Pensions and fertility: back to the roots

Abstract

Fertility has long been declining in industrialised countries and the existence of public pension systems is considered as one of the causes. This paper provides detailed evidence on the mechanism by which a public pension system depresses fertility, based on historical data. Our theoretical framework highlights that the effect of a public pension system on fertility is ex ante ambiguous while its size is determined by the internal rate of return of the pension system. We identify an overall negative effect of the introduction of pension insurance on fertility using regional variation across 23 provinces of Imperial Germany in key variables of Bismarck’s pension system, which was introduced in Imperial Germany in 1891. The negative effect on fertility is robust to controlling for the traditional determinants of the first demographic transition as well as to other policy changes.

Appendices

Appendix A: details on the theoretical model

1.1 A.1 Second order conditions

In the model of the Bismarckian pay as you go pension system the second derivatives of Eqs. 5, 6 and 7 are given by:

$$\begin{array}{@{}rcl@{}} V_{nn} &=&-U_{c}(1-\tau )w_{t}f^{\prime \prime }(n_{t})-U_{z}{\Omega}_{t+1}\tau w_{t+1}f^{\prime \prime }(n_{t}) \\ & & +U_{cc}\left[ (1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t}\right]^{2} \\ & & +U_{zz} \left[ b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime }\left( n_{t}\right) \right]^{2}+U_{nn}<0, \end{array} $$

(26)

$$ V_{ns}=V_{sn}=U_{cc}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t})+U_{zz}\left[ b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime }\left( n_{t}\right) \right] R_{t+1}, $$

(27)

$$ V_{ss}=U_{cc}+U_{zz}R_{t+1}^{2}<0, $$

(28)

$$ V_{bn}=V_{nb}=U_{cc}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t})<0, $$

(29)

$$ V_{bs}=V_{sb}=U_{cc}<0, $$

(30)

$$ V_{bb}=U_{cc}+n_{t-1}^{2}U_{z_{t}z_{t}}<0. $$

(31)

The second-order conditions for a maximum of problem (4) are satisfied since V _nn is negative and the following conditions hold true:

$$\begin{array}{@{}rcl@{}} V_{nn}V_{ss}-V_{ns}V_{sn} &=&(U_{cc}+U_{zz}R_{t+1}^{2}) \\ &&\cdot \left[ U_{nn}-U_{c}(1-\tau )w_{t}f^{\prime \prime }(n_{t})-U_{z}{\Omega}_{t+1}\tau w_{t+1}f^{\prime \prime }(n_{t})\right] \\ &&+U_{cc}U_{zz} \\ &&\cdot \left[ R_{t+1}((1\,-\,\tau )w_{t}f^{\prime }(n_{t})+\pi_{t})-\left( b_{t+1}\!-{\Omega}_{t+1}\tau w_{t+1}f^{\prime }(n_{t})\right) \right]^{2} \\ &>&0, \end{array} $$

(32)

$$\begin{array}{@{}rcl@{}} \left\vert \begin{array}{ccc} V_{nn} & V_{ns} & V_{nb} \\ V_{sn} & V_{ss} & V_{sb} \\ V_{bn} & V_{bs} & V_{bb} \end{array} \right\vert &=&\left[ U_{nn}-U_{c_{t}}(1-\tau )w_{t}f^{\prime \prime }(n_{t})-U_{z_{t+1}}{\Omega}_{t+1}\tau w_{t+1}f^{\prime \prime }(n_{t})\right] \\ &&\cdot \left( R_{t+1}^{2}U_{c_{t}c_{t}}U_{z_{t+1}z_{t+1}}+n_{t-1}^{2}U_{z_{t}z_{t}}\left( U_{c_{t}c_{t}}+U_{z_{t+1}z_{t+1}}R_{t+1}^{2}\right) \right) \\ &&+U_{c_{t}c_{t}}U_{z_{t+1}z_{t+1}}U_{z_{t}z_{t}}n_{t-1}^{2} \\ &&\cdot \left[ R_{t+1}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t})-\left( b_{t+1}-{\Omega}_{t+1}\tau w_{t+1}f^{\prime }(n_{t})\right) \right]^{2} \\ &<&0. \end{array} $$

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This demonstrates that the objective function V(n _t ,s _t ,b _t ) is strictly concave in the decision variables.

1.2 A.2 The savings effect of the Bismarckian PAYG pension system

The impact of extending the pension system on savings is given by:

$$ \frac{\partial s}{\partial \tau }=-\frac{\left\vert \begin{array}{ccc} V_{nn} & V_{n\tau } & V_{nb} \\ V_{sn} & V_{s\tau } & V_{sb} \\ V_{bn} & V_{b\tau } & V_{bb} \end{array} \right\vert }{\left\vert \begin{array}{ccc} V_{nn} & V_{ns} & V_{nb} \\ V_{sn} & V_{ss} & V_{sb} \\ V_{bn} & V_{bs} & V_{bb}. \end{array} \right\vert } $$

(34)

With the negative denominator savings decrease with a higher contribution rate to the PAYG system if the numerator is negative.

In the case of the Bismarckian pension system the numerator is given by

$$\begin{array}{@{}rcl@{}} \left\vert \begin{array}{ccc} V_{nn} & V_{n\tau } & V_{nb} \\ V_{sn} & V_{s\tau } & V_{sb} \\ V_{bn} & V_{b\tau } & V_{bb} \end{array} \right\vert &=&w_{t}(1-f(n_{t}))\left( U_{n_{t}n_{t}}-U_{c_{t}}(1-\tau )w_{t}f^{\prime \prime }(n_{t})-U_{z_{t+1}}{\Omega}_{t+1}\tau w_{t+1}f^{\prime \prime }(n_{t})\right) \\ &&\left[ n_{t-1}^{2}U_{c_{t}c_{t}}U_{z_{t}z_{t}}+{\Omega}_{t+1}R_{t+1}\left( U_{c_{t}c_{t}}U_{z_{t+1}z_{t+1}}+n_{t-1}^{2}U_{z_{t}z_{t}}U_{z_{t+1}z_{t+1}} \right) \right] \\ &&-U_{z_{t+1}}w_{t}f^{\prime }(n_{t})(R_{t+1}-{\Omega}_{t+1})\left[ U_{c_{t}c_{t}}U_{z_{t}z_{t}}n_{t-1}^{2}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t})\right. \\ &&\left. +\left( U_{c_{t}c_{t}}U_{z_{t+1}z_{t+1}}+n_{t-1}^{2}U_{z_{t}z_{t}}U_{z_{t+1}z_{t+1}} \right) R_{t+1}\left( b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime }(n_{t})\right) \right] \\ &&+U_{c_{t}c_{t}}U_{z_{t+1}z_{t+1}}U_{z_{t}z_{t}}w_{t}(1-f(n_{t}))n_{t-1}^{2} \\ &&\left[ R_{t+1}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t})-\left( b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime }(n_{t})\right) \right] \\ &&\left[ {\Omega}_{t+1}(w_{t}f^{\prime }(n_{t})+\pi_{t})-b_{t+1}\right]. \end{array} $$

(35)

Since the price of a child is positive, R _t+1((1−τ)w _t f ^′(n _t ) + π _t )−(b _t+1−Ω _t+1 τ w _t f ^′(n _t ))>0, the numerator is negative if the following condition for the intra-family transfer b _t+1 holds: $\tau w_{t}f^{\prime }(n_{t})<\frac {b_{t+1}}{{\Omega }_{t+1}} <w_{t}f^{\prime }(n_{t})+\pi _{t}$. If this condition holds, savings decrease with a higher contribution rate in the Bismarckian system. The condition is equivalent to: $-\frac {\partial p_{t+1}^{BIS}}{\partial n_{t}} <b_{t+1}<{\Omega }_{t+1}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi _{t})-\frac { \partial p_{t+1}^{BIS}}{\partial n_{t}}$. Note that the Bismarckian pension decreases in the number of children because the pension is proportional to contributions and income which decreases with more children. Thus, we have according to Eq. 9: $\frac {\partial p_{t+1}^{BIS}}{ \partial n_{t}}<0.$ We can rewrite the condition for a negative savings effect as ${\Omega }_{t+1}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi _{t})>b_{t+1}+ \frac {\partial p_{t+1}^{BIS}}{\partial n_{t}}>0$ which can be interpreted as follows.

Assume that a higher contribution rate reduces the number of children, i.e. the income effect is larger than the price effect. The second part of the inequality condition means that the loss of intra-family transfer due to fewer children in the second period is higher than the gain of a larger Bismarckian pension. Thus, having fewer children reduces income and decreases consumption in the second period. Then the first part of the condition implies that the discounted reduction of (opportunity) costs for children in the first period is higher than the loss of income in the second period due to fewer children. Hence, a lower number of children increases income and consumption in the first period by more than it reduces consumption in the second period. This implies that the parents react with lower savings in order to re-establish their preferred consumption profile and compensate the negative effect of the contribution rate on the number of children. If saved costs of fewer children in the first period are higher than the income loss in the second period a lower number of children induces lower savings.

Proposition 2

Savings effect The introduction or expansion of the PAYG system reduces savings if the lower number of children raises income in the first period to a larger extent than it lowers income in the second period.

1.3 A.3 The effect of a Bismarckian PAYG pension system on the intra-family transfer

The effect of a higher contribution rate on the intra-family transfer is given by:

$$ \frac{\partial b_{t}}{\partial \tau }=-\frac{\left\vert \begin{array}{ccc} V_{nn} & V_{ns} & V_{n\tau } \\ V_{sn} & V_{ss} & V_{s\tau } \\ V_{bn} & V_{bs} & V_{b\tau } \end{array} \right\vert }{\left\vert \begin{array}{ccc} V_{nn} & V_{ns} & V_{nb} \\ V_{sn} & V_{ss} & V_{sb} \\ V_{bn} & V_{bs} & V_{bb}. \end{array} \right\vert } $$

(36)

The numerator can be calculated as:

$$\begin{array}{@{}rcl@{}} \left\vert \begin{array}{ccc} V_{nn} & V_{ns} & V_{n\tau } \\ V_{sn} & V_{ss} & V_{s\tau } \\ V_{bn} & V_{bs} & V_{b\tau } \end{array} \right\vert &=&\left( U_{n_{t}n_{t}}-U_{c_{t}}(1-\tau )w_{t}f^{\prime \prime }(n_{t})-U_{z_{t+1}}{\Omega}_{t+1}\tau w_{t+1}f^{\prime \prime }(n_{t})\right) \\ &&w_{t}(1-f(n_{t}))U_{c_{t}c_{t}}U_{z_{t+1}z_{t+1}}R_{t+1}\left( R_{t+1}-{\Omega}_{t+1}\right) \\ &&-U_{z_{t+1}}w_{t}f^{\prime }(n_{t})(R_{t+1}-{\Omega}_{t+1})R_{t+1}U_{c_{t}c_{t}}U_{z_{t+1}z_{t+1}} \\ &&\left[ R_{t+1}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t}) -\left( b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime }(n_{t})\right) \right]\\ \end{array} $$

(37)

With R _t+1>Ω _t+1 and a positive price of a child, R _t+1((1−τ)w _t f ^′(n _t ) + π _t )−(b _t+1−Ω _t+1 τ w _t f ^′(n _t ))>0, the parents reduce the intra-family transfer if the PAYG system is extended: $\frac {\partial b_{t}}{ \partial \tau }<0.$ The intuition for this result is that a higher contribution rate together with R _t+1>Ω _t+1 reduces lifetime income since the contribution contains an implicit tax on wage income. The parents reduce their transfer to the grandparents in order to compensate for this loss in lifetime income. This reduces the old-age consumption of the grandparents.

Proposition 3

Effect on intra-family transfer The introduction or expansion of the PAYG system induces the parents to reduce the intra-family transfer to the grandparents.

1.4 A.4 Lack of capital markets

1.4.1 A.4.1 The fertility effect

If we assume that individuals have no possibility to provide for old age by savings the budget constraints in both periods are given by:

$$\begin{array}{@{}rcl@{}} c_{t} &=&w_{t}(1-f(n_{t}))(1-\tau )-\pi_{t}n_{t}-b_{t}, \\ z_{t+1} &=&p_{t+1}+b_{t+1}n_{t}, \end{array} $$

where the pension in a Bismarckian system is determined by Eq. 8. Again, the first-order condition (5) holds. The implicit function theorem yields:

$$\frac{\partial n}{\partial \tau }=-\frac{V_{n\tau }V_{bb}-V_{nb}V_{b\tau } }{V_{nn}V_{bb}-V_{nb}V_{bn}}, $$

and V _nn<0 by Eq. 26 and

$$\begin{array}{@{}rcl@{}} &&V_{\text{nn}}V_{\text{bb}}-V_{\text{{nb}}}V_{\text{b{n}}} \\ &=&\left[ -U_{c}(1-\tau )w_{t}f^{\prime \prime }(n_{t})-U_{z_{t+1}}{\Omega}_{t+1}\tau w_{t+1}f^{\prime \prime }(n_{t})\right. \\ &&\left. +U_{z_{t+1}z_{t+1}}\left( b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime }\left( n_{t}\right) \right)^{2}+U_{\text{nn}}\right] \end{array} $$

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$$\begin{array}{@{}rcl@{}} &&\left( U_{\text{cc}}+n_{t-1}^{2}U_{z_{t}z_{t}}\right) +n_{t-1}^{2}U_{z_{t}z_{t}}U_{\text{cc}}\left( (1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t}\right)^{2} \\ &>&0 \end{array} $$

(39)

satisfy the second-order condition. Hence, the fertility response with respect to an introduction or extension of the pension system is determined by the sign of V _{n τ} V _bb−V _nb V _{b τ} :

$$\begin{array}{@{}rcl@{}} V_{n\tau }V_{bb}-V_{nb}V_{b\tau } &=&(R_{t+1}-{\Omega}_{t+1})w_{t}f^{\prime }(n_{t})U_{z_{t+1}}\left( U_{cc}+n_{t-1}^{2}U_{z_{t}z_{t}}\right) \\ &&+w_{t}(1-f(n_{t})) \\ &&\left[ U_{cc}U_{z_{t}z_{t}}n_{t-1}^{2}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t})\right. \\ &&\left. +U_{z_{t+1}z_{t+1}}\left( b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime}(n_{t})\right) {\Omega}_{t+1}\left( U_{cc}+n_{t-1}^{2}U_{z_{t}z_{t}}\right) \right] . \end{array} $$

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If R _t+1>Ω _t+1, a higher contribution rate τ decreases the marginal price of a child which incites more children:

$$(R_{t+1}-{\Omega}_{t+1})w_{t}f^{\prime }(n_{t})U_{z_{t+1}}>0. $$

The second summand on the RHS is again the income effect. A higher contribution rate decreases income in the first period by w _t (1−f(n _t )) and raises pension income in the second period by Ω _t+1 w _t (1−f(n _t )). Reducing the number of children compensates the income loss in period 1 by the expenditure (1−τ)w _t f ^′(n _t ) + π _t per child and decreases the income in period 2 if b _t+1>Ω _t+1 τ w _t f ^′(n _t ), in other words, if the intra family transfer is larger than the Bismarck pension loss due to another child. Smoothing consumption across periods increases utility of the household so that due to the income effect fertility decreases with a higher contribution rate.

Hence, the size of the intra-family transfer determines the income effect and whether it is larger than the first (price) effect in which case fertility decreases with a higher contribution rate.

Corollary

Constrained investment effect in a pay as you go Bismarckian pension system In economies with lacking capital markets to provide for old-age the introduction or expansion of a Bismarckian pay-as-you-go pension scheme reduces the number of children if the intra-family transfers are sufficiently large.

1.4.2 A.4.2 The effect on intra-family transfer

In this case, the effect on intra-family transfer is given by:

$$\frac{\partial b_{t}}{\partial \tau }=-\frac{V_{nn}V_{b\tau }-V_{{n\tau } }V_{bn}}{V_{nn}V_{bb}-V_{nb}V_{bn}}. $$

With a positive denominator (39) the effect of intra-family transfer by a larger PAYG system depends on the sign of the numerator:

$$\begin{array}{@{}rcl@{}} &&V_{nn}V_{b\tau }-V_{{n\tau }}V_{bn} \\ &=&\left[ -U_{c}(1-\tau )w_{t}f^{\prime \prime }(n_{t})-U_{z_{t+1}}{\Omega}_{t+1}\tau w_{t+1}f^{\prime \prime }(n_{t})\right. \\ &&\left. +U_{z_{t+1}z_{t+1}}\left( b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime }\left( n_{t}\right) \right)^{2}+U_{nn}\right] w_{t}(1-f(n_{t}))U_{cc} \\ &&-\left[ (R_{t+1}-{\Omega}_{t+1})w_{t}f^{\prime }(n_{t})U_{z_{t+1}}+U_{z_{t+1}z_{t+1}}\left( b_{t+1}-{\Omega}_{t+1}\tau w_{t}f^{\prime }\left( n_{t}\right) \right) {\Omega}_{t+1}\right] \\ &&U_{cc}((1-\tau )w_{t}f^{\prime }(n_{t})+\pi_{t}). \end{array} $$

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The sign is ambiguous, in particular if b _t+1>Ω _t+1 τ w _t f ^′(n _t ).

Appendix B: supplementary (online) table on a comparison of estimators

Table 6 presents an OLS model in column (1), our baseline model in column (2) and a first differences estimator in column (3).

Table 6 Sensitivity: estimation approach

A standard OLS model would suffer from several endogeneity issues, such as clustered standard errors and serial, potentially also spatial correlation. Presenting the OLS model (with standard errors robust to at least serial correlation and clustering at the province level) in this context helps to illustrate the importance of controlling for the unobserved fixed effects. In particular, note that the OLS estimates differ in two important respects from our baseline model. First, the coefficients from our baseline model tend to be either overestimated or underestimated by the OLS approach. Second, even though standard errors are adjusted for some clustering as well as for serial correlation, the OLS model sometimes indicates significant estimates while the fixed effects model does not. At the same time, the OLS model is able to indicate the relative size of the different effects fairly well.

In theory, first differencing should yield exactly the same inference as a fixed effects model when the fixed effects model is applied to only two time periods. This is illustrated when comparing columns (2) and (3). The coefficients are the same while standard errors are larger in the model in first differences. This should not be surprising given that the first differences model is less efficient. Losing a degree of freedom in a model with only a small number of cross-sectional observations potentially has a big impact on the precision of the estimates. However, the coefficients in the first differences model are not substantially different from our baseline model and as conjectured.