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«Optimal Management of Indexed and Nominal Debt Robert Barro Department of Economics, Harvard University Cambridge, MA 02138-3001, USA E-mail: ...»

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4, 1–15 (2003)


Optimal Management of Indexed and Nominal Debt

Robert Barro

Department of Economics, Harvard University

Cambridge, MA 02138-3001, USA

E-mail: rbarro@harvard.edu

A tax-smoothing objective is used to assess the optimal composition of pub-

lic debt with respect to maturity and contingencies. This objective motivates

the government to make its debt payouts contingent on the levels of public

outlay and the tax base. If these contingencies are present, but asset prices of non-contingent indexed debt are stochastic, then full tax smoothing dictates an optimal maturity structure of the non-contingent debt. If the certainty- equivalent outlays are the same for each period, then the government should guarantee equal real payouts in each period, that is, the debt takes the form of indexed consols. This structure insulates the government’s budget constraint from unpredictable variations in the market prices of indexed bonds of various maturities. If contingent debt is precluded, then the government may want to depart from a consol maturity structure to exploit covariances among public outlay, the tax base, and the term structure of real interest rates. However, if moral hazard is the reason for the preclusion of contingent debt, then this con- sideration also deters exploitation of these covariances and tends to return the optimal solution to the consol maturity structure. The issue of nominal bonds may allow the government to exploit the covariances among public outlay, the tax base, and the rate of inflation. But if moral-hazard explains the absence of contingent debt, then the same reasoning tends to make nominal debt issue undesirable. The bottom line is that an optimal-tax approach to public debt favors bonds that are indexed and long term. c 2003 Peking University Press Key Words: Indexed debt; Nominal debt.

JEL Classification Numbers: H6

1. INTRODUCTION In standard macroeconomics, fiscal policy involves choices about expen- ditures, taxes, and debt issue. The kinds of public spending may be dis- tinguished with respect to their interactions with private decisions, for ex- ample, some public activities would influence private production and some would interact with households’ choices of consumption and leisure. The 1 1529-7373/2003 Copyright c 2003 by Peking University Press All rights of reproduction in any form reserved.


taxes may also be differentiated by types; levies may fall on labor income, capital income, consumption, bodies, and so on.

The fiscal authority also chooses its type of debt obligations. These choices include the maturity structure of the debt, whether to issue nominal bonds or bonds indexed to the price level or a foreign currency, whether debt payments should be contingent on other variables, such as government expenditures and the state of the business cycle, and so on. These kinds of decisions are less familiar as a part of macroeconomics, although some aspects have been studied by Lucas and Stokey (1983); Persson and Svensson (1984); Bohn(1988, 1990); Calvo and Guidotti (1990); Alesina, Prati, and Tabellini (1990); Giavazzi and Pagano (1990); Chari, Christiano, and Kehoe (1994); Missale and Blanchard (1994); and Marcet, Sargent, and Seppala (1996).

Optimal debt management can be thought of in three stages. First, if taxes are lump sum and the other conditions for Ricardian equivalence hold, as in Barro (1974), then the division of government financing between debt and taxes is irrelevant. Thus, the whole level of public debt will be indeterminate from an optimal-tax standpoint.

Second, if taxes are distorting-for example, because the amount paid depends on an individual’s labor income or consumption-then the timing of taxes will generally matter, as in Barro (1979). This consideration tends to motivate smoothing of tax rates over time and thereby can make determinate the levels of debt at various dates. However, this element does not pin down the composition of debt, say by maturity.

Finally, if there is uncertainty about levels of public outlay, the tax base (say aggregate consumption or GDP), and asset prices, then the kinds of debt that the government issues will matter. In particular, the government may want to smooth tax rates over states of nature, and this consideration may dictate an optimal structure of the public debt. As one example, it maybe desirable for debt payouts to be conditioned on the level of government spending. As another example, it may be possible to design the maturity structure of the indexed debt so as to insulate the government’s financing costs from shifts in riskless real interest rates.

The strategy in this paper is to assume that the government desires to smooth the path of taxes when confronted by a path of exogenous, but stochastic, outlays. Other analyses, such as Zhu (1992) and Barro (1995), show that this objective can be derived, under some conditions, from the more fundamental objective of expected utility maximization for the representative household. The analysis assumes that policymakers can make effective commitments about the form of future fiscal actions. Hence, unlike Lucas and Stokey (1983) and Persson and Svensson (1984), the debt composition is not set to ensure that policies are time consistent.




The real public outlay for period t is Gt. This outlay is exogenous and stochastic. The government sets its real tax revenue for period t at the value Tt. The precise nature of the taxes is left unspecified. However, these levies are assumed to be distorting in such a way that the policymaker wishes to minimize the overall expected deadweight loss, as given from the perspective of an initial date, time 0, by ∞ wj (Tj+1 − Tj )2 E0 (1) j=1 where the wj 0 are weighting factors. The idea here is that variations in taxes over time cause distortions that the government would like to avoid.

This objective will motivate smoothness in the Tj across time and states.1 If one think about levies on the tax base, such as income, consumption, or property, then distortions are likely to increase more than in proportion with the amount of taxes when expressed in relation to the tax base.

Therefore, Tt and Gt should be construed as ratios to the tax base. Uncertainty with regard to the tax base is analogous to uncertainty with respect to the level of public outlays, and a rise in Gt can be viewed alternatively as an increase in government expenditure or a decrease in the tax base.

Indexed public debt issued at time t pays the certain real amounts Bt1, Bt2,..., in periods t + 1, t + 2,... These payouts can represent coupons or principal. The real market prices of this debt at time t are Pt1, Pt2,....

These asset prices are taken to be exogenous and stochastic, although the model could be extended to allow the choices of debt policy to affect the asset prices.

The government will also wish, in this model, to issue debt with payouts that are contingent on the realizations of the Gt. The amount of this debt issued at date 0 and due at date t can be structured so that it pays off one unit less for each unit by which Gt exceeds its date-zero expectation, E0 Gt.2 This debt can also be set up so that it pays a (positive or negative) non-contingent amount at date t, expressed as β0t E0 Gt. This amount is assumed to be set so that the market value of contingent debt at date 0 is 1 The form of equation (1) is natural for consumption taxes in the absence of a laborleisure choice. In that case, distortions reflect only variations in tax rates over time, not the levels of tax rates. With a labor-leisure choice, terms involving the levels of consumption or labor-income tax rates would also appear. The tax-smoothing behavior considered below is sometimes optimal in this extended setting. See Zhu (1992) for a general discussion of the optimality of tax smoothing.

2 The debt therefore pays off badly when public outlays are surprisingly high or when the tax base is surprisingly low. The latter contingency is analogous to the GDP-linked bonds described by Shiller (1993).

4 ROBERT BARRO nil. That is, β0t is the premium (set at time zero) per unit of G-contingency.

The amount payable in each period t on the contingent debt is therefore

–  –  –

Since a high Gt represents bad times-because high public outlay and a low tax base will typically be associated with low consumption-and the contingent debt pays off badly at these times, the premium β0t tends to be positive.

The government can achieve perfect tax smoothing in this model, that is, it can minimize the sum in equation (1) by attaining T1 = T2 = · · · = T. First, the government issues the kind of G-contingent debt that has just been described. This issue effectively converts the path of uncertain ˆ ˆ outlays, G1, G2,... into a path of known outlays, G1 = E0 G1 (1+β01 ), G2 = E0 G2 (1+β02 ),.... This contingent debt issue ensures that the government’s tax smoothing will not be disturbed by surprises in the future levels of public outlays (and tax bases).

Second, the government has to manage its non-contingent debt to get the timing of taxes right, in other words, to ensure equal values of the ˆ Tt even when the certainty-equivalent outlays, Gt, vary over time. This problem would be simple if the future prices of non-contingent debt, Ptj, were known with certainty at date 0, that is, if riskless real interest rates were not subject to fluctuation. In that case, any maturity structure of the non-contingent debt — for example, one-period debt — could be used. The only concern, as in Barro (1979), would be to get the total quantity of debt issue correct in each period. However, this procedure does not work if the Ptj are subject to uncertainty. In this case, unanticipated shifts in these asset prices-and, hence, in the government’s refinancing costs-can affect the government’s budget constraint and thereby disturb the smoothing of taxes.

The quantities of non-contingent public debt of the various maturities at

date 0 must satisfy the constraint:

∞ B0j P0j = V0 (2) j=1

–  –  –

of G-contingent debt. If taxes are successfully smoothed, then the revenue in each period is the same value, T. If there is a gap between the full outlay and revenue in period 1, then the difference must be financed by noncontingent debt issue (plus or minus) at the prices, P1j, of non-contingent debt then prevailing. However, if each of these asset prices contains an independent random element, then any debt issue of this type will cause tax smoothing to fail, because the realizations of the asset prices will impact on required levels of future taxes.3 Hence, full tax smoothing requires a

balance between full outlay and revenue in period 1:

ˆ G1 + β01 = T (3)

–  –  –

One may to look at this answer, in terms of pure discount bonds, is that the maturity structure of the non-contingent debt has no holes:5 the government arranges the debt at the outset so that the real amounts to be paid in each future period (up to t = ∞) are the same.6 However, because of the discounting on future real payouts (that is, a declining time path of the P0j ), the current market value of the outstanding debt declines steadily with maturity.

From the standpoint of coupon bonds, the government should structure its debt as indexed perpetuities (consols).7 These issues pay a uniform and perpetual stream of real coupons but have no principal payments.

The prescription for consols may seem to entail a maturity structure of the public debt that is much longer than that observed in practice. However, when governments issue real bonds, the stated maturity-and, more pertinently, the average duration of the real payouts-tent to be long. For example, when Britain was on the gold standard in the eighteenth and nineteenth centuries, nominal obligations were effectively real. At that time, the public debt was mainly long term (funded) and often took the form of consols.8 TheU.S.debt issued under the gold standard before World War 4 If the one-period, non-contingent real interest rate is the constant r, then B0t = rV0.

5 Thisresult on the desirable maturity structure of the public debt therefore differs from the suggestion of Friedman (1959, p.63): I can find no valid argument for the present policy of issuing a wide variety of securities... The alternative suggestion follows...

Issue... debt in two standard forms, one short-term... the other moderately long-term, The short security might be a 90-day bill... The longer security might best be a consolthat is, a perpetuity... A less extreme break would be to make it, let us say, an eightor ten-year maturity. I do not myself believe that the precise maturity of the debt outstanding is of great significance.

6 Alesina, Prati, and Tabellini (1990) and Giavazzi and Pagano (1990) argue on different grounds-to avoid confidence crises-that similar amounts of public debt should come due in each period.

7 Lucas and Stokey (1983) argue that consol debt may also be desirable on time

–  –  –

I was also primarily long term; for example, most of the U.S. government bonds outstanding in 1916 had remaining maturities in excess of 20 years.9 Many developed countries have recently issued indexed bonds, and these securities tend to be long term. For example, the U.K. government has issued indexed coupon bonds with maturities as long as 38 years, which is nearly infinity. For other countries (as discussed in Bank of England [1995]), the issues of coupon bonds include Canada with 30-year maturity, Australia with 20 years, and Israel with 15 years. Sweden has issued discount bonds with maturities of 19 years — the duration of a consol would be 19 years if the real discount rate were around 5 − 1/2%. The United States, which began the issue of indexed bonds only in 1997, began with a 10-year maturity. More generally, the observed short maturity for public debt in modern times applies mainly to nominal bonds in the context of a paper monetary standard.10 Nominal debt is considered in a later section.

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