Social security reform and the support for public education

Abstract

Old age pensions and public education account for a large share of public budgets. We link both programs through a tax-transfer 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.

Appendix

1.1 Discussion of the concavity of y ⁱ(τ)

We compute the second derivative of y ⁱ(τ) with respect to τ:

$$ \frac{\partial ^{2}y^{i}(\tau )}{\partial \tau ^{2}}\!=\!\frac{1}{R}\frac{ \partial ^{2}b_{1}^{i}(\tau )}{\partial \tau ^{2}}\!=\!\frac{1}{R}\lambda _{1}^{P}\left[ D^{i}(\alpha )\frac{\partial ^{2}H_{1}(\tau )}{\partial \tau ^{2}}\!+\!2\frac{\partial H_{1}(\tau )}{\partial \tau }\frac{\partial D^{i}}{ \partial \tau }\!+\!H_{1}(\tau )\frac{\partial ^{2}D^{i}}{\partial \tau ^{2}} \right] . $$

(30)

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:

$$ \frac{\partial ^{2}D^{i}}{\partial \tau ^{2}}=-(1-\alpha )\frac{h_{0}^{i}}{ \left( H_{0}\right) ^{2}}\left[ \frac{\partial ^{2}H_{0}(\tau )}{\partial \tau ^{2}}-\frac{2}{H_{0}}\frac{\partial H_{0}(\tau )}{\partial \tau }\right] . $$

(31)

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:

$$ \frac{\partial y^{i}(\tau )}{\partial \tau }=-h_{0}^{i}+\frac{1}{R}\lambda _{1}^{P}\left[ D^{i}(\alpha )\frac{\partial H_{1}}{\partial \tau }+H_{1} \frac{\partial D^{i}(\alpha )}{\partial \tau }\right] . $$

(32)

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:

$$ -\frac{1}{R}\lambda _{1}^{P}(1-\alpha )\frac{H_{1}}{\left( H_{0}\right) ^{2}} \frac{\partial H_{0}}{\partial \tau }<1. $$

(33)

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 re-writing the above expression to include $E_{H_{0}}^{\tau }$ evaluated at τ = τ ^L , we find that this will be the case if:

$$ E_{H_{0}}^{\tau }>-\frac{\tau ^{L}H_{0}R}{\lambda _{1}^{P}(1-\alpha )H_{1}}. $$

(34)

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:

$$ \frac{\partial y^{i}(\tau )}{\partial \tau }=-h_{0}^{i}+\frac{1}{R}\frac{ \partial b_{1}^{i}(\tau )}{\partial \tau }=0. $$

(35)

By the implicit function theorem, we get:

$$ \frac{\partial \tau ^{i}}{\partial h_{0}^{i}}=-\frac{-1+\displaystyle\frac{1}{R}\frac{ \partial ^{2}b_{1}^{i}(\tau ^{i})}{\partial \tau \partial h_{0}^{i}}}{\displaystyle\frac{ \partial ^{2}y^{i}(\tau ^{i})}{\partial \tau ^{2}}}. $$

(36)

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:

$$ \frac{1}{R}\frac{\partial ^{2}b_{1}^{i}(\tau ^{i})}{\partial \tau \partial h_{0}^{i}}<1. $$

(37)

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:

$$ I+\frac{b_{1}^{\min }}{R}>(1-\tau -\lambda _{0})h_{0}^{i}-d+\frac{b_{1}^{i}}{ R}. $$

(38)

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:

$$ \frac{\partial \widehat{\tau }^{\,i}}{\partial h_{0}^{i}}=-\frac{(1-\widehat{ \tau }^{i}-\lambda _{0})-\frac{\partial b_{1}^{i}}{\partial h_{0}^{i}}}{ -h_{0}^{i}-\frac{\partial b_{1}^{i}}{\partial \tau }}. $$

(39)

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:

$$ \left( d+b_{1}^{i}\right) -\frac{\partial b_{1}^{i}}{\partial h_{0}^{i}} h_{0}^{i}>0, $$

(40)

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 saving-constrained worker satisfies the following first-order condition:

$$ V^{ir\prime }(\tau ^{ir})=-h_{0}^{i}u^{\prime }[\widehat{c_{0}^{i}}]+\rho \frac{\partial b_{1}^{i}\left( \tau ^{ir}\right) }{\partial \tau }u^{\prime }[\widehat{c_{1}^{i}}]=0. $$

(41)

Rewriting, we have:

$$ h_{0}^{i}\frac{u^{\prime }[\widehat{c_{0}^{i}}]}{u^{\prime }[\widehat{ c_{1}^{i}}]}=\frac{1}{R}\frac{\partial b_{1}^{i}\left( \tau ^{ir}\right) }{ \partial \tau }. $$

(42)

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 saving-constrained, 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 single-peaked 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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