Why the poor do not expropriate the rich: an old argument...

Journal of Public Economics 70 (1998) 399–424

Why the poor do not expropriate the rich: an oldargument in new garb

*John E. RoemerDepartment of Economics, University of California, Davis, CA 95616, USA

Received 31 December 1995; received in revised form 30 June 1997; accepted 23 March 1998

Abstract

We consider a political economy with two partisan parties; each party represents a givenconstituency of voters. If one party (Labour) represents poor voters and the other (ChristianDemocrats) rich voters, if a redistributive tax policy is the only issue, and if there are noincentive considerations, then in equilibrium the party representing the poor will propose atax rate of unity. If, however, there are two issues – tax policy and religion, for instance –then this is not generally the case. The analysis shows that, if a simple condition on thedistribution of voter preferences holds, then, as the salience of the non-economic issueincreases, the tax rate proposed by Labour in equilibrium will fall – possibly even to zero –even though a majority of the population may have an ideal tax rate of unity. 1998Elsevier Science S.A. All rights reserved.

Keywords: Political economy; Ideological parties; Political equilibrium

JEL classification: D72

1. The historical issue and a model preview

The framers of the US constitution extended suffrage only to (male) property-holders because they believed that, were the poor to be given the vote, they wouldsoon expropriate the rich. Property owners, it was believed, would behave‘responsibly.’ If all citizens have the vote, and median wealth is less than the mean(always true of actual wealth distributions), then a majority of voters (namely,

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400 J.E. Roemer / Journal of Public Economics 70 (1998) 399 –424

those whose wealth is less than the mean) should prefer a tax rate of unity, fullyredistributing all wealth to the mean.

Nevertheless, twentieth-century universal suffrage has not engendered theexpropriation of the rich through the tax system, and a variety of reasons havebeen offered in explanation, including the following. (1) Voters recognize thatthere would be adverse dynamic effects to expropriating the rich, who have scarceproductive talents which would cease to be supplied were their holders taxed tooharshly, and all would consequently suffer (trickle-down); (2) many voters whosewealth lies below the mean entertain the hope that they or their children willsomeday become richer than the mean, and they shun high tax rates for fear ofhurting their future selves or children; (3) even if there would be few dynamiceffects from high taxation, as described in (1), the rich convince the citizenry thatthere would be, with propaganda disseminated through the media, which theycontrol; (4) the citizenry believe that the rich person – and indeed everyone –deserves the wealth he /she receives, and hence high tax rates would be unethical.Marxists have called explanations (3) and (4) instances of ‘false consciousness.’Putterman (1997) has recently tried to assign degrees of importance to theexplanations here suggested, and some others.

In this article, I will propose another possible explanation for the non-expropria-tion of the rich in democracies, which depends upon there being party competitionon a policy space with two dimensions, the first being taxation, the second somenon-economic issue, such as slavery / integration, religion, nationalism, or ‘values.’The proposal I shall offer has nothing to do with incentives and trickle-down: werewealth simply manna from heaven, which fell unequally on the population, theargument I present would still hold.

The model behind the view that those with wealth less than the mean wouldvote for a tax rate of unity on wealth presupposes that political competition isunidimensional. But, indeed, political competition, in at least the US and Europe,is surely at least two dimensional. Poole and Rosenthal (1991) have shown thatroll call votes in the US Congress, going back to 1789, are best explained by atwo-dimensional model: knowing the position of congressmen on taxation and race(slavery before the Civil War and integration /civil rights after), one can explain85% of the variance in roll call votes, and adding a third dimension explains verylittle more. Laver and Hunt (1992) present empirical evidence that democraticpolitics are multi-dimensional in a set of over twenty countries. Somewhat moreschematically, Kitschelt (1994) argues that, in the main European countries,politics can be understood, in the past thirty years, as being two dimensional, overredistribution and a ‘communitarian’ dimension, whose poles he labels ‘au-thoritarian’ and ‘libertarian.’ The authoritarian voter wants more police, moredefense spending, illegalization of abortion, tough anti-drug legislation, the deathpenalty (in the US), and is pro-clerical. The libertarian voter wants the respectiveopposites, and is anti-clerical. Kitschelt argues that the ‘communitarian’ dimensionis quite orthogonal to the economic dimension: blue collar workers in manufactur-

J.E. Roemer / Journal of Public Economics 70 (1998) 399 –424 401

ing tend to be redistributionist and authoritarian, while some professional workersare anti-redistributionist and libertarian. On the other hand, many poor, minorityvoters are redistributionist and libertarian, while the ‘petty bourgeoisie’ areanti-redistributionist and authoritarian. Kalyvas (1996) and Przeworski andSprague (1986) together form a convincing argument that, in at least the period1880–1940, both religion and redistribution were important dimensions inEuropean politics.

Suppose, then, that voter preferences are defined over wealth and some non-wealth issue – I will call the second dimension ‘religion,’ for concreteness, fromnow on. The citizenry’s preferences are characterized by a joint probabilitydistribution over tax-religion policy space. Suppose there are two political partieswith policy preferences: one party represents constituents who are poor andanti-clerical (the Labour Party), and the other represents constituents who are richand pro-clerical (the Christian Democratic Party). It is important to understand thatthese parties are not Downsian – neither wishes to maximize the probability ofwinning the election per se, but rather to maximize its constituents’ expectedwelfare. (The distinction between these objectives will become clearer below.)

Given this political institution, here is a very rough intuition for why Labourmight not optimize in the electoral contest by proposing a tax rate of unity.Suppose that poor voters are generally anti-clerical, but there is a significantnumber of pro-clerical poor, and that richer citizens are generally pro-clerical, butthere is a significant number of anti-clerical voters among them. Indeed, there maybe a substantial number of voters, among the poor, who are so pro-clerical thatthey will not vote Labour even if Labour proposes a tax rate of unity, as long asthe Christian Democrats propose a more pro-clerical policy than Labour. Thus,Labour cannot necessarily win with high probability (more precisely, maximize theexpected welfare of its poor, anti-clerical constituents) if it proposes a tax rate ofunity, assuming it remains ‘principled’ on the religion issue. It may well be in theinterest of Labour’s constituents to propose a tax rate less than one, therebywinning the votes of some richer citizens who are quite anti-clerical. This, Marxmight well have said, is an instance of religion’s being the opiate of the masses –i.e., the poor, pro-clerical voters are acting against their ‘real’ interests. (If onethinks of the non-economic issue as race, which is perhaps the most appropriateone for the United States, one might paraphrase Marx by arguing that racism is theopiate of the masses.) But we are not here inquiring into why citizens have thesepreferences over a non-economic issue. The essential point is that, if voters caredeeply about some non-economic issue, and have widely disparate views on thatissue, it does not follow that all those whose wealth is less than the mean willnecessarily support a party which proposes a tax rate of unity.

To check whether this moderate redistributive policy of Labour can actually beoptimal – can be an equilibrium policy in the electoral game – will require amodel of party competition between Labour and the Christian Democrats – inparticular, a notion of equilibrium in the electoral contest between two parties,


1each representing constituents, when the policy space is two-dimensional. Thecentral technical problem facing the analyst is that the natural concept of politicalequilibrium – a Nash equilibrium in which each party plays a best response to theother’s policy – fails to exist, in pure strategies, with multi-dimensional issuespaces. There are two moves an analyst can usually make in such cases: either toconsider mixed strategy equilibria in the one-shot game, or to reconceive of thegame as one which takes place in stages, and then use some refinement of perfectNash equilibrium. The simplest example of the second option is Stackelbergequilibrium in a two-period game.

I do not believe we can reasonably think of parties playing mixed strategies, andso I reject the first option. I find the second option less objectionable, and I pursueit in Section 5 below.

But I believe that even the stage-game tack is a compromise with reality,because it can be argued that parties write their manifestos appoximatelycontemporaneously, and the manifestos determine their platforms. (Indeed, Budgeet al. (1993) argue, based on empirical analysis of ten countries, that parties’platforms adhere closely to their manifestos.) It is therefore advisable to find, if wecan, an equilibrium concept which works in the two-dimensional problem in asimultaneous move game between the parties. I introduce such a concept inSection 6 – the key is to alter the preferences of the parties from their usual form,based on modelling the intra-party struggle over policy, among its factions. Iname such Nash equilibria political unanimity Nash equilibria (PUNE).

My substantive question is: Is there a reasonable condition on the distribution ofvoter preferences (or traits), such that the equilibrium in the electoral contestbetween a Labour Party that represents a poor anti-clerical voter, and a ChristianDemocratic Party that represents a rich, clerical voter, entails Labour’s proposing atax rate which is significantly less than one?

What I discover, in Section 5 and Section 6, is such a condition, and moreover,that the same condition implies that, whether we model political competition asStackelberg or as PUNE, the desired result holds. In fact, under either con-ceptualization of political competition, if the religious issue is sufficiently salient,then the Labour Party will propose a zero tax rate in equilibrium.

In the process of answering the posed question, I will offer an answer to anotherquestion as well. Kitschelt has argued that the non-economic dimension (what hecalls the ‘communitarian’ issue) has increased in importance in western demo-cracies in the post-war period. Clearly, in a two-dimensional model, as thenon-economic issue becomes more salient for voters, we can expect bothcomponents of the equilibrium policies to change. Is there any reason to believethat, as the importance of the non-economic issue increases, the equilibrium tax

1A number of authors have studied electoral equilibrium between two parties, each of which haspolicy preferences (or represents constituents) when the policy space is one-dimensional: Wittman(1983); Calvert (1985); Alesina (1988), and Roemer (1994), (1997a), to name several. To myknowledge, there has been no analogous analysis of the two-dimensional model.


policies proposed by the Labour party should decrease, as opposed to increasing,or moving around non-monotonically? We can interpret the main results asanswering this question affirmatively, assuming that the key condition on thedistribution of voter traits holds.

Section 8 investigates, in a preliminary way, whether the key condition is trueof the US and British electorates, where we take the non-economic issue to be, inone case, racial attitudes, and in another, communitarian attitudes. Some tentativepredictions about US and British political behavior are drawn from the model.

2. Preliminaries

Here, I present the standard model of competition between partisan parties,applied to our context of a two-dimensional issue space.

Let the space of citizen traits be ! 5 W 3 R, with generic element (w, a), where¯W 5 [w,w ] is the set of wealth (or income) levels, and R is the set of religious

]views, taken to be the real number line. The utility function of a citizen with traits(w, a) over policies (t, z), where t is a uniform tax rate on wealth or income, and zis a religious position of the government, is given by v(t, z; w, a). The population

2is characterized by a probability distribution on !. There are two parties: Labour,or Left, represents a constituent with traits (w , a ) and the Christian DemocraticL L

Party, or Right, represents a constituent with traits (w , a ). Each party, i,R Ri i iproposes a policy pair t 5(t , z ). We suppose there is a stochastic element in these

1elections, which I will specify in Section 4, so that, given a pair of policies (t ,2 1 2

t ), there is only a probability that Left (Party 1) will win, denoted p(t , t ). Thefunction p is known to both parties. Then the pay-off functions of the Left andRight parties are:

1 1 2 1 2 1 1 2 2P (t , t ) 5 p(t , t )v(t ; w , a ) 1 (1 2 p(t , t ))v(t ; w , a )L L L L (2.1)2 1 2 1 2 1 1 2 2P (t , t ) 5 p(t , t )v(t ; w , a ) 1 (1 2 p(t , t ))v(t ; w , a ).R R R R

That is, the pay-off of a party at a policy pair is the expected utility of itsrepresentative constituent at that pair of policies.

It is generically the case that Nash equilibria in pure strategies, for the game in1 2 3which the payoff functions are P and P , do not exist.

2I take these parties to be historically given, just as in the Arrow–Debreu model, firms arehistorically given; I present no analysis which explains how these two particular parties have come tobe. I take the parties as representing particular voters, rather than coalitions of voters, as asimplification.

3In Roemer (1997a) I prove existence of Nash equilibrium for the one-dimensional electoral game,where parties face uncertainty and represent constituents. Even in that model, conditional payofffunctions are not quasi-concave. In the two dimensional model, however, the violation of quasi-concavity is so serious that, generically, pure strategy Nash equilibria do not exist.


3. Expropriation of the wealthy in a unidimensional contest

As a first (easy) exercise with the model of constituency representing parties, Ishow that, if the policy space is unidimensional (a single tax rate) and taxes arepurely redistributive, then a party representing a voter whose wealth is less thanthe mean will propose a tax rate of unity in Stackelberg equilibrium. Understand-ing this exercise should help the reader maintain his /her bearings in the morecomplicated two-dimensional problem to follow. Another reason to study this caseis that the analysis differs from that of the Downsian model, where parties have nopolicy preferences. Most readers will be familiar with the ‘median voter theorem’of the Downsian model.

Let W be an interval of real numbers, and let g(w) be a density on Wcharacterizing the society’s distribution of wealth. If t is a proportional tax rate onwealth, then per capita taxes collected will be t e wg(w)dw5tm, where m is thew

mean of g. Thus, ‘post-fisc’ wealth of a citizen with wealth w will be (12t)w1tm.Suppose von Neumann–Morgenstern preferences for wealth are universally risk-neutral: u(x)5x for all citizens. Then the indirect utility function of citizen w at taxrate t is

v(t; w) 5 (1 2 t)w 1 tm 5 w 1 t(m 2 w). (3.1)

Tax rates may be chosen in [0, 1].Now suppose that the distribution of voters, that is, of citizens who go to the

polls on election day, is g (w), where s is a random variable (state) uniformlys

distributed on [0, 1]. Alternatively, we may interpret the model as saying thateveryone votes, but that the distribution of preferences (i.e., of the parameter ‘w’)is subject to a stochastic element.

Denote the mean of g by m . Let G be the C.D.F. of g . We shall suppose thats s s s

G (m) is strictly decreasing in s. Interpretation: ‘s’ is the weather, with larger ‘s’s

meaning fouler weather. If the weather is foul, fewer poor people turn out to vote;thus G (m) is decreasing in s.s

1 2Let t .t be two tax policies. It is obvious from (3.1) that the set of citizens1 2 1 2who prefer t to t , denoted W(t , t ), is:

1 2W(t ,t ) 5 w , m . (3.2)h j

In state s the measure of this set is G (m). That is, G (m) is the fraction of voterss s1 2 1 2who vote for t over t in state s. Now t defeats t just in case this is a majority,

i.e., when

1]G (m) . . (3.3)s 2

By the italicized assumption of the previous paragraph, (3.3) is true just in cases,s*, where s* is defined by:


1]G (m) 5 . (3.4)s* 2

1Assuming that there is an s*[(0, 1) satisfying (3.4), then the probability that t2defeats t is just s*, since s is uniformly distributed on [0, 1].

Now suppose the L party represents a voter w ,m. That voter’s expectedL1 2 1 2 L 1utility, if party L proposes t and party R proposes t and t .t , is P (t ,

2 1 2 1t )5s*v(t ;w )1(12s*)v(t ; w ), since with probability s*, t wins, and withL L2probability 12s*, t wins. Similarly, if the R party represents a voter w .m, thenR

R 1 2 1 2its payoff function is P (t , t )5s*v(t ,w )1(12s*)v(t ; w ).R R

I next compute the Stackelberg equilibrium. Assume that L is the ‘incumbent’and R is the ‘challenger’, where by definition, the challenger moves first. AStackelberg equilibrium exists because the pay-off functions are continuous on the

2 2 2¯ ¯compact set [0, 1] . Let t be R’s equilibrium policy, and assume t ,1. Then LL 1 2 1 2¯ ¯obviously maximizes P (t , t ) at t 51. The same indeed holds if t 51.

1Alternatively, suppose R is the incumbent. Let t be any proposal; R maximizesR 2 1 1

P by choosing t 50. Then L’s problem is to choose t to maximize s*v(t ;1w )1(12s*)v(0; w ): the solution is t 51.L L

Hence, whether L is the incumbent or challenger, the equilibrium in the game ofparty competition involves the L party proposing a tax rate of unity. In sum:

Proposition 3.1. Let w ,m, let G (m) be strictly decreasing in s, and let u(x)5xL s

be the universal von Neumann–Morgenstern utility function. Suppose there exists1]s*[(0, 1) such that G (m)5 . Then, whether the party representing w is thes* L2

incumbent or challenger, the unique electoral equilibrium in the game of party1¯competition entails t 51.

Proposition 3.1 sets the stage for our study. Will two-dimensional politics causethe Left party to compromise the radical redistributive policy it advocates whenonly income is the issue?

4. The two-dimensional politico-economic environment

We now suppose there are two issues, taxation and ‘religion.’ A citizen withreligious view ‘a’ has a von Neumann–Morgenstern utility function u(x, z;

2a)5x2(a /2)(z2a) , where x is after-tax wealth and z is the government’sreligious policy. The positive number a shall be called the salience of thereligious issue. The joint distribution of wealth and religious views is representedby a density h(w, a) on !. The indirect utility function of voter (w, a) at policy (t,z), where t is a proportional tax rate, is

a 2]v(t, z; w, a) 5 (1 2 t)w 1 tm 2 (z 2 a) , (4.1)2


where m is mean wealth. From Eq. (4.1), we may compute that voter (w, a) preferspolicy t 5(t , z ) to t 5(t , z ), iff:L L L R R R

Dt(w 2 m)¯ ]]]z 1 . a if Dz . 0, (4.2a)

aDzDt

¯ ]]z 1 (w 2 m) , a if Dz , 0, (4.2b) (4.2)aDz5 6w , m if Dz 5 0 and Dt , 0, (4.2c)

w . m if Dz 5 0 and Dt . 0, (4.2d)

¯where Dz;z 2z , Dt;t 2t and z;((z 1z ) /2).R L R L L R

I will assume that h(w, a)5g(w)r(a, w), where g(w) is a density on W and, foreach w, r(a, w) is a density on R. The interpretation is that the wealth distributionof the population is given by g, and the distribution of religious views at wealth wis given by r(a, w). It shall be important that wealth and religious views are notindependently distributed.

The stochastic element in elections is as in Section 3. A random variable, ‘s,’which I shall assume is uniformly distributed on [0, 1], determines the distribution

4of traits among those who show up at the polls. I shall assume that, in state s, thedistribution of voters is given by:

h (w, a) 5 g (w)r(a, w); (4.3)s s

the interpretation is that s affects only the wealth distribution of the activeelectorate, but a representative sample of religious views shows up at each wealth

5level at the polls in every state of the world. Again, the interpretation may be thats is a measure of the weather’s foulness.

The coalition of voters W(t , t ) who prefer t to t is given by (4.2). Thus theL R L R

measure of voters who prefer t to t if, for instance, Dz.0, is, from (4.2a):L R

Dt(w2m )¯ ]]z1

aDz

H (W(t , t )) 5E E g (w)r(a, w) da dw, (4.4)s L R s

2`W

where H is the probability measure with density h .s s

4The reader might inquire: Why not analyze an easier problem, in which parties have no uncertaintyabout the distribution of voter preferences? Aside from the fact that it is more reasonable to assume thatparties are uncertain, it is easily observed that, in a two-dimensional political contest under certainty,equilibria usually fail to exist – Nash or Stackelberg. This is because there is usually no Condorcetwinner, and the best response correspondence is almost always empty.

5Indeed, nothing in what follows rests on the choice in (4.3) to place all the uncertainty on thedistribution of voter income. The assumption just simplifies some of the formulae to follow.


Let F(z, s) be the (cumulative) distribution function for religious views in states; that is,

z

F(z, s) 5E E g (w)r(a, w) da dw.s

2`W

We assume:

A1. For any z, F(z, s) is strictly decreasing in s.

A1 plays the role that the assumption that G (m) was decreasing in s played ins

Section 3. If the rich tend to be more religious than the poor, and the fraction ofrich voters increases with s (as when high s means foul weather on election day),

6then A1 will surely hold.1]Policy t defeats t in just those states s that H (W(t , t )). . (We needn’tL R s L R 2

1]worry about what happens if H (W(t , t ))5 , an event with zero probability.) Its L R 2

1]follows from A1 and (4.4) that H (W(t , t )). just in case s,s*(t , t ), wheres L R L R2

s*(t , t ) is defined uniquely by:L R

Dt(w2m )¯ ]]z1

aDz

1]E E g (w)r(a, w) da dw 5 . (4.5)s* 2

2`W

Thus, the probability that t defeats t is the probability of the event hs,s*j,1 2

which is s*(t , t ), since s is uniformly distributed on [0, 1].1 2

That is, letting p(t , t ) be the probability that t defeats t where z .z , weL R L R R L

have:

1]1 if H (W(t , t )) .1 L R 21

p(t , t ) 5 (4.6)]s*(t , t ) if H (W(t , t )) 5L R L R s* L R 25 1]0 if H ((t , t )) , .0 L R 2

More completely, we may write the function p(t , t ) for all possible cases,L R

using (4.2), as follows. Let l be Lebesgue (uniform) measure on [0, 1]. Then:

6Indeed, A1 can be relaxed, but at the cost of computational complexity.


Dt(w2m )¯ ]]z1

aDz

1]l su E E g (w)r(a, w) da dw . if Dz . 0,s15 622

2`W`

1]l su E E g (w)r(a, w) da dw . if Dz . 0,s 2 15 W 62Dt(w2m )

¯ ]]z1aDz

mp(t , t ) 5L R 1]l su E g (w) dw . if Dz 5 0 and Dt , 0,s 215 62

w] w]

1]l su E g (w) dw . if Dz 5 0 and Dt . 0,s15 622

m

1] if Dz 5 Dt 5 0. 2

It may be verified that, since g (w) is continuous in s and w and r(a, w) iss

continuous, the function p is continuous except on the subset V;hDz505Dtj ofthe domain T 3T, where T5[0, 1]3R is the issue space.

Let the Left party represent a voter (w , a ) and the Right party a voter (w ,L L R

a ), where w ,m ,w and a ,a . Recall that the parties’ pay-off functions areR L R L RR Lspecified by (2.1). It is easily verified that the functions P and P are

everywhere continuous on T 3T; the discontinuity of p on the subspace V of thedomain, defined above, turns out not to matter, since on V, v(t ; w, a)5v(t ; w, a)R L

for any (w, a).

5. Analysis of Stackelberg equilibrium

Now think of the salience parameter a in the utility function as variable, witha [[0, `]. It follows from the continuity of the payoff functions that, for any a,there is a Stackelberg equilibrium for the game & 5ka, (a , w ), (a , w ), g, r,a L L R R

7hg j, vl. We assume Left is the follower.s

We next assume:

7Although the strategy space for each player, [0, 1]3R, is not compact, one can show that the payoffL Rfunctions P (?, t ) and P (t , ?), are decreasing outside a compact set, and existence follows.R L


2A2. (a) In the game & (i.e., when u(x, z; w, a)52(z2a) ), there is a finite`

* *number of Stackelberg equilibria. For any such equilibrium (z , z ), weL R

* * * *have a ,z ,z , and 0,p(z , z ),1.L L R L R

*(b) For any equilibrium policy z in & , L’s best response is unique.R `

A2 is simply a non-degeneracy axiom about the one-dimensional game & . For`

the analysis of one-dimensional games, which justifies this claim, see Roemer(1997a).

LDenote the payoff to party L in the game & at the policy pair (t , t ) as P (t ,a L R L

t ; a) with the analogous notation for party R. Let Q(a) be the StackelbergR

equilibrium correspondence, which associates to any a the Stackelberg equilibriaof the game & . We have the following two facts:a

Proposition 5.1. Let A2(b) hold. Then Q(a) is upper-hemi-continuous at a 5`.

Proof: See Appendix A.Let (t (a), t (a)) be a continuum of equilibria for the games & , a ,`, whereL R a

t (a) 5 (t (a), z (a)).L L L

Proposition 5.2. Let A2(a) hold. For sufficiently large a :

(a) Dz(a).0 and Dz(a) is bounded away from 0;¯(b) z(a)2a is positive and bounded away from zero.L

Proof: See Appendix A.Our task is to find a condition under which, for sufficiently large a, at the

Stackelberg equilibria of & , t (a)50: that is, the Left will propose tax rates ofa L

zero! We next state that condition, and then our theorem.Let (z (`), z (`)) be any equilibrium in the game & , and Dz(`)5z (`)2L R ` R

z (`). Let s* be the probability of victory of party L at this equilibrium. DefineLs

¯ ] ¯ ¯ ¯ ¯the number m 5 , where s;e wg (w)r(z(`), w) dw, and r;e g (w)r(z(`),W s* W s*r¯w) dw. By definition, m is the mean wealth of the cohort of voters with religious

¯position z(`) in the state s*. Our condition is:

A3. For all Stackelberg equilibria in the game & , we have:`

(m 2 w )Dz(`)L¯ ]]]]]m 2 m . . (5.7)2(z (`) 2 a )L L

Theorem 5.1. Suppose A1, A2, and A3 hold. Then for all sufficiently large a, allStackelberg equilibria of the game & have t (a)50.a L


Proof: See Appendix A.

mDefinition 5.1. Let a (s) be the median religious view in state s. For any d .0, wemsay uncertainty is less than d iff there is a number n such that, for all s, a (s) lies

in a d interval around n.If uncertainty is sufficiently small, in the above sense, then Dz(`)¯0: in the

one-dimensional game & , both z (`) and z (`) will be very close to the median` L L

¯ ¯religious view in state s*, as will be their average z. Thus, m is approximatelyequal to the median wealth of the cohort of voters who have the median religious

¯view in state s*. But since Dz(`)¯0, (5.7) is true as long as m .m. Thus asufficient condition for the truth of (5.7) is that uncertainty be small and

the mean wealth of the cohort of voters with the median religious view in allstates is greater than mean wealth of the population. (*)

Thus, we have:

Corollary. If A1 and A2 hold, uncertainty is small, the mean wealth of the cohortof voters with the median religious view in all states is greater than mean wealthof the population, and the religious issue is sufficiently salient, then Labour willpropose a zero tax rate in all Stackelberg equilibria.

Although the analysis leading to this corollary is not the simplest, condition (*)is a simple one, which can be empirically tested, as I attempt in Section 8 below.We indeed need to know very little about the distribution of preferences to checkwhether (*) holds. The fact that, in the final analysis, we do not need to knowmuch about the joint distribution of (w, a) to decide whether increasing salience ofthe religious issue will lead to increasing economic conservatism of the Left partyhas been purchased by, inter alia, assuming a simple form for the utility function –that it be quasi-linear in income, and Euclidean in the religious dimension.Introducing a more complex utility function appears not to lead to a simplecondition like (*).

6. A new equilibrium concept based on internal party struggle

As I explained in Section 1, pure-strategy Nash equilibria do not generally existin the two-dimensional game between parties with pay-off functions specified in(2.1). I here introduce a new specification of party preferences, under whichpure-strategy Nash equilibria will exist.

The idea is based upon European party history. We will assume that there arethree factions in each party: reformists, opportunists, and militants. Reformists

L Rhave the preferences given by P and P : they wish to maximize the expectedutility of the party’s constituents. Opportunists have preferences given by p (for


Left), and 12p (for Right): they wish only to maximize the probability of victory.Opportunists are the characters who dominate Anthony Downs’s (Downs, 1957)view of political competition. Finally, militants are not concerned at all withwinning the election: Left’s militants which to maximize v and the Right’sL

militants wish to maximize v . Thus, the militants are interested in advertising theR

preferences of their constituent; they view elections as a pulpit for announcing andpropagating the party’s line. In this section, I shall assume that each party containsall three factions, and that each faction has the power to veto any proposal for theparty’s platform. The equilibrium concept that will follow from this assumption Icall party unanimity Nash equilibrium (PUNE).

I shall not here attempt to justify the historical basis of this approach, which I doin Roemer (1997b).

9Definition 6.1. We say that Left agrees to deviate from t [T to t [T at t iff allL L R

9factions in Left weakly prefer (t , t ) to (t , t ) and at least one faction strictlyL R L R

prefers the former to the latter.

Definition 6.2. (t , t ) is a party unanimity Nash equilibrium iff there is noL R

platform at t to which Left agrees to deviate and there is no platform at t toR L

which Right agrees to deviate.Formally, we can define a party’s (incomplete) preferences as the intersection of

the preference relations of its three factions; that is, for Left,

L9 9 9 9 9 9(t , t )K (t , t ) if and only if p(t , t ) $ p(t , t ) and P (t , t )L R L L R L R L R L R

L 9$ P (t , t ) and v (t ) $ v (t ),L R L L L L

with a similar definition for K . Then we can say that (t , t ) is a PUNE iff it is aR L P

Nash equilibrium with respect to the incomplete preferences K and K . DenoteL R

*the game with these preferences (of the parties) & , to be distinguished from thea

(Stackelberg) game & studied in the last section, where the players have differenta

preferences.In this section, I shall show that if condition (*) holds, then for large a,

*t (a)50 in all non-trivial PUNE of & , where a non-trivial equilibrium is one inL a

which neither party wins with probability one. Thus, our central result shall berobust with respect to a change in the equilibrium concept from Stackelberg toparty-unanimity-Nash. Finally, I will show that non-trivial PUNE exist in thesegames.

Before beginning, it is useful to observe that, indeed, we may ignore thereformists in the definition of PUNE. That is:

Lemma 6.1. A pair of platforms constitutes a PUNE iff the militants andopportunists, in both parties, do not agree to deviate.


Proof: Follows quickly from the definitions. (Or see Roemer (1997b).)To be precise, we must restate the critical axiom for the new equilibrium

concept:

*A3* For all non-trivial PUNE in the game & , we have:`

(m 2 w )Dz(`)L¯ ]]]]]m 2 m . . (6.7)2(z (`) 2 a )L L

Define Q*(a) as the PUNE correspondence; that is, Q*(a) is the set of all*PUNE for the game & .a

Our line of argument is as follows.

Lemma 6.2. Q* is upper-hemi-continuous at a 5`.Let h((t (a), z (a)), (t (a), z (a))j be a sequence of non-trivial PUNE for aL L R R

tending to infinity, which, by Lemma 6.2, converge to a PUNE, (z (`), z (`)) ofL R

* *& . (We can ignore the tax rates in & .)` `

Proposition 6.1. If A3* holds, then for all sufficiently large a, t (a)50.L

Proposition 6.2. Let ´.0. If uncertainty is sufficiently small, then z (`)2z (`)5R L

Dz(`),´.Now suppose that uncertainty is small and condition (*) holds. Then it follows

from Prop. 6.2 that (6.7) holds, and hence it follows from Proposition 6.1 thatt (a)50 for large a. This proves:L

Theorem 6.1. If uncertainty is small and condition (*) holds, and if a issufficiently large, then in any non-trivial PUNE, t (a)50.L

I shall not provide the proof of Lemma 6.2, but the proofs of Propositions 6.1and 6.2 are found in the appendix.

Finally, we prove that Theorem 6.1 is not vacuous:

Theorem 6.2. Let uncertainty be small and condition (*) hold. Then, for all large*a, non-trivial PUNE exist in the game & .a

Proof: See Appendix A.Before concluding this section, a further remark on the notion of PUNE is in

order. One might object to the assumption that each faction in the party has vetopower over the policy. Suppose, then, that each party works out a method ofinner-party bargaining, or compromise. The point is that, whatever policies the twoparties reach as a consequence of inner-party bargaining, that policy allocation is aPUNE. For it surely must be the case that, at that allocation, no parties’ factionswould agree to deviate to another policy. Therefore, any such (bargaining)


equilibrium inherits all the properties of a PUNE – in particular, if condition (*)holds, then for large a, Left proposes a tax rate of zero. What is called intoquestion, when we switch to a ‘bargaining’ concept of inner-party struggle, isexistence of equilibrium.

7. Further discussion

I have shown that, if there is a non-economic issue which is sufficientlyimportant to voters, if parties represent constitutents who have preferences overtaxation and the non-economic issue, and if assumption (*) holds and uncertaintyis small, then in two kinds of electoral equilibrium, the tax policy of the Left partywill be significantly less than unity. (Section 3 showed that when a 50, the Leftalways proposes a tax rate of one: so as a increases, the tax rate eventuallydecreases towards zero.) The result is striking because it may simultaneously betrue that the ideal tax rate for the majority of the population, in all states, is unity!This ‘paradox’ is due to the structure of political competition, which is partycompetition, in which the different dimensions of policy cannot be unbundled.While the ideal tax rate for the majority of a population may be unity, that tax ratewill not be observed in equilibrium, even when one party represents (a sub-constituency of) that expropriation-desiring majority.

I will try to give some intuition for how condition (*) drives our result. If a islarge, then the game & is essentially a one-dimensional game over religiousa

policy. If uncertainty is small, then the median religious view varies little acrossstates. In an equilibrium where both parties win with positive probability, bothparties must therefore play a religious policy close to that approximately constantmedian religious view. We may even say that the cohort of the population whohold approximately the median religious view are the decisive voters. But if thatcohort’s wealth is greater than mean population wealth, as condition (*) states,then their ideal tax rate is zero. Competition forces Left (and Right) to propose atax rate of zero, to attract the decisive cohort. If you object to some slippage in this‘argument,’ then read the proofs.

We may apply exactly the same analysis to determine when Right parties (whorepresent rich, religious voters) will, in Stackelberg or PUNE equilibrium, proposehigh tax rates. (In the Stackelberg case, we must assume that Right is thefollower.) The key condition now turns out to be:

(w 2 m)Dz(`)R¯ ]]]]]m 2 m , . (7.1)2(z (`) 2 a )R R

Note that the r.h.s. of (7.1) is negative, so (7.1) will be satisfied if:


uncertainty is small, and for all states, the mean wealth of the cohort ofvoters with the median religious view is less than mean population wealth.

Under these conditions, when a is sufficiently large, Right will propose a tax rateof unity in either kind of equilibrium.

88. Empirical tests

For the United States, I suggest that ‘race’ is the prominent non-economic issue.Using the National Election Surveys, we computed whether the average income ofvoters who hold the median view on the race issue is greater than mean populationincome – to see whether condition (*) holds. Among the many questions asked inthese Surveys is a ‘thermometer’ question on ‘Blacks.’ Respondents are asked tochoose a number between 0 and 100 telling how ‘warmly’ or ‘favorably’ they feelabout the issue. 100 is the warmest possible. In the question we used, the issue wassimply stated as ‘Blacks.’ The results, for 1974–1994, are presented in Table 1.

Not all respondents in the NES are voters; in particular, the respondent is askedif he voted. We took the mean population income (m) to be the mean reportedincome of all respondents in the survey (col. 1 of Table 1). Col. 2 of the tablegives the mean income of voters (which we do not use in our statistical test). Col.4 gives the median thermometer value of all voter responses on the Black issue,

Table 1Black issue (1974–1994)

Year Mean Mean Mean (*) Value of Std. Dev. Std. Dev. Std. Dev.Income Income Income black issue for Income Income IncomePopulation Voters Cohort median voter Population Voters Cohort

1974 $12 730 $14 296 $15 043 65.07 $9745 $10 104 $10 5721976 $14 628 $15 929 $17 964 61.08 $10 719 $11 051 $11 7741980 $20 955 $22 729 $23 357 64.46 $15.041 $15 236 $15 7921982 $22 734 $24 482 $25 054 63.7 $15 959 $15 905 $12 9371984 $25 402 $27 911 $29 458 65.01 $18 806 $19 375 $19 7151986 $28 412 $31 896 $33 089 67.37 $20 439 $21 143 $20 8601988 $29 927 $33 828 $37 597 62.92 $22 350 $23 170 $24 1571990 $31 262 $35 977 $38 233 71.31 $23 980 $24 575 $24 8101992 $35 751 $39 567 $40 277 65.57 $26 836 $27 209 $26 4791994 $37 727 $43 263 $46 087 64.33 $27 864 $28 713 $31 733

(*) Range [0, 100], where the higher the number the more favorable the agent feels toward blacks’issues.

8I thank research assistants Woojin Lee and Humberto Gonzalez–Llavador for carrying out the dataanalysis in this section.


¯and col. 3 gives the mean income of this median cohort (m ). It is evident thatm .m : in fact, using a central-limit-theorem test, we computed that for the last

¯four election years (1988 on), m .m at the 0.999 significance level.It is not surprising (compare columns 1 and 2 of Table 1) that the mean income

of voters is greater than population mean income. But it is not this fact alonewhich explains our result, since we note that, in every year, the mean income ofthe median cohort of voters is greater than the mean income of voters, as well.

Regarding the salience of non-economic issues for the American electorate,George Gallup (of the Gallup Poll) says: ‘‘[Americans] are more concerned aboutthe state of morality and ethics in their nation than at any time in the six decades

9of scientific polling.’’ We attempted to test whether the salience of non-economicissues has been increasing, as follows. The National Election Survey asks eachrespondent to list the three most important issues, in his view. There are hundredsof acceptable answers to this question, coded in the NES. We coded these issues as‘economic issues,’ ‘values issues,’ or ‘other issues,’ and defined the salience ofvalues, for a cohort, as the number of values issued mentioned divided by thenumber of economic issues mentioned in the answer to this question. Table 2 givesthe salience rate, so computed, for the election years 1974–1994.

Evidently, Gallup’s view is borne out: never, in the past twenty years, has thesalience of values issues been higher than in 1994. In fact, the salience of valuesissues has been rising steadily during this period, in the United States, except for adecline in the period 1988–1992, roughly corresponding to a recession.

The British Social Attitudes Survey is the annual counterpart, in the UK, of theUS General Social Survey. In 1993, the BSAS asked a series of questions designedto ascertain the respondents’ views on the authoritarian–libertarian dimension. Therespondent was asked to mark his degree of agreement, on a scale of one to five

Table 2Salience of values issue, US electorate

1974 0.3121976 0.3311980 0.3971982 0.5391984 0.6051986 0.9211988 1.2361990 0.9291992 0.5951994 2.109

9The Economist, November 11–17, 1995, p. 29.


(strongly agree, agree, neither agree nor disagree, disagree, strongly disagree) withthe following statements:

a) It is right that young people should question traditional British values;c) British courts generally give sentences that are too harsh;e) The death penalty is never an appropriate penalty;g) Schools should teach children to question authority;i) There are times when people should follow their conscience, even if it means

breaking the law.It is important to note that we do not have information on voters in the British

data, only on the general population.We coded the answers one to five, and assigned each respondent an average

value, including in the sample only respondents who answered at least three of thefive questions. We then computed the median cohort, whose response was 2.67,lying between ‘agree’ and ‘neither agree nor disagree’. We computed the meanincome of the median cohort, and the mean income of the sample.

In Table 3, I report the statistical features of the answers to these questions thatare relevant for us. This time, the mean income of the median cohort appears to beless than mean income of the sample; the central-limit-theorem test says that thisorder of the two means is correct with probability 0.78 – not a very highconfidence level. One must note, however, that we do not have the mean incomeof the median voter cohort, which may be greater than mean population income.

¯If, however, we assume that ‘m2m ,0’ is true, then, from the discussion ofSection 7, the relevant hypothesis is not about the behavior of the Labour Party butrather the Conservative Party. The inference is that, with probability 0.78,inequality (7.1) holds, and the model, in that case, implies that a ConservativeParty in power would move to the left in its economic policy as the salience of theauthoritarian–libertarian issue increases.

From these tests, the model suggests that, if the salience of the non-economicissue of race increases in the United States, Democrats would propose increasinglyconservative tax policies, while we have no reason to believe that Republicanswould propose increasingly liberal tax policy. We have somewhat weaker reason tobelieve that, as the salience of the authoritarian–libertarian issue increases inBritain, the Conservative Party would move to the left in its economic policies.

Table 3British Social Attitudes Survey, 1993 Authoritarian vs. Libertarian preferences

Sample size 2100Mean income of sample (m) 15 194Median view on issue 2.667Mean income of median cohort (m) 14 691Size of median cohort (n) 219S.D. of median cohort’s income (b ) 9777Prhm,mj 0.78


9. Concluding remark

We may finally reflect upon a view, which has often been held in Left circles,that the Right deliberately ‘creates’ a certain non-economic issue – or tries toincrease the salience of some such issue for voters – as a means of pullingworking-class voters away from Left parties, thereby driving economic policies tothe right. In this view, the Right party pretends to care about the ‘religious’ issue,while in fact being interested only in lowering tax rates (or rolling backnationalization, etc.). Right may implement this masquerade by attracting politicalcandidates who do, indeed, feel strongly on the ‘religious’ issue.

Our analysis certainly indicates that this can be a strategy to achieve moreconservative economic policy. Of course, Left can play the same game, and

¯attempt to increase the salience of an issue for which ‘m ,m’ holds, thus forcingRight to move to the left on economic policy. Our analysis, then, suggests a newway to read the history of the development of non-economic issues in electoralpolitics. Have Left and Right parties ‘chosen’ which non-economic issues toemphasize (i.e., increase the salience of) with an eye towards pushing electoralequilibrium on the economic dimension in a desired direction?

Whatever the verdict on that historical issue, our analysis suggests thatemerging new dimensions of citizen concern, which are addressed in competitive,party politics, can change the positions of parties on classical issues in surprisingways.

Acknowledgements

The idea for this paper was, I think, hatched in a discussion with IgnacioOrtuno–Ortin, during my visit to the University of Alicante in 1994. I am alsograteful to him for finding errors in an earlier version. I wish, as well, to thankanonymous referees for their advice, and Woojin Lee and Humberto Gonzalez–Llavador for expert research assistance.

Appendix A

Proof of Proposition 5.1: Let (t (a), t (a)) be a sequence of StackelbergL R

equilibria in the games & , and let z (a) and z (a) converge to z (`) and z (`),a L R L R

respectively. Suppose, contrary to the claim, that (z (`), z (`)) is not aL R10Stackelberg equilibrium in & . A standard continuity argument establishes that`

10In the game & the tax policies are irrelevant, so we do not refer to them in describing equilibria of`

& .`


z (`) must be a best response to z (`); so it must therefore be that there exists anL R

˜ ˜ ˜ ˜equilibrium pair (z , z ) such that z is a best response to z andL R L R

R R˜ ˜P (z , z ; `) . P (z (`), z (`); `).L R L R

ˆ ˜ˆ ˆLet (t (a), z (a)) be L’s best response to (t (a), z ) in & . Then z (`);limL L R R a L aR ˆ˜ ˜ˆ ˆz (a) is a best response to z in & . By A2(b), z (`)5z . Hence P ((t (a),L R ` L L L

R˜ ˜ ˜z (a)), (t (a), z )); a) approaches P (z , z ; `) as a approaches `. In particular,L R R L R

by the above inequality, for large a :

R Rˆ ˜ˆP ((t (a), z(a)), (t (a), z ); a) . P ((t (a), z (a)), (t (a), z (a)); a).L R R L L R R

This contradicts the fact that ((t (a), z (a)), (t (a), z (a))) is a StackelbergL L R R

equilibrium in & , which establishes the claim. ja

Proof of Proposition 5.2: By the upper-hemi-continuity of the equilibriumcorrespondence Q(a) at `, any converging subsequence of the continuum (t (a),L

t (a)) converges to an equilibrium of & . The claims follow immediately fromR `

A2(a). j

Proof of Theorem 5.1: Suppose to the contrary: that for a sequence of a’s tendingto infinity, there is a Stackelberg equilibrium of & in which t (a).0. We knowa L

that Dz(a).0 by Proposition 5.2; hence, for large a, p(t (a), t (a)) is indeedL R

given by (4.6), and hence, either p(t (a), t (a))5s*(t (a), t (a)), where s* isL R L R

defined by (4.5), or p(t (a), t (a))[h0, 1j. But by A2(a), since for all equilibriaL R

of the game & , p [⁄ h0, 1j, it follows that for sufficiently large a, p(t (a),` L

t (a))[⁄ h0, 1j, and therefore p(t (a), t (a))5s*(t (a), t (a)).R L R L R

Differentiating (4.5) implicitly w.r.t. t , we may write:L

Dt w 2 m¯ ]] ]]S DE g (w)r z 1 (w 2 m), w dws* aDz aDz

≠s* W] ]]]]]]]]]]]]5 , (Ap.1)Dt≠t ¯ ]z1 (w2m )L

aDz

≠gs*]]E E (w)r(a, w) da dw≠s

2`W

as long as the denominator in (Ap.1) does not vanish, where I have omitted the¯argument ‘a’ on the variables z, Dt, and Dz. But axiom A1 tells us that the

zexpression e e (≠g /≠s)(w)r(a, w) da dw,0, since this expression is just thew 2` s

derivative of F(z, s) w.r.t. s, and so the denominator of (Ap.1) does not vanish.We assume that L is the incumbent and R is the challenger (i.e., R moves first).

LSince s* is differentiable for large a, so is P (t , t ; a) differentiable at (t ,L R L

t )5(t (a), t (a)), for large a. Since t (a) is a best response to t (a), it thereforeR L R L RLfollows that (≠P /≠z ) (t (a), t (a), a)50, since z (a) is an interior solution (asL L R L


the domain of possible z ’s is the real line). This first-order condition can beL

solved to yield:

as*(z 2 a )≠s* L L] ]]]]]]]]5 . (Ap.2)¯≠z Dt(w 2 m) 1 aDz(z 2 a )L L L

LSimilarly, it follows that (≠P /≠t ) (t (a), t (a); a)$0, since by hypothesisL L R

t (a).0 for all (finite) a. The just stated inequality can be solved to yield:L

s*(w 2 m)≠s* L] ]]]]]]]]$ , (Ap.3)¯≠t Dt(w 2 m) 1 aDz(z 2 a )L L L

an expression whose derivation uses the fact that the denominator of (Ap.3) is11positive, which follows from Proposition 5.2.

Next, differentiating (4.5) w.r.t. z yields:L

Dt 1 Dt(w 2 m)¯ ]] ] ]]]S D2E g (w)r z 1 (w 2 m), w 1 dws* S 2 DaDz 2 a(Dz)≠s* W

] ]]]]]]]]]]]]]]]]5 . (Ap.4)Dt≠z ¯ ]z1 (w2m )LaDz

≠gs*]]E E (w)r(a, w) da dw≠s

2`W

Let the (common) denominator in the fractions on the r.h.s. of (Ap.4) and (Ap.1)be denoted ‘D.’ Using (Ap.4) and (Ap.2), we can solve for D, eliminating(≠s*/≠z ); substituting the expression for D into (Ap.1) yields:L

Dt 1 Dt(w 2 m)¯ ]] ] ]]]S DE g (w)r z 1 (w 2 m), w 1 dws* S 2 DaDz 2≠s* a(Dz)

] ]]]]]]]]]]]]]]]5 . (Ap.5)Dt≠t ¯ ]z1 (w2m )LaDz

≠g (w)s*]]E E r(a, w) da dw

≠s2`

In turn, (Ap.5) and (Ap.3) imply:

Dt w 2 m¯ ]] ]]S DE g (w)r z 1 (w 2 m), w a(z 2 a ) dws* L LaDz aDz

W]]]]]]]]]]]]]]]]$ 2 (m 2 w ),L

Dt 1 Dt(w 2 m)¯ ]] ] ]]]S DE g (w)r z 1 (w 2 m), w 2 2 dws* S 2 DaDz 2 a(Dz)

W

11Establishing the positivity of the denominator of (Ap.3) also uses the fact that Dt#0 atequilibrium, which is not proved here, though it is true.


or

Dt¯ ]]S D(z 2 a )aDz E g (w)r z 1 (w 2 m), w (w 2 m) dwL L s* aDz

W]]]]]]]]]]]]]]]]]]

2Dt 2 a(Dz)S D¯ ]] ]]]S DE g (w)r z 1 (w 2 m), w 2 Dt(w 2 m) dws* aDz 2

W

$ 2 (m 2 w ). (Ap.6)L

Letting a →`, (Ap.6) becomes, in the limit:

¯2(z (`) 2 a ) E g (w)r(z(`), w)(w 2 m) dwL L s*

W]]]]]]]]]]]]] # (m 2 w ), (Ap.7)L

¯Dz(`) E g (w)r(z(`), w) dws*

W

where we use the fact that Dz(a) is bounded away from zero (Proposition 5.2) soaDz(a)→`.

¯ ¯ ¯Using the definitions of r, s and m provided in the text, we can write thenegation of (Ap.7) as

(m 2 w )Dz(`)L¯ ]]]]]m 2 m . , (Ap.8)2(z (`) 2 a )L L

which is precisely condition (5.7). Hence, by A3, (Ap.7) does not hold, whichcontradicts the original suppostion – that there is a sequence of equilibria at whicht (a).0, and the theorem is proved. jL

Proof of Proposition 6.1: Suppose there is a sequence of a’s tending to infinity,with t (a).0. We shall show that, at each sufficiently large a, there is a directionL

in which Left’s militants and opportunists will agree to deviate, which, by Lemma6.1, contradicts the assumption that we are at a PUNE.

To be specific, we shall show the existence, for large a, of a direction (21,d(a)) such that:

=v ? (21, d(a)) . 0, (Ap.9)L

and

= s* ? (21, d(a)) . 0, (Ap.10)L

which means that both the militants and opportunists in Left can increase theirutility by moving in the direction (21, d(a)). Recall that the components of thegradient = s*5((≠s*/≠t ), (≠s*/≠z )) are given by equations (Ap.1) and (Ap.4).L L L


Since t (a).0, the direction (21, d(a)) is feasible at t , for any number d(a).L L

(Ap.9) expands to:

w 2 m 2 d(a)a(z (a) 2 a ) . 0,L L L

which we rewrite as:

w 2 mL]]]]d(a) , . (Ap.99)a(z (a) 2 a )L L

For the moment, let us choose d(a)5((w 2m) /a(z (a)2a )). Substituting thisL L L

value into the inequality (Ap.10), using the formulae for the components of = s*,L

and taking the limit of the derived expression as a goes to infinity, we maycompute that (Ap.10) holds for large a if:

w 2 mw 2 m L¯ ]] ]]]] ¯E g (w)r(z, w) dw 1 E g (w)r(z, w) dw . 0.s* s*Dz(`) 2(z (`) 2 a )L L

(Ap.11)

But (Ap.11) is equivalent to inequality (5.7): hence A3* implies the truth of (Ap.1210), for this choice of d(a).

It follows that if we choose d(a)5(w 2m) /(a(z (a)2a ))2´, for ´ suffi-L L L

ciently small, then both (Ap.9) and (Ap.10) hold, which is the desired contradic-tion. j

*Proof of Proposition 6.2: The game & is played on a one-dimensional strategy`

(issue) space, and the non-trivial PUNE for this game are easy to characterize.mConsider the interval defined by the values a (s), for s[[0, 1]: if uncertainty is

* *sufficiently small, then this interval becomes arbitrarily small. If (z , z ) is aL R

* *non-trivial PUNE, then z and z must both lie in the interior of this interval,L R

which proves the claim.

* * *Proof of Theorem 6.2: Let (z , z ) be non-trivial PUNE in the game & . I shallL R `

* * *argue that ((0, z ), (0, z )) is a non-trivial PUNE in the game & , for large a. It isL R `

immediate that, for large a, neither party wins with probability one at this policypair, which establishes the claim of non-triviality.

Suppose to the contrary, that for a sequence of a’s approaching infinity, ((0,* * *z ), (0, z )) is not a PUNE in & . There are two possibilities.L R a

Case 1. There is a subsequence of a’s such that Left’s militant and opportunist* *factions would agree to deviate from (0, z ) in the game & .L a

12 *There is a detail here. A3* only applies if the limit PUNE in the & game is non-trivial, and the`

limit of non-trivial PUNE could be a trivial PUNE. Nevertheless, we can deduce that inequality (5.7)will hold for such a limit PUNE, even if it is trivial.


Let, then, (t (a), z (a)) be a policy that Left’s militants and opportunists wouldL L

* *agree to deviate to from (0, z ) and that is a best response by Left to (0, z ), inL R

* *the game & . It follows that z (a) must be close to z for large a, or else thea L R

probability of Left victory would be zero, contradicting the supposition that Left’s*opportunists agreed to deviate to this point from (0, z ). It therefore follows, byL

condition (*), that, for large a :

*(z 2 z (a))(m 2 w )R L L¯ ]]]]]]m 2 m . . (Ap.12)2(z (a) 2 a )L L

But (Ap.12) plays exactly the role of (5.7): we can invoke the argument in theproof of Proposition 6.1 to conclude that, for large a, t (a)50, in any Left bestL

*response to (0, z ).R

* *Hence Left agrees to deviate to (0, z (a)) from (0, z ) when facing (0, z ) inL L R

*& . But, since both tax rates are zero, this means that Left would agree to deviatea

* * *from (0, z ) to (0, z (a)) in the game & when facing (0, z ) – which isL L ` R

* * *impossible, since ((0, z ), (0, z )) is a PUNE in & . The contradiction shows that,L R `

* * *for large a, (0, z ) is indeed a best response by Left to (0, z ) in & .L R a

*Case 2. There is a sub-sequence of a’s such that (0, z ) is not a Right bestR

* *reponse to (0, z ) in & .L a

* *Let, then, (t (a), z (a)) be a Right best response in & to (0, z ) to whichR R a L

*Right’s militants and opportunitists agrees to deviate, from (0, z ). We shallR

similarly prove that, for large a, it must be that t (a)50, and a contradiction willR

then follow, just as above. This time, however, we cannot invoke the argument ofProposition 6.1, for we did not study Right’s strategy in that proof. We thereforemust prove independently that t (a)50.R

We know, by condition (*), that, for large a :

*(z (a) 2 z )(m 2 w )R L R¯ ]]]]]]m 2 m . , (Ap.13)2(z (a) 2 a )R R

*because if z (a)2z did not become small, then Left would eventually win withR L

probability one, and (t (a), z (a)) would not be an attractive deviation to Right’sR R

* *militants from (0, z ) in & .R a

Suppose t (a).0. We shall construct a direction (21, d(a)) such thatR

= s* ? (21, d(a)) , 0 (Ap.14a)R

and

=v ? (21, d(a)) . 0, (Ap.14b)R

*where the gradients are evaluated at ((t (a), z (a)), (0, z )), which means thatR R L

*Right would agree to deviate in that direction, in & .a


By differentiating (4.5), we compute that the components of the gradient = s*R

are given by:

Dt w 2 m¯ ]] ]]S DE g (w)r z 1 (w 2 m), w dws* aDz aDz

≠s* W] ]]]]]]]]]]]]5 ,≠t DR

and

Dt 1 Dt(w 2 m)¯ ]] ] ]]]S D2E g (w)r z 1 (w 2 m), w 2 dwS Ds* 2aDz 2 aDz≠s* W

] ]]]]]]]]]]]]]]]]5 ,≠z DR

where, to recall, D is the denominator in Eq. (Ap.1) or (Ap.4). Using theseformulae to expand (Ap.14a), and letting a tend to infinity, we observe that(Ap.14a) holds for large a if and only if:

¯2(m 2 m )]]]d(a) , , (Ap.15)

aDz

*recalling here that Dz5z (a)2z .R L¯ ¯Let d(a)5(2(m 2m ) /aDz). Now suppose, contrary to (Ap.14b), that

ˆ=v ? (21, d(a)) # 0. (Ap.16)R

Expanding (Ap.16) yields:

*(z (a) 2 z )(m 2 w )R L R¯ ]]]]]]m 2 m # ,2(z (a) 2 a )R R

ˆwhich contradicts (Ap.13). Hence (Ap.14b) holds at the above choice for d(a).Consequently, for sufficiently small ´, (Ap.14b) holds for the direction (21,d(a)2´).

ˆBut inequality (Ap.14a) holds as well for the direction (21, d(a)2´), for anypositive ´, since (Ap.15) is true. Hence this case is impossible as well.

References

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Downs, A. 1957. An Economic Theory of Democracy, New York: Harper Collins.


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Science 35, 228–278.Przeworski, A., Sprague, J., 1986. Paper Stones. Chicago: University of Chicago Press.Putterman, L., 1997. Why have the rabble not redistributed the wealth? On the stability of democracy

and unequal property. In Roemer, J.E. (Ed.), Property Relations, Incentives, and Welfare. London:Macmillan.

Roemer, J.E., 1994. A theory of policy differentiation in single issue electoral politics. Social Choiceand Welfare 11, 355–380.

Roemer, J.E., 1997a. Political–economic equilibrium when parties represent constituents: Theunidimensional case. Social Choice and Welfare 14, 479–502.

Roemer, J.E., 1997b. The democratic political economy of progressive taxation. Dept of EconomicsWorking Paper, University of California, Davis. (In press, Econometrica).

Wittman, D., 1983. Candidate motivation: a synthesis of alternative theories. American PoliticalScience Review 77, 142–157.

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