Fiscal Policy and Debt Management with Incomplete
Markets∗
Anmol Bhandari
bhandari@umn.edu

David Evansdevans@uoregon.edu

Mikhail Golosov
golosov@princeton.edu

Thomas J. Sargent
thomas.sargent@nyu.edu
December 31, 2015

Abstract
A Ramsey planner chooses a distorting tax on labor and manages a portfolio of securities in anenvironment with incomplete markets. We develop a method that uses second order approximations of the policy functions to the planner’s Bellman equation to obtain expressions for theunconditional and conditional moments of debt and taxes in closed form such as the mean andvariance of the invariant distribution as well as the speed of mean reversion. Using this, we establish that asymptotically the planner’s portfolio minimizes an appropriately defined measureof fiscal risk. Our analytic expressions that approximate moments of the invariant distribution
can be readily applied to data recording the primary government deficit, aggregate consumption,and returns on traded securities. Applying our theory to U.S. data, we find that an optimal
target debt level is negative but close to zero, that the invariant distribution of debt is very
dispersed, and that mean reversion is slow.
Key words: Distorting taxes. Spanning. Transfers. Optimal Portfolio. Government debt.

1

Introduction

This paper analyzes a problem of a Ramsey planner who optimally manages a portfolio ofdebts and other securities to smooth fluctuations in tax distortions in an incomplete markets
economy hit by aggregate shocks. Within a production economy without capital, the government
raises revenue by issuing securities and imposing linear taxes on labor income, then spends
on exogenous government expenditures, payouts on government securities, and transfers. The
government and private agents trade a specified set of risky securities whose returns depend on
the aggregate state. Useful benchmarks like complete markets and one-period risk-free bonds
are special cases.
∗
We thank Fernando Alvarez, David Backus, V.V. Chari, Maryam Farboodi, Xavier Gabaix, Lars Peter
Hansen, Ali Shourideh, Pierre Yared, Stephen Zeldes and seminar participants at Columbia, Federal Reserve
Bank of Philadelphia, Kellogg School of Management, Princeton, University of Zurich for helpful comments1

We construct a type of second-order approximations of policy functions to the planner’s
Bellman equation around small aggregate shocks. These approximations allow us to capture theoptimal response to uncertainty at any given level of debt and under certain conditions can be
expressed as linear functions of aggregate shocks. This yields a very tractable formulation of
the planner’s optimal policies. We confirm that these quadratic approximations are accurate bycomparing them with reliable global approximations. An advantage of our quadratic approximations is that they enable analytic and interpretable expressions for the means, variances, andrates of convergence to the invariant distribution of debt, tax revenues, and tax rates.1 Empirical counterparts to our expressions for these objects can be easily constructed using data on
the primary government deficit, aggregate consumption, and returns on securities traded by thegovernment. We show that asymptotically the government’s optimal debt portfolio minimizes
an appropriately defined risk criterion.
To isolate principles that give rise to key findings, we start with a simple baseline setting inwhich agents have quasilinear preferences and the market structure is restricted to a single security whose payout we allow to be correlated with the government consumption process. The joint
distribution of returns and government expenditure is i.i.d. over time. Using properties of thegovernment’s Euler equation, we establish existence of an invariant distribution of government
debt. Up to third-order terms, we show that the drift in the dynamics of debt is proportional
to the covariance of total government spending (debt service plus exogenous government expenditures) and returns. A level of debt that minimizes the variance in total spending sets thiscovariance to zero; this level of debt becomes a point of attraction for the stochastic process
for debt. The speed of mean reversion is inversely proportional to the variance of the return on
the security, and the variance of the invariant distribution is proportional to the amount of riskthat the government bears at its risk-minimizing debt level. Later sections of the paper show
the tendency of debt to approach a level that minimizes the risk in the government’s portfolio
extends well beyond our baseline case.
Extending the analysis to multiple assets yields several insights. If returns satisfy a spanning
condition, the planner can replicate a complete markets allocation like the one that prevails in
Lucas and Stokey (1983). Allowing trade in more securities decreases the speed of convergenceto the invariant distribution. The reason for this outcome is that additional securities facilitate
better hedging and thereby lower the cost of being away from the long-run target level of debt.
By specializing the portfolio problem to two securities, a console and short term security, we canderive prescriptions for optimal maturity management. In this two-security case, the riskinessof the return on the short-maturity asset relative to that on the long-lived console drives theaverage maturity of the total debt. In particular, if the return on the long maturity bond is riskier
than the return on the short-maturity bill, then the optimal maturity of the planner’s portfolio
is inversely proportional to total public debt and most adjustment to aggregate shocks is donewith the short bill. We also extend the analysis to incorporate risk aversion and more generalshock processes. We show that insights from the baseline model continue to apply provided
that we use concepts of “effective returns” and “effective shocks” – returns on the government
debt portfolio and innovations in the present discounted value of the primary government deficit
1

We can also use our quadratic approximation to get analytic expressions for other moments.

2

adjusted by marginal utilities of consumption, respectively.
In the quantitative part of the paper, we focus on two goals: (i) to compare the quality of our
two numerical approximations to the Ramsey plan, one using a more accurate global method, theother using formulas derived from our quadratic approximation; and (ii) to study the predictionsof the model for realistic shock and return processes. To this end, we use plausible quantities
drawn from U.S. data to calibrate shock and return processes . Our analytical expressions
derived under simpler environments continue to be accurate in the calibrated version of themodel. We find that the optimal level of debt is close to zero and that the optimal policy displays
an extremely slow rate of mean reversion (half-life of 250 years). These results are driven by
the fact that a significant amount of variation in returns to the U.S. portfolio is uncorrelatedwith output; that means that holding large quantities of debt/assets would frustrate hedging
objectives.
Our paper obviously shuts down important forces emphasized in other theories of optimallevels of government debt. For example, the literature on sovereign debt, by allowing a government each period to choose whether or not to service its debt, focuses attention on how theadverse consequences of default endogenously generates incentives to repay debt obligations.
The government in our model has no such options and requires no such incentives. This eliminates the design of incentives to induce payment as determinants of government debt and its
maturity composition and puts the hedging considerations on which we focus front and center.
Our model describes optimal fiscal policy of a government that never contemplates dishonoring
its debts. (We like to think of the U.S. and some European governments as being in this situation.) Additionally, we focus on real debt and ignore nominal issues. Extending our approachto economies with possibilities of default and monetary economies is straightforward but spaceconsuming as it would require us to introduce several layers of additional complications to ourmodel. We leave that for future work.

1.1

Relationships to literatures

Our paper builds on a large literature about a Ramsey planner who chooses a competitive
equilibrium with distorting taxes once-and-for-all at time zero.2 Most of these papers use either
complete or approximately complete markets as in Lucas and Stokey (1983), Buera and Nicolini(2004), Angeletos (2002), or study a singe-period risk-free bond and quasilinear preferences asin Aiyagari et al. (2002) and Barro (1979). In contrast, our analysis allows for a more generalincomplete market structure and risk-aversion. In both complete market economies, as in Lucasand Stokey (1983), and quasilinear settings with a risk-free bond only, as in Barro (1979), anylevel of debt is optimal in the sense that the Ramsey planner sets expected public debt level in any
future period equal to initial debt. We show that this result is non-generic; small departures fromthose assumptions imply that for any initial debt, driven by hedging considerations, government
debt can be expected to converge to the unique risk-minimizing level.
In a related context, Barro (1999) and Barro (2003) study tax smoothing in an environment
in which revenue needs are deterministic but refinancing opportunities are stochastic. In Barro’s
setting it is optimal for the government to issue an infinitely lived console as a way to insulate2

For instance Lucas and Stokey (1983), Aiyagari et al. (2002), Chari et al. (1994), Farhi (2010).

3

inter-temporal tax smoothing motives from concerns about rolling over short maturity debt atuncertain prices. In contrast, our analysis allows both revenue needs and returns on the debt to
be stochastic. We fit empirically relevant properties of returns on debt and then find an optimal
government portfolio associated with those returns.
Technically, our paper is closely related to Aiyagari et al. (2002). Those authors include ananalysis of an economy in which a representative agent has quasilinear preferences. In addition
to a linear labor tax, they allow a uniform positive (but not negative) lump sum transfer. Thereis a continuum of invariant distributions for debt, all of which feature a zero labor tax rate anddebt levels that are negative and sufficiently large in absolute value to finance all government
expenditures from the government’s interest revenues, with transfers absorbing all aggregate
fluctuations by adjusting one-to-one with the aggregate shock. A key difference in our paper is
how we treat transfers. While Aiyagari et al. (2002) exogenously restrict transfers, we deducetransfers from an explicit redistribution motive by modeling agents who cannot afford to pay
positive lump sum taxes. We show that so long as the utility of such agents is strictly concave
and the planner puts positive weight on their well-being, the result of Aiyagari et al. (2002)
about properties of the invariant distributions goes away. A benevolent planner in such settings
wants to minimize fluctuations of both tax rates and transfers; he ultimately targets a (generally
unique) level of debt that minimizes risk. The invariant distributions studied by Aiyagari et al.
(2002) emerge only in the limit as the risk-aversion of all recipients of transfers goes to zero.
The equilibrium approximation tools that we apply in this paper are complementary to
Faraglia et al. (2012), Lustig et al. (2008), and Siu (2004) who numerically study optimal
Ramsey plans in specific incomplete markets settings. Our approximation method allows us
to derive closed form expressions for the invariant distribution of debt and taxes that enlightens
the analysis of the underlying forces.
Our theory of government portfolio management shares important features of the singleinvestor optimal portfolio theory of Markowitz (1952) and Merton (1969). Bohn (1990) andLucas and Zeldes (2009) use some insights from the single-investor literature to study portfolio
choices of a government in a partial equilibrium after having specified a loss function for thegovernment. Common to both Merton’s investor and our Ramsey planner are hedging motives
that shape portfolio and savings choices. In contrast, our analysis allows both revenue needs andreturns on the debt to be stochastic. However, unlike the Merton investor, our Ramsey planneris benevolent (i.e., it maximizes the utility of the agents with whom it trades) and it internalizes
the general equilibrium effects of his distorting tax rate choices on equilibrium asset prices. As
a consequence, the optimal portfolio does not feature the usual mean-variance trade-off but
instead attempts to minimize the appropriately defined measure of risk.
The remainder of this paper is organized as follows. In section 2, we analyze a streamlined
setting in which only one risky security can be traded and the representative agent has quasilinear
preferences. In sections 3 and 4 we extend the analysis to include multiple assets, persistentshocks, concerns for redistribution, and risk aversion. In section 5, we study a quantitative
example with parameters calibrated to U.S. data.

4

2

Quasilinear preferences

We begin with a streamlined setting with a representative household having quasilinear preferences. Each of a measure one of identical agents has preferences over consumption and labor∞
supply streams {ct }∞t=0 and {lt }t=0 that are ordered by

E0

∞
X


β ct −
t

t=0


1 1+γ
,
l
1+γ t

(1)

where Et is a mathematical expectations operator conditioned on time t information and β ∈(0, 1) is a time discount factor. One unit of labor produces one unit a nonstorable single goodthat can be consumed by the household or the government. Feasibility requires
ct + gt = lt ,

t≥0

(2)

where gt denotes government consumption.
The government imposes a flat tax at rate τ t on labor earnings and buys or sells a single
one-period security having an exogenous state-contingent payoff pt . Consumers sell or buy thesame security, so it is in zero net supply each period. Let Bt be the number of securities thatthe the government sells in period t at price qt . Government budget constraints are
gt + pt Bt−1 = τ t lt + qt Bt ,

t≥0

(3)

A probability measure π(ds) over a compact set S governs an exogenous i.i.d. shock st .
Government purchases and payoffs on the single security are random variables g, p represented
by bounded functions that map S → R+ . We order elements in S using the function p(·),
i.e., s ≥ s0 if p(s) ≥ p(s0 ) and, as such, we will use the phrase “p is increasing in s”. Let´ḡ = Eg ≡ s∈S π(ds)g(s) and p̄ = Ep. We use st = (s0 , ..., st ) to denote a history of shocks.
We often use zt to denote a random variable z with a time t conditional distribution that isa function of the history st . It is convenient to define Bt ≡ qt Bt and Rt+1 ≡ pt+1 /qt and to
re-write the government’s budget constraint (3) as
gt + Rt Bt−1 = τ t lt + Bt .
A representative agent’s budget constraint is
ct + bt = (1 − τ t ) lt + Rt bt−1 ,(4)where bt is the representative agent’s purchase of the single security. The period t market clearing
condition for the security is
bt = Bt .

(5)

We exogenously confine government debt to a compact set
Bt ∈ B, B .

5(6)

The assumption of compactness of the feasible debt simplifies the analysis. We make the bounds
sufficiently large that they do not affect the properties of the joint invariant distributions ofgovernment debt and the tax rate that we analyze below.
Definition 1. A competitive equilibrium given an initial government debt B−1 at t = 0 is a∞sequence {ct , lt , Bt , bt , Rt , τ t }∞t=0 such that (i) {ct , lt , bt }t=0 maximize (1) subject to the bud-get constraint (3); and (ii) constraints (2), (5), and (6) are satisfied. An optimal competitive
equilibrium given B−1 is a competitive equilibrium that has the highest value of (1).
Single-security incomplete market models such as Barro (1979) and Aiyagari et al. (2002)
assume that the security’s payout is risk-free, a special case of our setup in which p(s) is independent of s. Two purposes induce us to allow stochastic payoffs. First, it is well known thatstandard real business cycle models driven by productivity (and/or expenditure shocks) fail togenerate realistic holding period returns. Our more flexible choice for security markets allows
us to remedy this shortcoming in parsimonious way. To offer an example for just one place andtime, in section 5 we document how the real returns to U.S. government portfolios fluctuate anduse those findings to discipline payoffs. Second, as we show in section 4, the optimal process
for risk-free government debt when the representative consumer is risk-averse in consumption
resembles the optimal behavior of state-contingent debt when the represent agent is risk-neutral.
With risk-aversion and risk-free debt, a key object is an “effective” return on debt that takes
into account a shadow cost of raising revenues; this influential return is stochastic even when
the single security is risk-free. In this section, we identify the joint distribution of returns on
government debt and other variables as a key determinant of a long-run target government debt
level and also of the speed at which a government should approach it. We will use a notion ofan effective return on debt to extend that insight to more general settings in section 4.
The representative consumer’s first-order necessary conditions imply that1 − τ t = ltγ , Et−1 Rt =

1
.
β

(7)


t)
The security price qt satisfies qt = β p̄ so the return on the security Rt st = p(sβ p̄ . Since p isincreasing in st , so is R.
Substitute (7) into the consumer’s budget constraint to obtain
ct = lt1+γ + Rt B t−1 − B t .

(8)

Use (8) to eliminate ct from the feasibility condition (2) to obtain
lt − lt1+γ + B t = Rt B t−1 + gt .

(9)Lemma 1. {ct , lt , Bt , bt , Rt , τ t }∞t=0 is a competitive equilibrium given B−1 if and only if{lt , Bt−1 }∞t=0 satisfies (6) and (9).
The lemma allows us to obtain the optimal competitive equilibrium allocation and debt

6

process by solvingmax{lt ,Bt }∞
t=0

E0

∞
X
t=0

β

t




γ 1+γ
(Rt B t−1 − B t ) +
,
l
1+γ t

(10)

where the maximization is subject to constraints (6) and (9). The objective function in (10) is
a version of (1) in which we used (8) to eliminate ct .
For our purposes, instead of the labor supply lt , it is convenient to work with tax revenues
Zt = τ t l t :Zt = Z(lt ) ≡ lt − lt1+γ .

(11)

Tax revenues are bounded from above by Z, the level associated with the tax rate at the peakof the Laffer curve.3 We show in the appendix that it is not optimal to set the tax rate to avalue that is to the right of the peak of the Laffer curve, which implies that the map definedin (11) is invertible, in the sense that for each Z ≤ Z there are a unique labor supply l (Z)and associated tax rate τ (Z) = 1 − lγ (Z) that satisfy equation (7); l (Z) , −τ (Z) are each1decreasing functions of Z. Let Ψ (Z) ≡ 1+γl(Z)1+γ be the utility cost of supplying labor

required to raise Z units of tax revenues. We show in the appendix that Ψ is strictly decreasing,strictly concave, and differentiable and that it satisfies Inada conditions limZ→−∞ Ψ0 (Z) = 0
and limZ→Z Ψ0 (Z) = −∞.
The optimal value function V (B_) for problem (10) satisfies the Bellman equationˆ
V (B_) =π (ds) [(R (s) B_ − B (s)) + γΨ (Z (s)) + βV (B (s))]max{Z(s),B(s)}s∈S

(12)

s∈S

where maximization is subject to Z(s) ≤ Z, B (s) ∈ [B, B] andZ (s) + B (s) = R (s) B_ + g (s) for all s.

(13)Strict concavity and differentiability of Ψ implies that V is also strictly concave and differentiable.
Policy functions B̃ (s, B_) and Z̃ (s, B_) attain the right side of Bellman equation (12). Letτ̃ (s, B_) denote the associated optimal tax rate policy.
An important endogenous variable, gross government expenditures, E(s, B_) is defined as
E (s, B_) = R (s) B_ + g (s) ,(14)which equals government expenditures including interest and repayment of government debt.
Aggregate shocks have effects on E (s, B_) that depend partly on government debt B_.
We begin our analysis by stating a lemma that summarizes some key properties of the optimal
policy rules.
Lemma 2. There exists B max > B min such that the functions B̃, Z̃, and τ̃ are increasing in
B_ and decreasing in s if B_ ≤ B min and increasing in s if B_ ≥ B max .
Proof. The right side of expression (12) can be maximized separately for each s. The optimal
3

The expression for the maximum revenue is Z = γ



7

1
1+γ

1+1/γ.

values of {Z (s) , B (s)} are increasing in E (s, B_).4 As E (s, B_) is increasing in B_ for000g(s )−g(s )all s, both B̃ and Z̃ are increasing in B_ for all s. Let B max = maxs0 >s00 ∈S R(s0 )−R(s00 ) and00

0g(s )−g(s )B min = mins0 >s00 ∈S R(s0 )−R(s00 ) , then E (s, B_), and hence also B̃ and Z̃, is decreasing in s for

B_ ≤ B min , and increasing in s for B_ ≥ B max .5 Because τ̃ is an increasing function of Z̃, it
inherits the same properties.
The first part of Lemma 2 establishes that the optimum tax rate and debt level are bothincreasing in E. Total expenditures, E(s, B_) are higher when debt B_ is higher, which in turnimplies B and τ are increasing in B_. Since R is increasing in s, E is increasing in s if B_ issufficiently large (i.e., greater than B max ) and decreasing in s if B_ is sufficiently small (i.e.,
smaller than
B min
o ).
n
∞

Let

Z̃t , B̃tt=0

be an optimum process generated by iterations on policy functions thatattain the right side of Bellman equation (12). First-order conditions associated with the problem
on the right side of (12) imply that the marginal social value of assets V 0 (B̃t ) is a risk-adjusted
martingale, so that if B̃t is interior then6
V 0 (B̃t ) = βEt Rt+1 V 0 (B̃t+1 ) = Et V 0 (B̃t+1 ) + βcovt (Rt+1 , V 0 (B̃t+1 )).

(15)

Lemma 2 also allows us to restrict the sign of the covariance appearing in (15). Since
Rt+1 st , st+1 is increasing in st+1 , the covariance appearing on the right side of equation (15)is positive if B̃t ≤ B min and negative if B̃t ≥ B max .
We shall establish the principle that the level of debt that minimizes variation in E(s, B_),so thatB ∗ = argminB var R(s)B + g(s) ,(16)

directs the long run dynamics of the optimal plan. It is easy to verify thatB∗ = −cov (R, g).
var (R)(17)

We assume debt limits are loose enough that B ∗ ∈ [B, B].
Implications of Lemma 2 and equation (15) are particularly stark when R allows the government to hedge risk in g(s) perfectly. DefineR∗ ≡ R : there exists B ∈ [B, B] that makes R(s)B + g(s) be independent of s .

(18)

∗ + g(s) = 0 and theWhen R ∈ R∗ , the debt B ∗ defined by equation (17) satisfies
var
R(s)B
 
limits in Lemma 2 satisfy B min = B max . In this case, V 0 B̃t is a supermartingale if B̃t ≤ B ∗and a submartingale if B̃t ≥ B ∗ . Therefore, debt B̃t has an upward drift if it is less than B ∗and a downward drift if it is greater than B ∗ . The tax rate τ̃ t and tax revenues Z̃t follow
corresponding dynamics.
4
This follows from the objective function being additively separable and concave in (Z, B) as well as constraint(13) taking the form Z + B = E(s, B_).
5
Recall that we order elements in S using the payoff function p(s).
6The appendix provides an analysis of the situation in which B̃t is not required to be interior.

8

Proposition 1. The optimal process {B̃t } has a unique invariant distribution. If R ∈ R∗ , thenB̃t → B ∗ a.s.
Proposition 1 asserts that the invariant distribution of B̃t is unique. Moreover, if the returns
process R allows for perfect hedging of g, then the invariant distribution is easy to characterize:
the monotonicity of policy functions established in Lemma 2 and the martingale properties
of equation (15) immediately imply that B̃t converges to B ∗ defined in equation (17) withprobability 1.
A payoff vector R ∈ R∗ for which perfect hedging is possible is non-generic because it
requires perfect correlation between asset returns and government expenditures. This special
case nevertheless illustrates a general principle that an optimal policy drives government debt
to a level that minimizes fluctuations in E (·, B). When perfect hedging is impossible because
R ∈/ R∗ , shocks take government debt away from B ∗ . However, as we see shortly, that themartingale conditions will hold approximately and the mean of the invariant distribution ofdebt will still be B ∗ , which thus serves as a “target” debt. We shall provide good estimates ofthe speed of reversion to that target and of the variance of B̃t in the invariant distribution. A
one-to-one mapping between the debt B̃t and government revenues Z̃t allows us to characterize
probability distributions of tax revenues and the tax rate.
To analyze the more general case with R 6∈ R∗ , we start with the observation that any
random variable g can be expressed asg(s) = ḡ + σg (s)for some constant σ and a mean zero random variable g with E2g = 1. Similarly, we canrepresentp(s) = p + σσ gp p (s)for some constant σ gp and a mean zero random variable p with E2p = 1. Define σ 2p = σ 2 σ 2gp as
the variance of p.
We take a second order expansion of policy rules with respect to σ around σ = 0. Information about state s can be summarized by (p(s), g(s)), so we can write the policy functions
as B̃ (p(s), g(s), B_; σ), where the notation emphasizes that these are implicit functions of σ.
It is easy to check that B̃(p, g, B_; 0) = B_, which motivates us to approximate around B_,
the current debt. This approach differs from one that would approximate around a mean of anergodic distribution. We show in Appendix 7.2 that∗B̃ (g, p, B_; σ) = B + βĝ +B_ + σ 2p B ∗1 + σ 2p!

p̂1+
(B_ − B ∗ ) + O(σ 3 , (1 − β)σ 2 ), (19)p̄ 1 + σ 2pwhere p̂(s) = p(s) − p and ĝ(s) = g(s) − g. The second order expansion is linear in g and p up

9

to terms that appear in O(·).7 Dropping the higher order terms8 leads us to approximate B̃, Z̃
with

B_ + σ 2p B ∗1 + σ 2p∗

B̌ (g, p, B_; σ) = B + βĝ +

!

p̂
1
+
(B_ − B ∗ )p̄ 1 + σ 2p

(20)

and

Ž(g, p, B_; σ) =
+

1−β
β



B_ + σ 2p B ∗1 + σ 2p∗

B + ĝ +

!

p̂
β p̄


(21)


1
(B_ − B ∗ ) − B̌(g, p, B_) − B ∗ .
βThese approximate policy rules can be used to compute all conditional and unconditional moments of the stochastic process of debt and tax revenues. For brevity, we focus on three key
moments: the mean, variance and the speed of mean reversion to the invariant distribution of∞
debt and taxes. Let B̌t , Žt t=0 be debt and tax revenues generated by the approximated policy
functions.
Proposition 2. The invariant distribution of B̌t , Žt has:
• Mean
E B̌t = B ∗ ,1−β ∗E Žt ≡ Z ∗ = ḡ +B ,β• Variancevar (g + RB ∗ )var(B̌) =1 + β 2 var(R) ,var(R)1−ββvar(Ž) =2var(B̌),• Speed of reversion to mean
E0 B̌t − B ∗ = B̌0 − B ∗E0 Žt − Z∗= Ž0 − Z

∗





1
1 + β 2 var (R)12
1 + β var (R)

t
,

t
.

Proposition 2 shows that the risk-minimizing level of debt B ∗ , defined in equation (17), is themean of the invariant distribution of the approximating debt series B̌. It is positive if R and g
are negatively correlated and negative otherwise. Thus, it is better to have debt (positive B ∗ ) if
the bond return is high in states in which expenditures are low, and to have assets (negative B ∗ )otherwise. The lower is the variance of returns, the slower is the speed of reversion. This makes
sense because if Bt 6= B ∗ , then fluctuations in the rate of return put additional risk into E(s).
7Here O(xn ) denotes a member of a class of functions
f (x) ∈ O (xn ) if and only if lim kf (x)x−n k < ∞.
x→0

n

m

n

m

We denote O(x ) + O(y ) with O(x , y ).
8
Dropping terms of the order (1 − β)σ 2 from the second order expansion allows us to obtain closed form
solutions for moments of the ergodic distribution. Since standard parameterizations set the discount factor beta
fairly close to 1, dropping these terms is inconsequential.

10This risk is increasing in the volatility of R. Therefore, the more volatile is R, the more costly
it is for the government to stabilize its debt Bt and the higher is optimal speed to repay debt
when Bt > B ∗ (or to accumulate debt when Bt < B ∗ ). Debt dynamics approximate a random
walk when the security is nearly a risk-free bond, recovering a result of Barro (1979). Finally,
the dispersion of the invariant distribution of government debt depends on the magnitude of theresidual risk var(g + RB ∗ ) at the risk minimizing debt level B ∗ . The bigger is the variance of
this component, the less hedging the government can obtain from B ∗ , and the probability thatshocks push debt away from B ∗ becomes higher. If R ∈ R∗ , then the variance of the orthogonalcomponent is zero and the invariant distribution puts all mass on debt B ∗ , consistent with anassertion of Proposition 1.
To understand the striking finding that the mean of the invariant distribution of B̌t is B ∗ , itis useful to connect the martingale (15) and the static variance minimization problem (16). By
strict concavity of the value function, there is a one-to-one relationship between debt Bt and its
marginal value to the planner, V 0 (Bt ). An inspection of the martingale equation (15) shows thatthe covariance term covt (V 0 (Bt+1 ), Rt+1 ) is important in determining the drift of the dynamics
of debt in the long run. For a given Bt , the debt next period Bt+1 depends only on Et+1 andconsequently,
covt (V 0 (Bt+1 ), Rt+1 ) ∝ covt (Et+1 , Rt+1 ) + O(σ 3 ).
Bt +gt+1 )It is easy to verify that covt (Et+1 , Rt+1 ) = 12 ∂var(Rt+1. Thus, ignoring O(σ 3 ) terms, the∂Btcovariance term in the martingale equation (15) is proportional to slope of the variance of Et
with respect to government debt Bt . As B ∗ minimizes the variation in E the slope is zero at
B ∗ . The change in signs of the slope implies that, to second order, V 0 (Bt ) will be a sub(super)martingale when Bt > (<)B ∗ . This explains why the approximation B̌t of government debt B̃tdrifts towards B ∗ .
The preceding logic can be verified using equation (19). The drift in B̌t is Et [B̌t+1 − B̌t ] =σ 2p(B̌t − B ∗ ). Substituting for B ∗ , this simplifies to1+σ 2pEt [B̌t+1 − B̌t ] =β2cov(Rt+1 , Et+1 ),1 + σ 2p(22)


and thus, the drift Et [B̌t+1 − B̌t ] is proportional to the slope of the variance curve: var E(·, B̌t ) .
Proposition 2 also characterizes the invariant distribution of tax revenues Z. This is accomplished by first iterating equation (13) forward in time and taking expectations at t = 0. We
deduce thatE0 Bt X j=
β E0 (Z̃t+j − gt+j ).
βjAs t → ∞, conditional means converge to means of the invariant distribution andE(Z) = ḡ +

1−β
β


E(B).

The present value budget constraint implies that tax revenues scale with the annuity value ofdebt, so long run fluctuations in debt are associated with long fluctuations in tax revenues. The

11

9scaling factor 1−ββ indicates that tax revenues are much less volatile than debt. In appendix

7.2, we make this intuition precise by computing the variance of Zt in the invariant distribution
using the second order approximation to the optimal policy Z̃(g, p, B_) and deriving expressions
corresponding to those in Proposition 2.
As a natural extension, one can map the volatility of tax revenues Zt into the volatility of the
tax rate τ t . This mapping helps quantify the amount of tax smoothing when perfect spanning
is infeasible. Given that l = (1 − τ )1/γ , tax revenue as a function of τ is Z(τ ) = (1 − τ )1/γ τ .
The deviation of τ from the ergodic mean τ ∗ is then approximatelyτ − τ∗ =1
Z 0 (τ ∗ )(Z − Z ∗ ).

We can approximate the variance of τ as
var(τ ) ≈10Z (τ ∗ )2
var(Z).

The value of Z 0 (τ ∗ ) cannot be obtained in closed form but is easy to compute.10 To illustrate:
2

1
varies between 1 and 3.7. The standard
for γ = 2 and τ ∗ ∈ [0., 0.4], the coefficient Z 0 (τ∗)
deviation of τ is then up to twice that of tax revenues, but still much less than that of government
debt.
Figure I illustrates the accuracy of the quadratic approximation. Here we assume that γ = 2,
β = 0.98. Government expenditure g is a approximated using a 5 state Gaussian quadrature
with mean 20% and standard deviation 1.8% . The payoff vector p was defined byp(s) = 1 + χ(g(s) − 0.2) + σ e (s).
We solved where (s) is a 2 state i.i.d. process independent of g(s). The parameter χ was
set to 0.7 to make the asset payoff process be slightly correlated with g.11 Figure I plots theergodic distribution of the global solution, solved using projection methods, and our quadratic
approximation for 3 levels of σ e : 0.001,0.02, and 0.04. σ e is smallest in the topmost panel andlargest in the bottommost panel. The figure indicates that invariant distributions generated
by approximated policy rules (solid red lines) provide good approximations to the invariant
distribution generated by the global solution (dashed black lines).

3

Extensions

Forces that we isolated within the section 2 economy prevail under alternative assumptions
about numbers of securities, motives for redistribution, preferences that express aversion to
consumption risk, and persistence of shocks to government expenditures. We discuss them ingreater detail now.
1
For a β=0.98, 1−β≈ 50βSee the proof of Proposition 2 for more details.
11This matches the correlation that would occur if the representative agent were risk averse. We discuss thiscase more in section 4
9

10

12

3.1

More assets

K
We begin by allowing K one-period securities. Let p~ = pk k=1 be payoffs of the K securities~ = β −1 p~ be the corresponding matrix of returns. The return of asset k in state s is Rk (s).
and R
The government budget constraint at time t is
K
X

gt +

k
Rtk Bt−1= τ t lt +

k=1

K
X

Btk .

k=1

We restrict the value of the government portfolio to be in a compact set
Bt ≡

K
X



Btk ∈ B, B .

k=1

Temporarily suppose that the government portfolio weights are fixed, meaning that there areP
Bkkconstants α1 , ..., αK such that P tB k = αk for all t. Define R (s) ≡ K
k=1 αk R (s). Then thektoptimal policy problem is equivalent to the one analyzed in section 2. Thus, if the government
fixes its portfolio weights, then all insights about optimal debt management and fiscal policy
from section 2 apply.
Now suppose that the government optimally chooses portfolio weights each period. The
planners’ value function V (B_) satisfies a Bellman equationˆ
V (B_) =max{Z(s),B(s),B k _}s,k"!
X

π (ds)s∈Skk

R (s) B _ − B(s)#+ γΨ (Z (s)) + βV (B (s))k(23)
where maximization is subject to Z ≤ Z, B (s) ∈ [B, B], andB (s) + Z(s) =X

Rk (s) B k _ + g (s) ∀s,

(24)

k

X

B k _ = B_.

kHere B_ is the value of the government portfolio before portfolio choice in the current period.
Counterparts of the section 2 martingale equation (15) now hold for derivatives of the value function with respect to every payoff vector Rk . Paralleling Proposition 1, the invariant distributionof the total value of government portfolio B̃t is unique.
~ R]~ be the covariance of returns and C[R,~ g] be the vector of covariances between
Let C[R,~ ∗ that minimizes the variance of E(s, B_)
returns and exogenous expenditures g. The vector B
is

"
~∗

B = min var
~
B

!#
X

k

k

R (·) B + g (·)

.

(25)

k

~ R]
~ has full rank, the optimal portfolio problem (25) has a unique solution with
When C[R,~ ∗ = −C[R,~ R]~ −1 C[R,~ g].12 With multiple securities, we use the notation B ∗ introduced before
B
12
~ R]
~ fails to have full rank if there exists a linear combination of {Rk } that yields a risk free
The matrix C[R,bond. The minimum variance in equation (25) is then unchanged by the amount invested in a risk-free bond13

to denote value of the total portfolio,

P

kB

∗,k .

The next proposition is a counterpart of Proposition 2. B̌t now denotes an approximation
to the total value of government portfolio analogous to the one in section 2 and 1 denotes a K
dimensional (column) vector of ones.
Proposition 3. The invariant distribution of B̌t has• Means~ R]~ −1 C[R,~ g],
E B̌t = B ∗ = −1| C[R,E(Ž) = Z ∗ = g +

1−β ∗
B ,
β

• Variances




1
−1
2
|
|
−1 ~
~
~
~
~
~
+ [g − ḡ] ,var B̌t = (1 + β 1 C(R, R) 1)β var −1 C(R, g) C(R, R)
R−
β



1−β 2
var Žt =var(B̌t ),
β


−2 |

• Speeds of mean reversion
!t
−2 |
~ R)
~ −1 1β
1C(R,,
E0 (B̌t − B ∗ ) = (B̌0 − B ∗ )~ R)~ −1 1
1 + β −2 1| C(R,
!t
−2 |
~ R)
~ −1 1β
1C(R,.
E0 (Žt − Z ∗ ) = (Ž0 − Z ∗ )~ R)~ −1 1
1 + β −2 1| C(R,~For a government portfolio with total value B_, holdings of individual securities B(B_)
areapproximately~ ~ −1 ~~ R)~ −1 C(R,~ g) + C(R, R) 1 B_ + 1| C(R,~ R)
~ −1 C(R,~ g) .
B(B_)≈ −C(R,~ R)~ −1 1
1| C(R,(26)

When K > 1, the planner’s problem includes designing an optimal portfolio. However, moreis involved in the planner’s problem than the standard mean-variance trade-offs featured insingle-agent portfolio choice problems like those studied by Merton (1969) because the benevolent planner internalizes competitive equilibrium restrictions when he designs portfolios. Withquasilinear preferences, the planner knows that in equilibrium all assets bear the same expected
return β1 . That makes both the total savings decisions and the portfolio choice be driven by
the goal of minimizing risk rather than by the risk-return trade-offs emphasized by Merton.
In section 4, we show that this insight extends to settings in which consumers are averse to
consumption risk.
Next we illustrate Proposition 3 and formula (26) with some examples.
Example 1. Suppose that there are two securities and that security 1 allows perfect hedging inthe sense that R1 ∈ R∗ , where R∗ is defined in (18). Assume that the return on security 2 isand, consequently, the solution to problem (25) is undetermined up to

14

P

kB

∗,k

.

orthogonal to return on security 1. Proposition 3 implies that the invariant distribution has all∗of its mass on B ∗ = − cov(R,g)
var(R) and that the speed to convergence to B is

Et−1 [B̌t − B ∗ ] =

1
1+β

2

var(R2 )var (R1 )var(R1 )+var(R2 )(B̌t−1 − B ∗ ).

Formula (26) implies that the portfolio composition satisfies

var R2
var R1B̌ (Bt ) =Bt +B∗,var (R1 ) + var (R2 )var (R1 ) + var (R2 )1
var R1
var R1B̌ (Bt ) =Bt −B∗.
var (R1 ) + var (R2 )var (R1 ) + var (R2 )2In example 1, the government chooses a debt portfolio that in the long-run fully hedges its
shocks, just as in the section 2 one-security case. Because complete hedging can be achievedat the steady state with security 1 only, using security 2 there would introduce risk in E(s).
1 6= B ∗ , the government uses
So the government’s long run portfolio has only security 1. If B−1

security 2 to get some additional hedging while it gradually approaches B ∗ along a transition
path. Convergence to the long-run portfolio is slower than in section 2 case in which agents canonly trade security 1, since
1
1+β

2

var(R2 )var (R1 )var(R1 )+var(R2 )>1.
1 + β var (R1 )2

The availability of more securities slows convergence to the government’s long run target portfolio.
Example 2. Consider a setting in which it is possible perfectly to hedge government expenditures
shocks by using both of two securities, so that R1 , R2 ∈ R∗ and R1 6= R2 . In this case, markets
are complete. For any initial debt, the government can choose a portfolio to hedge aggregatek,g)shocks perfectly. Let B ∗,k = − cov(R. For an initial debt B−1 ∈ [B, B], find B 1 and B 2 thatvar(Rk )P Bk
P ksatisfy k B ∗,k = 1 and k B = B−1 .13 At this portfolio, government expenditures gross ofdebt service for any state s areE(s|B 1 , B 2 ) =X

B k Rk (s) + g(s) =k

X Bk hk

B ∗,ki
B ∗,k Rk (s) + g(s) .


Rk ∈ R∗ and equation (18) imply that B ∗,k Rk (s) + g(s) is constant and independent of s.14
Farhi (2010) studies a Ramsey problem for a government that invests claims to physicalcapital and one-period risk-free debt. The main difference between his setting and ours is thatFarhi’s Ramsey planner affects returns on the government’s portfolio by varying a distorting tax
rate on capital income. In Farhi’s model it is optimal for the government to hold large positionsof risk-free debt and equity of opposite signs. That lets the government implement a complete
A unique solution exists so long as B ∗,1 6= B ∗,2 which is guaranteed by R1 6= R2
It can be shown that the optimal policy satisfies B̃(s, B_) = B_ and a Z̃(s, B_) = Z ∗ , where Z ∗ is aconstant.
13
14

15

market allocation with arbitrarily small fluctuations in the capital tax rate. When we make oneof the two securities approach a risk-free bond in the example 2 setting, the optimal portfoliothat supports a complete markets allocation attains arbitrarily large gross positions.
Example 3. Suppose that every asset is an infinitely lived console with a per-period coupon
pk (s) so that the government budget constraint is

gt +

K 
X

K

X
pkt + qtk Bt−1 = τ t lt +qtk Bkt .

k=1

k=1

The price qtk of the long-lived asset satisfieshikqtk = βEt qt+1
+ pkt+1 ,which because s is i.i.d. implies that qtk = q̄ k ≡



β
1−β



Epk . Redefine Btk = q̄ k Bkt and Rtk =q̄ k +pk (s), so that the optimal policy problem with long-lived assets reduces to one with short livedq̄ kassets. The logic outlined in this example extends to assets that are in positive net supply, suchas Lucas trees.
Example 4. Example 3 can be interpreted as a theory of an optimal maturity of government
debt. Assume that there are two securities, a risk-free bond with one period maturity and an
infinitely lived console having a per-period stochastic coupon p(s). The amount of government
debt allocated to one security relative to another indicates the “maturity” of government debt.
Example 3 shows that in this setting the 
government
issues or holds two assets with returns
R1 (s) = β1 and R2 (s) = q̄+p(s)where q̄ ≡
q̄

β
1−β

Ep. The variance minimization problem (25)yields minimizers
B 1 (Bt ) = Bt − B ∗,2 and B 2 (Bt ) = B ∗,2 ,
2

∗,2

,g)
with B ∗,2 = − cov(R. The effective maturity of government debt is BtB−B ∗,2 . For any governvar(R2 )ment debt Bt with which it enters a period, the government issues a time-invariant amount B ∗,2of the console and adjusts its issues of the risk-free bond each period in response to shocks that
period. The effective maturity of government debt is an inverse function of total government
debt, a pattern consistent with empirical findings of Missale and Blanchard (1994), who studied
European countries during 1960-1990.

3.2

Persistent shocks

In this section we return to the settings of section 2 but assume that shocks are first orderMarkov rather than i.i.d. The planner’s optimal value function satisfies the Bellman equationˆV (s_, B_) =π (ds|s_) [(R(s, s_)B_ − B (s)) + γΨ (Z (s)) + βV (B (s) , s)]max{Z(s),B(s)}s∈S

s∈S

(27)
where maximization is subject to Z ≤ Z, B (s) ∈ [B, B] andZ (s) + B (s) = R(s, s_)B_ + g (s) for all s_, s,
16

(28)

where π(ds|s_) denotes a transition probability kernel andR(s, s_) =

p(s)
´
.
β s0 π(ds0 |s_)p(s0 )In the rest of the section we will generalize the static variance minimization problem inequation (16) to obtain counterparts of B ∗ when shocks are persistent. A useful construct isthe expected present discounted value of net of interest government expenditures, P Vg (s) =PE t β t g(st )|s0 = s .15 For a given level of Z, define a function B ∗∗ (Z, s) as
B ∗∗ (Z, s_) =βZ− βE (P Vg |s_) .
1−β

(29)

For initial conditions B−1 = B ∗∗ (Z, s_), and s−1 = s_, the policy of setting the tax rate
such that the total revenues are constant through time at Z is optimal under complete markets.
With returns R(s, s_), we can compute the residual in the budget constraint (28) when tax
revenues are Z(s) = Z and debt choices are given by B(s) = B ∗∗ (Z, s),D(Z, s_, s) ≡ B ∗∗ (Z, s_)R(s_, s)−(Z − g(s))−B ∗∗ (Z, s) = B ∗∗ (Z, s_)R(s_, s)−Z+P Vg (s),
1−β
(30)

where the last equality follows from equation (29).
We now discuss the appropriate variance minimization problem that generalizes (16). LetZ ∗ be the level of tax revenues that minimizes the ergodic variance of D. To calculate Z ∗ , define
λ as invariant distribution associated with the transition kernel π.16 We use Eλ (·) and varλ (·)
to denote the mean and variance of a random variable under the stationary measure λ. The
level Z ∗ solves
Z ∗ ∈ argminz Eλ (var(D(Z)|s_)) .

(31)

Using equation (30), and R̂(s_, s) = R(s_, s)−E (R|s_) and PˆV g (s_, s) = P Vg (s)−E (P Vg |s_)we can simplify the objective function in equation (31) to
 

2
∗∗
ˆ
E (var(D(Z)|s_)) = E E B (Z, s_)R̂(s_, s) + P V g (s_, s)
s_ .
λ

λ

Taking the first order condition with respect to Z and reorganizing terms we obtain,
Eλ (B ∗∗ (Z ∗ , s_)) = −covλ (R, P Vg ) covλ (var (R|s_) , βE (P Vg |s_))−.
varλ (R)varλ (R)(32)

When shocks are i.i.d, Eλ (B ∗∗ (Z ∗ , s_)) in equation (32) equals B ∗ defined in equation(17). To show that B ∗∗ (Z ∗ , s) captures the same long-run hedging motives when shocks are
persistent, we begin with a simple case in which perfect hedging is possible. We extend theWhen S is finite, π is an S × S stochastic
matrix P Vg = (I − βπ)−1 g.
´
The measure λ satisfies λ(A) = S λ(ds)π(A|s) for any measurable set A. For finite S, one can compute λusing the equation λ0 = λ0 Π and normalized to sum to 1.
15

16

17

notion of perfectly spanning returns and define
∗



R =


p(s)
R(s, s_) ≡ ´∃B ∈ [B, B] s.t Bp(s) + P Vg (s) is independent of s .
β s0 π(ds0 |s_)p(s0 )(33)

To explain why (33) is an appropriate extension of definition (18), we introduce,
Definition 2. A stochastic steady state is a function B : S → R that satisfies
B̃(s_, B(s_), s) = B(s) ∀s_, s.
With persistent shocks, perfect tax smoothing occurs when there exists a stochastic steadystate. Unlike the outcome with i.i.d. shocks, government debt is not be constant when tax
revenues are held constant. A stochastic steady state B is connected to the solution B ∗∗ (Z ∗ , s):
Proposition 4. For R ∈ R∗ , B ∗∗ (Z ∗ , s) is the stochastic steady state. Furthermore,B ∗∗ (Z ∗ , s_) = −cov(R, P Vg |s_)var(R|s_)∀s_.

We now extend Proposition 2 to allow for both persistence and R 6∈ R∗ . For the remainderof this section, we assume a continuous state space with s = (g, p) and that {gt , pt } obey
gt = (1 − ρ)ḡ + ρgt−1 + εg,t ,(34a)

pt = p̄ + εp,t ,(34b)where εg,t , εp,t are i.i.d over time with zero means and an arbitrary covariance structure. We∞
use the technique described in section 2 to construct an approximation B̌t t=0 of an optimaln o∞process B̃t.
t=0

Proposition 5. Let E and var represent the expectation and variance in the invariant distribution of B̌t . The invariant distribution of B̌t has• Mean
E B̌t |st = B ∗∗ (Z ∗ , s),• Variancevar(B̌t − Bt∗∗ ) =

1 + β 2 varλ (R)varλ (R)varλ (Dt∗ ) ,• Speed of mean reversion
E0 (B̌t − Eλ B ∗∗ ) = (B̌0 − Eλ B ∗∗ )

1
1 + β 2 varλ (R)t.

When net-of-interest government expenditures are persistent, inter-temporal revenue needs arelinked. In the long run, the planner uses asset income to hedge not only contemporaneous but
also future government expenditures. When shocks are AR(1) as in (34), equation (32) for themean of the ergodic distribution simplifies and becomes18

covλ (R, P Vg )E (B (Z , s_)) = −=−varλ (R)
λ

∗∗

∗



1
1 − ρβcovλ (R, g).
varλ (R)

Comparing this with equation (17) shows that planner accumulates more debt (or assets) relative
to the no persistence ρ = 0 case.

3.3

Transfers and redistribution

Optimal debt management in our model in section 2 differs significantly from that in otherincomplete markets models (e.g., Aiyagari et al. (2002) and Farhi (2010)). In our model, theinvariant joint distribution of debt and taxes is unique. In the long run, debt and tax rates
minimize fluctuations in gross government expenditures including debt service requirements,
E(s). By way of contrast, optimal plans in those other models have a continuum of invariant
distributions of debt levels. In all of them, tax rates are zero and debt levels are negativeand big enough in absolute value that to finance all net of interest government expendituresfrom earnings on the government’s portfolio. This results in large fluctuations in transfers in
the ergodic distribution. A key difference between those models and ours is that we prohibit
lump-sum taxes or transfers, while Aiyagari et al. (2002) and Farhi (2010) allow positive but
not negative lump-sum transfers. In this section, we extend our analysis to an economy withlump sum transfers. We show that our section 2 results carry over essentially unchanged. Wethen discuss the mechanism that drives the long-run tax rate in Aiyagari et al. (2002) to zeroand explain how to reconcile their results with ours.
A standard justification for ruling out lump-sum taxes in representative agent models is torefer to unmodelled “poor” agents who cannot afford to pay a lump-sum tax. In this section, westudy optimal anonymous transfers in an economy with such poor agents. We extend the section
2 economy to have just enough heterogeneity across agents to make the analysis interesting. In
particular, we assume that in addition to a measure 1 of agents of type 1 with quasilinear
1+γ

preferences U (c, l) = c − l1+γ , there is a measure n of agents (type 2 agents) who cannot workor trade securities and who enjoy utility

E0

∞
X

β t U (c2,t ) ,t=0

where c2,t is consumption of a type 2 agent in period t; U is strictly concave and differentiable
on R+ and satisfies the Inada condition limc→0 U 0 (c) = ∞.
The government and type 1 agent trade the same security described in section 2. The
government imposes a linear tax rate τ t on labor income and lump-sum transfers Tt that cannot
depend on the type of agent. Negative transfers are not feasible because a type 2 agent has no
income other than transfers. Tt denotes aggregate transfers, so each agent receives a per-capita
Tt. Since agent 2 lives hand-to-mouth, his budget constraint istransfer 1+nc2,t =

Tt
.
1+n

19

The planner ranks allocation according to

E0

∞
X
t=0


ct −
β
t

1 1+γ
l
1+γ t




+ ωU (c2,t ) ,for some ω > 0.
The time t government budget constraint is now
gt + Tt + Rt Bt−1 = τ t lt + Bt .
With only minimal modifications, the budget constraint of a type 1 agent, Definition 1 of thecompetitive equilibrium, and the section 2 recursive formulation of the optimal policy problem
extend to this environment. The planner’s optimal value function satisfies the Bellman equationˆ
V (B_) =max{Z(s),B(s),T (s)}s∈S



T (s)
π (ds) R (s) B_ − B (s) ++ γΨ (Z (s))
1+n
s∈S



T (s)
+ βV (B (s)) (35)+ωU
1+nsubject to,Z ≤ Z, B(s) ∈ [B, B],(36)

Z (s) − T (s) + B (s) = R (s) B_ + g (s) for all s.

(37)

Most section 2 results apply to this economy. We define B ∗ and R∗ exactly as in section 2.
 
Proposition 6. There exists a unique invariant distribution for B̃t . V 0 B̃t satisfies themartingale equation (15). If R ∈ R∗ , then B̃t → B ∗ almost surely and τ̃ t → τ ∗ , T̃t → T ∗ almost
surely for some constant τ ∗ , T ∗ , and τ ∗ 6= 0 generically. Both τ ∗ , T ∗ are increasing in ω.
Proposition 6 extends Proposition 1 to an economy with endogenous transfers. The invariant distribution of government debt is unique. If perfect hedging is feasible at some level ofgovernment debt, government debt eventually reaches that level. Because B ∗ is independent oftransfers and the Pareto weights, so is the long-run debt level. Long-run levels of transfers andtax revenues do depend on the Pareto weight and are generally not zero.
Following the same steps used to prove Proposition 2, we use a second order approximationsof policy functions to obtain
Proposition 7. The invariant distribution of B̌t satisfies properties stated in Proposition 2.
The invariant distribution of Žt − Ťt has the same properties as the invariant distribution of Žtin Proposition 2. Let F (Ť ; ω) be the cumulative distribution function of the ergodic distributionof Ťt . If ω > ω 0 then F (Ť ; ω) first order stochastically dominates F (Ť ; ω 0 ).
The insights in section 2 about optimal debt management carry over to this heterogeneous
economy. In this economy, fluctuations in both the tax rate and (non-agent-specific) lump-sumtransfers are costly, so an optimal policy smooths both. Adjusting the tax rate in response toaggregate shocks is costly because the dead-weight loss of taxation is convex in tax rates, as
stressed by Barro (1979). Adjusting transfers is also costly because that induces fluctuations in20inequality. For any fixed weight ω, there is an optimal level of transfers T ∗ (ω) that is increasingin ω and that achieves the socially optimal amount of inequality. Deviations from that level oftransfers reduce social welfare. That is why an optimal policy smooths transfers.
In Aiyagari et al. (2002) and Farhi (2010) the government eventually sets tax rates to zeroand thereafter adjusts transfers one-to-one with government expenditures. A main difference
between our setup and theirs is that they do not model heterogeneity explicitly but only appeal
to it informally as a justification for imposing a restriction Tt ≥ 0. This restriction puts a kink in
the cost of using transfers – the marginal cost of an increase in transfers is zero, while a marginal
cost of a decrease in transfers is infinite at T = 0. A high marginal cost of negative transfers
creates an incentive for the government in Aiyagari et al. (2002) and Farhi (2010) to accumulate
enough assets to ensure that the constraint Tt ≥ 0 is eventually slack. Since fluctuations inpositive transfers are costless, in the long-run the government uses those transfers to offset all
fluctuations in expenditures gt .
By way of contrast, in our economy, the cost of using transfers is endogenous. It is smooth(so that the marginal cost from both increasing and decreasing transfers around the optimal levelωT ∗ (ω) is the same, 1+n
U 0 (T ∗ (ω))) and welfare costs of departing from the optimal inequality

level are strictly convex. This implies very different long run dynamics than those emphasized
by Aiyagari et al. (2002). As we show in a companion paper, Bhandari et al. (2015b), thisinsight carries over to richer economies with much more heterogeneity and in which no agent is
excluded from the financial markets.

4

Risk aversion and persistent government expenditures

We extend our analysis to a setting in which a representative agent has preferences that display
risk-aversion and in which fluctuations in net-of-interest government deficits are persistent. We
assume that a representative agent’s one-period utility function , U : R2+ → R, is given by
U (c, l) =c1−σ − 1
l1+γ
−
.
1−σ
1+γ

(38)

The constant elasticity of substitution assumption simplifies exposition. Our results extend to
U ’s which are strictly concave in (c, −l) and twice continuously differentiable. We let Ux,t or
Uxy,t denote first and second derivatives of U with respect to x, y ∈ {c, l} . We assume thatnatural debt limits apply to the consumer, which ensures that first-order conditions are satisfied
off corners.
The state s ∈ S, where S is finite, is Markov with transitions described by a stochastic matrix
whose entries are π (st |st−1 ) . Both government expenditures g and a representative agent’s laborproductivity θ can depend on s. Feasibility at t requires
ct + gt = θt lt .

(39)

The government’s budget constraint at t is
gt +pt
Bt−1 = τ t θt lt + Bt ,
qt−1
21

(40)

where qt−1 =βEt−1 Uc,t pt
is the price of the security issued in period t − 1.
Ut,t−1

We employ a primal approach to formulate the optimal policy problem. We use the feasibility constraint and the household’s first-order necessary conditions to eliminate taxes andprices from either the government’s or the household’s budget constraint to deduce the following
implementability restrictions on competitive equilibrium allocations:Uc,t Bt + Uc,tUl,tpt Uc,t
θt lt +
lt − gt =Uc,t−1 Bt−1
Uc,tβEt−1 pt Uc,tt ≥ 1,(41)Ul,0l0 + p0 β −1 B−1 .
Uc,0P
The optimal policy problem is to maximize E0 t β t U (ct , lt ) by choosing {ct , lt , Bt }∞t=0 subject
c0 + b0 = −to implementability constraints (41) and the feasibility constraints (39).
The shadow cost to the planner of raising revenues varies with the marginal utility of
consumption. We define effective government debt Xt ≡ Uc,t Bt , an effective
h security return
i
U

U

p

t
l,t ct +gt
Rt ≡ βEt−1c,tUc,tpt , and an effective primary deficit Φt ≡ Uc,t [gt − Zt ] = −Uc,t ct + Uc,t ( θt ) ,where Zt denotes tax revenues. The variables Xt , Rt and Φt serve as counterparts to Bt , Rt andgt − Zt in the section 2 economy with quasilinear preferences.
It is possible to construct an optimal policy recursively by taking (X, s) as state variables,
meaning that an optimal allocation for τ ≥ t > 0 depends only on the values of (Xt−1 , st−1 ).
We can defineR(s, s_, c) =β

Uc (s)p(s).
00s0 ∈S π (s |s_) Uc (s)p(s )PFor t > 0, the planner’s optimal value function V (X_, s_) satisfies a Bellman equation
V (X_, s_) =max

X{c(s),X(s)}s∈S

s∈S

 


c(s) + g(s)π (s|s_) U c(s),+ βV (X (s) , s)
θ(s)

(42)

subject toX(s) = R (s, s_, c) X_ + Φ(s, c).

(43)

Paralleling the analysis in section 3.2, let c̃(s, X_, s_), X̃ (s, X_, s_) denote policy functions,and R̃(s, X_, s_), Φ̃ (s, X_, s_), B̃ (s, X_, s_), Z̃ (s, X_, s_) and τ̃ (s, X_, s_) denote effective returns, an effective net-of-interest deficit, and government debt, tax revenues and tax rates
implied by the policy functions that attain the right side of (42). We can use the first-orderconditions forthe problem
on the right side of (42) to show that the marginal cost of government
debt Vt0 = V 0 X̃t , st satisfies the martingale equation

V 0 (X̃t , st ) = βEt Rt+1 V 0 (X̃t+1 , st+1 ) = Et V 0 (X̃t+1 , st+1 ) + βcovt (Rt+1 , V 0 (X̃t+1 , st+1 )).

(44)

That effective returns depend on the marginal utility of consumption, an endogenous object,complicates the present economy relative to the section 2 economy with quasilinear preferences.
Nevertheless, many insights can be gleaned by proceeding along the lines of sections 2 and 3.2.
We extend the analysis of returns in set R∗ from the quasilinear environment and study properties of payoffs that allow perfect hedging. Since returns are endogenous, we will characterize22perfect spanning returns in terms of a set of exogenous payoffs P ∗ . Extending Definition 2 ofstochastic steady state to marginal utility adjusted debt, let {X (s)}s∈S be a vector satisfyingX̃(s, X (s_), s_) = X (s) ∀s, s_.
Proposition 8. There exists a non-empty set of exogenous payoffs P ∗ such that for each p ∈ P ∗there is a stochastic steady state X associated with a constant tax rate. The mean X ∗ of theinvariant distribution of X satisfies

cov R̃ (s, X (s_), s_) , P VΦ̃ (s, X (s_), s_) |s_,X ∗ = −Eλ 
var R̃ (s, X (s_), s_) |s_

Furthermore, there exists a class of economies for which the risk-free payoffs, prf (s) = 1 are in
P ∗.
Payoffs in P ∗ satisfy the spanning property similar to equation (33),p(s)Uc (s, X (s_), s_, ) X (s_) + P VΦ̃ (s, X (s_), s_, ) is constant for all s, s_.
As in the section 2 economy with quasilinear preferences, the tax rate is constant when government debt equals X (s_). Unlike the section 2 economy, here income effects cause tax revenues Z̃ (s, X (s_), s_) and consumption c̃ (s, X (s_), s_) to fluctuate. Therefore, debt choices{B̃ (s, X (s_), s_)}s∈S vary with s_ but in ways that make the vector of marginal utility adjusted debt choices [c̃ (s, X (s_), s_)]−σ B̃ (s, X (s_), s_) be constant for all s_, s. The tax rate,
consumption, and labor supply behave as in the complete market economy of Lucas and Stokey(1983).
The expression for X ∗ implies that sign B̃ (·, X ·, ) = −sign R̃ (·, X , ·) , P VΦ̃ (·, X , ·) .
This sign can be easily computed. For example, consider a simple economy with a risk-freebond in which shocks s affect only government expenditures g and but not the payoff p or
productivity θ and are i.i.d over time. It is easy to confirm that if perfect hedging is attained,
then cov R̃, P VΦ̃ > 0 and that therefore perfect hedging can be achieved only with negative
government debt. The intuition for this result is as follows. Even if the payoff p does not depend
on s, the effective return R is positively correlated both with g and with the effective primary
deficit Φ. Therefore, the government can hedge fluctuations in Φ by holding a positive quantityof this security.
Computing optimal policies in more general cases (for example when p 6∈ P) requires approximating the value function in Bellman equation (42) numerically. However, it is possible tofind the target level of debt, i.e., the (effective) level of debt that provides the maximum amountof hedging without having to solve Bellman equation (42) and then computing an invariant
distribution implied by the associated optimal policies. We describe the procedure next.
For a given τ solve for a consumption and labor, c(τ , s), l(τ , s) allocation using(1 − τ )θ(s)Uc (τ , s) + Ul (τ , s) = 0

23

(45a)

θ(s)l(τ , s) − c(τ , s) − g(s) = 0.

(45b)

From this consumption and labor allocation, construct effective primary deficits and effective
returns as
Φ(τ , s) = Φ(c(τ , s), l(c, s)) and R(τ , s_, s) =Uc (τ , s)p(s).
βE Uc (τ )p s_(46)

Under a policy that sets tax rate equal to τ every period, we can iterate on the budget
constraint to compute the marginal utility weighted debt as the expected present value of themarginal utility weighted primary surplus
X(τ , s_) = −βE (P VΦ (τ )|s_) .

(47)

As in equation (30) we compute the residual in budget constraint
D(τ , s_, s) = X(τ , s_)R(τ , s_, s)+Φ(τ )(s)−X(τ , s) = X(τ , s_)R(τ , s_, s)+P VΦ (τ )(s). (48)
The counterpart of the problem (31) isτ ∗ = arg min Eλ (var(D(τ )|s_)) ,

(49)

τ

where λ is the invariant distribution over s. As before, for p ∈ P ∗ , the stochastic steady statecoincides with X(τ ∗ , s). For more general payoffs we provide counterparts of the expressions in
Proposition 2 and Proposition 5. Let E and var be the expectation and variance taken over theergodic distribution generated by X̃ (s, X_, s_). The estimates for mean, speed of convergenceand variance of the ergodic distribution are
E[X̃t |st = s] ≈ X(τ ∗ , s),1
E0 [X̃t − X(τ , st )] ≈ (X0 − X(τ , s0 ))2
1 + β varλ (R(τ ∗ ))

1 + β 2 varλ (R∗ )∗var(D(τ ∗ )).
var(X̃t − X(τ , st )) ≈varλ (R(τ ∗ ))
∗

∗

(50a)
t
,

(50b)
(50c)

In section 5, we show that this procedure does a good job of approximating the mean, varianceand speed of convergence to the invariant distribution for X̃t in a variety of circumstances.

4.1

Optimal portfolio choice with risk aversion

In this section, we return to the connection with Merton (1969) to emphasize that, even withrisk aversion, the relevant objective for the government is minimizing a notion of risk and not amean-variance trade-off. We then apply this insight in a simple example with two securities, arisk free bond and a risky asset that mimics the “stock market”, to show that it is may not be
optimal for the government to go long in the stock market even if it yields a higher expected24

return.17
To extend problem (42) to K > 1 securities with {pk (st )}k as the vector of payoffs we only
Uc (s)pk (s)0k 0s ∈S π(s |s_)Uc (s)p (s )need to modify the implementability constraint (43). Let Rk (s, s_, c) = β P 0

be the effective returns on each of the K securities, the implementability constraint with K > 1
securities is given by
X(s) =X

Rk (s, s_, c) X k _ + Φ(s, c),

(51)

k

with

P

kX

k _ = X_ is the marginal utility adjusted total debt.

It is easy to check that theEuler equation for the planner holds with respect to each security
V 0 (X̃t , st ) = Et V 0 (X̃t+1 , st+1 ) + βcovt (Rkt+1 , V 0 (X̃t+1 , st+1 )).

(52)

We briefly describe the appropriate static risk-minimization problem in the case with K > 1
assets that captures dynamic hedging concerns. As in equation (46), construct effective returns~ k (τ , s_, s) =for the security with payoff pk (s) using the allocation defined by equation (45) and R
Uc (τ ,s)pk (s). We use a two step approach to compute the residual deficit in the governmentβE(Ūc (τ )pk |s_)budget constraint. Using X(τ , s) as in equation (47), we pose the following risk-minimization
problem that generalizes problem (25) in the quasilinear setting and extends problem (49). Let~ , s_) be a vector of size K that denotes how a given level of effective assets X(τ , s_) is splitX(τinto effective holdings of the K possible securities.
#2"~ , s_) = arg min
X(τ

X

X

π(s|s_)sX~ k (τ , s_)(s)X k + P V (τ )(s)
R
Φ̄

(53)

k

subject toX

X k = X(τ , s_).

k

~ R]
~ has full rank, the optimal portfolio is given by
When C[R,~ , s_) ∝ C[R(τ~, s_), R(τ~, s_)|s_]−1 C[R(τ~, s_), P VΦ (τ )|s_].
X(τTo pin down τ ∗ we compute the residual deficit in the government budget constraint as
D(τ , s_, s) =X

~ k (τ , s_)(s)X~ k (τ , s_) + P V (τ )(s)
R̄
Φ̄

k

and τ ∗ minimizes the ergodic variance of D(τ ) exactly as in equation (49).
The reason why the optimal portfolio is different from that of a typical Merton (1969) investor
is that our government is benevolent and operates in general equilibrium. Benevolence implies
that both the representative consumer and the government have the same attitudes towards
risk and end up discounting risky payoffs in exactly the same way. With general equilibrium,
17This result contrasts with partial equilibrium arguments of Lucas and Zeldes (2009) that the government
should always have a positive position in the security (stock market in their application) that has a positive risk
premium.

25

expected risk-adjusted returns adjusted by marginal utility are equalized across securities. Thus
any gains to the government by holding securities with higher expected revenues are offset bylosses to the consumer. This is reflected by effective holding period returns Rt and not thereturns Rt showing up in the government’s Euler equation.
We now construct a simple example to demonstrate that the government may not want tochase an equity premium. The representative agent is risk averse with period utility U (c, l) =2

log(c) − l2 with Frisch elasticity set to 0.5. For simplicity, we only have i.i.d. productivity andpayoff shocks. Productivity θt distributed as log(θ) ∼ N (0, 0.03). For the market structure, thegovernment can trade two assets. The first is a risk free bond with payoff pf = 1. The second
is a “stock market” asset which has a payoff with a loading, χ > 0 on log(θt ) plus an orthogonalcomponent
pS = 1 + χ log(θt ) + 0.01t ,where t ∼ N (0, 1) is independent of θ and i.i.d. over time.18
U

Uc,t pS
t+1
t t+1 Uc,t+1

S
Let Rtf = βEt Uc,tbe the risk-free rate and Rt,t+1= βE pS
c,t+1

be the holding periodS
return on the risky asset. This environment features an equity premium, Et Rt+1
> Rtf , but the

government does not choose invest in the stock market asset. Instead, when the government isinitially in debt, the government begins by shorting the stock market and in the long run holding
all of it’s assets in the risk free bond. Although the initial short position in the stock market
exposes the government to the orthogonal variation t , it provides a good (but temporary) hedge
by delivering higher returns in times of low TFP. Eventually the government only uses the risk
free bond to hedge. We see the dynamic portfolio re-balancing in Figure II. The two lines show
the marginal utility adjusted positions in the risk free security (blue line) and the risky security(green line).

5

A quantitative example
1−σ

We assume that a representative household has separable, isoelastic preferences U (c, l) = c1−σ −l1+γ1+γ . Labor productivity follows an AR(1) processlog θt = ρθ log θt−1 + σ θ θ,t ,(54)

where θ,t is an i.i.d. standard normal variable. A stochastic payoff pt from a single security is
pt = 1 + χp θ,t + σ p p,t ,(55)

where p,t is an i.i.d. standard normal variable; χp and σ p parameterize variance and correlation of asset returns with output. Analogous to equation (55), we parametrize government
expenditures aslog gt = log ḡ + χg θ,t + σ g g,t ,where g,t is another i.i.d. standard normal variable.
18

We approximate log(θ) and t with a finite state Markov process using Gaussian quadrature.

26

(56)

Baseline calibration
We set σ = 1 and γ = 2 to attain a Frisch elasticity of labor supply of 0.5. We set β = 0.98,
which implies a 2% annual interest rate in an economy without shocks. We calibrate parametersρθ , σ θ , χp , χg , σ g , σ p in equations (54)-(56) using moments from a competitive equilibrium withan exogenous government policy that fits some stylized features of actual U.S. debt in the post
war period.
We estimate policy rules that the U.S. government uses for (Bt , τ t ) as follows. We assume
that Bt follows a rulelog Bt = (1 − ρB,B ) log B̄ + ρB,B log Bt−1 + ρB,Y log Yt + ρB,Y _ log Yt−1 ,(57)

where we estimate coefficients ρB,B , ρB,Y , ρB,Y _ using market value of gross federal debt (annual)series for the period 1947-2010 published by the Federal Reserve Bank of Dallas for Bt .19 For Yt ,we use the annual aggregate labor earnings from the Bureau of Economic Analysis.20 Both Bt
and Yt are Hodrick-Prescott pre-filtered (after taking logs) using a smoothing parameter of 6.25.
Then we set τ t as a residual to make sure that the government budget constraint is satisfied.21
Table I reports the estimates and figure III shows the fit.22
Using the government policy described in (57), we set the parameters ρθ , σ θ , χp , χg , σ g , σ p
to jointly match moments for output, returns, and government expenditures for the period
1947-2010.
We impute returns Rt using data on the federal primary surplus P St 23 and market valueof debt as follows. In the data the duration of government debt is approximately constant,24
therefore the budget constraint of the government can be written as(pt + qt )Bt−1 = qt Bt + P St .

(58)

Multiply and divide the first term by qt−1 and use the fact that the holding period return for+ptlong term debt is given by Rt = qqtt−1(see example 3 in section 3.1) to rewrite equation (58) as

Rt =Bt + P St
,
Bt−1

(59)

where Bt = qt Bt is the market value of government debt observed in the data. We use thisformula to compute the holding period return.
The average return annual over the sample 1947-2010 is 5% with an annual standard devia19

The calculation of this series takes into account outstanding marketable and non-marketable debt of different
maturities issued by the Treasury and uses current market prices to convert par value to market value.
20
Since in our model we abstract from capital, our measure of output Y is aggregate labor earnings. The
results remain essentially unchanged if we use GDP per capita instead.
21
We obtain very similar result if we estimate policy rules for τ t along the lines of equation (57) using Barroand Redlick (2011) data on average marginal income tax rates
22
Our results are essentially unchanged by adding more lags debt (or taxes) or output. We report the resultswith the specification that has one lag of debt, contemporaneous output, and one lag of output because it fits
the data well (see figure III ) and also parsimoniously captures short-run dynamics of the optimal policy. This
allows us to compare an optimal policy and our approximation to the U.S. policy.
23
This is measured as total federal receipts net of federal consumption and transfer payments, both obtained
from the Bureau of Economic Analysis.
24
From 1985 to 2010 the average maturity of U.S. federal debt has been around 60 months.

27

tion of 5%. A key empirical fact that drives a lot of our results is that a substantial componentof fluctuations in returns is uncorrelated with fundamentals.25 For government expenditures, we
use federal government consumption expenditures plus transfer payments obtained from the Bureau of Economic Analysis. Table II summarizes parameters values and the fit of the competitive
equilibrium to U.S. data.
Using this calibration, we compute a global approximation to the Ramsey allocation.26
Results
Table III reports the long run properties of debt and taxes under the optimal plan. Convergenceis slow (half life of 250 years). In the long run under an optimal plan, government debt haslarge fluctuations (standard deviation of 20%) while there are small movements in tax rates andtax revenues (standard deviations of 0.5% and 0.6% respectively). Figures IV and V plot theergodic distribution and mean path across 30,000 simulations.
Sections 2 and 4 showed that an appropriately posed static variance minimization problem
anchors the long run debt level. Further, we used insights from quadratic approximations of thepolicy rules to obtain simple expressions that map moments of the joint distribution of returns
and deficits to the speed of convergence and variance around the long run levels. We now verify
that these descriptions apply to our calibrated economy.
Expressions in equations (50) are now compared against the mean, speed of convergence andvariance computed from long simulations using our global approximation to the optimal policy
rules. Table IV summarizes the results for effective debt, Xt . The simple formulas do well in
predicting the ergodic outcomes. Moreover, we see in Figure VI that formula (50b) gives a very
accurate approximation for the evolution of E0 Xt .
Formula (50) is evaluated at the optimal policy τ ∗ , and an intermediate step is to computeτ ∗ using (49). We show that a simple back-of-the-envelope calculation that constructs effective
returns and deficits directly from the data and that evaluates optimal debt level using (50)
directly (thus implicitly ignoring the fact that tax rates in the data may not be set optimally)
gives a good approximation to the target debt level, the speed of convergence, and the variancearound the target debt level.
To make this operational, we exploit the relationship between the one period holding returns
Rt and the effective return Rt ,
Rt = RtUc,t−1
Uc,t


(60)

as well the relationship between the primary surplus P St and the marginal utility weighted
surplus ΦtΦt = Uc,t P St .

(61)

Imputing Rt from equation (59) and time series on real personal consumption expenditure, we
25
Alternative ways of measuring returns yield similar conclusions. In Bhandari et al. (2015a), we study holding
period returns across debts of different maturities and document the presence of a component that is orthogonal
to fundamentals like productivity across several maturities.
26
We follow Kopecky and Suen (2010) to obtain a 5 state discrete approximation for the AR(1) process inequation (54). For the shocks p , g we use a 5 state Gaussian quadrature. The Bellman equation is solved using
a value function iteration method where we approximate the value function using a 30 point grid with linear
splines as basis functions.

28

can use (60) to infer Rt . For our sample, the returns Rt and Rt have a correlation coefficient of
0.91. We compute P VΦ by estimating a first order VAR using [log Yt , Φt ]. The estimates allow us
to express P VΦ,t = αy log Yt + αΦ Φt . We use these estimates in the formula X ∗ = −cov(Rt ,P VΦ,t )var(Rt )to obtain,X∗ = −

αy cov(Rt , Yt ) + αΦ cov(Rt , Φt )= −0.08,
var(Rt )where αy = 1.138 and αΦ = −0.80 from our VAR estimation. Similarly, using the sample
variance of effective returns in equation (50b) yields a half life of 244 years. These alternative
computations compares well with our estimates previous reported in Table IV
As our final exercise, we compare the optimal plan to actual U.S. policies. We use shortsample simulations of length 63 periods and compare the volatility and persistence of debt andthe tax rate for the optimal plan and U.S. data. Table V reports the comparison with U.S.
data. From this exercise we conclude that tax rates responded little to shocks under both the
optimal plan and in the U.S. data, but that they are more persistent under an optimal plan. As
a consequence, government debt is more volatile and is repaid more slowly under the optimal
plan.

6

Concluding remarks

This paper develops a theory of optimal government debt management. We show a generic insight that the dynamic hedging considerations are captured by a simple problem that minimizes
a particular measure of fiscal risk. Our analysis delivers simple formulas for the mean, varianceand speed of convergence to the ergodic distribution of debt. Here we also analyze some extensions of our basic environment, an endeavor we pursue more in Bhandari et al. (2015b) thatstudies economies whose substantial ex ante heterogeneity coming from permanent or persistentdifferences in skills that unleashes motives for redistribution and social insurance. Our analysis
in this paper sets the stage for such work – not only by providing appropriate tools for approximating equilibria well and for formulating Ramsey problems in mathematically convenient ways,
but also in terms of highlighting forces that ultimately determine transient and long-run dynamics of debt and taxes. For example, appropriately adjusted fiscal risk minimization problems
continue to shape the mean of an ergodic distribution of government debt, while the hedging
costs of being away from that fiscal risk-minimizing debt level shape the speed of convergence.
In another extension Bhandari et al. (2015a), we shall use the empirical properties of returns
across maturities to compute an optimal term structure of government debt.

29

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31

7

Appendix

7.1

Proof of Proposition 1

We first show preliminary results that we discussed in Section 2. Letto (10) and Z̃t = ˜lt − ˜l1+γ .

no∞be a solutionB̃t , ˜lt
t=0t1
Lemma 3. ˜ltγ ≥ 1+γ

for all t and there is one to one map between ˜lt and Z̃t with Z̃t ≤ Z for

all t. The function Ψ : (−∞, Z] → R is strictly decreasing, strictly concave, differentiable andsatisfies limZ→−∞ Ψ0 (Z) = 0 and limZ→Z Ψ0 (Z) = −∞.
1
1
for all t. Suppose there exists a k such that ˜lkγ < 1+γ.
Proof. First we show that ˜ltγ ≥ 1+γ

Let total revenues Z(l) = l − l1+γ . The maximum value of Z(·) is achieved at l∗ =11.

(1+γ) γSince Z(∞) = −∞ < Z(˜lk ) < Z(l∗ ), applying the Intermediate Value Theorem, we
n canofind
0
0
1
˜
˜
a lk >B̃t , ˆlt ∞1 > lk such that Z(lk ) = Z(lk ). Construct an alternative sequence of
t=0(1+γ) γno0where ˆlt = lk for t = k and ˆlt = ˜lt for t 6= k. This sequence B̃t , ˆlt ∞t=0 also satisfies constraints(9) for all t but has a strictly higher welfare as the objective function (10) is strictly increasing
in lt . Thus, we obtain a contradiction that {B̃t , ˜lt } is a optimal solution.
1
Since ˜ltγ ≥ 1+γ, it implies that Z̃t is bounded above byZ̄ ≡ γ1
1+γ

1+1/γ(62)

and the relationship between Z̃t and ˜lt is one to one in the relevant range of ˜lt . In the text weused the function, Ψ : (−∞, Z̄] → R that satisfies
Ψ(l − l1+γ ) =

1 1+γ
l
1+γ

to denote the utility cost of raising revenues Z. Differentiating with respect to l we getΨ0 (l − l1+γ ) =lγ.
[1 − (1 + γ)lγ ]1
1
On the domain where Ψ(·) is defined lγ ≥ 1+γ
and thus Ψ0 (·) < 0. As lγ → 1+γ
from above,

Z → Z̄ and Ψ0 (·) → ∞. Similarly lγ → ∞, Z → −∞ and Ψ0 (·) → 0. Differentiating twice withrespect of l we getΨ00 (·) [1 − (1 + γ)lγ ] − Ψ0 (·) γ(1 + γ)lγ−1 = γlγ−1 .
1Using both Ψ0 (·) ≤ 0 and lγ ≥ 1+γ
we get that Ψ00 (·) < 0.

We can now prove Proposition 1. The optimality conditions to (12) areˆ
0V (B_) = βR (s) V 0 B̃ (s, B_) π (ds) − κ̄ + κ

(63)

and




γΨ0 Z̃ (s, B_) = −1 + βV 0 B̃(s, B_) − κ̄ (s) + κ (s) ,
32

(64)

where κ̄ (s) and κ (s) are the Lagrange multipliers on B ≤ B and B ≥ B, and κ̄ =´´κ̄ (s) R (s) π(ds), κ = κ (s) R (s) π(ds). We begin with the following lemma that showsthat B̃(s, B_) is strictly increasing in E.
Lemma 4. If B̃(s, B_) ∈ (B, B) and E(s, B_) > (<)E(s0 , B_) then B̃(s, B_) > (<)B̃(s0 , B_)
Proof. We will prove the case where E(s, B_) > E(s0 , B_), the other case is symmetric. From
Lemma 2, B̃(s, B_) is weakly increasing in E(s, B_) , and thus, B̃(s, B_) ≥ B(s0 , B_). Suppose B̃(s0 , B_) = B̃(s, B_) then Z̃(s0 , B_) < Z̃(s, B_) from the budget constraint (13). Strict
concavity of Ψ(·) and an interior solution to (64) then implies−1 + βV 0 (B̃(s0 , B_)) = −1 + βV 0 (B̃(s, B_))= γΨ0 (Z̃(s, B_))< γΨ0 (Z̃(s0 , B_)).
As B̃(s0 , B_) ∈ (B, B), this violates (64) for (s0 , B_) yielding a contradiction. Thus B̃(s, B_) >B̃(s0 , B_).
First, suppose R ∈ R∗ . Since in this case B ∗ = B min = B max , Lemma 2 implies thatB̃ (s, B ∗ ) = B ∗ for all s. Since B̃ (s, ·) is increasing by Lemma 2, it satisfies B̃ s, B_ ≥ B ∗ for
all B_ ≥ B ∗ > B and B̃ s, B_ ≤ B ∗ < B for all B_ ≤ B ∗ , where strict inequalities follow
from the assumption that B ∗ ∈ B, B . If B0 ≥ B ∗ then B̃t ≥ B ∗ > B for all t implies thatκt = 0 and from (63)
 
 

V 0 B̃ t ≤ βER (·) V 0 B̃ ·, B̃t
 

 


= EV 0 B̃ ·, B̃t + βcov R (·) , V 0 B̃ ·, B̃t ≤ EV 0 B̃ ·, B̃t= EV 0 (B̃t+1 )
where
while
 the last inequality follows from the fact that R (s) is increasing by construction,
0
V B̃ ·, B̃t
is decreasing in s by concavity of V and the fact that B̃ s, B̃t is increasing 
in s by Lemma 2. B̃t ≥ B ∗ implies that V 0 (B̃t ) ≤ V 0 (B ∗ ) for all t. Therefore V 0 B̃t is abounded submartingale, which means it must converge by the Martingale Convergence Theorem.
Continuity of V 0 and strict concavity of V implies that there exists a B such that limt→∞ B̃t = B
a.s. We claim that B̃(s, B) = B for all s in the support of π. Suppose B̃(s0 , B) > B (the argument
for the other case is symmetric) for some s0 then B̃(s00 , B) > B for all s00 > s0 as B > B ∗ . As
B̃t → B almost surely there exists a subsequence tn with stn > s0 for all n. But aslim B̃tn = lim B̃(stn , B̃tn −1 ) ≥ lim B̃(s0 , B̃tn −1 ) = B̃(s0 , lim B̃tn −1 ) = B̃(s0 , B) > B,

n→∞

n→∞

n→∞

n→∞

we immediately obtain a contradiction. As B̃(s, B) = B for all s, we can conclude, from Lemma
4, that either B = B or E(s, B) is independent of s. From (63), B̃(s, B) = B for all s if and only
if κ = 0. The same arguments as in Lemma 4 then shows that B = B if and only if E(s, B) is
33

independent of s. We have therefore shown the limit B must satisfy the property that E(s, B)
is independent of s for all s. As R ∈ R∗ , there is a unique point B ∗ that satisfies this property.
Thus, if B0 > B ∗ then B̃t → B ∗ a.s. The case when B0 < B ∗ is symmetric.
Suppose R ∈/ R∗ , which implies that E (·, B_) is not a constant for all B_. We want toshow that there are sets S 0 , S 00 ⊂ S of positive measure such that B̃ (s0 , B_) ≥ B_ for all s0 ∈ S 0
and B_ ≥ B̃ (s00 , B_) for all s00 ∈ S 00 with at least one inequality strict and both inequalities
strict if B_ ∈ B, B . Suppose B > B_ ≥ B̃ (s, B_) for almost all s, which by strict concavity
of V and strict monotonicity of B̃ in E(s, B_) implies V 0 (B_) ≤ βER (·) V 0 B̃ (·, B_) withequality only if B_ = B̃(s, B_) = B for almost all s. In the previous paragraph we showedthat the latter can occur if and only if E(s,of s. As R ∈/ R∗ , this cannot be B) is independentthe case, and thus V 0 (B_) < βER (·) V 0 B̃ (·, B_) . On the other hand equation (63) impliesthat V 0 (B_) ≥ βER (·) V 0 B̃ (·, B_) , a contradiction . Thus, if B > B_ then there exists

S 0 ⊂ S with positive measure such that B̃(s0 , B_) > B_ for all s0 ∈ S 0 . Analogous arguments
show that if B_ > B then there exits S 00 with positive measure such that B̃(s00 , B_) < B_ forall s00 ∈ S 00 and that there exist s0 and s00 such that B(s0 , B) = B and B̃(s00 , B) = B.
The
results of the previous section imply that both
 Pick any small  > 0.
no
E B̃(s, B_) − B_ B̃(s, B_) ≥ B_ and P
B̃(s, B_) ≥ B_
are positive for all B_ ∈[B, B − ]. As both of these terms are continuous functions of B_, compactness of [B, B − ]
implies that there exists d, p > 0 such thatE B̃(s, B_) − B_ B̃(s, B_) ≥ B_ ≥ d

(65)

n
o
B̃(s, B_) ≥ B_
≥p

(66)

and
P

for all B_ ∈ [B, B − ]. As B̃(s, B_) − B_ is bounded above by D = B − B, we obtain that27
Po d/2nB̃(s, B_) − B_ > d/2≥
pD

for all B_ ∈ [B, B − ]. Therefore, there must exist an integer n and % > 0 that that probabilityof reaching B − , B starting from any B_ ∈ B, B −  in n steps is greater than %. Analogousarguments establish that probability of reaching B, B −  starting from B_ ∈ B − , B in
finite number of steps is finite. Since by Lemma 2 policy functions are monotone in B_, Theorem
2 in Hopenhayn and Prescott (1992) establishes the existence of an unique invariant distribution.
of (s, B_) − B_ > d2
while still satisnofying equations (65) and (66) is obtained by placing a mass α on B and a mass P
B̃(s, B_) ≥ B_
− α on27

The arbitrary function f (s, B_) that minimizes probability of Pd/2d
. As B − B_ ≤ D, α can be bounded from below by D−d/2
p.
2

34

n

7.2

Proof of Proposition 2

We first prove the result under the assumption of a continuous state space (as the arguments are
more transparent) and then show how to extend the results to settings with possibly discrete
state space. Let µ̃(B_) ≡ V 0 (B_). When the solution is interior, the first order condition withrespect to Z(s), equation (64) implies that


 

B̃(s,
B_)
−1
+
β
µ̃
0
 ≡ Φ µ̃ B̃(s, B_) .
Z̃ (s, B_) = Ψ −1 γSince shocks are i.i.d., s = (g, p) and equations (13) and (63) can be written as p
B_ + g = Φ µ̃ B̃(g, p, B_) + B̃(g, p, B_),
β p̄

(67)


p 
µ̃(B_) = E µ̃ B̃(g, p, B_) .
p̄

(68)

and



Equation (67) must hold for all g, p, B_ for which policy functions are interior. Therefore,
differentiating both sides with respect to p and g we have
h  



i
1
B_ = Φµ µ̃ B̃(g, p, B_) µ̃B_ B̃(g, p, B_) + 1 B̃p (g, p, B_)
β p̄

(69)

and
h

 



i
1 = Φµ µ̃ B̃(g, p, B_) µ̃B_ B̃(g, p, B_) + 1 B̃g (g, p, B_),(70)

where Φµ and µ̃B_ are derivatives of Φ and µ̃, B̃p and B̃g are partial derivatives of B̃ withrespect to p and g. The derivatives of (67) and (68) with respect to B_ give


i
h  

p
= Φµ µ̃ B̃(g, p, B_) µ̃B_ B̃(g, p, B_) + 1 B̃B_ (g, p, B_),β p̄µ̃B_ (B_) = Epµ̃B_ B̃(g, p, B_) B̃B_ (g, p, B_) .
p̄

(71)

(72)

Similarly, we can obtain all second order derivatives and their cross-partials by further differentiating these equations.
We will study the properties of the optimal debt B̃(g, p, B_) when shocks are small. To doso, without loss of generality, we parametrize variables p and g as p = p̄+σ pg σp and g = ḡ +σg ,for some σ, σ pg ≥ 0 where g and p are random variables with mean 0 and standard deviation
1. We study the behavior of the policy functions in the neighborhood of σ = 0. To emphasize
dependence on σ we use notation B̃(g, p, B_; σ) and µ̃ (B_; σ). B̃(g, p, B_; 0) = B_ and bydefinition p = p̄ and g = ḡ. Then equations (71) and (72) becomehi1= Φµ (µ̃ (B_)) µ̃B_ (B_) + 1 B̃B_ (B_),βµ̃B_ (B_) = µ̃B_ (B_) B̃B_ (B_),

35

where to simplify the notation we use (B_) as a shortcut for (ḡ, p̄, B_; 0). These equations
immediately giveB̃B_ (B_) = 1, Φµ (µ̃ (B_)) µ̃B_ (B_) + 1 =

1
.
β

(73)

Substitute these expressions into (69) and (70) to obtain
B̃p (B_) =B_, B̃g (B_) = β.
p̄Using the same steps for the second order derivatives we can show thatB̃B_B_ (B_) = B̃gg (B_) = B̃pp (B_) = B̃pg (B_) = 0,2
Φµµ (µ̃ (B_)) µ̃B_ (B_) + Φµ (µ̃ (B_)) µ̃B_B_ (B_) = 0.

(74)

For any σ we can write policy functions as B̃(ḡ + σg , p̄ + σ pg σp , B_; σ). Using the second
order Taylor expansion function B̃ with respect to σ around σ = 0, we getB̃(ḡ + σg , p̄ + σ pg σp , B_; σ) = B_ + B̃g (B_)σg + B̃p (B_)σσ pg p + B̃σ (B_)
1
+ B̃σσ (B_)σ 2 + B̃σp (B_)σ 2 σ pg p + B̃σg (B_)σ 2 g + O(σ 3 ), (75)2where we used the omitted the second derivatives that are equal to zero by (74). To evaluate
the derives of B̃ with respect to σ differentiate (67) with respect to σ:hiσ pg p
B_ + g = Φµ (µ̃ (B_)) µ̃B_ (B_) + 1 B̃g (B_)g + B̃p (B_)σ pg p + B̃σ (B_)β p̄+ Φµ (µ̃ (B_)) µ̃σ (B_)


1
B_
=
βg +
σ pg p + B̃σ (B_) + Φµ (µ̃ (B_)) µ̃σ (B_) ,βp̄where we used (73) to get the second equality. This implies that
0 = B̃σ (B_) + Φµ (µ̃ (B_)) µ̃σ (B_) .
Similarly, differentiate (68) with respect to σ and use Ep = 0 to getµ̃σ (B_) = B̃σ (B_) = 0.
Similar steps for these second order derivatives give"σ pg
B̃σσ (B_) = −2E
p
p̄



σ pg
p B_ + βgp̄Φ(B_)(1 − β)+βΦµ (µ(B_))2

andB̃σp (B_) = B̃σg (B_) = 0.

36



σ pg
p B_ + βg
p̄

2 #

Let σ p = σ pg σ, ĝ = σg , p̂ = σ p p . Then the expansion in (75) can be written asp̂B̃(g, p, B_; σ) =B_ + βĝ + B_ − βcov(p, g) − B_σ 2p
p̄
#
"
σ2
Φµµ (µ(B_))(1 − β)
2
−1
2 p
+
βσ + 2B_cov (p, g) + β B_ 2 + O(σ 3 )2Φµ (µ(B_))
pp̂=B_ + βĝ + B_ − βcov(p, g) − B_σ 2p + O(σ 3 , (1 − β)σ 2 ).
p̄
1
2
3
We can write (17) as B ∗ = − βcov(p,g)
var(p) . Furthermore, we have 1 − σ p = 1+σ 2 + O(σ ) andσ

σ p = 1+σp 2 + O(σ 3 ). Therefore we can write the expression above as

p

p

∗ p̂



1
B̃(g, p, B_; σ) = B +βĝ +B +p̄
1 + σ 2p∗p̂
1
(B_−B ∗ )+(B_−B ∗ )+O(σ 3 , (1−β)σ 2 ).
p̄
1 + σ 2p

Assuming that terms O(σ 3 , (1 − β)σ 2 ) are small, the approximated policy rules B̌ arep̂
B̌(g, p, B_) = B + βĝ + B ∗ +
p̄
∗



1
1 + σ 2pp̂
1
(B_ − B ∗ ) +(B_ − B ∗ ),p̄
1 + σ 2p

(76)

∞which induce the approximate solution B̌t t=0 . Using (76) we can immediately verify the mean
and the speed of convergence of the stochastic process for B̌t satisfies equations in Proposition
2. We next compute the expression for the ergodic variance.
Let F be the ergodic distribution of (B̌t , gt , pt ) generated by the approximated policy ruleB̌(g,h p, B_) along
i with the stochastic process for the exogenous shocks(g, p). Let varF (B̌) ≡
2EF B̌ − B ∗ , the ergodic variance of B̌t . As F is the ergodic distribution generated by B̌,2 i
B̌(g, p, B_) − B ∗
"


2 #
1
p̂
1
p̂ ∗
∗
∗
= EF
βĝ + B +(B_ − B ) +(B_ − B )p̄
1 + σ 2p p̄
1 + σ 2p
" "
##


2
p̂ ∗
1
p̂
1
βĝ + B += EF EF(B_ − B ∗ ) +(B_ − B ∗ )B_p̄
1 + σ 2p p̄
1 + σ 2p
( "
# 
"
#



2
p̂ ∗ 2p̂ ∗ p̂B_ +EFβĝ + B
B_ (B_ − B ∗ )= EF EFβĝ + Bp̄
1 + σ 2pp̄p̄!
)
1 + σ 2p+
(B_ − B ∗ )2 .
(1 + σ 2p )2varF (B̌) = EFhIn the last step we dropped
 terms that have a zero mean under F conditional on B_. Thus
1
varF (B̌) = var βĝ + p̄p̂ B ∗ + 1+σ
2 varF (B̌) and collecting terms we have,p
1 + σ 2pp̂ ∗var βĝ + B .
varF (B̌) =σ 2pp̄
Now we describe the approximations to the the total revenues Z̃t . Since Z̃ = Φ (µ̃) we can use

37

(67) to find the approximate law of motion Ž as
Ž(g, p, B_) =p
B_ + g − B̌(g, p, B_),β p̄which implies that∗EŽt ≡ Z =

1−β
β



B ∗ + ḡ.

1
3
Rewrite the budget constraint (67) and using p̂ = 1+σ2 p̂ + O(σ ) we get,pp̂Z̃(g, p, B_) − Z =(B_ − B ) +B ∗ + ĝβ p̄1
+ (B_ − B ∗ ) − B̌(g, p, B_) − B ∗ + O(σ 3 , σ 2 (1 − β)),
β
∗



1
1 + σ 2p



p̂
β p̄



∗



which after substituting for equation (76) simplifies further to1Ž(g, p, B_) − Z =
β
∗

=

!
σ 2p
∗
B̌(g, p, B_) − B −(B_ − B ) − (B̌(g, p, B_) − B ∗ )1 + σ 2p
∗

σ 2p
1−β
(B̌(g, p, B_) − B ∗ −(B_ − B ∗ ).
ββ(1 + σ 2p )Taking expectations at time 0,
##
"
"



σ 2p
1−β
∗
∗
∗
(B̌t+1 − B ) −(Bt − B )E0 Žt+1 − Z = E0 Et−1ββ(1 + σ 2p )
##
"
"

σ 2p
1
1−β
∗
∗
(B̌t − B ) −(Bt−1 − B )= E0 Et−1β
1 + σ 2pβ(1 + σ 2p )




1
∗
= E0
Žt − Z.
1 + σ 2p
Therefore Žt has the same speed of convergences as B̌t . Computing the variance of Ž using theergodic distribution of B̌ we obtain!2
σ 2p
covF (B̌, B_) +varF (Ž) =varF (B̌) − 2
varF (B̌)β(1 + σ 2p )β 2 (1 + σ 2p )
!




2
(1 − β)σ 2p
1
1−β 2
1
=
varF (B̌) − 2
var
(
B̌)
+
varF (B̌)
Fβ
1 + σ 2pβ(1 + σ 2p )β 2 (1 + σ 2p )!

1 − β 2 2(1 − β)σ 2p − σ 4p=−
varF (B̌).
ββ 2 (1 + σ 2p )2


1−β
β

2

(1 − β)σ 2p!

We drop terms that are of order O(σ 3 , (1 − β)σ 2 ) from the expression varF (Ž) to obtain theformula in the text.
Lastly we discuss how to extend the arguments in the proof when shocks are not continuous.
For all (g , p , B_) such that B̃ is interior, we can define continuous and differentiable function
B̊(g 0 , p0 , B 0 ) that solves (67) and (68) in an open neighborhood of (ḡ + σg , p̄ + σ pg σp , B_) and38

is equal to B̃(ḡ + σg , p̄ + σ pg σp , B_). By the chain rule, the derivatives in the expansion of B̃
with respect to σ solveB̃σ (g, p, B_, 0) = B̊g (B_)g + B̊p (B_)σ pg p + B̊σ (B_),
2and similarly for ∂∂σB̃2 . Since B̊ solves the same equations, (67) and (68), as B̃ before, theapproximation, B̌t is identical to (76).

7.3

Proof of Proposition 3

The arguments in the last part of proofof proposition
2 in appendix 7.2 allows us to work withnofunctions B̃(g, p~, B_), µ̃(B_) and B̃ k _(B_) that solve the optimality conditions (assuminginteriority) for problem (23) with K assets:
X pk B k _(B_)
k

β p̄k

= Φ(µ̃(B̃(g, p~, B_))) − g + B̃(g, p~, B_),pkµ̃(B_) = E[ k µ̃(B̃(g, p~, B_))] for all k,p̄XB̃ k _(B_) = B_.

(77a)

(77b)
(77c)

k

Equation (77b) is the counterpart of the martingale equation (15), where we have used µ̃(B_) ≡V 0 (B_). Equation (77a) is obtained by substituting for the tax revenues using a first ordercondition with respect to Z(s). The proof will follow the similar steps and notation convention
as used in appendix 7.2.

We parametrize the payoffs and expenditure as pk = p̄k + σ kpg σ kp and g = ḡ + σgwhere the subscript k denotes na particular security
and use p~ ≡ pk . Indexing the funco
tions B̃(g, p~, B_; σ), µ̃(B_; σ), B̃ k _(B_; σ) with the extra argument σ, we take a Taylor
approximation as σ → 0. Similar to thecase, B̃(g, p~, B_; 0) = B_. With K > 1
n K = 1 oequation (77) have multiple solution for B̃ k _(B_) . We will take an approximation around aparticular solution that is obtained as the limit σ → 0. For now, we let the B̃ k _(B_; 0) = B k _be arbitrary, but then show that the solution to the second order expansion of the system (77)will provide enough restrictions to pin down a limiting steady state portfolio.
The first order terms and the steps to obtain them are very similar to the single asset case.
The expressions for B̃B_ (B_), B̃g (B_), and µ̃B_ (B_) are unchanged. The derivatives withrespect to the pk are similar, now modified for the holdings of the particular securityB̃pk (B_) =

Bk_
. ∀k
p̄k

Also, the derivatives with σ,B̃σ (B_) = µ̃σ (B_) = B̃σk (B_) = 0 ∀kk (B_) are undetermined, but this does not affectare same as K = 1 case. The derivatives B̃B_

39

any future calculations or the approximated policy rules for total assets B̃.
Many of the second order terms are also identical to the K = 1 case. In particular,
B̃B_B_ (B_) = 0, µ̃B_B_ (B_) = −2Φµµ (B_) µ̃B_B_ (B_) ,Φµ (B_)

and the second order and cross terms with respect to g, pk
B̃pk pk = B̃gg = B̃pk g = 0 ∀k.
Differentiating (77b) twice with respect to σ, we obtainµ̃σσ (B_) = E





"

X
σ kpg
B̃pj (B_)σ jpg jp + B̃g (B_)g µ̃σσ (B_) + µ̃B_ (B_)B̃σσ (B_) + 2 k kp µ̃B_ (B_) 
p̄
j

2 #
X
+ µ̃B_B_ (B_) 
B̃pj (B_)σ jpg jp + B̃g (B_)g  .
j(78)
We can eliminate µ̃σσ (B_) from (78), substitute for B̃pj (B_) and solve out for B̃σσ (B_). For
all k we obtain,



2 
j
j
k
X
X
σ pg
Φ(B_)(1 − β) 
σ pg j j
σ pg j jB̃σσ (B_) = −2E  k kp 
p B̃ _(B_) + βg  + B̃ _(B_) + βg   .
jp̄βΦµ (B_)
p̄j p
p̄
j
j


(79)
Notice the system (79) has K equations, one for each security k. To satisfy all of them, there
must exist a constant λ such that



j
k
X
σ pg
σ pg j j
E  k kp  B̃ _(B_) + βg  = λ ∀k,
p̄j p
p̄

(80a)

j


~ = B k _(B_) such thatalong with a portfolio B_kX

B k _(B_) = B_.

(80b)

k


Let λ(B_) and B k _(B_) be the solution to the equations (80a) and (80b). Equations (80)corresponds exactly to the first order conditions of the problem minimizing the variance of
cash-in-hand given total asset holdings B_:min var {B k _}X B k _pk
k

β p̄k

!2 
+g



(81)

subject to the summing up constraint (80b) when 2βλ2 is the Lagrange multiplier on the constraint40

(80b). We now write equations (80) as a linear system of equations of the following form"C(~p, p~) 1
1|

#"
#
~
B_
−λ

0

=

#
"
−βC(~
p, g)

(82)

B_

where 1 is a K dimensional vector of ones. There are two possible types of solutions to (82).
If C(~
p, p~) is not of full rank, the minimization problem (81) has multiple solutions and further
the minimum is independent of B_. In these cases λ(B_) = 0 for all B_. The other case,~ as functions of B_. Define a scalarwhen C(~p, p~) is invertible, we can express λ(B_) and B_
η ≡ 1| C(~p, p~)−1 1, and using equation (82) we haveλ=

1
B_ + β1| C(~p, p~)−1 C(~p, g) ,
η

(83)



C(~
p, p~)−1 1B_
C(~p, p~)−1 11| C(~p, p~)−1−1~βC(~p, g) +
B_ = − C(~p, p~) −
η
η
−1

C(~
p, p~) 1= −βC(~
p, p~)−1 C(~p, g) +
B_ + β1| C(~p, p~)−1 C(~p, g) .
η(84)

Equation (84) gives us formula (26) in the text and pins down the steady state portfolio asa function of total assets B_. The second order Taylor expansion of B̃(g, p, B_; σ) with respectto σ around σ = 0 is
1B̃pk (B_)p̂k + B̃σσ (B_) + O(σ 3 ),2nowhere we have suppressed terms that are zero. Substituting for B̃g (B_), B̃pk (B_)
andB̃(g, p, B_; σ) = B_ + B̃g (B_)ĝ +XkB̃σσ (B_) we get
B̃(g, p, B_; σ) = B_ + βĝ +

X
k

p̂k
B k _(B_) k − λ(B_) + O(σ 3 , (1 − β)σ 2 ),p̄

where we drop the terms containing (1 − β) in equation (79) as we did in the K = 1 case.
Let B ∗ = −β1| C(~p, p~)−1 C(~p, g). Using equations (83) and (84), the approximated dynamics,
B̌(g, p~, B_; σ) are
∗

|

|



~
1 |
1
−1 p̂
∗
+ 1 C(~
p, p~)(B̌_−B )+ 1 −(B̌_−B ∗ ).
η
p̄~ η
p̄~
~

−1 p̂

B̌(g, p~, B̌_; σ) = B +βĝ−β1 C(~p, g) C(~p, p~)

ηηWe can replace 1 − η1 = 1+η+ O(σ 3 ) and 1+η
= 1 + O σ 3 to get,




1
η
p̂~
p̂~
B̌(g, p~, B_; σ) = B ∗ +βĝ−β1| C(~p, g)| C(~p, p~)−1 +1| C(~
p, p~)−1 (B̌_−B ∗ )+(B̌_−B ∗ ).
~
~
1
+
η
1
+
η
p̄
p̄
(85)
Using the stochastic process B̌t generated with B̌(g, p~, B_; σ), we can verify that the ergodic
mean for B̌(g, p~, B_; σ) is B ∗ and the speed of convergence is

41

η
1+η

. Applying the same steps

as in the proof of Proposition 2, we obtain the expression of the ergodic variance of B̌ as statedin Proposition 3, and mean, speed of convergence, variance of Žt .

7.4

Proof of Proposition 4

From the definition of R∗ , equation (33), for R ∈ R∗ there exists p(s) and B such that
R(s, s_) =

β

´

p(s)
π(ds0 |s_)p(s0 )

(86a)

Z(R)
1−β

(86b)

p(s)B + P Vg (s) =∀s.

We will show that the function B : S → R


ˆ
Z(R)
0
0
P Vg (s )π(ds |s)B(s) = β
−
1−β
s0 ∈S

(87)

is a stochastic steady state and then establish that B(s) = B ∗∗ (Z ∗ , s) where Z ∗ solves theminimization problem (31)
Using equations (86b) and (87), we obtainR(s, s_) =Z(R)/(1 − β) − P Vg (s).
B(s_)

Therefore,R(s, s_)B(s_) + g(s) = Z(R)/(1 − β) − P Vg (s) + g(s)ˆβZ(R)= Z(R) ++β
P Vg (s0 )π(ds0 |s)
1−β
0
s ∈S

(88)

= Z(R) + B(s).
Thus, for s_ = s̃ and B_ = B(s̃), the policy choices B(s) = B(s) and Z(s) = Z(R) satisfy
equation (28) and hence are feasible. We next show that this policy is also optimal.
Associated with the Bellman equation (27) define an operator T on the space of continuous,concave and bounded functions that map S × [B, B] → R as
T (V ) (s_, B_) =max{Z(s),B(s)}s∈S ,Z≤Z,B(s)∈[B,B]

E [(R (s) B − − B (s)) + γΨ (Z (s)) + βV (B (s) , s) |s_]subject to constraint (28). Standard arguments show that T is indeed an operator. Given
an initial value function V0 , define the sequence of value functions V1 , V2 , V3 , etc., as
T(V0 ), T(T(V0 )), T(T(T(V0 ))), etc. Let B̃n and Z̃n be the optimal policy functions associatedwith Vn .
Suppose Vn is continuous, concave, bounded function and differentiable function with its
derivative satisfying
βVn0 (s_, B(s_)) = γΨ0 (Z(R)) + 1 for all s_.

(89)

The stationary policy, B̃n (s_, B(s_), s) = B(s) and Z˜n (s_, B(s_), s) = Z(R) for all s_, s,42

satisfies the first order conditionγΨ0 (Z̃n (s_, B(s_), s)) = βVn0 (s, B̃(s_, B(s_), s)) − 1.
As the stationary policy is also feasible, we can conclude that it is optimal. Finally, Vn+1 = TVn
is concave and applying the envelope theorem (Benveniste and Scheinkman (1979)), satisfies
0βVn+1(s_, B(s_)) = γΨ0 (Z(R)) + 1 for all s_.

By induction, we can extend property (89) to all Vm with m ≥ n if it is satisfied by some Vn .
Choose V0 to be a linear function that satisfies (89), then B̃n (s_, B(s_), ) = B(s) for all n ≥ 1.
Problem (27) and V0 satisfy the assumptions of Theorem 9.9 in Stokey and Lucas (1989) andtherefore we can conclude that B̃n → B̃ pointwise. Hence,B̃(s_, B(s_), s) = lim B̃n (s_, B(s_), s) = B(s)n→∞for all s_.
Equation (87) and the definition in (29) implies B(s) = B ∗∗ (Z(R), s). From (88) note thatthat R(s, s_)B(s_) + P Vg (s) is constant with respect to s for all s_. Thus Z(R) solves thevariance minimization problem (31), concluding the proof of Proposition 4.

7.5

Proof of proposition 5

When gt and pt follow (34), the arguments in the last part of proof of proposition 2 in appendix
7.2 allow us to work with functions µ̃(B_, g_) and B̃(εg , εp , B_, g_), where B_, g_ be thedebt and expenditure level before shocks εg , εp are realized. These functions solve the following
optimality conditions (assuming interiority) for problem (27):

 

B_p
+ g = Φ µ̃ B̃(εg , εp , B_, g_), g + B̃(εg , εp , B_, g_),
β p̄
 


p
µ̃(B_, g_) = E µ̃ B̃(εg , εp , B_, g_), g g_ .
p̄

(90a)
(90b)

We parameterize εg = σg and εp = σσ pg p where variables (g , g ) are mean zero and with unit
standard deviations each and independent over time. We augment the functions µ̃(B_, g_)and B̃(g, p, B_, g_) with an extra argument σ: B̃(εg , εp , B_, g_; σ), µ̃(B_, g_; σ) and take a
Taylor expansion with respect to around σ = 0 and g_ = ḡ. We will use the notation (B_) asa shortcut for (ḡ, p̄, B_, ḡ; 0). Differentiating equations (90a) and (90b) with respect to B_ we
get,
B̃B_ (B_) = 1, µ̃B_ (B_) =1β −1

Φµ (B_)!
,

which are identical to the i.i.d case. Differentiating equations (90a) and (90b) with respect tog_ we find
ρ = Φµ (B_) µ̃g (B_)ρ + µ̃B_ (B_)B̃g_ + B̃g_,43

µ̃g_ (B_) = µ̃g_ (B_)ρ + µ̃B_ (B_)B̃g_ .
Substituting for µ̃B_ (B_) and B̃B_ (B_) obtained before and simplifying, we obtainρβ,
1 − ρβρβ1µ̃B_ (B_)B̃g_ =µ̃B_ (B_).
µ̃g_ (B_) =1−ρ
1 − ρβB̃g_ = (1 − ρ)

(91a)
(91b)

Differentiating with respect to g getB̃g (B_) = β

1−ρ
1 − ρβ.

The expressions for the derivativesB̃σ (B_) = 0, µ̃σ (B_) = 0,

B̃p (B_) =B_p̄

are same as in the i.i.d. case.
Next we discuss the second order terms. The expressions

B̃B_B_ (B_) = 0, µ̃B_B_ (B_) = −2
Φµµ (B_) µ̃B_ (B_)Φµ (B_)

are same as the i.i.d. case. We solve for the second order derivatives with respect to g_.
Differentiating (90) with respect to g_ and B_ we findih0 = Φµµ (B_) µ̃g_ (B_)ρµ̃B_ (B_) + µ̃B_ (B_)B̃g_ µ̃B_ (B_)
hi+ Φµ (B_) µ̃g_B_ (B_)ρ + µ̃B_B_ (B_)B̃g (B_) + µ̃B_ (B_)B̃g_B_ (B_)
andµ̃g_B_ (B_) = µ̃g_B_ (B_)ρ + µ̃B_B_ (B_)B̃g (B_) + µ̃B_ (B_)B̃g_B_ (B_).

(92)

Solving out for µ̃g_B_ from (92)iΦµµ (B_) hµ̃g_ (B_)ρµ̃B_ (B_) + µ̃B_ (B_)B̃g_ µ̃B_ (B_)Φµ (B_)iΦµµ (B_) h=−µ̃B_ (B_)µ̃g_ (B_) .
Φµ (B_)µ̃g_B_ (B_) = −We can substitute for µ̃g_B_ (B_) and µ̃B_B_ (B_) in (92) which, along with µ̃B_ (B_)B̃g_ =(1 − ρ)µ̃g_ (B_) implies thatB̃g_B_ (B_) = 0.
A similar procedure finds thatµ̃g_g_ (B_) = −2Φµµ (B_) µ̃g_ (B_) , B̃g_g_ = 0.
Φµ (B_)

44

As in the i.i.d. case it can be shown that the second (and cross) derivatives with respect to thep and g are all zero. The second order derivative with respect to σ is modified to"σ pg
B̃σσ = −2E
p
p̄



σ pg
1
p B_ + βp̄
1 − ρβΦµµ (B_)(1 − β)+βΦµ (B_)σ pgβ
p B_ +gp̄1 − ρβ2 #.

The second order Taylor expansion of B̃ around σ = 0, g_ = ḡ can then be written asB̃(εg , εp , B_, g_σ) = B_ + B̃g (B_)εg + B̃p (B_)εp + B̃g_ (g_ − g)1
+ B̃σσ (B_)σ 2 + O(σ 3 ),2gwhere we have dropped the terms that are zero. Define P Vg = 1−βρ
as the expected net present

value of g, and substituting for the expressions for derivatives computed above we obtainβ(1 − ρ)1 − ρβp̂ĝ + B_ − βcov (p, P Vg ) − B_σ 2p
p̄
"
#
2
2
σ
Φµµ (B_)(1 − β)βp
σ + 2βB_cov(p, g) + B_2 2+2βΦµ (B_)
1 − βρp̄B̃(εg , εp , B_, g_; σ) = B_ ++ O(σ 3 ),(94)

where cov (p, P Vg ) is the covariance under the ergodic distribution the shocks g, p. In the current
case, Eλ B ∗∗ (g) =σp
cov(p,P Vg )12
3
3
var(p) . We can replace 1 − σ p = 1+σ 2p + O(σ ) and σ p = 1+σ 2p + O(σ ) to

express (94) asλB̃(εg , εp , B_, g_; σ) =E B
+

∗∗

p̂
β(1 − ρ)+(g − g) + Eλ B ∗∗ +1 − ρβp̄11 + σ 2pp̂
(B_ − Eλ B ∗∗ )p̄1
(B_ − Eλ B ∗∗ ) + O(σ 3 , (1 − β)σ 2 ).
1 + σ 2p(95)

Inspecting the dynamics in equation (95) we see that the ergodic mean of the B̌t , the approxi1
mated process, equals EB ∗∗ (g_), and the speed of convergence to EB ∗∗ is given by 1+σ2.
p

We now derive the expression for the ergodic variance. Equation (95) can be expressed as,


β
p̂
1
p̂
∗
σg + B (g_) +(B_ − B ∗∗ (g_))
B̃(εg , εp , B_, g_; σ) − B (g) =2
1 − βρ
p̄
1 + σ p p̄1+
(B_ − B ∗∗ (g_)) + O(σ 3 , σ 2 (1 − β)).
1 + σ 2p∗∗

As E(B̃(εg , εp , B_, g_; σ) − B ∗∗ (g)) = 0, following the same steps as in proof for Proposition 2
in appendix 7.2 givesvar(B̌t ) =1 + σ 2p
σ 2p

!


var

β
p̂
σg + B ∗∗ (gt )1 − βρ
p

45


.

7.6

Proof of Proposition 6

Lemma 3 and bound Z̄ established in equation (62) applies to the case with transfers. The
bound Z̄ on total tax revenues, constraint (37) along with the bounds on B(s) imply that T (s)is bounded above. Substitute constraint (37) for Z(s) in the objective function (35) and split
the maximization problem by s to obtain the period gain function,maxR(s)B_ − B +T
1+nT ≤T̄ ,B∈[B,B]+ γΨ (E(s, B_) − B + T ) + ωU

T
1+n



Using strict concavity of U and Ψ and compactness of the choice set, standard arguments showthat V (·) is continuous, strictly concave and differentiable on B, B . The optimal choice forT (s) > 0 due to Inada conditions on U and ω > 0, and the upper bound will be slack for thesame reason the upper bound on Z(s) was slack in problem (12). Thus imposing interiority ofT (s), the first order conditions with respect to Z(s), T (s), B(s) areγΨ0 (Z(s)) = µ(s),
ω
Uc
1+n



T (s)
1+n



(96a)

1
+ µ(s) = 0,
1+n

(96b)

−1 + βV 0 (B(s)) − κ̄ (s) + κ (s) = µ(s),

(96c)

+

where µ(s) is the multiplier on constraint (37) and κ̄ (s) , κ (s) are the multipliers on the bound´´aries B, B . Let κ̄ = s π (ds) R (s) κ̄ (s) , κ = s π (ds) R (s) κ (s) , combining equation (96c)´with the envelope condition V 0 (B_) = β −1 + s π(ds)R(s)µ(s) and we get the martingale (63).
Problem (35) is a maximum of a concave and supermodular function in (B, Z, −T ) over the
pre-image of continuous, increasing and submodular function B + Z − T . Therefore by Corollary2(ii) in Quah (2007) the set of maximizers is increasing in E (s, B_). As E (s, B_) is increasing
in B_ for all s, and it is increasing in s if B_ ∈ B max , B , decreasing in s if B_ ∈ B, B min .
Thus functions B̃ (·, ·) , Z̃ (·, ·) and −T̃ (·, ·) are continuous and increasing in in B_ for all s,
decreasing in s for B_ ∈ B, B min , increasing in s for B_ ∈ B max , B . Lemma 2 extends
to this case, we have B̃t → B ∗ when R ∈ R∗ . As E(s, B ∗ ) = E ∗ is constant Z̃(s, B ∗ ) = Z ∗(and hence τ̃ (s, B ∗ ) = τ ∗ ) and T̃ (s, B ∗ ) = T ∗ . At B ∗ the steady state solves for µ∗ , Z ∗ , T ∗ thatsatisfyγΨ0 (Z ∗ ) = µ∗ ,
 ∗ 
ω
T
1
Uc
+
+ µ∗ = 0,
1+n
1+n
1+n
Z ∗ − T ∗ = E∗ − B∗.

(97a)
(97b)
(97c)

Substituting equation (97c) and (97a) in equation (97b) to eliminate µ∗ and Z ∗ we obtain
ω
Uc
1+n



T∗
1+n


+

1
= −γΨ0 (E ∗ − B ∗ + T ∗ ) .
1+n

46Differentiating with respect to ω
∂T ∗
∂ω

1
1+n Uc= −hωU[1+n]2 cc



T∗
1+n





T∗
1+n



+ γΨ00 (E ∗ − B ∗ + T ∗ )i.

∗Since Ucc < 0 and Ψ00 < 0, we get ∂T∂ω > 0. Next differentiating equation (97c) with respect to∗∗∂Tω we get ∂Z∂ω = ∂ω > 0.

7.7

Proof of Proposition 7

Equations (96a) and (96b) allow us to solve for I(s) = Z(s) − T (s) as a function of µ(s)0 −1I(µ) = Ψ



µ
1
1+n
−1
( ) − (1 + n)Uc
− −
µ .
γ
ω
ω

(98)

The equilibrium conditions for an interior solution to the planners problem can then be written
as
 

B_p
+ g = I µ̃ B̃(g, p, B_) + B̃(g, p, B_),β p̄  pµ̃(B_) = Eµ̃ B̃(g, p, B_) .
p̄

(99a)
(99b)

Note that the system of equations (99) exactly mirrors the system of equations (67) and (68)with Ψ replaced with I. Moreover the approximation B̌ in appendix 7.2 is independent of theproperties of Ψ, thus the ergodic distribution of B̌t will be identical to the representative agentcase. As before, we can approximate the law of motion of Iˇt using the budget constraint as
 σ 2p1−βIˇt − I ∗ =B̌t − B ∗ + (B̌t−1 − B ∗ ) + O(σ 3 )
β
β
with



∗

I =

1−β
β



B ∗ + g.

The law of motion for this process, and hence, the ergodic distribution is independent of ω.
Define the level of transfers when the government has a net tax revenue of I as T (I; ω).
Combine the first order conditions, (96a) and (96b) and substitute Z = I + T to express
transfers, T (I; ω) implicitly using
ω
Uc
1+n



T (I; ω)1+n

+ γΨ0 (I + T (I; ω)) = −

1
.
1+n

Differentiating with respect to ω we get,∂T(I; ω) = −∂ω

1
1+n UcωU(1+n)2 cc



T (I;ω)1+n

T (I;ω)1+n

+ γΨ00 (I + T (I; ω))ˇBoth Ucc < 0 and Ψ00 < 0, we can conclude that ∂T∂ω (I; ω) > 0. Thus Ťt = T (It ; ω) is increasing47

in ω. First order stochastic dominance follows immediately.28

7.8

Proof of Proposition 8

We begin by writing the sequential version of the t ≥ 1 recursive problem (42) asXXmax

ct (st ),Xt (st ),Wt (st )t≥1π(st |s0 )β t−1 Ust
ct (st ) + g(st )ct (st ),θ(st )(100)

s.t for all t ≥ 1

Xt (st ) = Wt (st ) + Φ(st , ct (st ))Wt (st ) =(101a)

Xt−1 (st−1 )p(st )Uc,t (st )βEt−1 [p(st )Uc,t (st )](101b)

given X0 = X_ and s0 = s_. Let X̃tS (st |X0 , s0 ), W̃tS (st |X0 , s0 ) and c̃St (st |X0 , s0 ) solve thesequential planner’s problem. The optimal policy functions X̃(s, X_, s_) and c̃(s, X_, s_)that solve the recursive planners problem (42) generate the solution to the sequential problem(100). For example: starting with X1S (s1 |X0 , s0 ) = X̃(s1 , X0 , s0 ), we can recursively define forall t > 1
S
XtS (st |X0 , s0 ) = X̃(st , Xt−1
(st−1 |X0 , s0 ), st−1 ).

(102)In order to characterize the stochastic steady states X and the set of payoffs that implement
them, we begin by solving the following relaxed problemXXmax

ct (st ),Xt (st ),Wt (st )t≥1π(st )β t−1 Ust
ct (st ) + g(st )ct (st ),θ(st )(103)

s.t. for all t ≥ 1
Xt (st ) = Wt (st ) + Φ(st , ct (st ))(104a)

Xt−1 (st−1 ) = βEt−1 [Wt (st )|st−1 ](104b)

for a given X0 and s0 .
Any sequence {ct (st ), Xt (st ), Wt (st )} that is feasible under constraints (101) isalso feasible under (104).29

We will denote the solution to the relaxed prob-

lem using X̃tR (st |X_, s_), W̃tR (st |X_, s_) andt
X̃tR (st |X_, s_), W̃tR (st |X_, s_) and c̃Rt (s |X_, s_)tc̃Rt (s |X_, s_).

Note

that

if

satisfy constraints (101a) and (101b)

then they will also optimal for the sequential problem (100).
Let β t π(st )µt (st ) be the Lagrange multiplier on equation (104a) . The first order conditions
28

Let x be a random variable and g(x) and h(x) be real valued functions. If g(y) > h(y) for all y, then g(x)first order stochastically dominates h(x)29Note that constraints (101a) and (104a) are the same any allocation that satisfies (101b) automaticallysatisfies (104b).

48

with respect to Wt (st ) and ct (st ) yield that µt (st , st−1 ) = µt−1 (st−1 ) = µ0 for all st , st−1 and thetR
optimal consumption c̃Rt (s |X0 , s0 ) = C̃ (st |X0 , s0 ) choice only depends on the current state st .

For a given X0 and s0 define∞X
W (s|X0 , s0 ) = E 
β j−1 Φ(sj , C̃ R (sj |X0 , s0 ))|s1 = s
j=1

andX (s|X0 , s0 ) = βXπ(s0 |s)W (s0 |X0 , s0 ).

s0The optimal choices for the planner are X̃tR (st |X0 , s0 ) = X (st |X0 , s0 ) and W̃tR (st |X0 , s0 ) =W (st |X0 , s0 ). Let st1 = (s1 , . . . , st ) Moreover for all t ≥ 1
X̃ R (s0 , st1 |X (s0 |X0 , s0 ), s0 ) = X̃ R (s0 , st1 |X0 , s0 )for all s0 (and likewise for c̃R and W̃ R ). If this were not the case then the policy R 1
if t = 1
ˆ R (st |X , s ) = X̃ (s |X0 , s0 )X̃0 0
X̃ R (st |X (s |X , s ), s ) otherwise
1
0 0
1
1
would be optimal and different from X̃ R , a contradiction.
0 ,s0 )
Define p∗ (s|X0 , s0 ) = U (WC̃(s|X. When p = p∗ equation (101b) is automatically satisfied
R (s|X ,s ))
c

0

0

t
0
0
by X̃tR (st |X (s0 |X0 , s0 ), s0 ), W̃tR (st |X (s0 |X0 , s0 ), s0 ) and c̃Rt (s |X (s ), s ), and thus these policies

are optimal for the sequential problem (100) when X0 = X (s0 ) and s0 = s0 for all s0 . Using themap between X̃(s, X_, s_) and XtR (st |X0 , s0 ) in equation (102), we obtainX̃(s, X (s_|X0 , s0 ), s_) = X (s|X0 , s0 )for all s_. Thus X is a stochastic steady state. The set of payoffs P ∗ that support a stochastic
steady state with risk aversion can be generated by varying the initial conditions X0 and s0
Lastly we construct an example to demonstrate such that prf (s) ∈ P ∗ : Suppose S = 2
1−σ

1+γ

shocks are i.i.d overtime, and U (c, l) = c1−σ − l1+γ . The productivity θ(s) = 1 and we orderg(s) such that g(2) ≥ g(1). Let cf b (s), lf b (s) s∈S be the first-best allocation that maximizes
 1−σ

fb
P
1+γ
E0 t β t c1−σ − l1+γ subject to the feasibility constraint (39). We will use Rf b (s) = Uc f(s)bβEUc (s)to denote the returns at the first best allocation. We show that the following condition−1g(2)1 − Rf b (2)>,g(1)
1 − Rf b−1 (1)(105)

is sufficient for existence of a X such that X̃(s, X ) = X and τ (s, X ) = τ ∗ for all s.
Let µ(s) be the Lagrange multiplier on equation (43). The first order conditions with respectto {c(s), X(s)} are49

c(s) : c(s)−σ − Φc (s, c)µ(s) = σ

X_c(s)−σ−1 µ(s) X_σc(s)−σ−1 X
π(s0 )µ(s0 )c(s0 )−σ−
βE[c−σ ]E[c−σ ]20s(106a)
X(s) : βV 0 (X 0 (s)) − µ(s) = 0(106b)In addition to these constraints we have the envelope condition
V 0 (X_) =

X

π(s)

s

µ(s)c(s)−σβE[c−σ ](107)

At X_ = X , imposing X̃(s, X ) = X we see that µ(s) = µ∗ satisfies equation (107) and(106b). For a given µ∗ , note that the right hand side of equation (106a) is zero and we can use(106a) and the resource constraint to solve c and l as functions of s, µ∗ . Denote these functions
as C(s, µ∗ ) and L(s, µ∗ ). Expanding Φc (s, c) = (1 − σ)c(s)−σ + (1 + γ)[c(s) + g(s)]γ , it is easy toverify from (106a) that τ (s) = 1 − [c(s) + g(s)]γ c(s)σ is independent of s and with slight abuseof notation, we use τ (µ∗ ) to denote the tax rate implied by C(s, µ∗ ) and L(s, µ∗ ). Let F(µ∗ ) be
defined as
F(µ∗ ) =Φ(1, C(1, µ∗ ) Φ(2, C(2, µ∗ )−R(1) − 1
R(2) − 1

Existence of steady state is equivalent to finding a solution to F(µ∗ ) = 0. To show this we need
the next lemma that allows us to order functions C(sµ∗ ) and L(s, µ∗ ) with respect to s.
Lemma 5. For all µ∗ such that τ (µ∗ ) < 1, the function C(s, µ∗ ) satisfies C(1, µ∗ ) > C(2, µ∗ )For τ (µ∗ ) < 1, the agents’ optimal labor choice satisfies(c + g(s))γ = (1 − τ (µ∗ ))c−σ .

(108)

The right hand side of equation (108) is decreasing in c. Furthermore, holding c constant wehave that the left hand of the equation (108) is increasing in g(s). Combining these two facts,
we obtain that C(1, µ∗ ) > C(2, µ∗ ).
Let µ1 be such that Φ(1, C(1, µ1 )) = 0.30 Using Lemma 2 we have that L(2, µ1 ) − L(1, µ1 ) <g(2) − g(1). Since τ (µ1 ) < 1, τ (µ1 ) [L(2, µ1 ) − L(1, µ1 )] < g(2) − g(1) and hence the marginalutility adjusted surplus is state s = 2, Φ(2, C(2, µ1 )) < 0. Combing this we have F(µ1 ) > 0.
The sufficient condition (105) implies that F(0) < 0. Since F is continuous we can apply theintermediate value theorem to conclude that there exists µ∗ such that F(µ∗ ) = 0.

30

Such a µ1 exists as long as it is feasible to raise revenues at least has high as g(1).

50Figure I: Using the quadratic approximation (red line) and a more accurate global approximator(black line), the top, middle, and bottom panels plot smoothed kernel densities (left side) anddecision rules (right side) associated with values of σ e = 0.001, 0.02, and 0.04. The right panel
displays policies B̃(s, B_)−B_ for states s that attain the extreme values for {g(s)} and {p(s)}.

51

Figure II: Marginal utility weighted assets held in the risk free bond (blue line) and “stock
market” asset (green line). Negative values implies that the government is shorting the asset.

52

Figure III: Fitted debt versus (H.P. filtered) U.S. debt [1947-2010]53

Figure IV: Ergodic distribution of effective debt Xt and tax rate τ t using 30000 paths of 15000
length each.

54

Figure V: Conditional mean path for effective debt, E0 Xt and tax rate E0 τ t obtained after
averaging across 30000 simulated paths.

55

Figure VI: The solid line is the conditional mean path for effective debt, E0 Xt after averaging across 1000 simulated paths. The dashed line is computed using the formula (X0 −tX ∗ ) 1+β 2 1var(R) + X ∗56

Parameter

Value

B̄

0.83 (0.004)

ρB,B

0.40 (0.13)

ρB,Y-0.33 (0.26)

ρB,Y _-0.62 (0.23)

Table I: OLS estimates for tax and debt policy rules. The numbers in brackets are standard
errors.

57

ParamValue

Moment

Model

Data

Log Output
σθ

0.020

std. dev

1.7%

1.70%

ρθ

0.160

auto corr

0.28

0.28

Returns
σp

0.05

std. dev

5.1%

5.02%

χp

0.650

corr with log Yt

-0.06

-0.08

G/Y
ḡ

0.230

mean

23%

23%

σg

0.040

std. dev

4.7%

4.7%

χg

-0.150corr with log Yt

-0.42

-0.41

Table II: Parameters and targeted moments in the competitive equilibrium with fitted U.S. tax
debt policies.

58

Moments

Ramsey Allocation

Tax Rate
Std. Dev.

0.5%

Mean

22.9 %

Debt/Output
Std. Dev.

20%

Mean

-6%

Half-life(years)

250

Table III: moments computed using 30,000 simulations of length 15,000 periods

59Effective debt: Xt

Using simulation

Using formula

Mean
Half life (years)
Std. deviation

-0.07
250
0.26

-0.06
257
0.26

Table IV: Ergodic moments and comparison with formula60

Moments

BenchmarkDatastd. dev

0.2%

0.7%

auto corr

0.97

0.24

std. dev

10%

3.3%

auto corr

0.95

0.33

Tax Rate

Log Debt

Table V: Short run: Moments computed using 3000 simulations of length 63 periods

61