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Wage and Public Debt Indexation

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1993
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After a period of relative neglect in the literature, the issue of public debt management—namely, policies regarding the characteristics of public debt instruments—has recently received increased attention.1 This sudden renewal of interest in public debt management reflects both empirical and theoretical developments. On the empirical side, the last decade has witnessed a marked growth of public debt in both industrial and developing countries (see Guidotti and Kumar (1991)). On the theoretical side, government policy is increasingly being characterized by economists as “endogenous” (see Persson and Tabellini (1990)), and more attention is being devoted to the consequences of incomplete markets. Policy endogeneity results from the presence of political constraints and from the recognition that government behavior may be subject to time inconsistency, particularly where rational private agents take into account policymakers’ reaction functions. One of the consequences of policy endogeneity and market incompleteness is that they provide a role for public debt management policies, which are largely irrelevant in the standard neoclassical model.2 Indeed, the focus of mainstream economics on the complete-markets neoclassical model may be the main reason for the past neglect of debt management issues.

An important aspect of public debt management, which has recently received attention, is that of debt indexation. As Calvo (1988) points out, when public debt contracts are written in nominal terms, policymakers have an incentive to resort to inflation as a means of reducing the real value of government liabilities.3 Thus, the perverse incentives attached to nominal public debt and policymakers’ time-inconsistent behavior provide a strong argument for indexing the public debt. Debt indexation breaks the link between public debt and inflation by making the real debt burden independent, ex post, of the inflation rate.4

This time-inconsistency argument identifies a significant danger associated with the presence of nominal government liabilities, and on this count alone optimal policy would call for 100 percent indexation of the public debt.5 However, Calvo and Guidotti (1990) pursue the argument further, noting that too much debt indexation may excessively limit policymakers’ ability to respond to unexpected shocks. In the Calvo and Guidotti model, the government faces the optimal-taxation problem of financing a stochastic level of government expenditure, plus the interest and amortization on the public debt, using conventional distortionary taxes and the inflation tax. Markets are incomplete because the government issues only two types of bonds: an indexed bond and a nominal bond, which pays a fixed, non—state contingent, nominal interest rate.

In the absence of a precommitment by policymakers, the following trade-off arises. On the one hand, the presence of nominal government liabilities induces policymakers to resort to suboptimally high inflation. The larger is the stock of nominal public debt, the higher is equilibrium inflation. On the other hand, too little nominal debt forces the government to rely excessively on distortionary conventional taxes to finance fluctuations in government expenditure. Since the stock of nominal debt is part of the unexpected-inflation tax base, it provides the policymaker with added flexibility when responding to unexpected shocks, and hence serves the purpose of smoothing taxes across states of nature.6 Thus, policymakers need to weigh the inflationary cost associated with nominal debt against the flexibility derived from it. This trade-off will yield the degree to which the public debt should be indexed.

These arguments for and against debt indexation focus essentially on fiscal policy, or the optimal-taxation problem. However, another important concern of policymakers may be whether debt indexation could lead to other forms of indexation, particularly that of wages, which is often regarded as a cause of inflation persistence. At first glance, it may be argued that a positive relationship between debt and wage indexation may arise simply because it is a matter of fairness to provide the same inflation protection to wage earners as to bondholders. At a more subtle level, it may also be argued that a higher degree of public debt indexation suggests that the government may expect higher inflation and hence signals workers to demand higher wage indexation for themselves. Although the relation between these two forms of indexation is of considerable relevance for policymakers, it has not yet been formally analyzed. This paper seeks to fill that gap.

What governs the optimal degree of wage indexation has long been a central issue in both macroeconomic theory and policy.7 The analysis pioneered by Gray (1976) and Fischer (1977) provides a clear and intuitive way of thinking about wage indexation. They focus on an eminently relevant issue: What is the optimal degree of wage indexation in a world where there are both monetary and real shocks and where a complete set of state-contingent wage contracts cannot be implemented? By focusing on incomplete markets—in particular, those where wage contracts are signed under imperfect information and, for the sake of realism, can be made contingent only on the price level—Gray and Fischer provide important insights for macroeconomic policy. Perhaps the most important insight is the relation of optimal wage indexation to the source of the shocks hitting the economy. In particular, as shown by Gray (1976), it is generally optimal to have partial wage indexation unless the only source of shocks is monetary. Moreover, the optimal degree of wage indexation is positively related to the variance of monetary shocks and negatively related to the variance of real shocks.

Policy is exogenous in Gray and Fischer, as well as in the related literature. This paper therefore blends the two approaches by introducing policy endogeneity into the Fischer-Gray framework and by linking public debt management—in particular, debt indexation—and the private sector’s degree of wage indexation. The focus on realistic forms of wage contracts in the Fischer-Gray approach is very similar in spirit to the public debt indexation approach taken by Calvo and Guidotti (1990).

The objective is to obtain basic insights by adding only a minimum of changes to the models of Fischer-Gray and Calvo-Guidotti. Nevertheless, blending the two approaches results in surprising richness. Since the wage and public debt indexation decisions are closely interdependent, the two models are extended significantly. The Calvo-Guidotti framework is enriched by accounting for output and employment fluctuations in the government objective function and by introducing imperfect control over inflation in the optimal-taxation problem.8 The Fischer-Gray approach—in particular, Gray’s model—is enriched by making the monetary shock endogenous. In particular, the monetary shock is made to reflect the rate at which money supply growth responds to shocks to government expenditure. The endogenous distribution of monetary growth is a function of the government’s choice of debt indexation and the private sector’s choice of wage indexation.

The analysis shows that as far as the government’s fiscal problem is concerned, wage and debt indexation are positively related: higher wage indexation increases unwanted inflation volatility and leads the government to increase public debt indexation. Since higher public debt indexation reduces the policymakers’ incentive to resort to inflation, it follows that a higher degree of wage indexation leads the government to a more anti-inflationary policy stance.9

As far as the private sector is concerned, the degrees of wage and debt indexation may be positively or negatively related. Whether higher debt indexation induces wage setters to choose a higher or lower degree of wage indexation depends on the effect of debt indexation on the variability of the rate of money growth. If higher debt indexation leads to increased monetary volatility, then wage setters will choose higher wage indexation. If higher debt indexation leads to decreased monetary volatility, the optimal degree of wage indexation will fall. Both public and private responses to indexation are discussed more fully in later sections.

Equilibrium wage and debt indexation represent a Nash equilibrium. The response of wage and debt indexation to exogenous changes—such as changes in the distribution of monetary and real shocks and in government expenditure as well as changes in the level of public debt—is analyzed. Whether wage and debt indexation respond to exogenous changes by moving in the same or opposite direction depends on the initial equilibrium, the nature of the change, and the policy response to it. The analysis shows that one form of indexation does not generally lead to other forms. Indeed, the two forms of indexation occasionally move in opposite directions.

Finally, it is shown that policy precommitment increases the role for nominal debt. Unlike Calvo and Guidotti (1990), where the optimal policy with precommitment calls for a large, unbounded swap of nominal debt for indexed government assets, it is shown that the optimal degree of debt indexation is bounded. The reason is that in this paper the government cannot fully control the rate of inflation. Hence, an excessively high stock of nominal debt—although not necessarily a high net stock of public debt—results in excessive variability of conventional taxes and inflation. The presence of policy precommitment is shown to alter some of the implications obtained when there is no precommitment. However, many qualitative features of the interaction between wage and public debt indexation remain unchanged.

I. Basic Model

This section sets up the basic analytical framework of the paper by integrating Gray’s (1976) analysis of wage indexation and Calvo and Guidotti’s (1990) analysis of debt indexation. Emphasis is placed on the interaction between the degree of wage indexation and government fiscal and monetary decisions.

Wage Contracts

Consider a small open economy in which the production function of the only produced good is given by

where Y denotes real output, L denotes labor (assumed to be the only variable input), α is a constant parameter [αϵ(0,1)],and u is a technological stochastic shock symmetrically distributed with a zero mean and constant variance, σu2.

It is assumed that there are two periods. In period 0, a wage contract is decided; in period 1, the production decision takes place. It is assumed that in period 1 the representative firm observes the realization of the technological shock, u. Thus, the firm demands labor according to the following first-order condition:

where W and P denote the nominal wage rate and the price level in period 1. Equation (2) yields the following demand for labor:

where ld,w, and p denote the logarithms of labor demand, the nominal wage, and the price level.

The supply of labor is given by

where ls is the logarithm of labor supplied, and h (> 0) is the labor supply elasticity, which is a constant.

With perfect information, equilibrium nominal wages, w*, would be obtained by settingld=ls. According to equations (3) and (4), therefore,

Equation (5) shows that, with perfect information, equilibrium real wages are directly related to the technological shock, u. Thus, real wages are high when productivity is high, and vice versa.

The realization of the technological shock, however, is assumed to be firms’ private information. Thus, wage contracts cannot be made contingent on u. Following Gray, it is assumed that wage contracts negotiated in period 0 index the nominal wage as follows:

where the superscript e denotes private expectations, and η is the wage indexation parameter [η(0,1)].. As will become clear below, the price level is a noisy signal of the technological shock. Therefore, the wage indexation decision entails a signal extraction problem.

Once the nominal wage contract is decided, employment becomes completely demand determined. Equation (6) can be written

where π denotes the inflation rate between period 0 and period 1. If there is full wage indexation (η = 1), then the real wage is fixed at the expected perfect-information equilibrium real wage. Note that, by equations (5) and (7), it follows that wp=E(w*p)=0,where E(.) denotes the expectations operator based on information available at time 0. If there is partial wage indexation (η < 1), then the real wage will vary with inflation around its expected perfect-information level. Note that E(wp)=E(w*p)=0.

Following Gray, it is assumed that the degree of wage indexation, η, is chosen to minimize the expected deviation of the logarithm of output, y, from its perfect-information level, y*. Using equations (1)—(7), wage setters would minimize the following expression for the loss function. λw:

Note that a distinction is being drawn between the mathematical expectations operator, E, and the concept of private expectations, superscript e. This distinction, which disappears in equilibrium because private expectations are assumed to be rational, highlights the fact that πe is embodied in the wage contract in period (1 and is therefore a predetermined variable for decisions taken in period 1. This factor becomes clear when the government’s decisionmaking process is analyzed.

Next consider the determination of domestic inflation. It is assumed that purchasing power parity (PPP) holds and that there is no foreign inflation. These conditions imply that the rate of domestic inflation is identical to the rate of nominal devaluation of the domestic currency. Furthermore, it is assumed that the small open economy operates under a flexible exchange rate and that the demand for money is interest inelastic. Hence, money market equilibrium implies that inflation satisfies

where μ denotes the rate of monetary growth and ϵ can be interpreted either as an unobservable shock to the demand for money or as a shock to the money supply that policymakers cannot control. It is assumed that ϵ is symmetrically distributed with a zero mean and constant variance, The sum of the last two terms in equation (9) equals the change in output. (For simplicity, but without loss of generality, it is assumed that the economic equilibrium in period 0 corresponds to a realization where both ϵ and u equal zero.) Equation (9) can be solved to yield:

whereβ1/(1α).Under rational expectations,πe=Eπin equilibrium. Thus, by equation (10),

Note that in equation (11) the rate of monetary expansion, μ is stochastic. The rate of monetary growth is endogenous, and its distribution will be derived in the next section on government decisionmaking. Government policy also determines the optimal degree of debt indexation.

The Government Problem

In period 0, the government inherits a stock of public debt, the real value of which is denoted by b and which is assumed exogenous. To focus on the issue of debt indexation, it is assumed that the only decision taken by government 0 is how much of the total public debt should be indexed to the price level; this proportion is denoted by 1 - θ. (θ = 0 implies full indexation.) Thus, most of the action occurs in period 1, in which the government decides how to finance the interest and amortization of the inherited public debt, as well as government expenditure, g.

In period 1, the government budget constraint is given by

where x denotes conventional (distortionary) taxes, i* is the (constant) international interest rate, i is the domestic nominal interest rate, and M/P isthe real quantity of money. Equation (12) states that conventional taxes equal government expenditure plus the service and amortization of the public debt, less the inflation tax on cash balances. The interest rate on indexed bonds equals the foreign interest rate.

It is assumed that individuals are risk neutral and that government bonds are pure assets. Hence, under perfect capital mobility,

Government expenditure is stochastic, g being symmetrically distributed with mean equal to g¯ and constant variance, σg2. As in Calvo and Guidotti (1990), it is assumed that the nominal interest rate on nominal government bonds is not state contingent. Hence, shocks to government expenditure cannot be financed by state-contingent variations in the interest paid on the public debt, which implies that markets are incomplete.10 The presence of market incompleteness focuses the analysis on a realistic menu of government bonds, and hence on meaningful debt management policies.

The presence of nominal debt implies that the government is subject to time inconsistency. This occurs because from the perspective of period I the amount of nominal debt, θ, as well as the nominal interest rate, i, are predetermined variables. Thus, government 1 has an incentive to use the inflation tax to reduce the real value of government liabilities beyond what would be optimal from the perspective of period 0.

Government 1 chooses taxes, x and the rate of monetary expansion, θ to minimize the following loss function:

where A and B are constants that denote the relative weights of taxes and output deviations in the government loss function, and E1(.) denotes the expectations operator based on information available to the government in period 1.11 It is assumed that government 1 knows the realization of g but cannot observe the realizations of ϵ and u. Equation (14) assumes that taxes, inflation, and output deviations are costly.

Government 0 chooses the degree of debt indexation, θ, to minimize

Government 0 does not know g but knows its distribution function. Government 0’s optimization takes into account the expected reaction of government 1 to government 0’s actions.

Two differences between the present analysis and that of Calvo and Guidotti (1990) are worth mentioning at this time. First, government 1 is subject to uncertainty, since it does not observe shocks ϵ and u. Thus, it does not have full control over the inflation rate. Second, policymakers do consider the effect that inflation has on output, in addition to the effect of taxes and inflation. (The government loss function is the same as in Calvo and Guidotti if B = 0.) In particular, the effect of inflation on output depends crucially on the degree of wage indexation chosen by the private sector. The degree of wage indexation in period 0, in turn, is chosen by the private sector takine account of the government’s behavior.

Before proceeding to a full discussion of wage indexation it is worth noting that in this model nominal wage contracts are contingent only on the price level. Yet, it may be argued that the nominal wage could also be made contingent on government expenditure. For the sake of realism, this analysis focuses only on price indexation, arguing that it is too costly to make wage contracts contingent on fiscal variables, because they are usually observed with substantial lags and are subject to frequent controversy and revision. Furthermore, this restriction is consistent with the assumption that the interest rate on the public debt is not state contingent.

II. Wage and Debt Indexation

In order to maintain analytical tractability, the government budget constraint in equation (12) and the interest parity condition in equation (13) are linearized. (This keeps the analysis closely comparable to Calvo and Guidotti.) Assuming—without loss of generality—that hereafter the international interest rate, i*, is equal to zero and taking the first-order terms of a Taylor series expansion of equations (12) and (13) around the point i=π=i*=0,it follows that12

where kπ is the seigniorage on cash balances, k is a constant, and

Let us now solve government l’s optimization problem. Government 1 chooses the pair of functions {E1x(g),μ(g)} in order to minimize

subject to equation (10) and budget constraint (16), taking it πe and i as predetermined; recall that πe is part of the wage contract. Note that since the government cannot fully control inflation—because of the unobservable shocks, ϵ and u —it cannot fully control taxes once inflation is realized. Government 1, however, can specify a function E1x(g) to indicate what taxes are expected to be—given the information available to government 1—as a function of government spending.

The above minimization problem yields the following first-order condition:

The left-hand side of equation (19) is the expected cost (as of period 1) of an increase in inflation triggered by a higher rate of monetary expansion, μ The expected cost of inflation is composed of a direct cost—the first term on the left-hand side of equation (19)—and a cost in terms of the expected deviation (as of period 1) of output from its optimal level— the second term on the left-hand side of the equation.13 The expected output deviation depends directly on the degree of wage indexation chosen by the private sector. With full wage indexation—when η = 1— the output deviation expected at time 1 is zero, independent of the inflation rate. The right-hand side of equation (19) equals the marginal cost reduction associated with government l’s expected conventional-tax saving obtained from higher inflation: higher inflation reduces the real value of government debt (in addition to raising seigniorage on cash balances), which is why the stock of nominal debt, θb, appears as part of the inflation tax in equation (19).

Calvo and Guidotti (1990) have shown that in equilibrium inflation, on average, produces no revenue from the stock of nominal debt. However, the inflation tax on nominal debt generates revenue on a state-by-state basis, so that it can he used to smooth conventional taxes across states of nature. This property is at the heart of Calvo and Guidotti’s (1989, 1990) concept of flexibility, which provides a role for nominal government debt.

In equilibrium, equations (11) and (17) hold. Using them (also see the Appendix details), the policy reaction functions of government 1 are obtained

and

Equation (20), the state-contingent policy for the rate of monetary expansion, shows that shocks to government expenditure are partly financed by government 1 with the inflation tax. The inflation-tax base that applies to unanticipated inflation includes the stock of nominal debt. Equation (21) shows that shocks to government expenditure are also partly financed by conventional taxes.14 Equation (22) shows that because of the presence of time inconsistency on the part of the government, expected inflation depends directly on the stock of nominal public debt. However, on average the inflation tax collects no revenue on the public debt (equation (23)).

As in the Appendix, equations (10), (12), (20), and (21) yield the following expressions for taxes and inflation:

In period 0. government 0 chooses the optimal degree of debt indexation—by choosing θ—while the private sector chooses the optimal degree of wage indexation, η. Both take into account the behavior of government 1 as described by equations (20)-(25). Formally, the resulting pair (θ, η) is a Nash equilibrium.

Using equations (20)-(25) and assuming that ϵ u, and g are uncorrelated, government 0’s loss function is given by

Similarly, the loss function of wage setters is given by

where

The loss function given in equation (27) looks exactly the same as that given in equation (14) in Gray’s (1976) paper. The difference between the two loss functions is that, in the present framework, the variance of monetary shocks, σM2, is endogenous since it is obtained from the government’s optimization problem. In particular, the variance of monetary shocks equals the sum of the variance of the exogenous money demand shock ϵσϵ2, and the variance of the rate of monetary growth, σμ2. The variance of the rate of monetary growth, in turn, is a function of both θ and η, as well as the variance of government expenditure, σg2.

Now the paper turns to the determination of equilibrium wage and debt indexation. To follow on intuition, the paper focuses on special cases that illuminate the essential elements driving the choice of θ and η. During the ensuing discussion, θ will be restricted to the interval (0, 1).15

CASE I: A = = a. If A = 0, then conventional taxes are not distortionary, implying no time-inconsistency problem in government behavior. It is easy to verify that in this case the government will finance government expenditure only by raising conventional taxes. Hence, the rate of monetary growth, μ, will be set equal to zero; the degree of debt indexation, θ, is irrelevant since it does not affect inflation (although it does affect the variability of conventional taxes). Since μ = 0, the variance of σM2 equals σϵ2.

The degree of wage indexation, η, follows from minimizing loss function (27), the exact problem studied by Gray (1976). It is well known that in the presence of both monetary and real shocks it is optimal to have partial wage indexation, 0<η0<1,where superscript o stands for “optimal.” Moreover, the optimal degree of wage indexation depends positively on the variance of monetary shocks, σM2, and negatively on the variance of real shocks, σu2. It can also be shown that ηο is positively related to the labor supply elasticity, h, and negatively to the labor demand elasticity,1/(1α).

The intuition behind the choice of wage indexation follows from Gray (1976). In the presence of shocks to productivity, it is optimal to have real wage flexibility. Indeed, if the variance σM2 equals zero, Gray shows that the optimal degree of wage indexation, ηο, equals h/(1+h),which is nonnegative and less than one. This degree of wage indexation achieves the optimal response of the real wage to real shocks. At the other extreme, if the only shocks are monetary—if σu2=0—then it is optimal to index wages fully—ηο = 0. This prescription reflects the fact that in the absence of shocks to productivity it is not optimal to have real wage variability.

CASE II: B = o. In this case the government carries out its monetary and fiscal policy without taking output deviations into account in its objective function.

Government 0 chooses θ to minimize

Loss function (29) is comparable to the government loss function given in equation (20) in Calvo and Guidotti (1990). In fact, both are equal when σϵ2=σu2=0. A new term appears in equation (29), relative to the government loss function of Calvo and Guidotti: the second term on the right-hand side of the equation, which contains the variances of ϵ and u and the expression [1+αβ(1η)].The latter, in particular, makes the government’s choice of 0 dependent on the degree of wage indexation set by the private sector.

These differences notwithstanding, it is shown in the Appendix that the choice of the optimal degree of debt indexation maintains the qualitative properties discussed by Calvo and Guidotti. In particular, it is shown that, for appropriate parameter values, optimal θ, θο, is greater than zero and less than one. Moreover, the optimal degree of debt indexation, 1 — θο, decreases with the variance of government expenditure, σg2, and increases with the expected value of government expenditure, g¯ and the level of public debt, b.

The intuition behind these results follows from considering the tradeoff faced by government 0. On the one hand, a higher level of nominal debt—less debt indexation—creates incentives for government 1 to generate a suboptimally high inflation rate because of the time-inconsistency problem. On the other hand, a higher level of nominal debt enlarges the unanticipated portion of the inflation tax base. (Recall that, since the nominal interest rate reflects the expected inflation rate point-for-point, no inflation tax can be collected on average on the nominal debt.) Since inflation is costly, a higher inflation tax base makes the inflation tax more efficient for smoothing out conventional taxes: it raises the amount of government revenue that can he generated under a given level of inflation distortion. Moreover, since nominal bonds are a part of the inflation tax base that corresponds to unanticipated inflation, the gains from the nominal debt accrue only to the extent that there is variability in the budget constraint. Thus, a higher variance of g increases the scope for nominal debt. Conversely, because of the time-inconsistency problem, a higher g¯ and a higher b increase the incentives to raise average inflation, thus inducing a reduction in θο.

It is apparent from equation (29) that the level of wage indexation, as well as the variances of shocks ϵ and u, affects the optimal degree of debt indexation. In particular, the Appendix shows that optimal debt indexation, 1 — θο, is positively related to η, as well as to the variances of ϵ and u.

The relation between θο and the variances of those shocks derives from the following considerations. The variability of unobservable shocks e and u induces output variability which, in turn, generates—through money market equilibrium—an inflation variability that cannot be controlled by the policymaker. This element of inflation variability is costly and plays no useful role in smoothing conventional taxes. In fact, since unanticipated inflation generates revenue fluctuations, it results in unwanted tax variability, which is uncorrelated with government expenditure. Since this effect is magnified by the presence of nominal debt, it follows that the higher the uncontrollable portion of inflation the larger the share of the public debt that should be indexed.

The presence of partial wage indexation implies that output variability reflects the variance of real wages, in addition to the variance of the technological shock, u. By equation (24) it can be observed that the higher the degree of wage indexation, the higher the variability of inflation, all other things equal. This relation reflects the fact that unexpected inflation is associated with an expansion of output—and hence an expansion of money demand. By equation (20), however, it can be observed that the variance of the rate of monetary growth, μ, takes into account the role of wage indexation: [1+αβ(1η)]appears in the numerator of the second term in the equation. Hence, the relationship between the variance of government expenditure and the variance of inflation is independent of the degree of wage indexation.

This amplification caused by wage indexation, however, applies to the uncontrollable—and, hence, unwanted—portion of inflation variability. The higher is the degree of wage indexation, the higher is the unwanted variability of inflation, for given σϵ2 and σu2. This explains why there is a positive relationship between the degrees of wage and debt indexation in the government’s optimization problem. More wage indexation generates higher unwanted inflation variability, thereby inducing the government to index more of the public debt. Thus, the government’s reaction function, with θο as a function of η, is downward sloping in the (η, θ) plane, as depicted by schedule G in Figure 1. Interestingly, the fact that higher wage indexation induces the government to choose higher debt indexation implies that higher wage indexation leads to a more anti- inflationary policy stance, which follows from the fact that higher debt indexation reduces government 1’s incentive to promote inflation.

Consider now the private sector’s choice of wage indexation. Wage setters choose η to minimize loss function (27), where σM2 is now given by

The full characterization of wage indexation is provided in the Appendix; the basic insights of Gray (1976) are shown to remain valid in this case. Equation (30) illustrates that the variance of the rate of monetary growth is a nonmonotonic function of θ. In particular, it is easy to show that σM2 increases with low values of θ and decreases with high values of θ. This nonmonotonicity reflects two factors. At low values of θ, an increase in θ makes the inflation tax more efficient: thus, it induces a substitution away from conventional taxes toward the inflation tax. For sufficiently high levels of θ, additional nominal debt decreases the variability of both conventional taxes and the inflation tax by enlarging the tax base for unanticipated inflation. Thus, a substitution effect dominates at low levels of θ, while a scale effect dominates at high levels of θ.

The nonmonotonicity of σM2 as a function of θ implies that the reaction of wage setters is nonmonotonic as well, as illustrated by schedule W in Figure 1. At low values of θ, an increase in θ increases the variance of monetary shocks, and hence increases the optimal degree of wage indexation, ηο At high levels of θ, an increase in θ decreases the variance of monetary shocks, and hence decreases ηο.

The intersection of schedules G and W provides the Nash equilibrium (ηο, θο). Depending on the parameter values, schedule G can intersect schedule W to the left of maximum η (shown in Figure 1.) or to its right.16

To study the effects of changes in various exogenous parameters on the equilibrium levels of wage and debt indexation, consider first an increase in the variance of government expenditure, σg2.17 A rise in σg2 shifts schedule G to the right, while it shifts schedule W upwards. Schedule G shifts to the right because, by previous considerations, a rise in σg2 increases the tax-smoothing role of nominal debt. Schedule W shifts up because an increase in σg2 increases the variance of monetary shocks, thus inducing the private sector to choose a higher degree of wage indexation. In the case where schedule G intersects schedule Won its upward-sloping segment, a rise in σg2 results in an increase in ηο an increase in wage indexation—but has an ambiguous effect on the degree of debt indexation, 1θ0.This ambiguous effect responds to two opposing elements. On the one hand, for a given degree of wage indexation the government will reduce debt indexation when government expenditure becomes more volatile. On the other hand, higher volatility of government expenditure increases the variance of monetary shocks and induces more wage indexation. Higher wage indexation, by previous arguments, induces more debt indexation.

Figure 1.Wage and Public Debt Indexation

When schedule G intersects schedule W on its downward-sloping part, an increase in σg2 has an ambiguous effect on both wage and debt indexation. The ambiguity with respect to ηο comes from two opposing effects. On the one hand, an increase in σg2 increases the variance of monetary shocks, other things being equal. On the other hand, the reduction in the equilibrium degree of debt indexation tends to reduce the variance of monetary shocks. This provides an example in which government policy regarding debt indexation dampens the private sector’s incentive to increase wage indexation. The rise in σg2 increases the variability of inflation and tends to reinforce the incentive to increase wage indexation. The government response to the rise in σg2 —indexing more of the public debt—reduces inflation volatility, and hence dampens the private sector’s incentive to raise wage indexation. This example shows that more debt indexation can reduce wage indexation.

Consider the effects of an increase in expected government expenditure, g¯. An increase in g¯ has no effect on schedule W and shifts schedule G to the left. Schedule G shifts to the left because an increase in expected government expenditure tends to raise expected inflation. Hence, it is optimal for government 0 to reduce the amount of outstanding nominal debt to reduce government l’s incentives to generate inflation. Thus, an increase in g¯ induces a fall in both ηο and θο if the initial equilibrium is on the upward-sloping portion of schedule W; it induces an increase in ηο and a fall in θο if the initial equilibrium is on the downward-sloping portion of schedule W. The change in ηο reflects the different effects that less debt indexation may have on the variance of monetary shocks affecting wage contracts.

Consider the effects of an increase in σϵ2. From the government’s point of view, an increase in σϵ2 raises the optimal degree of debt indexation for any given level of wage indexation. More debt indexation is optimal because a higher σϵ2 implies that the policymaker has less control over the variability of inflation. This implies that an increase in σϵ2 shifts schedule G to the left. Similarly, other things being equal, a higher σϵ2 increases the variance of monetary shocks, and hence shifts schedule W upward. As a result, a higher σϵ2 reduces θο but has an ambiguous effect on ηο if the initial equilibrium is on the upward-sloping portion of schedule W. It induces an increase in η73x03bf; and a fall in θ73x03bf; if the initial equilibrium is on the downward-sloping portion of schedule W.

Comparing the effects of an increase in σg2 with those of an increase in σϵ2 illustrates how important the policy response is to the equilibrium relation between wage and public debt indexation. For wage setters, an increase in both variances would increase monetary volatility, thereby resulting in higher wage indexation, which is why schedule W is affected in the same way in both cases. For the government, however, increases in σg2 and σϵ2 yield opposite policy responses—causing schedule G to shift in opposite directions. The higher volatility of government expenditure induces it to reduce debt indexation, but the higher volatility in the demand for money induces the government to increase debt indexation. Thus, exogenous changes that would have similar effects in the absence of a government response may have opposite equilibrium effects because of the different policy responses they induce.

Consider the effects of an increase in the variance of the technological shock, σu2. A rise in σu2 shifts schedule G to the left and shifts schedule W down. The shift of schedule G reflects the fact that higher σu2 increases the portion of inflation variability that cannot be controlled by the government. Schedule W shifts down because, other things being equal, an increase in the variance of real shocks makes it optimal for wage setters to reduce wage indexation. Hence, when schedule G intersects schedule W on its upward-sloping portion, an increase in σu2 results in a fall of ηο but has an ambiguous effect on θο. When schedule G intersects schedule W in its downward-sloping section, an increase in σu2 has ambiguous effects on both ηο and θο.

Consider the effects of an increase in the level of debt, b. Higher public debt shifts schedule G to the left since it induces the government to increase debt indexation. Schedule W shifts to the left: for given low levels of θ, wage setters choose a higher η and for given high levels of θ they choose a lower η. This reflects the different effects that an increase in b will exert on the variance of monetary shocks if an economy is situated on either the upward- or downward-sloping part of schedule W. As a result, an increase in b reduces θο but has an ambiguous effect on ηο if the initial equilibrium was on the upward-sloping part of schedule W, and has an ambiguous effect on both θο and ηο if the initial equilibrium was on the downward-sloping part of schedule W.

Finally, consider the effects of a change in the elasticity of the labor supply. An increase in labor supply elasticity, h has no effect on schedule G and shifts schedule W up. This implies that an increase in h results in higher indexation of both wages and debt.

The above analysis shows that, depending on the type of exogenous disturbance, wage and debt indexation may move in the same or in opposite directions. In Gray (1976), the effects on wage indexation are clearly classified by the source of shocks, but the present framework presents a less clear relationship between the source of shocks and the degrees of wage and debt indexation. There are two reasons for this. First, government policy responses are introduced into the analysis. Second, the effect of changes in debt indexation on the variance of the rate of monetary growth is shown to be nonmonotonic.

CASE III: THE GENERAL CASE.18 In addition to the effects considered in Case II, this final case incorporates the deviations of output from a perfect-information level in the government objective function.

Consider first the effect of introducing output deviations into the government’s loss function—that is, letting B be greater than zero. In particular, this case focuses on the effects of an increase in B from an initial equilibrium where B = 0. As indicated from equations (20)—(25), inflation and tax variability reflect the variability of shocks ϵ and u. and the government’s choice of the rate of monetary growth responds to shocks in government expenditure. More specifically, since the rate of monetary growth is uncorrelated with the technological shock, u. its variance adds to output variability. Thus, introducing output deviations into the government’s loss function leads the policymakers to reduce the optimal variance of the rate of monetary growth. By equations (20) and (21), it can be observed that an increase in B reduces the variance of monetary growth but increases the variance of conventional taxes, reflecting a substitution away from unanticipated inflation toward conventional taxes as means of financing shocks to government expenditure.

What is the effect on the optimal degree of debt indexation? Increasing B may be thought to make inflation variability more costly to the policy- maker. Thus, one would expect that debt indexation would move to reduce inflation variability. Indeed, the Appendix shows that, for a given η and for low initial values of θο, optimal debt indexation increases as B increases. For high values of ηο and a given η, optimal debt indexation declines following an increase in B. This implies that for low initial values of θο an increase in B shifts schedule G to the left and for high initial values of θο shifts schedule G to the right. The Appendix shows that a low initial value of θο refers to values of θ at which σM2/θ>0,implying that schedule W is upward sloping. Similarly, a high initial value of θο refers to values of θ at whichσM2/θ<0,for which schedule W is downward sloping.

Consider next the effects on η of letting B be greater than zero. Since, by previous considerations, an increase in B results in lower inflation variability, it should also reduce the optimal degree of wage indexation for a given θο. The Appendix shows that, starting from B = 0, an increase in B does lead wage setters to choose lower wage indexation, implying that an increase in B shifts schedule W down. The above considerations imply that if the initial equilibrium is on the upward-sloping portion of schedule W, then a higher B induces a fall in optimal wage indexation but has an ambiguous effect on debt indexation. If the initial equilibrium is on the downward-sloping portion of schedule W, then a higher B induces a fall in ηο and an increase in θο.

III. Wage and Debt Indexation Under Policy Precommitment

In this section the model is modified by allowing government 0 to precommit credibly to the policies of government 1. Calvo and Guidotti (1990) show that under full precommitment the flexibility of nominal debt becomes the dominant element in deciding the optimal degree of debt indexation. In their model, the absence of time-inconsistency costs implies that optimal policy calls for expanding the nominal public debt until, in the limit, both inflation and conventional taxes are completely smoothed out. In fact, in Calvo and Guidotti (1990), swapping an unboundedly large nominal stock of debt for an unboundedly large stock of indexed assets—note that net public debt is unaffected by such swap— attains, in the limit, the complete-markets solution.

It turns out that this remarkable result does not necessarily apply to the present framework. The reason is that in the present model there are unobservable shocks that impart unwanted volatility to inflation and conventional taxes. As long as this uncontrollability exists, a swap of nominal debt for indexed assets of the type described above would magnify the unwanted variability of inflation and taxes by enlarging without bound the tax base of the unexpected inflation. Hence, in the present framework, the optimal share of nominal debt, θο, will remain bounded as long as either σϵ2 or σu2 is positive.

As in Calvo and Guidotti (1990), it can be shown that optimal government policies under precommitment differ from those without precommitment only insofar as they concern the expected levels of monetary growth and taxes. The state-contingent part of the policy is the same with or without precommitment. The optimal policy regarding taxes and the rate of monetary growth under precommitment is given by equations (20), (21), (22), (24), and (25), where E1(.) is interpreted as the expectation over ϵ and u, and Eμ, is given by

Equation (31) shows that if government 0 can precommit to policies, then the optimal policy calls for expected inflation to be independent of the stock of nominal debt. This reflects the fact that, on average, the inflation tax collects no revenue on nominal debt because the nominal interest rate incorporates any expected inflation point-for-point.

In order to fix ideas, think of Case II, where B = 0. The government loss function under precommitment is given by

Equation (32) is the precommitment analog of equation (29). As indi- cated earlier, the only difference between the two loss functions involves expected inflation and taxes.19 If σϵ2andσu2 equal zero, then the minimization of equation (32) with respect to 0 reproduces the result of Calvo and Guidotti (1990): θ should be set unboundedly large. If either σϵ2or σu2is positive, then optimal θ<.Moreover, if (σϵ2+βσu2)/[1+αβ(1η)]2<σg2,then optimal θ>0.It is henceforth assumed that this condition holds.20

It is easy to verify that the optimal degree of debt indexation is obtained from the following first-order condition:

In accordance with intuition it can be shown that, other things being equal, optimal 0 is higher—optimal debt indexation is lower—in the presence of precommitment than in its absence.21 Furthermore, equation (33) shows that, as in the no-precommitment case, the government reaction function is negatively sloped in the (η,θ) plane. The Nash equilibrium with no policy precommitment (θ0,η0)is determined by equation (33) and the solution to the wage indexation problem in the previous section that is, the minimization over η of equation (27) whereσM2 is given by equation (30). The fact that the optimal degree of debt indexation is smaller in the presence of precommitment implies that, for given parameter values, the precommitment equilibrium (θ0,η0)is more likely to be situated on the downward-sloping portion of schedule W than the no-precommitment equilibrium is likely to be.

It can be shown that several comparative statics exercises yield the same qualitative (although not quantitative) results under policy precommitment as under no policy precommitment. This is the case for the effects of an increase in σg2 or σu2 on (θ0,η0). Other comparative statics exercises, however, yield different qualitative results depending on whether government 0 can precommit to its policies. Consider, for instance, the effects of an increase in expected government expenditure. g¯. Without precommitment, an increase in g¯ shifts schedule G to the left, because expected inflation is a function of debt indexation owing to the time-inconsistency problem. Under precommitment, however, expected inflation is independent of debt indexation because policymakers realize that, on average, the inflation tax collects no revenue on the stock of nominal debt. Thus, under precommitment, schedule G is independent of θ, and changes in g¯ have no effect on the equilibrium θ0,η0.

IV. Conclusions

This paper has explored the possible connections between government policy regarding the indexation of the public debt and the private sector’s choice of wage indexation. The analysis has been carried out in a framework where incomplete markets and policy incentives play central roles. The analysis has relied on existing theories of wage and public debt indexation. The insights obtained should be viewed as a first step toward understanding the macroeconomic interactions of different forms of indexation. Further analysis could be directed at exploring in greater depth the microfoundations of the problem, as well as the nature of wage contracts.

The analysis suggests that the relationship between wage and public debt indexation may be quite complex. As far as the government is concerned, the optimal degree of debt indexation is positively related to the degree of wage indexation chosen by the private sector. As far as wage setters are concerned, the relationship between the optimal degree of wage indexation and the level of debt indexation is nonmonotonic because of the different effects that debt indexation may have on inflation variability.

Wage and public debt indexation are determined as a Nash equilibrium. Whether equilibrium wage and debt indexation are positively or negatively related in response to changes in exogenous parameters depends on the nature of the changes, as well as on some characteristics of the initial equilibrium. It also may depend on whether or not the government has the ability to precommit to future policies. Hence, the analysis does not support the presumption that equilibrium wage and debt indexation move in the same direction. Moreover, the analysis provides examples in which more wage indexation may lead the government to adopt a more anti-inflationary policy stance. Therefore, the results of the analysis weaken the presumption that indexation encourages policymakers to tolerate inflation. Finally, this paper has shown that the uncontrollability of inflation as a tax instrument is an additional element that favors indexing the public debt. Introducing output fluctuations in the government objective function, however, does not have a clear-cut effect on optimal public debt indexation.

Problems Facing the Government and Wage Setters

Consider first the government’s situation. In period 1, the government chooses the pair of functions[E1x(g),μ(g)] to minimize loss function (18), subject to equations (10) and (16). The corresponding first-order condition is given by equation (19). Taking expectations of equation (16) based on information available in period 1, it follows that

where the equilibrium condition (17) has been taken into account.

Equations (22) and (23) are obtained by taking expectations of equations (16) and (19), based on information available at time 0. Adding A(θb+k)(g¯+b)and subtracting [1+Ak(θb+k)]Eπ—note that these expressions are equal by equation (22)—from equation (19), after replacing E1xwith the right-hand side of (AI), the following equation obtains:

From equations (10) and (11), it follows that

Hence, equation (20) follows from combining equations (A2) and (A3). Equation (24) follows from equations (10) and (1I). By taking expectations of equation (19), it follows that

Equation (21) follows from combining equations (19), (A3), and (A4). Equation (25) is obtained by combining equations (16), (Al), and (A3). Finally, loss function (26) is obtained by using equations (20)—(25).

Assume that B = 0. Then loss function (26) may be written as

wherez1+A(θb+k)2;ΩAσg2;Φ[A(σϵ2+β2σu2)]/[1+αβ(1η)]2;andrA(g¯+b)2Of course,z1.It can be verified thatf(1)=Ω+Φ+r<f()=.

where ω+A(z1)1/2.By equation (A6), lim(z1)f(z)=.Hence, argmin f(z)>1, and by previous considerations f(z)<.In particular, optimal a satisfies the following first-order condition:

It is assumed that

Hence, equations (A6) and (A7) imply that ω>kA,further implying that optimal θ>0.

Using the fact that, at a minimum, f(z)>0, one can use equation (A5) to compute comparative statics exercises. It can be easily shown that optimal 0 decreases with η,g¯,b,σϵ2and σu2 and increases with σg2. The comparative statics with respect to η, in particular, indicate that the government’s reaction function G is downward sloping in the (η,θ) plane.

Consider next the case in which B>0. In the general case, the first-order condition for the choice of θ is given by

where γ1+B(αβ)2(1η)2.By equation (A7), optimal θ>0.. Differentiat-ing equation (A9) one obtains

Recalling that f(z)>0,equation (A10) implies that, at B=0,dθ0/dB>0for z<2and dθ0/dB<0forz>2. It is easy to check that the condition that determines the signs of dθ0/dB at B = 0 is the same as the one that determines whether σM2 increases or decreases with 0. It is the same condition that determines whether schedule W is upward sloping or downward sloping. From equation (28), it can be verified that at B=0,σM2 increases with 0 as long as z<2and decreases with 0 when z > 2.

The wage setters’ loss function (27) is obtained from equations (8), (20), and (24). Equation (27) may be written as

where

The first-order condition for the choice of η is given by

Consider first the case where B = 0. This implies that Ψ/η=0in equation (Al2). It can be shown that, as in Gray (1976), η0=h/(1+h)when σϵ2=σg2=0,and η0=1when σu2=0.. When there are both monetary and real shocks, lies in the interval [h/(1+h,1)]. Moreover, it can be shown that ηο is positively related to σϵ2,σg2,and h, and it is negatively related to σu2. From equation (Al2), it can also be shown that n” is positively related to 0 when σM2/θ>0;it is negatively related to 0 when σM2/θ<0. This explains the form of schedule W. Finally, equation (Al2) can be used to show that, at B=0,dη0/dB<0.

REFERENCES

Pablo E. Guidotti is an Economist in the Research Department of the IMF and holds a Ph.D. from the University of Chicago. The topic of this paper emerged from conversations with Guillermo Calvo. The author is grateful to him for useful discussions and to Sophia Aguirre, Allan Drazen, José De Gregorio, Malcolm Knight, Guillermo Mondino, Gian Maria Milesi–Ferretti, Carmen Reinhart, and seminar participants at the International Monetary Fund, the University of Chicago, and Umversidad de San Andres (Argentina) for valuable comments.

1n some cases, as shown by Calvo (1988), nominal debt may even lead to multiple equilibria, and hence to the loss of nominal anchor.

Although the use of debt indexation as a tool to influence governments’ incentives to use the inflation tax has only recently been analyzed formally, it was recognized early by Bach and Musgrave (1941, p. 824), who stated that “by imposing upon the government a contingent liability dependent on its failure to check price inflation, the flotation of stable purchasing power bonds may exert a wholesome pressure upon Congress to adopt aggressive anti–inflationary policies.”

In Calvo (1988), optimal policy calls for 100 percent indexation, since the demand for money is assumed to be interest inelastic and the model is deterministic. In Persson,Persson, and Svensson’s (1987) perfect–foresight model, since the demand for money is interest elastic, the optimal policy calls for issuing nonindexed government assets (see, however, Calvo and Obstfeld (1990)).

The role of unanticipated inflation in smoothing taxes across states of nature as part of optimal monetary policy is developed in Calvo and Guidotti (1989).

It should be noted that, because of the assumption that the government’s targeted level of output coincides with that of the private sector, the inflation bias emphasized by Barro and Gordon (1983) is absent in this paper. The inflation bias emphasized here stems from the fiscal problem stressed by Calvo and Guidotti (1990).

This result provides an interesting insight into the discussion about whether wage indexation weakens the resolve of governments to fight inflation (see Fischer and Summers (1989) and De Gregorio (1991)).

See Calvo and Guidotti (1990) for more discussion.

Note that the target level of employment for the government is the same as that for wage setters. Therefore, the time–inconsistency considerations explored by Barro and Gordon (1983) are not present in this analysis. As will become clear in the next section, inflation occurs in this model because of a fiscal problem, as in Calvo and Guidotti (1990), and not because of the government’s attempt to push output above its equilibrium level, as in Barro and Gordon (1983).

In equation (16), it is implicitly assumed that the variation in seigniorage, which results from fluctuations in real money demand (because of output fluctuations), is absorbed through lump–sum taxes. Given that the focus of the paper is on the role of nominal debt in fiscal policy, adding those terms would make the algebra unnecessarily cumbersome. In addition, this assumption helps to maintain closer comparability with the analysis on debt indexation contained in Calvo and Guidotti (1990), where the demand for money is nonstochastic.

Note that actual output deviations also depend on the technological shock, u.

This follows from the assumption that unanticipated inflation is socially costly in this model. If only anticipated inflation were costly, it could be optimal to finance shocks to government expenditure by means of the inflation tax alone (see Calvo and Guidotti (1989)).

In theory, θ could take values outside the (0, 1) range, as discussed in Calvo and Guidotti (1990). When θ > 1, the government swaps nominal debt for indexed assets. Conversely, when θ <0 the government lends in nominal terms against indexed debt. Nothing essential is lost by assuming that θ ∈ (0, 1), since parameters can always he chosen to make optimal θ lie in the unit interval.

Existence of these two possible configurations was established by means of numerical simulations. Note that when schedule W is decreasing, it is less steep than schedule G.

The corresponding analytical derivations can be found in the Appendix.

The effects of comparative statics exercises in the general case were also explored by means of numerical simulations. These suggest that the qualitative response to shocks analyzed in Case II remains robust.

The derivation of equation (32) follows the same methodology set out in the appendix to Calvo and Guidotti (1990),

A stricter condition is assumed for the case of no precommitment in order to make θο > 0, as the Appendix shows.

The proof follows directly from comparing equation (33) to its analog—equation (A7) in the Appendix—in the absence of precommitment.

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