Optimal taxation with endogenously incomplete debt markets

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<ul><li><p>Journal of Economic Theory 127 (2006) 3673www.elsevier.com/locate/jet</p><p>Optimal taxation with endogenously incompletedebt markets</p><p>Christopher Sleeta, Sevin Yeltekinb,aDepartment of Economics, University of Iowa, Iowa City, IA 52242, USA</p><p>bMEDS, Kellogg School of Management, Northwestern University, 2001 Sheridan Road, 5th Floor, Evanston,IL 60208, USA</p><p>Received 9 January 2004; nal version received 30 July 2004Available online 23 November 2004</p><p>Abstract</p><p>Empirical analyses of labor tax and public debt processes provide prima facie evidence for im-perfect government insurance. This paper considers a model in which the governments inability tocommit to future policies or to report truthfully its spending needs renders government debt marketsendogenously incomplete. A method for solving for optimal scal policy under these constraints isdeveloped. Such policy is found to be intermediate between that implied by the complete insurance(Ramsey) model and a model with exogenously incomplete debt markets. In contrast to optimalRamsey policy, optimal policy in this model is consistent with a variety of stylized scal policy factssuch as the high persistence of labor tax rates and debt levels and the positive covariance betweengovernment spending and the value of government debt sales. 2004 Elsevier Inc. All rights reserved.</p><p>JEL classication: D82; E62; H21</p><p>Keywords: Optimal taxation; Fiscal policy; Dynamic contract theory</p><p>We thank two anonymous referees for many useful suggestions. We thank seminar participants at Duke, UCLA,Illinois Urbana-Champaign, Northwestern, the Cleveland and Richmond Feds, the 2003 SED Meetings and SITEfor their comments.</p><p> Corresponding author. Fax: +1 847 467 1220.E-mail address: s-yeltekin@kellogg.northwestern.edu (S. Yeltekin).</p><p>0022-0531/$ - see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2004.07.009</p></li><li><p>C. Sleet, S. Yeltekin / Journal of Economic Theory 127 (2006) 3673 37</p><p>1. Introduction</p><p>This paper considers the optimal design of scal policy under two sets of restrictions.The rst set is exogenous; it describes the technology by which the government can extractresources from agents. We follow the conventional Ramsey approach and suppose thatresources can be obtained by levying linear taxes or selling state contingent debt. We alsoassume that the government cannot lend. The second set of restrictions stem from incentiveproblems on the side of the government which we assume can neither commit to repayingits debt nor to truthfully revealing private information about its spending needs. Thesefrictions impede the governments ability to use asset markets to hedge scal shocks. Theyendogenously restrict the set of asset trades the government can make and this, in turn, hasimplications for the optimal setting of taxes. To analyze scal policy design in such settings,we embed the governments policy problem into a repeated game.We provide an equilibriumconcept that extends Chari and Kehoes [15,16] sustainable equilibrium to environmentswith private government information. We then give necessary and sufcient conditions foran allocation to be an equilibrium outcome of this game. These conditions are recursive andwe obtain a dynamic programming method for nding optimal equilibrium allocations thatexploits this recursivity. We back out the supporting scal policies from these allocations andanalyse optimal scal policy in this limited commitment-private information environment.</p><p>Our immediate motivation is a contrast between the benchmark Ramsey model of scalpolicy (as developed by Lucas and Stokey [22]) and the data. The former implies that scalpolicy variables should depend only upon the current realization of the shocks perturbing theeconomy and, consequently, should inherit their stochastic properties from these shocks. Incontrast, empirical evidence on labor tax rates and the public debt suggest that these variablesexhibit considerable persistence, much more than that for government spending and othercandidate shock processes. 1 To paraphrase Aiyagari, Marcet, Sargent and Seppl (AMSS)[2], the empirical labor tax rate process is smooth in the sense of being highly persistent,rather than smooth in the sense of having a small variance.</p><p>The data are suggestive of considerable intertemporal, but limited interstate smoothingof taxes. Thus, they provide prima facie evidence for incomplete government insurance.The papers of AMSS, Marcet and Scott [23] and Scott [27], which assume exogenouslyincomplete government debt markets, corroborate this view and suggest that a limited abilityto hedge against scal shocks may have signicant implications for the design and conductof scal policy. Given this, it becomes important to understand why this ability is limitedand in what circumstances it might be more or less restricted.</p><p>Many commentators have informally suggested that moral hazard problems of one sortor another might underpin incomplete government insurance (e.g. [9,10,25]). The privateinformation and limited commitment frictions that we incorporate into our model formalizethese ideas. Both are linked to familiar time consistency considerations. The repayment ofdebt requires the levying of distortionary taxes. Ex post the government, and all households,would be better off if the debt were cancelled, but if such cancellation is anticipated ex ante,</p><p>1 Marcet and Scott [23] show that even after capital, which is absent from the Lucas and Stokey model, isincorporated, the empirical processes for scal variables remain too persistent relative to those implied by theRamsey model.</p></li><li><p>38 C. Sleet, S. Yeltekin / Journal of Economic Theory 127 (2006) 3673</p><p>the government will be unable to sell any debt in the rst place. Our model gives thegovernment two channels via which it can avoid making debt repayments. The rst is anoutright repudiation of the debt. The second is more subtle; the government may exploitthe private information it has over its spending needs and the state contingency of debtrepayments to obtain a reduction in the latter. If, in order to smooth taxes, it has sold moreclaims against low relative to high spending needs states, the government can reduce itsdebt repayment by claiming its spending needs are high when they are really low. 2</p><p>We call allocations that can be supported as equilibrium outcomes of our game sus-tainable incentive-compatible competitive allocations (SICCAs). Our main focus is uponSICCAs that are optimal from the governments point of view. We show that optimalSICCAs are recursive in the value of the governments debt. The limited commitment con-straint translates into an upper bound on equilibrium debt values. Above this upper bound,the government cannot be given incentives to repay its debt. It is the scal policy analogueof the endogenous solvency constraints that Alvarez and Jermann [3] nd in a model ofhouseholds who are unable to commit ex ante to making debt repayments. There is also alower debt value limit that stems from our assumption that the government cannot lend. Inmaking this last assumption, we follow Chari and Kehoe [15]. We elaborate on its role andits justication in Section 3.</p><p>Our recursive method allows us to jointly solve for the debt value limits and for thegovernments optimal equilibrium payoff as a function of its current debt value. The methodis related to the approach of Abreu, Pearce and Stacchetti (APS) [1] and to the recentextensions of this approach to macroeconomic policy games provided by Chang [12], Phelanand Stacchetti [26] and Sleet [28]. We show that the governments value function satises aBellman equation on the set of debt values that lie between the limits. The policy functionsfrom this dynamic programming problem can be used to recursively construct optimalallocations and, hence, optimal scal policy. This policy has the following characteristics.Away from the debt value limits, it exhibits considerable intertemporal tax smoothing, amoderate degree of state contingency in debt returns and considerable persistence in bothtaxes and debt. These features are consistent with the empirical analyses of Bizer and Durlauf[8], Huang and Lin [20] and Kingston [21] (on taxes) and Marcet and Scott [23] (on debt).Close to the limits, there is much more volatility in tax rates.</p><p>The limited commitment, no lending and incentive compatibility frictions interact ininteresting ways. In a model with only the incentive compatibility friction, tax rates andthe excess burden of taxation tend to drift upwards over time. Sleet [29] shows that, undercertain assumptions on preferences, this drift continues until the government is maximallyindebted. At this point, it maximizes and uses all of its tax revenues to service debt. Theseverity of this outcome raises natural questions about the governments ability to committo implementing it ex ante. The endogenous upper debt value limit that stems from thecommitment friction arrests the drift before this severe outcome is attained. Moreover, boththe limited commitment and no lending frictions aggravate the incentive problem, especiallywhen the governments debt value is close to the debt value limits. This contributes to thegreater tax rate volatility in these regions.</p><p>2 The private information friction is less familiar than the limited commitment one. We provide an extendedmotivation for it in Section 3.</p></li><li><p>C. Sleet, S. Yeltekin / Journal of Economic Theory 127 (2006) 3673 39</p><p>In a model with only the no lending and commitment frictions, such as that of Chariand Kehoe [15], debt value limits are also present. As Chari and Kehoe show, there isa reduced scope for scal hedging and a motive for intertemporal smoothing of taxes inthe neighborhood of these limits. The addition of the incentive constraints further restrictsthe governments ability to hedge scal shocks and creates a motive for intertemporal taxsmoothing across the whole debt value domain. Overall, numerical calculations indicate thatthe three frictions result in a constrained optimal scal policy with properties somewherebetween those implied by the Ramsey model and a model with non-contingent debt andexogenously set debt limits. Corroborative evidence reported by Marcet and Scott [23]suggests that empirical scal policies are also somewhere between these benchmarks.</p><p>Our model is related to two recent contributions, Athey, Atkeson and Kehoe (AAK)[5] and Sleet [28], that have considered optimal monetary policy under private governmentinformation. InAAKs model the private information concerns the governments preferencesfor ination and the government implements policy contingent on the history of reports thatit has made concerning its attitude towards ination. A key result of AAK is that optimalmonetary policy is in fact static and does not respond to past reports. Although our modelshares some of the same structure as AAKs, we nd that optimal scal policy is not static.</p><p>Finally, Angeletos [4] presents an alternative decentralization of benchmark Ramseyallocations that relies upon non-contingent debt of varied maturities and a state contingentscal policy. It is important to emphasize that even though explicitly state contingent debtis absent from this model, the decentralization proposed by Angeletos does not immunizethe government from the frictions analyzed in this paper. In particular, if a governmentis privately informed about its spending needs, implementation of the Ramsey allocationunder this alternative arrangement would still require it to condition its policies on thisinformation. This would create an opportunity and an incentive for such a government tomisrepresent its spending needs in order to justify alternative policy actions.</p><p>The outline for the remainder of the paper is as follows. Section 2, describes the bench-mark environment with full commitment and without private information. Section 3, thenintroduces the incentive compatibility, limited commitment and no lending frictions andreformulates the model in game-theoretic terms. The next section provides a recursive for-mulation of the optimal SICCA that is amenable to computation. In particular, this sectionshows how limited commitment constraints can be recast as debt value limits. Section 5,gives a partial theoretical characterization of optimal SICCAs, while Section 6 providesillustrative numerical calculations.</p><p>2. The benchmark environment</p><p>The benchmark Ramsey environment is characterized by complete information and fullcommitment on the part of the government. The economy is inhabited by a governmentand a continuum of identical households, all assumed to be innitely lived. Taste shocksto the governments and, under some interpretations, societys preference for public goodsare the underlying source of uncertainty. Denote the associated shock process by ={t }t=0, with each t {1, . . . , N }, i+1 &gt; i . Assume that each t is distributedi.i.d. with probability distribution P = {P(i )}Ni=1 RN++. Denote histories of shocks</p></li><li><p>40 C. Sleet, S. Yeltekin / Journal of Economic Theory 127 (2006) 3673</p><p>(0, . . . , t ) t+1 by t and the probability distribution over such a history by P t(t ). Ahousehold allocation gives the consumption ct , and labor supply lt of a household at each tand conditional on all possible t . An allocation augments this with a sequence of functionsg = {gt }t=0 that gives government spending at each date and after all histories.</p><p>Denition 1. A household allocation is a collection of functions eh = {ct , lt }t=0 with,for each t, ct : t R+ and lt : t [0, T ]. A household allocation is interior if for allt, t , ct (</p><p>t ) &gt; 0 and lt (t ) (0, T ).An allocation is a collection of functions e = {ct , lt , gt }t=0 with {ct , lt }t=0 a household</p><p>allocation and for each t, gt : t R+. An allocation {ct , lt , gt }t=0 is interior if {ct , lt }t=0is an interior household allocation. Let denote the set of interior allocations.</p><p>Households value household allocations according to V h(eh) = E [t=0 tU(ct , lt )],with (0, 1), and U(c, l) = u(c) + y(l), where</p><p>u(c) ={</p><p>c11 (0, 1),ln c = 1, y(l) =</p><p>{(Tl)1</p><p>1 (0, 1),ln(T l) = 1. (1)</p><p>The arguments given below can be easily extended to the case = 0. To economize onspace they are omitted. There is a linear production technology that converts one unit oflabor supply into one unit of output. In each period, households can trade claims contingenton the next periods shock realization.</p><p>A scal policy is a collection of functionsx {st , St , t , qt }t=0. Here, st : t [0, 1],St : t R+ and t : t (, 1) denote, respectively, a tax on claim payouts, alump sum transfer and a labor income tax set by the government at t as a function of thehistory of shocks. qt : t+1 R+ is a pricing kernel, set by the government at t aftereach t , for one period ahead claims contingent on t+1. The government supplies claimson demand at these prices. Let Qst =</p><p>sr=1</p><p>[qt+r1/(1 st+r )</p><p>], s1, Q0t = 1, denote the</p><p>after-tax price of a unit of consumption at date t + s in terms of date t consumption. L...</p></li></ul>