THE LAFFER CURVE AND PRECAUTIONARY TAXATION: A RATIONALE FOR PAYING DOWN THE NATIONAL DEBT*

VIVEK H. DEHEJIA and NICHOLAS ROWE

Carleton University

I. INTRODUCTION

Why should governments ever attempt to pay down their national debts? In Barro’s (1 979) model of the optimal intertemporal choice of taxation, a government minimises the present value of the deadweight costs of taxation subject to its budget constraint by equalising marginal deadweight costs of taxation across time periods. With a time-invariant deadweight cost function, a government sets taxes equal to the annuity value of government expenditure (‘permanent’ government expenditure) plus (real) interest payments on the national debt. It does not try to pay down the principal. More generally, imagine two economies identical in all respects except that the second happens to have inherited a higher level of debt than the first. According to Barro’s model, the second will choose a higher level of taxes, but only high enough to pay the extra interest on the extra debt, not to pay any of the extra principal. This means that the second economy will maintain a permanently higher level of debt than the first. There is no history- independent long-run equilibrium level of the debt (or debt-income ratio’) in the model (as Barro himself recognises).

But this result runs counter to a common sense argument which suggests that there is a limit to any government’s capacity to tax, and so a limit on the size of the debt it can support. Governments running close to that limit have little flexibility or room to manoeuvre and would be hard pressed to meet some large and unexpected future need for government expenditure, like fighting a major war, or to cover a shortfall in revenue due to a large and prolonged recession. A prudent policy would require the building up of currently inadequate reserves as an insurance against such contingencies, even if unexpected losses are no more likely than unexpected gains. The government’s unused borrowing capacity is just such a reserve, as was the war chest in earlier years. What we do here is to formalise that common sense argument.

* Earlier versions of this paper were read at the annual conference of the WEA International, Seattle, WA, July 1991, and the Economics Seminar, Drexel University, Philadelphia, PA, October 1991. We thank two referees from this journal, and Mohamad L. Hammour and Bang Nam Jeon for their valuable help and suggestions. Responsibility for all errors is our own. Parts of this paper were written while Dehejia was a graduate student at Columbia University, and while Rowe was visiting the University of Adelaide.

1 The shift from talking about taxes and debt to tax rates and debt-income ratios is accomplished by supposing the period deadweight cost function to be linearly homogeneous in the current level of taxes and current real national income and using Euler’s theorem to obtain a new function of the tax rate alone.

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1995 THE LAFFER CURVE AND PRECAUTIONARY TAXATION 323

We present here a version of Barro’s model of the optimal intertemporal choice of taxation by a government wishing to minimise the present value of the deadweight costs of taxation subject to its budget constraint.2 As we will show, when uncertainty is added explicitly to the government’s problem by making the (exogenous) path of government expenditures stochastic, Barro’s famous tax-smoothing proposition - that taxes should follow a martingale - emerges only if the deadweight cost function is assumed to be quadratic, so that the marginal deadweight cost function is linear, so that certainty-equivalence prevails in the first order conditions.3 But a linear marginal deadweight cost function contradicts the quite reasonable presumption that there exists an upper bound on the economy’s capacity to generate tax revenue. If increasing tax rates eventually distort the economy enough that tax revenue actually begins to fall, then as we approach the top of the Laffer curve a first-order increase in tax rates and a first-order increase in deadweight costs yields only a second-order of small increase in tax revenue, which means the marginal deadweight cost of an additional unit of tax revenue becomes infinite at a finite level of tax revenue. If there is a Laffer curve, the marginal deadweight cost function must (at least eventually) be convex, so that marginal deadweight costs are increasing at an increasing rate.

With a convex marginal deadweight cost function, and uncertainty, we show that taxes, and government liabilities, are expected to decline over time. We dub this the ‘precautionary’ motive for taxation by analogy to the precautionary savings motive in the theory of consumption.4 But since we are able to demonstrate that the required convexity of the marginal deadweight cost function, given the Laffer curve, and no such analogous demonstration exists for the required convexity of the household’s marginal utility function, we consider the precautionary motive for taxation to be much more plausible a priori than the precautionary motive for saving.

Interestingly, our results are already implicit in some pioneering work on tax-smoothing by Prescott (1977) and Kingston (1984). Prescott’s model has government expenditure stochastic, as we do. Kingston uses Prescott’s model to derive a deadweight cost function which has increasing marginal deadweight costs and a Laffer curve. Kingston sees that this implies that tax rates are expected to decline over time, but does not go on to draw the next conclusion that government liabilities should also be expected to decline over time. On the contrary, he instead immediately downplays this result and says that tax rates will approximately follow a martingale. This is the sensible response of someone who wants to test for tax-smoothing and for whom precautionary taxation is an unneeded complication. But with both tax-smoothing and precautionary savings now established in the literature, it is time to deepen our understanding of optimal taxation by making explicit the precautionary taxation motive.

2 Unlike Lucas and Stokey (1983), Kingston (1991), and Chari, Christian0 and Kehoe (1991), we assume for simplicity that government debt yields a non state-contingent exogenous real return. The government might want to insure itself completely against shocks to government expenditure by issuing state-contingent debt, and if it did so, taxes would be constant. However, moral hazard and asymmetric information may prevent complete insurance, in which case taxes remain stochastic and our results should still hold.

3 Mankiw (1987). Bizer and Durlauf (1990) and Bohn (1990) all model optimal intertemporal taxation under uncertainty, but assume a quadratic deadweight cost function and so miss the precautionary taxation motive.

4 See Caballero (1990) and the references cited therein for a discussion of theory and evidence on the precautionary savings motive. While the analogy between optimal taxation and optimal consumption is not new, we h o w of no other work which draws the analogy between precautionary taxation and precautionary savings. Strangely, whereas there seems to be no good a priori reason for assuming either convexity or concavity of the marginal utility of consumption, we do have good reasons to assume convexity of the marginal deadweight costs of tax revenue, and yet the theory of precautionary savings preceded the theory of precautionary taxation.

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11. THE MODEL

Suppose a period deadweight cost function of the general form Z ( S ) , with Z, Z ' , Z" 0 for all S > 0, where S is the level of taxation and Z ( S ) is the corresponding deadweight loss (due to distortions and collection costs). Consider then the problem facing a government at time t , with stochastic expenditure, which wishes to minimise the expected present discounted value of deadweight costs from taxation over the infinite horizon

subject to the dynamic budget constraint

B,+,=R(B,+G,-S,), Z = t , t + l , ... (2)

with B, given, by choosing S, and where B is the level of the national debt, G is the level of government spending, and where the discount rate p and rate of return R are taken as constant. An interior optimum must satisfy the Euler equation

and the boundary or 'No Ponzi Game' condition

lim R-'B, = 0. ?+-=

(4)

We will first show that Barro's tax-smoothing result arises in this model only when two rather special assumptions are made. The first, which does not concern us here, is

that is, the government's discount rate equals the market interest rate, where R = ( 1 + r) and b = (1 + 6)-'. The second special assumption is

Z"' = 0 (ii)

or the deadweight cost function Z is quadratic, implying that the marginal deadweight cost function Z is linear. Imposing conditions (i) and (ii) imply by ( 3 ) , using the monotonicity of Z', that

that is, taxes obey the martingale property, since in this case the expectations operator can be passed through the linear function Z' . Now, condition (i) is perhaps unobjectionable, but condition (ii) is extremely restrictive; in imposing a quadratic objective, uncertainty per se ceases to play a role. We are in a world of certainty-equivalence. Furthermore, as we have noted above, a quadratic deadweight cost function, and its implied linear marginal deadweight cost function, are contradicted by the Laffer curve's upper bound to the government's taxing capacity.

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Far more interesting is the case in which uncertainty does matter, that is, when Z' is nonlinear. In particular, let us consider a case in which condition (i) continues to hold but now (ii) is replaced by

Z"' > 0. (iii)

We are now considering a case in which marginal deadweight costs are convex, so that (i) and (iii) immediately imply from (3), again using monotonicity and Jensen's inequality

Now we find that taxes are expected to be lower next period than this, even when the government's discount rate and the market interest rate are equal. It is no longer true that taxes follow a random walk; instead we now expect a falling tax profile.

What does a falling tax profile imply about the debt profile? Imposing the boundary condition - equation (4) - on the dynamic government budget constraint we can derive the intertemporal (long-run) government budget constraint, which requires the existing debt plus present value of government expenditures (which together we can think of as the government's gross liabilities) be exactly covered by the present value of taxes.

m - B, + C R-'Gr,, = C R-'Sr+,

r=O r=O

Leading (7) one period and taking expectations at time t gives

(7)

Subtracting (8) from (7) and rearranging gives

The right hand side of (9) is the present value of the sequence of expected decreases in taxes, each term of which must be positive as we have shown in (6) above, and so the left hand side of (9) must be positive also. This means that when the marginal deadweight cost function is convex, the government's gross liabilities of debt plus present value of expenditure are expected to decrease over time. So unless government expenditures are expected to decline, a falling tax profile implies a falling debt profile.

The intuition behind this result comes by noticing the structural similarity between this problem and the standard consumer's problem. In the consumption literature, it is well- established that when marginal utility is convex, the consumption profile is expected to rise over time, and hence the savings profile to fall, from a motive known as 'precautionary savings'. The assumption that marginal deadweight costs of taxation are convex, that Z"' > 0, seems reasonable

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given that a government that can use only distorting taxes presumably faces a Laffer curve, with an upper bound on its taxing capacity at which the marginal deadweight costs become infinite. Indeed, an everywhere linear marginal deadweight cost function would be inconsistent with an upper bound on taxing capacity. In the model here, with Z"' > 0, the government exhibits a desire for prudence by taxing relatively more today and building up a stock of reserves to meet future contingencies. Uncertainty about future outlays, as well as the expected value of future outlays, induces the government to build up a 'war chest' (either of assets or of unused borrowing capacity) by keeping current taxes relatively high.

We cannot, however, say any more than we have said in (6) and (9) without making more restrictive assumptions; in particular, to obtain a closed form solution for the path of taxes (and hence of debt), we need to specify the functional form of' Z ( S ) and the stochastic process governing government expenditure.

111. A SPECIAL CASE

Accordingly, we now present for illustrative purposes a very special case of the general model in which the precautionary taxation motive is exhibited in stark terms and in which we can get a closed form for the choice of taxation. This special case also has the attractive feature of yielding a simple and revealing expression for the evolution of the debt.

Suppose that the period deadweight cost function takes the exponential form

where 8 > 0 guarantees Z, Z', Z", and (our precautionary taxation condition) Z"' > 0. (Unfortunately, this function allows tax revenues to be unbounded, and so does not incorporate a Laffer curve, but we adopt it nevertheless for its neat results.)

To focus our discussion on the pure role of uncertainty, we continue to impose (i). Finally, we must impose a condition on the stochastic process of government spending. We suppose that it follows a random walk. (Again, this formulation unfortunately allows government expenditure to be unbounded, while a Laffer curve would make tax revenue bounded, but we adopt it nevertheless for its neat results.)

G,+, = G I + e,+l e,, I - iid N(0, 0 2 ) (iv)

Then, it is possible to derive (see appendix) the following expression for S,

Equation ( 1 1 ) is quite appealing since it allows us to see the separate influences of the tax- smoothing (Barro) and precautionary motives at work. Inspecting equation ( I I) , we see that taxation today equals the annuity value of inherited national debt plus present and future expected government expenditure (which because of the random walk assumption just equals current government expenditure) - j u s t as in Barro - plus an additional term which is proportional to the degree of uncertainty, as captured by 02 the variance of the shock term to

199s THE LAFFER CURVE AND PRECAUTIONARY TAXATION 321

government expenditure. This third term is precisely what captures the precautionary taxation motive. Consider two otherwise identical economies, except that in one the variability of government spending is higher. That economy would tax more today than the other, out of a desire to build up an unused borrowing capacity against future contingencies. Furthermore, if there were no uncertainty, i.e. if 0 2 = 0 then the third term would drop away and taxation today would exactly equal the annuity value of the inherited national debt and permanent government expenditure, just as in Barro. That the precautionary taxation term enters in an additive fashion, rather than changing the slope of the path of taxation, is a feature of our particular assumptions and (we suspect) is not general.

The other appealing feature of equation ( 1 1) is that it allows us to solve explicitly for the dynamics of the national debt. Substituting (1 1) into (2), the law of motion of the state variable, we obtain, after some rearranging

Equation (12) is a simple difference equation which tells us that the debt will fall by a constant amount each period, this amount being proportional to the degree of uncertainty; the greater the uncertainty, the faster will the debt fall. In Barro's world, with no uncertainty, debt would remain constant over time. With uncertainty the debt will fall, eventually becoming zero, and the government will then begin to accumulate assets, and will continue to do so into the infinite future. (The boundary condition, (4), will not be violated however, since the assets will grow by a constant arithmetic amount and so will eventually grow at a geometric rate smaller than the rate of interest.)

Iv. IS THERE A LOWER BOUND ON THE DEBT?

The special case of our model yields the extremely strong implication that debt will fall perpetually and go to minus infinity, and the general case has gross liabilities always expected to fall, which implies a falling debt if government expenditure is stationary. Fiscal conservatism seems eventually to lead to communism, since the government owns everything! We suspect that this result is tied to the simplicity of our set up. As we have shown in the general model and illustrated in the special case, taxes and hence gross liabilities are expected to fall whenever marginal deadweight costs are convex. Since the exponential deadweight cost function is everywhere convex, and is everywhere equally convex (in the sense that Z"/Z' is a constant), we get the strong implication that taxes and hence gross liabilities always decline, and decline by a constant amount, to minus infinity.

However, there are at least two reasons which are not captured in our set up for expecting gross liabilities to have a finite lower bound. Firstly, if we allowed the marginal deadweight cost function, Z' , to be linear or concave at low enough levels of taxation, we could have a steady state level of taxation - a situation in which S, = E,S,+ 1 - and hence a steady state level of gross liabilities (or debt) consistent with it. Suppose, for instance, that Z' starts out concave, has an inflection point at some positive level of taxation, and then becomes convex. In such a case, the inflection point would be our steady state level of taxation - since taxes would be falling to the right and rising to the left - which would imply a steady state equilibrium level of gross liabilities, consistent with those taxes, to which the economy would be driven irrespective of its starting point.

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Secondly, it seems reasonable to suppose that, when the government goes from being a borrower to a lender, i.e. when debt is just equal to zero, there is a drop in the interest rate facing the government, since in practice the interest rate at which the government (or for that matter an individual or firm) can lend is lower than the rate at which it can borrow. If this is true, then it is possible that as soon as the debt becomes negative the government’s rate of time preference will exceeds its lending rate, 6 > r, and hence from the Euler equation, (3), we see that the government has an incentive to adopt a rising tax profile and to decumulate assets or accumulate debt. If the difference between 6and r is big enough, this effect could outweigh the precautionary taxation effect and make zero debt the long run equilibrium. Both sets of considerations would suggest a lower bound on the level of gross liabilities.

V. CONCLUSION

In this concluding section, we shall not summarise our findings, but comment in a broader vein on the applicability of the set up we have used. We believe the framework to be an extremely flexible one, since the concept of the deadweight cost function is open to very broad interpretation. One obvious interpretation - and the one that we prefer - is that the function Z could represent the loss in utility to the representative agent from distorting taxes. However, from a political economy perspective, one could also interpret it as some political cost from taxation faced by the government, the objective then being to minimise the expected lifetime political losses from taxation. Or one could even give the model a Keynesian interpretation, supposing that the shocks to government expenditure arise from the need to increase spending to fight recessions. And the shocks could be introduced equally as revenue rather than expenditure shocks, shifting the function Z so that it could be written Z(S,-e,), which could also be done by simply reinterpreting the S in our model as ‘normal’ or ‘full-employment’ tax revenue.

A second issue is whether the model should be interpreted as a positive or normative model. Barro sees himself as providing a positive theory of how governments do actually behave, and in fact the model is amenable to very simple empirical tests analogous to tests of the permanent income hypothesis in the consumption literature. However - and this is the interpretation that we prefer - one can just as easily think of it as a normative model of how governments ought to behave, since minimising deadweight loss is presumably a necessary condition for efficiency.

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APPENDIX

Derivation of equation ( 1 1 )

The derivation proceeds by using Caballero's method, by analogy to the consumption literature. Given the specification in equations (7), (i), and (iv), we proceed by 'guessing' the solution and verifying that it is correct. Hence, conjecture a solution of the following form

where r is an (as yet) unspecified constant and V,+ I is a random variable with (as yet) unknown properties, other than its conditional mean. Insert (Al) into the Euler equation, (2), using the period deadweight cost function specified in (7) and imposing (i). We get

Taking logs and rearranging, we obtain the following expression for r

r = ( -1/8)1n~,exp(~y+,) .

But we know by Jensen's inequality that

lnE,exp(By+,)> E t l n e x p ( 8 ~ , , ) = 8 E , ~ + , = 0

which means that r i 0 and is a constant.

Now, recall the intertemporal budget constraint (2). Solving forward and ruling out explosive bubbles, we obtain the dynamic budget constraint

From (iv), we obtain

and from (Al) we obtain

7

S,,, = TI-+ S, + c v,,, ,=I

Now, taking conditional expectations over (A5), (A6), and (A7). we obtain

330 AUSTRALIAN ECONOMIC PAPERS

E,s,+, = rr+ s,

and EtGt+r = Gc.

Substituting (A9) and (A10) into (A8), we get

rn c RZ(rr+ s, - G, ) = B,. t=O

Since the above consists of a series of infinite sums, they can be easily evaluated. Rearranging yields

s-, = ((R - I ) / R)B, + G, - ( R/( R - i ) ) r . (A1 2)

Now, all we need do is compute the exact value of G. To do this, subtract (A8) from (A5), to obtain

But, (A13) implies that

for all t. But then

r = (-1 I @)lnE,exp(@e,+,).

But from the normality of e,, ,, we immediately obtain

r = (1/2)(-1/e)(0*0*)

which when substituted into (A12) gives us our desired expression, ( 1 1).

1995 THE LAFFER CURVE AND PRECAUTIONARY TAXATION 33 1

REFERENCES

Barro, Robert J. (1979), ‘On the Determination of the Public Debt’, Journal of Political Economy, vol. 87, no. 5.

Bizer, David S. and Durlauf, Steven N. (1990), ‘Testing the Positive Theory of Government Finance’, Journal of Monetary Economics, vol. 26.

Bohn, Henning (1990), ‘Tax Smoothing with Financial Instruments’, American Economic Review, vol. 80, no. 5.

Caballero, Ricardo J. ( 1 990), ‘Consumption Puzzles and Precautionary Savings’, Journal of Monetary Economics, vol. 25.

Chari, V.V., Christiano, Lawrence J. and Kehoe, Patrick J. (1991), ‘Optimal Fiscal and Monetary Policy: Some Recent Results’, Journal of Money, Credit and Banking, vol. 23, no. 3.

Kingston, Geoffrey H. (1984), ‘Efficient Timing of Income Taxes’, Journal of Public Economics, vol. 24, no. 2.

Kingston, Geoffrey (199l), ‘Should Marginal Tax Rates be Equalized Through Time?’ Quarterly Journal of Economics, vol. 106, no. 3.

Lucas, Robert E. and Stokey, Nancy L. (1983), ‘Optimal Fiscal and Monetary Policy in an Economy without Capital’, Journal of Monetary Economics, vol. 12.

Mankiw, N. Gregory (1987) ‘The Optimal Collection of Seigniorage: Theory and Evidence’, Journal of Monetary Economics, vol. 20.

Prescott, Edward C. (1977) ‘Should Control Theory be used for Economic Stabilization?’ Carnegie - Rochester Conference Seminar on Public Policy, vol. 7.