II. The choice of tax instruments for a predatory state
The choice to introduce an income tax can easily be analyzed within a rationalchoice framework. Levi (1988) argues that a society’s tax structure is a function of bargaining power between the ruler and the various types of ruled, of the transaction costs of each type of tax (including collection costs and search costs for taxable property), and of the underlying production technology and exchange opportunities, which influence the revenue available from each taxation instrument. It is helpful to investigate this tradeoff more thoroughly.
The model of the state that is chosen is a variant of the budgetmaximizing bureaucrat formulation of Niskanen (1971). In this case, the key decisionmaker is not a civil servant but the sovereign. He must choose a tax structure so as to maximize revenue, knowing that those who are taxed will react rationally so as to optimally avoid taxation. A ruler facing such a constraint will thus choose different instruments (e.g., income and sales) to the degree that citizens find it costly to avoid payment. It will be simplest to assume that the sovereign can either tax money income earned in market exchange or impose an excise tax on purchases of a consumption good.
A. Citizen optimum
Assume that there is a representative citizen who has the twicedifferentiable, strictly concave preferences u(x, e). x is a consumption good and e is leisure. x must be purchased out of the income earned from labor, and this income has no other value. The citizen has a time endowment of r, which must be allocated between labor and leisure.
In both his consumption and labor activities, the citizen has two choices: to participate in the legal market and pay the assessed taxes, or to avoid them by participating in the black market. Let the subscript 0 denote legally reported activity subject to taxation, which I will refer to as revealed activity, and 1 denote blackmarket activity. Revealed labor activity, before taxation, earns a return of w per unit, and blackmarket labor activity earns a return f(w, l_{1}), where f is strictly concave, f(w, 0) = 0 and f(w, l_{1}) < wl_{1} for all l_{1} > 0. Without an excise tax x can be purchased at a unit price of p. When x is purchased on the black market it comes at a cost of C(p, x_{1}), which is a strictly convex function subject to similar restrictions, i.e. C(p, 0) = 0 and C(p, x_{1}) > px_{1} for all x_{1} > 0.
Revealed and blackmarket purchases are thus perfect substitutes in consumption but imperfect ones in purchasing, and blackmarket and revealed labor are imperfect substitutes in the generation of income. The citizen must solve the following problem:
(1)
t and are the excise and income taxes, respectively, which are taken as given by the citizen. Constructing the Lagrangean and taking the firstorder conditions yields:
, (2a)
, (2b)
, (2c)
, (2d)
along with the two constraints. The restrictions on C(.), u(.) and f(.) guarantee that an interior solution will occur for any positive level of taxation.
B. Sovereign optimum
The FOCs in the prior section define a pair of reaction functions l_{1}(, w, p, r) and x_{1}(t, w, p, r). By the implicitfunction theorem and the concavity (convexity) of f (C), the citizen will supply less revealed labor and purchase fewer revealed goods as the tax rate on each increases. The task of the sovereign is to maximize revenue in light of these constraints. He will solve the unconstrained optimization problem
. (3)
g_{s}(.) and g_{i}(.) are the administrative costs associated with sales and income taxation, and are assumed to be strictly convex in the tax rates. The sovereign’s FOCs are
, (4a)
. (4b)
(4) implies a unique optimum for both taxation levels. In each case, there is a tradeoff in setting it between a greater revenue intake per unit of revealed commerce (goods purchased or labor income earned) and a lower amount of revealed commerce. But unlike the conventional criticism commonly known as supplyside thinking, in which a greater tax rate on income leads to substitution toward leisure, here there is additional substitution toward blackmarket activity. The magnitude of that substitution from an infinitesimal change in the tax rate, i.e. , depends by substitution using (2c) and (2d) on the responsiveness of blackmarket income to higher blackmarket labor. The more concave is f at the equilibrium, the more this blackmarket penalty from higher tax rates will be. In general, the sovereign will raise more money (in absolute terms) from income taxation as the productivity of blackmarket labor, i.e. the concavity of f(.) hence its departure from wl as l expands, is lower, and will raise more money from excise taxation as the cost of blackmarket purchases is farther removed from p.

Interpretation in light of U.S. tax structure
In the U.S. tax rates tax revenues are greater from income than sales taxation at the highest level of government. At the federal level in the U.S. there is no universal sales tax, although there are excise taxes. The latter are only imposed on select products, and in fixed monetary amounts rather than as percentages, but on some goods (especially tobacco and liquor) they may amount to substantial percentages of the product’s aftertax price. This is in contrast to federal income tax rates, which (according to the Tax Foundation) have a top marginal rate of 39.1 percent and yield an average tax rate for all who pay income tax of 14.4 percent. Sales tax rates in states are comparatively modest, with no combined state and local sales tax rates in excess of eleven percent as of 2001, roughly equivalent to income tax rates, which typically have multiple brackets, with no top rate higher than eleven percent.
As for tax revenue, for the U.S. federal government in 2000 income tax revenues were over 58 percent of total tax revenue, while excise taxes were less than four percent. At the state level, according to the Tax Foundation, in 1999 income taxes were 40.6 percent of all state tax revenue, while sales taxes were 33.2 percent. This is the combined sum for all fifty states, including the five that have no sales tax and the five that have no income tax. Of the remaining 40 states and the District of Columbia, 33 of them derive more revenue from income than sales taxes. State income and sales tax rates are listed in Table 2, and the revenue from each type of taxation is listed in Table 3.
There are thus two patterns worthy of note. First, the federal government and most states rely on income taxes as the largest source of revenue. Second, the federal government, which in some sense has first claim on tax design under the U.S. federal structure, is the level of government at which this reliance is most pronounced. Together, these findings provide at least an inference that income taxation may be the most lucrative source of revenue for a taxing government, at least in the U.S.
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