Optimal taxation in a habit formation economy

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1 Opmal axaon n a hab formaon eonomy Sebasan Koehne Morz Kuhn Deember 12, 2014 Absra Ths paper sudes hab formaon n onsumpon preferenes n a dynam Mrrlees eonomy. We derve opmal labor and savngs wedges based on a reursve approah. We show ha hab formaon reaes a move for subsdzng labor supply and savngs. In parular, hab formaon nvaldaes he well-known no dsoron a he op resul. We demonsrae ha he heoreal fndngs are quanavely mporan: n a paramerzed lfe-yle model, average labor and savngs wedges fall by more han one-hrd ompared wh he ase of me-separable preferenes. Keywords: opmal axaon; hab formaon; reursve onras JE Classfaon: D82, E21, 21 Ths paper s a revsed and exended verson of an earler manusrp led Opmal apal axaon for me-nonseparable preferenes. The auhors hank Erzo F. P. umer, hree anonymous referees, Arpad Abraham, Carlos da Cosa, John assler, Per Krusell, Eenne ehmann, Jean-Mare ozahmeur, Nola Pavon, Perre Peseau, Rk van der Ploeg, akk Yaz, and parpans a varous semnars and onferenes for many helpful ommens. Sebasan Koehne graefully aknowledges fnanal suppor from Torsen Söderbergs Sfelse, Ragnar Söderbergs Sfelse, and Knu oh Ale Wallenbergs Sfelse gran KAW Correspondng auhor a: Insue for Inernaonal Eonom Sudes IIES, Sokholm Unversy, SE Sokholm, Sweden, Phone: , sebasan.koehne@es.su.se, and CESfo, Munh, Germany. Unversy of Bonn, Deparmen of Eonoms, D Bonn, Germany, Phone: , mokuhn@un-bonn.de, and Insue for he Sudy of abor IZA, Bonn, Germany. 1

2 1 Inroduon Wha deermnes he opmal axes on labor nome and apal? Fundamenal o hs lass publ fnane queson s a desrpon of neremporal deson makng. Exsng sudes, followng Damond and Mrrlees 1978, have explored opmal axaon when deson makers aggregae aross me n a separable way. The presen paper proposes a model of deson makng movaed by evdene from maroeonoms, psyhology, and mro daa he hab formaon model. 1 Ths model onans me-separable preferenes as a speal ase bu allows for neremporal omplemenares n onsumpon. We nrodue hab formaon preferenes no an oherwse sandard dynam Mrrlees eonomy. Agens fae shoks o her ables o generae labor nome. abor nome s publly observed, bu ables and labor supply are prvae nformaon. In hs envronmen, we haraerze he soluon of he soal plannng problem n erms of labor and savngs wedges. As s ommon n hs leraure, posve wedges represen mpl axes and ndae ha deenralzaons of he soal plannng alloaon mus orre ndvdual labor or savngs reurns downward n one way or anoher. 2 To make he mulperod soal plannng problem raable for heoreal and numeral analyss, we ransform no a dynam programmng problem by generalzng nsghs from he reursve onra heory leraure. Ths approah s ommon n dynam prvae nformaon problems wh me-separable preferenes Spear and Srvasava, 1987; Phelan and Townsend, Our reursve formulaon exends beyond opmal axaon and apples o a large lass of prvae nformaon problems. We frs sudy opmal labor axaon. For hab formaon preferenes, labor wedges are shaped by wo ounervalng fores. Frs, as n any self-seleon problem wh me-separable preferenes, here s a move for downward dsorons o labor supply of all bu he mos produve ype. Ths move alls for posve labor wedges. Seond, hab formaon onnes presen and fuure self-seleon problems. Beause of omplemenary beween habs and onsumpon, self-seleon beomes easer n he fuure f he worker onsumes a lo n he presen. Ths hab effe alls for subsdes o labor supply for all ypes and ouneras he 1 See Messns 1999 for a summary of hab formaon n maroeonoms and Frederk and oewensen 1999 for a revew of hab formaon n he empral and behavoral eonoms leraure. 2 The deenralzaon of opmal alloaons s no unque; ompare Golosov, Koherlakoa, and Tsyvnsk 2003, Koherlakoa 2005, Albanes and Slee 2006, Golosov and Tsyvnsk 2006, Wernng 2011, Goard and Pavon 2011, and Abraham, Koehne, and Pavon

3 onvenonal self-seleon dsoron. As a onsequene, he no dsoron a he op resul breaks down, and he mos produve ype obans a negave labor wedge. For less produve ypes, labor wedges an be posve or negave, dependng on he mporane of he hab effe ompared wh he onvenonal self-seleon dsoron. We nex urn o opmal savngs axaon. Our deomposon of savngs wedges reveals hree axaon moves. Frs, savngs should be axed beause he agen has a beer nenve o supply labor n he nex perod f he sars he nex perod wh lower wealh wealh effe. Ths fore s well known from models wh me-separable preferenes. Seond, savngs should be axed, beause smulang presen onsumpon nreases he hab level n he nex perod. Ths effe makes hgh onsumpon n he nex perod more arave and hereby renfores he nenve o supply labor mmedae hab effe. Thrd, savngs should be subsdzed, beause smulang nex perod s onsumpon nreases he hab level n he remanng perods and hereby mproves labor supply nenves n hose perods subsequen hab effe. ab formaon hus affes savngs axaon n opposng ways, and s mpa wll depend on he relave magnude of mmedae versus subsequen hab effes. Our heoreal resuls denfy fores ha ounera he onvenonal Mrrleesan dsorons o labor supply and savngs. To demonsrae he quanave mporane of hese resuls, we evaluae hab formaon n a sylzed lfe-yle model. We paramerze he model aordng o empral fndngs for he U.S. eonomy. We fnd he mpa of hab formaon on opmal savngs and labor wedges o be negave and szable. Averaged over he lfe yle, opmal savngs wedges of a ypal worker fall by 40 peren, and opmal labor wedges by 35 peren, ompared wh he ase of me-separable preferenes. The negave mpa on labor wedges was already suggesed by our heoreal resuls. The negave mpa on savngs wedges s due o subsequen hab effes ha preval over mmedae hab effes. Inuvely, nenve provson beomes more osly when rewards an be smoohed over fewer perods. Therefore, relaxng nenve problems laer n lfe hrough subsequen hab effes s more mporan han relaxng nenve problems n he dre fuure hrough mmedae hab effes. Relaed leraure. Wh few exepons, mos exsng sudes of dynam axaon problems work wh me-separable preferenes. The onrbuon loses o ours s by Grohulsk and Koherlakoa 2010 and explores a Mrrlees framework wh me-nonseparable preferenes 3

4 smlar o he presen paper. Ther fous s deenralzaon, and hey show ha soal seury sysems wh hsory-dependen axes and ransfers upon reremen an be used o mplemen opmal alloaons when preferenes are me-nonseparable. Apar from a hree-perod example wh a negave savngs wedge, hey do no nvesgae savngs or labor wedges any furher. 3 Several papers sudy Mrrleesan models wh alernave forms of preferene nonseparables. Whle hab formaon dffers from oher nonseparables and requres an ndependen reamen, a general fndng s ha preferene nonseparables affe Mrrleesan wedges n magnude and sgn. Ths fndng apples o reursve preferenes Farh and Wernng, 2008, human apal effes Bohaek and Kapka, 2008; Grohulsk and Pskorsk, 2010; Sanheva, 2014, and nonseparables beween onsumpon and labor supply Farh and Wernng, 2013, for example. Anoher relaed paper s by Cremer, De Donder, Maldonado, and Peseau 2010 and explores opmal ommody axaon n a framework wh myop hab formaon. Ths framework gves rse o paernals axaon moves, beause ndvduals do no foresee he hab formaon relaon when makng onsumpon and savngs desons. Smlar effes arse when myop hab formaon s nrodued no a model of reremen; see Cremer and Peseau The presen paper s dfferen n several key aspes, beause we fous on labor and savngs axaon and sudy me-onssen deson makers ha anpae her fuure preferenes. Fnally, he paper bulds on he exensve leraure on hab formaon preferenes. ab formaon goes bak o he heory of adapaon formalzed n he psyhologal leraure by elson ab formaon posulaes ha ndvduals ompare her urren onsumpon wh a hsoral referene level and derve uly boh from onsumpon per se and from onsumpon growh. 4 een and Durham 1991 fnd suppor for hab formaon based on mro-level onsumpon daa. Frederk and oewensen 1999 revew he subsanal body of empral researh supporng he hab formaon hypohess. Moreover, hab formaon has reonled heory and evdene for several mporan quesons n he maroeonom leraure, 3 Our deomposon of savngs wedges shows ha he subsequen hab effe s responsble for her fndng. owever, we also reveal ha nenve problems n he mmedae fuure reae ounervalng fores beause of wealh and mmedae hab effes. Our quanave analyss herefore fnds ha, even hough s possble o onsru heoreal ases n whh savngs wedges are negave, hose ases are no represenave of ypal axaon envronmens. 4 In addon, here s he onep of exernal hab formaon, where he referene pon depends on he onsumpon levels of a peer group; see he dsusson of Cahng up wh he Joneses n Abel

5 suh as he equy premum puzzle Abel, 1990; Consanndes, 1990; Campbell and Cohrane, 1999, he relaonshp beween savngs and growh Ryder and eal, 1973; Carroll, Overland, and Wel, 2000, and reaons o moneary poly shoks Fuhrer, Model Ths seon ses up a dynam Mrrlees model of opmal axaon wh hab formaon preferenes. The eonomy onsss of a rsk-neural prnpal/planner and a un measure of rsk-averse agens fang a bnary sohas skll proess. Tme s dsree and ndexed by = 1, 2,..., T, wh T <. 2.1 Preferenes Agens have denal von Neumann-Morgensern preferenes and maxmze he expeed value of T β 1 u, h vl, =1 where, h, l represen he agen s onsumpon, hab, and labor supply n perod, and β 0, 1 s he agen s dsoun faor. 5 abor dsuly v : R + R s onnuous, srly nreasng, and weakly onvex. Consumpon uly u : R 2 + R s we onnuously dfferenable, srly onave, srly nreasng n s frs argumen, and srly dereasng n s seond argumen. Consumpon and hab are omplemens: u h subsrps o denoe paral dervaves. > 0. As usual, we use The omplemenary assumpon u h > 0 s sandard n he hab formaon leraure. I holds for he wdely used ase of lnear hab formaon: u, h = ũ γh, wh γ 0, 1] and ũ : R + R srly nreasng and srly onave; ompare Consanndes 1990 and Campbell and Cohrane 1999 among ohers. Anoher ommon spefaon of hab formaon s he Cobb-Douglas ase: u, h = ũ h γ ; ompare Abel 1990, Carroll, Overland, and Wel 2000, Fuhrer 2000, and Daz, Pjoan-Mas, and Ros-Rull ere, u h > 0 holds f he oeffen of relave rsk averson of ũ s bounded below by one. 6 5 The preferenes we use are me-onssen; see Johnsen and Donaldson 1985, for example. 6 Wre = h γ. Then u h, h = γh γ 1 ũ [ ũ /ũ 1]. 5

6 2.2 abs We assume from now on ha habs are shor-lved: h = 1, wh 0 beng exogenous. Ths assumpon smplfes he exposon and s emprally suppored by resuls n Fuhrer Our resuls generalze easly o he ase n whh habs are a funon of lagged onsumpon and lagged hab levels, h = 1, h 1. See Seon 3.1 for furher dsusson. 2.3 Sklls Agens dffer wh respe o her sklls. An agen wh hours l and skll realzaon produes y = l uns of oupu n perod. Oupu s publly observable, bu hours and sklls are prvae nformaon. For every, le Θ = {, } be he se of possble skll realzaons, wh 0 <. Defne Θ := Θ 1 Θ. A he begnnng of eah perod, a skll level Θ s drawn for eah agen. Draws are ndependen aross agens. For now, we assume ha draws are also ndependen aross me. In onlne Appendx C, we allow for skll proesses wh perssene. ene, here exs probably weghs π, wh Θ π = 1, suh ha he probably of a paral skll hsory = 1,..., Θ s gven by Π = π 1 1 π. Whou loss of generaly, we assume π > 0 for all Θ. We denoe he expeaon operaor wh respe o he unondonal dsrbuon of skll hsores T by E[ ]. As usual, he noaon E [ ] := E [ ] represens expeaons ondonal on he me- hsory. The ase of bnary sklls s a ommon smplfaon for dsree nome axaon problems; ompare Feldsen 1973, Sern 1982, and Sglz 1982, for example. Bnary sklls falae he exposon bu are no essenal o our resuls. Seon 3.1 provdes furher dsusson. 2.4 Soal planner We se up he soal plannng problem n s dual form: he soal planner mnmzes he oss of delverng a gven level of ex ane welfare o he agens. The planner dsouns fuure oss by a faor q < 1. Equvalenly, he planner has aess o a lnear savngs ehnology ha ransforms q uns of dae- oupu no 1 un of oupu a dae I would no be dfful o endogenze he reurn of he savngs ehnology by nrodung an expl produon funon ha depends on apal and labor. Ye, hs exerse would merely omplae he noaon and generae no addonal nsghs for he quesons addressed n hs paper. 6

7 2.5 Alloaons An alloaon s a sequene, y =, y =1,...,T of onsumpon plans : Θ R + and oupu plans y : Θ R +. A reporng sraegy s a sequene σ = σ =1,...,T of mappngs σ : Θ Θ. Denoe he se of all reporng sraeges by Σ and se σ := σ 1 1,..., σ. A he begnnng of every perod, he planner alloaes onsumpon and oupu aordng o he hsory of repored sklls. Beause of shor-lved habs, we have h = 1. ene, a reporng sraegy σ Σ yelds ex ane expeed uly aordng o w 1 σ, y σ; 0 [ T := u σ, 1 σ 1 1 y σ ] v Π. β 1 =1 Θ Sne sklls are prvaely observed, he planner needs o ensure ha all agens reveal her nformaon ruhfully. An alloaon ha sasfes he ruh-ellng onsran w 1, y; 0 w 1 σ, y σ; 0 σ Σ s alled nenve ompable. 2.6 Opmal alloaons The soal planner seeks o provde a gven level W 1 of ex ane welfare a mnmal oss. ene, an alloaon, y s alled opmal f solves he followng problem: C 1 W 1, 0 := mn,y T q 1 [ y ] Π 1 =1 Θ s.. w 1, y; 0 w 1 σ, y σ; 0 σ Σ 2 w 1, y; 0 = W Reursve formulaon We use a reursve approah o derve labor and savngs wedges and o sudy he quanave mporane of hab formaon n a paramerzed model. Ths subseon ses up he requred 7

8 noaon and saes he reursve formulaon of he problem. We show ha opmal alloaons have a reursve formulaon wh wo sae varables: promsed uly and he agen s hab level. Deals and proofs are relegaed o onlne Appendx B. Gven an alloaon, y and a hsory, 1 < T, he onnuaon alloaon T +1, y+1 T s defned as he resron of plans s, y s s=+1,...,t o hose hsores +1,..., T ha sueed. The onnuaon uly assoaed wh he onnuaon alloaon s defned as w +1 T +1, y+1 T ; T [ := u s s, s 1 s 1 v β s 1 s=+1 s Θ s ys s s ] Π s s. Noe ha he onnuaon uly w +1 depends no only on he onnuaon alloaon bu also on lagged onsumpon n order o apure he hab level a he begnnng of perod + 1. For any R + we defne dom +1 o be he se of onnuaon ules W wh he propery ha, gven hab level n perod + 1, here exss an nenve ompable alloaon T +1, yt +1 ha generaes uly E [ T s=+1 β s 1 u s, s 1 vy s / s ] = W, where =. Smlar o he fndngs for me-separable preferenes by Spear and Srvasava 1987 and Phelan and Townsend 1991, he onsran se and he objeve of he soal planner problem 1 an be gven a sequenal form. 8 Ths gves rse o he followng reformulaon of he problem. Proposon 1 Reursve formulaon. e W 1 dom 1 0. The value C 1 W 1, 0 of he soal planner problem 1 an be ompued by bakward nduon usng he followng equaon for all 8 Followng he approah by Fernandes and Phelan 2000, we an oban a smlar formulaon when skll shoks are perssen. 8

9 wh he onvenon C T +1 = W T +1 = 0: C W, 1 = mn,y,w +1 =, [ y + qc +1 W +1, ] π 4 s.. u, 1 v y / + βw +1 u =, j, 1 v y j / + βw j +1,, j =, 5 [ u, 1 v y / ] + βw +1 π = W 6 W +1 dom +1, =,. 7 Moreover, plans, y =1,...,T ha solve he sequene of problems 4 onsue an opmal alloaon. Conversely, any opmal alloaon solves he sequene of problems 4. Proposon 1 separaes he soal planner problem 1 no a sequene of smpler problems n whh he planner deermnes urren onsumpon, urren oupu, and onnuaon uly a every pon n me as a funon of he urren skll. Choes are onsraned by he emporary nenve ompably onsran 5, he promse-keepng onsran 6, and he doman resron 7. The only dfferene relave o he famlar reursve formulaon for nenve problems wh me-separable preferenes s ha he agen s hab level beomes an addonal sae varable. 9 In wha follows, we assume ha onnuaon ules are neror elemens of he doman. Ths assumpon an be jusfed by mposng approprae boundary ondons on preferenes abor and savngs wedges Ths seon derves he wedges ax dsorons mposed by opmal alloaons. As s well known n he dynam publ fnane leraure, he deenralzaon of opmal alloaons s no unque. ene, he robus nsghs from he presen analyss are no abou expl ax nsrumens bu abou wedges. In order o defne labor and savngs wedges, we frs examne he agen s margnal uly of onsumpon. Wh hab formaon, urren onsumpon nfluenes fuure hab levels. Gven 9 The reursve formulaon an be easly exended o he ase of perssen habs. See onlne Appendx B. 10 For nsane, dom = R for any and f onsumpon uly s unbounded below and above, or f onsumpon uly and labor dsuly are unbounded above. 9

10 a onsumpon hsory 1,..., T, he margnal uly of onsumng a dae s gven by u, 1 + βu h Ũ := +1, f < T, u T, T 1 f = T. If onsumpon n perod + 1 s uneran from he pon of vew of perod, margnal onsumpon uly beomes a random varable. We wre U := E [Ũ ] for he expeaon of hs random varable ondonal on dae- nformaon. Gven an alloaon, y, defne he labor wedge n perod as τ y, := 1 v y / U and he savngs wedge n perod as τ s, := 1 qu βe [U +1 ]. Noe ha τ y, and τ s, are random varables ha depend on he dae- hsory, even hough we have omed hs argumen for noaonal onvenene. Apar from he fa ha hab formaon hanges he formula for margnal onsumpon uly U, he above defnons are sandard. The labor wedge s he mpl ax rae ha equaes he agen s margnal rae of subsuon beween onsumpon and lesure o he afer-ax nome of an addonal un of labor supply. Smlarly, he savngs wedge s he mpl ax rae ha algns he agen s margnal rae of neremporal subsuon wh he relave pre of fuure onsumpon. We solve a relaxed problem n whh only downward nenve ompably onsrans are mposed. 11 emma 1 jusfes hs approah. The proof of emma 1 and all furher proofs are relegaed o onlne Appendx A. emma 1. The soluon o he soal planner problem 4 ondes wh he soluon o he 11 In addon, we assume ha onsumpon and oupu are nonzero. Ths assumpon an be jusfed by boundary ondons of he form v 0 = 0 and lm 0 u, h = for all h > 0, for nsane. 10

11 followng relaxed problem: C W, 1 = mn,y,w +1 =, [ y + qc +1 W +1, ] π 8 s.. u, 1 v y / + βw +1 u, 1 v y / + βw +1 9 [ u, 1 v y / ] + βw +1 π = W. 10 =, In wha follows, we fx he perod- sae veor W, 1. Equvalenly, we fx he assoaed skll hsory 1. We denoe he agrange mulpler for he nenve ompably onsran 9 by µ and he mulpler for he promse-keepng onsran 10 by λ. We begn our analyss wh he followng prelmnary nsgh. Remark 1 omogeneous sklls. e, y be an opmal alloaon and suppose = for 0. Then he labor and savngs wedges are zero: τ y, = τ s, = 0 for 0. Remark 1 mples ha ax dsorons n our model are enrely due o skll heerogeney, exaly as n he ase of me-separable preferenes. Thus, hab formaon does no reae a dre axaon move. owever, hab formaon does reae an mporan ndre axaon move beause hanges he sruure of he nenve problem o repor sklls ruhfully. Proposon 2 abor wedges. e, y be an opmal alloaon. For eah hsory 1, < T, here exs numbers A, B, B 0 and agrange mulplers µ, µ +1, µ +1 0 assoaed wh he nenve ompably onsrans n perods and + 1 suh ha τ y, 1, = µ +1 B 0, 11 τ y, 1, = µ A µ +1B For = T, equaons 11 and 12 hold wh µ +1, µ +1 replaed by zero. Fnally, n he lm ase of me-separable preferenes u h = 0, we have B = B = 0. For he agrange mulplers n he above resul, he supersrp refers o he urren perod s skll realzaon and me subsrps refer o he perod of he nenve ompably onsran. 11

12 In parular, µ +1 and µ +1 are he agrange mulplers for he nenve ompably onsran n perod + 1 when he skll realzaons n perod are and, respevely. For me-separable preferenes, Proposon 2 saes ha he labor wedge of he hgh-sklled worker s zero no dsoron a he op. The low-sklled worker faes he posve labor wedge µ A. As usual n self-seleon problems, hs downward dsoron s effen beause redues he nenve of he hgh-sklled worker o preend beng low-sklled. Wh hab formaon, he same self-seleon dsoron onnues o apply. In addon, here s a move for subsdzng he labor supply of hgh-sklled as well as low-sklled workers, apured by he erms µ +1 B for =,. As he agrange mulpler µ +1 ndaes, hs move s due o he nenve problem n perod + 1. The proof of Proposon 2 reveals ha B an be expressed as B = b [ u h +1, u h +1, ] = b u h ξ, [ +1 +1], 13 where b = b s a srly posve number, = 1,, whle = and +1 = +1, +1, +1 = +1, +1 are he onsumpon levels n perods and + 1, and ξ = ξ s some number beween +1 and +1. Sne hab and onsumpon are by assumpon omplemens, B s posve and eners negavely no he labor wedge. The nuon for hs fndng s as follows. A low labor wedge enourages work a dae. Ths nreases dae- onsumpon and resuls n a hgher hab level a dae + 1. Beause of omplemenary, he dfferene beween he uly of a hgh-sklled worker u +1, and a low-sklled worker u +1, nreases. Ths effe s soally desrable beause falaes self-seleon a + 1. A a more general level, Proposon 2 shows ha opmal nraperod dsorons ake no aoun neremporal preferene dependenes. Sne hgh hab levels are helpful for fuure nenve problems, hs generaes a move for subsdzng labor aross all skll ypes. We label hs he hab effe and denoe by B. As a onsequene, he labor wedge for hgh-sklled agens s negave subsdes a he op, whle he labor wedge for low-sklled agens onsss of he sandard axaon move for urren nenve provson A mnus he hab effe B. We now urn o he analyss of savngs wedges. For me-separable preferenes, savngs wedges an be analyzed by varaonal argumens ha perurb opmal alloaons n wo ad- 12

13 jaen me perods. The resul s he semnal Inverse Euler equaon. 12 Unforunaely, hs approah does no exend o he lass of hab formaon preferenes. The key problem s ha onsumpon a any gven pon n me affes fuure hab levels. Therefore, he onrbuon of onsumpon n perods and + 1 o he worker s lfeme uly depends on subsequen onsumpon levels and hene on subsequen skll realzaons. I s hus mpossble o fnd a onsumpon perurbaon ha s nenve-neural and uses only nformaon from perods and + 1 unless = T 1; see Grohulsk and Koherlakoa The agrangan ehnques adoped n hs paper delver nsghs on savngs wedges for he hab formaon ase. In he followng resul, he supersrp j {, } refers o he skll realzaon n perod + 1. Proposon 3 Savngs wedges. e, y be an opmal alloaon. For eah hsory, < T 1, here exs numbers D, E, F j 0, j {, }, and agrange mulplers µ +1, µ j +2 0, j {, }, assoaed wh he nenve ompably onsrans n perods + 1 and + 2 suh ha τ s, = µ +1 D + µ +1 E j=, π +1 j +1 µ j +2 F j. 14 For = T 1, equaon 14 holds wh µ +2, µ +2 replaed by zero. Fnally, n he lm ase of me-separable preferenes u h = 0, we have E = F = F = 0. Proposon 3 shows ha savngs wedges for hab formaon preferenes have hree omponens denoed by D, E, and F j. Inuvely, he hree omponens an be demonsraed by onsderng he followng hypoheal suaon. The agen, afer workng n perod and reevng he ransfer, saves one un of onsumpon for he followng perod. Three effes hen hange he agen s preferenes over fuure saes, and hereby he nenve o supply labor or, pu dfferenly, he nenve o repor ruhfully n he fuure. Frs, here s he famlar wealh effe D. Savng one onsumpon un a me yelds a fxed number of exra onsumpon uns n all saes a me + 1. Sne preferenes are onave n onsumpon, he value of exra onsumpon s hgher n saes wh low +1. owonsumpon saes hus beome relavely more arave, and he agen s nenve o supply 12 See Rogerson 1985 and Golosov, Koherlakoa, and Tsyvnsk 2003, for nsane. 13

14 labor n perod + 1 s redued. Ths onavy/wealh effe s apured by he erm D = d E [Ũ+1, +1 ] ] E [Ũ+1, +1, 15 where d = d s a srly posve number, and Ũ+1 s he margnal uly of onsumpon n perod +1. Sne he margnal uly of onsumpon s hgher n low-onsumpon low-skll saes, D s posve and alls for a posve ax on savngs. For me-separable preferenes, Proposon 3 shows ha D s n fa he only omponen of he savngs wedge. The seond omponen of he savngs wedge s he mmedae hab effe E. Savng n perod redues he agen s onsumpon and hereby dmnshes he hab level a me + 1. Beause of omplemenary beween hab and onsumpon, low-onsumpon saes a me + 1 beome relavely more arave. Ths resul redues he nenve o supply labor. Formally, he mmedae hab effe an be expressed as E = e [ u h +1, u h +1, ], 16 where e = e s srly posve, whle = and +1 = +1, +1, +1 = +1, +1 are he onsumpon levels n perods and + 1. Sne he ross dervave u h s posve by assumpon, E s posve. ene, he mmedae hab effe goes n he same dreon as he wealh effe and generaes an addonal move for axng savngs. Fnally, he savngs wedge has omponens F j ha apure a subsequen hab effe. As he agrange mulpler µ j +2 n equaon 14 suggess, hese omponens relae o he nenve problem n perod + 2 and an be wren as where f = f +1 s srly posve, +1 = F j = f [ ] u h +2, +1 u h +2, +1, 17, j +1, whle +2 = +2 +1, +2, +2 = +2 +1, +2 represens onsumpon n perod + 2. Complemenary beween hab and onsumpon mples ha F j s posve. Sne he subsequen hab effe eners wh a negave sgn n equaon 14, hs effe alls for savngs subsdes. The nuon s as follows. Savng a me nreases onsumpon a +1, and hereby he hab a +2. Beause of omplemenary beween hab and onsumpon, hs helps wh he nenve problem a + 2 by makng 14

15 onsumpon relavely more arave. Therefore, savng a should be enouraged n order o relax he nenve problem n perod + 2. In summary, Proposons 2 and 3 denfy fores ha ounera he onvenonal dsorons from me-separable Mrrlees models. Tme-separable reasonng generaes downward dsorons on labor supply arsng from presen self-seleon problems, whereas hab formaon adds a move o subsdze labor supply n order o falae self-seleon n he fuure. Smlarly, me-separable reasonng generaes savngs dsorons arsng from wealh effes, whereas hab formaon alls for savngs subsdes as a means of hangng he valuaon of onsumpon n he fuure. Noe ha he mplaons of hab formaon for savngs wedges are somewha less learu han hose for labor wedges, beause mmedae effes on preferenes have o be raded off agans subsequen effes. Ye, as long as nenve problems exaerbae over me, he fores pushng for savngs subsdes wll domnae. Fne-horzon models are a prme example of hs effe, beause he planner an spread rewards over fewer and fewer perods as me progresses. Ths makes nenve provson more osly over me and auses he ondonal onsumpon varane and he shadow os of he nenve onsran o grow over me, oher hngs beng equal. As equaons 14, 16, and 17 ndae, boh of hese fores nrease he subsequen hab effe relave o he mmedae hab effe. We demonsrae he quanave mporane of hs hannel n Seon Generalzaons of he bas model We made a number of smplfyng assumpons ha deserve a bref dsusson. Frs, nonbnary skll ypes would make he model mahemaally more edous bu do no hange he argumens underlyng our resuls. The effe of hab formaon on labor and savngs wedges s presely due o he fa ha he downward nenve ompably onsran 9 s relaxed f habs nrease. Nonbnary skll ypes generae a mulude of loal downward nenve ompably onsrans. Eah of hese onsrans s relaxed f habs nrease, and so we fnd he same hab effes on labor and savngs wedges ha we found above. Our resuls also generalze o he ase of perssen habs. Ye, n hs ase, he model 15

16 qukly beomes nraable. For nsane, f habs follow he weghed average spefaon 1 h = 1 η 1 + ηh 1 = 1 η η k 1 k + η 1 0, hen rasng he perssene parameer η from zero o a posve number enals ha he hab a any gven pon n me affes he habs for he remander of he agen s lfe. In ha ase, nreasng he hab level relaxes he nenve ompably onsrans n all remanng perods, and he exposon of our resuls beomes more nvolved beause we have o aoun for a large number of onsrans and agrange mulplers. Apar from hs omplaon, hab formaon modfes labor and savngs wedges n qualavely he same way as above. In parular, he mpa on savngs wedges sll nvolves a rade-off beween mmedae and subsequen effes: hab h +1 s a funon of, whle hab levels h +2, h +3,..., h T rea more srongly o +1 han o. k=1 Moreover, our resuls exend o he ase of perssen sklls Markov sklls. Ths ase may seem somewha less obvous han he prevous wo, sne skll perssene requres a novel reursve formulaon: beomes neessary o add promsed uly for devaors as well as he pas skll level o he veor of sae varables. Moreover, we oban an addonal promsekeepng onsran for agens who devaed n he pas perod. Ye, Proposons 2 and 3 hold rue f he wedge omponens are suably generalzed. Furher deals an be found n onlne Appendx C. 4 A paramerzed lfe-yle model By means of a paramerzed lfe-yle model, hs seon addresses he quanave mporane of our heoreal fndngs on labor and savngs axaon. The model apures several key feaures of he U.S. eonomy. In parular, he skll proess mahes he empral lfe-yle profle and he ross-seonal varane of wages. For ompuaonal reasons, he skll proess s ransory as n he heoreal model. All of our resuls are qualavely robus o perssen shoks, as he heoreal analyss n onlne Appendx C shows. owever, he quanave fndngs may depend on ha assumpon The ompuaonal dffules arsng from perssen shoks are beyond he sope of hs paper. See he onludng remarks for furher dsusson. 16

17 The reursve formulaon from Seon 2.7 gves rse o a sraghforward ompuaonal approah. We frs solve for he sequene of doman resrons dom =1,...,T. We hen explo he Bellman equaon 4 o oban he sequene of os funons C =1,...,T of he planner s problem usng sandard numeral opmzaon proedures. The assoaed poly funons are hen eraed forward o generae he opmal alloaon Parameers There are T = 11 perods wh a duraon of fve years eah. Agens ener he model a age 25, rere a age 65, and de a age 80. In eah perod before reremen, skll level s randomly drawn from a se {, }, where boh realzaons have equal probably and <. Draws are ndependen aross agens and me. We hoose he lfe-yle profle of expeed sklls n lne wh ansen 1993, Table II, who esmaes relave effeny profles of workers n he Uned Saes over he years 1955 o Expeed sklls are hump-shaped over he lfe yle and peak n perod 5 ages The varane of log-sklls s and mahes he ross-seonal varane of log-wages n he Uned Saes n he perod eahoe, Soresleen, and Volane, 2012, Table 3. Sklls are deermns afer reremen and amoun o one-half of he average skll pror o reremen. We nerpre he sklls afer reremen as sklls for home produon aves. We se up hab formaon n a Cobb-Douglas form: u, h = ũ h γ, where γ s a number beween zero and one ha onrols he mporane of habs. 15 In lne wh Daz, Pjoan- Mas, and Ros-Rull 2003, we hoose γ = Ths value orresponds o he ase of srong habs explored by Carroll, Overland, and Wel 2000 and s reasonably lose o empral resuls by Fuhrer 2000, who esmaes a value of 0.80 based on aggregae onsumpon daa. In lne wh our heoreal model and esmaons by Fuhrer 2000, habs are shor-lved: h = 1 for > 1. Perod uly s of he CRRA ype: ũx = x 1 σ /1 σ, wh σ = 3. The dsoun faor for agen and planner equals q = β = The labor dsuly funon s vl = αl 1+ 1 ψ /1 + 1 ψ, wh a Frsh elasy of labor supply of ψ = 0.5 and α = For ompuaonal reasons, we resr he spaes for onsumpon and oupu o ompa nervals. We verfy ex pos ha he quanave resuls do no depend on he hoe of he nerval bounds. 15 Anoher ommon spefaon of hab formaon s he lnear one: u, h = ũ γh. For our presen purposes, he Cobb-Douglas formulaon s more onvenen, sne perod ules are well defned whenever and h are posve. The lnear formulaon has he drawbak of rulng ou all pars, h wh < γh, whh makes he ompuaon of he doman resron and he opmal alloaon somewha more umbersome. 17

18 Fgure 1: Expeed onsumpon and oupu over he lfe yle 1.2 oupu onsumpon 1.2 oupu onsumpon age a ab formaon age b Tme-separable preferenes We se he nal uly promse W 1 suh ha he planner s budge s balaned, ha s, C 1 W 1, 0 = 0. We hoose he nal hab level 0 so ha ondes wh he agen s expeed onsumpon n he frs perod. 4.2 Resuls Fgure 1a presens he pahs of expeed oupu and onsumpon for he hab formaon ase γ = Expeed oupu follows he hump-shaped paern of he skll proess, omplemened by a moderae level of home produon oupu afer reremen. Expeed onsumpon nreases over he lfe yle and grows by abou 10 peren from ages 25 o 65. Toward he end of he lfe yle, onsumpon growh aeleraes as effes on fuure habs beome less of a onern. 16 Fgure 1b shows he orrespondng pahs for he ase of me-separable preferenes γ = The expeed oupu pah s very smlar o he hab formaon ase. Expeed onsumpon, however, s vrually fla bu slghly monoonally dereasng for me-separable preferenes. Ths shows ha hab formaon has a posve mpa on he opmal growh rae of onsumpon. 16 We aknowledge ha he onsumpon pah durng reremen s no well n lne wh empral fndngs. A more sophsaed model of reremen would allow for sohas moraly and poenally for a sruural hange n he hab formaon relaon a he me of reremen. Sohas moraly alone already mgaes onsumpon growh durng reremen o a large exen, beause he effes of onsumpon on fuure preferenes an never be fully gnored. 17 To make he alloaons omparable, we hoose a salng parameer of α = 4.3 for he me-separable ase, suh ha he dsouned value of lfeme oupu and onsumpon ondes wh he hab formaon ase. Ths adjusmen has a neglgble effe on labor and savngs wedges: averaged over he lfe yle, labor wedges are wh α = 1 and wh α = 4.3, whle average savngs wedges amoun o n boh ases. 18

19 Fgure 2: Expeed labor wedges 0.06 labor wedge onvenonal dsoron hab effe 0.08 labor wedge me separable preferenes labor wedge hab formaon age a Deomposon age b Comparson wh me-separable ase Fgure 3: Expeed savngs wedges savngs wedge wealh effe mmedae hab effe subsequen hab effe savngs wedge me separable preferenes savngs wedge hab formaon age a Deomposon age b Comparson wh me-separable ase Noes: The doed lnes n panel b dsplay he 10h and 90h perenles of he savngs wedges. Fgure 2a dsplays he omponens of expeed labor wedges for he hab formaon ase. The hab effe B alls for labor subsdes as oulned n our heoreal analyss. Ths effe s smaller n magnude han he onvenonal Mrrleesan move for labor axaon A. Thus, expeed labor wedges are posve hroughou he lfe yle bu sgnfanly smaller han n he ase of me-separable preferenes Fgure 2b. Averaged over he lfe yle, labor wedges n he hab formaon ase drop by approxmaely 35 peren ompared wh he me-separable ase. Fgure 3a deomposes expeed savngs wedges for he hab formaon ase no he wealh effe, mmedae hab effe, and subsequen hab effe. Boh hab effes are szable and n 19

20 fa are larger n magnude han he onvenonal axaon move aused by wealh effes. As argued n he heoreal seon above, he subsequen hab effe alls for savngs subsdes. Ths effe domnaes he mmedae hab effe allng for savngs axes, and hus he oal mpa of hab formaon on savngs wedges s negave. The lfe-yle average of he savngs wedge wh hab formaon s orrespondng o a 7.1 peren ax on ne neres. In he me-separable ase, s orrespondng o a 11.8 peren ax on ne neres; see Fgure 3b. 18 The varane of savngs wedges s relavely small, as he plos of he 10h and 90h perenles of he savngs wedges doed lnes n Fgure 3b ndae. ene, savngs wedges are lower n he hab formaon ase han n he me-separable ase for he vas majory of possble realzaons. Reall ha he subsequen hab effe enourages savng and hus nex perod s onsumpon n order o relax nenve problems n he subsequen fuure. The mmedae hab effe, by onras, dsourages savng n order o relax he nenve problem n he perod mmedaely followng. Over me, nenve provsons mus rely less on fuure promses and more on osly onsumpon rewards. Therefore, relaxng nenve problems laer n lfe s relavely more mporan, whh explans why he subsequen hab effe exeeds he mmedae hab effe. The only exepon o hs rule appears a he very end of he workng lfe, when he subsequen hab effe by defnon falls o zero. 4.3 Sensvy analyss Frs, we noe ha he problem of nenve provson beomes more nrae f here are more skll ypes. To explore he role of addonal skll ypes, we exend he quanave model o hree ypes. We se he lfe-yle profle of expeed sklls, he varane of log-sklls, and all oher parameers as n our baselne model n Seon 4.2. Table 1 repors he lfeyle averages of expeed labor and savngs wedges for hab formaon preferenes and meseparable preferenes. As n he ase wh wo skll ypes, he mpa of hab formaon on labor and savngs wedges s negave. In he 2-ype model, hab formaon auses labor wedges o fall by 35 peren, and savngs wedges by 40 peren. In he 3-ype model, hab formaon auses labor wedges o fall by 33 peren, and savngs wedges by 39 peren. In he 3-ype 18 The dfferene beomes even more pronouned f we fous on workers beween ages 25 and 50. For hose workers, he average savngs wedge wh hab formaon s roughly one-hrd of he average savngs wedge wh me-separable preferenes. 20

21 model, labor dsorons below he op apply o a larger fraon of agens and herefore labor wedges are hgher han n he 2-ype model. Savngs wedges are also hgher n he 3-ype model bu he dfferene s less pronouned. Table 1: Expeed wedges lfe-yle averages for skll proesses wh wo and hree ypes hab formaon me-separable skll ypes labor wedge savngs wedge labor wedge savngs wedge Seond, we noe ha he problem of nenve provson exaerbaes as he me horzon shrnks. To examne he role of he me horzon for he quanave resuls, we explore models wh dfferen me horzons and ompare wedges n he frs perod of hose models. We se T {5, 10, 20, 30} and paramerze he models as before, exep ha we replae he humpshaped profle of expeed sklls by a fla profle for he sake of omparably aross models. Table 2 shows he expeed labor and savngs wedges for hab formaon preferenes and meseparable preferenes. Table 2: Expeed wedges n he frs perod for dfferen me horzons hab formaon me-separable T labor wedge savngs wedge labor wedge savngs wedge For boh preferene spefaons, labor and savngs wedges fall as he me horzon nreases. Ths resul mrrors he observaon ha labor and savngs wedges rse over he lfe yle n our baselne model n Seon 4.2. The dependene of wedges on he lfe yle s a ypal fndng for dynam Mrrlees models Golosov, Troshkn, and Tsyvnsk, Qualavely, we fnd ha he mpa of hab formaon on labor and savngs wedges s negave a all me horzons. Quanavely, he mpa dmnshes as he me horzon nreases, bu remans szable for all spefaons. In he ase wh he longes me horzon T = 30, labor wedges wh hab formaon are approxmaely 25 peren lower, and savngs wedges 13 peren lower, han n he ase of me-separable preferenes. 21

22 5 Conludng remarks Fndngs from maroeonoms, psyhology, and mro daa provde evdene for hab formaon n onsumpon preferenes. Ths paper sudes he effe of hab formaon on opmal axaon n a model wh prvae nformaon. We haraerze opmal alloaons n erms of labor and savngs wedges and denfy several novel axaon moves. ab formaon generaes a move o subsdze labor supply n order o enourage work and ndrely onsumpon, beause hs move makes agens hungrer for onsumpon n he fuure and hereby relaxes fuure nenve problems. ene, opmal labor wedges end o be smaller n he presene of hab formaon preferenes. ab formaon also generaes a move for savngs subsdes. If he worker onsumes less n he presen and more n he followng perod, beause of hab formaon he agen wll be hungrer for onsumpon n subsequen perods. Thus, nenve problems n subsequen perods are relaxed f onsumpon n he presen perod beomes relavely more expensve.e., f savngs are subsdzed. Opmal savngs wedges rade off hs effe agans he move o ax savngs o make he agen hungrer n he perod mmedaely followng beause of wealh and mmedae hab effes. We demonsrae he quanave mporane of hab formaon n a paramerzed lfe-yle model. Averaged over he lfe yle, opmal labor wedges for hab formaon preferenes are 35 peren lower, and opmal savngs wedges 40 peren lower, han for me-separable preferenes. Our paramerzaon apures several key aspes of he U.S. eonomy. For ompuaonal reasons, we assume ha skll shoks are ransory. I s beyond he sope of hs paper o deal wh he ompuaonal hallenges ha arse when hab formaon s ombned wh perssen shoks. The reursve formulaon wll hen nvolve hree onnuous sae varables habs, promsed uly, hrea uly. The man dffuly, however, s ha he doman of feasble ules beomes a wo-dmensonal nonreangular se ha depends on me, he pas shok, and he hab level. To he bes of our knowledge, he reursve onrang leraure has no ye found numeral approahes o dealng wh suh problems. Kapka 2013 and Farh and Wernng 2013 ompue models wh me-separable preferenes and perssen shoks ha are onnuous. Relyng on he frs-order approah and balaned-growh preferenes, hey are able o redue he number of sae varables o wo. In prnple, he frs-order approah an also be appled n he hab formaon ase. Sne he 22

23 balaned-growh propery breaks down, he number of sae varables nreases o four and he urse of dmensonaly persss. Referenes Abel, A. B. 1990: Asse Pres under ab Formaon and Cahng up wh he Joneses, Ameran Eonom Revew, 802, Abraham, A., S. Koehne, and N. Pavon 2014: Opmal Inome Taxaon wh Asse Aumulaon, Insue for Inernaonal Eonom Sudes. Mmeo. Albanes, S., and C. Slee 2006: Dynam Opmal Taxaon wh Prvae Informaon, Revew of Eonom Sudes, 731, Bohaek, R., and M. Kapka 2008: Opmal human apal poles, Journal of Moneary Eonoms, 551, Campbell, J. Y., and J.. Cohrane 1999: By Fore of ab: A Consumpon-Based Explanaon of Aggregae Sok Marke Behavor, Journal of Polal Eonomy, 1072, Carroll, C. D., J. Overland, and D. N. Wel 2000: Savng and Growh wh ab Formaon, Ameran Eonom Revew, 903, Consanndes, G. M. 1990: ab Formaon: A Resoluon of he Equy Premum Puzzle, Journal of Polal Eonomy, 983, Cremer,., P. De Donder, D. Maldonado, and P. Peseau 2010: Commody Taxaon under ab Formaon and Myopa, BE Journal of Eonom Analyss & Poly, 101. Cremer,., and P. Peseau 2011: Myopa, redsrbuon and pensons, European Eonom Revew, 552, Damond, P. A., and J. A. Mrrlees 1978: A model of soal nsurane wh varable reremen, Journal of Publ Eonoms, 103,

24 Daz, A., J. Pjoan-Mas, and J. Ros-Rull 2003: Preauonary savngs and wealh dsrbuon under hab formaon preferenes, Journal of Moneary Eonoms, 506, Farh, E., and I. Wernng 2008: Opmal savngs dsorons wh reursve preferenes, Journal of Moneary Eonoms, 551, Farh, E., and I. Wernng 2013: Insurane and axaon over he lfe yle, Revew of Eonom Sudes, 802, Feldsen, M. 1973: On he opmal progressvy of he nome ax, Journal of Publ Eonoms, 24, Fernandes, A., and C. Phelan 2000: A reursve formulaon for repeaed ageny wh hsory dependene, Journal of Eonom Theory, 912, Frederk, S., and G. oewensen 1999: edon Adapaon, n Well-beng: The foundaons of hedon psyhology, ed. by D. Kahneman, E. Dener, and N. Shwarz, pp Russell Sage Foundaon Press. Fuhrer, J. C. 2000: ab Formaon n Consumpon and Is Implaons for Moneary- Poly Models, Ameran Eonom Revew, 903, pp Golosov, M., N. Koherlakoa, and A. Tsyvnsk 2003: Opmal Indre and Capal Taxaon, Revew of Eonom Sudes, 703, Golosov, M., M. Troshkn, and A. Tsyvnsk 2011: Opmal dynam axes, Dsusson paper, Naonal Bureau of Eonom Researh. Golosov, M., and A. Tsyvnsk 2006: Desgnng Opmal Dsably Insurane: A Case for Asse Tesng, Journal of Polal Eonomy, 1142, Goard, P., and N. Pavon 2011: Ramsey Asse Taxaon under Asymmer Informaon, European Unversy Insue. Mmeo. Grohulsk, B., and N. Koherlakoa 2010: Nonseparable preferenes and opmal soal seury sysems, Journal of Eonom Theory, 1456,

25 Grohulsk, B., and T. Pskorsk 2010: Rsky human apal and deferred apal nome axaon, Journal of Eonom Theory, 1453, ansen, G. 1993: The Cylal and Seular Behavour of he abour Inpu: Comparng Effeny Uns and ours Worked, Journal of Appled Eonomers, 81, eahoe, J., K. Soresleen, and G. Volane 2012: Consumpon and labor supply wh paral nsurane: An analyal framework, Federal Reserve Bank of Mnneapols and New York Unversy, Mmeo. een, D., and C. Durham 1991: A es of he hab formaon hypohess usng household daa, Revew of Eonoms and Sass, pp elson,. 1964: Adapaon-evel Theory. arper & Row New York. Johnsen, T.., and J. B. Donaldson 1985: The Sruure of Ineremporal Preferenes under Unerany and Tme Conssen Plans, Eonomera, 536, pp Kapka, M. 2013: Effen Alloaons n Dynam Prvae Informaon Eonomes wh Perssen Shoks: A Frs-Order Approah, Revew of Eonom Sudes, 803, Koherlakoa, N. R. 2005: Zero Expeed Wealh Taxes: A Mrrlees Approah o Dynam Opmal Taxaon, Eonomera, 735, Messns, G. 1999: ab Formaon and he Theory of Addon, Journal of Eonom Surveys, 134, Phelan, C., and R. M. Townsend 1991: Compung Mul-Perod, Informaon- Consraned Opma, Revew of Eonom Sudes, 585, Rogerson, W. P. 1985: Repeaed Moral azard, Eonomera, 531, Ryder, Jr.,. E., and G. M. eal 1973: Opmum Growh wh Ineremporally Dependen Preferenes, Revew of Eonom Sudes, 401, Spear, S., and S. Srvasava 1987: On repeaed moral hazard wh dsounng, Revew of Eonom Sudes, 544,

26 Sanheva, S. 2014: Opmal Taxaon and uman Capal Poles Over he feyle, MIT. Mmeo. Sern, N. 1982: Opmum axaon wh errors n admnsraon, Journal of Publ Eonoms, 172, Sglz, J. E. 1982: Self-seleon and Pareo effen axaon, Journal of Publ Eonoms, 172, Wernng, I. 2011: Nonlnear apal axaon, MIT. Mmeo. 26

27 Appendes for onlne publaon Opmal axaon n a hab formaon eonomy Sebasan Koehne Morz Kuhn Ths doumen s organzed as follows. Appendx A olles he proofs ha are omed from he man ex. Appendx B derves a reursve formulaon of he soal plannng problem wh hab formaon. The seup allows for general reursve hab proesses and onans he ase of one-perod habs dsussed n he man ex as a speal ase. Appendx C derves labor and savngs wedges when he skll proess s perssen. Appendx A: Proofs Proof of emma 1. Sne he onsran se of he unrelaxed problem s a subse of he onsran se of he relaxed problem, suffes o show ha he soluon of he relaxed problem s feasble for he unrelaxed problem. In oher words, suffes o show ha he soluon of he relaxed problem sasfes he upward nenve ompably onsran. Whou loss of generaly, we assume >. We frs show ha he downward nenve onsran s bndng for he relaxed problem. Suppose o he onrary ha he soluon of he relaxed problem has a slak downward nenve onsran. By nspeon of he Kuhn-Tuker ondons, he soluon hen akes he form =, W = W, and y > y. owever, alloaons of suh form volae he downward nenve onsran. ene he assumpon ha he soluon of he relaxed problem has a slak downward nenve onsran mus be false. We now show ha a bndng downward nenve onsran mples ha he upward nenve onsran s sasfed. Formally, a bndng downward nenve onsran mples u, 1 v y / + βw +1 = u, 1 v y / + βw Insue for Inernaonal Eonom Sudes IIES, Sokholm Unversy, SE Sokholm, Sweden, Phone: , sebasan.koehne@es.su.se, and CESfo, Munh, Germany. Unversy of Bonn, Deparmen of Eonoms, D Bonn, Germany, Phone: , mokuhn@unbonn.de, and Insue for he Sudy of abor IZA, Bonn, Germany. 1

28 Reall ha labor dsuly v s a onvex funon. Sne 1/ 1/, onvexy of v mples ha he dfferene v y/ v y/ nreases n y. Moreover, s easy o see ha a bndng downward nenve onsran mples y y. Combnng he las wo nsghs, we oban v y / v y / v y / v y /. 2 We rewre hs nequaly as v y / v y / v y / v y /. 3 We ombne he bndng downward nenve onsran wh he above nequaly and oban v y / v y / u, 1 u, 1 + βw +1 βw+1. 4 ene he upward nenve onsran s sasfed. Proof of Remark 1. Sne he nenve ompably onsran has a agrange mulpler of zero n all perods 0, we have µ = 0 for 0. Now he resul follows from Proposons 2 and 3. Proof of Proposon 1. See onlne Appendx B. Proof of Proposon 2. The fne horzon Bellman equaon of he soal planner problem s C W, 1 = mn,y,w +1 =, [ y + qc +1 W +1, ] π 5 s.. u, 1 v y / + βw +1 u, 1 v y / + βw +1 6 [ u, 1 v y / ] + βw +1 π = W. 7 =, Problem 5 has he followng frs-order ondons for onsumpon 0 = π [ 1 + qc+1,h W +1, ] λ u, 1 π µ u, 1, 8 [ 0 = π 1 + qc+1,h W +1, ] λ u, 1 π + µ u, 1, 9 2

29 for oupu 0 = π v y + / λ π v y + / µ, 10 0 = π v y + / λ π v y / µ, 11 and for onnuaon ules 0 = π qc+1,w W +1, λ βπ µ β, 12 0 = π qc+1,w W +1, λ βπ + µ β. 13 We begn wh he labor wedge of he hgh-sklled worker. Combne he frs-order ondon for y wh ha for o oban 1 + qc +1,h W +1, = u, 1 v y /. 14 By he envelope heorem, appled o he Bellman equaon 5 a dae + 1, we have C +1,W W +1, C +1,h W +1, = λ +1, 15 = λ +1 u h j +1, π +1 j +1 µ [ +1 uh +1, uh +1, ] 16. j ene we an rewre 14 as v y / u, 1 = 1 qc +1,W W +1, qµ [ +1 uh +1, j u h j +1, uh +1, ]. π +1 j Combne he frs-order ondon for W +1 wh he frs-order ondon for y o oban qc +1,W W +1, = β v y /. 18 Use hs o rewre 17 as follows: E [Ũ 1, ] = v y / 1 qµ +1 [ uh +1, uh +1, ]. 19 3

30 Therefore he labor wedge s τy, = µ qv y / +1 E [Ũ 1, ] [ u h +1, uh +1, ]. 20 Usng he frs-order ondon for y and he deny qπ λ +1 = β λ π + µ, and defnng B = β [ ] u h +1, uh +1, λ +1 [Ũ E ], 21 1, he labor wedge s τ y, = µ +1B. We now urn o he labor wedge of he low-sklled worker. Frs we wre he frs-order ondon for as [ ] λ π µ u, 1 π = qπ C+1,h W +1,. 22 The envelope heorem, appled o he Bellman equaon 5 a dae + 1, yelds C +1,W W +1, C +1,h W +1, = λ +1, 23 = λ +1 u h j +1, π +1 j +1 µ [ +1 uh +1, uh +1, ] 24. j Combned wh he frs-order ondon for W +1, we oban qπ C+1,h W +1, = λ π β j u h j +1, π +1 j +1 +µ β u h j +1, π +1 j +1 j µ [ +1π q uh +1, uh +1, ]. 25 We subsue hs n he frs-order ondon for o oban ] λ π E [Ũ 1, π 26 ] = µ E [Ũ 1, µ [ +1π q uh +1, uh +1, ]. 27 Now we use he frs-order ondon for y o replae π : { ] } λ π E [Ũ 1, v y / ] = µ {E } [Ũ 1, v y / 28 µ [ +1π q uh +1, uh +1, ]. 29 4

31 Ths an be rewren as { ] λ π µ E [Ũ 1, { v y / } = µ v y / } v y / 30 µ [ +1π q uh +1, uh +1, ]. 31 Usng he deny π qλ +1 = β λ π µ, and defnng [ A β v y / ] = ] qπ λ +1 E [Ũ 1, v y /, 32 B = β [ ] u h +1, uh +1, λ +1 [Ũ E ], 33 1, he labor wedge s hene τ y, = µ A µ +1B. Ths omplees he proof. Proof of Proposon 3. We begn wh he savngs wedge for he hgh-sklled worker. Combne he frsorder ondon for onsumpon 8 and he envelope ondon 16 o oban λ π + µ u π, = qλ +1 u h j +1, π +1 j +1 qµ [ +1 uh +1, uh +1, ]. j Usng he deny qπ λ +1 = β λ π + µ, we an rewre he prevous equaon as qλ +1 [Ũ β E 1, ] = 1 qµ [ +1 uh +1, uh +1, ]. 35 The frs-order ondons for onsumpon n perod + 1 are [ 0 = π qc+2,h W +2, ] +1 λ +1 u +1, [ 0 = π qc+2,h W +2, ] +1 λ +1 u +1, π+1 +1 µ +1 u +1, π µ +1 u +1, 36,.37 Summng up hese ondons and subsung he resul no he prevous equaon yelds qλ +1 [Ũ β E 1, ] = π qc+2,h W +2, +1 π+1 +1 qc+2,h W +2, +1 ] π λ [ +1 u +1, π u +1, 38 µ [ +1 u +1, u +1, ] qµ +1 [ uh +1, uh +1, ]. 5