Front Page Titles (by Subject) 7.3.: The Surplus Maximizer: Algebraic Treatment - The Collected Works of James M. Buchanan, Vol. 9 (The Power to Tax: Analytical Foundations of a Fiscal Constitution)
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7.3.: The Surplus Maximizer: Algebraic Treatment - 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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The Surplus Maximizer: Algebraic Treatment
The basic relationships inherent in our central proposition, along with the limits within which these relationships must operate, may be more fully captured in a simple algebraic model.
In our discussion, the king is taken to maximize
When the maximum revenue rate t* is applied to base B, total expenditure on base B, gross of tax, is depicted by B*. For example, in Figure 7.4, when D1 is the demand curve for B, B* is the area ASTO. We can, on this basis, specify revenue R* as
The parameter a in (5) can be rewritten as
Now, we have specified that the tax base is chosen so as to depend on the level of public outlay on the public good, G. So
Complementarity between B and G in the relevant range implies that
In fact, under plausible assumptions, it seems as if da/dG will be positive. To see this, consider Figure 7.4. As the level of G rises, the demand curve for B depicted as D1 in Figure 7.4 moves outward by virtue of the complementarity relation. Suppose that when there is a particular increase in G, it moves in a parallel fashion to take up the position D2. The new revenue-maximizing equilibrium will be at 0C, which is half of 0B2. The new revenue-maximizing tax rate, t2*, is the distance JK. We need to show that JK exceeds SM, that is, that the increase in G has led to an increase in the revenue-maximizing tax rate and hence in the parameter a. Now, TC is exactly half of B0B2 and hence exactly half of SH. It follows that K must lie on D2 above and to the left of H, so that KJ must exceed SM; that is, t2* > t1*.
If t2* > t1*, then
Let us suppose, however, that condition (10) is not satisfied. Is there a simple way of increasing the likelihood that it may be met? It would be possible to relate government’s (or the king’s) receipt of revenue from general sources, unrelated to B, to the amount of revenue raised from the single source, tax base B, that is known to be tied to the provision of G. In such a case, R* could be set as any multiple, β, of its value defined in (5). Hence, in lieu of (5), we have
While accepting this emendation analytically, it may be challenged on the grounds that it seems inconsistent with the underlying institutional assumptions. While one can imagine the possibility that the king’s ability to raise general revenue might be tied to the revenue from base B, it does seem as if, once he has been allowed access to some more general tax source, he would use that source exclusively and spend all the revenue on private goods. In a more realistic institutional setting, however, it may be possible to establish a bureau whose sole function is to raise revenue from some general source, under the constraint that it be handed over directly to other public-goods-supplying bureaus in direct relation to the latters’ revenue-raising activities from the assigned complementarity tax bases.5
If even this seems implausible, roughly the same effect might in any case be achieved by assigning several tax bases to government, all of which are complements to the public good, G. Suppose that there should exist a whole set of potential bases, B1, B2, ... , Bn. Consider assigning both B1 and B2 for usage as possible tax bases to the surplus-maximizing king. In this case,
[4. ] Given the second-order conditions implied by the shapes of CC′ and NN′ in Figure 7.1 (i.e., ∂2B*/∂G2 < 0).
[5. ] There is an analogy of sorts between such an arrangement as that described here and the return of bloc grants or revenue shares to local units based on “fiscal-effort” criteria. The purpose in the two cases could, however, scarcely be more opposed. With the fiscal-effort criteria, the purpose is to ensure that local governments levy sufficiently high taxes on citizens. With our model, by contrast, the underlying purpose is to ensure that tax money is expended on public goods rather than on bureaucrats’ perks.
[6. ] Although their normative emphasis is quite different from that of this chapter, Atkinson and Stern introduce the complementarity between public goods and the tax base as a determinant of the allocatively optimal budget. See A. B. Atkinson and N. H. Stern, “Pigou, Taxation, and Public Goods,” Review of Economic Studies, 41 (April 1974), 119-28.