Front Page Titles (by Subject) Appendix: Progression in the Multiperson Setting - The Collected Works of James M. Buchanan, Vol. 9 (The Power to Tax: Analytical Foundations of a Fiscal Constitution)
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Appendix: Progression in the Multiperson Setting - 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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Appendix: Progression in the Multiperson Setting
The question of interest here is this: Can a progressive tax system raise more revenue than the revenue-maximizing proportional system, when there are many taxpayers? We have already shown that the revenue-maximizing proportional system always gives more revenue in the single-taxpayer case, or equivalently in the many-taxpayer case when all taxpayers are identical. Is this true more generally?
To investigate this question, we examine a simple two-person example, denoting the two individuals by A and B. We shall assume that B is “richer” than A—that Yb exceeds Ya at all tax rates. We further assume, for analytic convenience, that the individuals’ demand curves for nonleisure activity have constant elasticities. We can therefore write
In fact, recalling that Yb exceeds Ya by assumption, a necessary condition for progression to yield more revenue than the revenue-maximizing proportional rate is that
We should note that if ηa is too large relative to ηb, then the revenue-maximizing arrangement in the proportional tax case may ignore A entirely and simply levy tb*, and the progressive tax system becomes “proportional” anyway. That is, we require that t* < 1/ηa (since at t = 1/ηa, individual A ceases to pay tax entirely). In other words, we require that
It is however clear that, for any value of y, there is a value of ηb/ηa that satisfies (14). Progression can therefore yield more revenue than proportionality, under the appropriate relative values of ηb and ηa.
It is interesting at this point to contrast the results under the simple two-tier rate structure with those that emerge under a progressive tax system in which the marginal tax rate is a linear function of Y1. Such a “linear” progressive tax is depicted in Figure 3.4 by the line SM. Now, under such a rate structure, the largest amount of revenue in the two-person case that could conceivably be obtained is exactly one-half of the maximum revenue obtainable under a regime in which the revenue-maximizing proportional rates, ta* and tb*, are imposed on A and B, respectively. This situation is depicted in Figure 3.4—the revenue obtained from B is at most ½Rb*, from A at most ½Ra*.
Let the revenue under this “linear progressive” rate structure be RL. Then
In the two-person case, then, the linear progressive tax derives unambiguously smaller revenue than the revenue-maximizing uniform proportional system.
We can add a third party, C, where Rc* > Rb* > Ra* by construction. Then, by analogous reasoning,
And similarly, if Rc* ≥ 3Ra*, 2Rb*, or 2Rb* ≥ 3Ra*, Rc*. More generally, it can be shown by induction that
Now, Rpn+1 must exceed Rpn, since Rpn is feasible (one could simply ignore individual n + 1). Moreover, Rpn+1 must exceed Rn+1*, since Rn+1* is feasible. Therefore,
A third variety of progression is worth mentioning here. This is the degressive system mentioned in Chapter 3. Any such system, combining a flat exemption with a uniform proportional rate, must raise less revenue than the revenue-maximizing uniform proportional rate structure, since less revenue is obtained from each and every taxpayer.
The Taxation of Commodities
But what is government itself, but the greatest of all reflections on human nature? If men were angels, no government would be necessary. If angels were to govern men, neither external nor internal controls on government would be necessary. In forming a government which is to be administered by men over men, the great difficulty lies in this: you must first enable the government to control the governed; and in the next place oblige it to control itself.
—James Madison, The Federalist No. 51, The Federalist Papers, p. 160
In Chapter 3, we examined the question of how we might expect the individual citizen-taxpayer to choose among alternative bases and rate structures at the constitutional level under ignorance concerning his future position. The analysis in that chapter can be most directly applied to an income tax. In this chapter, we use essentially the same analytic framework to examine commodity taxation. As in Chapter 3, the analysis will be conducted in a setting that abstracts entirely from the time dimension. All taxes are current, and all tax rates are known with certainty in advance, permitting the appropriate behavioral response. We shall relax these time-dimension restrictions in Chapters 5 and 6.
Our purposes in this chapter are twofold. First, we wish to examine questions concerning the commodity-tax arrangements that would be selected by the potential taxpayer, given our constitutional setting and our assumptions about the motivations and actions of government. Because of the directional similarity in behavioral responses on the sources (income) and the uses (expenditure) side of the taxpayer’s account, many of the implications here tend to mirror those derived in Chapter 3. This varied reiteration of the earlier analysis may be useful in itself but it also offers a springboard for discussion of a different issue—which brings us to our second purpose. Here and elsewhere, the focus of our discussion is primarily on the selection of tax instruments that are suitably constraining. But clearly there may be a variety of tax instruments that are equally constraining—instruments that will, when exploited to the fullest by government, yield identical maximum revenue. Are some of these arrangements to be preferred to others by the potential taxpayer? Are the standard techniques for “equi-revenue” comparison applicable to the subset of tax instruments that yield identical maximum revenue? Specifically, can excess burden considerations be used to rank these instruments? If so, how do these considerations intervene in evaluating alternative tax institutions more generally? Is it possible to determine some “optimal” trade-off between smaller excess burden and larger welfare loss due to excessive spending? To answer such questions and to predict what tax institutions might emerge from the constitutional contract, we need to evaluate alternative means of achieving desired constraints. This part of our discussion has the added heuristic advantage of allowing us to indicate how our basic analysis incorporates, or may incorporate, the standard equi-revenue construction.