Abstract
Old age pensions and public education account for a large share of public budgets. We link both programs through a taxtransfer system that is also sensitive to labor market distortions. We analyze the impact that alternative pension reforms have, through the political process, on publicly financed education. We explain how changes in the pension system design affect the link between the two programs and also labor market incentives. These effects, if they exist, act in opposite directions. Overall, we find that most proposals that entail a partial privatization of pensions reduce the willingness of the society to fund public education.
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Notes
See the book by Gruber and Wise (1999) for an excellent description of the problems that pension systems face in several OECD countries.
Poutvaara (2003) suggests an alternative mechanism through the market of land to explain the provision of intergenerational goods.
In this model, we abstract from the issue of intergenerational altruism to stress the importance of the interrelation between both transfer schemes when evaluating reforms to the social security.
Sommacal (2006) analyzes the consequences of a pension reform for lifetime income inequality. In particular, he shows that reallocating pension funds from an earningsrelated pension to a flatrate pension will reduce lifetime inequality with exogenous labor supply. Such an effect may, however, disappear once labor supply responses are considered.
See, among others, Diamond and Orszag (2002). To give an idea of the controversy about reforming social security and that the social security tax is much more politically sensitive than taxes for other public programs (Medicare, education), the huge amount of articles in the media devoted to covering this issue in the last presidential American campaigns is remarkable (see, for example, Kessler (2008) at washingtonpost.com).
We are implicitly assuming that this productivity level was acquired when young through schooling.
We model labor supply as a binary choice for two reasons. First, for simplicity. Second, because labor supply responses at the extensive margin (whether or not to work) seem to be much more important than at the intensive margin (hours worked for those who are working). Indeed, this is a central finding in the empirical literature of Labor Economics.
As the stock of parents’ human capital is given, replacing Eq. 6 by an specification where \(h_{1}^{i}\) is independent of \(h_{0}^{i}\) would not change our results.
\(T_{1}^{P}=\int_{i\in W_{1}}\lambda _{1}^{P}h_{1}^{i}dG(h_{1}^{i})=\lambda _{1}^{P}H_{1}\), where W _{1} and H _{1} denote the set of workers and their average human capital in period 1, respectively.
This result is similar to the one obtained by Cigno (2008).
Since expenditure in the pensions of those currently retired in period 0 is \( T_{0}^{P}=\lambda _{0}^{P}H_{0}\), the value of \(T_{0}^{P}\) is also maximized when τ = 0.
In the “Appendix”, we discuss the strict concavity of \(b_{1}^{i}(\tau )\).
This effect will be small if α is close to 1.
See the “Appendix” for a formal proof.
Also note that our assumption that φ ^{′}(0) = + ∞ implies that τ ^{*} > 0.
The human capital threshold value C can take any value in the interval [a, b]. Nevertheless, the more interesting case to be analyzed is when C > B.
Provided that λ _{1} = λ _{0}. If they are different, the return is λ _{1} H _{1}/λ _{0} H _{0} − 1.
Here the fact that labor supply is endogenous is crucial. If labor supply is exogenous, the cost is entirely borne by workers.
We skip the detailed analysis of this point as it is exactly the same as in the first reform.
However, making the pension system more redistributive can have a negative impact on labor supply as we mentioned in Section 3.
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Acknowledgements
We would like to thank Georges Casamatta, M. Dolores Collado, Ignacio CondeRuiz, Ana Guerrero, Carmen Herrero, Xavier Raurich, Pablo Revilla, and William Thomson for helpful comments. We are also especially thankful to two anonymous referees and the editor in charge whose comments helped us to improve the quality of the manuscript and to shape the exposition. Financial support from the Ministerio de Ciencia e Innovación and FEDER funds (projects SEJ200762656 and ECO200803674/ECON), Junta de Andalucía (SEJ02905 and SEJ426), and Instituto Valenciano de Investigaciones Econó micas (IVIE) is gratefully acknowledged.
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Appendix
Appendix
Discussion of the concavity of y ^{i}(τ)
We compute the second derivative of y ^{i}(τ) with respect to τ:
The sign of \(\frac{\partial ^{2}H_{1}}{\partial \tau ^{2}}\) is negative because H _{1}(τ) is concave with respect to τ. The sign of \( \frac{\partial ^{2}D^{i}}{\partial \tau ^{2}}\) is also negative:
Since H _{0}(τ) is a decreasing and convex function of τ, this above expression is negative. Finally, we have the term \(2\frac{\partial H_{1}(\tau )}{\partial \tau }\frac{\partial D^{i}}{\partial \tau }\), which is positive for τ < τ ^{L}. This term must be small enough for the expression to be negative.
Proof that all workers prefer a tax lower than τ ^{L}
To see this, we need to prove that the derivative of y ^{i}(τ) evaluated at τ = τ ^{L} is negative.
Rewriting Eq. 23, we have:
As H _{1}(τ) is maximized when τ = τ ^{L}, \(\frac{\partial H_{1} }{\partial \tau }\) \(\mid _{\tau =\tau ^{L}}=0\). Thus, \(\frac{\partial y^{i}(\tau )}{\partial \tau }\mid _{\tau =\tau ^{L}}=h_{0}^{i}+\frac{1}{R} \lambda _{1}^{P}H_{1}\frac{\partial D^{i}(\alpha )}{\partial \tau }\).
This is negative provided that:
Defining the elasticity of H _{0} with respect to τ as \( E_{H_{0}}^{\tau }=\frac{\partial H_{0}}{\partial \tau }\frac{\tau }{H_{0}}\) and rewriting the above expression to include \(E_{H_{0}}^{\tau }\) evaluated at τ = τ ^{L} , we find that this will be the case if:
Just to give a quick intuition for this result, consider that R = H _{1}/H _{0}, \(\lambda _{1}^{P}=0.25,\alpha =\tau ^{L}=0.5\). The condition becomes \(E_{H_{0}}^{\tau }>4\). The condition requires that the elasticity of labor supply with respect to the tax cannot be very large.□
Proof that τ ^{i} decreases with \(h_{0}^{i}\)
The ideal tax rate for an individual, τ ^{i}, is defined implicitly by:
By the implicit function theorem, we get:
As \(\frac{\partial ^{2}y^{i}(\tau ^{i})}{\partial \tau ^{2}}<0\), the sign of \(\frac{\partial \tau ^{i}}{\partial h_{0}^{i}}\) will be the sign of the term in the numerator. This is negative if:
Since R > 0 , \(\frac{\partial ^{2}b_{1}^{i}}{\partial \tau \partial h_{0}^{i} }<R\) is a sufficient condition for \(\frac{\partial \tau ^{i}}{\partial h_{0}^{i}}<0\). □
Proof that \(\widehat{h}_{0}\) cannot decrease when borrowing is not allowed
Consider all those individuals with human capital below \(\widehat{h }_{0}\). Their savings are described by Eq. 15. If \( I\geqslant b_{1}^{\min }\), \(s_{0}^{\ast }\geqslant 0\). The constraint \( s_{0}^{i}\geq 0\) is not binding for them and not working is still optimal. If \(I<b_{1}^{\min }\), the constraint \(s_{0}^{i}\geq 0\) is binding for them. If they do not work, their preferred choice is \((\widehat{c}_{0},\widehat{c} _{1})=(I,b_{1}^{\min })\), while it is \((\widehat{c}_{0},\widehat{c} _{1})=((1\tau \lambda _{0})h_{0}^{i}d,b_{1}^{i})\) if they choose to work. Given that these individuals prefer not to work when borrowing is allowed, for all of them it must be true that:
Given that \(b_{1}^{i}\geq b_{1}^{\min }\), it must be true that \(I>(1\tau \lambda _{0})h_{0}^{i}d\). This implies that the bundle \(((1\tau \lambda _{0})h_{0}^{i}d,b_{1}^{i})\) is always strictly contained in the budget set corresponding to the choice of not working. Then, the bundle \((I,b_{1}^{\min })\) is always strictly preferred, meaning that the individual still prefers not to work.□
Proof that \(\widehat{\tau }^{\,i}\) increases with \( h_{0}^{i}\)
Applying the implicit function theorem in Eq. 25, we get:
As \(\frac{\partial b_{1}^{i}}{\partial \tau }>0\), the sign of \(\frac{ \partial \widehat{\tau }^{\,i}}{\partial h_{0}^{i}}\) will be the sign of the term in the numerator. Since \((1\widehat{\tau }^{\,i}\lambda _{0})h_{0}^{i}=d+b_{1}^{i}\), we have:
since \(b_{1}^{i}>\frac{\partial b_{1}^{i}}{\partial h_{0}^{i}}h_{0}^{i}\).□
Proof of Proposition 1
Properties (i), (ii), and (iv) are immediate. We prove property (iii). To simplify notation, we write \(V^{ir}(\tau )=u[\widehat{c_{0}^{i}} ]+\rho u[\widehat{c_{1}^{i}}]\), where \(\widehat{c_{0}^{i}}=(1\tau \lambda _{0})h_{0}^{i}d\) and \(\widehat{c_{1}^{i}}=b_{1}^{i}\left( \tau \right) \). The preferred tax rate for a savingconstrained worker satisfies the following firstorder condition:
Rewriting, we have:
Due to the perfect consumption smoothing, we have that \(\frac{u^{\prime }[c_{0}^{i\ast }]}{u^{\prime }[c_{1}^{i\ast }]}=1\). At \((\widehat{c_{0}^{i}}, \widehat{c_{1}^{i}})\), the worker is savingconstrained, which means that \( \widehat{c_{0}^{i}}<c_{0}^{i\ast }\) and \(\widehat{c_{1}^{i}}>c_{1}^{i\ast }\). But then, \(\frac{u^{\prime }[\widehat{c_{0}^{i}}]}{u^{\prime }[\widehat{ c_{1}^{i}}]}>1\). This implies that τ ^{ir} < τ ^{i}.□
Proof of Proposition 2
The function W ^{i}(τ) can have three different shapes, depending on the relationship between τ ^{i} (the preferred tax rate without restrictions on savings) and \(\widehat{\tau }^{\,i}:\)

Case 1:
\(\tau ^{i}\!<\!\widehat{\tau }^{\,i}\). Then, W ^{i}(τ) is singlepeaked since, to the right of \(\widehat{\tau }^{\,i}\), W ^{i}(τ) = V ^{ir}(τ) and V ^{ir}(τ) is a decreasing function to the right of \(\widehat{\tau }^{\,i}\). The preferred tax rate is still τ ^{i}.

Case 2:
\(\tau ^{i}>\widehat{\tau }^{\,i}\). To the right of \(\widehat{\tau } ^{\,i}\), W ^{i}(τ) = V ^{ir}(τ). As both V ^{i}(τ) and V ^{ir}(τ) are strictly concave and \(V^{i}(\widehat{\tau }^{\,i})=V^{ir}( \widehat{\tau }^{\,i})\), the preferred tax rate of i is τ ^{ir}, and not τ ^{i}. Moreover, τ ^{ir} < τ ^{i}. This follows from the first order conditions 23 and 28 (see Proposition 2).

Case 3:
\(\tau ^{i}=\widehat{\tau } ^{i}\). In this case, we also have τ ^{i} = τ ^{ir}. The preferred tax rate of this agent is τ ^{i}.
□
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IturbeOrmaetxe, I., Valera, G. Social security reform and the support for public education. J Popul Econ 25, 609–634 (2012). https://doi.org/10.1007/s0014801003384
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DOI: https://doi.org/10.1007/s0014801003384
Keywords
 Social security reform
 Public education
 Voting
JEL Classification
 D72
 H55
 I22