Front Page Titles (by Subject) 4.3.: Alternative Forms of Commodity Tax: The Choice of Base - The Collected Works of James M. Buchanan, Vol. 9 (The Power to Tax: Analytical Foundations of a Fiscal Constitution)
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4.3.: Alternative Forms of Commodity Tax: The Choice of Base - 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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Alternative Forms of Commodity Tax: The Choice of Base
We classify the alternative forms of commodity taxation to be examined in terms of the restrictions on allowable arrangements—whether base restrictions, restrictions on rate structure, requirements for uniformity across commodities, or requirements for uniformity across individuals and over units.
The discussion can be conducted conveniently in terms of the diagrammatics of Figure 4.1. In this diagram, Da and Db represent the aggregate (market) demand curves for two private, partitionable goods, A and B. These demand curves are assumed to be linear for expositional convenience, and cost curves are horizontal to reflect constant average (and hence marginal) costs. Quantity units are defined so that the initial marginal cost (assumed equal to price) is unity.
Let us suppose that A and B represent the only potential tax bases that might be assigned to government. The individual’s constitutional choice between them depends directly on the estimated needs for financing the quantity of public goods and services that he expects to desire over the sequence of budgetary periods during which the selected tax rule is to remain in force. If government is assigned the authority to tax A, it will, under Leviathan assumptions, maximize the revenue it can obtain from taxes on this base. The power to tax commodity or good A is identical analytically to the assignment of a monopoly franchise for the sale of commodity A. As depicted in Figure 4.1, the revenue maximum occurs at Qa*, the quantity defined by the intersection between the marginal revenue curve, MRa, and the marginal cost curve, MC. The revenue-maximizing tax rate is ta*; revenue collected is ta*Qa*, shown by the shaded area.
It is clear from the construction in Figure 4.1 that if the tax on commodity A is replaced by a tax on commodity B, the same amount of revenue could be obtained at a lower tax rate and a smaller excess burden. Let r be a rectangular hyperbola with the ordinate and the horizontal MC line as its axes, and let r pass through F. Every price-quantity combination along r, by construction, yields the same tax revenue (“monopoly profit”) as at F. Since F is the point at which maximum tax revenue is obtainable from A, r must be tangent to Da at F. Now, consider point L, at which r intersects with Db (to the right of F).5 At L, the same revenue that is obtained by the maximum revenue tax on A could be obtained by a tax of rate LS imposed on B. The welfare loss imposed by such a tax on B is LST, which is much smaller than the welfare loss FCE imposed by the tax on A. Hence, the traditional efficiency or welfare argument for the broader-based tax seems to emerge.
Under the assumptions about the behavior of government that we have made, however, it is clear that a tax rate of LS will not be imposed on commodity B. If government is granted access to base B, with no accompanying restriction on rates, this change will ensure that there will be an increase in the revenue that government will appropriate. Government will secure the maximum revenue obtainable from the tax on B. This revenue is tb*Qb* in Figure 4.1, where Qb* is determined by the intersection of MRb and MC, and tb* is the excess of demand price over MC at that quantity. It is clear that, under these conditions, the excess burden will be larger under the tax on B than the tax on A. Under the assumptions of proportional rate structure and linear demand curves,6 the excess burden induced by a tax exploited to its maximum revenue potential is exactly one-half the maximum revenue raised. A simple geometric proof of this proposition suffices at this point in the argument. All we need to note is that in the linear case, the marginal revenue curve lies exactly halfway between the demand curve and the vertical axis. In Figure 4.1, Qa* is exactly one-half Qa. The area of the triangle CEF in Figure 4.1 is given by ½ (ta* · CE); and since CE equals Qa*, the welfare-loss triangle CEF is ½ (tb*Qa*), or one-half of maximum revenue. Analogously, since Qb* is exactly one-half of Qb, the welfare loss attributable to the maximum revenue tax on B is ½ tb* · (Qb - Qb*) or ½ (tb*Qb*). We can conclude therefore that the ratio of excess burdens induced by alternative taxes will be the same as the ratio of their maximum revenue yields: if tax B generates larger maximum revenue than tax A, then it will generate a concomitantly larger excess burden when maximum revenue yields are obtained.
An immediate corollary of this result is that if we compare two taxes with identical maximum revenue yields under a proportional rate structure, they will generate identical excess burdens (given the appropriate linearity assumptions). Consider, for example, a commodity-tax base, X, which, when used to generate maximum revenue, yields the same maximum revenue as does tax base Y. Any number of such tax bases may exist, but a simple direct comparison of any two of them will suffice to make the point. As before, let Dx and Dy depict the aggregate demand curves for these alternative tax-base commodities in Figure 4.2. The maximum revenue rates produce identical revenues. The solutions represent differing points along the rectangular hyperbola, r. Is either one of these taxes to be preferred on the grounds that it generates a lower excess burden? Since both generate the same maximum revenue, and since the welfare loss at maximum revenue rates is one-half of maximum revenue in both cases, the excess burdens as well as the maximum revenues are identical. In other words, in this construction if we compare proportional taxes that raise the same maximum revenue, there can be no preference between them on excess burden grounds. Maximum revenue itself becomes a sufficient indicator of excess burden, as conventionally measured.7
A further corollary of this result is that if we compare two tax bases which are of identical magnitude pretax but for which elasticities of demand differ, the one with the lower elasticity of demand will give rise to the larger excess burden, because it gives rise to a larger maximum revenue. (Geometrically, this comparison could be depicted in a diagram similar in construction to Figure 3.3, with the demand curves of persons replaced by those for commodities.) This result is diametrically opposed to that which emerges from the conventional equi-revenue approach, which relates excess burden directly with elasticity.
If there can be no preference for broad-based taxes derived from excess burden comparisons (rather the contrary) within our Leviathan setting, the choice between broad- and narrow-based taxes—say, between a tax on A and a tax on B, as depicted in Figure 4.1—depends solely on the level of public-goods supply which the citizen expects to want. This anticipated desired level of public-goods supply depends in turn on both the predicted demand for public goods and on the total cost of providing them.
Consider total cost first. These costs are composed of three elements, as illustrated geometrically in Figure 4.3. First, there is the physical cost of production of G, depicted in marginal terms by MCg (which again for convenience we assume to be constant). Second, there is the additional cost imposed by the Leviathan government’s appropriation of some proportion, 1 - α, of tax revenue as pure surplus. If, out of each dollar of revenue, only 100α cents are expended on public goods, then the total revenue cost of $1’s worth of G is 1/α dollars. If government, for example, spends only one-half of each revenue dollar on providing G, then each dollar’s worth of G will cost taxpayers $2 in tax revenue. This “α effect” raises the per unit cost of G to 1/α MCg, as indicated in Figure 4.3. Third, there is the excess burden generated by the tax itself, which is also a genuine cost. As indicated above, given linear demand schedules and constant costs, and with proportional rates imposed at maximum revenue levels, this excess burden will be exactly one-half of (maximum) revenue. The aggregate cost per dollar’s worth of G will then be 3/2 1/α MCg, depicted in Figure 4.3.
The desired level of G, given this aggregate cost, can now be determined by appeal to the typical citizen’s constitutional predictions as to his demand for public goods. We should emphasize at this point that there is no necessary requirement here that all citizens have identical predicted demands for public goods (although in a strict veil-of-ignorance setting such an assumption would not be particularly implausible: to the extent that each individual is totally uncertain as to his future tastes and income level, we might expect that predicted demands for public expenditures on G would be roughly identical across individuals). It is sufficient here, however, to focus on the calculus of a single typical citizen. To consider the additional problems involved in deriving some constitutional agreement among citizens on the issue of appropriate tax allocations to Leviathan in the case where those citizens have differing predicted demands for G would complicate the discussion and divert us from our main purpose.
Let us depict the typical citizen’s predicted demand for G by Dg in Figure 4.3. Where this demand curve cuts the aggregate marginal cost line, 3/2 1/α MCg, will determine the desired level of G, depicted by G*. The level of revenue required to generate this output level, G*, is 1/α MC · G*, because MC · G* is the physical cost of producing G* and a is the proportion of revenue that Leviathan spends on G. This revenue level is depicted by the shaded rectangle in Figure 4.3. We can now, finally, construct a rectangular hyperbola through Er, depicted by r*, which depicts the desired level of (maximum) revenue.
The virtue of this diagrammatic treatment is that it permits us to show on a single diagram and in a neat and simple way the interrelationships among the citizen’s demand for public goods; the cost of physically producing those public goods; the “exploitative” dimensions of Leviathan’s surplus generation, as captured by the parameter, α the excess burden generated by the tax system;8 and the desired level of (maximum) revenue potential to which the constitution will permit Leviathan access.
Since our main interest here is with the constitutional selection of tax instruments, we focus directly on that issue. What we seek in the selection of the appropriate commodity-tax base is a base that will, when exploited to its maximum revenue potential, yield exactly that revenue which is required to generate G* (i.e., a revenue level subtended by any rectangle under r*). Suppose, for the purposes of argument, that the tax base A does this. In other words, suppose that the level of tax revenue indicated by r* in Figure 4.3 is exactly the same as that indicated by r in Figure 4.1. (Geometrically, this requires that if Figure 4.3 were superimposed on Figure 4.1 so that the abscissa in Figure 4.3 lay along the MC line in Figure 4.1 and so that the vertical axes were collinear, then r and r* would be coincident.) Then, if tax base A is in use, it follows that any broadening of the tax base would be undesirable. Such a broadening would lead to a higher level of tax revenues and of public-goods supply than the citizen-taxpayer desires, given the particular values of a and MCg as shown in Figure 4.3. To select, for example, the tax base B would generate a maximum revenue level above that depicted by r*, a corresponding level of G above G*, and leave the citizen predicting excessive exploitation by government in postconstitutional periods, along with unnecessarily high welfare losses in the tax system. At the same time, any tax base that generates the same maximum revenue as can be derived from a tax on A is equally acceptable with A. Within that set of tax options which yield identical maximum revenue, the citizen-taxpayer is genuinely indifferent.
What this discussion shows can be encapsulated in two statements: first, greater broadness of coverage under commodity taxation is not unambiguously desirable and it is positively undesirable beyond some point; and second, excess burden varies directly and linearly with maximum revenue, so that taxes which have identical maximum revenue have identical excess burdens when maximum revenue rates are applied, and taxes with larger maximum revenue yield a larger excess burden when maximum revenue rates are used (and vice versa). Base limitation therefore emerges as a legitimate instrument in the design of tax constitution for a Leviathan government. The general thrust toward broader-based tax institutions, characteristic of the orthodox analysis, cannot be sustained within the alternative perspective on tax institutions that we postulate in this book.
[5. ] We ignore the intersection above and to the left of F, although its existence has been a source of some controversy in the literature on tax progression and leisure consumption. See Robin Barlow and Gordon R. Sparks, “A Note on Progression and Leisure,” American Economic Review, 54 (June 1964), 372-77; and John G. Head, “A Note on Progression and Leisure: Comment,” American Economic Review, 66 (March 1966), 172-79.
[6. ] The linearity assumption is, of course, special. We should note, however, that this assumption is embedded in conventional measures of excess burden; by examining only first- and second-order terms in the relevant Taylor series expansion of the utility function, those measures of welfare loss are in fact only linear approximations. See note 7 below.
[7. ] Conventional measures of excess burden focus on the second term of a Taylor series expansion of utility functions. That is, if we can represent individual utility functions as U = f (X1, X2, ... , Xn), then which on manipulation can be shown to yield [See Harberger, “Taxation, Resource Allocation and Welfare”; and Harold Hotelling, “The General Welfare in Relation to Problems of Taxation and of Railway and Utility Rates,” Econometrica, 6 (July 1938), 242-69.] By taking higher terms of the Taylor series expansion, one can of course get more accurate measures of utility change. This is tantamount to allowing for differential curvature of demand (or marginal valuation) curves. So doing would permit a ranking of equi-maximum-revenue taxes according to total welfare loss but would involve measuring “excess burden” with a degree of refinement not used elsewhere in the literature and would in any case involve dealing with a high order of “smalls.”
[8. ] See David B. Johnson and Mark V. Pauly, “Excess Burden and the Voluntary Theory of Public Finance,” Economica, 36 (August 1969), 269-76.