Front Page Titles (by Subject) 11.: Simple Collective Decision Models - The Collected Works of James M. Buchanan, Vol. 4. Public Finance in Democratic Process: Fiscal Institutions and Individual Choice
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11.: Simple Collective Decision Models - James M. Buchanan, The Collected Works of James M. Buchanan, Vol. 4. Public Finance in Democratic Process: Fiscal Institutions and Individual Choice 
The Collected Works of James M. Buchanan, Vol. 4. Public Finance in Democratic Process: Fiscal Institutions and Individual Choice Foreword by Geoffrey Brennan (Indianapolis: Liberty Fund, 1999).
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Simple Collective Decision Models
In an effectively democratic political order, collective decisions emerge from a process that takes individual expressions of preference as inputs and somehow combines these to produce outcomes. Fiscal institutions affect these preferences. The influence on individual behavior is not, however, equivalent to an influence on collective outcomes. Such an extension requires a crossing of the bridge between individual participation and the final outcome of the collective choice process. It becomes necessary to translate the effects of institutions on individual behavior into effects on political results. To accomplish this, we must examine the rules that serve to combine individual “votes.”
Complete discussion would require a volume. Here we can only construct very simple models that abstract from the complexities of actual political process in order to concentrate on those elements that will be of assistance in making predictions. The models used are those of direct democracy. It is assumed that fiscal decisions are made directly through voting processes in which all citizens participate. Such models are, of course, highly unrealistic in any descriptive sense. Common observation tells us that collective decisions are not made in this manner. The underlying realism of the models depends, however, not on their apparent correspondence with observed reality, but upon their assistance in developing hypotheses about political choices that can be conceptually tested. If the models allow us to do this, they are of some significance for an understanding of the fiscal process as it actually exists in its complex institutional setting. Some of the general problems involved in moving from the models of the theorist to the real world are discussed in Chapter 12.
A Three-Person Model of Equals
Return to the initial models of individual demand for a single public good that were presented in Chapter 2. Recall that, in those models, elements of uncertainty, ignorance, and illusion were neglected. Initially, we stay within the same limitation here. The situation confronting the single individual is that shown in Figure 11.1. Provided only that he is not required to consider alternatives on some all-or-none basis, the individual depicted will “vote for” an amount, 0X, of the public good, for which he is charged a tax-price (determined externally) of 0P per unit. To the individual, 0X is the “optimal” quantity of the public good to be supplied. He will not be allowed, individually, to determine whether or not the community will supply more than, less than, or just this amount, since the collective decision will result from a political choosing process in which he is only one among several participants. To discuss the reaching of collective outcomes, which, once reached, must be imposed upon all members, it is necessary to examine the behavior of more than one person.
We begin with the simplest model that may be constructed. Assume that there are only three persons in the community, and that these three persons are identical in all respects. (The second of these assumptions makes the model applicable for any number of persons, but it will be useful to stay within the three-person restriction for purposes of comparison with later models.) How much will this group, acting as a collective unit, decide to devote to the supply of the public good, given the structure of tax-prices as indicated?
This model is interesting, even if the results are trivially obvious, because it is the only one in which neither the decision rule nor the tax institution, as normally considered, exerts an influence on the final outcome. Provided only that the tax is a general one, any decision rule and any tax scheme will yield the same result, which will be that shown in Figure 11.1. This result will also satisfy the necessary marginal conditions for Pareto optimality, although this welfare implication is not our primary concern at this point.
The conclusions may be demonstrated by postulating a tax institution. Suppose that a system of equal-per-head taxes has been agreed on in some “constitutional” setting before the particular fiscal decision is to be made, with the agreement stipulating that the total tax bill shall be residually determined as a result of the voting process on the amount of the public good to be supplied. The good is available to the community at constant cost; this assumption is common to all of the models introduced in this chapter. Under such conditions, each person, were he given his own “private” choice, would desire that the collectivity supply a quantity, 0X, of the good, which is, of course, equally available to all members. Each person would choose to have the collectivity expend the same total outlay. Hence, unanimity could be secured on this outcome without difficulty. Simple majority voting would in no way modify the result. Any less-than-unanimity voting rule will, in this model, produce the same outcome as the unanimity rule because the predetermined agreement on the tax institution removes any opportunity that either a single dictator or a majority coalition may have of exploiting other members of the group. Thus, any possible rule for making group choices will yield the same result, provided only that the tax is general and not discriminatory. Other general taxes will yield equivalent results. Since all persons are identical by assumption, a proportional or a progressive tax on income, or any other tax that is not specifically discriminatory, will impose equal tax-prices on the separate members of the group. Public spending programs are always “ideally efficient,” and institutional influences on outcomes are absent.
A Three-Person Model with Unequal Evaluations for the Public Good
As a first step toward making the model less restrictive, let us allow for only one difference among the three persons, a difference in their evaluation or their demand for the single public good. This change will enable us to isolate the effects of varying decision rules on collective outcomes independently of the effects of tax institutions. Since here we retain the assumption that the three persons are identical in all respects relevant to the levy of any general tax, any such tax will still impose the same tax-price on each person.
The three individual demand curves are shown in Figure 11.2(a), along with the privately preferred adjustments to the common tax-price. If he could choose independently, for the whole group, Individual A would have the collectivity provide 0XA; Individual B would have the collectivity finance, instead, 0XB units; and Individual C’;s most preferred quantity is 0XC. Clearly the delegation of decision-making power to a single person under these circumstances would produce different results with different “dictators.” Figure 11.2(b) depicts the preferences of the three persons in a slightly different manner. Here on the ordinate we measure the ordinal preferences of the persons and a point standing higher on this scale indicates that the individual prefers this quantity (given the tax-price) to any other point standing lower on the scale. Note that, when viewed in this way, the preference schedule for each person is single-peaked, with his most preferred outcome, for the group, being that shown by his “private equilibrium” position in Figure 11.2(a). The fact that the schedules are single-peaked is important, since this characteristic insures that under a simple majority voting rule, there will be some determinate outcome. There will be no cyclical majority.1
If no limits are put on the number of proposals that may be put forward for a vote, the outcome represented by the single-peak for the median preference member of the group will be selected under a decision rule of simple majority. Individual A will prefer the quantity 0XB to any output that is larger. Similarly, Individual C will prefer 0XB to any smaller quantity. Hence, Individual B will become controlling in the majority decision, just as if he were delegated privately with exclusive decision-making authority for the group. The predicted outcome will be that indicated to be “most preferred” by the person whose preferences are median for the whole group. This analysis suggests that, because of the median-man construction, some analysis of collective decision-making under majority rule is possible even if we remain at the level of the individual decision calculus. If fiscal institutions are predicted to influence the preferences expressed by the median voter, they can be predicted to influence the final collective results in the same direction.
Under a voting rule of unanimity, the outcome becomes indeterminate within wide limits. In the three-man model here, unanimity may produce a result that is confined only within the limits between quantities 0XA and 0XC. If votes are taken on successive additions to output starting from small numbers, no consensus can be attained for going beyond 0XA. On the other hand, if the choosing process starts with some quantity larger than 0XC, agreement could be reached only on a reduction to this level and no more. No quantity falling between these limits could be modified by general agreement of all parties.
A Three-Person Model with Equal Evaluation but Unequal Tax-Prices
It will now be useful to isolate, to the extent that is possible, the differences that arise from changes in the fiscal institutions. To do this, we shall now assume that the separate individual demand schedules are identical, as drawn in Figure 11.3(a). However, we shall assume that the individuals differ in some respect that may be relevant to the determination of tax-price. Let us say that they differ only in income.
Under the equal-per-head tax, the model becomes the same as that discussed two sections above in connection with Figure 11.1. The same quantity would be selected under any voting rule since both demand and tax-price are identical for all participants. If, however, we introduce a tax system that relates liability for tax to income level, the individuals will confront different tax-prices. Assume that Individual A has the lowest income, B the median income, and C the highest income in the group. Proportional income taxation, let us say, will confront the three persons with a set of tax-prices shown as PA, PB, PC in Figure 11.3(a). Despite the equivalence in demand, the three individuals will now prefer different quantities of the public good because of the discrimination in tax-price. In this model, as in the unequal evaluation model above, the particulars of the decision rule will affect the outcome of the political process. Under majority rule, the man with the median income tends to exert controlling influence.
Progressive income taxation does not differ from proportional taxation in terms of general results. It tends to widen the differentials between the higher and the lower tax-prices, but, under majority rule, the median-income receiver tends to remain controlling. Progression may, however, exert important effects on outcomes through shifting the preferred outcome for this median voter. To clarify this point, it is useful to introduce the notion of symmetrical and nonsymmetrical shifts in the structure of tax-prices. Assume that a proportional rate structure is in being and that progression is introduced. One means of introducing progressive rates would be that of increasing the tax-price charged to the “rich” man while reducing the tax-price charged to the “poor” man, leaving the middle or median man’;s tax-price unchanged. This is defined here as a symmetrical shift in rate structure, and it will not affect the outcome in this model. The median voter retains control over the majority-rule outcome, and his preferred results are not changed. However, suppose that progression is introduced by increasing the tax-price for the “rich” man, reducing the tax-price for the “poor” man and for the median-income man. When tax-price is modified for the median-income man, the shift is defined to be nonsymmetrical. If the shift serves to reduce this critical tax-price, the introduction of progression will have the effect of increasing the quantity of public good supplied under simple majority-voting rules. The median man is confronted with a lower tax-price, and he will desire a larger quantity. Nonsymmetrical shifts need not be unidirectional; progression might be introduced by increasing the tax-price for the “rich” man and, also, for the median man, while reducing tax-price for the “poor” man. In this case, the effects on simple majority-rule outcomes are the reverse of those traced above. The quantity of public good supplied will tend to be reduced.
This analysis suggests that even in the highly restricted model of equal evaluation, the differential effects of progression and proportion on the supply of public goods under simple majority-voting rules cannot be predicted until and unless the effects on the relative tax-price of the median voter are determined. This is an empirical fact, and useful research into the nature of real-world rate structures and changes in these structures within the context of the collective choice models discussed here can be undertaken.
The effects of progression, as compared with proportion, under a decision rule of unanimity can be readily observed from Figure 11.3(b). The limits are extended beyond those applicable under proportion. The “solution” set of points is larger. This generalization may be more important than it initially seems. Decision-making in democratic political structures may well require either more or less than the equivalent of simple majority approval. Insofar as some greater-than-majority support is, in fact, required to secure decision, the unanimity model can yield helpful predictions. The range over which a solution, under all qualified majority rules, may be found is larger under a progressive rate structure than under a proportional rate structure. The observed degree of discontent about the “proper” size of the public sector should, therefore, be greater. This seems an implication that could, in some proximate sense, be empirically tested.
A Three-Person Model with Unequal Evaluation and Unequal Incomes, but with Equal Preference Patterns
Less restrictive models are necessary if we are to develop hypotheses of extended interest. In any real-world fiscal setting the evaluations of different persons for public goods will be different and, also, persons will differ in other respects, some of which will be relevant for determining their tax liabilities. Given such a world of unequals, is there any orderly theorizing that can be carried out? Clearly if we impose no structure on the direction and extent of the variations among individuals, few predictions can be advanced. Some order may be introduced, however, if we impose restrictions on the model, but restrictions that are considerably less severe than those hitherto employed.
We propose to adopt a variant on the world-of-equals model. We assume that the preference patterns of the separate persons are identical, but that incomes differ. That is to say, the individuals in the model would be identical in their demand behavior if their incomes should be the same, but, since their incomes are different, their demands for public goods will differ. This allows marginal evaluation or demand schedules to vary among individuals due to the influence of income effects on individual choice behavior. While still highly restricted, this model is considerably more general than those previously introduced. If income effects should be absent or relatively unimportant in influencing the choices between public and private goods, this model reduces to that which has just been discussed; in this case, the separate demand curves become identical. If, however, income effects are significant, this new model, which we may call the equal-preference model, allows these to be incorporated into the analysis.
In general terms, any conceivable values for the income elasticity coefficients for public goods can be analyzed with this model. We shall, however, limit the scope of the analysis by imposing the further restriction that these coefficients are positive. This means that the demand schedules of the three persons may be ordered by the levels of income over relevant quantity ranges. This is shown by Figure 11.4, where DA, the demand schedule for the individual with the lowest income, falls below DB, which, in turn and for the same reason, falls below DC.
The tax-price confronted by each of the three persons also varies with income under either proportional or progressive income taxation. In terms of an ordering, therefore, tax-prices correspond with marginal evaluation or demand so long as the income elasticity coefficient is positive. This suggests that there may exist some structure of ordered tax-prices that will generate a unique collective outcome for which the decision rule is not influential. That is to say, for any given ordering of marginal evaluations for the public good, there should be some ordering of tax-prices that will guarantee what we may call “full neutrality.” This result will satisfy the necessary conditions for Pareto optimality, and, also, will not depend critically on the nature of the rules for the reaching of political choices. The condition that must be satisfied for “full neutrality” to be achieved is as follows: The income elasticity of the tax-price schedule must be equal to, but opposite in sign, the income elasticity of demand for the public good divided by the relative price elasticity of demand for the public good. When this condition holds, the decision rule is unimportant, and the outcome generated under any rule is “optimal.”
This principle has already been demonstrated for the case in which income elasticity of demand is zero; here the required structure of tax-prices must also have zero income elasticity. In other words, only when tax-prices are equal for all persons will the condition be met. If the income elasticity of demand for the collective good is unitary, a strictly proportional tax on income will generate neutral results only if the price elasticity of demand for the good is also unitary. Note that the elasticity of the tax-price schedule under proportional rates is unitary.
We are able to utilize the familiar concepts of income and price elasticity here because of our assumption that the underlying preference patterns of the separate persons are identical. This allows us to consider the shift from the choice situation of one individual to that of another at a different income level as equivalent, analytically, to a change in the income of a single person. Price elasticity of demand is normally expected to be negative; therefore, as the formula suggests, if the income elasticity of demand is positive, “full neutrality” must require that tax-price increase as income increases. Suppose, for instance, that the income of a person is increased by 10 per cent. With, say, an income elasticity of demand over this range of two, the preferred quantity of public good would increase by 20 per cent. Suppose, further, that over the relevant range, price elasticity of demand is unitary. What increase in tax-price would be just sufficient to keep the individual at the same preferred quantity as before the income change? Twenty per cent is clearly the answer. Hence, if the income elasticity of the schedule of tax-prices, over this range, is also two (a progressive rate structure), a shift in his income position from the first to the second status would not influence his choices as to the most preferred quantity of public-goods supply. Applying this reasoning to two persons at separate income levels rather than to one person at two separate income levels, which the equal-preference assumption allows us to do, we may say that if the tax-price schedule exhibits an elasticity of two in this case, both persons will be “satisfied” with the same public-goods quantity, which is, of course, a precondition for “full neutrality” in the sense that we have defined this latter term. If the formula is satisfied over the whole range of possible incomes, each member of the group, regardless of his income level, will “choose” the same quantity of the public good. If each person were made dictator in turn there would be no change in the amount of public good supplied as the decision-making power shifts. Given a tax system that satisfies the “full neutrality” formula, dictatorship, simple majority voting, and unanimity would guarantee the same, and Pareto-optimal, result.
What does this “full neutrality” conception suggest in regard to the actual structure of tax-prices among individuals? The relationship between income and price elasticity of demand is important in determining the rate structure that will satisfy the required condition, or, more appropriately stated for our purposes, in determining the effects of any specific rate structure that is postulated. If the income elasticity of the demand for the public good tends to be high, and positive, while the price elasticity of demand tends to be low, a progressive rate structure would be necessary to achieve the sort of neutrality noted. Or, to say the same thing somewhat differently, if these elasticity conditions prevail, a given structure of progression need not produce over-all inefficiency in the supply of public goods, and need not make the outcome so directly dependent on the political decision rule as might otherwise be the case. On the other hand, any shift downward in the income elasticity coefficient relative to that for price elasticity, tends to reduce the progressivity of the neutral tax-price schedule. And, if the income elasticity of demand should be negative, the elasticity of the tax-price schedule must also be negative to achieve neutrality. This means, of course, that persons with low incomes would in this case actually have to be charged higher tax-prices than persons with higher incomes.
Where does a “regressive” rate structure fit in this picture? By normal usage, this term characterizes systems in which the tax-price increases with income but not proportionately so. It is relatively easy to see that, for public goods possessing relatively low income elasticity coefficients, a regressive tax schedule may be necessary if neutrality is to be reached. This suggests that the whole notion of “full neutrality” be examined somewhat more carefully. Strictly speaking, the formula means only that the system is neutral with respect to the rule for making political choices. By implication, any system meeting this requirement will also be “neutral” or “efficient” in the more familiar sense that emerges from an application of the Pareto welfare criteria. Only through meeting this condition can a point on the Pareto welfare surface be attained in a Pareto-optimal manner. Note that this does not suggest that a position on the welfare surface cannot be attained nonoptimally; it may well be so attained. And, if some net redistribution is desired through the financing of the public good, the Pareto welfare surface (neglecting for now other possible violations of the necessary conditions) may be attained without satisfying the formula above, provided that the several departures from individual “optimality” are mutually canceling or offsetting. Note, however, that this nonoptimal attainment of the surface can never be inferred directly from individual choice behavior, and note, also, that in such cases, the political rule again becomes all important in determining the particular outcome that is likely to be generated. In other words, only the omniscient and benevolent despot is likely to be able to move the group to the welfare surface nonoptimally.
Our interest here is not primarily that of analyzing fiscal institutions for their effects on “efficiency” in the standard Pareto sense.2 It is, instead, that of attempting to make rudimentary predictions concerning the direction of effects on total spending for public goods that various fiscal institutions exert, via their influence on individual choice behavior. Returning to this primary emphasis, let us remain for the time being in the equal-preference model and assume a decision rule of simple majority voting. Is it possible to make any predictions concerning the differential effects of regressive, proportional, and progressive taxation, without regard to the question as to whether or not “full neutrality” is present? Once again the critical position assumed by the median voter-taxpayer must be stressed. If any shift is symmetrical with respect to this median man, there will be no direct effect on the majority solution. The structure of tax-prices can be modified without changing the political result. So long as symmetry with respect to the position of the median man is maintained, the rate structure can be “tilted” within wide limits without affecting the quantity of public goods supplied under majority-rule institutions. If nonsymmetrical changes are made in the structure of tax-prices, the majority solution will tend to be changed and the efficiency of the system modified.
Symmetry or nonsymmetry is defined with reference to the median voter, as preferences are arrayed along some public-goods quantity scale. In the simple cases that we have to this point discussed, we have assumed that the individual evaluations are ordinally related to income, and, also, that the structure of tax-prices is ordinally related to income levels. Even within these assumptions, however, it may be possible that the median or decisive voter in a majority-rule model is not the median-income recipient. To this point, we have implicitly assumed that this possibility did not exist. It should be admitted, however, that, under certain structures, the array of individuals by public-goods preferences may not correspond with their array by incomes; this should be especially noted since some of the empirical evidence to be cited suggests the presence of this pattern. In local communities that are characterized by nonprogressive tax structures, especially over the middle-upper income ranges, political coalitions may combine the upper- and lower-income classes in opposition to the middle-income classes. As we shall note in Chapter 13, there is considerable empirical evidence which suggests precisely this situation for American municipalities. In this case, the median voter may stand at either the lower or the upper end of the middle-income range. The analysis with respect to symmetry and nonsymmetry continues to apply, even here, although its implications with respect to actual effects of changes in rate structures cannot be so readily advanced. The situation can arise only when the “progressivity” in evaluation among persons exceeds the “progressivity” in tax-price structure.
The Relevance of the Equal-Preference Model
The equal-preference model is highly restrictive. Individual demands for public goods, as for private goods, differ for reasons other than differences in incomes. If this is admitted, however, are there any restrictions that can be placed on a model of behavior that will still allow conceptual predictions to be made? Here it is, I think, necessary to rise to the partial defense of the equal-preference model. When properly considered, the model is less restrictive than it at first appears. Individual tastes differ, one from the other; this may be, and must be, acknowledged. But is there a pattern to such differences or must they be assumed random? If we examine particular goods, private or public, a widely scattered pattern of demand would surely be observed. Some people just do not like garlic; others do. Similarly, for foreign aid. If, on the other hand, we examine the whole package of private goods, or the whole package of public goods, would such wide differences in tastes be observed? Differences would remain, but these may be relatively narrow, except as related to income levels. The final and critical test is provided when income effects are examined. If, for the consumption of the whole package of public goods relative to private goods, the differences in incomes among persons tends to overwhelm or to swamp the differences in tastes, the model that has been introduced here retains considerable relevance for our purposes.
If individual differences in the demand for public goods, on the average, are not related to individual differences in income or wealth, the levy of taxes on the basis of these characteristics makes little economic sense and surely leads to serious distortions in the allocation of resources. Implicit in the development of the familiar institutions of general taxation, which have used personal incomes or assets as the bases for computing individual tax liabilities, has been the assumption that all members of the group, generally, share in the common benefits of public services, and that these benefits may be, in some way, related to income-asset positions. This is not to suggest that modern tax institutions have evolved out of a “benefit principle” of taxation, as such. But even the so-called “ability-to-pay” principle carries with it some implied “willingness-to-pay” which, in turn, implies that general charges should be related to income-asset levels presumably because individual demands are so ordered.
The appropriateness of any general tax principle, or the possible efficiency of any general tax institution, depends on the effective limitation of the collective sector. The services financed through tax funds must be appropriately “chargeable to the whole community,” which is to say, they must provide general, nondiscriminatory benefits. If this “principle” is not followed, and the public sector is used to provide services that are designed to benefit specific subgroups in the community, the model of equal-preference is clearly inapplicable, as are general tax institutions. It would, for example, clearly be inappropriate to apply an equal-preference model when discussing the financing of irrigation projects by the United States government. And, more importantly, it is also inappropriate, on the basis of efficiency considerations, to utilize general tax institutions for the financing of such special benefit services.
Ignorance, Uncertainty, and Illusion
In this chapter individual participants have been assumed to act on the basis of complete information concerning alternatives. As earlier chapters have shown, such an assumption is untenable, and individuals must make voting choices under conditions of ignorance, uncertainty, and illusion, with these factors varying significantly from one institution to another. The effects are to make the outcomes of any political decision process less predictable for the simple reason that individual choices are less predictable.
The process of reaching agreement may actually be somewhat less costly than it would be under conditions of more information. An individual faced with genuine uncertainty as to alternative prospects may tend to agree more readily with his fellows than he would under certainty where his own interest is more sharply identified. This appears to be a positive advantage, and it suggests that fiscal institutions which embody considerable uncertainty and which create illusion possess attributes of “efficiency” that are often overlooked. This is no doubt correct, but the efficiency involved in reducing the costs of reaching agreement, under any decision rule, will tend to be offset by the greater costs, or inefficiency, in an allocative sense. Despite the fact that the individual does not know with certainty the effects of alternative outcomes, there continues to exist some “optimal” outcome, for him, if he could know what this is. And departures from this “optimal” outcome, viewed ex post, reflect allocative inefficiency. Thus, institutions that generate uncertainty in the mind of individual choosers tend, at the same time, to reduce the costs of reaching political agreement and to increase the costs of the “mistakes” made in some allocative sense. These two elements would have to be compared in each particular instance to determine the over-all effects of the separate institutions.
The problems that arise in the construction and use of political decision models are evident from the discussion. Despite these, the critical position assumed by the median taxpayer-voter in most of the models of majority voting allows an important step to be taken toward converting the analysis of individual choice behavior into one that retains relevance for group choice. If we can, in some fashion, locate the median voter, we are then able to predict the direction of effect that the various institutions will exert on fiscal choice through an analysis of the decision calculus of this individual. This device enables us to utilize much of the theory of individual choice behavior developed in previous chapters while crossing the bridge to collective choice.
The relevance of the whole analysis depends on the appropriateness of the simple majority-voting models as reflections of real-world political process in democratic governments. Obviously decisions on taxes and public spending are not made in glorified town meetings, even at the local government level. The critical question is whether or not the simplified town meeting can serve as a model with which we can analyze the much more complex process through which fiscal decisions get made. There is no way in which this question can be answered other than through the testing of hypotheses that emerge from the model against observed experience. The fact that, in some superficially descriptive sense, decisions do not seem to be made in this manner, tells us relatively little about the predictive power of the models.
Appendix to Chapter 11
Pareto Efficiency Under Equal-Preference Models
In this Appendix the relationship between schedules of tax-prices and variations in individual marginal evaluations over income changes will be examined more carefully. The derivation of the formula for full neutrality presented in the text of Chapter 11 will be clarified, and some of the efficiency implications of familiar taxing institutions will be suggested. The analysis is restricted to the equal-preference model, and the framework assumptions made in the discussion of the main text continue to hold here.
Assume that there exists a proportional tax on income and that this is the only means used to finance a single public good. What characteristics of individual preference patterns (and, by assumption, these are the same for all individuals) would have to be present in order for full neutrality to be guaranteed? The problem is illustrated in Figure 11.5. On the ordinate are measured private goods, on the abscissa, public goods. A system of proportional income taxation is represented by the fan-like array of “budget lines” intersecting on the abscissa at G. For an individual with private goods (income) of Y1, the “budget line” that he confronts is T1, and the slope of this line is the tax-price that he faces in any decision as to the amount of the public good to be supplied. Similarly, the individual at income Y2 faces the “budget line” T2, etc.
By definition of a purely collective good, all members of the group must consume or have available the same quantity of the public good. Assume this quantity to be shown as Q in Figure 11.5. The question becomes: What characteristics of the preference map must be present to insure that Q units of public good, financed by the schedule of tax-prices indicated, satisfies the conditions required for full neutrality? The answer is now evident from the construction. The indifference curves must be tangent to the successive budget lines along the vertical drawn from Q. As an individual is moved from E1 to E2 to E3, both income and tax-price increase. The increase in income tends, normally, to make him prefer a larger quantity of the public good; the increase in tax-price tends to make him prefer a smaller quantity. The necessity that the same Q satisfy the individual at the different income levels (or different individuals at different income levels in this model) requires that the income effect on his demand for the public good be precisely and fully offset by the price effect.
If proportional income taxation is to meet this condition, the income-elasticity coefficient must be the same as the price-elasticity coefficient, with reversed sign, since we know that the income elasticity of the tax-price schedule under a proportional rate structure is equal to one.
A progressive tax is illustrated in Figure 11.6. Note that, as drawn, the tax-price to the individual remains constant over the variations in the quantity of public goods; the “budget lines” remain linear. This assumption is not essential to the analysis, but is used here for convenience only.3 The elasticity of the tax-price schedule under progression is greater than unity; the budget lines increase in slope more than proportionately with income increases. In this configuration, if the price effect is to be completely offset by the income effect, the income elasticity of demand must exceed the price elasticity in absolute value, the ratio being just equal to the elasticity of the tax-price schedule. This is the formula presented in the text of Chapter 11, and it retains general validity.
The analysis may be extended by constructing schedules of tax-price and of marginal evaluation. In Figure 11.7, income (or private goods) is now measured along the abscissa and tax-price along the ordinate. For any specific quantity of public good, say Q, which we assume is supplied to the community at constant cost, there will be a schedule of tax-prices that will confront the individual as he moves along the income scale. Thus, under proportional taxation at, say, 10 per cent, the individual will pay a total tax-price of $100 if his income is $1000, and a total tax-price of $1000 if his income is $10,000. These totals can be translated readily into tax-prices per unit, once we know the cost per unit of the public good and the number of individuals in the community along with their appropriate income levels. If some greater Q must be supplied, the 10 per cent will have to be increased, imposing thereby a higher tax-price per unit on all members of the group.
In Figure 11.7, the line labeled R is drawn to represent a schedule of tax-prices under proportional taxation, for a given quantity of public good. If the full neutrality position depicted in Figure 11.5 is to hold, the line, R, must also represent the schedule of marginal evaluation. This schedule or curve is derived by plotting the slopes of the successive indifference curves along the vertical from Q in Figure 11.5 against income. If a progressive tax structure is to accomplish full neutrality, as in Figure 11.6, the line of marginal evaluation must lie along, and correspond with, a line of tax-price such as R’; in Figure 11.7. A regressive system is depicted as R’;’;.
The analysis remains limited, however, unless departures from full neutrality are introduced, and the construction of Figure 11.7 facilitates this extension. Assume that the marginal evaluation schedule lies along R, and that a system of proportional taxation would accomplish full neutrality. Let us examine what will happen when a decision is made to shift to a system of progressive taxation. As we have noted in connection with earlier models, it is necessary to distinguish between symmetrical and nonsymmetrical changes in the tax-price schedule with respect to the position of the median voter-taxpayer. If the change from proportion to progression is accomplished with the tax-price confronted by this median voter remaining unchanged, the total revenue collected from the group remains unchanged, and the “political equilibrium” that prevails under simple majority voting is not modified. The same quantity of the public good, Q, is supplied. The outcome remains Pareto optimal despite the fact that the shift is not, in itself, Pareto optimal. If we neglect the other necessary conditions (e.g., effects on the supply of effort) we can say that such a symmetrical shift from proportional to progressive taxation amounts to a movement from one point on the Pareto-welfare surface to a different point, the differences being exclusively distributional. The shift is equivalent, in effect, to a set of income transfers between the rich and the poor. Therefore, if the tax-price schedule is shifted from R to R’; in Figure 11.7, the supply of public goods will remain unchanged, and the excess of tax-price over marginal evaluation for the “rich” is just equal to the excess of marginal evaluation over tax-price for the “poor.” There is no way that the “rich” man can overcompensate the “poor” man, or vice versa. The point may be illustrated by supposing a three-man group, one each at income levels Y1, Y2, and Y3, in Figure 11.7.
A symmetrical shift to a regressive structure of tax-prices is analytically similar in all respects to a move to progression. One such shift is shown by R’;’; in Figure 11.7. The “poor” man is exploited under this structure, but he cannot bribe the “rich” man to change, and the position of the median man remains unaffected.
Symmetry need not characterize a change from one tax-price schedule to another. Let us now assume that for the given quantity of the public good, Q, the tax-price schedule in existence is that shown by R in Figure 11.7, and, as before, assume that this schedule guarantees full neutrality. That is to say, given cost and distributional conditions, the marginal evaluation schedule is also shown by R, for this Q. Now suppose that a change to a progressive rate structure is made, but that the change is nonsymmetrical with respect to the tax-price confronted by the median voter. In this case, the quantity, Q, will no longer remain the “equilibrium” quantity, as determined by simple majority voting.
Nonsymmetrical shifts can, of course, be weighted in either of the two directions. The tax-price faced by the median voter may be increased or decreased. If it is increased, the equilibrium quantity of the public good will be reduced; if it is decreased, the equilibrium quantity will be expanded. Other things equal, therefore, a change from a system of proportional taxation to one of progression may decrease, leave unchanged, or increase the quantity of public goods that will tend to be supplied in response to the desires of majorities. The direction of effect here will depend on whether or not, relative to the marginal evaluation schedule, the rate change is symmetrical or nonsymmetrical and, if the latter, the direction of the weighting. These results can be shown readily in a table, as illustrated in Table 11.1, where a five-person group is considered. Incomes are shown in Column 2. A proportional rate structure of 10 per cent yields a total revenue of $750, and, for simplicity, assume that the public good is available to the community at a cost of one dollar, allowing a quantity of 750 units to be initially supplied. Assume further that at the tax-price of 20 cents, the median man, C, is in “private” equilibrium, as indicated by his demand schedule, shown in Table 11.1 (b). Column 5 of the Table represents a symmetrical progressive structure. Columns 6 and 7, by contrast, represent nonsymmetrical progressive structures relative to the proportional structure in being at the outset. In the first, Column 6, note that the tax-price to C is reduced, and C will, therefore, desire 900 units of the public good instead of the 750 previously provided. Similarly, in the rate structure shown in Column 7, C will demand only 500 units of the public good because the tax-price that he confronts will have increased. In neither of these situations is 750 an equilibrium quantity.
These results can be depicted geometrically in Figure 11.8. Assume that the marginal evaluation schedule exhibits unitary income elasticity, as before, and that it is shown by R. In a new political equilibrium reached under majority voting, the position of the median voter, with income Y2, will be more favorable than under proportional taxation if the nonsymmetrical shift is weighted to the right. In the construction, if the new tax schedule is that shown by R’;, the median voter pays less than the average amount of tax; his tax-price is lower than under proportion, and the equilibrium quantity of the public good is increased. The result is nonoptimal in the Pareto sense. The “rich” man can now afford to bribe the “poor” man into modifying his vote, were such bribery possible, something which he could not do under symmetrical progression. He can do so because the excess tax that he now pays is greater than the excess benefit, at the margin, that the “poor” man receives. Thus, in shifting from proportion to progression nonsymmetrically, the supply of public goods has been shifted, and a position off the welfare surface is the result.
If progression is introduced nonsymmetrically, but is weighted to the left, the opposite results hold. The median man now faces a higher tax-price than under proportion, and he will exercise his influence on majority outcome through demanding fewer public goods. The situation, after the new equilibrium is established, is shown by the configuration, R’;’;, in Figure 11.8.
The same analysis that has been applied to the introduction of a progressive rate structure could be applied to the introduction of a regressive structure. This extension will not be carried out here. The analysis can, of course, be applied to any conceivable configuration of tax-price and marginal evaluation schedules.
To what extent is the analysis useful in helping to answer real, rather than hypothetical, questions? Although, once again, heroic assumptions are required, plausible predictions can be made. It seems that the income elasticity of demand for public goods is positive and probably not greatly different from unity in value. For purposes of analysis, we can also assume that the price elasticity is unitary. In this case, strict satisfaction of the conditions for full neutrality requires a system of proportional taxation. If the effective structure of tax-prices is also roughly proportional, when all tax institutions are considered jointly, we can conclude, with some degree of accuracy, that majority-voting rules probably generate roughly an “optimal” outlay on public goods, provided these goods include only those that are genuinely collective in the general sense. If, on the other hand, the effective structure of tax-prices is observed to be sharply progressive, and, also, if the median voter is observed to receive less than the average income, and to pay less than the average amount of tax, the situation becomes comparable to that noted with curve R’; in Figure 11.8. The result is probably nonoptimal because, relatively, the quantity of public goods is in excess of that which is “efficient” in the Pareto sense. By contrast, if the effective structure of tax-prices, under the same assumed conditions of income and price elasticities, should be regressive, the situation is nonoptimal due to an undersupply of public goods. Empirical research into several aspects of these relationships can, of course, establish better grounds for making over-all judgments.
The elasticity assumptions can, of course, be questioned, along with parts of the model, and changes in these will lead to different general predictions. Recall, also, that the whole analysis has been based on the equal-preference model. This need not be so restrictive as it appears, however, when real-world predictions are attempted. The empirical data that may be secured on the marginal evaluation of public goods will be drawn from cross-section statistical surveys. At best, some composite preference map, typical of or approximating that for the “average” or representative taxpayer-voter, may be derived. To this sort of data, the equal-preference model can be appended without difficulty.
Observation of political experience can yield helpful suggestions as to the direction of divergence between schedules of tax-price and of marginal evaluation. If there seems to be no observable relationship between income levels and the reactions of individual citizens to proposed extensions in public spending programs, there may be little divergence here, and full neutrality may be closely approximated. If, on the other hand, the “poor” are observed, generally, to approve extensions in spending, while the “rich,” generally, oppose them, the direction of “tilt” between marginal evaluation and the tax-price schedule is suggested, or vice versa in the opposing case. One extremely interesting case, which is suggested to be relevant by some of the empirical work that will be reported in Chapter 13, involves the “poor” and the “rich” combining forces to approve extensions in public spending programs over the opposition of the “middle”-income groups. This can be “explained” by the models developed here in plausible fashion. If the elasticity of the marginal evaluation schedule is unitary, as shown in Figure 11.9 by R, and the tax-price schedule takes the form of the curve shown by R’; in the same figure, this political result will follow. Note that this seems a possible shape for the tax-price schedule in municipalities where the lowest-income groups largely escape tax, and where the bulk of the revenues are collected from general property taxes. Note that, in the configuration of Figure 11.9, the man with income Y1, not Y2, is the median voter, since preferences for the public good are not arrayed in the same ordering as incomes. Other “explanations” for such results are also possible, of course, but the fact that the tools developed here can be extended to cover such results is perhaps indicative of their power on the one hand and their limitations on the other.
[1. ]The construction and the use of single-peaked preference schedules is based on the work of Duncan Black. See Black, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958).
[2. ]For an analysis similar to that of this chapter which places somewhat more emphasis on such efficiency aspects, see my paper “Fiscal Institutions and Efficiency in Collective Outlay,” American Economic Review, LIV (May, 1964), 227-35; and, also, see the Appendix to this Chapter where the equal-preference model is examined in further detail.
[3. ]For a discussion of nonlinear “budget lines” of this nature, see R. A. Musgrave, The Theory of Public Finance (New York: McGraw-Hill, 1959), p. 122. Note that Musgrave’;s figure is similar to the constructions introduced here, but that he uses this for a somewhat different purpose.