## Abstract

When there are collection lags in the tax system, inflation reduces the real revenues. This is often offered as an argument for less reliance on the inflation tax. But the optimal rates of other taxes should also be reconsidered in the light of collection lags. When this is done, the focus shifts from the revenues (which can be recouped by changing the rates of these taxes), to the associated costs of collection. In a benchmark case where the average costs of collection are constant, the optimal inflation tax is independent of the collection lag.

## I. Introduction

When there is a lag between the accrual and the payment of taxes, inflation erodes the real value of the government’s revenues. This insight was developed, and its importance for less developed countries stressed, by Tanzi (1977). This has since been developed by Tanzi (1978) and many others, including most recently Choudhry (1990). The argument runs as follows. Inflationary finance has a twofold harmful effect on the economy when there are collection lags. Not only does it generate the usual deadweight loss by driving a wedge between the marginal benefit and cost of real balances, it also reduces the revenue from other kinds of taxes. Therefore, the marginal cost of using inflationary finance is higher when we take collection lags into account. The inference is that inflationary finance should be used less than would be indicated by an analysis that ignores collections lags.

But this is a partial equilibrium insight. It examines the dead-weight burden of inflationary finance at given rates of other kinds of taxes. When we recognize that collection lags exist, we should expand the analysis to consider the effect of these lags on the optimal mix of the inflation tax and other taxes. Since the marginal cost of the inflation tax rises at given rates of other taxes it becomes desirable to increase the rates of these other taxes. In the process, equilibrium prices in the economy will also change. The question is whether less reliance on inflationary finance is indicated even when all such responses are taken into account and the new optimal mix is established.

The optimal mix of inflation and commodity taxation was examined by Phelps (1973), and his analysis was extended and refined by several others, including Aghevli (1977), Kimbrough (1986), Frenkel (1987), and Végh (1989). A positive inflation tax can be optimal only if other feasible taxes have distortions of their own, and many of the above papers model these as “collection costs.” But they do not consider lags, so the Tanzi link between inflation and other taxes is missing. In this paper I shall modify the latest of these models, that of Végh, to incorporate such lags.

The results do not support the partial equilibrium intuition stated above. In a benchmark case where the average cost of collection of commodity taxes is constant, I find that the optimal inflation tax is independent of the length of the lag in the collection of commodity taxes. The optimal rate of commodity taxation takes the full burden of adjustment to offset the revenue loss. This is as if commodity taxation were indexed for inflation, a solution that is indeed adopted in many high-inflation countries. An equivalent procedure is to charge interest for the period of the collection lag on the taxes due.

More generally, if the average cost of collecting commodity taxes is a stable and rising function of the real revenue, then a longer collection lag implies a __higher__ optimal inflation tax rate. This runs counter to the intuition outlined above. But the revenue erosion can be made good by changing the rates of commodity taxes. It is only to the extent that the adjustment of rates changes the cost of collection of those taxes, that the optimal rate of inflation is altered. If the inflation tax is kept unchanged and an indexation or interest scheme adopted for commodity taxes, a longer lag now makes it more costly to collect the commodity tax. Therefore, it becomes optimal to shift some of the tax burden toward the inflation tax. Correspondingly, there should be less than full indexation of commodity taxes, or less than full interest charged for the duration of the collection lag.

I should emphasize that the analysis does nor negate the importance of collection lags, or the erosion of tax revenues due to inflation when such lags are present. But it does suggest that the optimal policy response to the recognition of these issues is in general not what the earlier partial intuition had led economists to believe..

## II. Végh’s Model with Collection Lag

Most of Végh’s notation will be retained, but the notation for interest and inflation rates will be generalized somewhat. Let *π*_{t} denote the inflation rate between periods t and (t+1), and Π(t,s) the cumulative price multiple between periods t and s, that is

Π(t,s) = (1+

π_{t})(1+π_{t+1}) … (1+π_{s-1}).

Similarly, i_{t} is the nominal interest rate and I(t,s) the cumulative nominal interest multiple, and r_{t} the real rate of interest and R(t,s) the corresponding cumulative real interest multiple.

These satisfy

I(t,s) = R(t,s) Π(t,s) for all t,s

and

I(t

_{1},t_{3}) = I(t_{1},t_{2}) I(t_{2},t_{3}) for all t_{1}< t_{2}< t_{3}

and similarly for R, Π.

I will assume that the whole profile of real interest multiples R(t,s) is exogenous, and is equal to the reciprocals of the corresponding profile of the consumer’s utility discount factors. As in Végh, this is done to eliminate intrinsic dynamics and allow the model to settle down to a steady state at once.

In a steady state, where the real interest rate is constant and equal to r, the nominal interest rate constant and equal to i, and the inflation rate constant and equal to *π*, we have

R(0,t) = (1+r)

^{t}, I(0,t) = (1+i)^{t}, Π(0,t) = (1+π)^{t}.

There is one perishable consumption good produced in each period using labor. It takes a unit of labor to produce a unit of the good. Let P_{t} denote the price of the good relative to labor in period t. Then, taking money in period t. Then, taking money in period 0 as the unit of account, the nominal prices in period t are

Π(0,t) for labor, P

_{t}Π(0,t) for consumption.

There is a tax on period-t consumption at rate *θ*_{t}. If the quantity of consumption at time t is c_{t}, then the nominal tax accrued at time t is *θ*_{t}c_{t}P_{t}Π(0,t). This amount is paid by the consumer to the firm at the time of purchase, but turned over by the firm to the government at time (t+k). Thus, k is the collection lag. It is assumed that the lag is fixed and known, as in the case of VAT payments made at fixed dates.

Now the firm’s profit from producing and selling one unit of the consumption good at time t is

Competition will reduce this to zero. Simplifying the resulting equation, we find

(1+

θ_{t}) P_{t}-θ_{t}P_{t}/I(t,t+k) - 1 = 0,

or

We can then define an ‘effective’ tax rate

or using (1),

We will see that all the consumer’s decisions are made as if, instead of the tax rate *θ*_{t} and the price P_{t}, he faces the tax rate *θ*_{t} > 0 and I(t,t+k) > 1, we have

This is the first of the general equilibrium effects of the collection lag we must consider. As the collection lag reduces the real tax revenues of the government, it gives the firms the benefit of holding on to these revenues for k periods. In equilibrium, competitive firms must pass on this benefit to consumers in the form of lower prices. If we had assumed that consumers directly pay the tax revenue, then they would benefit directly, and again would face a lower effective tax rate.

Next consider the consumer’s transaction technology. If he holds an amount of money M_{t}, and spends a nominal amount E_{t}, or a real amount e_{t} = E_{t}Π(0,t), it is assumed that the transaction requires time

s

_{t}-ν/(M_{t}/E_{t}) e_{t}.

Observing that

E

_{t}- (1+0θ_{t}) c_{t}P_{t}Π(0,t),

using (2), and defining real balances m_{t} = M_{t}/Π(0,t), we have

As in Végh, I shall write the argument of *ν* as X_{t} for brevity. The function is defined over the range [0,X_{s}], where X_{s} corresponds to the saturation level of real balances. The function satisfies *ν*(X_{s}) = 0, and *ν*’(X) < 0, *ν*”(X) > 0 for all X in [0,X_{S}].

Suppose the consumer has an initial endowment of bQ units of account, and an endowment of a unit of time each period. Let h_{t} denote his leisure. Define

the rate of tax on the consumer’s real balances, that is, the inflation tax. (Végh’s symbol for my *γ* is I; my I is the cumulative interest multiple.) With all this notation, the consumer’s budget constraint is

The s_{t} in (6) is given by (4).

Subject to (6), the consumer chooses the sequence (c_{t},h_{t},m_{t}) to maximize

This problem is exactly the same as in Végh (1989), except that *θ*_{t} is replaced by

and

Now turn to the government’s problem. Assume that if it receives T_{t} units of account at time t, its collection costs are *ϕ*(T_{t} /A(0,t))T_{t}, where the average cost function is *ϕ*. The function A(0,t) represents technological progress in collection. Thus A(0,t+1)/A(0,t)-1 is the rate of improvement in the collection technology between periods t and (t+1). This is a very general formulation, which seems appropriate since we have no precise theory of the nature of these costs. At one extreme, one might think the cost consists literally of counting nominal cash. Then there is zero technical progress, and A(0,t) is constant. At another extreme, if auditing is costly but the cost is independent of the size of the audit, then the technology improves in pace with the nominal interest rate. More interestingly, Végh defines collection costs as a function of the real revenues. In my notation, this amounts to assuming that the nominal collection technology exactly keeps pace with inflation, or A(0,t) = Π(0,t) for all t.

We know that T_{t+k} = *θ*_{t}c_{t}P_{t}Π(0, t). Therefore, the discounted present value of the government’s receipts at tirae (t+k) is

The discounted present value of the collection costs at (t+k) is

For brevity, write

The government purchases exogenously specified quantities (g_{t}) of the good through time. Then the government’s budget constraint becomes

and the economy’s overall resource constraint is

The government can be regarded as choosing (c_{t},h_{t},m_{t}) subject to these two constraints and the equilibrium price functions, to maximize the utility sum in (7). The policy is assumed to be committed in advance; thus issues of time-’neonsistency are not addressed.

## III. The Invariance Result

The collection lag k enters this problem only through B_{t,k}, which is an argument of the average collection cost function *ϕ*. Therefore, the simplest benchmark cases to consider are the ones where the value of *ϕ* is not affected by its argument, and where B_{t, k} does not depend on k. In either of these cases, the solution to the government’s optimization problem determines *γ*_{t} and *θ*_{t}, will be adjusted to maintain the same values of the effective rates

This can be seen as a kind of indexation of the tax system.

A better interpretation is that interest is charged on accrued tax payments over the period of the collection lag. A firm that collects one unit of account from the consumer at time t must pay I(t,t+k) units to the government at time (t+k). Let

Setting this equal to zero for a competitive equilibrium, we get __as if__ into a reality.

Under either system, the same inflation rate as before will remain optimal as the collection lag changes, and the consumption tax will be adjusted to offset the effect on the revenue.

We saw two kinds of circumstances where k does not affect B_{t,k} and therefore the optimal values of *τ*_{t}: (1) The average cost of collection is constant, that is *ϕ*(z) = *ϕ*_{0}, constant, for all values of the argument z. This is also the benchmark case considered by Végh. (2) The rate of progress in the collection technology always equals the nominal rate of interest, so I(t,t+k)/A(0,t+k) and therefore B_{t,k} is independent of k. This seems an exceptional or ‘razor’s edge’ case, but it helps us develop the intuition behind the result.

As k changes, suppose we hold *γ*_{t} at their old optimal values, but charge interest on tax revenues held by firms for the longer collection lag. Then the amount collected goes up at the rate of interest. But if the collection technology improves exactly in step, the cost of collecting each unit of revenue is unchanged, and therefore so is the present value of the collection costs. Then the optimality conditions are unaffected, so it remains optimal to keep _{t} unchanged.

## IV. The General Case

When the average cost of collection *ϕ* is an increasing function, and k affects B_{t,k}, the length of the collection lag does alter the optimal inflation tax rate. To examine this in more detail, I specialize the analysis to a steady state, as does Végh at the corresponding point in his paper. Now the time subscripts can be dropped from all the variables, and a constant capitalization factor (l+r)/r dropped from the present value constraints. I follow Végh in setting b_{0} = 0 and choosing the utility function

both are matters of algebraic convenience that do not affect the basic results.

Now the consumer’s optimization yields h = ½, and then the equilibrium price functions can be written

and

where

If a steady state is to exist, B_{t,k} must be independent of t. In a steady state, (10) becomes

B

_{t,k}- (l+i)^{k}(l+π)^{t}/(l+a)^{t+k},

where a is the constant rate of improvement of the collection technology. For this to be independent of t, we need a = π, and then

B

_{t,k}= (l+i)^{k}/(l+π)^{k}= (l+r)^{k}.

This is exactly the implicit assumption about technical progress in Végh. He defines the real average costs of collection as a stable function of the real revenues. In my notation, this amounts to assuming that the collection technology improves in step with inflation, that is, A(0,t) = Π(0,t), or with constant rates in a steady state, a = *π*. Therefore, I make this assumption to proceed with the steady state analysis.

Write B_{t,k} as B for short. Since the real rate of interest must be positive to ensure convergence of the consumer’s and the government’s objective functions, b is increasing in k.

Next write z = *θ*’c, so c(l+*θ*’) - (c+z). Then the government’s budget constraint (11) becomes

The overall resource constraint (12) reduces to (15). The objective function is simply ln(c).

We can regard c, z, and X as the choice variables. Since these are subject to the two constraints (IS) and (18), there is one degree of freedom, and one independent first-order condition emerges. After some algebra, it can be written as

The solution can be shown diagrammatically, following Végh’s method.’ It is most convenient to do this in terms of two variables, a compound variable *ζ* = zB, and the inflation tax rate *γ*, which is negatively related to X by (16).

The two equations in terms of *ζ* and X are the Fiscal Equilibrium condition (FE)

and the Optimality Condition (OC)

These are generalizations of the correspondingly numbered and labelled equations of Végh, incorporating collection lags.

Figure 1 shows the two curves in (*γ*,*ζ*) space. Following Végh’s argument, the curve FE is downward-sloping and the curve OC is upward-sloping. Their intersection determines the optimal values of *γ* and *ζ*. The former is just the inflation tax; the consumption tax rate *θ*’ can be computed from *ζ*.

At the optimum, the left-hand side of (20) is an increasing function of *ζ* and a decreasing function of X; this is because the tax revenue is increasing in the tax rates. New an increase k raises B, which lowers the first term in (20). To restore equality, we must increase *ζ* or reduce X (increase *γ*). Thus the increase in k shifts the FE curve upward and to the right to the new position FE’. The optimal *γ* increases.

The intuition is as follows. If *θ*’ and *γ* are kept unchanged, an increase in *ζ* raises *ζ* equiproportionately which increases the average cost of collection of taxes on consumption. This disturbs the balance of benefits and costs of the two kinds of taxes. The consumption tax now has a higher marginal cost per unit of revenue than the inflation tax, making it desirable to switch some of the burden of taxation away from consumption and toward inflation. At the new optimum, the inflation tax is levied at a higher rate, and the consumption tax is less than fully indexed, or less than full interest is charged for the tax revenues held by the firms for the length of the collection lag. Note that the conventional argument based on the loss of revenue is not the driving force; instead, the general equilibrium reasoning rests on the ‘secondary’ effects on collection costs.

A restatement of the intuition further clarifies the issue. Suppose we proceed as suggested by the invariance result, keeping *γ*_{t} unchanged, and charging nominal interest on the taxes due for the period of the collection lag. Since the real interest rate is positive, when the revenues are finally received, they are larger in real terms. Under Végh’s specification, the average cost of collection is a stable function of the real revenues. Therefore, the average cost of collecting the larger real revenues is now higher.

## V. Concluding Comments

The particular model studied in this paper embodies many specific assumptions. The ‘regular’ tax is levied on consumption, it is collected by the firms in advance and held for the duration of the collection lag, and so on. But similar analysis comparing other kinds of taxes (for example, on labor income) and the inflation tax can be constructed. The specific results will depend on the nature of the collection or distortion costs associated with the tax system. For example, the costs of the regular tax system are a function of the effective rate *θ*’, then the invariance result will prevail.

But as a general point, the analysis emphasizes the need for looking at the issue in a wider framework than has been common. When we recognize the existence of a collection lag, our response should be to re-calculate the whole tax structure, and not merely the inflation tax. When this is done, the important channel of interaction is the effect of the inflation tax on the collection (or distortion) costs associated with the other taxes, not on the revenues from those taxes.

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*Working Paper No. 90/48*, (International Monetary Fund, 1990).Frenkel, Jacob, “Comment on Zvi Hercovitz and Efraim Sadka’s ‘Optimal Currency Substitution Policy and Public Finance’,” in

, eds. Asaaf Razin and Efraim Sadka, (London: Macmillan, 1987), 165–169.__Economic Policy in Theory and Practice__Kimbrough, Kent S., “The Optimum Quantity of Money Rule in the Theory of Public Finance,”

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, 75, (March 1973), 67–82__Swedish Journal of Economics__Tanzi, Vito, “Inflation, Lags in Collection, and the Real Value of Tax Revenue,”

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^{}*

Professor of Economics, Princeton University, and Visiting Scholar, International Monetary Fund, June-July 1930. I thank Matthew Canzoneri and Nurun Choudhry for useful conversations, and Pablo Guidotti, Lars Svensson, and Carlos Vegh for valuable comments on the first draft. The paper does not necessarily represent any views of the Fund.