Voting on pensions with endogenous retirement age

Transcription

1 Voting on pensions with endogenous retirement age Georges Casamatta, Helmuth Cremer and Pierre Pestieau November 2001 Abstract People tend to retire earlier and the main determinant of such a phenomenon seems to be the implicit tax imposed on continued activity of elderly workers. In this paper we study the relevance of such a distortion in a political economy model with endogenous age of retirement. The setting is that of a two-period overlapping generations model. Individuals differ in their productivity. In the first period they work a fixed amount of time; in the second, they choose when to retire and then receive a flat rate pension benefit. Pensions are financed by a payroll tax on earnings in the first and in the second period of life. Such a tax is non distortionary in the first period; it is in the second period. We allow for some rebating of the second period tax which thus make the effective rate lower. Individuals vote on the level of the payroll tax given the rebate which can range from 0 to 100%. Given the majority voting equilibrium tax rate, we consider whether of not a full rebate is desirable. It is generally not so because even though this implies some allocative distortion it also generates redistribution. 1

2 1 Introduction Over the last forty years, labor force participation of the elderly has been dramatically decreasing in almost all industrialized countries. Participation rates for men aged 60 to 64 were above 70% in the early 60s; they have fallen to 57% in Sweden and to below 20% in Belgium, France, Italy and the Netherlands by the mid 90s. 1 At the same time, people are living longer and longer. In the European Union, life expectancy at age 65 has increased by more than one year per decade since This puts an enormous pressure on the financial viability of Pay-As-You-Go PAYG) pension systems and the situation will become even more problematic when the baby boomers will come to retirement. Gruber and Wise 1997) attribute this large decline in the labor force participation to the incentives created by social security systems. Continued work at later ages may be subject to two burdens: the traditional payroll tax which is nowhere age-dependent and forgone benefits when social security wealth decreases with the age of retirement. This double burden which Gruber and Wise calls an implicit tax represents an allocative distortion that induces early retirement. As they show it is higher in France, Belgium and Italy than in Japan, Sweden or the US. Average retirement is also quite earlier in these first countries than in the second. Why do we have such an implicit tax on continued activity? On pure efficiency grounds, we would like to avoid any distortion in the labor retirement choice of aged workers and let them choose the age of retirement such that the marginal utility of retiring is equal to the worker productivity times marginal utility of consumption. This is not only the efficient solution but also the laissez-faire equilibrium. As argued by Cremer, Lozachmeur and Pestieau 2001) as soon as one wants to introduce some redistributive objective in the pension system one cannot avoid downward distortions even with non-linear optimal schemes. In this paper the concern is not normative but positive. We want to see whether or not a majority of voters will push for such an implicit tax in a setting where the bulk of pensions is financed by a non distortionary payroll tax on earnings at earlier ages. Anticipating on the results we show that in general a majority of voters, young and old, with productivity below average, will push for such a tax. They will do so to a certain extent, namely to the point where the redistributive gain redistribution from elderly workers with high productivity) is offset by the efficiency loss the deadweight loss of such a distortive taxation). 1 A notable exception is Japan. 2

3 To develop those ideas we consider the following model. Each individual lives two periods. He works one unit of time in the first, pays a proportional tax to the pension system and saves. In the second period he works for some time and then retires. His consumption then is financed by disposable earnings, gross returns of savings and a flat pension benefit. Individuals differ according to age, they are young or old, and according to productivity. In the second period, only a fraction of the payroll tax is imposed on elderly workers, the remaining being implicitly rebated. Given such a fraction which implies a distortion, a bias, on the retirement decision, young and old vote on their preferred payroll tax rate. We show that a majority voting equilibrium tax exists under rather plausible assumptions. Given that tax rate, we then try to see whether or not a majority of voters is in favor of increasing the implicit tax bias. The mechanics of the model is quite intuitive. All old workers are in favor of a payroll tax as high as possible and old workers with income below average are in favor of some bias. Young workers with productivity below average are in favor of some payroll tax, the level of which is limited by the liquidity constraint. The same individuals are also in favor of some implicit taxation in the second period. The effect of the introduction of a pension system on the retirement decision has been studied by Sheshinski 1978) and Crawford and Lilien 1981). These authors argue that introducing a pension system actuarially fair total benefits are equal to total contributions) does not affect the retirement decision when there are no borrowing constraints. In this case, private savings are just replaced by public pensions. Introducing borrowing constraints may cause individuals to retire earlier but not later. When the level of public pension contributions are high, forced saving may result. This induces an income effect leading to early retirement retirement leisure being a normal good). However, if pension benefits cannot be collected before a given age, constrained individuals may want to delay retirement in order to smooth consumption before and after retirement. The introduction of a pension system which is not marginally fair 2 leads to an increase of the price of leisure with respect to consumption. If the substitution effect dominates the income effect, it induces people to retire earlier. 3 When the system is marginally fair but give benefits that outweighs contributions, the income effect will imply early retirement. 2 Departure from actuarial fairness can be of three types. First, the pension system may redistribute across individuals. Second, the aggregate level of benefits may outweigh the aggregate level of contributions, which is typically the case in a non mature PAYG system. Third the system may not be marginally fair which is the case when there is an implicit tax as defined above. 3 This takes an extreme form in our model where income effects are assumed away. 3

4 We can thus expect that with a mature unbalanced PAYG system people tend to retire later. To our knowledge, only two papers deal with the retirement decision in a political economy environment. Lacomba and Lagos 1999) study the problem of a direct vote on the mandatory) retirement age. More closely related to our study, Conde Ruiz and Galasso 2000) develop a model in which the vote takes place simultaneously on the payroll tax rate and on the decision to introduce or not an early retirement provision. They show that the early provision may be sustained at equilibrium by a coalition of the poor workers, who want to retire early, and old people with incomplete earnings history, who would receive no pension without this provision. This analysis and ours can be considered as complementary. Indeed, we do not investigate the issue of introducing an early retirement age. 2 The model Individuals live for two periods and they are differentiated according to their wage level per unit of time productivity). The distribution of productivities has support [, w + ], density function f.), and cumulative distribution function F.). We assume that the median productivity, w m, is lower than the mean, w. The intertemporal utility function is: ) U c, x, z) = u c) + βu x z 2 /2, where c is the first period consumption and x is the second period consumption; β is a factor of time preference, which is, by assumption, equal to 1/ 1 + r) where r is the interest rate. The utility function, u.) is increasing and concave: u.) > 0, u.) < 0. Moreover, we assume that lim x 0 u x) = + and that the coefficient of relative risk aversion is lower than 1: R r x) = xu x) /u x) 1. In the following, we denote d = x z 2 /2. First and second periods are of equal length, normalized to 1. labor supply is assumed to be inelastic in the first period. In the second period, individuals decide which fraction of time, denoted by z [0, 1], they spend working. This variable is interpreted as the retirement age. The disutility of work, z 2 /2, is expressed in monetary terms and represents the intensity of the disutility of work. With this particular form of the labor disutility function, we get rid of any income effect: labor supply decisions only depend on the relative price of leisure and consumption. First and second period consumptions for an individual with productivity w are respectively 4

5 given by: c = w 1 τ) s x = s 1 + r) + wz 1 θτ) + P, where τ [0, 1] is the payroll tax rate and s 0 is the amount of savings; P corresponds to the total pension received and, by assumption, does not depend on z. The parameter θ [0, 1] measures the bias of the pension system. Indeed, we define a neutral system as a system that does not distort individual decisions concerning retirement age. In other words, such a system does not modify the relative price of leisure and consumption, compared to the situation with no pension scheme. In a neutral system, the marginal benefit of working one more year is then w. This is the case in our setting when θ = 0. When θ > 0, the relative price of leisure and consumption becomes w 1 θτ). Consumption is therefore more expensive and individuals are induced to retire earlier. 4 Note that P does not depend on w. This means that the pension system considered operates income redistribution across individuals of the same generation. Everyone contributes for an amount proportional to his labor income but the benefit received does not vary across individuals. 3 Individual saving and retirement decisions In this section, we characterize the optimal savings and retirement decisions of old and young individuals, for given τ, P and θ. We denote z y, s y ) the optimal decisions of young individuals, where z y is the retirement age and s y is the amount of savings. Decisions concerning savings have been made in the past for old people. Their only decision is to choose when to retire. The optimal retirement decision of an old individual is denoted z o. 3.1 The old The program of old individuals is the following: subject to max z s 1 + r) + wz 1 θτ) + P z2 2 0 z 1. 4 A pension system might also induce people to retire earlier when the amount of pension benefit forgone if working one more year is not compensated by a corresponding increase in the pension level. This effect would be taken into account in our model if P was decreasing with z. 5

6 The first-order condition for an interior value of z is: This leads to w 1 θτ) z o = 0. z o = w 1 θτ). In order to ensure that z o 1 for everyone, we assume w +. All the individuals choose to work in the second period except when θτ = 1). The higher the productivity of an individual, the later he retires: consumption being cheaper for more productive individuals, they choose to work and consume more, provided of course that there is no income effect. On the other hand, increasing the bias of the system or the payroll tax rate contributes to increase the price of consumption with respect to leisure and consequently induces people to retire earlier. It should be noted that with no distortions, z o is equal to w/ and corresponds to the first best level. Finally, a higher disutility of work yields lower retirement ages. 3.2 The young The program of young individuals is the following: subject to max z,s [ ] u [w 1 τ) s] + βu s 1 + r) + wz 1 θτ) + P z 2 /2 0 z 1 and 0 s 1. The choice of z is the same as for old individuals and we have: This yields z y = w 1 θτ). 1) d = s 1 + r) + w2 1 θτ) 2 + P. 2 Recalling that β 1 + r) = 1, the first order condition for an interior solution of s is: u c) + u d) = 0. Individuals want to equalize first and second period consumptions net of the disutility of labor). For individuals choosing an interior solution, we obtain: w 1 τ) w2 1 θτ) 2 s y 2 = 2 + r) P. 2) 6

7 3.3 Budget constraint A feasible pension scheme must satisfy the government budget contraint: w+ N o P f w) dw = N y τ P = 1 + n) τw + θτy = θτ 1 θτ) 1 + n) τw + w+ w+ wf w) dw + N o θτ w+ wzf w) dw w 2 f w) dw, 3) where N y and N o are respectively the numbers of young and old individuals, n is the rate of population growth, y = wz and y = w + yf w) dw. We denote E w 2) = w + w 2 f w) dw and assume that 1 + n) / 1 + r) E w 2 )/w. The total pension received by a given individual is the sum of tax revenues on first and second period incomes. The tax base in the first period, 1 + n) w, is fixed whereas it depends on θτ in the second period. When the level of taxation is increased, individuals choose to retire earlier and as a consequence, there is less income to tax. Put differently, taxation only gives rise to distortions on second period income. Differentiating 3), we have: and P τ) = 1 + n) w + θ 2θ2 τ E w 2) 4) P τ) = 2θ2 E w 2) 0. 5) The budget curve is concave, always above the line τ 1 + n) w and P is equal to τ 1 + n) w when τ = 0 and θτ = 1. This is represented on the picture below. 4 Optimal solution: first- and second-best Even though our approach is mainly positive, it is worth looking at the solution chosen by a utilitarian social planner. This allows one to study the rationale if any) of a distortive tax biased system) from a normative optimal taxation perpective. s.t. Consider the first-best solution, namely the solution of w+ [ )] max u c w)) + βu x w) z w) 2 /2 f w) dw w+ [ c w) + x w) 1 + n w 1 + n 1 + n + z w)) ] f w) dw = 0. 7

8 Figure 1: The budget curve The first order conditions imply: u c w)) = β 1 + n) u d w)) = µ and z w) = w where µ is the Lagrange multiplier associated with the resource constraint. We now consider the problem of a social planner who has the same objective but whose instruments are limited to the parameters τ, θ and P. Without loss of generality we will introduce a new variable θ θτ so that the social planner acts as if he determines a tax rate for the first period τ and another one for the second period θ. The optimality problem can now be rewritten: [ w+ )] ) max = u w 1 τ) s y ) + βu s y 1 + r) + P + wz y 1 θ zy2 f w) dw 2 8

9 s.t. P = 1 + n) τ w + θ w+ w z w). Using 1), the utility in the second period is: u s y 1 + r) + w2 1 θ ) n) τ w + θ 1 θ ) E w 2)). 2 We can now derive the FOC: and w+ τ = [ wu c) β 1 + n) wu d) ] f w) dw θ = w+ u d) w 2 1 θ ) 1 2 θ E w 2)) f w) dw. To interpret, assume first that there is no liquidity constraint savings can be negative). τ Then, c w) = d w) for all w and τ = 1. As soon as there is a liquidity constraint the lowest segment of the distribution of w will restrict τ to be less than 1 but surely positive. The condition is equivalent to the standard formula for an optimal linear tax: θ w+ w θ = u d) [ w 2 E w 2)] f w) dw u d) [w 2 2 E w 2 )] f w) dw w+ where the numerator is the covariance between the marginal utility of d w) and the square of the productivity levels. This covariance is negative. We thus have: namely 0 < θ < 1. θ = cov u d), w 2) cov u d), w 2 ) E w 2 ) w + u d) f w) dw, Note that if θ was imposed on w rather than on wz, in other words if there was no distortion, θ would be equal to 1. This is useful for what follows. With a utilitarian objective, what prevents the tax rates to be maximal is the liquidity constraint for τ and the labor disincentive for θ. These two sources of inefficiency are the gist of our political economy model. 5 Majority voting equilibrium tax rate 5.1 Preferred tax rates Let define [ ] V y τ, θ; w) = u [w 1 τ) s y ] + βu s y 1 + r) + P τ, θ) + w 2 1 θτ) 2 /2 6) 9

10 and [ ] V o τ, θ; w) = u s o 1 + r) + P τ, θ) + w 2 1 θτ) 2 /2, 7) which represent the utility levels attained by type w individuals, young or old, for given τ and θ. Preferred tax rates for young and old individuals, denoted respectively τ y and τ o, are obtained by solving the following programs: max V i τ, θ; w), i = y, o. τ [0,1] We prove in appendix 1 the following proposition: Proposition 1 i) Preferred tax rates of young individuals are decreasing with productivity. ii) No young individual chooses a corner solution at τ = 1. The preferred tax rate of young individuals with productivity w w 1 + n) / 1 + r) is positive. iii) Old individuals choose corner solutions: the poor old choose τ o = 1 and the rich old choose τ o = 0. The intuition for the first result is the following. Consider for simplicity the case θ = 0. When w increases, first period income also increases. Second period consumption being a normal good, it is increased through an increase of the tax rate. On the other hand, the relative price of first and second periods consumptions, w 1 + n) /w decreases. Put differently, it costs less in terms of first period consumption) to buy one unit of second period consumption for a low productivity individual than for a high one. By this substitution effect, high productivity individuals are induced to buy less second period consumption. For utility functions such that R r.) < 1, the substitution effect dominates and low productivity individuals want tax rates larger than high productivity individuals. Note that when R r.) = 1 logarithmic utility function), income and substitution effects neutralize and preferred tax rates are constant with respect to productivity. Finally, there is a third effect of an increase in w. Because second period income increases with productivity, high productivity individuals raise their first period consumption which is a normal good) by reducing the payroll tax rate. This effect reinforces the second one. As a consequence, preferred tax rates are decreasing with productivity when R r.) 1. The first part of point ii) is obvious. If an individual chooses a tax rate equal to 1, he consumes nothing in the first period. When the marginal utility of consumption tends to infinity, this individual has an incentive to reduce marginally the tax rate. To illustrate the second part, 10

11 let us write the first-order derivative of a young individual life cycle utility at the point τ = 0 savings being optimally chosen): dv y dτ = wu c) + β 1 + n) wu d) + β θ τ=0 = E w 2) w 2) u d) w + β 1 + n) w + β θ E w 2) w 2)) u d), where we have used the fact that savings are positive when τ = 0 which implies that u c) = u d). A first observation is that, in a neutral system θ = 0), individuals choose a positive tax rate if and only if w βw 1 + n) = w 1 + n) / 1 + r). Indeed, in such a system there is no redistribution of second period incomes. Individuals favoring a positive tax rate are those for whom the rate of return of the PAYG system, 1 + n) w/w is higher than the rate of return of private savings, 1 + r. Moreover, these individuals do not want to save at their optimal tax rate. Now, if one introduces a bias in the system, second period incomes are redistributed from individuals with a productivity level higher than E w 2 ) towards individuals with a lower productivity. 5 Therefore, individuals such that w w 1 + n) / 1 + r) still want a positive tax rate but some individuals with a higher productivity also do. It should be noted that contrarily to the neutral case, it may be the case that some individuals make savings when their optimal tax rate is implemented. In this model, the payroll tax rate serves two objectives: intertemporal consumption smoothing and second period) income redistribution. An individual may then have some incentive to reduce his optimal tax rate with respect to the neutral case in order to limit tax distortions and to benefit from an increased redistribution. In a such a case, savings might constitute a useful instrument to transfer resources between first and second periods. The last point of the proposition says that old individuals choose corner solutions for the tax rates. Consider the first-order derivative with respect to the tax rate of the old objective function given in 7)): dv o dτ = 1 + n) w + θ 1 2θτ) E w 2) ) θ 1 θτ) w 2 u d). In a neutral system and by continuity in a slightly biased system), every old has an increasing objective function and chooses the maximal possible tax rate, τ = 1. When θ is increased, one can see, by evaluating the above expression at τ = 0, that old people with productivity w 2 < E w 2) n) w/θ want a positive tax rate. We prove in the appendix that they in 5 Second period income is less than the average if and only if w < E w 2 ). 11

12 fact most prefer τ = 1. Starting from τ = 0, old individuals with a higher productivity dislikes a marginal increase in the tax rate. However this does not mean that their optimal tax rate is 0. Indeed, we show that their objective function may be convex. To see this, evaluate the above expression at τ = 1 when θ = 1: it is positive. When θ = 1 and τ approaches 1, everyone stops working and the old rich do not suffer anymore from the redistribution towards the poor. On the other hand, their pension increases with the tax rate. They thus favor a marginal increase in the tax rate. 5.2 Voting equilibrium We now want to determine the payroll tax rate chosen in a majority vote, namely the Condorcet winner. Conditions ensuring the existence of an equilibrium are stated in the following proposition. Proposition 2 If θ/ 1 + n) w/ w 2 + E w 2)), a voting equilibrium on τ exists. Proof. We proved in appendix 1 that, for utility functions such that R r.) 1, preferences of the young over tax rates satisfy the single-crossing condition established by Gans and Smart 1996). This means that we can order young individuals and alternatives in such a way that if an individual prefers the higher of two alternatives, all the individuals ranked to the right of this individual display the same preference. This is not however sufficient to guarantee the existence of a Condorcet winner. The reason is that the preferences of the old may be convex, as argued in the last paragraph of the previous section. In these circumstances, it is possible that an old individual favors a slight decrease in the tax rate even though his preferred tax rate is 1. To overcome this difficulty, we restrict our attention to cases where the utility of the old is monotonically increasing with the tax rate. This is true when the marginal utility of the richest old individual is positive at τ = 0. The discussion in the last paragraph of the last section makes it clear that this is the case when θ/ 1 + n) w/ w 2 + E w 2)). This condition is satisfied when the bias parameter, θ, is small enough, or when is large enough. The pivotal voter, who is a young individual, is then implicitly determined by the following condition: N o + N y F w piv ) = N o + N y 2 n F w piv ) = n). 8) 12

13 For any n 0, n/ n)) is lower than 1/2. It follows that the pivotal voter has a productivity level below w m. The pivotal voter is such that half of the total population the old and the young with a lower productivity) want a higher tax rate and the other half the young with a higher productivity) want a lower tax rate. The majority voting equilibrium, τ mv, is then τ y θ; w piv ). If n r, the majority voting tax rate is positive. Indeed, we know that if n r, preferred tax rates are positive for all the individuals with a productivity below the mean. The median productivity being lower than the mean, more than half of the young population favor a positive tax rate. This implies that the pivotal voter chooses a positive tax rate. It is important to keep in mind that the condition stated in the above proposition is only sufficient and is adopted for expositional convenience. A less stringent condition would require that the utility of the richest old be higher at the point τ y θ; w piv ) than at τ = 0, that is: V o [τ y θ; w piv ), θ; w + ] V o [0, θ; w + ] [1 1 θτ y θ; w piv )) 2] w n) τ y θ; w piv ) w + θτ y θ; w piv ) 1 θτ y θ; w piv )) E w 2). 9) If this condition is satisfied then it is also true for individuals with a lower productivity. It follows that any tax rate lower than τ y θ; w piv ) is rejected by the coalition of all the old and the young to the left of w piv. Moreover a tax rate higher than τ y θ; w piv ) is rejected by the young to the right of w piv. Therefore τ y θ; w piv ) is a Condorcet winner even though the preferences of the old are not monotonically increasing. 6 Determination of θ We now turn to the political determination of θ, which was until now given. The natural approach would be to determine the pair θ, τ) chosen jointly in a majority vote. However, it is well known that a Condorcet winner is unlikely to exist when the issue space is multidimensional and this model is not an exception. The procedure we adopt is the majority voting determination of θ and τ one issue at a time. We first describe the sequential procedures, in which θ and τ are chosen one after the other and then present the case in which these two policies are chosen simultaneously and independently. 13

14 6.1 Sequential voting Vote on θ first In this case, θ is determined first by a majority vote. Then, knowing the value of θ, people vote on τ. Unfortunately, the usual sufficient conditions for a voting equilibrium to exist singlepeakedness, single-crossing) are extremely difficult to verify at the first stage and there is no guarantee that an equilibrium exists. We therefore adopt a less ambitious, tax reform, approach which consists in looking at the impact on the individuals welfare of introducing a small bias in the pension system. This allows us to evaluate the political support for a slightly biased system, compared to a neutral one. Impact of the bias on the equilibrium tax rate In order to determine the impact of introducing a bias in the pension system on welfare, we first have to determine its impact on the equilibrium tax rate. We obtain the following result, proved in appendix 2. Proposition 3 Starting from a neutral system and considering a small increase in θ, the majority voting tax rate increases. The increase in θ results in a redistributive effect that benefit people such that y < y w < E w 2 )), which is the case of the pivotal voter. Second period consumption becomes cheaper for him with respect to first period consumption. When the coefficient of relative risk aversion is lower than 1, the substitution effect dominates the income effect and he chooses a higher tax rate in order to consume more in the second period. Impact on welfare The next proposition, proved in appendix 3, examines the welfare consequences of introducing a small bias in the system. Proposition 4 Starting from a neutral system and following a small increase in θ, the welfare of the young individuals to the left of the pivotal voter and the old individuals such that w 2 < E w 2) increases if the coefficient of relative risk aversion is lower than 1. Increasing θ has two consequences. First, it induces some redistribution from people with income above the mean towards people with income below the mean. Second, it yields a higher voting tax rate. 14

15 Young people to the left of the median voter benefit from the two effects. Those with productivity between w piv and E w 2 ) benefit from the first, redistributive, effect but want a tax rate lower than the one chosen by the median voter and are worse off when the equilibrium tax rate rises. Therefore, the net effect is ambiguous. Finally, the young with productivity higher than E w 2 ) suffer from the two effects. Concerning the old, they all benefit from the increase of the tax rate induced by the introduction of the bias. However, only those with productivity lower than E w 2 ) benefit from the redistributive effect. From the discussion above, it is clear, recalling that w m < E w 2 ), that more than one half of the old favor the introduction of the bias. Nevertheless, this is not necessarily the case for the young. As a consequence, we cannot conclude that the majority of the voters are better off in the biased system. In order to get more insights, we resort in the next section to numerical simulations Vote on τ first In this setting, the timing of decisions is reversed with τ being chosen first. We proceed backward and determine the voting outcome on θ for a given τ second stage). The slope of an indifference curve in θ, P ) plane is: dp dθ = τw2 1 θτ) > 0. This expression is increasing with w. Consequently, preferences are single-crossing so that a majority voting equilibrium on θ exists. Individually optimal levels of θ are obtained by solving: This yields and max V i τ, θ; w), i = y, o. θ [0,1/τ] θ i = E w 2) w 2 τ 2E w 2 ) w 2 ), i = y, o dθ i dw = 2τwE w 2 ) [τ 2E w 2 ) w 2 )] 2 < 0. Optimal levels of θ are decreasing with productivity and does not depend on age. The Condorcet winner is then the level of θ most preferred by the individuals with median productivity: θ mv = E w 2) w 2 m τ 2E w 2 ) w 2 m). 10) 15

16 We now turn to the first stage of the game, that is the voting decision on τ. One can show that, as soon as R r.) 1, the slope of indifference curves of the young in the τ, P ) plane are increasing with productivity. Therefore preferences of the young satisfy the single-crossing condition and their preferred tax rates are decreasing with productivity. Moreover, the utility of every old is increasing with the value of the tax rate. This leads to the conclusion that a voting equilibrium exists at the first stage of the game and consequently that an equilibrium of the full game also exists. The decisive voters at the first stage are such that: F w piv ) = The majority voting tax rate is the solution of n n). 6.2 Simultaneous voting max τ V y τ, θ mv τ) ; w piv ). In this setting, the parameters θ and τ are chosen simultaneously and independently. The equilibrium is given by the intersection of the two reaction curves τ mv θ), determined in section 5, and θ mv τ), such as defined in 10). It can be easily shown that the two curves τ y θ; w piv ) and θ mv τ), represented on the picture below, intersect. However, this does not mean that a simultaneous, issue by issue, voting equilibrium exists.this is true if condition 9) is satisfied at the point θ = θ mv τ ), τ = τ y θ ; w piv ), that is w 2 + [1 1 θ τ ) 2] n) τ w + θ τ 1 θ τ ) E w 2), where θ τ = E w 2) w 2 m 2E w 2 ) w 2 m). 11) 16

17 If an equilibrium exists, the value of θτ is given by 11). It is the same as in the previous section. 7 Numerical examples In these simulations, productivities are distributed on [1, 100] and = 100. We consider two possible distributions: a distribution skewed to the right with w m = < w = < E w 2 ) = and a uniform distribution function with w m = w = 50.5 < E w 2 ) = The utility function is isoelastic: u x) = x 1 ε / 1 ε), where ε is the coefficient of relative risk aversion. We present results in the cases ε = 0.2, 0.5 and 1 and r = n = 1. We were not able to derive, even numerically, the equilibrium of the sequential voting game with θ determined first. We therefore adopt a tax reform approach and compare individual welfare levels for θ = 0 and θ = The second column in the table below gives the productivity of the young individuals indifferent between θ = 0 and θ = 0.01 as well as between brackets) the proportion of the young individuals prefering the higher of these two values. The same analysis is conducted for old individuals and the results are presented in the third column. The last column then indicates the proportion of the total population favoring θ = 0.01 over θ = 0. 17

18 Skewed distribution young indif. old indif. total prop. ε = %) %) % ε = %) %) % ε = %) %) % Uniform distribution young indif. old indif. total prop. ε = %) %) 55.1 % ε = %) %) % ε = %) %) % With a skewed distribution, a large fraction of the total population favors a slight increase in θ, whatever the value of the coefficient of relative risk aversion. In the case of a uniform distribution, the political support for an increase in θ is much lower than in the previous case. In some cases, it is possible that the young do not favor the introduction of a small) bias. However, due to the influence of the old, the overall population support such a policy. The next table gives the values of θ, τ and θτ chosen respectively at the political equilibria sequential voting with τ decided first and simultaneous voting) and by a utilitarian social planner. Skewed dist. Sequ. τ θ Shepsle Util. Opt. θ τ θτ θ τ θτ θ τ θτ ε = ε = ε = Unif. dist. Sequ. τ θ Shepsle Util. Opt. θ τ θτ θ τ θτ θ τ θτ ε = ε = ε = With a skewed distribution, the value of the implicit tax rate chosen at the political equilibrium is larger than the one chosen by a utilitarian social planner. With a uniform distribution, it is however possible that the level of the implicit tax rate chosen at the political equilibrium be lower than the optimal one. This occurs when the utility function is sufficiently concave, which means that the social planner has a great concern for inequality. 18

19 8 Conclusion [to be completed] 19

20 References [1] Casamatta, G., Cremer, H. and P. Pestieau, 2000, The political economy of social security, Scandinavian Journal of Economics, 102, [2] Conde Ruiz, J.I. and V. Galasso, 2000, Early retirement, CEPR discussion paper n [3] Crawford, V. and D.M. Lilien, 1981, Social security and the retirement decision, Quarterly Journal of Economics, 95, [4] Cremer, H., J.-M. Lozachmeur and P. Pestieau, 2001, Social security and variable retirement schemes: an optimal taxation approach, mimeo. [5] Gans, J.S. and M. Smart, 1996, Majority voting with single-crossing preferences, Journal of Public Economics, 59, [6] Gruber, J. and D. Wise, 1997, Social security programs and retirement around the world, NBER working paper [7] Lacomba, J.A. and F.M. Lagos, 1999, Social security and political election on retirement age, mimeo. [8] Sheshinski, E., 1978, A model of social security and retirement decisions, Journal of Public Economics, 10,

21 Appendix A Proof of proposition 1 i) The equation of an indifference curve is derived by solving: [ ] u [w 1 τ) s y ] + βu s y 1 + r) + P + w 2 1 θτ) 2 /2 = c, where c is a constant. Differentiating this expression, the slope of an indifference curve is If savings are positive, u c) = u d). This leads to w 2 dp wu c) + β dτ = θ 1 θτ) u d) βu > 0. 12) d) w + β dp dτ = w 2 θ 1 θτ). 13) β Indifference curves are increasing and concave. Moreover, d 2 P/dτdw > 0: the slope of indifference curves is increasing with productivity. If the individual does not want to save, the differentiation of 12) with respect to w leads to w d 2 P u c) + w 1 τ) u c)) βu d) βwu c) dτdw = 1 θτ)2 u d) βu d)) 2 + 2w θ 1 θτ) = u c) 1 R r c))) βu d) βwu c) w 1 θτ)2 u d) βu d)) 2 + 2w θ 1 θτ). If R r.) 1, d 2 P/dτdw is positive. It follows that the slope of indifference curves is increasing with productivity. This leads to our conclusion that preferred tax rates are decreasing with productivity. ii) Differentiating 6) with respect to τ and using 4), we have dv y dτ ) = wu c) + β P τ) w2 θ 1 θτ) u d) = wu c) + β 1 + n) w + θ ) 2θ2 τ E w 2) w2 θ 1 θτ) u d). 21

22 At τ = 1, dv y dτ = wu 0) τ=1 +β u 1 + n) w + θ 2θ2 E 1 + n) w + E w 2) θ 1 θ) ) w 2) w2 θ 1 θ) ). It is clear that, if lim u x) = +, dv y /dτ x 0 τ=1 < 0. At τ = 0, dv y dτ = wu c) + β τ=0 We argue now that s y > 0 when τ = 0. From 2), s y τ=0 = 1 + n) w + θ E w 2) w2 θ w2 2 P 0) = 2 + r) > 0 w < 2. w r) ) u d). Because w +, the condition w < 2 is satisfied for any w. It follows that u c) = u d) and dv y dτ = u c) w + β τ=0 1 + n) w + θ E w 2) w2 θ > 0 w + β 1 + n) w + θ ) E w 2) w2 θ > 0 w n 1 + r w + 1 ) θ 1 + r E w 2) w2 θ > 0. If w < w 1 + n) / 1 + r) then, by assumption, w < E w 2 ) and the above expression is positive. iii) Indifference curves for old individuals are derived by solving u s 1 + r) + P + w2 1 θτ) 2 ) = c. 2 )) Differentiating, we have and dp dτ = w2 θ 1 θτ) > 0 d 2 P dτ 2 = θ 2 w2 < 0. 22

23 Comparing with 5), we find that the slope of indifference curves decreases more quickly than the slope of the budget curve if and only if w 2 > 2E w 2). At τ = 0, the productivity level such that the slope of the indifference curve equals the slope of the budget curve is given by: w 2 θ = 1 + n) w + θ E w 2) w 2 s = θ 1 + n) w + E w 2). Noting that and recalling that w +, E w 2) w+ w+ = w 2 f w) dw < w + wf w) dw = w + w ws 2 > 2E w 2). Following the discussion above, this indifference curve is always below the budget curve. Therefore, individuals with productivity w s have a preferred tax rate equal to 1. Observing that the slope of indifference curves is increasing with productivity, all individuals with a productivity less than w s want also a tax rate equal to 1. Individuals with a higher productivity do not choose an interior solution for τ. Indeed, their indifference curves being more concave than the budget curve, a point of tangency between an indifference curve and the budget curve corresponds to a minimizing tax rate. The individuals indifferent between τ = 0 and τ = 1 are such that V o 0; w o ) = V o 1; w o ) ) u s 1 + r) + P 0) + wo2 = u s 1 + r) + P 1) + wo2 1 θ) 2 ) 2 2 wo2 2 w o2 = θ 1 θ) = 1 + n) w + E w 2) + wo2 1 θ) n) w θ 2 θ) θ 2 θ E w 2), if θ 0. 14) Note that this equation does not always have a solution. In particular, when θ = 0, every old individual most prefer a tax rate equal to 1. Observe also that the indifferent old individual has a productivity level higher than w. Finally, dw o2 /dθ < 0. B Proof of proposition 2 The majority voting tax rate, when positive, is determined by the following first-order condition: wu c) + β 1 + n) w + θ ) 2θ2 τ E w 2) w2 θ 1 θτ) u d) = 0, 23

24 where c = w 1 τ) s y d = s y 1 + r) n) τw + θτ 1 θτ) E w 2) + w2 1 θτ) 2 2 and w = w piv. Differentiating this expression with respect to θ, we obtain w dsy 1 4θτ dθ u c) + β dτ y w piv ) dθ = E w 2) w2 1 2θτ) ) u d) D τ β 1 + n) w + θ 2θ2 τ E ) w 2) w2 θ 1 θτ) + D τ ) ds y τ 2θτ r) + E w 2) w2 τ 1 θτ) u d) dθ where D τ = w 2 u c) + w dsy w 2 θ 2 2E w 2) θ 2 ) dθ u c) + u d) +β 1 + n) w + θ ) 2 2θ2 τ E w 2) w2 θ 1 θτ) u d) +β dsy 1 + r) 1 + n) w + θ ) 2θ2 τ E w 2) w2 θ 1 θτ) u d). dθ When θ = 0, we know that the pivotal voter does not want to save. Moreover, the identity of the pivotal voter does not change when θ stays close to 0. All this leads to D τ θ=0 = w 2 u c) + β 1 + n) w) 2 u d) < 0 and dτ mv dθ = θ=0 = E w 2 ) w 2 ) E w β u 2 ) w 2 ) d) + βτ 1 + n) w u d) D τ E w 2 ) w 2 ) ) β u d) 1 R r d)) w2 2 u d). D τ Recalling that w 2 piv < w2 m w 2, we have that w 2 piv < E w 2) by Jensen s inequality w 2 < E w 2) ). We can conclude that this expression is positive if R r.) 1. 24

25 C Proof of proposition 3 We differentiate 6) with respect to θ: dv y dθ + = β τ 2θτ 2 dτ mv dθ E w 2) wu c) + β ) τ 1 θτ) w 2 u d) 1 + n) w + θ 2θ2 τ E w 2) ) ) θ 1 θτ) w 2 u d). This becomes at θ = 0 dv y dθ τ = β w θ=0 E 2) τ ) w2 u d) dτ mv + dθ wu c) + β 1 + n) wu d) ). θ=0 It is clear that all the individuals to the left of the pivotal voter are such that w 2 < E w 2). The first term is then positive. Besides, these individuals want a tax rate higher than the median voter. The majority voting tax rate being increasing when the coefficient of relative risk aversion is lower than 1, the second term is also positive. It follows that every individual to the left of the pivotal voter benefits from an increase in θ. We now turn to old individuals. From 7): dv o dθ = β τ 2θτ 2 dτ mv + dθ β E w 2) ) τ 1 θτ) w 2 u d) w 2) 1 + n) w + θ 2θ2 τ E ) θ 1 θτ) w 2 u d). Therefore dv o dθ τ = β w θ=0 E 2) τ ) w2 u d) dτ mv + dθ β 1 + n) wu d). θ=0 This is positive for individuals with productivity w 2 < E w 2) as soon as relative risk aversion is lower than 1. 25