Income averaging and progressive taxation

Journal

of Public Economics

INCOME

12 (1979) 387-397.

AVERAGING

AND

0 North-Holland

Publishing

PROGRESSIVE

Company

TAXATION

John CREEDY* University

of Durham,

Received September

Durham

DHl

1978, revised version

3HE: England

received

August

1979

This paper exammes a simple scheme for the taxation of income on the basis of an average of income over three years, and where no tax is paid in arrears. It is shown that individuals have greater liquidity than under a conventional system. Longitudinal data on incomes of several cohorts of males in the U.K. are used to estimate the possible effects on total revenue. The effect on inequality is also briefly examined.

1. Introduction It is well known that in a progressive tax system higher taxes are paid on fluctuating incomes. In the United Kingdom income is cumulated over the financial year, but only in certain exceptional cases are there any provisions is that the for averaging over a number of years. ’ An obvious implication burden of taxation falls more heavily on individuals with more highly fluctuating incomes, so that the question arises of how this burden is distributed over the population. If those with less-stable incomes are compensated by higher absolute incomes, when averaged over a longer period, then it may be thought that no serious equity issues are raised. Furthermore, there may be an incentive for individuals to attempt to achieve a smooth earnings stream, so that the difficult question of labour supply is also raised. There has, however, been very little systematic analysis of the possible effects of income averaging for tax purposes. Considerably more attention has been devoted to the question of the appropriate taxation of ‘permanent’ and ‘temporary’ incomes; the classic treatment being that of J.S. Mill (1848)’ *This paper forms part of a larger project carried out under the aegis of the National Institute of Economic and Social Research. I am grateful to the Department of Health and Social Security for providing the data and for financing the project. I should also like to thank Michele Foot for computing assistance, participants in seminars at the universities of Reading, Warwick and Goteborg, and the editor and referees of this Journal for very helpful comments. ‘Contributions for sickness and unemployment insurance, and pensions, are actually based on weekly earnings, but since there is an upper limit some people with fluctuating earnings will pay lower contributions. ‘See the Ashley edition of Mill (1920, pp. 810416) for the many changes which Mill made after giving evidence to a number of select committees. The question of differentiation of income

388

J. Greedy, Income meraging

and progressice

taxatio,l

There has, however, been more interest in the United States.3 Early analyses are by Simons (1938) and by Vickrey (1938) who both favoured some method of averaging. Vickrey (p. 79) suggested that, ‘the discounted value of the series of tax payment made by any taxpayer should be independent of the way in which his income has been allocated to the various income years’.4 The scheme proposed by Simons (1938, p. 154) was more complex, and allowed for an ex post adjustment (after a period of, say, five years) whereby a rebate would be given in cases where the amount which a person would have paid on a constant stream (equal to the average value of the actual income stream) is lower than the amount actually paid by more than 10 7;. In the United Kingdom the most serious discussions of this subject can be found in the reports of the Royal Commissions of 1920 and 1952.5 In 1920 the commission suggested that all employees should be taxed on the basis of each year’s income, and pointed out that, ‘the revenue is likely to benefit by the change’.6 The possible effect on total revenue will be considered in more detail later in this paper. The 1952 commission did not clearly distinguish between income averaging and systems which involve collection in arrears. The latter may provide an interest-free loan whenincome is rising but may involve hardship in the case of an unanticipated fall in income.’ Although some averaging schemes have involved collection in arrears, the next section of this paper considers a scheme which specifically avoids this difficulty. The plan of the paper, then, is as follows. Section 2 suggests a simple scheme for income averaging over three years, and examines the consequences for tax liability of changes in income, using a simple hypothetical tax schedule. Section 3 then uses longitudinal data to examine the implications of this simple schedule for the possible changes in total tax revenue which would result from the adoption of an averaging scheme, and the changes in tax rates required to maintain an unchanged total revenue. Finally a number of issues concerning inequality are briefly examined.

tax according to the source of income is discussed in Shehab (1953, pp. 92-94, IO7- 111). Shehab with a sophistry comparable only to that of later notes that the issue. ‘was discussed,. scholastic logic’ (p. 5). ‘An averaging scheme was introduced in Wisconsin in 1928, but lasted only four years. A further scheme was introduced in 1964 and heavily criticrsed m David et al. (1969). There has also been interest in Canada. see Merkies (1965). See also Steger (1956) and Ttbbetts (1940). 4More recently Vickrey (1972) has suggested lifetime averaging. ‘For a summary of the 1920 report by Pigou, one of the commissroners, see Pigou (1923, pp. 126 140). The chairmen in 1920 and 1952 were Colwyn and Radcliffe, respectively. ‘(1920, para. 479). At that time wseekly wage earners were assessed quarterly, while many salaried people were assessed under schedule D, involving three-year averagmg. All employees were moved to schedule E in 1922, however. ‘The system operating in 1940 in the U.K., which involred collection in arrears, 1s discussed rn Barr, James and Prest (1977. p. 23).

J. Greedy.

Income

ureraging

and progressive

tarution

389

2. A simple averaging system 2.1. The assessment

of tax liability

In order to avoid collection in arrears it is necessary that at the end of each period the total tax liability is met in full by the taxpayer. Consider the following stages in the assessment of taxation for each individual. (i) In each period calculate the average income over the current and previous two periods. The resulting three-year moving average is denoted .u. (ii) Calculate the total tax which would have been paid had income been constant at the value of X. Thus, if the ‘nominal’ tax schedule involves taxation of T(x) on an income of x, the appropriate figure is simply 3T(.f). (iii) Calculate the total tax actually paid over the previous two periods. For convenience of exposition at this stage, consider the case where income had been constant in previous years at the value of, say s. (This assumption is not critical for the following discussion.) The tax paid would then be 2T (x). (iv) Subtract the value obtained in (iii) from that in (ii). The resulting figure is the tax to be paid in the current period. If income in the third period is x3, after changing by 1006 “;,, the tax to be paid may be denoted T* (x3). In the simple case considered here T* (.Y~) = 3 T(X) - 2T (.u). 2.2. Changes

in income and the tax 1iabilitJ

It is then necessary to consider how the tax liability changes. under a given nominal tax schedule, as a result of a change in income. As in section 2.1, it is assumed that income up to the current period is constant at s, and changes to xj = (1+6)x in the current period. However, the opening sentence of this paper must first be qualified. If a progressive scheme is regarded as one in which the marginal tax rate (MTR) exceeds the average tax rate (ATR), then progression alone is not sufficient to ensure that fluctuating incomes pay more tax. In the case of the linear negative income tax, which is progressive but has a constant MTR, it is easy to show that the above system of averaging makes no difference to the tax liability. The question of averaging is only relevant where marginal tax rates continually increase with income. The following analysis is therefore restricted to a simple tax formula which has an increasing MTR. In order to obtain results which can readily be interpreted, it is most convenient to use a simple tractable function which contains only one parameter, k. This single parameter then affects both the ‘progressivity’ of the system and, combined with the distribution of income, the total tax revenue. The average tax rate t(x)= T (x)/x, is given by t (.x ) = 1 - x k,

k< 1,

(1)

390

J. Greedy, Income

averuging

und pr-ogressiw

taxation

so that post-tax income, J, is equal to XI-~ and the marginal tax rate is l-(l-k)x-k. This tax formula was suggested by Edgeworth in his discussion of the report of the 1920 Royal Commission,’ and has often been used since then. The choice of this function is necessary in order to focus attention specifically on the effects of averaging, rather than other features of any actual tax system (it does not correspond to the complex U.K. scheme but is more relevant to other countries). Using the definitions and assumptions mentioned above, and noting that X=x(3+6)/3, then

The relevant question here is whether the tax paid in the third period under income averaging is less than the amount which would otherwise be paid where the tax is based on each separate year’s income, i.e. if T*(.u,) < T (x3). In order for this condition to hold it is necessary that

1-k

3 3_+2

( ! 3

>2+:(1

+zi)’ -:k.

(2)

Let A and B be the left-hand side and right-hand side, respectively, of the inequality in (2). Now when 6 =0 it can be seen that A =B, giving the obvious result that there is no difference in the tax liability. In order to show that the inequality always holds it is therefore only necessary to show that L’A/S >L’i?/&? for 6 >O (that is, for an increase in income of 1006 T,,), and vice versa for 6 < 0. For 6 > 0 the required condition is that 3+s

-k

(--I 3

> (1 +?jmk

or 3+36

k>l

( > 3+6

%ee Edgeworth (1925, p. 268), who also considered the more general form t(x)= I -hx k and found that rt gave a good lit to a number 01 actual tax schedules (with b very close LO unity). Dalton (1954, p. 68) showed that the above form can be derived from the principle of equal proportional sacrifice combined with a logarithmic utility function. Tlus deribatlon would not have been supported by Edgeworth. who rejected both proportional sacrifice and the Bernoulli utility function. See also Doresamiengor (1929, p. 327).

J. Creedp, Income averaging

and progressive

taxation

391

whence

so that (3) always holds. The same steps also confirm that when 6
3. Further analysis using longitudinal

data

This section uses information provided by the Department of Health and Social Security (U.K.) about the earnings of three cohorts of males over the three-year period 1971-73. The cohorts were born in 1923, 1933 and 1943.” It is important to note that these data are used for illustrative purposes, since no information is available about the family circumstances and allowances claimed by each individual, and of course a hypothetical tax function is used. ‘This complex problem cannot be discussed here. but it is of interest that the question of annual cumulation and incentives has been considered in the contexts of strikes [see Cole (1975) and Gennard and Lasko (1975)], and unemployment duration [see Atkinson and Flemmmg (197811. ‘“It should be stressed that complete anonymity has been maintained, and no individual can be identified from the records. The sample sizes for the 1943, 1933, and 1923 cohorts were 1346. 1157, and 1252 individuals, respectively. Individuals were included in the samples irrespective of the number of weeks of sickness or unemployment they experienced. so long as their earnings and contributions records were complete and consistent. For furthei- analysis of these data see Creedy (1979).

J. Crrcdy, Income uwruging

392

and progressice

tuxation

They do, however, accurately reflect earnings mobility, and from the point of view of the present paper this is the most critical element in the analysis.” So long as the conditions which affect allowances change slowly relative to earnings changes, the results should still be of interest.

3.1. Changes

in total

recewe

The total tax revenue per person is simply the difference between average pre-tax and average post-tax income. The analysis is therefore considerably simplified if a convenient transformation exists between the two distributions of x and J. Such a transformation exists for the tax schedule used in section 2, and the case where income is lognormally distributed. Thus, if the distribution function, F(x), is A(p,a’), where p and 0’ are the mean and variance respectively of the logarithms of income, then it can be shown that12 J’=xl

Using

-k is A(/l(l -k),

this result the means

(1 - k)2~2).

of s and 1; s and r respectively,

and

(4) are given by

(5) f=expjp(l

-k)++(l

-/~)‘a~).

Eqs. (5) can be used to obtain the tax per person when each year’s income is taxed separately, given the annual values of p and u2. The problem remains of obtaining the corresponding values for the distribution of average income over three years. This problem has been discussed extensively elsewhere, but for present purposes it is assumed that the distribution of the sum of income over three years can also be approximated by the lognormal distribution.‘” Thus, if the sum. X, is distributed as A(nz,.s’), average income. X 3. is distributed as A(m - log 3, s’ ). Under an averaging system the distribution of post-tax income can be obtained using (4). so that average post-tax incamc.

y* say, is given by ~*=exp~(1-k)(m-log3)+~(1-k)20”j.

(6)

The values of ~1and 0’ for each year, and m and .s7 for each cohort are given in table 1. These are used to obtain table 2 which shows the tax paid for various periods for two values of k. I

Table Summary

measures

of alternal~+e

distrihurions.

Tax year 1973

sum of three years

1971

1972

Cohort lY43 Geomerric Mean Variance of logs

IS16 0.240

1776

2010

0.256

0.254

5443 0.146

Cohorr 1933 Geometric Mean Variance of logs

1707 0.303

2024 0.203

2223 0.243

607X 0.170

Cohort 1923 Geometrlc Mean Variance of logs

1645 0.247

1850 0.259

2082 0.222

5680 0.174

Table 2 Total tax revenue

over three years.”

x-y

!:

Time period

k =0.05

k=O.lO

k =0.05

k=O.lO

Cohort 1943 1971 1972 1973 (1971+ 1Y72+ 1973)/3

1709 2018 2282 1951

1172 1371 1541 1331

803 932 1041 909

537 647 741 620

906 1026 1241 IO42

Cohort 1933 1971 1972 1973 (1971+ 1Y72+ 1973)/3

1986 2240 2510 2205

1349 1516 1687 1495

917 1026 1135 1013

637 724 823 711

1069 1214 1375 1191

Cohort 1923 1971 1972 1973 (1Y71+ lY72+ 1973)/3

1861 2016 2326 2065

1270 1427 1570 1404

867 968 1061 955

591 673 756 661

1137 1265 1110

“Note: X IS obtained using exp(p+$ta*), rather Thus, the lognormal assumption is consistently lations were made before the figures were rounded

994

than the actual value of the arithmetic mean. used throughout the calculations. All calcuto the nearest pound.

For cohort 1943 it can be seen that a conventional progressive tax system with k=0.05 would raise f1925 per person over the three years, whereas the alternative averaging system would raise &1860. In each case the average tax rate works out at approximately 32’%,, but the fall in total revenue as a consequence of ‘changing’ to an averaging scheme is of the order of 3.4%. For the more progressive system with k =O.lO, implying an overall tax rate of about 53 y,;, the fall in total revenue is of a similar order oi magnitude, i.e. 3.3%. Similarly for cohort 1933 the reduction in total revenue would be 2.3 % for both values of k; while for cohort 1923 the reduction would be 2.1 “/:, with k=0.05, falling to 1.97; for k=O.lO. The fact that the percentage revenue loss which would result from a change in the system is lower for the higher age groups is an expected consequence of the higher earnings mobility within the younger cohort. It is perhaps worth noting that these results may ‘overstate’ the revenue loss if averaging does have a significant effect on labour supply. 3.2. Equal revenue taxes The previous section considered the revenue loss which would result from a change to an averaging system of taxation, with the tax schedule unchanged. It is also of interest to examine the required change in the tax schedule (in this case the parameter k) which would be required to maintain a constant total revenue. Eq. (6) gives the average post-tax income where tax is assessed on the average of three years’ income, and eq. (5) gives the average post-tax income where tax is based on each year independently. Each are functions of the value of k, and may be written as j*(k), and ji(k) for the ith year. For total revenue to remain unchanged it is required to choose a value of k, k’ say, such that14 r*(k’)=f

;

y,(k).

(7)

i=l

The value of the right-hand side of (7), and therefore the required value of j*(k’), can easily be obtained from the values given in table 2. For example, consider cohort 1933 with k=0.05, where it is equal to &1517. The value of k’ may then be obtained by using the appropriate values of m and s2 from table 1 and solving the quadratic equation in (6). In this example it is found that in order to ensure a constant value of El517 for y*, the parameter of the tax function should be raised to k’ =0.058. This result may be used to 14Notice that the means are additive.

of the x’s and of X do not enter into (7) because

arithmetic

means

calculate the difference in the annual tax which has to be paid by an individual with a constant income, as a result of a change to an averaging system which maintains total revenue at its former level. For someone with a constant income of, for example, El400 per year. the difference is &55 per year; while an individual with a constant income of &2500 per year would pay additional taxes of El03 per year. This difference increases to El70 when pre-tax income is constant at &4000. 3.3. Ineyudity

cfpost-(ax

incomes

It has already been mentioned that equity issues are raised insofar as more highly fluctuating earnings are not compensated by relatively higher average earnings. If this kind of compensation does occur, then a change to averaging would obviously shift the burden of taxation towards the relatively lower paid, and would increase the inequality of post-tax earnings measured over a three-year period.15 Longitudinal data are therefore very useful here, and a direct approach (rather than considering mobility separately) is simply to compare a measure of the inequality of post-tax earnings over three years, under the two different systems. Measures of discounted earnings were also considered, but the period is too short for discounting to have a significant effect. Thus. with the individual data it is possible to calculate the sum of posttax earnings over three years when each year is taxed independently; that is, c .x! mk (table 3). Th e d’ISt rt‘b u t’ton of post-tax earnings when tax is based on Table 3 Inequality

of post-tax Cohort

incomes

over three years (variance Cohort

1943

of logarithms). Cohort

1933

1923

Tax period

k =0.05

k=O.lO

k = 0.05

k=O.lO

k=0.05

k=O.lO

One year Three years

0.133 0.132

0.120 0.118

0.154 0.153

0.139 0.138

0.158 0.157

0.142 0.141

the three-year average can be obtained directly from the values of m and s* in table 1, using the transformation in eq. (4). However, the calculations reveal little difference between the two sets of measures. These results therefore suggest that there is very little variation in relative mobility over the range of income distribution observed here, and that there ISThis kind of statement incentive effects.

may of course

need to be qualified

by consideration

of the possible

would not be expected to be any systematic from the introduction of an averaging system.”

effect on inequality

resulting

4. Conclusions The purpose of this paper has been to consider a simple scheme for the assessment of income tax which is based on an average of income over three years. The scheme has the advantage that no tax is paid in arrears, although individuals have greater liquidity under the averaging system than under the conventional system. Using longitudinal data for individuals over three consecutive years, tentative estimates of the possible effects on total tax revenue were obtained. It was also shown that an averaging system would have a negligible systematic effect on the inequality of post-tax earnings. “Further indirect evidence concerning relative mobility and earnings over long periods in various occupations is given rn Creedy (1979). It may. however, be suggested that the comparison should not use the same value of k for both types of system, but should be between schemes whrch raise the same total revenue. For cohort 1933 with k=0.05 it was noted that with avjeraging k would have to be raised to 0.058 for constant revenue. This would gave a value of 0.151 for the variance of logarithms of post-tax earnings over three years, implytng a slightly larger reduction than shown m table 3.

References social security and incentrves, Midland Atkinson, A.B. and J.S. Flemming, 1978, Unemployment, Bank Review, Autumn. Aitchison, J. and J.A.C. Brown, 1975, The lognormal distrtbutron (Cambrtdge University Press). Barr, N.A., S.R. James and A.R. Prest, 1977, Self assessment for income tax (Heineman for ICAEW and IFS). Cole. W.J.. 1975, The financing of the individual striker: A case study in the Building Industry, British Journal of Industrial Relations 13, 94-97. Creedy, J., 1977, The distribution of lifetime earnings, Oxford Economic Papers 29, 412-429. Greedy, J., 1979, The rnequality of earnings and the accounting period, Scottish Journal of Political Economy 26. 89996. Dalton, H., 1954, Principles of public finance, 4th edn. (Routledge & Kegan Paul). David, M. et al., 1969, The operation of the 1964 averaging provisions of the internal revenue code: A simulation study and recommended changes, University of Wisconsin Social Systems Research Institute. Doresamiengor, M.R., 1929, Some recent developments in the mathematical theory of taxation due to Edgeworth, Indian Journal of Economics 8, 317-330. Edgeworth, F.Y., 1925, Papers relating to political economy, vol. II (Macmillan). Gennard, J. and R. Lasko, 1977, Financing strikes (Macmillan). Merkies, A.H.Q.M.. 1968, On an optimal use of the income averaging facilities in the proposals of the Carter Commission, Canadian Journal of Economics 3, 255-267. Mill, J.S., 1920, edited by Ashley, T.S., Principles of political economy (Macmillan). Pigou, A.C., 1923, Essays in applied economics (P.S. King & Son). Shehab, F., 1953, Progressive taxation (Oxford University Press). Simons, H., 1938, Personal income taxatton (Chicago University Press). Steger, W.A., 1956, Averaging income for tax purposes: A statrstical study, National Tax Journal 9, 97-115.

J. Greedy,

Incontr

nw~~~ging and

progressiw

tuvution

397

Tibbetts, F.C., 1940, The accounting period in federal income taxation. Souther-n Economic Journal 7, 362--379. Vickrey, W., 1938, Averaging of income for income tax purposes, Journal of Political Economy 47; reprinted in: Musgrave and Shoup, eds., Readings in the economics of taxation (Allen and Unwin for the AEA, 1959). Vickrey, W., 1972, Cumulative averaging after thnty years, in: R.M. Bird and J.G. Head, eds., Modern fiscal issues (Toronto Press). Report of the Royal Commission on the Income Tax, 1920, Cmd. 615.