Front Page Titles (by Subject) 3.2.: Tax-Base and Tax-Rate Constraints in a Simple Model - The Collected Works of James M. Buchanan, Vol. 9 (The Power to Tax: Analytical Foundations of a Fiscal Constitution)
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3.2.: Tax-Base and Tax-Rate Constraints in a Simple Model - 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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Tax-Base and Tax-Rate Constraints in a Simple Model
Initially, we restrict the analysis to a single individual who is assumed to be exercising his constitutional choice between only two potential definitions of the tax base—one that is fully “comprehensive” and another which falls short of this limit. We shall relax these assumptions later, but at this point the simplification is convenient. It is immaterial for our argument precisely what the noncomprehensive base is and whether the tax is levied on the “uses” or the “sources” side (i.e., whether it might be an income tax or an expenditure tax). Let us consider a simple model in which labor is the only factor of production. Suppose, further, that the noncomprehensive tax base is money income derived from labor effort in the market, and that the comprehensive base includes such money income and also the imputed money equivalent of the individual’s nonmarket production of valued end products, including leisure; in other words, the comprehensive base is full income or potential income. The question to be examined is whether the person would prefer a tax constitution that embodies the comprehensive base over the one that restricts the base of tax to money income.8
The situation may be depicted as shown in Figure 3.1. The indifference curves, labeled with i’s, indicate the individual’s preferences as between money-income-earning activity, Y, and, say, leisure activity, L. These preferences exhibit the standard properties.9 As is customary in orthodox tax analysis, in this introductory discussion we ignore income-effects feedbacks generated by the provision of public goods. The pretax situation is characterized by a relative trade-off between L and Y that reflects the productivity of income-earning activity. The initial pretax equilibrium is at E (Y0, L0) on i0.
Consider next the prospect that the individual would face if the government acquires access to the fully comprehensive tax base. In such an event, the individual would be exploitable up to the full limits of his potential income-earning ability over and beyond some minimal subsistence. Apart from this minimum, all of the “income equivalent,” 0Ya, is potentially available for governmental use. The government bent on maximizing revenue could levy a tax that expropriated the individual’s maximum potential earnings beyond the allowed subsistence level.10
Since it is inconceivable that anyone could ever anticipate an “efficient” public-sector-private-sector mix that would require all potential income above subsistence for governmental purposes, it seems clear that a potential taxpayer-beneficiary would not select the comprehensive tax base if he predicts postconstitutional governmental behavior of the type that we have postulated. He will seek instead to impose constitutional constraints on the fisc, on the ability of government to tax. He can do so, in our simple case, by allowing the government to levy an income tax only on the ordinary sources of earnings—only on money incomes. The maximum revenue that can be secured from this narrowed tax base is depicted by YmYa in Figure 3.1. Clearly, if the government imposes a tax on money income with revenue in excess of YmYa, the individual would be better off by ceasing to earn income at all; he would improve his position by switching to position La. If limited to the money-income base, therefore, the government can secure revenues only up to this new maximum limit, YmYa, and it can secure this amount only if it levies an “ideally” structured regressive tax, in which the rate for each level of Y is equal to the slope of im. This would involve creeping down im to the maximum revenue equilibrium shown in the limit at Em, allowing the taxpayer a minute slice of surplus to ensure that his final equilibrium in the neighborhood of Em is preferred to La.
Recognizing this prospect, the potential taxpayer may wish to impose the further constitutional constraint that the rate schedule should not exhibit regressivity. This choice would clearly emerge if the money-income base, together with the predicted value for α and the revenue-maximizing regressive rate schedule, should be predicted to generate outlays on desired public goods and services in excess of predicted efficient levels of provision. If, for example, the government should be required to stay within the confines of a rate structure that exhibits proportionality, at the least, it would effectively be confronted with a locus of potential equilibria along the individual’s “price-consumption” curve for varying “prices” of Y, depicted by LaKE in Figure 3.1. The revenue-maximizing arrangement in this case is shown where a line drawn parallel to LaYa is tangent to the price-consumption curve, indicated at K, with the associated revenue-maximizing proportional rate of tax on Y being YkYa/0Ya, and the revenue collected being YpYa. The precise characteristics of this case and the analytic resemblance to familiar results in price theory can be isolated by appeal to the corresponding partial equilibrium diagram shown in Figure 3.2.11
Curve DD in Figure 3.2 indicates the individual’s demand for the income-yielding activity; this curve might be derived from a preference mapping exhibiting the properties depicted in Figure 3.1. Confronted with the requirement that it must levy a proportional tax, what tax rate will the revenue-maximizing government select? The question is clearly analogous to that asked about the behavior of the monopoly firm that seeks to maximize profits, with the same answer. We derive a “marginal revenue” curve, MR, in Figure 3.2, and the quantity of Y at which revenue is maximized is determined by the intersection of this curve with the horizontal dollar-price line (which is marginal cost), indicating a posttax equilibrium level of money income at Y1 and a revenue-maximizing tax rate of t*. (Note that, when evaluated in the money-income numéraire, the cost of earning a dollar of income is simply a dollar. The “consumer’s surplus” area between the demand curve and the cost curve in Figure 3.2 measures the utility value of money income relative to that of leisure, again evaluated in the numéraire, over and above that of leisure.)
The construction reveals the precise analogy between our model of postconstitutional governmental process and monopoly theory—in an analytic as well as a conceptual sense our model is appropriately designated a “monopoly theory of government.” The revenue-maximizing tax rate, t*, can be derived algebraically as follows. We know that R = tY1, where t is the proportional tax rate and R is tax revenue. Further,
As we have indicated, the revenue raised from the given base under a proportional tax is less than that which might be raised under an ideally regressive rate structure. We are then led to ask what might be the influence of a progressive rate structure on revenue. In its dealings with a single taxpayer, the revenue-maximizing government will have no incentive to shift from the equilibrium proportional rate to any rate structure that embodies progression, since this latter would imply increasing rather than declining marginal rates of tax with income. The revenue effect can be demonstrated most easily by thinking of the simplest of all progressive rate structures, one that involves only two marginal rates, with the first being zero. Consider such a structure, sometimes called a degressive one, where income over some initial range, Ye, is wholly exempted from tax. With this additional constraint, the revenue-maximizing proportional rate on remaining units of income falls and total revenue collections fall correspondingly.
Diagrammatically, this result can be indicated by drawing the new marginal revenue curve, MRd, over the range where the nonzero proportional rate is to be applied—as in Figure 3.2, with the maximum rate being td*, with equilibrium total income at Yd. Observation of Figure 3.2 reveals that the revenue-maximizing degressive structure generates less total revenue than under the strict proportional tax and, also, that the excess burden is smaller. Under proportionality, the excess burden is measured in Figure 3.2 by the area ABC. Under the postulated degressive structure, excess burden falls to AHF.
Not all forms of progression yield this result for the change in excess burden. For example, a linear progressive rate schedule (of the form shown by line ST in Figure 3.2) will yield a revenue-maximizing marginal rate, t*, that is equal to the revenue-maximizing proportional rate, with the same posttax equilibrium income at Y1. Note that, in this case, the total revenue obtained under progression is a constant share of that which would be obtained under proportionality where the marginal rate levied at income Y1 would be applied over the entire income range. Hence, under the rate structure, ST, total revenue raised under progression is one-half that raised under proportionality. Note that the excess burden in the two cases is identical.
It may be useful to summarize the basic arguments of this section. We have observed that the constitutional decision-making calculus of the taxpayer-beneficiary, operating under the expectation of a Leviathan-like postconstitutional fiscal process, involves his opting for institutional devices that will limit the revenue-raising potential of the tax system. We have explored in some detail two ways that might accomplish this purpose. One is by limiting the size of the tax base—increasing the size of the tax base will be, beyond a point, clearly undesirable. The other is by imposing constitutional constraints on allowable rate structures on any given base. These constraints may rule out the imposition of regressive rate schedules. The argument stems, of course, both from the constitutional perspective within which our whole analysis is developed, and from the unconventional, but uncomfortably plausible assumptions that we have made about the predicted working properties of the political process.
[8. ] The analysis holds equally for the case where the comparison is between a more comprehensive and a less comprehensive base, even when the former is itself less comprehensive than the “ideal,” when defined in the orthodox terms.
[9. ] By the assumptions of the model, the individual cannot predict his precise preference pattern as between money-income-earning and alternative activity in postconstitutional periods. All that is required for our analysis here is that these preferences are predicted to be standard.
[10. ] It should perhaps be noted at this point that we are assuming that the revenues collected, as these accrue to the agents of government, are not included in the comprehensive base of the tax itself. In the simple conception of Leviathan as the ruling class or monarch, the assumption here is tantamount to allowing the king to be exempt from taxation. In a dominant majority conception of political process, the comprehensive tax may be presumed to fall on all income, but not on the special benefits, transfers, and so on, that accrue to members of the majority coalition as a result of disbursements from total revenues collected.
[11. ] The construction in Figure 3.1 can be used to demonstrate how the constitutional choice setting under our political assumptions transforms the familiar excess-burden argument made in support of a comprehensive tax base. A solution at point K, in the traditional argument, is shown to be inferior to that which might be attained with a comprehensive base or general tax that will yield the same revenue, producing an ideal solution at a point such as H in Figure 3.1, which may lie on a higher indifference curve than K. This argument must presume, however, that the government, once empowered to levy the comprehensive base tax, will, in fact, restrict its attempt to raise revenue to the collections dictated by the equi-yield comparisons.
The partial equilibrium version, based on the Marshallian demand-curve construction, can be used to illustrate the revenue-maximizing regressive rate structure, but because of income-effect feedbacks on demand, more restrictive assumptions must be made. For similar reasons, the area under the demand curve cannot accurately reflect consumer or taxpayer surplus; nor can the standard welfare triangle measure welfare loss accurately, except in the case where the income-consumption curve in Figure 3.1 is a horizontal line. We set such problems as these aside in the analysis here since they have no particular relevance to our discussion.