Front Page Titles (by Subject) 4.6.: Discrimination by Means of the Rate Structure - The Collected Works of James M. Buchanan, Vol. 9 (The Power to Tax: Analytical Foundations of a Fiscal Constitution)
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4.6.: Discrimination by Means of the Rate Structure - 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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Discrimination by Means of the Rate Structure
We may extend this analogy between the taxing power and the monopoly franchise even further. If the ability to tax X is identical with the power to sell X at monopoly prices, it is clear that the best of all possible situations for Leviathan would be a situation in which tax rates might be set to mirror the price structure of the perfectly discriminating monopolist. In both cases, the idealized objective is to appropriate the full consumer surplus of each and every consumer. For purposes of analysis, let us ignore the practical difficulties noted above and examine two questions:
The “perfectly discriminatory” tax system. We consider initially a simple two-person community, in which government has power to tax some good, X, which we assume, as before, is produced under constant costs. The marginal evaluations of the two individuals for X are depicted in Figure 4.5 by DI and DII.12 Under perfectly discriminatory taxes, each individual will face a different and regressive rate structure that traces out his or her marginal evaluation curve. The purchase-offer schedule will be designed to leave each individual with just enough consumer surplus to induce purchases up to the point where marginal evaluation equals marginal price, which will be set equal to marginal cost. For example, the tax-rate structure for I will begin with a rate marginally below AK for the first unit of X, and the rate will fall linearly with increases in X consumption by I until the rate is zero at QI. Similarly, for II the tax-rate structure will begin with a rate marginally below AJ for the first unit of X that II buys, and the rate will decline linearly to zero at QII. Total revenue is given by the area under the aggregate demand curve for X, the shaded area determined from the aggregate “average-revenue” schedule depicted as ARx in Figure 4.5. This revenue is obtained without the imposition of any excess burden.13
This relationship between rate discrimination and excess burden has implications that are worth exploring, even given the idealized confines of the model. To do so, we can ask the following question: If, under a uniform proportional tax on X, the level of revenue is precisely that desired, could we be certain that allowing discrimination by means of the rate structure would be undesirable? For any other form of discrimination so far considered, the answer would be an unambiguous affirmative: discrimination would increase maximum revenue, increase excess burden, and push the level of public expenditure beyond the desired level. In this case, where the discrimination involves differentiation in rates over separate quantities of commodities purchased, however, the conclusion is not definitive. The move from the uniform proportional rate to the perfectly discriminatory set of regressive rates would, in the linear case, exactly double the revenue collected from each individual and hence the total revenue derived. It would, however, increase the costs endured by each citizen-taxpayer by only one-third (again assuming a linear demand curve for the taxed commodity). These propositions can be demonstrated by appeal to Figure 4.6. In the uniform nondiscriminatory tax case, the revenue derived from X is the area MNSW, representing · Qx′. Under this tax regime, the individual enjoys the net consumer surplus depicted by triangle LMN. Consider the move to the discriminatory tax regime: the total revenue is here the full area of triangle LWEx because all consumer surplus is appropriated by government in tax. Now, triangles LMN and NSEx are congruent, because MN is equal to WS, which is in turn equal to SEx (i.e., Q′x is exactly half of Qx), and MN and SEx are parallel. Further, we know that NSEx is exactly one-half the area of MNSW. Hence, LWEx has twice the area of MNSW: revenue is twice as large in the perfectly discriminatory regime. And LMN has an area equal to one-third of MNExW. We can therefore conclude that, in the move from uniform to perfectly discriminatory taxation, total revenue (and the level of public-goods supply) doubles, while aggregate costs rise by one-third. There has clearly been a reduction in the per unit cost to the citizen of public-goods supply. We know that
Thus, the average cost, that is, the cost per unit of public good, which in this simple model is equal to marginal cost, varies in the two cases as follows:
In short, perfect discrimination both increases the level of public-goods supply and reduces the per unit cost of public-goods supply. In order to determine whether the citizen-taxpayer would prefer the perfectly discriminating solution to the simple uniform outcome with the same tax base, we need to examine the value that he places on the additional units of public-goods supply he obtains. We can do this in a simple way by appeal to Figure 4.7. In this diagram, the revenue derived from tax base, X, under a uniform proportional rate structure is depicted by rx, and the corresponding level of public-goods supply by Gx, where rx and the horizontal line (1/α)MCg intersect. The aggregate marginal cost, including excess burden, of this expenditure level is 3/2(1/α)MCg and the desired level of G, depicted by G*, occurs where the citizen’s-taxpayer’s predicted demand for G, Dg, intersects this aggregate marginal-cost line. Now, let us suppose that X is an appropriate tax base, given a uniform proportional rate structure. Then, Gx and G* will represent identical levels of public-goods supply; Figure 4.7 is drawn on this basis.
In this sense, the combination of tax base X with the simple proportional rate structure seems to represent a constitutionally optimal “fiscal rule.” However, suppose that perfectly discriminatory taxation of X is now allowed. Revenue doubles, and public-goods output increases to 2Gx, where 2rx cuts (1/α)MCg. The relevant cost curve, however, is now lower than before: it is (1/α)MCg, not 3/2(1/α)MCg, because the tax involves no excess burden. (We note that this represents a reduction in per unit costs of one-third, as derived above.) Because costs are lower, the level of G desired under the perfectly discriminatory tax regime is correspondingly larger: it occurs where Dg cuts the new marginal cost curve, shown as Gd in Figure 4.7. Depending on the elasticity (or slope) of Dg, the point Gd may be to the right or the left of 2Gx. In fact, Gd will be to the right of 2Gx if the point elasticity of demand evaluation at G* has an absolute value greater than 3. For, if Gd and 2Gx were coincident, that elasticity, η, would be given by
However, even where Gd lies to the left of 2Gx, as in Figure 4.7, the perfectly discriminating solution may still be preferred. The gain from the move to the perfectly discriminating solution is the shaded area to the left of Dg between 3/2(1/α)MCg and (1/α)MCg, and has to be compared with the area of the shaded triangle between Gd and 2Gx below (1/α)MCg, which is the loss from possibly excessive government spending. (It can, in fact, be shown that the perfectly discriminating solution will be preferred if the point elasticity of demand at G* is greater than ¾.)
Care must be taken in the analysis throughout to avoid confusion between the taxpayer’s anticipated demand for G, the public good, and his anticipated demand for tax-base commodities, A, B, or X in our discussion, and depicted in Figures 4.1, 4.2, 4.4, 4.5, 4.7, and 4.8. Only in Figures 4.3 and 4.7 and the associated discussion are demand schedules for the public good depicted. In the discussion immediately above, the perfectly discriminatory solution serves to eliminate all potential consumer’s surplus in the purchase-use of the tax-base commodity, X. But, by lowering the effective cost-price for the public good, G, through the elimination of all excess burden in taxing X, consumer’s surplus involved in the “purchase-use” of G is increased. And, given plausible values of relative elasticity coefficients, it is conceivable that the taxpayer-consumer may prefer the discriminatory tax solution to the nondiscriminatory one.
At this point, we note an important difference between the model of the monopoly firm in price theory and our model of Leviathan as tax gatherer. In the former, the effective value of the analogue of α is zero—that is, the consumer is not conceived as being a beneficiary of any part of the profits that the monopolist secures, and hence no value is placed by the consumer on any addition to such profits. The consumer would never voluntarily give up the net surplus that a shift from simple to discriminating monopoly would involve. In the Leviathan model, by contrast, the taxpayer does place some positive value on increments to monopoly tax revenues, because some proportion, a, of them is spent on things (public goods) that the taxpayer values; and the parameter a must take a positive value if any taxing power is to be granted to government at all.
It is interesting to note, however, that beyond the requirement that a be positive, its value plays no direct role in the foregoing analytics at all: everything hinges on the elasticity of the demand curve for public goods. In some ways, this is a slightly surprising result because it seems to imply that the value of the increment to revenue that the move from proportional to perfectly discriminatory taxation generates is independent of the proportion of that revenue expended on public goods. However, this apparent anomaly is explained by the fact that the parameter α exercises a similar influence on both the value of the incremental units of G and the units originally allowed for under the proportional tax structure alternative. To the extent that a does exercise an influence on this calculation, it does so indirectly and in a slightly surprising direction. The smaller a is, ceteris paribus, the higher the price and hence the smaller the quantity of G within the neighborhood of which the extension of public-goods supply occurs. It seems reasonable to presume that, as in the case of a linear demand curve, the elasticity of demand will be higher in this range. In this sense, the smaller the proportion of revenue spent on public goods, the greater the likelihood that the citizen will desire the extension of revenue that the shift from a proportional to a perfectly discriminatory rate structure makes possible.
The results of this section contrast strikingly with those which emerged under other forms of discrimination examined earlier. If there is discrimination among individuals or among commodities, but not over units (because, say, of a requirement that rate structures be proportional), excess burden is directly related to maximum revenue, and the comparison of alternative tax bases can be conducted in terms of maximum revenue comparisons alone. Once discrimination over units is allowed, however, excess burden issues do obtrude. An increase in maximum revenue yield when the initial revenue yield is “appropriate” may indeed be desirable if that revenue increase results from a change in rate structure, from proportionality to regression.
Analogously, a reduction in revenue when revenue is “excessive,” achieved by restrictions on the rate structure—requiring progression, perhaps, or outlawing regression—cannot be unambiguously designated as desirable.
A second aspect of these results merits emphasis. It is clear that, at the constitutional level, the individual would prefer a perfectly discriminatory tax regime to a proportional tax that generates the same revenue yield. Suppose, for example, in Figure 4.6, that there is some potential tax base, Y, the demand curve for which is identical with ARx. The proportional tax on Y would achieve the same maximum revenue as the perfectly discriminatory tax on X, but would do so at 50 percent higher cost in terms of surplus forgone.
As in the earlier case with our discussion of possible discrimination in rates among taxpayers but not over quantities, the practical relevance of the whole analysis should not be overly emphasized. For the same reasons there mentioned, and others, effective rate discrimination over quantities may prove impossible to enforce for ordinary commodities. Even if interpersonal trades to implement indirect purchases could be prevented, the prospect for storage could allow single purchasers to take full advantage of the quantity discounts that the idealized differential rate structure would offer to the taxpayer. The possibility of any enforceable and effective differentiation over quantities purchased would be limited to nonstorable commodities.
Uniformity among persons, discrimination over units of commodity. To complete the discussion of this section, it is interesting to examine briefly the case in which there is uniformity of treatment as among individuals, but where regression in the rate structure is allowed. We might conceptualize this setting as one in which constitutional-legal rules dictate uniformity of treatment among persons, but where differentiation in rates of tax over units is allowed as long as all persons face the same “offer schedule.” What rate structure will generate maximum revenue under these conditions? To answer this question, consider Figure 4.8. The simple two-person setting is again used, with DI and DII being the demand curves of individuals I and II for commodity X. Under the constraint that I and II must be treated identically (i.e., they must face the same rate structure), it is clearly possible for II to be treated “as if” his demand curve for X is DI; it is equally clearly impossible for I to be treated as if his demand curve were DII, because DI lies inside DII throughout. The revenue to be obtained from a single perfectly discriminating rate structure based on DI is given by the area between the new curve RI and MC; RI is constructed so that the vertical distance between RI and MC at any output is twice the vertical distance between DI and MC. The revenue to be obtained from a single perfectly discriminating rate structure based on DII is the area under DII above MC. A rate structure geared to discriminate ideally along DII would, of course, have the effect of eliminating the first person (I) from the market completely. The revenue-maximizing Leviathan will, if limited to these two options, choose between these solutions on the basis of revenue yield.
However, government will not in general find it revenue maximizing to focus on one or the other demand curve entirely. What is relevant here is the upper “envelope” combining the RI and DII curves. Up to the quantity level Qs, where RI and DII intersect, more revenue is obtained by pricing according to DI; beyond that point, revenue will increase if there is a switch to DII and government levies a tax rate that just induces II to purchase additional units of X. For instance, consider the output unit immediately beyond Qs. If government levied taxes according to DI, the marginal revenue would be RI, which is less than DII in this range. The rate structure is, then, that depicted by the heavy line in Figure 4.8. It follows DI to Qs and DII thereafter. It is therefore “regressive” over the ranges zero to Qs and Qs to QII, but there will be a sudden discontinuous jump in rates at Qs.
It is of some interest to compare the welfare loss in terms of excess burden per dollar of revenue in this solution with that in the various other discriminatory solutions previously examined, over commodities or individuals. We can show easily that the excess burden is indeed smaller here. For it will be recalled that, under maximum revenue assumptions, in all the cases where discrimination over quantities to individuals is not allowed, excess burden is exactly one-half of maximum revenue. In the case under examination here, however, such discrimination is all that is allowed. For individual I the excess burden is the shaded area under DI between Qs and QI (Figure 4.8). Because RI lies above DI but below DII over this range, this excess burden is less than one-half of the revenue raised from II over that range (from Qs to QI); since more revenue than this is raised in toto, excess burden must a fortiori be less than one-half of revenue over the entire range zero to QII. And there is no excess burden on individual II since he purchases the same quantity under the tax structure postulated and under the no-tax market solution. The general conclusion, therefore, is that, under maximum revenue assumptions, the possibility of discrimination over units of the taxed good by means of the appropriately “regressive”14 rate structure provides a means of achieving revenue “more efficiently” (i.e., at smaller welfare loss per dollar of revenue raised) than does the completely nondiscriminatory alternative. Discrimination among goods and individuals in the absence of discrimination over units of output does not exhibit this property. Furthermore, this “efficiency” advantage of discrimination within the rate structure confronting each person does not depend on the presence or absence of other forms of discrimination.
It seems, on this basis, that the same conclusions hold here as with the perfectly discriminating regressive rate structure examined above. A uniform proportional tax may be dominated in the constitutional calculus by a tax that gains the same maximum revenue by means of an appropriately chosen rate structure that declines over units but is uniform among persons, applied to a smaller base. The desired level of revenue will be larger in the latter case, because total costs of public goods are lower due to the lower excess burden. It is possible that a tax which raises “too much” revenue under a uniform proportional rate will raise “too little” under a uniform “regressive rate” structure, even though revenue in the second case will be larger than in the former.
[12. ] As elsewhere, the demand curves and the marginal evaluation curves for X are taken to be identical—we abstract from income effects—strictly for analytic convenience.
[13. ] For an early analysis of the analogous construction depicting a private monopolist’s profit-maximizing quantity discount offers, see James M. Buchanan, “The Theory of Monopolistic Quantity Discounts,” Review of Economic Studies, 20 (1952), 199-208.
[14. ] Note that the rate structure is violently progressive in the neighborhood of Qs, but regressive elsewhere.