
==== Front
Proc Natl Acad Sci U S A
Proc Natl Acad Sci U S A
PNAS
Proceedings of the National Academy of Sciences of the United States of America
0027-8424
1091-6490
National Academy of Sciences

38451950
202318365
10.1073/pnas.2318365121
research-articleResearch Articleecon-sciEconomic Sciences415
Social Sciences
Economic Sciences
Managing government debt
Jiang Wei a 1
Sargent Thomas J. thomas.sargent@nyu.edu
b c 1 2
Wang Neng d e f 1 https://orcid.org/0000-0001-7895-6502

Yang Jinqiang g 1 https://orcid.org/0000-0001-9067-4970

aDepartment of Industrial Engineering and Decision Analytics, Hong Kong University of Science and Technology, Hong Kong 999077, China
bDepartment of Economics, New York University, New York, NY 10003
cHoover Institution, Stanford University, Stanford California 94305
dCheung Kong Graduate School of Business, Beijing 100738, China
eNational Bureau of Economic Research, Cambridge, MA 02138
fGraduate School of Business, Columbia University, New York, NY 10027
gSchool of Finance, Shanghai University of Finance and Economics, Shanghai 200433, China
2To whom correspondence may be addressed. Email: thomas.sargent@nyu.edu.
Contributed by Thomas J. Sargent; received October 24, 2023; accepted January 16, 2024; reviewed by Fernando Alvarez, William Fuchs, and Hanno Lustig

1W.J., T.J.S., N.W., and J.Y. contributed equally to this work.

7 3 2024
12 3 2024
7 9 2024
121 11 e231836512124 10 2023
16 1 2024
Copyright © 2024 the Author(s). Published by PNAS.
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

Significance

A government that starts out either owing or owning debt should trade risky Shiller GDP securities that hedge its primary surplus risk. The indirect utility functional from a government debt and tax control problem that maximizes the expected utility of a representative consumer is a government loss function that appears in widely used models of tax smoothing.

To construct a stochastic version of [R. J. Barro, J. Polit. Econ. 87, 940–971 (1979)] normative model of tax rates and debt/GDP dynamics, we add risks and markets for trading them along lines suggested by [K. J. Arrow, Rev. Econ. Stud. 31, 91–96 (1964)] and [R. J. Shiller, Creating Institutions for Managing Society’s Largest Economic Risks (OUP, Oxford, 1994)]. These modifications preserve Barro’s prescriptions that a government should keep its debt-gross domestic product (GDP) ratio and tax rate constant over time and also prescribe that the government insure its primary surplus risk by selling or buying the same number of shares of a Shiller macro security each period.

tax smoothing
Ricardian equivalence
risk premium
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pmcWe construct a normative model of taxation and government debt management that adds risks and opportunities to Barro (1)’s nonstochastic model of optimal tax-debt policies. We begin by deducing Barro’s government loss function as the indirect utility functional that emerges from maximizing the expected utility of a representative consumer. Having added risk to Barro’s environment, we also add markets in one-period ahead Arrow (2) securities that allow decision makers to insure those risks. Given the fiscal risks that it chooses to take, the government chooses to trade a single Shiller (3) security whose payoff is proportional to GDP growth.*

We adopt a setup close to one that Lucas (7, 8, section III) used to measure potential benefits from improving countercyclical macroeconomic policies. We incorporate insights of Hansen et al. (9), Alvarez and Jermann (10), and Barillas et al. (11) about the information about costs of business cycles that is contained in a stochastic discount factor (SDF) process. Like Lucas (7, 8, section III), an SDF process is determined outside our model. We specify the SDF process to be a simple generalization of the time-discount factor process assumed by Barro (1).

Our model of optimal taxation and debt management preserves the constant tax rate and constant debt-GDP ratio prescriptions of Barro (1), and adds the prescription that the government sells shares in a Shiller (3) claim to GDP. Our government uses a formula of Jiang et al. (12, 13) for discounting risk-free government debt. It adds a risk premium to the formula used by Blanchard (14), the ultimate source of which is shortfalls in the government’s risky primary government surplus stream as a sole source of backing for its risk-free debt.

The Setting

GDP {Yt} follows[1] Yt+1=expg−σ22+σεt+1Yt,

where εt+1∼N(0,1) is an i.i.d. process. Government expenditures Γt=γYt are perpetually proportional to Yt. Government debt B0 is due at time 0. Total tax collections Tt at time t≥1 are a measurable function of the history εt=[εt,εt−1,⋯,ε0]; T0 is an initial condition chosen by the government. A stochastic discount factor process {Mt} is determined outside our model and has multiplicative increments†[2] Mt+1Mt≡mt+1=exp−r+η22−ηεt+1,

where η is the price of GDP growth risk εt+1 and M0>0 is given. To price Arrow (2) securities, we can interpret the multiplicative increment m(εt+1) of the SDF process as an exogenous time t state-price density of a claim bought at time t that pays off at time t+1.‡ To understand the sense in which m(ε) is a density, where Φt(·) is the standardized normal cumulative distribution function, an Arrow security that pays off 1 unit of GDP at time t+1 whenever the realized GDP shock lies in interval (εt+1,εt+1+dεt+1) is priced at time t by m(εt+1)dΦ(εt+1). Let a function Π(·):(−∞,∞)→R represent a bundle of εt+1-contingent payoffs. The price at time t of bundle Π is ∫Π(ε)m(ε)dΦ(ε).§

It is useful to view claims on GDP as a security and to use the SDF process with increments defined by Eq. 2 to price this Shiller (3) macro security:[3] St=Et∑u=t+1∞MuMtYu=e−δYt1−e−δ,

where λ=ησ and δ=r+λ−g. The one-period gross return on this Shiller security is[4] Rt+1≡St+1+Yt+1St=expr+λ−σ22+σεt+1,

with expected returnEt[Rt+1]=er+λ.

Since the SDF process given in Eq. 2 implies that the price of a one-period risk-free bond is Et(mt+1)=e−r, λ=ησ is the risk premium component of the continuously compounded return (r+λ) on the Shiller security; it equals the price η of risk εt+1 times the Shiller security’s exposure to that risk: σ. To indicate the dependence of Rt+1 on εt+1, we’ll often write Rt+1=R(εt+1).

We assume that the government instead manages risks in a way recommended by Shiller (3). Appendix A shows that trading the Shiller security lets the government attain the same optimal outcomes as it can by trading Arrow securities.¶

Optimal Fiscal Policy

We provide a theory of how the government chooses stochastic processes for its tax and portfolio management policy {τt,Δt}t=0∞. We follow Barro (1) and assume that raising revenues Tt brings distortions measured by Θ(Tt,Yt), where[5] Θ(Tt,Yt)=θ(τt)Yt

and the scaled deadweight loss function θ(τ) is increasing, convex, and smooth. The positive derivative θ′(·) plays a key role in inducing the government to make total tax collections Tt be homogeneous of degree one in GDP so that primary surplus Tt−Gt also becomes homogeneous of degree one in GDP and subject to the same risk εt+1 that affects GDP growth. If it wants to issue risk-free bonds, the government must insure that risk. A way to do that would be to purchase or sell an appropriate package of one-period Arrow (2) securities. The Shiller security is that appropriate package because the government faces primary surplus risk that is perfectly correlated with risk in the price of the Shiller security at time t+1.#

At time t the government purchases Δt shares of the Shiller security. Consequently, starting with an initial risk-free debt balance B0, risk-free government debt {Bt} evolves as[6] Bt+1=erBt+er(γ−τt)Yt−ΔtRt+1−erSt,t≥0,

where (γ−τt)Yt is the government’s primary deficit at time t and −ΔtRt+1−erSt describes how the government’s purchase of Δt shares of the Shiller security at time t affects Bt+1.

Let Ct be time-t consumption of a representative consumer and let a one-period felicity function be U(C)=C1−ψ1−ψ, where ψ>0 is a coefficient of relative risk aversion.|| A benevolent government wants a tax-portfolio policy {τt,Δt}t=0∞ that maximizes an indirect utility function F0 of a representative household defined by[7] F0=max{Ct}E0∑t=0∞e−ρtU(Ct).

The household’s maximization is subject to the intertemporal budget constraint[8] E0∑t=0∞MtCt≤W0+E0∑t=0∞MtYt−Tt−Θt,

where W0 is the household’s initial exogenously endowed wealth that is unaffected by forces active in our model. The household takes SDF {Mt}, tax collection {Tt}, and deadweight loss {Θt} processes as given. By modifying a Lagrangian method used by Cox and Huang (15) to allow for taxes and deadweight losses, we can deduce[9] F0=(β(W0+X0))1−ψ1−ψ,

where β=1−exp−1−1ψr+ρψ+121−1ψ1ψη2−ψ1−ψ and[10] X0=E0∑s=0∞MsM0Ys−Ts−Θs.

To maximize the household’s utility function F0 given in Eq. 9 over a taxation-portfolio policy {τt,Δt}t=0∞, it suffices for the the government to choose a joint {τt,Δt}t=0∞ process to maximize the present value of the households’ payoffs Ys−Ts−Θs given in Eq. 10 subject to tax distortions Eq. 5, initial condition (B0,Y0) and the government budget constraint[11] B0≤E0∑u=0∞MuM0Tu−Γu.

Evidently, maximizing the households’ value given in Eq. 10 is equivalent to minimizing the present value of taxes Tt plus deadweight losses Θt; and both are equivalent to maximizing the representative consumer’s expected utility functional F0 defined in Eq. 7.

Before approaching this problem, we state:

Proposition 1 To recover findings of Barro (16), suppose that the τ(·) function satisfies θ′(·)=0 for all tax rates τ. Then, any taxation-portfolio strategy {τt,Δt}t=0∞ that satisfies the government’s budget constraint solves the government’s problem.

Proposition 1 is Barro (16)’s Ricardian equivalence theorem. To make an optimal taxation-portfolio strategy profile determinant, Barro (1) injected tax distortion function θ(·) with θ′(·)>0.

Now turning to the government’s problem in the presence of an increasing, convex, and smooth θ(·) function, we formulate Eq. 10 as a dynamic programming problem in which X0∗=P(B0,Y0) is the maximal attainable X0 that satisfies Eq. 10. Value function P(B0,Y0) satisfies the Bellman equation:[12] P(Bt,Yt)=maxτt,ΔtEtYt−τtYt−θ(τt)Yt+mt+1P(Bt+1,Yt+1).

Substituting Eq. 6 into Eq. 12 gives[13] P(Bt,Yt)=maxτt,ΔtYt−τtYt−θ(τt)Yt+Etmt+1PerBt+er(γ−τt)Yt−ΔtRt+1−erSt,Yt+1.

First-order necessary conditions for τt and Δt, respectively,** are:[14] 1+θ′(τt)=−Etmt+1erPB(Bt+1,Yt+1),

and[15] −Etmt+1(Rt+1−er)PB(Bt+1,Yt+1)=0.

Let bt=Bt/Yt denote the debt-GDP ratio. We guess and verify that P(Bt,Yt)=p(bt)Yt and that bt+1−bt=0 for all t. First-order condition Eq. 14 for tax rate τt implies[16] 1+c′(τt)=−p′(bt)=−p′(b0),

which equates the marginal cost 1+c′(τt) of taxing the household with the marginal benefit −p′(bt) of reducing debt. Eq. 16 implies that the tax rate τt is constant over time. Combining this outcome with the government’s budget constraint Eq. 6 implies[17] τt=(1−e−δ)bt+γ.

The optimal tax rate is constant over time and independent of the deadweight cost function θ(·). To verify bt+1−bt=0 for all t≥0, it is necessary that[18] Bt+1Yt+1=erBt+er(γ−τt)Yt−ΔtRt+1−erStYt+1=BtYt,

which implies:††[19] Δt=−1−e−δbt=−1−e−δb0.

If b0=0, then the government sets τt=γ and bt=b0 for all t≥0, there is no need to manage risk, holding of the Shiller security Δt=0 for all t≥0. If b0>0, then for the government to honor its debt, τt−γ>0 for all t≥0. Because it permanently runs a primary surplus at a rate proportional to Yt, its net liability Bt+1 is risky. So the government lowers the exposure to risk that Bt+1 presents by taking a short position in the Shiller security. Thus, it sets Δt<0 as prescribed by Eq. 19. Doing that makes the net risk exposure of (Bt+1−Bt)/Bt equal that of (Yt+1−Yt)/Yt, enabling the government to sustain bt=b0>0 for all t≥0. Symmetrically, if b0<0, the government permanently runs a primary deficit at a rate proportional to Yt. That makes the stock of Bt+1 less risky than the government wants, inducing it to take a long position in the Shiller security by setting Δt>0 as prescribed by Eq. 19. Doing that sets the net risk exposure of (Bt+1−Bt)/Bt equal to that of (Yt+1−Yt)/Yt, which allows the government to sustain bt=b0<0 for all t≥0.

The random vector [Rt+1,mt+1]⊤ is bivariate normal, so[20] Etmt+1Rt+1=er+λ−σ22Etmt+1eσεt+1=Etermt+1.

Outcome Eq. 19 and bt+1−bt=0 for all t≥0 confirm that the first-order condition Eq. 15 holds.

In summary, we have established:

Theorem 2 The optimal fiscal plan is described by bt=b0 and the following three equations:

Optimal tax rate: [21] τ(bt)=τ(b0)=(1−e−δ)b0+γ.

Optimal purchase of Shiller security: [22] Δt=Δ0=−1−e−δb0.

The GDP-scaled value X0/Y0 of after-tax, after tax-distortions GDP flowing to households: [23] p(bt)=p(b0)=1−τ(b0)−θ(τ(b0))1−e−δ.

Theorem 2 tells the government how to smooth taxes and to manage its debt.‡‡ Eqs. 21 to 23 can be solved recursively. Use Eq. 21 to compute a tax rate τ(b0) and then set τt=τ(b0) for all t≥0. This tax rate suffices to fund government expenditures and to service the government’s risk-free debt, including costs that arise from selling the Shiller security in the amount recommended by Eq. 22. The optimal Shiller security position Δt is negative and constant over time for b0>0. Finally, p(b0) in Eq. 23 is the (scaled) value of revenues after taxes and after deadweight losses from taxation flowing to households: p(b0)=X0/Y0, where X0 is defined in Eq. 10 under the optimal policy.

Who Owns GDP?

By applying some asset pricing formulas, we can summarize how an optimal government policy distributes costs and benefits. The time-0 value of GDP is[24] V0=E0∑t=0∞MtYt=Y01−e−δ.

Our “value distribution formula” is:[25] V0=P(B0,Y0)+B0+PV(Γ)+PV(Θ),

where

the value of after-tax, after tax-distortions GDP flowing to households is [26] P(B0,Y0)=E0∑t=0∞MtYt−τtYt−Θ(Tt,Yt)=1−τ0−θ(τ0)Y01−e−δ.

the value of risk-free government debt is [27] B0=E0∑t=0∞MtTt−Γt=(τ0−γ)Y01−e−δ.

the value of government spending is [28] PV(Γ)=E0∑t=0∞MtΓt=γY01−e−δ.

the value of deadweight taxation loss is [29] PV(Θ)=E0∑t=0∞MtΘt=θ(τ0)Y01−e−δ.

Evidently, after government expenditures, taxes, and debt-servicing costs are taken into account, three shareholders own GDP: private households own 1−τ0−θ(τ0) shares; government creditors own share (τ0−γ); and the government tax and spending authority, being a pass-through, owns and spends share γ+θ(τ0). All three owners hold cum-dividend shares of the Shiller security. The three claimants have equal priority.

Remark 3 An increasing, convex, and smooth deadweight cost θ(·) function induces the government to manage its exposure to GDP risk εt+1 by keeping both the tax rate τt and the debt-GDP ratio bt constant.§§ Investors are willing to hold risk-free one-period debt at a gross interest rate er. The government’s risky primary surplus process {(τ0−γ)Yt;t≥0} constitutes the ultimate “backing” behind all of its debts. To minimize deadweight costs of taxes, the government insures against primary surplus risk risk εt+1 by selling 1−e−δb0Yt shares of the Shiller security each period. The government pays a “portfolio management cost” in the form of the Shiller security’s risk premium λ per unit of time.

Remark 4 The government services Bt with the fiscal surplus stream Tt−Γt and by trading Δt shares of the Shiller security. The government could instead issue τ0−γ shares of the (cum-dividend) Shiller security at time 0 and use the proceeds to retire B0. By issuing shares in the Shiller security in that way, the government would in effect be selling a constant fraction of the country’s perpetual stream of output that it commandeers by taxing; these shares of the Shiller security would be fully backed by the government’s perpetual stream of fiscal surpluses. Consequently, government debt B0 has the same value and risk as τ0−γ shares of the (cum-dividend) Shiller security. Selling shares of the Shiller security in this way at time 0 would in effect be conducting a “debt-equity swap.”

Concluding Remarks

By allowing the government to trade either a complete set of one-period Arrow securities or a single Shiller (3) security, we have extended Barro (1) model in a way that preserves salient prescriptions: It is optimal for the government to keep its initial debt-GDP ratio constant forever and to levy a time-invariant tax rate sufficient to finance a constant ratio of its primary surplus to GDP. The government issues risk-free debt and sells a Shiller security each period. A Bellman equation discounts after-tax, after tax-distortions GDP flowing to households at a rate r+λ−g that includes the risk premium λ on the Shiller (3) security in addition to the r−g term that appears prominently in the analysis of Blanchard (14).

We have retained an assumption shared by Arrow (2) and Barro (1) that financial contracts are perfectly enforced and have withheld from government debt an additional “convenience yield” occasionally included in recent positive, as opposed to normative, macro-finance papers. In subsequent work (17), we plan to add features that bring our model closer to observed government debt/GDP series by letting a government default if it is willing to accept consequences; we will also endow government debt with a convenience yield. Adding those features promises to shed light on forces that can impart positive drifts to tax rates and to debt-GDP dynamics, features whose absence is a notable and sometimes counterfactual feature of Barro (1)’s normative model as well as ours.¶¶

Materials and Methods

Arrow Securities Can Replace Shiller’s Security.

Instead of using Shiller’s macro asset to insure GDP shocks, the government can support the same optimal outcomes for the tax rate and risk-free government debt processes by using a complete set of one-time-ahead state-contingent Arrow securities. Either financial arrangement supports outcomes described by Theorem 2.

As noted in Section 1, to obtain Π(εt+1) at time t+1 contingent on εt+1 being realized, the government would have to pay ∫Π(ε)m(ε)dΦ(ε) at time t. When the government trades Arrow securities, the counterpart to Eq. 6 for the evolution of risk-free government debt {Bt} is[30] Bt+1=erBt+er(γ−τt)Yt−Π(εt+1)+er∫Π(ε)m(ε)dΦ(ε).

Substituting Eq. 30 into Eq. 12 gives[31] P(Bt,Yt)=maxτt,ΠYt−τtYt−θ(τt)Yt+Et[mt+1PerBt+er(γ−τt)Yt−Π(εt+1)+er∫Π(ε)m(ε)dΦ(ε),Yt+1].

Choosing Π(εt+1) for each εt+1 to maximize P(Bt,Yt) is equivalent to choosing Π(εt+1) for each εt+1 to maximize[32] ∫[m(εt+1)PerBt+er(γ−τt)Yt−Π(εt+1)+er∫Π(ε)m(ε)dΦ(ε),Yt+1]dΦ(εt+1).

The first-order necessary condition for Arrow security demand Π(εt+1) in state εt+1 is[33] −m(εt+1)PB(Bt+1,Yt+1)dΦ(εt+1)+m(ε)dΦ(ε)|ε=εt+1∫m(εt+1)erPB(Bt+1,Yt+1)dΦ(εt+1)=0.

Once again guessing that P(Bt,Yt)=p(bt)Yt and that bt+1−bt=0 for all t to simplify Eq. 33, we confirm that 1=Etmt+1er.

First-order necessary condition for τt agrees with Eq. 14. Combining this outcome with bt+1−bt=0 for all t, we obtain Eq. 16 for τt. In conjunction with budget constraint Eq. 30, we obtain tax-smoothing result given in Eq. 21 To verify bt+1−bt=0 for all t≥0, we can show that for all εt+1:##[34] Bt+1Yt+1=erBt+er(γ−τt)Yt−Π(εt+1)+er∫Π(ε)m(ε)dΦ(ε)Yt+1=BtYt.

We thank Fernando Alvarez, V.V. Chari, Marco Bassetto, Anmol Bhandari, William Fuchs, and Hanno Lustig for helpful criticisms of earlier drafts.

Author contributions

W.J., T.J.S., N.W., and J.Y. designed research; performed research; and wrote the paper.

Competing interests

The authors declare no competing interest.

Data, Materials, and Software Availability

There are no data underlying this work.

Reviewers: F.A., University of Chicago; W.F., The University of Texas at Austin; and H.L., Stanford University.

*This is a consequence of a “dynamic trading” argument in the spirit of Harrison and Kreps (4), Black and Scholes (5), and Merton (6).

†This is a counterpart to and generalization of an assumption of Barro (1), who took a time-invariant risk-free interest rate as given.

‡We abuse notation by using εt+1 to represent both GDP shock and its realization.

§Consider a claim whose payoff is 1 for all εt+1 at t+1 so that Π(εt+1)=1. The time-t price of this claim equals ∫Π(ε)m(ε)dΦ(ε)=∫m(ε)dΦ(ε)=e−r.

¶That is, as long as the country can use the Shiller security to manage risk, there is no need to have a complete set of Arrow securities at each date t.

#Appendix A.

||The key result that the household’s utility maximization requires that the maximization of the market value of income flows holds for any well-behaved increasing and concave utility function.

**The second-order condition with respect to τt holds because θ(τ) is convex. The second-order condition with respect to Δt isEtmt+1(Rt+1−er)2PBB(Bt+1,,Yt+1)<0

because PBB<0, as we verify later.

††Substituting τt=(1−e−δ)bt+γ and Eq. 3 into Eq. 18, we obtain:Δt=erBt−er(1−e−δ)BtRt+1−erSt−BtRt+1−erStYt+1Yt=(1−e−δ)erBtRt+1−erYt−(1−e−δ)eδBtRt+1−erYteg−12σ2+σεt+1=−(1−e−δ)BtYteδ+g−12σ2+σεt+1−erRt+1−er=−(1−e−δ)bt.

‡‡To connect with findings of Barro (16) and Barro (1), note that when taxation brings no deadweight losses, i.e., when θ(τ)=0, Ricardian equivalence holds because p(b)=1−γ1−e−δ−b and p′(b)=−1 for all b.

§§The increasing, convex, and smooth θ(·) function plays a key role via −PB(Bt+1,Yt+1)=−p′(bt+1)=−p′(b0)=1+θ′(τ0)>1.

¶¶Studies in applied macroeconomics use a normative model like Barro’s to “rationalize” an observed government policy and thereby explain it. Various histories of national fiscal policies have used the Barro (1) model as a benchmark positive model. Sargent (18) compares various normative models as possible explanations of post-WWII inflation history.

##After substituting the tax policy Eq. 21 into Eq. 34, we obtain:er−δBt−Π(εt+1)+er∫Π(ε)m(ε)dΦ(ε)Ytexpg−12σ2+σεt+1=BtYt⇔er−δ−eg−12σ2+σεt+1Bt=Π(εt+1)−er∫Π(ε)m(ε)dΦ(ε)⇔−e−δR(εt+1)Bt−er∫−e−δR(ε)Btm(ε)dΦ(ε)=Π(εt+1)−er∫Π(ε)m(ε)dΦ(ε),

which impliesΠ(εt+1)=−e−δR(εt+1)Bt=ΔtR(εt+1)St,

where the second equality uses the expression for Δt given in Eq. 22.
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