Public Debt in Economies with Heterogeneous Agents

Transcription

1 Publc Debt n Economes wth Heterogeneous Agents Anmol Bhandar, Davd Evans, Mkhal Golosov, Thomas J. Sargent October 20, 2016 Abstract We study publc debt n an economy n whch taxes and transfers are chosen optmally subject to heterogeneous agents dverse capactes to pay. We assume a government that commts to polces and can enforce tax and debt payments. If the government enforces perfectly, asset nequalty s determned n an optmum compettve equlbrum but the level of government debt s not. In addton, welfare ncreases f the government commts not to enforce prvate debt contracts and ntroduces borrowng frctons. By dong so, t reduces competton on debt markets and gathers monopoly rents from provdng lqudty. Regardless of whether the government chooses to enforce prvate debt contracts, the level of ntal government debt does not affect an optmal allocaton, but the dstrbuton of net assets does. Key words: Dstortng tax. Transfers. Government debt. Rcardan equvalence. We thank Perre Yared for frutful dscussons.

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.... 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 1 Introducton To understand 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 a workable descrpton of 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 n the quotaton above. The logc apples to structures wth and wthout physcal captal, wth complete or ncomplete asset markets, and wth more general tax structures. The role of government debt crucally depends on how well prvate debt contracts are enforced. If both tax and debt oblgatons are enforced perfectly, then agents borrowng s 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 1

3 debts s optmal and a verson of Rcardan equvalence holds despte the fact that taxes dstort prvate agents decsons. The dynamcs of asset nequalty, however, share some of the qualtatve features of the dynamcs of debt n representatve agent models n whch transfers are ruled out. We also show that welfare ncreases f the government commts not to enforce prvate debt contracts so 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. The gans come from monopoly power on the asset market that the government acqure 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 nablty of agents to use antcpated transfers to relax ther current borrowng constrants. An optmal level of government debt s determned by a trade-off between obtanng monopoly rents and dstortng agents ntertemporal margnal rates of substtuton. There s a szable lterature n macro about debt and Rcardan equvalence, gong back at least to Barro It s well understood that n representatve agent economes the role of debt hnges on whether lump sum taxes are allowed. But n the context of those models, there s no nherent economc reason to rule them out. Proportonal labor taxes, often assumed n such models, are counterfactual snce transfers are a large part of modern tax systems see, e.g., Fgure 1. In our settngs, agents are heterogeneous, taxes are restrcted by agents ablty to pay, and the government chooses taxes to maxmze a weghted average of lfetme utltes of agents. 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. 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 for market structures beyond complete markets. Wernng 2007 characterzed optmal allocatons and dstortons n complete market economes, whle Bhandar et al and Wernng 2012 nvestgate how precautonary savngs motves that 2

4 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 n Yared 2012, 2013, who showed that t may not be optmal to undo agents borrowng frctons even though the government has ablty to do so. Bassetto 1999 studed the role of taxaton and debt lmts n the economes wth heterogenety n whch transfers are ruled out. The rest of our paper s organzed as follows. In Secton 2, we lay out a baselne envronment n whch taxes are restrcted to be affne functons of labor ncome and agents are heterogeneous n labor earnngs but do not face dosyncratc uncertanty. In Secton 3, we study an economy n whch agents borrowng s restrcted only by ther ablty to pay. In Secton 4, we study an economy n whch agents face more strngent borrowng constrants. We show that our results extend to dosyncratc shocks n Secton 5 and to rcher taxes constraned only by nformatonal frctons n Secton 6. 2 Envronment Tme s dscrete and nfnte. There s a government and 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. We use boldface letters x to denote 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 1 Other recent pertnent papers nclude Azzmont et al. 2008a,b and Correa These authors study optmal polcy n economes n whch agents are heterogeneous n sklls and ntal assets. 3

5 s t consttute s t. When t does not cause confuson, we use x t to denote a random varable wth a tme t for all 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 productvty of each ndvdual {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 n c,t + g t = =1 I n θ,t l,t. 2 =1 Agents and the government ssue and trade rskless one perod dscount bonds. 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 the government. Households and the government begn wth 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 mposes a tax on labor earnngs T t y t. To be comparable to the lterature, we assume throughout most of ths secton that T t are affne functons T t y t = T t + τ t y t. 4 Such affne tax functons approxmate actual tax and transfer programs pretty well; see Fgure 1 from Heathcote et al As wll be ndcated from our proofs, our results drectly extend to more general non-lnear ncome tax schedules T t y t and to even rcher tax systems. We dscuss these later. The government budget constrant wth affne taxes s g t + q t B t = τ t I n θ,t l,t + B t 1 T t. 5 =1 2 We have truncated the plot at 162K USD, whch s the 95th ncome pre-tax percentle, to emphasze the role of transfers specally for poor agents. 4

6 Fgure 1: Representaton of U.S. tax and transfer system from the PSID and TAXSIM. Source: Heathcote et al The dotted lne s the 45 degrees lne and the sold lne s a lnear ft. A government s preferences over stochastc process for consumpton and work are ordered by E 0 I n ω =1 t=0 where ω 0, I =1 ω = 1 s a set of Pareto weghts. A type agent s budget constrant at t 0 s β t U t c,t, l,t 6 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 further restrctons on debt holdngs, ths problem s ll-posed because t allows agents to acheve nfnte utlty by runnng Ponz schemes. To rule out explosve debt paths, we requre 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 are nterested n two questons 5

7 How does the level of the ntal government debt B 1 affect welfare n the optmal equlbrum? What determnes propertes of an optmal path of government debt B? The frst of these questons allows us to thnk about legacy costs of past debt. We want to understand the mplcatons of our neoclasscal model for such costs. The answer to the second queston also allow us to shed lght on what determnes an optmal level of debt f t exsts and how quckly the government should converge to t. The assumpton that agents are heterogeneous affects our answers. In a representatve agent economy t s well understood that the answers to these questons depend on whether lump-sum taxes are avalable see Barro If agents are dentcal, there s lttle reason to thnk that lump sum taxes are nfeasble. Much of the macro lterature rules out such taxes by mplctly alludng to unmodeled heterogenety and the presence of a subset of poor agents who cannot afford to pay such taxes. By modelng such poor agents explctly, we can study the optmal tax polcy wthout relyng on ad hoc restrctons on transfers. Instead, snce our transfers T are anonymous, the budget constrants of the poorest agents endogenously restrct ther sgn and magntude. Our answers depend n part on borrowng constrants. We nterpret these constrants as arsng from the ablty or nclnaton of a government to punsh agents who default on ther oblgatons. As a benchmark, we start wth the loosest borrowng lmts: these allow agents to borrow amounts that are feasble for them to repay n all future states. These lmts are ratonalzed by the government beng wllng and able to mpose the harshest punshments on agents who ever default. We then dscuss strcter lmts on prvate borrowng. 3 Optmal debt under natural debt lmts We start wth the stuaton n whch consumers face the loosest possble borrowng constrants. We call these the natural debt lmts and begn wth some standard defntons Defnton 1. An allocaton s a sequence {c, l }. An asset profle s a sequence 6

8 {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 maxmzes6. A short dscusson of our termnology s n order. The natural debt lmt termnology was popularzed by Ayagar He consdered an economy wth fnte after tax endowment and utlty defned over non-negatve consumpton. When the equlbrum nterest rate 1 q t s strctly greater than one ths condton can be relaxed to requre that Q s a strctly postve and summable sequence 3 Ayagar requred that an agent s debt does not exceed the present value of hs maxmum after-tax ncome n the worst shock sequence. 4 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 n the 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 an 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 The followng lemma ndcates that our defnton of compettve equlbrum smply 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. 3 That s, t Q ts t exsts for all s. 4 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 Proof. Snce Y s bounded above and Q s a strctly postve summable sequence, the sequence D t Y s bounded above and therefore 9 mples that b s bounded below. Suppose that b s bounded below but does not satsfy the natural debt lmt at some hstory s t. In partcular, for some ɛ > 0 suppose that b,t s t = D t Y ; s t ɛ. 10 Use agent s budget constrant and mpose non-negatvty of consumpton to get q t+1 s t+1 b,t+1 s t+1 Y,t+1 s t+1 D t Y ; s t ɛ. Usng the defnton of D t Y ; s t, there exsts s t+1 that occurs wth strctly postve probablty such that Repeatng ths process, we get b,t+1 s t, s t+1 D t+1 Y ; s t, s t+1 b,t+n s t, s t+1... s t+n D t+1 Y ; s t, s t+1... s t+n ɛ q t+1 s t, s t+1. ɛ Π N j=0 q t+1+js t, s t+1... s t+1+j. Snce Q s summable, lm N Π N j=0q t+1+j s t, s t+1... s t+1+j = 0 and hence for any constant B there exsts an N B suffcently large that b,t+n s t, s t+1... s t+n B ɛ N N B, and thus we obtan a contradcton. Our defnton of an optmal compettve equlbrum allows the government to optmze over taxes and transfers τ, T. Snce compettve equlbrum s defned only over taxes that all consumers can afford to pay.e., for whch each consumer s budget set s nonempty, ths defnton endogenously mposes restrctons on feasble 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] 8

10 there exst sequences ˆT, ˆB such that {c, l, ˆb } equlbrum wth natural debt lmts gven Proof. For any bounded {ˆb } {ˆb, 1 }, ˆB, q, τ, ˆT, ˆB 1. s a compettve 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 to show that c, l, ˆb s the optmal choce gven q, τ, ˆT. Observe that c, l, ˆb satsfes budget 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 gve consumer hgher utlty gven q, τ, ˆT. The sequence c, l, b satsfes 7 and 9 gven q, τ, T and gves 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 summarze the answers to two questons posed n Secton 2 by means of 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 that an optmum equlbrum allocatons wth natural debt lmt startng from {b } {b }, 1 and from, 1 are the same. These asset profles satsfy, B 1, B 1 b, 1 b I, 1 = b, 1 b I, 1 for all. Proposton 1 asserts that t s not total government debt but how ts ownershp s dstrbuted that matters for equlbrum allocatons. To understand why, suppose that we ncrease the ntal level of government debt from 0 to some arbtrary level B 1. If transfers T were to be held fxed, the government would want to ncrease 9

11 taxes τ to collect a present value of revenues suffcent to repay B 1. Snce deadweght 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 consder how ths concluson would change f we were to allow the government to 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 new debt. 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 households meanng 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 households. 5 Proposton 1 cautons aganst comparng debt burdens across countres based purely on aggregate quanttes lke debt to GDP ratos. Assumng that governments generally want to redstrbute from hgh-earnng to low-earnng households, 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 cautons aganst lumpng both explct debt and mplct promses such as Socal Securty oblgatons nto one headlne number wthout adjustng for heterogenety across holdngs of varous types of debts. Another mplcaton of Lemma 2 s that the 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 5 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 a natural debt lmt gven {b, 1 }, B 1 exsts. Then any bounded B s part of an optmal compettve equlbrum. Although Proposton 2 asserts that the level of government debt s ndetermnate, there are close parallels between optmal allocatons n our economy and representatve agent Ramsey models of debt. To llustrate ths, assume that I = 2 and that both U s are dfferentable and satsfy Inada condtons. Then usng standard arguments, one can show that {c, l, b }, B, q, τ, T s a compettve equlbrum f and only f t satsfes c 2,t + U l2,tl 2,t U c2,t c 1,t + U l1,tl 1,t + q t b 2,t b 1,t = b 2,t 1 b 1,t 1, 12a U c1,t U l1,t θ 1 U c1,t = U l2,t θ 2 U c2,t, c 1,t + c 2,t + g t θ 1 l 1,t + θ 2 l 2,t, 12b 12c U c,t E t U c,t+1 = β q t. 12d An nspecton of these equatons ndcates that we can defne a net asset poston b t b 2,t b 1,t and represent ts dynamcs recursvely usng standard technques wth b t beng a state varable. By normalzng b 2,t = 0 we can nterpret b t to be government debt. Ths structure bears a close resemblance to some representatve agent Ramsey models e.g. Ayagar et al. 2002; Farh 2010, although t mght have substantally dfferent qualtatve and quanttatve mplcatons, some of whch we nvestgate n Bhandar et al fothcomng and Bhandar et al Lemma 2 and ts mplcaton n the form of Propostons 1 and 2 hold n more general envronments too. For example, we could allow agents to trade all concevable Arrow securtes and stll show that equlbrum allocatons depend only on agents net asset postons. Smlarly, our results hold n economes wth captal, or wth arbtrary non-lnear ncome tax schedule T t y t. 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 11

13 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 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 decde to make alternatve 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 pursung such polces offequlbrum. 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. 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 from default are suffcently severe from falng to do so. Thus, the equlbrum defnton n Secton 3 ndrectly requres not only that the government has the ablty to enforce payments, but also that t exercses ths ablty to enforce both tax and debt payments. In ths secton, we stay wthn the boundares of a conventonal Ramsey analyss and 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. Formally, we capture ths decson n the smplest form by assumng that agents can borrow up to an ad hoc debt lmt b,t b 13 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 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 12

14 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 lablty 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 13 for all ; 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 more flexblty to the government n managng ts own debt servce costs. In fact, the optmal path of debt s pnned down by these consderatons n contrast 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 any 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 wth 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. 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. 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. We hope to pursue ths lne of work. 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 the planner may fnd t optmal not to undo agents borrowng constrants even when that s feasble. 13

15 Moreover, by constructon ˆb,t b = b,t + t b 0. Therefore, ˆb satsfes 13. Snce agents budget sets are smaller n the economy wth ad hoc debt lmts and {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. For the proof of the second part, 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 we can show that {c, l, ˆb, ˆB, q, τ, ˆT s a compettve equlbrum wth lmt b by usng arguments from Lemma 2. 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 optmum equlbrum wth ad hoc debt lmts s hgher than welfare n the optmum equlbrum wth natural debt lmts and does not depend on the value of the debt lmt b. A crucal dfference between 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 polcy-dependent n the Secton 3 dscusson of natural debt lmts, t s exogenous n ths secton where we have ad hoc debt lmts. Polcy nvarance of the debt lmt here mples that changng the tmng of transfers can change the set of agents who are up aganst ther borrowng lmts. Ths gves the government the ablty to ncrease welfare. The prevous dscusson also hghlghts a crtcal assumpton underlyng the results of ths secton: asymmetrc enforcement of taxes and prvate debt. If the government allowed agents to use transfers as collateral for prvate borrowng, postponng transfers nto the future would relax agents borrowng constrants and undo a government s ablty to gan from pushng people aganst ther borrowng constrants. Asymmetry between enforcement of debt oblgatons and tax oblgatons s common n practce. For example, n the U.S. t s llegal to use future socal securty payments 14

16 as collateral and t s typcally easer to dscharge unsecured debt than tax labltes through bankruptcy. Varous authors have studed Ramsey polces n economes wth ad hoc constrants 13 and ponted out that Rcardan equvalence fals and consequently that the optmal debt s determned. 9 Thus, n the context of the results of the prevous secton, our Proposton 2 would generally not hold when agents are subject to the ad hoc constrant 13. In the followng example we nvestgate the sources of welfare gans that came 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 has preferences c 1,t. Agent 2 has preferences 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 so that agents face a natural debt lmt. In ths case, agent 1 s preferences mply that the only feasble equlbrum nterest rate n ths economy satsfes q t = β for all t. The government s objectve functon makes t want to maxmze the present value of tax revenues, evaluated at ths nterest rate. Gven the assumpton about agent 2 s preferences, the optmal tax rate s τ t = τ for all t, where τ s the top of the Laffer curve Z. 1 β 2 tax rate. 10 Ths mples that welfare wth natural debt lmts s t βt T t = 1 2 The tmng of transfers s ndetermnate by Lemma 2. Wthout loss of generalty the 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 14 and u 1 τ l + 2βB 2t 1 1+γ l = u T 2t+1 2B 2t. 1 + γ 9 For nstance, see Woodford 1990; Ayagar and McGrattan 1998; Azzmont et al Some commentators observed that ths constructon mplctly requres 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 ths outcome even f the same technology s avalable for enforcng both types of payments. 10 It s easy to verfy that they satsfy τ = γ 1+γ, l = γ 1/γ, Z = γ 1+1/γ. 1 1+γ

17 Agent 2 s budget constrant and 2 mply that B 2t < 0 for all t. Thus, the government ssues debt n even perods that agent 2 uses to smooth margnal utlty ntertemporally. The government repays ths debt n odd perods by levyng negatve lump sum transfers. We denote that optmum equlbrum wth natural debt lmts usng 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 agent s debt must satsfy b,t 0 for all, t. 15 Observe that {c nat, l nat, b nat }, B nat, q nat, τ nat, T nat stll satsfy the agents budget constrants wth ths addtonal debt lmts so that t s also an equlbrum n the economy wthout prvate borrowng. We now construct a welfare mprovng equlbrum. One can show that {c, l, b }, B, q, τ, T s part of an equlbrum wth ad hoc lmts 15 f and only f budget constrants 7 holds for both agents wth θ 1,t = 0 for all t, feasblty 2 and 3 and borrowng constrants 15 are satsfed, and the followng equatons hold: 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, 16a 1 + γ l1+γ 2,t βu c 2,t γ l1+γ 2,t+1, 16b βu c 2,t+1 1 ] 1 + γ l1+γ 2,t+1 b 2,t = 0 16c q t β, 16d [q t β] b 1,t = 0. 16e Equaton 16a s the optmalty condton for labor of agent 2; equatons 16b- 16e 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 equlbrum q t that are hgher than the dscount factor β when assets of agent 1 are at zero. We show n the appendx that for any ϱ β one can construct an equlbrum n whch τ t = τ and b 1,t = 0 for all t and annverse of nterest rate sequence qϱ = ϱ, β, ϱ, β,... Ths 16

18 equlbrum s supported by transfers and debt sequence T ϱ, Bϱ that satsfes T 2t+1 ϱ = B 2t ϱ, T 2t ϱ + ϱb 2t ϱ = Z, B 2t+1 ϱ = 0, 17 whch generalzes 14. Dfferentate to obtan ϱ T 2t+1ϱ = ϱ B 2tϱ ϱ T 2tϱ + ϱ ϱ B 2tϱ + B 2t ϱ = 0 Snce welfare s smply t βt T t ϱ, t follows that for ϱ close to β lowerng equlbrum nterest rates ncreasng ϱ mproves welfare: β t T t ϱ = ϱ t ϱ=β t β 2t+1 B nat 2t > 0. 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 the key drvng force behnd the determnaton of the optmal quantty of debt. If the government ssues debt n equlbrum, t s generally better off f nterest payments on that debt are lower. In the economy wth natural debt lmts, equlbrum nterest rates are determned mplctly by 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, as soon as the nterest rate drops below the nverse β 1 of hs rate of tme preference. When prvate debt contracts are unenforceable, agent 1 cannot ssue rskless debt and the government becomes a monopolst suppler of lqudty to agent 2. By usng ts monopoly power, t can extract addtonal surplus from agent 2 by ssung debt at a lower nterest rate. The results n Woodford 1990, Ayagar and McGrattan 1998 are often nterpreted as justfyng a role for government debt to ncrease lqudty of savngs nstruments. Our dscusson suggests that the government should decrease the aggregate supply of lqudty by lmted enforcement of prvate debt contracts and use the addtonal market power thereby acqured n provdng lqudty to extract monopoly rents. Ths force runs counter to a common opnon that the role of publc debt s to ncrease lqudty of agents who are borrowng constraned. 17

19 Whle the optmal contnuaton level of government debt s often determned n equlbrum, the ntal level of government debt s rrelevant for welfare n the same sense as n Proposton 1. Proposton 4. Proposton 1 holds n the economy wth ad hoc debt lmts. If B {b } s the optmal path of debt gven, 1, B 1, then B s also the optmal path of {b } debt gven, 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, B 1. Defne sequence T as T 0 = T 0 + b I, 1 b I, 1 and T t = T t for all t > 0. Followng the same steps as n the proof of Lemma 2 we can verfy that {b } τ, T are optmal taxes n the economy wth ntal assets, 1., B 1 To understand why the ntal level of government debt s welfare-rrelevant, note that the welfare gans n the 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. the ntal debt level does not play a role dfferent from that n Secton 3. Thus, Note that Proposton 4 shows that not only welfare but also the optmal debt path s ndependent of the level of ntal government debt B 1 though they generally do depend how the ntal assets are dstrbuted across agents. Thus, transtons to an optmal debt level take exactly one perod, ndependently of the sze of the ntal debt. 5 Idosyncratc rsk It s relatvely straghtforward to allow for dosyncratc ncome rsk. 11 There are I groups of agents each wth a measure n of ex-ante dentcal ndvduals. We mantan the aggregate shocks s t S and ntroduce dosyncratc shocks that are drawn from s S { } for all I where S and S are fnte sets. We use S t, S t to denote t I perod Cartesan products and S, S as the space of nfnte sequences of aggregate and dosyncratc shocks for each group. We use π t to denote the probablty measure over Borel sets of S t and π as the extenson to the Borel sets of S. In an analogous 11 See for nstance Bewley 1986, Ayagar 1994, Huggett 1993 or Krusell and Smth

20 fashon, we defne µ,t and µ to be probablty measures over Borel sets of S t and S, respectvely, for each I. We mpose that π t s t > 0 for all s t S t and µ,t s t > 0 for s t S t for all t. Next, we assume that condtonal on the hstory of aggregate shocks s t, the dosyncratc shocks s t are dstrbuted ndependently and dentcally across members of group. We assume a Law of Large numbers: for any Borel subset set B of S, t the measure µ,t B denotes the fracton of type agents that have hstores s t B. The productvty of an agent n group s now descrbed by functons θ s, s whle, as before, government expendtures are gs. Tme t realzatons after hstory s t, s t are denoted θ,t s t, s t and g t s t. Bond prces, tax polcy, and government debt polcy are measurable wth respect to aggregate shocks s t and ndvdual consumpton, lesure, and asset choces depend on s t, s t. Indvdual budget constrants reman as n equaton 7. We modfy the resource constrant to reflect that there s a contnuum of agents: for any hstory s t of aggregate shocks, n c,t s t, s t dµ,t s t + g t s t = n θ,t s t, s t l,t s t, s t dµ,t s t s t S t s t S t n b,t s t, s t dµ,t s t + B t s t = 0 s t S t Intal condtons consst of {b, 1, s, 1 } I, s 1. The natural debt lmt n equaton 8 s constructed by takng the nfmum over jont hstores s, s S S. 12 The defntons of compettve equlbrum and optmal compettve equlbrum wth ether natural debt lmts or ad hoc debt lmts are unchanged. The arguments n Lemma 2 preval here too. Any perturbaton t such that ˆB t = B t + t and ndvdual assets ˆb t = b,t t preserve parwse dfferences n assets; then adjustng transfers as n equaton 11 keeps all budget sets unaltered. Ths mples that Lemma 2 and also Proposton 1-4 contnue to hold when we allow for dosyncratc ncome rsk. 12 More formally we would have D t Y ; s t, s t nf s, s S S st, s t s s k>t,s t, s t s s Q s k 1 Q s t Y,ks k, s k. 19

21 6 Informatonally-constraned optmal taxes The analyss of the prevous sectons followed the Ramsey tradton when we a pror restrcted attenton to taxes of a partcular form, such as affne taxes 4. An alternatve approach s to start wth explct nformaton constrants on the government and to derve mplcatons for government polcy. Ths approach orgnated n the work of Mrrlees 1971 and was ntroduced to macro by Golosov et al and Wernng In ths secton, we nvestgate the role of debt and taxes when government actons are restrcted by nformatonal frctons only. An nformatonally-constraned optmum s a sequence {c, l } that maxmzes 6 subject to feasblty 2 and the constrants that specfy nformaton about agents that s avalable to the government. Informatonally-constraned taxes are tax functons that use observable varables as ther arguments; optmal nformatonallyconstraned taxes mplement an nformatonally-constraned optmum as a compettve equlbrum. A standard assumpton snce Mrrlees 1971 s that the government does not observe 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 an ndvduals assets and consumpton are observable. 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. The reason s smple. Let { } c ob, l ob be an nformatonally-contaned optmum wth observable assets and let y ob be defned by y,t ob θ,t l,t. ob The government can mplement { } c ob, l ob by offerng agents 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 = 1. One constraned optmum tax schedule smply sets T 0 y 0, b 1, = θ,0 l ob,0 c ob,0 b 1 T t y t, b t 1, b 1, = θ,t l ob,t c ob,t so long as y t = y ob,t, b 1 = b 1, and b t 1 = 0 and T t y,t, b,t 1, b 1, = or any suffcently hgh number for any other y t, b t 1. Ths tax sequence de facto shuts down all assets markets by penalzng agents for choosng anythng other than b,t = 0 20

22 and restrctng agents budget sets to sequences { } c ob, l ob. Incentve compatblty then ensures that an optmum s mplemented. Ths s not a unque way to mplement t. Instead of shuttng down asset markets entrely, one can choose any sequence {b } and re-defne taxes approprately to ensure that n an optmal equlbrum agent chooses b. 13 When agents assets are unobservable, the problem becomes more nterestng. We 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 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. 14 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, 18 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 = are satsfed. n y r,t 19 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 13 It s easy to smooth the constructed tax functon to mplement the optmum wth contnuous or dfferentable tax functons see, for example, Kocherlakota 2005, Wernng 2009, Grochulsk and Kocherlakota The concluson that nether publc nor prvate assets are pnned down contnues to hold n such mplementatons. 14 See Golosov and Tsyvnsk 2007 for detals. 21

23 contracts. Gven that, we frst analyze ths economy when prvate borrowng s subject to the ad hoc lmt 13. Then we dscuss how our conclusons would change f debt enforcement on prvate markets s 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, θ,t 20 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 20. In prncple, agent can further ncrease hs utlty from bundle x j, 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 and the ncentve constrant 20. 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 for all, j. 21 c adhoc j,0 + b, 1 b j, 1, yadhoc j,0 θ,0 c adhoc,t θ,1 q t b = x adhoc,t + b,t, 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 22 } t>0. Constrant 20 ensures

24 that the optmal report 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,t. Observe that the 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 tax payments 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 the 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 21 would hold wth equalty n 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 hand sde of 20. It s straghtforward to generalze these results to economes wth dosyncratc shocks, rcher asset markets, and captal. 15 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 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. 15 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. In a related paper, Bassetto and Kocherlakota 2004 show that allowng tax functons that are suffcently flexble n terms of hstory dependence makes the path of debt rrelevant. 23

25 7 Concludng remarks The sprng of 2013 wtnessed a lvely debate n newspapers and economc magaznes about the accuracy and meanng of emprcal correlatons between output growth rates and ratos of government debt to GDP nferred from data sets assembled by Renhart and Rogoff From the perspectves of ths paper and of Wernng 2007, those correlatons and some of the contrbutons to those debates are dffcult to nterpret because our models tell us that total government debt per se does not mpnge on allocatons, government transfers, or tax rates. 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. To nterpret those Renhart-Rogoff facts country-by-country, we would want to know much more about how the dstrbutons of net assets across people have vared across 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 debt s determned when agents abltes to borrow are restrcted because that allows prospectve publc debt ssues to affect nterest rates. However, ths path s ndependent of the current debt level. We restrcted our analyss to economes n whch the government commts to future polces. We beleve that a promsng drecton for research s to explore the role of debt n economes n whch a government cannot commt. As our dscusson n Secton 4 suggests, mperfect commtment may mpose addtonal restrctons on transfers and debt that are feasble n equlbrum. We leave ths extenson to future work. 24

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