Front Page Titles (by Subject) 4.4.: Uniformity of Rates over Commodities - The Collected Works of James M. Buchanan, Vol. 9 (The Power to Tax: Analytical Foundations of a Fiscal Constitution)
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4.4.: Uniformity of Rates over Commodities - 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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Uniformity of Rates over Commodities
Let us now suppose that, maintaining the basic assumptions of the previous section, the maximum revenue realized from taxation of A or B is insufficient to provide the level of public-goods supply anticipated to be desired during the postconstitutional sequence. The question then arises as to how government might be granted authority to tax both A and B, in order to secure the required revenue. If we assume that specific rates for each commodity cannot be set in advance, essentially two tax-structure possibilities present themselves. Government may be constrained to set the same rate on both A and B, or it may be allowed to set the rates on commodity A and commodity B independently and separately. In other words, Leviathan may be allowed access to a “uniform” sales tax which includes A and B in the base; or it may be permitted to apply a set of discriminatory excises, with rates of its own choosing.
In the latter case, if we assume that A and B are two of a large number of commodities and neither complements or substitutes one for another, a change in the price of A does not directly affect the price of B, and vice versa. In this setting, it is clear that a revenue-maximizing Leviathan will set the tax rate on A at ta*, as depicted in Figure 4.4, and the tax rate on B at tb*.
If, by comparison, government is constrained to set a uniform rate for both commodities, it is necessary to aggregate the demand curves in order to depict its revenue-maximizing solution. Our construction in which quantity units for the two goods are normalized in the dollar’s worth dimension facilitates the aggregating of heterogeneous commodities. We can, quite simply, add the demand curves for the two commodities horizontally, to get the curve Dab, as shown in Figure 4.4. Given the rate-uniformity constraint, government will set the tax rate so as to generate the quantity determined by the intersection between the marginal revenue curve relevant to the aggregated demand curve and the marginal cost curve. This solution is depicted in Figure 4.4 at quantity ½(Qb + Qa), with tax rate t* imposed uniformly on each of the two commodities. Given the assumptions of linear demand and constant marginal cost curves, total quantity, over the two goods, remains constant as between rate uniformity and rate differentiation. This constancy result emerges from the simple geometrical configuration. Since Qa* (the quantity of A under discriminatory rates) is ½Qa, and Qb* (the quantity of B under discriminatory rates) is ½Qb, the aggregate quantity under the discriminatory solution is ½ (Qa + Qb). But this quantity is identical to that which emerges in the nondiscriminatory solution. However, given that tb* > ta*, the quantity of B will be lower in the discriminatory solution than in the uniform-rate solution; conversely, the quantity of A will be higher in the discriminatory solution than in the uniform-rate solution. The revenue-maximizing uniform rate of tax, t*, will lie between the two separate revenue-maximizing rates, ta* and tb*, in the discriminatory solution.
The maximum revenue yield to government is higher in the discriminatory than in the uniform-rate situation as long as the elasticities of demand for the two commodities differ and as long as positive revenues are obtained from both commodities in the uniform-rate solution. Under such conditions, by our earlier analysis, it follows that excess burden is also lower under the uniform-rate requirement than in the situation where differential revenue-maximizing rates may be set. These results are, of course, familiar and become obvious once one accepts the analogy between the granting of the power to tax a commodity and the granting of a monopoly franchise. The requirement for uniform rates of tax for separate commodities is precisely analogous to the prevention of discrimination in monopoly price over separate segments of a market.9
The connection between these aspects of our analysis and the optimal-tax literature here is interesting. Reinterpreted within the setting of our model, what the so-called “optimal-taxation” literature provides is a specific set of instructions to government as to how it might exploit its power to tax in the most effective way. The analysis provides precisely that tax-price rule which would lead a monopolist to discriminate perfectly between markets.10 In this context, the optimal-tax rules are precisely the maximum revenue-maximum profit rules. The analogy is evident here, but it has not, to our knowledge, been specifically noted.
The diagrammatic analysis we have employed is, to be sure, strictly partial—no allowance is made for the interaction between the demand for A and the price of B, and vice versa. It could therefore be claimed that it hardly bears on the optimal-tax literature, which typically adopts a general equilibrium framework. But the foregoing discussion can easily be extended to a general equilibrium setting; such an extension is provided in the appendix to this chapter.
We have, in this section, referred to differing rates of tax on differing commodities as “discrimination,” partially to emphasize the analogy with discriminatory monopoly. This construction is made more tenable by our normalization of quantity units for separate commodities in the “dollar’s worth” dimension. Nonetheless, we should acknowledge that economists conventionally do not apply the term “discrimination” with reference to relative tax treatment of observably different commodity bases. The language of this section can be modified to conform with conventional usage by referring only to uniform and nonuniform rates of tax.
There is an additional argument that may be adduced in support of conventional language here, an argument that has some relevance for the analysis of the following sections. As we know from orthodox price theory, a private monopoly firm may find it difficult to differentiate among customers or over units because of resale-retrading prospects. However, a monopoly firm that markets two separate and unrelated commodities would face little difficulty in setting different prices, even if quantity units are normalized in dollar-cost terms. In considering the differential rates of tax among commodities that governments may levy, we are analyzing a situation analogous to the monopoly firm in the second of these situations. By contrast, when we come to examine the prospects of tax-rate differentiation among persons or over units of goods, we look directly at the analogues to various forms of discriminatory monopoly.
[9. ] See Joan Robinson, The Economics of Imperfect Competition (London: Macmillan, 1933), chap. 15.
[10. ] Although not among consumers or over units of output; see below.