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We characterize the optimal sequential choice of monetary policy in economies with either nominal or indexed debt. In a model where nominal debt is the only source of time inconsistency, the Markov-perfect equilibrium policy implies the progressive depletion of the outstanding stock of debt, until the time inconsistency disappears. There is a resulting welfare loss if debt is nominal rather than indexed. We also analyze the case where monetary policy is time inconsistent even when debt is indexed. In this case, with nominal debt, the sequential optimal policy converges to a time-consistent steady state with positive—or negative—debt, depending on the value of the intertemporal elasticity of substitution. Welfare can be higher if debt is nominal rather than indexed and the level of debt is not too high.

Fiscal discipline has often been seen as a precondition for price stability. Such is, for example, the rationale behind the Growth and Stability Pact in Europe. The underlying policy debate shows the concern regarding a timeinconsistency problem associated with high levels of nominal debt that could be monetized. In this paper we analyze the implications for the optimal sequential design of monetary policy when public debt is nominal and when it isindexed. We characterize the optimal sequential policy choices with both nominal and indexed debt and assess the relative performance of the two in terms of welfare. The model is a cash-in-advance production economy where agents start the period with predetermined money balances, which are used for transactions during the period, as in Svensson (1985). The government’s problem is to finance exogenous government expenditures in the least distortionary manner. In this economy, an increase in the price level decreases the real value of outstanding money and nominal debt and therefore reduces the need for distortionary taxation. However, this also induces a fall in present consumption because of the cash-in-advance constraint. As shown by Nicolini (1998), who analyzes the same class of economies, the incentives to inflate, or deflate, depend on preferences and on whether debt is nominal or real. If debt is indexed, the decision on whether to use the inflation tax, to tax today or tomorrow, hinges on the intertemporal elasticity of substitution. If the elasticity is one then it is equal to the implicit elasticity of the cash-in-advance constraint and the optimal plan is time consistent. However, with nominal debt, there is a reason to monetize the debt, and the optimal policy plan is no longer time consistent. We show that in a Markov-perfect equilibrium path the debt is asymptotically depleted, and therefore the path for the nominal interest rate is decreasing. In this case of unitary elasticity, the fact that debt is nominal rather than indexed introduces a dynamic distortion that lowers welfare unambiguously. For the general case of non unitary elasticity, the optimal policy plan is time inconsistent even with indexed debt. Optimal taxation principles dictate whether current or future consumption should be taxed more. In particular, if the intertemporal elasticity of substitution is higher than one—that is, higher than the implicit elasticity of the cashin- advance constraint—it is efficient to tax more current consumption; along a sequentially optimal path, indexed debt is depleted all the way to the first best, where it is negative and large enough in absolute value to finance all expenditures without the need to collect distortionary taxes. If the intertemporal elasticity is, instead, lower than one, future consumption is taxed more and debt increases asymptotically. With nominal debt, the incentives to inflate when debt is positive can compensate the incentives to deflate when the intertemporal elasticity is lower than one. Similarly, the incentives to deflate when debt is negative can compensate the incentives to inflate when the intertemporal elasticity is higher than one. At the debt level where these conflicting incentives cancel out there is a steady state. This stationary level of debt is negative for elasticity higher than one, and positive for elasticity lower than one. For different levels of initial debt, optimal sequential paths of nominal debt converge to this steady state. When the elasticity is different from one, in contrast with the unitary elasticity case, nominal debt solves—in the long-run—a time-inconsistency problem present in the indexed-debt case; in particular, if the elasticity is higher than one, there is no need to accumulate so many assets in order to achieve the first best, as in the indexed-debt case; if the elasticity is lower than one, debt does not increase asymptotically. A central contribution of this paper is the welfare comparison of the two regimes, nominal or indexed debt. If the intertemporal elasticity of substitution is one, indexed debt unambiguously dominates nominal debt in terms of welfare. In contrast, if the elasticity is non-unitary, the fact that the incentive to monetize the debt can compensate the distortions present with indexed debt can result in nominal debt dominating indexed debt. In particular, as our computations show—when debt is relatively low—nominal debt can be a blessing, rather than a burden, to monetary policy. Related work includes Calvo (1988), Obstfeld (1997), Nicolini (1998), Ellison and Rankin (2007), Martin (2006), Persson et al. (2006), and Reis (2006). Calvo (1988) addressed the question of the relative performance of nominal versus indexed debt, considering a reduced form model with two periods, where nominal debt creates a time inconsistency. There is an ad hoc cost of taxation and an ad hoc cost of repudiation that depends on the volume of debt. The focus of Calvo (1988) is on multiple equilibria, which result from his assumption on repudiation costs. With such a model, it is not possible to understand how debt, either nominal or indexed, can be used as a state variable affecting future monetary policy; how optimal equilibrium paths should evolve, or why different welfare rankings of indexed versus nominal debts are possible. Obstfeld (1997) and Ellison and Rankin (2007) assume that debt is real, and focus on monetary policy. They computeMarkov- perfect equilibria when the source of the time inconsistency of monetary policy is related to the depletion of the real value of money balances. Obstfeld (1997) uses a model where money balances are not predetermined and therefore must consider an ad hoc cost of a surprise inflation. Ellison and Rankin (2007) use the model in Nicolini(1998) with a class of preferences for which the level of real debt matters for the direction of the time inconsistency problem. Martin (2006) studies a version of the same model we analyze although in Martin (2006) in this paper, as well as in Díaz-Giménez et al. (2004), the government only issues nominal debt, not indexed. He provides an analytical characterization of the long-run behavior of Markov-perfect equilibria in the case of nominal debt, and shows that the long-run behavior depends on the intertemporal elasticity of substitution. Our paper analyzes and contrasts both types of debt regimes, providing a numerical comparison of Markov-perfect equilibrium outcomes, characterizing the equilibria and comparing the indexed and nominal debt regimes in terms of welfare. A different strand of related literature studies how optimal policies under commitment can be made time consistent by properly managing the portfolio of government assets and liabilities. The closest paper to ours in this literature is Persson et al. (2006).1 They use a structure similar to Nicolini (1998) and assume that the government can use both nominal and real debt as well as that there are no restrictions on debt being positive or negative. Although we use as benchmark economies with full commitment, our main focus is on Markov-perfect equilibria. In fact, the full characterization and computation of the optimal policy in such equilibria—with debt as a state variable—is an additional contribution of our work.2 Finally, there is a recent related literature on the characterization of the best sustainable equilibrium in similar optimal taxation problems, which also reaches the conclusion that optimal policies should, asymptotically, eliminate time inconsistency distortions (see, for example, Reis, 2006). The paper proceeds as follows: in Section 2, we describe the model economy and define competitive equilibria with nominal and indexed debt. In Section 3, we characterize the optimal allocations and policies under commitment, for the purpose of understanding the sources of time inconsistency. In Section 4, we analyze and compute the Markovperfect equilibria with indexed and nominal debt. Section 5 contains the main results of the paper: we compare the different regimes in terms of welfare. Finally in Section 6, we show that considering alternative taxes does not change the analysis, as long as taxes are set one period in advance.

This paper has discussed the different ways in which nominal and indexed debt affect the sequential choice of optimal monetary and debt policies. To this purpose, we have studied a general equilibrium monetary model where the costs of unanticipated inflation arise from a cash-in-advance constraint with the timing of Svensson (1985), and where government expenditures are exogenous. In our environment, as in Nicolini (1998), when the utility function is logarithmic in consumption and linear in leisure and debt is indexed, there is no time-inconsistency problem. In this case, the optimal monetary policy is to maintain the initial level of indexed debt, independently of the level of commitment of a benevolent government. In contrast, for the same specification of preferences, when the initial stock of government debt is nominal, a timeinconsistency problem arises. In this case, the government is tempted to inflate away its nominal debt liabilities. When the government cannot commit to its planned policies, to progressively deplete the outstanding stock is part of an optimal sequential policy. Optimal nominal interest rates in this case are also decreasing and converge asymptotically. In the rational expectations equilibria of our economies there are no surprise inflations. Still, for a given initial real value of outstanding debt, the sequential optimal equilibrium with indexed debt provides higher welfare. In this sense nominal debt can be a burden on optimal monetary policy. When we consider CRRA preferences with the intertemporal elasticity of substitution different from one, it is still true that in a Markov-perfect equilibrium the path of nominal debt converges to a stationary level of debt. However, it is not zero, but negative or positive depending on the intertemporal elasticity being greater or lower than one. With such general preferences, optimal sequential policy is time inconsistent even when debt is indexed. The interaction of the two sources of dynamic distortions, resulting from the differing elasticities and from nominal debt, can overturn the above efficiency result and it may actually be the case that nominal debt provides higher welfare than indexed debt. In fact, our computations show that, for relatively low values of debt, welfare is higher when debt is nominal. This is one more illustration of the principle that in a second best, adding a distortion may actually increase welfare. However, our computations also show that, for large levels of debt, indexed debt dominates in terms of welfare and therefore nominal debt is a burden to monetary policy. The introduction of additional forms of taxation further clarifies the interplay between the various forms of debt and commitment possibilities. Under the natural assumption that fiscal policy choices are predetermined, we have shown that the optimal policy problem has the same characterization, provided that the revenues levied through seigniorage are enough to allow for an optimal monetary policy with non-negative interest rates. If there is full commitment to an optimal fiscal policy, the fiscal authorities, anticipating monetary policy distortions, may choose to fully finance government liabilities and—provided the elasticity of substitution is greater or equalto one—the resulting monetary policy follows the Friedman rule of zero nominal interest rates. Moreover, this policy results in the equilibrium that obtains in the economy with full commitment with indexed debt, even if debt is nominal and the monetary authority cannot commit. Ours is a normative (second-best) analysis that takes into account the commitment problems which are at the root of institutional design in many developed economies (such as Central Bank independence, constraints on public indebtedness, etc.). As such, it throws new light on the ways in which the possibility of monetizing nominal debts can affect monetary policy (a central concern in policy design), and on how optimal debt and monetary policies should be designed. We do not claim that our results on optimal-equilibrium debt paths match—or should match—observed data. Still, it is the case that the prescriptions of our model could be used to provide a more detailed positive analysis of existing monetary policies and some insights on how monetary and debt policies should be redesigned if necessary.