Optimal time-consistent taxation with default

Optimal time-consistent taxation with default Anastasios G. Karantounias Karen A. Kopecky February 15, 2015 Abstract We study optimal time-consistent distortionary taxation when the repayment of overnment debt is not enforceable. The overnment taxes labor income or issues non-continent debt in order to finance an exoenous stream of stochastic overnment expenditures. The overnment can repudiate its debt subject to some default costs. Our setup blends elements of time-consistent fiscal policy and the soverein default literature. Keywords: Labor tax, soverein default, Markov-perfect equilibrium, time-consistency, eneralized Euler equation. JEL classification: D52; E43; E62; H21; H63. Preliminary and incomplete. Do not circulate without permission. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree St NE, Atlanta, GA 30309, USA. Email: anastasios.karantounias@atl.frb.or. Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree St NE, Atlanta, GA 30309, USA. Email: karen.kopecky@atl.frb.or. 1

1 Introduction This paper studies the optimal time-consistent allocation of tax distortions and the optimal issuance of debt in an environment where overnment debt can be defaulted on. We consider a overnment that has to finance an exoenous stream of stochastic overnment expenditures and maximizes the utility of the representative household. The overnment can use distortionary labor taxes or issue non-continent debt. The overnment can default on its debt subject to a default cost. Our setup is fully time-consistent; neither tax nor debt promises are honored. Our analysis builds on the notion of Markov-perfect equilibrium (MPE) of Klein et al. (2008). Optimal policy is time-consistent in the payoff-relevant state variables, which for our case are overnment debt and overnment expenditures. Furthermore, we model default as in the work of Arellano (2008) and Auiar and Gopinath (2006), that builds on the debt repudiation setup of Eaton and Gersovitz (1981). This setup allows to observe default in equilibrium. In most of the soverein default literature, overnment debt is assumed to be held only by foreiners whereas domestic households are hand-to-mouth consumers. However, Reinhart and Rooff (2011) find that, on averae, domestic debt accounts for nearly two-thirds of total public debt for a lare roup of countries. We consider a closed economy in which domestic households hold overnment debt. Thus, our model takes into account that default events often involve default on debt held by domestic households. This assumption is supported by the empirical literature. While domestic default events are more difficult to identify than external default events, Reinhart and Rooff (2011) document 68 cases of overt default on domestic debt since 1800. Moreover, often even when default is only on external debt (which we do not model), a sinificant portion of the external debt is held by domestic investors (Sturzeneer and Zettelmeyer 2006). For these reasons it is of interest to understand the tradeoffs overnments face when considerin whether to default on domestic households. Our purpose it to analyze optimal tax-smoothin and debt issuance in such an environment. The lack of state-continent insurance markets hinders the ability of the overnment to smooth taxes. Default can in principle make debt partially state-continent. In particular, the overnment affects both the pricin kernel of the aent and the payoff of overnment debt. Default risk is reflected in equilibrium prices and alters the optimal allocation of tax distortions over states and dates. The overnment has an incentive to default when either overnment debt or overnment expenditures are hih. By defaultin the overnment can avoid hih distortionary taxation. However, default entails either direct costs in terms of output losses, or indirect costs, in terms of a limited functionin of the market of overnment debt after a default event. In particular, we follow Arellano (2008) and assume that the market for overnment debt pauses to function for a random number of periods after a default event. Optimal policy is characterized in our model by a eneralized Euler equation (GEE) that balances the dynamic costs and benefits that the overnment is facin. The averae welfare loss that 2

is incurred by an increase in debt issuance (since hiher debt has to be accompanied with hiher future taxes) has to be balanced with the benefits of relaxin the overnment budet and allowin less taxes today. Our GEE reflects the fact that interest rates increase when debt increases, due to a hiher probability of default. However, hiher debt can also lead to reduction in interest rates by increasin marinal utility. The increase in marinal utility is comin from the fact that future consumption decreases in the event of repayment. This second channel is particularly important in our setup because it reflects the interest rate manipulation throuh the pricin kernel that is essential for our time-consistent setup. Our overnment chooses a debt and tax policy for the future, that will find optimal to follow in the followin periods. Related literature. The basic paper that analyzes optimal taxation in incomplete markets is Aiyaari et al. (2002). They solve for optimal policy under commitment and without default. In the time-consistent literature, Krusell et al. (2004) and Debortoli and Nunes (2013) analyze timeconsistent taxation and debt in deterministic setups without default. The closest paper to ours is Pouzo and Presno (2014), which has been the first to consider the possibility of debt repudiation à la Arellano in the optimal taxation problem. These authors alter the Aiyaari et al. (2002) setup only in one dimension; they allow the overnment to default but they retain a notion of commitment. 1 In their setup, the overnment cannot commit to repay debt, but as lon as the overnment decides to repay, it honors the marinal utility promises of the plan devised in previous periods and commits therefore to the tax sequence and the evolution of interest rates. In contrast, we treat debt and taxes symmetrically and derive the fully time-consistent policy in terms of the payoff -relevant endoenous state variable which is debt. Our setup delivers obviously identical results with theirs if we consider a utility function that is linear in consumption, a feature that would eliminate the time-inconsistency of the commitment solution. 2 A two-period economy To make thins concrete, we will start our analysis with a two-period version of our model, t = 0, 1, and proceed in a later section with an infinite horizon economy. The only uncertainty in the economy is comin from exoenous overnment expenditure shocks G at t = 1 with probability π(). There is a representative household that consumes c, works h, pays linear taxes τ on its labor income and trades in overnment debt. Government debt b is non-state continent and trades at price q. At t = 1 the overnment can default on its promise to repay subject to an output loss. If the overnment defaults, it runs a balanced budet. Thus, overnment debt is a security that provides one unit of consumption next period at each state for which the overnment is not defaultin. 1 Pouzo and Presno (2014) consider also the possibility of secondary markets in the event of default, a feature we do not share. 3

Notation. Let D denote the set of shocks at t = 1 for which the overnment is defaultin. Let A D c denote the repayment set. We will not specify yet what these sets depend on, since the representative household is a a price-taker. Let d() be an indicator variable that takes the value 1 if the overnment defaults and zero otherwise, so d() = 1, D and d() = 0 if A. Resource constraints Output is produced by labor. The resource constraint at t = 0 reads c 0 + 0 = h 0 (1) At t = 1 we have A : c() + = h() (2) D : c() + = zh(), (3) where z < 1. We model the cost of default as an adverse technoloy shock. Household. The household is derivin utility from consumption and leisure. The total amount of leisure is unity. Its preferences are u(c 0, 1 h 0 ) + β π()u(c(), 1 h()). (4) We assume that initial debt is zero. The household s budet constraint at t = 0 reads c 0 + qb = (1 τ 0 )w 0 h 0 (5) At t = 1, at the household s budet constraints read A : c() = (1 τ())w()h() + b (6) D : c() = (1 τ())w()h(). (7) Note that labor taxes depend on the realization of the shock. Government. Similarly, the overnment budet constraint at t = 0 reads 0 = τ 0 w 0 h 0 0 + qb 4

and at t = 1 we A : τ()w()h() = b D : τ()w()h() = 0. Firms. Competitive firms maximize profits iven the linear technoloy and the default costs. The equilibrium wae is w 0 = 1 and w() = 1, A, w() = z < 1, D. Household s problem. Given {q, τ 0, τ(), w 0, w(), D} the household is choosin {c 0, h 0, c(), h(), b} to maximize (4) subject to (5-7). The labor supply condition at t = 0 is whereas at t = 1 we have u l0 u c0 = 1 τ 0, A : D : u l () u c () = 1 τ() u l () = (1 τ())z u c () Note that we have already used the equilibrium waes in these conditions. Furthermore, the Euler equation with respect to b is q = β = β A π() u c(c(), 1 h()) (1 d()) u c (c 0, 1 h 0 ) π() u c(c(), 1 h()). (8) u c (c 0, 1 h 0 ) The Euler equation depicts the possibility of overnment default. If the default set is empty, D =, then (8) simplifies to the standard Euler equation with risk-free debt. Competitive equilibrium. The definition of the competitive equilibrium iven overnment policy (b, τ, D) is obvious. 5

3 Optimal policy in the two-period economy The overnment is choosin the optimal amount of taxes, debt and to default or repay. We will analyze optimal policy in two staes usin backwards induction: At t = 1, iven issued debt b, the overnment is decidin to default or not and how much to tax. The overnment takes into account the optimal reaction of the household to the tax rate, so it acts as a Stackelber leader within the period. At t = 0, the overnment is choosin {b, τ 0 }, takin into account the decision of the household in the current period and the overnment s optimal decisions next period. 3.1 Default and repayment sets Let u d and u r denote the equilibrium utility of default and repayment respectively. Define the default set, which depends on b, D(b) { G u d > u r }, (9) and the repayment set A(b) D(b) c = { G u d u r }. (10) In the definition of the sets we assume that if the overnment is indifferent about repayin or defaultin, then it is repayin. The default and repayment sets depend on the amount of debt throuh the default and repayment allocation. Default allocation. The default allocation at is determined by the followin equations, c + = zh u l u c = (1 τ)z τzh =. The first is the resource constraint (takin into account the default costs), the second the labor supply and the third the balanced budet requirement. From these equations we et the default consumption-labor allocation and the default tax rate as functions of, {c d (), h d (), τ d ()}. The equilibrium utility of default is u d = u(c d (), 1 h d ()). Note that we can use the primal approach 6

of Lucas and Stokey (1983) and eliminate the tax rate throuh the labor supply condition. This leads to a system of consumption and labor only, c + = zh Ω(c, h) = 0, (11) where Ω(c, h) u c (c, 1 h)c u l (c, 1 h)h. (12) Ω stands for consumption net of after-tax labor income, in marinal utility units. Equivalently, it is equal to overnment surplus in marinal utility units. For future reference, note that Ω c u c = 1 ɛ cc ɛ ch (13) Ω h u l = 1 ɛ hh ɛ hc (14) where ɛ cc u cc c/u c, ɛ ch u cl h/u c, ɛ hh u ll h/u l, ɛ hc = u cl c/u l, the own and cross elasticities of the marinal utility of consumption and the marinal disutility of labor. Repayment allocation. If the overnment repays, A, we have c + = h u l u c = 1 τ τh = + b which determines the repayment allocation and the repayment tax rate as functions of the debt and the shock, {c r (b, ), h r (b, ), τ r (b, )}, and the repayment utility u r u(c r (b, ), 1 h r (b, )). As before, the above system can be reduced to c + = h Ω(c, h) = u c (c, 1 h)b, (15) where the second constraint expresses the budet of the overnment in terms of consumption, labor and debt. 7

3.2 Properties of the default decision We will make now two claims about the structure of the default set. We will see later the proofs. Property 1. If b > b, then D(b) D(b ). Default sets increase in debt. Property 2. Let >. If D(b), then D(b). Default incentives increase with adverse fiscal shocks. Property 1 will be easy to prove. We will see later about property 2. Assume for the moment that the first property is true. Define the followin borrowin limits: b inf{b D(b) = G} b sup{b D(b) = }. b is the maximum amount of debt so that the overnment is repayin with certainty. b is the amount of debt above which the overnment is defaultin with probability 1. We have b b. Furthermore, if b (b, b), then D(b) and A(b), so for intermediate values of debt there is always a shock for which the overnment defaults and a shock for which the overnment repays. Lemma 1. The utility of repayment is decreasin in debt. If b > b then u(c r (b, ), 1 h r (b, )) < u(c r (b, ), 1 h r (b, )). Proof. To be written. Corollary 1. Property 1 holds. Proof. Let b > b and D(b). Then u d (c d (), 1 h d ()) > u(c r (b, ), 1 h r (b, )) > u(c r (b, ), 1 h r (b, )), thus D(b ). So property 1 is based on the fact that if debt increases, the overnment has to increase taxes, which leads to a reduction in utility. Lemma 2. The overnment never defaults if b = 0 for any value of the shock, u(c r (0, ), 1 h r (0, )) u(c d (), 1 h d ()) D(0) =. Proof. The lemma seems obvious since the overnment would never default and incur the default costs. It needs some elaboration thouh. To be written. Corollary 2. b 0. 8

Threshold. Define now ω() as the amount of debt for which the overnment is indifferent between repayin and defaultin at, u(c d (), 1 h d ()) = u(c r (ω(), ), 1 h r (ω(), )) (16) Note that this equation has a solution in [b, b] (which is unique since the utility of repayment is decreasin in b). Since the repayment utility is decreasin in debt, we have d() = 1 if b > ω() and d() = 0 if b ω(). Furthermore, the monotonicity of the threshold in is equivalent to property 2. Lemma 3. Let ω( ) ω() for > Property 2 holds. Proof. 1.( ) Let D(b). This implies that b > ω() ω( ). Therefore, D(b). 2. ( ) Rephrase property 2 as follows: if A(b) then A(b) for > (if I repay for the bad shock, I repay for the ood shock). Assume now that ω( ) > ω(). Since u(c d ( ), 1 h d ( )) = u(c r (ω( ), ), 1 h r (ω( ), )), we have A(ω( )). This implies that A(ω( )) by property 2. Thus, u(c d (), 1 h d ()) = u(c r (ω(), ), 1 h r (ω(), )) u(c r (ω( ), ), 1 h r (ω( ), )) ω() ω( ), which is a contradiction. Thus, property 2 is equivalent to a non-increasin debt threshold in overnment expenditures. Its validity is not clear. We will show later that it holds if utility is linear in consumption. Furthermore, it holds numerically. To understand where this property depends on, note that if we could show that the difference in utility u u d u r is increasin in, i.e. if u(c d ( ), 1 h d ( )) u(c r (b, ), 1 h r (b, )) > u(c d (), 1 h d ()) u(c r (b, ), 1 h r (b, )), >, then property 2 follows immediately. 2 It is easy to show that default and repayment utility fall if increases. u increasin in is stroner: it means that the loss in default utility due to larer is smaller in absolute value than the loss in repayment utility. 2 An increasin u is the exact condition for a decreasin threshold. Use the implicit function theorem in the threshold equation to see that. 9

3.3 Problem at t = 0 At t = 0 the overnment is choosin (c 0, h 0, b) to maximize u(c 0, 1 h 0 ) + β π()u(c d (), 1 h d ()) + π()u(c r (b, ), 1 h r (b, )) subject to D(b) A(b) Ω(c 0, h 0 ) + β [ π()u c (c r (b, ), 1 h r (b, )) ] b = 0 A(b) c 0 + 0 = h 0 Assume now that the shock follows a continuous distribution with density f() and support [, ḡ]. Furthermore, assume that 2 is true and that the threshold is strictly decreasin in. Then we can rewrite the default decision in terms of ω 1 (b), which is the level of for which the overnment is indifferent between repayment and default for a particular level of b. Apparently, we have u(c d (ω 1 (b)), 1 h d (ω 1 (b))) = u(c r (b, ω 1 (b)), 1 h r (b, ω 1 (b))) (17) The repayment and default sets become respectively A(b) = [, ω 1 (b)] and D(b) = (ω 1 (b), ḡ] for b [b, b]. In the analysis later we will also assume that ω 1 is differentiable, i.e. that the implicit function theorem applies to (17). The purpose of these assumptions is to derive an optimality condition for the optimal debt issuance of the overnment. We will not make them in any numerical treatment of the problem. Given the structure of the default sets the optimization problem becomes: choose (c 0, h 0, b) to maximize u(c 0, 1 h 0 ) + β subject to [ ω 1 (b) u(c r (b, ), 1 h r (b, ))f()d + ḡ ω 1 (b) ] u(c d (), 1 h d ())f()d Ω(c 0, h 0 ) + β [ ω 1 (b) u c (c r (b, ), 1 h r (b, ))f()d ] b = 0 (18) c 0 + 0 = h 0 (19) 10

The overnment is takin into account how increasin debt affects the equilibrium price q. Hiher debt affects the equilibrium price by both increasin the default reion (reducin therefore the price) and by increasin the aent s marinal utility since repayment consumption falls (which increases the equilibrium price). The marinal utility effect is not present in Arellano (2008) due to risk-neutral forein lenders. Analysis. Assin the multiplier Φ on the implementability constraint (18) and λ on the resource constraint (19). First-order necessary conditions for (c 0, h 0 ) are c 0 : u c0 + ΦΩ c0 = λ 0 h 0 : u l0 + ΦΩ h0 = λ 0 which delivers the familiar wede expression u l0 ΦΩ h0 u c0 + ΦΩ c0 = 1. This expression can be rewritten in terms of the tax rate τ 0 by usin (13) and (14) as Optimal debt issuance. followin first-order condition: τ 0 = Φ(ɛ cc + ɛ ch + ɛ hh + ɛ hc ) 1 + Φ(1 + ɛ hh + ɛ hc ). (20) Turn now to the optimal choice of b. Use Leibnitz s rule to et the dω 1 db f(ω 1 (b)) [ u ( c r (b, ω 1 (b)), 1 h r (b, ω 1 (b)) ) u ( c d (ω 1 (b)), 1 h d (ω 1 (b)) )] ω 1 (b)[ + u r c r c b h r ] ur l f()d b { ω 1 (b) +Φ u r cf()d + b [ u c (c r (b, ω 1 (b)), 1 h r (b, ω 1 (b)))f(ω 1 (b)) dω 1 db + ω 1 (b) ( u r cc c r b ur cl h r ) ] } f()d = 0 b This expression can be simplified as follows. The terms in the first and second line correspond to the chane in expected utility triered by an increase in debt. An increase in debt has two effects on expected utility. At first it reduces the repayment reion, by decreasin the threshold value, dω 1 /db < 0. Second, it decreases expected utility because an increase in debt decreases the utility 11

of repayment. The term in the first line corresponds to utility differential due to the reduction in the repayment reion. This utility differential is equal to zero at the threshold value of spendin ω 1 (b), where the overnment is indifferent between repayment and defaultin. GEE in the two-period economy. Thus, the optimality condition reduces to ω 1 (b) [ u r c r c b h r ] { ω 1 (b) ur l f()d = Φ u r b cf()d } {{ } } {{ } I II(+) +b [u c (c r (b, ω 1 (b)), 1 h r (b, ω 1 (b)))f(ω 1 (b)) dω 1 + } {{ db } III( ) ω 1 (b) ( u r c r cc b h r ) ]} ur cl f()d (21) b } {{ } IV (+) The LHS denotes the expected marinal utility loss due to an increase in debt. There is a utility cost because repayment utility falls with an increased amounted of debt due to the increase in taxes next period, u r / b = u r c cr b ur l hr = b (ur c u r l ) cr / b < 0, since c r / b = h r / b < 0 and u r c > u r l (we need to tax in order to repay). This is term I. The RHS denotes the welfare benefit of increasin debt, which comes form relaxin the budet constraint of the overnment and allowin less taxes today (Φ > 0). The riht-hand side is essentially the welfare benefit of the marinal revenue of debt issuance. The overnment would never find it optimal to issue a level of debt that would deliver a neative marinal revenue, so the riht-hand side is positive. This essentially will imply a stricter borrowin limit than b. The riht-hand side has three terms: Term II is proportional to the price q. By issuin debt by one unit, the overnment ets revenue proportional to q. The third and fourth term essentially correspond loosely to q (b). The overnment takes into account that increasin debt will affect the price of debt throuh two channels: 1. Term III: by increasin debt the overnment reduces the repayment reion, which decreases the prices. Term III is neative (dω 1 /db < 0). 2. Term IV: By increasin debt, the repayment consumption and labor become lower. As a result, marinal utility increases (and recall that u cl 0), so the price increases. Term IV is positive. The hiher the curvature in c, the more important we expect this term to be quantitatively. Note that (21) is a eneralized Euler equation (GEE) with default. 3.4 Quasi-linear example Consider now an example with quasi-linear utility 12

u = c 1 2 h2. (22) This utility allows a simpler characterization of the default set. In particular we prove property 2 for this utility function and prove also differentiability of ω 1. Furthermore, the absence of the marinal utility channel implies equilibrium prices are determined only by the probability of repayment. In other words, the planner is manipulatin the equilibrium price of overnment debt only throuh the size of the repayment reion and not throuh repayment consumption. This element eliminates time-inconsistency issues in the infinite horizon problem so it is of limited interest for us. Nevertheless, it provides an easier interpretation of (21) by eliminatin term IV. Proposition 1. ( Default in the quasi-linear case ) Assume the period utility function (22). Define λ z 2 < 1 and assume that the shocks are not too lare, < 1/4λ and that the debt position is not too lare, b + < 1/4. Then, 1. The default allocation is h d = z(1 τ d ) and c d = zh d. The default tax rate and respective utility are is τ d () = 1 1 4/λ 2 u d () = 1 2 λ(1 τ 2 d ). 2. The repayment allocation is h r = 1 τ r, c r = h r. The repayment tax rate and respective utility are τ r (b, ) = 1 1 4(b + ) 2 u r (b, ) = 1 2 (1 τ 2 r ) Note that for b > (1/λ 1) > 0 we have τ r > τ d. 3. Default and repayment decision: d() = 1 if τ 2 r > λτ 2 d + 1 λ d() = 0 if τ 2 r λτ 2 d + 1 λ 13

4. The threshold ω() or ω 1 (b) is defined implicitly τr 2 (b, ) = λτd 2 () + 1 λ. The threshold is monotonically decreasin, dω 1 (b) db = τ r(1 2τ d ) τ r τ d < 0 Note also that the slope of the threshold is larer than unity in absolute value, dω 1 (b) db < 1. Note that the indifference condition that determines the threshold requires that the square repayment tax is a weihted averae of the square default tax and unity (λ < 1). Therefore, for the overnment to be indifferent between repayment and default the repayment tax rate has to be reater than the default tax rate (τ r > τ d ). Thus, even if the repayment tax is larer than the default tax, which superficially would lead to the false conclusion that the overnment has to default, the overnment may still want to repay. The reason behind that is comin from the fact that default entails output costs that reduce utility. Thus, for a iven overnment expenditure shock, debt and the associated repayment tax has to be sufficiently hih to lead to default. Therefore, a necessary (and not sufficient) condition for indifference (or defaultin) is that b > (1/λ 1). The formula also shows that if λ = z 2 = 1, so if there are zero default costs, the overnment will never issue any debt. The overnment defaults if τ r > τ d b > 0 and repays if b 0. Equilibrium price and debt issuance. price as a function of b, In the quasi-linear case we can write the equilibrium q(b) = βp rob(repayment) = βf (ω 1 (b)), b [b, b] q (b) = βf(ω 1 (b)) dω 1 db < 0. Note that q(b) = 0 for b > b and q(b) = β for b < b. Let R(b) q(b)b denote the revenues from debt issuance. Expressin consumption and labor in terms of the tax ate, the implementability constraint simplifies to τ 0 (1 τ 0 ) 0 + q(b)b = 0, (23) which furnishes an initial tax rate as function of debt revenue τ 0 (b, 0 ) = 1 1 4( 0 R(b)) 2, as lon as the initial expenditures adjusted for any revenue from debt issuance are not too lare, 0 R(b) < 1/4. The larer the revenue from debt issuance, the smaller the initial tax rate, which shows the tradeoffs that the overnment is facin. 14

The optimality equation with respect to debt (21) for the quasi-linear case simplifies to 3 where β ω 1 (b) τ r τ r b f()d = Φ[q + bq (b)] = Φq(1 ɛ), ɛ q (b)b q = b f(ω 1 (b)) dω 1 F (ω 1 (b)) db > 0, the elasticity of the equilibrium price with respect to b. The riht-hand side depicts the social value of the marinal revenue from debt-issuance, ΦR (b). The marinal revenue from debt issuance has to be positive, otherwise issuin more debt will have only a welfare effect loss, since it is associated with hiher taxes, as the left-hand side of the optimality equation shows. Therefore, we need ɛ(b) < 1 in order to have a positive marinal revenue from debt issuance. Note that this implies a stricter upper bound for borrowin, above which the overnment never borrows. In particular, let b the level of debt for which the revenue from debt issuance is maximal. If the solution is in (b, b), this corresponds to ɛ(b ) = 1. We expect that for b < b we have R (b) > 0. A sufficient condition for this (and for which we have a unique maximum) is that the price elasticity of debt is strictly increasin in b, ɛ (b) > 0. This obviously depends on the assumptions on the cumulative distribution function of the shocks and the slope of the threshold dω 1 /db. If it is true, we can restrict attention to the quadrant where τ 0 < 1/2 and b < b. Furthermore, b has to be as follows. Lemma 4. The revenue-maximal level of debt satisfies b [b, b). If lim b b + R (b) > 0, then b (b, b). Proof. We cannot have b b since revenue is zero for this interval. Furthermore, we cannot have b < b since marinal revenue for this interval is positive. Therefore b [b, b). There is the possibility thouh that the maximum revenue is at the lower boundary point, b. The riht derivative of the revenue schedule is lim R (b) = β + βf(ḡ) dω 1 (b) b > 0, b b + db accordin to the claim. Unless f(ḡ) = 0, there is a downward jump in marinal revenue. If the marinal revenue thouh still remains positive, then we cannot have an optimum at b. Thus, b (b, b). 3 We multiply with β in order to express the condition in terms of the price q. 15

0.2 Default and repayment reions, for z = 0.99 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 b Fiure 1: The fiure plots ω 1 (b). For each level of debt the overnment is defaultin if > ω 1 (b). An interior b, which corresponds to ɛ(b ) = 1, implies that there is the possibility of an equilibrium with default (the planner may still optimally choose amounts of debt below b). We would never have an equilibrium with default if b = b. This could happen if the reduction in prices was such so that the marinal revenue at b + is neative. This would make b a local maximum and if revenue falls for larer debt, a lobal maximum. So there is the possibility for an equilibrium without default only if the price schedule is extremely steep. Remark 1. Even if b > b, optimal debt issuance does not necessarily entail default. The planner may run a deficit at the initial period to be financed by debt that matures at t = 1, but he may find it optimal to issue b b. In that case there is no default in equilibrium. The optimality condition captures these tradeoffs. The larer the default costs (low z), the more probable this scenario. Furthermore, this possibility depends also on the initial level of overnment expenditures 0. If they are too small, then the planner may run a small deficit that does not require an optimal debt that falls in the default reion. At the extreme, when 0 = 0, the planner runs a surplus at the initial period and uses the proceeds to lend to the private sector, b < 0. 3.5 Numerical illustrations for the quasi-linear case Calibration of shocks: = 0, ḡ = 0.2. To et an idea of their size, note that the first-best output it unity, so overnment expenditures vary from 0 till 20% of first-best output. We use 2, 000 ridpoints and a uniform distribution. Furthermore, we set β = 0.95, z = 0.99. 16

0.2 0.18 Default and repayment reions for various default costs z=0.95 z=0.98 z=0.99 z=1 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.05 0.1 0.15 0.2 0.25 b Fiure 2: The fiure plots ω 1 (b) for varyin default costs. Default and repayment reions. Fiure 1 depicts the default/repayment reion for the baseline calibration. Note the monotonically decreasin threshold ω 1. To understand the impact of default costs z, on the deault/repayment reions, fiure 2 plots the correspondin sets for varyin default costs. The case of no default costs z = 1 corresponds to a vertical line at b = 0, i.e. the overnment defaults with certainty if there is positive debt and repays with certainty if b 0. Besides this extreme case (for which ω 1 is not well-defined), note that an increase of default costs shifts the threshold curve to the riht (thus for a iven level of overnment expenditures, the overnment can sustain a larer debt without defaultin). Furthermore, the threshold becomes flatter. Equilibrium price and revenues. Fiure 3 plots the equilibrium price q and the correspondin revenues R(b). For b b the price is β, whereas for b > b we have q = 0. Note that the level of debt for which revenues are maximal, b is larer than b. Optimal debt issuance. As noted in remark 1, the optimal debt that the overnment issues is not necessarily lare enouh so that it entails default. For the particular calibration I use 0 = ḡ, and it turns out that b > b, as fiure 4 shows. The probability of default is 20.85%. Fiure 5 depicts how the optimal debt issuance depends on the initial shock. For each level of 0 we calculate the optimal debt. As noted earlier, at the extreme where the initial shock is zero, the overnment is lendin to the private sector, b < 0. There is a positive relationship between the initial shock and 17

Price of debt, minb = 0.016159, maxb = 0.12117 b * = 0.047865 1 0.025 Revenue from debt issuance R(b) 0.9 0.8 0.02 0.7 0.6 0.015 q 0.5 0.4 0.01 0.3 0.2 0.005 0.1 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 b 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 b Fiure 3: The left raph plots the price of debt, q(b). The riht raph plots the revenues from debt issuance. 0.6374 Welfare, b0 = 0.026763, bmin = 0.016159, b * = 0.047865 0.6373 0.6372 0.6371 0.637 0.6369 0.6368 0.6367 0.6366 0.6365 0.015 0.02 0.025 0.03 0.035 0.04 0.045 b Fiure 4: The raph depicts expected discounted utility at t = 0 when 0 = ḡ. The red vertical line denotes b. The upper level of debt is b. The optimal debt issuance b 0 is larer than b. The probability of default is 0.2085. optimal b. Note that there is a reion of initial shocks for which the optimal debt issuance is always at b. 18

0.04 Optimal b as function of the initial shock 0.03 0.02 0.01 0 b 0.01 0.02 0.03 0.04 0.05 0.06 0 0.05 0.1 0.15 0.2 0 Fiure 5: The raph depicts the optimal debt issuance as function of the initial shock 0. The reen dashed line depicts b. Any level of optimal debt that is smaller or equal than b entails repayment with certainty. We set the initial shock 0 = κḡ with κ {0, 0.5, 0.6, 0.8, 0.9, 0.95, 0.98, 0.99, 1, 1.1}. 3.6 Constant Frisch elasticity Assume that the period utility is h 1+φ h U = c1 ρ 1 1 ρ a h. (24) 1 + φ h The quasi-linear example we analyzed earlier corresponds to the case of (ρ, φ h, a h ) = (0, 1, 1). [To be completed]. 4 Infinite horizon economy Consider now an infinite horizon model, where the overnment can default on issued debt. The uncertainty is comin from overnment expenditures shocks t that take values in G, with probability of the partial history t equal to π t ( t ). We assume that there is no uncertainty at the initial period, so π 0 ( 0 ) 1. We use d t = 1 to denote default and d t = 0 when there is repayment. The resource constraint in the economy when the overnment does not default is 19

c t + t = h t. (25) If the overnment defaults there are default costs that are captured as a technoloy shock. The resource constraint in the event of default is c t + t = zh t, where z < 1. We will explore also more elaborate default costs as in Arellano (2008). Household. The household s preferences are E 0 t=0 β t u(c t, 1 h t ). The household trades with the overnment a discount bond that ives one unit of consumption next period at any state of the world where the overnment is not defaultin and zero in the event of default. The price of the bond is q t. The household pays a linear tax τ t on labor income w t h t. The household s budet constraint when the overnment is not defaultin reads c t + q t b t+1 (1 τ t )w t h t + b t. Note that the bond position b t+1 is function of information at time t. Furthermore, the household may have some initial debt b 0. Default entails direct and indirect costs. The direct ones are in terms of output losses due to the neative technoloy shock. The indirect costs are comin from the exclusion from the market for overnment debt. When the overnment defaults at time t, b t is wiped out and the household is also excluded from the market for new overnment debt at the same period. This can be thouht of as a collapse of the market. At every period after default, the household can enter the market with probability α or stay excluded with probability 1 α. When α = 1, the implicit cost of default is small, since the exclusion lasts only one period, whereas when α = 0, the cost is lare, since the overnment has to run a balanced budet forever. In the open economy literature as in Arellano (2008), α is calibrated in order to match the averae duration of exclusion from international markets. The international market justification is obviously not relevant for the closed economy, thus our market collapse interpretation. Therefore, the household s budet constraint in the event of default, or for any period where there is exclusion from the market is 20

c t = (1 τ t )w t h t. The household is also subject to some borrowin limits that we assume that are lare enouh so that they do not bind. Waes. Note that in equilibrium the wae rate is w t = 1 if d t = 0 and w t = z if d t = 1 or if there is exclusion after a default event. Note that the assumption that the direct output costs are relevant also for any period that the household is excluded from the market for overnment debt compounds the implicit default cost. Government. The budet constraint of the overnment in the event of repayment is B t = τ t w t h t t + q t B t+1. B t > 0 means that the overnment borrows and B t < 0 that the overnment lends. If there is default or for any period after a default event for which there is exclusion, the overnment runs a balanced budet, τ t w t h t = t. Equilibrium. A competitive equilibrium with taxes and default is a price-tuple {q t, w t }, a overnment policy {τ t, d t, B t }, and a household s allocation and bond holdins {c t, h t, b t } such that 1) Given prices and overnment policies, the household maximizes his utility subject to the budet constraint. 2) Given waes, firms maximize profits. 3) Prices and overnment policies are such so that markets clear: the resource constraint and the overnment budet constraint hold. Furthermore, the bond market clears, b t = B t. Remark 2. We have used different notation for overnment debt and then imposed the equilibrium condition b t = B t. This is really redundant. Furthermore, iven b t = B t and the rest of equilibrium conditions, the overnment budet constraint is redundant. Optimality conditions. The labor supply condition is u lt u ct = (1 τ t )w t. 21

The Euler equation for overnment bonds is q t = βe t u c,t+1 u ct (1 d t+1 ). The household is aware of the default decision of the overnment but is not able to affect it. The equation shows that the equilibrium price of debt is zero, if the overnment defaults with certainty. If the overnment repays with certainty, then it reduces to the standard Euler equation without default. 5 Markov-perfect policy The policymaker decides how much to tax, how much debt to issue and if he will repay or not. His objective is to maximize the utility of the representative household. The constraints are the optimality conditions, the budet, and resource constraints comin from the competitive equilibrium. We are usin the primal approach of Lucas and Stokey (1983) to eliminate tax rates and equilibrium prices. We are assumin a Markov-perfect timin protocol as in Klein et al. (2008), so the solution to the policy problem will be time-consistent in the payoff-relevant state variables. Our Markov-perfect equilibrium (MPE) has two state variables, overnment debt B and the exoenous shock, which we also assumes that is Markov. Let V r (B, ) denote the value function if the overnment decides to repay and V d () the value function if the overnment defaults. The value function of the overnment is V (B, ) = max{v r (B, ), V d ()}. Value of default. When the overnment default the consumption and labor allocation is (c d (), h d ()) for each value of the shock is determined by the resource constraint, the labor supply condition and the balanced budet requirement. Thus, it has to satisfy Ω(c, h) = 0 c + = zh, as in the two-period model. Given (c d, h d ) we can immediately deduce the default tax rate, τ d () = 1 u d l /(zud c). The value of default is 22

V d () = u(c d (), 1 h d ()) + β π( ) [ αv (0, ) + (1 α)v d ( ) ] Note that if α = 0, i.e. if the market for overnment debt seized to exist forever after a default event, and if G is finite, we could calculate immediately the value of autarky as V d = (I βπ) 1 u d. Boldface variables denote vector columns, I the identity matrix and Π the transition matrix of the shocks. Default decision. Define the default set as D(B) { G V d () > V r (B, )} and the repayment set as the complement of D(B), A(B) D(B) c = { G V d () V r (B, )}. Given an amount of debt B at the beinnin of the period, the default set denotes the set of values of for which the overnment decides to default, so d(b, ) = 1 if D(B). The repayment set corresponds to d(b, ) = 0 if A(B). Given the default and repayment set we have V (B, ) = V r (B, ), A(B) or V (B, ) = V d (), D(B). Value of repayment. In a Markov-perfect equilibrium the planner takes into account at the current period that he will follow an optimal policy from next period onward, iven the value of debt next period. To capture this requirement, let C(B, ) and H(B, ) denote the consumption and labor policy functions in the event of repayment. They satisfy C(B, ) + = H(B, ). The current planner takes into account that by choosin debt B, he affects the consumption-labor choice of the future planner thouh C and H. The value of repayment is V r (B, ) = max c,h,b u(c, 1 h) + β π( )V (B, ) 23

subject to u c (c, 1 h)b Ω(c, h) + βb π( )u c (C(B, ), 1 H(B, )) A(B ) c + = h c 0, h [0, 1] We have used the Euler equation and the labor supply condition in order to rewrite the budet constraint of the household in terms of allocations. Takin into account the optimal policy functions of next period has a bite only in the case of curvature in the utility function. If the utility was linear in consumption, so if there was no room for manipulation of interest rates, C and H would not be relevant and the commitment solution would be time-consistent. Note that iven the definition of the default sets we can rewrite the problem as V r (B, ) = max c,h,b u(c, 1 h) + β[ subject to MPE requirement. A(B ) u c (c, 1 h)b Ω(c, h) + βb c + = h c 0, h [0, 1] π( )V r (B, ) + A(B ) D(B ) π( )V d ( ) ] π( )u c (C(B, ), 1 H(B, )) Let c(b, ), h(b, ) and B (B, ) be the policy functions of the above problem. The Markov-perfect requirement is that c(b, ) = C(B, ) and h(b, ) = H(B, ). 4 Note that there may be multiple solutions for the policy functions. We are oin to focus on the MPE that is the limit of a finite horizon problem. So we are oin to solve for T periods and increase T till there is no difference in the policy and value functions. 6 Analysis We can et two lemmata. Lemma 5. The value of repayment is decreasin in B. Proof. This is obvious since for B 1 < B 2 the constraint correspondence increases, and therefore the repayment value is larer at B 1, V r (B 1, ) V r (B 2, ). 4 A more precise MPE requirement would be that C(B, ) and H(B, ) are maximizers of the stated problem in order to account for the existence of multiple solutions. This is for example what Klein et al. (2008) do. 24

Since the repayment value decreases in debt we have property 1 of the two-period model, B 1 > B 2 D(B 2 ) D(B 1 ). We can define as in the two period model the upper and lower debt limit, B inf{b D(B) = G} B sup{b D(B) = }. Lemma 6. V (0, ) = V r (0, ),. If the overnment has no debt, it does not default. Thus D(0) = and B 0. Proof. To be written. As in the two period model, let ω() denote the amount of debt iven the value of spendin such that the overnment is indifferent between defaultin and repayin, V d () = V r (ω(), ). The overnment defaults if B > ω() and repays if B ω(). We only need Claim I for that. Assume now that property 2 of the two-period model is true, i.e. that if D(B) then D(B) for >, which as we saw is equivalent to a monotonically decreasin threshold. If it is strictly decreasin we can define ω 1 (B) as the value of overnment spendin that makes the overnment indifferent between repayin or defaultin iven B, so the overnment defaults if > ω 1 (B) and repays if ω 1 (B) and we obviously have V d (ω 1 (B)) = V r (B, ω 1 (B)). Assume aain a continuous distribution of shocks in [, ḡ] with conditional density f( ). We can write the the value function of repayment as ω 1(B ) V r (B, ) = max u(c, 1 h) + β[ c,h,b subject to V r (B, )f( )d + ḡ ω 1 (B ) V d ( )f( )d ] ω 1 (B ) u(c, 1 h)b Ω(c, h) + βb u c (C(B, ), 1 H(B, ))f( )d c + = h 25

6.1 Optimal tax rate We will assume now differentiability and take first-order conditions. This is only to develop intuition for the tradeoffs that the overnment is facin. We will not make any differentiability assumption in our numerical treatment of the problem. Note that non-differentiabilities arise from two sources: a) the default decision b) the MPE requirement. Assin multiplier Φ and λ on the implementability and resource constraint respectively. The first-order conditions with respect to consumption and labor are c : h : u c + Φ[Ω c u cc B] = λ u l + Φ[Ω h + u cl B] = λ Eliminatin λ we et u l Φ[Ω h + u cl B] u c + Φ[Ω c u cc B] = 1 (26) Given the resource constraint and (26) we can write c, h as functions of (Φ, B, ). Debt has two effects: a direct one throuh B and an indirect one thouh Φ since at the optimum Φ = Φ(B, ). Furthermore, we can derive the optimal tax rate as 5 τ = Φ(ɛ cc(1 B/c) + ɛ ch + ɛ hh + ɛ hc (1 B/c)). (27) 1 + Φ(1 + ɛ hh + ɛ hc (1 B/c)) This expressions shows the dependence of the tax rate on the marinal cost of taxation, captured by Φ, on debt and the particular elasticities of the period utility function. For the constant Frisch elasticity case (24) it takes the form 6.2 Generalized Euler equation Consider now the optimality condition with respect to B. τ = Φ(ρ(1 B/c) + φ h). (28) 1 + Φ(1 + φ h ) 5 Bear in mind also the two non-neativity conditions from the positivity of λ, 1 + Φ[1 ɛ cc (1 B/c) ɛ ch ] > 0 1 + Φ(1 + ɛ hh + ɛ hc (1 B/c)) > 0 26

ḡ { ω 1 (B ) V (B, )f( )d = Φ B u c (C(B, ), 1 H(B, ))f( )d +B [ u c (C(B, ω 1 (B )), 1 H(B, ω 1 (B )))f(ω 1 (B ) ) dω 1 db + Note that C/ B = H/ B and that ω 1 (B ) [u cc C B H ]} u cl B ]f( )d ḡ V (B, )f( )d B = f(ω 1 (B ) ) dω 1 [V r (B, ω 1 (B )) V d (ω 1 (B ))] db ω 1 (B ) V r (B, ) + f( )d B Thus, we have = ω 1 (B ) V r (B, ) B f( )d. Proposition 2. ( GEE ) The eneralized Euler equation in an environment with incomplete markets and default takes the form ω 1 (B ) V r (B, ) { ω 1 (B ) f( )d = Φ u B c (C(B, ), 1 H(B, ))f( )d [ +B u c (C(B, ω 1 (B )), 1 H(B, ω 1 (B )))f(ω 1 (B ) ) dω 1 + ω 1 (B ) [u cc u cl] C B f( )d ]} db (29) Each term of the GEE has exactly the same interpretation as in the two-period model. The GEE equates the marinal cost of increasin debt, with the marinal benefit comin from the relaxation of the overnment budet constraint at the current period. The relaxation of the overnment budet constraint is comin from increasin debt revenue and bein able therefore to decrease the current tax rate. The marinal revenue expression reflects the way the default reion increases with increased debt, a fact which decreases equilibrium prices, and the way equilibrium prices increase due to the the increase of marinal utility, in the case of C B < 0. The envelope condition under the differentiability assumption takes the form V r B = Φu c, (30) which allows the rewritin of the GEE (29) in terms of the multipliers on the implementability 27

constraint as, ω 1 (B ) { ω 1 (B ) u cφ(b, )f( )d = Φ u c (C(B, ), 1 H(B, ))f( )d [ +B u c (C(B, ω 1 (B )), 1 H(B, ω 1 (B )))f(ω 1 (B ) ) dω 1 db + ω 1 (B ) [u cc u cl] C B f( )d ]}. (31) This form of the GEE is potentially helpful in order to contrast our analysis with Aiyaari et al. (2002) and Pouzo and Presno (2014). 7 Numerical results [To be completed.] 8 Concludin remarks [To be completed.] References Auiar, Mark and Gita Gopinath. 2006. Defaultable debt, interest rates and the current account. Journal of International Economics 69 (1):64 83. Aiyaari, S. Rao, Albert Marcet, Thomas J. Sarent, and Juha Seppala. 2002. Optimal Taxation without State-Continent Debt. Journal of Political Economy 110 (6):1220 1254. Arellano, Cristina. 2008. Default Risk and Income Fluctuations in Emerin Economies. American Economic Review 98 (3):690 712. Debortoli, Davide and Ricardo Nunes. 2013. Lack of commitment and the level of debt. Journal of the European Economic Association 11 (5):1053 1078. Eaton, Jonathan and Mark Gersovitz. 1981. Debt with Potential Repudiation: Theoretical and Empirical Analysis. The Review of Economic Studies 48 (2):289 309. Klein, Paul, Per Krusell, and José-Víctor Ríos-Rull. 2008. Time-Consistent Public Policy. The Review of Economic Studies 75 (3):789 808. 28

Krusell, Per, Fernando M. Martin, and José-Víctor Ríos-Rull. 2004. Time-consistent debt. Mimeo, Institute for International Economic Studies. Lucas, Robert Jr. and Nancy L. Stokey. 1983. Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12 (1):55 93. Pouzo, Demian and Inacio Presno. 2014. Optimal taxation with endoenous default under incomplete markets. Mimeo, UC Berkeley. Reinhart, Carmen M. and Kenneith S. Rooff. 2011. The Forotten History of Domestic Debt. The Economic Journal 121 (552):319 350. Sturzeneer, F. and J. Zettelmeyer. 2006. Debt Defaults and Lessons from a Decade of Crises. The MIT press. 29