Inflation, Lags in Collection, and the Real Value of Tax Revenue
Author: VITO TANZI

The literature dealing with the impact of inflation on taxation is so extensive that it may suggest that it would be difficult to write anything novel on this subject. Yet a close perusal of this literature shows that it has been biased by the recent experiences of the industrialized countries. For these countries, inflation has generally been associated with increases in the real value of tax revenues, so that many authors have been led to believe that the main inflation-induced problems are the prevention of this supposedly unwanted, or at least unlegislated, increase in revenue and the neutralization of the inevitable effects on the redistribution of the tax burden among income groups. The increase in real revenue is likely to occur mainly when (a) the lags in the collection of taxes are short, and (b) the tax systems are elastic. However, while these conditions seem to characterize many industrialized countries, they are not common to all countries.

Abstract

The literature dealing with the impact of inflation on taxation is so extensive that it may suggest that it would be difficult to write anything novel on this subject. Yet a close perusal of this literature shows that it has been biased by the recent experiences of the industrialized countries. For these countries, inflation has generally been associated with increases in the real value of tax revenues, so that many authors have been led to believe that the main inflation-induced problems are the prevention of this supposedly unwanted, or at least unlegislated, increase in revenue and the neutralization of the inevitable effects on the redistribution of the tax burden among income groups. The increase in real revenue is likely to occur mainly when (a) the lags in the collection of taxes are short, and (b) the tax systems are elastic. However, while these conditions seem to characterize many industrialized countries, they are not common to all countries.

The literature dealing with the impact of inflation on taxation is so extensive that it may suggest that it would be difficult to write anything novel on this subject. Yet a close perusal of this literature shows that it has been biased by the recent experiences of the industrialized countries. For these countries, inflation has generally been associated with increases in the real value of tax revenues, so that many authors have been led to believe that the main inflation-induced problems are the prevention of this supposedly unwanted, or at least unlegislated, increase in revenue and the neutralization of the inevitable effects on the redistribution of the tax burden among income groups. The increase in real revenue is likely to occur mainly when (a) the lags in the collection of taxes are short, and (b) the tax systems are elastic. However, while these conditions seem to characterize many industrialized countries, they are not common to all countries.

When one deals with countries having somewhat longer lags in the collection of taxes, and tax systems with money-income elasticities not much greater, or even less, than unity, the consequences of inflation can be very different, especially when the rate of inflation becomes high. Unfortunately, these alternative situations are not products of an economist’s imagination but, on the contrary, have either existed or continue to exist in many developing countries, and perhaps even in some industrialized countries. For these countries, the problem has not been an increase, but rather an inflation-induced fall in real revenue. In many cases, this fall has, in itself, become a contributing factor in the inflationary process when the affected governments have financed the fiscal deficits through the printing of new money.

There has been hardly any analysis of what happens to real tax revenue when the tax systems are not elastic, the lags in tax collection are not short, and the rate of inflation becomes high. The main objective of this paper is to show that when the rate of inflation becomes high, the inevitable lags in the collection of taxes become very important and, unless compensated for by high elasticities, may often lead to a decrease in real revenue. The impact of lags has generally been ignored.1 After a theoretical discussion of the issues, the paper will use Argentina as a concrete example of a country in which the combination of high inflation, a relatively long average lag in tax collection, and a low elasticity of the tax system has recently brought about a drastic fall in real revenue. The paper will focus on the effects of the lags, and will thus ignore the inflation-induced distortions in taxable bases that may also affect real revenues.

I. Theoretical Analysis

taxable events and de jure elasticity

This paper defines a taxable event as one that creates a legal financial liability on the part of a taxpayer (individual, retailer, corporation, etc.) toward the state. For example, the earning of income may create an obligation on the part of the income earner to pay an income tax; the spending of income may create a legal obligation on the part of the buyer or seller to pay a sales tax on that expenditure;2 or the ownership of a real asset may create an obligation to pay a property tax. This paper will assume that a seller sells X dollars worth of commodities that are subject to a sales tax at an ad valorem rate of α. Or, alternatively, it will assume that an individual earns Y dollars and that this income is subject to a proportional income tax levied at the rate of β. In the first case, the taxable event (i.e., the sale) has created a liability of αX in the second, it has created a liability of βX. However, while these liabilities have been created at the moment the taxable event has occurred (i.e., at the moment when income has been earned or spent), the government will receive the payment only at a later time. In the first case, the seller may not transfer the tax to the government until some later period. In the second case, the taxpayer himself, or, alternatively, the employer who may have withheld the tax will transfer the tax money after some delay.

These lags introduce complications in the proper definition of tax elasticity, and even that of the average tax rate. In a theoretical world in which payments were made at the same time that taxable events occurred, there would be a clearly definable elasticity of the tax payment with respect to income, or some other taxable base, that would not be affected by any lag in the payments. As national income changed, depending on the legal characteristics of the tax, the percentage change in the tax collection would be equal to, or more or less than, the percentage change in national income. This theoretical, or de jure elasticity would consequently be equal to, or more or less than, unity. In this lagless world, if the elasticity of the tax system were greater than unity, inflation would, ceteris paribus, bring about a real increase in tax payments, so that the ratio of taxes to national income would, in fact, rise.3

An alternative way of looking at this theoretical or de jure elasticity is to relate the tax collection at a given time to the income (or, perhaps, the expenditure) at the time when the event that created the legal liability occurred (in other words, to its legal base). In this case, if the earning of an income Y0—where 0 indicates the period when the taxable event occurred—creates a tax liability equal to β0, which, because of a lag in collection, is paid, say, two periods later when nominal income has risen to Y2, one could calculate the elasticity (or even the average tax rate) by relating the tax payment βY0 to the original income Y0, rather than to the present higher income Y2, as is generally done. However, the elasticity—or even the average tax rate—calculated on this basis would be different from the effective one normally estimated statistically, which relates revenue received in a given period to income received in the same period, and thus ignores the legal connection defined above.4 In conclusion, whenever nominal income is growing and there are lags in tax collection, both the elasticity of the tax and the average tax rate will be different, if they are estimated with respect to current income rather than with respect to the income prevailing at the time of the taxable event. This difference can become substantial whenever there is considerable inflation associated with sizable lags. The nature of these lags will be discussed later. For the time being, it is sufficient to assume that these lags exist and that somehow they can be measured.

inflation, lags, and real tax revenue

The impact of different lags and rates of inflation on the real value of one dollar of tax revenue can be estimated by solving the equation

R=1(1+P)n(1)

where R is the real value of a dollar of tax revenue collected today but measured in prices of the period when the taxable event occurred; p is the monthly rate of inflation; and n is the size of the lag, expressed in months.

Table 1 has been calculated by solving equation (1) in relation to various rates of inflation and lags. The table assumes that taxes are collected successively, with lags that may be zero, 1 month, 2 months, 3 months … up to 12 months. Surely, these alternatives embrace the realities of the tax systems of most countries, although particular taxes may be, and are at times, collected with even longer lags. These lags are shown horizontally at the top of the table. Vertically, on the left, the table indicates selected monthly rates of inflation. It starts with the assumption of a zero inflation rate per month, then it considers 1 per cent per month, 2 per cent, 3 per cent, and selective rates all the way up to 50 per cent per month.5 All of these rates have been experienced by some countries, at least for some months, in recent years. The alternative lags and rates of inflation provide a matrix that is likely to include the experience of most countries.

Table 1.

Impact of Lags in Tax Payments and of Rates of Inflation on the Real Value of Tax Revenue

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If the price elasticity of a country’s tax system were one, the table could also be used to raise questions about the effect of inflation on total tax revenue. In other words, it would allow one to answer the following question: Assuming that taxes in a given country are collected with an average lag of x months, and that the country is experiencing a monthly rate of inflation of y, what will be the impact on the real value of its tax revenue? 6 Each row in the table will then tell us what happens to the real value of the tax revenue when, given a certain rate of inflation, the length of the lag changes from zero to 12 months. Thus, for example, if the monthly rate of inflation were 10 per cent, and the country collected its taxes with a zero lag, it would not experience any fall in the real value of its revenue; however, if it collected its taxes with a 1-month lag, it would experience a 9 per cent fall; if it collected its taxes with a lag of 5 months, it would experience a 38 per cent fall, and so on. The higher the monthly rate of inflation and the longer the lag in payment, the greater will be the reduction in the real value of tax revenue that a country will experience. This is seen clearly in the table by reading down the columns and across the rows.

Alternatively, the columns in the table show what happens when, given a certain average lag in collection, the rate of inflation is assumed to become progressively higher. Thus, for example, a country that had an average lag of four months would lose 4 per cent of the real value of its tax revenue if, after a period of price stability, it entered an inflationary period in which prices increased at the rate of 1 per cent per month; it would lose 18 per cent if the rate of inflation rose to 5 per cent per month, and 32 per cent if it rose to 10 per cent. If the rate of inflation should become extremely high, say 50 per cent per month, the real value of taxes would be reduced to 20 per cent of what they would be in the absence of inflation. If the rate of inflation is zero, there is no fall in the real value of taxes, since the value of the dollar collected would remain unchanged over time.7 Alternatively, if the lag should be zero, then there would be no decrease in the real value of tax revenue regardless of the rate of inflation.8

In summary, the main conclusions so far are as follows: first, given the rate of inflation, and assuming that the price elasticity of the tax system is one, the longer the lag in the collection of taxes, the greater will be, ceteris paribus, the net inflation-induced real reduction in the tax revenues that the government receives. Second, given the lag in the collection of taxes, the higher the rate of inflation, the lower, ceteris paribus, will be the real value of the tax revenue. Finally, the table also shows the gains that are possible to a country from the reduction of the lags in payments. For example, a country experiencing an inflationary period could replace an income tax with a value-added tax collected with a much shorter lag, and would benefit from the change even if, in the absence of inflation, that change would have been associated with a zero net change in revenue.

inflation, lags, and revenue when the elasticity is not one

An assumption used above was that of a unitary de jure elasticity of the tax system. This implies that inflation per se, in the absence of any other factors, would not generate any real increase or fall in revenues. The assumption of unitary elasticity for the tax system in toto is a realistic one for most developing countries (as well as for some developed countries), as their personal income tax is not of overwhelming importance and often is not particularly progressive. However, if a country were highly dependent on personal income taxes, and if these taxes were collected with a short lag,9 then, as the current literature on the impact of inflation on taxes has emphasized, the inflationary conditions would bring about, ceteris paribus, an increase in the real value of revenues by shrinking the real size of the exemptions and the brackets. Thus, if taxes are collected with a lag, the gain coming from the above-mentioned progressivity (and possibly from distortions in components of income other than wages) would have to be balanced against the losses emphasized in the table. Whether revenues in real terms would increase or decrease over a given period would depend on the interrelationship among the elasticity, the rate of inflation, and the lag in collection.

Given the existence of a lag, and given an elasticity that exceeds unity, a steady increase in the general price level (following a period of price stability) will, at first, lead to the same fall in real revenue shown in the table; then, as the average price level becomes progressively higher, and as the increase in the price level is accompanied by a greater than proportional increase in nominal tax revenue, real tax revenue (and thus the share of revenue in the income of the current period) will start increasing and will continue increasing as long as the price level keeps rising. In time, the initial loss in revenue will be made up and, if inflation continues, real revenue will rise. The shorter the lag in collection, and the higher the elasticity, the more quickly will the real level of tax revenue regain and exceed the preinflationary level. The relationships between real revenue and the rate of inflation, the level of prices, the elasticity of the tax system, and the size of the lag are analyzed mathematically in the Appendix.

Once the issue of elasticity is introduced (in other words, once one assumes that the de jure elasticity of the tax system is different from one), the arithmetic computations needed to recalculate the results given in the table become more complex; furthermore—and this is more important—the results in the table can no longer be used as estimates of what happens to the total real tax revenue of a country as a result of changes in the rate of inflation.10 It must, however, be emphasized that the issue of elasticity is important mainly for the personal income tax, and may be compensated for in part by the existence of other taxes (such as those on property and on sales that are levied with specific rates) that are likely to have an elasticity of less than one. Therefore, as a working hypothesis for developing countries, a unitary elasticity for the whole tax system would often be a defensible one. This is the hypothesis used in connection with Argentina in Section III.

II. The Measurement of the Average Lag

The lag in the collection of a given tax—that is, the lapse of time between the taxable event and the tax collection connected with that event—is made up of two parts, which could be called, respectively, the legal lag and the delinquency lag. The first is the government-sanctioned delay in payment, which carries no penalty. For example, a self-employed individual is normally required to pay his income tax some time after the earning of the income; similarly, a retailer transfers the receipts from sales taxes to the authorities some time after the sales have taken place. The delinquency lag exists when the payment is made after the time it falls due. In most cases the legal lag is the most important, although under particular circumstances the delinquency lag can become very significant.

In order to deal with the whole tax system of a specific country, one needs to determine the size of the lag for that tax system. This lag, of course, is likely to differ among countries; it will be affected by the lag of each tax used and by the particular tax structure of the country. If the size of the lag for each tax is known, then the lag for the whole system can be determined by taking a weighted average of all the lags related to the specific taxes. The weights in this calculation are determined by the relative importance of each tax in the total tax revenue. It follows that the fewer the tax sources on which the country depends, the easier will be the estimation of the overall lag. The lag that applies to each tax can often be determined only by having discussions with the authorities in charge of tax collection, as the tax laws are often not very helpful on this matter. These discussions should cover the method of payment, the time when payment is made, the variability of the lag, etc.

Let us assume that Ti refers to the proportion of the total tax revenue generated by a particular tax, and that Li is the lag between the time when the liability for that tax payment is created (i.e., the time of the taxable event) and the time when the tax payment is actually made.11 The i,…,n refer to the specific taxes (on income, wealth, etc.) that are being used in the country to collect revenue. This paper defines the tax system’s total lag Lt as

Lt=Σi=1nTiLi(2)

This is the lag used in Table 1.

III. An Application to Argentina

In Argentina, the number of taxes being collected at any particular moment is truly very large. For example, in June 1976 the Tax Bureau (Dirección General Impositiva) was collecting revenue in connection with 34 existing taxes and 48 defunct taxes; furthermore, the Tax Bureau was not the only agency responsible for tax collection, since part of total tax revenue was collected by the customs administration, by the local governments, and by the social security administration. However, the specific taxes can be grouped by main categories, and the average lag for each one of these categories can be estimated. Discussions with the authorities responsible for tax collection indicated that in 1975 and for the first half of 1976, the sales or value-added tax was collected with a lag of 90 days, stamp taxes with a lag of 30 days, fuel taxes with a lag of 45 days, import and export taxes with lags of 45 days, excise taxes with a lag of 120–50 days, social security taxes with a lag of 30 days, and income and property taxes with lags of at least a year.12

The basic information for estimating the average lag for the tax system of Argentina has been summarized in Table 2.13 By using this information to solve equation (2), one obtains, alternatively, a lag of 4.3 months when social security taxes are included (column 2), and a lag of 5.7 months when social security taxes are excluded (column 3).

Table 2.

Argentina: Tax Revenue by Type of Tax (1974) and Lags in Collection (1975)

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Sources: The revenue data were provided by the Argentine Ministry of Economics, and the information on lags by officials from the different collecting agencies.

Lag between the creation of the liability for payment and the actual payment.

Proportion of total revenue generated by a particular tax.

Including social security taxes.

Excluding social security taxes.

These are partly stamp taxes collected with a 30-day lag and partly local property taxes collected with at least a 12-month lag. The specific breakdown is not available.

In 1974 the ratio of total taxes, including social security, to gross domestic product (GDP) for Argentina was 17.74. By 1975 this ratio had fallen to 12.24, or by 31 per cent. If social security taxes were excluded, then the ratio fell from 12.60 to 7.75, or by 38 per cent. It would be interesting to see to what extent one can approximate this reduction in taxes by looking at the behavior of prices between 1974 and 1975. In 1974 the wholesale price index (WPI), which is the most reliable one in the case of Argentina, rose at an average monthly rate of 1.5 per cent. In 1975 the monthly increase in the WPI was 9.4 per cent. Using the above-determined lags in connection with Table 1, it can be calculated that an increase in the monthly rate of inflation from 1.5 per cent to 9.4 per cent would have been associated with a fall in real revenues of 26 per cent if the lag were 4.0 months, 31 per cent if the lag were 5.0 months, and 36 per cent if the lag were 6.0 months. Since, as indicated above, the estimated lag for the tax system was at least 4.3 months with social security and 5.7 months without social security, it can be concluded that in 1975 the actual behavior of tax revenues followed very closely the behavior that would be expected from the theoretical analysis. In fact, if lags of 5.0 months (including social security) and 6.0 months (excluding social security) are taken, the expected fall in real revenues turns out to be almost exactly equal to the actual fall.14

In the early part of 1976, the rate of inflation accelerated and for some months became very high. For example, it was 19.1 per cent in January, 28.5 per cent in February, 53.8 per cent in March, and 26.3 per cent in April. At this rate of inflation, lags of 5.0 months would have reduced the real revenues by about 75 per cent. The preliminary information available indicates, in fact, that revenues fell by at least this percentage. By the end of 1975 and early 1976, other factors came into play, resulting in the distortion of the theoretically estimated relationship between the behavior of taxes and the behavior of prices. These other factors, a discussion of which is beyond the scope of this paper, concern the increasing rate of tax evasion and particularly the increasing lag in payment owing to the fact that many taxpayers faced with very low penalties for tax delay often resorted to a postponement of their taxes as the cheapest form of available credit. Therefore, by 1976 one would expect a fall in revenues that exceeds the theoretically estimated fall. At least for the first few months of 1976 this, in fact, happened.

IV. Concluding Remarks

The foregoing analysis indicates that countries that face highly inflationary pressures, or that are likely to face them at some stage, should pay much more attention to the impact that lags in the payment of taxes may have on real revenue. No country, of course, can collect taxes without any delay, and some taxes require longer lags than others. However, for most countries the necessary delay—that is, the legal lag—in tax collection can be substantially reduced while still recognizing that taxpayers require time to gather all the information needed to calculate their tax payments and to make them.

Policy should be aimed at trying to reduce the legal lag to some “optimal” level, and the delinquency lag to zero. This delinquency lag can be eliminated by stiff penalties, applied on top of tax payments, that have been adjusted for the change in the price level that has occurred during the delay. Stiff penalties alone are not sufficient, as a number of countries have learned, since what appears to be a stiff penalty when the rate of inflation is low may become insignificant when the rate of inflation becomes very high.

Argentina has recently passed a law that will require the indexation of payments due to the Government, in addition to the penalties. As to the legal lag—that is, the one between the taxable event and the time when the tax is due—governments should attempt to reduce it without creating difficulties of an accounting nature for the taxpayer. Citing again the case of Argentina, the lag in the value-added tax, which was 90 days, is being reduced to 60 days, which still appears to be relatively long. It is reported that in Chile the value-added tax is now being paid twice a month in order to reduce to a minimum the erosion in real value associated with inflation. Other taxes on production, transactions, imports, and exports can also be collected with lags that can be reduced to perhaps not much more than one month.

For property and income taxes, however, the question will remain much more complex, as long as these taxes are paid on an annual, rather than on a monthly, basis, as is the case with income taxes collected at the source on wages and salaries. For these taxes there is the problem of estimating the base on which they are calculated. For example, in Argentina the advance payments of taxes on the income of enterprises were estimated on the basis of the previous year’s income. In a situation where prices have increased significantly between one year and the next, an advance payment based on the previous year’s nominal income becomes much too low in relation to present income. A solution that is being considered in Argentina, and that seems to be a reasonable one, would be to base the advance payment on the previous year’s income adjusted for the change in prices over the year. An alternative solution could be to make the taxpayer estimate the income that he expects to receive in the current year, and make him pay monthly or quarterly installments based on it. At the time when he settled the final payment, he would have to escalate the payment for the price change since the middle of the taxable year if such a payment was due to the Government. For property taxes, the same problem occurs. With high rates of inflation, it would be necessary to escalate the previous year’s assessment by the rate of inflation before the tax was calculated for the current year.

APPENDIX15

Define:

  • R = real value of tax revenue accruing in period 0 but paid n periods later, in terms of resources in period 0 (i.e., the period the taxable event took place)

  • T = nominal value of accruals in period 0

  • P0 = price level at time 0

p˙=1P0dP0dt=instantaneousrateofinflationattime0
R=TP0(1+p˙)n(3)

To give an economic interpretation to equation (3), one needs to take small discrete values of n and p.

E = elasticity of tax accrual with respect to changes in the price level (de jure elasticity)

E=dTdP0.P0T(4)

Equation (4) can be rewritten as dT ET,

dTdP0=ETP0and(5)
P0dTETdP0=0(6)

Dividing by POT, we get

dTTEdP0P0=0fromwhich,takingtheintegral,weget(7)
1TdTEP0dP0=C(8)

where C is the constant of integration. From (8) we get

LnTELnP0=Cor(9)
TP0E=ec=KandT=KP0E(10)

Substituting in equation (3), we get

R=KP0EP0(1+p˙)n=KP0E1(1+p˙)n(11)

Equation (11) can be differentiated partially with respect to the rate of inflation , the absolute price level P, the elasticity E, and the lag n. The equations so derived are as follows:

Rp˙=nKP0E1(1+p˙)n+1(12)
RP=(E1)KP0E2(1+p˙)n(13)
RE=KP0E1LnP0(1+p˙)n(14)
Rn=KP0E1Ln(1+p˙)(1+p˙)n(15)

Equation (12) indicates that the real value of tax revenue R will fall with an increase in the rate of inflation as long as the lag n is greater than zero. If n = 0, R will not change with regardless of the size of elasticity E.

Equation (13) indicates that RP0 as E ⋛ 1. If the elasticity E is equal to unity, the real value of revenue R will not depend on the absolute price level, but only on the rate of inflation and the length of the lag.

Equation (14) indicates that, given the initial price level P0 and the rate of inflation then the higher elasticity E is, the higher real revenue R will be.

From equation (15) it can be seen that, given a positive rate of inflation, the longer the lag, the smaller real revenue will be.

*

Mr. Tanzi, Chief of the Tax Policy Division of the Fiscal Affairs Department, is a graduate of Harvard University. He was formerly a professor and Chairman of the Economics Department at American University. He is the author of The Individual Income Tax and Economic Growth and of numerous chapters in books and articles in professional journals.

1

One important exception is the paper by Teruo Hirao and Carlos A. Aguirre, “Maintaining the Level of Income Tax Collections Under Inflationary Conditions,” Staff Papers, Vol. 17 (July 1970), pp. 277–325. Hirao and Aguirre limit the discussion to the income tax, while the discussion in this paper refers to all taxes. Of interest also is the paper by Pedro Rado, “Income Payment Systems and Inflation” (mimeographed, July 2, 1975).

2

Legally the tax may be levied either on the seller or on the buyer, although, except for an expenditure tax, it is the seller who generally transfers the tax payment to the authorities.

3

By the same token, the real value of revenues would not be affected by inflation if the elasticity were one. Thus, if indexation of the tax system succeeded in making the elasticity equal to one, there would not be, m this lagless world, any inflation-induced increases in the ratio of taxes to national income. But, of course, this theoretical situation is not practically possible.

4

During this whole discussion the underlying legal structure is assumed to remain unchanged. This paper is, therefore, talking about automatic, or built-in, elasticity that excludes the effects of discretionary changes.

5

In order to use the table in connection with the Argentine situation, the results associated with monthly inflation rates of 1.5 per cent and 9.4 per cent are also shown.

6

To simplify the analysis and emphasize the impact of price changes, real growth during the inflationary period is assumed to be either zero or insignificant.

7

Even with zero inflation, the taxpayer gets some advantage owing to the postponement of taxes. The higher the rate of discount, the greater the advantage related to a given lag.

8

Even in this case, real revenue would be affected to the extent that inflation distorted the taxable bases (capital gains, interest, profits, property values, etc.). However, the change in real revenue would not be induced by the lag. It should be recalled that this paper has assumed a unitary price elasticity of the tax system. For an analysis of the inflation-induced distortions of interest income, see Vito Tanzi, “Inflation, Indexation and Interest Income Taxation,” Banca Nazionale del Lavoro, Quarterly Review, Vol. 29 (March 1976), pp. 64–76.

9

In most countries, income taxes, except for those withheld at the source, are collected with considerable lags.

10

But they still indicate how lags reduce the real value of each revenue dollar collected compared with the real value of a tax liability of a dollar at the time when it accrues—in other words, in relation to what real revenue would be in a lagless world.

11

It will be necessary to assume that the variance for these specific lags is so small that it can be ignored. If the variance is not small, the whole issue becomes much more complex and the method suggested in this section may give results that are no longer reliable.

12

The Government is, at the present time, contemplating the possibility of reducing some of these lags.

13

The revenue figures relate to 1974, which, by Argentine standards, can be considered a good year.

14

Since the change in real GDP was overwhelmed by the price change, the ratio of taxes to GDP is a close indicator of the behavior of real revenue.

15

The author wishes to thank William J. Byrne, economist in the Tax Policy Division, for his help with this mathematical appendix.