Front Page Titles (by Subject) 7.2.: Public-Goods Supply under a Pure Surplus Maximizer: Geometric Analysis - The Collected Works of James M. Buchanan, Vol. 9 (The Power to Tax: Analytical Foundations of a Fiscal Constitution)
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7.2.: Public-Goods Supply under a Pure Surplus Maximizer: Geometric Analysis - James M. Buchanan, The Collected Works of James M. Buchanan, Vol. 9 (The Power to Tax: Analytical Foundations of a Fiscal Constitution) 
The Collected Works of James M. Buchanan, Vol. 9 The Power to Tax: Analytical Foundations of a Fiscal Constitution, Foreword by Geoffrey Brennan (Indianapolis: Liberty Fund, 2000).
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Foreword and coauthor note © 2000 Liberty Fund, Inc. © 1980 Cambridge University Press.
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Public-Goods Supply under a Pure Surplus Maximizer: Geometric Analysis
The characteristic feature of our Leviathan model is that, in the absence of any constraints that force him to act differently, the king will set a at zero. That is, he will provide none of the public good, G, valued by citizens. He will simply maximize tax revenues, R, and he will utilize all of these for the funding of his own privately consumed goods and services. The question to be posed is as follows: Is there any way in which tax institutions may be selected, at the constitutional stage, so that a will not be set at zero—so that at least some G will be provided?
By our Leviathan-like assumptions about political process in postconstitutional periods, the potential taxpayers-beneficiaries have no direct control over the quantity of G provided by government. How can the king be induced to supply some positive quantity as a part of his own utility-maximizing behavior? Such inducement may be introduced if, by supplying G, total revenue collections are increased sufficiently to increase Yk. That is, an increase in a, the proportion of revenue devoted to the financing of G, may, in certain cases, increase the value of (1 - a)R, provided that there is a positive relationship between a and R.
Total revenues, R, are a function of the tax base and rate structure. In order to generate the required positive relationship between R and a, therefore, the base and rate structure, the essential determinants of R, must be variable and somehow related to the provision of G. This suggests that the tax base, whether it be an expenditure item or an item of income, must be complementary with the provision of G, as reflected in the independent behavioral adjustments of the taxpayers-beneficiaries.
The tax-base variable, B, may be arranged so that it is subject to some direct control by the taxpayers-beneficiaries. The public-goods variable, G, is, by our assumptions, under the direct control of government. Hence, we have a reaction-function sequence that may be illustrated in familiar diagrammatics. In Figure 7.1, we measure G along the abscissa and B along the ordinate, both in dollar units. Consider now the curve NN′, which is drawn to be horizontal over the range out to some production constraint. This represents the locus of equilibrium consumption levels of B as the quantity of G increases or, alternatively, the reaction curve (line of optima, ridge line) traced out by the utility-maximizing reaction of taxpayers-beneficiaries in “supplying” B for each possible level of G. Over the relevant range along NN′, note that the “supply” of taxable base, B, is invariant with the provision of G. In such a situation, the government would have no incentive at all to use any tax revenues collected to provide a positive quantity of G. It can maximize R by levying the highest allowable tax rate on B, and then maximize Yk by using all of the R to satisfy its own strictly private needs.
Contrast this situation with one in which B is highly complementary with G. The curve CC′ in Figure 7.1 depicts this case. Note that here the amount of taxable base “supplied” by the taxpayers-beneficiaries increases with the amount of G provided by government, at least over a substantial relevant range. And as CC′ in Figure 7.1 suggests, there may be situations where any revenue collection is impossible without some positive provision of the public good: individuals will simply not spend money on B unless there is some G to consume with it.
In order to determine how much G will be provided, it is necessary to specify the relationship between tax revenues and the tax base. For this purpose, we assume that the government is limited to a specific rate structure—which for ease of treatment we take to be proportional. This allows us, in Figure 7.2, to depict a relationship between the equilibrium amount of B consumed by individuals and the level of G, in the presence of the revenue-maximizing proportional tax rate, t*, applied to the designated base, B. This is shown by QQ, which will in general differ from C′C in Figure 7.1. The curve QQ traces out the behavioral adjustments of taxpayers-beneficiaries in generating taxable base under the imposition of the maximum revenue tax. (QQ may lie above, below, or be coincident with CC′ over any part of the relevant range, with the precise relationship being primarily dependent here on the income elasticity of demand for the base variable.) The curve QT in Figure 7.2 relates the tax collections derived from the revenue-maximizing tax on B to levels of G provision. For each level of G, the vertical distance from the abscissa to QT represents total tax revenue. The vertical distance between QT and QQ represents net-of-tax expenditures on B.
On the basis of the set of relationships indicated in Figure 7.2, what level of spending on G will the surplus-maximizing king opt for? Given that he is restricted to tax base B and a proportional rate structure, we can answer this question by constructing a 45° ray, 0Z, from the origin. Since all variables are measured in dollar units, the location of a position on the 45° line implies that all revenues collected from the tax are required for spending on the provision of G. There is no net surplus. (Here, as elsewhere, we assume that government has no access to revenue-raising instruments other than those being analyzed.) Clearly, if B is the only tax source available, points to the left of M′ are infeasible: the maximum revenue that can be raised from taxing B cannot, over this initial range, sustain the levels of G that are required to generate such revenues in the first place. Positions to the right of M′ and below M are feasible in the sense that the levels of outlay on G required can be financed by levies on the designated tax base, B. If the relationships are as depicted in Figure 7.2, the king’s surplus is maximized at E, where the “marginal cost” of producing more G is equal to the “marginal revenue” generated by that provision (where the slope of QT is unity). At this point, spending on G is measured by 0L (equal to LC), and total revenue collections are LE, with a maximum surplus of EC. The proportion of revenue spent on G, the a previously noted, is LC/LE. This illustration demonstrates that tax institutions—and specifically the selection of an appropriate tax base—may serve to ensure that the king (or, more generally, the monopoly government) will spend a share of tax revenues on financing valued output. He will do so to maximize his own utility, without any enforcing agency, in a setting where, if there were no such relation between the tax base and G, spending on G would be nil.
The surplus-maximizing solution for government or the king may, however, generate varying levels of G, depending on the tax base selected and on the precise shape of the complementary relationship between the base and the public good. Suppose, for example, that a tax base, B*, is selected such that QT shifts to the shape shown by QT* in Figure 7.2. Net surplus is maximized at E*. But G* may not be the predicted efficient level of outlay on the valued public good; such a constitutional arrangement may succeed in raising a only to ensure that an unduly restricted level of outlay be undertaken by government.
The construction does suggest, however, that, if there should exist an unconstrained choice among possible tax bases, with varying degrees of complementarity between these and the public good, an optimum optimorum solution might be imposed constitutionally. This would require that the tax base be selected such that, when the king levies the allowable revenue-maximizing proportional tax rate on this base, the only viable budgetary position requires that virtually all funds collected be spent on providing the good and, further, that these funds will purchase precisely the efficient quantity, as predicted at the constitutional level. Such a solution is shown at E′, where G′ is the predicted efficient level of outlay on the public good, and where Q′T′ suggests that the position at E′ is the only possible position for viable budgetary behavior on the part of the government. In the limit, there is no surplus left over for exploitation by the revenue-maximizing, perquisite-seeking king. Under such constitutional “fine tuning” as this, the problem of ensuring the predicted efficient level of outlay is incorporated into the problem of ensuring that revenues collected will be disposed efficiently.
Possible criticism of the analysis at this point involves the unconstrained-choice assumption. Such fine tuning may not be possible, especially when it is recognized that the complementarity relationships for feasible tax-base usage may be severely restricted in number, and, even among the feasible set, the relationships may be narrowly confined. Constrained optimization for the potential taxpayer-beneficiary will in general require trade-offs between allowing the king additional surplus, on the one hand, and accepting levels of public-goods outlay which differ from that desired.
Indeed, far from there being an unconstrained choice over tax base, each generating a different level of G and amount of king’s surplus, we must face the possibility that there will be no tax base available that constrains the king to produce any G at all. To demonstrate some of the effective limits on the central proposition, consider Figure 7.3. If a tax base selected is too narrow in relation to the public good in question, a viable budgetary solution may prove impossible. For example, consider a situation like that depicted by the curve Q3T3 in Figure 7.3, which lies entirely below the 45° line. As an illustration, suppose that an attempt were made to finance highways exclusively by taxes on automobile air conditioners. It is probable that the revenue-maximizing tax on such a narrow base would generate far less revenue than would be required even to maintain a road network, much less construct it. A second possibility might be that the complementarity between a selected base and the public good might be insufficiently strong to offer any incentive for public-goods provision by the surplus-maximizing king. Consider a situation as depicted in the Q4T4 curve in Figure 7.3, which, as drawn, has a slope less than unity over its entire range. Unless otherwise constrained, and despite the complementarity between the tax base and the public good, the king will maximize his own surplus by providing none of the good, by keeping a at zero.
What is required for the disciplinary influence of selected tax-base constraints on governmental fiscal behavior in disposing of tax revenues is a tax base that exhibits a strongly complementary relationship with the public good and is sufficiently broad to finance its provision. It is not entirely obvious that such a tax base will be available for each of the public goods that the taxpayer-voter might demand. We can, however, think of some examples where the required relation does hold—and the highway-public road case is one such. In the absence of a road network, few automobiles would be privately purchased and used. In the presence of a road network, automobile usage is “supplied.” Hence, a general constitutional requirement that roads be financed exclusively by taxes levied on automobiles (perhaps along with other privately purchased road-using inputs—gasoline, oil, tires, etc.) will ensure that the government, even in the model of the pure surplus-maximizing king, will spend some part of its tax revenues on road construction and maintenance.