Fiscal policy and debt management withincomplete markets
Anmol Bhandari
Minnesota

David Evans
OregonMikhail Golosov
Princeton

Thomas Sargent
NYU

Classical question in macro public …nance

How should the government manage its debt over the business cycle?

Classical question in macro public …nance

How should the government manage its debt over the business cycle?

What is the "right" level of public debt?
How quickly debt should be repaid?
How much debt and taxes should be used to respond to agg shock?

Classical question in macro public …nance

How should the government manage its debt over the business cycle?

What is the "right" level of public debt?
How quickly debt should be repaid?
How much debt and taxes should be used to respond to agg shock?

Renewed interest in the aftermath of 2008 crisis
Concerns that current debt levels are "too high" for rich countries...

Classical question in macro public …nance

How should the government manage its debt over the business cycle?

What is the "right" level of public debt?
How quickly debt should be repaid?
How much debt and taxes should be used to respond to agg shock?

Renewed interest in the aftermath of 2008 crisis
Concerns that current debt levels are "too high" for rich countries...
And "too low" (negative) for China, Norway,...

This paper

A theory of optimal public debt management
Ramsey planner with distortionary taxation and incomplete markets

Contribution: develop quadratic approximations that characterize
moments of the invariant distribution in closed form
Derive explicit formulas ("su¢ cient statistics") for the moments of
the invariant distribution

This paper

Most of the focus:
mean ("target") debt level
speed of reversion to the target
variance of debt in the invariant distribution

Key insight: optimal debt minimizes risk for the gov’t
Other questions that our framework addresses
what is the optimal composition of portfolio of gov’t debts?
how should gov’t debt respond to shocks?
how should government set taxes, transfers, tax rates over the cycle?

Results

Main formulas:target debt

=speed of convergence=cov (returns, de…cit)var (returns)
1
1 + β2 var (returns)

Here:
returns: MU-adjusted returns on gov’t portfolio of debts/assets
de…cit: MU-adjusted present value of primary de…citsSu¢ cient statistics: can be easily computed given observed dataCalibration: US 1947-2010

Optimal debt level keeping maturity constant:target debt level: -7% of GDP
speed of mean reversion: 250 years (half life)
std. deviation: 0.26

Tax rates are peristent and smooth
Taxes and debt have similar volatility in the data but are lesspersistentRelated literature
1. Complete markets: Lucas-Stokey, Chari-Christiano-Kehoe,
Angeletos, Buera-Nicolini
any debt level is optimal, all …scal hedging through (equivalent of)Arrow securities
hard to see how to achieve that with real world instruments

2. Incomplete markets: Barro, Bohn, Faraglia-Marcet-Scott,
Lustig-Sleet-Yeltekinmostly numerical, often for models with counterfactual returns
analytics (Barro): any debt level is optimal

3. Accumulate enough assets to never use taxes: Aiyagari et al (2002),Farhi (2010)
can get their results in the limit, knife-edge cases

4. Portfolio theory: Markowitz, Merton, ...
GE, benevolence, interaction of portfolio decisions with taxation

5. Nominal debt, possibility of default
have not studied, but our approach should work there too

The simplest modelContinuum of identical agents with preferences∞E0 ∑ βt ct
t =01 1 +γl1+γ t

No capital + exogenous gov’t expenditures
ct + gt = lt
Gov’t can use proportional tax τ t and trade with agents one-period
security (in zero net supply) at price qt with stochastic payo¤ pt
gt + pt Bt

1 = τ t lt + qt Bt

iid shocks for (gt , pt ) , Bt is in a compact set
Let Btqt Bt , Rt
pt /qt 1Characterization
Lemma

fct , lt , Rt , Bt , τ t gt∞=0 is a competitive equilibrium if and only if
flt , Bt gt∞=0 satis…es
1 +γ+Bt = Rt Bt
lt lt| {z }

1 + gt= τ t lt

Easier to express hours as a function of tax revenues ZZ
Ψ (Z )=l (Z ) l (Z )1 + γ
1l (Z )1 + γ1+γConsumption is a residual
ct = (1 + γ) Ψ (Zt ) + Rt Bt

1Bt

Ramsey problem in recursive form

Bellman equation (state s = (g , p )) :V (B ) =maxfZ (s ),B 0 (s )gE RB

B 0 + γΨ (Z ) + βV B 0subject toZ (s ) + B 0 (s ) = R (s ) B + g (s ) for all s|{z}E (B ,s )Policy functions B̃ (B, s ) , Z̃ (B, s ) , τ̃ (B, s ) induce optimumB̃t , Z̃t , τ̃ t t

Optimal policy

Monotonicity: B̃, Z̃ , τ̃ are increasing in E
Distortion smoothing:
V 0 B̃t = Et V 0 B̃t +1 + βcovt Rt +1 , V 0 B̃t +1Uniqueness: B̃t converges to a unique invariant distributionOptimal policy
Our goal: characterize properties of the invariant distribution
Amount of risk depends on debt level:
E (B, s ) = R (s ) B + g (s )Let B be the debt level that minimizes var (E (B, )) :Bcov (R, g )var (R )Let Z be the level of tax revenues that satis…es budget constraintin expectation
1 β
Zḡ +BβSpecial case: p and g are perfectly correlated

If corr (p, g ) =1 then E (B , s ) is independent of srisk is completely eliminated if Bt = B

Monotonicity of policy rules:BBB< B =) cov R ( ) , V 0 B̃ (B, )>0= B =) cov R ( ) , V0B̃ (B, )=0> B =) cov R ( ) , V0B̃ (B, )<0Euler equation and Martingale convergence theorem implyB̃t ! B , Z̃t ! Z , var (τ̃ t ) ! 0

Imperfect hedging

If shocks are imperfectly correlated, complete elimination of risk isimpossible, invariant distribution of B̃t , Z̃t is not degenerate
Our approach: take quadratic approximation of B̃ (B, s ) around B
as variance of shocks goes to zero
Simple linear policy rules
hiB̃ (s, B ) = B + β [g (s ) ḡ ] + β R (s ) β 1β2 var (R ) B

β2 cov (R, g ) + O ks k3 , (1β ) ks k2

Main result: moments of invariant distribution
Proposition: the mean, variance and mean reversion of B̃t , Z̃t
up to order O (ks k , (1 β)):The mean of the invariant distributionEB̃t = B , EZ̃t = Z
Speed of mean reversion
Et

1B̃t

B̃t
1B
B=Et

1Z̃tZ̃tZZ1=11 + β2 var (R )Variance of the invariant distributionvar (E (B ))var (R )var B̃t=var Z̃t= 0satisfy,

Intuition
Back to Euler equation:
cov Rt +1 , V 0 B̃t +1∝

cov (Rt +1 , Et +1 ) + O ks k3∝∂var (Rt +1 , Et +1 (B, )) + O ks k3∂Bvar (Rt +1 , Et +1 (B, )) is minimized at B = B :BBB< B =) cov (Rt +1 , Et +1 (B, )) > 0= B =) cov (Rt +1 , Et +1 (B, )) = 0> B =) cov (Rt +1 , Et +1 (B, )) < 0

The optimal policy is to revert to risk-minimizing position

Main insights

Target debt level: minimizes risktarget level is positive if cov (R , g ) < 0
target level is negative (accumulate assets) if cov (R , g ) > 0

Speed of mean reversion is determined by var (R )var (R ) = 0 implies debt is random walk as in Barro (1979)

The less hedging B o¤ers, the bigger the variance of the invariant
distribution isFor β close to one, var Z̃t and var (τ̃ t ) is close to 0 =) alladjustment to shock is done via debt

Reliability of approximations

Extensions

Richer asset structure
Persistence, other shocks
Risk aversion

Extension 1: richer market structure

Suppose there are K assets with arbitrary payo¤s, durationnote that …xed portfolio weights are isomorphic to one securityhiNotation: R = R 1 , ..., R K ; C [R, R] and C [R, g ] are covariances
matrices
assume that C [R, R] is non-singular

Risk-minizining total debt level and porfolio are(B , B )arg min var
B =1T B=

∑ Rk Bk + g

1T C [R, R] 1 C [R, g ] , C [R, R] 1 C [R, g ]Optimal portfolio with active debt management

Mean debt level:
E B̃t = B
Mean reversion:
Et

1B̃t

B̃t

1

B
B

=β 2 1T C [R, R] 1 1
1 + β 2 1T C [R, R] 1 1

Optimal portfolio:
Bt = B +C [R, R] 1 1
1T C [R, R] 1 1

B̃t + 1T C [R, R] 1 C [R, g ]Some insights

Optimal portfolio chosen to minimize risk
unlike Merton’s investor’s, no risk-return trade-o¤
gov’t benevolent + general equilibrium implies that not optimal tochase returns for gov’t

Speed of mean reversion is slower with more asset: can hedge risks
better when Bt 6= B
Higher debt Bt =) higher weight of securities with small var R k

Extension 2: persistent shocks

Suppose that shocks are …rst order Markov + TFP shocks θ +discount factor shocks
For any random variable x let"#PV (x; s ) = E∞

∑ β t xt s 0 = s .

t =0Optimal policy with persistent shocks

Optimal debt satis…esVt0 B̃t = Et Vt0+1 B̃t +1 + βcovt Rt +1 , Vt0+1 B̃t +1
Our quadratic approximations imply that in invariant distributioncov (R, PV (g ))EB̃tmean reversion:=ḡ covvar (R )1
1 + β2 var (R )

R, PV

1 +γ

θ γ

Intuition: risk minimization
Planner wants to minimize ‡uctuations in τ tPrimary de…cit, holding τ constant isXτ1 +γ

θ γ Zτ = gg

1 +γ

θ γ τ (11τ) γ

Mean level of debt B and τ related through budget constraint:
1ββ

B = ḡ

τ (111 +γτ ) γ Eθ γ

The mean of invariant distribution is risk-minimizing debt:
arg min var RB + PV Xτ (B )BBE¤ect from τ (B ) is second order:B

cov R, Xτ (B )var (R )for any B

Extension 3: Risk aversion

Same environment as extension 1 but utility isc1 σ
1 σl 1 +γ1+γNew implementability constraint
Uc ,t Bt + Uc ,t lt +Ul ,tlt
Uc ,tgt =pt Uc ,tUc ,tβEt 1 pt Uc ,t1 Bt 1

E¤ective debt and return

De…nee¤ective debt: Bt = Uc ,t Bt
e¤ective return: Rt = βEtpt 1Upct,tU c ,te¤ective primary de…cit: Xt = Uc ,t Xt

All can be written as functions of ct

Recursive problem
Bellman equation

V (B , s_) =max

fc (s ),X 0 (s )gE U

c (s ),c (s ) + g (s )
θ (s )+ βV (B , s ) js_

subject toB 0 (s ) = R (s ) B + X (s ) for all sSimilar to recursive formulation in quasi-linear case, same optimality
condition for e¤ective debt:
Vt B̃t = Et Vt0+1 B̃t +1 + βcovt Rt +1 , Vt0+1 B̃t +1

Risk-minimizing e¤ective debt

Planner wants to minimize ‡uctuations in τThe risk-minizing e¤ective debt isB̃ =cov (R, PV (X ))var (R)

Terms on the r.h.s. are endogenous but, up to the second order, do
not depend on τCan be easily computed without doing dynamic programing
Risk-free R =) R is when X is high =) optimal to hold negative
quantity of risk-free debt
Easy to generalize to K assetQuantitative exercise

Apply our analysis to the U.S. economy
Since formulas are approximation, also evaluate how well they doModel speci…cation

Preferences

1 3l
3
1 asset, return are matched to returns of the U.S. gov’t portfolio
3 shock process:ln cln θ tln gt
ln pt= ρθ θ t 1 + σθ eθ ,t= ln ḡ + χg eθ ,t + σg eg ,t= χp eθ ,t + σp ep,tCalibration
Target statistics:
dynamics of GDP
dynamics of returns to U.S. gov’t portfolio

Returns computed from budget constraint:(qt + pt ) Bt 1
Rt= Xt + qt Bt=)
market value of debtt + primary de…citt=market value of debtt 1

GDP and returns are endogenous, depend on tax policy. We estimate
τ t = (1ρτ ) τ t

1 + ρτ τ̄ + ρY ln Yt + ρYln Yt1

Model …tOptimal policy: computed and analytical

Correlation of returns and output is close to 0:
correlation with e¤ective returns is negative
accumulate assets

Variability of e¤ective returns is quite low, provides bad hedge
slow convergence to the mean
large variance of debt

Simple back of envelope
Run VARXtYt=AXt 1
Yt 1= Iβ 1A

LetαXαY+ εt1

10ThenPVt (X ) = αX X t + αY Yt
Risk minimizing e¤ective debt

B =cov (Rt , PVt (X ))=var (Rt )αY cov (Rt , Yt ) + αX cov (Rt , Xt )cov (Rt )Applying to the U.S. data

B =0.08

Comparison to the U.S. policy

Similar orders of magnitude
Debt in the U.S. too smooth, reverts to the mean too quickly

Conclusion

Portfolio theory for government assets
general equilibrium e¤ects
benevolence

Easily extend to other countries
open economy and accumulating foreign debt (e.g. China)
investing in stocks (e.g. Norway, sovereign funds)