A normative justification of compulsory education

Abstract

Using a household production model of educational choices, we characterise a free-market situation in which some agents (high wagers) fully educate their children and spend a sizable amount of resources on them, while others (low wagers) educate them only partially. The free-market equilibrium is iniquitous, both because the households have different resources and because the children have different access to education. Public policy is thus called for, for vertical as well as horizontal equity purposes. Conventional wisdom has it that both objectives could be achieved using price control instruments, i.e. income taxes and price subsidies. We find instead that income taxes reduce equality of opportunity and that price subsidies cannot remedy this. Quantity controls become necessary: a compulsory education package, financed by a redistributive tax system, achieves both types of equity. Redistributive taxation and compulsory education are therefore best seen as complementary policies.

Appendices

Appendix A: A model of consumer choice with bequests

In the main text, we assume that parents can transfer resources to their children only through investments in education. Here, we allow the parents to transfer resources that can be directly used for consumption, such as, for example, bequests. Let then the parent choose the transfer c. Parent’s preferences are

$$ \left[ U\left( C\right) +F\left( y\left( H,h\right) +z\right) \right] +\left[ u\left( c+x\left( e,d\right) \right) +f\left( y\left( H,h\right) +z\right) \right] , $$

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and the budget constraint is

$$ C+pz+c+e=wl. $$

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The time constraints for the parent and the kid, respectively, are

$$ H+l=1\text{\ and }h+d=1. $$

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Using these elements, we write the parent’s problem as one of choosing C, H, z, c, e and h so as to

$$\begin{array}{@{}rcl@{}} \text{Max}~\left[ U\left( C\right) +F\left( y\left( H,h\right) +z\right) \right] &&+\left[ u\left( c+x\left( e,1-h\right) \right) +f\left( y\left( H,h\right) +z\right) \right] \\ &&{\kern-6.2pc}~\text{s.t.}~C+pz+e+c-w(1-H)=0; \\ &&{\kern-6.2pc}H\geq 0;~z\geq 0;~c\geq 0;~e\geq 0;~h\geq 0. \end{array} $$

The first-order conditions (FOCs) are as follows:

$$ U^{\prime }=\lambda ~~~\left( C\right) ; $$

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$$ \left( F^{\prime }+f^{\prime }\right) y_{H}\leq \lambda w,~\text{plus complementary slackness~~}~\left( H\right) ; $$

(63)

$$ F^{\prime }+f^{\prime }\leq \lambda p,~\text{plus complementary slackness~~} ~\left( z\right) ; $$

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$$ u^{\prime }=\lambda \text{ }~~~\left( c\right) ; $$

(65)

$$ u^{\prime }x_{e}=\lambda \text{ ~~}~\left( e\right) ; $$

(66)

$$ \left( F^{\prime }+f^{\prime }\right) y_{h}\leq u^{\prime }x_{d}\text{ plus complementary slackness~~}~\left( h\right) \text{.} $$

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Our first question is whether the parent prefers to transfer resources to the child using the consumption good or investing in education. From Eqs. 65 and 66, it is immediate to see that

$$\begin{array}{@{}rcl@{}} \left( x_{e}>1\right) &\longrightarrow &\left( u^{\prime }<\lambda ;~u^{\prime }x_{e}=\lambda \right) \longrightarrow \left( c=0,e>0\right) ; \end{array} $$

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$$\begin{array}{@{}rcl@{}} \left( x_{e}<1\right) &\longrightarrow &\left( u^{\prime }=\lambda ;~u^{\prime }x_{e}<\lambda \right) \longrightarrow \left( c>0,e=0\right) , \end{array} $$

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while x _e =1 would lead to an undetermined result. The outcome is very sharp because c and x are (reasonably) perfect substitutes. The intuition is clear: each unit of consumption that the parent transfers forward becomes, within the assumptions of the present model, exactly one unit of extra consumption for the child, while each unit of consumption that the parent transforms into one unit of educational expenditure may become more or less than one unit of extra consumption for the child depending on the returns to said expenditure. The result can clearly be generalised to more complicated settings: the parent will always choose the most efficient way of transferring resources to the child.

The model that we consider in the paper is basically one in which x _e >1 everywhere for all households. In the opposite extreme case in which x _e <1 everywhere for all households, we would have e=0, which implies d=0 because both inputs are essential to the production of x; we would not have any education at all, which would make the model uninteresting for our purposes.

Mixed situations could be discussed, though. If some agents choose to educate their children while others rely on consumption transfers, the horizontal equity problem would be exacerbated. An especially clear case would emerge if, for example, the marginal returns to education x _e were to depend on the parent’s wage: we might assume, in line with the empirical literature (e.g. Mayer 1997; Blau 1999) that the parent’s marketable skills (her wage in our model) and the child’s ability to learn and profit from what she has learned (her $x\left (\cdot \right )$ function in our model) are positively, although not necessarily perfectly, correlated. If so, it might well be the case that most of the high wagers in our model send their kids to school while most of the low wagers do not—thereby worsening the horizontal equity issues.

Appendix B: An example in which w ^∗ is unique

From Eqs. 5 and 6, the parent’s maximization problem is

$$ \underset{H,e,d,z}{\max }~U\left( w-e-pz-wH\right) +F\left( y\left( H,1-d\right) +z\right) +u\left( x\left( e,d\right) \right) +f\left( y\left( H,1-d\right) +z\right) , $$

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where U, u and y are concave, F, f and x are strictly concave. We consider the special case where (i) y _{H H} = y _{h h} =0 and (ii) $U^{\prime \prime }=u^{\prime \prime }=0$; also, recall that x _{e d} >0 and y _{h H} >0 by assumption. In order to prove that w ^∗ corresponds to a point and not to an interval, we must prove that the FOCs can simultaneously hold as equalities for only one value of w. Focusing on an interior solution for all of the four variables H, e, d, z, the FOCs are

$$ -U^{\prime }w+\left( F^{\prime }+f^{\prime }\right) y_{H}=0~~~\left( H\right) ; $$

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$$ -U^{\prime }+u^{\prime }x_{e}=0~~~\left( e\right) ; $$

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$$ -\left( F^{\prime }+f^{\prime }\right) y_{h}+u^{\prime }x_{d}=0~~~\left( d\right) ; $$

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$$ -U^{\prime }p+F^{\prime }+f^{\prime }=0~~~\left( z\right) . $$

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By totally differentiating, we have:

$$\begin{array}{@{}rcl@{}} &&\left[ -U^{\prime }\right] \text{d}w+\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}\right] \text{d}H+\left[ 0\right] \text{d}e \\ &&+-\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{H}y_{h}+\left( F^{\prime }+f^{\prime }\right) y_{Hh}\right] \text{d}d+ \left[ (F^{\prime \prime }+f^{\prime \prime })y_{H}\right] \text{d}z=0; \end{array} $$

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$$\begin{array}{@{}rcl@{}} &&-\left[ 0\right] \text{d}w+\left[ 0\right] \text{d}H+\left[ u^{\prime }x_{ee} \right] \text{d}e+\left[ u^{\prime }x_{ed}\right] \text{d}d+\left[ 0\right] \text{d}z=0; \end{array} $$

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$$\begin{array}{@{}rcl@{}} &&\left[ 0\right] \text{d}w-\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}y_{H}+\left( F^{\prime }+f^{\prime }\right) y_{hH}\right] \text{d}H+\left[ u^{\prime }x_{de}\right] \text{d}e \\ &&+\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{h}\right)^{2}+u^{\prime }x_{dd}\right] \text{d}d-\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}\right] \text{d}z=0; \end{array} $$

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$$\begin{array}{@{}rcl@{}} &&\left[ 0\right] \text{d}w+\left[ (F^{\prime \prime }+f^{\prime \prime })y_{H} \right] \text{d}H+\left[ 0\right] \text{d}e-\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}\right] \text{d}d+\left[ F^{\prime \prime }+f^{\prime \prime }\right] \text{d}z=0. \end{array} $$

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Now, let

$$\begin{array}{@{}rcl@{}} \left[ \begin{array}{ll} a_{11} \\ a_{21} \\ a_{31} \\ a_{41} \end{array} \right] =\left[ \begin{array}{ll} \left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}<0\\ 0 \\ -\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}y_{H}+\left( F^{\prime }+f^{\prime }\right) y_{hH}\right] \\ (F^{\prime \prime }+f^{\prime \prime })y_{H}<0 \end{array} \right] ,~\left[ \begin{array}{ll} a_{12} \\ a_{22} \\ a_{32}\\ a_{42} \end{array} \right] =\left[ \begin{array}{ll} 0 \\ u^{\prime }x_{ee}<0\\ u^{\prime }x_{ed}>0\\ 0 \end{array} \right] , \\ \left[ \begin{array}{ll} a_{13}\\ a_{23}\\ a_{33}\\ a_{43} \end{array} \right] =\left[ \begin{array}{ll} -\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{H}y_{h}+\left( F^{\prime }+f^{\prime }\right) y_{Hh}\right]\\ u^{\prime }x_{de}>0\\ \left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{h}\right)^{2}+u^{\prime }x_{dd}<0 \\ -\left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}>0 \end{array} \right] ,~\left[ \begin{array}{ll} a_{14}\\ a_{24}\\ a_{34}\\ a_{44} \end{array} \right] =\left[ \begin{array}{ll} \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{H}<0 \\ 0 \\ -\left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}>0\\ F^{\prime \prime }+f^{\prime \prime }<0 \end{array} \right], \end{array} $$

where the sign of a ₁₃ = a ₃₁ is not determined. Then

$$ \left[ \begin{array}{llll} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right] \left[ \begin{array}{lll} \text{d}H/\text{d}w\\ \text{d}e/\text{d}w\\ \text{d}d/\text{d}w\\ \text{d}z/\text{d}w \end{array} \right] =\left[ \begin{array}{ll} U^{\prime } \\ 0 \\ 0 \\ 0 \end{array} \right] $$

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and the comparative statics signs are as follows. Take d H/d w first:

$$ sgn~\text{d}H/\text{d}w=sgn~U^{\prime }\left\vert \begin{array}{llll} a_{22} & a_{23} & a_{24}\\ a_{32} & a_{33} & a_{34}\\ a_{42} & a_{43} & a_{44} \end{array} \right\vert <0, $$

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as the determinant is negative because of the maximization condition. Consider then d e/d w:

$$ sgn~\text{d}e/\text{d}w=-sgn~U^{\prime }\left\vert \begin{array}{llll} a_{21} & a_{23} & a_{24}\\ a_{31} & a_{33} & a_{34}\\ a_{41} & a_{43} & a_{44} \end{array} \right\vert . $$

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Given that a ₂₁ = a ₂₄=0, it is

$$\begin{array}{@{}rcl@{}} sgn~\text{d}e/\text{d}w&=&-sgn~U^{\prime }(-a_{23})(a_{31}a_{44}-a_{41}a_{34}) \\ &=&sgn~U^{\prime }u^{\prime }x_{de}\left[ -(F^{\prime \prime }+f^{\prime \prime })\left( F^{\prime }+f^{\prime }\right) y_{Hh}\right] >0. \end{array} $$

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Consider then d d/d w:

$$ sgn~\text{d}d/\text{d}w=sgn~U^{\prime }\left\vert \begin{array}{lll} a_{21} & a_{22} & a_{24}\\ a_{31} & a_{32} & a_{34}\\ a_{41} & a_{42} & a_{44} \end{array} \right\vert . $$

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Given that a ₂₁ = a ₂₄=0, and that $a_{31}a_{44}-a_{41}a_{34}=-(F^{\prime \prime }+f^{\prime \prime })\left (F^{\prime }+f^{\prime }\right ) y_{Hh}>0$, it is

$$ sgn~\text{d}d/\text{d}w=sgn~U^{\prime }a_{22}(a_{34}a_{41}-a_{31}a_{44})=u^{\prime }x_{ee}(F^{\prime \prime }+f^{\prime \prime })\left( F^{\prime }+f^{\prime }\right) y_{Hh}>0. $$

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Consider finally d z/d w:

$$ sgn~\text{d}z/\text{d}w=-sgn~U^{\prime }\left\vert \begin{array}{lll} a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ a_{41} & a_{42} & a_{43} \end{array} \right\vert . $$

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Given that a ₂₁ = a ₄₂=0, it is

$$\begin{array}{@{}rcl@{}} sgn~\text{d}z/\text{d}w&=&-sgn~\left\{ U^{\prime }a_{41}(a_{22}a_{33}-a_{32}a_{23})-a_{31}a_{22}a_{43}\right\} \\ &=&-sgn~U^{\prime }\left\{ \begin{array}{ll} \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{H}\left[ u^{\prime }x_{ee}\left( \left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{h}\right)^{2}+u^{\prime }x_{dd}\right) -\left( u^{\prime }x_{ed}\right)^{2}\right] + \\ -\left[ -\left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}y_{H}+\left( F^{\prime }+f^{\prime }\right) y_{Hh}\right] u^{\prime }x_{ee}\left[ -\left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h} \right] \end{array} \right\} \\ &=&-sgn~U^{\prime }\left\{ \left( F^{\prime \prime }+f^{\prime \prime }\right) \left[ y_{H}x_{dd}-y_{H}\left( u^{\prime }x_{ed}\right)^{2}+\left( F^{\prime }+f^{\prime }\right) y_{Hh}(u^{\prime })^{2}x_{ee}y_{h}\right] \right\} <0\\ \end{array} $$

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We then have:

$$ \text{d}H/\text{d}w<0,\quad \text{d}e/\text{d}w>0,\quad \text{d}d/\text{d} w>0,\quad \text{d}z/\text{d}w<0. $$

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We can use this comparative statics to prove that it is not possible to have the four FOCs holding as equalities over an interval of values of w. In fact, for the four FOCs to hold as equalities it must be the case that

$$ \frac{w}{y_{H}}=p, $$

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over the whole interval but this would imply that h has to increase as w increases, contradicting d d/d w>0. Hence, with this specification, w ^∗ is unique.

Appendix C: Comparative statics in laissez-faire

Recall that we are using a separable utility function throughout and that we require $U\left (\cdot \right )$ and $U\left (\cdot \right )$ to be concave, $F\left (\cdot \right )$, $F\left (\cdot \right )$ and $x\left (\cdot \right )$ to be strictly concave.

1.1 High wagers (w>w ^∗)

In the case of high wagers, we can write the maximisation problem as

$$ \underset{z,e}{\max }~\left( U\left( w-pz-e\right) +F\left( z\right) \right) +\left( u\left( x\left( e,1\right) \right) +f\left( z\right) \right) . $$

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The FOCs are

$$ -U^{\prime }p+F^{\prime }+f^{\prime }=0~~~\left( z\right) ; $$

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$$ -U^{\prime }+u^{\prime }x_{e}=0~~~\left( e\right) . $$

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By totally differentiating, we have:

$$ -U^{\prime \prime }p\text{d}w+\left[ U^{\prime \prime }p^{2}+F^{\prime \prime }+f^{\prime \prime }\right] \text{d}z+U^{\prime \prime }p\text{d}e=0; $$

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$$ -U^{\prime \prime }\text{d}w+U^{\prime \prime }p\text{d}z+\left[ U^{\prime \prime }+\left( u^{\prime \prime }\left( x_{e}\right)^{2}+u^{\prime }x_{ee}\right) \right] \text{d}e=0. $$

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Therefore:

$$ \left[ \begin{array}{cc} U^{\prime \prime }p^{2}+F^{\prime \prime }+f^{\prime \prime } & U^{\prime \prime }p\\ U^{\prime \prime }p & U^{\prime \prime }+\left( u^{\prime \prime }\left( x_{e}\right)^{2}+u^{\prime }x_{ee}\right) \end{array} \right] \left[ \begin{array}{c} \text{d}z/\text{d}w \\ \text{d}e/\text{d}w \end{array} \right] =\left[ \begin{array}{c} U^{\prime \prime }p \\ U^{\prime \prime } \end{array} \right] . $$

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Then, the signs are as follows

$$ sgn~\text{d}z/\text{d}w=sgn~\left[ U^{\prime \prime }p\left( U^{\prime \prime }+\left( u^{\prime \prime }\left( x_{e}\right)^{2}+u^{\prime }x_{ee}\right) \right) \right] -\left( U^{\prime \prime }\right)^{2}p\geq 0; $$

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$$ sgn~\text{d}e/\text{d}w=sgn~\left[ \left( U^{\prime \prime }p^{2}+F^{\prime \prime }+f^{\prime \prime }\right) U^{\prime \prime }-\left( U^{\prime \prime }p\right)^{2}\right] \geq 0. $$

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1.2 Low wagers (w<w ^∗)

In the case of low wagers, we can write the maximisation problem as

$$ \underset{H,e,d}{\max }~U\left( w-e-wH\right) +F\left( y\left( H,1-d\right) \right) +u\left( x\left( e,d\right) \right) +f\left( y\left( H,1-d\right) \right) . $$

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The FOCs for an interior solution are:

$$ -U^{\prime }w+\left( F^{\prime }+f^{\prime }\right) y_{H}=0~\left( H\right) ; $$

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$$ -U^{\prime }+u^{\prime }x_{e}=0~\left( e\right) ; $$

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$$ -\left( F^{\prime }+f^{\prime }\right) y_{h}+u^{\prime }x_{d}=0~\left( d\right) . $$

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Totally differentiating, we have:

$$\begin{array}{@{}rcl@{}} &&-\left[ U^{\prime \prime }\left( 1-H\right) w+U^{\prime }\right] \text{d}w+ \left[ U^{\prime \prime }w^{2}+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}\right] \text{d}H+\left[ U^{\prime \prime }w\right] \text{d}e\\ &&-\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{H}y_{h}+\left( F^{\prime }+f^{\prime }\right) y_{Hh}\right] \text{d}d=0; \end{array} $$

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$$\begin{array}{@{}rcl@{}} &&-\left[ U^{\prime \prime }\left( 1-H\right) \right] \text{d}w+\left[ U^{\prime \prime }w\right] \text{d}H+\left[ U^{\prime \prime }+\left( u^{\prime \prime }\left( x_{e}\right)^{2}+u^{\prime }x_{ee}\right) \right] \text{d}e\\ &&+\left[ u^{\prime \prime }x_{e}x_{d}+u^{\prime }x_{ed}\right] \text{d}d=0; \end{array} $$

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$$\begin{array}{@{}rcl@{}} &&0\text{d}w-\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}y_{H}+\left( F^{\prime }+f^{\prime }\right) y_{hH}\right] \text{d}H+ \left[ \left( u^{\prime \prime }x_{d}x_{e}+u^{\prime }x_{de}\right) \right] \text{d}e\\ &&+\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{h}\right)^{2}+\left( F^{\prime }+f^{\prime }\right) y_{hh}+\left( u^{\prime \prime }\left( x_{d}\right)^{2}+u^{\prime }x_{dd}\right) \right] \text{d}d=0. \end{array} $$

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Now, let

$$ \left[ \begin{array}{c} a_{11} \\ a_{21} \\ a_{31} \end{array} \right] =\left[ \begin{array}{c} U^{\prime \prime }w^{2}+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}<0 \\ U^{\prime \prime }w\leq 0 \\ -\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{h}y_{H}+\left( F^{\prime }+f^{\prime }\right) y_{hH}\right] \end{array} \right] ; $$

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$$ \left[ \begin{array}{c} a_{12} \\ a_{22} \\ a_{32} \end{array} \right] =\left[ \begin{array}{c} U^{\prime \prime }w\leq 0 \\ U^{\prime \prime }+u^{\prime \prime }\left( x_{e}\right)^{2}+u^{\prime }x_{ee}<0 \\ u^{\prime \prime }x_{e}x_{d}+u^{\prime }x_{ed} \end{array} \right] ; $$

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$$ \left[ \begin{array}{c} a_{13} \\ a_{23} \\ a_{33} \end{array} \right] =\left[ \begin{array}{c} -\left[ \left( F^{\prime \prime }+f^{\prime \prime }\right) y_{H}y_{h}+\left( F^{\prime }+f^{\prime }\right) y_{Hh}\right] \\ u^{\prime \prime }x_{d}x_{e}+u^{\prime }x_{de} \\ \left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{h}\right)^{2}+\left( F^{\prime }+f^{\prime }\right) y_{hh}+u^{\prime \prime }\left( x_{d}\right)^{2}+u^{\prime }x_{dd}<0 \end{array} \right] , $$

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where the signs of a ₃₁, a ₁₃, a ₃₂ and a ₂₃ are not determined. Then

$$ \left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right] \left[ \begin{array}{c} \text{d}H/\text{d}w \\ \text{d}e/\text{d}w \\ \text{d}d/\text{d}w \end{array} \right] =\left[ \begin{array}{c} \left[ U^{\prime \prime }\left( 1-H\right) w+U^{\prime }\right] \\ U^{\prime \prime }\left( 1-H\right) \\ 0 \end{array} \right] . $$

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Further assuming that

$$ -\frac{U^{\prime \prime }}{U^{\prime }}<\frac{1}{\left( 1-H\right) w}, $$

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the comparative statics signs are as follows. Take d H/d w first:

$$\begin{array}{@{}rcl@{}} sgn~\text{d}H/\text{d}w &=&-sgn\left\{ \left[ \underset{+}{\underbrace{ U^{\prime \prime }\left( 1-H\right) w+U^{\prime }}}\right] \left( \underset{+ }{\underbrace{a_{22}a_{33}-a_{32}a_{23}}}\right) \right. \\ &&\left. +\underset{-}{\underbrace{U^{\prime \prime }\left( 1-H\right) }} \underset{?}{\underbrace{\left( \underset{?}{a_{23}}\underset{?}{a_{31}}- \underset{-}{a_{21}}\underset{-}{a_{33}}\right) }}\right\} . \end{array} $$

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Since $\left (a_{22}a_{33}-a_{32}a_{23}\right )$ is positive because of the maximization condition, the first term is also positive. For $U^{\prime \prime }<0$, the sign of the second term is ambiguous. To sign it, we assume that there is a moderate complementarity both between d and e and between h and H, namely we suppose that

$$\begin{array}{@{}rcl@{}} -\frac{u^{\prime \prime }}{u^{\prime }} >\frac{x_{de}}{x_{d}x_{e}}, -\frac{F^{\prime \prime }+g^{\prime \prime }}{F^{\prime }+g^{\prime }} >\frac{y_{hH}}{y_{h}y_{H}}, \end{array} $$

which imply that a ₂₃<0 and a ₃₁>0, respectively. Since $\left (a_{23}a_{31}-a_{21}a_{33}\right )$ is then non positive, it follows that

$$ \text{d}H/\text{d}w<0\text{ and d}l/\text{d}w>0. $$

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Consider then d e/d w:

$$ sgn~\text{d}e/\text{d}w= -sgn\left\{ \underset{-}{\underbrace{U^{\prime \prime }\left( 1-H\right) }} \underset{+}{\underbrace{\left( \underset{}{a_{11}}\underset{}{a_{33}}- \underset{}{a_{31}}\underset{}{a_{13}}\right) }}+\underset{+}{\underbrace{ \left[ U^{\prime \prime }\left( 1-H\right) +U^{\prime }\right] }}\underset{?} {\underbrace{\left( \underset{?}{a_{32}}\underset{?}{a_{13}}\underset{-}{ -a_{12}}\underset{-}{a_{33}}\right) }}\right\} $$

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We know that the term a ₁₁ a ₃₃−a ₃₁ a ₁₃ is positive because of the maximization conditions, and because of our previous assumptions, we are considering the case where a ₃₂ a ₁₃−a ₁₂ a ₃₃≤0; then,

$$ \text{d}e/\text{d}w>0. $$

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Finally, let us consider d d/d w:

$$ sgn~\text{d}d/\text{d}w=-sgn\left\{ \underset{+}{\underbrace{\left[ U^{\prime \prime }\left( 1-H\right) +U^{\prime }\right] }}\underset{?}{\underbrace{\left( \underset{-} {a_{12}}\underset{?}{a_{23}}-\underset{?}{a_{13}}\underset{-}{a_{22}}\right) }}+\underset{-}{\underbrace{U^{\prime \prime }\left( 1-H\right) }}\underset{? }{\underbrace{\left( \underset{-}{a_{21}}\underset{?}{a_{13}}\underset{-}{ -a_{11}}\underset{?}{a_{23}}\right) }}\right\} $$

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Since a ₂₃<0 and a ₃₁>0, it immediately follows that

$$ \text{d}d/\text{d}w<0\text{ and d}h/\text{d}w>0. $$

(114)

Appendix D: Equivalence of revenue constraint

To see that public budget constraint (40) is equivalent to budget constraint (41), let us first write the government budget if E is paid for by the government itself and then if E is paid by the parent:

$$ \tau {\int}_{0}^{\overline{w}}w\left( 1-H\left( w\right) \right) G(w) \text{d}w+\tau {\int}_{0}^{\overline{w}}x\left( E+e,D+d\right) G(w) \text{d}w-E=\widehat{T}, $$

(115)

$$ \tau {\int}_{0}^{\overline{w}}w\left( 1-H\left( w\right) \right) G(w) \text{d}w+\tau {\int}_{0}^{\overline{w}}x\left( E+e,D+d\right) G(w) \text{d}w=T. $$

(116)

To check the equivalence, integrate (40) to yield

$$ {\int}_{0}^{\overline{w}}\left( C+pz+e\right) G(w)\text{d}w-\left( 1-\tau \right) {\int}_{0}^{\overline{w}}w\left( 1-H\right) G(w)\text{ d}w=\widehat{T}, $$

(117)

and substitute the revenue constraint (115); then integrate (41) to yield

$$ {\int}_{0}^{\overline{w}}\left( C+pz+e\right) G(w)\text{d}w+E-\left( 1-\tau \right) {\int}_{0}^{\overline{w}}w\left( 1-H\right) G(w)\text{ d}w=T, $$

(118)

and substitute (116) (recall that the agents have unit mass). It is immediate to see that the resource constraints computed using the two procedures coincide:

$$ \int \left( C+pz+e\right) G(w)\text{d}w+E+\tau {\int}_{0}^{ \overline{w}}x\left( E+e,D+d\right) G(w)\text{d}w ={\int}_{0}^{\overline{w}}w\left( 1-H\right) G(w)\text{d}w. $$

(119)

Appendix E: Comparative statics under compulsory education

1.1 High wagers (w>w ^∗)

Having no need to employ their time in home production, the high wagers set H = h=0: all the parent’s time goes into working and all the kid’s time goes into education. The constraint that D=1 is of no consequence because that is what the parents would have chosen anyway. On the contrary, E is an actual constraint (e=0). Then, high wagers choose z to maximise

$$ \left[ U\left( \left( 1-\tau \right) w+T-pz-E\right) +F\left( z\right) \right] +\left[ u\left( \left( 1-\tau \right) x\left( E,1\right) \right) +f\left( z\right) \right] . $$

(120)

The FOC is:

$$ -U^{\prime }p+F^{\prime }+f^{\prime }=0, $$

(121)

and it follows that

$$ \frac{\partial z}{\partial w}=-\frac{-U^{\prime \prime }p\left( 1-\tau \right) }{U^{\prime \prime }p^{2}+F^{\prime \prime }+f^{\prime \prime }}\geq 0;~\frac{\partial z}{\partial \tau }=-\frac{U^{\prime \prime }wp}{U^{\prime \prime }p^{2}+F^{\prime \prime }+f^{\prime \prime }}\leq 0; $$

(122)

$$ \frac{\partial z}{\partial T}=-\frac{-U^{\prime \prime }p}{U^{\prime \prime }p^{2}+\alpha F^{\prime \prime }+f^{\prime \prime }}\geq 0;~\frac{\partial z }{\partial E}=-\frac{U^{\prime \prime }p}{U^{\prime \prime }p^{2}+F^{\prime \prime }+f^{\prime \prime }}\leq 0. $$

(123)

1.2 Low wagers (w<w ^∗)

Low wagers are constrained by both E and D (d=0, e=0). Consequently, they only choose H to maximise

$$ \left[ U\left( \left( 1-\tau \right) w+T-E-\left( 1-\tau \right) wH\right) +F\left( y\left( H,0\right) \right) \right] +\left[ u\left( \left( 1-\tau \right) x\left( E,1\right) \right) +f\left( y(H,0\right) \right] . $$

(124)

The FOC is

$$ -\left( 1-\tau \right) U^{\prime }w+\left( F^{\prime }+f^{\prime }\right) y_{H}=0. $$

(125)

Hence, since

$$ -\frac{U^{\prime \prime }}{U^{\prime }}<\frac{1}{\left( 1-\tau \right) w(1-H) }, $$

(126)

from (108), we have that

$$\begin{array}{@{}rcl@{}} \frac{\partial H}{\partial w}&=&-\frac{-\left( 1-\tau \right) U^{\prime }-\left( 1-\tau \right)^{2}w\left( 1-H\right) U^{\prime \prime }}{U^{\prime \prime }\left( 1-\tau \right)^{2}w^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}}\\ &=&-\frac{-\left( 1-\tau \right) \left( U^{\prime }+\left( 1-\tau \right) w\left( 1-H\right) U^{\prime \prime }\right) }{U^{\prime \prime }\left( 1-\tau \right)^{2}w^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}}<0 ; \end{array} $$

(127)

$$ \frac{\partial H}{\partial \tau }=-\frac{w\left[ U^{\prime }+\left( 1-\tau \right) w\left( 1-H\right) U^{\prime \prime }\right] }{U^{\prime \prime }\left( 1-\tau \right)^{2}w^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}}>0; $$

(128)

$$ \frac{\partial H}{\partial T}=-\frac{-\left( 1-\tau \right) wU^{\prime \prime }}{U^{\prime \prime }\left( 1-\tau \right)^{2}w^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}}\geq 0; $$

(129)

$$ \frac{\partial H}{\partial E}=-\frac{\left( 1-\tau \right) wU^{\prime \prime }}{U^{\prime \prime }\left( 1-\tau \right)^{2}w^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}}\leq 0. $$

(130)

1.3 w ^∗-wagers

The agents at w ^∗ are constrained by both E and D (d=0, e=0 ); they choose H and z to maximise

$$\begin{array}{@{}rcl@{}} &&U\left( \left( 1-\tau \right) w^{\ast }+T-E-pz-\left( 1-\tau \right) w^{\ast }H\right) +F\left( y\left( H,0\right) +z\right) \\ &&+u\left( \left( 1-\tau \right) x\left( E,1\right) \right) +f\left( y\left( H,0\right) +z\right) . \end{array} $$

(131)

The FOCs are

$$\begin{array}{@{}rcl@{}} &&-U^{\prime }\left( 1-\tau \right) w^{\ast }+\left( F^{\prime }+f^{\prime }\right) y_{H}=0~~~\left( H\right) ; \end{array} $$

(132)

$$\begin{array}{@{}rcl@{}} &&-U^{\prime }p+F^{\prime }+f^{\prime }=0~~~\left( z\right) . \end{array} $$

(133)

Hence

$$ \frac{\left( 1-\tau \right) w^{\ast }}{y_{H}}=p, $$

(134)

as expected. Total differentiation yields

$$\begin{array}{@{}rcl@{}} &&-\left[ \left( 1-\tau \right) U^{\prime }+\left( 1-\tau \right)^{2}\left( 1-H\right) w^{\ast }U^{\prime \prime }\right] \text{d}w^{\ast }+\left[ U^{\prime }w^{\ast }+\left( w^{\ast }\right)^{2}\left( 1-\tau \right) \left( 1-H\right) U^{\prime \prime }\right] \text{d}\tau \\ &&+ -\left[ \left( 1-\tau \right) w^{\ast }U^{\prime \prime }\right] dT+\left[ \left( 1-\tau \right) w^{\ast }U^{\prime \prime }\right] \text{d}E\\ &&+ \left[ \left( 1-\tau \right)^{2}\left( w^{\ast }\right)^{2}U^{\prime \prime }+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}\right] \text{d }H \\ &&+\left[ \left( 1-\tau \right) w^{\ast }pU^{\prime \prime }+(F^{\prime \prime }+f^{\prime \prime })y_{H}\right] \text{d}z=0; \end{array} $$

(135)

$$\begin{array}{@{}rcl@{}} &&-\left[ \left( 1-\tau \right) \left( 1-H\right) pU^{\prime \prime }\right] \text{d}w^{\ast }+\left[ w^{\ast }\left( 1-H\right) pU^{\prime \prime } \right] \text{d}\tau -\left[ U^{\prime \prime }p\right] dT+\left[ U^{\prime \prime }p\right] \text{d}E\\ &&+\left[ \left( 1-\tau \right) w^{\ast }pU^{\prime \prime }+(F^{\prime \prime }+f^{\prime \prime })y_{H}\right] \text{d}H+\left[ p^{2}U^{\prime \prime }+F^{\prime \prime }+f^{\prime \prime }\right] \text{d}z=0. \end{array} $$

(136)

Now, let

$$\begin{array}{@{}rcl@{}} \left[ \begin{array}{l} a_{11} \\ a_{21} \end{array} \right] &=&\left[ \begin{array}{l} \left( \left( 1-\tau \right)^{2}\left( w^{\ast }\right)^{2}\right) U^{\prime \prime }+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}<0 \\ \left( 1-\tau \right) w^{\ast }pU^{\prime \prime }+(F^{\prime \prime }+f^{\prime \prime })y_{H}<0 \end{array} \right] , \\ \left[ \begin{array}{l} a_{12} \\ a_{22} \end{array} \right]&=&\left[ \begin{array}{l} \left( 1-\tau \right) w^{\ast }pU^{\prime \prime }+\left( F^{\prime \prime }+f^{\prime \prime }\right) y_{H}<0\\ p^{2}U^{\prime \prime }+F^{\prime \prime }+f^{\prime \prime }<0 \end{array} \right] . \end{array} $$

Then:

$$ \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] \left[ \begin{array}{c} \text{d}H/\text{d}w \\ \text{d}z/\text{d}w \end{array} \right] =\left[ \begin{array}{c} \left( 1-\tau \right) \left( U^{\prime }+\left( 1-\tau \right) \left( 1-H\right) w^{\ast }U^{\prime \prime }\right) \\ \left( 1-\tau \right) \left( 1-H\right) U^{\prime \prime }p \end{array} \right] ; $$

(137)

$$ \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] \left[ \begin{array}{c} \text{d}H/\text{d}\tau \\ \text{d}z/\text{d}\tau \end{array} \right] =\left[ \begin{array}{c} -w^{\ast }\left( U^{\prime }+w^{\ast }\left( 1-\tau \right) \left( 1-H\right) U^{\prime \prime }\right) \\ -w^{\ast }\left( 1-H\right) U^{\prime \prime }p \end{array} \right] ; $$

(138)

$$ \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] \left[ \begin{array}{c} \text{d}H/\text{d}T \\ \text{d}z/\text{d}T \end{array} \right] =\left[ \begin{array}{c} \left( 1-\tau \right) w^{\ast }U^{\prime \prime } \\ U^{\prime \prime }p \end{array} \right] ; $$

(139)

$$ \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] \left[ \begin{array}{c} \text{d}H/\text{d}E \\ \text{d}z/\text{d}E \end{array} \right] =\left[ \begin{array}{c} -\left( 1-\tau \right) w^{\ast }U^{\prime \prime } \\ -U^{\prime \prime }p \end{array} \right] . $$

(140)

Using (108) and the fact that $\left (1-\tau \right ) w^{\ast }=py_{H}$ by (134), we have:

$$\begin{array}{@{}rcl@{}} sgn~\text{d}H/\text{d}w&=&sgn~\left[ \begin{array}{c} \left( w^{\ast }\left( \left( 1-\tau \right) \left( U^{\prime }+\left( 1-\tau \right) \left( 1-H\right) U^{\prime \prime }\right) \right) \right) a_{22}+ \\ -\left( \left( 1-\tau \right) \left( 1-H\right) U^{\prime \prime }p\right) pa_{12} \end{array} \right] <0; \end{array} $$

(141)

$$\begin{array}{@{}rcl@{}} sgn~\text{d}z/\text{d}w&=&sgn\left[ \begin{array}{c} a_{11}\left( \left( 1-\tau \right) \left( 1-H\right) U^{\prime \prime }p\right) + \\ -a_{21}\left( \left( 1-\tau \right) \left( U^{\prime }+\left( 1-\tau \right) \left( 1-H\right) w^{\ast }U^{\prime \prime }\right) \right) \end{array} \right] \geq 0; \end{array} $$

(142)

$$\begin{array}{@{}rcl@{}} sgn~\text{d}H/\text{d}\tau &=&sgn~\left[ \begin{array}{c} \left( -w^{\ast }\left( U^{\prime }+w^{\ast }\left( 1-\tau \right) \left( 1-H\right) U^{\prime \prime }\right) \right) a_{22}+ \\ -\left( -w^{\ast }\left( 1-H\right) U^{\prime \prime }\right) pa_{12} \end{array} \right] >0; \end{array} $$

(143)

$$\begin{array}{@{}rcl@{}} sgn~\text{d}z/\text{d}\tau &=& sgn~\left[ \begin{array}{c} a_{11}\left( -w^{\ast }\left( 1-H\right) U^{\prime \prime }p\right) + \\ -a_{21}\left( -w^{\ast }\left( U^{\prime }+w^{\ast }\left( 1-\tau \right) \left( 1-H\right) U^{\prime \prime }\right) \right) \end{array} \right] \leq 0; \end{array} $$

(144)

$$\begin{array}{@{}rcl@{}} sgn~\text{d}H/\text{d}T &=& sgn~\left\{ \left( 1-\tau \right) w^{\ast }U^{\prime \prime }\left( U^{\prime \prime }p^{2}+F^{\prime \prime }+f^{\prime \prime }\right) -U^{\prime \prime }p\left[ \left( 1-\tau \right) w^{\ast }pU^{\prime \prime }+\left( F^{\prime \prime }+f^{\prime \prime }\right) y_{H}\right] \right\} \\ &=& sgn~\left\{ U^{\prime \prime }\left( F^{\prime \prime }+f^{\prime \prime }\right) \left[ \left( 1-\tau \right) w^{\ast }-py_{H}\right] \right\} =0; \end{array} $$

(145)

$$\begin{array}{@{}rcl@{}} sgn~\text{d}z/\text{d}T&=& sgn~\left\{ \begin{array}{c} \left[ \left( 1-\tau \right)^{2}\left( w^{\ast }\right)^{2}U^{\prime \prime }+\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}+\left( F^{\prime }+f^{\prime }\right) y_{HH}\right] U^{\prime \prime }p+ \\ -\left( 1-\tau \right) w^{\ast }U^{\prime \prime }\left[ \left( 1-\tau \right) pw^{\ast }U^{\prime \prime }+(F^{\prime \prime }+f^{\prime \prime })y_{H}\right] \end{array} \right\} \\ &=&sgn~\left[ \begin{array}{c} \left( U^{\prime \prime }\right)^{2}\left( 1-\tau \right)^{2}\left( w^{\ast }\right)^{2}p+U^{\prime \prime }p\left( F^{\prime \prime }+f^{\prime \prime }\right) \left( y_{H}\right)^{2}+ \\ +\left( F^{\prime }+f^{\prime }\right) y_{HH}U^{\prime \prime }p-\left( 1-\tau \right)^{2}\left( w^{\ast }\right)^{2}\left( U^{\prime \prime }\right)^{2}p-\left( 1-\tau \right) w^{\ast }U^{\prime \prime }(F^{\prime \prime }+f^{\prime \prime })y_{H} \end{array} \right] \\ &=&sgn~\left[ \left( F^{\prime }+f^{\prime }\right) y_{HH}U^{\prime \prime }p \right] \geq 0. \end{array} $$

(146)

Also, clearly, d H/d E=0 and d z/d E≤0.