The Online Library of Liberty

A project of Liberty Fund, Inc.

• Foreword
• Geoffrey Brennan
• Preface
• Acknowledgments
• The Power to Tax
• 1.: Taxation In Constitutional Perspective
• 1.1.: The Notion of a “constitution”
• 1.2.: The Logic of a Constitution
• 1.3.: The Means of Constitutional Constraint
• 1.4.: The Wicksellian Ideal and Majoritarian Reality
• 1.5.: The Power to Tax
• 1.6.: The Enforceability of Constitutional Contract
• 1.7.: Normative Implications
• 2.: Natural Government a Model of Leviathan
• 2.1.: Leviathan As Actuality and As Contingency
• 2.2.: Monopoly Government and Popular Sovereignty
• 2.3.: The Model of “leviathan”: Revenue Maximization
• 2.4.: The Model of Leviathan As Monolith
• 2.5.: The Constitutional Criteria
• 3.: Constraints On Base and Rate Structure *
• 3.1.: Government As Revenue Maximizer Subject to Constitutional Tax Constraints
• 3.2.: Tax-base and Tax-rate Constraints In a Simple Model
• 3.3.: One Among Many
• 3.4.: Tax Limits and Tax Reform
• Appendix: Progression In the Multiperson Setting
• 4.: The Taxation of Commodities
• 4.1.: The Conventional Wisdom
• 4.2.: Constitutional Tax Choice
• 4.3.: Alternative Forms of Commodity Tax: the Choice of Base
• 4.4.: Uniformity of Rates Over Commodities
• 4.5.: Uniformity of Rates Over Individuals
• 4.6.: Discrimination By Means of the Rate Structure
• 4.7.: Summary
• Appendix
• 5.: Taxation Through Time Income Taxes, Capital Taxes, and Public Debt
• 5.1.: Income Taxes, Capital Taxes, and Public Debt In Orthodox Public Finance
• 5.2.: The Timing of Rate Announcement
• 5.3.: Income and Capital Taxes Under Perpetual Leviathan
• 5.4: Leviathan’s Time Preference
• 5.5.: The Time Preference of the Taxpayer-citizen With Respect to Public Spending
• 5.6.: The Power to Borrow
• 5.7.: Conclusions
• 6.: Money Creation and Taxation
• 6.1.: The Power to Create Money
• 6.2.: Inflation and the Taxation of Money Balances: a “land” Analogy
• 6.3.: Inflation and the Taxation of Money Balances
• 6.4.: Inflationary Expectations Under Leviathan
• 6.5.: Inflation, Wealth Taxation, and the Durability of Money
• 6.6.: The Orthodox Discussion of Inflation As a Tax
• 6.7.: The Monetary Constitution
• 6.8.: Inflation and Income Tax Revenue
• 6.9.: Monetary Rules and Tax Rules
• 7.: The Disposition of Public Revenues *
• 7.1.: The Model
• 7.2.: Public-goods Supply Under a Pure Surplus Maximizer: Geometric Analysis
• 7.3.: The Surplus Maximizer: Algebraic Treatment
• 7.4.: The Nonsurplus Maximizer
• 7.5.: Toward a Tax Policy
• 8.: The Domain of Politics
• 8.1.: Procedural Constraints On Political Decision Making
• 8.2.: The Rule of Law: General Rules
• 8.3.: The Domain of Public Expenditures
• 8.4.: Government By Coercion
• 9.: Open Economy, Federalism, and Taxing Authority
• 9.1.: Toward a Tax Constitution For Leviathan In an Open Economy With Trade But Without Migration
• 9.2.: Tax Rules In an Open Economy With Trade and Migration
• 9.3.: Federalism As a Component of a Fiscal Constitution
• 9.4.: An Alternative Theory of Government Grants
• 9.5.: A Tax Constitution For a Federal State
• 9.6.: Conclusions
• 10.: Toward Authentic Tax Reform Prospects and Prescriptions
• 10.1.: Taxation In Constitutional Perspective
• 10.2.: Tax Reform As Tax Limits
• 10.3.: Tax-rate Limits: the Logic of Proposition 13
• 10.4.: Tax-base Constraints
• 10.5.: Aggregate Revenue and Outlay Limits: Ratio-type Proposals For Constitutional Constraint
• 10.6.: Procedural Limits: Qualified Majorities and Budget Balance
• 10.7.: Toward Authentic Tax Reform
• Epilogue

7.3.

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
Y_k = R* − G,     (4)
where R* is the maximum revenue that can be derived from the assigned tax base, B.

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
R* = a · B*,     (5)
where a is the proportion of gross of tax expenditure B* represented by tax revenues.

The parameter a in (5) can be rewritten as
     (6)
where t* is the revenue-maximizing tax rate, expressed as a proportion of net-of-tax expenditure (as in Chapters 3 through 6).

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
B* = B*(G),     (7)
and
     (8)
In general, the revenue-maximizing tax rate, t*, and hence the parameter, a, will also depend on the level of public-goods supply. Consequently, we can rewrite (4) as
Y_k = a(G) · B*(G) − G.     (9)
We can now examine how Y_k, the government’s maximand, responds to changes in expenditure on G. Consider
     (10)
If the selection of the tax base B is to exert a constraining effect on the government’s disposition of revenues, then (10) must be greater than or equal to zero over the relevant range.

Complementarity between B and G in the relevant range implies that
     (11)
but this is not sufficient, clearly, to ensure that (10) is positive. Since a is always less than 1 [see equation (6)], we would seem to require both that B* be very responsive to changes in G and that da/dG also be positive. In fact, since B* is potentially large, it does seem possible that the second term in (10) may predominate. Therefore, the sign of da/dG may be crucial.

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

and da/dG > 0. We can examine nonparallel shifts in D₁ in response to increases in G supply, but in all cases in which D₂ lies entirely above D₁, given our linearity assumptions, the revenue-maximizing tax rate will increase. Only in the special and somewhat implausible case in which D₁ and D₂ are coincident on the vertical axis will t* not increase: in this special case, t* and hence a remain invariant with respect to G (i.e., da/dG = 0). If this is accepted, then both the first and second terms in (10) will be nonnegative when B and G are complementary: there is therefore some presumption that increases in G may lead to increases in Y_k, and hence be desired by Leviathan.

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
R* = (1 + β)a(G) · B*(G),     (12)
where β > 0 and
     (13)
Clearly, if both ∂B*/∂G and ∂a/∂G exceed zero, (13) exceeds (10) for β > 0, for a given value of G in the relevant range; and there exists some value for β which will guarantee that (13) is positive for positive values for G. Moreover, the higher the value of β, the larger the value of G for which (13) is zero.4 Therefore, by increasing the value of β, we can both ensure that the king will want to provide some G, and increase the amount of G thereby obtained (at least up to the point where the complementarity relationship ceases).

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, ... , B_n. Consider assigning both B₁ and B₂ for usage as possible tax bases to the surplus-maximizing king. In this case,
     (14)
and
     (15)
where
     (16)
As before, (15) exceeds (10) for values of G in the relevant range, and the value of G for which (15) is zero (if it exists) exceeds the comparable value of G in (10). Hence, by adding bases to the government’s taxing retinue, all of which are complements to G, we both increase the possibility that it will prove profitable to provide some public goods, some G, and increase the level of G that will be provided.6

[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.