## 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.

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## Notes

This is not necessarily true for higher education, see for example Oei and Ring (2014) on the US case.

An exception is represented by Cigno (2012) who recommends compulsory school enrolment in developing countries where child labour is an issue in order to remedy for problems deriving from asymmetric information on the use of children’s time. For a comparison with our results see section IV on Alternative policy frameworks.

For example, see Go (2009) for a paper presenting a political economy explanation for the American achievement of universal free public schooling in a historical perspective.

According to Cipolla (1969), opportunity costs appear to be the single most important factor behind the development of education systems. Thus, as long as the kid’s time was valuable for help in the farm or for employment in the Dickensian factories of the early industrial revolution, large-scale education programs were not undertaken in the Western world, but became important starting from the second half of the XIX century, when productivity began to be high enough to make the kids’ work dispensable.

This literature started with Rawls (1971), and nowadays equality of opportunity can be considered ‘the prevailing conception of social justice in contemporary western societies’ (Ferreira and Peragine 2014, p. 2). Beyond the case of education, such approach has been applied to different areas of public policy such as health, anti-poverty schemes, income taxation and redistribution. For a recent survey on this topic, see Roemer and Trannoy (2015).

The model could be recast in an OLG framework. All the results reached in the simpler case treated here would carry over, and we would have to add many unnecessary details, with the consequent risk of making the model lose its focus.

Appendix A briefly discusses the case in which parents transfer to the children resources that can be directly consumed as well as resources meant to be invested in education.

This seems a reasonable assumption. Our qualitative results would carry over also in its absence but the exposition would be less straightforward.

If we were to consider an equilibrium with

*x*=0, it could be proved that the households where*d*=*e*=0 would be either all those that produce the public good at home or the ones with a relatively higher wage among them. The existence of this equilibrium would not affect our argument.Having already excluded equilibria where

*z*=0 and*h*=1 (*d*=0), this means that we do not consider the case where Eq. 9 is binding and Eq. 12 is slack, i.e. where \(\left (F^{\prime }+f^{\prime }\right ) y_{H}=U^{\prime }w\) and \(\left (F^{\prime } + f^{\prime }\right ) y_{h}|_{h=0}<u^{\prime } x_{d}|_{d=1}\), implying*H*>0 and*h*=0 (*d*=1). Since we know from Appendix C that d*d*/*d**w*<0 (and d*h*/*d**w*>0) when*h*>0, this could happen either for very low values of*w*, or for all the low wagers (i.e. all those households that produce*y*at home). In the former case our qualitative results would not be affected. In the latter case, instead, children are never employed in home production and thus equality of opportunity would be achieved. This however means that it is never profitable to employ children in home production, independently of the level of*w*. Since*H*is decreasing in*w*for the low wagers, as the comparative statics will tell us, it must be the case that \(\left (F^{\prime }+f^{\prime }\right ) y_{h}|_{h=0}<u^{\prime }x_{d}|_{d=1}\) holds even for those low wagers that have a relatively high*w*and thus a high level of \( F^{\prime }+f^{\prime }\). This does not seem to represent a general case although it might, for example, represent an economy rapidly shifting from a rural to an industrial structure.For a similar assumption, see Becker and Murphy (2007).

This is due to the ‘price’ of

*y*being nonlinear and increasing, so that it is profitable to produce*y*only up to the point where its marginal price equals*p*.As

*H*is decreasing in*w*for the households that produce*y*at home, we cannot exclude the case where*H*=0 at*w*^{∗}so that the public good will be completely purchased on the market. This however would not affect our analysis.If one should make an assumption, it would probably be safe to postulate that increasing expenditure will entail little compensation for a reduced time at school, which is probably mostly irreplaceable due to the teacher’s inputs, the peer effects, etc.. An alternative approach is that proposed by Glewwe (2002) for less developed countries where years of schooling and school quality are considered as alternative inputs in the production of the child cognitive skills. In that set-up, when the learning efficiency of the child increases or the cost of education decreases, parents prefer to increase the quality of the children education rather than the quantity because the latter has a greater opportunity cost in terms of less time for the child market work.

Incidentally notice that also a tax on

*z*, which we do not consider in the model to keep things simple, would have a negative effect on the threshold wage rate—much in the same way as an increase in the income tax rate. Since the tax would have mostly a redistributive purpose (hitting a good that is only consumed by the high-wagers) the logic is the same: vertical equity conflicts with horizontal equity.Part (or all) of the low wagers will see their income increase thanks to the poll subsidy. This will translate in a higher level of educational expenditure. Given the complementarity of

*e*and*d*, this will make both*d*and*x*increase for those households who were below the threshold even without the policy. Those households that switch from high to low wagers however do not necessarily see their income increase (they could be net contributors to the tax system). In any case they will face a strong reduction in*d*(recall that there is a discountinuity in*d*at*w*^{∗}) which is likely to imply a reduction in*x*even if their income and consequently*e*were to increase.A parallel can be drawn with health economics, where some theories of justice analyse the interpersonal distribution of health by establishing different types of condition upon outcomes that need to be satisfied before solving any social planner maximisation problem. For example, a common condition is that a minimum decent level of health has to be fulfilled for some specified groups, and no trade-off is allowed between such a goal and any other (Williams and Cookson 2000).

As far as the effects of the education policies are concerned, the conclusions in the other cases are analogous to those discussed here. The results for the other cases are available from the authors.

Alternatively, the horizontal equity requirement could be accounted for by assuming a different social welfare function which could also capture a measure of inequality of educational opportunities as a negative externality for the society. For example, social welfare could be negatively affected by a higher variance in the distribution of

*x*. In this case, the argument in favour of a compulsory education scheme reducing such a variance would find an even higher support. A different approach is followed by Gasparini and Pinto (2006) where the choice of the optimal policy in terms of cash versus in-kind transfers depends, among the others, on the degree of the rich people’s concern about education quality dispersion. In this work, income redistribution serves to counter a negative externality coming from the fact that rich people’s utility negatively depends on the difference between the average education quality levels of the rich and the poor groups in the society.It may be worth noting that, had we chosen

*D*optimally rather than have it fixed at the outset, we would have reached a similar conclusion to that for*E*, that is, we would have found that it is socially desirable, on vertical equity grounds, to establish mandatory schooling. Of course, there would not have been, at this level of generality, any guarantee that the optimal level of*D*were exactly unity, so in this sense we would not have necessarily achieved a full equality of opportunity.As we have discussed in footnote 17, the redistributive policy will benefit the children of those households that would be low wagers even without the policy but would damage the children of the households that switch from high to low wagers.

This result can be linked to the recent empirical literature that tries to analyse how exogenous changes in parents’ education due to variations in compulsory schooling laws may affect the intergenerational transmission of education. For example, Piopiunik (2014) provides evidence that individuals with more schooling (and thus on average higher wages) value their kids’ education more highly.

Another possible reason for public intervention, which we do not explore but just mention briefly here, is the fact that the children, despite all having the same ability, are educated at different level: this might indeed have efficiency implications. The

*laissez-faire*equilibrium is clearly efficient from the point of view of the parents (or of the families as a whole), but if we look at it from the point of view of the children (e.g. if the social welfare function were given by the sum of children’s sub-utility functions) this is no longer the case.

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## Acknowledgments

We are very grateful to two anonymous referees, the editor and Domenico Menicucci for helpful comments. Previous drafts of this paper have been presented at the 2013 SIEP conference (Pavia, Italy), at the 2014 IIPF conference (Lugano, Switzerland) and at the 2014 ASSET conference (Aix-en-Provence, France) as well as at seminars at the University of Paris I and at the University of Pisa. We thank the audiences for the comments, and Elena Del Rey for her insightful discussion.

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*Responsible editor*: Alessandro Cigno

## 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

and the budget constraint is

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

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

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

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

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

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

By totally differentiating, we have:

Now, let

where the sign of *a*
_{13} = *a*
_{31} is not determined. Then

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

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

Given that *a*
_{21} = *a*
_{24}=0, it is

Consider then d *d*/*d*
*w*:

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

Consider finally d *z*/*d*
*w*:

Given that *a*
_{21} = *a*
_{42}=0, it is

We then have:

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

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

The FOCs are

By totally differentiating, we have:

Therefore:

Then, the signs are as follows

### 1.2 Low wagers (*w*<*w*
^{∗})

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

The FOCs for an interior solution are:

Totally differentiating, we have:

Now, let

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

Further assuming that

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

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

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

Consider then d *e*/*d*
*w*:

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,

Finally, let us consider d *d*/*d*
*w*:

Since *a*
_{23}<0 and *a*
_{31}>0, it immediately follows that

### 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:

To check the equivalence, integrate (40) to yield

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

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:

### 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

The FOC is:

and it follows that

### 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

The FOC is

Hence, since

from (108), we have that

### 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

The FOCs are

Hence

as expected. Total differentiation yields

Now, let

Then:

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

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

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Balestrino, A., Grazzini, L. & Luporini, A. A normative justification of compulsory education.
*J Popul Econ* **30**, 537–567 (2017). https://doi.org/10.1007/s00148-016-0619-7

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DOI: https://doi.org/10.1007/s00148-016-0619-7