Optimal Debt-Maturity Management

Optimal Debt-Maturity Management

Saki Bigio

UCLA

Galo Nuño

Banco de España

Juan Passadore∗

EIEF

October 20th 2017

PRELIMINARY AND INCOMPLETE

Abstract

We solve the problem of a government that wants to smooth financial expenses

by choosing over a continuum of bonds of different maturity. The planner takes into

account that adjusting debt too fast can affect prices. At the same time, it wants to

insure against several sources of risk: (a) income risk, (b) interest rate (price) risk, (c)

liquidity risk (prices can become more sensitive to issuance’s), and (d) the risks in the

cost of default. We characterize this infinite dimensional control problem to aid the

design of the debt-maturity profile in response to these forms of risk.

Keywords: Maturity, Debt Management, Open Economies

JEL classification: E32, E34, E42

∗The views expressed in this manuscript are those of the authors and do not necessarily represent theviews of the European Central Bank or the Bank of Spain The authors are very grateful to Manuel Amador(discussant), Anmol Bhandhari, Alessandro Dovis, Hugo Hopenhayn, Boyan Jovanovic, Francesco Lippi,Pierre-Olivier Weill, Pierre Yared, Raquel Fernandez and the participants of SED Meetings at Edinburghand NBER Summer Institute for helpful comments and suggestions. All remaining errors are ours.

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1 Introduction

The Treasury Office of every Government or Corporation faces a large-stakes problem:how to design a strategy for the optimal issuance, pre-payment or purchases of debts ofdifferent maturities? A large literature in Economics and Finance has proposed alterna-tive environments to study this debt management problem. The literature on sovereigndebt studies a small open economy that borrows in the international debt market tosmooth income shocks;1 the literature on optimal taxation studies the debt managementproblem as one in which the government uses debt to smooth the tax burden in a closedeconomy; 2corporate finance studies the debt management problem of a corporation thattrades off the cost of debt dilution against the cost of issuing new debt.3 One of challengesin all of these areas is to deal with multiple assets: as we increase the number of assets thestate space grows exponentially. Our main contribution is to develop a methodology tosolve the debt-maturity management problem in an incomplete-markets economy wherea continuum of maturities are available.

We consider the following problem: a government in a small-open economy choosesthe issuance, pre-payment, or purchase, of bonds among a continuum of maturities. Hisfinancial counterpart are international investors. The planner’s objective is to smoothhis expenditures, i.e., income after financial expenses. Several features complicate theplanner’s problem. First, issuing debt of one particular maturity impacts the issuanceprice.4 Second, the planner faces several sources of risk; (a) he faces income-risk becausehis income is risky, (b) interest-rate (duration) risk because the international yield curvecan unexpectedly change to any shape, (c) liquidity risk: the price impact of issuance’smay become more sensitive. Finally, (d) default risk: the planner can face an increase indefault spreads because the process for exogenous default shocks can change. This paperdevelops the technology to solve this problem.

We proceed in layers by adding one trade off at a time. We start by by setting upthe planner’s problem as an infinite dimensional control problem under perfect foresight.

1This literature builds on the setting developed by Arellano (2008); Aguiar and Gopinath (2006). Exam-ples are Chatterjee and Eyigungor (2012); Arellano and Ramanarayanan (2012) and more recently Aguiaret al. (2016). See Aguiar and Amador (2013) for a recent review of the literature on sovereign debt.

2See for example Bhandari et al. (2016), Angeletos (2002); Buera and Nicolini (2004) and the seminalcontributions of Barro (1979); Stokey et al. (1989); Aiyagari et al. (2002).

3The seminal contribution is Leland and Toft (1996). For recent examples in this literature see He andMilbradt (2014); Chen et al. (2014).

4A recent micro-foundation of price impact for each maturity is in Vayanos and Vila (2009). Greenwoodand Vayanos (2014) test the implications of a version of the preferred habitat theory of the interest rates,finding that the supply of bonds is a predictor of the interest rates the government pays. Preferred habitattheory of the interest rates dates back at least to Culbertson (1957) and Modigliani and Sutch (1966); for aclassic application to debt management see Modigliani and Sutch (1967).

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The trade off for the planner is to consume and issue debt of different maturities to min-imize the interest rate and adjustment cost of achieving a particular path. To solve thisproblem we adapt tools from the literature on Mean Fields games5 and characterize thenecessary conditions for an optimal debt management and consumption policy, and de-velop an algorithm to compute this solution. For this case we can characterize full transi-tional dynamics: from any initial distribution of debt towards a steady state. One of thereasons why solving this benchmark is useful if because it provides us with the tools toanalyze, numerically and analytically, unexpected-permanent shocks.

Besides the methodological contribution of solving this perfect foresight case, one in-sight that emerges from this problem is that it can be solved as if the government had acontinuum of subordinate traders. Each trader is in charge of managing debt of a sin-gle maturity. Each trader behaves as if he was risk-neutral, but takes as given the pro-cess for the international yield curve and an “internal” discount factor. The behavior ofeach trader is then characterized by an individual Hamilton-Jacobi-Bellman (HJB) equa-tion that determines the value for each type of debt. That value for the trader problemcorresponds to the marginal value of debt of a particular maturity for the planner. Theissuance policy for the trader problem is the optimal issuance of debt of each maturitygiven the assigned planner’s discount factor. The induced evolution of the debt profilethen determines the planner’s disposable income, which in turn, must be consistent withthe internal discount factor. As it turns out, we can solve for the HJB equation of eachtrader analytically, taking as given the planner’s discount factor. Hence, the only numeri-cal requirement is to solve for a fixed-point problem in the planner’s discount factor, andas a consequence, the numerical algorithm converges in seconds.

The optimal issuance of the traders is given by a very simple formula that extends tothe cases with fluctuation in interest rates and income and the case with default incen-tives. Issuances are given by:

ι(τ, t)︸︷︷︸optimal issuance

=

value discrepancy︷︸︸︷ψ(τ, t) + v(τ, t)

λ(τ, t)︸︷︷︸price impact

.

In this equation, ι(τ, t) is the optimal issuance at date t of a constant coupon debt whoseface value matures at τ. The optimal issuance depends on the discrepancy between theinternational price of debt of that maturity at that date, ψ(τ, t), and the internal valuationof the cost of debt of that maturity, v(τ, t). Naturally, if ψ+ v is positive, it is as if the trader

5See Bensoussan et al. (2016) and for an application to Economics see Nuño and Moll (2015) .

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issuing that debt receives a higher amount than his net-present valuation of the cost ofthat debt. Without a price impact, the trader would issue an infinite amount of that debtto exploit that arbitrage. When the planner aggregates among all traders, he would noticethat either he gave the traders the wrong discount factor, in which he would have to givea new instruction, or there is indeed an arbitrage. In our framework, the function λ isa measure of the liquidity cost associated with that issuance. As long as it is a positivenumber, this parameter controls how quickly issuance of one type will occur.

We then introduce risk in either income, interest-rate, or liquidity. We extend the per-fect foresight characterization to account for permanent and expected shocks. The impor-tance of introducing this case is that, even though the perfect foresight case is useful inunderstanding the price vs cost of issuances trade off, this portfolio of debt is not de-termined by risk considerations. There are three features of this problem that are worthnoting. First, risk does not alter the issuance margin. However, valuations for the gov-ernment and the international investor are affected. Second, as opposed to the perfectforesight case, we cannot solve for the exact transitional dynamics; even in a case withPoisson shocks the problem becomes numerically intractable to solve exactly. Therefore,we study shocks that are not recurrent and we characterize exactly the risky steady statedistribution of debt before the shock, and then the full transitional dynamics after theshock has hit. Third, the fact that we can model debt of different maturities introduces in-teresting margins. For example, when a negative income shock hits, the desire to smooththe shock by the government, added to the liquidity cost and finite maturity, producesan issuance cycle; this implies that a large portion of debt will be concentrated in a smallinterval of maturities, and will be due all together.

We finally turn to the case where there is risk of default. In particular, we study a setupin which there are no shocks to income, interest rates, or liquidity, but the governmentcannot commit to repay debt and receives an option to default with Poisson intensity.The value of the option is drawn from a distribution as in Aguiar et al. (2016). We againprovide necessary conditions and an algorithm to compute the solution. To characterizethe solution we study the limit of a perturbed problem in which the government has fullcommitment over an internal of time, as in the deterministic case. The solution of the gov-ernment problem is the limit when the interval converges to zero. This solution approachprovides the Markov perfect stackelberg Equilibria of the game between the governmentand the international investors.6 As in the case with shocks to income, interest rate, or

6Finite-dimensional Markov Perfect Stackelberg equilibria have been studied both in continuous anddiscrete time. See for example Basar and Olsder (1998). An example in Economics of Markov Stackelbergequilibrium is Klein et al. (2008).

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liquidity, we study a risky steady state. We find that the planner tilts maturities towardsshort term debt, in line with the findings of Aguiar et al. (2016).

2 A Sovereign Borrower

Environment. Time is continuous. There is a single, freely-traded consumption goodwhich has an international price normalized to one. The economy is endowed with a flowof yt units of the good where ytt≥0 is a continuous Markov process.7 The householdpreferences over paths for consumption c (t) are given by

V0 =∫ ∞

0e−ρtU (c (t)) dt,

where the instantaneous utility U (·) is increasing, concave, and the discount factor ρ

is a positive constant. Households receive transfers from the government and do notintervene in the financial market.

Government. The benevolent government (the planner) wants to maximize the utilityof the representative household. To do so, the planner trades a continuum of bonds withrisk-neutral competitive foreign investors. These bonds differ in their expiration dates τ ∈(0, T] where T is the maximum maturity. At the maturity date, the principal is repaid. Thestock of outstanding bonds owed by the Government at time t with a time-to-maturity τ isdenoted as f (τ, t). The law of motion of the stock of maturities satisfies the Kolmogorov-Forward equation

∂ f∂t

= ι (τ, t) +∂ f∂τ

, (2.1)

where ι (τ, t) is the new issuance of bonds of time-to-maturity τ at time t. The issuancesι (τ, t) are chosen from a space of functions I : [0, T]× (0, ∞)→ R that satisfies technicalconditions.8 By construction we have that f (T+, t) = f (0−, t) = 0. Finally, we let f0 (τ)

be the initial stock of of debt of maturity τ. The budget constraint of the government is:

c (t) = y (t)− f (0, t) +∫ T

0[q (τ, t, ι) ι (τ, t)− δ f (τ, t)] dτ, (2.2)

7The stochastic process is defined on a filtered probability space(Ω,F , Ftt≥0 , P

).

8In particular I =L2 ([0, T]× (0, ∞)) is the space of functions on [0, T] × (0, ∞) with a square that isLebesgue-integrable.

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where f (0, t) is the repayment of the principal of the bonds at maturity,∫ T

0 qιdτ is theamount of funds collected by issuing new debt −or spent in purchases of assets− and−∫ T

0 δ f dτ is the financial expenditure repayments of the loan coupons. The problem ofthe government is

V [ f (·, t)] = maxι(·)∈I

Et

[∫ ∞

te−ρ(s−t)U (c (s))

](2.3)

subject to the law of motion of debt (2.1) and the budget constraint (2.2). Here V [ f (·, t)]is the optimal value functional, which maps the distribution f (·, t) at time t into the realnumbers.

International investors. The government sells bonds to competitive risk-neutral inter-national investors at a price q (τ, t, ι). This price has two separate components:

q (τ, t, ι) = ψ(τ, t) + λ (τ, t, ι) .

The first component, ψ(τ, t), is the valuation by the international investor of the domesticbond. This price ψ(τ, t) is given by

ψ(τ, t) = Et

[e−∫ t+τ

t r(u)du + δ∫ t+τ

te−∫ t+τ

t r(u)dudτ

]. (2.4)

The idea is that at every t international investors can invest elsewhere in bonds of matu-rity τ ∈ [0, T] that they buy at a price ψ(τ, t). These bonds are priced by arbitrage giventhe stochastic process for the short rate r(t). The second component, λ (τ, t, ι), representsa liquidity cost associated with issuing or purchasing ι bonds of time-to-maturity τ. Theliquidity cost λ is convex in ι and the idea is that it captures the impact of many issuancesof a given bond at a point in time. For the rest of the paper we work with a tractablefunctional form: λ (τ, t, ι) = −1

2 λι(τ, t).9

Equilibrium. We study a Markov Equilibrium with state variable f (τ, t); it is definedas follows. A Markov equilibrium is a value functional V [ f (·, t)] , a issuance policyι (τ, t, f ) , bond prices q (τ, t, ι, f ) , a stock of debt f (τ, t) and a consumption path c (t)

9Note that the fact that the bond obtains q(τ, t, ι) < ψ(τ, t) does not mean that there is an arbitrage.One way to micro found the yield curve that the government is confronting is to follow Vayanos and Vila(2009). In this paper there is a downward slopping demand curve for bonds of the government at time t formaturity τ coming from a preferred habitat. An alternative is to micro found λ as an inter mediation costof issuance that is paid to a broker dealer.

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such that: 1) Given c (t) , q (τ, t, ι f ) and f (τ, t) the value functional satisfies governmentproblem (2.3) and the optimal control is ; 2) Given ι (τ, t, f ) the debt stock f (τ, t) evolvesaccording to the KPE equation (2.1); 3) Given ι (τ, t, f ) , q (τ, t, ι, f ), f (τ, t) and c (t) thebudget constraint (2.2) of the government is satisfied.

3 Perfect Foresight

We start with the study of the problem of a government that faces a constant interest rate rand output y, and begins with an initial condition f (·, 0). In particular, the planner solvesP1:

V [ f (·, t)] = maxι(·)∈I

Et

[∫ ∞

te−ρ(s−t)U (c (s)) ds

](3.1)

subject to the law of motion of debt (2.1), the budget constraint (2.2), an initial conditionf (·, 0), and prices of debt as given. We study this case because it reveals some interestingforces. It is also the simplest case to illustrate our solution approach. For this case wecan characterize the steady state distribution of debt f ∗(τ) , issuances ι∗(τ), and theirdeterminants; an exact path for the transition from any initial distribution f (·, 0) to thedistribution in steady state f ∗(τ); and finally, the response to permanent and unexpectedshocks, f ∗(τ, Θ) → f ∗(τ, Θ′) where Θ, Θ′ are two different sets of parameters. Oneof the advantages of the methodology we are introducing is that even though there isa continuum of bonds, the numerical solution is simple to implement and converges inseconds.

3.1 Optimal Paths: Necessary Conditions

We now characterize the solution of P1. The solution strategy for this problem involvessetting up an infinite dimensional Lagrangian. We reproduce here the main ideas of theproof. First, note that the Lagrangian is given by:

L (ι, f ) =∫ ∞

0e−ρtU

(y (t)− f (0, t) +

∫ T

0[q (t, τ, ι) ι (τ, t)− δ f (τ, t)] dτ

)dt

+∫ ∞

0

∫ T

0e−ρt j (τ, t)

(−∂ f

∂t+ ι (τ, t) +

∂ f∂τ

)dτdt.

An optimal path of issuances and debt for the government cannot be improved up to firstorder. This implies that taking a variation of the control ι would imply no increase in the

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Lagrangian. This holds if:

U′ (c)(

∂q∂ι

ι (τ, t) + q (t, τ, ι, f ))= −j (τ, t) .

That is, the gateaux-derivative of the Lagrangian with respect to ι has to be equal to zero.The intuition is that, by issuing or reducing debt, the government is changing the con-sumption of the household directly, and indirectly by changing the prices of debt. Sec-ond, there has to be no improvement up to first order with respect to the stock of debt,the state variable. Therefore, issuances equalize the marginal value of issuing debt andthe marginal cost, a change in the continuation value. Taking the gateaux-derivative withrespect to the state, f , we obtain a PDE for the Lagrange multipliers. In particular:

ρj (τ, t) = −U′ (c (t)) δ +∂j∂t− ∂j

∂τ, if τ ∈ (0, T] (3.2)

First, note that consumption is changing due to the change in debt; this is measured bythe term −U′ (c (t)) δ. Second, note that the variation in the stock of debt over time andmaturities will imply that the multipliers associated with the stock of debt, j(τ, t), willbe changing over time. This is accounted by the terms ∂j

∂t ,∂j∂τ , ρj (τ, t).10 The tradeoff is

clear: a change in debt will imply a change in consumption, but also, an alternative lawof motion for the debt, captured by the evolution of the multipliers. It will be useful toredefine the multiplier j (τ, t) in terms of units of consumption. Define it as:

v (τ, t) := j (τ, t) /U′ (c (t)) .

Effectively we re-express the multipliers j(τ, t) in terms of consumption goods. This im-plies that we can re-express the first order conditions with respect to ι as

∂q∂ι

ι (τ, t) + q (t, τ, ι) = −v (τ, t) ,

and first order conditions with respect to f , the PDE equation of the multipliers (3.2), by:(ρ− U′′ (c (t))

U′ (c (t))

.c(t)c(t)

)v (τ, t) = −δ +

∂v∂t− ∂v

∂τ, if τ ∈ (0, T).

The following proposition summarizes the necessary conditions for a solution to P1.

10Note that the PDE for j is the analog of the ODE of the co-state when the costate is unidomensional. Inthis case, for example if the government only had access to an instantaneous bond, the PDE for the costatewould be given by ρj (t) = −δU′ (c (t)) + ∂j

∂t .

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Proposition 3.1. If a solution to P1 with e−ρt f , e−ρtι ∈ L2 ([0, T]× [0, ∞)),e−ρtc ∈ L2[0, ∞),given by ι(τ, t), c(t)∞

t=0 exists, it satisfies the PDE

r (t) v (τ, t) = −δ +∂v∂t− ∂v

∂τ, if τ ∈ (0, T]

v (0, t) = −1,

limt→∞

e−ρtv (τ, t) = 0

where v (τ, t) is the marginal value of a unit of debt with time-to-maturity τ, the interest rate r (t)is given by r (t) = ρ− U′′(c(t))

U′(c(t))

.c(t)c(t) and e−ρtv ∈ L2 ([0, T]× [0, ∞)); the optimal issuance ι (τ, t)

is given by∂q∂ι

ι (τ, t) + q (t, τ, ι) = −v (τ, t)

Proof. See Appendix.

3.2 Understanding Optimal Paths: Theory

In this subsection we discuss the result in Proposition 3.1 and the behavior of the bench-mark when there are no adjustments costs.

Benchmark: λ = 0. Given that we introduce adjustment costs in debt issuances, we startwith the discussion for the case in which there are no adjustment costs. The proof of thenecessary conditions is mechanically similar to the case with adjustment costs; however,the solution is qualitatively different. To be precise formally state it.

Proposition 3.2. If a solution to P1 with e−ρt f , e−ρtι ∈ L2 ([0, T]× [0, ∞)),e−ρtc ∈ L2[0, ∞),given by ι(τ, t), c(t)∞


r (t) v (τ, t) = −δ +∂v∂t− ∂v

∂τ, if τ ∈ (0, T]

v (0, t) = −1,

limt→∞

e−ρtv (τ, t) = 0


U′(c(t))

.c(t)c(t) and e−ρtv ∈ L2 ([0, T]× [0, ∞)); valuations are such that:

ψ (t, τ) = −v (τ, t)

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In this benchmark the solution coincides with the solution of a standard consump-tion savings problem with one bond. First, note that because the yield curve is arbi-trage free, intuitively, the government should be indifferent regarding issuing debt ofany maturity. The cost would be the same. This emerges as a feature of the solution in:q (t, τ) = −v (τ, t). This means that the valuation of any bond for the investors and forthe government is the same. If this were not to hold, then the government would wantto issue an infinite amount of one maturity. Second, note that for this equality will holdif only of r(t) = r for all t. This implies that ψ (τ) = −v (τ) , so as long as the interestrate is constant, the valuations are equal and constant. Third, note that (for the CRRAcase) we get from the definition of r(t) that

.c(t)c(t) = ρ−r

σ . This implies that the growthrate of consumption is constant as well. Finally, we need to pin down the initial value ofconsumption. This will be pinned down by the budget constraint

c(t) = y (t)− f (0, t) +∫ T

0[ψ (t, τ) ι (τ, t)− δ f (τ, t)] dτ.

Note that the total amount of debt that is issued at each point in time is pinned down,∫ T0 ψ (t, τ) ι (τ, t), but not on what particular maturity.

Two Cases with λ > 0. After discussing the benchmark we now go back to the maincase that we focus on this paper. It is woth noting that there are three casees in terms ofλ. The first one is λ = 0, that we just discussed. The second, for a low level of lambda, iswhen λ ∈ (0, λmin]. This case is characterized by consumption decaying over time, andthe growth rate of decay depends on lambda. This case will be detailed in the Appendix.The final case, is the one we focus in this section, λ > λmin, the solution converges (infinite time) to

.c(t)c(t) = 0 and limt→∞ c(t) = css; ie, this case is defined by a steady state with

positive consumption and zero consumption growth.

Main Case: λ > λmin: Steady State. Regarding the steady state, we can characterizevaluations, issuances, and the stock of debt in closed form. Discussing the steady stateequations is useful to understand the role of liquidity frictions in the model. How muchthe government issues of each maturity τ? Note that because

.c(t)c(t) = 0, r(t) = ρ. Therefore,

the valuation of debt of maturity τ for the government is given by:

v∗(τ) = −δ1− e−ρτ

ρ− e−ρτ.

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Issuances, therefore, are given by

ι∗(τ) =v∗(τ) + ψ(τ)

λ;

and the stock of debt is the sum of past flows:

f ∗(τ) =∫ T

τι∗(s)ds.

Recall that the valuation of international investors is given by:

ψ(τ) = δ1− e−rτ

r+ e−rτ.

The trade off is the following: the benefit of issuing debt of a particular maturity is ψ, theincome; the cost is the tightening of the budget constraint measured in units of consump-tion today v(τ), and the issuance marginal cost λι.

The are four points regarding steady state debt issues that are worth noting. First,there is a tilt towards long term debt as long as the government is more impatient thaninternational investors; that is, when ρ > r. The idea is that because the government ismore impatient than the investors, as in a consumption savings problem, the governmentwill accumulate debt. However, debt has a cost of issuance. The cost of issuance favors is-suing debt less often. Note that because of the decreasing returns to scale of the issuance,the issues are not only in the long maturities. Second, note that for the case in which ρ = rthe government issues zero debt in steady state. In a standard consumption savings prob-lem total debt in this case would be indeterminate. However, because there are issuancecosts, the government does not want to issue any more debt that the one it has inherited.Third, note that as the government becomes more patient, the total debt is going down. Ifthe government is more patient than the international investor the same logic applies, butthe government accumulates assets. This is standard logic that comes from consumptionsavings problems. Fourth, note that the trade off in steady state is, given a stock debt, tofinance it minimizing the expenditure in adjustment costs. The amount to be issued ateach maturity depends on the difference in valuations of the debt by the government andthe international investors. And this difference in valuations depends on their discountfactor.

Figure A.1 displays the steady state of the model for the calibration with ρ > r. Asteady state solves the ODE

ρv (τ) = −δ− ∂v∂τ

,

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with boundary condition v (0) = −1. All quantities are expressed in percentage of thesteady state output (that is equal to 1). In steady state, the country devotes 6 percentof GDP to debt service; 4.4 percent of GDP to the payment of bond principals and 1.6percent of GDP to coupon payments. Liquidity costs, that in the current calibration are0.3 percent of GDP, that is, about 5% of total financial expenses. New debt issuance’s arealso 4.4. of GDP, since at steady state, they compensate for the payment of the principal.Consumption is 97.6 percent of GDP.

Main Case: λ > λmin: Transitional Dynamics. Besides the steady state, the model hasinteresting insight regarding the transitional dynamics. First, lets focus on the PDE forthe value of each vintage. For each one of the maturities, the value evolves over timeaccording to

r(t)v (τ, t) = −δ +∂v∂t− ∂v

∂τ.

This is the value measured in goods, of issuing a particular vintage τ. This value has acash flow that is −δ, a discount rate r(t) that is the internal rate for all the vintages, andan evolution over time because the vintages are closer to maturity as time goes by. Thisvalue is negative and the solution is given by

v(τ, t) = −∫ t+τ

te∫ t+τ

u r(s)dsδdu− e−r(τ+t).

The planner values the bonds only taking into account the internal valuation. The bound-ary conditions for the PDE are: v (τ, t) = −1 and limt→∞ e−ρtv (τ, t) = 0. The first onestates that a bond maturing instantaneously has a marginal cost in terms of coupons of-1. All the cost comes from the principal. The second one, states that the valuation of avintages is not exploding. If this were to be the case there would be a feasible variationthat the government could exploit to increase its welfare. Second, note that the character-ization provides naturally an algorithm to compute the transitional dynamics. We startfrom a sequence of consumption that yields an interest rate: r(t) = ρ + σ

c(t)c(t) . Plug r(t)

into value PDE:r(t)v (τ, t) = −δ +

∂v∂t− ∂v

∂τ

Build issuances:ι (τ, t) =

ψ(τ, t) + v(τ, t)λ

12

The tradeoff regarding issuances is the same one as in steady state. Obtain f (t, τ):

f (τ, t) =∫ minT,τ+t

τι(t + τ − s, s)ds + I[T > t + τ] · f (τ + t, 0)

Construct a new path for consumption c(t) given by

cout(t) = y (t)− f (0, t) +∫ T

0[q (τ, t, ι) ι (τ, t)− δ f (τ, t)] dτ

And iterate until convergence. In the Numerical Appendix we discuss the numericalmethod to find a solution and we discuss the code.

3.3 Understanding Optimal Paths: The Dual

In order to state Proposition 3.1 we transformed the Lagrange multipliers j(τ, t) intov(τ, t) and because they solved a PDE that resembled a bond price, we referred to themas value of issuances. We now show this formally: how v(τ, t) measures a marginal costof issuing a unit of maturity τ at moment t. We do so by solving the dual of P1. Thisproblem finds the lowest cost of achieving a particular path of consumption with thelowest amount of resources. Given a desired path of consumption c(t) the objective is tominimize the resources needed to achieve that path. More precisely, D1 is given by:

D [ f (·, 0)] = minι(τ,t),ytt∈[0,∞),τ∈[0,T]

∫ ∞

0e−∫ t

0 r(s)dsytdt s.t.

c (t) = y (t)− f (0, t) +∫ T

0[q(τ, t, ι)ι(τ, t)− δ f (τ, t)] dτ

∂ f∂t

= ι(τ, t) +∂ f∂τ

; f (τ, 0) = f0(τ)

r(t) = ρ + σc(t)c(t)

Turns out that the bond prices of the traders, v(τ, t) are the Lagrange multipliers of thisoptimization problem. The next Proposition establishes this result.

Proposition 3.3. Suppose that for a given income path y(t) and initial debt f0 the solution to P1is c∗(t), ι∗(τ, t), j∗(τ, t). Then, y(t), ι∗(τ, t), j∗(τ,t)

u′(c(t)) solves D1 given the path c∗(t).

Proof. See Appendix on Duality.

13

3.4 Understanding Optimal Paths: Numerical Results

Coming back to the solution of P1 in this section we want to illustrate numerically twomain forces that drive the solution: consumption smoothing versus the smoothing ofadjustment costs. We will study a permanent and unexpected shock to output and theinterest rate that reverts to the long run initial value of them. In particular, we study anAR(1) income path for y(t) to illustrate the issuance cycle and an AR(1) path for the shortrate r(t). Transitional dynamics reveal two forces: issuance cycles and consumption vs.price smoothing.

A Shock to Output. In Figures A.2 and A.3 we analize the response of issuances, con-sumption and total debt from a shock to output of 5% that reverts to the long run steadystate as an AR(1) process. The main take out is that to smooth the shock the governmentincreases issuances on impact, and this will generate a wave of payments concentrated inT years. Upon impact, we observe two things. We see an immediate increase of issuancesand a pronounced increase on impact of the internal discount factor. The liquidity costprevents a perfect smoothing of consumption. This is why the internal discount factorsjump. Also, there is a cycle of payements. As the initial vintage of borrowings matures,and it is particularly pronounced for long-term bonds, consumption growth slows down,but then accelerates again as the wave passes. This is an interesting phenomenon becauseit suggests that in presence of liquidity costs, we should expect waves of debt refinancing.

A Shock to the Interest Rates. In Figures A.5 and A.6 we analyze the response ofissuances, consumption and total debt from a shock to the interest rate, that goes to zero,and returns to the long run steady state as an AR(1) process. We compare the responseswhen σ = 2 and σ = 0. When the IES is not infinite, the model shows that when ratesare unusually low, the country increases it’s borrowing. This is captured by a spike inconsumption beyond it’s steady state level. Then, as rates begin to increase, the issuancerate declines. Eventually, there’s a period low consumption were debt is being repaid.The reason for this repayment phase is the liquidity cost. As rates return to normality,while the stock is higher due to the past issuances, the country is making higher interestand principal payments, which take consumption to a lower level than at steady state. Asthe debt is repaid and issues return to steady state, consumption converges back to steadystate. Turning on a consumption smoothing motive tampers this effect. There’s a trade-offbetween exploiting the low interest rate environment and smoothing consumption.

14

Take Away Deterministic Dynamics. First, Steady-State: preference for long-termdebt. Second, the importance of maximal maturityT: produces an issuance cycle. Third,the role of the IES: Adjustment cost vs. consumption smoothing

4 Risky Steady State

We now study the role of risk in the dynamics of government debt. The goal is to studyhow the small-open economy should behave while expecting a shock. In particular, thecountry is expecting a jump to a new path of output or short-term rates. The new pathconverges to the same or a different steady state. To shorten the exposition, our derivationhere focuses on the case of a sudden drop in output; however, the same argument appliesto a change in the short interest rate. Thus, for now, we assume that the endowment pro-cess follows a two-state Markov process with values

yH, yLwhere state yL is absorbing.

If the economy is in state yH, it may jump to state yL at a rate φ. This might be interpretedas if the economy is initially in a state yH (“normal times”) and may receive an aggregateshock that permanently reduces its output to yL. The problem of the government, P2, inthis case is:

V[

f (·, t) , yH]= maxι(·)∈I

Et

[∫ τC


+ e−ρ(τC−t)V[

f(·, τC

), yL] ∣∣∣yt = yH

]before the shock hits and

V[

f (·, t) , yL]= maxι(·)∈I

Et

[∫ ∞


∣∣∣yt = yL]

,

after the shock hits, subject to the law of motion of the debt distribution, the budgetconstraint and the price of debt. The price of debt will not depend on the state, becausethere is not risk of default. Here τC is the first arrival time of the aggregate shock, and it isdistributed according to an exponential random variable with parameter φ. The dynamicsof ι(

f (·, t) , yL) are given by the equations described the previous Section as the economyconverges to the steady state with endowment yL and remains there.

15

4.1 Optimal Paths: Theory

We provide an intuition for the proof of the necessary conditions for an optimal path.Note that from the principle of optimality we can show:

ρV [ f (·, t)] = maxι(·,t)U (c (t)) +

∫ T

0

δVδ f

∂ f (·, t)∂t

dτ + φ[V [ f (·, t)]−V [ f (·, t)]

], (4.1)

where V [ f (·, t)] := V[

f (·, t) , yL] and V [ f (·, t)] := V[

f (·, t) , yH]. The flow value ofutility, the left hand side, has to be equal to the static utility of consumption plus theexpected change in the continuation value. The only non-standard term is coming fromthe fact that the distribution is changing not only because of the shock, but also becausethe distribution of maturities can be changing over time; the second term on the righthand side of the HJB. The necessary conditions come from taking first order conditionsin the HJB. First, note that from the first order condition with respect to issuance is givenby:

U ′ (c)(

∂q∂ι

ι (τ, t) + q (τ, t, ι)

)= −δV

δ f.

and if we define j (τ, t) := δVδ f , then the first order condition results in

U ′ (c)(

∂q∂ι

ι (τ, t) + q (τ, t, ι)

)= −j (τ, t) .

So, as in the perfect foresight case, the marginal cost of issuances, that is the sum of themultiplier and the adjustment cost, equalizes to the the marginal revenue benefit. Second,from the first order conditions with respect to f :

(ρ + φ) j (τ, t) = U ′ (c (t)) (−δ) +∂j∂t− ∂j

∂τ+ φ (τ, t) , if τ ∈ (0, T] (4.2)

j (0, t) = −U ′ (c (t)) , if τ = 0.

If we define the variables

v (τ, t) = j (τ, t) /U ′(

cH (t))

,

v (τ, t) = (τ, t) /U ′(

cL (t))

,

we can spell out our main result where cH (t) is consumption before the shock hits, andcL (t) is consumption after the shock hits. The following proposition states the main re-

16

sult of this section. Note the terms we had before there is an additional one that takesinto account the fact that with probability φ the shock to output arrives; this is the usualcorrection for risk of a Poisson event. We are now ready to state the main result of thesection.

Proposition 4.1. If a solution to P2 with e−ρt f , e−ρtι ∈ L2 ([0, T]× [0, ∞)),e−ρtc ∈ L2[0, ∞)

given by ι(τ, t), c(t)∞t=0 exists, it satisfies the PDE(

ρ−U ′′(cH (t)

)U ′ (cH (t))

dcH

dt

)v (τ, t) = −δ +

∂v∂t− ∂v

∂τ+ φv (τ, t)

U ′(cL (t)

)U ′ (cH (t))

, if τ ∈ (0, ∞),

v (0, t) = −1, if τ = 0,

limt→∞

e−ρtv (τ, t) = 0,

the optimal issuance ι (τ, t) is given by

∂q∂ι

ι (τ, t) + q (τ, t, ι) = −v (τ, t) ,

before the shock hits, and after the shock hits is given by

r (t) v (τ, t) = −δ +∂v∂t− ∂v

∂τ, if τ ∈ (0, T]

v (0, t) = −1,

limt→∞

e−ρtv (τ, t) = 0

where v (τ, t) is the marginal value of a unit of debt with time-to-maturity τ, the interest rate

r (t) is given by r (t) = ρ− U′′(c(t))U ′(c(t))

.cL(t)

cL(t) and e−ρtv ∈ L2 ([0, T]× [0, ∞)); the optimal issuanceι (τ, t) is then given by

∂q∂ι

ι (τ, t) + q (τ, t, ι) = −v (τ, t) .


4.2 Understanding Optimal Paths: Theory

The solution to Proposition 4.1 is a fixed-point in a suitable space of functions with do-main [0, T] × [0, ∞). To see this, notice that the value function v (τ, t) before the shockarrival depends on the continuation value v (τ, t) , which in turn is a function of the valueof the distribution f (τ, t) at the (unknown) time of the shock arrival. As the distributiondepends on the issuances ι (τ, t) and hence on the value function itself v (τ, t). The nu-

17

merical solution to this fixed point problem would be extremely complex, as it requiresiterating over spaces of functions. However, in a particular case of interest (the risky-steady state) the system can be analyzed in a tractable way, as we describe next. Morespecifically, given an initial distribution of debt f0, the solution involves the governmentchoosing a distribution of debt at each point in time just in case the shock hits. In turn,this choice depends on the continuation values, so it implies a fixed point in a sequenceof distributions.

Benchmark: λ = 0. We start again with the case in which there are no adjustment costs.We state the Proposition precisely below.

Proposition 4.2. If a solution to P2 with e−ρt f , e−ρtι ∈ L2 ([0, T]× [0, ∞)),e−ρtc ∈ L2[0, ∞)

given by ι(τ, t), c(t)∞t=0 exists, it satisfies the PDE(

ρ−U ′′(cH (t)

)U ′ (cH (t))

dcH

dt

)v (τ, t) = −δ +

∂v∂t− ∂v

∂τ+ φv (τ, t)

U ′(cL (t)

)U ′ (cH (t))

, if τ ∈ (0, ∞),

v (0, t) = −1, if τ = 0,

limt→∞

e−ρtv (τ, t) = 0,

the optimal issuance ι (τ, t) is given by

q (τ, t) = −v (τ, t) ,

before the shock hits, and after the shock hits is given by

r (t) v (τ, t) = −δ +∂v∂t− ∂v

∂τ, if τ ∈ (0, T]

v (0, t) = −1,

limt→∞

e−ρtv (τ, t) = 0

where v (τ, t) is the marginal value of a unit of debt with time-to-maturity τ, the interest rate r (t)is given by r (t) = ρ− U

′′(c(t))U ′(c(t))


is then given byq (τ, t) = −v (τ, t) .

The characterization of an optimal path in this case, again follows the one for the caseof one bond. First, note that because the yield curve is arbitrage free and there is no riskof default, investors will be indifferent among bonds of different maturities. The shock is

18

to output. At the same time, the government is also indifferent. This emerges as a featureof the solution in: q (t, τ) = −v (τ, t). This means that the valuation of any bond for theinvestors and for the government is the same. Second, note that for this equality to holdif only of r(t) = r for all t. This implies that ψ (τ) = −v (τ) and ψ (τ) = −v (τ) so aslong as the interest rate is constant, the valuations are equal and constant. Third, note thatfrom the definition of

.c(t)c(t) = ρ−r

σ and this implies that the growth rate of consumption isconstant as well. Fourth, finally, we need to pin down the initial value of consumption.

Risky Steady State. We will characterize debt, issuances and consumption, for the casein which the economy is waiting for the shock but the shock has not arrived yet; that is,in histories where the shock has not occurred yet and the economy already convergedto a steady state. In this case the solution of the system is composed of the followingequations:

(ρ + φ) v (τ) = −δ− ∂v∂τ

+

[φU ′(cL (0)

)U ′ (cH)

]v (τ, 0) , if τ ∈ (0, ∞)

v (0) = −1 if τ = 0(ρ−U ′′(cL (t)

)U ′ (cL (t))

dcL

dt

)v (τ, t) = −δ +

∂v∂t− ∂v

∂τ, if τ ∈ (0, ∞), v (0, t) = −1, if τ = 0,

cH = yH − f H (0) +∫ T

0

[q(

τ, ιH)

ιH (τ)− δ f H (τ)]

dτ,

cL (t) = yL − f L (0, t) +∫ T

0

[q(

τ, ιL)

ιL (τ, t)− δ f L (τ, t)]

dτ,

∂q∂ι

ιH (τ, t) + ψ(

τ, ιH)= −vH (τ) ,

∂q∂ι

ιL (τ, t) + ψ(

τ, t, ιL)= −vL (τ, t) ,

0 = ιH (τ) +∂ f H

∂τ∂ f L

∂t= ιL (τ, t) +

∂ f L

∂τ,

f L (·, 0) = f H (·) .

We study this case because we can obtain an exact solution for the transitional dynam-ics. The case in which the shock can arrive at any moment involves computing the fixedpoint in a sequence of distributions and is numerically unfeasible without an approxima-tion. An analogous challenge appears in models of incomplete markets and one proposedsolution is an approximation as in Krusell and Smith (1998).

19

4.3 Optimal Paths: Risky Steady State Numerical Results

Next, we present two numerical exercises for the general case. In the first exercises, outputis expected to drop by 5% on impact, and then a recovery back to steady state. Expectinga 5% y(t) drop. First, we compare the risky steady state (red), with the steady state(blue). There are two patterns to be dissected from Figures A.7 and A.8. On one hand, thepresence of income risk produces an overall decline in issuances and debt outstandingof all maturities. On the other hand, we see a relative decline in shorter maturities. Thereason for the decline in overall borrowing is the presence of risk, which captured by theratio of marginal utilities:

φU ′(cL (0)

)U ′ (cH)

,

present in the valuation formulas. This ratio tells us how costly an outflow of paymentsis once the shock is realized. This penalizes all bonds, and in the valuation equations theeffect is analogous to an expected increase in coupon payouts. The second observation isthat the decline is more pronounced on short term assets. It’s better to explain the logicin the next exercise.

Shock 4% to 8% on impact, ρ = 6%. In this experiment, depicted in Figures A.9and A.10, we present an expected shock where short term rates are expected to increasesuddenly, to 8%. This is a rate above the discount factor of the small economy. As in theprevious exercise, the effect of this source of risk is to shrink the issuances at all maturities.However, the exercise results in a more extreme reversal of positions. For very short-termbonds, the country actually begins buying back and eventually accumulating short termbonds. This means that the way the country reacts to risk is by issuing long-term debt butholding short term assets. Again, all of the logic is captured by the valuation equations. Along-term bond will be long lived. With a high probability, it will experience a cycle thatbegins with a positive spread between r(t) and r(t), but then, when the shock is realized,with a converse relationship. When international rates are high, the country is worseoff by holding those assets, because paying out coupons when discounts are high is notdesirable. However, there are two other phases where the country spread is positive. Thismakes long-term debt desirable, despite the fact that with a high probability there will bea period where the debt will be particularly costly because marginal utility is high. Theprinciple, on the other hand, will be likely to be repaid once marginal utility is close tosteady state again. Short-term debt is different. Before the shock is realized, the short-term bond yields a benefit because rrss − rss > 0. However, being a short-term bond, itis likely to experience a period of high internal discounting. The chances of entering thisperiod are the same as for the long-term bond. However, the short bond doesn’t have the

20

length to cover the reversal of the cycle. Furthermore, it’s principle is likely to be paid inperiods of high marginal utility. This makes the short-term bond be valuable as an asset.

Both exercises have the same forces behind. However, the calibration of interest-raterisks are strong enough to reverse the positions.

5 Defaultable Debt

We now introduce the option to default for the government. The nature of the problemchanges since this interaction is a game between the government and the internationalinvestors: the prices, as opposed to the first two cases, will depend on the actions of thegovernment, and these actions will in turn depend on prices. Our strategy to solve thisstrategic interaction will be to spell out a problem where the government can only defaultin fixed moment in time, being precise about the timing of actions, and then take the limitwhen the time interval between default options goes to zero. We start this section bydiscussing how the setup of the first sections of the paper is modified to include the optionto default. Then we state and show the main result: a characterization of the consumptionand debt policies when there is risk of default. We finally study a risky steady state andnumerically solve the model.

Government. The setup is analogous to the one in the main section but now the gov-ernment has the option to default on debt. This option arrives as a Poisson process withintensity θ. Once the government receives the option to default it draws a value VD from[V, V] according to some distribution Φ. It decides to default or not by comparing thisvalue with the value of repayment. When it defaults, it defaults on all loans. We denoteby τD the time to default. The latter is a stopping time with respect to the filtration Ft.If we define τn as the n−th arrival time of the Poisson shock that grants the governmentthe options to default, then

τD := min

τn, VD > V [ f (·, t + τn)]

.

International Investors. The government sells bonds to competitive risk-neutral foreigninvestors that can invest elsewhere at the risk-free real rate r (τ, t). Notice that we allowfor a (stochastic) time-varying (risk free) yield curve where r (τ, t) is the discount factorfor a cash flow of maturity τ at time t. The price of each bond q is, again, composed by

21

two separate components:

q (τ, t, ι) = ψ(τ, t) + λ (τ, t, ι) ,

where ψ(τ, t) is the net-of-liquidity cost price of a bond of time-to-maturity τ in the inter-national market and λ (τ, t, ι) is the liquidity cost associated with issuing or purchasing ι

bonds of time-to-maturity τ. The value of the defaultable bonds ψ(τ, t) is given by

ψ(τ, t) = Et

[∫ t+minτD,τt

e−∫ s

t r(s,u)duδds + 1(

τ < τD)

e−∫ t+τ

t r(τ,u)du

]. (5.1)

The first term is the coupon payment received by investors up until the government de-faults on debt, or up until maturity; that is why the integration limit is t + min

τD, τ

.

The second term is the value of the principal that investors will receive if the governmentdoes not default before the maturity of the bond; that is, if τD the time to default is largerthan the time to maturity.

Government Problem. The problem of the government, PD, is

V [ f (·, t)] = maxι(·)∈I

Et

[∫ t+τD

te−ρ(s−t)U (c (s)) ds + e−ρτD

VD

], (5.2)

subject to the law of motion of debt (2.1), the budget constraint (2.2), and the prices (5.1).Here, again, V [ f (·, t)] is the optimal value functional, which maps the distribution f (·, t)at time t into the real numbers.11

5.1 Optimal Paths: Theory

We focus on the case that the government faces a constant interest rate r and output y.To solve PD we proceed in two steps. First, we setup a discrete commitment problem. Inthis problem the government can default but only at ∆ time intervals. Second, we take thecontinuous time limit of this problem when ∆→ 0, that is, the government can default atany moment in time.

Discrete Commitment Problem. In this problem ∆ is defined as an arbitrary time stepsuch that the government receives the option to default at the end of the interval (t, t+∆].

11There set I is given by I =L2 ([0, T]× [0, ∞)) the space of functions on [0, T]× [0, ∞) with a square thatis Lebesgue-integrable.

22

This implies that the riskless bond price results in:

ψ(τ, t) =∫ t+min(∆,τ)

te−r(s−t)δds + 1τ≤∆e

−rτ

+ 1∆<τe−r∆

[P(τ1 ≤ ∆)Φ (V [ f (·, t + ∆)])ψ(τ, t + ∆)

+P(τ1 > ∆)ψ(τ, t + ∆)]

(5.3)

where and τ1 is the first arrival time of the option to default, characterized by an expo-nential random variable with parameter θ. Note that the first two components are deter-ministic. Applying the Feynman-Kac formula to the bond pricing equation (5.3), wherethe stopping time is the time to default, yields the following PDE for the price of the bond:

rψ(τ, t) = δ +∂ψ

∂t− ∂ψ

∂τ, if τ ∈ (0, T) (5.4)

ψ(0, t) = 1, if τ = 0,

ψ(τ, t + ∆) = P(τ1 ≤ ∆)Φ (V(t))ψ (τ, t + ∆) + P(τ1 > ∆)ψ (τ, t + ∆) , τ ∈ (0, T)

whereΓ (x) :=

∫ ∞

xξdΦ (ξ) .

The problem of the government is then given by:

V (t) : = V [ f (·, t)] = maxι(·)∈I

∫ t+∆


+ e−ρ∆[

P(τ1 ≤ ∆) [Γ (V (t + ∆)) + V (t + ∆)Φ (V (t + ∆))]

+P(τ1 > ∆)V(t + ∆)]

(5.5)

subject to the law of motion of debt (2.1), the budget constraint (2.2) and the bond pricingequation (5.4). Problem (5.5) describes a deterministic commitment problem over an ar-bitrary finite-length interval (t, t + ∆]. Default may only happen at the end of the intervalprovided that an option to default shock has arrived (with probability P(τ1 ≤ ∆)) andthat the value after default is higher than the continuation value in the case of no defaultat the beginning of the next interval (t + ∆, t + 2∆] : VD > V (t + ∆) . In the case of adefault, the expected terminal value is Γ (V (t + ∆)) . Notice that V (t) := V [ f (·, t)] is theaggregate value functional of the problem, which depends on time through its depen-dence on the distribution.

23

Main Result. The solution to the government problem (5.2) is the symmetric MarkovPerfect Stackelberg Equilibrium that is defined as the limit as ∆ → 0 of the family ofproblems in (5.5). The necessary conditions of a solution to (5.2) are given by the followingproposition.

Proposition 5.1. If a solution to problem 5.2 exists with e−ρt f , e−ρtι ∈ L2 ([0, T]× [0, ∞)) ande−ρtc ∈ L2[0, ∞) it should satisfy the PDE

r(t)v (τ, t) = −δ +∂v∂t− ∂v

∂τ− θ [1−Φ (V (t))] v (τ, t) , if τ ∈ (0, T],

v (0, t) = −1, if τ = 0,

with the transversality condition limt→∞ e−ρtv (τ, t) = 0, where v (τ, t) is the marginal valueof a unit of debt with time-to-maturity τ, e−ρtv ∈ L2 ([0, T]× [0, ∞)) and the optimal issuanceι (τ, t) is given by the optimality condition

∂q∂ι

ι (τ, t) + q (t, τ, ι, f ) = −v (τ, t) . (5.6)

The value of V (t) is the solution of the HJB equation

ρV [ f (·, t)] = U (c (t)) +∫ T

0U ′ (c (t)) v (τ, t)

∂ f∂t

dτ

−θ [1−Φ (V [ f (·, t)])]V [ f (·, t)]− Γ (V [ f (·, t)])

and the price of the bond

rψ(τ, t) = δ +∂ψ

∂t− ∂ψ

∂τ− θ [1−Φ (V (t))]ψ(τ, t), if τ ∈ (0, T)

ψ(0, t) = 1, if τ = 0.


There are four points worth noting regarding the necessary conditions for optimaldebt and consumption policies. First, the introduction of the option to default on debt af-fects the valuation of debt for the investor but also for the government. For internationalinvestors, the fact that the possibility of default implies a lower value of debt is commonto any model of credit risk. This is well understood. However, for the government, thecost of issuing a bond of maturity τ and time t is now lower, because now the discountfactor is r(t) + θ [1−Φ (V (t))] . This is intuitive: the government takes into account thata unit of debt will be defaulted under some states of nature, and will as a consequence,

24

imply lower expenses in terms of resources honoring this debt. This is a unique feature ofour model since we can compare the cost in terms of resources of the different possibilitiesfor the government. Second, note that even though the two valuations decrease unam-biguously, this has an ambiguous effect over issuances. For example, in the quadraticcase, issuances are given by ι(τ, t) = (v(τ, t) + ψ(τ, t))/λ, it is clear that the total effectwill depend on the relative changes.12. Third, the value functional for the government inequilibrium is now modified. To the terms in the perfect foresight model, we need to addone that takes into account the fact that the government is defaulting under some statesof nature. Fourth, and finally, note that the stochastic option to default again introduces acomputational challenge to compute exact transitional dynamics. This because we needto compute a fixed point in a sequence of distributions and it is numerically unfeasiblewithout an approximation. The government wants to find the optimal threshold to de-fault at each point in time, but the threshold at t depends on the threshold at t′, and soon and so forth. In order to overcome this problem, we will analyze the particular casein which the economy is in the risky steady state before the arrival of the option to de-fault. As in Section 4, we study this case because we can obtain an exact solution for thetransitional dynamics.

5.2 The risky steady state

In this case, the value functional V [ f (·)] is a constant V satisfying:

[ρ + θ [1−Φ (V)] V − Γ (V)] V = U (c) ,

and the individual value and bond prices are

v(τ) = −δ(

1− e−ρ+θ[1−Φ(V)]τ)

ρ + θ [1−Φ (V)]− e−ρ+θ[1−Φ(V)]τ

ψ(τ) =δ(

1− e−r+θ[1−Φ(V)]τ)

r + θ [1−Φ (V)]+ e−r+θ[1−Φ(V)]τ.

For the quadratic case, issuance’s are again given by

ι(τ) =ψ(τ) + v(τ)

λ.

12In the risky steady state that we will analyze below, default will have an un-ambiguous effect.

25

The first result that we can show is that, as opposed to the case when there are no de-fault opportunities, the maturity of debt shortens. Denote by ιθ=0(τ) the optimal debtissuance’s for a model when there are no default opportunities. Denote by ιθ>0(τ) theoptimal debt issuance’s for that same model with a positive probability of a default op-portunity. Then we can show the following.

Corollary 5.1. When issuance costs are quadratic, issuance’s are given by

ιθ>0(τ) ≈ e−θ[1−Φ(V)]τ ιθ=0(τ)

with equality when δ = 0.

5.3 Numerical results

Consider the same calibration as in the deterministic case and assume that θ = 0.02 andVD is uniformly distributed between [−b,−a], with a, b > 0. We calibrate the distribution

such that Φ (V) = V+ba−b = 0.5. In this case Γ (V) := 1

a−b

∫ −aV ξdξ = 1

2(a2−V2)

a−b .

5.4 The case with a single default opportunity

In order to have an easier comparison with the model in Section 4, where the shocks tooutput and the interest rate can hit only one time, we now consider instead the case inwhich the opportunity to default only arrives once. In this case, if the country does nodefault when the opportunity arrives, it just follows a deterministic path from then on,characterized by the solution of the model with Perfect Foresight. Let’s define V (t) :=V [ f (·, t)] as the aggregate value functional in the deterministic case if the debt distri-bution is f (·, t) . The expected value if the shock arrives between t and t + ∆ is given byΓ(V (t + ∆)

)+ V (t + ∆)Φ

(V (t + ∆)

). The necessary conditions for an optimal plan are

given by the following proposition.

Proposition 5.2. If a solution to problem with one-off exists with e−ρt f , e−ρtι ∈ L2 ([0, T]× [0, ∞))

and e−ρtc ∈ L2[0, ∞) it should satisfy the PDE

r(t)v (τ, t) = −δ +∂v∂t− ∂v

∂τ− θ

[v (τ, t)−Φ

(V [ f (·, t)]

)v (τ, t)

U ′ (c (t))U ′ (c (t))

], if τ ∈ (0, T],

v (0, t) = −1, if τ = 0,

with the transversality condition limt→∞ e−ρtv (τ, t) = 0, where v (τ, t) is the marginal valueof a unit of debt with time-to-maturity τ, e−ρtv ∈ L2 ([0, T]× [0, ∞)) and c (t) and v (τ, t) are

26

respectively consumption and the marginal value of a unit of debt with time-to-maturity τ in theno-default case. The optimal issuance ι (τ, t) is given by the optimality condition (5.6). The valueof V (t) is the solution of the HJB equation in the no-default case

ρV (t) = U (c (t)) +∫ T

0U ′ (c (t)) v (τ, t)

∂ f∂t

dτ. (5.7)


Remarks. There are three points worth noting. First, note that with intensity θ a defaultoptions arrives. In this case, the government will default in case it draws a value lowerthan V [ f (·, t)]. This occurs with probability Φ

(V [ f (·, t)]

). In this case, the government

valuation of the bond is given by v (τ, t), the valuation of a bond given by the solution ofthe perfect foresight model. Second, note in addition, that there is a correction term thataccounts for the change in the marginal utilities. This term is given by U

′(c(t))U ′(c(t)) , because

consumption will in general jump after the shock has hit. Third, note that the bond pricesin this case are given by

rψ(τ, t) = δ +∂ψ

∂t− ∂ψ

∂τ− θ

[ψ(τ, t)−Φ

(V (t)

)ψ(τ)

], if τ ∈ (0, T)

ψ(0, t) = 1, if τ = 0,

where

ψ(τ) =δ (1− e−rτ)

r+ e−rτ,

is the riskless bond price with a flat yield curve. The prices take into account that witharrival θ, a default opportunity arises, and the government takes it only if the value ofcontinuation is lower than the realized outside options. This occurs with probabilityΦ(V (t)

), where V (t) is the value of repayment if no more shock can arrive; this value if

characterized by Section 3, Perfect foresight.

Risky Steady State. The risky steady state valuations are given by

(ρ + θ) v (τ) = −δ− ∂v∂τ

+ θ

[Φ(V [ f (·)]

)v (τ, 0)

U ′ (c (0))U ′ (c)

]τ ∈ (0, T],

v (0) = −1, if τ = 0,

27

and a bond price

(r + θ)ψ(τ) = δ− ∂ψ

∂τ+ θ

[Φ(V (t)

)ψ(τ)

], if τ ∈ (0, T)

ψ(0, t) = 1, if τ = 0.

and an aggregate value functional

V [ f (·)] = 1ρ

U (c (0)) + U ′ (c (0))

∫ T

0v (τ, 0)

∂ f∂τ

dτ

.

As opposed to the case in which there is a recurrent default arrival, the difference is inthe continuation payoffs, both for the government and the international investors, if thegovernment does not default. In this case, they are given by the values of the bonds, forboth agents, with an initial distribution and if there are no more default opportunitiesarriving; the Perfect foresight case.

28

References

Achdou, Yves, Jiequn Han, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll,“Heterogeneous agent models in continuous time,” Preprint, 2014.

Aguiar, Mark and Gita Gopinath, “Defaultable debt, interest rates and the current ac-count,” Journal of International Economics, 2006, 69 (1), 64 – 83.

and Manuel Amador, “Sovereign debt: a review,” Technical Report, National Bureauof Economic Research 2013.

, , Hugo Hopenhayn, and Iván Werning, “Take the short route: Equilibrium defaultand debt maturity,” Technical Report, National Bureau of Economic Research 2016.

Aiyagari, S Rao, Albert Marcet, Thomas J Sargent, and Juha Seppälä, “Optimal taxationwithout state-contingent debt,” Journal of Political Economy, 2002, 110 (6), 1220–1254.

Angeletos, George-Marios, “Fiscal policy with noncontingent debt and the optimal ma-turity structure,” The Quarterly Journal of Economics, 2002, 117 (3), 1105–1131.

Arellano, Cristina, “Default Risk and Income Fluctuations in Emerging Economies,” TheAmerican Economic Review, 2008, 98 (3), 690–712.

and Ananth Ramanarayanan, “Default and the Maturity Structure in SovereignBonds,” Journal of Political Economy, 2012, 120 (2), pp. 187–232.

Barro, Robert J, “On the determination of the public debt,” Journal of political Economy,1979, 87 (5, Part 1), 940–971.

Basar, Tamer and Geert Jan Olsder, Dynamic noncooperative game theory, SIAM, 1998.

Bensoussan, Alain, MHM Chau, and SCP Yam, “Mean field games with a dominatingplayer,” Applied Mathematics & Optimization, 2016, 74 (1), 91–128.

Bhandari, Anmol, David Evans, Mikhail Golosov, and Thomas J Sargent, “Fiscal policyand debt management with incomplete markets,” The Quarterly Journal of Economics,2016, p. qjw041.

Buera, Francisco and Juan Pablo Nicolini, “Optimal maturity of government debt with-out state contingent bonds,” Journal of Monetary Economics, 2004, 51 (3), 531–554.

Chatterjee, Satyajit and Burcu Eyigungor, “Maturity, indebtedness, and default risk,”The American Economic Review, 2012, 102 (6), 2674–2699.

29

Chen, Hui, Rui Cui, Zhiguo He, and Konstantin Milbradt, “Quantifying liquidity anddefault risks of corporate bonds over the business cycle,” Technical Report, NationalBureau of Economic Research 2014.

Culbertson, John M, “The term structure of interest rates,” The Quarterly Journal of Eco-nomics, 1957, 71 (4), 485–517.

Greenwood, Robin and Dimitri Vayanos, “Bond supply and excess bond returns,” TheReview of Financial Studies, 2014, 27 (3), 663–713.

He, Zhiguo and Konstantin Milbradt, “Endogenous liquidity and defaultable bonds,”Econometrica, 2014, 82 (4), 1443–1508.

Klein, Paul, Per Krusell, and Jose-Victor Rios-Rull, “Time-consistent public policy,” TheReview of Economic Studies, 2008, 75 (3), 789–808.

Krusell, Per and Anthony A Smith Jr, “Income and wealth heterogeneity in the macroe-conomy,” Journal of political Economy, 1998, 106 (5), 867–896.

Leland, Hayne and Klaus Toft, “Optimal Capital Structure, Endogenous Bankruptcy, andthe Term Structure of Credit Spreads,” The Journal of Finance, 1996, 51 (3), 987–1019.

Modigliani, Franco and Richard Sutch, “Innovations in interest rate policy,” The Ameri-can Economic Review, 1966, 56 (1/2), 178–197.

and , “Debt management and the term structure of interest rates: an empirical anal-ysis of recent experience,” Journal of Political Economy, 1967, 75 (4, Part 2), 569–589.

Nuño, Galo and Benjamin Moll, “Controlling a Distribution of Heterogeneous Agents,”2015.

Stokey, Nancy, Robert Lucas, and Edward Prescott, “Recursive Methods in EconomicDynamics,” Harvard University Press, 1989.

Vayanos, Dimitri and Jean-Luc Vila, “A preferred-habitat model of the term structure ofinterest rates,” Technical Report, National Bureau of Economic Research 2009.

30

A Figures

Maturity, =0 5 10 15 20

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Outstanding Debt f(=)

Maturity, =0 5 10 15 20

#10-3

0

1

2

3

4

5

6

Issuances 4(=)

Maturity, =0 5 10 15 20

#10-3

0

1

2

3

4

5

6

7

Maturity Distribution

Maturity, =0 5 10 15 20

0.75

0.8

0.85

0.9

0.95

1

A(=) vs. !v(=)

A(=)!v(=)

Figure A.1: The figure describes the steady state values of debt, issuances, maturity dis-tribution and the wedges in valuations.

31

Time, t0 50 100 150 200

%

5.9

6

6.1

6.2

6.3

6.4

6.5

Time Discount r(t) (bps)

Time, t0 50 100 150 200

0.85

0.9

0.95

1

c(t) vs. y(t)

c(t) y(t)

Time, t0 50 100 150 200

%y s

s

0

0.01

0.02

0.03

0.04

Issuances Rate, 4(t)

Time, t0 50 100 150 200

%y s

s

0

0.2

0.4

0.6

0.8

Debt Outstanding, f(t)

0-5 (years) 5-10 10-15 T-5 to T

Figure A.2: The figure describes the response of the government discount factor, is-suances, consumption, and total debt, from an unexpected shock to output of 5% thatreverts to the long run mean as an AR(1) deterministic process.

32

Time, t0 50 100 150 200

%

5.9

6

6.1

6.2

6.3

6.4

6.5


Time, t0 50 100 150 200

0.85

0.9

0.95

1

c(t) vs. y(t)

c(t) y(t)

Time, t0 50 100 150 200

%y s

s

0

0.01

0.02

0.03

0.04


Time, t0 50 100 150 200

%y s

s

0

0.2

0.4

0.6

0.8


0-5 (years) 5-10 10-15 T-5 to T

Figure A.3: The figure describes the response of the government discount factor, is-suances, consumption, and total debt, from an unexpected shock to output of 5% thatreverts to the long run mean as an AR(1) deterministic process, when T increases from 20to 30 years.

33

Figure A.4: The figure describes the response of the yield curve to a shock in the shortrate that reverts as an AR(1) deterministic process.

34

Time, t0 50 100 150 200

%

4

5

6

7

8


Time, t0 50 100 150 200

0.9

0.95

1

1.05

1.1

1.15

c(t) vs. y(t)

c(t) y(t)

Time, t0 50 100 150 200

%y s

s

0

0.02

0.04

0.06

0.08


Time, t0 50 100 150 200

%y s

s

0

0.2

0.4

0.6

0.8


0-5 (years) 5-10 10-15 T-5 to T

Figure A.5: The figure describes the response of the government discount factor, is-suances, consumption, and total debt, from an unexpected shock to the short interestrate that reverts to the long run mean as an AR(1) deterministic process.

35

Time, t0 50 100 150 200

%

4

5

6

7

8


Time, t0 50 100 150 200

0.9

0.95

1

1.05

1.1

1.15

c(t) vs. y(t)

c(t) y(t)

Time, t0 50 100 150 200

%y s

s

0

0.02

0.04

0.06

0.08


Time, t0 50 100 150 200

%y s

s

0

0.2

0.4

0.6

0.8


0-5 (years) 5-10 10-15 T-5 to T

Figure A.6: The figure describes the response of the government discount factor, is-suances, consumption, and total debt, from an unexpected shock to the short interestrate that reverts to the long run mean as an AR(1) deterministic process, for the case inwhich the households are risk neutral.

36

Maturity, =0 5 10 15 20

0

0.01

0.02

0.03

0.04

0.05

0.06


Maturity, =0 5 10 15 20

#10-3

0

1

2

3

4

5

Issuances 4(=)

Maturity, =0 5 10 15 20

#10-3

0

1

2

3

4

5

6

7


Maturity, =0 5 10 15 20

0.75

0.8

0.85

0.9

0.95

1

A(=) vs. !v(=)

A(=; ss)A(=; rss)!v(=; ss)!v(=; rss)!v(=; 0)

Figure A.7: Response, in the risky steady state, of the maturity distribution, total debt,issuances, and valuations to an expected 10 percent drop in output that reverts back tonormal in the long run as an AR(1) deterministic process.

37

Time, t0 50 100 150 200

%

5.8

5.85

5.9

5.95

6

6.05

6.1

6.15


Time, t0 50 100 150 200

0.94

0.95

0.96

0.97

0.98

0.99

1

c(t) vs. y(t)

c(t)

y(t)

Time, t0 50 100 150 200

%y s

s

0

0.005

0.01

0.015

0.02

0.025


Time, t0 50 100 150 200

%y s

s

0

0.05

0.1

0.15

0.2

0.25

0.3


0-5 (years) 5-10 10-15 T-5 to T

Figure A.8: Dynamics total debt, issuances, consumption and the internal rate after a10 percent shock in output that reverts to the long run mean as an AR(1) deterministicprocess.

38

Maturity, =0 5 10 15 20

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06


Maturity, =0 5 10 15 20

#10-3

-1

0

1

2

3

4

5

Issuances 4(=)

Maturity, =0 5 10 15 20

-0.015

-0.01

-0.005

0

0.005

0.01

0.015


Maturity, =0 5 10 15 20

0.7

0.75

0.8

0.85

0.9

0.95

1

A(=) vs. !v(=)

A(=; ss)A(=; rss)!v(=; ss)!v(=; rss)!v(=; 0)

Figure A.9: Response, in the risky steady state, of the maturity distribution, total debt,issuances, and valuations to an expected permanent increase in the short rate.

39

Time, t0 50 100 150 200

%

5.6

5.8

6

6.2

6.4

6.6

6.8

7


Time, t0 50 100 150 200

0.96

0.97

0.98

0.99

1

1.01

c(t) vs. y(t)

c(t)

y(t)

Time, t0 50 100 150 200

%y s

s

-0.005

0

0.005

0.01

0.015

0.02

0.025


Time, t0 50 100 150 200

%y s

s

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3


0-5 (years) 5-10 10-15 T-5 to T

Figure A.10: Dynamics total debt, issuances, consumption and the internal rate to anexpected permanent increase in the short rate.

40

B Proofs

B.1 Proof of Proposition 3.1

First we construct a Lagrangian in the space of functions g such that∥∥e−ρt/2g (τ, t)

∥∥L2 <

∞. The Lagrangian is

L (ι, f ) =∫ ∞

0e−ρtU

(y (t)− f (0, t) +

∫ T

0[q (t, τ, ι, f ) ι (τ, t)− δ f (τ, t)] dτ

)dt

+∫ ∞

0

∫ T

0e−ρt j (τ, t)

(−∂ f

∂t+ ι (τ, t) +

∂ f∂τ

)dτdt,

where j (τ, t) is the Lagrange multiplier associated to the law of motion of debt. TakingGateaux derivatives, for any suitable h (τ, t) such that e−ρth ∈ L2 ([0, T]× [0, ∞)) :

limα→0

∂

∂αL (ι, f + αh) =

∫ ∞

0e−ρtU′ (c)

[−h (0, t) +

∫ T

0

(∂q∂ f

ι (τ, t)− δ

)h (τ, t) dτ

]dt

−∫ ∞

0

∫ T

0e−ρt ∂h

∂tj (τ, t) dτdt

+∫ ∞

0

∫ T

0e−ρt ∂h

∂τj (τ, t) dτdt,

The last two terms can be integrated by parts

∫ T

0

∂h∂τ

j (τ, t) dτ = h (T, t) j (T, t)− h (0, t) j (0, t)−∫ T

0h

∂j∂τ

dτ,

−∫ ∞

0e−ρt ∂h

∂tj (τ, t) dt = − lim

s→∞e−ρsh (τ, s) j (τ, s) + h (τ, 0) j (τ, 0) +

∫ ∞

0e−ρth(τ, t)

(∂j∂τ− ρj

)dτ.

As the initial distribution f0 is given the value of h (τ, 0) = 0. The Gateaux derivativeshould be zero for any suitable h (τ, t)

0 =∫ ∞

0e−ρtU′ (c)

[−h (0, t) +

∫ ∞

0

(∂q∂τ

ι (τ, t)− δ

)h (τ, t) dτ

]dτ

+∫ ∞

0

∫ T

0e−ρt

(−ρj− ∂j

∂τ+

∂j∂t+

)h (τ, t) dτdt∫ ∞

0e−ρt (h (T, t) j (T, t)− h (0, t) j (0, t)) dt

−∫ ∞

0lims→∞

e−ρsh (τ, s) j (τ, s) dτ.

41

Therefore, as f (T+, t) = 0 then h (T, t) = 0 and we have

ρj (τ, t) = U′ (c (t))(

∂q∂ f

ι− δ

)+

∂j∂t− ∂j

∂τ, if τ ∈ (0, T] (B.1)

j (0, t) = −U′ (c (t)) , if τ = 0,

limt→∞

e−ρt j (τ, t) = 0.

Proceeding similarly in the case of ι

limα→0

∂

∂αL (ι + αh, f ) =

∫ ∞

0e−ρtU′ (c)

[∫ T

0

(∂q∂ι

ι (τ, t) + q (t, τ, ι, f ))

h (τ, t) dτ

]dt

+∫ ∞

0

∫ T

0e−ρth (τ, t) j (τ, t) dτdt,

and henceU′ (c)

(∂q∂ι

ι (τ, t) + q (t, τ, ι, f ))= −j (τ, t) .

If we define the variablev (τ, t) = j (τ, t) /U′ (c (t)) ,

the PDE equation (C.1) results in(ρ− U′′ (c (t))

U′ (c (t))dcdt

)v (τ, t) =

∂q∂ f

ι− δ +∂v∂t− ∂v

∂τ, if τ ∈ (0, ∞),

v (0, t) = −1, if τ = 0,

limt→∞

e−ρtv (τ, t) = 0,

and the first order condition is

∂q∂ι

ι (τ, t) + q (t, τ, ι, f ) = −v (τ, t) .

42

B.2 Proof of Proposition 4.1

To compute the dynamics of ι[

f (·, t) , yH] we need to employ dynamic programming.The value functional can be expressed as

V[

f (·, t) , yH]= maxι(·)∈I

Et

[∫ ∞

te−ρ(s−t)

[e−φ(s−t)U (c (s)) +

(1− e−φ(s−t)

)V[

f (·, s) , yL]]

ds∣∣∣yt = yH

]

We first apply Bellman’s Principle of Optimality

V[

f (·, t) , yH]

= maxι(·)∈I

Et

[∫ t′

te−ρ(s−t)

[e−φ(s−t)U (c (s)) +

(1− e−φ(s−t)

)V[

f (·, s) , yL]]

ds∣∣∣yt = yH

]+e−ρ(t′−t)Et

[e−φ(t′−t)V

[f(·, t′)

, yH]+(

1− e−φ(t′−t))

V[

f(·, t′)

, yL] ∣∣∣yt = yH

],

for an arbitrary t < t′ and then we take the derivative with respect to t′ and the limitt′ → t :

ρV [ f (·, t)] = maxι(·,t)U (c (t)) +

1dt

dV [ f (·, t)] + φ[V [ f (·, t)]−V [ f (·, t)]

], (B.2)

where V [ f (·, t)] := V[

f (·, t) , yL] and V [ f (·, t)] := V[

f (·, t) , yH]. We work with theHJB (B.2) in the space of functions L2 ([0, T]× [0, ∞)) . If we apply the Riesz representationtheorem the Gateaux derivative of V with respect to f in the direction h (τ) ∈ L2 ([0, T])can be expressed as

∂

∂αV [ f (·, t) + αh]

∣∣∣∣α=0

=∫ T

0

δVδ f

h (τ) dτ,

where δVδ f ∈ L2 ([0, T]). Therefore

1dt

dV [ f (·, t)] =∫ T

0

δVδ f

∂ f (·, t)∂t

dτ,

43

where we have applied the chain rule.13 The HJB (B.2) can thus be expressed as

(ρ + φ)V [ f (·, t)] = maxι(·,t)U (c (t)) +

∫ T

0

δVδ f

∂ f∂t

dτ (B.3)

= maxι(·,t)U(

y (t)− f (0, t) +∫ T

0[q (τ, t, ι) ι (τ, t)− δ f (τ, t)] dτ

)+∫ T

0

δVδ f

(ι (τ, t) +

∂ f∂τ

)dτ + φV [ f (·, t)] .

where we have substituted the budget constraint and the KFE.The first order condition with respect to ι can be obtained by computing the Gateaux

derivative in (B.3):

0 =∂

∂αU(

y (t)− f (0, t) +∫ T

0[q (τ, t, ι + αh) (ι + αh)− δ f ] dτ

)∣∣∣∣α=0

+∂

∂α

∫ T

0

δVδ f

(ι + αh +

∂ f∂τ

)dτ

∣∣∣∣α=0

,

and

0 = U ′ (c)[∫ T

0

(∂q∂ι

ι (τ, t) + q (τ, t, ι)

)h (τ) dτ

]dt

+∫ T

0

δVδ f

h (τ) dτ.

The Gateaux derivative should be zero for any suitable h (τ) ∈ L2[0, T)

U ′ (c)(

∂q∂ι

ι (τ, t) + q (τ, t, ι)

)= −δV

δ f.

If we definej (τ, t) :=

δVδ f

, (B.4)

13Notice that

1dt

dV [ f (·, t)] = limα→0

V [ f (·, t + α)]−V [ f (·, t)]α

= limα→0

V [ f (·, t) + α∂ f (·, t) /∂t]−V [ f (·, t)]α

=∂

∂αV [ f (·, t) + α∂ f (·, t) /∂t]

∣∣∣∣α=0

.

44

then the first order condition results in

U ′ (c)(

∂q∂ι

ι (τ, t) + q (τ, t, ι)

)= −j (τ, t) .

If we compute Gateaux derivatives with respect to f in the HJB equation (B.3) we obtain

0 =∂

∂αU(

y (t)− ( f (0, t) + αh (0)) +∫ T

0[qι +−δ ( f + αh)] dτ

)∣∣∣∣α=0

+∂

∂α

∫ T

0

δVδ ( f + αh)

(ι +

∂ ( f + αh)∂τ

)dτ

∣∣∣∣α=0

− (ρ + φ)∂

∂αV [ f (·, t) + αh]

∣∣∣∣α=0

+ φ∂

∂αV [ f (·, t) + αh]

∣∣∣∣α=0

and

0 = U ′ (c)[−h (0, t) +

∫ T

0(−δ) h (τ) dτ

]+∫ T

0

∫ T

0

δ2Vδ f 2

(ι +

∂ f∂τ

)h(τ′)

dτdτ′

+∫ T

0

δVδ f

∂h∂τ

dτ − (ρ + φ)∫ T

0

δVδ f

h (τ) dτ

+φ∫ T

0

δVδ f

h (τ) dτ,

where (τ, t) := δVδ f corresponds to the value function in the case of yt = yL. The term∫ T

0δ2Vδ f 2 h (τ′) dτ′ is the second Gateaux derivative of V. There is a mapping between δ2V

δ f 2

and ∂j∂t , where j has been defined in (B.4):

∂j∂t

=∂

∂tδVδ f

=∫ T

0

δ2Vδ f 2

∂ f∂t

dτ =∫ T

0

δ2Vδ f 2

(ι +

∂ f∂τ

)dτ,

where we have applied again the chain rule and the Riesz representation theorem. Thefirst order condition with respect to f then results in

0 = U ′ (c)[−h (0, t) +

∫ T

0(−δ) h (τ) dτ

]+∫ T

0

[∂j∂t− (ρ + φ) j (τ, t) + φ (τ, t)

]h (τ) dτ

+∫ T

0j (τ, t)

∂h∂τ

dτ

45

Integrating by parts we obtain

0 = U ′ (c)[−h (0, t) +

∫ T

0(−δ) h (τ) dτ

]+∫ T

0

[∂j∂t− ∂j

∂τ− (ρ + φ) j (τ, t) + φ (τ, t)

]h (τ) dτ

+j (T, t) h (T)− j (0, t) h (0) .

As f (T+, t) = 0 then h (T) = 0. As the Gateaux derivative should be zero for anysuitable h (τ):

(ρ + φ) j (τ, t) = U ′ (c (t)) (−δ) +∂j∂t− ∂j

∂τ+ φ (τ, t) , if τ ∈ (0, T] (B.5)

j (0, t) = −U ′ (c (t)) , if τ = 0.

The transversality condition derives from the fact that limt→∞ e−ρtV [ f (·, t)] = 0 and thus,if we take Gateaux derivatives at both sides:

limt→∞

e−ρt j (τ, t) = 0.

Finally, if we define the variables

v (τ, t) = j (τ, t) /U ′(

cH (t))

,

v (τ, t) = (τ, t) /U ′(

cL (t))

,

where cH (t) and cL (t) correspond to the optimal consumption in the case of y (t) = yH

or y (t) = yL respectively, the PDE equation (B.5) results in(ρ−U ′′(cH (t)

)U ′ (cH (t))

dcH

dt

)v (τ, t) = −δ +

∂v∂t− ∂v

∂τ+ φv (τ, t)

U ′(cL (t)

)U ′ (cH (t))

, if τ ∈ (0, ∞),

v (0, t) = −1, if τ = 0,

limt→∞

e−ρtv (τ, t) = 0,


∂q∂ι

ι (τ, t) + q (τ, t, ι) = −v (τ, t) .

46

A. Proof of Proposition 5.1

The solution strategy proceeds in three steps. We start by solving the problem with com-mitment in an arbitrary interval (t, t + ∆], then we take the limit as ∆→ 0, and finally weemploy dynamic programming to obtain the value of the value functional V [ f (·, t)].

Step 1. The problem with commitment

Assume first that the time interval [0, ∞) is divided in subintervals [0, ∆] ∪ (∆, 2∆] ∪(2∆, 3∆] ∪ ... such that the shock may potentially arive at the begining of each interval(except at time t = 0) and that the government solves a commitment problem in eachsubinterval. The solution procedure to problem (??) follows the same lines as in the per-fect foresight case. The Lagragian is given by:

L [ι, f , ψ] =∫ t+∆

te−ρ(s−t)U

(y− f (0, s) +

∫ T

0[ψ(τ, s) + λ (τ, s, ι)] ι (τ, s)− δ f (τ, s) dτ

)ds

+e−ρ∆(

1− e−θ∆)[Γ (V [ f (·, s + ∆)]) + V [ f (·, t + ∆)]Φ (V [ f (·, t + ∆)])]

+e−ρ∆e−θ∆V [ f (·, t + ∆)]

+∫ t+∆

t

∫ T

0e−ρ(s−t) j (τ, s)

(−∂ f

∂s+ ι (τ, s) +

∂ f∂τ

)dτds

+∫ t+∆

t

∫ T

0e−ρ(t−s)µ (τ, s)

(−rψ(τ, s) + δ +

∂ψ

∂s− ∂ψ

∂τ

)dτds

where j (τ, t) and µ (τ, t) are the Lagrange multipliers associated to the law of motion ofdebt (2.1) and bond pricing equation (??), respectively. We integrate by parts the terms

47

that involve derivatives of f and ψ:

−∫ t+∆

t

∫ T

0e−ρ(s−t) ∂ f

∂sj (τ, s) dsdτ = −

∫ T

0e−ρ∆ f (τ, t + ∆) j (τ, t + ∆) dτ

+∫ T

0f (τ, t) j (τ, t) dτ

+∫ t+∆

t

∫ T

0e−ρ(s−t) f

(∂j∂s− ρj

)dsdτ∫ t+∆

t

∫ T

0

∂ f∂τ

j (τ, s) dτds =∫ t+∆

te−ρ(s−t) f (T, s) j (T, s) ds

−∫ t+∆

te−ρ(s−t) f (0, s) j (0, s) ds

−∫ t+∆

t

∫ T

0e−ρ(t−s) f

∂j∂τ

dτds∫ t+∆

t

∫ T


(−∂ψ

∂τ

)dτds =

∫ t+∆

te−ρ(s−t)µ (T, s)ψ (T, s)− e−ρtµ (0, s)ψ (0, s)

+∫ t+∆

t

∫ T

0e−ρtψ (τ, s)

∂µ

∂τdτds,∫ t+∆

t

∫ T


∂ψ

∂sdτds =

∫ T

0

[e−ρ∆µ (τ, t + ∆)ψ (τ, t + ∆)− µ (τ, t)ψ (τ, t)

]dτ

−∫ t+∆

t

∫ T

0e−ρtψ

(∂µ

∂s− ρµ

)dτds

and substitute the terminal condition for the prices and distribution:

ψ (τ, t + ∆) =[(

1− e−θ∆)

Φ (V [ f (·, t + ∆)]) + e−θ∆]

ψ [τ, f (·, t + ∆)] , (B.6)

ψ (0, s) = 1,

f (T, s) = 0.

48

Going back to the Lagrangian, taking Gateaux derivative with respect to f for any suitableh (τ, t) such that e−ρth ∈ L2 ([0, T]× (t, t + ∆]) : ∂

∂α L [ f + αh]|α=0 equals

∫ t+∆

te−ρ(s−t)U ′ (c)

[−h (0, s) +

∫ T

0(−δ) h (τ, s) dτ

]ds

+e−ρ∆(

1− e−θ∆) ∂

∂αΓ (V [ f (·, t + ∆) + αh (τ, t + ∆)])|α=0

+e−ρ∆(

1− e−θ∆) ∂

∂αV [ f + αh]Φ (V [ f + αh])|α=0

+e−ρ∆e−θ∆ ∂

∂αV [ f (·, t + ∆) f (·, t + ∆) + αh (τ, t + ∆)]|α=0

−∫ T

0e−ρ∆h (τ, t + ∆) j (τ, t + ∆) dτ +

∫ T

0h (τ, t) j (τ, t) dτ

+∫ t+∆

t

∫ T

0e−ρ(s−t)h

(∂j∂s− ρj

)dsdτ

+∫ t+∆

te−ρ(s−t) [h (T, s) j (T, s)− h (0, s) j (0, s)] ds−

∫ t+∆

t

∫ T

0e−ρ(t−s)h

∂j∂τ

dτds,

+∫ T

0e−ρ∆

(1− e−θ∆

)µ (τ, t + ∆)Φ′ (V [ f (·, t + ∆)])

∂

∂αV [ f (·, t + ∆) + αh (·, t + ∆)]|α=0 ψ (τ, t + ∆) dτ

+∫ T

0e−ρ∆

[(1− e−θ∆

)Φ (V [ f (·, t + ∆)]) + e−θ∆

]µ (τ, t + ∆)

∂

∂αψ [τ, f (·, t + ∆) + αh (·, t + ∆)]|α=0 dτ,

We can express

∂

∂αV [ f (·, t + ∆) + h (·, t + ∆)]|α=0 =

∫ T

0

δVδ f(τ, τ′, t + ∆

)h(τ′, t + ∆

)dτ′,

∂

∂ατ, ψ [ f (·, t + ∆) + αh (·, t + ∆)]|α=0 =

∫ T

0

δψ

δ f(τ, τ′, t + ∆

)h(τ′, t + ∆

)dτ′.

49

As the initial distribution f (τ, t) is given, the value of h (τ, t) = 0. The Gateaux derivativeshould be zero for any suitable h (τ, t)

0 =∫ t+∆


[−h (0, s) +

∫ T

0(−δ) h (τ, s) dτ

]ds

−e−ρ∆(

1− e−θ∆)

V [ f (·, t + ∆)] φ (V [ f (·, t + ∆)])∫ T

0

δVδ f

(τ, t + ∆) h (τ, t + ∆) dτ

+e−ρ∆(

1− e−θ∆)

V [ f (·, t + ∆)] φ (V [ f (·, t + ∆)])∫ T

0

δVδ f

(τ, t + ∆) h (τ, t + ∆) dτ

+e−ρ∆(

1− e−θ∆)

Φ (V [ f (·, t + ∆)])∫ T

0

δVδ f

(τ, t + ∆) h (τ, t + ∆) dτ

+e−ρ∆e−θ∆∫ T

0

δVδ f

(τ, s) h (τ, s) dτ

−∫ T

0e−ρ∆h (τ, t + ∆) j (τ, t + ∆) +

∫ t+∆

t

∫ T

0e−ρ(s−t)h

(∂j∂s− ρj

)ds

+∫ t+∆

te−ρ(s−t) [h (T, s) j (T, s)− h (0, s) j (0, s)] ds−

∫ t+∆

t

∫ T

0e−ρ(t−s)h

∂j∂τ

dτds

+e−ρ∆(

1− e−θ∆)

φ (V [ f (·, t + ∆)])∫ T

0µ (τ, t + ∆) ψ (τ, t + ∆)

∫ T

0

δVδ f(τ′, t + ∆

)h (τ, t + ∆) dτ′dτ

+e−ρ∆[(

1− e−θ∆)

Φ (V [ f (·, t + ∆)]) + e−θ∆] ∫ T

0µ (τ, t + ∆)

∫ T

0

δψ

δ f(τ, τ′, t + ∆

)h(τ′, t + ∆

)dτ′dτ


ρj (τ, s) = U ′ (c (s)) (−δ) +∂j∂s− ∂j

∂τ, if τ ∈ (0, T] (B.7)

j (0, s) = −U ′ (c (s)) , if τ = 0,

j (τ, t + ∆) =(

1− e−θ∆)

V [ f (·, t + ∆)]Φ (V [ f (·, t + ∆)])δVδ f

(τ, t + ∆) + e−θ∆ δVδ f

(τ, t + ∆) (B.8)

+(

1− e−θ∆)

φ (V [ f (·, t + ∆)])[∫ T

0µ(τ′, t + ∆

)ψ(τ′, t + ∆

)dτ′]

δVδ f

(τ, t + ∆)

+(

1− e−θ∆) [(

1− e−θ∆)

Φ (V [ f (·, t + ∆)]) + e−θ∆] ∫ T

0µ(τ′, t + ∆

) δψ

δ f(τ′, τ, t + ∆

)dτ′.

Taking Gateaux derivative with respect to ι the results are the same as in the deterministiccase:

U ′ (c)(

∂q∂ι

ι (τ, t) + q (t, τ, ι, f ))= −j (τ, t) .

50

Finally, we need to the Gateaux derivatives with respect to ψ :

∂

∂αL [ψ + αh]|α=0 =

∫ t+∆


[∫ T

0ι (τ, s) h (τ, s) dτ

]ds

+∫ t+∆

te−ρ(s−t)µ (T, s) h (T, s) ds− e−ρtµ (0, s) h (0, s)

+∫ t+∆

t

∫ T

0e−ρth (τ, s)

∂µ

∂τdτds

−∫ T

0µ (τ, t) h (τ, t) dτ

−∫ t+∆

t

∫ T

0e−ρt

(∂µ

∂s− ρµ

)h(τ, s)dτds,

where h (0, s) = 0 as the value of ψ (0, s) is fixed to 1. And the optimality condition thenresults in

∂µ

∂s= ρµ (τ, s) + U ′ (c) ι (τ, s) +

∂µ

∂τ, if τ ∈ (0, T), s ∈ (t, t + ∆)

µ (T, s) = 0, if τ = T, s ∈ (t, t + ∆) (B.9)

µ (τ, t) = 0, if τ ∈ (0, T).

Step 2: the instantaneous limit

The second step is to take the limit as ∆ → 0. Before that, we express the HJB equation(B.7) as

j (τ, t) =∫ t+∆

te−ρ(t−s) U ′ (c (s)) (−δ)

ds (B.10)

+e−ρ∆Et

[δVδ f

(τ, t + ∆) +∫ T

0µ(τ′, t + ∆

) δψ

δ f(τ′, τ, t + ∆

)dτ′]

, (B.11)

j (0, s) = −U ′ (c (s)) , if τ = 0,

where the term δVδ f (τ, t + ∆) depends on the arrival of a shock with probability

(1− e−θ∆)

that decreases the value by[Φ (V [ f (·, t + ∆)]) + φ (V [ f (·, t + ∆)])

∫ T

0µ (τ, t + ∆)ψ (τ, t + ∆) dτ′

].

Analogously, the term∫ T

0 µ (τ′, t + ∆) δψδ f (τ

′, τ, t + ∆) dτ′may decrease by Φ (V [ f (·, t + ∆)])if the shock arrives. As ∆ → 0, the Lagrange multiplier µ (τ, t) collapses to zero given

51

(B.9):µ (τ, t) = 0, for all τ ∈ (0, T], t ∈ (0, ∞), (B.12)

reflecting the lack of commitment over finite intervals. Notice that conditional on noshock arrival, the the limit case ∆→ 0 in equation (B.8) yields to

j (τ, t) =δVδ f

(τ, t) , for all τ ∈ (0, T], t ∈ (0, ∞), (B.13)

which is the same equation as in the deterministic case. Taking into account (B.12) and(B.13), the limit as ∆→ 0 of equation (B.10) can be expressed as an HJB of the form

ρj (τ, t) = U ′ (c (t)) (−δ) +∂j∂t− ∂j

∂τ+ θ [Φ (V [ f (·, t)])− 1] j (τ, t) , if τ ∈ (0, T]

j (0, t) = −U ′ (c (t)) , if τ = 0.

If we substitute these values in (B.10) we obtain

ρj (τ, t) = U ′ (c (t)) (−δ) +∂j∂t− ∂j

∂τ− θ [1−Φ (V [ f (·, t)])] j (τ, t) , if τ ∈ (0, T],

j (0, t) = −U ′ (c (t)) , if τ = 0,

limt→∞

e−ρt j (τ, t) = 0.

Defining again the variable

v (τ, t) = j (τ, t) /U ′ (c (t)) ,

the HJB results in(ρ− U

′′ (c (t))U ′ (c (t))

dcdt

)v (τ, t) = −δ +

∂v∂t− ∂v

∂τ− θ [1−Φ (V [ f (·, t)])] v (τ, t) , if τ ∈ (0, T],

v (0, t) = −1, if τ = 0,

limt→∞

e−ρtv (τ, t) = 0.


∂q∂ι

ι (τ, t) + q (t, τ, ι, f ) = −v (τ, t) .

52

Step 3. The aggregate HJB

The last step is to construct the aggregate HJB in order to obtain the value of V [ f (·, t)] .The idea is to compute the derivative with respect to t + ∆ in the dynamic programmingequation (??) and then to take the limit as ∆→ 0 :

ρV [ f (·, t)] = U (c (t)) +∫ T

0

δVδ f

∂ f∂t

dτ

−θ [1−Φ (V [ f (·, t)])]V [ f (·, t + ∆)]− Γ (V [ f (·, t + ∆)]) ,

or equivalently

ρV [ f (·, t)] = U (c (t)) +∫ T

0U ′ (c (t)) v (τ, t)

∂ f∂t

dτ

−θ [1−Φ (V [ f (·, t)])]V [ f (·, t + ∆)]− Γ (V [ f (·, t + ∆)]) .

B. Proof of Corollary 5.1

The proof, follows from plugging in the valuation in the risky steady state. In particular,

ι(τ) =ψ(τ) + v(τ)

λ

=1λ

−δ(

1− e−ρ+θ[1−Φ(V)]τ)

ρ + θ [1−Φ (V)]− e−ρ+θ[1−Φ(V)]τ

+δ(

1− e−r+θ[1−Φ(V)]τ)

r + θ [1−Φ (V)]+ e−r+θ[1−Φ(V)]τ

= e−θ[1−Φ(V)]τ 1

λ

−δ(

eθ[1−Φ(V)]τ − e−ρτ)

ρ + θ [1−Φ (V)]+

δ(

eθ[1−Φ(V)]τ − e−rτ)

r + θ [1−Φ (V)]

+(−e−ρτ + e−rτ

)From Section 3, we know that the second term in square bracket are the optimal issuanceswhen there are no shocks and the coupon rate is zero. The first term vanishes, when thecoupon rate is zero.

C. Proof of Proposition 5.2

The proof is similar to that of the Proposition 1 above. The main difference is that, giventhe default-free value V [ f (·, t)] , the continuation value after the default opportunity ar-rives is now Γ

(V [ f (·, t + ∆)]

)+ Φ

(V [ f (·, t)]

)V [ f (·, t + ∆)] . If we proceed as in the

53

proof of Proposition 1 we obtain

ρj (τ, t) = U ′ (c (t)) (−δ) +∂j∂t− ∂j

∂τ− θ

[j (τ, t)−Φ

(V [ f (·, t)]

)j (τ, t)

], if τ ∈ (0, T],

j (0, t) = −U ′ (c (t)) , if τ = 0,

limt→∞

e−ρt j (τ, t) = 0,

where

j (τ, t) =δVδ f

(τ, t) , for all τ ∈ (0, T], t ∈ [0, ∞),

j (τ, t) =δVδ f

(τ, t) , for all τ ∈ (0, T], t ∈ [0, ∞).

Defining the variables

v (τ, t) = j (τ, t) /U ′ (c (t)) ,

v (τ, t) = j (τ, t) /U ′ (c (t)) ,

where c (t) is the consumption in the riskless path, the HJB before the arrival of the shockresults in(

ρ− U′′ (c (t))U ′ (c (t))

dcdt

)v (τ, t) = −δ +

∂v∂t− ∂v

∂τ− θ

[v (τ, t)−Φ

(V [ f (·, t)]

)v (τ, t)

U ′ (c (t))U ′ (c (t))

], if τ ∈ (0, T],

v (0, t) = −1, if τ = 0,

limt→∞

e−ρtv (τ, t) = 0.

The aggregate HJB after default is the one in the deterministic case:

ρV [ f (·, t)] = U (c (t)) +∫ T

0U ′ (c (t)) v (τ, t)

∂ f∂t

dτ.

54

C Appendix: Duality

The Primal. Given a path of resources y(t), the problem is:

V [ f (·, 0)] = maxι(τ,t),c(t)t∈[0,∞),τ∈[0,T]

∫ ∞

te−ρ(s−t)u(c(s))ds s.t.

c (t) = y (t)− f (0, t) +∫ T

0[q (τ, t, ι) ι(τ, t)− δ f (τ, t)] dτ

∂ f∂t

= ι(τ, t) +∂ f∂τ

; f (τ, 0) = f0(τ)

Definition C.1. Given a path of income y(t) and initial debt f0 a solution to P1 is a pathof consumption c(t) and debt issuances ι(τ, t) such that 1) the budget constraint holds forevery t 2) the evolution of debt satisfies the KFE 3) the no ponzi condition holds, there isnot other path of consumption and debt issuances c, ι that is feasible and yields strictlyhigher utility at zero.

Let j(τ, t) be the lagrange multiplier associated with the KFE. It measures the changemaginal utility of issuing more debt.

Proposition C.1. If a solution to P1 with e−ρt f , e−ρtι ∈ L2 ([0, T]× [0, ∞)),e−ρtc ∈ L2[0, ∞),given by ι(τ, t), c(t)∞


ρj (τ, t) =∂q∂ f

ι− δ +∂j∂t− ∂j

∂τ, if τ ∈ (0, T]

j (0, t) = −u′(c(t)),

limt→∞

e−ρt j (τ, t) = 0


U′(c(t))


is given by (∂q∂ι

ι (τ, t) + q (t, τ, ι)

)u′(c(t)) = −j (τ, t)

The Dual. This problem finds the lowest cost of achieving a particular path of consump-tion with the lowest amount of resources. Given a desired path of consumption c(t) theobjective is to minimize the resources needed to achieve that path. More precisely, P2 isgiven by:

D [ f (·, 0)] = minι(τ,t),ytt=τ∈[0,∞),τ∈[0,T]

∫ ∞

0e−∫ t

0 r(s)dsytdt s.t.

55

c (t) = y (t)− f (0, t) +∫ T

0[q(τ, t, ι)ι(τ, t)− δ f (τ, t)] dτ

∂ f∂t

= ι(τ, t) +∂ f∂τ

; f (τ, 0) = f0(τ)

r(t) = ρ + σc(t)c(t)

Definition C.2. Given a path of consumption c(t) and initial debt f0 a solution to P2 isa path of income y(t) and debt issuances ι(τ, t) such that 1) the budget constraint holdsfor every t 2) the evolution of debt satisfies the KFE 3) the no ponzi condition holds, thereis not other path of consumption and debt issuances y, ι that is feasible and has lowerresources associated.

Let v (τ, t) be the Lagrange multiplier associated with KFE. it measure marginal re-sources needed if a unit of debt is issued. The necessary conditions are the following:

Proposition C.2. If a solution to P2 with e−ρt f , e−ρtι ∈ L2 ([0, T]× [0, ∞)),e−ρtc ∈ L2[0, ∞),given by ι(τ, t), c(t)∞


r (t) v (τ, t) =∂q∂ f

ι− δ +∂v∂t− ∂v

∂τ, if τ ∈ (0, T]

v (0, t) = −1,

limt→∞

e−ρtv (τ, t) = 0


U′(c(t))


is given by∂q∂ι

ι (τ, t) + q (t, τ, ι) = −v (τ, t)

Proof. See below. .

The Connection.

Corollary C.1. Suppose that for a given income path y(t) and initial debt f0 the solution to P1is c∗(t), ι∗(τ, t), j∗(τ, t). Then, y(t), ι∗(τ, t), j∗(τ,t)

u′(c(t)) solves P2 given the path c∗(t).

56

C.1 Proof of Proposition

First we construct a Lagrangian in the space of functions g such that∥∥e−ρt/2g (τ, t)

∥∥L2 <

∞. The Lagrangian is

L (ι, f ) =∫ ∞

0e−r(t)t

(c (t) + f (0, t)−

∫ T

0[q (t, τ, ι, f ) ι (τ, t)− δ f (τ, t)] dτ

)dt

+∫ ∞

0

∫ T

0e−r(t)tv (τ, t)

(−∂ f

∂t+ ι (τ, t) +

∂ f∂τ

)dτdt,

where j (τ, t) is the Lagrange multiplier associated to the law of motion of debt. TakingGateaux derivatives, for any suitable h (τ, t) such that e−ρth ∈ L2 ([0, T]× [0, ∞)) :

limα→0

∂

∂αL (ι, f + αh) =

∫ ∞

0e−r(t)t

[h (0, t)−

∫ T

0

(∂q∂ f

ι (τ, t)− δ

)h (τ, t) dτ

]dt

+∫ ∞

0

∫ T

0e−r(t)t ∂h

∂tv (τ, t) dτdt

−∫ ∞

0

∫ T

0e−r(t)t ∂h

∂τv (τ, t) dτdt,

The last two terms can be integrated by parts

−∫ T

0

∂h∂τ

v (τ, t) dτ = − h (T, t) v (T, t) + h (0, t) v (0, t) +∫ T

0h

∂v∂τ

dτ,

+∫ ∞

0e−r(t)t ∂h

∂tv (τ, t) dt = lim

s→∞e−r(t)sh (τ, s) v (τ, s)− h (τ, 0) v (τ, 0)−

∫ ∞

0e−r(t)th(τ, t)

(∂v∂t− r(t)v

)dτ.

As the initial distribution f0 is given the value of h (τ, 0) = 0. The Gateaux derivativeshould be zero for any suitable h (τ, t)

0 =∫ ∞

0e−r(t)t

[+h (0, t)−

∫ (∂q∂τ

ι (τ, t)− δ

)h (τ, t) dτ

]dτ

−∫ ∞

0

∫ T

0e−r(t)t

(−r(t)v− ∂v

∂τ+

∂v∂t

)h (τ, t) dτdt

−∫ ∞

0e−ρt (h (T, t) v (T, t)− h (0, t) v (0, t)) dt

+∫ ∞

0lims→∞

e−ρsh (τ, s) v (τ, s) dτ.

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r(t)v (τ, t) =

(∂q∂ f

ι− δ

)+

∂v∂t− ∂v

∂τ, if τ ∈ (0, T] (C.1)

v (0, t) = −1, if τ = 0,

limt→∞

e−r(t)tv (τ, t) = 0.

Proceeding similarly in the case of ι

limα→0

∂

∂αL (ι + αh, f ) =

∫ ∞

0e−r(t)t

[−∫ T

0

(∂q∂ι

ι (τ, t) + q (t, τ, ι, f ))

h (τ, t) dτ

]dt

−∫ ∞

0

∫ T

0e−r(t)th (τ, t) v (τ, t) dτdt,

and hence (∂q∂ι

ι (τ, t) + q (t, τ, ι, f ))= −v (τ, t) .

The PDE equation (C.1) results in

r(t)v (τ, t) =∂q∂ f

ι− δ +∂v∂t− ∂v

∂τ, if τ ∈ (0, ∞),

v (0, t) = −1, if τ = 0,

limt→∞

e−r(t)tv (τ, t) = 0,


∂q∂ι

ι (τ, t) + q (t, τ, ι, f ) = −v (τ, t) .

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D Computational Method Deterministic Dynamics

We describe the numerical algorithm used to jointly solve for the equilibrium value func-tion, v (τ, t), bond price, q (t, τ, ι) , consumption c(t), issuance ι (τ, t) and density f (τ, t) .The equilibrium is characterized by the HJB equation

r (t) v (τ, t) = −δ +∂v∂t− ∂v

∂τ, if τ ∈ (0, T] (D.1)

v (0, t) = −1, if τ = 0, (D.2)

where the interest rate r (t) is given by:

r (t) = ρ +γ

c (t)dcdt

, (D.3)

where γ is the risk coefficient in U (c) := c1−γ−11−γ . The optimal issuance ι (τ, t) is given by

ι =1λ

(δ (1− e−rτ)

r+ e−rτ + v (τ, t)

). (D.4)

The law of motion of the density of maturities is given by the Kolmogorov Forward equa-tion

∂ f∂t

= ι (τ, t) +∂ f∂τ

, (D.5)

and consumption by the budget constraint

c (t) = y− f (0, t) +∫ T

0

[(δ (1− e−rτ)

r+ e−rτ − 1

2λι (τ, t)

)ι (τ, t)− δ f (τ, t)

]dτ. (D.6)

The parameters are T, δ, y, γ, λ, ρ and r = ρ. The initial distribution is f (τ, 0) = f0 (τ) .The algorithm proceeds in 3 steps. We describe each step in turn.

Step 1: Solution to the Hamilton-Jacobi-Bellman equation The HJB equation (D.1) issolved using an upwind finite difference scheme similar to Achdou et al. (2014). We ap-proximate the value function v (τ) on a finite grid with step ∆τ : τ ∈ τ1, ..., τI, whereτi = τi−1 + ∆τ = τ1 + (i− 1)∆τ for 2 ≤ i ≤ I. The bounds are τ1 = ∆τ and τI = T, suchthat ∆τ = T/I. We use the notation vi := v(τi), and similarly for the policy function ιi,.Notice first that the HJB equation involves first derivatives of the value function. At eachpoint of the grid, the first derivative can be approximated with a forward or a backwardapproximation. In an upwind scheme, the choice of forward or backward derivative de-

59

pends on the sign of the drift function for the state variable. As in our case, the drift isalways negative, we employ a backward approximation in state:

∂v(τi)

∂τ≈ vi − vi−1

∆τ. (D.7)

The HJB equation is approximated by the following upwind scheme,

ρvi = −δ +vi−1

∆τ− vi

∆τ,

with terminal condition v0 = v(0) = −1. This can be written in matrix notation as

ρv = u + Av,

where

A =1

∆τ

−1 0 0 0 · · · 01 −1 0 0 · · · 00 1 −1 0 · · · 0... . . . . . . . . . . . . . . .

0 0 · · · 1 −1 00 0 · · · 0 1 −1

, v =

v1

v2

v3...

vI−1

vI

(D.8)

u =

−δ− 1/∆τ

−δ

−δ...−δ

−δ

.

The solution is given by

v = (ρI−A)−1 u. (D.9)

Most computer software packages, such as Matlab, include efficient routines to handlesparse matrices such as A.

To analyze the transitional dynamics, define tmax as the time interval considered,which should be large enough to ensure a converge to the stationary distribution andtime is discretized as tn = tn−1 + ∆t, in intervals of length

∆t =tmax

N − 1,

60

where N is a constant. We use now the notation vni := v(τi, tn). The value function at tmax

is the stationary solution computed in (D.9) that we denote as vN.14 We choose a forwardapproximation in time. The dynamic HJB (D.1) can thus be expressed

rnvn = u + Avn +

(vn+1 − vn)

∆t,

where rn := r (tn). By defining Bn =(

1∆t + rn

)I−A and dn+1 = u + vn+1

∆t , we have

vn = (Bn)−1 dn+1, (D.10)

which can be solved backwards from n = N − 1 until n = 1.The optimal issuance is given by

ιni =1λ(ψi + vn

i ) ,

where

ψi =δ (1− e−ρτi)

ρ+ e−ρτi .

Step 2: Solution to the Kolmogorov Forward equation Analogously, the KFE equation(2.1) can be approximated as

f ni − f n−1

i∆t

= ιni +f ni+1 − f n

i∆τ

,

where we have employed the notation f ni := f (τi, tn). This can be written in matrix nota-

tion as:fn − fn−1

∆t= ´n + ATfn, (D.11)

where AT is the transpose of A and

fn =

f n1

f n2...

f nI−1

f nI

.

14You may begin directly by employing the analytical solution from equation (??) as vN .

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Given f0, the discretized approximation to the initial distribution f0(τ), we can solve theKF equation forward as

fn =(

I−∆tAT)−1

(´n∆t + fn−1) , n = 1, .., N. (D.12)

Step 3: Computation of consumption The discretized budget constraint (??) can beexpressed as

cn = y− f n−11 ∆τ +

I

∑i=1

[(ψi −

12

λιni

)ιni − δ f n

i

]∆τ, n = 1, .., N.

Compute

rn = ρ +γ

cncn+1 − cn

∆t, n = 1, .., N − 1.

Complete algorithm The algorithm proceeds as follows. First guess an initial path forconsumption, for example cn = y, for n = 1, .., N. Set k = 1;

Step 1: HJB. Given ck−1 solve the HJB and obtain ι.

Step 2: KF. Given ι solve the KF equation with initial distribution f0 and obtain the dis-tribution f .

Step 3: Consumption. Given ι and f compute consumption c. If ‖c− ck−1‖ = ∑Nn=1

∣∣cn − cnk−1

∣∣ <ε then stop. Otherwise compute

ck = ωc + (1−ω) ck−1, λ ∈ (0, 1) ,

set k := k + 1 and return to step 1.

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Optimal Debt-Maturity Management

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