## Abstract

This paper analyzes simultaneous voting on the wage tax rate and investment in public education with three overlapping generations and ability differences inside each cohort. Wage tax revenue finances public education and social security benefits. The presence of ability differences introduces a time-consistency problem with repeated voting. This can be solved by trigger strategies, which do not punish upward deviations in the wage tax rate. If there are multiple equilibria, then higher tax rates are associated with more education. Surprisingly, the median voter may be a young citizen, even when cohorts are of the same size.

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## Notes

The share is at least 80% also in Belgium, the Czech Republic, France, Germany, Ireland, Italy, Portugal, the Slovak Republic, and Sweden. In Hungary, the Netherlands, Spain, and the United Kingdom, the share of public financing is 70 to 80%.

Voting equilibrium may not be established by general non-linear taxation because of the possibility of Condorcet cycles.

Miles and Timmermann (1999) report that the gross replacement rate was more than 10 percentage points higher for low-income workers in Belgium, Denmark, France, Greece, Ireland, Luxembourg, the Netherlands, Spain, Sweden, and the United Kingdom in 1997.

A related concept developed to study repeated majority voting is the notion of a Dynamic Condorcet Winner, developed by Bernheim and Nataraj (2004). They assume infinitely lived agents. Using the same framework, Nataraj (2004) considers dividing a fixed surplus with overlapping generations. This paper is more general than Nataraj (2004) in endogenizing the surplus arising from public education and in allowing for ability differences, while Nataraj (2004) is more general in allowing for arbitrary divisions of the fixed pie between all cohorts.

With identical voters inside each cohort and three overlapping generations of equal size, no single voter or a group of voters smaller than a half of a cohort could cause a deviation. With a continuum of types, even a single individual could cause a deviation.

Untaxed home production may also take form of working in ordinary production and evading taxes. Here the assumption corresponding to the declining marginal productivity in home production would be that an increasing fraction of time spent in tax-evading work goes to avoiding detection.

Allowing consumption in the first period would not change the results, as shown in the previous version of this paper, which is available as IZA DP 1408.

Assuming that there are no citizens with ability \(\overline{a}\) simplifies the notation in Section 5. Voting for an ability threshold \(\overline{a}\) is there equivalent to voting for a zero provision of public education.

This ability threshold is either the politically decided minimum ability required to enter the higher education or the minimum ability with which an individual would like to enter higher education, whichever is higher. Section 5 shows that in all equilibria, the politically decided minimum ability required to enter the higher education is higher than the minimum ability with which an individual would like to enter higher education.

An alternative to simultaneous voting would be voting separately and sequentially over each dimension. In case of analyzing sequential voting, assumptions made on the timing of the choices may affect political outcome.

A “hat” denotes the equilibrium outcome of a variable, while \(\tau _{t}\) and \(\widetilde{a}_{t}\) denote any possible values for the wage tax rate and the ability threshold above which citizens receive higher education. In period \( t \), \(\widehat{a}_{t}\) and \(\widehat{\tau }_{t}\) refer to the expected equilibrium outcomes arising from the political process, while \(\widetilde{a} _{t-1}\) is the threshold ability level that was in place in the previous period.

Note that if \(q<ru_{1}\), then citizens with \(a<(1+r)u_{1}+u_{2}\) would not want to receive public education in case this would be offered to them. The model could be solved in this case by replacing \(\widetilde{a}_{t}\in \lbrack 0,\overline{a}]\) in Eqs. 17, 21 and 24 by \( \widetilde{a}_{t}\in \lbrack (1+r)u_{1}+u_{2},\overline{a}]\). This would be without loss of generality as choosing \(\widetilde{a}_{t}\) below \( (1+r)u_{1}+u_{2}\) would result in the same take-up of education on the part of potential students as choosing \(\widetilde{a}_{t}=(1+r)u_{1}+u_{2}\).

Note that the policy conclusions do not rely on the assumption that voting would take place only once in every three periods. These are just simplifying steps to derive the results with repeated voting in the following subsection.

Conde Ruiz and Galasso analyze simultaneous voting on the tax rate and early retirement provisions. They restrict the decision on early retirement provisions to a binary choice between full benefits or no benefits at all, whereas, this paper analyzes a two-dimensional policy space with a continuum of alternatives in both dimensions.

An alternative trigger strategy would specify that all agents vote for \( \widetilde{a}=\overline{a}\) and \(\tau =0\) if the contract has been violated in the previous period. While this would lead to the same outcome as the IICVS, voting for \(\widetilde{a}=\overline{a}\) is weakly dominated by voting for any \(\widetilde{a}\in \lbrack 0,a^{i}]\) for the young with \( a^{i}>(1+r)u_{1}+u_{2}\). Likewise, for the elderly, voting for \(\tau =0\) is weakly dominated by voting for the rate of \(\tau \), which maximizes the wage tax revenue for the social security benefits. With IICVS, no citizen votes for a weakly dominated strategy.

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## Acknowledgements

I am indebted to Vesa Kanniainen, Katarina Keller, Tapio Palokangas, Jukka Pirttilä, Steinar Strøm, Andreas Wagener, conference participants, and three anonymous referees for useful comments. Financial support from the Yrjö Jahnsson Foundation is gratefully acknowledged.

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*Responsible editor*: Gil S. Epstein

The first draft of this paper appeared as part of my doctoral dissertation at the University of Helsinki. An earlier version was written while I was a Marie Curie Fellow at CES at the University of Munich, whose hospitality I gratefully acknowledge. The paper was revised while I was employed full-time at the Centre for Economic and Business Research at the Copenhagen Business School. Earlier versions of this paper were presented at the ENTER Jamboree in Mannheim, February 14–17, 2001, the European Public Choice Society Annual Meeting in Paris, April 18–21, 2001, the 57th International Institute of Public Finance Congress in Linz, August 27–30, 2001, the CEPR/EPRU Workshop “Dynamic Aspects of Public Expenditure” in Copenhagen, September 28–30, 2001, and at the EEA Annual Meeting in Venice, August 22–24, 2002.

## Appendix

### Appendix

### Proof of Lemma 1

for those young with \(a\geq \widetilde{a}_{t}^{y}(a,\widehat{\tau }_{t})\) and for those with \(a\leq u_{1}(1+r)+u_{2}.\) Those young with the ability between these values vote either for this same ability threshold or for their own ability level as threshold, whichever would give them higher utility. The young never vote for a lower ability level than Eq. 28, as if they would, then they could increase the value of their old-age benefits by switching to this level. By Eqs. 7, 8, 19, 20 and 21, the preferred choice of all the middle-aged is given by

By Eq. 7, 8, 23 and 24, the elderly vote for \( \widetilde{a}_{t}^{o}(a,\widehat{\tau }_{t})=\overline{a}.\) As the right-hand sides of Eqs. 28 and 29 are identical, and the elderly (who are one-third of the population) vote for a higher and none of the young for a lower cutoff level of ability, this is the median voter choice. Note that there is no single median voter; rather, the coalition consists of all the middle-aged and at least those young with \(a\geq \widetilde{a}_{t}^{y}(a,\widehat{\tau }_{t})\) or \(a\leq u_{1}(1+r)+u_{2}.\)

An individual with ability \(a\) is willing to receive higher education, if and only if this results in higher after-tax income, that is

By assumption \(q\geq ru_{1},\) any \(a\) higher than (or equal to) \(\widetilde{a }_{t}^{y}\) implied by Eq. 28 satisfies Eq. 30. Therefore, all citizens for whom public education is available prefer to receive it.

###
*Proof of Lemma 2*

Insert Eq. 9 into Eq. 7 and then the resulting expression into Eq. 8. Then insert Eq. 27. By Eq. 13, \(b_{t+1}=\,\,\,b_{t+2}=\tau _{t}(1-\tau _{t})\omega _{t+1}-(1-F(\widehat{a}_{t}))q.\) Insert this into Eqs. 15, 16, 19 and 20. The solutions for \( \tau _{u,t}^{y}(a,\widehat{a}_{t})\) and \(\tau _{e,t}^{y}(a,\widehat{a}_{t})\) follow from the first-order conditions of Eq. 18, subject to the non-negativity constraint, and those for \(\tau _{u,t}^{m}(a,\widehat{a} _{t-1})\) and \(\tau _{e,t}^{m}(a,\widehat{a}_{t-1})\) follow, respectively, by the first-order conditions of Eq. 22. The solution for the elderly follows by maximizing Eq. 25, using Eqs. 7, 9 and 14.

###
*Proof of Proposition 1*

As the provision of education chosen is given by Lemma 1 and is that preferred by the claimed median voter in each regime, it suffices to analyze voting on the wage tax rate. By Lemma 2, the middle-aged uneducated prefer higher taxes than the young uneducated, and these prefer higher taxes than anyone who is going to become educated. Furthermore, the preferred tax rate of the educated is declining in their ability and is the same whether they are young or middle-aged. With mass of each cohort normalized to unity, note that in scenario (1), the elderly and the middle-aged uneducated alone form more than half of the population. In scenario (2), the elderly and the middle-aged uneducated alone are not enough to form a majority. Together with the young uneducated, they form the majority. In scenario (3), the elderly and the uneducated are not alone in majority. As the young to-be-educated and the middle-aged educated prefer the same tax rate with the same ability, the median voter is a citizen with ability \(a_{iii},\) whether young or middle-aged.

###
*Proof of Proposition 2*

In each tentative equilibrium, the equilibrium tax rate can be solved from Lemma 2 and Proposition 1, given the assumption on the provision of education. This tentative equilibrium tax rate can then be inserted into Eq. 26 to verify whether the associated ability threshold is consistent with the initial assumption on who becomes educated. Assume first that \(30\)% of population have \(a=1.7,\) \(40\)% have \(a=1.5,\) and \(30\)% have \(a=0.8,\) and that \(u_{1}=0.1,\) \(u_{2}=0.8,\) \(q=0.1,\) and \(r=0.1.\) There are three SSSIEs. In the first one, both citizens with ability level \( 1.7\) and those with ability level \(1.5\) receive education, and \(\tau _{t}=0.215.\) In the second one, only citizens with ability level \(1.7\) receive education, and \(\tau _{t}=0.186.\) In the third equilibrium, no citizen receives education, and \(\tau _{t}=0.022.\) In the first equilibrium, the median voter is a young citizen who remains uneducated, and in the second and third equilibria, a middle-aged uneducated. Assume next that \(30\)% of population have \(a=2,\) \(40\)% have \(a=1.8,\) and \(30\)% have \(a=0.8,\) and that \(u_{1}=0.5,\) \(u_{2}=0.5,\) \(q=0.1,\) and \(r=0.1.\) Now the only equilibrium which exists is the one in which citizens with \(a=2\) and \(a=1.8\) become educated. Inserting the tentative equilibrium tax rate associated with the other equilibria into Eq. 26 shows that the associated ability threshold violates the initial assumption on who becomes educated.

###
*Proof of Proposition 4*

Assume that \((\widehat{a},\widehat{\tau })\) is a SSSIE with voting with commitment. It is sufficient to prove that with IICVS, the citizens either do not want to deviate or, if they would like to deviate, then their deviation does not change the outcome of the voting in a way that would result in the collapse of the intergenerational contract. The elderly have clearly no interest in deviating from voting for the \(\tau \) and \(\widetilde{ a},\) which would maximize their current social security benefits. Neither do the middle-aged have any incentive to deviate from the \(\tau \) and \( \widetilde{a} \) they would prefer with commitment. A deviation downward in the wage tax rate or the provision of education would only result in them losing their social security benefits in the following period. The young uneducated, on the other hand, already vote for the \(\widetilde{a}\) and \( \tau \) that would maximize their lifetime utility, so they have no incentive to deviate. As for the young citizens who are going to become educated, they are, in any case, in minority when voting on the provision of education, so any deviation in that dimension by them would have no effect on the voting outcome. When voting on the wage tax rate, the young who are going to lose from income redistribution would prefer to have the wage taxation and public provision of education abolished in future. However, they are already voting for a lower wage tax rate than the median voter, so that any deviation downward would not affect the outcome of the voting. The only way in which the young who prefer a lower wage tax rate to that preferred by the median voter can change the outcome of voting is by voting for a higher wage tax rate than that preferred by the median voter. By the definition of IICVS, a deviation upward would not cause the abolition of wage taxation and public provision of education.

Therefore, the young who will become educated cannot gain anything by deviating from voting for their preferred wage tax rate with voting with commitment. The threat point of the voting equilibrium (\( \overline{a},0\)) following a punishable deviation is also a subgame perfect Nash-equilibrium. If the young and the middle-aged expect that social security benefits will not be maintained in the future, they have no interest in maintaining them after a deviation. This implies that the middle-aged would join the elderly in opposing any investment in public education. The elderly would still vote for \(\tau >0\) and the young, whose ability exceeds \((1+r)u_{1}+u_{2},\) for \(\widetilde{a}<\overline{a},\) but both are in minority.

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Poutvaara, P. On the political economy of social security and public education.
*J Popul Econ* **19**, 345–365 (2006). https://doi.org/10.1007/s00148-006-0063-1

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DOI: https://doi.org/10.1007/s00148-006-0063-1

### Keywords

- Social security
- Public education
- Voting

### JEL Classification

- H52
- H55
- D72