benevolent government offers menu of student grnts to the gents. These grnts G re conditionl on eduction. In the second period, there is tx schedule in plce, which, importntly, does not only condition on ernings but lso on eductionl investment. Further, svings txes re high enough to mke privte svings dispper from the mrket; the definition of the svings tx builds on Werning (2011). We summrize this in the following proposition: Proposition 4.1. Any constrined Preto optiml lloction in two period economy cn be implemented by grnt schedule G(z), n eduction dependent income tx T (y, z) nd svings tx T s (s), where G(z(θ)) = z(θ) + c 1 (θ) T (y(θ, ), z(θ)) = y(θ, ) c 2 (θ, ) T s (s) s defined in Appendix A.5. Proof. See Appendix A.5 Implementtion of svings wedges: The svings function T s (s) is prohibitively high such tht ll gents choose s = 0, hence in this implementtion there re no privte svings. However, s shown in Werning (2011) this comes without loss of generlity: by Ricrdin equivlence rgument, we cn djust G(z(θ)) nd T (y(θ, ), z(θ)) with lump-sum trnsfers nd deductibles to rrive with non-liner svings tx 23
schedule, which produces non-zero privte svings for every gent nd the sme lloction with the sme distortion of consumption cross periods. The full rgument is found in Werning (2011). Implementtion of lbor wedges: Agents enter the second period with no svings s rgued bove. Their budget constrint is then: T (y(θ, ), z(θ)) = y(θ, ) c 2 (θ, ). From the gents optimlity conditions for y nd c 2 it follows tht mrginl tx rtes T y (y(θ, ), z(θ)) re equl to lbor wedges τ y (θ, ) s chrcterized in Section 3.2.1. Implementtion of eduction wedges: To fix ides, the budget constrints of n gent in both periods re given by: c 1 (θ) + z(θ) G(z(θ)) c 2 (θ, ) y(θ, ) T (z(θ), y(θ, )), where we lredy imposed the zero svings. In contrst to the optiml lbor wedge, which equls the optiml lbor tx, there is no single policy instrument for which the eduction wedge equls the mrginl distortion of the policy. Insted, the government uses two instruments: i) the non-liner grnt schedule G(z), which depends on eduction chosen nd ii) the lbor tx code in the second period. Using the gents optimlity conditions in the proposed implementtion one cn show tht the wedge equls: τ z (θ) = G (z) u (c 2 (θ, )) u (c 1 (θ)) g( z(θ), θ)t z(y(θ, ), z(θ))d A positive vlue of τ z (θ) encourges eduction t level θ. The incentive for gents to increse their eductionl ttinment comes from: i) An increse in their grnt mesured by G (z) 17 nd ii) n increse or decrese in their lbor income tx burden for ll sttes, i.e. T z (y(θ, ), z(θ)). 4.1.2 T Periods The possible extension to the life-cycle cse is summrized s follows: 17 Theoreticlly it could be the cse tht G is (prtly) decresing in z if c 1 (θ) is sufficiently decresing. However, this is rther unlikely nd in ll our numericl exmples we hve c 1(θ) > 0. 24
Corollry 4.2. Any constrined Preto optiml lloction in T-period economy cn be implemented by grnt schedule G(z), n eduction dependent income tx T t (y t, z) tht conditions on the history of incomes nd svings tx T s (s). Proof. See Appendix A.6 The corollry is n ppliction of insights from Werning (2007), who chrcterizes possible implementtions in the dynmic deterministic Mirrlees problem. The problem here is similr to his model with the only difference tht there is not only one tx schedule conditioning on the history of incomes y t, but rther one for ech θ-type or ech eduction level z. One might wonder why the optimum is not simply implementble with historyindependent schedules. The reson is tht with such tx system it might be possible for n individul to profit from tx rbitrge if tx schedules re sufficiently concve. Insted of erning the ssigned income y every period, it cn, e.g., then be fvorble for n individul to ern y ɛ tody nd y +ɛ tomorrow. Averge gross income would be the sme, however, due to concvity of T (y) the tx burden would decrese; if this effect is strong enough, it might compenste for the higher disutility of lbor. In relity such strtegic behvior to exploit the non-linerity of the tx system seems unlikely to occur. One reson is tht shifting lbor income between periods in such wy is often very costly or simply infesible due to djustment costs nd hours restrictions. 18 Formlly, let C(y ) denote the present vlue of totl djustment costs of n individul tht chooses income history y, nd if y T e+1 = y T e+2 =... = y T, then C(y ) = 0, so if lbor supply is constnt, there re no djustment costs. Definition For given djustment cost function C( ), n income tx schedule T (y, z) is tx rbitrge resistnt if V (y, c (y, T (y, z(θ)), C(y )) V (y truth (θ, ), c truth (θ, )) y nd (θ, ), where c (y, T (y), C(y )) is the optiml consumption sequence given y, T (y) nd V (, ) is the respective (deterministic) working life utility conditionl on the reliztion of. With this in mind, we mke the following ssumption stting tht strtegic devitions re never fesible for worker becuse of djustment costs: 18 For recent tretment of hours constrint in the public economics literture see Chetty, Friedmn, Olsen nd Pistferri (2011); for the implictions of djustment costs on hours choices nd the lbor supply elsticity see Chetty (2012). In ddition to djustment costs, there re lso resons outside our model, which mke sophisticted intertemporl devitions unlikely, for exmple, idiosyncrtic or ggregte uncertinty, imperfect credit mrkets or bounded rtionlity. 25