The Re-distributive Role of Child Benefits Revisited

Abstract

In this paper, we reexamine the commonly invoked argument that due to the existence of a negative correlation between earning ability and family size, the latter can be used as a “tagging” device, justifying subsidizing children (via provision of child allowances) to enhance egalitarian objectives. Employing a benchmark setting where the quality–quantity paradigm holds, we show that the case for subsidizing children is far from being a forgone conclusion. We demonstrate that the desirability of subsidizing children crucially hinges on whether benefits are means-tested or are offered on a universal basis.

Appendices

Appendix A: the additively separable case

We show that in the absence of taxes ($z_n =0\hbox { and }z_y =1)$ when the utility in (1) is taking an additively separable form, $g(l,n,e,\alpha )=\psi (l)+v(n)+u(e)+\Phi (\alpha )$ and, in addition, satisfies the property that $-\alpha \cdot \Phi ^{\prime \prime }(\alpha )/\Phi ^{\prime }(\alpha )>1$, then $\partial [y(w)/w]/\partial w>0$ and $\partial n(w)/\partial w<0$.

We first turn to show that $\partial n(w)/\partial w<0$. The $w$-household is seeking to maximize the utility in (1) subject to the budget constraint in (2). Formulating the household, first-order conditions (assuming no taxes in place) yields

$$\begin{aligned}&1-\psi ^{\prime }(1-y/w-n\cdot \alpha )/w=0\end{aligned}$$

(15)

$$\begin{aligned}&v^{\prime }(n)-\alpha \cdot \psi ^{\prime }(1-y/w-n\cdot \alpha )-e=0,\end{aligned}$$

(16)

$$\begin{aligned}&u^{\prime }(e)-n=0,\end{aligned}$$

(17)

$$\begin{aligned}&\Phi ^{\prime }(\alpha )-n\cdot \psi ^{\prime }(1-y/w-n\cdot \alpha )=0. \end{aligned}$$

(18)

Substituting from (15) into (16) and (18) yields

$$\begin{aligned} v^{\prime }(n)-\alpha \cdot w-e(n)=0, \end{aligned}$$

(16')

$$\begin{aligned} \Phi ^{\prime }(\alpha )-n\cdot w=0, \end{aligned}$$

(18')

where $e(n)$ is given by the implicit solution to (17).

The system of two Eqs.  [(16’) and (18’)] implicitly defines the optimal solution for the number of children and the level of parental time invested per child, as a function of the wage rate [$n(w)$ and $\alpha (w)$]. Fully differentiating the two first-order conditions in (16’) and (18’) with respect to $w$ yields

$$\begin{aligned}&v^{\prime \prime }(n)\cdot n^{\prime }(w)-[\alpha +w\cdot \alpha ^{\prime }(w)]-e^{\prime }(n)\cdot n^{\prime }(w)=0,\end{aligned}$$

(19)

$$\begin{aligned}&\Phi ^{\prime \prime }(\alpha )\cdot \alpha ^{\prime }(w)- [n+w\cdot n^{\prime }(w)]=0. \end{aligned}$$

(20)

Applying Cramer’s Rule, one then obtains

$$\begin{aligned} n^{\prime }(w)=\frac{\alpha \cdot \Phi ^{\prime \prime } (\alpha )+n\cdot w}{[v^{\prime \prime }(n)-e^{\prime }(n)]\cdot \Phi ^{\prime \prime }(\alpha )-w^{2}}, \end{aligned}$$

(21)

where $[v^{\prime \prime }(n)-e{\prime }(n)]\cdot \Phi ^{\prime \prime }(\alpha )-w^{2}>0$, by the household’s optimization second-order conditions. Substituting for the term $n\cdot w$ from (18’) into the numerator on the right-hand side of (21) implies that $\partial n(w)/\partial w<0$ if-and-only-if $\Phi ^{\prime }(\alpha ) +\alpha \cdot \Phi ^{\prime \prime }(\alpha )<0\Leftrightarrow -\alpha \cdot \Phi ^{\prime \prime }(\alpha )/\Phi ^{\prime }(\alpha )>1$. The latter follows from our assumption. This completes the first part of the proof.

We turn next to show that $\partial [y(w)/w]/\partial w>0$. Fully differentiating the condition in (15) with respect to $w$ yields

$$\begin{aligned} 1+\psi ^{\prime \prime }(1-y/w-n\cdot \alpha ) \cdot [\partial (y/w)/\partial w+\partial (n\cdot \alpha )/\partial w]=0. \end{aligned}$$

(22)

By the strict concavity of $\psi$, it suffices to show that $\partial (n\cdot \alpha )/\partial w < 0$ in order to establish that $\partial [y(w)/w]/\partial w>0$.

Employing (16’) and (20) and re-arranging, one obtains

$$\begin{aligned} \partial (n\cdot \alpha )/\partial w=n^{\prime }(w)\cdot \alpha +\alpha ^{\prime }(w)\cdot n=\frac{n^{2}+n^{\prime }(w)\cdot [\Phi ^{\prime }(\alpha )+\alpha \cdot \Phi ^{\prime \prime }(\alpha )]}{\Phi ^{\prime \prime }(\alpha )}<0.\nonumber \\ \end{aligned}$$

(23)

The sign of the inequality follows from the concavity of $\Phi$, the fact that $n^{\prime }(w)<0$ by the first part of the proof and our assumption that $-\alpha \cdot \Phi ^{\prime \prime }(\alpha )/ \Phi ^{\prime }(\alpha )>1$. This concludes the proof.

Appendix B: proof of proposition 1

Assuming that the second-order conditions for the government optimization are satisfied, to determine whether taxing or subsidizing children at the margin would be optimal, it suffices to examine the sign of the derivative of the Hamiltonian [given in (10)] with respect to $n$, starting from a system where the marginal child tax/subsidy is set to zero and the income tax is set at the optimum. A positive sign would imply that social welfare will rise by introducing a small marginal child subsidy (thereby increasing the number of children), whereas a negative sign would suggest taxing children at the margin would be desirable.

Formulating the first-order condition for the Hamiltonian in (10) with respect to $U$ yields

$$\begin{aligned} \frac{\partial H}{\partial U}=G^{\prime }(U)\cdot f-\lambda \cdot f=-\mu ^{\prime }. \end{aligned}$$

(24)

The transversality conditions are given by

$$\begin{aligned} \mu (\underline{w})=\mu (\overline{w})=0, (\lim _{\overline{w} \rightarrow \infty } \mu (\overline{w})=0, \hbox {when the distribution of skills is unbounded).}\nonumber \\ \end{aligned}$$

(25)

Integrating condition (24), employing the transversality condition, $\mu (\overline{w})=0$, yields

$$\begin{aligned} \mu (w)=\int \limits _w^{\overline{w}}{\left[ {G^{\prime } [U(t)]-\lambda }\right] dF(t)}. \end{aligned}$$

(26)

Employing the second transversality condition, $\mu (\underline{w})=0$, yields

$$\begin{aligned} \lambda =\int \limits _{\underline{w}}^{\overline{w}} {G^{\prime }[U(t)]dF(t)}. \end{aligned}$$

(27)

Now define the function $D$ by

$$\begin{aligned} D(w)=\frac{1}{1-F(w)}\int \limits _w^{\overline{w}} {G^{\prime }[U(t)]dF(t)}. \end{aligned}$$

(28)

In words, the function $D$ measures the average social marginal utility of income over the interval $[w,\overline{w}]$. Moreover, employing (26)–(28) yields

$$\begin{aligned} \mu (w)&= [1-F(w)]\cdot [D(w)-D(\underline{w})] < 0\end{aligned}$$

(29)

$$\begin{aligned} \lambda&= D(\underline{w}), \end{aligned}$$

(30)

where the negative sign of the expression on the right-hand side of (29) follows from the fact that $D(w)$ [defined in (28)] is decreasing by virtue of the concavity of $G$.

Setting the marginal child tax/subsidy to zero $(z_n =0)$ and differentiating the Hamiltonian with respect to $n$ yields

$$\begin{aligned} \left. \begin{array}{l} \frac{\partial H}{\partial n}\\ \end{array}\right| _{z_n =0}=\lambda \cdot f\cdot h_1 (n,y/w)- \mu \cdot h_{12} (n,y/w)\cdot \frac{y}{w^{2}}<0. \end{aligned}$$

(31)

The negative sign follows from (i) the fact that $h_1 (n,y/w)=0$, by virtue of the household’s first-order condition in (4), (ii) the fact that $h_{12} (n,y/w)<0$ by virtue of the assumptions on the utility function ensuring a negative relationship between family size and labor supply, and (iii) the fact that $\mu (w)<0$, by virtue of (29).

Appendix C: derivation of Eq. (11)

In this appendix, we derive the necessary and sufficient condition for the social desirability of providing a marginal child subsidy given in Eq. (11). As in the means-tested case (analyzed in appendix B), assuming that the second-order conditions for the government optimization are satisfied, to determine whether taxing or subsidizing children at the margin would be optimal it suffices to examine the sign of the derivative of the Hamiltonian [given in (10’)] with respect to $n$, starting from a system where the marginal child tax/subsidy is set to zero and the income tax is set at the optimum.

Formulating the necessary first-order condition for the Hamiltonian in (10’) with respect to the state variable $U$, employing the transversality conditions and re-arranging (repeating the same steps as in appendix B, which are therefore omitted), one obtains

$$\begin{aligned} \mu (w)&= [1-F(w)]\cdot [D(w)-D(\underline{w})]<0\end{aligned}$$

(32)

$$\begin{aligned} \lambda&= D(\underline{w}), \end{aligned}$$

(33)

where $D(w)=\frac{1}{1-F(w)}\int _w^{\overline{w}}{G^{\prime } [U(t)]dF(t)}$.

Notice that the conditions in (32) and (33) replicate the conditions in (29) and (30). The negative sign of the expression on the right-hand side of (32) follows from the fact that $D(w)$ is decreasing by virtue of the concavity of $G$.

Setting the marginal child tax/subsidy to zero ($z_n =0)$ and differentiating the Hamiltonian with respect to $n$ yields

$$\begin{aligned} \left. \begin{array}{l} \frac{\partial H}{\partial n}\\ \end{array}\right| _{z_n =0}&= \lambda \cdot f\cdot [y^{\prime }+h_1 +h_2 \cdot y^{\prime }/w]-\mu \cdot [h_{12} \cdot y/w^{2}+h_{22} \cdot y^{\prime }\cdot y/w^{3}\nonumber \\&+h_2 \cdot y^{\prime }/w^{2}], \end{aligned}$$

(34)

where we omit some of the arguments to abbreviate notation.

The household’s first-order conditions with respect to $y$ and $n$ (replicated for convenience), assuming that the marginal child tax/subsidy is set to zero, are given, respectively, by

$$\begin{aligned} a_y +h_2 /w&= 0,\end{aligned}$$

(35)

$$\begin{aligned} h_1&= 0. \end{aligned}$$

(36)

Fully differentiating the condition in (35) with respect to $n$ yields

$$\begin{aligned} a_{yy} \cdot y^{\prime }+h_{12} /w+h_{22} \cdot y^{\prime }/w^{2}=0. \end{aligned}$$

(37)

Re-arranging yields

$$\begin{aligned} y^{\prime }=-\frac{h_{12} /w}{a_{yy} +h_{22} /w^{2}}<0, \end{aligned}$$

(38)

where the negative sign follows from the fact that $h_{12} <0$, by our earlier assumptions regarding the household utility function, and the household second-order condition [differentiation of the first-order condition in (35) with respect to $y$ implies that $a_{yy} +h_{22} /w^{2}<0$]. That is, an increase in the number of children (say in response to offering a marginal subsidy) results in a reduction in the labor supply, and consequently the gross level of income, as expected.

Substituting for $\lambda \hbox { and }\mu$ from (32) and (33) into (34), employing the conditions in (32)–(35) and following some re-arrangements yield the following necessary and sufficient condition for the desirability of providing a marginal child subsidy:

$$\begin{aligned} \left. \begin{array}{l} \frac{\partial H}{\partial n}\\ \end{array}\right| _{b_n =0} >0\Leftrightarrow \left[ {(1\!-\!a_y )+\left[ {1\!-\!\frac{D(w)}{D(\underline{w})}}\right] \cdot \frac{1-F(w)}{f(w)\cdot w} \cdot \left[ {-a_{yy} \cdot y-a_y}\right] }\right] \cdot y^{\prime }>0.\nonumber \\ \end{aligned}$$

(39)

By virtue of (38), the condition in (39) holds if-and-only if the following condition [equivalent to the one given in Eq. (11)] is satisfied:

$$\begin{aligned} \left. {\begin{array}{l} \frac{\partial H}{\partial n}\\ \end{array}_{b_n =0}}\right| >0\Leftrightarrow \left[ {1-\frac{D(w)}{D(\underline{w})}}\right] \cdot \left[ {\frac{1-F(w)}{f(w)\cdot w}}\right] \cdot \left[ {a_{yy} \cdot y+a_y}\right] >(1-a_y).\qquad \quad \end{aligned}$$

(40)

Appendix D: the correlation between family size and earning ability and the properties of the income tax schedule

In this appendix, we state and prove the following claim:

Claim

$n^{\prime }(w)>=<0\hbox { if-and-only-if }a_{yy} \cdot y+a_y < = >0$

Proof

We reproduce, for convenience, the $w$-household’s first-order conditions, given in Eqs. (4) and (5), assuming a universal system, namely $z(y,n)=a(y)+b(n)$ and setting the marginal child subsidy to zero $(b_n = 0)$:

$$\begin{aligned} K(y,n,w)&{\,\equiv \,}&a_y +h_2 /w=0,\end{aligned}$$

(41)

$$\begin{aligned} H(y,n,w)&{\,\equiv \,}&h_1 =0. \end{aligned}$$

(42)

The systems of two Eqs. [(41) and (42)] provide an implicit solution for $n(w)$ and $y(w)$, the optimal choices of the $w$-household.

Fully differentiating the two conditions in (41) and (42) with respect to $w$ yields

$$\begin{aligned}&\partial K/\partial y\cdot y^{\prime }(w)+\partial K/\partial n\cdot n^{\prime }(w)+\partial K/\partial w=0,\end{aligned}$$

(43)

$$\begin{aligned}&\partial H/\partial y\cdot y^{\prime }(w)+\partial H/\partial n\cdot n^{\prime }(w)+\partial H/\partial w=0. \end{aligned}$$

(44)

Employing Cramer’s Rule then yields

$$\begin{aligned} n^{\prime }(w)=\frac{-\partial K/\partial y\cdot \partial H/\partial w+\partial H/\partial y\cdot \partial K/\partial w}{\partial K/\partial y\cdot \partial H/\partial n-\partial H/\partial y\cdot \partial K/\partial n} \end{aligned}$$

(45)

By the second-order conditions of the household’s optimization, it follows that $\partial K/\partial y\cdot \partial H/\partial n-\partial H/\partial y\cdot \partial K/\partial n >0$. Thus, it follows

$$\begin{aligned} Sign[n^{\prime }(w)]=Sign[-\partial K/\partial y\cdot \partial H/\partial w+\partial H/\partial y\cdot \partial K/\partial w]. \end{aligned}$$

(46)

Differentiating the household’s first-order conditions in (41) and (42) yields

$$\begin{aligned}&\partial H/\partial w=-h_{12} \cdot y/w^{2},\end{aligned}$$

(47)

$$\begin{aligned}&\partial H/\partial y=h_{12} /w,\end{aligned}$$

(48)

$$\begin{aligned}&\partial K/\partial y=a_{yy} +h_{22}/w^{2},\end{aligned}$$

(49)

$$\begin{aligned}&\partial K/\partial w=-h_{22} \cdot y/w^{3}-h_2/w^{2}. \end{aligned}$$

(50)

Substituting from (47)–(50) into the expression on the right-hand side of (46) and re-arranging yields

$$\begin{aligned} -\partial K/\partial y\cdot \partial H/\partial w+\partial H/\partial y\cdot \partial K/\partial w=\frac{h_{12} }{w^{2}}\cdot [a_{yy} \cdot y-h_2 /w]. \end{aligned}$$

(51)

The result follows by substituting $a_y$ for the term $-h_2/w$ by virtue of the first-order condition in (35) and by noting that $h_{12} <0$ by our assumption regarding the utility function.

Appendix E: derivation of Eq. (13)

In this appendix, we derive the formula for the optimal marginal income tax given in Eq. (13). The Hamiltonian for the government program is given by

$$\begin{aligned} \begin{array}{ll} H=&{}\left[ {G(U)+\lambda \cdot \left[ {y-U+h[n(y),y/w]-R}\right] } \right] \cdot f \\ &{}-\mu \cdot h_2 [n(y),y/w]\cdot y/w^{2}, \\ \end{array} \end{aligned}$$

(52)

where $n(y)$ is implicitly given by the household’s first-order condition in (5).

Formulating the necessary first-order conditions for the Hamiltonian in (52), omitting the arguments to abbreviate notation, yields

$$\begin{aligned} \frac{\partial H}{\partial y}&= \lambda \cdot f\cdot \left[ {1+h_1 \cdot n^{\prime }+h_2 /w} \right] \nonumber \\&-\mu \cdot [(h_{12} \cdot n^{\prime }+h_{22} /w)\cdot y/w^{2}+h_2 /w^{2}]=0, \end{aligned}$$

(53)

$$\begin{aligned} \frac{\partial H}{\partial U}&= G^{\prime }(U)\cdot f-\lambda \cdot f=-\mu ^{\prime }. \end{aligned}$$

(54)

The transversality conditions are given by

$$\begin{aligned} \mu (\underline{w})=\mu (\overline{w})=0, (\lim _{\overline{w} \rightarrow \infty } \mu (\overline{w})=0, \hbox {when the distribution of skills is unbounded)}.\nonumber \\ \end{aligned}$$

(55)

Integrating condition (54), employing the transversality conditions and repeating the steps made in appendix B (which are hence omitted) yield

$$\begin{aligned} \mu (w)&= [1-F(w)]\cdot [D(w)-D(\underline{w})]<0,\end{aligned}$$

(56)

$$\begin{aligned} \lambda&= D(\underline{w}), \end{aligned}$$

(57)

where $D(w)=\frac{1}{1-F(w)}\int \nolimits _w^{\overline{w}}{G^{\prime } [U(t)]dF(t)}$.

The negative sign of the expression on the right-hand side of (56) follows from the fact that $D(w)$ is decreasing by virtue of the concavity of $G$.

Substituting from (56) and (57) into (53), employing the household’s first-order conditions in (4)–(5) yields, after some re-arrangements, the following expression [which is identical to Eq. (13)]:

$$\begin{aligned} \frac{1-a_y}{a_y}=\left[ {1-\frac{D(w)}{D (\underline{w})}}\right] \cdot \frac{1-F(w)}{f(w)\cdot w}\cdot \left[ {1+\frac{1}{\varepsilon _L}}\right] , \end{aligned}$$

(58)

where

$$\begin{aligned} \varepsilon _L =-\frac{w\cdot a_y }{y\cdot (n^{\prime }\cdot h_{12} +h_{22} /w)}. \end{aligned}$$

(59)

It remains to show that $\varepsilon _L$ denotes the labor supply elasticity.

To see this, let $w_{net} {\,\equiv \,} w\cdot a_y$ denote the after-tax wage rate and let $m {\,\equiv \,} y/w$ denote labor supply. Reproducing the household first-order conditions in (4)–(5) obtains

$$\begin{aligned} w_{net} +h_2 (n,m)&= 0,\end{aligned}$$

(60)

$$\begin{aligned} h_1 (n,m)&= 0. \end{aligned}$$

(61)

Fully differentiating the system of two equations given by (60) and (61) with respect to $w_{net}$ yields

$$\begin{aligned}&1+h_{12} (n,m)\cdot \frac{\partial n}{\partial w_{net}} +h_{22} (n,m)\cdot \frac{\partial m}{\partial w_{net}}=0,\end{aligned}$$

(62)

$$\begin{aligned}&h_{11} (n,m)\cdot \frac{\partial n}{\partial w_{net}} +h_{12} (n,m)\cdot \frac{\partial m}{\partial w_{net}}=0. \end{aligned}$$

(63)

Employing cramer’s rule then yields

$$\begin{aligned} \frac{\partial m}{\partial w_{net}}=\frac{-h_{11}}{h_{11} \cdot h_{22} -h_{12}^{2}}. \end{aligned}$$

(64)

The labor supply elasticity is given by

$$\begin{aligned} \varepsilon _L =\frac{\partial m}{\partial w_{net}} \cdot \frac{w_{net}}{m}=-\frac{h_{11}}{h_{11} \cdot h_{22} -h_{12}^{2}} \cdot \frac{w^{2}\cdot a_y}{y}. \end{aligned}$$

(65)

Differentiating the first-order condition in (5) with respect to $y$ and re-arranging yields

$$\begin{aligned} n^{\prime }(y)=\frac{-h_{12}}{w\cdot h_{11}}. \end{aligned}$$

(66)

Substituting for $n$’ from (65) into (59) and re-arranging yields the expression in (65). This concludes the derivation of (13).