Fairness, self-interest, and the politics of the progressive income tax

Journal

of Public

Economics

FAIRNESS,

36 (1988) 197-230.

SELF-INTEREST, PROGRESSIVE

James M. SNYDER

North-Holland

AND THE POLITICS INCOME TAX

and Gerald

OF THE

H. KRAMER*

Uniwrsity of Chicago, Chicago, IL 60637, USA

Received

April 1986, revised version

received

March

1988

Virtually all advanced democracies have adopted income taxes with considerable progression in marginal tax rates. To explain this we examine the nature of individual and collective preferences over alternative tax schedules, in the context of a simple two-sector model. Our results suggest that marginal-rate progression is not the result of a society’s desire to achieve a more ‘fair’ distribution of income, but rather, it is due to the desire of middle-income citizens to reduce their own tax burden.

1. Introduction Redistributional issues are the most divisive and potentially destabilizing a democratic society can face, and clearly the taxation of incomes is the most direct and transparent, and hence potentially most explosive, redistributive mechanism. Yet in practice the potential instabilities do not seem to arise often. A highly stylized but broadly accurate summary of the salient facts might run as follows: All advanced industrial democracies impose direct taxes on incomes, and the income tax typically serves redistributive as well as revenue-raising ends. The incidence of the tax burden seems quite stable over time; while income tax schedules are revised from time to time, the changes are typically rather modest and incremental (e.g. readjusting brackets for inflation), and hardly of the chaotic and large-scale kind we might expect if the political process underlying the changes were driven by majority coalitions of ‘have-nots’ getting together to tax away the incomes of the minority of ‘haves’. The effective (as distinct from statutory) incidence across

*This paper has benefited greatly by comments from the editor and two anonymous referees, as well as the participants of the Caltech Theory Workshop and the University of Chicago Theory Workshop. 0047-2727/88/$3.50

0

1988, Elsevier Science Publishers

B.V. (North-Holland)

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J.M. Snyder and G.H. Kramer, The progressive income tax Table

1

Marginal tax rates applicable to taxpayers at selected positions the before-tax income distributions, various countries. Marginal

tax rate for taxpayers

at

Tenth percentile

Median

Ninetieth percentile

United Kingdom Australia

0.34 0.335

0.34 0.335

0.34 0.335

Austria Germany Denmark

0.28 0.22” 0.144

0.28 0.22 0.144

0.33 0.38 0.288

Italy Canada Netherlands United States France

o.:o 0.18 0.20 0.16 0.15

0.13 0.21 0.25 0.22 0.20

0.22 0.28 0.39 0.36 0.40

Norway Sweden

0.06 0.02

0.16 0.19

0.33 0.31

Ireland Belgium

0.20 0.21

0.35 0.37

0.45 0.44

Source: OECD (1981). Data from table noted. “Taken from country graph on p. 28.

in

2, p. 19, except

as

income classes is even more stable. Income tax paid as a fraction of beforetax income typically increase somewhat with income. The aoerage effective tax rate is thus mildly progressive, and achieves some redistribution of incomes. The degree of redistribution is quite modest, however, and falls far short of complete equalization of after-tax income. At the same time, however, statutory marginal rates increase more rapidly with income, in many cases dramatically. In Australia, Belgium and France, for example, the marginal tax rate on taxable (declared) income varies from zero in the lowest tax bracket to 60 percent or more in the highest bracket.’ Statutory marginal rates show considerable progression in all the advanced industrial democracies, with maximum rates ranging from 39.6 percent in Denmark to a high of 83 percent in Great Britain. The extreme rates are to some degree misleading, of course, since often they apply to only a minuscule portion of the taxpaying population. A better idea of the effective progression of rates can be obtained from table 1, which shows the

‘Data from OECD (1981), for central government income tax schedules in effect during the mid-seventies. Inter-country comparisons are subject to the usual caveats concerning differing definitions of taxable income, joint versus individual taxation, the role of social security contributions and local income taxes. and so forth.

J.M. Snyder and G.H. Kramer, The progressive income tax

199

statutory marginal rates for taxpayers at the tenth, fiftieth and ninetieth percentiles of the before-tax income distribution, for various countries. The countries are grouped, somewhat subjectively, according to the degree and nature of marginal-rate progressivity. In Australia and Great Britain the marginal rate is constant across all incomes in this range; despite the nominal progressivity of their tax schedules (from 0 to 61.5 percent in Australia, 34 to 83 percent in Great Britain), the income tax in these countries is, effectively, a linear tax for most of the population. In Austria, Germany and Denmark the marginal rate is constant across low and middle incomes, but then shows some progression in the upper-income range. In the next group of countries there is some modest increase in rates over the lower half of the distribution, but most of the progression again occurs at higher income levels. The progression in rates extends down through all income levels in Norway and Sweden, and in the last two countries, Ireland and Belgium, the increase is actually greater over the lower half of the income distribution. The tax schedules chosen by different countries thus vary considerably. All, however, seem committed to some degrees of progression; with one exception, marginal rates always increase, and never decrease, with income. Of course, income taxation is only part of the redistributive story. In most industrialized democracies, social security contributions, other taxes (e.g. state, local and sales taxes) and various cash payments also affect net disposable income. Including these complicates the picture, making it very difficult to calculate ‘full’ marginal tax rates, especially since these marginal rates may differ widely even across households with the same gross income, depending on the households’ circumstances. Nonetheless, the available evidence suggests that the basic fact is unchanged. For example, taking into account income taxes, social security payments and cash transfers, the marginal tax rate rises with income, on average, in 18 out of 23 OECD countries, at least over the range of incomes between half and twice that of the average worker.2 There has been little formal theoretical work done to explain this empirical regularity. Most positive analyses of income taxation have focused on linear tax schedules, and thus do not address the issue of marginal-rate progressivity. Foley (1967), Romer (1975) and Roberts (1977), for example, study majority-rule choice of an income tax by rational, self-interested citizentaxpayers, but restrict the set of allowable schedules to linear functions. Introducing nonlinear schedules in a world of (heterogeneous) selfinterested voters is problematic because once such schedules are allowed on the political agenda there is generally no majority-rule equilibrium - the ‘Data from OECD (1980, tables 32, 33 and 35). The exceptional countries are Austria, France, Spain, the United Kingdom and the United States, where the implied average marginal rates actually fall at the higher end, due to regrzssivity in the social security contribution schedules.

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alternative space becomes multidimensional, and only in exceptional circumstances will there exist a schedule which is majority-preferred to all others.3 Aumann and Kurz (1977) avoid this difficulty by applying a different solution concept, the Harsanyi-Shapley-Nash value, and thus are able to analyze a majoritarian income-redistribution game that includes nonlinear income taxes. Their results do not help explain the prevalence of marginalrate progression, however, as the tax schedule predicted by their model may have either increasing or decreasing marginal (and average) rates. Hamada (1973) and Roberts (1977) explore an alternative possibility, namely that voters view the income distribution as a pure public good [Thurow (1971)], and judge redistributive tax schemes primarily in terms of fairness and social justice. Hamada studies majority voting over a large class of tax schedules, under the assumption that pretax incomes are fixed (i.e. there are no ‘incentive effects’ of taxation).4 In this setting there is a unique majority-rule equilibrium, but it is a linear schedule, with a 100 percent tax rate that results in complete equalization of after-tax incomes. Thus again we cannot account for increasing marginal rates. Roberts also considers majority rule in the context of voters interested in social welfare, but restricts the set of alternatives to linear taxes. Another body of work which sheds light on the issue is the literature on optimal income taxation. While generally normative, the work has positive implications to the extent that government acts as a benevolent dictator, and is also useful to characterize the preferences of ‘sociotropic’ voters discussed above. The results there suggest, however, that the tax chosen might be approximately linear [e.g. Mirrlees (1971)], or even have decreasing marginal rates [Sadka (1976), Tuomala (1984)]. In this paper we offer an explanation for the prevalence of increasing-rate schedules: they reflect the success of the middle class in reducing its tax burden. We analyze the problem in the context of a simple mode1 which incorporates incentive effects that are similar in spirit, but different in detail, to those of the optimal taxation literature. We assume, conventionally, that individuals vary in their potential income-earning abilities. However, in our mode1 an individual faced with a high tax rate on his labor income responds, not by substituting untaxable leisure for taxable work effort, but rather by working in an untaxed, ‘underground’ economy, at a lower, but tax-free, wage rate. Individual welfare is measured by total after-tax income. While this is a highly stylized model, it captures some incentive effects of taxation, thus making it more genera1 than the models of Foley and Hamada where pretax incomes are fixed, and it is considerably more tractable than the labor-leisure framework, allowing us to obtain clear, intuitive results. 3See, for example, Plott (1967) or Kramer (1973). 4Foley (1967) also assumes that pretax incomes are exogenous.

J.M. Snyder and G.H. Kramer, The progressive income tax

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Also, from an empirical standpoint, many modern democracies have large, which may play an even more importproductive underground economies5 ant role in the labor supply decisions than the consumption of leisure. To the extent that underground labor markets are competitive so that one wage prevails, our model may be a reasonable description of the world for the purpose at hand. We first show (Proposition 3.1) that for any marginal-rate progressive schedule T we can always find a less progressive schedule T’ which is fairer according to any fairness criterion; thus, the relationship between fairness and marginal-rate progressivity is not necessarily positive, and may well be a perverse one. We then consider the choice of a fair tax schedule from the point of view of an individual decision-maker (a benevolent despot, elected representative, or ‘sociotropic’ voter) who views distributional issues solely in public-goods terms. In Propositions 3.2 and 3.3 we show that any such individual will always choose a linear schedule, or one that is equivalent to a linear schedule, no matter what particular fairness criterion he employs (although the parameters of the preferred linear schedule will depend on the criterion). In particular, a schedule which has increasing marginal rates is never optimal. We then turn to the issue of collective choice of a tax schedule by majority rule. In Proposition 3.5 we show that with sociotropic voters who judge tax proposals according to their fairness, there necessarily exists a majority equilibrium. The equilibrium schedule is, once again (up to equivalence), linear. It would appear then that a sociotropic electorate would tend to choose linear income tax. This leaves unexplained the apparent preference, in virtually all modern democracies, for increasing-rate schedules. We thus turn to the study of self-interested citizen-taxpayers who are concerned only in maximizing their own welfare, rather than promoting distributional fairness. In Proposition 4.1 we show that a majority of such taxpayers, consisting of those in the middle- and upper-income ranges, may indeed prefer more marginal-rate progressivity in income taxation. This preference has nothing to do with fairness ~ the more-progressive tax schedule they prefer over the status quo is unambiguously less fair - but rather arises from the fact that greater progression in marginal rates can actually reduce the tax burden on middleand upper-income taxpayers, at the expense of the poor. In Proposition 4.2 we show that a self-interested middle-income taxpayer, given a free choice of the form of the tax schedule, would choose a sharply progressive one, which imposes a low (in fact, zero) marginal rate on lower incomes, and a high rate on large incomes. Individual preferences over different schedules of this form of course vary, since the parameters of the ‘For example, the underground economy may account for 20 percent or more of GNP in Italy and Spain, and 10 percent or more of GNP in Belgium, France, Sweden, Switzerland and the United States. See Boeschoten and Fase (1984).

J.M. Snyder and G.H. Kramer,

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The progressive income tax

individually optimal schedule depend on the individual’s own position in the income (or ability) distribution; nevertheless, as we show in Proposition 4.3, there exists a majority equilibrium within the set of such schedules. The equilibrium is the most-preferred schedule of the median income, i.e. typical middle-class, taxpayer. Our results thus suggest that the observed progressivity of income taxation in democratic societies has little to do with fairness or equity considerations, but rather arises from the success of the middle class in minimizing its own tax burden, at the expense of upper- and low-income taxpayers. This conclusion is, of course, quite reminiscent of Director’s Law of Income Redistribution [Stigler (1970)].

2. The model and some preliminary

results

2.1. The model We assume that individuals vary in their ability to earn income and that this ability is indexed by a single number n. The distribution of ability in the population is described by a distribution function F, which has a continuous density f = dF/dn. The ability index is normalized to lie between zero and one for all individuals in the population, and we assume there are at least a few individuals of every ability level in this range [i.e. f(n) >O for all n~(0, l)]. The mean ability level is N =I; ~dF(n).~ Individuals are worker-consumers in a simple one-good economy with two productive sectors, a ‘legal’, taxable sector, and an ‘underground’, untaxable sector. We can think of an individual of ability n as possessing n units of standardized labor, which he can allocate as he chooses between the two sectors. We again normalize so that the wage rate in the taxable sector is always unity. Suppose that the wage rate in the underground economy is w. Then an individual of ability n who supplies x units of labor to the taxable sector and n-x units to the untaxed sector earns a pretax taxable income of x, an untaxable income of w(n -x), and hence total pretax income x + w(n-x). If he is taxed T(x) on his taxable income, his after-tax taxable income is x- T(x), and his total after-tax income is y=x-T(x)+w(n-x).

(2.1)

The wage rate in the untaxed sector depends on the aggregate labor supply L to that sector. We assume the wage schedule (inverse labor demand function) wn is a strictly decreasing, twice contmuously differentiable function, with w,(O) 5 1 and w,(N) > 0. Since wn is strictly decreasing, it has an 6Henceforth,

when the limits of integration

are 0 and 1 we shall omit them

J.M. Snyder and G.H. Kramer, The progressive income tax

203

inverse (the labor demand function) on the interval [wn(N),w,(O)], which we denote by LD.l These assumptions can be thought of as the reduced form of a one-good, one-factor, two-sector general equilibrium model. Labor in each sector is paid its marginal product, and productivity in the taxed sector exhibits constant returns to scale, while that in the untaxed sector is lower than that in the taxed sector, and suffers decreasing returns. For example, the economy might have a plentiful supply of fertile land in a valley which the government can easily monitor, the taxed sector, and increasingly less fertile land in the surrounding mountains which the government cannot monitor, the untaxed sector. A tax schedule is a function T which specifies for any level of (taxable) income x the amount of tax T(x) to be paid. We restrict attention to schedules which are nondecreasing, continuous, and continuously differentiable except possibly at some finite number of income levels.8 Thus, for any T the marginal-rate schedule t = dT/dx is defined except possibly at some finite number of points (and continuous wherever defined). For points where t is not defined, the left-hand and right-hand derivatives of T are defined, which we denote by t- and t +, respectively. We also require that T(x) 5 x for all x, i.e. tax liabilities cannot exceed taxable income. Note that these conditions place few restrictions on the shape of allowable tax schedules; for example, the marginal and average tax rates may be increasing, decreasing, or even nonmonotonic functions of income. As will be clear shortly, allowing such a broad range of tax schedules makes the analysis somewhat complicated, and it would be much simpler if we could restrict attention to the class of strictly convex, differentiable schedules, or, at least the class of convex schedules. Unfortunately, we cannot, for three reasons. Firstly, as shown in section 3 below, the socially optimal tax schedule is linear, so we must admit linear schedules, which are ‘Alternatively, endowing all individuals ‘productivity’, we can write (2.1) as nz - T(w) + nw( 1 -z),

with one unit of time and letting

n be a measure

of

(N.1)

where ZE [0, I] is the fraction time spent working in the taxed sector. Note that if we intepret untaxed-sector income as ‘leisure’ then (N.l) is similar to the utility of an individual in the ‘labor-leisure’ framework of Mirrlees (1971) with a linear utility function of the form u(x,z)=x-wwz. Note the difference, however, that in (N.l) an individual’s productivity n directly affects both his marginal utility of working and his marginal utility of leisure, whereas in Mirrlees, n directly affects only the marginal utility of working. More importantly, our specification is different because the marginal product of labor in the untaxed sector (‘marginal product of leisure’) w is not fixed exogenously, but determined by market equilibrium. Thus, our model is not a special case of the ‘labor-leisure’ model used in much of the literature on optimal income taxation. 81n fact, for many of the results below, most notably Propositions 2.1 and 3.1, we need only assume that tax schedules are lower-semicontinuous functions of income. Thus, for example, Propositions 2.1 and 3.1 hold even if we allow tax liabilities to decrease with income.

204

J.M. Snyder and G.H. Kramer,

The progressive

income tax

not strictly convex. Moreover, since linear taxes are in some sense on the ‘boundary’ of the set of convex schedules, to restrict attention only to convex schedules would leave some doubt about the robustness of the linear optimality result. The critical reader would be quite justiticd in asking whether the so-called optimal linear tax might not be a corner solution, which, if one relaxed the constraint and allowed concave schedules, would cease to be a global optimum. Finally, as shown in section 4, the optimal schedules for self-interested individuals are ‘kinked’, with just two marginal rates, and hence are neither strictly convex nor differentiable. Some descriptive terms for different kinds of schedules will be useful. A schedule T is linear if it is of the form T(x) =a+lJx, so its marginal rate t(x) = p is constant for all incomes. A schedule T is marginal-rate progressive (or simply, progressive) over some interval of incomes I if it is weakly convex, but not linear, on I. Thus, if T is progressive on I then its marginal-rate schedule t is nondecreasing on I, and strictly increasing on some subset of I. Schedule T is more progressive than schedule T’ if the marginal-rate schedule t crosses t’ from below, i.e. if there is some income level x, such that t(x) 2 t’(x) for x < x, and t(x) 2 t’(x) for x > x,, with strict inequalities holding on J - {x,} for some open interval J which contains x,. Note that our use of the term ‘more progressive than’ refers to progression in marginal tax rates, as distinct from progression in average tax rates, or progression in residual income studied by Jacobsson (1976). A feasible schedule is one which raises just enough revenue to meet an exogenously given revenue target. Since we are interested in redistributional issues, we shall suppose that the target is zero, so that the income taxation is purely redistributional. Thus, denoting by R(T) the total tax collected under IT; T is feasible if and only if R(T) = 0.

2.2. Individual

labor supply and labor market

equilibrium

Let the tax schedule T and untaxed-sector wage rate w be fixed. The problem for an individual of type n is to choose the amount he works in the taxed sector, x, to maximize his after-tax consumption as given by (2.1) taking the wage w and tax schedule T as given. Suppose, initially, that the schedule T is continuously differentiable and strictly convex, so the marginal rate schedule t is continuous and strictly increasing on [0,11. Then the optimal labor choice is unique for all n, and is easily characterized, as follows (see fig. 1). There exists some quantity of labor, call it n,(w, T), which is the maximum amount that any individual will supply to the taxed sector. Individuals with labor endowments n 5 n,(w, T) work fulltime in the tax sector, while those with n>n,(w, T) work n,(w, T) in the taxed sector, and supply the balance of their endowment, n-n,(w, T), to simple characterization is possible the untaxed sector. This

205

J.M. Snyder and G.H. Kramer, The progressive income tax

X

-n

I

nsP4T)

1

Fig. 1

because of our assumption that the labor market in the untaxed sector is competitive, so that one wage prevails, together with the assumption that individual utility is measured by consumption (since the economy has only one good). The proof is straightforward. Comment 2.1. Let n,(w, T) be defined as follows: If w > 1 -t(O), n,(w,T)=O; if w
x,(n, w, T) = ” i ns(w, T),

gives the uniquely

optimal

n < n&w,T), n L ns(w, 0,

allocation

then then

(2.2)

of labor to the taxed sector, for all n.

Proof: Note first that, since t is continuous and strictly w E [l -t(l), 1 -t(O)], then there exist a unique value of w = 1 -t(n). Thus, n,(w, T) is well defined for all w.

increasing, if n such that

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J.M. Snyder and G.H. Kramer,

The progressive

income tax

Differentiating (2.1) with respect to x yields dy/dx= 1 -t(x) - w. If w> 1 -t(O), then w > 1 -t(x) for all XE [0, 11, so the optimal choice of x is xs(n, w, T) =0 for all n. This is exactly what (2.2) says for this case. Similarly, if w < 1 -t(l) then w < 1 -t(x) for all XE [0, 11, so x,(n, w, T) = n for all n, again consistent with (2.2). If w E [l -t(l), 1 -t(O)], then for all n-c&w, T), w< 1 -t(n), so all w< 1 -t(x) for all x~[O,n], and the optimal labor allocation is x,(n, w, T) =n. For n>=n,(w, T), w = 1 -t(x) if and only if x = ns(w, T), so x,(n, w, T) = ns(w, T). Again, these choices correspond exactly to (2.2). Q.E.D. Let X,(w, T) = j xs (n, w, T) dF(n) be the aggregate supply of labor to the taxed sector, and let L&w, T) = N - X,(w, T) be the aggregate labor supply to the untaxed sector. A market equilibrium for the tax schedule T is a wage w* and an aggregate level of untaxed labor L* satisfying w*= wD(L*) and L* =Ls(w*, T). It is straightforward to show that for strictly convex, continuously differentiable tax schedules, a unique market equilibrium exists. Comment 2.2. If T is continuously differentiable and strictly convex, then there exists a unique market equilibrium (w*, L*).

Using Comment sector can be written:

Proof:

2.1, the aggregate

labor supply to the untaxed

wl-t(0). Let M be defined by M(m) =j,!, (n - m) dF(n). Then M is continuous and strictly decreasing on [0, l] [differentiating yields M’(m) = -Sk dF(n), which is negative for all mE(0, l)], with M(0) = N and M( 1) =O. Also, n(., T) is continuous on [0, l] and strictly decreasing on (1 -t(l), 1 -t(O)) [since t is continuous and strictly increasing, and n,(w, T) solves w= 1- t(n&w, T)) for wE(l-$1),1--_(O))], with n,(w,T)=l for wsl--t(l), and n,(w,T)=O for w -2 1- t(0). Thus, Ls( ., T), is continuous on [0, l] and strictly increasing on (I-t(l),l-t(O)), with L,(w,T)=O for wsl-t(1) and L,(w,T)=N for wz 1 -t(O). (See fig. 2.) By assumption, the aggregate labor demand Lo is continuous and strictly decreasing on [w,(N), w,,(O)]. Thus, if 1 -t(l) 2 wD(0), then in equilibrium all labor must be supplied to the taxed sector, and the unique equilibrium is (w*, L*) =(w,(O), 0). Similarly, if 1 -t(O) 5 w,(N), then the unique equilibrium

J.M. Snyder and G.H. Kramer, The progressive

201

income tax

--w

w’(T)

WD(N) l-t(l)

WD

(0) l-t(o)

1

Fig. 2

is (w*,L*)=(wu(N),N). Otherwise, there is a unique value of w in the interval (w,(N), w,(O)), call it wo, such that Ls(w,, T) =L,(w,), and the equilibrium is (w*,L*)=(wo,Ls(wo, T)). Q.E.D. These results can be extended to tax schedules which are not differentiable, and to schedules which have constant or even decreasing marginal rates. The main complication that arises when a tax schedule is linear over part of its domain is that optimal labor supply choices are no longer unique for all individuals at all untaxed-sector wage rates. For example, suppose that the tax schedule is weakly convex, and the marginal tax rate is constant over some interval [x,X] of income levels, at to (see fig. 3). Then, if the untaxed-sector wage rate is w= 1 -to, then for all individuals with labor endowment n E [5,X], any labor allocation x in the interval [s, n] is optimal, and for individuals with n>_C, any labor allocation in the interval [x,X] is optimal. Thus, when dealing with schedules that are only weakly convex, the optimal labor choices, xs(., ., T), and hence the aggregate supplies X,(., T) and L,( ., T), must be defined as correspondences, rather than functions. The situation is complicated further still if marginal tax rates are decreasing over some income levels, since the solutions to the first-order conditions may then yield local minima, rather than local maxima, to some individuals’ optimization problems. And, if the tax schedule is not differen-

J.M. Snyder and G.H. Kramer, The progressive income tax

208

Fig. 3

tiable, then we must keep track of both the left-hand and right-hand derivatives. Nonetheless, we can in fact treat all these cases without too much difftculty. In particular, we can characterize the optimal labor choices in a straightforward fashion, and prove the existence and uniqueness of a market equilibrium. As the details are somewhat tedious, we relegate them to an appendix. The key result is the following generalization of Comment 2.2. Proposition 2.1. Proof.

For any tax schedule 7; a unique market equilibrium exists.

See appendix

A.

Denote the equilibrium let X*(T)=N-L*(T).

associated

with schedule

T by (w*(T), L*(T)),

and

2.3. Further results If

T

is not

strictly

convex,

then

the

optimal

labor

choices

at

the

J.M. Snyder and G.H. Kramer, The progressive income tax

209

equilibrium need not be unique for all n. In particular, if w*(T) = 1 -t,, where t, is the marginal tax rate over some flat portion of 7; then some individuals will have a continuum of optimal labor allocations. However, each individual’s equilibrium after-tax income is unique (since after-tax income is what individuals maximize), and we denote it by y*(n, T). That is, y*(s T) =x’ - 7(x’) +(n -x’)w*( T), where x’ is any optimal labor choice for an individual of type n, given w*(T) and 7: Thus, aggregate after-tax income, which we denote Y*(T), is unique. And, since aggregate pretax income, Z*(T) =X*(T) + w*(T)L*( T), is also unique, the total revenue raised by T is unique, since total revenue can be calculated as R(T) =Z*(T) - Y*(T). Finally, note that the ‘deadweight loss’ associated with a tax schedule that raises zero net revenue is equal to N -Z*(T) = (1 - w*( T))L*( T). These remarks imply that, even when the individual labor choices (and, hence, individual pretax incomes and taxes paid) associated with a tax schedule are to some extent indeterminate, the welfare implications of the tax, both for individuals and for society, are not. To calculate the levels of individual and social welfare associated with a tax schedule, we need only know the distribution of after-tax incomes and the total revenue raised under the tax, and these are uniquely determined. Thus, when more than one optimal labor allocation exists for some individuals, we are rather free to choose an allocation that is convenient for our analysis. For the class of convex tax schedules, it is particularly convenient to choose labor allocations of the form given by eq. (2.2) of Comment 2.1, since this allows us to treat strictly and weakly convex schedules together in a straightforward fashion. Thus, let n*(T) be the unique solution to L*(T)=M(n*(T)), where M is defined by M(m)=li(n-m)dF(n). A unique value exists since, as noted in the proof of Comment 2.2, M is strictly decreasing and takes on all values in [0, N]. Then we have: Comment 2.3.

If T is any convex

x*(n, T) =

schedule,

n,

n -=cn*( T),

n*(T),

n 2 n*(T),

then the labor allocation,

is optimal for all n, given w*(T) and 7: and supports (w*(T),L*(T)). (See fig. 3.)

(2.3) the market

equilibrium

Proof: Note first that if T is differentiable and strictly convex, then n*(T) = ns(w*( T), T), where n,( ., T) is as defined in Comment 2.1 above. Thus, eq. (2.3) describes the uniquely optimal labor allocation for T at wage w*(T), and hence clearly supports the market equilibrium. If T is convex, but not differentiable and strictly convex, then there are

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J.M. Snyder and G.H. Kramer, The progressive income tax

two possibilities. First, suppose that w*(T) does not coincide with a flat portion of ?: that is, suppose there is a unique value of n, say n,, such that 1 -t+(n,)s w*(T)5 1 -t-(no). Then it is straightforward to verify that the optimal labor choices given w*(T) and T are unique for all n, and can be written as

xn=

i Thus,

n,

n5no,

no,

n>n,.

(2.4)

the

(unique) aggregate supply of labor to the untaxed sector is M(no), so no = n*( T) by the definition of n*(T), and thus J,‘,(n-no)dF(n)= (2.4) is equivalent to (2.3), proving the result for this case. Next, suppose that w*(T) does coincide with a flat portion of 7; that is, suppose t(x) = 1 -w*(T) for some range of x values. Clearly, it T is convex then the set of x such that t-(x) 5 1 - w*(T)st+(x) is a closed interval, say [x,X]. Then, for individuals with n>x, the set of optimal labor allocations given w*(T) is the interval [x,X]; for nE [x,X], the set of optimal allocations is b, n]; and for n >x, the unique optimal allocation is n. Thus, for any no E [&,I?], eq. (2.4) describes labor allocations that are optimal for all n given w*(T). Also, the set of possible aggregate labor supply levels in the untaxed wage, sector given w*(T) is [M(Z), M($], and since w*(T) is the equilibrium the equilibrium level of labor in the untaxed sector, L*(T), lies in [M(Z), M(x)]. Thus, by the definition of n*(T) and the monotonicity of M, n*(T) E b, X], so the labor allocations given by (2.3) are optimal for all n and support the market equilibrium. Q.E.D. We assume from now on that, for convex tax schedules, the equilibrium labor choices satisfy (2.3). Then the taxes paid by each individual are given by E(n, T) =

T(n),

n
T(n*(T)),

nZn*(T),

(2.5)

and the total revenue raised is R(T)=jA E(n, T)dF(n). Notice that the ‘effective’ tax schedule, E(=, T), depends only on the shape of T for income levels x 5 n*(T). Also, we can write after-tax incomes conveniently as

y*(n, T) =

n - T(n),

n
n-T(n)+w*(T)(n-n*(T)),

nzn*(T).

(2.6)

Perhaps the most important feature of our model, and the feature that makes the model so much easier to analyze than the labor-leisure model, is

J.M. Snyder and G.H. Kramer, The progressive income tax

211

that it is very easy to find tax schedules that produce the same market equilibrium. This is because we only need to worry about the marginal tax rate at one income level, n*(T). Comment 2.4. Let T and Tl be convex tax schedules. Then T, induces same market equilibrium as T if one of the following holds: (i) w*(T) = w,(O) and t;( 1) 5 1 -w,(O), or (ii) w*(T)=w,(N) and t:(O)zl-w,(N), or (iii) w*(T)E(w~(N), w,(O)) and t;(n*(T))s 1 -w*(T)gtt:(n*(T)).

the

Proof. It is straightforward to show that if (i), (ii), or (iii) holds, then the equilibrium labor allocations for T given by eq. (2.3) of Comment 2.3 are also optimal for all n at T’ and w*(T). Thus, L*(T) is one of the possible levels of aggregate untaxed-sector labor supply given T’ and w*(T), so (w*(T), L*(T)) is the unique market equilibrium for T’. Q.E.D. Note that (i) and (ii) are ‘corner’ cases, in which only one of the two sectors operates, while in case (iii) the equilibrium is in the ‘interior’. Comment 2.4 plays a crucial role in proving the results in section 3 and section 4 below. Another useful fact is the following. Comment 2.5. Let T and Tl be convex c1+ T(x) for all x. Then R( Tl) = a + R( T).

tax schedules

Proof: By Comment 2.4, T and Tl induce Thus, n*(T) = n*( T,), and using eq. (2.5),

the same

such

market

that

T,(x)=

equilibrium.

n’(T)

R(T)=

s Town)+ 0

j Tl(n*(T))df’(n) n’(T)

n*(T)

=a+

s T(n)dF(n)+ 0

i

T(n)dF(n)=cr+R(T).

Q.E.D.

n*(T)

This makes intuitive sense, since vertical shifts of the tax schedule cause ‘lump-sum’ changes in people’s tax burdens. In fact, the results in Comment 2.5 holds for all tax schedules, not just convex schedules. Before proceeding, we state one more definition which will simplify the exposition below. Two tax schedules, TI and T2, are equivalent if they induce the same market equilibrium, raise the same amount of revenue, and produce the same distribution of after-tax income, i.e. if (i) w*(T,) = w*(Tz), (ii) R(T,) = R(T,), and (iii) y*(n, Tl) = y*(n,T,) for all n. Evidently, the welfare implications of equivalent schedules are the same.

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J.M. Snyder and G.H. Kramer, The progressice income tax

3. Fairness and progressivity 3.1. Fairness and the degree of progressivit4 We assume that individual welfare depends only on after-tax income or consumption. Hence the ‘fairness’ of any tax schedule T can be judged solely in terms of the fairness of the after-tax income distribution it induces. We assume that the fairness of any income distribution can be assessed in terms of a social welfare function of the form S,(T) =l W(y*(n, T)) dF(n). As Atkinson (1970) has pointed out, the usual inequality measures can be rationalized by social welfare functions of this kind. We shall refer to W as the evaluation function. We assume W is twice continuously differentiable, strictly increasing, and strictly concave on [0,11. (The second property implies that the welfare of every individual, no matter how wealthy, is given some weight, and the third implies that income equalization is positively valued by the social welfare function.) If S,( T,)>S,(T), the tax schedule Tl is conditionally fairer than 7: conditional upon the particular evaluation function k+! Any admissible W induces a complete, transitive ordering of the feasible schedules. If no feasible T is conditionally fairer than T, (and if T, itself is feasible), then the schedule Tl is optimal for W; as it turns out, in the structure we consider here there exists an optimal schedule for each K and with a weak restriction on the wage function w,,, all schedules which are optimal for W are equivalent. If T1 is conditionally fairer than T according to every evaluation function w then we shall say that T, is unconditionally or unambiguously fairer than 7: Evidently the ‘unambiguously fairer’ relation is a partial, transitive ordering of the admissible schedules, and is equal to the intersection, over all w of the ‘conditionally fairer’ relations. The conditional relationship is the one which would guide a single decision-maker, such as a benevolent despot or elected representative interested in promoting social justice, or a voter who judges tax proposals from a ‘sociotropic’ point of view. We examine the relationship between progressivity and conditional fairness in subsection 3.3 below. Propositions 3.2 and 3.3 show that any such decision-maker, given a free choice of tax schedules, will always select a linear schedule, rather than a progressive one. Alternatively, in a democracy citizens may have their own views on what constitutes fairness, and may vote accordingly (for specific tax reforms proposals, or for candidates who advocate such proposals). Since different citizens may employ different criteria w their individual orderings of the admissible tax schedules will generally differ, so majority rule may well not yield a consistent social ordering of tax schedules. As we show in Proposition 3.5, however (again, under the weak restriction on the wage function w,), there does in fact exist a majority equilibrium schedule TE, which is linear. This equilibrium, it should be noted, assumes that voters judge tax changes solely in terms of whether they lead to a fairer after-tax income distribution. They thus view the income distribution as a pure public good, in Thurow’s

J.M. Snyder and G.H. Kramer, The progressive income tax

213

(1971) sense, and are not influenced by considerations of their own tax burden. We consider the other extreme, that citizens vote in a self-interested fashion to minimize their own tax burden, in section 4. It will be useful to first note some useful facts about the ‘unambiguously fairer’ relationship. Comment 3.1 (Pareto). If Ti and T are two schedules such that y*(n, Tl)z y*(n, T) for all n, with strict inequality for some (non-null, measurable set of) n, then Tl is unambiguously fairer than IT:

Comment 3.1 is an immediate consequence of the fact that W is a strictly increasing function. The second proposition, from Atkinson (1970, pp. 2455 248), says that one income distribution is unambiguously fairer than another if it can be obtained from the latter by redistributing income from the richer to the poorer. In the present context this assertion can be more precisely stated as follows: Comment 3.2. If if there exists an y*(n, T,)sy*(n, T) measurable set of)

Finally,

Tl and T are two schedules such that Y*( Tl) = Y*(T), and ability level n’ such that y*(n, T,)zy*(n, T) for nn’, with strict inequality holding for some (non-null n, then Tl is unambiguously fairer than 7:

we should

note

y*(n, Tl) for all n so S,(T)

schedules equivalent

by any social schedules.

if T is equivalent to T,, then y*(n, T)= T,) for all u! Thus, anyone ranking tax welfare function S, will be indifferent between that

=S,(

3.2. Fairness and the degree of progressivity The first result below expresses one relationship degree of marginal-rate progressivity; it shows somewhat perverse.

between fairness and the that the relationship is

Proposition 3.1. If T is a feasible, differentiable schedule which is progressive over [0, n*(T)] [i.e. t(0) < t(n*( T)], with O t(x) for x n*(T). Evidently we can always find such a schedule, and clearly Tl is less progressive than 7: By Comment 2.4, both schedules induce the same equilibrium, so n*(T) =n*(T,). If T,(n*(T)) 5

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J.M. Snyder and G.H. Kramer, The progressive income tax

T(x)

Fig. 4

T(n*( T)) then the effective tax satisfies E(n, Tr) = TI(n*( Tr)) = E(n*( T,), 7’) for all n 2 n*(T), and E(n, Tr) = T’,(n) < T(n) = E(n, T) for all n t(n) over this range], implying that R( Tr) < R( T) =0 and hence that TI is not feasible. Thus, T,(n*(T)) > T(n*(T)), so E(n, Tr) > E(n, T) for all n > n*(T). Since tl(n) > t(n) for nE(n,T)=T(n) for all n>n,, with equality at n=n,. Now x*(n, T) = x*(n, T) for all n, so y*(n, Tl) > y*(n, T) for all n < n, and y*(n, Tl) < y*(n, T) for all n > n,. Also, Y*( Tl) = Y*(T) (from feasibility), so Comment 3.2 applies, implying that Tl is unambiguously fairer than 7: Q.E.D. The restriction to differentiable schedules is clearly not essential, and the inequalities on n*(T) and t(n*(T)) are needed only to ensure that Tl is less progressive than T in the precise sense of the earlier definition [without these conditions, we could still find an unambiguously fairer schedule Tl such that tl(n)Zt’(n) for nn*(T), although unless both equalities were strict Tl would not be less progressive than T in the

J.M. Snyder and G.H. Kramer, The progressive income tax

sense defined earlier]. Proposition 3.1.

In fact, we can prove

the following

strengthening

215

of

Proposition 3.2. Let T be any tax schedule. If T is not equivalent to any linear schedule, then there exists a linear schedule T, that is unambiguously fairer than 7: Proof

See appendix

B.

Thus, only linear schedules, or schedules that schedules, can be optimal.9 To show what this progressivity, we present the following. Comment

[0, n*(T)],

are equivalent to linear implies for marginal-rate

3.3. If T is a convex tax schedule that is progressive then T is not equivalent to any linear schedule.

over

Proof. By Comment 2.4, if T’ is a linear schedule with marginal rate p, then T is equivalent to T’ only if w*(T) = 1 --jK But in that case, by Comment 2.3, y*(n, T) =n- T(n) and y*(n, T’) =n- T’(n) for all n5 n*(T). Thus, T and T are equivalent only if T(x) = T’(x) for all xsn*( T). But then T is linear on Q.E.D. [0, n*(T)], contrary, to hypothesis. Thus, no schedule for any IV

that is marginal-rate

progressive

over [0, n*(T)] is optimal

3.3. Individual and collective choice of a fair tax schedule Proposition schedule T,, straightforward

3.2 implies that for any evaluation function W the optimal if one exists, is equivalent to a flat-rate schedule. It is to show that such an optimum does in fact exist.

Proposition 3.3. For any evaluation function optimal feasible schedule Tw which is linear,

W there exists a conditionally

any feasible linear schedule T defined by T(x) =/3x + c(. feasible, government net revenues are zero, so R(T) =/3X*(T) +cl=O, or c1= -pX*(T). Thus, c( is determined uniquely as a function of /I, so there is a one-to-one correspondence between feasible linear tax schedules and tax rate parameters. For T defined by T(x) = l?x - /3X*( T), let j(n, p) =y*(n, T). Then define 3,: [0, l] -rR by s,(p) =s W(j(n, /?)) dF(n) = Proof: Since

Consider T is

‘Note that Proposition 3.2, and Comment 3.3 and Proposition 3.3 below, are also relevant to the optimal income taxation literature, since they provide a characterization of the ‘optimal’ tax in the context of an economy with a significant, productive underground sector (see footnote 7).

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J.M. Snyder and G.H. Kramer, The progressive income tax

of one real variable, j WY*(C TN dF(4 = S,(T). &v is a simple function rather than a functional on tax schedules as is SW. Also, s”,(p) zs,(Bi) if and only if S,(T) zS,(T,), where T and TI are feasible linear tax schedules with tax rates /I and pi, respectively. Since F has a continuous density, and W and j are continuous functions, 9, is continuous [see Apostol (1974, p. 281, theorem 10.38)]. The interval [0, l] is compact, so SW attains its maximum for some flW=[O, 1). Then the tax schedule Tw defined by T,(x) = f?wx - p,,,X*( T) is a conditionally optimal schedule for IY Q.E.D. It is straightforward also to show that the tax rate bW of any conditionally optimal linear schedule must lie in the interval [ 1 -w,(O), 1 -w,(N)]. Of course, any schedule T, equivalent to T, is also optimal for K so the set of optimal schedules is infinite. The following proposition establishes that under a weak condition on the wage function w,,, the optimal linear schedule is unique; then all optimal schedules are equivalent, so they all induce the same effective schedule. Proposition 3.4. If the labor demand function L, satisfies Lb(w)> [2/(1 -w)]LL(w) for all WE [w,(N), w,(O)], then for any evaluation function W the function s”, is strictly concave. Hence, the optimal linear schedule Tw is unique, and any (possibly nonlinear) optimal schedule T, is equivalent to Tw. Moreover, voter preferences over the set of linear schedules are single-peaked, if every voter i judges schedules according to some social welfare function S,,. Proof.

Recall that s”,+,,introduced in Proposition 3.3, is defined by $,&I)= onsider all linear taxes with tax rates /l satisfying j W?(n, P)) dF(n). C 1 -/IE [w,(N), w,(O)]. If s”, is concave over this set of 8, then S,(T) has a unique maximum T, among such taxes, and hence, by Proposition 3.2, any optimal schedule must be equivalent to T,. Single-peakedness follows directly if s”, is concave for every W - simply order the linear taxes on the real line by their tax rate parameter. Since W is concave, s, will be concave if j is concave in /l for each n. We now show this. For T defined by T(x) = /?x -pX*( T) the equilibrium wage rate in the untaxed sector is w*(T) = 1 -/I. Then j(n, j3) = (1 - f3)n+ fiX*( T) = (1 - P)n + j is also, and we P(N-LL,(l -fl)). Since L, is twice differentiable, have aj/a/I= -n+(N-L,(l -/))+bLb(l -/I), and a*j/~?/I*=2Lb(l -p)pLn( 1 -fi). This is negative for all PIZ [l -w,(O), 1 -w,(N)] if and only if Lb(w) > [2/( 1 - w)]Lb(w) for all w E [w,(N), wn(O)]. Q.E.D. Note that since L&(w) ~0 for all w, the condition on L, is that it not be ‘too concave’. For example, if L,, is linear or convex, the conditions will be satisfied.

J.M. Snyder and G.H. Kramer, The progressiveincome tax

217

In view of the propositions above, we can easily extend the analysis from the individual case to group decision-making under majority rule. Define a majority equilibrium TE as a tax schedule which no other schedule can defeat in a pairwise majority vote. If voters have different weighting functions w, their preferences over tax schedules will be different, so the possibility of voting cycles arises and a majority equilibrium may not exist. Proposition 3.4, along with the well-known result on single-peakedness and majority rule [Black (1958)], implies that majority votes over linear schedules will be consistent (i.e. different pairwise votes will be transitive), and that at least one such schedule T, will satisfy the median voter condition and be able to defeat any other such schedule in a pairwise vote. However, TE is only a restricted equilibrium, within the set of linear schedules; Proposition 3.4 does not preclude the possibility that some nonlinear schedule T, could defeat TE and hence that there is no majority equilibrium within the set of admissible schedules. However, Proposition 3.2 does preclude this possibility, and implies that T, is a global equilibrium. Proposition 3.5. exists a majority

If L, satisfies the condition of Proposition 3.4, then there equilibrium. Furthermore, any such equilibrium is equivalent

to a linear schedule. Proof: Let TI be any feasible schedule. By Proposition 3.2, either TI is equivalent to some linear schedule T2, or there exists a linear schedule T, that is unambiguously fairer than TI. In either case, we know by Proposition 3.4 that TE is (weakly) majority-preferred to T,. By transivity of individual preferences, TE is then (weakly) majority-preferred to TI. Thus, TE is a majority equilibrium over the set of all feasible schedules. The fact that any such equilibrium must be equivalent to a linear schedule follows directly from Proposition 3.2. Q.E.D.

As a final observation, we note that if the distribution of voters has a unique median then all majority equilibrium tax schedules must be equivalent.

4. Self-interest

and progressivity

4.1. A preliminary

result

We now consider the problem interested in promoting their own pursuing social ends. Individual consumption; thus an individual schedule T, if and only if y*(n, T) >

from the viewpoint of citizen-taxpayers welfare, narrowly interpreted, rather than welfare is again measured by after-tax of ability n will prefer schedule T to y*(n, TI). In general, citizens with differing

218

J.M. Snyder and G.H. Kramer,

The progressive

income tax

earning abilities will have have differing preferences over such schedules, and in particular may view more or less progressive schedules differently. We first present a preliminary result which gives some insight into this relationship; it is something like Proposition 3.1 ‘in reverse’. Let n, denote the median ability level. (Note that the result could easily be strengthened in various ways, e.g. by relaxing the differentiability assumption.) Proposition 4.1. If T is a feasible, progressive, differentiable schedule such that 0~ t(n*(T))< 1, then there exists a more progressive feasible schedule TI which is favored by upper-income and opposed by lower-income taxpayers. If 0 c t(n,) < 1, then we can find such a schedule TI which is preferred to T by a majority, consisting of middle- and upper-income citizens. Proof: Clearly, we can find a feasible schedule TI that satisties tt(n*(T)) = t(n*(T)), t,(n)< t(n) for all n t(n) for all n>n*(T). Then TI is more progressive than ?; and by Comment 2.4, TI induces the same equilibrium as 7: Using similar arguments to that in the proof of Proposition 3.1, we can show that there exists n, e(O,n*(T)) such that T,(x)> T(x) for all and T,(x)< T(x) for all x>nr. Thus, y*(n, T,) y*(n, T) for all n > n,, [the latter condition holds for n > n*( T) since T,(n*(T)) < T(n*( T)) and w*( TI) = w*(T)]. So TI is opposed by lowerincome (low n) citizens, and favored by higher-income (high n) citizens. Since n,>O and we can choose TI to be arbitrarily close to T over [0, n*(T)], we constitutes a can insure that n, nr> majority. Q.E.D. Thus, an increase in marginal-rate progressivity may well redistribute incomes upwards, producing an unambiguously less fair income distribution; nevertheless, if the more progressive schedule TI is properly chosen, it may be preferred by a majority of taxpayers, and thus prevail over the fairer schedule 7: 4.2. Individual and collective choice of a tax schedule We say that a feasible schedule T, is optimal for an individual of ability n, if it maximizes the individual’s after-tax income y*(n,, T) over the set of feasible schedules, i.e. if y*(n,, T,) 2 y*(n,,, T) for all feasible T. Heuristically, an optimal schedule is one which shifts as much as possible of the tax burden to other taxpayers. Since feasibility implies that total revenue is constant, this will minimize his own burden, and, ceteris paribus, maximize his after-tax income. To formally characterize the optimal schedules, fix no and consider the following one-parameter class of schedules:

J.M. Snyder and G.H. Kramer, The progressive income tax

Ti(x) =

0,

for x 5 n,,

/?(x - n,),

for x > n,,

219

(4.1)

where PZO. Then we have: Proposition 4.2.

Fix n,, and define the schedule TO by for for

where PO maximizes the revenue collected from {nln 2 no> over all schedules of the form given by (4.1), and ~1~is chosen to ensure feasibility. Then To is optimal for no, and any optimal schedule for no is equivalent to To. Proof.

See appendix

C.

Individuals with a (potential) income level no will prefer a schedule whose kink is located at no. Thus, if lower-income taxpayers can control the choice, the kink will be located far to the left, and the schedule will resemble a linear or flat-rate schedule over most of its range. Conversely, upper-income taxpayers would choose a schedule which is essentially a per-capita tax for most of the population. If the political process is controlled by middleincome citizens, however, the resulting schedule will be a sharply progressive one, in which major segments of the population confront quite different marginal rates. lo Progressive income taxation of this kind is an effective means for the middle class to minimize their own tax burden, at the expense of lower- and upper-income taxpayers. Proposition 4.1 and the results of the previous section strongly suggest that the observed social preference for progressive income taxation has much more to do with individual selfinterest and the desire to maximize personal welfare, rather than any attempt to promote social justice. We now turn to the question of majority voting over alternative tax schedules. The following preliminary result will be useful. Comment 4.1. If L, satisfies the condition of Proposition 3.4, then there exists a unique To for each no. Also, the parameters x0 and PO are continuously differentiable functions of no, with da,/dn, > 0 and d/?,/dn, < 0. “Snyder (1985) also considers citizens with more general preferences over tax schedules, that combine both narrow self-interest and social welfare considerations. In this case, the optimal tax for an individual of ability n, is again a progressive, two-rate schedule, with the ‘kink’ at income level no. but the tax rate on incomes lower than n, may be positive rather than zero.

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J.M. Snyder and G.H. Kramer, The progressive income tax

Proof. As noted in the proof of Proposition 4.2, /I0 maximizes g(p)= fi[M(n,)-M(fi*(fi))], where fi*(/?)=n*(Tp) for the schedule TB given by (4.1). interval straightforward to show that &, lies in the It is [l -w,(O), 1 - w,(A4(n,))]. For PE( 1 -w,(O), 1- w,(M(n,))), labor market equilibrium implies M(F?*@)) = Z_.,(1 -/I), so g(p) =/?[M(ne)-L&l -fl)]. Differentiating, g’(p) = M(n,) - L,( 1 -b) + /I&( I- /I) and g”(p) = 2Ln( 1 -b) /I&(1 --/I), which is negative by the condition of Proposition 3.4. Thus, g is strictly concave over (1 -w,(O), 1 - w,(M(n,))), so the maximum PO is unique. Differentiating the first-order condition M(n,) - L,( 1 -/I,,) + ,$,Ln( 1- &,) = 0 totally with respect to no yields:

M’(n,)+[2Lb(1-~,)-~L;;(1--li,)]~=0 0

or dBo -= dn,

M’(no) /?L;;(l -Be)-2Ln(l

which is negative schedule To is

since

M’ ~0.

-PO)’ Since

R(T,)=cro+BoCM(no)-M(ii*(Bo))l

the

total

tax

collected

=~o+PoCM(no)-L,(l

feasibility requires that a0 = - flo[M(no) - L,( 1 - flo)], so using theorem, dcc,/dn, = - fioM’(no) > 0. Q.E.D.

under

the

-Poll, the envelope

Denote by 0 the set of schedules To that are individually optimal no, and by T, the optimal schedule of the voter with median ability.

for some Then

Under the condition of Proposition 3.4, all taxpayer’s Proposition 4.3. preferences are single-peaked on 0; thus the schedule T,,, is a majority equilibrium within this set. Proof: Denote by y,(no) the after-tax income of an individual with ability n, under the schedule To which is optimal for no. We show that (dy,/dn,)(n,)< 0 for n < no and (dy,Jdn,)(n,) > 0 for n > no, which implies that n’s preferences are single-peaked over 0, with (from Proposition 4.2 and Comment 4.1) his most-preferred schedule at no = n. Under the schedule To, yn=n-tlo for nO from Comment 4.1. For nE(n,,n*(T,)], x*(n,T,)=n, so yn=n-aoBo(n - no). For n > n*(T), x*(n, To) = n*( To), and since w*( To) = 1 -PO, again y, = n - a0 - po(n - no). Thus, for all n > no:

J.M. Snyder and G.H. Kramer, The progressive income tax

dyn -= dno =[l

221

-~+Bo-(n-no)~=~oH’(no)+/?o-(n-nn)~ 0 0 0 +H'(no)]/?o-(n-no)~. 0

Since dfi,/dn, < 0 from Comment positive if [l +H’(no)]20. And Q.E.D. 0 for all no >O.

4.1, and PO> 0 and -[l -F(no)],

H'(n,)=

the expression so 1 +H’(n,)=F(n,)>

n > no,

is

The schedule T, is only an equilibrium within the restricted subset 0 of feasible schedules. Unlike Proposition 3.4, the above result cannot be extended to show that T, is a majority equilibrium within the entire set of admissible schedules. The reason for this is that the set of admissible schedules defines a multidimensional choice space. As is well known, if the choice space is multidimensional, then majority equilibria generally do not exist. The fact that a global majority equilibrium exists when voters are sociotropic derives from the special nature of sociotropic voters’ preferences in this model, in particular the fact that nonlinear tax schedules are always dominated (i.e. less-preferred by all sociotropic voters) by some linear schedule. If voters are self-interested, there is no such ‘similarity’ in preferences. Indeed, for virtually any pair of tax schedules, there will be individuals whose preferences are opposed. Given the very large class of tax schedules that we allow, with self-interested voters the situation is much like a majority-rule ‘divide the dollar’ game, and no global equilibrium will exist.” Proposition 4.3 is nevertheless suggestive as to the likely outcome under pure self-interested voting over the set of individually optimal schedules; the equilibrium schedule is the one which is optimal for the voter with median ability, i.e. middle-income, and is sharply progressive, as noted earlier. The result can also be given an alternative interpretation in a representative democracy framework; in most democracies citizens normally cannot vote directly on alternative tax schedules; rather, they vote for representatives, and delegate the various decisions of government, including those on taxation, to these elected delegates. Representatives of varying backgrounds will have differing attitudes toward taxation and redistribution, and their views may well reflect the preferences or interests of their own ‘class’ (i.e. in the context of our simple model, those of similar ability levels). To the extent this is so, and that such distributional issues are important electoral considerations, the “Although, if we are willing to put more restrictions on the voting process, then T,, might be an equilibrium within the entire set of admissible schedules. For example, if we assume that, because of communication networks within and across income classes, only ‘connected’ coalitions (i.e. coalitions made up of intervals [t~,ri] of voters) may form to overturn the status quo, then T, is a global equilibrium.

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J.M. Snyder and G.H. Kramer, The progressive

income tax

equilibrium outcome will be election of a median-ability representative, whose own inclinations on taxation issues will tend toward the kind of progressive taxation which favors the middle class. Appendix A Here, we prove that for any tax schedule 7; a unique market equilibrium exists. First, we characterize the optimal labor supply choices for genera1 tax schedules. Rewriting (2.1) as y = w, - [T(x) - (1 - w)x], we have: x maximizes

y if and only if x minimizes

T(x) -( I-

w)x.

64.1)

Let a(n w, T) denote the set of optima1 choices for any individual of type n, given w and 7: Since T is continuous, Z(n, w, T) is nonempty and compact for all n, w, and 7: To characterize i more explicitly, we make the following definition: suppose the function T(x) -( 1- w)x possesses a minimum x’ on some interval of the form [0,x”]. Let x be the smallest such minimum [by continuity, x=min {x’lx’= argmin,,t,,,.ZI T(x) - (1 - w)x} is well defined], and let X define the largest interval on which x’ is still a minimum, i.e. Z=max {x”lT(x)-(1 -w)xz T(x’)-(1 -w)x’ for all x~[O,x”]}. We shall say the interval [x,X] is w-critical for 7: Clearly, any T and w define a unique (possibly empty) set of disjoint critical intervals (see fig. A.l). Then Lemma (i) (ii)

A.I.

The correspondence

a(., ., T) is as follows:

w-critical some interval i-5, X], nEC34 if for ~(n,w,T)=(x’E[~,n](T(x’)+(l-w)x’=T(x)+(l-w)?}, and if n 6 b, X] for every w-critical interval b, X), then i(n, w, T) = {n}.

then

Proof. For individuals with endowments n E j&X], (A.l) implies that x’~i(n, w, T) if and only if T(x’)-(1 -~)x’=rnin~~t,,.~ T(x)+(l -w)x= T($+(lw)?; thus 5 is always optimal (though not necessarily uniquely so) for all such n. On the other hand, if n does not belong to any critical interval, then it must be true that T(x) -(l - w)x has no minimum on [O, n). By continuity it must have a minimum on [O,n], which must therefore be at n, so n(n, w, T) = {n} uniquely [again by (A.l)]. Q.E.D. Let s(w, T) = {X(X={x’(n) dF( n) , w h ere x’E~?(.,w, set of possible aggregate labor supplies to the taxed tax schedule T, and let E(w, T) = N-g(w, T) be the supplies to the untaxed sector. Clearly, E(w, T) is (and convex, from Richter’s theorem) for all w and

T) is integrable) be the sector given wage w and possible aggregate labor nonempty and compact 7: A market equilibrium

J.M. Snyder and G.H. Kramer, The progressive income tax

223

Fig. A.1

for the tax schedule T is a wage w* and an aggregate untaxed labor supply L* satisfying w,(L*) = w* and L* E e(w, T). We break up the proof that a unique market equilibrium exists into Let ~((n,w,T)=min~(n,w,T), Z(n,w, T)=maxf(n,w,T), several parts. X(w, T) =min X(w, T), and X(w, T) =max X(w, T). (Note, the use of x and X here is not to be confused with their use in the definition of critical intervals above.) Lemma (i) (ii)

A.2.

For each n E [0, 11, .xr(n;.)

and Xr(n;.)

satisfy:

ifw,Zw,, then~(n,w,,T)~~(n,w,T)~~(n,w,,T)~~(n,w,T), and _x(n, ., T) is right-hand continuous and %(n, ., T) is left-hand continuous.

Proof: For any w, x(n, w, T) z%(n, w, T) by definition. For notational con venience, write x,=x(n, wl, T). We show that T(x) -( 1- wz)x > T(x,) (1 - w2)x, for all x E(x,, n] and thus Z(n, w2, T) 5x,,,, from which (i) follows.

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J.M. Snyder and G.H. Kramer, The progressive income tax

If x~(x,,n), and w,>w,, then x>x, implies so T(x)-T(x,)+x,-x+w,x-w,x,>T(x)-T(x,)+x,-x+w,x-w,x,,

w2x-ww2w,>w1x-wiw,,

C~(x)-(1-wz)xl-CT(x,)-(1-w,)x,l>CT(x)-(1-w,)xl-CT(x,);;-w&J. B u t x, minimizes T(x’)-(lwi)x’ over [O,n], so the right-hand side is non-negative, and T(x) -( 1- wZ)x > T(x,) -( 1- wJx, as desired. We now show that .$n, ., T) is left-hand continuous; the proof for -(n, ., T) is analogous. Let w ~(0, l), write Z(n, w, T) =xm, and fix E>O. We must find 6>0 such that W’E(W-8,~) implies x(n,w,, T)-X” n, then clearly Z(n, w, T)
S(E)=

min [x” fe. b]

T(x) - T(x”) X-xm

.

Clearly, S(E) exists, since [xm+s, n] is compact and [T(x) - T(x”)]/(x-x”) is continuous. Also, for all x E (xm, n), T(x) -( 1- w)x > T(x”) - (1 - w)xm, or [T(x)-T(x”)]/(x-x”)>l-w, so s(E)>~-w. Let ~=s(E)-l+w>O, and W’E(W-&W)=(l-S(E),W). Then for all let XE[Xm+E,n], or T(x)-(1-w’)x>T(x”)-(1-w’)x”. [T(x)-7-(x”)]/(x-x”)~S(&)>1-w, So, x(n, w’, T)
A.3.

X(., T), and X(., T) satisfy the following:

(i)

ifw2>wI,

(ii)

X(., T) is right-hand continuous,

Proof.

For

then X(w,,T)~~(w,,T)~i;X(w,,T)~~(w,,T), and and x(., T) is left-hand continuous,

WE(O, l), X(w, T)=J&n,w, T)dF(n) and X(w, T)= so (i) follows directly from Lemma A.2. To see that X(., T) continuous, let w E (0,l) and let {wi} be a sequence in (0, w] with all

j%(n, w, T) dF(n),

is left-hand wi+w. Then

lim X(w,, T)= lim 1 x-( n, wi, T) dF(n) =j lim .f(n, wi, T) dF(n) W,‘W W.-w w,+w =f$n,

w, T)dF(n)

=x(w,

T).

The second equality follows from the Lesbegue Dominated Convergence Theorem, and the third follows from Lemma A.2. A similar argument shows that X(., T), is right-hand continuous. Q.E.D. Proposition

2.1.

For any tax schedule

7: a unique market equilibrium

exists.

J.M. Snyder and G.H. Kramer, The progressive income tax

225

N] has nonempty compact, Proof: As noted above, X(., T): (0,l) -[O, convex values. Hence, X(w, T) = [X(w, T), X(w, T)] for all w, so by Lemma A.3, X(., T) clearly has closed graph (it is upper hemicontinuous and has compact range), so e( ., T) = N - X( ., T) does also. By assumption, wn: [0, N]+(O, 1) is continuous, so by the Von Neumann Intersection Lemma, an equilibrium (w*,L*) exists. To prove uniqueness, let (w’,L’) be another equilibrium, and suppose w’ # w*.If w’ > w *, then L’ CL* since wn is a strictly decreasing function, so X*
Appendix B Here we prove Proposition 3.2. For simplicity, suppose that the tax schedule T has at most one w*( T)-critical interval, [x,X]. The generalization to more than one w*(T)-critical interval is straightforward. Lemma

B.I.

T’(x) =

(seejg.

T is equivalent to the schedule

W+C1--*(THCx-xl, T(x),

T’ defined by

x~Cz,d, x$Cx,d

A.2):

Proof. Evidently, T’(x) - [ 1 - w*( T)]x = T’(xJ - [ 1 -w*(T)]& for all x E &, X], and [x,X] is the unique w*(T)-critical interval of T’. Comment 2.2 then implies that

a(n, w*(T), T”) =

[x, n] s i(n, w*(T), T),

for n E [zx, X],

{n} = i(n, w*(T), T),

for n $ [x, X].

Thus, if i(n) is optimal for n at w*(T) under 7; then it is also optimal for n under T’ at the same wage, so *(w*(T), T)sg(w*(T), T’), implying w*(T) = w*(T) and X*(T’) =X*(T). This implies that [x,X] is the unique w*( T’)-critical interval of T’. Now we show that both T and T’ induce the same after-tax income distribution. By (i) of Lemma A.l, for nE @,X1, a(n) is optimal under T at w*(T) and under T’ at w*( T’) (since [x,2] is critical in either case) and T’(z) = T(x) by definition, so

226

J.M. Snyder and G.H. Kramer, The progressive income tax

Fig. A.2

Similarly, for n 6 [x,X], a(n) = n is (uniquely) optimal under T at w*(T) and under T’ at w*(T’), and T(n)= T’(n), so y*(n, T)=nT(n) =nT’(n)= y*(n, T’) for all n. Y*(T) = Y*(T’), and since X*(T) =X*(T’), it follows that Hence, Z*(T) = Z*( T’) and R(T) = R( T’). So, T’ is equivalent to 7: It remains only to show that T’ is an allowable tax schedule, i.e. that T’(x)sx for all x (continuity is obvious). But this follows immediately, since T’(x) 5 T(x) for all x by the definition of T’, and T(x) 5x since T is, by assumption, an allowable tax schedule. Q.E.D. Proposition 3.2. Let T by any tax schedule. If T is not equivalent to any linear schedule, then there exists a linear schedule TI that is unambiguously fairer than 7: Proof. Let T’ be as in Lemma B.l, and let TI be the linear schedule defined by T,(x) =a+/Ix for all x, where p= 1 -w*(T) and a= w*(T)NY*(T). It is

J.M. Snyder and G.H. Kramer, The progressive income tax

227

easy to verify that Ti induces the same equilibrium as T and T’ (the interval w*( T)-critical for T,, so 8(w*( T), Tl) = [0, N], whence [0,1] is X*(T) E$(w*(T), T,)); thus, w*(T,) =w*(T) and X*(T,) =X*(T). Also, the choice of c( makes R(T,) =R(T). If T is not equivalent to any linear schedule, then T is not equivalent to T,, and thus (since T is equivalent to T’) Tl #T’. Using Lemma B.1 and the definition of T,, one can easily show that T’(.x2)_2T’(xl)+[1-w*(T)][x2-xi] for all xi and x2 such that Ozx,_2 02x, 1x22 1 implies x,21. Thus, T,(x,)-T’(x,)~;(x,)-T’(xl)+ C~-W*(T)ICX~-X~~~~(~,)-T’(~,), so Tl -T’ is nondecreasing (and continuous) on [0,11. Now we show that T,(n,)= T’(n,) for some n,. First, looking back at the proof of Lemma B.l, we see that n ei(n, w*(T’), T’), thus y*(n, T’) =n- T’(n), for all n. Thus R( T’) = z*( rl) - Y *( T’) =Nw*(T’)+[l

-w*(T’)]X*(T’)-N

+I T’(n)dF(n)

=fT’(n)dF(n)-[lw*(t’)JL*(T’). Similarly, it is clear that n Ei(n, w*(T,), T,), so y*(n, Tl) =n- T,(n) for all n, and R(T,) = j T,(n) dF(n) - [ 1 - w*( T,)]L*( Tl). Thus, since Tl and T’ induce the same equilibrium, and R( T,) = R( T) = R( T’), j T,(n) dF(n) = j T’(n) dF(n). Evidently, then, we cannot have T,(n) > T’(n) for all n or T,(n) < T’(n) for all n, so Tl (n,) = T’(n,) for some n,. Now, since Tl - T’ is nondecreasing and continuous on [O, 11, and T, #T’, T,(n) 5 T’(n) for all n < n, and T,(n) 2 T(n) for all n > n,, with strict inequality for some open neighborhoods of 0 and 1. Thus, y*(n, T,)2y*(n, T’) for all n in, and y*(n, Tl) 5 y*(n, T’) for all n>n, with strict inequality for open neighborhoods of 0 and 1. Noting that Y*(T,)=N-R(T,)=N-R(T’)= Y*(T’) and applying Comment 3.2, the proof is complete. Q.E.D.

Appendix C

Here we prove Proposition 4.2. We first show that a feasible schedule of the form To exists. Fix n,, and let 5 be a schedule of the form given by (4.1). The aggregate supply of labor to the untaxed sector under T, is

0, L(w,T,)=

[O,M(n,,)], i M(n&

forwl-/?,

where M(n,) = I,& (n - no) dF(n). Thus, letting

ii*(j) = n*( TB), the equilibrium

is

J.M. Snyder and G.H. Kramer,

228

The progressive income tax

easily characterized: for b 5 1 -w,(O), w*( TB)= w,(O) and s*(p) = 1; for /?E( 1 - w,(O), 1- w,(M(n,))), the equilibrium is ‘interior’ with w*(q) = 1 -/I= wn(M(ii*(fi))) and G*(p)= M-‘(L.,(l -fl))~(n,, 1); and for bs 1 -w,(O), w*(TB)= wD(M(no)) and Z*(/?)=n,. Since wu is continuous and strictly decreasing on [0, N], and M is continuous and strictly decreasing on [0,11, fi* is continuous on [O,l], and strictly decreasing on (1 -w,(O)> 1- w,(M(n,))). The tax collected from an individual

E(n, T,) =

of type n is

0,

for nsn,,

P(n - 4,

for n E (no, fi*(8)),

0*(P)

- 4,

for n 2 fi*(B),

so the total revenue collected is R( TB)= /l[M(n,) - M(ii*(p))], and maximizing the revenue raised from {n[nLn,} is equivalent to maximizing R( TB). Let g be defined by g(p) = /?[M(n,) - M(ii*(fl)))]. Evidently, g is a continuous, bounded function on [0,11, so it has a maximum, which is clearly strictly positive. Let g achieve its maximum at PO. Then R( Tp,) > 0, so we can choose cl,-,~0 so that the schedule T, defined by T,(x)=cr,+ TBO(x) is feasible [note that the /l which maximizes g must be less than 1, so T, will satisfy T’,(x)~x for all x]. Thus, a feasible schedule of the indicated form exists. We now show that there is no feasible schedule T such that y*(n,, T) > y*(n,, To), and that y*(n,, T)=y*(n,, To) only if T is equivalent to To, thus proving the proposition. Let T be any feasible schedule, and let T’ be the schedule derived from T by linearizing over its w*(T)-critical intervals (as done in appendix B). That is, let T’(x) = T(x) + [l -w*(T)] [x--xi] for all x E bi, Xi], where ki, Xi] is some w*(T)-critical interval of 7: and let T’(x) = T(x) if x is not in any w*(T)-critical interval. By Lemma B.l of appendix B, T’ is equivalent to T. Now, define T” by T”(x) =

TYkJ,

for n 5 n,,

T’(n,)+[l-w*(T’)][n-n,],

for n>n,.

Then E(w, T”) =

0,

w < w*( T’),

[0, M(q,)],

w = w*( T’),

M(nd

w>w*&q,

so either w*( T”) = w*( T’) and L*( T”) = L*( T’) (i.e. T” and T’ induce the same equilibrium), or w*( T”) > w*( T’) and L*( T”) = M(n,) < L*( T’). In the former case we can use the same argument as in the proof of Proposition 3.1 in

J.M. Snyder and G.H. Kramer, The progressive income tax

229

B to show that R( T’) = j T’(n) dF(n) - [ 1 - w*( T’)]L*( T’), and appendix R( T”) =s T”(n) dF(n) - [ 1- w*( T”)]t*( T”). Clearly, T”(x) 2 T’(x) for all x, so R(T”) zR( T’), with equality holding only if T’= T”. In the latter case, we [since R( T”) = T’(n,) R( T’) =I T’(n) dF(n) - [ 1- w*( T’)]L*( T’) and have x*(n, T”) = n, for all n > no]. Now T’(n) 5 T’(n,) + [l - w*( T’)] (n - n,) for all n>no, so we can write R(T’)=yT’(n)dF(n)+i 0

T’(n)dF(n)-[l-w*(T’)]L*(T’) “0

syT’(n,)dF(n)+i 0

T’(n,)dF(n)+[l-w*(T’)][M(n,)-L*(T’)] “0

so R(T’)
unless T’= T”, we can (by Comment 2.5) shift T” downward to produce a feasible schedule T”’ with T”‘(n,)< T”(n,) = T’(n,) and hence y*(n,, T”‘)> y*(n,, T’). Also, unless 1 - w*(T’) maximizes the revenue collected from {nlnz no}, we could find another feasible schedule (of the form To) that n, prefers to T”‘. Thus, T”’ must be of the form To described in the proposition, and if T is optima1 for no then T is equivalent to T”‘. Q.E.D. vertically

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