Voting equilibrium in a simple tax model

JOURNAL

OF ECONOMIC

Notes, Voting

THEORY

147-154 (1975)

11,

Comments,

and Letters

Equilibrium

in a Simple

to the

Editor

Tax Model*+

The chief obstacle to the existence of a voting equilibrium in a general model appears to be the ability of majority coalitions to form and improve upon any outcome by confiscating wealth from, or by levying increased taxes upon, the complementary coalitions. Take, for example, the problem of determining a mix x of public projects along with a tax assignment y to pay for these projects. If y leaves each individual with a positive amount of wealth (no matter how small) any majority coalition can find an alternative assignment y’ which reduces the taxes payable by its members and increases the tax burden of nonmembers and yields the same revenue as y. Hence every pair (x, v) is dominated by some other pair (x, y’) and no voting equilibrium exists. In this paper the ability of majority coalitions to tax individual citizens is restricted in order to permit the existence of an undominated tax assignment. The mix x of public outputs is not allowed to vary, so the results will provide only a necessary condition for the existence of a voting equilibrium when the determination of x is part of the decision problem. The existence condition uncovered is a simple inequality involving: a, that fraction of each individual’s wealth which may not be taxed away; /3, the cost of forming and maintaining a majority coalition, expressed as a fraction of total national wealth; and A, the fraction of total national wealth which must be collected in tax revenues to pay for x. If individual utility depends only on total individual taxes payable, the necessary and sufficient condition for the existence of an equilibrium tax assignment is essentially 213+ cx+ A 3 1 for large numbers of voters. Section I of the paper sets out the model employed. Section II presents the results and Section III offers some concluding remarks. I.

THE

MODEL

N = l,..., n} is the set of IZ voters. The set W of winning coalitions is precisely the collection of all subsets of N with m or more members. The

* This paper has benefitted much from the comments of a referee at an earlier stage. I would like to express my gratitude and claim responsibility for all errors. Financial support from the Canada Council is gratefully acknowledged. + Currently visiting at the Graduate School of Business, Stanford California,

147 Copyright AI1 rights

0 1975 by Academic Press, Inc. of reproduction in any form reserved.

148

DONALD

E. CAMPBELL

positive integer m can be interpreted as the size of the smallest majority coalition. The proofs are valid, though, for any positive integer m and we needn’t specify exactly the value of m. For each individual i there is a vector wi specifying the market value of i’s holdings of each of the k types of wealth. A tax assignment y = (n ,..., yn) is an nk-vector which requires each i to pay the vector yi of taxes; one for each type of wealth. Throughout this paper the wi and x are fixed so we may suppose that utility depends only on y. Each individual i’s preference order is represented by a continuous, real-valued function Ui on the Euclidean space of tax assignments. The mix x of public projects requires the collection of c(x) dollars of tax revenue. No winning coalition, no matter how large, can tax away more than olwi of the wealth of each i in the complementary coalition. The cost of forming and maintaining a winning coalition is the fraction /I of total national wealth. To render the notation less cluttered, for each function f on N set f(Z) = C, fi for Z C N. Let e = (I,..., 1) be a k-vector and set h = c(x)/ew(N). It is assumed throughout that ewi is strictly positive for each i. A tax assignment y* is dominated by assignment y through coalition Z If and only if: for

yi < wi Y,,<(l

for

-a>wi

v(N)

iEN.

(1)

iEN-Z.

b 44 + Bew(N). ZE w.

U,(Y) > ui(Y*)

(2) (3) (4)

for

i e I.

(5)

Condition (3) implies that y must yield enough revenue to pay for x and the cost of organizing coalition Zin opposition to y*. Condition (2) implies that the cost of organization cannot be shifted on to the members of N - I; the after-tax wealth of each i in N - Zmust be at least olwi no matter how high are the costs of coalition formation. The other conditions are standard.l 1 The yi are not required to be nonnegative; redistribution of wealth is permitted if it is consistent with (l)-(5). The reader can verify that nonnegativity of the yi can be imposed without invalidating any of the proofs. The requirement (4) that y* can be dominated only if at least m voters stricly prefer some y to y* is not as restrictive as it might seem. If more voters prefer y to y* than prefer y* to y, voters who are indifferent between y and y* can shift a tiny amount of their tax burden to anyone preferring y toy* creating a coalition of at least m voters each of whom strictly prefers y (as modified) toy*. All that is required is continuity of Vi and some type of monotonicity of preference for after-tax wealth.

149

VOTING EQUILIBRIUM

A feasible tax assignment is an assignment y satisfying (1) and

ey(N) t 4.4.

(6)

A voting equilibrium is a feasible tax assignment which is rational for N (i.e., Pareto optimal) and which is not dominated by any other tax assignment. Note that wealth is always evaluated at the same prices even though a change in the tax structure will change market prices. This is justified if a winning coalition will always use current prices to evaluate the benefit of a proposed change in the tax assignment. The following condition E is sufficient for the existence of a voting equilibrium if each Vi is quasiconcave in after-tax wealth Wi - yi , and is also necessary if Vi depends only on ey, , the total amount of tax payable by i (assuming that utility falls as taxes increase).

If m is the number of members in the smallest majority coalition (i.e., if m is the smallest integer which exceeds n/2) then as n gets very large E reduces to E*:2/3+X+cw>l. It is assumed throughout that X + OL< 1 since otherwise is is impossible to finance x and leave after-tax wealth at least as large as olwi for each i.

II. THE RESULTS In order to establish the sufficiency of E we need to define a game, the core of which is precisely the set of voting equilibria. The game will be defined by means of the characteristic sets V,(Z C N). (See Scarf [2] and [3] for a discussion of this form of a game.) We assume that each Vi depends only on yi (really wi - yi , but wi is fixed). The vector u is an n-vector of individual utility levels. If Z $ W, V, = {u; ui < U,((l - a) wi) Vi E Z}. IfZE WandZ# Nthen V, = {u; ui < U&J Vi E Z for some y satisfying l-3). V, = (u; ui < U&J Vi E N for some feasible v}. a Definitions of quasiconcavity, balanced coalitions characteristic set form can be found in Scarf [2].

and games, and games in-

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DONALD

E. CAMPBELL

If U* is in the core of this game and ui* < Ui( yi*) for all i E N then y* is a voting equilibrium since y* may be assumed to be feasible and y* is dominated by some ZE W if and only if U* is blocked by I. Theorem 1 makes use of Scarf’s result [2] that a balanced game has a nonempty core to prove that a voting equilibrium exists. THEOREM 1. Zf E holds and each Ui is quasiconcave in wi - yi then the game is balanced and therefore the core and the set of voting equilibria are nonempty.

Proof. Let T be a collection of coalitions which is balanced by weights tj (ZE T). First, show that Cs tj < n/m holds for S = {ZE T, ZE W}. This follows from & trZi = 1 (Vi E N, where Zd= 1 if i E Z and Zi = 0 if i $ I)

and C tI I Z I = 5: tI ; Zi = ; T h4 = n T

and therefore n=Ct,lZl

3CtrlZI

T

amEt*.

s

s

Now suppose that u is an n-vector and u E V, for all Z E T. We must show that u E VN . For each Z E T let y’ be a tax vector which qualifies u for membership in V’ . If Z $ W we may suppose that y’ = (1 - a) w. Since CT tlZi = 1 (Vi E N) we have wi - yi = CT t,Zi(wi - yi’), a convex combination of the wi - yi’ and therefore ui < Ui( y) since Ui is quasiconcave and ui < min{ Ui( yil); Z E T and i E Z}. y satisfies (1) as a convex combination of vectors satisfying (1). It remains to show that y satisfies (6). From the definition of y

y(N) = 1 c MiY, 3 c N

T

tIYV>-

(7)

s

For ZES y’(Z) = yyN) - yyiv - Z) 3 y’(N) - (1 - ol)(w(N) - w(Z))

03)

and eyW

3 c(x) + Pew(N) = (A + f0 ew(N).

(9)

For Z$S y’(z) = (1 - a) w(Z).

(10)

a It might appear that since the Ui are quasiconcave the theorem in Scarf [3] can be applied directly. Closer examination will reveal that the characteristic sets are not related in a way that permits ready application of Scarf [3].

VOTING EQUILIBRIUM

151

Combining (8) and (9) yields, for all Z E S, d(Z)

>, (A + B>ew(N) - (1 - a) ew(N) + (1 - CL)ew(Z)

(11)

From (7) and (11) and the definition of y one obtains q(N)

= C f~v’(Z> 3 (A + f4 4N T

C 11+ (1 - 4 4W. s

Since & tI < n/m we have ey(JO 3 (fdm)(~ + B + a - 1) ew(N) + (1 - a) ew(N) > Xew(N) = c(x). This last inequality holds since E is equivalent to (n/m)@ + p + a - 1) + (1 - a) 3 A.

Q.E.D.

If utility depends only upon total taxes payable, so that U&yJ = -eyi , tax assignment y satisfying (6) and E is necessary and sufficient for the existence of an equilibrium

yi < (1 - a) wi for i E N.

U’14

In this case the set Y of voting equilibria can be easily characterized. (Assume that X + 01< 1 since otherwise (1 - a) w is the unique equilibrium tax assignment.) First, define A, the set of functions from N into the nonnegative reals satisfying for each a E A: a(N) = n and a(Z)>n(l --a-/I--X)/(1 -A---~)forallZEW. Y = { y*; for some a E A, yi* = (1 - a) wi - (a&2)(1 - X - a) w(N) Vi E N}5. THEOREM 2.

(12)

Zf iJi( yJ = -eyi for each i then Y is the set of voting

equilibria. Proof.

satisfied.

(i). Suppose that y* E Y is defined by a E A. Certainly (1’) is Also ey*(N) = (1 - a) ew(N) - (1 - h - CL)ew(N) =

4 If it is only known that the U, are quasiconcave then an equilibrium y* gives each voter at least as much utility as (1 - +vi (otherwise (i) could block u* in Theorem 1). To go further and insist upon (1’) requires some type of monotonicity. 5 One can think of all of (1 - a)wi being taxed away from each i and then the surplus (1 - oi - A) is returned to the voters by means of the weighting function a. This bears some resemblance to the fair division problem analysed by Kuhn [l].

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E. CAMPBELL

hew(N) = c(x) so (6) holds also. Suppose there exists a y and Z satisfying (l)-(5). Recalling (2) and (3), ey(Z) > (A + /3) ew(N) - (1 - a) ew(N) + (1 - CY)ew(Z).

(13)

However, (5) implies ey(Z) < ey*(Z) = (1 - IX)ew(Z) - (a(Z)/n)(l - A - LX)ew(N) and therefore ey(Z) < (1 - LX)ew(Z) - (1 - 01- /3 - A) ew(N). But (13) and (14) are contradictory (ii). If y* is a voting equilibrium eYi

(14)

so y* must be undominated. set

* = (1 - CL)ewi - (a&)(1 - A - a) ew(N)

and solve for ai . ey*(N) 2 c(x) implies a(N) < II and Pareto optimality implies a(N) 2 n. Therefore a(N) = n. It remains to show that (12) holds. Define y : yi = (1 - a) wi for i $ Z and yi = yi* for i E I. Then ey(N) = (1 - CL)ew(N - Z) + (1 - CX)ew(Z) - (a(Z)/n)(l - X - CX)ew(N), and if (12) is violated ey(N) > (1 -

a) ew(N) - (1 - (Y - /3 - A) ew(N) = c(x) + pew(N).

In this case Z could reduce the tax burden of y for each of its members and still satisfy (3). But this is impossible since y* is undominated so that a Q.E.D. must satisfy (12). THEOREM 3. Zf Ui( y) = -eyi for each i then Y is nonempty if and only if E is true.

Proof.

(i). If E holds and ai = 1 for i E N then a E A so Y is not empty.

(ii). Suppose y* satisfies (1’) and (6) and 1Z 1 = m. ey(Z) = (1 - a) ew(Z) - (m/n)(l - h - a) ew(N)

(15)

and ey(N - Z) = (1 - a) ew(N - Z)

(16)

153

VOTING EQUILIBRIUM

are consistent with (l)-(3). also satisfy

If E is false, any y satisfying (15) and (16) will

ey(N) = (1 - a) ew(N) - (m/n)(I - h - a) ew(N) > (A + p> ew(N).

(17)

If also ey(Z) < ev*(Z) then we can find a y satisfying (l)-(5) which means that y* is not an equilibrium. It remains to find an Z for which (15) and (16) hold, i.e., an Z for which ey*(Z) >, (1 - CC)ew(Z) - (m/n)(l - h - a) ew(N). It will suffice to choose the m members of N with the largest values of eyi* - (1 - a) ewi (ties may be broken in any fashion). Then since ey*(N) 2 Xew(N) we have q*(Z) - (1 - CL)ew(Z) 2 (m/n)[ey*(N)

- (1 - a) ew(N)]

2 (m/n)(A - 1 + a) ew(N). III.

Q.E.D.

CONCLUSION

The condition E is quite strong, unless there is reason to believe that only large coalitions are likely to form and vote down proposed tax assignments. (If E holds for p = IYI and q > p then E holds for q = m.) The instability caused by the possibility of winning coalitions redistributing wealth by shifting tax burdens can only be overcome, in this model, if the cost of coalition formation (or the cost of organizing support for an election), the fraction of individual wealth not subject to tax, and the amount of tax revenue which must be collected are all relatively large. If one makes the grand assumption that the electoral process underlying the above model will ultimately generate an equilibrium pair (x*, y*) of public outputs and tax assignment, condition E places a lower bound on the cost of x* since /3 will be given independently of the process and CLwill be determined by convention or constitution. The relative cost of x* is bounded from above by 1 - a, the proportion of individual wealth which may be taxed. If /3 is small very narrow bounds are placed on the cost of x* and public expenditure can grow only as wealth grows. The relative cost of x* will still be bounded and can change only if the cost of organizing coalitions changes or there is a change in the community norm governing the extent to which individual wealth may be taxed. REFERENCES 1. H. W. KUHN, On games of fair division, in “Essays in Mathematical Economics in Honor of Oskar Morgenstern,” (M. Shubik ed.). Princeton University Press, Princeton, NJ, 1967, pp. 29-37.

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E. CAMPBELL

2. H. E. SCARF, The core of an n-person game, Econometrica 35 (1967), 50-69. 3. H. E. SCARF, On the existence of a cooperative solution for a general class of n-person games, J. Econ. Theory 3 (1971), 169-181. RECEIVED:

November

16, 1972 DONALD

E. CAMPBELL*?

University of Toronto Toronto, Ontario, Canada