Why mandate young borrowers to contribute to their retirement accounts?

Abstract

Many countries, in an effort to address the problem that many retirees have too little saved up, impose mandatory contributions into retirement accounts, that too, in an age-independent manner. This is puzzling because such funded pension schemes effectively mandate the young, the natural borrowers, to save for retirement. Further, present-biased agents disagree with the intent of such schemes and attempt to undo them by reducing their own saving or even borrowing against retirement wealth. We establish a welfare case for mandating the middle-aged and the young to contribute to their retirement accounts, even with age-independent contribution rates. We find, somewhat counterintuitively, that even though the young responds by borrowing more, that too at a rate higher than offered by pension savings, their lifetime utility increases.

Appendices

Appendix

Borrowing by the young

The savings decision of the middle-aged ($s_{m}$) depends on borrowing as young ($b_{y}$), and this is perceived by the (sophisticated) young. Hence, the two decisions are interconnected. The main text works with the case where agents in the absence of mandatory pension savings are borrowers as young ($b_{y}>0$) and savers as middle-aged ($s_{m}>0$), since this is the case where the welfare rationale for mandatory savings (especially for the young) may be called into question. If the young are savers, there is no essential difference between being young and middle-aged.

In the absence of mandatory savings, it is trivial that the middle-aged are saving ($s_{m}>0$) since this is the only way of ensuring consumption when old. From the main text—see Eq. (10), we have that

$$\begin{aligned} \frac{\partial \Omega }{\partial b_{y}}=u^{\prime }\left( c_{y}\right) +{\widetilde{\delta }}\left[ \delta -{\widetilde{\delta }}\right] Ru^{\prime }\left( c_{o}\right) \frac{\partial s_{m}}{\partial b_{y}}-\widetilde{\delta }R_{b}u^{\prime }\left( c_{m}\right) . \end{aligned}$$

If $\frac{\partial \Omega }{\partial b_{y}}\mid _{b_{y}=0}>0$ it follows that $b_{y}>0$ which in turn requires

$$\begin{aligned} u^{\prime }\left( w_{y}\right) >{\widetilde{\delta }}R_{b}u^{\prime }\left( w_{m}-s_{m}\right) -{\widetilde{\delta }}\left[ \delta -\widetilde{\delta }\right] Ru^{\prime }\left( Rs_{m}\right) \frac{\partial s_{m}}{\partial b_{y}}, \end{aligned}$$

or using ${\widetilde{\delta }}Ru^{\prime }\left( Rs_{m}\right) =u^{\prime }\left( w_{m}-s_{m}\right) ,$

$$\begin{aligned} u^{\prime }\left( w_{y}\right) >u^{\prime }\left( w_{m}-s_{m}\right) \left[ {\widetilde{\delta }}R_{b}-\left( \delta -{\widetilde{\delta }}\right) \frac{\partial s_{m}}{\partial b_{y}}\right] . \end{aligned}$$

Note that ${\widetilde{\delta }}R_{b}-\left( \delta -{\widetilde{\delta }}\right) \frac{\partial s_{m}}{\partial b_{y}}<\delta R_{b}$ since $\frac{\partial s_{m}}{\partial b_{y}}>-R_{b}$. Thus, a sufficient condition ensuring $b_{y}>0$ is

$$\begin{aligned} u^{\prime }\left( w_{y}\right) >u^{\prime }\left( w_{m}-s_{m}\right) \delta R_{b}, \end{aligned}$$

(18)

where $s_{m}$ is determined from $u^{\prime }\left( w_{m}-s_{m}\right) =R{\widetilde{\delta }}u^{\prime }\left( Rs_{m}\right)$; hence $0<s_{m}<w_{m}$. Hence, condition (18) holds if the wage as young is not too high, i.e.,

$$\begin{aligned} w_{y}\le {\overline{w}}_{y}(w_{m},\delta ,{\widetilde{\delta }},R,R_{b}) \end{aligned}$$

(19)

Overborrowing by the young and undersaving by the middle-aged

This appendix proves that agents “overborrow” when young ($b_{y}>b_{y}^{*}$) and “undersave” when middle-aged ($s_{m}<s_{m}^{*}$) in the absence of any pension scheme ($\tau _{y}=\tau _{m}=0$). The savings decision of the middle-aged is determined by: $-u^{\prime }\left( w_{m}-R_{b}b_{y} -s_{m}\right) +{\widetilde{\delta }}Ru^{\prime }\left( Rs_{m}\right) =0$ and since $u^{\prime }\left( Rs_{m}\right) \rightarrow \infty$ for $s_{m} \rightarrow 0$, it follows that $s_{m}>0$. Intuitively, the only way by which the individual can ensure consumption as old is by saving as middle-aged. Under the true preferences we have that savings is determined by $-u^{\prime }\left( w_{m}-R_{b}b_{y}^{*}-s_{m}^{*}\right) +\delta ^{*} Ru^{\prime }\left( Rs_{m}^{*}\right) =0$. The savings as middle-aged depends on borrowing as young and we have $s_{m}=\psi (b_{y}),$ $s_{m}^{*}=\psi ^{*}(b_{y}^{*})$. Notice, that $\psi (b_{y})<\psi ^{*}(b_{y})$ since ${\widetilde{\delta }}R<\delta ^{*}R$, and both functions are decreasing in their argument.

The first-order condition for the borrowing decision by the young (11)

$$\begin{aligned} -u^{\prime }\left( c_{y}\right) +{\widetilde{\delta }}R\left[ \left( {\widetilde{\delta }}-\delta \right) \left( \frac{\partial D\left( b_{y}\right) }{\partial b_{y}}\right) +R_{b}{\widetilde{\delta }}\right] u^{\prime }\left( c_{o}\right) =0 \end{aligned}$$

can by use of $u^{\prime }(c_{m})={\widetilde{\delta }}Ru^{\prime }\left( c_{o}\right)$ be written

$$\begin{aligned} -u^{\prime }\left( w_{y}+b_{y}\right) +\left[ \left( \widetilde{\delta }-\delta \right) \left( \frac{\partial D\left( b_{y}\right) }{\partial b_{y}}\right) +R_{b}{\widetilde{\delta }}\right] u^{\prime }\left( w_{m} -R_{b}b_{y}-\psi (b_{y})\right) =0; \end{aligned}$$

denote the solution by $b_{y}^{c}$.

For the true preferences, lifetime utility depends on borrowing as young as

$$\begin{aligned} \frac{\partial \Omega ^{*}}{\partial b_{y}^{*}}=u^{\prime }\left( w_{y}+b_{y}^{*}\right) -\delta ^{*}R_{b}u^{\prime }\left( w_{m} -R_{b}b_{y}^{*}-\psi ^{*}(s_{y}^{*})\right) . \end{aligned}$$

Evaluate this derivative, for the borrowing decision of the young under the choice preferences $b_{y}^{c}$

$$\begin{aligned}&\frac{\partial \Omega ^{*}}{\partial b_{y}^{*}}\mid _{b_{y}^{c}}=u^{\prime }\left( w_{y}+b_{y}\right) -\delta ^{*}R_{b}u^{\prime }\left( w_{m} -R_{b}b_{y}-\psi ^{*}(b_{y})\right) \nonumber \\&\quad =\left[ -\left( {\widetilde{\delta }}-\delta \right) \left( -\frac{\partial D\left( b_{y}\right) }{\partial b_{y}}\right) +R_{b}\widetilde{\delta }\right] u^{\prime }\left( w_{m}-R_{b}b_{y}-\psi (b_{y})\right) \nonumber \\&\qquad -\delta ^{*}R_{b}u^{\prime }\left( w_{m}-R_{b}b_{y}-\psi ^{*}(b_{y})\right) \end{aligned}$$

(20)

Note that the largest possible value of $-\frac{\partial D\left( b_{y}\right) }{\partial b_{y}}=R_{b}$ which implies

$$\begin{aligned} \left[ -\left( {\widetilde{\delta }}-\delta \right) \left( -\frac{\partial D\left( b_{y}\right) }{\partial b_{y}}\right) +R_{b}\widetilde{\delta }\right] \le -\left( {\widetilde{\delta }}-\delta \right) \left( R_{b}\right) +R_{b}{\widetilde{\delta }}=\delta R_{b}<\delta ^{*}R_{b}\nonumber \\ \end{aligned}$$

(21)

Since $\psi ^{*}(b_{y})>\psi (b_{y})\Leftrightarrow$ $u^{\prime }\left( w_{m}-R_{b}b_{y}-\psi (b_{y})\right) <u^{\prime }\left( w_{m}-R_{b}b_{y} -\psi ^{*}(b_{y})\right) ,$ using (21) in (20) yields

$$\begin{aligned} \left. \frac{\partial \Omega ^{*}}{\partial b_{y}^{*}}\right| _{b_{y}^{c}}<0. \end{aligned}$$

It follows $b_{y}^{*}<b_{y}$ and, therefore, by implication $s_{m}^{*}>s_{m}$.

Appendix: Age-dependent contribution rates

The following considers whether the young should be mandated to contribute to pension savings, when the middle-aged are doing so ($\tau _{m}>0$). To this end, we first need to establish that it is optimal to make the middle-aged contribute, and next consider whether the young should. Similar to above, define ${\underline{\tau }}_{m}^{a}$ and ${\overline{\tau }}_{m}^{a}$ as the critical contribution rates which delineates the positive savings, savings corner and negative savings for the middle-aged, i.e.,

$$\begin{aligned} {\underline{\tau }}_{m}^{a}&:u^{\prime }\left( \left( 1-{\underline{\tau }} _{m}^{a}\right) w_{m}-R_{b}b_{y}\right) ={\widetilde{\delta }}Ru^{\prime }\left( R{\underline{\tau }}_{m}^{a}w_{m}\right) \\ {\overline{\tau }}_{m}^{a}&:u^{\prime }\left( \left( 1-{\overline{\tau }} _{m}^{a}\right) w_{m}-R_{b}b_{y}\right) ={\widetilde{\delta }}R_{b}u^{\prime }\left( R{\overline{\tau }}_{m}^{a}w_{m}\right) \end{aligned}$$

Case I Middle-aged are saving ($s_{m}>0$)

When the middle-aged are savers ($s_{m}>0$), we have that savings ($b_{y} ,s_{m}$) for $\tau _{y}=0$,$\tau _{m}>0$ are determined by

$$\begin{aligned}&u^{\prime }\left( w_{y}+b_{y}\right) ={\widetilde{\delta }}R\left[ \left( {\widetilde{\delta }}-\delta \right) \left( \frac{\partial D\left( b_{y}\right) }{\partial b_{y}}\right) +R_{b}{\widetilde{\delta }}\right] u^{\prime }\left( Rs_{m}+R\tau _{m}w_{m}\right) \\&u^{\prime }\left( w_{m}-R_{b}b_{y}-\tau _{m}w_{m}-s_{m}\right) ={\widetilde{\delta }}Ru^{\prime }\left( Rs_{m}+R\tau _{m}w_{m}\right) \end{aligned}$$

This system determines borrowing as young $b_{y}$ and total savings as middle-aged $D\equiv s_{m}+\tau _{m}w_{m}$, hence $\frac{\partial s_{m} }{\partial \tau _{m}}=-w_{m}$ as long as $s_{m}>0$ holds, and therefore $\frac{\partial \left( s_{m}+\tau _{m}w_{m}\right) }{\partial \tau _{m}}=0$ which in turn implies $\frac{\partial b_{y}}{\partial \tau _{m}}=0$. To obtain the comparative statics with respect to $\tau _{y},$ differentiate each of the optimality conditions above. Tedious algebra shows

$$\begin{aligned}&-u^{\prime \prime }\left( c_{y}\right) \frac{\partial b_{y}}{\partial \tau _{y} }+R\Lambda _{a}u^{\prime \prime }\left( c_{o}\right) \frac{\partial s_{m} }{\partial \tau _{y}}=-w_{y}u^{\prime \prime }\left( c_{y}\right) -u^{\prime \prime }\left( c_{o}\right) R^{2}w_{y}\Lambda _{a}\\&R_{b}u^{\prime \prime }\left( c_{m}\right) \frac{\partial b_{y}}{\partial \tau _{y}}+\left[ u^{\prime \prime }\left( c_{m}\right) +\widetilde{\delta }R^{2}u^{\prime \prime }\left( c_{o}\right) \right] \frac{\partial s_{m} }{\partial \tau _{y}}=-{\widetilde{\delta }}RR^{2}w_{y}u^{\prime \prime }\left( c_{o}\right) \end{aligned}$$

from where the signs of $\frac{\partial b_{y}}{\partial \tau _{y}}$ and $\frac{\partial s_{m}}{\partial \tau _{y}}$ are established.

Case II Middle-aged at the corner ($s_{m}=0$)

For ${\underline{\tau }}_{m}^{a}\le \tau _{m}\le {\overline{\tau }}_{m}^{a},s_{m}=0$ and a change in the contribution rate for the middle-aged affect true lifetime utility as

$$\begin{aligned} \frac{\partial \Omega ^{*}}{\partial \tau _{m}}\mid _{s_{m}=0}=-u^{\prime } (c_{m})w_{m}+\delta ^{*}u^{\prime }(c_{o})w_{m}R. \end{aligned}$$

Assessing the marginal welfare effect for $\tau _{m}={\underline{\tau }}_{m}^{a}$ using $u^{\prime }(c_{m})=R{\widetilde{\delta }}u^{\prime }(c_{o})$ we find

$$\begin{aligned} \frac{\partial \Omega ^{*}}{\partial \tau _{m}}\mid _{s_{m}=0,\tau _{m}={\underline{\tau }}_{m}}=\left( \delta ^{*}-{\widetilde{\delta }}\right) u^{\prime }(c_{o})w_{m}R>0 \end{aligned}$$

i.e., true welfare can be improved by setting a contribution rate $\tau _{m}>{\underline{\tau }}_{m}^{a}$.

Case III Middle-aged as borrowers ($s_{m}<0$)

Assume that $\tau _{m}^{*}>{\overline{\tau }}_{m}^{a}$ in this case the middle-aged are borrowers. The lifetime utility as perceived when young reads

$$\begin{aligned} \text { }u\left( (1-\tau _{y})w_{y}+b_{y}\right) +{\widetilde{\delta }}\left[ u\left( (1-\tau _{m})w_{m}-R_{b}b_{y}-s_{m}\right) +\delta u\left( b+R_{b}s_{m}\right) \right] \end{aligned}$$

and the first-order condition reads

$$\begin{aligned} u^{\prime }\left( c_{y}\right) +{\widetilde{\delta }}\left[ -u^{\prime }\left( c_{m}\right) +\delta R_{b}u^{\prime }\left( c_{o}\right) \right] \left. \frac{\partial s_{m}}{\partial b_{y}}\right| _{s_{m}<0}-\widetilde{\delta }R_{b}u^{\prime }\left( c_{m}\right) =0 \end{aligned}$$

by use of $u^{\prime }\left( c_{m}\right) ={\widetilde{\delta }}R_{b}u^{\prime }\left( c_{o}\right)$ it can be written

$$\begin{aligned} u^{\prime }\left( c_{y}\right) =\Lambda _{b}u^{\prime }\left( c_{o}\right) \end{aligned}$$

where

$$\begin{aligned} \Lambda _{b}\equiv \left[ \left( {\widetilde{\delta }}-\delta \right) \left. \frac{\partial s_{m}}{\partial b_{y}}\right| _{s_{m}<0}+\widetilde{\delta }R_{b}\right] R_{b}{\widetilde{\delta }} \end{aligned}$$

Note that

$$\begin{aligned} \left. \frac{\partial s_{m}}{\partial b_{y}}\right| _{s_{m}<0} =-\frac{R_{b}u^{\prime \prime }\left( c_{m}\right) }{u^{\prime \prime }\left( c_{m}\right) +{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{o}\right) } \end{aligned}$$

implying that $\frac{\partial s_{m}}{\partial b_{y}}>-R_{b},$ and hence it follows that $0<\Lambda _{b}<\delta R_{b}R_{b}{\widetilde{\delta }}$. With homothetic preferences $\Lambda _{a}$ is independent of $c_{m}$ and $c_{o}$.

Writing the first-order condition for the borrowing and savings decisions, we have

$$\begin{aligned} u^{\prime }\left( \left( 1-\tau _{y}\right) w_{y}+b_{y}\right)&=\Lambda _{b}u^{\prime }\left( R^{2}\tau _{y}w_{y}+R\tau _{m}w_{m}+R_{b} s_{m}\right) \\ u^{\prime }\left( \left( 1-\tau _{m}\right) w_{m}-R_{b}b_{y}-s_{m}\right)&={\widetilde{\delta }}R_{b}u^{\prime }\left( R^{2}\tau _{y}w_{y}+R\tau _{m} w_{m}+R_{b}s_{m}\right) \end{aligned}$$

Differentiating w.r.t $\tau _{y}$ yields

$$\begin{aligned} u^{\prime \prime }\left( c_{y}\right) \left[ -w_{y}+\frac{\partial b_{y} }{\partial \tau _{y}}\right]&=\Lambda _{b}u^{\prime \prime }\left( c_{o}\right) \left[ R^{2}w_{y}+R_{b}\frac{\partial s_{m}}{\partial \tau _{y} }\right] ,\\ u^{\prime \prime }\left( c_{m}\right) \left[ -R_{b}\frac{\partial b_{y} }{\partial \tau _{y}}-\frac{\partial s_{m}}{\partial \tau _{y}}\right]&={\widetilde{\delta }}R_{b}u^{\prime \prime }\left( c_{o}\right) \left[ R^{2}w_{y}+R_{b}\frac{\partial s_{m}}{\partial \tau _{y}}\right] , \end{aligned}$$

and w.r.t $\tau _{m}$

$$\begin{aligned} u^{\prime \prime }\left( c_{y}\right) \left[ \frac{\partial b_{y}}{\partial \tau _{m}}\right]&=\Lambda _{b}u^{\prime \prime }\left( c_{o}\right) \left[ Rw_{m}+R_{b}\frac{\partial s_{m}}{\partial \tau _{m} }\right] ,\\ u^{\prime \prime }\left( c_{m}\right) \left[ -w_{m}-R_{b}\frac{\partial b_{y}}{\partial \tau _{m}}-\frac{\partial s_{m}}{\partial \tau _{m}}\right]&={\widetilde{\delta }}R_{b}u^{\prime \prime }\left( c_{o}\right) \left[ Rw_{m}+R_{b}\frac{\partial s_{m}}{\partial \tau _{m}}\right] . \end{aligned}$$

Writing this in matrix form we have

$$\begin{aligned} \left[ \begin{array}[c]{cc} u^{\prime \prime }\left( c_{y}\right) &{} -\Lambda _{b}u^{\prime \prime }\left( c_{o}\right) R_{b}\\ -u^{\prime \prime }\left( c_{m}\right) R_{b} &{} -\left[ u^{\prime \prime }\left( c_{m}\right) +{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{o}\right) \right] \end{array} \right] \left[ \begin{array}[c]{c} \frac{\partial b_{y}}{\partial \tau _{m}}\\ \frac{\partial s_{m}}{\partial \tau _{m}} \end{array} \right] =\left[ \begin{array}[c]{c} \Lambda _{b}u^{\prime \prime }\left( c_{o}\right) Rw_{m}\\ {\widetilde{\delta }}R_{b}u^{\prime \prime }\left( c_{o}\right) Rw_{m} +u^{\prime \prime }\left( c_{m}\right) w_{m} \end{array} \right] , \end{aligned}$$

implying

$$\begin{aligned} \frac{\partial b_{y}}{\partial \tau _{m}}=\frac{\left| \begin{array}[c]{cc} \Lambda _{b}u^{\prime \prime }\left( c_{o}\right) Rw_{m} &{} -\Lambda _{b}u^{\prime \prime }\left( c_{o}\right) R_{b}\\ {\widetilde{\delta }}R_{b}u^{\prime \prime }\left( c_{o}\right) Rw_{m} +u^{\prime \prime }\left( c_{m}\right) w_{m} &{} -\left[ u^{\prime \prime }\left( c_{m}\right) +{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{o}\right) \right] \end{array} \right| }{\left| \begin{array}[c]{cc} u^{\prime \prime }\left( c_{y}\right) &{} -\Lambda _{b}u^{\prime \prime }\left( c_{o}\right) R_{b}\\ -u^{\prime \prime }\left( c_{m}\right) R_{b} &{} -\left[ u^{\prime \prime }\left( c_{m}\right) +{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{o}\right) \right] \end{array} \right| }<0. \end{aligned}$$

It follows from the expressions above

$$\begin{aligned}&sign\left[ -w_{m}-R_{b}\frac{\partial b_{y}}{\partial \tau _{m}}-\frac{\partial s_{m}}{\partial \tau _{m}}\right] =sign\left[ Rw_{m}+R_{b}\frac{\partial s_{m}}{\partial \tau _{m}}\right] \\&sign\left[ \frac{\partial b_{y}}{\partial \tau _{m}}\right] =sign\left[ -w_{m}-R_{b}\frac{\partial b_{y}}{\partial \tau _{m}}-\frac{\partial s_{m} }{\partial \tau _{m}}\right] \end{aligned}$$

Since $\frac{\partial b_{y}}{\partial \tau _{m}}<0$ it follows that $-w_{m}-R_{b}\frac{\partial b_{y}}{\partial \tau _{m}}-\frac{\partial s_{m} }{\partial \tau _{m}}<0$ and $w_{m}R+R_{b}\frac{\partial s_{m}}{\partial \tau _{m}}<0$. Next note

$$\begin{aligned} \frac{\partial \Omega ^{*}}{\partial \tau _{m}}=\left[ \Lambda _{a} \frac{\partial b_{y}}{\partial \tau _{m}}-\delta ^{*}{\widetilde{\delta }} R_{b}\left[ w_{m}+R_{b}\frac{\partial b_{y}}{\partial \tau _{m}}+\frac{\partial s_{m}}{\partial \tau _{m}}\right] +\left( \delta ^{*}\right) ^{2}\left[ w_{m}R+R_{b}\frac{\partial s_{m}}{\partial \tau _{m}}\right] \right] u^{\prime }(c_{o}) \end{aligned}$$

and hence $\frac{\partial \Omega ^{*}}{\partial \tau _{m}}<0$ for $\tau _{m}>{\overline{\tau }}_{m}^{a}$. This proves the optimal $\tau _{m}\in \left[ {\underline{\tau }}_{m}^{a},{\overline{\tau }}_{m}^{a}\right]$.

1.1 Pension contributions by the young

Assume that the optimal $\tau _{m}^{*}\in \left[ {\underline{\tau }}_{m} ^{a},{\overline{\tau }}_{m}^{a}\right] .$ Can welfare be improved by setting a $\tau _{y}>0$? We have (recall $s_{m}=0$) or

$$\begin{aligned} \frac{\partial \Omega ^{*}}{\partial \tau _{y}}=-u^{\prime }(c_{y})w_{y}-\left[ \delta ^{*}u^{\prime }(c_{m})R_{b}-u^{\prime }(c_{y})\right] \frac{\partial b_{y}}{\partial \tau _{y}}+\left( \delta ^{*}\right) ^{2}u^{\prime } (c_{o})R^{2}w_{y} \end{aligned}$$

Since $u^{\prime }(c_{y})={\widetilde{\delta }}R_{b}u^{\prime }(c_{m})$ it follows that

$$\begin{aligned} \delta ^{*}u^{\prime }(c_{m})R_{b}-u^{\prime }(c_{y})=\left( \delta ^{*}-{\widetilde{\delta }}\right) R_{b}u^{\prime }(c_{m})>0 \end{aligned}$$

and therefore

$$\begin{aligned} \frac{\partial \Omega ^{*}}{\partial \tau _{y}}>-u^{\prime }(c_{y})w_{y}-\left[ \delta ^{*}u^{\prime }(c_{m})R_{b}-u^{\prime }(c_{y})\right] w_{y}+\left( \delta ^{*}\right) ^{2}u^{\prime }(c_{o})R^{2}w_{y} \end{aligned}$$

since $\frac{\partial b_{y}}{\partial \tau _{y}}<w_{y}$. Hence a sufficient condition that $\frac{\partial \Omega ^{*}}{\partial \tau _{y}}>0$ is

$$\begin{aligned} -u^{\prime }(c_{y})w_{y}-\left[ \delta ^{*}u^{\prime }(c_{m})R_{b}-u^{\prime }(c_{y})\right] w_{y}+\left( \delta ^{*}\right) ^{2}u^{\prime } (c_{o})R^{2}w_{y}>0, \end{aligned}$$

or

$$\begin{aligned} \delta ^{*}w_{y}\left[ \delta ^{*}u^{\prime }(c_{o})R^{2}-u^{\prime } (c_{m})R_{b}\right] >0. \end{aligned}$$

This inequality holds if

$$\begin{aligned} \frac{\delta ^{*}R^{2}}{R_{b}}u^{\prime }(c_{o})>u^{\prime }(c_{m} )\Leftrightarrow \frac{\delta ^{*}R^{2}}{{\widetilde{\delta }}R_{b}R_{b} }u^{\prime }(c_{o}){\widetilde{\delta }}R_{b}>u^{\prime }(c_{m}) \end{aligned}$$

For the middle-aged to at the zero savings corner we require

$$\begin{aligned} u^{\prime }(c_{o}){\widetilde{\delta }}R_{b}>u^{\prime }(c_{m})>\widetilde{\delta }Ru^{\prime }(c_{o}) \end{aligned}$$

and hence a sufficient condition that $\frac{\partial \Omega ^{*}}{\partial \tau _{y}}>0$ is

$$\begin{aligned} \frac{\delta ^{*}R^{2}}{{\widetilde{\delta }}R_{b}R_{b}}>1\Leftrightarrow \delta ^{*}R^{2}>{\widetilde{\delta }}R_{b}^{2} \end{aligned}$$

1.2 Implementing ($b_{y}^{*},s_{m}^{*}$)

Is it possible to implement the optimal choice under the true preferences ($b_{y}^{*},s_{m}^{*}$)and the associated consumption levels ($c_{y}^{*},c_{m}^{*},c_{o}^{*}$) by some choice of $\tau _{y}$ and $\tau _{m}$? To address this question, first write the consumption levels for the optimal choices under the true preferences, i.e.,

$$\begin{aligned} c_{y}^{*}&=w_{y}+b_{y}^{*}\\ c_{m}^{*}&=w_{m}-R_{b}b_{y}^{*}-s_{m}^{*}\\ c_{o}^{*}&=Rs_{m}^{*} \end{aligned}$$

and under the choice preference for given values of $\tau _{y}$ and $\tau _{m}$.

$$\begin{aligned} c_{y}&=(1-\tau _{y})w_{y}+b_{y}\\ c_{m}&=(1-\tau _{m})w_{m}-R_{b}b_{y}-s_{m}\\ c_{o}&=R^{2}\tau _{y}w_{y}+R\tau _{m}w_{m}+Rs_{m} \end{aligned}$$

For $c_{y}$ $=c_{y}^{*}$, we require

$$\begin{aligned} w_{y}+b_{y}^{*}=(1-\tau _{y})w_{y}+b_{y} \end{aligned}$$

or

$$\begin{aligned} b_{y}-b_{y}^{*}=\tau _{y}w_{y} \end{aligned}$$

(22)

For $c_{m}=c_{m}^{*}$ we require

$$\begin{aligned} w_{m}-R_{b}b_{y}^{*}-s_{m}^{*}=(1-\tau _{m})w_{m}-R_{b}b_{y}-s_{m} \end{aligned}$$

or

$$\begin{aligned} s_{m}-s_{m}^{*}=-\tau _{m}w_{m}-R_{b}\left( b_{y}-b_{y}^{*}\right) \end{aligned}$$

(23)

Finally, for $c_{o}=c_{o}^{*}$ we require

$$\begin{aligned} Rs_{m}^{*}=R^{2}\tau _{y}w_{y}+R\tau _{m}w_{m}+Rs_{m} \end{aligned}$$

or

$$\begin{aligned} s_{m}-s_{m}^{*}=-\left( R\tau _{y}w_{y}+\tau _{m}w_{m}\right) \end{aligned}$$

(24)

Combining (23) and (24) gives

$$\begin{aligned} -\left( R\tau _{y}w_{y}+\tau _{m}w_{m}\right) =-\tau _{m}w_{m}-R_{b}\left( b_{y}-b_{y}^{*}\right) \end{aligned}$$

or $R\tau _{y}w_{y}=R_{b}\left( b_{y}-b_{y}^{*}\right)$ and using (22) this requires $R\tau _{y}w_{y}=R_{b}\tau _{y}w_{y}$ which cannot hold for $R<R_{b}$, showing that it is not possible to implement ($b_{y}^{*},s_{m}^{*}$)and the associated consumption levels ($c_{y}^{*},c_{m}^{*},c_{o}^{*}$) by some choice of $\tau _{y}$ and $\tau _{m}$.

Age-independent contribution rates

When the middle-aged are savers, savings decisions are determined by

$$\begin{aligned} u^{\prime }\left( (1-\tau )w_{y}+b_{y}\right)&=\Lambda _{i}u^{\prime }\left( Rs_{m}+R^{2}\tau w_{y}+R\tau w_{m}\right) \\ u^{\prime }\left( (1-\tau )w_{m}-R_{b}b_{y}-s_{m}\right)&=\widetilde{\delta }Ru^{\prime }\left( Rs_{m}+R^{2}\tau w_{y}+R\tau w_{m}\right) \end{aligned}$$

where

$$\begin{aligned} \Lambda _{I}\equiv {\widetilde{\delta }}R\left[ \left( {\widetilde{\delta }} -\delta \right) \left( \frac{\partial D\left( b_{y}\right) }{\partial b_{y}}\right) +R_{b}{\widetilde{\delta }}\right] \end{aligned}$$

Note that $0<\Lambda _{I}<RR_{b}\delta {\widetilde{\delta }}$ and $\Lambda _{I}$ is independent of $b_{y}$ and $s_{m}$ under homothetic preferences.

Hence, differentiating the optimality conditions above with respect to $\tau$, we get

$$\begin{aligned} u^{\prime \prime }\left( c_{y}\right) \left[ -w_{y}+\frac{\partial b_{y} }{\partial \tau }\right]&=\Lambda _{I}u^{\prime \prime }\left( c_{o}\right) \left[ R\frac{\partial s_{m}}{\partial \tau }+R^{2}w_{y}+Rw_{m}\right] \\ u^{\prime \prime }\left( c_{m}\right) \left[ -w_{m}-R_{b}\frac{\partial b_{y}}{\partial \tau }-\frac{\partial s_{m}}{\partial \tau }\right]&={\widetilde{\delta }}Ru^{\prime \prime }\left( c_{o}\right) \left[ R\frac{\partial s_{m}}{\partial \tau }+R^{2}w_{y}+Rw_{m}\right] \end{aligned}$$

implying

$$\begin{aligned} \text {sign}\left[ w_{y}-\frac{\partial b_{y}}{\partial \tau }\right]&=-\text {sign}\left[ R\frac{\partial s_{m}}{\partial \tau }+R^{2}w_{y} +Rw_{m}\right] ,\\ \text {sign}\left[ -w_{m}-R_{b}\frac{\partial b_{y}}{\partial \tau } -\frac{\partial s_{m}}{\partial \tau }\right]&=\text {sign}\left[ R\frac{\partial s_{m}}{\partial \tau }+R^{2}w_{y}+Rw_{m}\right] . \end{aligned}$$

To solve for $\frac{\partial s_{y}}{\partial \tau }$ and$\frac{\partial s_{m} }{\partial \tau }$ the above expression can be written in matrix form as which implies that

$$\begin{aligned} \frac{\partial b_{y}}{\partial \tau }=\frac{\left| \begin{array}[c]{cc} u^{\prime \prime }\left( c_{y}\right) w_{y}+\Lambda _{I}u^{\prime \prime }\left( c_{o}\right) \left[ R^{2}w_{y}+Rw_{m}\right] &{} -\Lambda _{I}u^{\prime \prime }\left( c_{o}\right) R\\ u^{\prime \prime }\left( c_{m}\right) w_{m}+{\widetilde{\delta }}Ru^{\prime \prime }\left( c_{o}\right) \left[ R^{2}w_{y}+Rw_{m}\right] &{} -\left[ u^{\prime \prime }\left( c_{m}\right) +{\widetilde{\delta }}R^{2}u^{\prime \prime }\left( c_{o}\right) \right] \end{array} \right| }{\left| \begin{array}[c]{cc} u^{\prime \prime }\left( c_{y}\right) &{} -\Lambda _{I}u^{\prime \prime }\left( c_{o}\right) R\\ -u^{\prime \prime }\left( c_{m}\right) R_{b} &{} -\left[ u^{\prime \prime }\left( c_{m}\right) +{\widetilde{\delta }}R^{2}u^{\prime \prime }\left( c_{o}\right) \right] \end{array} \right| }, \end{aligned}$$

implying

$$\begin{aligned} \frac{\partial b_{y}}{\partial \tau }=\frac{u^{\prime \prime }\left( c_{y}\right) u^{\prime \prime }\left( c_{m}\right) +R^{2}\widetilde{\delta }u^{\prime \prime }\left( c_{y}\right) u^{\prime \prime }\left( c_{o}\right) +\Lambda _{I}u^{\prime \prime }\left( c_{o}\right) u^{\prime \prime }\left( c_{m}\right) R^{2}}{u^{\prime \prime }\left( c_{y}\right) u^{\prime \prime }\left( c_{m}\right) +R^{2}{\widetilde{\delta }}u^{\prime \prime }\left( c_{y}\right) u^{\prime \prime }\left( c_{o}\right) +\Lambda _{I} u^{\prime \prime }\left( c_{o}\right) u^{\prime \prime }\left( c_{m}\right) RR_{b}}w_{y}<w_{y}. \end{aligned}$$

From this, it can be inferred by use of the above sign-relations: $w_{y} -\frac{\partial b_{y}}{\partial \tau }>0;$ $w_{m}+R_{b}\frac{\partial b_{y} }{\partial \tau }+\frac{\partial s_{m}}{\partial \tau }>0;$ $R\frac{\partial s_{m}}{\partial \tau }+R^{2}w_{y}+Rw_{m}<0$. Considering the welfare effects from a change in $\tau$ we have

$$\begin{aligned} \frac{\partial \Omega ^{*}}{\partial \tau }= & {} -u^{\prime }(c_{y})\left[ w_{y}-\frac{\partial b_{y}}{\partial \tau }\right] -\delta ^{*}u^{\prime }(c_{m})\left[ w_{m}+R_{b}\frac{\partial b_{y}}{\partial \tau }+\frac{\partial s_{m}}{\partial \tau }\right] \\&\quad +\left( \delta ^{*}\right) ^{2}u^{\prime }(c_{o})\left[ R\frac{\partial s_{m}}{\partial \tau }+R^{2}w_{y}+Rw_{m}\right] <0, \end{aligned}$$

and it follows that $\frac{\partial \Omega ^{*}}{\partial \tau }<0$ for $\tau <{\underline{\tau }}$.

Consider next $\tau \in \left[ {\underline{\tau }},{\overline{\tau }}\right]$ where the middle-aged are at a corner. In this case, borrowing as young is determined by

$$\begin{aligned} u^{\prime }\left( (1-\tau )w_{y}+b_{y}\right) ={\widetilde{\delta }} R_{b}u^{\prime }\left( (1-\tau )w_{m}-R_{b}b_{y}\right) \end{aligned}$$

and the corner condition reads

$$\begin{aligned} u^{\prime }(R^{2}\tau w_{y}+R\tau w_{m}){\widetilde{\delta }}R_{b}>u^{\prime }((1-\tau )w_{m}-R_{b}b_{y})>{\widetilde{\delta }}Ru^{\prime }(R^{2}\tau w_{y}+R\tau w_{m}). \end{aligned}$$

It follows straightforwardly that

$$\begin{aligned} \frac{\partial b_{y}}{\partial \tau }=\frac{u^{\prime \prime }\left( c_{y}\right) w_{y}+{\widetilde{\delta }}R_{b}u^{\prime \prime }\left( c_{m}\right) w_{m}}{u^{\prime \prime }\left( c_{y}\right) +\widetilde{\delta }R_{b}^{2}u^{\prime \prime }\left( c_{m}\right) }. \end{aligned}$$

Note for later

$$\begin{aligned} w_{y}-\frac{\partial b_{y}}{\partial \tau }=\frac{{\widetilde{\delta }}R_{b} ^{2}u^{\prime \prime }\left( c_{m}\right) w_{y}-{\widetilde{\delta }} R_{b}u^{\prime \prime }\left( c_{m}\right) w_{m}}{u^{\prime \prime }\left( c_{y}\right) +{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{m}\right) }. \end{aligned}$$

For $\tau \in \left[ {\underline{\tau }},{\overline{\tau }}\right]$ and thus $s_{m}=0$ we have that welfare is affected by a change in the contribution rate as:

$$\begin{aligned} \frac{\partial \Omega ^{*}}{\partial \tau }&=-u^{\prime }(c_{y})\left[ w_{y}-\frac{\partial b_{y}}{\partial \tau }\right] -\delta ^{*}u^{\prime }(c_{m})\left[ w_{m}+R_{b}\frac{\partial b_{y}}{\partial \tau }\right] +\left( \delta ^{*}\right) ^{2}u^{\prime }(c_{o})\left[ R^{2}w_{y} +Rw_{m}\right] \\&=\left[ -{\widetilde{\delta }}R_{b}\left[ w_{y}-\frac{\partial b_{y} }{\partial \tau }\right] -\delta ^{*}\left[ w_{m}+R_{b}\frac{\partial b_{y} }{\partial \tau }\right] \right] u^{\prime }(c_{m})+\left( \delta ^{*}\right) ^{2}u^{\prime }(c_{o})\left[ R^{2}w_{y}+Rw_{m}\right] \\&=\left[ -{\widetilde{\delta }}R_{b}w_{y}-\delta ^{*}w_{m}-\left[ \delta ^{*}R_{b}-{\widetilde{\delta }}R_{b}\right] \frac{\partial b_{y} }{\partial \tau }\right] u^{\prime }(c_{m})+\left( \delta ^{*}\right) ^{2}u^{\prime }(c_{o})\left[ R^{2}w_{y}+Rw_{m}\right] , \end{aligned}$$

evaluated for ${\underline{\tau }}$ and using $u^{\prime }(c_{m} )={\widetilde{\delta }}Ru^{\prime }(c_{o})$ we get

$$\begin{aligned} \left. \frac{\partial \Omega ^{*}}{\partial \tau }\right| _{{\underline{\tau }}}=\left[ \left[ -{\widetilde{\delta }}R_{b}w_{y}-\delta ^{*}w_{m}-\left[ \delta ^{*}R_{b}-{\widetilde{\delta }}R_{b}\right] \frac{\partial b_{y}}{\partial \tau }\right] {\widetilde{\delta }}R+\left( \delta ^{*}\right) ^{2}\left[ R^{2}w_{y}+Rw_{m}\right] \right] u^{\prime }(c_{o}), \end{aligned}$$

or

$$\begin{aligned} \frac{\left. \frac{\partial \Omega ^{*}}{\partial \tau }\right| _{{\underline{\tau }}}}{u^{\prime }(c_{o})}&=-{\widetilde{\delta }}R_{b} {\widetilde{\delta }}Rw_{y}-{\widetilde{\delta }}R\delta ^{*}w_{m} -{\widetilde{\delta }}R\left[ \delta ^{*}R_{b}-{\widetilde{\delta }}R_{b}\right] \frac{\partial b_{y}}{\partial \tau }+\left( \delta ^{*}\right) ^{2}\left[ R^{2}w_{y}+Rw_{m}\right] \\&=\left[ \left( \delta ^{*}\right) ^{2}R^{2}-{\widetilde{\delta }} R_{b}{\widetilde{\delta }}R-{\widetilde{\delta }}R\left[ \delta ^{*} R_{b}-{\widetilde{\delta }}R_{b}\right] \frac{u^{\prime \prime }\left( c_{y}\right) }{u^{\prime \prime }\left( c_{y}\right) +{\widetilde{\delta }} R_{b}^{2}u^{\prime \prime }\left( c_{m}\right) }\right] w_{y}\\&\quad +\left[ R\delta ^{*}\left[ \delta ^{*}-{\widetilde{\delta }}\right] +{\widetilde{\delta }}R\left[ \delta ^{*}-{\widetilde{\delta }}\right] \frac{{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{m}\right) }{u^{\prime \prime }\left( c_{y}\right) +{\widetilde{\delta }}R_{b}^{2} u^{\prime \prime }\left( c_{m}\right) }\right] w_{m}, \end{aligned}$$

where

$$\begin{aligned} \left( \delta ^{*}\right) ^{2}R-{\widetilde{\delta }}R\delta ^{*}+{\widetilde{\delta }}R\left[ \delta ^{*}-{\widetilde{\delta }}\right] \frac{{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{m}\right) }{u^{\prime \prime }\left( c_{y}\right) +{\widetilde{\delta }}R_{b}^{2} u^{\prime \prime }\left( c_{m}\right) }>0. \end{aligned}$$

Consider the term

$$\begin{aligned} \left( \delta ^{*}\right) ^{2}R^{2}-{\widetilde{\delta }}R_{b} {\widetilde{\delta }}R-{\widetilde{\delta }}R\left[ \delta ^{*}-\widetilde{\delta }\right] R_{b}\frac{u^{\prime \prime }\left( c_{y}\right) }{u^{\prime \prime }\left( c_{y}\right) +{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{m}\right) }. \end{aligned}$$

Since $\frac{u^{\prime \prime }\left( c_{y}\right) }{u^{\prime \prime }\left( c_{y}\right) +{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{m}\right) }<1$ it follows that

$$\begin{aligned} \left( \delta ^{*}\right) ^{2}R^{2}-{\widetilde{\delta }}R_{b} {\widetilde{\delta }}R-{\widetilde{\delta }}R\left[ \delta ^{*}-\widetilde{\delta }\right] R_{b}\frac{u^{\prime \prime }\left( c_{y}\right) }{u^{\prime \prime }\left( c_{y}\right) +{\widetilde{\delta }}R_{b}^{2}u^{\prime \prime }\left( c_{m}\right) }>R\delta ^{*}\left[ \delta ^{*}R-{\widetilde{\delta }} R_{b}\right] . \end{aligned}$$

Hence, this expression is positive if $\delta ^{*}R-{\widetilde{\delta }} R_{b}>0$, and it follows that a sufficient condition that$\frac{\partial \Omega ^{*}}{\partial \tau }\mid _{{\underline{\tau }}}>0$ is $\delta ^{*}R-{\widetilde{\delta }}R_{b}>0$.