Public Debt in Economies with. Heterogeneous Agents

Transcription

1 Publc Debt n Economes wth Heterogeneous Agents Anmol Bhandar Unversty of Mnnesota Mkhal Golosov Unversty of Chcago Davd Evans Unversty of Oregon Thomas J. Sargent New York Unversty September 6, 2017 Abstract We study publc debt n compettve equlbra n whch a government chooses transfers and taxes optmally and n addton decdes how thoroughly to enforce debt contracts. If the government enforces perfectly, asset nequalty s determned n an optmum compettve equlbrum but the level of government debt s not. Welfare ncreases f prvate debt contracts are not enforced. Borrowng frctons let the government gather monopoly rents that come from ssung publc debt wthout facng competng prvate borrowers. Regardless of whether the government chooses to enforce prvate debt contracts, the level of ntal government debt does not affect an optmal allocaton. Ths paper extends and supersedes results from a prevously crculated paper wth ttle Taxes, Debt and Redstrbutons wth Aggregate Shocks NBER WP no We thank the edtor and an anonymous referee for helpful crtcsms. We thank Marco Bassetto, Drk Nepelt, and Perre Yared for frutful dscussons. Correspondng author. Emal address: bhandar@umn.edu, phone: , malng address: Hanson Hall 1925 Fourth Street South,Mnneapols, MN 55455

2 Key words: Dstortng tax. Transfers. Government debt. Rcardan equvalence. 1 Introducton 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.... f t were 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) Understandng whether a government s debt s too hgh or too low requres knowng who owes what, when, to whom. That mpels studyng balance sheets of both credtors and debtors as well as the budget sets that appear n a coherent economc model and leads to dstngushng superfcal from substantve features by trackng and properly consoldatng assets and labltes. We seek features of government debt that affect contnuaton allocatons and prces. For that purpose, ths paper studes an economy wth people who dffer n ther productvtes and a government that admnsters a non-lnear tax on labor earnngs. Agents and the government trade one-perod bonds. There s no captal. The economy starts wth an exogenously gven dstrbuton of debt across agents and the government. Taxes are restrcted by agents abltes to pay. Publc polces are chosen at tme 0.e., the government commts. The structure of budget constrants mples that the cross-secton dstrbuton of ntal net assets, not gross assets, affects the set of feasble allocatons that can be mplemented n compettve equlbra. An ncrease n ntal government debt that s shared equally among all agents leaves the dstrbuton of net assets unchanged and therefore also leaves an equlbrum allocaton unaltered. Ths outcome embodes deas proclamed by Smon Newcomb (1865). The same logc apples to models wth and wthout physcal captal, wth complete or ncomplete asset markets, and wth more general tax structures. On the other hand, f the government cares about redstrbuton, an ncrease n ntal government debt 1

3 that s held mostly by agents wth hgh labor earnngs decreases welfare. In our settng, the correlaton of ntal debt holdngs and labor earnngs affects the welfare cost of publc debt. The role of government debt depends on how well prvate debt contracts are enforced. If both tax and debt oblgatons are enforced perfectly, then agents abltes to borrow are restrcted only by ther abltes to repay ther debts and an optmal level of government debt s ndetermnate. In ths case, any sequence of government debts s optmal and a verson of Rcardan equvalence holds even though taxes dstort prvate agents decsons. Nevertheless, t turns out that n ths case the dynamcs of asset nequalty share qualtatve features wth the dynamcs of publc debt n representatve agent models that exogenously rule out transfers. We also show that welfare under an optmal polcy ncreases f the government commts not to enforce prvate debt contracts n ways that produce the outcome that agents can borrow only up to an exogenous ad hoc debt lmt. In ths case, government debt provdes an addtonal nstrument to affect equlbrum allocatons. Welfare gans under an optmal polcy come from monopoly power on the asset market that the government acqures by restrctng the ablty of prvate agents to provde lqudty. What matters for ths result s not the sze of the debt lmt per se, but the agents nabltes to use antcpated transfers to relax current borrowng constrants. An optmal government debt s determned by a trade-off between gans from explotng monopoly rents and costs from dstortng agents ntertemporal margnal rates of substtuton. A szable lterature about government debt and Rcardan equvalence goes back at least to Barro (1974). It s well understood that n representatve agent economes the role of government debt hnges on whether lump sum taxes are allowed. But there s no nherent economc reason to rule out lump sum taxes n those models. Furthermore, the proportonal labor taxes often assumed n representatve agent models do not approxmate data well because transfers are such a large part of modern tax systems (see, e.g., Fgure 1). In our model, agents are heterogeneous, taxes are restrcted by agents resources, and the 2

4 government chooses taxes to maxmze a weghted average of agents lfetme utltes. Wernng (2007) obtaned counterparts to our results about net versus gross asset postons n a complete markets economy wth heterogeneous agents, an affne tax structure, and transfers that are unrestrcted n sgn. Because he allowed unrestrcted taxaton of ntal assets, the ntal dstrbuton of assets played no role n the model. Our lemma 2 and ts corollares extend Wernng s results by showng that all dstrbutons of gross assets among prvate agents and the government that mply the same net asset postons lead to the same equlbrum allocaton, a concluson that holds beyond complete markets. Whle Wernng (2007) characterzed optmal allocatons and dstortons n complete market economes, Wernng (2012) nvestgated how precautonary savngs motves that ncomplete markets mpart both to prvate agents and to a benevolent government affect optmal allocatons. 1 Our results on desrablty of weak enforcement of prvate debt contracts buld on nsghts of Yared (2012, 2013), who showed that t may be optmal not to undo agents borrowng frctons even when a government can undo them. Bassetto (2014) studed the roles of taxaton and debt lmts n heterogeneous agent economes n whch transfers are ruled out. Broner et al. (2010) and Broner and Ventura (2011, 2016) explored ncentves of the government to enforce prvate debt contracts n the context of nternatonal fnance. Ths paper s organzed as follows. In Secton 2, we descrbe a baselne envronment n whch taxes are restrcted to be affne functons of labor earnngs and agents are heterogeneous n labor earnngs but face no dosyncratc uncertanty. In Secton 3, we study an economy n whch agents abltes to borrow are restrcted only by ther abltes to pay. In Secton 4, we study an economy n whch agents face more strngent borrowng constrants. We show that our results extend to rcher tax systems constraned only by nformatonal frctons n Secton 5. 1 Other recent pertnent papers nclude Azzmont et al. (2008a,b) and Correa (2010). These papers study optmal polces n economes wth agents heterogeneous n sklls and ntal assets. 3

5 2 Envronment Tme s dscrete and nfnte. There are I types of agents each of mass n for {1, 2,..., I} 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 ordered 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 and β a dscount factor. We assume that U s ncreasng and concave n (c, l). The labor supply of agent les n a set [ 0, L ]. We allow L to be nfnte. Uncertanty s summarzed by a shock s t governed by an rreducble Markov process that takes values n a fnte set S. We let s t = (s 0,..., s t ) denote a hstory of shocks havng jont probablty densty π t (s t ). A boldface letter x denotes a sequence {x t (s t )} t 0,s t. We wrte s t s t for t > t f the frst t elements of s t consttute s t. When t does not cause confuson, we use x t to denote a random varable that depends on s t. Fnally, we defne a set of nfnte hstores S such that s S satsfes π t (s t ) > 0 for all s t s. Shock s t affects government expendtures g t (s t ) and ndvduals productvtes {θ,t (s t )}. An agent of type who supples l unts of labor produces y θ (s t ) l unts of output. Feasble allocatons satsfy I I n c,t + g t = n θ,t l,t. (2) =1 =1 Agents trade rskless one-perod zero coupon bonds wth each other and the government. At date t, hstory s t the prce s denoted by q t (s t ). Let the cumulaton of past prces at t, s t be Q t (s t ) k t,s k s q t k (s k ). We denote asset holdngs of agents and the government n perod t by {b,t } and B t, respectvely. We use a conventon that negatve values denote net ndebtedness of the agent or of the government. Agents and the government begn wth 4

6 assets {b, 1 } I =1 and B 1, respectvely. Asset holdngs satsfy market clearng condtons I n b,t + B t = 0 for all t 1. (3) =1 In each perod, the government collects T t (y,t ) from agent, where y t, = θ,t l,t. To be comparable to the lterature, we assume throughout most of ths secton that T t s an affne functon T t (y t ) = T t + τ t y t. (4) Affne tax functons approxmate actual tax and transfer programs pretty well; see Fgure 1, adapted from Heathcote et al. (2017 forthcomng). [Fgure 1 about here.] As wll be ndcated n our proofs, our results extend to more general non-lnear ncome tax schedules T t (y t ) and to even rcher tax systems. We dscuss these later. Wth affne taxes, the government budget constrant s g t + q t B t = τ t I n θ,t l,t + B t 1 T t. (5) =1 A government s preferences over stochastc process for consumpton and work are ordered by E 0 I n ω =1 t=0 β t U t (c,t, l,t ), (6) where ω 0, I =1 ω = 1 s a set of Pareto weghts. A type agent s budget constrant at t 0 s c,t + q t b,t = (1 τ t ) θ,t l,t + b,t 1 + T t. (7) In compettve equlbrum, agent maxmzes utlty (1) by choosng sequences (c, l, b ) that satsfy budget constrants (7). Wthout restrctons on debt holdngs, ths problem s 5

7 ll-posed because t allows agents to acheve nfnte utlty by runnng Ponz schemes. To rule out explosve debt paths, we restrct sequences b to be bounded from below. Later we consder more strngent constrants on prvate debt. In the sprt of Lucas and Stokey (1983), we study government polces (τ, T, B) that maxmze welfare crteron (6) n a compettve equlbrum, gven an ntal dstrbuton of assets ( ) {b, 1 }, B 1. We want to answer two questons: (a) how does the level of the ntal government debt B 1 affect welfare n an optmal equlbrum and (b) what determnes propertes of an optmal path of government debt B. The frst queston s about legacy costs of past debt. The second queston s about whether an optmal level of government debt exsts and, f t does, how quckly the government should converge to t. That agents are heterogeneous affects our answers. In a representatve agent economy, the answers to these questons depend on whether lump-sum taxes are avalable (see Barro (1974)). If agents really are dentcal, there s lttle reason to rule out lump sum taxes. Authors of representatve agent models typcally justfy rulng out lump-sum taxes by explctly or mplctly alludng to unmodeled heterogenety n the form of the presence of a subset of poor agents who cannot afford to pay lump-sum taxes. We model the presence of such poor agents explctly and are able to study an optmal tax polcy whle allowng lump-sum taxes and transfers. Because our transfers T are anonymous, the budget constrants of the poorest agents restrct the sgn and magntude of lump-sum taxes or transfers. Answers to our two questons depend partly on borrowng constrants. We nterpret such constrants as arsng from the nablty or dsnclnaton of a government to punsh agents who default on ther oblgatons. As a benchmark, we start wth the loosest borrowng lmts: the so-called natural borrowng lmts that allow agents to borrow any amounts that are feasble for them to repay n all future states. We nterpret these lmts as ndcatng the presence of a government that s wllng and able to mpose the harshest punshments on agents who default. We then dscuss mplcatons of strcter lmts on prvate borrowng. 6

8 3 Optmal debt under natural debt lmts We begn wth defntons. Defnton 1. An allocaton s a sequence {c, l }. An asset profle s a sequence ({b }, B). A prce process s a sequence q. A tax polcy s a sequence (τ, T ). Defnton 2. A compettve equlbrum wth natural debt lmts gven ntal assets ( {b, 1 }, B 1 ) s a ({c, l, b }, B, q, τ, T ) such that () (c, l, b ) maxmze (1) subject to (7) and b s bounded below for all ; () constrants (2), (3), and (5) are satsfed. Defnton 3. An optmal compettve equlbrum wth natural debt lmts gven ntal asset ( ) {b, 1 }, B 1 s a compettve equlbrum wth natural debt lmts that maxmzes (6). A dscusson of our termnology s useful. Ayagar (1994) popularzed the natural debt lmt termnology. He consdered an economy wth fnte after tax endowments and utlty functons defned over non-negatve consumpton. When the equlbrum nterest rate 1 q t s strctly greater than one 2 Ayagar requred that an agent s debt not exceed the present value of hs maxmum after-tax ncome at a worst shock sequence. 3 Formally, the maxmum ncome of agent n state s t s Y,t (s t ) max { (1 τ t (s t )) θ,t (s t ) L, 0 } + T t (s t ) and the present value of hs maxmum ncome at a worst shock sequence s D t ( Y ; s t) Q k 1 nf s S :s t s k>t,s k s ( s k 1 ) Q t (s t ) Y,k (s k ). (8) The natural debt lmt requres that f agents s consumpton s bounded below and Q s a summable sequence, then agent s assets are constraned by b,t (s t ) D t ( Y ; s t) for all t, s t. (9) 2 Ths condton can be relaxed to requre that Q s a strctly postve and summable sequence, meanng that t,s Q t(s t ) exsts. t 3 When the gross nterest rate s less than one so that the present value of ncome s nfnte he mposed an explct lower bound on debt. 7

9 The followng lemma ndcates that our defnton of compettve equlbrum extends Ayagar s noton of borrowng constrants to stuatons n whch hs defnton of a natural debt lmt s ll-posed. Lemma 1. Suppose that U s defned only for c 0, Y s bounded above and bounded below away from zero, and Q s a strctly postve summable sequence. Then b satsfes the natural debt lmt f and only f b s bounded below. Proof. See onlne appendx 1 Our defnton of an optmal compettve equlbrum allows the government to optmze over taxes and transfers (τ, T ). Snce compettve equlbra are well defned only over (τ, T ) systems under whch all consumers can afford to pay (.e., for whch each consumer s budget set s nonempty), ths defnton endogenously mposes restrctons on admssble tax polces. We start wth an mportant result. Lemma 2. Gven ( ) {b, 1 }, B 1, let ({c, l, b }, B, q, τ, T ) be a compettve equlbrum } } wth natural debt lmts. For any bounded sequences {ˆb and {ˆb, 1 that satsfy ˆb,t ˆb I,t = b,t b I,t for all t 1, [1, 2,..., I 1] (10) ( ) there exst sequences ˆT, ˆB wth natural debt lmts gven } Proof. For any bounded {ˆb } such that ({c, l, ˆb ({ˆb, 1 }, ˆB 1 )., ˆB, q, τ, ˆT ) s a compettve equlbrum let t ˆb I,t b I,t for all t 1. Defne, for all t 1, ˆT t = T t + q t t t 1, ˆBt = B t + t. (11) The sequence ({c, l, ˆb }, ˆB, q, τ, ˆT ) satsfes (2), (3), and (5), so t remans only to show that (c, l, ˆb ) ( ) s the optmal choce gven q, τ, ˆT. Observe that (c, l, ˆb ) satsfes budget 8

10 constrant c,t = (1 τ t ) θ,t l,t + b,t 1 q t b,t + T t = (1 τ t ) θ,t l,t + (b,t 1 b I,t 1 ) q t (b,t b I,t ) + T t + b I,t 1 q t b I,t = (1 τ t ) θ,t l,t + (ˆb,t 1 ˆb ) I,t 1 q t (ˆb,t ˆb ) I,t + T t + b I,t 1 q t b I,t = (1 τ t ) θ,t l,t + ˆb,t 1 q tˆb,t + ˆT t. ) Suppose that (c, l, ˆb s not an optmal choce for consumer, n the sense that there exsts ( ) some other sequence (c, l, b ) that provdes consumer hgher utlty gven q, τ, ˆT. The sequence (c, l, b ) satsfes (7) and (9) gven (q, τ, T ) and provdes strctly hgher utlty than (c, l, b ). Therefore, (c, l, b ) cannot be a part of a compettve equlbrum ({c, l, b }, B, q, τ, T ), a contradcton. We answer the two questons posed n Secton 2 wth two propostons that follow from Lemma 2. Proposton 1. For any par B 1, B 1, there are asset profles { } b, 1 and { } b, 1 such ( {b } ) that optmum equlbrum allocatons under natural debt lmts startng from, 1, B 1 ( {b } ) are the same as those startng from, 1. These asset profles satsfy, B 1 b, 1 b I, 1 = b, 1 b I, 1 for all. (12) Proposton 1 asserts that t s not total government debt but how ts ownershp s dstrbuted that affects equlbrum allocatons. To understand why, suppose that we ncrease an ntal level of government debt from 0 to some arbtrary level B 1. If transfers T were held fxed, the government would want to ncrease tax rates τ to collect a present value of revenues suffcent to repay B 1. Snce dead-weght losses are convex n the tax rate, hgher levels of debt would then mpose dsproportonately larger dstortons, whch makes hgher levels of debt partcularly bad. But ths concluson changes f we allow the government to 9

11 adjust transfers. To fnd optmal transfers, we need to know how holdngs of government debt B 1 are dstrbuted. Suppose that agents hold equal amounts of the addtonal debt B 1. In ths case, each unt of debt repayment acheves the same redstrbuton as one unt of transfers. Snce the orgnal level of transfers at zero government debt s optmal, the best polcy for the government wth debt B 1 s to reduce transfers by exactly the amount of the ncrease n per capta debt. As a result, both the dstortng taxes τ and allocatons reman unchanged. Ths example llustrates deas expressed by Smon Newcomb (1865, p. 85) n the quotaton wth whch we began ths paper. Ths logc s senstve to the assumpton that holdngs of addtonal government debt are equal across agents. Suppose nstead that the government debt s owned dsproportonately by hgh-earnngs agents so that nequalty s hgher n economes wth hgher government debt; the optmal fscal response would typcally call for an ncrease n both tax rates τ and transfers T. The concluson would be the opposte f government debt were to be dsproportonately owned by low-earnngs agents. 4 Proposton 1 cautons aganst comparng debt burdens across countres based purely on aggregate quanttes lke debt to GDP ratos. If governments want to redstrbute from hgh-earnng to low-earnng agents, publc debt that s held wdely by prvate agents or government agences typcally wll be less dstortng than publc debt held by agents n the rght tal of the earnng dstrbuton or by foregn nvestors. Smlarly, our result warns aganst lumpng together explct debt and mplct debt (such as Socal Securty oblgatons) nto one aggregate number wthout adjustng for heterogenety across holdngs of the varous types of debts. Another mplcaton of Lemma 2 s that a path of government debt n the optmal compettve equlbrum wth natural debt lmts s ndetermnate. Proposton 2. (Rcardan equvalence) Suppose that an optmal equlbrum wth a 4 It s straghtforward to extend our analyss to an open economy wth foregn holdngs of domestc debt. The more government debt s owned by the foregners, the hgher are the dstortons the government wll need to mpose. 10

12 natural debt lmt gven ( ) {b, 1 }, B 1 exsts. Then any bounded B s part of an optmal compettve equlbrum. Lemma 2 and ts mplcatons n the form of Propostons 1 and 2 are true n more general envronments too. For example, we can allow agents to trade all concevable Arrow securtes and stll show that equlbrum allocatons depend only on agents net asset postons. Our results also hold n economes wth captal and wth arbtrary non-lnear ncome tax schedules T t (y t ). Results n ths secton suggest that the presence or absence of dstortng taxes or ncomplete markets s by tself nsuffcent to mply anythng about the level publc debt or ts welfare costs. In contrast to representatve agent models such as Barro (1974, 1979), n our heterogeneous agent settng, both the slope τ t and the ntercept T t are dstortng, but the path of debt n the optmal allocaton s ndetermnate. The dscusson also sheds lght on the role of debt n Woodford (1990) and Ayagar et al. (2002) who allow for lump-sum taxes but feature ncomplete markets. In those models, addtonal restrctons n the form of borrowng lmts and ad hoc costs of lump-sum taxes generate motves that ultmately pn down a path of debt. We nvestgate the role of such assumptons n Secton Numercal example To llustrate that t s not total government debt but the dstrbuton of debt across people that matters, we study how the correlaton between debt holdngs and labor earnngs affects an optmal tax rate and output. Labor earnngs and holdngs of government debt are hghly correlated n the U.S. We use data from the 2013 wave of the Survey of Consumer Fnance (SCF) wth the sample restrcted to marred households. The SCF provdes nformaton on households total labor earnngs, as well as hours worked by prmary and secondary earners. From these data, we construct average household wages. To measure households holdngs of government debt, we sum drect holdngs plus ndrect holdngs through government bond mutual funds (taxable and 11

13 nontaxable), savng bonds, money market accounts, and components of retrement accounts that are nvested n government bonds. We use these varables to compute a least squares regresson debt = ˆδ 0 + ˆδ 1 wage + ɛ (13) and obtan ˆδ 0 = wth s.e. of 2.15, ˆδ 1 = 0.93 wth s.e. of 0.024, and an R 2 of 7.8%. 5 To study how the correlaton between debt and wages affects an optmal tax rate and output, we assume soleastc preferences ( ) U(c, l ) = c1 σ l 1+γ 1 σ ψ 1 + γ (14) and no uncertanty. We set set utlty functon parameters σ, γ, β equal to 1, 2, We choose I = 20 wth equal measures of agents n each group and fx a tme-nvarant dstrbuton {θ } that replcates average wages per quntle for each of 20 wage quntles n the 2013 SCF. We summarze the dstrbuton of debt holdngs {b } wth an affne functon b, 1 = δ 0 + δ 1 θ. As a baselne, we set (δ 0, δ 1 ) equal to our estmated regresson coeffcents, (ˆδ0, ˆδ 1 ). We evaluate welfare usng Pareto weghts ω θ α. We assume that government expendtures are tme-nvarant and pck (g, α, ψ) jontly to match a federal government expendtures to output rato of 12%, the average federal labor tax rate of 24% estmated by Barro and Redlck (2011), and a total debt to total labor earnngs rato of 0.92 that we nfer from the 2013 SCF. Gven our assumptons, for any (δ 0, δ 1 ) the optmal tax rate and output are constant for all t 1. It follows from Proposton 1 that nether the optmal tax rate nor optmal output depends on the value of δ 0 because any change n debt holdngs due to changes n δ 0 leaves net asset nequalty unchanged. But net asset nequalty s affected by changes n δ 1. Fgure 2 shows comparatve statcs of optmal tax rates and outputs as functons of δ 1. Hgher 5 We measure wages n unts of thousand dollars per annual hours and average hours per year suppled by the household are measured usng the 2013 SCF. Bond holdngs are measured n thousand dollars. 12

14 values of δ 1 correspond to dstrbutons of bond holdngs that are more concentrated wth productve agents. We see that an optmal plan responds to a more top heavy dstrbuton of bond holdngs by rasng the tax rate n order to redstrbute towards poor agents. A hgher tax dstorton makes output declne. We can use these graphs to nterpret effects from changes n government debt that are dstrbuted proportonally to ntal benchmark debt holdngs. Suppose that, relatve to the benchmark economy, government debt s ncreased by percent and that ths addtonal debt s dstrbuted to households proportonally to ther benchmark debt holdngs. The new dstrbuton of debt s descrbed by parameters (δ 0, δ 1 ) = (ˆδ0 (1 + ), ˆδ 1 (1 + )). Snce δ 0 does not affect the tax rate or output, the rato δ 1 /ˆδ 1 captures the effect of a δ 1 /ˆδ 1 -fold ncrease n government debt dstrbuted proportonally. In ths experment, the level of debt seems to matter, but that s because we are smultaneously changng both the total amount of debt and the net dstrbuton of debt holdngs. If government debt doubles and debt s dstrbuted proportonally to the holdngs of debt n the data, fgure 3 ndcates that output drops by 2 percentage ponts and that the tax rate ncreases by 4 percentage ponts as we vary [ 0.5, 0.5]. [Fgure 2 about here.] [Fgure 3 about here.] 4 Imperfect debt enforcement and ad hoc borrowng constrants The analyss of the prevous secton closely follows the Ramsey tradton of answerng normatve questons. At the outset we specfy sequences of nstruments avalable to the government (τ, T, and B n our case) and assume that the government commts to those sequences n perod 1. Optmzng over a set of compettve equlbra assocated wth those sequences 13

15 mplctly assumes that the government has the ablty to pck the equlbrum wth the hghest welfare from that set. That s, the government has a technology that allows t perfectly to mplement an equlbrum allocaton assocated wth ts polces. To elaborate the mplementaton ssue, consder a stuaton n whch agents make choces that render some budget constrants volated, for example, by some agents not workng enough to be able to meet ther tax labltes. An mplct enforcement technology assumpton would requre the government to mpose punshments suffcently harsh to prevent agents from makng such choces off-equlbrum. If consumpton s bounded by 0 and lm c 0 U (c, l) = for all, l, t s suffcent to specfy that the government commts to sezng all of an agent s labor and asset ncome n a perod n whch he cannot pay ts prescrbed taxes. But f the utlty functon s bounded from below, addtonal non-pecunary punshments may be needed to mplement an allocaton. The same assumpton of perfect enforcement would extend to repayment of prvate debts agents never fal to repay ther debts n equlbrum presumably because the punshments for not dong so are suffcently severe. Thus, the equlbrum defnton n Secton 3 ndrectly requres not only that the government has the ablty to enforce payments, but also that t uses ts ablty to enforce both tax and debt payments. In ths secton, we stay wthn the boundares of a conventonal Ramsey analyss but focus on whether t s desrable for the government to enforce both tax and debt oblgatons and whether t can mprove welfare by commttng to enforce some type of payments and not others. We represent the government s enforcement choce n a smple form by assumng that agents can borrow up to an ad hoc debt lmt b,t b (15) for some exogenously gven b 0. We nterpret these constrants as arsng from mperfect government debt enforcement: the government mposes an arbtrary hgh punshment on 14

16 agents f they default on any debt less than b and no punshment for any default on debt over b; the case b = 0 s nterpreted as the government s refusng to enforce any prvate debt contracts. The natural debt lmt consdered n the prevous secton s a lmt that arses when agents are punshed for any debt default. 6 Note that we mantan the assumpton that the government enforces tax labltes perfectly: thus, we study whether t s optmal to enforce taxes and debt contracts dfferentally. 7 Defnton 4. A compettve equlbrum wth an ad hoc debt lmt gven ntal assets ( {b, 1 }, B 1 ) s a ({c, l, b }, B, q, τ, T ) such that () (c, l, b ) maxmze (1) subject to (7) and (15) for all ; and () constrants (2), (3), and (5) are satsfed. To understand what determnes the path of debt, we frst show that, n general, t s optmal for the government not to enforce prvate contracts. Restrctng prvate borrowng allows the government more flexblty n managng ts debt servce costs. Indeed, an optmal path of debt s pnned down by these consderatons. Ths outcome contrasts wth to alternatve accounts that emphasze that a government should ssue debt to ncrease lqudty because there s a lack of other means of savngs. We begn wth our man proposton for ths secton. 8 Proposton 3. If there are tax polces that support an allocaton (c, l) as a compettve equlbrum allocaton wth a natural debt lmt, then there are tax polces that support (c, l) as a compettve equlbrum allocaton wth an ad hoc debt lmt for any b. If (c, l) can be supported as a compettve equlbrum allocaton wth an ad hoc debt lmt b, t can also be supported as a compettve equlbrum allocaton wth ad hoc debt lmt b for any b. 6 We beleve that another frutful way to study the role of debt s to drop the full commtment assumpton and explctly specfy strateges for all hstores for agents and the government as was done by Bassetto (2002) n a closely related context of monetary economcs and the fscal theory of prce level. See also Bassetto (2005). 7 Bryant and Wallace (1984) descrbe how a government can use borrowng constrants as part of a welfare-mprovng polcy to fnance exogenous government expendtures. Sargent and Smth (1987) descrbe Modglan-Mller theorems for government fnance n a collecton of economes n whch borrowng constrants on classes of agents produce the rate of return dscrepances that Bryant and Wallace manpulate. 8 Our proposton bulds on Yared (2012, 2013), who showed that a planner may fnd t optmal not to undo agents borrowng constrants even when dong so s feasble. 15

17 Proof. Let {c, l, b } be a compettve equlbrum allocaton and debt wth a natural debt lmt. Let t max {b b,t }. Defne ˆb,t b,t + t for all t. By Lemma 2, {c, l, ˆb } s also a compettve equlbrum allocaton wth natural debt lmts. Moreover, by constructon ˆb,t b = b,t + t b 0. Therefore, ˆb satsfes (15). Snce agents budget sets are smaller n the economy wth ad hoc debt lmts and snce {c, l, ˆb } les n ths smaller budget set, then {c, l, ˆb } s also an optmal choce for agents n the economy wth exogenous borrowng constrants b. Snce all market clearng condtons are satsfed, {c, l, ˆb } s a compettve equlbrum allocaton and asset profle. To prove the second asserton, let ({c, l, b }, B, q, τ, T ) be a compettve equlbrum ( ) wth debt lmt b. Defne t b b and construct ˆT, ˆB as n (11), ˆb,t = b,t + t for all, t. Then by usng arguments from Lemma 2 we can show that ({c, l, ˆb }, ˆB, q, τ, ˆT ) s a compettve equlbrum wth debt lmt b. A remarkable mplcaton of Proposton 3 s not only that the government fnds t optmal to treat transfers and debt dfferently, but that the weakest possble enforcement of prvate debt contracts s optmal. Wthout loss of generalty, we can assume that agents cannot borrow. Corollary 1. Welfare n an optmum equlbrum wth ad hoc debt lmts s hgher than welfare n the optmum equlbrum wth natural debt lmts. Ths s true for any debt lmt b. A crucal dfference between outcomes wth the ad hoc debt lmts studed n ths secton and natural debt lmts n the prevous secton s how they depend on the tax polcy. Whle the lower bound on debt s endogenous and depends on the government tax-transfer polcy Secton 3 dscusson of natural debt lmts, t s exogenous wth the ad hoc debt lmts of ths secton. The presence of a polcy nvarant debt lmt here mples that changng the tmng of transfers can change the set of agents who are up aganst ther borrowng lmts. Ths power lets the government ncrease welfare. 16

18 The crtcal feature beng exploted here s an asymmetrc enforcement of taxes and prvate debt. If the government allowed agents to use prospectve transfers as collateral for prvate borrowng, then by postponng transfers the government would relax agents borrowng constrants and undo ts ablty to ncrease welfare by pushng some people aganst ther borrowng constrants. Asymmetry between enforcement of debt oblgatons and tax oblgatons seems to be common n practce. For example, n the U.S. t s llegal to use future socal securty payments as collateral and t s typcally easer to dscharge unsecured debt than tax labltes through bankruptcy. Others have also studed Ramsey polces n economes wth ad hoc constrants (15) and ponted out that Rcardan equvalence fals and consequently that the optmal debt s determned. 9 Thus, n the context of the results of secton 3, our Proposton 2 would generally not hold when agents are subject to the ad hoc constrant (15). In the followng example we nvestgate the sources of welfare gans that come from from lmtng agents opportuntes to borrow. Example 1. Suppose that there are two types of agents wth equal mass. Agent 1 cannot ( ) work and orders preferences by c 1,t. Agent 2 orders preferences by u c 2,t 1 1+γ l1+γ 2,t wth γ > 0. Agent 2 s productvty satsfes θ 2,t = 1 f t s even and θ 2,t = 0 f t s odd. There are no government expendtures. The government puts Pareto weght 1 on agent 1 s utlty. All agents start wth no ntal assets. Consder frst the optmum equlbrum when debt enforcement s perfect and agents face a natural debt lmt. In ths case, agent 1 s preferences mply that the equlbrum sets q t = β for all t. The government s objectve functon makes t want to maxmze the present value of tax revenues, evaluated at the prce system mpled by q t = β for all t. Gven agent 2 s preferences, the optmal tax rate s τ t = τ for all t, where τ s the top of the Laffer curve tax 9 For nstance, see Woodford (1990), Ayagar and McGrattan (1998), and Azzmont et al. (2014). Some commentators observed that these breakdowns of Rcardan equvalence mplctly requre that t s easer to extract a dollar from an agent n taxes than n debt servce. Our analyss ndcates that t s an optmal choce for the government to choose arrangements that produce that outcome even f the same technology s avalable for enforcng both types of payments. 17

19 rate, namely, τ = γ 1+γ whch mples a labor supply l = welfare wth natural debt lmts s t βt T t = 1 2 Z 1 β 2 ( ) 1 1/γ. 1+γ These fndngs mply that where Z ( ) 1 1+1/γ. = γ 1+γ By Lemma 2, the tmng of transfers s ndetermnate. For example, the welfare optmum can be attaned by settng b 1,t = 0 for all t and (B, T ) that jontly solve the followng equatons for all t T 2t+1 = B 2t, T 2t + βb 2t = Z, B 2t+1 = 0 (16) and ( u (1 τ) l + 2βB 2t 1 ) 1+γ l = u (T 2t+1 2B 2t ). (17) 1 + γ Agent 2 s budget constrant and (2) mply that B 2t < 0 for all t. Thus, the government ssues debt n even perods. The government repays ths debt n odd perods by levyng (negatve) lump sum transfers. Agent 2 holds government debt to smooth margnal utlty ntertemporally. We denote an optmum equlbrum wth natural debt lmts and these transfer and debt sequence as ({c nat, l nat, b nat }, B nat, q nat, τ nat, T nat ). Now consder the economy n whch prvate debt constrants are not enforced, so that agents debts must satsfy b,t 0 for all, t. (18) Observe that ({c nat, l nat, b nat }, B nat, q nat, τ nat, T nat ) stll satsfes agents budget constrants under debt lmts (18), so t s also an equlbrum n the economy wthout prvate borrowng. But now we can construct an equlbrum wth hgher welfare. Thus, t s possble to show that ({c, l, b }, B, q, τ, T ) s part of an equlbrum wth ad hoc lmts (18) f and only f budget constrants (7) holds for both agents (wth θ 1,t = 0 for all t), feasblty (2), market clearng (3), and borrowng constrants (18) are satsfed, and the followng equatons also hold: 18

20 q t u ( c 2,t 1 ( [q t u c 2,t γ l1+γ 2,t l γ 2,t = (1 τ t ) θ 2,t, (19a) ) ( 1 + γ l1+γ 2,t βu c 2,t+1 1 ) 1 + γ l1+γ 2,t+1, (19b) ) ( βu c 2,t+1 1 )] 1 + γ l1+γ 2,t+1 b 2,t = 0, (19c) q t β, (19d) [q t β] b 1,t = 0. (19e) Equaton (19a) s the optmalty condton for labor of agent 2; equatons (19b)-(19e) are optmalty condtons for savngs that hold wth nequalty only f the agent s assets are zero, and wth equalty otherwse. A key observaton about these condtons s that there exst equlbrum q t that are hgher than the dscount factor β when the assets chosen by agent 1 are zero. We show n the onlne appendx 2 that for any ϱ β we can construct an equlbrum n whch τ t = τ and b 1,t = 0 for all t and an (nverse of) gross nterest rate sequence q(ϱ) = (ϱ, β, ϱ, β,...). Ths equlbrum s supported by transfer and debt sequences (T (ϱ), B(ϱ)) that satsfes T 2t+1 (ϱ) = B 2t (ϱ), T 2t (ϱ) + ϱb 2t (ϱ) = Z, B 2t+1 (ϱ) = 0, (20) whch generalzes (16). Dfferentate to obtan ϱ β t T t (ϱ) = ϱ=β t t β 2t+1 B nat 2t > 0. (21) As welfare s smply t βt T t (ϱ), t follows that, for ϱ close to β, lowerng equlbrum nterest rates (ncreasng ϱ) mproves welfare. Snce ϱ = β corresponds to welfare n the optmum equlbrum wth natural debt lmts, ths also proves that welfare wth ad hoc lmts s strctly hgher. Example 1 llustrates what determnes an optmal quantty of debt. If the government ssues debt n equlbrum, t s generally better off f nterest payments on that debt are lower. 19

21 In the economy wth natural debt lmts, equlbrum nterest rates are determned mplctly by a competton between the government and agent 1 to supply savngs ( lqudty ) to agent 2. Even though n that equlbrum agent 1 does not supply lqudty, he would, by ssung prvate rsk-less debt whenever the nterest rate drops below the nverse of hs rate of tme preference, namely, β 1. When prvate debt contracts are unenforceable, agent 1 cannot ssue rskless debt, so the government becomes a monopoly suppler of lqudty to agent 2. The government can use ts monopoly power to extract addtonal surplus from agent 2 by ssung debt at a lower nterest rate. 10 Results of Woodford (1990) and Ayagar and McGrattan (1998) are often nterpreted as justfyng a benefcal role for government debt by ts ncreasng the supply of savngs nstruments. Our analyss nstead suggests that the government should decrease the aggregate supply of lqudty by lmtng the enforcement of prvate debt contracts and usng market power thereby acqured to extract monopoly rents from provdng lqudty. Whle an optmal contnuaton level of government debt s determned n the equlbra we have been analyzng, the ntal level of government debt s rrelevant for welfare n the same sense as n Proposton 1. Proposton 4. Proposton 1 holds n an economy wth ad hoc debt lmts. If B s the ( {b } ) optmal path of debt gven, 1, B 1, then B s also the optmal path of debt gven ( {b } ), 1 f b, 1 b, 1 s ndependent of., B 1 Proof. Suppose (τ, T ) are the optmal taxes n the economy wth ntal assets ( {b } ), 1. Defne a sequence T by T 0 = T 0 + b I, 1 b I, 1 and T t = T t for all, B 1 t > 0. Followng the same steps as n the proof of Lemma 2 we can verfy that (τ, T ) are ( {b } ) optmal taxes n the economy wth ntal assets, 1., B 1 10 Ths example llustrates a more general prncple. In an economy wth a natural debt lmt, observe that the resource flow between agents 1 and 2 s determnate, but that the level of borrowng s not n order to let agent 2 smooth consumpton, ether agent 1 or the government can borrow from agent 2. Lower nterest rates n a perod t beneft the agent who experences the net resource nflow n that perod, whether he borrows hmself or receves ths nflow through transfers. Lowerng nterest rates s desrable when the government favors such agents. Bassetto (2014), Nepelt (2004), and Yared (2012) apply ths prncple n other contexts. 20

22 To understand why the ntal level of government debt s welfare-rrelevant, note that the welfare gans n example 1 are obtaned from the government s ablty to nfluence prces of future debt. The value of legacy debt wth whch the government enters perod 0 was set n the past and s not affected by future polces. Thus, the ntal debt level plays a role no dfferent from that n Secton 3. Note that Proposton 4 shows not only that welfare but also that the optmal debt path s ndependent of the level of ntal government debt B 1, though they generally do depend on how ntal assets are dstrbuted across agents. Thus, transtons to an optmal debt level take exactly one perod, ndependently of the ntal debt. As a fnal remark, Lemma 2 and Proposton 1-4 contnue to hold when we allow for dosyncratc ncome rsk. More detals for economes wth dosyncratc rsk are provded n the onlne appendx 3. 5 Informatonally-constraned optmal taxes The analyss of prevous sectons follows the Ramsey tradton by a pror restrctng the tax-transfer system to take a partcular form, n our case the affne tax system (4). An alternatve approach s to put explct constrants on the government s nformaton and then to derve optmal government polces that respect them. Ths approach orgnated n the work of Mrrlees (1971) and was ntroduced to macro by Golosov et al. (2003) and Wernng (2007). In ths secton, we nvestgate the role of debt and taxes when government actons are restrcted only by such nformatonal frctons. An nformatonally-constraned optmum s a sequence {c, l } that maxmzes (6) subject to feasblty (2) and constrants that specfy the government s nformaton about agents. Informatonally-constraned taxes are tax functons that use observable varables as ther arguments; optmal nformatonally-constraned taxes mplement an nformatonallyconstraned optmum as a compettve equlbrum. 21

23 Snce Mrrlees (1971), a standard assumpton s that the government does not observe an ndvdual s labor supply l,t or productvty θ,t but that t does observe labor earnngs y,t. We mantan ths assumpton throughout ths secton. The role of publc and prvate debt depends crtcally on whether the government observes ndvduals assets and consumptons. If agents assets are observable, publc or prvate debt plays no nterestng role: any sequence (B, {b } ) that satsfes feasblty (3) can be supported n an optmal compettve equlbrum for the followng smple reason. Let { } c ob, l ob be an nformatonally-contaned optmum wth observable assets and let y ob { } c ob, l ob be defned by y ob,t θ,t l ob,t. The government can mplement by offerng a menu of I tax schedules of the form {T t (y t, b t 1, b 1, )} t and lettng agents permanently self-select nto one of them n perod t = The problem becomes more nterestng when agents assets are unobservable. Assume that nteractons n asset markets are anonymous and that agents and the government can ssue and buy debt, but that t s mpossble for the government to ascertan an ndvdual agent s asset holdngs. Ths assumpton also requres that ndvdual consumpton s not observable. The nformatonally-constraned optmum can be characterzed by nvokng the Revelaton prncple and settng up a mechansm desgn problem. 12 We now defne an nformatonally-constraned optmal allocaton wth unobservable assets assocated wth a mechansm desgn problem that determnes labor ncome {y } and payments {x } as well as a debt sequence B. A reportng strategy s a functon r : I I. A mechansm {x, y } and B s feasble f there exsts an allocaton {c, l }, asset choces {b }, a reportng strategy r and bond prces q such that each agent chooses {c, l }, b, r() to maxmze (1) subject to the budget constrant c,t + q t b,t = x r(),t + b,t 1, (22) 11 It s easy to mplement an optmal allocaton wth smooth tax functons. See, for example, Kocherlakota (2005), Wernng (2009), and Grochulsk and Kocherlakota (2010). The concluson that nether publc nor prvate assets are pnned down contnues to hold n those mplementatons. 12 See Golosov and Tsyvnsk (2007) for detals. 22

24 wth b,t satsfyng ether natural or ad hoc debt lmts. Prces q are such that debt market clearng (3) and feasblty n c,t = n y r(),t (23) are satsfed. A feasble mechansm {x, y } and B s ncentve compatble f the assocated reportng strategy r() =. An nformatonally constraned optmum s an ncentve compatble mechansm {x, y } and B such that the assocated allocaton {c, l } maxmzes (6). The ablty of agents to trade assets anonymously lowers welfare, whch mples that the government would fnd t optmal to mnmze enforcement of prvate debt contracts. Gven that, we frst analyze ths economy when prvate borrowng s subject to the ad hoc lmt (15). Then we dscuss how our conclusons would change f debt enforcement on prvate markets were perfect. Consder any ncentve compatble mechansm {x, y } and B and let {b, c } be the assocated optmal asset and consumpton choces and let q be bond prces. A necessary condton for ncentve compatblty s E 1 t=0 ( β t U c,t, y ) (,t E 1 [U c j,0 + b, 1 b j, 1, y ) j,0 + θ,t θ,0 t=1 ( β t U c j,t, y ) ] j,t, (24) θ,t for all pars, j. The left sde s the utlty of agent when he receves allocaton {x, y }. Ths should be at least as hgh as utlty from clamng a bundle ( x j, y j ) and choosng asset profle b j on the anonymous market at the same prces q. The payoff from that choce s the rght sde of constrant (24). In prncple, agent can further ncrease hs utlty from bundle ( xj, y j ) f he chooses some other asset profle b ; but as we show below, f tradng s subject to ad hoc debt lmts, an optmally chosen debt sequence B prevents such retradng. Let { } c adhoc, y adhoc be a maxmzer of the objectve functon (6) subject to feasblty (3) 23

25 and ncentve constrant (24). Let B t = b for all t and choose any q that satsfes E t Uc q t β ( q 0 β U c U c ( ) c adhoc j,t+1, yadhoc j,t+1 θ,t+1 c adhoc j,t Choose sequence { } x adhoc such that, yadhoc j,t θ,t ) for t > 0, all, j, ( ) E 0 Uc c adhoc j,1, yadhoc j,1 θ,1 ( ) for all, j. (25) c adhoc j,0 + b, 1 b j, 1, yadhoc j,0 θ,0 c adhoc,t q t b = x adhoc,t + b,t, (26) where b,t = b for t > 0 and b,t = b, 1 for t = 0. For an agent of type who clams sequence x adhoc j and faces debt prces q, t s optmal to borrow up to the maxmum debt lmt b and therefore obtan the after-tax consumpton allocaton c adhoc j,0 +b, 1 b j, 1 and { } c adhoc j,t. Constrant (24) ensures that the optmal report t>0 s r() = for all, verfyng that { } c adhoc, y adhoc s ndeed an nformatonally-constraned optmum. Ths optmum can be mplemented by a sequence of tax functons of the form {T t (y t, )},t. Observe that government debt B plays the same role here as t dd n Secton 4. When agents face ad hoc borrowng constrants the government can affect nterest rates by choosng the level of ts debt. As n Secton 4, the government explots monopoly power on asset markets and lowers nterest rates. The sze of the borrowng constrant b s rrelevant for welfare because the government covers ts nterest expenses by adjustng the stream of transfers to agents wthout affectng fnal allocatons. Lke our dscusson n Secton 4, t s crucal for ths result that prvate debt contracts are enforced mperfectly. If agents can trade on anonymous markets subject only to a natural debt lmt, the government loses ts ablty to nfluence nterest rates through B, so the Rcardan equvalence result of Proposton 2 reemerges. Snce equaton (25) would hold 24

26 wth equalty n an equlbrum wth natural debt lmts, welfare would be lower. The role of the ntal debt level and ntal asset nequalty also mrrors that descrbed n Propostons 1 and 4. The absolute level of government debt B 1 per se does not affect welfare n the constraned optmum, but asset nequalty does, as can be seen from the rght sde of (24). We summarze our analyss n the followng proposton. Proposton 5. The ntal level of debt B 1 does not affect welfare wth optmal nformatonally-constraned taxes, but the level of ntal asset nequalty b, 1 b I, 1 generally does. Necessary condtons for the optmal path of government debt B to be determnate are anonymous asset trades and ad hoc borrowng constrants. Welfare s hgher n the economy wth ad hoc debt lmts than n the economy wth natural debt lmts. It s straghtforward to generalze these results to economes wth dosyncratc shocks, rcher asset markets, and captal Concludng remarks A prncpal message of ths paper s that wthout exogenous restrctons on transfers, the level of government debt doesn t matter. What matters s how ownershp of government debt s dstrbuted. Dependng on socety s atttudes toward unequal dstrbutons of consumpton and work, the cross-secton dstrbuton of government debt across assets can matter very much. Ths means that n order to nterpret emprcal correlatons between output growth rates and ratos of government debt to GDP as n Renhart and Rogoff (2010), we would want to know much more about how the dstrbutons of net assets across people have vared across 13 Under mld techncal assumptons (e.g., assumpton 1 n Kocherlakota (2005)) one can mplement nformatonally-constraned optmal allocatons usng tax schemes that depend on past hstores of ndvdual ncomes. Bassetto and Kocherlakota (2004) show that allowng tax functons that are suffcently flexble n terms of hstory dependence makes the path of government debt rrelevant. Gonzalez-Eras and Nepelt (2015) extend analyss of Rcardan equvalence to poltcal economy by studyng condtons under whch dfferent polcy regmes (nsttutons) are poltco-economcally equvalent n the sense that both regmes gve rse to poltco-economc equlbra wth dentcal allocatons. 25

27 countres and how they have nteracted wth rsks to nterest rates and to the underlyng sources of unequal productvtes across people. An optmal path of government debt s determned when agents abltes to borrow are restrcted because that allows prospectve publc debt ssues to affect nterest rates. We restrcted our analyss to economes n whch the government commts to future polces. A promsng topc for research s the role of debt when a government cannot commt. As our dscusson n Secton 4 suggests, mperfect commtment can mpose addtonal restrctons on transfers and debt that are feasble n equlbrum. We leave ths extenson to future work. References Ayagar, S. Rao Unnsured Idosyncratc Rsk and Aggregate Savng. The Quarterly Journal of Economcs, 109(3): Ayagar, S. Rao, Albert Marcet, Thomas J. Sargent, and Juha Seppala Optmal Taxaton wthout State-Contngent Debt. Journal of Poltcal Economy, 110(6): Ayagar, S. Rao, and Ellen R. McGrattan The Optmum Quantty of Debt. Journal of Monetary Economcs, 42(May 1997): Azzmont, Marna, Eva de Francsco, and Per Krusell. 2008a. Aggregaton and Aggregaton. Journal of the European Economc Assocaton, 6(2-3): Azzmont, Marna, Eva de Francsco, and Per Krusell. 2008b. Producton Subsdes and Redstrbuton. Journal of Economc Theory, 142(1): Azzmont, Marna, Eva de Francsco, and Vncenzo Quadrn Fnancal Globalzaton, Inequalty, and the Rsng Publc Debt. Amercan Economc Revew, 104(8):