## Abstract

This paper examines the possibility of ascertaining the welfare changes that occur when a consumption tax replaces an equal-yield income tax. It finds that those with saving/income ratios greater than the social saving/income ratio under the income tax will surely benefit and those with ratios smaller than the social rate under the consumption tax will surely be harmed. These conditions are in each case sufficient to guarantee these results—but not necessary. Some can be better off and others worse off without these conditions holding. It is thus theoretically impossible to predict the welfare effects on many taxpayers of the tax shift from the observable data.

## I. Introduction

This paper attempts to ascertain who will benefit and who will be hurt when a consumption tax replaces an equal-yield income tax. A succession of distinguished economists since J. S. Mill have examined and debated which of the two taxes ought to be used to satisfy some normative criteria of taxation. Some argue that efficiency, equity, and administrative simplicity would be enhanced if the consumption tax replaced the income tax, while the defenders of the income tax argue that the case for consumption tax is seriously flawed. ^{1/} Recently, the optimum taxation approach has been extended to the analysis of the consumption tax and the income tax as a way of maximizing intergenerational welfare under flexible labor supply. ^{2/} These efforts have considerably clarified, if not clearly answered, the questions that are relevant to the choice of a tax system on normative grounds.

However, these efforts are perhaps of little practical consequence. In a democratic society, where the tax system is determined by the choice at the voting booth, taxpayers’ preferences for a tax system play a far more critical role than the views of academic analysts, and the taxpayer will make the choice on the basis not of the normative advantages of a given system but on the basis of its impact on his welfare, given his taste and income. This paper, therefore, does not examine the normative but the positive question: which of the two tax systems is more likely to benefit and hurt which taxpayers?

Rather surprisingly, a few have ever addressed this question and the very few answers that have been given to it invariably treat the conclusions as more or less self-evident and deserving little further analysis. Professor R. A. Musgrave, for instance, stated simply:

Now there can be no question that a tax on income differs from a tax on consumption. The consumption tax is preferable to the saver, and the income tax to the consumer, the same yield being raised in both cases (1959, p. 162).

^{3/}

Professor A. R. Prest echoed a similar view when he stated:

The change of basis [from the income tax to the equal-yield expenditure tax] will obviously improve the post-tax position of people with a high average saving/income ratio and will worsen the post-tax position of those with low average saving/income ratio” (1959, p. 485).

One of the problems of these statements is that they fail to define precisely the “savers” and “consumers” or those who have “high” and “low” average ratios of saving to income. Needless to say, the major difference between the income and the consumption tax is the presence and the absence, respectively, of taxation on interest income. Accordingly, the replacement of the income tax with the consumption tax changes the net-of-tax interest rate (the take-home rate) facing taxpayers, assuming the market rate of interest remains unchanged (as almost all analyses on this subject assume). Hence some of the consumers (or borrowers) under the income tax regime (where the take-home interest rate is lower than the market rate) may well turn out to be the savers under the consumption tax regime (where the take-home interest rate is higher than that prevailing under the income tax). Thus it is not clear from Musgrave’s statement whether by savers he means those who save under the income tax, under the consumption tax, or under both tax regimes. The same ambiguity exists about the definition of consumers. In short, savers and consumers can only be defined unambiguously when the interest rate or the tax system is specified under which saving and consumption decisions are made. Professor Prest’s definition of high and low average ratio of saving to income suffers from the same ambiguity. A taxpayer who maintains a high average ratio under one tax regime (one interest rate) may be the one who maintains a low average ratio under another tax regime (another interest rate).

This difficulty is magnified when the social net saving under the income tax is large. When an equal-yield consumption tax replaces an income tax, those who saved little under the income tax must then bear a higher tax burden than before because the government must recoup the revenue lost from giving up the tax on capital income by increasing the tax rate on consumption above the rate applied on the labor income under the income tax. The ratio of saving to income would respond not only to the net-of-tax interest rate but also to this income change. The larger the “income effect,” the greater the probability of a taxpayer’s changing his saving behavior. The simple distinction between “savers” and “consumers” fails to take these possible changes into account.

We shall first examine the effects on savings and then present alternative ways to predict changes in taxpayers’ welfare from the shift in the tax regime. Finally, we shall compare our analysis with the problems associated with construction of the correct cost of living indexes.

## II. The Model

### 1. The economy

Let us assume that an individual’s life is divided into periods 1 and 2 and that he (or she) receives a given amount of income over the two periods. ^{4/} His utility is a function of the amounts of his consumption in the two periods, and he attempts to maximize his lifetime utility subject to his income constraint and leaves neither savings nor liabilities when the second period expires.

The economy consists of population n, and the utility function of the jth individual (j = 1, 2, 3, … n) in the absence of the government sector is given by:

where

where ^{j} is the present value of his income discounted by the market rate of interest.

The role of government in our model is to impose either an income tax or a consumption tax on the population to absorb resources into the public sector and return them to the population as publicly supplied goods and services each period. We assume for simplicity that, following the conventional approach in this type of analysis, the taxation and expenditure process does not alter the market rate of interest. ^{5/} We assume accordingly that (1) the government can borrow from and lend to the private sector at the market interest rate, and (2) under either one of the two taxes the aggregate private demand for consumption in each period is not greater than the sum of the citizenry’s aggregate private disposable income and government revenue. We assume that an individual would receive the same amount of utility from government expenditure regardless of its timing as long as the total present value of this expenditure is the same under both tax regimes. Accordingly, the jth person’s utility function in the presence of the government sector is:

where

G_{1} and G_{2} above are government spending in periods 1 and 2, respectively. Thus we assume that

### 2. The income tax

Suppose now that the government imposes a comprehensive proportional income tax with rate t. This tax is a composite tax on labor incomes and on interest earnings from saving, and allows interest payments on borrowing to be deducted from the tax base. The budget constraint for the jth individual under the income tax becomes:

where

### 3. The equal-yield consumption tax

Suppose now that the government changes the tax base from income to consumption and that the rate e is chosen so that the present value of total consumption tax revenue is the same as that of the income tax.

The post-consumption-tax budget constraint for the jth individual, having wealth equal to I*, becomes as follows:

where

The jth individual’s budget line can be shown in Figure 1 by sketching a line parallel to the pretax budget line, IJ. The exact location of such a post-expenditure-tax budget line depends on the consumption tax rate, e, and the pretax labor income and consumption vectors. We must find out how the rate e is related to the income tax rate, t. In order to do so it is necessary to know the consumption patterns under the income tax of all n individuals in the economy.

The present value of aggregate income tax revenue, denoted by T_{i}, is the sum of n individuals’ present values of tax payments on given labor income and on earnings from savings (or their tax deduction on borrowed funds), and is given (with the market interest rate r used for discounting) as follows:

The present value of total consumption tax revenue, denoted T_{e}, is the sum of the present value of tax payments on consumption by n individuals in both periods and is given by:

By equating T_{i} to T_{e}, and from (1.6), we obtain the following:

where:

Here s_{i} is the ratio of the aggregate amount saved in period 1 by the private sector under the income tax regime to the present value of the pretax aggregate income. We shall refer to s_{i} as the ratio of social savings to income (under the income tax).

Incidentally, the rate of a consumption tax is usually expressed as a tax-exclusive rate, while the income tax is a tax-inclusive rate. Let e and e’ be the two types of rates of a consumption tax which are levied on a tax-exclusive base and on a tax-inclusive base, respectively. Then

From equation (1.10), we can see that when s_{i}=0, eʹ = t; and when s_{i} > 0, e′ > t, and when s_{i} < 0, eʹ < t; ∂eʹ/ ∂s_{i} > 0.

By substituting formula (1.9) for e in (1.11), we obtain the jth individual’s budget line, as representative of all individuals with wealth equal to I* under the equal-yield consumption tax, as follows:

Thus, depending on the ratio of social savings to income or the value of s_{i}, the location of the post-consumption tax budget line is different. If s_{i} equals zero, the budget line for individuals with I* wealth becomes line M_{1}N_{3} in Figure 1. If s_{i} is positive, the budget line shifts to a position parallel to M_{1}N_{3} and left of it; if s_{i} is negative, the shift is to the right. ^{6/}

## III. Analysis

### 1. Effects of different earning patterns

Consider a group of individuals with the same before-tax wealth, I*. From (1.2) and (1.5), the budget constraint of this group can be written as:

Thus, although the pretax budget constraints were the same for all people in this group, the post-tax budget constraints differ among them: as first-period income, _{1} and small y_{2}) while it favors individuals whose labor income is lower in the first and higher in the second period (small y_{1} and large y_{2}) This differential treatment is shown in Figure 1 by the different post-income-tax budget lines of individuals with the same pretax budget line: lines M_{1}N_{1}, M_{2}N_{2}, and M_{3}N_{3} with slopes equal to r(l-t) - 1 represent budget lines under an income tax for three individuals with the same before-tax income I* but different earning patterns. The smaller is y_{1}, the further out is his budget line from the origin under the income tax. ^{7/}

### 2. Effects of different consumption patterns

Assume an economy consisting of individuals with the same income and the same earning time path but different consumption patterns. If the income vector is given as (y_{1}, y_{2}) in Figure 1 and the economy’s net saving is zero, the budget line under the consumption tax is shown by line M_{1}N_{3}, which is parallel to line IJ, and that under the equal-yield income tax is given by line M_{2}N_{2}, which crosses the M_{1}N_{3} line at point [(l-t)y_{1}, (l-t)y_{2}]. Take one person, the jth person, and assume that his taste is such that under both income and consumption tax regimes he prefers to consume more than his income in the second period, as represented by his indifference curves, j_{1}, J_{2}, … He is then better off under the consumption tax than under the income tax. Take another person, the kth person, and assume that his taste is such that under both tax regimes he would like to consume more than his income in the first period as represented by his indifference curves, k_{1}, k_{2}, … This person would be worse off under the consumption tax than under the income tax.

These two cases suggest that (1) one who borrows (either because he has a higher income in the second period or he prefers to consume more than his income in the first period) is better off under the income tax than under the equal-yield consumption tax; and that (2) one who saves (either because his income is higher in the first period or he prefers to defer most of his consumption in the second period) is better off under the consumption than under the equal-yield income tax. These cases may have prompted most economists to conclude that the savers prefer consumption taxes and consumers (or borrowers) prefer income taxes. However, it is important to note that the results above are obtained because we assumed special tastes and income earning patterns for these taxpayers. That is, the taxpayers’ interest elasticities of savings are small and income elasticities of savings are nearly positive constants, so that with given income patterns those who save under the income tax will still save under the consumption tax and those who consume under the income tax will continue to do so under the consumption tax. The results obtained by assuming these special types of individuals cannot be generalized.

### 3. Substitution effects

The traditional view asserts rather casually that an individual’s ratio of savings to income determines his preference between income and consumption taxes, though the definition of such a ratio is usually not precise. To show this view is incorrect we present here first an example showing that two individuals who have an identical ratio of savings to income, no matter how such a ratio is defined, could prefer different tax systems.

Consider two individuals, say the ith and kth persons, with the same pretax income vector (y_{1}, y_{2}, the identical equilibrium consumption vector under the income tax and the identical equilibrium consumption vector under the equal-yield consumption tax. In Figure 3, which is reproduced from Figure 1, points y’, F, and H represent such an identical income vector, consumption vector under the income tax, and consumption vector under the consumption tax, respectively. At point F the indifference curves of both persons are tangent to the budget line under the income tax, MN; and at point H, to the budget line under the consumption tax, DD’. Because the net-of-tax interest rates, or the slopes of the budget lines, are different for the two tax regimes, these two budget lines must intersect each other at a point (y’) and the kinked line My’D’ becomes convex to the origin. The location of the point of intersection depends upon the net social savings, as we shall explain later; point y’ in Figure 3 shows the point of intersection when the net social savings rate is zero. (It is located on the line connecting the origin and the pretax income vector.) When point F is located below the intersection on the budget line under the income tax (the My’ segment) and point H, above the intersection on the budget line under the consumption tax (the y’D’ segment), because the kinked line My’D’ is convex to the origin, it is possible to construct an indifference map for that diagram for which H (the equilibrium under the consumption tax) is inferior to F (the equilibrium under the income tax), and another map for which H is superior to F. In Figure 3, the jth person’s indifference map shows that point F is inferior to point H and the kth person’s indifference map shows the reverse. Clearly, the jth person is better off under the income tax and the kth person better off under the consumption tax. Yet these two persons have, by assumption, an identical ratio of saving to income, no matter how such a ratio is defined, for they have an identical pretax income vector and an identical consumption vector under both the income and the consumption taxes. Note that the situation depicted in Figure 3 indicates that those who would borrow under the income tax regime will save under the consumption tax regime, because the take-home interest rate under the consumption tax becomes higher than that under the income tax. The greater the interest elasticity of saving, the greater chances of such borrower-to-saver switchings.

### 4. Income effects

Besides the differences in taste or interest elasticities of saving among people, there is another factor that influences the direction of the changes in the welfare of taxpayers when the income tax replaces the consumption tax: changes in real disposable incomes.

As seen in equation (1.9), the greater the social saving under the income tax, the greater the difference between the consumption tax rate (e’) and the income tax rate (t) with a given market rate of interest (r). Consequently, those who saved less than the average under the income tax would find their lifetime tax liabilities increased when the consumption tax replaces the income tax. This fact may be depicted in Figure 4, where the relevant section of Figure 3 is reproduced.

Take a group of m individuals who possess an identical pretax wealth when their incomes are discounted by the market rate of interest (r), and, therefore, whose pretax income vectors are located on the same line IJ. When an income tax of rate t is imposed and the tax rate (t) is DI/OI, their disposable income vectors must be located on a point on line DD’. The budget line under the income tax of anyone of the individuals in this group is a line with slope equal to -[1+(1-t)r] passing through his disposable income vector located on line DD’. Line MN shows one of these budget lines. Each person in the group with the pretax income vector y (and hence the post-tax disposable income vector y’) faces this budget line. Line EE’ is the budget line under the consumption tax that yields revenue equal to that from the income tax t when social saving is positive. Obviously, it has a slope equal to -(1+r) and intersects the horizontal axis at point E, where EI/OI = e’. From equation (1.9), it is known that the greater the social saving ratio, s_{i}, the greater e’, and hence the closer the budget line to the origin. (Note that under the consumption tax, unlike under the income tax, everyone with the same pretax wealth has the same budget line.)

Suppose that an indifference curve (a_{1}) of one person (person A) in the jth group is tangent to his budget line (MN) under the income tax at H. Because his disposable income vector is y’ he saves an amount equal to the horizontal distance between y’ and H in the first period. This person’s indifference map is so constructed in Figure 4, that the welfare at H is superior to that at K, where his lower indifference curve (a_{0}) is tangent to the budget line under the consumption tax (EE’). This construction demonstrates that with positive saving even a saver is worse off when the consumption tax replaces the income tax. It also demonstrates that this does not preclude the possibility that another person in the group with the same ratio of savings to income under the income tax, say person B, with indifference curves b_{1}, b_{2} could be better off when the consumption tax replaces the income tax.

In short, it is not possible to infer from a person’s ratio of savings to income the welfare change arising from the shift in the tax regime. A more important implication of the comparison of Figure 4 with Figure 3 is that if the social saving ratio were zero, the welfare of a person A would not have improved by replacing the consumption tax with the income tax because his budget line under the consumption tax was DD’ and it would intersect MN line at y’. In other words, whether one is better off does not depend on preference patterns alone, but also on individual saving rates relative to the social saving rate. This may be seen from Figure 4 where the larger the shift in line EE’ toward the origin, the greater the chances of a person in a similar position as the person A being made better off under the income tax, even though his elasticity of substitution of C_{1} for C_{2} is not as large (in the absolute value) as that of the person A. Moreover, the extent of the shift of EE’ depends upon the size of the social saving rate; the greater this is, the smaller is the amount of the tax that the labor income must yield under the income tax. Hence when the equal-yield consumption tax replaces the income tax, it reduces the net-of-tax incomes of those with smaller ratios of savings to income than the social rate under the income tax. We shall refer to this effect as the “income effect,” which can be measured by the distance ED in Figure 4. The greater the income effect, the greater the chances are for some low savers to be made better off under the income tax than the consumption tax. The traditional views, such as those represented by Professor Prest’s statement, thus seem to agree with this income effect. But they have apparently neglected the “substitution effect” altogether.

### 5. Classification of taxpayers

Can we identify which taxpayers will be better off under the income tax than under the consumption tax and vice versa by using observable data as saving/income ratios? The answer is no. We can, however, identify the following two groups using these ratios: those who are surely better off under one tax than the other and those who may or may not be better off but whose welfare positions cannot be determined from the obversable data.

In Figure 4, it can be seen that the taxpayers whose consumption vectors under the income tax are located on the GN segment of budget line MN would be definitely better off under the consumption tax than the equal-yield income tax, for their indifference curves tangent to EE’ must represent higher utility levels than those tangent to the MN line. This condition is sufficient to determine the direction of improvement, but is not necessary. Those whose consumption vectors under the income tax are located on the MG segment can be better off without this condition holding. But welfare changes for these people cannot be determined from their saving/income ratios alone; further information is needed on the precise shapes of their utility functions.

Similarly, by using the saving/income ratios prevailing under a consumption tax, taxpayers can be classified into those who are definitely better off under the income than the consumption tax and those who may or may not be better off but their welfare changes are indeterminable from the observable data. Those whose consumption vectors under the consumption tax are located on the EG segment of budget line EE’ belong to the former group, and those located on the GE’ segment belong to the latter.

Obviously, point G, the intersection of the budget lines under the two types of taxes, is the key in this classification. It is the consumption vector at which the jth person’s saving/income ratio equals the social/saving income ratio. This is shown as follows.

The jth person’s ratio of savings to his pretax income is given as:

Let

By substituting

Thus, if a taxpayer’s equilibrium consumption vector happens to be at point G, his rate of saving is equal to that of the economy.

This finding enables us to state that those whose equilibrium saving/income ratios under the income tax are greater than the social saving/income ratio, will definitely find their welfare improved by a shift from an income to a consumption tax. This does not imply, however, that the welfare of all of those with saving/income ratios below the social saving/income ratio will not be improved. Some can be better off without this condition holding, depending upon their elasticities of substitution of present for future consumption.

On the other hand, those whose equilibrium saving/income ratios under the consumption tax are below the social saving/income ratio (those whose indifference curves are tangent to budget line EE* at points between E and G) must be worse off if an equal-yield consumption tax replaces the income tax. This does not imply, however, that those whose saving/income ratios are above the social saving/income ratio under the expenditure tax will __not__ be made worse off by that replacement. Some can be worse off without this condition holding. Thus we can identify the sufficient conditions to determine the direction of change, but not the necessary conditions.

The number of people who become definitely better off under the consumption tax tends to decline as the level of social savings deviates from zero, because the higher are social savings (dissavings), the more the budget line EE’ will shift to the left (right) of line DD’ (which is the budget line under the consumption tax when the social savings are zero). A shift of the budget line to the left reduces the length of the GN segment and hence the number of people who will be definitely better off under the consumption tax than the income tax. (These people may be termed super-savers, in the sense that their saving ratios are above the social ratio under either tax system.) Similarly, a shift of the budget line to the right (that is the social saving/income ratio becomes smaller or negative) reduces the number of people who will definitely become worse off under the income tax than under the consumption tax. (They may be termed the super-consumers, in the sense that they consume more than the social consumption/income ratio under either one of the two tax regimes.)

## IV. Discussions in Terms of Index Numbers

The problems analyzed in this paper are logically the same as those encountered in constructing a correct cost of living index when relative prices change over the periods. It is known that the Paasche and Laspeyres indexes will differ if the relative prices differ in the periods being compared, and that they yield in each case only the sufficient and not necessary conditions to determine the direction of improvement in the standard of living. Our problems are just the same. To make this point clear, the discussion can be couched in terms of the theory of index numbers.

Let the consumption tax regime be taken to be the “base year,” and the income tax regime as the “given year,” and let “cost of living” indexes be constructed that relevant for each individual, say the jth person, these indexes can then be compared with an index of net-of-tax money income change between the two regimes relevant for that individual.

In this framework the indexes that correspond to the Laspeyres indexes of prices are given as:

Here, and in the expressions below, P^{0} and P^{1} are price vectors of the base year and the given year, respectively, and Q^{0} and Q^{1} are the quantity vectors of the base year and the given year, respectively. The indexes that correspond to the Paasche indexes of price are given as

Let us define an index of net-of-tax income change as

This income index corresponds to the ratio of the present value of the lifetime net-of-tax income under the income tax regime to that under the consumption tax regime. It is the same as the ratio OM/OE in Figure 4. From the earlier discussion, it is known that the greater the social saving/income ratio, the larger the income index.

It is obvious that a consumer is surely better off under the income tax (in the given year) than under the consumption tax (in the base year), if:

so that the Paasche quantity index, p_{q} ≡ ΣP^{1}Q^{1}/ΣP^{1}Q^{0} is greater than one. This is so because this inequality implies that the budget line that enables to purchase the basket of goods, C_{1} and C_{2} our case, purchased in the consumption tax regime (the base year) at the prices prevailing in the income tax regime (the given year), is located inside the income tax regime’s (given year’s) budget line. Dividing both sides of this inequality by ΣP^{0}Q^{0}, we have:

Thus, if the income index ɛ exceeds the Laspeyres index of prices, the consumer is better off under the income tax regime (in the given year) than under the consumption tax regime (in the base year).

The interpretation of the relation between the income index and price index in our context is that if the net-of-tax income has risen because of the “income effect” of the change in the tax regime, but more than in proportion to prices of incomes, which are reciprocals of the net-of-tax interest rates, on the average, then the average quantity consumed (real income) must have increased. Nevertheless, as is known, the condition that the Paasche quantity index is greater than unity is sufficient for improvement in the given year but not necessary. The taxpayer can be better off without this condition holding. (It was possible to show this in terms of Figure 3.) The indexes that correspond to the Laspeyres quantity index are given as:

If the Laspeyres quantity index is less than unity, the observer knows that the taxpayer must have become worse off in the income tax regime (the given year). Then the income index is less than the Paasche price index. But again a taxpayer can be worse off without this condition holding.

In conclusion, two groups of people can be clearly classified by their observable consumption (or savings and income) vectors: those who will surely be better off and those who will surely be worse off from a change in tax regime. But there is also a group of people whose welfare changes cannot be ascertained from observable data.

## V. Conclusions

The popular but rather casual proposition that replacing an income tax with an equal-yield consumption tax would improve the welfare of savers and worsen that of consumers (or borrowers) turns out to be not generally correct. Some of those who save under an income tax would be worse off and even some who borrow under the income tax could be better off. The only group whose welfare will definitely be improved under the income tax is that with saving/income ratios greater than the social saving/income ratio; and the only group whose welfare will definitely be worse under the consumption tax is that with saving/income ratios smaller than the social saving/income ratio.

The reasons for the difficulties with classifying welfare improvements according to the saver and the consumer criteria or the saving/income ratio criteria are twofold. (1) The substitution effect makes savers and consumers ambiguous concepts. Because the net-of-tax interest rates differ between the two tax regimes, some borrowers under the income tax regime may well turn out to be savers under the consumption tax regime. (2) Even if the net-of-tax interest rate faced by taxpayers were unchanged, the income effect that arises from the difference in tax rates applicable to the labor incomes under the two regimes could change a taxpayer’s saving/income ratio substantially.

By defining the saving/income ratios of individuals precisely and comparing them with the social saving/income ratio, it is possible to identify those taxpayers who would be surely better off and those who would be worse off from the shift in the tax regime. Nevertheless, they are only a subset of those affected and the impact of the change in taxation on the welfare of the rest of the population cannot be identified. Thus, a fundamental difficulty exists in identifying individual welfare effects of replacing the income tax with an equal-yield consumption tax. The nature of the difficulty has been compared with that of index numbers. The Laspeyres and Paasche indexes identify the conditions sufficient to determine the direction of improvement, but these are not necessary when relative prices change between the two periods being compared. So do our criteria. In our case the relevant prices are the prices of incomes, namely, the reciprocals of the net-of-tax interest rates. The greater the revenue being raised, the larger the difficulty, as the more people are likely to fall into the category of taxpayers whose welfare changes cannot be ascertained objectively when income taxes are replaced by consumption taxes.

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^{}*

The authors are Professor of Economics at Osaka University and Professor of Economics, Tezukayama University, Nara, Japan, respectively. This paper was written when Hirofumi Shibata was a visiting scholar in the Fiscal Affairs Department. The views expressed represent the opinions of the authors, who gratefully acknowledge the helpful comments of Richard Goode, Vito Tanzi, Howell Zee, and other participants in the seminar at the International Monetary Fund where an earlier draft of this paper was presented. Remaining errors are of course the authors’ sole responsibility.

^{}1

For a modern debate, see J. Pechman, ed. (1980).

^{}2

See for example A. B. Atkinson and A. Sandmo (1980); M. A. King (1980); A. P. Diamond (1965), and M. Feldstein (1984).

^{}3

Musgrave expressed a similar view in a little more precise terms in an earlier publication: “We secondarily assume equal incomes but varying saving-consumption ratios for various income receivers. Now compare case A, where we have a general income tax, with case B, where we have a spending tax yielding the same total revenue. Under B, the “savers” (people who save and invest a large part of their income) are better off, relative to the “spenders” (people who consume a large part of their income) than they would be under A. While this is true, it fails to prove that under A, the savers are the victims of undue double taxation” (1939, p. 549). Recent writing shows he may have changed his views (1983).

^{}4

The origin of this model is Samuelson’s (1958).

^{}5

This assumption is for simplicity only. Our conclusions in this paper hold even if the market rate of interest changes, unless the change is exactly the same amount as the tax so that the net-of-tax rates facing taxpayers are the same under both the income and consumption tax regimes.

^{}6

This fact can be shown geometrically as follows: Suppose that the economy’s aggregate labor income vector is represented by point y in ^{Figure 2}. When a 50 percent income tax is imposed, the private sector’s aggregate post-income-tax budget line becomes a line containing point yʹ (which is the midpoint of the ray connecting y and the origin, 0) with slope equal to r(1-t) - 1 (line Aʹjʹ). If under this tax, the private sector saves Fyʹ of period l’s net-of-tax income (the economy’s consumption vector is G), it generates interest income and yields revenue from a tax on interest income corresponding to GE in the second period whose present value is CB. Consequently, the present value of the tax revenue of the 50 percent income tax is CA, the sum of the revenue from the tax on labor income (BA) and the revenue from the tax on interest income (CB). Accordingly, the budget line under an expenditure tax that yields present-value revenue equal to CA is the line containing point G, with slope equal to r-1 (line CK). The tax rate of the consumption tax eʹ (eʹ= CA/OA) becomes larger than the rate of the equal-yield income tax rate t (t = BA/OA) whenever the social saving rate is positive.

^{}7

The jth individual’s budget line under the income tax is obtained by rewriting (2.1) as follows:

Thus, among m individuals, the smaller y_{1} is (and thus, the larger y_{2} is), the larger becomes the value of the point at which their budget lines cut the vertical axis, though the slopes of their budget lines are the same. Line M_{1}N_{1} is the post-income tax budget line for an individual with no period-2 given income (y_{2} = 0 and I*=y_{1}). M_{1}’s coordinates are given as [(l-t)I*, 0], and point N_{1}’s coordinates as [0, (l-t)I*[l + r(l-t)]]. For an individual with no period-1 given income [y=0 and I* = y2/(l+r)] the post-income-tax budget line becomes M_{3}N_{3}, at which point M_{3}’s coordinates are _{3}’s coordinates are [0, (1-t)I*(1+r)]. The post-income-tax budget line for an individual who has some given income in both periods is above line M_{1}N_{1} but below line M_{3}N_{3}, such as at line M_{2}N_{2}.