A Note on Progression and Public Sector Size

A Note on Progression and Public Sector SizeAuthor(s): Geoffrey BrennanSource: Public Choice, Vol. 32 (Winter, 1977), pp. 123-129Published by: SpringerStable URL: http://www.jstor.org/stable/30022998 .

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NOTES 123

1 Notes

A NOTE ON PROGRESSION AND PUBLIC SECTOR SIZE

Geoffrey Brennan

It is occasionally suggested, in public choice circles, that progression in the tax structure makes for a larger public sector.1

The reasoning behind such an assertion seems obvious. Greater progression implies a shift in cost-sharing arrangements so that richer individuals share a greater proportion of the cost of public spending than poorer individuals do. If, as is

typical, the distribution of income is such that the median income is less than

average income, and if demands for public goods rise with income level, then

progression reduces the tax liability of the median voter. Under majority rule, the

single peaked preferences, the median voter's judgement is decisive, so that it seems as if a reduction in the tax-price of public spending to the median voter must lead to a higher level of public spending ceteris paribus.

The aim of this note is to show that this is not necessarily so, and that under what are perhaps the most plausible assumptions will not be so. The argument depends on a distinction, commonly overlooked in such discussions, between

marginal and average cost-shares.2 Such a distinction is, it seems to me, an

*Visiting Research Fellow, VPI&SU; Senior Lecturer in Public Finance, Australian National University. I am indebted to my colleagues, James Buchanan and Gordon Tullock, for comments on earlier versions.

1The most analytically articulate statement of this view is to be found in Buchanan (1964), and (1967 Ch. 11 appendix). The suggestion is also aired in Buchanan & Tullock (1962), and Head (1964).

2The distinction between marginal and average cost-shares is to be found in Buchanan (1964) pp. 38-43, specifically drawn in the context of progressive income taxation. Buchanan does not observe, however, that these are systematic reasons for believing that marginal cost-shares will lie above average cost-shares for the median voter; indeed, in the discussion of progression and public sector size in a later chapter of the same book, he seems to revert to the constant cost-share approach.



124 PUBLIC CHOICE

absolutely fundamental one: attempts to explain phenomena within the public choice/fiscal exchange paradigm which ignore it and which, implicitly or explicitly, assume tax-prices to be invariant over levels of public spending seem to me to be extremely dubious. In this sense, the present exercise can be looked on as an illustration of the significance of the average/marginal tax-price distinction in public choice analysis-as well as an exploration of an issue which is of some interest in its own right.

For the purposes of the present discussion we assume: (i) that all public spending is on a pure Samuelsonian public good G; (ii) that G is produced under conditions of constant costs; (iii) that all individuals have identical preference maps between X and G,

and that these preference maps are such that neither X nor G are inferior;

(iv) that any individual's demand curve for G takes a position such that his demand for G at any price is proportional to his initial endowments of X;

(v) that we can abstract from income effects due to changes in

infra-marginal tax prices, so that we can treat the demand for G as a function of marginal tax-prices only;3

(vi) that total tax payments for any individual, and marginal tax increases, are monotonic functions of income.

These assumptions are sufficient to ensure that the median voter is the median income-recipient, irrespective of the precise rate structure implied by the tax system4, so that it is possible to focus on the median income-recipient alone in determining public goods supply.

Let the median income-recipient's demand curve for public goods be M'M in

figure 1, and let T'T be M's tax-price under proportional taxation. Given constant costs, T'T will be a horizontal straight line. The tax-price of public goods to M,Cm, is simply

Y

Cm Y MC (1) 11

where Ym is median income

3Technically, this assumption is inconsistent with (iv). Its virtue is that it does not require us to redraw the individual's demand curve for G in response to infra-marginal tax changes: to do this would complicate the discussion, but would not alter the basic thrust of the argument.

4An exception of no consequence is that there exist some tax arrangements under which all individuals' demands for public goods supply will be the same (the "Lindahl" tax solution is one of these). In this case the median income-earner remains decisive, but there is technically no particular median voter.



NOTES 125

E

E4r

T,- i El-

73 G3

fic l



126 PUBLIC CHOICE

.Yi is total income

1

and MC is the (constant) marginal cost of producing G. Clearly, Cm is unaffected by changes in G.5 The level of public goods supply will thus be G1 in

figure 1.

Suppose now that the proportional tax regime is replaced by a progressive one, and let the rate structure change be such that M benefits. Then, the share of revenue collected from M will decline. The resultant change in M's tax-price might be expressed as a shift to T2T2, indicating that M's average cost-share/tax-price has been reduced. But this total tax burden for M--M's averages, tax-price--does not in itself, indicate anything concerning the median voter's perceptions of how

marginal increases in public spending might be financed, or of how marginal reductions in public spending may be reflected in reduced tax burdens for

taxpayers in different income positions. Yet it is these marginal cost-shares/tax-prices that are crucial for determining M's demand for levels of

public spending, and changes in them. Once this is recognized, then it becomes clear that figure 1 must depict M's

perceptions of these marginal cost-shares, for various levels of G, if it is to be useful in making predictions about M's decision on public sector size. One possible depiction of these perceptions, consistent with an average cost-share of T2T2, is indicated by the line T3T3: with T3T3 as the lows of marginal tax-prices, the average tax-price is area T3E3G3 over OG3, and is drawn to equal T2T2. But of course, it is the marginal tax-price E3G3 that is crucial for determining the quantity of G that M will opt for. In the case cited, G3 is clearly less than the quantity of G which M would have opted for under proportional taxation. Thus, it is not

necessarily true that a reduction in (average) tax payment for M leads to an increase in G.

Acknowledging that a reduction in average tax paid might lead to an increase in marginal tax-price does not of course prove the point: what remains to be argued is that under progressive taxation, this is, in fact, likely to be the case. One way of

showing this is to indicate that, under progression, marginal expenditures are

typically financed in a less progressive fashion than infra-marginal expenditures are financed-that is, that the degree of progressivity typically falls as the level of

spending rises. As a point of departure, observe that under a progressive tax system

maximum marginal tax rates are not unbounded-maximum rates are usually set somewhat below 100 percent (although rates in excess of 100 percent are not altogether unknown). In fact, efficiency considerations (disincentive effects,

5In other words, we abstract explicitly from the possibility that changes in public goods supply may affect factor prices and, hence, either the identity of the median income-recipient, or the income position of the median voter [see Brennan (1975)]. We also abstract from the possibility of proportionally differential adjustment in income-leisure choices at different income levels. A sufficient assumption to handle the latter is that the elasticity of labour supply is zero for all individuals.



NOTES 127

incentives to evade and so on) limit-or ought to limit--maximum rates to well below that level. Indeed, even in a completely exploitative tax system, marginal rates will not rationally be set above the level where revenue elasticity is unity.6

Whatever upper bound on marginal rates happens to be set, it is obvious that once it is so determined, the degree of progression, as conventionally measured, must be inversely related to revenue requirements. Suppose, for example, that the maximum rate is 60 percent. Then as the public sector expands to tkae a portion of GNP closer and closer to 60 percent, the tax system must necessarily appear to be more and more proportional. The last units of expenditure must, almost of

necessity, be financed in a lump-sum manner. For example, an increase in tax rates on income over the range 0-$1000 increases the tax bill on all incomes from $1000

up in equal absolute amounts.

And, of course, what is true in the limit is also true to a lesser extent in less

limiting cases. As a pure matter of fact, the growth of public spending in most countries of the world over the past decade or so has been financed largely out of new money creation and automatic increases in personal income tax revenue due to inflation. Both involve, necessarily, less "progressive" tax arrangements than prevail over the tax-expenditure system as a whole.

The general conclusion that follows is that taxation arrangements within a

progressive tax system typically involve a system of cost-shares/tax-prices which decline with higher levels of public spending for the richest, and which rise with

higher levels of public spending for all others. And it is quite clear that marginal cost-shares can be higher in the progressive

than in the proportional tax case. If, for example, the additional revenue is raised

by allowing inflation to cause the real level of the personal exemption under the income tax to evaporate, the marginal tax is virtually lump-sum; the median voter

SMC Ym faces a cost-share of , rather than ~ MC which he would face if the revenue

source were an increase in the proportional income tax rate prevailing. Since most tax systems are progressive, at least in spirit, this matter is of

interest not only in the particular case of isolating the effect of tax changes on

public sector size, but also within the public choice paradigm more generally.7 There is simply no a priori case for assuming that tax-prices/cost-shares are likely to be invariant with respect to changes in the level of public spending. And in the fiscal environment in which we find ourselves, there seems to be a good presumptive case for the contrary view.

6See Brennan & Buchanan (1977) for a more detailed treatment of this point, and of "exploitative" political institutions more generally.

7Browning (1976), in a neat paper directed towards a different issue, simply assumes that marginal expenditures will be financed by maintaining the degree of liability progression (ratio of marginal to average votes) for all taxpayers, an assumption that is central to his specific results. As I have here argued, there seems to me to be excellent reasons for believing that marginal expenditures are most unlikely to be financed in this way.



128 PUBLIC CHOICE

It is perhaps worth noting in passing that Galbraith, virtually the only contemporary economist who has argued extensively for an expansion of the public sector, has also argued for the "...greatly multiplied use of the sales tax..." (my italics) as the means for financing this expansion. The reasoning which Galbraith advancess for the use of the sales tax (inherently proportional) as opposed to the

progressive income tax is not particularly clear or persuasive. But Galbraith does seem to imply that the interests of "equality" and "social balance" (i.e. increased

public spending, as he perceives it) tend in opposite directions--a view that contrasts somewhat with the standard public choice position. As I have argued here, Galbraith may well be right, in this at least: a shift to a more proportional tax

system may expand popular demand for public spending, in that it achieves a reduction in the marginal tax-price of public spending for the median voter.

Of course, none of this seeks to argue that constant, or declining, cost-shares are inconceivable: marginal expenditures may be financed as progressively as (or more progressively than) infra-marginal ones. For example, if marginal revenues are acquired by broadening the base of the progressive income tax, with no change in the nominal rate structure, the effect will tend to be towards increased progression (i.e., marginal tax-prices for lower income individuals, including possibly the median, will tend to decline as public spending increases). Whether or not marginal revenues have in fact been acquired in this way over the recent past, and whether or not the taxpayer-voter believes that future marginal expenditures will be financed in this way are ultimately empirical questions. All that I have sought to argue here is that these are good reasons for believing that progression will decline as public sector size increases. At the very least, the analysis should establish that the "constant tax-price" assumption is entirely arbitrary and that its slavish application is, potentially, enormously misleading.

REFERENCES

Brennan, Geoffrey. "Public Goods and Factor Prices." Public Finance/Finances

Publiques, no. 1, 1975. and Buchanan, James M. "Towards A Tax Constitution for

Leviathan." (mimeo) 1977.

Browning, Edgar K. "The Marginal Cost of Public Funds." Jour. Pol. Econ., April, 1976.

Buchanan, James M. "Fiscal Institutions and Efficiency In Collective Outlay." Am. Econ. Rev., May, 1964.

. Public Finance In Democratic Process. Uni. of North Carolina Press, 1967.

Buchanan, James M. and Tullock, Gordon. The Calculus of Consent. Ann Arbor 1962, Uni. of Mich. Press. 8See Galbraith (1958), Ch. XXII and particularly pp. 313-314. I am grateful to Gordon

Tullock for bringing Galbraith's "sales tax" argument to my attention.



NOTES 129

Galbraith, John K. The Affluent Society. Boston: Houghton/Mifflin, 1958.

Head, John G. "Lindahl's Theory of the Budget." Finanzarchiv, 1964, reprinted in Public Goods and Public Welfare. Duke University Press 1975.



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A Note on Progression and Public Sector Size