Despite the best intuitions of policymakers and the towering contribution by Tobin (1963) to the issue of the optimal maturity structure of government debt, it is fair to say that until recently the subject has not attracted much attention in the mainstream economic literature. One probable reason for this surprising neglect is that under the assumptions of complete markets and policy precommitment—which until recent times were both important assumptions in mainstream economics—debt maturity does not change the set of equilibrium solutions. However, when either one of those key assumptions is relaxed, debt maturity has a role to play.
Lucas and Stokey (1983) forcefully brought the debt maturity issue to center stage. They showed that by relaxing the assumption of policy precommitment, debt maturity could be used to affect the incentives of future policymakers. In fact, they offered a special example in which a careful choice of debt maturity neutralized all the time-inconsistency problems, thus permitting decentralization of the present government’s first-best plan. This finding was a decided turnabout in the thinking on the role of debt maturity.
Persson, Persson, and Svensson (1987) extended the discussion to a monetary economy. They showed that in such a context the maturities of both indexed (to the price level) and nonindexed government debt mattered, even in a perfect-foresight context.1 The assumption of complete markets was relaxed by Calvo and Guidotti (1989a, 1989b); they assumed, instead, that bonds could be indexed to the price level, but not to any other characteristic of the state of nature, such as, government expenditure or the terms of trade.
So far, the results obtained with these models reflect two different approaches. Either proofs are offered for general propositions—with little insight into the characteristics of the associated debt maturity structure (for example, Persson, Persson, and Svensson (1987), Calvo and Obstfeld (1989))—or the results are based on more specific maturity structures, but are obtained in the context of special examples, as in Lucas and Stokey (1983) and Calvo and Guidotti (1989a). Both approaches have yielded useful insights, although more research is needed to close the gap between the two.
In the present paper we will pursue, in a more general context, some of the issues raised in Calvo and Guidotti (1989a). However, we concentrate on nominal debt—which is still the dominant form debt takes in practice—and set aside the question of indexation. For the sake of tractability, only the perfect-foresight case will be studied.
The model aims at capturing the implications of nominal debt maturity in the simplest possible setup. We assume three periods. The government in period 0 has a given stock of debt that has to be rolled over (that is, shifted to periods 1 and 2). Thus, assuming that nominal debt is the only form the debt can take, government 0 has the relatively simple task of having only to choose the maturity structure of its outstanding debt. As stated above, if there were full precommitment on the part of government 0—that is, if future governments were bound by the announcements made by government 0—then the maturity structure would be irrelevant. However, in the present paper we assume that government 0 has, at best, limited control over the behavior of future governments. Government 0 therefore has to take into account the incentives of future governments to “inflate away” inherited debt.
We show that the nature of optimal policy is heavily dependent on the type of precommitment. Thus, if, on the one hand, government 0 can precommit the actions of government 1, but neither one can precommit those of government 2, then, once again, maturity structure is shown to be irrelevant. However, the lack of perfect precommitment leads governments to accelerate the rate of debt repayment beyond what would prevail under full precommitment. We call that situation the case of “debt aversion.” This phenomenon, first noticed by Obstfeld (1989), arises because without full precommitment the cost of nominal debt is larger than the regular interest rate charges. With imperfect precommitment, a higher nominal debt leads to higher inflation due to the future governments’ futile (in equilibrium) attempt to inflate it away. This is the reason for the above-mentioned extra costs.
On the other hand, if government 1 can precommit the actions of government 2, but government 0 cannot place any constraint on any future government, then there exists a well-defined optimal maturity structure from the perspective of government 0. For example, if the demand for high-powered money is nil, we show that it is optimal to have only one-period debt. A positive demand for high-powered money induces a richer maturity spectrum, but the demand has to be sufficiently large to change the structure in favor of long-run debt. This is an interesting finding because it provides a rationale for the observed shortening of the maturity structure in several countries (see, for instance, Alesina, Prati, and Tabellini (1990)). This phenomenon is shown to be independent of debt aversion, because due to the assumption that government 1 can commit the actions of government 2, there is no extra social benefit associated with early repayment. Therefore, government 1 always finds it optimal to smooth conventional taxes completely over time, independently of the choice of different maturities by government 0.
Finally, the paper examines the case in which there is no precommitment. The analytical results are much less clear-cut here than in the previous situations, because debt aversion interacts with optimal maturity. The absence of precommitment on the part of government 1 introduces debt aversion whenever government 1 has to issue new debt. This occurs because the nominal debt issued by government 1 is part of the inflation tax base of government 2. The choice of different maturities by government 0 interacts with debt aversion by affecting the amount of new debt to be issued by government 1 and, hence, alters the time profile of conventional taxes. In addition, the choice of maturities affects the time profile of incentive-compatible inflation by altering the time profile of the inflation tax base. The problem is explored with the help of some numerical simulations. The main implication that emerges from our simulations is that lack of precommitment is associated with a more balanced maturity structure. The numerical simulations also suggest that optimal maturity lengthens in response to an increase in the stock of debt and to an increase in the demand for high-powered money, and shortens in response to an increase in government spending.
The paper is organized as follows. Section I presents the basic model and characterizes the equilibrium under full precommitment. Section II considers the first example of partial precommitment and shows how time inconsistency in government behavior distorts the intertemporal choice of conventional taxes leading to debt aversion. Section III presents the second example of partial precommitment and shows how time inconsistency introduces a role for debt maturity. Section IV analyzes the case of no precommitment where debt aversion interacts with optimal maturity. Section V contains the conclusions.
I. The Basic Model
Output value, at time i of nominal public debt issued in period i with maturity in period j is denoted by bij. The government is assumed to have a three-period horizon. In the last period (period 2), the government inherits a given stock of maturing nominal debt issued in the previous two periods. In addition to repaying the maturing debt, the government finances a constant (exogenous) flow of expenditure, g, by levying a distorting tax, x, and by using the revenue from inflation. Hence, the flow budget constraint in period 2, expressed in output values, is given by
where Iij denotes the nominal interest factor (that is, 1 plus the corresponding interest rate) between periods i and j, and ∏i denotes the inflation factor in period i (that is, Pi = P0∏1∏2, …, ∏i, where P is the price level), and S(∏) is the inflation tax on money balances. We assume throughout the paper that S(∏) = k(∏–1)/∏, where k denotes a constant (interest-inelastic) demand for money.2
In period 1, the government’s flow budget constraint is given by
Equation (2) states that the government may finance expenditure plus the amortization of debt issued in period 0 maturing in period 1 by resorting to conventional taxation, or to the inflation tax, or by issuing new nominal debt.
In order to focus sharply on the debt maturity issue, we assume that at period 0 the only decision faced by the government is choosing the maturity structure of a given initial stock of public debt, whose output value is denoted by b:3
where b0j (j = 1, 2) may, in principle, take negative as well as positive values. Positive values show a debt position.
The government’s optimal choice of instruments responds to the objective of minimizing the value of the following intertemporal cost function:
To characterize government behavior it is necessary to assume something about its ability to precommit future actions at any given point in time. The first-best situation is one in which the government at period 0 has the ability to fully commit future policies. In such a case, policies chosen at period 0 are, by defintion, time-consistent (Kydland and Prescott (1977) and Calvo (1978)).
In many instances, however, the government is unable to make credible precommitments; in such a case, policies that are optimal in a first-best situation in period 0 would not be optimal to follow in the future, and, hence, would be infeasible. In the present framework, a time-inconsistency problem arises because the existence of nominal debt induces future governments to attempt to partially repudiate those obligations through inflationary means. Such partial repudiation, is, however, just an illusion. In a world of rational agents, any incentive the government has to inflate away its nominal obligations is reflected in higher nominal interest rates at the time those obligations are contracted. Ex post, the nominal value of those obligations is predetermined and provides the basis for those incentives. But if individuals know the government’s incentives, the equilibrium interest rate would exactly cover them against the resulting opportunistic inflation.6
Since no real gains result from excess inflation, it is in the interest of the government to take into account, in period 0, those future incentives to liquidate nominal debt through inflation. Accounting for those incentives provides the basis for the formulation of time-consistent, or second-best, policies in a world where precommitment is not possible.
Before analyzing the government’s behavior under time inconsistency and characterizing the optimal role of debt maturity in formulating time-consistent policies, we discuss briefly the first-best case where complete precommitment on the part of the government is possible and where, therefore, policies chosen at period 0 are credible.
The intertemporal government budget constraint in period 0 is given by
Equation (5) is obtained by combining equations (l)–(3), taking into account that under perfect foresight, I0.1 = R∏1, I0.2 = R2∏1∏2, and I1.2 = R∏2 (recall that R is the real interest factor, which has been assumed to be constant over time). Equation (5) simply says that the government is constrained to make the present value of expenditure (including debt obligations) equal to the present value of taxes (including the inflation tax).
With full precommitment, the government minimizes social loss in equation (4), subject to budget constraint (5). It is obvious that, for this problem, the maturity structure of the public debt is irrelevant. Moreover, the first-best choice of tax and inflation rates implies Equations (6) and (7) imply that it is optimal to achieve perfect smoothing of taxes and inflation over time. Equation (8) implies that the marginal cost of inflation equals, at the optimum, the cost reduction from the (conventional) tax cut that is induced by the associated larger inflation tax.
In the following sections we discuss how the above problem is modified in the presence of time inconsistency of government behavior. As mentioned earlier, the potential for time inconsistency exists because the presence of nominal debt provides the government with an incentive to resort in the future to inflation, in order to reduce the real value of nominal debt obligations. Time inconsistency has two major implications: first, it alters the optimal intertemporal allocation of taxes, and second, it introduces a role for the maturity structure of nominal public debt.
II. Partial Precommitment I: A Case of Debt Aversion
Consider a simple case of partial precommitment where the government in period 0 is able to make commitments regarding period 1 variables but cannot precommit period 2 variables. This case implies that there are only two interesting decision points: at period 2 where x2 and ∏2 are decided, and at period 0 where the period 1 policy variables (that is, x1, ∏1, and bl.2), as well as the maturity structure of initial debt, are chosen. One way to think about partial precommitment is to consider a situation where government administrations are able to make precommitments during the periods in which they are in power, but cannot make precommitments about the policies of future administrations. This case could reflect a situation where there is a change of government at the end of period 1 and the incoming administration is in power in period 2.
The government in period 2 (government 2, for short), faces the budget constraint given in equation (1), where the only nonpredetermined variables are ∏2 and x2. Furthermore, since planning is done at period 2, the planner is not constrained by the perfect-foresight conditions that I0.2 = R2∏1∏2 and I1.2 = R∏2.7 Given the objective of minimizing the value of the cost function H(∏2) + V(x2), subject to equation (1), the optimal choice of period 2 inflation is given by
Equation (9) differs from equation (8), because the absence of precommitment of period 2 variables in previous periods leaves government 2 free to resort to inflation in order to reduce the output value of its obligations (recall that in period 2 the nominal value of outstanding debt, as well as the nominal interest rate factors, and are predetermined variables). In addition to the “genuine” revenue gain associated with a higher inflation tax on money balances, which also shows up in equation (8), the right-hand side of equation (9) takes into account the (conventional) tax cut associated with the fall in the real value of debt obligations maturing in period 2. The right-hand side of equation (9) shows that, from the perspective of period 2, the base of the inflation tax is not just high-powered money but also the stock of nominal debt obligations maturing in period 2.
The gains resulting from reducing the real value of government obligations are, however, an illusion. The market perceives the future incentive to inflate on the part of the government, and at the time the nominal debt is being issued, nominal interest rates reflect, point for point, future inflation. This implies that for government bonds issued in period 0, we have in equilibrium
Furthermore, the equilibrium interest rate for debt issued in period 1 satisfies
Government 0 can precommit x1, and ∏1 but cannot precommit period 2 variables. The formulation of its time-consistent policy takes into account that in equilibrium government 2 chooses ∏2 according to equation (9’), which shows that ∏2 is a function of the stock of nominal debt maturing in period 2. The problem in period 0 is to minimize social loss in equation (4), subject to budget constraint (5) and the incentive-compatibility constraint (9’). This problem involves only the choice of x1,x2, ∏1, and ∏2, and is entirely independent of the maturity structure of initial debt; that is, debt maturity is irrelevant. The first-order conditions for optimization imply
where µ is the Lagrange multiplier associated with the incentive-compatibility constraint (9’). Equations (12) and (13) imply that, compared to the first-best optimum, it is no longer optimal to completely smooth out taxes over time. In particular, equation (12) shows that altering the intertemporal distribution of taxes—which requires changing the amount of nominal obligations maturing in period 2—has an effect on period 2 inflation through the incentive-compatibility constraint. The left-hand side of equation (12) corresponds to the cost reduction from the cut in x1 associated with a marginal increase in b1.2. The first term in the right-hand side of equation (12) is the cost induced by the higher x2 called for by the increase in b1.2. The second term in the right-hand side of equation (12) is the effect on the incentive-compatibility constraint (9’). The multiplier µ, which we will show must be positive, represents the marginal social loss derived from increasing the gains from inflation as perceived by government 2.
Since the presence of nominal debt introduces a distortion by providing an incentive to attempt (in vain) to reduce its real value through inflation, it is reasonable to expect that, in a time-consistent equilibrium, the government in period 0 will find it optimal to reduce the amount of nominal obligations left in period 2 by raising tax revenues in period 1 relative to what was optimal under full precommitment. This phenomenon is what we call debt aversion.
To verify that the presence of nominal debt induces debt aversion, consider the simple case where k = 0; that is, S = 0 (the proof for the case where k > 0 is presented in Appendix I). If k = 0, we claim that x1 > x2, which implies the existence of debt aversion, since in the first-best optimum, x1 = x2 = x. To prove the claim, consider equations (9’), (12)-(14), and (5), where the terms involving the inflation tax on cash balances are set to zero. We will show by contradiction that x1 must be different from x2, and that x1 cannot be smaller than x2. The same argument also shows that µ > 0.
We will first show why, at the optimum, x1 ≠ x2. Suppose that x1 = x2. Then, from equation (12), this would imply that µ = 0, which also implies, from equation (14), that ∏1 = ∏2. However, the only case in which the equality between tax and inflation rates is consistent with equations (9’) and (13) is when x2 = g. This, however, is inconsistent with budget constraint (5), as long as b > 0 (as assumed), since tax revenues would suffice only to finance government spending.
Next, suppose that x2 > x1. Since V”(x) > 0, the latter implies V’(x2)>V’(x1). Thus, by equation (12)
First, if x2 > x1, it must be true that V’(x2) > 0 (that is, x2 > 0 to be able to raise revenue). Second, x2 > g, because otherwise tax revenues would not be enough to finance expenditures. Hence, µ<0. From equation (14), this implies that H’(∏2)<0, which in turn implies, from equation (9’), that x2<g. Again, this is inconsistent with the budget constraint, because we started with x1 < x2.
Therefore, the time-consistent equilibrium exhibits higher taxes in period 1 relative to period 2. Moreover, since when k = 0 the second-order condition for the choice of ∏2 requires b0.2R + b1.2>0, then, by equation (9’), at equilibrium, H’(∏2) > 0. Since ∏1, is at its first-best level (that is, where H’= (∏1), ∏2>∏1= 1.8
The fact that x1 > x2 implies that it is optimal to lower the amount of debt to be left for the government in period 2 relative to the case of full precommitment. The intuition is clear; by equation (9’), period 2 taxes are more costly than with full precommitment because they are linked to period 2 inflation. Therefore, if it was optimal to smooth taxes completely over time with full precommitment, now it is optimal to use lower period 2 taxes relative to period 1 taxes. The lower amount of debt maturing in period 2—because x2 is lower than its first-best level9—has the effect of reducing (but not eliminating) the future incentive to increase inflation above the first-best optimum. Notice that the distortion of the intertemporal distribution of taxes occurs even though the discount rate in the government’s objective function is equal to the real interest rate.10 Moreover, in this example, whereas the time profile of conventional taxes is downward-sloping, the opposite holds true for the time profile of inflation.
III. Partial Precommitment II: Optimal Debt Maturity
Assume that the government in period 1 has the ability to precommit period 2 policy variables,11 but let us also assume—in order to differentiate this case from the one in the previous section—that at period 0 the government is unable to make any commitments about future policies. This case could reflect a situation where there is a change of government at the end of the first period (that is, period 0) and the incoming administration stays in power for the following two periods.
Government 1 minimizes the value of cost function (4), subject to the following intertemporal budget constraint:
The first-order conditions for an interior optimum imply
As in the full precommitment case, equation (16) indicates that complete tax smoothing over time is optimal. This result is the consequence of a “separation” property that exists between the optimal choice of conventional taxes and inflation rates. This separation property derives from two facts. First, the cost function is separable in x and ∏. Second, the government is able to borrow (or lend) between periods 1 and 2, without being subject to a time-inconsistency problem. As a result, changes in the intertemporal distribution of taxes can be achieved through the use of b1.2, without at all affecting the choice of inflation rates.
It is important to notice, however, that whereas the above-mentioned separation property makes it optimal to completely smooth out taxes over time, as in the first-best optimum, the level of x differs from the first-best optimum, because as will be argued, the choice of ∏1 and ∏2 generally differs from that of the first-best optimum. However, if k = 0 (that is, there is no inflation tax on cash balances), then x would be the same as in the full precommitment case of Section I.
Equations (17) and (18), as was the case with equation (9) in Section II, reflect the fact that since no precommitment is possible in period 0, government 1 is free to resort to inflation in order to reduce the output value of its obligation (recall that in period 1 the nominal value of outstanding debt, b0.1 and b0.2, as well as the nominal interest rate factors, I0.1 and I0.2, are predetermined variables). Equations (17) and (18) state that, at the optimum, the marginal costs of increasing inflation in periods 1 and 2 are equated to a decline in the tax costs, due to the revenue gains of the associated inflation tax. The base of the inflation tax, however, includes debt obligations maturing in periods 1 and 2. In particular, whereas period 1 inflation reduces the real value of total government debt issued in period 0, period 2 inflation may only be used to reduce the real value of nominal debt maturing in period 2. This implies that, from the point of view of government 1, it is always optimal to set ∏1 greater than ∏2 whenever b0.1 is positive.
Since in equilibrium there are no revenue gains from excess inflation—that is, conditions (10) and (11) hold—the government’s budget constraint (15) reduces to equation (5); thus, at equilibrium revenue from inflation is given by the inflation tax on cash balances only.
It is interesting that unlike the examples in Sections I and II, the maturity structure of debt in period 0 is no longer irrelevant. The maturity structure of nominal debt has a role to play because it influences the optimal choice of inflation rates in periods 1 and 2 by government 1 (which by assumption, is able to precommit future policies).
Dividing (18’) by (17’), and assuming away division by zero, the following expression obtains:
Equation (19) implies that, if b0.2R <(=)b, then ∏1 > (=) ∏2. This results from the fact that f(∏) = ∏[H’(∏)–V’(x)S’(∏1)] is an increasing function, given that H”(∏)>0 and S”(∏)<0.
Hence, a change in the maturity structure that increases b0.2 / b generates a fall in x and ∏1 and an increase in ∏2. This implies that the increase in b0.2/b has two effects. First, it induces a substitution between ∏1 and ∏2; more specifically, there is a stronger incentive to use ∏2 relative to ∏1. Second, the substitution of ∏2 for ∏1 generates an increase in the present value of seigniorage, S(∏1) + R-1 S(∏2), generating a fall in x.13
An increase in both initial debt, b, and government expenditure induces government 1 to use the three instruments—that is, x, ∏1, and ∏2—to meet the higher revenue needs. In addition, both the increase in b and the increase in x raise the incentive for government 1 to use inflation.
The time-consistent choice of the optimal maturity structure of the initial stock of public debt by government 0 responds to the objective of minimizing social loss in equation (4), subject to the incentive-compatibility constraints (20a)–(20c).
A few things can be said on an intuitive level before looking at the general solution of this problem. Suppose that k = 0, In this case it can be seen that equations (5) and (17’) determine x and ∏1 independently from b0.2/b; namely, issuing long-term debt affects only the choice of inflation in period 2. Since the tax base for period 1 inflation is the total stock of debt, ∏1 is independent of the debt maturity. Moreover, since in equilibrium no revenue gains are associated with inflation, the optimal choice of maturity is clear: government 0 should issue only short-term debt (that is, b0.2 = 0). Issuing long-term debt would only raise the inflation tax base in period 2—resulting in higher period 2 inflation—with no effect on the choice of x and ∏1.
One important aspect of this solution should be noted. In this model debt has two functions. First, it is used to smooth out conventional taxes and inflation over time. Second, it affects the future incentive to resort to inflation. From the point of view of tax smoothing, given that short-term debt can be issued in every period, the possibility of issuing long-term debt adds nothing to the opportunity set. From the point of view of the incentives to inflate, since the government in period 1 can issue b1.2 without being subject to the problem of time inconsistency, long-term debt is always dominated by the alternative of issuing successively the equivalent amount of short-term debt. Hence, the solution of setting b0.2 = 0 reflects the principle that it is optimal to use the comparative advantage that the government has in period 1 to make precommitments about ∏2.
Let us tackle the more general case of k >0 and consider intuitively why it is no longer optimal in period 0 to issue only short-term debt (that is, it is no longer optimal to set b0.2 = 0). Assume for the sake of argument that R = 1. If all debt is issued with short-term maturity, it can be seen from equations (5), (17’), and (18’) that, compared to the first-best optimum, ∏1 is “too high,” while x and ∏2 are “too low.” These observations reflect the fact that the incentive to inflate boosts up the period 1 (conventional) inflation tax relative to its first-best level, and, thus, lower levels of x and ∏2 are needed to balance the budget. In addition, from the intratemporal first-order conditions (17’) and (18’), it can be seen when b0.2 = 0, that whereas the cost of ∏1 exceeds the true associated gains (that is, H’(∏1) > V’(x)S’(∏1)), this is not the case for ∏2 (where the above relation holds with equality). Therefore, given the convexity of the cost function, an increase in ∏2 coupled with a fall in ∏1, could improve welfare. This is precisely what an increase in b0.2 (from b0.2 = 0) does; it induces the government to increase ∏2 and to reduce ∏1 and x. The fall in x and ∏1 occurs because when k >0, the increase in ∏2 has a positive effect on government revenues that was not present in the case where k = 0. This shows that the presence of a positive demand for money provides the link between changes in ∏2, induced by different debt maturity structures and changes in x and ∏1. When k = 0, we have shown that debt maturity could not be used to affect x and ∏1.
Two important points follow from the above discussion. First, it is optimal to issue both short- and long-term debt. Second, since when k = 0 it was optimal to set b0.2 = 0, the presence of a positive (conventional) inflation tax base lengthens the optimal maturity of government debt.
The first-order condition that, along with the set of incentive-compatibility constraints (20a)-(20c), characterizes the optimal time-consistent policy equilibrium is given by
Equation (21) shows that, at the time-consistent equilibrium, H’(∏2) > V’(x)S’(∏2) from equation (18’), and H’(∏1) / H’(∏2) > S’(∏1)/ S’(∏2) from equations (17’) and (18’). These inequalities imply that 0 < (b0.2/b)°< 1, where (b0.2/b)° denotes the optimal debt maturity structure.14
IV. Optimal Debt Maturity Under No Precommitment
In contrast to Sections II and III, we now assume that no government has the ability to precommit future policies. Therefore, the formulation of a time-consistent policy in period 0 requires that all future reaction functions with respect to present policies be taken into account. An interesting feature of this example is that it combines the role of debt maturity with the issue of debt aversion discussed in Section II.
In period 2 the government’s behavior is analogous to that described in Section II. Government 2 minimizes the value of cost function H(∏2) + V(x2), subject to equation (1), where the only nonprede-termined variables are x2 and ∏2. The optimal choice of period 2 inflation is given by equation (9), which reflects the government’s incentive in period 2 to use inflation to reduce the real value of nominal debt obligations maturing in period 2. Since market interest rates fully reflect equilibrium inflation, the choice of ∏2 is given by equation (9’) from Section II, which for convenience is reproduced below:
Government 1 can choose ∏1, x1, and b1.2, but cannot make commitments about period 2 variables. The formulation of its time-consistent policy takes into account the fact that government 2 decides ∏2 according to equation (9’). It is important to note that the reaction function of government 2, given by equation (9’), is affected by the nominal debt issued by government 1, b1.2.
The time-consistent policy for government 1 minimizes the value of cost function (4), subject to the intertemporal budget constraint (15) and the incentive-compatibility constraint (9’) for government 2. By substituting equations (1) and (2)—where condition (11) holds—for x1, and x2 in the cost function and in the incentive-compatibility constraint (9’), the problem reduces to the choice of bl.2, ∏1, and ∏2 to minimize social loss in equation (4), subject to equation (9’).
The first-order conditions for b1.2, ∏1, and ∏2 are given by
First-order conditions (22)-(24) boil down to the first-order conditions under the case of partial precommitment in Section III when µ equals zero. The analysis in Section II, however, is useful for interpreting the additional terms appearing in equations (22)-(24), which represent the effects of the decision variables on the incentive-compatibility constraint (9’). Equation (22), which corresponds to the choice of b1,2, is the same as equation (12). The first term reflects the cost reduction from the tax cut in period 1 made possible by the increase of bl,2. Similarly, the second term in (22) reflects the cost from the higher period 2 conventional taxes necessary to finance the repayment of bl,2. The third term, where µ. is the Lagrange multiplier associated with equation (9’), is the effect of borrowing on the government’s incentive to inflate in period 2; namely, higher taxes associated with higher b1,2, as well as the larger stock of nominal obligations in period 2, increase the government’s incentive to raise future inflation.
The first and third terms in equation (23) represent the cost reduction from the tax cut allowed by the fall in the real value of nominal obligations maturing in period 1 associated with an increase in ∏1. The second term is the direct cost of increasing ∏1. The third term in equation (23) reflects the effect of ∏1, via a change in x2, on the incentive-compatibility constraint.
The first term in equation (24) is the direct cost of increasing ∏2, and the second term reflects the tax cut associated with the reduction in the real value of public debt maturing in period 1. The remaining terms represent the effects of changes in ∏2 on the incentive-compatibility constraint.
Since the government’s incentive to resort to inflation in period 1 is recognized by the market, the nominal interest rates applying to debt issued in period 0 satisfy condition (10). Using (10), the first-order conditions (22)–(24) can be written as
Equation (26) indicates that, as with the first case of partial precommitment discussed in Section II and unlike the second case described in Section III, it is usually not optimal to smooth taxes completely over time. As noted in Section III, when government 1 was able to make precommitments, the result of complete tax smoothing was dependent on the government’s being able to issue b1.2 without a time-inconsistency problem. Therefore, changes in the intertemporal distribution of taxes could be supported by borrowing (or lending) without affecting the choice of inflation rates. In the present case, changes in bl.2 affect the choice of ∏2, as can be seen from the breaking down of the separation result encountered in Section III by the incentive compatibility constraint (9’).
Further analysis of equation (26) shows that the intertemporal distribution of taxes under no precommitment is directly related to whether the government borrows or lends in period 1. Using equation (9’), equation (26) can be written as:
where (SOC} > 0 is the second-order condition for the choice of ∏2 in period 2. Since the right-hand side of (26’) is positive, the following relationship obtains:15
if and only if
b1,2 ≥ 0.
The above inequality indicates that the only time-inconsistency problem that matters to government 1 for altering the intertemporal distribution of taxes is the one concerning b1.2. In particular, if b1.2 > 0, government 2 is provided with an additional (to b0.2) incentive to increase ∏2. As a result, if without the time-inconsistency problem it was optimal for government 1 to choose x1 = x2, now government 1 internalizes part of the cost of period 2 inflation and finds it optimal to increase taxes in period 1 and decrease taxes in period 2 to reduce its borrowing in period 1. The opposite reasoning applies if b1.2 < 0, since a negative b1.2 provides an incentive to deflate.
The possibility expressed in equation (27) that in equilibrium period 1 conventional taxes could be lower than period 2 conventional taxes may appear quite surprising if one expects the debt-aversion result discussed in Section II to carry over to the present case. In particular, if k = 0, then x1 < x2 implies that it is optimal to postpone, instead of anticipating, tax revenue collection. The underlying intuition, however, is clear. Consider the differences between the problem government 1 faces here and the one that it faced in the earlier case (Section II). The only difference lies in the government’s budget constraint, which in Section II was given by equation (5) while in this section it is given by equation (15). To sharpen intuition let us focus on the case where k = 0. In the case detailed in Section 11, debt aversion always occurs because x2, being positively linked to ∏2 through equation (9’), is more costly than x1 relative to the first-best case. Moreover, if k = 0, then budget constraint (5) is independent of ∏1 and ∏2. In the problem studied in this section, ∏2—in addition to being linked to x2 by equation (9’)—affects budget constraint (15) by altering the real value of nominal debt maturing in period 2. From government 1’s perspective, this effect, which exists only if b0.2 > 0, goes in the direction of making ∏2, and therefore x2, less costly. Hence, depending on the amount of long-term debt inherited by government 1, it is perfectly plausible to encounter a situation in which the debt aversion result is overturned because government 1 has a sufficiently large incentive to raise x2 (and ∏2) relative to x1. It is interesting to note that, by equation (27), we know that government 1 has debt aversion only when it wants to issue new debt.
To study the optimal choice of the maturity structure of initial debt by government 0, let us examine how conventional taxes and inflation rates respond to changes in debt maturity in the reaction functions summarized by equations (28a)-(28d). Notice that equation (9’) implies that ∏2 and x2 always move together. Similarly, equation (25) implies that ∏1 and x1 also move together. Moreover, the government’s budget constraint (5), along with the above two statements, implies that ∏2 and ∏1, must move in opposite directions. The intuition behind these relationships should be clear by now. Therefore, the only relationship that needs to be understood is that between b0.2 and ∏2. Interestingly, the effect of b0.2 on ∏2 entails a relationship between governments 0 and 2, through government 1. An increase in b0.2 may not necessarily imply a higher incentive to inflate for government 2 if government 1 responds by reducing b1.2 enough to generate a fall in the inflation tax base of government 2. If an increase in b0.2 is not offset by a reduction in b1.2, and results in a higher incentive to inflate for government 2, then it generates an increase in ∏2, and, by previous considerations, a fall in ∏1, an increase in x2, and a fall in x1.17 In Appendix II we show that if the incentive-compatibility constraint (9’) is linearized—that is. only first-order effects are taken into account—it is always the case that a higher b0.2 induces an increase in ∏2.
The optimal maturity structure for the initial stock of government debt is the one that minimizes social loss in equation (4), subject to the incentive-compatibility constraints (28). The first-order condition, which along with (28) determines the optimal maturity structure, (b0.2/b)*, is given by
To investigate how optimal debt maturity depends on exogenous variables like b, k, and g, we numerically simulate the model. For the numerical example we use the following quadratic cost function:
The results are summarized in Table 1. The benchmark case is characterized by the following parameter values: the initial stock of debt, b, is assumed to be equal to 20 percent of gross national product (GNP), the demand for high-powered money, k, equals 10 percent of GNP, and government expenditure, g, equals 40 percent of GNP. The real interest rate (as well as the discount rate in cost function (30)) is assumed to be equal to zero; that is, R = 1.
The simulations show that it is optimal to issue both short- and long-term debt. In the cases considered, optimal debt maturity calls for a share of long-term-to-total debt of between 30 and 45 percent. The resulting inflation rates (denoted by π1 and π2) show that in every instance, π1 > π2, a result that is consistent with Obstfeld’s (1989) dynamic analysis of seigniorage. In this model, however, the downward-sloping time profile of inflation follows from the presence of nonindexed debt and it is not necessarily linked to inflation tax revenue considerations or to the presence of debt aversion.18 In fact, even if k = 0 (that is, the inflation tax collects no revenue) and x1 ≤ x2 (there is no debt aversion), it can be shown that π1 > π2, because the base for ∏1 (that is, bR) exceeds the base for ∏2 at the optimum (that is, b0.2R2 + b1.2R).
The simulations presented in Table 1 illustrate the effects of changes in b, k, and g. They indicate that optimal debt maturity lengthens with increases in b and k, while it shortens with increases in g. The intertemporal distribution of inflation also appears to be affected in a systematic way. An increase in b raises the ratio π1/π2, whereas an increase in k reduces it. An increase in g appears to cause only a slight upward movement in the ratio π1/∏2.
This paper has provided relatively simple examples of the importance of the maturity structure of public debt when the government has incentives to use inflation ex post to reduce the real value of its nominal liabilities. The nature of the optimal policy was found to be quite sensitive to the type of precommitment enjoyed by the government and, hence, to the prevailing type of incentive-compatibility constraints.
The analysis suggests that when governments are partially able to precommit policies, the optimal maturity structure of nominal debt tends to be short. However, when no precommitment exists, our simulations suggest that it is optimal to have a more balanced maturity structure. The simulations also suggest that optimal maturity is longer the higher is the level of public debt, and shorter the higher is government spending.
The examples provided in this paper should be viewed as first steps toward a more general characterization of the role of maturity in managing incentive-compatible inflation over time. These first steps, however, show that the problem of characterizing the time-consistent policy becomes analytically complex even when the basic structure of the economy is kept at a minimum.
In Appendix I we prove that debt aversion occurs for the general case of Section II, in which k > 0. If k > 0, debt aversion implies that period 1 revenues—including the inflation tax—in the second-best equilibrium are higher than in the first-best optimum, implying that less debt is left in period 2 compared to the first-best case. Define the first-best allocation by (x*, ∏*). Since there is complete smoothing over time of conventional taxes and inflation, period 1 (and 2) revenues are equal to
Hence, debt aversion is characterized by
where (x1, x2, ∏1, ∏2) is the second-best (time-consistent) optimum.
For convenience, we rewrite (and renumber) below the equations that characterize the time-consistent equilibrium:
Proposition 1. x1 > x2.
(1) x1 ≠ x2. Suppose x1= x2. Then, equation (34) implies that
which can only be true if μ = 0, since the term in braces depends on V”(x2), which may be chosen arbitrarily. If μ = 0, then equations (35) and (36) imply that ∏1—∏2. Moreover, equations (34) and (36) imply that
x–g + S(Π) = 0,
which is inconsistent with budget constraint (37) if b >0. Hence, x1 ≠ x2.
(2) If k = 0, then x1 > x2. This has already been proven in the text.
(3) Using (1) and (2) above, the property that the second-best equilibrium is continuous in k, we conclude, by the mean value theorem, that if k >0, then x1 > x2.
Proposition 2. Debt Aversion: x1 + S(∏1) > x* + S(∏*) = c.
(1) x1 + s(∏1) ≠ c, x2 + S(∏2) ≠ c. Suppose, instead
Consider the system of equations (39), and (34)-(37). If α = 0, then this system yields the first-best allocation (x*, ∏*). In particular, we know that the first-best allocation satisfies (38a) and (38b). Since we are supposing that (38b) holds, which implies that x2 = c—S(∏2), the left-hand side of equation (39) is an increasing function of ∏2.
Now let α > 0 at the value at which the above system of equations yields the second-best equilibrium. Since the left-hand side of equation (39) is an increasing function of ∏2, then ∏2 >*, which implies by equation (38b) that x2<x*.
Consider what happens with x1 and ∏1. By equation (35) we know that x1 ∏1 and move together. Hence, if x1 ≠ x*, then equation (38a) does not hold, providing a contradiction, since it would be inconsistent with (38b) and budget constraint (37).
To establish that in fact x1 ≠ x*, suppose that x1 = x* and prove by contradiction. Since equation (33) implies that x2 and ∏2 move together, the only way in which the budget constraint could be satisfied is if x2 = x* and ∏2 = ∏* as well. But we have already established that ∏2 > ∏*.
(2) If k = 0, then x1 > x2 by Proposition 1, Also, since k =0, the inequality x1 + S(∏1) > c > x2 + S(∏2) is met.
(3) Using (1) and (2) above, the property that the equilibrium is continuous in k, we conclude by the mean value theorem that if k > 0, then x1 + S(∏1) > c.
Optimal Maturity Under No Precommitment
In Appendix II we show that, by linearizing the incentive compatibility constraint (9’), an increase in b0.2 in equations (28a)–(28d) generates, other things being equal, an increase in ∏2 and x2 and a fall in ∏1 and x1. After linearizing, equation (9’) can be written as
where θ is a positive constant and Z is a linear function of k and g. Using equation (40) instead of (9’), the reaction function (28a)-(28d) would be characterized by equation (40) and the following set of equations:
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Guillermo A. Calvo is Senior Advisor in the Research Department and holds a Ph.D. from Yale University.
Pablo E. Guidotti is an Economist in the Research Department and holds a Ph.D. from the University of Chicago.
Calvo and Obstfeid (1989) have shown, however, that the stronger claim of Persson, Persson, and Svensson (1987) that the today-planner’s optimum can be decentralized by choosing an appropriate maturity structure does not hold in general.
As will become clear below, the presence of a “genuine” revenue inflation effect is important. The assumption that the demand for money is interest-inelastic introduces this effect in the simplest way. Assuming that the demand for money is interest-elastic would make the analytical presentation more complex without providing additional insights.
It is assumed that the price level at period 0 is a predetermined variable. This makes the output value of the initial stock of debt an exogenous variable.
In addition, V’(0) = 0 and H’(1) = 0.
Notice that we are making the cost of inflation a function of actual inflation, not just expected inflation. In this we follow Barro and Gordon (1983). For some microfoundations, see Calvo (1988), Persson, Persson, and Svensson (1989). and Calvo and Guidotti (1989b).
The assumption of perfect certainty helps to show in a dramatic way the revenue-ineffectiveness of expected inflation when applied to the stock of nominal bonds. Under uncertainty, however, the government would be able to collect inflation tax on nominal bonds as long as inflation is unanticipated. On average, of course, unexpected inflation would anyway collect nothing from nominal bonds. For a discussion, see Calvo and Guidotti (1989a, 1989b).
Recall that ∏ = 1 is the first-best level of inflation when k = 0.
The fact that x2 is lower than its first-best level follows from the fact that since we have shown that x1 > x2, the opposite would imply excess government revenues.
See Obstfeld (1989) where debt aversion is shown to arise even though bonds are fully indexed to the price level.
All that is needed is the ability to make precommitments about inflation in period 2.
We assume that the conditions of the implicit function theorem obtain. Moreover, the results assume existence of a regular minimum (that is, a minimum where the second-order sufficient conditions are satisfied).
The intuition behind the effects of a change in b0.2 can be obtained through contradiction. Using the fact that H” > 0 and S” < 0, we can observe that equation (17’) implies that ∏1 and x always move in the same direction. Consider now the effects of an increase in b0.2, other things being equal. If an increase in b0.2 increases x, it must also increase ∏1. However, from equation (18’), it can be seen that the increase in both x and ∏1 implies an increase in ∏2. which is inconsistent with budget constraint (5). Therefore, an increase in b0.2 must reduce x and ∏1. It is clear that, if x and ∏1 fall, only an increase in ∏2. is consistent with both equations (18’) and (5).
Notice that if b0.2 = b, then ∏1 = ∏2 and H’(∏1)/H’(∏2) = S’(∏1)/S’(∏2).
Again, we assume that the conditions of the implicit function theorem obtain, and we assume existence of a regular minimum.
In the numerical simulations presented in the next section an increase in b0.2 always has a positive impact on ∏2.
Note that if debt were indexed in this model, as in Obstfeld (1989), there would be no time-inconsistency problem because we are assuming that the demand for money is interest-inelastic.