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

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 Y_b exceeds Y_a 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
     (1)
where
Y₀i is i’s pretax income
Y₁i is i’s posttax income (expressed in dollars net of tax)
t_i is i’s tax rate expressed as a rate on net income
η_i is i’s elasticity of demand for “income,”
so that

and Δp = t, p₀ = 1, by assumption. The individually revenue-maximizing proportional rates, t_i*, can be derived by maximizing t_iY₁ⁱ in each case by simple calculus. On this basis, we obtain
     (2)
The uniform proportional tax rate that maximizes revenue, t*, is obtained by maximizing the expression
     (3)
which yields
     (4)
From (4), it is clear that t* is a weighted average of sorts of t_a* and t_b* and must lie between t_a* and t_b*.

In fact, recalling that Y^b exceeds Y^a by assumption, a necessary condition for progression to yield more revenue than the revenue-maximizing proportional rate is that
t_b* > t_a*.     (5)
If this is not so, a departure from t* by lowering the rate for earlier units of income or raising the rate on later units of income (to B) must reduce revenue. From this it follows directly, using (2), that
η_a > η_b.     (6)
To derive sufficient conditions, consider the revenue-maximizing “progressive” rate structure, which consists of two rates: t₁ over the range in which both pay tax and t₂ in the range where B only pays tax. Any additional progression must lose revenue. We can examine aggregate revenue in this case and determine the conditions that must hold for t₂ to exceed t₁. Accordingly, we examine
     (7)
     (8)
     (9)
Setting (8) and (9) at zero, we have the equation system
     (10)
where y = Y₀b/Y₀a (> 1). Thus, we have
     (11)
For t₂* > t₁*, therefore, we require that
4η_by + η_ay − η_b < 4η_ay − 4η_a + 2η_a
or
     (12)
Since 1 < y < ∞ (by construction), (12) requires that η_b/η_a be less than three-fourths, or that η_a be at least one-third larger than η_b.

We should note that if η_a is too large relative to η_b, then the revenue-maximizing arrangement in the proportional tax case may ignore A entirely and simply levy t_b*, and the progressive tax system becomes “proportional” anyway. That is, we require that t* < 1/η_a (since at t = 1/η_a, individual A ceases to pay tax entirely). In other words, we require that

or
     (13)
That is,

Combining (12) and (13), we can specify the general requirement on η_b and η_a as
     (14)
For y in the neighborhood of unity, we require η_a to be more than twice as large as η_b. As y becomes very large, we require only that η_a be more than one-third larger than η_b—but not more than twice as large, since then the tax system will force A to earn no taxable income at all.

It is however clear that, for any value of y, there is a value of η_b/η_a that satisfies (14). Progression can therefore yield more revenue than proportionality, under the appropriate relative values of η_b and η_a.

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, t_a* and t_b*, are imposed on A and B, respectively. This situation is depicted in Figure 3.4—the revenue obtained from B is at most ½R_b*, from A at most ½R_a*.

Let the revenue under this “linear progressive” rate structure be R_L. Then
     (15)
Let the revenue derived from the revenue-maximizing uniform proportional rate structure be R_p. Then we know that B can be treated as if he were identical with A, so that R_p must be greater than or equal to 2R_a*; and A could be ignored entirely, so that R_p must be at least as great as R_b*. In other words,
     (16)
Suppose that 2R_a* ≥ R_b*. Then

so that
     (17)
If, on the other hand, R_b* ≥ 2R_a*, 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 R_c* > R_b* > R_a* by construction. Then, by analogous reasoning,

and

Let 3R_a* ≥ 2R_b* ≥ R_c*. Then

And similarly, if R_c* ≥ 3R_a*, 2R_b*, or 2R_b* ≥ 3R_a*, R_c*. More generally, it can be shown by induction that

(where R_Lⁿ is the linear progressive rate for n taxpayers). Assume that R_Lⁿ < R_pⁿ. Then

Now, R_pⁿ⁺¹ must exceed R_pⁿ, since R_pⁿ is feasible (one could simply ignore individual n + 1). Moreover, R_pⁿ⁺¹ must exceed R_n+1*, since R_n+1* is feasible. Therefore,

So ; and since the result is true for n = 2, and n = 3 it is true for all n, by induction. Thus, the “linear progressive” schedule always derives less revenue than the revenue-maximizing proportional rate structure.

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.

4.

The Taxation of Commodities

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.