Power, majority voting, and linear income tax schedules

Journal

of Public

POWER,

Economics

36 (1988) 53-67.

MAJORITY

VOTING, AND SCHEDULES Richard

University

North-Holland

LINEAR

INCOME

TAX

M. PECK*

of Illinois at Chicago, Chicago, IL 60680, USA

Received April 1987, revised version

received April 1988

We compare the linear income tax solutions of a version of the Aumann-Kurz ‘Power and Taxes’ game and the majority voting model. We establish that an agent whose type is in the majority receives a lower payoff in this model than in the majority voting model. This reflects the possibilities for bargaining and coalition formation considered in the Aumann-Kurz model. The solutions of an example are computed and compared with the optimal utilitarian, the optimal Rawlsian and the majority voting equilibrium linear income tax schedules.

0. Introduction

An important concern of public choice is explaining society’s redistribution of income. The majority voting model considered by Foley (1967), Romer (1975) and Roberts (1977), for instance, provides a depiction of a democratic society’s selection of a redistributive tax schedule. In a pioneering alternative approach, Aumann and Kurz (1977a, b, 1978) also model a democratic process of majority control which can give rise to redistributive taxation. Unlike the majority voting model, however, the Aumann and Kurz approach takes into account the possibilities in a democracy for coalition formation and bargaining. Peck (1986) presented a version of the Aumann and Kurz model which required equilibrium redistributive income tax schedules to be linear and allowed for incentive effects .I In this paper, we compare this version of the Aumann-Kurz model with the majority voting model. Comparison with the majority voting model of tax selection provides insights into the role bargaining and threats, as specified in our model, play in determining equilibrium outcomes. Such a comparison also highlights the effect that the assumed absence of such possibilities for the majority voting *I wish to thank Robert Anderson, Martin Osborne, and, for providing numerous suggestions. Remaining errors are my ‘Other interesting variants of the Aumann-Kurz model are and Osborne (1979, 1984). Aumann, Kurz and Neyman framework to the provision of public goods. 004772727/88/$3.50

0

1988, Elsevier Science Publishers

in particular, Hugo Sonnenschein, responsibility. presented in Gardner (1981, 1984) (1983) apply the Aumann-Kurz

B.V. (North-Holland)

54

R.M. Peck, Majority voting and linear income tax

model has on its outcomes. We argue that this exercise also sheds light on actual institutions. In particular, the majority voting model can be thought of as a depiction of referendum voting. On the other hand, the possibilities for bargaining and coalition formation taken into acrom,; by our version of the Aumann-Kurz model are more naturally associated with a representative legislature. Our principle focus is on an economy with two types of agents: low-wage and high-wage agents. We use a result of Peck (1986) characterizing linear tax schedules which are non-transferable utility value solu~i~)us (the NTU solutions) of our amended Aumann-Kurz model. Our observations can be summarized as follows. The solution of our version of the Aumann-Kurt model is less preferred to the solution of the majority voting model by individuals whose type is constitutes a majority.* Therefore, tax selection processes allowing for bargaining and coalition formation may better serve the interests of individuals whose type is in the minority. Hence, the majority voting model, to the extent that it excludes the possibility of bargaining and coalition formation, may yield misleading results. In particular, the use of the majority voting model to depict the selection of a redistributive tax by a legislature could be inappropriate. Our results have interesting normative implications as well, particularly for the design and evaluation of political institutions. Under conditions similar to those considered by Sheshinski (1972), equilibrium linear tax schedules of our model always redistribute income from high-wage agents to low-wage agents. In contrast, for an otherwise identical economy, majority voting equilibria may not involve such redistribution, if high-wage agents are in the majority. Thus, the possibility of bargaining and coalition formation may lead to more income redistribution. Indeed, the outcome of our amended Aumann-Kurz model is closer to the Rawlsian solution [Rawls (1971)] than the equilibrium majority voting tax when high-wage agents are a majority. On the other hand, for the economy considered, the outcome of the majority voting model coincides with the Rawlsian solution when low-wage individuals are in the majority. In comparison, the equilibrium tax rates of our model are lower. The more limited income redistribution indicated by our model is attributable to the ability of high-wage individuals to join and shift a coalition into the majority, a possibility taken into account by the NTU value solution concept. Thus, in some instances, a Rawlsian would prefer tax selection processes limiting possibilities for bargaining and coalition formation yet retaining majority rule, such as secret ballot referenda. The organization of this paper is as follows. In section 1 we briefly review the version of the Aumann-Kurz model considered. In section 2, results *In an economy with more than two types, this generalizes median voter’s ability to affect influence is reduced.

to indicate

that, in our model,

the

R.M. Peek, Majority

voting and linear income tax

55

comparing the solutions of our model to solutions of the majority voting model of tax selection are presented. We consider an example in section 3. For this example, we compute the linear tax rate solutions of our model, along with optimal Rawlsian, optimal utilitarian and majority voting linear tax rates. This provides a closer comparison of our model with alternative normative and positive models of tax selection than is possible analytically. In section 4, we prove our results.

A version We assume

Aumann-Kurz that the reader

framework

is familiar

with the Aumann-Kurz

exposition and overview of their More formal presentations Aumann-Kurz model are provided in Peck (1983, 1986). We begin this section by the characteristics individual agents. Agents have the same continuous,

increasing and strictly quasi-concave, utilities. Finally, the utility function vanishes when either of its arguments equal zero. Each agent a chooses leisure 1* and consumption, c*, to solve: V,(t, b) = max u(l, C) subject to c=(l-1)(1 011s --

1,

--)w,+b, osc.

(1.1)

V,(t, b) is the agent’s indirect utility function. The budget constraint (1.1) depends on an exogenously determined wage rate, w,, and on the tax schedule (t,b), where t is the tax rate and b is the lump-sum transfer. The price of the consumption good, c, has been set equal to one. Endogenously determined income, y,(t,b), is equal to w&l -I,*(t,b)), where &t,b) is the individual’s leisure choice. In our model, an agent’s type is determined by his wage rate w,. Leisure is assumed to be a normal good, that is,3

‘As I*+l, 1* may fail to be differentiable I* is differentiable.

at [=l,

but otherwise

we assume

u to be such that

56

R.M. Peck, Majority

voting and linear income tax

In addition, the agent’s leisure choice is assumed to be a non-increasing, differentiable function of the net wage, so that as taxes increase, the amount of leisure chosen is non-decreasing, that is, lU,*(t, b)/i?t2 0. This assumption also implies that an agent’s income will rise as the wage rate increases. Therefore, it is possible by means of a progressive linear income tax to redistribute income from high-wage agents to low-wage agents. We now turn to an informal discussion of the strategies available to coalitions. Instead of arbitrary lump-sum taxes considered by Aumann and Kurz, the only redistributive mechanism assumed here are linear income tax schedules. A solution to our version of the Aumann-Kurz model is a linear income tax schedule (t, b). The simplicity of linear tax schedules facilitates comparison with other models and the computation of examples. This restriction is also informally supported by Kurz (1977) who shows that when agents strategically misrepresent preferences in the Aumann-Kurz singlecommodity framework, linear tax schedules result. The model retains aspects of Aumann and Kurz’s notion of majority control along with similar possibilities for coalition formation and bargaining.4 Coalition strategy spaces are modified, however, to take into account the restriction that solutions must be linear income tax schedules. A majority coalition can select a linear tax (t, b) to impose upon its own members. The majority coalition may also threaten the corresponding minority coalition with 100 percent taxation. In other words, majority coalitions can threaten minority coalitions with confiscatory taxation, as in the Aumann-Kurz model.5 In a democracy, minority coalitions also enjoy certain rights. These rights include the right to assemble and the prohibition of compulsory labor. Accordingly, minority coalitions may strike, that is, members of a minority coalition can agree to withdraw from the labor force. Therefore, members of 4Following Aumann and Kurz, the population 1 is a continuum. In particular, 1, along with collection of permitted coalitions, form a measure space isomorphic to the unit interval with its Bore1 subsets. 5Our formal specification of coalition strategy spaces [see Peck (1983)], after allowing for restriction to linear tax schedules, corresponds closely to that of Aumann and Kurz (1977b). The specifications are interpreted differently, however. In the original Aumann and Kurz model, their specification is consistent with a society which allows majority coalitions to choose for the entire population essentially any feasible lump-sum tax structure. This power of majority coalitions permits the possibility of confiscatory taxation. Our interpretation must be different to account for the absence of non-linear (or piecewise linear) tax schedules in equilibrium. In the society we consider, majority coalitions have some discretion over tax selection, but less than Aumann and Kurz assume. In particular, a majority coalition can select a feasible tax schedule to impose upon itself and it can set the tax rate on the corresponding minority coalition equal to one. We do not necessarily suppose, though, that majority coalitions may impose any feasible linear tax on minority coalitions, which would be the analog of Aumann and Kurz’s interpretation.

R.M. Peck, Majority voting and linear income tax

57

a minority coalition may block the transfer of income to a majority coalition by ceasing to work. This threat corresponds in the Aumann-Kurz framework to the minority coalition’s threat to destroy its initial endowment. The solution concept employed is the Nash-Harsanyi-Shapley nontransferable utility value (NTU solution), recently axiomatized by Aumann (1985). A formal definition of the NTU solution concept, appropriately modified for our model, is given in Peck (1986). A linear tax schedule (t, b), which is an NTU solution of our model, has the following interpretation, due to Martin Osborne (1979). An NTU solution is regarded as a compromise based on the threat possibilities available to the set of possible coalitions. Threats, in the model considered here, are mechanisms to disrupt the economy. In the view of society presented here, an observed tax is an equilibrium between the disruptive pressures existing in society. If a signiticant group does not approve of a tax, it has the means to disrupt the economy. In our model, a majority coalition can threaten to set the tax rate on the complementary minority coalition equal to one. A minority coalition can threaten to cease working. Therefore, in our amended version of the Aumann-Kurz model, a linear tax solution is viewed as emerging from a bargaining process which balances threats of contiscatory taxation with counter threats of ‘strike’.

2. Comparison with the majority voting model of tax selection The majority voting model is a non-cooperative game in which there is no bargaining. Given a pair of feasible linear tax schedules, an agent votes noncollusively, according to his true preferences, for the tax schedule which yields the highest utility level. A linear tax is a majority voting equilibrium (MVE) tax schedule if it receives a majority of votes when paired with any other feasible tax. A sufficient condition for the existence of a MVE tax schedule is that preferences be single peaked.6 With only two types of agents, an MVE tax is the solution to the following maximization problem:’ max K(t, b),

if pi > l/2

subject to

6See Roberts (1977) for a discussion of existence of majority voting equilibria over tax schedules. ‘Tax parameters are restricted to be non-negative in order to facilitate verification of the conditions specified in footnote 4 of Peck (1986). This assumption does not affect or qualitative conclusions.

R.M. Peck, Majority

58

voting and linear income tax

pi is the proportion of low-wage agents and p2 is the proportion of high-wage agents; pi and pL2sum to one. (Throughout this and the following section, the subscript 1 refers to low wage agents while the subscript 2 refers to high wage agents.) Constraint (2.1) ensures the feasibility of the selected tax. If high-wage agents are in the majority, that is p2 > f, then t = b = 0. This is because, under the assumptions of section 1, positive tax parameters redistribute income from high-wage agents to low-wage agents. Under the conditions specified in section 1, when low-wage agents are in the majority, that is, pi >$, the MVE tax rate, t,,,, will be positive: income is redistributed from high-wage agents to low-wage agents. This is a simple consequence of Lemma 2, stated in section 4. Before stating Theorem 1, we define the following notation. Suppose the set of feasible b can be expressed as a continuous function of t, b(t), that is,

iilPi(tYi(t,b(t))-b(t)) ~0, implicitly defines solution of

b as a continuous

function

of t. We define

t,,,

as the

max b(t)

subject

to

tE[O, 1-J voting tax schedule Finally, tMVE denotes the majority rate solution of our model is denoted as tNTU.

while the NTU

tax

Theorem 1. Suppose an NTU solution exists, an agent’s characteristics are as described in section 2, and pI and pLz are positive. If the set offeasible b can be expressed as a continuous function of t, b(t), so that q(t, b(t))= c(t), i= 1,2, is strictly concave and twice dtfferentiable

O=tM”E
over the interval [0, t,,,],

then

if/k>;.

The proof of Theorem 1 is in section 4. There are two initial to make. First, the proposition indicates that the comparison

observations of t,,, and

R.M. Peck, Majority voting and linear income tax

59

tMvE depends on the income distribution of the population, that is, on the proportion of the population that consists of high-wage agents and the proportion of the population consisting of low-wage agents. Second, the solution of our model is less preferred to the MVE tax solution by agents of the type which is in the numerical majority, while the converse is true for the type in the numerical minority. What accounts for this last observation? Part of the explanation lies in the following difference between our model and the majority voting model of tax selection. In our bargaining model, majority coalitions receive a larger payoff than minority coalitions; thus, an agent’s ability to pivot a coalition from a minority coalition to a majority coalition is important. Since each agent has one vote, all agents, in our model, have an equal ability to pivot a coalition. Thus, in our game, situations can be contemplated where an agent of the minority type alters an outcome by pivoting a coalition from a minority coalition to a majority coalition. The ability of a minority type to change the outcome in some situations means that a minority type agent has some power; this is taken into account by the NTU solution concept.8 In the majority voting model, however, an agent of the minority type can never affect the outcome; in a loose sense a minority type is powerless. Comparing the NTU tax schedule and the MVE tax schedule is comparing the solution of a game where the minority type agent has some power, to the outcome of a game where the minority type agent is powerless. Viewed in this way, it is not surprising that solutions of our model are closer to the preferred tax schedule of the minority type than is the MVE tax schedule. Alternatively, in our model the majority type has less power than in the majority voting model. This is reflected in our outcome. What can we conclude about actual institutions from these observations? The majority voting model is a depiction of referendum voting, while the model considered here more closely corresponds to a legislative process. A legislature is a forum which reduces transaction costs and thus facilitates bargaining9 and coalition formation. In addition, legislative voting is public, permitting prior agreements between coalitions and the members of coalitions to be monitored and presumably enforced. By contrast, in referendum voting the absence of a forum means that transactions costs are higher so that the possibilities for bargaining and coalition formation are reduced. Finally, since referendum balloting is secret, adherence to agreements cannot be monitored. Individuals may not have an incentive to vote according to 8Aumann and Kurz (1977a) point out the importance of an agent’s ability to shift minority coalitions to majority solutions in determining NTU solutions. See in particular Aumann and Kurz (1977a, pp. 1154-l 155). 91n assessing the realism of coalition threats note that, in equilibrium, threats are neither made nor executed; in equilibrium, the role of threats is only implicit.

60

R.M. Peck, Majority voting and linear income tax

prior agreement. Given this characterization of our model and the majority voting model, our analysis suggests that certain minority interests in redistributive taxation can be better served by legislative processes while majority interests are better served by direct referendum voting on redistributive taxes. While the simplicity of an economy with only two types of agents makes it an instructive special case, its relevance to understanding actual economies is obviously limited. What can we say when there are a finite number of types exceeding two? First, as in the two type case, the equilibrium linear tax schedule of our modified Aumann-Kurz model is progressive (see Proposition 1 of section 4). Second, in our model the power of the median voter to influence outcomes is reduced in the following way. The NTU solution concept assigns to each type an endogenously determined social welfare function weight. The weights are non-negative and sum to one. The equilibrium linear tax schedule maximizes, subject to a government budget constraint, an additive social welfare function using these weights. Proposition 2 of section 4 indicates that in our variant of the Aumann-Kurz model each equilibrium weight is strictly less than one.‘O In contrast, the majority voting model assigns the median voter a weight of one, while all other types are assigned a weight of zero. Therefore, in determining the equilibrium tax schedule, the median voter is given less weight than in the majority voting model.

3. An example A closer comparison of the solution of our modified Aumann-Kurz model, the optimal utilitarian and Rawlsian linear income tax schedule” and the majority voting tax, is provided by considering an example which illustrates Theorem 1. While the conclusions that can be drawn from this type of exercise are naturally limited, the computational results presented here are suggestive. In the example, the population of agents is a continuum loIn an economy with only two types of individuals, the median voter’s type and the type in the numerical majority coincide. In addition, since weights sum to one, the equilibrium weight of the type in the minority must be strictly greater than zero in such an economy. Hence, the outcome of our version of the Aumann-Kurz model improves the welfare of the minority type over that of the majority voting model in an economy with two types. With more than two types of agents, however, the outcome of our amended Aumann-Kurz model may not necessarily improve the welfare of each minority type. The following example confirms this. There are three types of agents: low-wage agents, w1 = 1, ‘upper middle class’ agents, wz = 1.80 and high-wage agents, w,=2; wi, i= 1,2,3, refers to the type’s wage rate. The population proportion of the agents’ types are p, =0.35, pc,=O.10 and ~,=0.55. Agents have CobbDouglas preferences, (3.1), with LX=:. The equilibrium tax rate is t=0.143. Upper middle class agents prefer the outcome of the majority voting model over this NTU solution. ‘I See Stern (1976) for references and a discussion of optimal Rawlsian and utilitarian income tax schedules.

R.M. Peck, Majority

voting and linear income tax Table

a=$;

61

1

W, = 1.00.

PI

1

hTU

hlVE

Lil

tRsvlsia”

IV,= 1.25

0.2 0.4 0.6 0.8

0.5270 0.5266 0.5266 0.5270

0.0154 0.0226 0.0227 0.0154

0.0000 0.0000 0.1548 0.0871

0.0076 0.0116 0.0119 0.0082

0.2540 0.2092 0.1548 0.0871

W* = 1.50

0.2 0.4 0.6 0.8

0.5457 0.5436 0.5440 0.5456

0.0469 0.0669 0.0672 0.0474

0.0000 O.OQOO 0.2540 0.1548

0.0240 0.0367 0.0381 0.0270

0.3765 0.3241 0.2540 0.1548

w, = 1.75

0.2 0.4 0.6 0.8

0.5578 0.5533 0.5531 0.5575

0.0814 0.1125 0.1136 0.0833

OKKKI 0.0000 0.3241 0.2092

0.0435 0.0664 0.0698 0.0509

0.4503 0.3982 0.3241 0.2092

w, = 2.0

0.2 0.4 0.6 0.8

0.5653 0.5581 0.5578 0.5645

0.1136 0.1531 0.1554 0.1180

0.0000 O.oooO 0.3765 0.2540

0.0637 0.0966 0.1024 0.0767

0.5000 0.4503 0.3765 0.2540

consisting of high- and low-wage agents. Agents have the same Douglas utility function ‘2 defined over leisure, 1 and consumption, c:

Cobb

U(l,C)=l=Ct-=, l,c,ZO. Initially, each agent has one unit of leisure which may be sold at exogenously specified wage rates. The low wage, wI, is equal to 1 and the high wage w2, is specified to be greater than 1 but less than or equal to 2. Table 1 sets out for selected parameter values of our example the NTU solution tax rate, t,,,, as well as tMvE, tuti,, and tla\vlsian, the majority voting, the optimal utilitarian, and optimal Rawlsian tax rates, respectively. In the table 1 computations, both the high-wage, w2, and the low-wage proportion of the population, pl, have been varied. For each parameter value of table 1, the conditions specified in Theorem 1 hold. This is verified in Peck (1983).13 12This utility function is not differentiable at the origin. Thus, the argument developed by Sheshinski (1972) which considers first-order conditions and is invoked in our Lemma 2 of section 4 is not strictly applicable. Peck (1983), however, verities directly that the conclusion of Lemma 2 holds in this instance. Hence, our other results remain applicable for this example. 13For this example, it is not possible to analytically determine the sign of the second derivative, c,,, i= 1,2, on [O, t,,,] and verify strict concavity. Instead, for each parameter value of table 1, the negative sign of the second derivatives are verified by the following numerical procedure. Since the third derivative, &,,, i= 1,2, has a computable upper bound Ki on [0, tmax], VI,,satisfies a Lipshitz condition. Choosing a suitable positive number d less that tmarand setting L=int[t,,,/dl+ 1, we verify that ~,,(t,)i -Kid, i=l,2, at L+l points, t,=nd, n=O,l,..., L. This implies vi,,, i= 1,2, is, for the specified parameter values, less than zero on [O, t&J. Thus, c. i= 1.2, is striictly concave on [O,t,,,] for the specified parameter values. Additional discussion is provided in Peck (1983).

J.P.E.-

C

62

R.M. Peck, Majority voting and linear income tax

Table 1 also includes a column labeled I, which can be explained briefly as follows. As we mentioned in section 2, t,,, maximizes the weighted sum of indirect utility functions, subject to a government budget constraint which are given as expressions (4.1) and (4.2), respectively, of section 4. The weights are endogenously determined along with tNTU as part of the solution of our model. A1 is the weight assigned at the solution to low-wage agents while l-1, is the weight assigned at the solution to high-wage agents. In interpreting table 1, it is useful to keep in mind that optimal utilitarian taxes are computed by setting 1, equal to f in expression (4.1) while optimal Rawlsian taxes are computed by setting A1 equal to one. Equilibrium majority voting tax rates are computed by setting R, equal to one in expression (4.1) when pL1exceeds f; otherwise, J., is set equal to zero. Comparing the tax rates, tNTU, t,,,, tuli, and tRawlsianin table 1, two clear patterns emerge.14 First, for all parameter values, the NTU tax rate solution of our model, tNTU, is greater than the optimal utilitarian linear tax rate, tuti,. This reflects the fact that 1, is greater than f; the solution of our model gives more weight to the utiiity of low-wage agents than optimal utilitarian tax calculations. As table 1 indicates, this means that the tax rate solution of our model lies between the optimal utilitarian and optimal Rawlsian tax rates. Second, for the computed values of the parameters we have:

when pL1>f. The right-hand side of this inequality follows from Theorem 1 which we have stated earlier. Thus, for this set of computations, when lowwage agents are in the majority, the tax solutions of our model are closer to the optimal linear tax schedule than are majority voting tax rates, tMVE. This suggests that, depending on the distribution of income, a tax selection process which facilitates bargaining and coalition formation, such as a legislature, may result in equilibrium tax schedules closer to optimal utilitarian tax schedules than referendum voting where such possibilities are absent.15 Finally, table 1 indicates that A1 increases as w1 rises. This observation can be interpreted by appealing to Aumann and Kurz’s (1977a) ‘fear of ruin’. Fear of ruin measures the willingness of an individual to choose, over a certain consumption level, a lottery with two outcomes: with small probability q, the individual’s consumption level is zero, and with probability 1 -q, he receives a small increase in consumption. If fear of ruin rises with 14For computations using alternative values of a, similar patterns emerged. 151ndeed, for the case when high-wage agents are in the majority the utilitarian social welfare function is higher when evaluated at tNTuthan when evaluated at t,,,, for all parameter values of table 1, except w2 = 1.25 and ~1 =0.20.

R.M. Peck, Majority voting and linear income tax

63

consumption, an individual should be willing to pay a higher tax to avoid the possibility of contiscatory taxation, that is, ruin. For CobbDouglas preferences, fear of ruin is an increasing linear function of consumption’6 Consequently, fear of ruin increases as the wage rate rises, since for Cobb-Douglas preferences consumption rises with the wage rate. This means that the threat of confiscatory taxation is a more potent threat vis-a-vis high-wage individuals. This, all else equal, increases the relative bargaining power of low-wage agents which is reflected in their higher equilibrium weight 2,. 4. Proof of Theorem 1 In this section we prove the results discussed in section 2. Our proofs use Theorem 1 of Peck (1986) which characterizes equilibrium linear tax schedules of our version of the Aumann-Kurz model. There are a finite number of types, where an individual’s type is determined by his wage rate. Types are indexed so that if i< j for two types i and j, then wi
max i f,b

~ini~(t, b)

(4.1)

i=l

subject to

(4.2) (b) The following

nj~(t,b)-

n equations are satisfied:

$ uiJ.iK(t,b)=G(tyj(t,b)-b),

j=l,...,n,

(4.3)

i=l

16Using u(LO)=O, it is straightforward

to show that an agent’s fear of ruin at (1,~) is given by

u(l, c)/du(l, c)/&. For CobbDouglas

preferences,

fear of ruin is c/( 1 -a).

Sec. Peck (1983) for further

details.

64

R.M. Peck, Majority voting and linear income tax

where 6 is the Lagrange multiplier for the constraint (4.2). Proof.

See Peck (1983).

In addition to Lemma 1, we need a result, stated here as Lemma 2, concerning optimal linear income tax schedules. Lemma 2 is a corollary of a result on optimal linear income tax schedules due to Sheshinski (1972). Lemma 2 [Corollary to Sheshinski’s Theorem (1972)]. Suppose the utility function, u, satisfies the conditions of section 1, and pi, i = 1,. . . , n, is positive. If the non-negative weights, summing to one, satisfy

then the t and b which solve the maximization problem specified in (a) of Lemma 1 are both positive. Proof: The conditions on utility functions specified in section 1 are the properties of strict concavity used by Sheshinski in the proof of his optimal linear income tax schedule theorem. The proof of Lemma 2 follows directly from the proof of Sheshinski’s theorem given by Sheshinski (1972). The following proposition indicates, under the conditions the NTU solutions are non-trivial, that is, if (t, b) is an NTU and b are positive.

specified, that solution, then t

Proposition I. Suppose a solution characterized by Lemma 1 exists and the conditions of Lemma 2 obtain. Then NTU solutions are non-trivial, that is, t and b are positive. Proof. Proof is by contradiction. Suppose t = b=O is an equilibrium schedule satisfying the conditions of Lemma 1. Then from (4.3): A, V,(O,O)= ... =2.1/,(0,0). By assumption (types V”(O,0). This implies:

are

ordered

(4.4) by wage

rate)

V,(O, 0) 5 Vz(O,0) 5 ... 5

A,~&~...~&. From Lemma 2, however, t, b must be positive contradiction completes the proof of Proposition 1. Proposition 2.

tax

(4.5) when

(4.5)

holds.

This

Suppose that a solution (t, b) characterized by Lemma 1 exists

R.M. Peck, Majority voting and linear income tax

65

and the conditions of Lemma 2 obtain. Then each of the equilibrium nonnegative weights lli, i = 1,. . . , n, described in Lemma 1, are strictly less than one.. Proof: Proof is by contradiction. Suppose that for an equilibrium tax schedule characterized by Lemma 1, the corresponding equilibrium weights are I,=1 for some k=l,..., n and Aj=O, j=l,..., n, j#k. This implies, from (4.3) that (t, b) must satisfy: (4.6) -pt&(t,b)=G(tyj(t,b)-b),

j= I,...,

n, jfk.

(4.7)

Under our assumptions about preferences, the left-hand side of (4.6) is positive. Note that if tyk(t,, b)-b is positive, then (t, b) could not be a solution of (4.1) with the weights we have assumed. Therefore, we conclude from (4.6) that 6 is negative and tyk(t, b) - b < 0.

(4.8)

Using these observations, ty,(t,b)-b>O, By assumption so that

we conclude

from (4.7) that

n, j#k.

j=l,...,

higher wage agents

(4.9)

do not work less than lower wage agents

Y15Y,S.~.5Y,.

(4.10)

Together (4.8), (4.9) and (4.10) imply k equals 1. Equality occurs in (4.10) only when agents withdraw from the labor market and earn zero income. (4.7) implies, though, that

so that each type’s income is zero. This means contradiction completes our proof of Proposition Proof of Theorem I.

Consider

the maximization

that (4.9) cannot 2.

hold. This

problem: (4.11)

where

1~ [O, l]

and

t 20.

Under

the hypothesis

of the theorem,

for each

66

R.M. Peck, Majority

voting and linear income tax

AE [0, 11, (4.11) has a unique solution, denoted t(A), which belongs to [0, t,,,]. First we note that t(0) = 0; the correspponding first-order condition in this case is

Since Pis strictly concave, for t>O we have: (4.12)

Similarly, for t(l), the necessary first-order condition is:

m(l))=o, where VI;,= aQat. Since vr is strictly concave, for t < t(l), we have: Qt)

> 0.

(4.13)

We now show that if t(A) is positive and IE(O, l), then t(A) is an increasing function. For a fixed AE(O, l), the first-order condition for (4.11) is:

Implicitly differentiating the first-order conditions, we have, using (4.12) and (4.13):

at(nya2

(4.14) where

i+azQat*, From Proposition IINTU corresponding

i= 1,2. 2, if an NTU solution exists, the comparison weight to t,,,, that is, 1 such that t(n)= tNTU, lies strictly

R.M. Peck, Majority voting and linear income tax

67

between 0 and 1. Let IZMvE denote the 1 associated with the MVE tax solution. If p1 >$, the MVE tax schedule is a solution of (4.1 l), for lhlvE equal to one. When p2 zf, the MVE tax schedule is a solution of (4.11) for &,vE equal to 0. Summarizing, we have: (4.15) O=AMvE<&ru,

if pL2>f.

From (4..14), (4.19, (4.16) and Proposition completes the proof of Theorem 1.

(4.16) 1, Theorem

1 follows.

This

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