The Sustainability and Optimality of Government Debt

The sustainability and optimality rules for government debt are derived within an intertemporal optimizing framework in which both capital and government debt are endogenous, driven by utility and profit-maximizing behavior of private agents and tax and expenditure policies of the government. The rules are expressed purely in terms of familiar economic parameters, and their ready applicability in an operational context is illustrated by instructive numerical examples. A discussion of the relationship between the optimality rule and the “golden rule” of savings in the literature is also provided.

Abstract

The sustainability and optimality rules for government debt are derived within an intertemporal optimizing framework in which both capital and government debt are endogenous, driven by utility and profit-maximizing behavior of private agents and tax and expenditure policies of the government. The rules are expressed purely in terms of familiar economic parameters, and their ready applicability in an operational context is illustrated by instructive numerical examples. A discussion of the relationship between the optimality rule and the “golden rule” of savings in the literature is also provided.

An increase in government spending must be financed by an increase in taxes. The optimal choice between tax finance and debt finance is really a choice about the timing of those taxes. A permanent increase in government spending must be matched by at least an equally large permanent increase in taxes unless taxes are increased by even more in the short run … [T]here is no way to choose between a permanently higher level of taxes and a permanently higher level of debt (Feldstein (1985, p. 244)).

IN THE VOLUMINOUS literature on economic growth and capital accumulation, no result has richer and more enduring positive and normative implications than the “golden rule” of savings (hereafter referred to as the golden rule)1 as a guide for the conduct of fiscal policy. In addition to being the rule that governs the steady-state, welfare-maximizing level of capital intensity in the economy, it has also become the crucial benchmark against which to assess the role government debt plays in the steady state. This latter point was made particularly clear in the exchange between Barro (1976) and Feldstein (1976) on Barm’s (1974) earlier restatement of the Ricardian equivalence principle, which had assumed a stationary economy. The basic point was that in a growing economy a positive level of government debt in the steady state could be either a net burden on or a net addition to private wealth, depending on whether the steady-state growth rate is, respectively, less than or greater than the interest rate.2

The possibility that government debt could be regarded as net wealth by bond holders in the steady state is intriguing. The prevailing view in the public finance and growth literature, however, is that a steady-state equilibrium characterized by a rate of growth that exceeds the interest rate is somehow inconsistent with rational behavior, although it has long been recognized that in a competitive economy there is really no reason, on pure theoretical grounds, to presume that the steady-state level of capital intensity is necessarily below that implied by the golden rule.3 If one agrees with the prevailing view, then the conclusion reached by Feldstein (1985), quoted at the beginning of this paper, is inescapable. In a steady state where the level of per capital debt is constant, the total amount of debt must be increasing at the same rate as the labor force. But if the interest rate exceeds this rate, taxes must be raised in order to service the debt. The choice between debt and taxes as a source of financing the government budget is therefore illusory in the long run under these circumstances.

In writing off the government’s ability to reduce taxes in the long run simply by further increasing its borrowing,4 the literature has apparently overlooked the fact that steady states with an interest rate greater than the growth rate could be incompatible with stability, if the level of debt is positive. This observation is superficially obvious under the simplifying assumption that the interest rate can be treated as a constant. For under such circumstances, either per capita taxes must be forever increasing to keep the level of per capita debt constant, or with constant taxes per capita debt would rise without limit. Although these consequences do not immediately follow in a more complete model in which the interest rate, or equivalently the stock of capital, is allowed to vary, the dynamic properties of a model whose path of evolution is government by simultaneous adjustments in debt and capital deserve more than the rather scant attention they have received so far.5

The necessary and sufficient conditions for stability are studied here based on an intertemporal optimizing model in which both capital and government debt are endogenous, driven by the utility- and profit-maximizing behavior of private agents and tax and expenditure policies of the government. In such a model, it turns out that a necessary (but not sufficient) condition for stability in the neighborhood of a steady-state equilibrium with a constant positive per capita debt is that the growth rate of the economy be greater than the interest rate.6 Under very general assumptions regarding utility and production functions, an operational rule expressed purely in terms of familiar economic parameters for the determination of the critical level of debt above which it cannot be sustained is also derived. Because the model, though simple, is sufficiently rich in economic structure, the sustainability rule can be conveniently used to ascertain in a meaningful way the margin between an economy’s existing and sustainable debt levels, given unchanged existing economic conditions.

An important implication of the above sustainability analysis is that, if one restricts one’s attention only to stable equilibria (as is customary), then debt could be considered as a legitimate source of revenue for the government to finance its budget. Assuming lump-sum taxes are not available under realistic economic settings, the choice between debt and (distortionary) tax financing by the public sector therefore has substantive economic content and provides the motivation for the analysis of the optimal level of government debt.

In this paper, the government’s optimization calculus is defined in terms of its objective to maximize the steady-state utility level of the representative individual, subject to its budget constraint as well as the technological and behavioral constraints of the private sector. Three different policy experiments are analyzed: (1) the determination of the optimal level of expenditures for a given tax rate, (2) the determination of the optimal tax rate for a given level of expenditures, and (3) the simultaneous determination of the optimal levels of expenditures and the tax rate. In all cases, taxation is taken to be in the form of a general income tax at a constant ad valorem rate. Formulating the government’s problem in this way implies that the optimal debt level is a derived concept (as it should be) through the government budget constraint.

In each of the policy experiments described above, the impact of a given government action is transmitted to the private sector by its effect on the latter’s level of capital intensity. The long-run incidence effects of taxing incomes from productive factors are the subject of a large body of economic literature (see, for example, Diamond (1970) and Feldstein (1974a, 1974b)), and it is also generally recognized that variations in the level of debt can produce incidence (interest rate) effects (unless private savings behavior responds to negate them, such as under complete Ricardian equivalence). Seen in this light, debt and taxes are essentially fiscal tools by which the government alters the path of private capital accumulation to achieve its objective.7

In a decentralized economy in which the government has no direct control over the economy’s resources, and the optimal conduct of fiscal policy necessitates the use of debt or distortionary taxes or both, the optimal level of capital intensity in the steady state no longer always coincides with that implied by the golden rule.8 To see the possible negation of the golden rule as an optimality condition requires a basic understanding of the implication of the rule for fiscal policy. When the rate of growth of the labor force is positive, the maintenance of a steady-state capital stock of any given size requires the sacrifice of some consumption. Hence, even though a higher capital stock would necessarily produce more output, steady-state consumption is maximized only if the marginal product of capital is equated with the marginal unit of foregone consumption. However, in a life-cycle model such as the one employed in this paper (see Section I), not only the steady-state level of consumption matters but also its distribution between the young and the old. Hence, the optimality of the golden rule in this context is implicitly predicated on the ability of the government to peg the interest rate to the rate of population growth and then clear the capital market by varying the level of government debt. (No taxes and subsidies would be required, however, if there were no autonomous government expenditures.)9 This is the fundamental reason why the golden rule can be characterized, paradoxically, entirely by the given rate of population growth alone, with other parameters in the production and utility functions playing no role in the optimality condition.

Now suppose the government can increase or decrease the economy’s capital intensity only by changing the level of its outstanding debt through discretionary tax or expenditure or both.10 In such circumstances, the constraint imposed by the capital market equilibrium condition of a decentralized system becomes binding, and one would expect that the optimal level of capital intensity could either exceed or fall short of that implied by the golden rule. Since variations in capital intensity produce changes in the ratio of wages to interest rates, which in turn would alter the lifetime income profile of the representative individual, the exact location of the optimum relative to the golden rule should now depend, among other things, on the nature of both the utility and the production functions—and it does. This paper derives and interprets the optimality condition in the presence of government debt resulting from the optimization of the level of government expenditures for a given tax rate (the first policy experiment stated above) as the rational rule of savings. It turns out that, under fairly mild restrictions, the rational rule can be characterized in a relatively simple way.11 Hence, it can conveniently be used as an operational optimality rule for the determination of government debt.

Two important implications follow from the above optimality analysis. First, if the level of capital intensity dictated by the rational rule is below that corresponding to the golden rule, a positive level of government debt (in order to reduce the capital intensity) is required, resulting in an interest rate in the steady state that is greater than the rate of growth. The relationship is exactly reversed should the rational rule dictate a level of capital intensity that exceeds that implied by the golden rule. In either case, therefore, a nonzero optimal debt level gives rise to an item of outlay in the government’s budget constraint in the steady state. This means that when the government’s budget is optimally conceived, debt issuance in itself cannot be a source of revenue, and therefore, sufficient positive resources at the disposal of the government are required to attain the optimum. Furthermore, it can be seen from the earlier sustainability analysis, that the optimal level of government debt, should it be positive, is necessarily incompatible with a stable steady-state equilibrium.

The second important implication is that, if the government has the freedom to choose the optimal level of expenditures and the tax rate simultaneously, then government debt plays no role in its optimization calculus. The reason is simply that since the optimal level of capital intensity is always achievable through an appropriately chosen level of expenditures (and therefore of debt), there is never a reason to use the income tax as a tool for this purpose because it necessarily entails an excess burden.12 However, to conduct an expenditure or debt policy without positive tax revenues is infeasible, unless the government has access to other nontax revenues. Hence, the third policy experiment stated above is essentially an experiment to determine the optimal tax rate, not the optimal level of government debt. The optimality condition in this case, which simply balances the benefits to be gained by moving a competitive economy toward the golden rule with the cost of the excess burden of the tax, is also derived here as the taxation rule.

The paper reaches two basic conclusions. First, in order for any (arbitrarily given) amount of positive debt to be sustainable, the rate of economic growth must be greater than the net-of-tax interest rate in the steady state. It follows that a sustainable positive debt must also be a source of revenue in the government’s budget. Second, the optimal amount of debt, determined from a maximization of the steady-state utility of a representative consumer, depends on the nature of both the utility and production functions. It is positive (negative) if the optimal level of the economy’s capital intensity is below (above) that implied by the golden rule of savings. Hence, under no circumstances can such debt, when optimally derived, be a source of revenue for the government.

The present analysis of the sustainability and optimality of government debt has been carried out within the context of a closed economy. In view of the large external imbalances experienced by most countries recently, the impact of a change in the level of government debt on an economy’s external position is clearly of importance.13 The theoretical framework employed in this paper has already been extended to an open-economy setting by Persson (1985) for a one-good model, and by Zee (1987) for a two-good world. The open-economy implications of the issues analyzed here are therefore clearly on the agenda for future research.

I. The Model

The basic framework of analysis is the familiar overlapping-generations model, in which each generation lives for two periods, working during the first (when young), and retiring in the second (when old). When young, a member of the generation born in period t (henceforth, individual t) is endowed with one unit of labor. There is no labor-leisure choice.14 The size of each generation—that is, the total labor force in each period—Lt, grows at the rate n:

Lt=(1+n)Lt1.(1)

Individual t wishes to maximize his utility over his life cycle, given by a strictly quasiconcave, twice-differentiable, increasing real-valued function

Ut=u(ct,c¯t),(2)

where c, and

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denote, respectively, his consumption when young and when old, subject to the budget constraint

Ωt=ct+Ptc¯t,(3)

where Ωt = wt(1 – τt) is his net-of-tax wage income (also his lifetime net wealth), pt= 1/[1 + rt+1 (1 - τt+1)] is the current price of next period’s consumption, and τ is the ad valorem rate of income tax. The term rt+1 is the gross rate of return he can earn in period t + 1 for postponing one unit of consumption in period t. The first-order condition of the above maximization is

u1=u2/Pt,(4)

where subscripts on u denote its partial derivative with respect to its corresponding arguments. Together with the budget constraint (3), equation (4) can be used to solve for the individual’s current consumption when young as

ct=c(Ωt,Pt).(5)

It is convenient at this point to define three useful elasticities relating to consumption. First, along any indifference curve in the

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space, the elasticity of substitution between consumption when the individual is young and when he is old is

σtdln(c¯t/ct)/dlnpt>0,(6)

Second, the income elasticity of current consumption is

ηtdlnc¯t/dlnΩt>0,(7)

where the sign restriction in equation (7) rules out consumption (in either period) as an inferior good. Third, from the Slutsky equation, the elasticity of savings with respect to the net-of-tax rate of return [rt+1(1 – τt+1)] is related to σt, and ηt, according to

δtdln(Ωtct)/dln[rt+1(1τt+1)]=(1st)(1pt)(σtηt),(8)

where stptc¯t/Ωt is the share of income devoted to the individual’s consumption when old—that is, the rate of savings. equation (8) gives the well-known result:

δt<>0asσt<>ηt.

The production side of the model displays all the standard neoclassical characteristics. Let xt be the per capita output and kt the capital/labor ratio. The production function,

xt=f(kt),(10)

is assumed to satisfy the Inada conditions. Competition in factor markets yields the usual marginal conditions for factor rewards:

wt=fktf(10)
rt=f.(11)

It is again convenient at this point to define two useful elasticities relating to production. The first is the elasticity of output:

Φtdlnxt/dlnkt>0,(12)

and the second is the interest elasticity of the demand for capital intensity:

ϵtdlnkt/dlnrt<0.(13)

In a model such as this one, where the desired capital intensity is always immediately realized,

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is equally interpretable as the interest elasticity of gross investment.15

The government finances its budget by an ad valorem tax on income (wages and interest) at the rate τt and by issuing debt, measured in per capita terms as bt. These are one-period bonds paying a rate of return equal to that on capital prevailing in the period in which they mature. Outlays for the government in each period are current expenditures gt, plus the redemption cost (with interest) of debt issued in the previous period. Consequently, its budget constraint can be stated as

(1+n)gt+bt1/pt1=(1+n)(τtxt+bt).(14)

It should be noted from equation (14) that taxes are paid on debt interest income to bond holders, thus ensuring that debt and capital are perfect substitutes in the individual’s decision to save.

The intertemporal equilibrium for the entire economy is established when the asset market is cleared in each period according to

(1+n)kt+1=(Ωtct)bt.(15)

Given initial conditions and the time profiles of gt, and τt equations (14) and (15) allow one to trace out the complete path along which the economy will evolve over time.

II. Sustainability

Sustainability is a term that has been used with increasing frequency in the academic literature and recent multilateral policy discussions on the current world debt situation, but unfortunately with different connotations under different circumstances. There has also been a tendency to inject normative considerations, without at the same time being explicit about the underlying objective function, into what should otherwise be a purely positive concept. To avoid ambiguity and confusion, sustainability is taken here to mean stability (and stability only). A sustainable level of public debt is therefore one that allows the economy, in the absence of unanticipated exogenous shocks, to converge to a steady state.

The dynamics of the model set out in the previous section are governed by a simultaneous difference equation system in which changes in the per capita debt level bt and the capital/labor ratio kt (and therefore the rate of interest rt) from one period to the next are fully described by the government budget constraint (14) and the economy-wide intertemporal equilibrium condition (15). For given and constant levels of g and τ, total differentiation of equation (14) yields, after leading it by one period,

dbt+1=dbt/[(1+n)pt]+γ1tdrt+1,(16)

where

γ1r[(1τ)bt(1+n)τϵt+1kt+1]/(1+n)>0.

Total differentiation of equation (15) yields

y2tdrt+1=[γ3tdrt+rt+1dbt],(17)

where

γ2t(1+n)kt+1ϵt+1Ωtδtst<0γ3t[1ηt(1st)]ktrt+1(1τt+1)0.

In signing γ2t and γ3t, it has been assumed that δ, ≥ 0—that is, the interest elasticity of savings is nonnegative16—and that 1≥ηt(1 – st)≥0—that is, normality in current and future consumption.17 Substituting equation (17) into equation (16) to eliminate the drt+1 term from the latter enables the dynamic system to be written as

[dbt+1drt+1]=[a21a11a22a12][dbtdrt],(18)

where

a11γ3t/γ2t0a12rt+1/γ2t>0a13γ1tγ3t/γ2t0a221/[(1+n)pt]rt+1γ1t/γ2t>0.

With aij≥0 ∀ i, j in equation (18), the necessary and sufficient conditions for (local) stability are (1): (1 – a11) > 0—that is,

γ2t+γ3t<0,(19)

and (2): [(1 – a11) (1 – a22) – a12a21]>0—that is,

[nrt+1(1τ)](γ2t+γ3t)+(1+n)rt+1γ1t<0.(20)

Because γ1t>0, by equation (19) a necessary (but not sufficient) condition for the satisfaction of equation (20) is

n>rt+1(1τ),(21)

which says that stability requires that the growth rate of the labor force (also the real rate of growth of the economy in the steady state) be greater than the net-of-tax interest rate, or that the economy be overcapitalized relative to the level associated with the (tax-modified) golden rule of savings.18 This necessary condition establishes the important implication that, in the steady state, debt is a source of revenue rather than an item of expenditure for the government.

If equation (19) is expanded and evaluated at the steady state, this “low-level” stability condition can be expressed as

[1η(1s)]+(1+n)ϵ/[r(1τ)]sδ(1Φ)/ΦΦ<0,

which is independent of the level of per capita debt. It also clearly reveals how the inherent dynamic stability capacity of an economy can be enhanced (hampered) by a high- (low-) interest elasticity of savings.19

By a similar expansion of equation (20), again evaluating it at the steady state, one obtains the “high-level” stability condition:

[nr(1τ)]Φk(1+n)ϵkτ/(1τ)+b<0.(23)

Let λk and λb be the capital/output and debt/output ratios, respectively. The critical value of λb for which equation (23) can be negated is

λb*=λk{ϵ(1+n)τ/(1τ)[nr(1τ)]Φ}.(24)

Hence, a given debt level is sustainable if and only if

λb<λb*.(25)

Equation (25) states the sustainability rule. Note that the right-hand side of equation (24) comprises only terms that have transparent economic interpretations and whose numerical magnitudes are usually either readily available or can be easily estimated. Ascertaining the value of

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is therefore a straightforward matter.

Numerical calculations showing the sensitivity of

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to changes in the values of important economic parameters are provided in Table 1. In the baseline scenario, parameter values are chosen so that they are more or less consistent with either their existing magnitudes or empirical findings in the literature for the U.S. economy.20 Alternative scenarios are identical to the baseline except for the particular parameter value(s) identified in the left column of the table.

Table 1.

Sustainable Levels of Government Debt

(In percent)

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Parameter values are η=1.0, s =0.17, δ = 0.3, φ = 0.25,

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= –1.33, τ = 0.15, n=0.03 (per year), and r= 0.03 (per year). Values for both n and r are adjusted in each of the calculations to account for the length of the time period in the model, which corresponds roughly to half a generation (30 years).

Alternative scenarios are identical to the baseline except for the parameter value(s) identified below for each calculation.

As can be seen from Table 1, the value of

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is quite sensitive to η (income elasticity of current consumption), n (the natural growth rate), r (the interest rate), and τ (the proportional income tax rate). A 1 percentage point decrease in the growth rate or increase in the interest rate, for example, would reduce the sustainable level of debt relative to the baseline by approximately 10 percent of output. A 2 percent decrease in the income elasticity of consumption would also reduce the sustainable debt level by roughly 5 percent of output. An increase of almost 20 percent of output in the sustainable debt level can be obtained by increasing the tax rate by 5 percentage points. Changes to the values of the other parameters do not seem to have an appreciable impact on the value of
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.

While the above calculations are only intended to be illustrative, the sustainability rule as stated in equation (25) is nevertheless useful in providing a first-order approximation of the margin between an economy’s existing debt level and the level at which it can be sustained, given unchanged existing economic conditions.21

III. Incidence

The incidence of taxation is a subject of long-standing interest in public finance, stemming from the recognition that the entity upon which a tax is levied is not necessarily the one that bears the burden of the tax. Since the effectiveness and feasibility of any tax are predicated on both the degree and direction by which its burden can be shifted from one entity to another, the determination of the incidence of a tax in a decentralized market economy is probably one of the most important analyses to be performed in the conduct of fiscal policy.

The effects of taxing incomes from factors of production have been investigated extensively in the literature. As is well known, such taxes, apart from their short-run effects (where capital is held fixed), have significant positive and normative implications for the economy in the long run as well through their impact on capital accumulation. Traditional analyses in this area, however, have not properly taken government debt into consideration and have not examined carefully the stability properties of growth models when such debt exists. The model set out here makes it possible to demonstrate precisely the role government debt plays in the determination of whether the capital intensity of an economy in the steady state is increased or reduced by a change in the tax rate (here, the rate of a general income tax).

The impact of debt and taxes on the equilibrium value of the interest rate in the steady state can be obtained by totally differentiating the intertemporal equilibrium condition (15) to get

(γ2+γ3)dr=rw[δs+1η(1s)]dτrdb.(26)

For a given level of per capita government expenditures, g, changes in taxes cannot be independent of changes in debt. In the steady state, the government budget constraint can be written as

(1+n)g=(1+n)τx+[nr(1τ)]b.(27)

Total differentiation of equation (27) provides the important relationship between debt and taxes, according to

[nr(1τ)]db=[(1+n)τϵk(1τ)b]dr[(1+n)x+rb]dτ.(28)

From equation (28), it can be seen that, for a constant r, a one-unit increase in τ allows a decrease of [(1 + n)x + rb]/[nr(1 – τ)] units in b, a trade-off value that is not independent of the existing debt level. Substituting equation (28) into equation (26) to eliminate the term db, one obtains

drdτ=r2k{(1+n)(1Φ)[nr(1τ)][δs+1η(1s)]}/Φ+r2b[nr(1τ)](γ2+γ3)+(1+n)rγ1(29)

By the stability condition (20), the denominator of equation (29) is negative. The sign of equation (29) therefore depends on the sign of its numerator.22 The critical value of debt for which the numerator is zero is defined by

λb**=λk{(1Φ)[nr(1τ)][δs+1η(1s)](1+n)}/Φ.(30)

Hence, it immediately follows that

dr/dτ<¯0asλb>¯λb**,(31)

which stipulates the manner by which a change in the tax rate would affect the interest rate (and therefore the capital intensity) in the steady state. The a priori ambiguity of the sign of equation (29) stems from the fact that, while an increase in τ lowers the net-of-tax income and interest rate (for constant r) and, therefore, the savings (and also investment) in the economy, it allows, at the same time, a lowering of the debt level in the financing of a constant government budget. The lowered debt level permits a higher level of investment and capital accumulation. Thus, the overall impact on the steady-state capital intensity as a result of any given change in the tax rate would depend on the net outcome of these two opposing forces.

Although from equation (31) it is seen that an increase in τ would increase the capital intensity only if the existing debt level, expressed as a ratio to output, exceeds the critical value

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calculated from equation (30),23 a little algebraic manipulation reveals that
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if the interest elasticity of savings is zero (that is, δ = 0), which then unambiguously implies dr/dτ < 0. Indeed, it can be shown that an implausibly high value of δ is required in order to produce the opposite outcome. The reason for this is clear. Since the tax impact on savings works through the utility function, but the debt impact on capital accumulation is a direct one-to-one trade-off, only a high δ can lead to a reduction in savings that is sufficient to more than offset the increase in investment afforded by the lowering of the debt level.

IV. Optimality

Once it is established that in a stable steady state a given government budget could be financed by either taxation or debt issuance, an interesting and important question immediately arises: how to set the level of expenditures or the tax rate or both so as to maximize the steady-state utility level of a representative individual?24 Through the government budget constraint, the solution to this problem implies an optimal level of government debt.

In a decentralized economy in which the government has no direct control over the economy’s resources, the individual’s budget constraint and first-order condition for utility maximization are constraints in the government’s optimization calculus. To take them into account, the government needs to maximize the individual’s indirect utility function, obtained by substituting equations (3) and (5) into equation (2)—that is,

maxu{c(Ω,p),[Ωc(Ω,p)/p]},(32)

subject, of course, to its own steady-state budget constraint (equation (27)) and the intertemporal capital market equilibrium condition (equation (15)). Three conceptually different policy experiments to achieve equation (32) can be analyzed: (1) the determination of the optimal level of expenditures for a given tax rate, (2) the determination of the optimal tax rate for a given expenditure level, and (3) the simultaneous determination of the optimal expenditure level and the tax rate.

In solving the above optimization problem, one must first recognize the importance of equation (15) as a constraint in the optimization process. This constraint deprives the government of its ability to choose directly the level of capital intensity at which the economy is to operate. Instead, the government can now only affect the economy’s capital intensity through the market mechanism by varying the amount of its outstanding debt. The binding nature of this intertemporal capital market equilibrium condition (equation (15)) causes the government’s optimization calculus to be different from one that does not take this constraint into account. Indeed, without this binding constraint, the standard golden rule result would follow, even in the context of an overlapping-generations model (such as the one employed in the present paper), in which the distribution of factor incomes between wages and interests matters in the individual’s utility function.

To clarify further the consequence of requiring the government to abide by the market mechanism in its optimization calculus, and to anticipate analytical results to follow later, Figures 13 provide a graphical illustration of the crucial relationship between equation (15) and the golden rule.

In Figure 1, the production space and the consumption space are represented respectively by the right-hand and left-hand quadrants. Consider the arbitrarily chosen capital/labor ratio k* and the associated level of per capita output y*. Since the slope at point I on the production function equals r*, income to capital (r*k*) is measured by the distance By* on the vertical axis. The distance OB then measures the wage rate w*. Suppose there is no debt or taxes. To maximize utility, the representative individual would choose a point on his budget constraint BJ where it is tangent to one of his indifference curves.25 Not all points on BJ are compatible, however, with a long-run equilibrium for the economy as a whole. A feasible steady state must be such that the amount of savings in the economy, such as that measured by the distance Bc*, just equals (1 + n)k*. Applying the same procedure for alternative budget constraints, one can therefore ascertain a different feasible steady state for every point on the production function. The curve 0UHGZ traces out the locus of all such feasible steady states (LFSS). The underlying capital/labor ratio increases as it moves from the origin 0 toward Z.

For a general constant-returns-to-scale production function, the shape of the LFSS is not easy to characterize. From the individual’s budget

Figure 1.
Figure 1.

Mapping of Consumption Possibilities from the Production Function

Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A006

constraint (3) and the economy-wide equilibrium condition (15), one can obtain the slope of the LFSS as

dc/dc¯=[(1+n)+r/ϵ]/{(1+n)[(1+r)+r/ϵ]}.(33)

Since

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is a function of k in general, the LFSS may contain more than one inflection point. However, for a production function with a constant
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(such as the Cobb-Douglas), the LFSS is well-behaved and everywhere concave towards the horizontal axis. This can be seen by noting that

dc/dc¯=[(1+n)(1+ϵ)]1>0asr=(point0)=0asr=ϵ(1+n)(pointH)=1<0asr=0(pointZ).(34)

In Figure 1, the LFSS is drawn on the assumption that

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is a constant. An important implication of equation (33) is that, at the golden rule where n = r, the slope of the LFSS equals –(1 + n)-1 (point G). Hence, ail points on the 0HG and GZ portions of the curve are associated with, respectively, r > n and n > r.

A competitive steady state that is feasible is one in which the chosen consumption vector, such as (c*,

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), happens to lie on the LFSS. This can occur on either side of point G, depending on the utility function. For illustrative purposes, it is shown to be at point U in Figure 1, which corresponds to a level of capital intensity below that corresponding to point G. If the government has direct control over the economy’s resources and its objective is to maximize the level of steady-state utility, then it clearly pays for the government to move the economy to point G—that is, the golden rule. However, should the tangency between the indifference curve and the individual’s budget constraint associated with point G not coincide with the latter (such as point V), to clear the capital market the government would need to provide an instrument (such as debt) to satisfy the desired intertemporal consumption pattern of the individual. The validity of the foregoing analysis is predicated on the government’s ability to choose G directly, bypassing a decentralized market mechanism altogether. Once there, the injection of government debt into the economy would have no real consequence, since at n = r, debt of any size is self-financing. It is for this reason that the golden rule is optimal and can be characterized even in a life-cycle model without reference to the nature of the utility and production functions (see Buiter (1979)).

The analysis becomes substantially different if the government must respect the market mechanism and can change the economy’s level of capital intensity only indirectly through debt and taxes. The LFSS in Figure 1 is reproduced in Figure 2 (with the axes rotated clockwise by 90 degrees) as the broken curve labeled LFSS (τ = 0). For r > 0, the feasible consumption vector associated with any given output level must lie to the southwest of the vector for τ = 0, except at the end-points (points 0 and Z). Hence, for positive taxes the LFSS (τ > 0) shrinks relative to the LFSS with no taxes.26 The curvature properties of the LFSS (τ > 0) continue to be characterized by those stated in equation (34) if the term r is replaced by the term r (1 – τ), with the implication that at the golden rule (point G’), n = r (1 – τ).

If the individual’s indifference curve is tangent to the LFSS (τ > 0) at G’, then the golden rule is optimal and no government debt would be required. But such an outcome is purely coincidental. As discussed earlier, in a competitive economy the steady-state level of capital intensity could either exceed (point M) or fall short (point N) of the level that corresponds to the golden rule. In such circumstances, steady-state welfare could be improved with the injection of government debt (positive or negative), the optimal amount of which is to be determined by the nature of both the production and utility function.

Figure 2.
Figure 2.

Locus of Feasible Steady States

Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A006

Figure 3.
Figure 3.

Distribution of Income Between Young and Old

Citation: IMF Staff Papers 1988, 004; 10.5089/9781451930733.024.A006

Consider the competitive feasible steady state M in Figure 2. At point M, n>r(1 – τ), and the government could increase the individual’s steady-state utility level by introducing a positive amount of debt in order to lower the economy’s level of capital intensity,27 which in turn increases the interest and decreases the wage rates. Figure 3 illustrates the impact of such an action on the individual’s budget constraint and his level of utility. At the initial equilibrium, the consumption vector is point U (corresponding to point M in Figure 2) on the budget constraint AB. A positive amount of government debt shifts the budget constraint to CD,28 resulting in the new consumption vector V, which is on a higher indifference curve. A further increase in the debt level could lower welfare, however, if it were to shift the budget constraint to a line such as EF. At the optimum, therefore, a marginal change in the slope and location of the budget constraint, induced by a marginal change in the debt level, must leave the individual’s level of utility unchanged. The same analysis applies should the initial competitive feasible steady state be at point N in Figure 2; only in this case, the amount of government debt to be introduced would be negative.

The optimal amount of debt determined in the manner described above in fact corresponds to the first policy experiment stated earlier—that is, the determination of the optimal level of expenditures for a given tax rate. For this experiment, the solution to equation (32) is satisfied (using equation (4)) by

(dΩ/dg)/Ω=s(dp/dg)/p,(35)

which says that at the optimum, the proportionate change in lifetime net wealth and the price of future consumption (the latter being weighted by the savings rate) must be equal. Because a unique relationship exists between the wage rate and the interest rate along the factor-price frontier implied by equations (10)-(11), equation (35) can be manipulated to yield

(1τ)ρ(dr/dg)=0,(36)

where ρ ≡ [(Ω – c) – k/p] measures how a given change in the interest rate affects the utility level of the individual. Since dr/dg does not equal zero (see Section III), it is necessary and sufficient to have ρ = 0 for equation (36) to hold. The intuition behind this result is the same as that discussed earlier in graphical terms: the change in g alters the time profile of the individual’s income stream (in the form of a change in the wage relative to interest incomes); at the optimum the change in utility induced by this effect must be zero. By use of the economy-wide equilibrium condition (15) in the steady state, ρ = 0 implies

b/k=[nr(1τ)],(37)

which states that the debt/capital ratio in the economy at the optimum is equal to the deviation of the net-of-tax interest rate from the (tax-modified) golden rule of savings. An important implication of equation (37) is that the optimum can be characterized by either the golden rule [n = r (1 – τ)], in which case b = 0, or an over-capitalization relative to the golden rule [n > r (1 – τ)], in which case b < 0. A positive optimal debt level is incompatible with the stability requirement; it is also an item of expenditure rather than a source of revenue in the government’s budget in the steady state, since such debt must be associated with the net-of-tax interest rate being greater than the population growth.29

The exact location of the optimum relative to the golden rule depends on both the nature of the production function and the utility function. Substituting equation (37) into equation (15) yields the optimal net-of-tax interest rate as given by

r(1τ)=[s(1Φ)/Φ1]1,(38)

which can then be compared with n. Since equation (38) is an implicit equation in k alone, its solution can be obtained in a straightforward manner. It is also of interest to note that the existence of a meaningful solution to equation (38)—that is, r (1 – τ)>0, requires the existence of a region along the production function for which the output elasticity (φ) is compatible with such a solution. Rearranging equation (38) allows one to express the savings rate at which utility is maximized as

s=[1/(1p)][Φ/(1Φ)],(39)

which shows that, everything else being equal, the higher the elasticity of output or the current price of future consumption, the higher the rate of savings. equation (39) states the rational rule of savings.

The optimal debt level

article image
for the economy, given the tax rate τ, can now be stated as

λb***=λk{Φ/[s(1Φ)Φ]n}.(40)

Equation (40) states the optimality rule of government debt. The substitution of equation (40) into equation (27) then solves for the optimal expenditure level as

λg=τ+[nr(1τ)]λb***/(1+n),(41)

where λg denotes the expenditure/output ratio. equation (41) can be used for the determination of the optimal response of the level of government expenditures to a given arbitrary change in the tax rate, although the analytical expression for this response is unwieldy, primarily because it involves the second-order (excess burden) effects of taxation.30

As an illustrative example, consider the particularly simple case of a Cobb-Douglas production function:

x=Ψka,Ψ>0,1>α>0(42)

and a CES utility function:

u=(c11/σ+βc11/σ)/(11/σ),1σ>0=lnc+βlnc¯,σ=1,(43)

where β > 0 is a measure of the individual’s preference for future relative to current consumption and subsumes his pure rate of time preference. For the above production and utility functions, it is easily verified that φ = α,

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= 1/(α – 1),η = 1, and the rate of savings implied by equation (43) is

s=11/(p1σβσ+1).(44)

To solve for the rational rule of savings, equation (44) is equated with equation (39), and an economically meaningful root to the following implicit equation in p is sought:

11/(p1σβσ+1)[1/(1p)][Φ/(1Φ)]=0.(45)

Note that when α = 1, s becomes a constant, and p (and, therefore, r) can be solved explicitly from equation (45).

Calculations for the optimal levels of government debt under different sets of parameter values are provided in Table 2. As with Table 1, alternative scenarios are identical to the baseline scenario except for the particular parameter value identified on the left-hand column of the table.

Employing plausible parameter values, Table 2 strikingly illustrates the fundamental theoretical result derived earlier that the optimal level of government debt can be either positive or negative—which is equivalent to saying that the optimal capital intensity of the economy can be either smaller or greater than that implied by the golden rule. The numerical calculations also clearly show the influence of various parameters in the utility and production functions on the optimal debt level. A high debt level would be optimal for the economy, for example, all other things being equal, if the individual’s intertemporal elasticity of substitution in consumption (a) is low, or if his preference for future relative to current consumption (β) is low, since either tends to lead to a low savings rate. A low elasticity of output in the production function, in contrast, would lead to a low level of optimal debt.

Table 2.

Optimal Levels of Government Debt

(In percent)

article image

Parameter values are σ=1.0, β=0.9, φ = 0.25, φ = 0.15, and n=0.03 (per year). Values for both n and (solved) r are adjusted in each of the calculations to account for the length of the time period in the model, which corresponds roughly to half a generation (30 years).

Alternative scenarios are identical to the baseline except for the parameter value(s) identified below for each calculation.

Table 2 also provides some interesting results illustrating the trade-off between debt and taxes when the level of government expenditures is optimally determined. In general, because of the interest rate effect of debt, a change of 1 percentage point in the tax rate leads to a less than proportionate change in the optimal debt level in the opposite direction.

In characterizing the optimum under the second policy experiment—that is, in determining the optimal tax rate for a given expenditure level—one should note that any change in the tax rate will have a direct revenue effect on the individual’s budget, in addition to its incidence effect (which works only through the interest rate channel). To see this clearly, note first that the first-order condition for this experiment is of the same general form as equation (35), except that the dg term is now replaced by dτ. However, similar manipulations as before yield

(1τ)ρ(dr/dτ)θ=0,(46)

where θ ≡ [r(Ω – c) + w/p] measures the tax revenue effect per unit change in the tax rate. A comparison between equations (36) and (46) reveals that the two conditions for optimum have a similar structure, except for the θ term in the latter. Because of the presence of this term, the magnitude of the derivative dr/dτ now has a direct bearing on the solution for the optimum. Because, as given by equation (29), it is a complex analytical expression, the optimum of this experiment cannot in general be characterized in a simple manner. To solve for the optimal tax rate and the debt level in this case would typically involve a numerical evaluation of equation (46).

Even though the solution to the second policy experiment does not lend itself to an elegant characterization, when taken together with the first experiment, it does shed some light on the nature of the government’s budget constraint if g and τ can be chosen simultaneously. As formulated, the optimal solution for this third experiment requires the concurrent satisfaction of equations (36) and (46). A closer inspection of these equations reveals, however, that ρ = 0 (for equation (36) to hold) implies θ = 0 (for equation (46) to hold); that is, no taxes should be levied at all. Although this result follows directly from the fact that any desired incidence effect that can be achieved by varying the tax rate (which entails an excess burden) is equally achievable by varying the expenditure level, it is not a feasible policy option. To carry out an optimal expenditure program, the government needs positive resources at its disposal, unless the optimal capital intensity implied by the rational rule coincides with that of the golden rule.31

A moment’s reflection on the nature of the above policy dilemma uncovers a basic fallacy in the use of equation (27) as the appropriate form of the government budget constraint when g and τ are to be optimally chosen at the same time. The fallacy arises because in the present model in which g is not intrinsically valued (that is, it does not enter directly into either the utility or production function), the choice of g cannot be made independent of the choice of τ; thus, the correct government budget constraint is, instead, g = xτ. With this reformulation, the optimal choice of τ (and, therefore, g) becomes purely a matter of balancing the benefit of moving the economy toward the golden rule of savings with the cost of the excess burden of the tax. One can see this most clearly if income effects are ruled out by assuming that tax revenues are returned as lump-sum transfers to the individual and distributed over the two periods of his life in exactly the same proportions as the shares of the wage and interest income taxes in total revenues raised.

More formally, let T and

article image
be the transfer payments the individual receives when young and when old, respectively, where T = wτ and
article image
. The maximization of the individual’s indirect utility function, now written as

u[c,(Ω+Tc)/p+T¯],(47)

yields the first-order condition

ρ+τ(1+n)kε=0.(48)

The term [τ(1+n)k

article image
] measures the excess burden of the tax, and vanishes at τ = 0, a well-known result in the taxation literature. The optimal tax rate can be solved from equation (48) as

τ=(rn)/[r+(1+n)ε].(49)

Equation (49) states the taxation rule. It shows that the optimal tax rate is dependent on the configuration of parameter values as well as on the location of the initial equilibrium (with no taxes) relative to the golden rule. The optimal tax rate would be zero if the right-hand-side expression in equation (49) is negative. In this policy experiment, therefore, government debt has no role to play in the optimal conduct of fiscal policy.

V. Concluding Remarks

The applicability and robustness of the various theoretical results and operational rules developed in the present paper would be enhanced if the model were extended in two important directions. First, allowance should be made for government expenditures to have intrinsic value in the private economy, either through public-good benefits in the utility function of the representative individual, or through productive benefits (say, the provision and maintenance of the private economy’s infrastructure) in the production function. This extension would render a more complete characterization of the government’s optimization problem, expanding the role of fiscal policy to more than its customary incidence effects on capital accumulation.

Second, the response of private savings to changes in the level of government debt should be taken into account. The basic challenge of this extension is to incorporate such behavioral responses without producing the degenerate case of complete Ricardian equivalence.32 If successfully developed, these two extensions would allow one to differentiate in a meaningful way between the effects of changes in government debt arising from changes in expenditures and those stemming from changes in taxes.

References

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*

Mr. Zee, an economist in the Special Fiscal Studies Division of the Fiscal Affairs Department, was in the External Adjustment Division of the Research Department when an initial draft of this paper was prepared. He is a graduate of St. John’s University, Collegeville, Minnesota, and the University of Maryland, College Park.

For helpful comments and discussions, the author thanks Joshua Aizenman, Martin J. Bailey, Atish R. Ghosh, and his colleagues in the Fund.

1

The classic work on the golden rule of savings is Phelps (1966).

2

It turns out that, as was shown later by Carmichael (1982), the validity of the Ricardian equivalence principle, at least within the theoretical framework employed by Barro (1974), is not affected by these complications.

3

It is worth noting, however, that both Buiter (1979) and Carmichael (1982) have shown that in an overlapping-generations model with operative transfers from children to parents (the gift motive), the steady-state equilibrium is necessarily one in which the growth rate exceeds the interest rate.

4

Buiter (1983), for example, likens it to forbidding the government to play Ponzi games forever.

5

Notable exceptions are studies by Blinder and Solow (1973, 1974, 1976), which explicitly recognized stability complications arising from the presence of government debt. Their major focus of attention, however, revolved around the wealth effects of debt in a traditional IS-LM-type macroeconomic model, a concern that has certainly faded in importance in the last decade.

6

Masson (1985) also discussed this condition under a different model setup and with the utility function restricted to the Cobb-Douglas form.

7

This observation is in striking contrast to the framework used by Barro (1979), where under complete Ricardian equivalence debt serves only the function of tax smoothing.

8

If interest income is taxed, as it would be under a general income tax, the proper definition of the golden rule must be modified to one that equates the growth rate of the economy with the net-of-tax interest rate faced by the consumer.

9

When the rates of interest and population growth are equal, any level of per capita debt is self-sustaining. A more detailed discussion on this and other issues relating to the golden rule is provided in Section IV.

10

In an analysis of optimal fiscal policy, there is, of course, no reason why the level of government debt has to be exclusively positive.

11

In contrast, the optimality condition under the second policy experiment—that is, the determination of the optimal tax rate for a given level of expenditures—cannot be characterized in a simple manner, due to the excess burden of the tax, as will be shown later in Section IV.

12

The present analysis excludes from consideration the public goods or productivity aspects of government expenditures.

13

A comprehensive study of the international effects of fiscal policies can be found in Frenkel and Razin (1987).

14

Available empirical evidence suggests that the uncompensated elasticity of labor supply is on the whole quite small (see Killings worth (1983)).

15

It can easily be shown that the elasticity of substitution (kt) between capital and labor in the production function is related to φt, and

article image
by the relationship kt=εt(1Φt)..

16

The stipulated sign for γ2t holds even if δt < 0, as long as its absolute value is not large enough for the second term in γ2t to overwhelm the first. Available empirical evidence on savings behavior does not support large and negative interest elasticity of savings, however.

17

This follows from the fact that the weighted average of income elasticities of consumption (current and future) must add to unity, the weights being the relative shares of each period’s consumption in total net wealth.

18

The per capita debt level is not well-defined if n = r (1 – τ) in the steady state.

19

If the utility function (equation (2)) takes the Cobb-Douglas form, then δ = 0 and η = 1, which simplifies equation (22) to ε<sr(1τ)/(1+n).. For given equilibrium values of the tax, interest, and savings rates, this condition states the minimum required interest responsiveness of investment in order for the economy to be stable, provided that the “high-level’” stability condition is also satisfied (see below).

20

See Atkinson and Stiglitz (1980) and Auerbach and Kotlikoff (1987) for useful discussions of some of the available empirical evidence.

21

Stability conditions are, of course, highly model-specific. Policy implications derived from them should therefore always be interpreted with caution.

22

In contrast, the incidence of a change in government expenditures, given the tax rate, is unambiguously positive. It can be shown that the derivative dr/dg has the same denominator as that of equation (29), but its numerator is [ – (1 + n)r]. Hence, an increase in g necessarily reduces the capital intensity of the economy, as expected.

23

Note again that the calculation of this critical value only involves terms whose numerical magnitudes are usually readily obtainable.

24

This is not the only possible form of the objective function of the government. An alternative but much less tractable formulation would be to include in the function the sum of utilities of all generations along the path of transition from one steady state to another.

25

The slope of the individual’s budget constraint is –(1 + r*)-1 with respect to the horizontal axis. Hence, in general it will be different from the slope of line IBJ. To reduce clutter in the diagram, it has been assumed, without loss of generality, that the two lines coincide at the interest rate r*. This would occur, of course, only if r*≈0.62.

26

It is possible for LFSS (τ > 0) to intersect LFSS (τ = 0) along the positively sloped portion of the latter.

27

Alternatively, this could be interpreted as a way to return part of the tax revenues to the private sector.

28

This shift corresponds to the fact that wages (measured along the horizontal axis) are now lower, and the sum of interest income and capital (measured along the vertical axis) is now higher, than before the injection of debt.

29

For b<0, the condition [n > r (1 – τ)] is neither necessary nor sufficient for stability.

30

The foregoing analysis on the determination of the optimal level of government expenditures could be criticized as incomplete, since such expenditures are assumed to yield neither utility nor productivity. Although this criticism is strictly valid and extensions to the model in that direction are well worth taking, it should be pointed out that the focus of the analysis here is not on the formulation of a complete normative theory of government expenditures, but rather on their consequences on the economy’s capital intensity by altering the amount of available resources to the private sector. Nevertheless, the inclusion of g in the individual’s utility function, for example, would result in an extra term (relating to the marginal utility of g) in equation (36) and a more complicated expression for the rational rule of savings. A more substantive consequence of not ascribing an intrinsic value to g is discussed below in connection with the third policy experiment.

31

Substituting equation (37) into equation (27) and setting τ = 0 yields g = –(nr)2k/(1 + n)<0. A negative g is equivalent to a government subsidy. When the optimal capital intensity is less than that of the golden rule, the subsidy is used to service and maintain a positive amount of debt. When the optimal capital intensity is greater than that of the golden rule, the subsidy increases the private capital stock through a negative amount of debt. In either case, the achievement of the rational rule of savings, when it deviates from the golden rule, requires that the government have sufficient resources at its disposal.

32

Blanchard’s (1985) uncertain lifetime approach, although conceptually intriguing, remains somewhat ad hoc in spirit.

IMF Staff papers: Volume 35 No. 4
Author: International Monetary Fund. Research Dept.