Online Appendix for Optimal Taxation and Human Capital Policies over the Life Cycle

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1 Online Appendix for Opimal Taxaion and Human Capial Policies over he Life Cycle Sefanie Sancheva Firs version: Ocober 22 This version: Augus, 26 A Implemenaion Proofs A sequenial reformulaion of he recursive allocaions: Any soluion o he recursive social planner s problem can be mapped ino a soluion which depends on he full pas hisory, using a recursive consrucion. To see his, denoe he soluions o he recursive problem a ime, for each realized ype, as a funcion of all sae variables by: v v,, s,,, v,, s,,, ω v,, s,,, y v,, s,,, s v,, s,,, c v,, s,, and he soluions o he planner s sequenial problem by { x } = { y, s, c }. The dependence on iniial promised uiliies in period, due o iniial heerogeneiy, is dropped for noaional convenience; hey can be jus carried as an addiional condiioning variable. This allocaion gives rise o a sequence of uiliies for he agen, generaed recursively by: ω = u c y φ w, s + β ω +, + f + + d + v = ω+, + f + + d + = ω+, + f + + d + To iniialize he allocaions, se ω = ω v,, s,, = U if here is iniial heerogeneiy in, wih v arbirary in ha case, =, y = y v,, s,,, s = s v,, s,,, and consruc ieraively he full, hisory dependen allocaion for all hisories using he difference equaions: ω = ω v,, s,, where v and are wrien as funcions of ω, i.e., v = β [ ω u c y ] + φ w, s

2 Consruc also: = β y s c [ ω = y = s = c φ y w s, y w s, 2 w v,, s, v,, s, v,, s, Using hese sequenial opimal policies in he planner s problem we can rewrie he coss as a funcion of he hisory, i.e., K = K v, s,, or, if agens also have heerogeneous iniial wealh levels, K b,. Noe ha in he planner s problem i.e., in he direc revelaion mechanism iniial wealh holdings are irrelevan since he planner can fully observe and allocae consumpion. Since agens can borrow a he same rae as he governmen, wihou loss of generaliy, agens can do all he borrowing and saving on heir own. The implici wealh levels are hen defined recursively as: R b b, = y b, c b, M e b, + b b,. ] Proof of Proposiion 4 In he firs sep, we consruc he hisory-independen savings ax, in he spiri of Werning 2, wih added human capial. Consider an incenive compaible allocaion expressed as a funcion of he full hisory c, y, e,s, and is associaed coninuaion uiliy ω, and suppose i is implemened as he oucome of a direc revelaion mechanism wih no savings. Allow agens o save any desired amoun, wih he resricion b T end of period T asse level. Consider a general ax funcion T K b, r as a funcion of savings and he hisory of repors up o period. Given he repor of he agen up o period, and he repor of he curren shock, he planner assigns c r, r, y r, r, e r, r and he agen chooses savings b. Le V b, r, be he coninuaion value of an agen wih beginning of period savings b, a hisory of repors r, and a realized shock. The agen s problem is: V b, r, = max r,b Ṽ b, b, r, r, wih Ṽ b, b, r, r, defined as he value from saving an amoun b and reporing r : Ṽ b, b, r u c r, r + b, r, = R b + T K b, r, r φ +βe V + b, r, + yr,r w,s r,r In period T, define for each ype realizaion T, curren asse level b T 2, hisory of repors r T, and savings levels b T a ficiious ax T K which makes an agen jus indifferen beween saving b T and saving zero. ω T T = u T c T r T + b T 2 R b T T KT bt, r T, T y T r T φ T w T T, s T r T + βe V T bt, r T, T T 2

3 Taking he sup over all ypes T yields a repor hisory-dependen, bu ype-independen savings ax T K r bt, r T = sup T T KT bt, r T, T. By inducion, suppose ha in period he agen is faced wih a coninuaion value funcion V + b, r, +. Define he ax funcion for period as TK b, r = sup T K b, r, wih T K b, r, o equae: ω = u c r + b R b T K b, r, φ y r w, s r + βe V + b, r, + Work backwards o define he ax funcions in his fashion for all periods. The sequence of ax funcions { TK b, r } T hus defined implemen zero savings each period for all sequences of repors. Nex, = ake he supremum over all hisories of repors r o obain a hisory independen ax which implemens zero savings. T K b = sup r TK b, r In he second sep, we consruc he loan and repaymen schedule ha mimics he direc mechanism above. Noe ha L can direcly be mapped ino a hisory of human capial and educaion levels e recall ha s =, using ha e = M L Hence, L, y and e, y will be used inerchangeably. Firs, se implici finie bu poenially very large upper and lower limis on asse holdings, b > and b <. This can be done eiher by exending he proposed savings ax so ha for b > b and b < b,, i is confiscaory e.g., ax away all wealh and imposes a large penaly on borrowing or by direcly seing borrowing and saving limis. Le B [ b, b ]. In each period, all allocaions which can arise as oucomes in he opimum are made affordable: D L, y, e,, y, + T Y y, = y, c, L e, = M e, 2 for all L, y such ha Θ { M L,..., M L }, y and all Θ. To mimic he direc revelaion mechanism, we need o exclude pairs e, y which would no be assigned o any ype in he social planner s problem afer a hisory, and, consequenly, exclude hisories y, e, which do no correspond o any hisory. Call non-allowed a choice which is no assigned in he social planner s problem for any ype afer hisory, i.e., such ha e, y Q e,y. The repaymens a non-allowed levels have o be sufficienly dissuasive o make hem sricly dominaed by allowed choices. Then he hisory of repors would exacly be racked by r = Θe,y y, e, and he agen s problem becomes equivalen o making a repor r Θ in each period, by choosing a pair e, y designed for some afer. We know ha in his case, he previously consruced hisory-independen savings ax would enforce zero savings. There are several ways o rule ou non-allowed allocaions, and he goal here is jus o provide a possible one, which is o simply se he repaymen prohibiively high, so ha irrespecive of savings, i is never opimal o chose such allocaions. For insance, afer a hisory Θ e, y and for any choice e, y Q e,y, se D L, y, e, y + TY y > [ b b + y ] 3

4 i.e., he repaymen plus income ax a leas confiscae income and impose an addiional penaly such ha all wealh is confiscaed and agens can never borrow sufficienly o reain posiive consumpion. This leaves he agen wih zero consumpion, and will never be chosen. More generally, arbirarily large repaymens can be se. The second and less draconian way is o ake he envelope of he repaymen schedules which, afer each hisory, for any possible beginning of period wealh and opimal savings choice, would make he agen jus indifferen beween any non-allowed allocaion and his opimal allocaion. Whaever he mehod chosen, once he non-allowed choices are ruled ou, each period, afer every hisory, he agen faces a problem equivalen in oucomes o he direc revelaion mechanism, i.e., he faces only allocaions which are also available o him in he social planner s problem afer ha hisory. Accordingly, he savings ax ensures ha he will find i opimal no o save. By emporal incenive compaibiliy, he will chose he allocaion designed for him. Proof of Proposiion 5 : The proof is an exended and modified version of he proof of Proposiion in Albanesi and Slee 26, adding human capial. Wih iid shocks, he recursive formulaion of he relaxed program no longer requires he saes and : K v, s, = subjec o: min [c + M s s w, s l + R ] K v, s, + f d c,l,ω,s,v ω = u c φ l + βv. ω = w, l φ l, l w v = ω f d Denoe by K,s and K,v he parial inverse funcions of K v, s, wih respec o is argumens s and v respecively. Define he se Q e,y b, s = { e, y : e = e v, s,, y = y v, s, for some Θ, wih v = K,v b, s } o be he se of oupu levels y and educaion levels e which are available o an agen wih promised uiliy v and previous human capial level s in he social planner s problem. The value funcion of he agen who sars a period wih wealh b and human capial s is denoed by V b, s, as in he main ex. In each period, he agen s problem can be spli ino wo sages, because of he separabiliy beween consumpion and labor. In sage, he chooses labor supply l equivalenly, oupu y and human capial expenses e. He pays a ax T b, s, y, e and is lef wih a oal resource amoun b m = y T b, s, y, e M e + b. In sage 2, he chooses consumpion and nex-period To exend he repaymen scheme s domain o hisories L, y for which Θ T e T, y T =, se for all e, y afer such hisories: D L, y, e, y + TY y > b b + y 4

5 bond holdings, b o maximize V m b m, s + e, he inermediae value funcion from resource level b m, defined as: V m b m, s = max c,b u c + βv + b, s s.. : b m = c + R b wih VT m bm T, s T = u T b m T. Denoe he marke oucomes by ˆb b m, s and ĉ b m, s. In sage, he problem of he agen is: V b, s = max {y,e,b m } φ y /w, s + e + V b m, s f d Θ s.. : b m = y T b, s, y, e M e + b Le he marke oucomes be denoed by ŷ b, s,, ê b, s,, and ˆb m each ype. In each period, he planner solves a wo-sage problem as well. b, s, for In he firs sage, he allocaes human capial expenses e, oupu y, and an inermediae promised uiliy v m. In he second sage, he allocaes consumpion c, and coninuaion uiliy v. In sage 2, given an inermediae promised uiliy v m, and an acquired human capial level s + e = s, he planner solves he program wih inermediae coninuaion cos funcion K m v m, s, : K m v m, s, = min c + c,v R K v, s, + s.. : u c + βv = v m wih K v T, s T, T +. Denoe he soluions o his problem by c v m, s and v v m, s. In sage, he problem of he planner is hence: K v, s, = min {v m,e,y} M e y + K m v m, s + e, f d Θ s.. : v m φ y /w, s + e v m φ y /w, s + e, v m φ y /w, s + e f d = v Θ Denoe he soluions o he planner problem by v m v, s,, e v, s,, and y v, s,. In sage 2, he problem of an agen who has acquired human capial s and who sars wih inermediae wealh b m = K m v m, s, is exacly he dual of he planner s problem who has promised uiliy v m = K m,v b m, s, for an agen wih human capial s where K m,v is he parial inverse of K m wih respec o is firs argumen, i.e., he resource cos for he planner of providing uiliy v m o an agen wih human capial s a ime, so ha he consumpion choice of he agen and planner will coincide if mapped appropriaely, i.e., ĉ K m v m, s,, s = c v m, s. Furhermore, ˆb K m v m, s,, s = K v v m, s, s, +. To see his, suppose insead ha here was anoher pair c, b c, K such ha c + R b = b m, bu which yields higher uiliy: u c + βv + b, s > 5

6 v m = K m,v b m, s, Noe ha he choice of s, already made in he previous sage is now fixed. Under he assumpion ha V +., s is increasing and coninuous in is firs argumen, and given ha u c is increasing and coninuous in c, here is also a pair c, b such ha c c, b K wih one or c, b boh of hese inequaliies sric and such ha u c + βv + b, s = v m. Bu hen is beer han c, K in he planner s problem and hence, c, K could no have been opimal, a conradicion. For he firs sage, consider an agen wih iniial wealh b and human capial level s. Firs, map he allocaions from he social planner s problem o allocaions defined on he sae space b, s, : y b, s, = y K,v b, s,, s, e b, s, = e K,v b, s,, s, b m b, s, = K m v m v, s,, s + e v, s,, Then, se he ax level such ha for all y, e which are consisen wih wealh level b and human capial level s i.e., can arise for some hisory : T b, s, y b, s,, e b, s, = y b, s, b m b, s, +b M e b, s, ha is, make all allocaions consisen wih an allocaion in he planner s problem exacly affordable. To exend he definiion of he ax funcion o he full domain of allocaions, even hose which would no arise in he social planner s problem, hree seps are aken. Firs, o rule ou wealh levels b no observed along he equilibrium pah, i.e., such ha b K v, s, for any, se he borrowing limis o be b = min v,s K v, s, where he min is aken over he possible values of v and s a ime in he planner s program. Second, if b = K v, s, for some bu s s ha is, he levels of wealh and human capial from he pas period are no muually consisen, se he ax T such ha, for all e, y : T b, s, e, y = y + max {b, } min { b, } Finally, if he agen is on he equilibrium pah wih b and s, and leing v = K,v b, s,, se T such ha for all pairs y, e which are no consisen wih b and s and for all : y φ w, s + e y v, s, < φ w, s + e v, s, + βv b m + y T b, s, y, e M e, s + e + v m v, s, In he firs sage, given he dissuasive axes on choices which never arise in he planner s problem, he agen can eiher choose he full allocaion y and e desined for some ype ha could be his own rue ype, which will hen lead him o choose also he coninuaion wealh opimal for ha same ype, or he could choose y opimal for some ype bu e opimal for some ype 2. This will however leave him wih lower value given he axes on he off-equilibrium pahs. Hence, if a ype deviaes, he mus deviae o he full allocaion of anoher ype. By he emporal incenive compaibiliy consrain, he will choose no o do so. 6

7 B Addiional Figures and Numerical Resuls B. Tesing he calibraion: Baseline economy The following figures illusrae how he baseline economy behaves, when calibraed as in Table. In his case, axes and subsidies are calibraed as explained in Secion 5 o mimic he curren U.S. economy and no se opimally. Figure depics he allocaions of income, consumpion, and human capial expenses. Figure 2 depics he variance of he logs of consumpion, wage and income, while Figure 3 shows he variances of wages, income, and consumpion. Figure 4 depics asse holdings, while Figure 5 shows he mean asses-o-income raio. Figure : Allocaions a ρ s =.2 b ρ s = income consumpion HC expenses income consumpion HC expenses Panel a shows he allocaions of consumpion, human capial expenses and income for he case ρ s =.2, while panel b shows he allocaions for ρ s =.2. 7

8 Figure 2: Log variances of wages, income and consumpion a ρ s = b ρ s =.2.4 log wage variance.2 log consumpion variance log income variance.4 log wage variance.2 log consumpion variance log income variance Panel a shows he variance of he logs of consumpion, wage and income for he case ρ s =.2, while panel b shows he variances for ρ s =.2. Figure 3: Variances of wages, income, and consumpion.5 a ρ s =.2.25 b ρ s = wage variance consumpion variance income variance.5 wage variance consumpion variance income variance Panel a shows he variance of consumpion, wage and income for he case ρ s =.2, while panel b shows he variances for ρ s =.2. 8

9 Figure 4: Asse holdings a ρ s =.2 Asses Asses 6 b ρ s =.2 Asses Asses Panel a shows he mean asses accumulaed by agens for he case ρ s ρ s =.2. =.2, while panel b shows he asses for Figure 5: Mean asses-o-income raio a ρ s =.2 b ρ s = asses o income raio asses o income raio Panel a shows he mean asses-o-income raio for he case ρ s =.2, while panel b shows he raio for ρ s =.2. 9

10 B.2 Opimal Policies B.2. Properies of he Opimal Allocaions Figure 6: Opimal allocaions a ρ s =.2 b ρ s = oupu consumpion human capial 2.5 oupu consumpion human capial Agens opimally inves in human capial early in life. Consumpion is a maringale; hence, average consumpion is perfecly fla. Oupu and consumpion are higher when human capial disproporionaely benefis higher abiliy agens. Allocaions and Insurance: Figure 6 plos he average allocaions over ime. Average human capial invesmens are highes early in life, before declining wih age. Mean consumpion is consan, a resul due o he Inverse Euler equaion in 3, and log uiliy wih β = R, which imply ha consumpion is a maringale: E c = c. Mean oupu is increasing, despie he rising labor wedge, because of he growing produciviy of agens driven by heir endogenous human capial invesmens. Figure 7 panel a shows ha lifeime spending on human capial is more ighly linked o lifeime income when human capial disproporionaely benefis high abiliy agens. The causaliy goes boh ways: higher abiliy agens boh acquire more human capial and earn more. In urn, human capial increases earnings even furher. When ρ s >, boh effecs are amplified. However, he fac ha higher abiliy people also acquire more human capial does no mean ha here is no insurance. Panel b highlighs he exen of lifecycle insurance a he opimum by ploing he ne presen value of lifeime spending consumpion plus human capial expenses agains he ne presen value of lifeime income. Clockwise pivos of he line represen higher insurance. Finally, figure 8 describes he cross-secional variances of oupu, human capial, consumpion, and abiliy over ime. The variance of oupu is driven no only by sochasic abiliy, bu also by differenial invesmens in human capial a differen abiliy levels. Oupu is much more volaile han consumpion. Hence, pre-ax income inequaliy grows a an increasing rae, bu he provision of insurance prevens his from ranslaing fully ino consumpion inequaliy. In addiion, while consumpion

11 Figure 7: Human Capial and insurance over he life cycle a Human capial expenses and lifeime income b Consumpion and educaion expenses a Lifeime income is posiively correlaed wih lifeime human capial expenses, he more so when ρ s >. There is a wo-way causaliy: Higher abiliy people boh acquire more human capial and have higher earnings poenial. A he same ime, human capial increases earnings. b The figure shows he presen value of consumpion and human capial expenses agains he presen value of lifeime income. The laissez-faire oucome is represened by he 45 degree line. Clockwise pivos of he line represen more insurance. Figure 8: Variance and Risk a ρ s =.2 b ρ s = produciviy oupu consumpion human capial produciviy oupu consumpion human capial The figure shows cross-secional variances over ime. Oupu is more volaile han consumpion. Is volailiy grows a an increasing rae, driven boh by abiliy shocks and differenial invesmens in human capial. Bu pre-ax income inequaliy does no fully ranslae ino consumpion inequaliy.

12 variance grows, i does so a a decreasing rae, echoing he ax and subsidy smoohing resuls described above. B.2.2 Income Coningen Loan Implemenaion In his Secion, we presen some addiional feaures of he income coningen loan schedule ha were referred o in he ex. The ineres rae in period is defined as he raio of he repaymen in period o he ousanding loan balance oal pas loans minus all pas repaymens up o period. Figure 9: Ineres Rae agains he Ousanding Loan Balance a ρ s =.2 b ρ s = Ineres rae.5.4 Ineres rae Ousanding loan balance Ousanding loan balance This figure plos a snapsho in period = 5 of he implied ineres rae agains he Ousanding Loan Balance each do represens one agen, i.e., one of he, Mone Carlo simulaions. Panel a plos he resuls for ρ s =.2 while panel b plos he resuls for ρ s =.2. 2

13 Figure : Ineres Rae again he Conemporaneous Loan a ρ s = b ρ s =.2 Ineres rae Conemporaneous loan Ineres rae Conemporaneous loan This figure plos a snapsho in period = 5 of he implied ineres rae agains he conemporaneous loan given each do represens one agen, i.e., one of he, Mone Carlo simulaions. Panel a plos he resuls for ρ s =.2 while panel b plos he resuls for ρ s =.2. 3

14 References Bohacek, Radim and Marek Kapicka, Opimal human capial policies, Journal of Moneary Economics, 28, 55, 6. Findeisen, Sebasian and Dominik Sachs, Educaion and opimal dynamic axaion: The role of incomeconingen suden loans, Technical Repor, Working Paper Series, Deparmen of Economics, Universiy of Zurich 22.