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Voting over income tax schedules
Journal of Public Economics 8 (1977) 329-340. 0 North-Holland
Publishing Company
VOTING OVER INCOME TAX SCHEDULES Kevin
W.S. ROBERTS*
St. John’s College, Oxford OX1 3JP, England Received February 1977, revised version received June 1977 In many voting situations, preferences over options may fail to be single-peaked. This is especially true when options consist of different amounts of a good which is provided through distortionary taxation. In this paper, voting over linear income tax schedules is considered. Although preferences may fail to be single-peaked, a choice set is shown to exist when only mild restrictions are imposed. For many choices in the public domain, the conditions required for this result are likely to be satisfied.
1. Introduction and summary It may appear unrealistic to consider that choices made in the public domain are the outcome of some voting process. However, if political parties make choices so as to maximise the likelihood of being elected then it is possible to view the chosen options as being determined, albeit indirectly, by a voting process. The point is not whether choices in the public domain are made through a voting mechanism but whether choice procedures mirror some voting mechanism. The problem then arises as to whether there will exist a ‘best’ policy. If individual preferences are unrestricted then it is well known [see Arrow (1963)] that choice sets may fail to exist under many mechanisms, e.g. majority rule. But it is also well known [see Black (1958)] that if individual preferences are single-peaked then majority rule will operate in a satisfactory manner. The following question arises: How likely is it that preferences over issues in the public domain will be single-peaked? Consider a decision being made about the level of provision of some public good. If each individual must pay a constant percentage ot the cost of provision then single-peakedness will follow from the assumption of convex production sets and quasi-concave preferences. In many instances, both of these assumptions are questionable. For instance, consider the case where a municipal body is deciding upon the level of public transport facilities. If an individual owns an automobile then his preferences over the level of provision may be expected *This paper is based upon chapter 4 of Roberts (1975). I am grateful to two anonymous referees for their comments.
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K. W.S. Roberts, Voting over income tax
to be single-peaked. Similarly if he does not own an automobile. But at some level of provision, the individual may switch from owning to not owning an automobile. It is then quite possible for individual preferences to be doublepeaked. Various other examples of a similar nature could be cited, e.g. public replacing private schooling - on which see Stiglitz (1974). The results of this paper could be applied to the situation just mentioned, but our main interest will centre upon cases where individual preferences over goods are well-behaved although preferences over options can fail to be single-peaked. To be more specific, we shall consider the problems that arise when provision is made through distortionary taxation and, in particular, preferences over linear income tax schedules will be considered; these may be viewed as a (distortionary) tax on income which is used to finance a lump-sum subsidy. It is not immediately clear why single-peakedness over tax schedules should fail to hold, given that preferences are well-behaved. At the risk of oversimplification, the following example illustrates the nature of the problem: an individual may dislike a small increase in the marginal tax rate because the increase in the tax rate causes a large reduction in the work effort of others and so the lump-sum subsidy must decrease. However, further increases in the tax rate may cause little shifting and so the individual may agree with these further increases in the tax rate. Finally, with a marginal tax rate of 100x, nobody will choose to work and the lump-sum subsidy will be at its minimum level so that, eventually, the individual will dislike further increases in the tax rate. Thus, because there is distortionary taxation of a heterogeneous population, preferences for tax rates may fail to be single-peaked, even though each individual’s preferences map, considered in isolation, is well-behaved. In fact, restrictive assumptions may be necessary to ensure single-peakedness in these circumstances. Even when individuals have identical preferences over consumption and leisure and differ with respect to the wage rate that they receive, Itsumi (1974) and Romer (1975) have shown that single-peakedness is not guaranteed. It is not the purpose of this paper to investigate the likelihood of singlepeakedness. Instead, the problem to be investigated is whether, in the circumstances that have been described, it is likely that voting mechanisms will give rise to a most preferred outcome, irrespective of whether single-peakedness is satisfied or not. It will be shown that if preferences are such that the ordering of individuals by income is independent of the tax schedule in operation, an assumption which is termed Hierarchical Adherence in the text, then a most preferred outcome will exist under a wide class of voting mechanisms. For the class of income tax models first developed by Mirrlees (1971), hierarchical adherence is a consequence of the assumptions which are generally imposed. Thus relative to other work in this area, hierarchical adherence may be viewed as a mild assumption.
K. W.S. Roberts, Voting over income tax
331
The results obtained in this paper are applicable to many situations. Essentially, if, say, the provision of some public good is being considered then an assumption similar in effect to hierarchical adherence is that the ordering of individuals in terms of their marginal preference for the public good is independent of the level of provision of the public good. This would seem to be a reasonable assumption for the example of public transport provision mentioned earlier. It may be noted that the assumption requires that there be a ‘natural’ ordering of individuals, whereas single-peakedness requires that there be a ‘natural’ ordering of options. In section 2, individual preferences for tax schedules are considered. The class of voting mechanisms being admitted is considered in section 3. The main results of the paper are obtained in section 4 and most preferred tax schedules under the majority decision rule are briefly considered in section 5.
2. Individual preferences for tax schedules It will be assumed throughout that all prices are unaffected by changes in tax rates and that the only tax in operation is one on income. Consider an individual who is deciding how much labour to supply. As prices are fixed, it may be assumed that he has preferences over post-tax income (consumption) c and pre-tax income (labour supply) y. Preferences are assumed to be representable by a differentiable utility function u(c, y) with u, > 0 and tlY < 0. The government, or municipal body, implements a linear income tax. The tax paid by an individual with income y is given by T(y) = ty-a,
(1)
where t is the marginal tax rate and a is a lump-sum subsidy. Usually, the tax implemented will have to satisfy the property that total receipts can finance some fixed revenue requirement G, i.e.
(2) where yi is the income of individual i (if there is a continuum of individuals then summation could be replaced by integration). If yi is continuous in LY then we may expect that for any t, a is made as large as possible so that (2) is satisfied with equality. In this case, combining (1) and (2): ct = ty-G/n,
(3)
where J is the mean pre-tax income and n is the number of individuals in the community. If leisure is a normal good so that J decreases with CI, for any value of t there will be a unique value of a which satisfies (3).
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K. W.S. Roberts, Voting over income tax
It will be assumed that for all tax schedules to be considered, all individuals can attain a position inside their consumption sets. An individual chooses y so as to maximise u(c(+(l - t)y, y). First-order conditions of maximisation give (l-t)u,+u, with equality an individual.
5 0,
(4)
if y > 0. Consider how a small change in the tax schedule affects Totally differentiating the utility function with respect to t and ~1: du = dy[u,(l-t)+u,,]+u,[da-ydt].
Making
(5)
use of (4): du = u,[dcl-ydt],
which is basically of t, then
Roy’s identity.
da -& =y+tz.
dj
(6) If (3) defines
CI as a differentiable
function
(7)
This equation may be used in conjunction with (6) (notice that it is an implicit relationship because jj and dJ/dt are influenced by changes in LY). When incomes are unaffected by a change in the tax schedule, (6) and (7) combine to give: du - = u,[jj--y]. dt Thus an individual with an income y less than the mean income jj will always prefer a higher to a lower tax rate and preferences will be reversed if y is greater than 7. In this situation, the consequences of majority voting, say, are easily analysed. If the median income is less than the mean income, so that the income distribution is positively skewed (using Pearson’s second coefficient of skewness), then majority voting will lead to the tax schedule with the highest marginal tax rate being adopted. This result has been obtained by Foley (1967). 3. Voting mechanisms The underlying structure of the problem being analysed is best brought out if a more general class of voting mechanisms than majority rule is considered. The existence of a most preferred outcome (choice set) for mechanisms satisfying decisiveness, neutrality, and nonnegative responsiveness will be investigated. A
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K. W.S. Roberts, Voting over income tax
complete description of the economy is provided when the two parameters CI and t are specified. In fact, if (3) is imposed, say, then it will generally only be necessary to specify t but such an assumption is not required for the results that will be obtained. Individual preferences over tax schedules may be obtained by integration of (6). Ri will denote individual i’s ordering of tax schedules, i.e. 1, t,)R,(a,, tz) iff individual i weakly prefers schedule (al, tl) to (cI~, t2). (M For notational convenience the CIterm will be suppressed and so t,Rit, will be written. Corresponding to Ri, Pi and Zi will denote strict preference and indifference respectively. The voting mechanism is assumed to provide a binary relation R over a set of tax schedules r, and R will be some function f of the list (Ri)i=l,, over z. We have: Decisiveness : R is decisive it is complete (t,Rt, : t,Rt,)
A.2. Neutrality:
Consider
t,Ritz for two lists (Ri) t,Rt,
and
and (Rf),
then:
where R = f((R,))
then t,Rt,
and
t,R,t,
t,Rt,
possible
list
t,, t,, tJ, t4. If for all i:
f+ t,R,!t3,
t) t,R’t,,
and R’ = f((R:)).
A.3. Nonnegative t,Rit,
any quadruple
t) tJRft4
++ t,R’t,
iff for every logically and reflexive (tIRt,).
responsiveness: -+ t,Rft,
and
Consider t,P,t,
any pair t, , t,. If for all i:
+ &Pit,,
-+ t,R’t,.
These three assumptions are much used in the theory of social choice [see Sen (1970) and Pattanaik (1971)]. Mechanisms that satisfy A.l-A.3 include the method of majority decision, the method of nonminority decision, and variants of these methods which do not treat individuals in an anonymous manner. 4. Analysis So far, we have only considered a single individual’s preference over tax schedules. No assumption has yet been imposed upon the class of utility functions which is permissible. Instead of pinning down any one individual’s
334
K. W.S. Roberts, Voting over income tax
utility function, we impose a condition which restricts the set of preferences that individuals in society may have, relative to each other: A.4. Hierarchical adherence: For all i, j: either yi(a, t) 6 uj(E, t) for all CY, t, or yi(cL,t) 2 yj(a, t) for all M,t, where yi(a, t) is the pre-tax income that i chooses to obtain under the tax schedule (~1,t).
Thus hierarchical adherence states that there exists an ordering of individuals such that pre-tax income is monotonically increasing irrespective of the tax schedule which is in operation. If all individuals have identical preferences over consumption and leisure but differ with respect to the wage rate that they receive then A.4 will be satisfied if the elasticity of labour supply (in terms of hours worked) is not less than minus unity (pre-tax income will increase with the wage rate). Further, in this case, A.4 holds even when, under some tax schedules, there are individuals who choose not to work, a situation which creates problems when single-peakedness is being analysed [see Romer (1975, section 5)]. It is a remarkable feature of the problem under consideration that A.4 is a sufficient condition on preferences for the existence of a choice set to be proved. The analysis proceeds in stages, the implications of A.4 for individual preference orderings being considered first. With A.4 satisfied, it will be useful to adopt the labelling convention that if i is greater thanj then yj(a, t) is no greater than yi(a, t) for all (a, t). Lemma 1.
(1) (2) (3)
Under A.4, if t2 > t, then for all i:
t,Rjt, t,Rj, t,P,t, --) t,P& t,P,t, +
vit,
--)
tzPjt,
for for for for
all j > i, all j < i, all j > i, all j < i.
Proof
(1) Consider i’s indifference map in (~1,t) space which is derived from his indirect utility function ui(q t) z z&c1+ (1 - t)yi(cr, t), yi(cc, t)). An indifference curve connects (a,, tl) to (Q, t,l and, from (6), its slope is given by da - = yi(ol, t). dt Consider j > i. Moving along the indifference curve, (6) gives du’ - = U&(c(, t)-yj(cr, t)] 5 0 dt
K. W.S. Roberts, Voting over income tax
335
by A.4. Hence,
u’$x2,t,)-~&~,
I,) = j;:
u~[yi(cl,t)-yj(cc, t)]dt $ 0,
i.e. tlRjt2. When j < i, the inequalities are reversed (2) AS t,Pit,, there exists an E > 0 such that for j > i:
z&q Applying
-&,
t1) <
and t,Rjt,. (c~i-E, tl)Zi(CIZ)t2). NOW,
u.“(c(l,t1).
the first part of the lemma:
uj(cI1-&,
t1) 2
u’(cQ, tz).
Combining these inequalities we obtain (3) In a symmetrical manner to (2).
tIPit,. Q.E.D.
Thus A.4, by itself, imposes an interesting set of restrictions on individual preferences. For instance, if everybody has a unique most preferred tax schedule then lemma 1 implies that individuals with higher incomes have most preferred schedules with lower marginal tax rates. A result similar to lemma 1 has interesting implications in welfare theoretic analyses. For such analyses, interest centres not upon individuals as such but upon their utilities. Consider replacing A.4 by the following adherence condition : A.5:
For all i, j, for all (c(, t):
Yi(“, t> 2 where v is the indirect
_Yj(m,
t>
utility
f+
2)i(u,t) 2
uj(as
t),
function.
Under AS, lemma 1 holds with Ri replaced by R’, where R’ is the ordering by positions in the income (or utility) hierarchy, i.e. (~1~)t,)R’(cr,, tJ iff the individual who is the ith worst-off when the schedule (txi, tl) is in operation is at least as well off under (c(i, tl) as the individual who is the ith worst-off under (a,, tz) when (cx~, tJ is in operation. For although as we integrate around a positional indifference curve the relevant individuals may be changing, A.5 ensures that there is no change in utility just because of this. Thus the amended lemma 1 now states that if a schedule with a higher marginal tax rate is adopted, the utility distribution will alter so that the lowest utilities will increase, utilities in the middle of the distribution will remain constant, and the highest utilities will decrease. It is therefore to be expected that more inequality averse SWFs will prefer schedules with higher marginal F
336
K. W.S. Roberts, Voting over income tax
tax rates. Following on from the work of Rothschild and Stiglitz (1970) and Diamond and Stiglitz (1974), define a SWF W = ~~=I G(u’) to be more inequality averse than W’ = ~~=I ui if G is increasing and strictly concave (u’ could obviously be viewed as a function of individual utilities). Let p be an index of inequality aversion and let R, be the ordering of schedules by a SWF W, with inequality aversion p. Assume that
above we know Define r(t) so that (a(t), t)l,,(cl,, t 1). From the discussion that if W,, = cu’, then an increase in t, with a change in a given by a(t), will lead to a mean preserving reduction in the inequality of the ui, to adopt the terminology of Rothschild and Stiglitz (1970). By their well-known equivalence theorem (theorem 2), a more inequality averse SWF will prefer the increase in the marginal tax rate. Thus we shall have
where pz > pl. We may thus obtain Theorem (1)
tlJp,t,
(2) (3)
t,P,,t, t,P,,t,
1.
Under AS,
t,R,,t, l t2R,,,tl -+ t,P,,t, -i t,P,,t, +
a dual result to lemma
1:
if t, > t, then for all p1 : for for for for
all all all all
p2 < ply p2 > pl,
p2 < ply p2 > pl.
Theorem 1 also has interesting implications. For instance, the theorem implies that SWFs with a greater degree of embodied inequality aversion will choose schedules with higher marginal tax rates [see Roberts (1975) and Helpman and Sadka (1976)]. It is important to notice that the driving force of this result is condition A.5. Further, if individuals’ voting behaviour is based upon some form of ethical preference, as captured by a SWF, then the structure of individuals preference will be basically identical to the structure implied by A.4 as long as it is possible to order individuals in terms of the inequality aversion that is embodied in their preferences. Returning explicitly to voting procedures, the outcome of voting mechanisms may be considered by utilizing lemma 1: Lemma 2. If the voting mechanism satisfies A.l-A.3 A.4, then for all t,, t,, t3, tl < t2 < t,: (1) (2)
t,Rt, t,Rt,
and and
t2RtJ -+ t,Rt,, t,Rt, + t,Rt,,
and preferences
satisfy
K. W.S. Roberts, Voting over income tax
(3) (4)
r,pt, t3pt2
and and
t,Pt, t,Pt,
337
-+ tlPtJ, -9 t3Ptl,
where R = f(( R,)). Proof. (2) is proved in a symmetrical manner to (1) ; (3) and (4) may be proved as in (1) and (2), strict replacing weak preference. Thus we concentrate upon (1). Consider individual preferences over the pair t, , t, . By lemma 1, preferences must be of the form: t,Pit,Vi 5 fi; tl_litlVfi < i < y; tlPit2Vi 2 y, for some 5 6; t,Zit,V6 < i < E; /?, y, jI < y. Similarly, over the pair t,, t,: t,Pit,Vi t,P,t,Vi >= E, for some 6, E, 6 < E. There are six cases to be considered: (i)p < y 6 6 < e;(ii)/3 5 6 6 y 5 .s;(iii)fi 5 6
2 may now be applied
and the main result of the paper obtained:
Theorem 2. If the voting mechanism satisfies A.l-A.3 A.4 then the social preference relation is quasi-transitive, t,pt,
and
t,Pt,
and preferences satisfy i.e. for all t, , t2, t3 :
-+ t,Pt,.
Proof. Let the antecedent hold but assume that t,Rt, . There are six cases where the marginal tax rates differ to be considered: t, < t, < t, ; 1, < t2 < t, ;
ti < 5, < t, ; t, < t, < t,;
t, < t, < t3 ; t, < t, < I,.
The last three cases are essentially the same as the first three so that we concentrate upon the first three cases. For each case we obtain a contradiction:
338
K. W.S. Roberts, Voting over income tax
Case (i) Case (ii) Case (iii)
by lemma 2, applying lemma 2, applying lemma 2,
t,Pt, ; tzPtS and tJRtl + t,Rt,; t3Rtl and t,Pt, + t,Rt,.
When t, = t,, say, as t,Pt, it must be the case, by A.2 and A.3, that ~1~> CI~ and tIPit, for all i. The desired result then follows from a simple application of A.2 and A.3. The other cases where two or three of the marginal tax rates are equal may be treated in a similar way. Q.E.D. A sufficient condition for a choice set to exist is that the underlying binary relation is complete, reflexive and quasi-transitive, and the set of social states is finite [see Sen (1970, ch. l*)]. Completeness and reflexivity follow from A.1 so that as a corollary to the above theorem we have: Corollary. If the voting mechanism satisfies A.I-A.3, preferences satisfy A.4, and z is jinite then C(z), the choice set of z under the voting mechanism, is nonempty. Given this result, the question arises as to whether there is any connection between the structure of preferences implied by A.4 and the structure implied by other preference restrictions used in the social choice literature. Consider the condition of value restriction (VR) which ensures the existence of a choice set. VR states that over any triple, all concerned individuals agree that some option is either not best (NB), not medium (NM), or not worst (NW) (an individual is concerned if he is not indifferent between the three options). Single-peakedness corresponds to NW [see Sen (1970, ch. IO)]. Although the restrictions imposed by A.4 do not imply VR, consider the following strengthening of A.4: A.6. Yj(a,
t)
For all i, j, i # j: either yr(a, t) > yj(a, t) for all ~1,t, or yi(a, t) < for all Cc,t.
A.6 is clearly an unattractive condition in the present context; for instance, it is compatible with only at most one individual not working. Applying A.6 instead of A.4 in lemma 1, the effect of the change is that if (x1, tl), (czz, t2), t, -c t,, are such that t,lit, then tIPit, for allj > i, and t,Pjt, for all j < i. The following result provides the reason for looking at A.6: Theorem 3.
If preferences satisfy A.4 then they satisfy VR.
Proof. Consider any triple (a,, tl), (I+, tz), (CQ, t3). If tl = t2, say, then with tli # CI~there will be strict domination and NB and NW will be satisfied. Thus assume that t, < t, < t, . It will be shown that (aZ, tJ is either NB or NW. Assume not so that there exist i, j, who are concerned, such that: i finds
K. W.S. Roberts, Voting over income tax
339
(Q, tz) best, i.e. tZRit3 and tZRitl; jfinds (uZ, tz) worst, i.e. t,Rjtz and t3RjtZ. With A.4 replaced by A.6 in lemma 1, we have: as t, < t3, t2Rit3 and t3RltZ + t,R,t, and t,Rjt, -+ i $ j. Therefore, i = j and t,lit,Zit, ilj; as t,
adherence
is weaker than one particular
form of value
5. Majority voting As an example of a voting mechanism which satisfies A.1-A.3, the method of majority decision may be briefly considered. Assume that the number of individuals is odd or individuals form a continuum so that, under A.4, there will exist an individual m with a median income level. Utilizing lemma 1, it is easy to see that if m shows a strict preference between two schedules then a majority will have the same preference. Thus the choice set under majority voting will be a non-empty subset of m’s choice set. To consider m’s most preferred schedule, assume that c1and t are related as in eq. (7). Combining (6) and (7):
If y, < Jo, i.e. the distribution of income is positively skewed, and dy/dt < 0 so that the adoption of a more progressive tax schedule leads, because of disincentive effects, to a reduction in national income, then the median voter will most prefer a schedule with a positive marginal tax rate. Of course if G, the government revenue requirement, is positive, then it is still possible that the lump-sum subsidy c1will be negative. From lemma 1, the median individual has a most preferred tax schedule with a marginal tax rate below that which is desired by an individual with zero income; from (6), this will be the schedule where a is maximised. As nobody will work when the marginal tax rate is 100x, CIwill be maximised at a tax rate below 100%. Hence, the median individual will most prefer a schedule with a marginal tax rate below 100%. To be able to say anything further about the tax schedule which will be chosen under a majority voting rule, further assumptions must be imposed. Assume that aggregate labour supply is given by the constant elasticity function Y =
ny = nk(l- f)kY 9
(10)
where k is a constant and /I and y are the relevant elasticities. This equation is clearly unacceptable when CIis small. However, it may be assumed to hold
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K. W.S. Roberts, Voting over income tax
locally around m’s most preferred tax schedule. Assuming that G = 0, m’s most preferred tax schedule will be given by l-(l+y)6 t = l-(l+y)6+p’
(11)
where 6 = y,,,/J (assuming second-order conditions are satisfied). As an example, let b = 0.5, y = 0.2 and 6 = +. Then the tax schedule which will be chosen will have a marginal tax rate of 29%. Assuming that the worst-off individual does not work, this may be compared to the maximin tax rate which will be given by (11) with 6 set at zero. Thus the maximin tax rate will be 67%. By lemma 1, no individual will prefer a schedule with a higher marginal tax rate to this schedule. References Arrow, K.J., 1963, Social choice and individual values, 2nd ed. (Yale University Press, New Haven, CT). Black, D., 1958, The theory of committees and elections (Cambridge University Press). Diamond, P.A. and J.E. Stiglitz, 1974, Increases in risk and in risk aversion, Journal of Economic Theory 8,337-360. Foley, D., 1967, Resource allocation and the public sector, Yale Economic Essays 7, 45-98. Helpman, E. and E. Sadka, 1976, The optimal income tax: Some comparative statics results, Foerder Institute for Economic Research Working Paper No. 15-76. Itsumi, Y., 1974, Distributional effects of linear income tax schedules, Review of Economic Studies 41,371-381. Mirrlees, J.A., 1971, An exploration in the theory of optimum income taxation, Review of Economic Studies 38,175-208. Pattanaik, P.K., 1971, Voting and collective choice (Cambridge University Press). Roberts, K.W.S., 1975, Aspects of the theory of optimal income taxation, B.Phil. thesis, Oxford University. Romer, T., 1975, Individual welfare, majority voting, and the properties of a linear income tax, Journal of Public Economics 4,163-186. Rothschild. M. and J.E. Stialitz. 1970. Increasing risk: 1. A definition. Journal of Economic Theory 2,225-243. ’ ’ Sen, A.K., 1970, Collective choice and social welfare (Holden Day, San Francisco). Stiglitz, J.E., 1974, Demand for education in public and private school systems, Journal of Public Economics 3,349-386.
Publishing Company
VOTING OVER INCOME TAX SCHEDULES Kevin
W.S. ROBERTS*
St. John’s College, Oxford OX1 3JP, England Received February 1977, revised version received June 1977 In many voting situations, preferences over options may fail to be single-peaked. This is especially true when options consist of different amounts of a good which is provided through distortionary taxation. In this paper, voting over linear income tax schedules is considered. Although preferences may fail to be single-peaked, a choice set is shown to exist when only mild restrictions are imposed. For many choices in the public domain, the conditions required for this result are likely to be satisfied.
1. Introduction and summary It may appear unrealistic to consider that choices made in the public domain are the outcome of some voting process. However, if political parties make choices so as to maximise the likelihood of being elected then it is possible to view the chosen options as being determined, albeit indirectly, by a voting process. The point is not whether choices in the public domain are made through a voting mechanism but whether choice procedures mirror some voting mechanism. The problem then arises as to whether there will exist a ‘best’ policy. If individual preferences are unrestricted then it is well known [see Arrow (1963)] that choice sets may fail to exist under many mechanisms, e.g. majority rule. But it is also well known [see Black (1958)] that if individual preferences are single-peaked then majority rule will operate in a satisfactory manner. The following question arises: How likely is it that preferences over issues in the public domain will be single-peaked? Consider a decision being made about the level of provision of some public good. If each individual must pay a constant percentage ot the cost of provision then single-peakedness will follow from the assumption of convex production sets and quasi-concave preferences. In many instances, both of these assumptions are questionable. For instance, consider the case where a municipal body is deciding upon the level of public transport facilities. If an individual owns an automobile then his preferences over the level of provision may be expected *This paper is based upon chapter 4 of Roberts (1975). I am grateful to two anonymous referees for their comments.
330
K. W.S. Roberts, Voting over income tax
to be single-peaked. Similarly if he does not own an automobile. But at some level of provision, the individual may switch from owning to not owning an automobile. It is then quite possible for individual preferences to be doublepeaked. Various other examples of a similar nature could be cited, e.g. public replacing private schooling - on which see Stiglitz (1974). The results of this paper could be applied to the situation just mentioned, but our main interest will centre upon cases where individual preferences over goods are well-behaved although preferences over options can fail to be single-peaked. To be more specific, we shall consider the problems that arise when provision is made through distortionary taxation and, in particular, preferences over linear income tax schedules will be considered; these may be viewed as a (distortionary) tax on income which is used to finance a lump-sum subsidy. It is not immediately clear why single-peakedness over tax schedules should fail to hold, given that preferences are well-behaved. At the risk of oversimplification, the following example illustrates the nature of the problem: an individual may dislike a small increase in the marginal tax rate because the increase in the tax rate causes a large reduction in the work effort of others and so the lump-sum subsidy must decrease. However, further increases in the tax rate may cause little shifting and so the individual may agree with these further increases in the tax rate. Finally, with a marginal tax rate of 100x, nobody will choose to work and the lump-sum subsidy will be at its minimum level so that, eventually, the individual will dislike further increases in the tax rate. Thus, because there is distortionary taxation of a heterogeneous population, preferences for tax rates may fail to be single-peaked, even though each individual’s preferences map, considered in isolation, is well-behaved. In fact, restrictive assumptions may be necessary to ensure single-peakedness in these circumstances. Even when individuals have identical preferences over consumption and leisure and differ with respect to the wage rate that they receive, Itsumi (1974) and Romer (1975) have shown that single-peakedness is not guaranteed. It is not the purpose of this paper to investigate the likelihood of singlepeakedness. Instead, the problem to be investigated is whether, in the circumstances that have been described, it is likely that voting mechanisms will give rise to a most preferred outcome, irrespective of whether single-peakedness is satisfied or not. It will be shown that if preferences are such that the ordering of individuals by income is independent of the tax schedule in operation, an assumption which is termed Hierarchical Adherence in the text, then a most preferred outcome will exist under a wide class of voting mechanisms. For the class of income tax models first developed by Mirrlees (1971), hierarchical adherence is a consequence of the assumptions which are generally imposed. Thus relative to other work in this area, hierarchical adherence may be viewed as a mild assumption.
K. W.S. Roberts, Voting over income tax
331
The results obtained in this paper are applicable to many situations. Essentially, if, say, the provision of some public good is being considered then an assumption similar in effect to hierarchical adherence is that the ordering of individuals in terms of their marginal preference for the public good is independent of the level of provision of the public good. This would seem to be a reasonable assumption for the example of public transport provision mentioned earlier. It may be noted that the assumption requires that there be a ‘natural’ ordering of individuals, whereas single-peakedness requires that there be a ‘natural’ ordering of options. In section 2, individual preferences for tax schedules are considered. The class of voting mechanisms being admitted is considered in section 3. The main results of the paper are obtained in section 4 and most preferred tax schedules under the majority decision rule are briefly considered in section 5.
2. Individual preferences for tax schedules It will be assumed throughout that all prices are unaffected by changes in tax rates and that the only tax in operation is one on income. Consider an individual who is deciding how much labour to supply. As prices are fixed, it may be assumed that he has preferences over post-tax income (consumption) c and pre-tax income (labour supply) y. Preferences are assumed to be representable by a differentiable utility function u(c, y) with u, > 0 and tlY < 0. The government, or municipal body, implements a linear income tax. The tax paid by an individual with income y is given by T(y) = ty-a,
(1)
where t is the marginal tax rate and a is a lump-sum subsidy. Usually, the tax implemented will have to satisfy the property that total receipts can finance some fixed revenue requirement G, i.e.
(2) where yi is the income of individual i (if there is a continuum of individuals then summation could be replaced by integration). If yi is continuous in LY then we may expect that for any t, a is made as large as possible so that (2) is satisfied with equality. In this case, combining (1) and (2): ct = ty-G/n,
(3)
where J is the mean pre-tax income and n is the number of individuals in the community. If leisure is a normal good so that J decreases with CI, for any value of t there will be a unique value of a which satisfies (3).
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K. W.S. Roberts, Voting over income tax
It will be assumed that for all tax schedules to be considered, all individuals can attain a position inside their consumption sets. An individual chooses y so as to maximise u(c(+(l - t)y, y). First-order conditions of maximisation give (l-t)u,+u, with equality an individual.
5 0,
(4)
if y > 0. Consider how a small change in the tax schedule affects Totally differentiating the utility function with respect to t and ~1: du = dy[u,(l-t)+u,,]+u,[da-ydt].
Making
(5)
use of (4): du = u,[dcl-ydt],
which is basically of t, then
Roy’s identity.
da -& =y+tz.
dj
(6) If (3) defines
CI as a differentiable
function
(7)
This equation may be used in conjunction with (6) (notice that it is an implicit relationship because jj and dJ/dt are influenced by changes in LY). When incomes are unaffected by a change in the tax schedule, (6) and (7) combine to give: du - = u,[jj--y]. dt Thus an individual with an income y less than the mean income jj will always prefer a higher to a lower tax rate and preferences will be reversed if y is greater than 7. In this situation, the consequences of majority voting, say, are easily analysed. If the median income is less than the mean income, so that the income distribution is positively skewed (using Pearson’s second coefficient of skewness), then majority voting will lead to the tax schedule with the highest marginal tax rate being adopted. This result has been obtained by Foley (1967). 3. Voting mechanisms The underlying structure of the problem being analysed is best brought out if a more general class of voting mechanisms than majority rule is considered. The existence of a most preferred outcome (choice set) for mechanisms satisfying decisiveness, neutrality, and nonnegative responsiveness will be investigated. A
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K. W.S. Roberts, Voting over income tax
complete description of the economy is provided when the two parameters CI and t are specified. In fact, if (3) is imposed, say, then it will generally only be necessary to specify t but such an assumption is not required for the results that will be obtained. Individual preferences over tax schedules may be obtained by integration of (6). Ri will denote individual i’s ordering of tax schedules, i.e. 1, t,)R,(a,, tz) iff individual i weakly prefers schedule (al, tl) to (cI~, t2). (M For notational convenience the CIterm will be suppressed and so t,Rit, will be written. Corresponding to Ri, Pi and Zi will denote strict preference and indifference respectively. The voting mechanism is assumed to provide a binary relation R over a set of tax schedules r, and R will be some function f of the list (Ri)i=l,, over z. We have: Decisiveness : R is decisive it is complete (t,Rt, : t,Rt,)
A.2. Neutrality:
Consider
t,Ritz for two lists (Ri) t,Rt,
and
and (Rf),
then:
where R = f((R,))
then t,Rt,
and
t,R,t,
t,Rt,
possible
list
t,, t,, tJ, t4. If for all i:
f+ t,R,!t3,
t) t,R’t,,
and R’ = f((R:)).
A.3. Nonnegative t,Rit,
any quadruple
t) tJRft4
++ t,R’t,
iff for every logically and reflexive (tIRt,).
responsiveness: -+ t,Rft,
and
Consider t,P,t,
any pair t, , t,. If for all i:
+ &Pit,,
-+ t,R’t,.
These three assumptions are much used in the theory of social choice [see Sen (1970) and Pattanaik (1971)]. Mechanisms that satisfy A.l-A.3 include the method of majority decision, the method of nonminority decision, and variants of these methods which do not treat individuals in an anonymous manner. 4. Analysis So far, we have only considered a single individual’s preference over tax schedules. No assumption has yet been imposed upon the class of utility functions which is permissible. Instead of pinning down any one individual’s
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K. W.S. Roberts, Voting over income tax
utility function, we impose a condition which restricts the set of preferences that individuals in society may have, relative to each other: A.4. Hierarchical adherence: For all i, j: either yi(a, t) 6 uj(E, t) for all CY, t, or yi(cL,t) 2 yj(a, t) for all M,t, where yi(a, t) is the pre-tax income that i chooses to obtain under the tax schedule (~1,t).
Thus hierarchical adherence states that there exists an ordering of individuals such that pre-tax income is monotonically increasing irrespective of the tax schedule which is in operation. If all individuals have identical preferences over consumption and leisure but differ with respect to the wage rate that they receive then A.4 will be satisfied if the elasticity of labour supply (in terms of hours worked) is not less than minus unity (pre-tax income will increase with the wage rate). Further, in this case, A.4 holds even when, under some tax schedules, there are individuals who choose not to work, a situation which creates problems when single-peakedness is being analysed [see Romer (1975, section 5)]. It is a remarkable feature of the problem under consideration that A.4 is a sufficient condition on preferences for the existence of a choice set to be proved. The analysis proceeds in stages, the implications of A.4 for individual preference orderings being considered first. With A.4 satisfied, it will be useful to adopt the labelling convention that if i is greater thanj then yj(a, t) is no greater than yi(a, t) for all (a, t). Lemma 1.
(1) (2) (3)
Under A.4, if t2 > t, then for all i:
t,Rjt, t,Rj, t,P,t, --) t,P& t,P,t, +
vit,
--)
tzPjt,
for for for for
all j > i, all j < i, all j > i, all j < i.
Proof
(1) Consider i’s indifference map in (~1,t) space which is derived from his indirect utility function ui(q t) z z&c1+ (1 - t)yi(cr, t), yi(cc, t)). An indifference curve connects (a,, tl) to (Q, t,l and, from (6), its slope is given by da - = yi(ol, t). dt Consider j > i. Moving along the indifference curve, (6) gives du’ - = U&(c(, t)-yj(cr, t)] 5 0 dt
K. W.S. Roberts, Voting over income tax
335
by A.4. Hence,
u’$x2,t,)-~&~,
I,) = j;:
u~[yi(cl,t)-yj(cc, t)]dt $ 0,
i.e. tlRjt2. When j < i, the inequalities are reversed (2) AS t,Pit,, there exists an E > 0 such that for j > i:
z&q Applying
-&,
t1) <
and t,Rjt,. (c~i-E, tl)Zi(CIZ)t2). NOW,
u.“(c(l,t1).
the first part of the lemma:
uj(cI1-&,
t1) 2
u’(cQ, tz).
Combining these inequalities we obtain (3) In a symmetrical manner to (2).
tIPit,. Q.E.D.
Thus A.4, by itself, imposes an interesting set of restrictions on individual preferences. For instance, if everybody has a unique most preferred tax schedule then lemma 1 implies that individuals with higher incomes have most preferred schedules with lower marginal tax rates. A result similar to lemma 1 has interesting implications in welfare theoretic analyses. For such analyses, interest centres not upon individuals as such but upon their utilities. Consider replacing A.4 by the following adherence condition : A.5:
For all i, j, for all (c(, t):
Yi(“, t> 2 where v is the indirect
_Yj(m,
t>
utility
f+
2)i(u,t) 2
uj(as
t),
function.
Under AS, lemma 1 holds with Ri replaced by R’, where R’ is the ordering by positions in the income (or utility) hierarchy, i.e. (~1~)t,)R’(cr,, tJ iff the individual who is the ith worst-off when the schedule (txi, tl) is in operation is at least as well off under (c(i, tl) as the individual who is the ith worst-off under (a,, tz) when (cx~, tJ is in operation. For although as we integrate around a positional indifference curve the relevant individuals may be changing, A.5 ensures that there is no change in utility just because of this. Thus the amended lemma 1 now states that if a schedule with a higher marginal tax rate is adopted, the utility distribution will alter so that the lowest utilities will increase, utilities in the middle of the distribution will remain constant, and the highest utilities will decrease. It is therefore to be expected that more inequality averse SWFs will prefer schedules with higher marginal F
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K. W.S. Roberts, Voting over income tax
tax rates. Following on from the work of Rothschild and Stiglitz (1970) and Diamond and Stiglitz (1974), define a SWF W = ~~=I G(u’) to be more inequality averse than W’ = ~~=I ui if G is increasing and strictly concave (u’ could obviously be viewed as a function of individual utilities). Let p be an index of inequality aversion and let R, be the ordering of schedules by a SWF W, with inequality aversion p. Assume that
above we know Define r(t) so that (a(t), t)l,,(cl,, t 1). From the discussion that if W,, = cu’, then an increase in t, with a change in a given by a(t), will lead to a mean preserving reduction in the inequality of the ui, to adopt the terminology of Rothschild and Stiglitz (1970). By their well-known equivalence theorem (theorem 2), a more inequality averse SWF will prefer the increase in the marginal tax rate. Thus we shall have
where pz > pl. We may thus obtain Theorem (1)
tlJp,t,
(2) (3)
t,P,,t, t,P,,t,
1.
Under AS,
t,R,,t, l t2R,,,tl -+ t,P,,t, -i t,P,,t, +
a dual result to lemma
1:
if t, > t, then for all p1 : for for for for
all all all all
p2 < ply p2 > pl,
p2 < ply p2 > pl.
Theorem 1 also has interesting implications. For instance, the theorem implies that SWFs with a greater degree of embodied inequality aversion will choose schedules with higher marginal tax rates [see Roberts (1975) and Helpman and Sadka (1976)]. It is important to notice that the driving force of this result is condition A.5. Further, if individuals’ voting behaviour is based upon some form of ethical preference, as captured by a SWF, then the structure of individuals preference will be basically identical to the structure implied by A.4 as long as it is possible to order individuals in terms of the inequality aversion that is embodied in their preferences. Returning explicitly to voting procedures, the outcome of voting mechanisms may be considered by utilizing lemma 1: Lemma 2. If the voting mechanism satisfies A.l-A.3 A.4, then for all t,, t,, t3, tl < t2 < t,: (1) (2)
t,Rt, t,Rt,
and and
t2RtJ -+ t,Rt,, t,Rt, + t,Rt,,
and preferences
satisfy
K. W.S. Roberts, Voting over income tax
(3) (4)
r,pt, t3pt2
and and
t,Pt, t,Pt,
337
-+ tlPtJ, -9 t3Ptl,
where R = f(( R,)). Proof. (2) is proved in a symmetrical manner to (1) ; (3) and (4) may be proved as in (1) and (2), strict replacing weak preference. Thus we concentrate upon (1). Consider individual preferences over the pair t, , t, . By lemma 1, preferences must be of the form: t,Pit,Vi 5 fi; tl_litlVfi < i < y; tlPit2Vi 2 y, for some 5 6; t,Zit,V6 < i < E; /?, y, jI < y. Similarly, over the pair t,, t,: t,Pit,Vi t,P,t,Vi >= E, for some 6, E, 6 < E. There are six cases to be considered: (i)p < y 6 6 < e;(ii)/3 5 6 6 y 5 .s;(iii)fi 5 6
2 may now be applied
and the main result of the paper obtained:
Theorem 2. If the voting mechanism satisfies A.l-A.3 A.4 then the social preference relation is quasi-transitive, t,pt,
and
t,Pt,
and preferences satisfy i.e. for all t, , t2, t3 :
-+ t,Pt,.
Proof. Let the antecedent hold but assume that t,Rt, . There are six cases where the marginal tax rates differ to be considered: t, < t, < t, ; 1, < t2 < t, ;
ti < 5, < t, ; t, < t, < t,;
t, < t, < t3 ; t, < t, < I,.
The last three cases are essentially the same as the first three so that we concentrate upon the first three cases. For each case we obtain a contradiction:
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K. W.S. Roberts, Voting over income tax
Case (i) Case (ii) Case (iii)
by lemma 2, applying lemma 2, applying lemma 2,
t,Pt, ; tzPtS and tJRtl + t,Rt,; t3Rtl and t,Pt, + t,Rt,.
When t, = t,, say, as t,Pt, it must be the case, by A.2 and A.3, that ~1~> CI~ and tIPit, for all i. The desired result then follows from a simple application of A.2 and A.3. The other cases where two or three of the marginal tax rates are equal may be treated in a similar way. Q.E.D. A sufficient condition for a choice set to exist is that the underlying binary relation is complete, reflexive and quasi-transitive, and the set of social states is finite [see Sen (1970, ch. l*)]. Completeness and reflexivity follow from A.1 so that as a corollary to the above theorem we have: Corollary. If the voting mechanism satisfies A.I-A.3, preferences satisfy A.4, and z is jinite then C(z), the choice set of z under the voting mechanism, is nonempty. Given this result, the question arises as to whether there is any connection between the structure of preferences implied by A.4 and the structure implied by other preference restrictions used in the social choice literature. Consider the condition of value restriction (VR) which ensures the existence of a choice set. VR states that over any triple, all concerned individuals agree that some option is either not best (NB), not medium (NM), or not worst (NW) (an individual is concerned if he is not indifferent between the three options). Single-peakedness corresponds to NW [see Sen (1970, ch. IO)]. Although the restrictions imposed by A.4 do not imply VR, consider the following strengthening of A.4: A.6. Yj(a,
t)
For all i, j, i # j: either yr(a, t) > yj(a, t) for all ~1,t, or yi(a, t) < for all Cc,t.
A.6 is clearly an unattractive condition in the present context; for instance, it is compatible with only at most one individual not working. Applying A.6 instead of A.4 in lemma 1, the effect of the change is that if (x1, tl), (czz, t2), t, -c t,, are such that t,lit, then tIPit, for allj > i, and t,Pjt, for all j < i. The following result provides the reason for looking at A.6: Theorem 3.
If preferences satisfy A.4 then they satisfy VR.
Proof. Consider any triple (a,, tl), (I+, tz), (CQ, t3). If tl = t2, say, then with tli # CI~there will be strict domination and NB and NW will be satisfied. Thus assume that t, < t, < t, . It will be shown that (aZ, tJ is either NB or NW. Assume not so that there exist i, j, who are concerned, such that: i finds
K. W.S. Roberts, Voting over income tax
339
(Q, tz) best, i.e. tZRit3 and tZRitl; jfinds (uZ, tz) worst, i.e. t,Rjtz and t3RjtZ. With A.4 replaced by A.6 in lemma 1, we have: as t, < t3, t2Rit3 and t3RltZ + t,R,t, and t,Rjt, -+ i $ j. Therefore, i = j and t,lit,Zit, ilj; as t,
adherence
is weaker than one particular
form of value
5. Majority voting As an example of a voting mechanism which satisfies A.1-A.3, the method of majority decision may be briefly considered. Assume that the number of individuals is odd or individuals form a continuum so that, under A.4, there will exist an individual m with a median income level. Utilizing lemma 1, it is easy to see that if m shows a strict preference between two schedules then a majority will have the same preference. Thus the choice set under majority voting will be a non-empty subset of m’s choice set. To consider m’s most preferred schedule, assume that c1and t are related as in eq. (7). Combining (6) and (7):
If y, < Jo, i.e. the distribution of income is positively skewed, and dy/dt < 0 so that the adoption of a more progressive tax schedule leads, because of disincentive effects, to a reduction in national income, then the median voter will most prefer a schedule with a positive marginal tax rate. Of course if G, the government revenue requirement, is positive, then it is still possible that the lump-sum subsidy c1will be negative. From lemma 1, the median individual has a most preferred tax schedule with a marginal tax rate below that which is desired by an individual with zero income; from (6), this will be the schedule where a is maximised. As nobody will work when the marginal tax rate is 100x, CIwill be maximised at a tax rate below 100%. Hence, the median individual will most prefer a schedule with a marginal tax rate below 100%. To be able to say anything further about the tax schedule which will be chosen under a majority voting rule, further assumptions must be imposed. Assume that aggregate labour supply is given by the constant elasticity function Y =
ny = nk(l- f)kY 9
(10)
where k is a constant and /I and y are the relevant elasticities. This equation is clearly unacceptable when CIis small. However, it may be assumed to hold
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K. W.S. Roberts, Voting over income tax
locally around m’s most preferred tax schedule. Assuming that G = 0, m’s most preferred tax schedule will be given by l-(l+y)6 t = l-(l+y)6+p’
(11)
where 6 = y,,,/J (assuming second-order conditions are satisfied). As an example, let b = 0.5, y = 0.2 and 6 = +. Then the tax schedule which will be chosen will have a marginal tax rate of 29%. Assuming that the worst-off individual does not work, this may be compared to the maximin tax rate which will be given by (11) with 6 set at zero. Thus the maximin tax rate will be 67%. By lemma 1, no individual will prefer a schedule with a higher marginal tax rate to this schedule. References Arrow, K.J., 1963, Social choice and individual values, 2nd ed. (Yale University Press, New Haven, CT). Black, D., 1958, The theory of committees and elections (Cambridge University Press). Diamond, P.A. and J.E. Stiglitz, 1974, Increases in risk and in risk aversion, Journal of Economic Theory 8,337-360. Foley, D., 1967, Resource allocation and the public sector, Yale Economic Essays 7, 45-98. Helpman, E. and E. Sadka, 1976, The optimal income tax: Some comparative statics results, Foerder Institute for Economic Research Working Paper No. 15-76. Itsumi, Y., 1974, Distributional effects of linear income tax schedules, Review of Economic Studies 41,371-381. Mirrlees, J.A., 1971, An exploration in the theory of optimum income taxation, Review of Economic Studies 38,175-208. Pattanaik, P.K., 1971, Voting and collective choice (Cambridge University Press). Roberts, K.W.S., 1975, Aspects of the theory of optimal income taxation, B.Phil. thesis, Oxford University. Romer, T., 1975, Individual welfare, majority voting, and the properties of a linear income tax, Journal of Public Economics 4,163-186. Rothschild. M. and J.E. Stialitz. 1970. Increasing risk: 1. A definition. Journal of Economic Theory 2,225-243. ’ ’ Sen, A.K., 1970, Collective choice and social welfare (Holden Day, San Francisco). Stiglitz, J.E., 1974, Demand for education in public and private school systems, Journal of Public Economics 3,349-386.