Optimal Government Debt Maturity

Transcription

1 Optimal Government Debt Maturity Davide Debortoli Ricardo Nune Pierre Yared October 13, 214 Abtract Thi paper develop a model of optimal government debt maturity in which the government cannot iue tate-contingent bond and cannot commit to fical policy. If the government can perfectly commit, it fully inulate the economy againt government pending hock by purchaing hort-term aet and iuing long-term debt. Thee poition are quantitatively very large relative to GDP and do not need to be actively managed by the government. Our main reult i that thee concluion are not robut to the introduction of lack of commitment. Under lack of commitment, large and tilted poition are very expenive to finance ex-ante ince they exacerbate the problem of lack of commitment ex-pot. In contrat, a flat maturity tructure minimize the cot of lack of commitment, though it alo limit inurance and increae the volatility of fical policy ditortion. We how that the optimal maturity tructure i nearly flat becaue reducing average borrowing cot i quantitatively more important for welfare than reducing fical policy volatility. Thu, under lack of commitment, the government actively manage it debt poition and can approximate optimal policy by confining it debt intrument to conol. Keyword: Public debt, optimal taxation, fical policy JEL Claification: H63, H21, E62 We would like to thank Marco Baetto, Fernando Broner, V.V. Chari, Iabel Correia, Mike Goloov, Patrick Kehoe, Aleandro Lizzeri, Jean-Baptite Michau, Facundo Piguillem, Andrew Scott, Pedro Tele, Joaquim Voth, Iván Werning, and eminar participant at Bank of Portugal, Columbia, CREI, EIEF, European Univerity Intitute, Fed Board of Governor, London Buine School, NBER, Pari School of Economic, Univeridade Católica Portuguea, and Univeridade Nova de Liboa for comment. The view expreed in thi paper are olely the reponibility of the author and hould not be interpreted a reflecting the view of the Board of Governor of the Federal Reerve Sytem or of any other peron aociated with the Federal Reerve Sytem. Univeritat Pompeu Fabra and Barcelona GSE: davide.debortoli@upf.edu. Board of Governor of the Federal Reerve Sytem: ricardo.p.nune@frb.gov. Columbia Univerity and NBER. pyared@columbia.edu. 1

2 1 Introduction How hould government debt maturity be tructured? Two eminal paper by Angeleto 22) and Buera and Nicolini 24) argue that the maturity of government debt can be optimally tructured o a to completely hedge the economy againt hock. Thi reearch conclude that optimal debt maturity i tilted long, with the government purchaing hort-term aet and elling long-term debt. Thee debt poition allow the market value of outtanding government liabilitie to decline when pending need and hort-term interet rate increae. Moreover, quantitative exercie imply that optimal government debt poition, both hort and long, are large in abolute value) relative to GDP. Finally, thee poition are contant and do not need to be actively managed ince the combination of contant poition and fluctuating bond price deliver full inurance. In thi paper, we how that thee concluion are enitive to the aumption that the government can fully commit to fical policy. In practice, a government chooe taxe, pending, and debt equentially, taking into account it outtanding debt portfolio, a well a the behavior of future government. Thu, a government can alway purue a fical policy which reduce increae) the market value of it outtanding newly-iued) liabilitie ex-pot, even though it would not have preferred uch a policy ex-ante. We how that once the lack of commitment by the government i taken into account, it become cotly for the government to ue the maturity tructure of debt to completely hedge the economy againt hock; there i a tradeoff between the cot of funding and the benefit of hedging. 1 Our main reult i that, under lack of commitment, the optimal maturity tructure of government debt i quantitatively nearly flat and i actively managed by the government. We preent thee finding in the dynamic fical policy model of Luca and Stokey 1983). Thi i an economy with no capital and with public pending hock in which the government chooe linear taxe on labor and iue public debt to finance government pending. Our model feature two important friction. Firt, a in Angeleto 22) and Buera and Nicolini 24), we aume that tate-contingent bond are unavailable, and that the government can only iue real non-contingent bond of all maturitie. Second, and in contrat to Angeleto 22) and Buera and Nicolini 24), we aume that the government lack commitment to policy. We focu on the Markov Perfect Competitive Equilibrium in which the government dynamically chooe it policie at every date a a function of payoff relevant variable: the fical hock and it outtanding debt poition at variou maturitie. Neither of thee friction on it own lead to any inefficiency, ince debt maturity can be tructured o a to addre each friction eparately. Firt, the work of Angeleto 22) and Buera and Nicolini 24) how that, even in the abence of contingent bond, an optimally tructured portfolio of non-contingent bond can perfectly inulate the government from all 1 Our framework i conitent with an environment in which the legilature equentially chooe a primary deficit and the debt management office equentially minimize the cot of financing ubject to future rik, which i what i done in practice ee the IMF report, 21). 1

3 hock to the economy. Second, the work of Luca and Stokey 1983) how that, even in the abence of commitment by the government, an optimally tructured portfolio of contingent bond can perfectly induce a government without commitment to purue the ex-ante optimally choen policy ex-pot. While each of thee two friction in iolation i irrelevant for welfare, the combination of the two lead to a non-trivial tradeoff between market completene and commitment in the government choice of maturity. To get a ene of the importance of each friction, it i ueful to conider for implicity environment which only feature one friction and illutrate how maturity tructure can be deigned to addre each friction eparately. Suppoe for intance that the government lack commitment, but there are no hock, implying that the government doe not need to worry about inurance. In thi cae, the ex-ante optimal policy under full commitment i perfectly mooth taxation. Moreover, there are many maturity tructure under full commitment which can lead to thi perfectly mooth outcome. However, under lack of commitment, a government today can only guarantee commitment to thi mooth policy by future government by chooing a flat maturity tructure. A tilted debt poition would caue a future government to deviate from the optimal mooth path. Suppoe, for example, that a future government were to chooe policy while entering the period with zero hort-term debt and poitive long-term debt. Rather than puruing mooth taxation a it i uppoed to, the government ha an incentive to deviate to a non-mooth tax policy which reduce hort-term conumption and increae long-term conumption. Thi deviation benefit the government by reducing the market value of it outtanding liabilitie by increaing hort-term interet rate). 2 Analogouly, if a future government chooe policy while entering the period with poitive hort-term debt and zero long-term debt, then it ha an incentive to deviate to a non-mooth tax policy which increae hort-term conumption and decreae long-term conumption. Thi deviation benefit the government by increaing the market value of it newly iued hort-term liabilitie by reducing hort-term interet rate). Thu, only a flat maturity tructure can guarantee that taxe remain mooth, ince in thi cae the government doe not have any beneficial deviation ex-pot i.e, any deviation marginal effect on the market value of outtanding debt equal that on the market value of newly iued debt). Importantly, while a flat maturity tructure minimize the cot of lack of commitment, thi cot would increae the larger and more tilted are the debt poition. Large and tilted poition are very cotly to finance ex-ante if the government cannot commit to policy ex-pot. To ee why, note that baed on our above dicuion, the larger and more tilted the debt poition, the greater a future government benefit from puruing a non-mooth tax policy ex-pot to relax it budget contraint. Ex-ante, houehold purchaing government bond internalize the fact that ex-pot, the government will chooe non-mooth tax policie relative to thoe under commitment. 2 Our obervation that long-term debt poition lead to lower fical dicipline i conitent with other argument in the literature on debt maturity ee Miale and Blanchard, 1994; Miale et al., 22; Chatterjee and Eyigungor, 212; and Broner et al., 213). 2

4 A a conequence, ex-ante, they require a greater premium for lending to the government. For example, if houehold are primarily buying long-term bond ex-ante, then they appropriately anticipate that the government lacking commitment will purue future policie which increae future hort-term interet rate, thereby diluting their claim. In thi cae, houehold require a higher ex-ante interet rate relative to commitment) to induce them to lend long-term to the government. An analogou reaoning hold if houehold are primarily buying hort-term bond ex-ante. Therefore, even though the maturity of debt iuance doe not affect the cot of financing under full commitment, under lack of commitment, larger and more tilted debt poition are more expenive to finance. To get a ene of how maturity can be ued to reduce the cot of lack of inurance, uppoe for implicity now that the government ha full commitment, but there are fical hock againt which the government mut inure. The optimal policy under commitment ue debt to mooth taxation in the preence of thee hock. If fully contingent claim were available, there would be many maturity tructure which would upport the optimal policy. However, if the government only ha acce to non-contingent claim, then there i a unique maturity tructure which can be choen which replicate full inurance. A ha been hown in Angeleto 22) and Buera and Nicolini 24), uch a maturity tructure i tilted in a manner which guarantee that the market value of outtanding government liabilitie decline when the net preent value of future government primary urplue alo decline. If thi occur when hort-term interet rate rie a i the cae in quantitative example then the optimal maturity tructure require that the government purchae hort-term aet and ell long-term debt. Becaue interet rate movement are mall quantitatively, the tilted debt poition required for hedging are large. In uch an environment, contraining the government to iuing a flat debt maturity i.e., in the form of a conol) i cotly. The reaon i that the market value of debt doe not fluctuate enough to provide full inurance, and thi induce more volatility in fical policy ditortion than would be achieved under perfect inurance. Thu, a flat maturity tructure minimize the cot of commitment, wherea a large and tilted maturity tructure minimize the cot of volatility. In the preence of both lack of commitment and lack of inurance, the government face a tradeoff. If it chooe a large and tilted debt poition a it would under full commitment it would reduce the volatility of fical policy ditortion, but becaue of lack of commitment, uch large and tilted poition would be very expenive to finance and would entail large average tax ditortion. To explore where the government poition itelf in thi tradeoff, we imulate a two-hock economy with both friction in which the government iue a one-year bond and a conol, and we characterize optimal policy. Our main reult i that, under lack of commitment, the optimal maturity tructure of government debt i quantitatively nearly flat. In our benchmark imulation, the hort-term bond the one-year bond plu the annual conol payout due in one year) i 2.2% of GDP and the market value of the conol i 59.7% of GDP, with annual payout equal to 2.21% of GDP. Thu, the optimal maturity tructure i eentially flat, and optimal policy under lack of commitment 3

5 can be approximated with a conol. 3 Thi reult contrat with the cae of full commitment, in which the hort-term bond i -269% of GDP and the market value of the conol i 276% of GDP, with annual payout equal to 11.8% of GDP. Moreover, in contrat to the cae of full commitment, we find that under lack of commitment, debt i actively managed, and both taxe and debt are volatile and repond peritently to fical hock. Our quantitative reult emerge becaue of the combination of two force. Firt, ubtantial hedging require maive tilted debt poition, a ha been hown in Angeleto 22) and Buera and Nicolini 24). Due to their ize, financing thee poition can be very expenive in term of average tax ditortion becaue of the lack of commitment by the government. Second, under empirically plauible level of volatility of public pending, the cot of lack of inurance under a flat maturity tructure i mall. Therefore, the optimal policy puhe in the direction of reducing average tax ditortion veru reducing the volatility of tax ditortion, and the reult i a nearly flat maturity tructure. Thu, in the preence of lack of commitment by the government, optimal government debt policy can be approximated by active conol management. 4 Thi paper i connected to everal literature. A dicued, we build on the work of Angeleto 22) and Buera and Nicolini 24) by introducing lack of commitment. 5 In thi regard, our work i related to that of Arellano and Ramanarayanan 212) and Aguiar and Amador 213), but in contrat to thi work, we ignore the poibility of default and focu purely on lack of commitment to taxation and debt iuance. Our work i alo complementary to that of Arellano et al. 213), but in contrat to thi work, we ignore the preence of nominal friction and the lack of commitment to monetary policy. In thi regard, our work i mot applicable to economie in which the rik of default and urprie inflation are not alient, but the government i till not committed to a path of deficit and debt maturity iuance. 6 More broadly, our paper i alo tied to the literature on optimal fical policy which explore the role of non-contingent debt and lack of commitment. A number of paper have tudied optimal policy under full commitment but non-contingent debt, uch a Barro 1979) and Aiyagari 3 Though thi policy precription differ from current practice in advanced economie, it ha been purued hitorically, mot notably by the Britih government in the Indutrial Revolution, when conol were the larget component of the Britih government debt during thi time period ee Mokyr 211)). Confining debt iuance to conol i alo a policy which receive ome upport in the popular pre e.g., Leitner and Shapiro 213) and Ygleia 213)). 4 It hould be mentioned that the concluion that the welfare benefit of moothing economic hock i mall relative to that of raiing economic level i more generally tied to the inight in Luca 1987). 5 Additional work explore government debt maturity while continuing to maintain the aumption of full commitment. Shin 27) explore optimal debt maturity when there are fewer debt intrument than tate. Faraglia et al. 21) explore optimal debt maturity in environment with habit, productivity hock, and capital. Lutig et al. 28) explore the optimal maturity tructure of government debt in an economy with nominal rigiditie. Guibaud et al. 213) explore optimal maturity tructure in a preferred habitat model. 6 Chari and Kehoe 1993a,b) and Sleet and Yeltekin 26) alo conider the lack of commitment under full inurance, though they focu on etting which allow for default. Niepelt 28) alo focue on default rik. Alvarez et al. 24) and Peron et al. 26) conider problem of commitment in an environment with longterm debt where the poibility of urprie inflation arie. 4

6 et al. 22). 7 A in thi work, we find that optimal taxe repond peritently to economic hock, though in contrat to thi work, thi peritence i due to the lack of commitment by the government a oppoed to the ruling out of long-term government bond. Other work ha tudied optimal policy in etting with lack of commitment, but with full inurance e.g., Kruell et al., 26 and Debortoli and Nune, 213). We depart from thi work by introducing longterm debt, which in a etting with full inurance implie that the lack of commitment friction no longer introduce any inefficiencie. Our paper proceed a follow. In Section 2, we decribe the model. In Section 3, we define the equilibrium and characterize it recurively. In Section 4, we review the optimal policy in the abence of lack of commitment and lack of inurance. In Section 5, we dicu the cot of lack of commitment in a determinitic environment, and we how that thi cot increae the larger and more tilted i the debt maturity. In Section 6, we dicu the cot of lack of inurance in an environment with full commitment, and we how that thi cot i mall for empirically plauible volatilitie of public pending. In Section 7, we combine lack of commitment and lack of inurance and perform our main quantitative exercie and preent our main reult. Section 8 conclude and the Appendix provide all of the proof and additional reult not included in the text. 2 Model 2.1 Environment We conider an economy identical to that of Luca and Stokey 1983) with two modification. Firt, we rule out tate-contingent bond. Second, we aume that the government cannot commit to fical policy. There are dicrete time period t = {1,.., } and a tochatic tate t S which follow a firt-order Markov proce. i given. Let t = {,..., t } S t repreent a hitory, and let π t+k t) repreent the probability of t+k conditional on t for t + k t. There i a continuum of ma 1 of identical houehold that derive the following utility: E β t uc t,n t ) + θ t t )v g t )), β,1). 1) t= c t i conumption, n t i labor, and g t i government pending. u ) i trictly increaing in conumption and trictly decreaing in labor, global concave, and continuouly differentiable. v ) i trictly increaing, concave, and continuouly differentiable. Under thi repreentation, θ t t ) i high low) when public pending i more le) valuable. In contrat to the model of Luca and Stokey 1983), we have allowed g t in thi framework to be choen by the government, a oppoed to being exogenouly determined. We allow for thi poibility to alo conider that 7 See alo Farhi 21). 5

7 the government may not be able to commit to the ex-ante optimal level of public pending. In our exercie, we conider commitment to taxe and commitment to pending eparately, o our analyi ubume the Luca and Stokey 1983) environment in which there i no dicretion over government pending. Houehold wage are normalized to 1 and are taxed at a linear tax rate τ t. b t+k t repreent government debt purchaed by a repreentative houehold at t, which i a promie to repay 1 { unit of conumption { } at t+k > t, and qt t+k i it price at t. At every t, the houehold allocation } c t,n t, mut atify the houehold dynamic budget contraint B t+k t b t+k t k=1 c t + qt t+k k=1 b t+k t ) b t+k t 1 = 1 τ t )n t + b t t 1. 2) repreent debt iued by the government at t with a promie to repay 1 unit of { { } } conumption at t + k > t. At every t, government policie τ t, mut atify the government dynamic budget contraint g t + B t t 1 = τ t n t + qt t+k k=1 B t+k t The economy i cloed and bond are in zero net upply: B t+k t k=1,g t ) Bt 1 t+k. 3) b t+k t which combined with 2) and 3) implie that { Initial debt cheme: B k 1 1 } = B t+k t t,k, 4) c t + g t = n t. 5) k=1 i exogenou.8 We aume that there exit debt limit to prevent Ponzi B t+k t [ B,B ]. 6) We let B be ufficiently low and B be ufficiently high o that 6) doe not bind in our theoretical and quantitative exercie. A key friction in thi environment i the abence of tate-contingent debt, ince the value of outtanding debt Bt t+k i independent of the realization of the tate t+k. If tate-contingent bond were available, then at any date t, the government would own a portfolio of bond 8 Our model implicitly allow the government to buy back the long-term bond from the private ector. While ruling out bond buyback i intereting, a 212 urvey by the OECD found that 85 percent of countrie conduct ome form of bond buyback and 32 percent of countrie conduct them on a regular bai ee the OECD report by Blommetein et al., 212). Note furthermore, that even if bond buyback i not allowed in our environment, a government can replicate the buyback of a long-term bond by purchaing an aet with a payout on the ame date ee Angeleto, 22). 6

8 { {Bt 1 t+k )} t+k S t+k} k=, where the value of each bond payout at date t + k would depend on the realization of a hitory of hock t+k S t+k. In our dicuion, we will refer back to thi complete market cae. The government i benevolent and hare the ame preference a the houehold in 1). We aume that the government cannot commit to policy and therefore chooe taxe, pending, and debt equentially. 3 Markov Perfect Competitive Equilibrium 3.1 Definition of Equilibrium We conider a Markov Perfect Competitive Equilibrium MPCE) in which the government mut optimally chooe it preferred policy at every date a a function of current payoff-relevant variable. The government take into account that it choice affect future debt and thu affect the policie of future government. Houehold rationally anticipate thee future policie, and their expectation are in turn{ reflected } in current bond price. Formally, let B t Bt t+k. In every period t, the government enter the period and k=1 { { } } chooe a policy {τ t,g t,b t } given { t,b t 1 }. Houehold chooe an allocation c t,n t, given the policy and { t,b t 1 }. An MPCE conit of a government trategy ρ t,b t 1 ) which depend on t,b t 1 ), and a houehold allocation trategy ω t,b t 1 ),ρ t ) which depend on t,b t 1 ) and on the government policy ρ t = ρ t,b t 1 ) uch that 1. The government trategy maximize 1) given t,b t 1 ), the houehold allocation trategy ω ), bond price, and government budget contraint 3), 2. The houehold allocation trategy maximize 1) given t,b t 1 ) and policy ρ t, the government trategy ρ ), bond price, and houehold budget contraint 2), and 3. Bond price atify 4) given the government trategy ρ ) and the houehold allocation trategy ω ). While we have aumed for generality that the government can freely chooe taxe, pending, and debt in every period, we focu throughout our draft on the cae in which the government doe not have dicretion in either etting pending or in etting taxe. Thee pecial cae highlight how the right choice of government debt maturity can induce future government to chooe the commitment policy. The exact manner in which we do thi i decribed in Section Primal Approach Any MPCE mut be a competitive equilibrium. We follow Luca and Stokey 1983) by taking the primal approach to the characterization of competitive equilibria ince thi allow u to b t+k t k=1 7

9 abtract away from bond price and taxe. Let { {ct t ),n t t ),g t t )} t S t } t= 7) repreent a tochatic equence, where the reource contraint 5) implie c t t ) + g t t ) = n t t ). 8) We can etablih neceary and ufficient condition for 7) to contitute a competitive equilibrium. The houehold optimization problem implie the following intratemporal and intertemporal condition, repectively: 1 τ t t ) = u n,t t ) u c,t t ) and qt+k t β k π t+k t) u c,t+k t+k ) t ) = t+k S t+k u c,t t. 9) ) Subtitution of thee condition into the houehold dynamic budget contraint implie the following condition: u c,t t ) c t t ) + u n,t t ) n t t ) + k= t+k S t+k β k π β k π t+k t) u c,t+k t+k) B t+k t t ) = 1) t+k S t+k t+k t) u c,t+k t+k) Bt 1 t+k t 1 ). k=1 Forward ubtitution into the above equation taking into account the abence of Ponzi cheme implie the following implementability condition: k= t+k S t+k β k π t+k t) u c,t+k t+k) c t+k t+k) + u n,t+k t+k) n t+k t+k)) = 11) k= t+k S t+k β k π t+k t) u c,t+k t+k) Bt 1 t+k t 1 ). By the argument jut made, if a tochatic equence in 7) i generated by a competitive equilibrium, then it necearily atifie 8) and 11). We prove in the Appendix that the convere i alo true, which lead to the below propoition which i ueful for the ret of our analyi. Propoition 1 competitive equilibrium) { A tochatic equence 7) i a competitive equilibrium if and only if it atifie 8) t and Bt 1 t+k {{ t 1 )} } } which atifie 11) t. k= t S t t= 8

10 A ueful corollary to thi propoition concern the relevant implementability condition in the preence of tate-contingent bond, B t t+k ), which provide payment conditional on the realization of a hitory t+k. Corollary 1 In the preence of tate-contingent debt, a tochatic equence 7) i a competitive equilibrium if and only if it atifie 8) t and 11) for t = given initial liabilitie. If tate-contingent debt i available, then the atifaction of 11) at guarantee the atifaction of 11) for all other hitorie t, ince tate-contingent payment can be freely choen o a to atify 11) at all future hitorie t. 3.3 Recurive Repreentation of MPCE We can ue the primal approach to repreent an MPCE recurively. Recall that ρ t,b t 1 ) i a policy which depend on t,b t 1 ), and that ω t,b t 1 ),ρ t ) i a houehold allocation trategy which depend on t,b t 1 ) and on the government policy ρ t = ρ t,b t 1 ). A uch, an {{ MPCE in equilibrium i characterized by a tochatic equence in 7) and a debt equence Bt t+k t )} }, where each element depend only on t through t,b t 1 ), the payoff k=1 t= relevant variable. Given thi obervation, in an MPCE, one can define a function h k ) h k t,b t 1 ) = β k E [u c,t+k t,b t 1 ] 12) for k, which equal the dicounted expected marginal utility of conumption at t + k given t,b t 1 ) at t. Thi function i ueful ince in making a policy deciion at date t, the government mut take into account how it affect future expectation of policy which in turn affect current bond price through expected future marginal utility of conumption. Note furthermore that chooing {τ t,g t,b t } at date t i equivalent to chooing {c t,n t,g t,b t } from the perpective of the government, and thi follow from the primal approach delineated in the previou ection. Thu, we can write the government problem recurively a V t,b t 1 ) = max c t,n t,g t,b t u c,t ct Bt 1) t + un,t n t + β uc t,n t ) + θ t t ) v g t ) + β k=1 t+1 S.t. t+1 S π t+1 t )V t+1,b t ) 13) c t + g t = n t, 14) π t+1 t )h k 1 t+1,b t ) ) Bt t+k Bt 1 t+k =, 15) where 15) i a recurive repreentation of 1). Let f t,b t 1 ) correpond to the olution to 13) 15) given V ) and h k ). It therefore follow that the function f ) necearily implie 9

11 a function h k ) which atifie 12). An MPCE i therefore compoed of function V ), f ), and h k ) which are conitent with one another and atify 12) 15). 4 Full Commitment and Full Inurance Benchmark Before conidering how the lack of commitment and lack of inurance interact in our framework, it i ueful to firt characterize optimal policy in the abence of thee two friction. To facilitate expoition, in thi ection a well a in Section 5 and 6, we aume that the government ha zero initial debt liabilitie. In the quantitative exercie of Section 7, we relax thi aumption and take into account that the government ha non-zero initial liabilitie. Given Corollary 1, in the preence of tate-contingent debt and full commitment, optimal policy olve the following program: max {{c t t ),n t t ),g t t )} t S t} t= t= β t π t ) u c t t ),n t t )) + θ t t )v g t t ))) 16) t S t.t. 8) t and 11) for t =. 17) Letting µ correpond to the Lagrange multiplier on the implementability contraint 11), firt order condition yield: u c,t t ) = u n,t t ) [ µ u cc,t t ) + u cn,t t )) c t t ) )] µ u nn,t t ) + u cn,t t )) n t t ) and 18) θ t t ) v g,t t ) = u c,t t ) [ 1 + µ) + µ u cc,t t ) c t t ) + u cn,t t ) n t t ) ] u c,t t. 19) ) Propoition 2 Luca-Stokey benchmark) In the preence of full commitment and full inurance, optimal policy atifie 18) and 19) for a given µ. There are two important point to keep in mind from thi benchmark. Firt, 18) and 19) imply that in the preence of full inurance, taxe and pending are independent of hitory and depend only on the tate t. Thi follow from the preence of perfect inurance market. A in Luca and Stokey 1983), we aume that the program in 16) 17) i globally concave o that firt order condition are ufficient to characterize the optimum. Thi implie that we can define {c t ),n t ),g t ),τ t )} t S 2) a the level of conumption, labor, pending, and taxe which depend only on the tate t in the full inurance and full commitment benchmark. We will refer back to thee quantitie when dicuing the cot of lack of commitment and lack of inurance. 1

12 A econd point to notice about the olution to the program i that 18) and 19) imply the poible preence of fical policy ditortion. For example, if the term in bracket in 18) i poitive, then thi implie that u c,t t ) > u n,t t ), which mean that the marginal utility of conumption exceed the marginal utility of leiure and taxe are poitive, which i ditortive. In other word, if given the opportunity, the government would prefer to reduce thee ditortion by increaing conumption and increaing labor. Analogouly, if the term in bracket in equation 19), exceed 1, then thi implie that θ t t ) v g,t t ) > u c,t t ), which mean that the marginal utility of public pending exceed the marginal utility of conumption. Thu, if given the opportunity, the government may chooe to increae public pending while reducing conumption. A government deire to reduce ditortion i ueful to note ince it come into play once we take into account the government incentive to deviate from the commitment policy. We now move to illutrate how lack of commitment and lack of inurance alter the equilibrium decribed in Propoition 2. In Section 5 and 6, we how how, under only one friction, maturity can be tructured to fully alleviate the cot of the friction. In Section 7, we combine the two friction and preent our main reult. 5 Cot of Lack of Commitment In thi ection, we conider the problem of the lack of commitment by the government in a determinitic environment in which the maturity tructure can be deigned to addre thi problem. We how that the optimal maturity tructure in thi cae i flat. We then how how chooing a tilted maturity tructure can entail welfare loe in thi etting, and we do thi uing a imple three-period example with a quantitative exercie. Before proceeding, we want to note that we conider the problem of lack of commitment to taxe and lack of commitment to pending eparately. We do thi both to remain in line with the model of Luca and Stokey 1983) which doe not allow any dicretion over the choice of pending and alo to explore the role of maturity when there i dicretion over pending. We will ay that the government lack commitment to taxe if the government chooe taxe and debt equentially but it cannot alter pending which i exogenou et at g t t ) = g t ) defined in 2) i.e., the tate-contingent level of pending under commitment). Analogouly, we will ay that the government lack commitment to pending if the government chooe pending and debt equentially but it cannot alter taxe which are exogenouly et at τ t t ) = τ t ) i.e., the tate-contingent level of taxe under commitment). The ituation in which the government lack commitment to taxe and pending correpond to the ituation in which taxe, pending, and debt are choen equentially. A we will ee, our main reult hold a long a the government lack commitment to at leat one intrument. 11

13 5.1 Flat Maturity Solve Lack of Commitment Conider a determinitic environment in which 1 and t = 1 t 1, with θ ) = θ H > θ 1 ) = θ L. A in Section 4, uppoe there are zero initial government debt liabilitie for expoitional implicity. In thi environment, the government at date finance a one time pending increae by iuing debt into the future, and the marginal value of pending remain contant in all future date. To ee how maturity can be ued to olve the problem of lack of commitment, conider firt the program under full commitment. From Propoition 2, it i clear that the optimal policy et ome policy {c,n,g,τ } followed by a contant policy {c 1,n 1,g 1,τ 1 } for t 1, and thi i becaue the tate of the economy doe not change from t = 1 onward. The government iue ome debt at date to finance a pending increae, and in all future period, it run a contant primary urplu to pread out the repayment of that debt. Under commitment, thi policy equence can be implemented with a large number of different debt maturitie iued at date. To undertand the intuition, conider the implementability condition 11) at date 1 in thi determinitic economy. It i clear that a large number of different maturity tructure can atify the right hand ide of 11) given an optimal equence conumption and labor equence {c 1,n 1 } which pin down the left hand ide. Clearly, one uch maturity tructure i flat. Specifically, let B = τ 1 g 1 for t 1, 21) where B repreent the primary urplu in the full commitment optimum from t = 1 onward. The government can thu atify the implementability contraint with a flat maturity with B t = B t 1. Can the government lacking commitment chooe the optimal commitment policy ex-pot? The anwer to thi quetion i ye, if the government only lack commitment to either taxe or to pending but not both), and thi can be enured by iuing debt with a flat maturity tructure at date. To gain an intuition to thi reult, uppoe that a government entering the period at t = 1 were to perform a one-time reevaluation of policy, fully committing to it new policy equence thereafter. If it were entering date 1 with a flat maturity tructure, it would olve the following problem: max {c t,n t,g t} t= β t 1 uc t,n t ) + θ L v g t ) ) 22) t=1.t. c t + g t = n t t, and 23) β t 1 u c,t ct B ) ) + u n,t n t =. 24) t=1 12

14 Under lack of commitment to taxe, uch a government face the additional contraint that g t = g 1, 25) and under lack of commitment to pending, uch a government face the additional contraint that u n,t u c,t = 1 τ 1). 26) It i clear that under both lack of commitment to taxe and lack of commitment to pending, the tradeoff faced by the government are the ame in all future period, and if the problem i globally concave, then it will chooe a contant allocation and policy. Moreover, it can be hown that thi contant allocation and policy correpond to the optimum under commitment. In other word, a flat maturity induce the government to chooe the ex-ante optimum ex-pot. Thi inight generalize further to a ituation in which the government reoptimize in all future period a it would in an MPCE. Thi i expreed formally in the below propoition. Propoition 3 optimality of flat maturity) In the determinitic environment with 1 and t = 1 t 1, there exit an MPCE in which B t = Bt+k t 1 and k where the allocation under the MPCE i identical to the commitment optimum if 1. There i lack of commitment to taxe and the program in 22) 24) and 25) i globally concave, or 2. There i lack of commitment to pending and the program in 22) 24) and 26) i globally concave. In the abence of commitment to both taxe and to pending, it i not poible to olve the problem of lack of commitment by uing debt maturity becaue the government ha too many tool at it dicretion. While a government reevaluating policy under a flat maturity would chooe a mooth policy a it doe in the example above it may not necearily chooe the ame mooth policy a it would have prefered ex-ante. 9 In our quantitative exercie, we do conider the poibility that the government lack commitment to both taxe and to pending, and we can how that, though a flat maturity tructure doe not remove the cot of lack of commitment altogether, it doe minimize thi cot and i therefore alo optimal Three-Period Example We turn to a imple three-period example to provide further intuition for the reult in Propoition 3. The advantage of thi example i that it can be eaily olved by hand and it highlight 9 See Roger 1989) for more dicuion. 1 In principle, there are many poible MPCE, and our reult characterize one uch MPCE. It hould be noted however that the MPCE which we characterize alo correpond to the limit of the finite period economy with end date T a T. 13

15 the reaon for the theoretically very high welfare cot of chooing a large and tilted maturity tructure. The diadvantage of thi example i that it i too tylized to be able to make quantitative tatement, and in the next ubection we conider uch a computed example to calculate the welfare cot of lack of commitment. Let t =,1,2, θ = θ H, and θ 1 = θ 2 = θ L < θ H. Suppoe that the government lack commitment to pending and that taxe and labor are exogenouly fixed to ome τ and n, repectively, o that the government collect a contant revenue in all date. 11 Aume that the government welfare can be repreented by β t 1 ψ) log c t + ψθ t g t ) 27) t=,1,2 for ψ [,1]. We conider the limiting cae in which ψ 1. In thi environment, the government doe not have any dicretion over tax policy, and any ex-pot deviation by the government i driven by a deire to increae pending ince the marginal benefit of additional pending alway exceed the marginal benefit of conumption The Optimality of a Flat Maturity We now preent the main reult in Propoition 3 in the context of thi model and dicu the intuition for it. The analog of the implementability condition at date in 1) can be written a a weak inequality contraint ince it bind in the optimum): which after ubtitution yield the analog of 11): c n 1 τ) c + β B1 c 1 + β 2B2 c 2 28) c n 1 τ) c + β c 1 n 1 τ) c 1 + β 2c 2 n 1 τ) c 2. 29) The optimum under commitment maximize 27) ubject to 29), which lead to the following optimality condition: ) θ H 1/2 c = c 1 = c 2 = n 1 τ) + B 3) θ L for ome B which atifie 29) when it bind. In thi ituation, houehold lend to the government at t = to finance the initial pending urge, and the government provide them with net tranfer of B at t = 1 and t = 2. Clearly, under full commitment, any date iuance of { B 1,B2 } atifying B 1 + βb2 = B 1 + β) 31) 11 Such a ituation would prevail for example if taxe are contant and the underlying preference atify thoe of Greenwood et al. 1988). 14

16 can implement the equilibrium, where we have taken into account that the one-period bond price at date 1 i β. However, thi i not the cae under lack of commitment. In thi cae, one mut ue backward induction to olve for the behavior of the government. At t = 2, the government policy i trivially determined by the implementability condition which et c 2 = n 1 τ)+b1 2. At t = 1, the government take thi future policy into account, and given outtanding debt { B 1 },B2, it maximize it continuation welfare ubject to the date 1 implementability condition, 11), which can be written a a weak inequality contraint: c 1 n 1 τ) c 1 + β c 2 n 1 τ) c 2 B1 c 1 + β B2 c 2. 32) It can be hown that in thi cae, optimal government policy atifie c 1 c 2 = ) n 1 τ) + B 1 1/2. 33) n 1 τ) + B 2 Equation 33) implie that if B 1 < >)B2, then c1 < >) c2. A uch, it i only when debt maturity i flat with B 1 = B2 that thi guarantee that c1 = c2, o that the olution under lack of commitment in 33) coincide with that under full commitment in 3). The government at date take thi outcome into account and therefore chooe a flat maturity tructure o a to guarantee that the date 1 government will follow the ex-ante optimal policy. Lemma 1 In the three-period example, the unique MPCE i identical to the commitment optimum and the optimal debt maturity et B 1 = B2. Why doe a flat maturity tructure induce the commitment olution? A way to ee thi i to conider the fact that the date 1 government which care only about raiing pending would like to reduce the market value of what it owe to the private ector which from the intertemporal condition can be repreented by B 1 + β c 1 c 2 B 2. 34) Moreover, the government would alo like to increae the market value of newly iued debt which can be repreented by β c 1 c 2 B ) If debt maturity were tilted toward the long end, then the date 1 government would deviate from a mooth policy o a to reduce the value of what it owe. For example. uppoe that B 1 = and B2 = B 1 + β) /β. Clearly, under commitment, it would be poible to achieve the optimum under thi debt arrangement. However, under lack of commitment, 33) implie that the government deviate from the mooth ex-ante optimal policy by chooing c 1 < c2. Thi deviation, which i achieved by iuing higher level of debt B 2 1 relative to commitment erve 15

17 to reduce the value of what the government owe in 34), therefore freeing up reource to be utilized for additional pending at date 1. Analogouly, if debt maturity were tilted toward the hort end, then the government would deviate from a mooth policy o a to increae the value of what it iue. For example, uppoe that B 1 = B 1 + β) and B2 =. A in the previou cae, thi debt arrangement would implement the optimum under commitment. However, rather than chooing the ex-ante optimal mooth policy, the date 1 government chooe policy according to 33) with c 1 > c2. Thi deviation, which i achieved by iuing lower level of debt B1 2 relative to commitment erve to increae the value of what the government iue in 34), therefore freeing up reource to be utilized for additional pending at t = 2. It i only when B 1 = B2 = B that there are no gain from deviation. In thi cae, it follow from 3) that B1 2 = B2, and therefore any deviation marginal effect on the market value of outtanding debt i perfectly outweighed by it effect on the market value of newly iued debt. For thi reaon, a flat debt maturity tructure induce commitment What i the Welfare Cot of Lack of Commitment? One way to invetigate thi quetion i to conider the ource of the welfare loe which would enue if the government were to chooe ome debt maturity tructure { B 1,B2 } atifying 31) o that it i optimal under commitment) but with B 1 B2 o that it doe not coincide with the MPCE and doe not induce the optimal commitment olution. One can how that the further apart are B 1 and B2 in thi ituation, the lower i q1 B1 + q2 B2, and therefore, the higher i the cot of financing for the government from a uboptimal debt maturity policy. Lemma 2 Suppoe that the date government chooe ome { B 1 },B2 atifying 31) and that the date 1 government chooe policy given { B 1 },B2 according to 33). The higher i B 1 B 2, the lower i q 1B1 + q2 B2. Thi lemma tate that, even though borrowing cot do not repond to debt maturity in the cae of full commitment that i, holding B 1 + βb2 fixed borrowing cot do increae under lack of commitment the further apart are B 1 and B2. Therefore, a large and highly tilted debt poition can be very cotly to finance. To undertand thi lemma, recall that at t = 1, the government purue policie which reduce the market value of outtanding debt and increae the market value of newly iued debt. More 12 One naturally wonder how the concluion of thi example would change if θ 1 θ 2. In thi cae, it can be hown that the MPCE again coincide with the full commitment optimum under a flat maturity with B 1 = B 2. For example, uppoe θ 1 > θ 2. The date 1 government i then not only concerned about the market value of what it owe and what it iue, but it alo ha a deire to borrow in order to increae pending at date 1 at the current hort-term interet rate). Thi deire i offet by the fact that borrowing more would raie hort-term interet rate, thu deteriorating the government financial poition ince B 2 1 > B 2. Note that the optimality of a flat maturity if θ 1 θ 2 i not generally true theoretically in other environment, but it i approximately true quantitatively in the computed example of thi paper. 16

18 pecifically, the date 1 government i intereted in relaxing the implementability condition 32) by reducing the right hand ide of 32) a much a poible. Thi i why if B 1 < >) B2, it chooe to et c 1 < >)c 2. If B 1 and B2 are very different, then there i a greater cope for a deviation from a mooth policy at t = 1. For example, uppoe that B 1 < B2 with c 1 < c 2 choen according to 33). Clearly, if B 1 were to be reduced by ome ǫ > and B2 increaed by ǫ/β o a to keep 31) atified, then 32) could be relaxed even further, and the date 1 government would chooe a policy which further reduce the right hand ide of 32). The greater cope for deviation ex-pot i very cotly from an ex-ante perpective. Thi i becaue if the right hand ide of 32) i lower, then the left hand ide of 28) i alo lower. Therefore, by relaxing the implementability condition at date 1, the date 1 government i tightening the implementability condition at date, which directly reduce the ex-ante welfare at date. The imple example of the previou ubection yield ome generalizable inight regarding the welfare cot of lack of commitment. A government lacking commitment at date 1 will alway chooe to deviate ex-pot in order to relax it budget contraint. By relaxing the budget contraint, the government can reduce any ex-pot ditortion which were impoed ex-ante, either on the conumption-pending tradeoff or on the conumption-leiure tradeoff. The way the government relaxe the budget contraint i by puruing policie which increae conumption in the ame direction a the maturity of outtanding debt. Thi reduce the market value of what the government owe while increaing the market value of what the government iue. Importantly, the cope for deviation i larger if outtanding debt i highly tilted. Houehold lending to the government ex-ante at date anticipate thi behavior from the government. Thu, they expect their conumption to be higher relative to the cae of full commitment) in future period when currently iued debt i due. Specifically, they anticipate that the future government will take meaure to reduce the market value of what they are owed or increae the cot to them of any future newly-iued aving. A uch, houehold at date require higher interet rate relative to the cae of full commitment) to induce them to lend to the government. For example, if houehold are primarily buying long-term bond ex-ante, then they appropriately anticipate that the government lacking commitment will purue future policie which increae future hort-term interet rate, thereby diluting their claim. In thi cae, houehold require a higher ex-ante interet rate relative to commitment) to induce them to lend long-term to the government. An analogou reaoning hold if houehold are primarily buying hort-term bond ex-ante. Therefore, the larger and more tilted are the debt poition, the cotlier they are to finance under lack of commitment. In um, even if large and tilted poition entail no additional funding cot in the cae of full commitment, in the cae of lack of commitment, they ignificantly increae the cot of borrowing. Thi tighten the date budget of the government and therefore lead to larger date ditortion, either on the conumption-pending tradeoff or on the conumption-leiure tradeoff. For thi reaon, a flat debt maturity minimize the funding cot of the government and minimize the cot of lack of commitment. 17

19 5.2.3 Quantitative Cot of Lack of Commitment In the previou ubection, we preented a determinitic example in which the optimal maturity tructure i flat. We alo howed that a tilted maturity tructure which would otherwie be optimal under full commitment lead to riing funding cot the more tilted i thi tructure. In thi ubection, we explore how large thee cot are quantitatively. To do o, we compute a three-period example following the ame parameterization a in Chari et al. 1994). More pecifically, we et the per period payoff of houehold to c 1 σc t 1 1 σ c + η 1 n t) 1 σl 1 1 σ l + θ t log g t, 36) with σ c = σ l = 1. A in the three-period example of the previou ubection, we aume that θ = θ H and θ 1 = θ 2 = θ L. We make the following parametric aumption: β =.9644, η = 3.33, θ L =.2195, and θ H =.236. Our choice of β implie that each period can be interpreted a repreenting a year. The choice of η implie that hour worked n =.23 under commitment at date 1 and 2, and coincide with the teady tate value in Chari et al. 1994). Our choice of θ L and θ H implie that in the commitment benchmark, g /y =.19, and that g 1 /y 1 = g 2 /y 2 =.18. Thi implie that the ize of the initial hock i of 7% around the date 1 and 2 mean. Thee parameter imply that the mean of g/y and the tandard deviation of g match that in Chari et al. 1994). Thi model i analogou to that of the previou ubection. The government iue debt at date in order to finance a pending increae at date. Under full commitment, there are many debt maturitie which are conitent with the optimal policy. More pecifically, any uch debt maturitie { B 1,B2 } mut atify 31) for ome B appropriate in thi example. However, under lack of commitment, only a debt maturity which et B 1 = B2 i optimal. How cotly i it to chooe a tilted maturity under lack of commitment? To anwer thi quetion, we conider the date value of ocial welfare ubject to the date government chooing a combination of { B 1,B2 } atifying 31) in three different environment: i) the government cannot commit to taxe o pending i exogenou and fixed at the commitment optimum), ii) the government cannot commit to pending o taxe are exogenou and fixed at the commitment optimum), and iii) the government cannot commit to taxe and pending both taxe and expenditure are endogenou). Figure 1 meaure the welfare cot of a tilted debt maturity in each of thee cenario. The x-axi repreent the value of B 2 a a fraction of GDP with GDP meaured at date in the commitment optimum). Thu, the axi repreent a long-term debt to output ratio. Clearly, the higher thi value, the lower i B 1 relative to output and thi follow from the fact that the value of { B 1 },B2 atify 31). The y-axi capture the welfare cot aociated with thi uboptimal policy. It compare welfare under full commitment which i contant in all maturity cenario) to that under lack of commitment, and it repreent the lo in total conumption required under 18

20 Figure 1: Maturity Structure and the Cot of Lack of Commitment Exogenou g Exogenou τ Endogenou g and τ 3 Cot of Lack of Commitment CEV %) Long Term Debt % of GDP) Note: The figure how the cot of lack of commitment of different maturity tructure i.e. changing the quantity of long-term debt x-axi) while keeping contant the value of total debt iued in period. The welfare cot y-axi) are expreed a percentage point of conumption equivalent variation CEV) with repect to the commitment cae, for three cae: no-commitment to taxe olid line), no-commitment to expenditure dahed line), and no-commitment to both taxe and expenditure line with dot). 19