Taxes, debts, and redistributions with aggregate shocks

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1 Taxes, debts, and redstrbutons wth aggregate shocks Anmol Bhandar Davd Evans Thomas J. Sargent Mkhal Golosov Aprl 2015 Abstract Ths paper models how transfers, a tax rate on labor ncome, and the dstrbuton of government debt should respond to aggregate shocks when markets are ncomplete. A planner sets a lump sum transfer and a lnear tax on labor ncome n an economy wth heterogeneous agents, aggregate uncertanty, and a sngle asset wth a possbly rsky payo. Lmts to redstrbuton comng from ncomplete tax nstruments and lmts to hedgng comng from ncomplete asset markets aect optmal polces. Two forces shape long-run outcomes: the planner's desre to mnmze the welfare cost of uctuatng transfers, whch calls for a negatve correlaton between agents' assets and ther sklls; and the planner's desre to use uctuatons n the return on the traded asset to compensate for mssng state-contngent securtes. In a mult-agent model calbrated to match facts about US booms and recessons, the planner's preferences about dstrbuton make polces over busness cycle frequences der markedly from Ramsey plans for representatve agent models. Key words: Dstortng taxes. Transfers. Redstrbuton. Government debt. Interest rate rsk. JEL codes: E62,H21,H63 We thank Mark Aguar, Stefana Albanes, Manuel Amador, Andrew Atkeson, Marco Bassetto, V.V. Char, Harold L. Cole, Fath Guvenen, Guy Laroque, Francesco Lpp, Robert E. Lucas, Jr., Al Shourdeh, Perre Yared, Ellen Mcgrattan, and semnar partcpants at Boccon, Chcago, EIEF, the Federal Reserve Bank of Mnneapols, IES, Prnceton, Stanford, the Federal Reserve Bank of Atlanta, UCL, Unversdade Catà lca,wharton, Swss Fnance Insttute, 2012 Mnnesota macro conference, Monetary Polcy Workshop NY Fed for helpful comments.

2 If, ndeed, the debt were dstrbuted n exact proporton to the taxes to be pad so that every one should pay out n taxes as much as he receved n nterest, t would cease to be a burden.possble, there would be [no] need of ncurrng the debt. For f a man has money to loan the Government, he certanly has money to pay the Government what he owes t. Smon Newcomb 1865, p.85 1 Introducton What are the welfare costs of publc debt? What determnes whether and how quckly a government should retre ts debt? How should tax rates, transfers, and government debt respond to aggregate shocks? We study how answers to these questons depend on a government's eagerness and ablty to redstrbute and to hedge uctuatons n dects. We restrct tax collectons to be ane functons of labor ncome. Agents der n ther productvtes and wealth. They trade a sngle securty whose payo possbly depends on aggregate shocks. A Ramsey planner attaches a vector of Pareto weghts to derent types of agents' dscounted utltes and adjusts the proportonal labor tax rate, transfers, and asset purchases n response to aggregate shocks. A dstrbuton of assets gves rse to ows of asset earnngs across agents that depend on how returns on the asset comove wth aggregate shocks. These ows requre the government to adjust labor taxes and transfers to acheve ts dstrbutve and nancng goals. Labor taxes dstort labor supples, but uctuatons n transfers also lower welfare. A decrease n transfers n response to adverse aggregate shocks aects agents who have low present values of earnngs especally. Secton 2 sets out preferences and possbltes that dene the economc envronment. Secton 2.1 descrbes a Rcardan property for our envronment that we explot n concsely formulatng a Ramsey problem. An ndvdual's net asset poston equals hs assets mnus the assets held be a partcular benchmark agent. The vector of all agents' net asset postons aects the set of allocatons that can be mplemented n compettve equlbra wth ane taxes. The nsght that net and not gross asset postons matter reduces the dmenson of a state sucent to descrbe a Ramsey plan recursvely. It also provdes a way to separate roles played by government debt and transfers despte the Rcardan property. We do ths by settng the asset holdngs of the least productve agent always to zero and than backng transfers out from the derence between ths agent's consumpton and hs after-tax labor ncome. A Ramsey plan nduces an ergodc dstrbuton of transfers, the labor tax rate, and the dstrbuton of assets that s determned by nteractons between a the government's ablty to hedge aggregate shocks by takng advantage of uctuatons n the return on the asset t trades; and b the government's preferences about redstrbuton that aect how costly t s to use uctuatons n transfers to hedge those shocks. The analyss n sectons 3, 4, and 5 shows that these nteractons shape the ergodc dstrbuton of government debt n the followng ways. If equlbrum outcomes make the return on the asset low when the net-of nterest government dect s hgh, then the government wll run up debt. But f the return on the asset s hgh when the net-of-nterest government dect s hgh, the government wll accumulate assets. The long run varances of government assets and the tax rate are both lower and rates of convergence to the ergodc dstrbuton are hgher n economes n whch the comovement between the net-of nterest dect and the return on the asset s bgger n absolute magntude. Other thngs equal, governments that want more redstrbuton eventually hold fewer assets. The comovement between the aggregate shock that drves government expendtures and re- 2

3 turns on the sngle asset s a key ntermedatng object that shapes the ergodc jont dstrbuton of the tax rate and government debt. In our general settng, one-perod utltes are concave n consumpton. That makes the return on the sngle asset an equlbrum object partly under the control of the Ramsey planner. To help us understand forces shapng both the ergodc jont dstrbuton and rates of convergence to t, we begn by studyng a specal settng n whch returns on the sngle asset and ther correlaton wth the aggregate shock are exogenous, namely, a settng wth one-perod utltes that are quaslnear, meanng lnear n consumpton. Secton 3 completes ths analyss n two parts. We begn by analyzng a representatve agent economy wth quaslnear preferences, a government restrcted to set transfers equal to zero always, and..d aggregate shocks to publc expendtures. Our man ndng here s that for a large class of payo structures, debt drfts towards an ergodc set that n a sense maxmzes the government's ablty to hedge expendture shocks, a set that prmarly depends on how the payo on the asset correlate wth uctuatons n the government's net-of-nterest dect. For specal cases n whch the asset payo s ane n government expendture shocks, we show that the ergodc dstrbuton s degenerate. For other assumptons about the asset payo, shock correlaton, we develop tools to approxmate the ergodc dstrbuton and tell how the spread of the ergodc dstrbuton of government debt and the tax rate ncreases wth how far the payos are from allowng perfect scal hedgng. The representatve agent analyss wth transfers restrcted to zero s nformatve about outcomes n multple agent economes wth no restrctons on transfers but wth Pareto weghts that make the welfare costs of transfers so hgh that the Ramsey planner chooses never to use them, or at least not to use them eventually. 1 We show ths by analyzng an economy wth unrestrcted transfers and two types of agents both of whom have quaslnear one-perod utltes and one of whom s not productve. We study the eects of the presence or absence of a nonnegatvty constrant on the consumpton of the unproductve agent as a determnant of the asymptotc level of government debt and whether and when transfers are used. The asymptotc level of assets s decreasng n the planner's desre for redstrbuton. Ths comes from the fact that welfare costs of usng transfers are lower for a more redstrbutve government. Consequently t reles more on transfers and has less cause to accumulate assets to hedge aggregate shocks. In secton 4, we study economes more general n ther heterogenety, preferences, and shock structures. We formulate the Ramsey problem recursvely n terms of two Bellman equaton, one for tme 0 and another for tmes t 1. That these Bellman equatons have derent state varables expresses the tme nconsstency of plans that as usual attrbute to the Ramsey planner the ablty to commt to an ntertemporal plan at tme 0. Subsectons 4.2 and?? derve condtons under whch the planner can eventually acheve complete hedgng,.e., both constant labor tax rate and consumpton shares even when the return on the asset s endogenous. In secton 5, we numercally verfy that the forces solated n the more analytcally tractable models of sectons 3 and 4 preval n a verson of the model wth several types of agents calbrated to match US data. Our calbraton captures 1 the ntal heterogenety wages and assets; 2 the observaton that n recessons the left tal of the cross-secton dstrbuton of labor ncome falls by more than rght tal; and 3 how naton and asset return rsk comove wth labor productvty. We use ths model to quantfy the channels dscovered n our theoretcal analyss of smpler envronments. We also descrbe features of optmal government polcy, especally n booms and recessons at hgher frequences. Durng recessons accompaned by hgher nequalty, 1 The analyss also augments what s known about representatve agent economes n whch a sngle rsk-free bond s traded e.g., Ayagar et al. 2002, Farh 2010, and Faragla et al

4 t s optmal to ncrease taxes and transfers and to ssue government debt. These outcomes der both qualtatvely and quanttatvely from those n ether a representatve agent model or n a verson of our model n whch a recesson s modelled as a pure TFP shock that leaves the dstrbuton of sklls unchanged. Two techncal contrbutons facltate our analyss. Frst, for the quaslnear preferences studed n secton 3, we obtan sharp characterzatons of the ergodc dstrbuton of debt and taxes by approxmatng transton dynamcs around economes that permt perfect scal hedgng. Ths approach also enables a local stablty analyss for economes wth the more general preferences of secton??. Second, quanttatve applcatons as ambtous as those n secton 5 are made possble by takng a sequence of polynomal approxmatons about steady states of a sequence of determnstc economes and evaluatng the polynomals at a current dstrbuton of dosyncratc state varables of the ncomplete markets economy. Ths approach allows us to study transton paths of the optmal allocaton that express the dynamcs of the jont dstrbuton of ndvdual wealths and past consumptons, a hgh dmensonal endogenous state vector. These techncal contrbutons, whch buld on deas developed n Evans 2014, wll be useful n many other settngs wth heterogeneous agents and aggregate rsk. Secton 6 oers concludng remarks. 1.1 Relatonshps to lteratures A large lterature on Ramsey problems exogenously restrcts transfers n the context of representatve agent, general equlbrum models. Lucas and Stokey 1983, Char et al. 1994, and Ayagar et al henceforth called AMSS Fgure 1.1 shows that an ane structure better approxmates the US tax-transfer system than just proportonal labor taxes. 4

5 In contrast to those papers, our Ramsey planner cares about the dstrbuton of welfare among agents wth derent sklls and wealths. Except for not allowng them to depend on agents' personal denttes, we leave transfers unrestrcted and let the Ramsey planner set them optmally. We nd that some of the same general prncples that emerge from the representatve agent, notransfers lterature contnue to hold, n partcular, the prescrpton to smooth dstortons across tme and states. However, t s also true that allowng the government to set transfers optmally changes the optmal polcy n mportant respects. 2 Some of our results n secton 3 relate to Angeletos 2002, Buera and Ncoln 2004, and Shn These papers show that for a gven ntal government debt, the same Ramsey allocaton that emerges from a complete markets lke the one n Lucas and Stokey 1983 can also be supported n a wth non-contngent debts wth an approprate set of maturtes. In contrast, we characterze propertes of payos on government debt that cause the optmal plan wth ncomplete markets eventually to converge to a complete markets allocaton. That lmtng allocaton depends on model prmtves but not on the ntal level of government debt. We use ths ndng to buld tools that allow us to say more about the ergodc dstrbuton of debt and taxes n economes n whch only mperfect scal hedgng s possble. In a settng wth self-nterested poltcans who ssue debt and tax captal and cannot commt to future government actons, Aguar and Amador 2011 nd condtons under whch optmal allocatons feature no tax dstortons n the lmt. Our results about condtons for eventual complete scal hedgng n our envronment have a smlar avor to Aguar and Amador's. We delneate condtons under whch uctuatons n the tax rates vansh n the lmt, although the constant tax rate may not be zero. Also, for a determnstc settng wth lnear utlty, Aguar and Amador characterze speeds of convergence of debt and tax rates n neghbourhoods of steady states and they analyze how poltcal economy frctons can arrest the rate of convergence. In secton 3, we too characterze the speed of convergence and show how t depends on the covarance structure of the payos and aggregate shocks. Our paper extends both Barro 1974, whch showed Rcardan equvalence n a representatve agent economy wth lump sum taxes, and Barro 1979, whch studed optmal taxaton when lump sum taxes are ruled out. In our envronment wth ncomplete markets and heterogeneous workers, both forces dscovered by Barro play large roles. But the dstrbutve motves that we nclude alter optmal polces. Several other papers mpute dstrbutve concerns to a Ramsey planner. Three papers most closely related to ours are Bassetto 1999, Shn 2006, and Wernng Lke us, those authors allow heterogenety and study dstrbutonal consequences of alternatve tax and borrowng polces. Bassetto 1999 extends the Lucas and Stokey 1983 envronment to nclude N types of agents wth heterogeneous tme-nvarant labor productvtes. There are complete markets. The Ramsey planner has access only to proportonal taxes on labor ncome and state-contngent borrowng. Bassetto studes how the Ramsey planner's vector of Pareto weghts nuences how he responds to government expendtures and other shocks by adjustng the proportonal labor 2 A dstnct strand of lterature focuses on optmal polcy n settngs wth heterogeneous agents when a government can mpose arbtrary taxes subject only to explct nformatonal constrants see Golosov et al for a revew. A strkng result from that lterature s that when agent's asset holdngs are perfectly observable, the dstrbuton of assets among agents s rrelevant and an optmal allocaton can be acheved purely through taxaton see, e.g. Bassetto and Kocherlakota In the prevous verson of the paper we showed that a mechansm desgn verson of the model wth unobservable assets generates some of the smlar predctons to the model wth ane taxes that we study, n partcular, the relevance of net assets and hstory dependence of taxes. We leave further analyss along ths drecton to the future. 5

6 tax and government borrowng to cover expenses whle manpulatng compettve equlbrum prces to redstrbute wealth between `renters' who have low productvtes and whose man ncome s from ther asset holdngs and `workers' who have hgh productvtes whose man ncome source s ther labor. Shn 2006 extends the AMSS Ayagar et al ncomplete markets economy to two rsk-averse households who face dosyncratc ncome rsk. When dosyncratc ncome rsk s bg enough relatve to government expendture rsk, the Ramsey planner chooses to ssue debt so that households can engage n precautonary savng, thereby overturnng the AMSS result that a Ramsey planner eventually sets taxes to zero and lves o ts earnngs from assets thereafter. Shn emphaszes that the government does ths at the cost of mposng tax dstortons. Constraned to use proportonal labor ncome taxes and nonnegatve transfers, Shn's Ramsey planner balances two competng self-nsurance motves: aggregate tax smoothng and ndvdual consumpton smoothng. Wernng 2007 studes a complete markets economy wth heterogeneous agents and transfers that are unrestrcted n sgn. He obtans counterparts to our Rcardan results about net versus gross asset postons, ncludng the legtmacy of a normalzaton allowng government assets to be set to zero n all perods. Because he allows unrestrcted taxaton of ntal assets, the ntal dstrbuton of assets plays no role. Our corollary 1 and corollary?? generalze Wernng's results by showng that all allocatons of assets among agents and the government that mply the same net asset poston lead to the same optmal allocaton, a concluson that holds for market structures beyond the complete markets structure analyzed by Wernng. Wernng 2007 provdes an extensve characterzaton of optmal allocatons and dstortons n complete market economes, whle we focus on precautonary savngs motves for prvate agents and the government that are absent when markets are complete. 3,4 Fnally, our numercal analyss n Secton 5 s related to McKay and Res Whle our focus ders from thers McKay and Res study the eect of a calbrated verson of the US tax and transfer system on stablzaton of output, whle we focus on optmal polcy n a smpler economy both papers conrm the mportance of transfers and redstrbuton over busness-cycle frequences. 2 Envronment We consder an nnte horzon economy n dscrete tme. There s a mass n of type I nntely lved agents wth I =1 n = 1. Preferences of an agent of type over stochastc processes for consumpton {c,t } t and labor supply {l,t } t are represented by E 0 t=0 β t U c,t, l,t, 1 where E t s a mathematcal expectatons operator condtoned on tme t nformaton, β 0, 1 s a tme dscount factor, c,t and l,t are consumpton and labor supply of type n perod t, and U s the per-perod utlty functon of type. 3 Wernng 2012 studes optmal taxaton wth ncomplete markets and explores condtons under whch optmal taxes depend only on the aggregate state. 4 More recent closely related papers are Azzmont et al. 2008a,b and Correa Whle these authors study optmal polcy n economes n whch agents are heterogeneous n sklls and ntal assets, they do not allow aggregate shocks. 6

7 Exogenous fundamentals nclude a stochastc cross secton of sklls {θ,t },t, government expendtures g t, and the payo p t on a sngle asset traded between the government and the prvate sector. These are all functons of a shock s t S governed by an rreducble Markov process that takes values n a nte set S. We let s t = s 0,..., s t denote a hstory of shocks havng jont probablty densty πs t. To smplfy exposton, we often use notaton z t to denote a random varable z wth a tme t condtonal dstrbuton that s a functon of the hstory s t. Occasonally, we use the more explct notaton z s t to denote a realzaton at a partcular hstory s t. An agent of type who supples l unts of labor n state s t produces θ s t l unts of output, where θ s t Θ s a non -negatve state-dependent scalar. Feasble allocatons satsfy I n c,t + g t = =1 I =1 The tme t payo p t on the traded asset s descrbed by p t = Ps t s t 1, n θ,t l,t. 2 where P s an S S matrx normalzed to satsfy E t p t+1 = 1. Specfyng the asset payo n ths way lets us nvestgate consequences of the correlaton between asset returns, on the one hand, and government expendtures or shocks to the skll dstrbuton, on the other hand. Purchasng ˇbt unts of the asset at a prce of q t unts of tme t consumpton per unt of the asset enttles the owner to p t+1ˇbt unts of tme t + 1 consumpton. Consequently, R t+1 = p t+1 /q t s the gross rate of return on the asset measured n unts of tme t + 1 consumpton good per unt of tme t consumpton good. We let b t q tˇbt denote a tme t value of ˇb t unts of the asset, measured n unts of tme t consumpton. From now on, we express budget sets n terms of the gross rate of return R t and counterparts of the value of assets b t. To facltate a uned notaton for budget constrants for dates t 0, we dene R 0 p 0 β 1. Households and the government begn wth assets {b, 1 } I =1 and B 1, respectvely. Asset holdngs satsfy the market clearng condton I =1 n b,t + B t = 0 for all t 1. 3 There s a proportonal labor tax rate τ t and common lump transfer T t. The tax bll of an agent wth wage earnngs l,t θ,t s T t + τ t θ,t l,t. A type agent's budget constrant at t 0 s and the government budget constrant s c,t + b,t = 1 τ t θ,t l,t + R t b,t 1 + T t, 4 g t + B t = τ t I =1 n θ,t l,t + R t B t 1 T t. 5 Denton 1. An allocaton s a sequence {c,t, l,t },t. An asset prole s a sequence {{b,t }, B t } t. A returns process s a sequence {R t } t. A tax polcy s a sequence {τ t, T t } t. 7

8 We mpose debt lmts on asset proles. In general they can depend on the hstory of exogenous shocks s t and the tax polcy of the government {τ t, T t } t and are descrbed by functons {b s t ; {τ t, T t } t } and B s t ; {τ t, T t } t. Ths allows both for natural debt lmts, whch correspond to the maxmum debt that an agent can repay almost surely, and for tghter, ad-hoc lmts. So for all t 1 we mpose, and b,t b s t ; {τ t, T t } t 6 B,t Bs t ; {τ t, T t } t. 7 Denton 2. For a gven ntal asset dstrbuton {b, 1 }, B 1, a compettve equlbrum wth ane taxes s a sequence {{c,t, l,t, b,t }, B t, R t } t and a tax polcy {τ t, T t } t such that {c,t, l,t, b,t },t maxmze 1 subject to 4 and 6; and constrants 2, 3, 5, and 7 are satsed. A Ramsey planner's preferences over compettve equlbrum allocatons are ordered by E 0 =1 I ω t=0 where the Pareto weghts satsfy ω 0, I =1 ω = 1. β t U t c,t, l,t, 8 Denton 3. Gven {b, 1 }, B 1, an optmal compettve equlbrum wth ane taxes s a compettve equlbrum wth an allocaton that maxmzes Relevant and rrelevant aspects of the asset dstrbuton Our economy features both dstortonary taxaton snce agent-specc lump sum taxes are not avalable and debt lmts. Despte these frctons, there s a sense n whch the value of the government debt by tself provdes lttle nformaton about welfare costs of servcng t. The mportant statstc s not the level of government debt per se, but who holds t. To formalze ths pont, we dene agents' net assets n perod t as agents' asset holdngs relatve to those of some benchmark agent, {b,t b 1,t } >1. The next proposton shows that t s the ntal dstrbuton of the net assets that determnes welfare n the optmal compettve equlbrum. Proposton 1. For any par of ntal dstrbutons satsfy the debt lmts and the values of 8 at the optmal allocatons are the same. {b, 1 }, B 1 and {b, 1 }, B 1 that b, 1 b 1, 1 = b b 1, 1 9 We relegate the proof to the appendx 7.1. The ntuton conrms the quote of Newcomb 1865 wth whch we began our paper. To see ths, magne an ncrease an ntal level of government debt from 0 to some arbtrary level B 1 < 0 when agents asset holdngs are equal across agents. The optmal way to nance ths ncrease n publc debt s to reduce transfers and keep dstortonary taxes {τ t } t unchanged. If asset holdngs were equal to begn wth, each unt of debt repayment acheves the same redstrbuton as one unt of transfers. The stuaton would be derent f, say, rcher people ntally own dsproportonately more government debt. A 8

9 government wth Pareto weghts {ω } that favor equalty would want to ncrease both transfers {T t } t and the dstortng labor tax rate {τ t } t to oset the ncrease n nequalty assocated wth the ncrease n government debt. 5 Proposton 1 holds for arbtrary debt lmts. In general, the absolute levels of government debt wll be determnate. As long as there are agents who face debt lmts that are bndng along the equlbrum path, changes n government debt can relax or tghten borrowng constrants and alter welfare. 6 As a way of contrast, n Corollary 1 we show that wth natural debt lmts, only net asset postons are meanngfully determned n any compettve equlbrum. Corollary 1. Gven {b, 1 }, B 1, let { {c,t, l,t, b,t }, B t, R t }t and {τ t, T t } t be a compettve equlbrum wth natural debt lmts. For bounded sequences there exst sequences {{c,t, l,t, ˆb,t }, ˆB } t, R t {ˆb, 1 }, ˆB 1. {ˆb,t },t 1 that satsfy ˆb,t ˆb 1,t = b,t b,t b 1,t for all t 1, 2, 10 t { ˆTt } and t { } and ˆBt that satsfy 3 and that make t 1 {τ t, ˆT } t consttute a compettve equlbrum gven t In some parts of the dscusson below we wll nd t useful to normalze assets of the least productve agents to zero. Wth such normalzaton the usual ntuton that hgher level of government debt leads to hgher dstortons and lower welfare s recovered, due to the fact that t eectvely corresponds to hgher asset nequalty. Ths normalzaton s helpful to relate our ndngs to studes of government debt n representatve agent models. 3 Quaslnear preferences In our general secton 2 settng, the endogenety of the return on the sngle asset and the multplcty I of types of agents makes t dcult to parse the forces that govern how transfers should be used to smooth tax rate dstortons and to redstrbute across people. In ths secton, we analyze a specal case that helps us solate key forces. We assume two types of agents who have one-perod utlty functons that are quas-lnear n consumpton. Ths makes the return on the asset exogenous and allows us to focus on how the comovement between returns and aggregate rsk aects how transfers are used to smooth tax dstortons and to redstrbute. Assumpton 1. Quaslnear preferences: uc, l = c l1+γ 1+γ. Assumpton 2. θ 1 > θ 2 = 0. Assumpton 3. IID shocks to expendture: gs t s..d over tme. 5 Ths logc mples that government debt that s wdely and evenly dstrbuted e.g., mplct Socal Securty debt s less dstortng than government debt owned mostly by people whose ncomes are at the top of the ncome dstrbuton e.g., government debt held by hedge funds. It s possble to extend our analyss to open economy wth foregn holdngs of domestc debt. The more government debt s owned by the foregners, the hgher are the dstortng taxes that the government needs to mpose. 6 Ths ntuton s smlar to that n Woodford The government can relax the ad-hoc debt lmts because of the mplct assumpton that t s easer to collect taxes than to enforce debt payments by ctzens. 9

10 Assumng one type of agent who cannot work makes redstrbuton between rch and poor transparent, and..d shocks help wth analytcal tractablty. Wthout loss of generalty we can restrct our attenton to payo matrces P that have dentcal rows, denoted by a vector P of dmenson S such that s S πsp s = 1. At nteror solutons, the rst-order condtons for the household's problem for l 1,t and b,t mply that l γ 1,t = 1 τ t θ, R t = p t β, where we set R 0 = β 1 p t. Substtutng these nto the budget constrant 4 gves c 1,t + b 1,t = l 1+γ 1,t + p t β b 1,t 1 + T t, t 0, 11a c 2,t + b 2,t = p t β b 2,t 1 + T t, t 0. 11b Subtract the budget constrant of the unproductve agent 11b from 11a to elmnate transfers T t and get c 1,t c 2,t + b 1,t b 2,t = l 1+γ 1,t + p t β b 1,t 1 b 2,t 1, t Equaton 12 presents a recursve verson of the mplementablty constrants approprate for the quaslnear settng. Note from equaton 12 that only relatve assets, b 1,t b 2,t matter, so we adopt a normalzaton where we set the assets b 2,t of the unproductve agent always to zero. Ths makes the asset market clearng condton mply that nb 1,t = B t and also mples that transfers T t = c 2,t. Next we make Assumpton 4. c 2,t 0. Requrng that c 2t 0 s an easy way to make transfers too costly to a Ramsey planner who cares too lttle about the unproductve agent. Snce the only ncome of the unproductve agents comes from transfers, the non-negatvty constrants ensures that such transfers are always weakly postve. We mpose bounds on the planner's asset postons B t. Gven our normalzaton b 2,t = 0, bounds on planner's asset are equvalently restrctons on net assets postons. Imposng these bounds turns out to be useful as they map to debt lmts n a representatve agent economy, a specal case of our quaslnear economy as we shall show below. Households are assumed to operate under debt lmts that are looser than those mpled by the bounds on the planner's assets. Let ω, n [0, 1] [0, 1] be the Pareto weght and mass assgned to the productve type 1 agent. We can explot the recursve nature of constrant 12 to wrte the recursve problem that characterzes the optmal polcy as a functon of the begnnng of the perod government assets B_. { V B_ = max πs ω {c 1 s,c 2 s,l 1 s,bs} s where the maxmzaton s subject to s c 1 s l 1s 1+γ 1 + γ c 1 s c 2 s n 1 Bs = l 1 s 1+γ n 1 β 1 P sb_, 10 } + 1 ωc 2 s + βv Bs 13 14a

11 nc 1 s + 1 nc 2 s + gs = nθl 1 s, 14b c 2 s 0, 14c B Bs B. 14d As we shall see n later sectons, the government has two tools to smooth aggregate shocks: use uctuatons n asset returns to obtan state-contngent payo or adjust taxes and transfers n response to a shock. Typcally t s optmal to use both nstruments smultaneously. In the quaslnear economy analyss smples because n the long run the planner only uses one of these tools. Whch tool s used depends on how much the government cares about the redstrbuton. We show that f government's concern for redstrbuton s sucently low,.e. the Pareto weght assgned to the productve agent s sucently hgh, then the government uses only uctuatons n assets returns to hedge the aggregate rsk. Conversely, wth sucently hgh concerns for redstrbuton the government eventually uses only uctuatons n transfers to smooth aggregate shocks. The threshold Pareto weght s dened by ω = n and the two regons exsts as long as n 1+γ γ Pareto weghts ω > ω 1+γ γ < 1. If t s not satsed, we only have the second regon. We start our analyss by consderng the problem of the government whch has a sucently low concern for redstrbuton. The key ntermedate step s the followng lemma Lemma 1. Suppose that n < γ 1+γ. If ω > ω then T t = 0 t 0. The proof s relegated to appendx 7.2. For Pareto weghts attached to the productve agent above ω, usng transfers s too costly for the planner because postve transfers subsdze the unproductve type 2 agent whose welfare the planner values too lttle. Ths makes the Ramsey outcomes dentcal to those for an economy n whch a Ramsey planner s restrcted to use a lnear tax schedule he cannot use transfers and n whch there s a representatve agent whose allocaton equals the allocaton to the productve agent n our economy. The characterzaton of ths case s of ndependent nterest as we can compare our results wth the lterature, Lucas and Stokey 1983 and Ayagar et al. 2002, both of whch studed a representatve agent economy wth lnear taxes. The key force at play here s how market ncompleteness, as captured by the structure of payo on the asset P, mpedes tax-smoothng n a jont ergodc dstrbuton for the tax rate and government debt. To organze analytcal results n ths secton, t s useful to collect some P vectors that are perfectly correlated wth expendture shocks g n the followng set { P = P : P s = 1 + β } B gs Eg sfor some B [B, B]. 15 Payos n the set P have a property that t s feasble for the government to perfectly hedge uctuatons n net-of nterest dects by holdng debt level B. Note that for each P P there s a unque B P that satses condtons of equaton 15. The relatonshp between any such par of P, B P can be wrtten as vargs B P = β covp s, gs. 16 Fnally, observe that the set of payos P s non-generc: t has a measure 0 n the space of possble asset payos. 11

12 Proposton 2. Suppose that n < γ 1+γ. If ω > ω, the value functon V B_ n equaton 13 s strctly concave and the behavor of government assets under a Ramsey plan s characterzed as follows: 1. If P P then government assets converge to a degenerate steady state lm t B t = B P a.s B 1. There s τ P such that lm t τ t = τ P a.s B If P s P there exsts an nvarant dstrbuton of government assets wth the property, ɛ > 0, Pr{B t < B + ɛ and B t > B ɛ.o} = 1. There s a τb such that the tax rate τ t = ˆτB t and ˆτ < 0. Moreover, f P s P s > β B [gs gs ] B 1, B 2 such that s, s and B B s sucently large, there exst EV Bs > V B_ for B_ > B 2, EV Bs < V B_ for B_ < B 1. The rst part of Proposton 2 shows that f there s any asset level whch allows the government to hedge ts rsk, t converges to that level almost surely. In that steady state the tax level s constant and the government uses uctuatons n asset return to nance stochastc expendtures. Ths dynamcs resembles that of a complete market economy a-la Lucas and Stokey 1983, except the long-run level of debt and taxes n pnned down by vector P n our ncomplete market economy, whle wth complete market economy t s pnned down by the ntal value of government assets. Expresson 16 tells us that whether the government eventually holds assets or owes debt s determned by the sgn of the covarance of P s wth gs. In partcular, the government accumulates postve negatve assets f returns are hgh low n the states when gs s hgh. As we dscussed above, the payos that allow the government to hedge ts rsks, are not generc. The second part of Proposton 2 shows that n general, when perfect hedgng s mpossble, the support of the nvarant dstrbuton s wde n the sense that almost all asset sequences recurrently revst small neghborhoods of any arbtrary lower and upper bounds on government assets. Wth ncomplete markets a long enough sequence of hgh or low shocks takes government assets arbtrarly far from any startng level of debt. Because the labor tax rate s decreasng n government assets, t vares too. These outcomes contrast sharply wth those n a correspondng complete market benchmark lke Lucas and Stokey's, where both debt and tax rates would be constant sequences, and wth those n the ncomplete markets economy of Ayagar et al., where government assets would approach levels that allow the lmtng tax rate to be zero and the tal allocaton to be rst-best. Fnally, the last part of the proposton shows that wth ncomplete markets government assets exhbt a form of mean-revson: f the government accumulated sucently hgh or low level of assets, future debt levels revert towards the mddle. 12

13 To acqure more nformaton about the nvarant dstrbuton of government assets when the payo vector P does not allow for perfect hedgng, we consder lnear approxmatons of the polcy rules. For any P consder the closest perfect hedgng payo P that solves mn P P Use ths P to nd a component ˆP of P that s orthogonal to g: s πs [P s P s] P s = ˆP s + P s. We lnearze polcy rules around ˆP = 0 and study the propertes of the ergodc dstrbuton generated by such rules. Proposton 3. The lnearzed polcy rules nduce a unque ergodc dstrbuton of government debt wth the followng propertes. Mean: The ergodc mean of asset dstrbuton, EB, satses EB = B P. Varance: The ergodc coecent of varaton of government assets B s σb EB = varp s covgs, P s 1 + covgs, P s covgs, P s var ˆP s varp s. Convergence rate: The rate of convergence to of the mean to ts ergodc value s descrbed by, E t 1 B t B = varp corr 2 P, g B t 1 B, where corrp, g s the correlaton coecent between P and g We relegate the proof to appendx 7.4, where we descrbe how we take a rst-order Taylor approxmaton to the decson rules and laws of moton for the state varables of our economy around complete market counterparts assocated wth P P. Proposton 3 descrbes how devatons from P map nto larger varances for government debt and the tax rate under the ergodc dstrbuton. The last part of Proposton 3 decomposes the the speed of convergence of assets B t nto two components: varance of P s and the correlaton between P s and gs. If P were to belong to P, the second term relatng to correlaton would be equal to one, and ths bounds the exponental rate of convergence of the condtonal mean for payo P s close to P by varance of P s. Fgure 1 llustrates how the ergodc dstrbuton of government debt and the tax rate spread as we exogenously alter the covarance of P s wth gs. 7 7 For the purposes of these graphs we used the global approxmaton to the polcy rules usng standard projecton methods rather than the lnear approxmatons appealed to n Proposton 3. The ergodc dstrbuton assocated wth the approxmate lnear polces s smlar. 13

14 Fgure 1: Ergodc dstrbuton for government assets B t and the labor tax rate τ t n the representatve agent quaslnear economy for three derent asset payo vectors P. Pareto weghts ω ω The prevous subsecton descrbed the optmal plan n the case where the Pareto weght on the productve type 1 agent was so hgh that the Ramsey planner chose not to use transfers to hedge aggregate shocks. Proposton 4 now completes the characterzaton by descrbng the case when the Ramsey planner has larger redstrbutve concerns and the Pareto weght attached to the productve agent are lower. Ths nduces the planner to use transfers to hedge shocks. Proposton 4. For ω < ω, and mn s {P s} > β, there exst a Bω satsfyng B ω > 0 and the optmal tax rate, transfer, and government asset polces {τ t, T t, B t } t are characterzed as follows: 1. If B 1 > B ω then 2. If B 1 B ω then T t > 0, τ t = τ ω, and B t = B 1 t 0. a If P P or P P and B P > Bω b Otherwse, T t > 0.o., lm t τ t = τ ω and lm t B t = Bω a.s. Pr{lm t T t = 0, lm t τ t = τ, lm t B t = B P or T t > 0.o. and lm t τ t = τ ω, lm t B t = Bω} = 1. 14

15 Greater concerns for redstrbuton lower costs of transfers and ths expands the planner's hedgng possbltes aganst aggregate shocks. Proposton 4 asserts that the ergodc dstrbuton s degenerate and long run level of assets n general depends on both the ntal assets and how much the planner values the welfare of the unproductve agent. If the government begns wth enough assets as n part 1 of Proposton 4, the planner chooses an nteror allocaton n whch all uctuatons n net-of nterest dects are always nanced by uctuatng transfers T t. Here government assets reman at ther ntal level and the tax rate s constant at a level that s ndependent of the ntal assets. Part 2 descrbes a settng n whch government assets are ntally too low to valdate the part 1 outcome. Part 2a shows that f perfect spannng s not feasble, then the planner eventually accumulates government assets untl they reach the threshold Bω at whch the welfare costs of transfers are low enough to nduce the planner to keep the tax rate constant. 8 Pareto weghts that ndcate that the planner cares more about the unproductve agent lower the margnal welfare costs of collectng revenue from labor taxes pad by the productve agent. Ths nduces the planner to ncrease the labor tax rate, lowerng the threshold level of assets that are requred to nance all shocks by transfers. In ths way, a more redstrbutve planner reles more on transfers and consequently s less motvated to accumulate assets to hedge aggregate shocks. Part 2b shows that f P P, then assets converge ether to the bound Bω or to the perfect spannng asset level B P. 4 Optmal allocaton wth rsk averson In ths secton we extend our analyss to economes wth rsk averson. The man derence from the analyss n the prevous secton s that the return Rs depends both on the structure of asset payos and on the margnal utltes of consumpton of agents. The latter term n general, depends on the realzaton of shocks and government polcy. We begn our analyss wth a par of Bellman equatons that formulate the Ramsey plan recursvely for the general settngs wth I > 2 agents and one-perod utlty functons that are concave n consumpton. We assume that U : R 2 R s concave n c, l and twce contnuously derentable. We let Ux,t or Uxy,t denote rst and second dervatves of U wth respect to x, y {c, l} n perod t. Fnally, we mpose natural debt lmts, whch ensures that all rst order condtons are nteror. In ths secton, we restrct attenton to nteror solutons and standard steps see e.g. Char et al ensure that { } {c,t, l,t, b,t }, R t, τ t, T t t s a compettve equlbrum allocaton f and only f they satsfy the constrants, c,t + b,t = U l,t Uc,t l,t + T t + 1 τ t θ,t U c,t = U l,t, 18 U c,t = βe t R t+1 U c,t+1, 19 p tu c,t 1 βe t 1 p t U c,t b,t 1 1, t 1, 20 8 The planner's choce under the condtons of part 2 to accumulate enough government assets so that eventually the government can use earnngs on ts assets together wth uctuatng postve transfers to hedge scal shocks s remnscent of outcomes n the Ayagar et al economy. There, wth a representatve agent and non-negatvty constrants on transfers, the planner accumulated enough assets to nance shocks wth zero dstortonary labor taxes whle costlessly usng transfers to dspose of any excess earnngs on ts asset holdngs. Wth multple agents, uctuatng transfers may brng welfare costs that depend on the Ramsey's planner atttude about redstrbuton. 15

16 c,0 + b,0 = U l,0 Uc,0 l,0 + T 0 + p 0 β 1 b, 1 1, 21 together wth feasblty 2. We can smplfy 20 and 21 by substtutng for T t and dervng the mplementablty constrants n terms of net asset postons as n of equaton 12: c,t c I,t + b,t 22 = U l,t Uc,t l,t + U l,t I Uc,t I l I,t + p tuc,t 1 βe t 1 p t Uc,t b,t 1 for < I and t 1. We are now ready to wrte the problem recursvely. Let x = Uc I b 1,..., Uc I b I 1, ρ = U 1 c /Uc I,..., Uc I 1 /Uc I, and R s s_ P s s_ U I c s E s_p U. Followng Kydland and Prescott C I 1980 and Farh 2010, we splt the Ramsey problem nto a tme-0 problem that takes { b, 1 } I 1 =1, s 0 as state varables and a tme t 1 contnuaton problem that takes x, ρ, s_ as state varables. For t 1, let V x, ρ, s_ be the planner's contnuaton value gven x t 1 = x, ρ t 1 = ρ, s t 1 = s_. Let as = {c s, l s} denote the allocaton, the optmal allocaton solves: V x, ρ, s_ = max {as,x s,ρ s} where the maxmzaton s subject to [ ] πs s_ ω U s + βv x s, ρ s, s s U Uc I s [c s c I s] + l sl s Ucs Uc I s l I sul I s + x s = x Rs s_ β E s_ P Uc E s_ P Uc I for all s, < I 23 24a = ρ for all < I 24b Ul s θ sucs = U l Is for all θ I suc I s, < I s 24c n c s + gs = n θ sl s s ρ s = U cs U I c s for all s, < I 24e 24d Constrants 24b and 24e mply 19. The denton of x t and constrants 24a together mply equaton 22 scaled by U I c. Perod 0 maxmzaton problem s smlar, except that t does not have constrants 24b. For completeness we provde the tme t = 0 Bellman equaton n appendx A formula for optmal taxes We summarze the man tradeo that the planner faces n settng labor taxes and transfers n the followng proposton. Let {µ s} <I, ξs be the multplers on constrants 24a and 24d. Proposton 5. Suppose preferences are separable and have a constant Frsch elastcty, U c, l = U c l1+γ 1+γ. Then the optmal labor tax rate can be expressed as, 16

17 1 1 τs = wsȳs + cov w s, y s ξsȳs, 25 where w s = 1 + γˆµ s + γ [ n 1 ω Ucs ], ws = n w s, ȳs = n y s and ˆµ s = Uc I sµ s n 1 ω Ucs <I for < I and ˆµ I s = ˆµ sn n I n 1 I ω I Uc I s. The weghts {w } I =1, dened n ths Proposton, show how the socal planner evaluates uctuatons n nequalty n ths economy. It conssts of Pareto-weght adjusted margnal utlty of consumpton of the agents adjusted to the cost of rasng revenues, represented by the Lagrange multpler {ˆµ } I =1. 9 Ths equaton shows the man trade-o that the socal planner faces wth heterogeneous agents. The planner can respond to an aggregate shock ether by adjustng transfers, whch leads to uctuaton n equalty represented by { ω Uc} I, or by changng taxes, the deadweght =1 loss of whch s represented by {ˆµ } I =1. In the optmum the planner equlzes dstortons from the two polces. All thngs beng equal, labor taxes are hgh f nequalty n consumpton s hgh or the deadweght loss of taxaton s low. We dscuss two scenaros where thngs smplfy and we can nterpret these weghts more tghtly. The rst case s when preferences are quaslnear and our formula 25 reduces to 1 1 τs = 1 covn 1 ω, y s γ 1 ȳs. 26 Snce margnal utltes are constant, planner's concerns for equty are captured by the covarance between exogenous Pareto weghts and pre-tax ncome. The covarance s negatve when Pareto weghts are hgher on agents wth low pre-tax labor ncome and ths calls for hgher labor taxes. The covarance and the optmal taxes vary across states f the dstrbuton of sklls {θ s} changes wth aggregate shocks, otherwse we have constant tax rates. 10 Formula 25 s closely related to classcal results on optmal commodty taxaton and redstrbuton derved Damond 1975 and Atknson and Stgltz These papers studed optmal commodty taxes n statc settngs and derved formulae for optmal taxes smlar expressed n terms of average elastctes on one hand and cross sectonal covarances that captured equty consderatons on the other. In absence of savngs, our problem 23 s a specal case of ther settng and formula 25 reduces to where σ = U cc sc s U cs 1 covn 1 1 τ = [1 T s c s net of transfers for agent, and ε = 1 σ σ +γ ω U c, y ȳ1 + γγ 1 ξ 1 ξ ε, 27 ] as the elastcty of substtuton wth respect to consumpton n y. 9 In partcular ˆµ s s agent 's contrbuton to the planner's total cost of reducng agent asset holdngs and smultaneously ncreasng transfers by one unt. 10 In secton 3, we mposed non-negatvty constrants on the consumpton of the unproductve agent as a way of modelng costs of transfers that typcally come from spreadng of margnal utltes for more general preferences n adverse aggregate shocks. As summarzed by Theorems 2-4, these can vary across states even f the skll dstrbuton s unchanged. 17

18 4.2 Eventual complete hedgng wth bnary shocks In secton 3 wth quaslnear preferences, we studed scenaros where optmal polcy perfectly hedged aggregate shocks. Although the condtons requred were restrctve, speccally that the payo vectors had to be perfectly correlated wth expendture shocks, these economes served as useful benchmark and a pont of approxmaton to study optmal polcy n more general cases where perfect hedgng was not feasble for any asset levels. In ths secton we extensons of those nsghts to the economy wth rsk-averson. Wth rsk averson, perfect spannng corresponds to an allocaton such that the adjusted U l s U c sui c s l I su I l s for all payos Rs s_ oset movements n Uc I s [c s c I s] + agents keepng xs = x and at the same tme provde full nsurance.e ρs = ρ. For a gven state x, ρ, s_, let Ψ s; x, ρ, s_ = x s, ρ s solve the value functons 23 so that Ψ s; x, ρ, s_ s the law of moton for the state varables under a Ramsey plan at t 1. Denton 1. A steady state satses x SS, ρ SS = Ψ s; x SS, ρ SS, s for all s, s_. At such steady states, outcomes resemble those n the complete market economy of Wernng The tax rate and transfers both depend only on the current realzaton of shock s t. 11 Arguments of Wernng can be adapted to show that the tax rate s constant when preferences have the CES form c 1 σ /1 σ l 1+γ /1 γ and relatve sklls are constant across aggregate shocks. We dscuss the condtons for exstence of steady states and then study a smple example that wll allow us to dentfy forces also present n the quaslnear economy of secton 3 and the more general economes to be analyzed wth numercal methods n secton 5. Lemma 2. When utlty s strctly concave n consumpton, S = 2 s necessary for a steady state to exst genercally. The exstence of steady state depends on the soluton to a non-lnear system, whch can be vered only numercally. Appendx 7.8 dscusses the structure of these equatons n more detal. Here we present an example wth rsk averse agents where exstence and comparatve statcs can be establshed analytcally. Our example s an economy consstng of two types of households wth θ 1 > θ 2 = 0 and common one-perod utltes ln c 1 2 l2. The shock s takes two values {s L, s H } that are..d across tme. We assume that g s H > g s L. Wth natural debt lmts, Corollary 1 apples and we are free to normalze the assets of the unproductve agent to zero and nterpret xs = U I c sbs as the margnal utlty adjusted government's assets. In ths example we show, Proposton 6. Suppose that gs < θ 1 for all s and let Rs x, ρ P su c 2s x,ρ be the adjusted EP Uc 2 x,ρ payo vector. If P s = 1 for all s, then there exsts a steady state x, ρ wth x > 0 and R s L x, ρ < R s H x, ρ. In addton, there exsts a payo vector P such that x < 0 and R s L x, ρ > R s H x, ρ. Proposton 6 establshes the exstence of steady states and solates how the comovement of the return on the asset wth expendture shocks predcts the sgn of government assets. 12 In the quaslnear case we could dsentangle the consderatons for spannng and redstrbuton 11 However note that the dstrbuton of assets n the steady state for us s endogenous, t only depends on the technology and preference parameters, as aganst n Wernng 2007 where t s pnned down by the ntal dstrbuton of assets. 12 Ths parallels the quaslnear case, see expresson 16 18

19 explctly. Wth rsk averson, costs of transfers come from uctuatons n margnal utltes and are nterlnked wth uctuatons n endogenous returns. A decrease n transfers n states wth hgh expendture dsproportonately aects the low-sklled agent, so hs margnal utlty falls by more than the margnal utlty of a hgh-sklled agent. Choosng polces such that the net assets of the hgh-sklled agent decrease over tme makes the two agents' after-tax and after-nterest ncome become closer, allowng decreases n transfers to have smaller eects on nequalty n margnal utltes. Ths force pushes the government to accumulate assets n the long run. On the other hand, f adjusted payos are sucently low n states wth hgh expendtures, holdng negatve assets or debt s optmal from hedgng perspectve; as dong so lowers the nterest rate burden. The sgn of long run assets ultmately depends on the balance of these forces. 5 Quanttatve nvestgaton In sectons 3 and 4, we studed steady states as a way of summarzng the asymptotc behavor of Ramsey allocatons n partcular dstrbuton of assets and taxes. In ths secton, we use a calbrated verson of the economy a to revst the magntude of these forces; and b to study optmal polcy responses at busness cycle frequences when the economy s possbly far away from a stochastc steady state. We calbrate parameters and shock process to reect stylzed facts about dynamcs of U.S. dstrbuton of earnngs, assets, and returns on government assets for the perod We match those facts n a compettve equlbrum wth an arbtrary government polcy chosen to reect the U.S. tax-debt polcy for the same perod. We then solve for the optmal Ramsey allocaton for the same ntal dstrbuton of relatve assets wth Pareto weghts chosen to match the average tax level n the compettve equlbrum. We can then compare outcomes predcted under the optmal polcy to actual U.S. data. The numercal calculatons use methods adapted from Evans 2014 and descrbed n the appendx Calbraton To facltate calbraton, we construct a compettve equlbrum wth ncomplete asset markets usng an arbtrary tax-debt polcy that ts US tax, debt and expendture data for the perod We set the tax and debt polcy to follow log B t = 1 ρ B log B + ρ B,B log B t 1 + ρ B,Y log Y t + σ B ɛ B,t, τ t = τ + ρ τ,y log Y t + σ τ ɛ τ,t. To estmate the parameters that descrbe these rules we use HP ltered annual data on debt, aggregate labor earnngs, and the tme seres for average margnal federal tax rates from Barro and Redlck Government expendtures are exogenous and follow log g t = log ḡ + σ g ɛ g,t. Transfers are obtaned as a resdual from the government budget constrant. Table 1 summarzes the estmates. We use I = 9 so that our agents represent 10th to 90th quantles of workng US households labor earnngs dstrbuton. To calbrate the wealth nequalty, we use Survey of Consumer Fnances [ ] to compute average net assets by earnngs quantles. We set ntal assets 19

20 such that the relatve assets match the SCF and they add up to the total federal debt held by the publc. We assume that all households have soelastc preferences uc, l = c1 σ 1 σ l1+γ. We 1+γ set σ = 2.5 and choose γ = 2 to target a Frsch elastcty of labor supply of 0.5. We set the tme dscount factor β = 0.98, whch mples the annual nterest rate n an economy wthout shocks would be 2% per year. Our theoretcal analyss emphaszed two forces that determne cyclcalty of the optmal polcy: the correlaton of ncome nequalty and asset returns wth aggregate shocks. To parsmonously capture the behavor of labor ncome nequalty, we assume that agents productvty follows a two-parameter famly of stochastc processes log θ,t = log θ + ɛ t [1 +.9 Qm], 28 where Q s the percentle of agent. In ths speccaton ɛ t s a common aggregate productvty shock, and parameter m governs relatve volatlty of wages of agents n derent quantles of labor ncome dstrbuton. For example, when m = 1./0.8 > 0 the percentage declne n productvty n a recesson for agents n the 10th percentle s twce that of the 90th percentle. Aggregate productvty shock followsan AR1 process ε t = ρ ε ε t 1 + σ ε ɛ θ,t. As n the Ramsey taxaton problem, the agents trade a sngle asset wth exogenous payos p t. We assume p t = 1 + χɛ t, where χ captures the ex-post comovement n returns on government assets and aggregate shocks. The parameters θ, m, ρ ɛ, σ ɛ and χ are calbrated so the a compettve equlbrum matches the followng moments for the U.S. economy n the perod. We set { θ } such that the dsperson earnngs n the compettve equlbrum matches the average dsperson per quantle as reported by Guvenen et al To calbrate m use use the ndng n Guvenen et al that n the past four recessons, the average fall n ncome for agents n the rst decle of earnngs was on an average three tmes that experenced by the 90th percentle. Furthermore, between the 10th and the 90th percentles, the change n the percentage drop n earnngs was almost lnear. From these facts we nfer a slope m = The persstence and standard devaton of the productvty shock, ρ ɛ, σ ɛ are set to match auto-correlaton and the standard devaton of aggregate labor earnngs. Fnally the real return on 3 month treasury 13 These authors use recently avalable hgh qualty admnstratve data to document labor ncome loss across for derent percentles of earnng dstrbuton n the last four US recessons These nclude , , , recessons. Parameter Value Parameter Value B 0.6 τ 0.24 ρ B,B, ρ B,Y 0.87, ρ τ,y 0.09 σ B σ τ Table 1: Estmates for US tax-debt polces 20