Francesca Barigozzi
email: francesca.barigozzi@unibo.it.

Helmuth Cremer
email: helmuth.cremer@tse-fr.eu.

Kerstin Roeder
email: kerstin.roeder@wiwi.uni-augsburg.de.

"Having it all, for all: child-care subsidies and income distribution reconciled"

Keywords: JEL-Classi…cation: D13, H23, J16, J22 Child care, women's career choices, child care subsidies, redistribution, social norms

This paper studies the design of child-care policies when redistribution matters. Traditional mothers provide some informal child care, whereas career mothers purchase full time formal care in the market. The sorting of women across career paths is endogenous and shaped by a social norm about gender roles in the family. Via this social norm traditional mothers' informal child care imposes an externality on career mothers, so that the market outcome is ine¢cient. Informal care is too large and the group of career mothers is too small so that ine¢ciency and gender inequality go hand in hand.

In a …rst-best, full information word redistribution across couples and e¢ciency are separable. Redistribution is performed via lump-sum transfers and taxes which are designed to equalize utilities across all couples. The e¢cient allocation of child care is obtained by subsidizing formal care at a Pigouvian rate.

However, in a second-best settings, we show that a trade-o¤ between the reduction of gender inequality and redistributive considerations emerge. The optimal uniform subsidy is lower than the "Pigouvian" level. Under a nonlinear policy the …rst-best "Pigouvian" rule for the (marginal) subsidy on informal care is reestablished. While the share of high career mothers continues to be distorted downward for incentive reasons, this policy is e¤ective in reconciling the objectives of reducing the child care related gender inequalities and achieving a more equal income distribution across couples.

Introduction

While female labor force participation has been increasing steadily over the last decades [START_REF] Goldin | The quiet revolution that transformed women's employment, education, and family[END_REF]and 2014b[START_REF] Kleven | Gender inequality and economic development: fertility, education and norms[END_REF]) mothers continue to be the main providers of child care within the family (e.g., [START_REF] Paull | Children and women's hours of work[END_REF][START_REF] Ciccia | Parental leave regulations and the persistence of the male breadwinner model: using fuzzy-set ideal type analysis to assess gender equality in an enlarged Europe[END_REF]. Maternity leave and other child related career breaks or part-time work contribute in a signi…cant way to the persistence of gender inequalities in the labor market. The so called "child penalty" appears to explain up to about 80% of the gender wage gap; see [START_REF] Kleven | Children and gender inequality: evidence from Denmark[END_REF].

As a possible reason for the persistence of child-care compatible (part-time) work and child penalties, many studies point to social norms shaping women's preferences over family and career During the last …ve decades, most developed countries have put into practice multiple child policies with various declared goals, including gender equity, higher fertility, and child development. The policies who seem to have been the most e¤ective in reducing gender disparities are child care provision and subsidization. Evidence indicates that early childhood spending contributed substantially to enabling women to combine working life and motherhood, and to altering social norms regarding gender roles; see [START_REF] Olivetti | The economic consequences of family policies: lessons from a century of legislation in high-income countries[END_REF].

Reducing gender disparities in the labor market is not the unique concern which is relevant for child care policies. Redistribution across income levels has been the major issue for the design of tax and expenditures policies, it has lead to the emergence of the concept of "welfare state" which applies to all developed countries albeit to a di¤erent degree; see [START_REF] Boadway | Public Goods, Self-Selection and Optimal Income Taxation[END_REF].

Unfortunately, the objective of reducing gender disparities in the labor market and redistributive concerns may be con ‡icting goals. Speci…cally, child care provision and subsidization may be regressive if the parents who bene…t more from the policy are the ones with relatively higher income. This seems to be the case in most OECD countries, where very young children (aged 0-2) are more likely to use early childhood education and care services when they come from relatively advantaged socio-economic backgrounds; see OECD (2017). 1Surprisingly, the interplay between child care provision/subsidization and redistribution has so far to a large extent been ignored in the literature. 2 We o¤er a fresh new look at this issue and propose a theoretical model whose crucial ingredient is an ine¢cient child penalty created by a gender norm. We then investigate the interaction between child penalties, child care policies and redistribution. Our research questions are the following. To what extent reducing the child care related gender inequalities and achieving a more equal income distribution are con ‡icting objectives? How can this potential con ‡ict be mitigated by an appropriate design of the child care policies?

We consider a model in which spouses' career prospects are perfectly correlated. However, while fathers always enter a high-career path, mothers can either enter the same high-career path or a low-career one. In the latter case mothers are "traditional" because they are able to provide some informal child care. "Career mothers" instead need to purchase full-time formal care in the market. The sorting of women across career paths is endogenous and shaped by a social norm about gender roles in the family. Via this social norm traditional mothers' informal care imposes an externality on career mothers, who feel guilt if they provide less informal care than the average amount provided by woman. Hence, in the laissez-faire informal care is too large and the share of career mothers is too small. This translates in ine¢ciently high child penalties so that ine¢ciency and gender inequality go hand in hand. Furthermore, career choices exacerbate inequalities (as measured for instance by the Gini coe¢cient) because higher incomes are concentrated on a smaller share of the population, which further decreases social welfare. We study the optimal design of linear and non-linear child care policies when the government is concerned with both e¢ciency, corresponding to a reduction of gender inequality, and redistribution.

In a recent paper, [START_REF] Barigozzi | Women's Career Choices, Social Norms and Child Care Policies[END_REF] have examined the interplay between social norms, career choices and child-care decisions. We build on their model but adopt a di¤erent modeling strategy for the social norm. The research questions addressed in the two papers are completely di¤erent. [START_REF] Barigozzi | Women's Career Choices, Social Norms and Child Care Policies[END_REF] study whether eradicating or mitigating gender norms is socially optimal and how the design of speci…c policies (a uniform subsidy on child care, a women quota and parental leave) helps to achieve either one or the other objective. Redistribution is not a concern of the government in their model. In this paper we focus on the design of child care 2 Two exceptions are the literature on in-kind transfers and optimal taxation (Cremer and Pestieau 1996) and the literature on optimal taxation with endogenous fertility. In the latter, low-ability families may choose to 'specialize' in quantity, that is, to raise more children relative to higher-ability households. Child-related subsidies can, therefore, be used to enhance re-distribution: family size can be employed as an indicator for the earning capacity of the household [START_REF] Cigno | Fertility and the Tax-Bene…t System: A Reconsideration of the Theory of Family Taxation[END_REF]). We totally depart from that literature because the number of children is exogenous in our model. In addition, we do not solve a model of optimal income taxation, we instead design non-linear child care subsidies.

subsidies when income redistribution is relevant.

We show that, in a …rst-best, full information word e¢ciency and equity are separable.

Redistribution is performed via lump-sum transfers and taxes which are designed to equalize utilities across all couples. Child care policies, on the other hand, are designed to achieve the appropriate level of informal child care and the e¢cient share of high career couples. Since the underlying problem is an externality, it is not surprising that the e¢cient policy involves a Pigouvian subsidy on market child care, which acts like a Pigouvian tax on informal care.

And once child care levels are e¢cient, the induced career choices are also e¢cient. However, since this policy taxes away all extra earnings of high-career couples, it is of course not incentive compatible and it cannot be implemented when the spouses' earning opportunities in the high career path are not observable. This leads to the study of feasible second-best policies.

We consider two types of second-best settings. First, we study a linear subsidy and we show that it involves a trade-o¤ between the reduction of child penalties and redistributive considerations. Consequently, the optimal subsidy is lower than in the case where the government is concerned with e¢ciency (or gender inequalities) only.

More interestingly, we then show that this trade-o¤ depends on the linearity of the policy.

To see that we characterize the optimal incentive compatible policy, that is the non-linear policy constrained by the information structure. We show that this policy reestablishes the …rst-best "Pigouvian" rule for the (marginal) subsidy on informal care. In other words there is no longer a trade-o¤ between child care subsidies and income redistribution. High-career couples enjoy positive rents and their share has to be reduced (compared to the …rst-best) to mitigate these rents. Consequently the outcome remains second-best. Still the policy is e¤ective in reconciling at least to some degree the objectives of reducing child penalties and achieving a more equal income distribution across couples. Note that the subsidy on formal care can be implicit in the case where child care is provided in kind.

The information requirement to implement this policy is rather minimal. It is su¢cient that career paths or levels of formal child care are publicly observable. Amongst these the …rst one appears to be the least restrictive. When consumption of formal child care is observable for each couple, "topping-up" of child care provided in kind can be prevented, which in practice may appear di¢cult. But our analysis shows that when career paths are observable, topping up, is not a problem anyway. High career couples will then receive full time care (in kind or subject to a non linear subsidy) and they do not want to supplement this level by care paid at full market prices anyway. And due to the implicit or explicit subsidy, low career couples consume already more formal care then they would at market prices.

From a practical perspective, the non linearity or the policy introduces a measure of meanstesting into our policies because child-care fees e¤ectively di¤er across income levels. Because of the information limitations, means-testing remains quite basic and couples within a given career path cannot be distinguished. Still even this basic screening device has a rather dramatic impact in reconciling redistribution and child care policies (see also Sections 7 and 8 on this point).

The model

Consider a population of couples with children, the size of which is normalized to one. Each couple consists of a mother 'm', a father 'f ', and a given number of children. Couples choose their career path, the mode of child care, and their consumption.

There exist two types of career paths (indexed by j). First, a full engaging high-career path, j = h, where individuals who take up this career path have to work an entire day which we normalize to one. Second, a less demanding low-career path, j = `, o¤ering ‡exible working hours, where individuals can freely choose how much time to spend in the labor market. The time not spent at work can be used for child care c i , where i = f; m. Both jobs pay the wage rate y, but the high-career path comes with additional future earning possibilities q i . We let q f 2 [0; Q] and q m = q f 2 [0; Q]; with 2 (0; 1]. An < 1 captures pure discrimination: unequal pay for equally quali…ed workers, as it continues to be documented in nearly all developed countries. 3Observe that while < 1 adds a measure of realism to the descriptive part of our model, it will not be essential for our results that all continue to hold when = 1. Future revenue q f is distributed according to the density function f (:) ; with the cumulative distribution being F (:).

Future earning opportunities are perfectly correlated in a couple. Consequently, there is a single level of q m associated with each level of q f . 4Care for children provided by the spouse(s) is denoted by c i (i = f; m), while that bought in the private market is denoted by c p . The latter costs p per unit of time. We let p = y, meaning that the current salary of one member in the couple exactly covers the costs of buying full-time child care on the private market. 5 The children must be taken care of for the entire day, implying c f + c m + c p = 1. Couples in which both parents choose the high-career path thus have to fully rely on private child care. When parents enter a ‡exible job their salary decreases proportionally to the time devoted to care. Informal and private care constitute a family public good and its value to the parents is given by:

G (c f ; c m ; c p ) = v(c f + c m ) + v(c p );
where v 0 > 0; v 00 < 0 and v(0) = 0. Care provided by the father and mother are thus perfect substitutes while informal and private care are imperfect substitutes, with private care being (weakly) less welfare-enhancing than informal care, 2 (0; 1]. 6 Apart from child care, each parent derives utility from consumption of a numeraire commodity x.

Following [START_REF] Akerlof | Economics and identity[END_REF]2010), individuals may su¤er a disutility by deviating from the social categories that are associated with their identity (that is, an individual's sense of self), which causes behavior to conform toward those norms. We assume that individuals desire to conform to the behavior of the group they belong to, namely the behavior of women for mothers and the behavior of men for fathers. Mothers feel guilt if they provide less informal care than the average amount of care provided by woman in the society. 7 Fathers, by contrast, su¤er from social stigma when they devote more time to informal care than the average amount of time devoted to care by man in the society. 8 Given our assumption on the ‡exibility associated with the two available career paths, the social norm for mothers corresponds to the cost of the full-time job given by m (maxf0; c m c m g), where c m is the average time spent with children by mothers in the society. For fathers, the social norm translates into the cost of the ‡exible job given by f (maxf0; c f c f g), where c f is the average time spent with children by fathers. The parameter i 2 [0; 1]; i = f; m; re ‡ects the costs of norm deviations.

The timing of couples' decisions is as follows: …rst, parents choose their career path and then, in the second stage, they choose consumption and the amount of child care (be it formal or informal). Parents act cooperatively and maximize the sum of their utilities:

W = x m + x f + G(c f ; c m ; c p ) m (maxf0; c m c m g) f (maxf0; c f c f g): (1)

Couple's optimization

We …rst analyze the choice of child care activities for a given career path. Then, by proceeding backward, we consider the choice of career path made by the couple. This allows us to determine the average child care provided in the society and thus to de…ne the cost of the social norm both for fathers and for mothers. We consider only decisions made at the second stage by the couples that turn out to be relevant for our analysis, namely the couples where (i) only the father enters the high-career path while the mother enters the ‡exible job market (traditional couples), and those where (ii) both parents take up the high-career path; see Appendix A.1 for the dominated couples' decisions. 9 Traditional couple. We denote welfare of this couple by W h`, where the …rst subscript refers to the father's career choice and the second subscript refers to the mother's career choice.

Since the father took up the high-career path he is not able to take care of the children, and c f = 0. Hence, c f c f 0 and the father does not su¤er any cost associated with the social norm. Noting that c m + c p = 1; the couple chooses child care private provision to maximize [START_REF] Akerlof | Economics and identity[END_REF] where x h`= x m + x f = y + q because p = y. Optimal level of formal child care is thus implicitly determined by

v 0 (c p ) = v 0 (1 c p ): (2)
First-order condition [START_REF] Akerlof | Identity economics[END_REF] indicates that traditional mothers purchase formal care, c p , in the market up to the point where marginal utility from formal care equals the marginal bene…t from informal care, 1 c p .

The marginal norm cost for traditional mothers, m ; does not enter the FOC (2); they do not su¤er any norm cost because by de…nition we have

c hh = 0 so that c h`= 1 c p > c > c hh = 0.
The indirect utility of this h` couple as a function of private child care c p writes:

W h`= y + q + v(1 c p ) + v(c p )
High-career couple. High-career couples have no child care decision to make; they have to buy the full amount of private care on the market. Since c f = 0; the father does not su¤er any cost associated with the social norm. As a result, the social norm for fathers is binding neither in traditional nor in career-couples. Thus, we can simplify the notation writing c m = c and m = : High-career mothers su¤er the cost from deviating from the norm and the couple's welfare amounts to:

W hh = y + q(1 + ) + v(1) c:
Note that high-career couples who exclusively have to rely on private child care are those with higher consumption levels, that is x h`= y + q < x hh = y + q(1 + ).

We are now in the position to analyze the couple's decision about the two partners' career paths. Families have to choose whether to be a high-career hh couple fully relying of formal 9 Only the mother in the high-career path is dominated by having both parents entering the high-career path which involves no norm costs for the father and higher future bene…ts. Similarly, having both parents entering the low-career path can never be optimal since then the couple forgoes future bene…ts q f . As a result, the social norm for fathers is never binding in equilibrium.

child care, or to be a traditional h` couple where the mother provides some informal care. A couple will become a high-career couple if it is bene…ciary to do so, that is if W hh W h`, or if

q q 1 v(1 c p ) + v(c p ) v(1) + c :
The marginal couple q is the couple where parents are indi¤erent between belonging to a traditional and to a career couple. Given q we can now de…ne average informal child care in society:

c = Z q 0 c h`f (q)dq = F (q )c h`= F (q )(1 c p ):

Market outcome

An allocation is given by the identity of the marginal couple and by the amount of child care provided by traditional couples. The following proposition characterizes the laissez-faire allocation.

Proposition 1 (Characterization of the laissez-faire) When mothers who do not provide child care su¤er from deviating from the social norm, i.e. > 0, and/or the job market su¤ers from gender discrimination, < 1, then:

(i) it is never optimal for fathers to take up the low-career path;

(ii) the marginal couple is given by

q = 1 v(1 c p ) + v(c p ) v(1) + F (q )(1 c p ) ; (3)
couples with future job opportunities higher or equal to the threshold q choose the highcareer path for both parents;

(iii) private care purchased by traditional couples, c p , satis…es equation [START_REF] Akerlof | Identity economics[END_REF].

There are both traditional and career couples in the economy if q 2 (0; Q). From (3), an interior solution requires that q exists such that q = (1= 1) > 0 holds so that the previous inequality is always met provided that Q is su¢ciently large and F (q) is concave, which we assume in the remainder of the paper.

)[v(1 c p ) + v(c p ) v(1) + F (q )(1 c p )] < Q: Due to the concavity of v ( ) ; v(1 c p ) + v(c p ) v(
The gender wage gap (GWG) is de…ned as the di¤erence in total income earned by mothers and fathers in equilibrium and is given by:

GW G = Z Q 0 [y + q]f (q)dq F (q )yc p + Z Q q [y + q]f (q)dq = F (q ) 1 c p y | {z } child penalty + Z q 0 qf (q)dq | {z } adverse sorting + Z Q q (1 )qf (q)dq | {z } plain discrimination (4)
The GWG decomposes in the gap between the hours worked because of family duties, and in the di¤erent return to labor supplied in sectors where man and women are employed. The …rst term in (4) thus represents "child penalty" (see Blau Before turning to the design of child-care policy, we de…ne the social planner's objective function and the optimal allocation.

The optimal allocation

The social planner is interested both in e¢ciency and in redistribution. Speci…cally, the social welfare function is assumed to be a concave transformation, ( ); of the families' welfare functions in order to capture inter-family inequality aversion. 10 Thus, a …rst-best (f b) allocation is de…ned by aggregate consumption levels x f b h`( q) and x f b hh (q), by the indi¤erent couple, qfb (which determines the share of female participation in the high-career path), and by the level of formal child care chosen by traditional couples, c f b p (q) for q < qfb (recall that, by de…nition, c f b p (q) = 1 for q qfb ).

Speci…cally, the social planner chooses fx hh (q) ; x h`( q) ; c p (q) ; qg to maximize the following welfare function:

SW = Z q 0 x h`( q) + v(1 c p (q)) + v(c p (q)) f (q)dq + Z Q q x hh (q) + v(1) c f (q)dq (5)
subject to the budget constraint:

y + Z Q 0 qf (q)dq + Z Q q qf (q)dq = Z q 0 x h`( q)f (q)dq + Z Q q x hh (q)f (q)dq; (6)
1 0 In [START_REF] Barigozzi | Women's Career Choices, Social Norms and Child Care Policies[END_REF], redistribution across income levels is not relevant because they assume quasi-linear preferences with a constant marginal utility of income. While the excessive share of traditional couples does also a¤ect the income distribution by making it more concentrated this in itself does not a¤ect welfare in their setting.

where c = R q 0 (1 c p (q)) f (q) dq. In Appendix A.2 we derive the optimal allocation that is characterized as follows.

Welfare is constant irrespective of the couple's career path and their future earning possibilities:

W f b h`( q) = W f b hh (q) = W f b 8q;
Formal child care is such that c f b p (q) = c f b p 8q and is implicitly given by:

v 0 (c f b p ) + [1 F (q f b )] = v 0 (1 c f b p ): (7)
The left-hand side denotes the social marginal bene…t of formal child care while the righthand side denotes the social marginal cost of informal care. Note that the above equation is independent of a traditional couple's q. Compared to the laissez-faire described in (2), the marginal bene…t contains an additional term [1 F (q f b )] which re ‡ects the negative externality of informal care provision on type-hh couples whose share is 1 F (q f b ). Informal child care is thus ine¢ciently high in the laissez-faire, which translates in underconsumption of formal care:

c p < c f b p .
Not surprisingly c f b p and qfb do not depend on the social welfare function . This is due to the quasi-linearity of preferences. All Pareto-e¢cient allocations imply the same levels of c p and q, but may di¤er in consumption levels. But since we use a symmetric social welfare function any concave implies that in the …rst-best utility levels are equalized. However, the degree of concavity will matter in the second-best settings considered below.

Interestingly, W f b h`( q) = W f b hh (q) and c f b p (q) = c f b p 8q imply that the consumption of couples is constant in each career-path:

x f b h`( q) = x f b h`a nd x f b hh (q) = x f b hh 8q
. This in turn implies that:

x f b hh x f b h`= v(1 c f b p ) + [v(c f b p ) v(1)] + F (q f b )(1 c f b p ) > 0 (8)
The above expression shows that high-career couples do not get higher consumption because of their higher q (as it was the case in the laissez-faire), but because the government compensates them for their utility loss due to the cost of the social norm and to the purchase of full private care (whose utility is mitigated by the parameter ). Finally, in Appendix A.2 we show that the FOC wrt q can be rewritten as:

qfb f (q f b ) = f (q f b )[v(1 c f b p )+ (v(c f b h`) v(1))+ F (q f b )(1 c f b p )] [1 F (q f b )](1 c f b p )f (q f b ) (9) so that qfb 1 f[v(1 c f b p ) + (v(c f b h`) v(1))] + F (q f b )(1 c f b p ) [1 F (q f b )](1 c f b p )g (10)
Comparing ( 3) and [START_REF] Boadway | Public Goods, Self-Selection and Optimal Income Taxation[END_REF] and recalling that c p < c f b p , we observe that q > qfb , that is the share of high-career couples is ine¢ciently low in the laissez-faire. Expression (9) has a simple interpretation in terms of cost and bene…ts of decreasing q (that is moving f (q) couples from traditional to high-career). The LHS measures the marginal bene…ts in terms of extra future earnings. In the RHS, the …rst two terms in brackets represent the net lost utility from formal care and the norm cost, respectively. The last term is the Pigouvian term which is negative because the externality imposed on all high-career couples decreases because the average informal care falls. Formally, we have @c=@ q = (1 c p )f (q). Since a negative cost is e¤ectively a bene…t this term could have been moved to the LHS, but since the interpretation of ( 9) also shows that of [START_REF] Boadway | Public Goods, Self-Selection and Optimal Income Taxation[END_REF] this presentation is more telling. 11 Observe that qfb does not depend on ; it is the same in all Pareto e¢cient allocations. The …rst-best level qfb is set purely on e¢ciency grounds-to maximize the size of the cake which is then redistributed according to social preferences (which in our case involves equalization of utilities).

The following propositions characterizes the optimal allocation: Proposition 2 (The optimal allocation) The optimal allocation fx f b hh ; x f b h`; c f b p ; qfb g maximizes the social welfare function [START_REF] Bertrand | Gender identity and relative income within households[END_REF] subject to the budget constraint [START_REF] Blau | The gender wage gap: extent, trends, and explanations[END_REF] and is characterized as follows:

(i) Couples' welfare does not depend on the career choice of the mother nor on career prospects: W f b h`( q) = W f b hh (q) 8 q: High-career couples get higher consumption because they are compensated for their utility loss due to full private care and due to the cost of the social norm.

(ii) Formal child care c f b p (q) = c f b p is the same for all traditional couples and satis…es [START_REF] Blinder | Wage discrimination: reduced form and structural estimates[END_REF]. It is chosen such that the negative externality induced by the social norm is fully internalized.

(iii) The share of high-career couples is given by 1 F (q f b ) where the marginal couple qfb is de…ned in [START_REF] Boadway | Public Goods, Self-Selection and Optimal Income Taxation[END_REF].

(iv) The optimal level of the GWG entails a child penalty and a sorting di¤erential equivalent to F (q f b ) 1 c f b p y and R qfb 0 qf (q)dq; respectively.

Point (iv) directly follows from substituting (c f b p ; qfb ) into equation (4).

Welfare analysis of the laissez-faire allocation

By comparing the optimal allocation and the market outcome we can establish in which sense the laissez-faire allocation is ine¢cient.

Proposition 3 (Welfare analysis of the laissez-faire) In the laissez-faire allocation:

(i) Within each career path, welfare di¤ers across couples; it increases with career prospect q:

(ii) Formal child care, c p , is ine¢ciently low and informal care, c h`, is too high. This is due to the negative externality that informal care exerts on high-career mothers through the social norm.

(iii) Female participation in the high-career path is ine¢ciently low, qfb < q .

(iv) In the GWG, both the child penalty and adverse sorting are ine¢ciently high.

In the …rst-best all couples receive the same welfare. Proposition 3(i) shows that, in the laissez-faire, welfare is increasing in q both among traditional couples and among career couples.

Thus, welfare is equalized neither across couples belonging to di¤erent career paths nor across couples within the same career path.

Point (ii) shows that the negative externality translates into under consumption of formal child care by traditional couples in laissez-faire (c p < c f b p ). Point (iii) concerns the share of women entering the high-career path which is ine¢ciently low in laissez-faire. When the negative externality is internalized, formal child care increases and the cost of the social norm falls. As a result the high-career path is more attractive in the …rst-best, or q > qfb .

Finally, point (iv) requires some explanations. For any given q; in the laissez faire, the female spouse's earnings are less than or equal to her …rst-best earnings. Indeed, child penalty is lower in the …rst-best because women's labor income is higher due to the higher formal child care (c p < c f b p ). The optimal level of child penalty is thus obtained when the negative externality exerted by traditional mothers on career mothers is properly taken into account. This clari…es why, in the model, e¢ciency is reached via the appropriate reduction of child care related inequalities. Finally, adverse sorting is lower because more women enter the high-career path and bene…t from future prospects (q > qfb ).

Decentralizing the …rst-best allocation

Decentralization of the …rst-best solution requires a subsidy s on formal child care and individualized lump-sum taxes or transfers T h`( q) and T hh (q). When a subsidy s is in place, the net price of private child care is p n = p s = y s; and a traditional couple's optimal child care decision solves:

v 0 (1 c p ) s = v 0 (c p ): (11)
Comparing ( 11) with [START_REF] Blinder | Wage discrimination: reduced form and structural estimates[END_REF] shows that a subsidy of

s f b = [1 F (q f b )] (12)
implements the …rst-best level of child care. Since formal and informal care sum up to one, a subsidy on market care is e¤ectively a tax on informal care. According to equation ( 12) s f b corresponds to a Pigouvian tax on informal child care; it equals the marginal social cost of the externality informal care imposes on high-career couples.

The lump-sum transfers T h`( q) and T hh (q) must be chosen such that welfare levels between all couples are equalized, that is

W h`( q) = y + q + s f b c f b p + v(1 c f b p ) + v(c f b p ) + T h`( q) = W f b when q qfb ; W hh (q) = y + (1 + )q + s f b + v(1) c + T hh (q) = W f b when q qfb :
Decentralizing qfb further requires T h`( qfb ) = T hh (q f b ). To see this note that when T h`( q) = T hh (q) the marginal couple de…ned by W h`( q) = W hh (q) is determined by

y + q + s f b c f b p + v(1 c f b p ) + v(c f b p ) = y + (1 + )q + s f b + v(1) c , q = 1 [v(1 c f b p ) + v(c f b p ) v(1) + c s f b (1 c f b p )] (13)
Using [START_REF] Casarico | Intergenerational transmission of skills during childhood and optimal public policy[END_REF] together with c = F (q)(1 c f b p ) shows that ( 13) and (10) coincide once formal child care is subsidized at the Pigouvian rate.

Hence, with su¢ciently powerful instruments e¢ciency and redistribution can be addressed separately: the Pigouvian subsidy s f b on private child care optimally reduces informal child care provision (and thus child penalties) while the transfers T h`( q) and T hh (q) assure equal welfare to all couples. Note that the individualized transfers redistribute from high to low q couples but also compensate the high-career couples for their utility losses due to full private care and to their cost of the social norm. 12 We now turn to the study of second-best policies.

Linear policy

First, we consider a simple policy under which instruments are restricted in an ad hoc way.

In other words, we remain agnostic about the information structure. We assume that the instruments necessary to implement the …rst-best are not available (speci…cally the individualized transfers) and consider a simple policy which is empirically appealing and e¤ectively used in practice. 1 2 In [START_REF] Barigozzi | Women's Career Choices, Social Norms and Child Care Policies[END_REF], the social norm is determined by child-care decisions made by the median couple of the preceding generation. With this di¤erent speci…cation of the norm it turns out that a Pigouvian subsidy does not restore e¢ciency but reduces informal care too much. Hence the optimal subsidy must be set below the Pigouvian rule.

The considered policy consists of a uniform (linear) subsidy s on market child care, …nanced by a uniform lump-sum tax . The government's budget constraint is then given by

= sF (q(p n ))c p (p n ) + s[1 F (q(p n ))]:
Recall that p n = p s = y s is the net, after subsidy, price of market care. Let us denote c s p = c p (p n ) consumption of formal care under the linear subsidy s: As before it is implicitly determined by:

v 0 (1 c s p ) s = v 0 (c s p ) (14)
The social welfare function can be written as:

SW (s; ) = Z q(p n ) 0 y + q + sc p (p n ) + v(1 c p (p n )) + v(c p (p n )) f (q)dq + Z Q q(p n ) y + (1 + )q + s + v(1) c(p n ) f (q)dq; (15)
where qs = q(p n ) and c(p n ) = F (q s )(1 c p (p n )). The FOC wrt is given by:

= Z Q 0 0 (q)f (q)dq E[ 0 ]; (16)
where E is the expectation operator and where (q) is de…ned as (W h`( q)) for h`couples and as (W hh (q)) for hh couples. This equation has a familiar ‡avor from linear taxation models, in particular [START_REF] Sheshinski | The Optimal Linear Income Tax[END_REF]. It states that the social marginal cost of raising an additional dollar, , should be equal to its social marginal bene…t, E[ 0 ]. Now de…ne:

E h`[ 0 ]
R qs 0 0 (q)f (q)dq

F (q ) and E hh [ 0 ] R Q qs 0 (q)f (q)dq 1 F (q ) ; (17)
which represent the average marginal utilities of income by traditional and high-career couples respectively.

The FOC with respect to s is given by:

F (q s )E h`[ 0 ]c s p + (1 F (q s ))E hh [ 0 ] 1 + F (q s ) dc s p dp n (1 c s p )f (q s ) dq s dp n F (q s )c s p sF (q s ) dc s p dp n + s(1 c s p )f (q s ) dq s dp n + 1 F (q s ) = 0: (18)
Noting that E[c s p ] = F (q s )c p + 1 F (q s ) we show in Appendix A.4 that the optimal linear subsidy on formal child care, s o , amounts to:

s o = (1 F (q s ))E hh [ 0 ] E[ 0 ] cov[ 0 ; c s p ] E[ 0 ] @E[c s p ] @p n (19)
The …rst expression is the Pigouvian term and the second term is the redistributive term. When 00 = 0 so that social welfare is not concave and there is no concern for redistribution and the above expression reduces to s o = [1 F (q s )] , which is the …rst-best Pigouvian rule. From expression [START_REF] Card | Bargaining, sorting, and the gender wage gap: quantifying the impact of …rms on the relative pay of women[END_REF] this also yields q = qs so that we return to the …rst-best allocation. When the social welfare function is concave, we have cov[ 0 ; c s p ] < 0 since families with higher formal care have a higher welfare. In the Appendix we show that @E[c s p ]=@p n < 0 so that the second term on the RHS in expression ( 19) is negative (a positive fraction is preceded by a negative sign). Redistributive concerns thus decrease optimal child care subsidies since it is mainly the high-career couples who pro…t from such subsidies. Furthermore, we have

E hh [ 0 ] < E[ 0 ]
so that the Pigouvian term is also reduced compared to its …rst-best counterpart. This is because the externality a¤ects high career-couples who in the second-best have a lower social marginal utility. The marginal social damage of the externality is determined by converting their (marginal) utility into social (marginal) utility, which is achieved by the term E hh [ 0 ]=E[ 0 ]. 13 Consequently, we have s o < s f b ; see Appendix A.4 for the formal proof. Proposition 4 (Linear child care subsidy) The optimal linear policy when redistribution is relevant ( 00 > 0) implies: (i) s o < s f b because it is mainly the high-career couples who pro…t from this policy. Thus, formal child care purchased by traditional couples, c s p , is ine¢ciently low c f b p > c s p ;

(ii) and qs > qfb so that there are more traditional couples in the second best than in the …rstbest. The marginal couple is distorted upwards to reduce the share of high career couples receiving the subsidy for full-time formal care which improves redistribution.

(iii) In the GWG, both child penalties and adverse sorting are ine¢ciently high.

The intuition for (iii) is the same as for the corresponding point in Proposition 3. As expected, the linear subsidy mitigates the ine¢ciency of the laissez-faire informal care provision but does not fully restore e¢ciency. However, welfare is obviously higher with the linear policy than in the laissez faire.

Nonlinear policy

Now, we take a di¤erent approach and assume that the available policies are not restricted in an ad hoc way. Instead, we study the design of the best policy that is available given the information structure. This is not just a matter of theoretical interest. The important underlying practical question is whether the distortions characterized in the previous section are unavoidable once redistribution under asymmetric information is involved, or whether they are simply artifacts of the linearity of the considered policy.

Under full information this approach yields the …rst-best, but this supposes that all relevant variables, including a couple's high-career earning opportunities q are publicly observable. We shall now assume that q is not publicly observable but that both the career path and the level of market care are observable at the individual (couple's) level. The government can then o¤er two contracts conditioned on the reported type q denoted by fJ(q); c g p (q); T (q)g ; where J 2 fh`; hhg indicates the career path, T is the transfer that households have to pay and c g p (q) is the amount of formal child care provided by the government. Since c g p (q) is observable at the couple's level, the distinction between in-kind provision and a nonlinear taxation of market care is not relevant; see [START_REF] Cremer | In-kind transfers, self-selection and optimal tax policy[END_REF]. To be more precise, this is simply a matter of practical implementation of the underlying optimal contract. This implies, in particular, that when c g p (q) is interpreted as in-kind provision, topping up is not possible. 14 As usual we shall, without loss of generality, concentrate on incentive compatible contracts.

Given that no topping up is possible it must be c g p (q) = 1 for all hh couples. In addition, given that, conditional on the career path, all families have the same preferences for child care, it is impossible to separate families according to q once the career path has been assigned. Hence, the government o¤ers only two contracts: fT h`; c g p g for h`-couples and fT hh ; 1g for hh-couples. In other words, all traditional couples consume the same level of market care and face the same tax or transfer. The same is true for all high-career couples. 15 The average externality now is c = F (q g ) 1 c g h` ; where qg indicates future prospects of the marginal couple, or the couple such that welfare is the same in the two career paths, qg : W hh (q g ) = W h`( qg ).

The government maximizes the following welfare function:

max T h`; c g p ;T hh ;q g SW = Z qg 0 (y + q + c g p y T h`+ v 1 c g p + v c g p | {z } ) W h`f (q) dq + Z Q qg (2y + (1 + ) q T hh + v (1) F (q g ) 1 c g p | {z } ) W hh f (q) dq ( 20
)
subject to the budget constraint

F (q g ) T h`+ [1 F (q g )] T hh p F (q g ) c g p + 1 F (q g ) 0; (21)
and subject to the following incentive constraint:

2y + (1 + ) qg T hh + v (1) F (q g ) 1 c g p y + qg + c g h`y T h`+ v 1 c g p + v c g p = 0: (22)
1 4 With the considered information structure it can be prevented and nothing can be gained by allowing it. 1 5 This is a well known property in contract theory and we skip the proof. To establish the results formally one has to maximize social welfare subject to the budget and incentive constraints. A simple …rst-order approach will show that the solution involves pooling within each career group.

Since there is pooling in both groups, incentive compatibility requires simply that qg is indi¤erent between the two career paths. This follows because @W hh (q)=@q = 1 + > @W h`( q)=@q = 1 so that W hh increases faster in q than W h`. Consequently, condition [START_REF] Goldin | The quiet revolution that transformed women's employment, education, and family[END_REF] ensures that no highcareer couple with future earnings q qg should have an incentive to mimic a traditional couple, that is W hh (q) W h`( q) 8 q 2 [q g ; Q]. Similarly, it implies that no traditional couple wants to mimic a high career couple.

We denote the Lagrangian multipliers associated with the budget constraint and the incentive constraint b and respectively. Using the expectation operators de…ned in [START_REF] Cornelissen | Who Bene…ts from Universal Child Care? Estimating Marginal Returns to early Child Care Attendance[END_REF] we can write the FOCs with respect to the transfers T h`a nd T hh as:

F (q g )E h`[ 00 ]F (q g ) + + b F (q g ) = 0 ( 23
)
(1 F (q g ))E hh [ 00 ] [1 F (q g )] + b [1 F (q g )] = 0 (24)
Combining ( 23) and ( 24) and rearranging yields:

b = Z qg 0 0 ( ) f (q)dq + Z Q qg 0 ( ) f (q)dq = E 0 : (25)
This equation simply states that the marginal cost of raising additional revenue, b , must be equal to its marginal social bene…t, E[ 0 ]. The FOC with respect to formal child care for traditional couples, c g p , is given by:

Z qg 0 0 ( ) y v 0 1 c g p + v 0 c g p f (q) dq + Z Q qg 0 ( ) F (q g ) f (q)dq b pF (q g ) + F (q g ) y + v 0 1 c g p v 0 c g p = 0 (26)
In Appendix A.5 we show that by using ( 24) and ( 25) the ( 26) reduces to:

v 0 (1 c g p ) + v(c g p ) = [1 F (q g )] : (27)
Comparing this expression to [START_REF] Bursztyn | Acting wife': marriage market incentives and labor market investments[END_REF] shows that the level of child care c g p can be decentralized by a subsidy on market care given by:

s g = [1 F (q g )] : (28)
Consequently, the public provision of c g p corresponds to an implicit subsidy on market care which is set according to the Pigouvian rule de…ned by [START_REF] Casarico | Intergenerational transmission of skills during childhood and optimal public policy[END_REF]. In other words, it re ‡ects the marginal social damage which is here measured by the extra norm cost imposed on all career couples. This is an interesting result because it implies that the downward distortion on s implied by the redistributive bias obtained in the previous section indeed appears to be an artifact of the ad hoc restrictions imposed on the policy, namely its simple linear speci…cation. When the policy is constrained only by the information structure this distortion vanishes. However, while s g is set according to the …rst-best Pigouvian rule, its actual level will di¤er from s f b , unless qg = qfb . This brings us to the next question namely the comparison between qg and qfb . This amounts to studying whether the solution under asymmetric information involves a distortion on the marginal couple and if yes in which direction.

The FOC with respect to qg can be written as:

(W h`( qg )) (W hh (q g )) f (q g )(1 c g p )(1 F (q g ))E hh [ 0 ] + b f (q g ) (T h` T hh ) + pf (q g ) 1 c g p + f (q g ) 1 c g p = 0; (29)
where the …rst two terms vanish because of the incentive constraint.

The approach is to evaluate the FOC for qg at qfb while adjusting all the other endogenous variables according to their respective FOCs. 16 When qg = qfb we have from ( 27) that c g p = c f b p . In Appendix A.6 we show that

T h` T hh = [1 F (q f b )](1 c g p ) y(1 c g p ): (30)
Solving [START_REF] Guendouzi | The guilt thing': balancing domestic and professional roles[END_REF] for and inserting [START_REF] Oecd | Doing better for families[END_REF] in ( 29), we have:

@L @ qg qg =q f b = E hh [ 0 ](1 F (q g )) f (q f b )(1 c g p ) + E[ 0 ] h f (q f b )( y(1 c g p ) + (1 F (q f b ))(1 c g p )) + f (q f b )y(1 c g p ) i + [ E hh [ 0 ](1 F (q g )) + E[ 0 ](1 F (q f b ))][ f (q f b )(1 c g p )] =(1 F (q f b ))(E[ 0 ] E hh [ 0 ]) > 0: (31)
So that we have qg > qfb . In words, the second-best solution implies an upward distortion of the marginal couple qg . Consequently, there are more traditional couples in the second-best solution than in the …rst-best.

To understand this expression note that a couple with q b q enjoys an informational rent of (q b q) = W hh (q) W hh (b q). Total rents are thus given by: R =

Z Q b q (q b q)f (q)dq and we have: 17 @R @ b q = Z Q b q f (q)dq = [1 F (b q)]:
1 6 If the other variables were held constant the sign of the derivative would be inconclusive. However, adjusting all the other variables in an optimal way reduces the problem to a single dimension so that the derivative is informative. As an example, consider the maximization of f (x; y) and denote the solution (x ; y ). Showing that at any given point (x; y), @f =@x > 0 is not enough to show that x > x . However, by using the FOC for y we reduce the problem to the maximization of f (x; y (x)) and the derivative of this expression allows us to compare

x and x , as long as the problem is concave which we have to assume anyway. 1 7 Note that the derivative wrt the lower bound is zero.

Under full information these rents can be extracted and redistributed. Under asymmetric information they cannot because of the incentive constraint. As b q increases the extra amount [1 F (b q)] can be extracted and redistributed which implies a social bene…t of (E[ 0 ]

E hh [ 0 ]) (1 F (q f b )).
In words, the second-best solution involves an upward distortion in the marginal couple in order to reduce "informational rents" of the high-career couples. This means that by increasing the level of q of the marginal couple more tax revenue can be extracted from the high-career couple and redistributed to the traditional couples with lower income, so that welfare increases.

We can now also return to the levels of the implicit subsidy implied by the policy. Equation ( 12) and ( 28) together with qg > qfb imply s g < s f b , so that asymmetric information leads to a lower implicit subsidy on formal care. Intuitively, the strict Pigouvian rule applies in both cases but with qg > qfb the group of high-career couples a¤ected by the externality is smaller so that its marginal social damage is also smaller. Consequently, using [START_REF] Bursztyn | Acting wife': marriage market incentives and labor market investments[END_REF] we'll also have c g p < c f b p . As in the linear case all these results emerge as long as 00 < 0 so that social welfare is concave and there is a concern for redistribution. When 00 = 0 we return to the …rst-best solution.

To sum up, while the nonlinear policy brings us back to the …rst-best Pigouvian rule for the marginal subsidy, it continues to imply a downward distortion on formal care and there will be more traditional couples than e¢cient. Consequently, the potential con ‡ict between child care provision and redistribution does not solely arise with linear instruments.

Finally, let us revisit the underlying information structure. We have assumed for simplicity that a couple's formal care and career path are observable. We have made this assumption for the ease of exposition, but the arguments and results we presented make clear that the observability of the career path is e¤ectively not necessary. The policy we characterize here can be implemented as long as a couple's level of formal care is observable. This is because highcareer couples need full-time care so that their choice of child care would reveal any attempt to mimic a traditional couple. Similarly, a traditional couple mimicking a high-career one would have to choose full-time day care so that mimicking involves the same consumption bundle with or without observable career paths.

The main results of this section are summarized in the following proposition.

Proposition 5 Assume that couples' formal child care is observable and can be provided publicly at level c g p (q) or subject to a nonlinear tax or subsidy. The optimal incentive compatible policy when redistribution is relevant ( 00 < 0) implies: (i) that there is pooling within the traditional and the high career couples groups: all traditional couples receive the same level of formal care and pay the same tax and similarly for all high career couples.

(ii) that high-career couples receive full-time formal care, while the level of c g p implies an implicit marginal subsidy which is determined by the Pigouvian rule: it equals s g = [1 F (q g )] which re ‡ects the marginal social damage represented by the norm cost imposed on the high-career couples.

(iii) qg > qfb so that there are more traditional couples in the second best than in the …rst-best.

The marginal couple is distorted upwards to reduce the high-career couples' informational rents which improves redistribution.

(iv) s g < s f b ; while both levels are set according to the Pigouvian rule, the inequality follows because there are less high-career couples in the second best so that the marginal social damage of the norm cost is smaller.

(v) that, in the GWG, both child penalties and adverse sorting are ine¢ciently high.

The intuition for (v) is the same as for the corresponding part in Propositions 3 and 4.

Again, the policy mitigates the ine¢ciency of the laissez-faire informal care provision but while welfare is obviously higher with the nonlinear policy than with the linear one …rst-best e¢ciency is not restored.

Policy discussion

Elizabeth Warren (a democratic candidate for the US presidential elections) has included "universal child care" as a main pillar in her electoral platform. Similarly, in recent Bavarian elections a new "Free voters of Bavaria" movement managed to unsettle the traditional Christian Democratic majority in the regional parliament with a program aiming at o¤ering free child care to all families. Whether or not these are realistic policy options or utopian visions that are impossible to …nance (and mainly a boon for well o¤ couples) remains to be seen. But these two examples (which could be completed by many others) show how signi…cant these issues are in practice.

Policy choices that are made in the coming years may a¤ect gender roles (and even fertility decisions) for many decades to come.

In most countries the current situation is not the laissez-faire allocation used as reference in our analysis. Various policies already provide child care and early childhood education. In the majority of countries, education now begins for most well before 5 years old: 71.5% of young children aged 3 and 4 years are enrolled in education across OECD countries as a whole, and this rises to 79.8% in the OECD countries that are part of the European Union. 18 Publicly-funded pre-primary provision tends to be more strongly developed in the European than in the non-European countries of the OECD. In Europe, the concept of universal access of 3-to 6-year-olds is generally accepted. Most European countries provide all children with at least two years of free, publicly-funded provision before they begin primary provision. Public expenditure on early childhood and educational care, in cash or in kind, represents today on average 0.8 percent of GDP in OECD countries. 19 Typically, neither the nursery school nor the primary school provide a form of child care which fully covers the needs of full-time working parents. Apart from Scandinavian countries, the demand for day-care centers is signi…cantly larger than the available capacity, even in countries with long parental leave. 20 In countries where public funding for such provision is limited, most working parents must either seek solutions in the private market, where ability to pay signi…cantly in ‡uences accessibility to quality services, or else rely on informal arrangements with family, friends and neighbors; see OECD (2010).

Fees charged to parents for publicly provided early child-care are often high. Parents in Ireland, the Netherlands, Switzerland and the United Kingdom face some of the highest out-ofpocket costs for centre-based care in Europe. Even though all countries except Ireland provide additional …nancial support for families on very low incomes, net fees often remain high in absolute terms. 21 This explains, at least to some extent, the fact that children are more likely to use early childhood education and care services when they come from relatively advantaged socio-economic backgrounds; see OECD (2017).

To sum up, currently most child care systems are not designed in such a way to accommodate parents' working hours. The supply of day-care facilities is rationed in terms of spots available, opening hours are generally too short and fees tend to be quite high for children of 0-3 years of age. As long as this remains the case, child care policies notwithstanding, the current situation su¤ers from the same de…ciencies as the laissez-faire in our model. The policies we present, though only second best, would represent a step in the right direction.

Our model shows that a "free for all" approach would be neither e¢cient nor fair. This would 1 9 It attains 2 percent in Denmark, and is above 1 percent in the rest of Scandinavia, the United Kingdom, and France. North American and Southern EU countries have the lowest rates of early childhood public spending. In the United States, early childhood public spending is 0.4 percent of GDP; see OECD (2014).

2 0 Average hours in early childhood education and care di¤er substantially across countries. In most OECD countries, children (0-to-2-year-olds) in early childhood education and care use it for an average of somewhere between 25 and 35 hours during a usual week, with the OECD average just under 30 hours per week. However, in some countries (e.g. Iceland, Latvia and Portugal) average hours approach 40 hours during a usual week. In others, such as the Netherlands and the United Kingdom, 0-to 2-year-olds in early education centers are there for an average of less than 20 hours during a usual week. 2 1 In the Netherlands and the United Kingdom the out-of-pocket cost of full-time centre-based care for two children (aged 2 and 3) in a low-earning dual-earner family works out at around 20% of family disposable income, and at 35% in Ireland.

be overshooting in the opposite direction. Our analysis suggests that attendance can be used as a devise to screen high-and low-income families because the number of hours children spend in day-care represents a proxy for the family's income. Hence, fees contingent on the time children spend in the facility are e¢ciency enhancing. However, in OECD countries, day-care fees are generally based on enrolment and possibly on family's income but to a much lower extent on hours of attendance. 22 In addition, information about hours of attendance is typically easier to collect than information about family income or wealth and is not falsi…able. Which again points at children's hours of attendance as a practical screening instrument.

Conclusion

We have studied the design of child-care policies when women's career choices are endogenous.

High career mothers su¤er from a norm cost caused by "mothers' guilt". Through their child care choices low career mothers create a negative externality via the norm cost. Consequently, the laissez faire solution is ine¢cient; it implies too much informal child care and a share of high-career mothers which is too low.

Child-care policies are e¤ective in enhancing e¢ciency and reducing gender inequalities.

However, since they provide larger bene…ts to high income couples, they tend to be regressive.

Under full information, this e¤ect can be o¤set by lump-sum transfers and the optimal policy is a Pigouvian subsidy on formal child care. A uniform subsidy, on the other hand, involves a trade-o¤ between e¢ciency and redistribution across couples and should be set below the Pigouvian level. Under a nonlinear policy the …rst-best "Pigouvian" rule for the (marginal) subsidy on informal care is reestablished. While the share of high career mothers continues to be distorted downward for incentive reasons, this policy is e¤ective in reconciling the objectives of reducing the child care related gender inequalities and achieving a more equal income distribution across couples.

From a practical perspective a non-linear policy can be implemented through in-kind provision of child care, at di¤erent levels, depending on the mothers career path, and …nanced with non-linear taxes. Alternatively non-linear subsidies on market care can be used. 23 Either way, day-care fees should be contingent on the amount of time children spend in the facility.

2 2 A …nancial contribution is demanded of all parents who use the public child-care services before compulsory school; this is rarely as much as the unit cost of the service and may be adjusted in accordance with criteria, such as size and income of the family. In most OECD countries, the contribution demanded of parents only depends on their income; see OECD (2017). More generally, our model indicates that providing "free child care to all" is problematic. While universal provision of preschool child care is desirable, free access is never optimal because it represents a too regressive policy.

Our model has ignored a certain number of important aspects. For instance, we do not consider the welfare of children and the impact of early education on their human capital. There is now ample evidence that high quality formal child care yields better outcome for the children than informal care by less advantaged mothers (see Duncan and Sojourner, 2013, Cornelissen et al., forthcoming). This is likely to call for an even more generous child care policy and tend to increase subsidies. We have also ignored the issue of explicit means testing. This would require a more complex information structure to keep the problem interesting. In essence we would have to combine our approach with a more traditional optimal tax model inspired, for instance, by [START_REF] Casarico | Intergenerational transmission of skills during childhood and optimal public policy[END_REF]. These and further extensions are on our research agenda.

Optimal child care provision, c ``= c m + c f ; is implicitly determined by the two conditions:

v 0 (c f ) v 0 (1 c ``) + I f v 0 (c m ) v 0 (1 c ``) + I m
Welfare W ``now is:

W ``= y + v(c ``) + v(1 c ``) f (maxf0; c f c f g) m (maxf0; c m c m g):

A.2 The optimal allocation

Denoting the Lagrangean multiplier with respect to the budget constraint, the FOCs of [START_REF] Bertrand | Gender identity and relative income within households[END_REF] with respect to the couples' consumption levels can be rewritten as:

@SW @x h`( q) = 0 (W h`( q))f (q) f (q) = 0 8q q @SW @x hh (q) = 0 (W hh (q))f (q) f (q) = 0 8q > q:

so that: 0 (W f b hh (q)) = 0 (W f b h`( q)) = , W f b h`( q) = W f b hh (q) 8q:

Equalizing welfare levels across career paths, we can write:

x f b h`( q) + v(1 c f b p (q)) + v(c f b p (q)) = x f b hh (q) + v(1) F (q f b )(1 c f b p (q)) 8q: (A.1)

We now consider the point-by-point derivative of the social welfare with respect to c p (q) : Given that c p (q) exerts a negative e¤ect on all hh couples we have:

0 (W f b h`( c f b p (q)))
@W f b h`( c f b p (q)) @c f b p (q) f (q) + Z Q q 0 (W f b hh (")) @W f b hh (") @ c @ c @c f b p (q) f (") d" = 0 which gives:

0 (W f b h`) h v 0 (1 c f b p (q)) v 0 (c f b hp (q)) i f (q) + Z Q q 0 (W f b hh ) ( f (q)) f (") d" = 0
Considering that W f b h`= W f b hh , we can simplify the previous equation as follows:

v 0 (1 c f b p (q)) v 0 (c f b p (q)) Z Q q f (") d" = 0
showing that it must be c f b p (q) = c f b p 8q: Rearranging, the above equation we obtain [START_REF] Huerta | Early maternal employment and child development in …ve OECD countries[END_REF] in the main text.

Taking the derivative of the social welfare function with respect to the marginal couple q and rearranging, yields:

qf (q f b ) = f (q f b ) h x f b hh (q f b ) x f b h`( qfb ) [1 F (q f b )](1 c f b p ) i : (A.2)
Given that c p (q) = c p 8q; we observe that x f b h`( q) = x f b h`a nd x f b hh (q) = x f b hh 8q. Hence, equation (??) can be rewritten as:

A.3 Comparative statics

Child care, c p , and the marginal couple, q, are implicitly determined by the following two equations:

f 1 (c p ; q; p n ) y p n v 0 (1 c p ) + v 0 (c p ) = 0 f 2 (c p ; q; p n ) y q + c p y + p n (1 c p ) + v(1 c p ) + [v(c p ) v( 1)] + m F (q)(1 c p )

When we want to know the e¤ect in price changes of formal child care, we have to solve:

2 4 @f 1 @cp @f 1 @ q @f 2 @cp @f 2 @ q 3 5 " dc p dq # = " @f 1 @p n @f 2 @p n # dp n :
Inserting the derivatives and inverting the …rst matrix, we have:

A.4 Uniform subsidies

The FOC wrt s can be written as

E[ 0 c p ] + (1 F (q ))E hh [ 0 ] F (q ) dc p dp n (1 c p )f (q ) dq dp n E[ 0 ]E[c p ] E[ 0 ]s F (q ) dc p dp n + (1 c p )f (q ) dq dp n = 0;
where E[c p ] = F (q(p n ))c p (p n ) + 1 F (q(p n )). Noting that @E[c p ] @p n = F (q ) dc p dp n (1 c p )f (q ) dq dp n < 0 and cov[ 0 ; c p ] = E[ 0 c p ] E[ 0 ]E[c p ], we can write @SW @s = cov[ 0 ; c p ] (1 F (q ))E hh [ 0 ] @E[c p ] @p n + E[ 0 ]s @E[c p ] @p n : (A.6)

Setting this expression equal to zero and solving for s yields equation [START_REF] Duncan | Can intensive early childhood intervention programs eliminate income-based cognitive and achievement gaps?[END_REF]. Further evaluating (A.6) at the Pigouvian level s f b = [1 F (q f b )] m , which from (13) implies q = qfb yields @SW @s s=s f b = cov[ 0 ; c p ] (1

F (q f b ))E hh [ 0 ] @E[c p ] @p n +E[ 0 ][1 F (q f b )] m @E[c p ] @p n = cov[ 0 ; c p ] < 0 (A.7)
so that assuming concavity we must have s o < s f b .

A.5 Proof of equation ( 27)

The FOC wrt c g h`i s given by: Z qg 0 0 ( ) y v 0 1 c g h` + v 0 c g h` f (q) dq + Z Q qg 0 ( ) F (q g ) f (q)dq b pF (q g ) + F (q g ) y + v 0 1 c g h`

v 0 c g h` = 0:
With equations ( 24) and ( 25) and the following de…nitions:

E h`[ 0 ] =
R qg 0 0 ( )f (q)dq F (q g ) and E hh [ 0 ] = R Q qg 0 ( )f (q)dq 1 F (q g ) we can rewrite the above FOC as: E h`[ 0 ]F (q g )[y v 0 (1 c g h`) + v 0 (c g h`) ] + F (q g )E hh [ 0 ](1 F (q g )) E[ 0 ]yF (q g ) + [ E hh [ 0 ](1 F (q g )) + E[ 0 ](1 F (q g ))] F (q g ) y + v 0 1 c g h` v 0 c g h` = 0:

Noting that E h`[ 0 ]F (q g ) + E hh [ 0 ](1 F (q g )) = E[ 0 ], we can write:

E[ 0 ][y v 0 (1 c g h`) + v 0 (c g h`) ] E[ 0 ]yF (q g ) + E[ 0 ](1 F (q g ))[ F (q g ) y + v 0 1 c g h` v 0 c g h` ] = 0
which reduces to:

[1 F (q g )] v 0 (1 c g h`) + v 0 (c g h`) = 0:

A.6 Proof of equation [START_REF] Oecd | Doing better for families[END_REF] Solving the IC constraint for T h` T hh yields T h` T hh = y qg v (1) + F (q g ) 1 c g h` + c g h`y + v 1 c g h` + v c g h`

From [START_REF] Card | Bargaining, sorting, and the gender wage gap: quantifying the impact of …rms on the relative pay of women[END_REF] we have the …rst-best marginal couple:

qfb 1 h v(c f b h`) + v(1 c f b h`) v(1) + F (q f b )c f b h` [1 F (q f b )]c f b h`i
We now substitute c g h`= 1 c f b h`a nd qg = qfb :

T h` T hh = y v(1 c g h`)
v(c g h`) + v(1) F (q f b )(1 c g h`) + [1 F (q f b )](1 c g h`)

v (1) + F (q f b ) 1 c g h` + c g h`y + v 1 c g h` + v c g h` :

The above equation simpli…es to:

T h` T hh = [1 F (q f b )](1 c g h`)
y(1 c g h`) :

(

  see Fortin 2015, Farré and Vella 2013, Bertrand et al. 2015, Bursztyn et al. 2017 and Kleven et al. 2018, among others). Social norms may contribute to the di¤erential sorting of men and women across occupations with women entering low pay occupations that allow for shorter working days or more ‡exible working hours (see Goldin, 2014 and Card et al. 2016).

  and Kahn 2017; Kleven et al. 2018): mothers in traditional couples do not work full time, but spend part of their time to provide informal child care. Child penalty thus depends on average informal care, c = F (q ) 1 c p ;provided by traditional mothers. The second term accounts for the fact that women forego the extra earning opportunities associated with the high-career path. Interestingly, both child penalty and "adverse sorting" are a¤ected by social norms and child care decision through q . They decrease when the share of career mothers in the society increases. The model thus o¤ers a clean explanation of how social pressure determines women sorting and thus their low participation in leading positions together with lower wages. Finally, the last term in (4) captures the unexplained component of the GWG of the Oaxaca-Blinder decomposition, or the plain discrimination part; it vanishes when = 1.

2 3

 2 See also[START_REF] Cremer | In-kind transfers, self-selection and optimal tax policy[END_REF] who show that when individual consumption levels are observable, inkind transfers and nonlinear subsidies are equivalent. This information structure also di¤erentiates our model from the extensive literature on in-kind transfers of which child care is of course a prime example (seeBlomquist and Chirstiansen, 1995, or Blomquist et al., 2010, for two examples amongst many).

  v(1 c f b p ) + [v(c f b p ) v(1)] + F (q f b )(1 c f b p ) > 0 (A.3)With (A.3) we can rewrite (A.2) as (9) in the main text.

F 1 ( 1

 11 (q) v 00 (1 c p ) + v 00 (c p ) # " c p ) # dp n ; where D = [ + f (q)(1 c p )][v 00 (1 c p ) + v 00 (c p )] > 0.We thus have:dc p dp n = 1 v 00 (1 c p ) + v 00 (c p ) < 0 (A.4) dq dp n = [v 00 (1 c p ) + v 00 (c p )](1 c p ) + F (q) [ + f (q)(1 c p )][v 00 (1 c p ) + v 00 (c p )] > 0 (A.5)

In Ireland, the participation rate for children in low-income families is, at about

20%, less than one-third of that for children from high-income families (66%). In Belgium, France and the Netherlands, participation rates for children from low-income backgrounds are generally a little higher (around 30-40%), but are still only about half those for children from the richest families (roughly 60-75%). Similarly, in a number of OECD countries children are also more likely to use early childhood education and care when their mother is educated to degree-level.In the United Kingdom, the participation rate for children with a mother that has attained tertiary education is at 41%, 17 percentage points higher than the rate for children with mothers that have not attained tertiary education (24%). In Switzerland, the gap is as large as 30 percentage points.

The parameter generates the unexplained component in the Oaxaca-Blinder decomposition of the GWG; see[START_REF] Blinder | Wage discrimination: reduced form and structural estimates[END_REF] and[START_REF] Oaxaca | Male-female wage di¤erences in urban labor markets[END_REF]. Equation (4) below presents the decomposition of the GWG obtained in our model.

Assortative mating is commonly observed and has been increasing over the last decades; see Chiappori et al.(2017) and references within.

This assumption is simply a normalization that has no relevance for our results. Without it we would obtain a term proportional to (p y) in the …rst-order conditions with respect to child care. This would a¤ect the equilibrium levels of child care but otherwise all other results are not a¤ected.

[START_REF] Akerlof | Economics and identity[END_REF] Similarly, multiplying both sides of (9) by 1, would be more in line with the original FOC, because it then measures the cost and bene…ts (reversed from the interpretation discussed) of increasing b q.

[START_REF] Barigozzi | Women's Career Choices, Social Norms and Child Care Policies[END_REF] In the FB, utilities are equalized so that this term is equal to one.

[START_REF] Blomquist | Public Provision of Private Goods as a Redistributive Device in an Optimum Income Tax Model[END_REF] Enrolment rates for early childhood education at this age range from over 90% in Belgium, Denmark, France, Germany, Iceland, Italy, New Zealand, Norway, Spain, Sweden and the United Kingdom, at one end of the spectrum, to less than a third in Australia, Greece, Korea, Switzerland and Turkey.

Appendix

A.1 Couples' optimization A.1.1 Only the mother enters the high-career path Since the mother is in the high-career path, she is not able to take care of the children, and c m = 0. Hence, c m c h`> 0 and the mother su¤ers the cost of not conforming to the norm. If the father provides some child care he su¤ers because of the norm too, so that here both social norms are potentially binding.

Welfare of this couple is denoted by

Optimal child care provision, c `h; is implicitly determined by:

where I is an indicator function which takes value 1 when the social norm for fathers is binding, namely when c `h > c f , and 0 otherwise.

Indirect welfare W `h writes:

Both couples enter the low-career path

Here again, if the father provides some child care, he su¤ers because he deviates from the norm.

Both social norms are potentially binding. Welfare of this couple is denoted by W ``. Noting that c m + c f + c p = 1 and p = y; the couple chooses c ``= c m + c f to maximize:


Patrick Touron

Christian Siatka

Amaury Pussiau

Sébastien Follot

Thibaud Fritz

Mikael Petit

Noussair Latifa

Jean-Louis Herrmann

Martin Rottman

Antoinette Lemoine

Mikaël Petit

& Sylvain Hubac
email: sylvain.hubac@gendarmerie.interieur.gouv.fr

A mobile DNA laboratory for forensic science adapted to coronavirusSARS-CoV-2 diagnosis

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

On March 11, 2020, the World Health Organization (WHO) classified the outbreak due to severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) as a public health emergency. This epidemic has indeed taken on the proportions of a pandemic (WHO, March 2020) [START_REF]Coronaviridae Study Group of the International Committee on Taxonomy of Viruses (2020) The species severe acute respiratory syndrome-related coronavirus: classifying 2019-nCoV and naming it SARS-CoV-2[END_REF]. The viral pathology linked to this Coronavirus was described in China for the first time in December 2019 and named COVID-19 (coronavirus disease 2019) [START_REF] Huang | Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China[END_REF]. Quickly, this disease fuelled a massive and legitimate interest, especially in terms of specific molecular diagnosis capabilities. The clinical consequences are multiple and variable but most importantly feature an acute respiratory distress syndrome (ARDS), which can be associated with multi-organ failure with myocardial, renal, enteric and hepatic lesions, possibly leading to death, particularly in elderly patients with comorbidities [START_REF] Chen | Epidemiological and clinical characteristics of 99 cases of 2019 novel coronavirus pneumonia in Wuhan, China: a descriptive study[END_REF]. SARS-CoV-2 transmission has affected more than 2.85 million people and caused more than 196,000 deaths worldwide as of April 27, 2020, including more than 22,500 in France (www.who.int). For these reasons, monitoring COVID-19 is important and strategic [START_REF] Cdc | Coronavirus Disease 2019[END_REF].

Département Médical Universitaire de Biologie-Génétique-Pharmacie, AP-HP, Université Paris Saclay, 78 rue du Général Leclerc, 94270 Le Kremlin-Bicêtre, France In 2015, the French "Gendarmerie Nationale" patented (FR3050945B1) a Mobil'DNA lab for genetic forensic analysis which allows magistrates and investigators to quickly obtain genetic results. This mobile laboratory is transportable as close as possible to scenes of disasters, complex crime scenes or to places where people in police custody are gathered [START_REF] De Boer | The role of forensic anthropology in disaster victim identification (DVI): recent developments and future prospects[END_REF]. It has already been used for the identification of the victims of the GermanWings (2015) air crash and the 2016 Nice terror attack [START_REF] Gaudry | L'identification de victimes de catastrophe : une approche scientifiquepluridisciplinaire[END_REF] where it carried out several hundred analyses per day.

This Mobil'DNA lab makes it possible to implement an analytical process for determining genetic profiles from a biological sample with a configuration which makes it possible to respect the principle of "forward movement" of the processing of a sample described as:

-A sampling area, to manage and analyse the sample

COVID-19 diagnosis: SARS-CoV-2 detection

Since March 2020, French public health authorities stressed the need to rapidly expand the diagnostic capabilities for COVID-19. This global health crisis shows the need to mobilize a diagnostic force quickly at national and international level. These solutions must interface with clinical diagnostic laboratories to meet the quality standards for medical biology (ISO 15189) [START_REF] Ho | Practical application of ISO 15189 by accreditation bodies -: a comparison with ISO/IEC 17025[END_REF] and should be mobile to best address the needs. The French National Gendarmerie, as an interior security force serving the protection of the French people, has taken the initiative to adapt the forensic Mobil'DNA. The objective is to propose an autonomous mobile biomolecular laboratory for rapid tests at high-throughput level for the endemic pathogen SARS Cov-2, in support of local hospitals and health systems, in rural or confined areas and overseas.

To be efficient for this mobile force for microbiological diagnosis, the IRCGN combined its expertise in mobile and high-speedanalysis ofbiological samples withthe expertise of a French medical biology structure in the diagnosis of endemic pathogens.

Eur J Clin Microbiol Infect Dis

The creation of a mixed unit composed of IRCGN and the Microbiology Department of Hospital Raymond Poincaré (GHU Paris Saclay, Assistance Publique-Hôpitaux de Paris) allowed the rapid adaptation IRCGN mobile DNA analysis laboratory in order to guarantee optimal conditions for operator safety, sample traceability and analysis sensitivity in order to detect the presence of COVID-19 by viral RNA analysis [START_REF] Gao | Structure of the RNA-dependent RNA polymerase from COVID19 virus[END_REF][START_REF] Wölfel | Virological assessment of hospitalized patients with COVID-2019[END_REF]. This rapid adaptation is greatly facilitated by the convergence of analytical methods and ISO 17025/ISO 15189 normative requirements between the analysis of a human biological sample for DNA analysis and the analysis of this same biological sample for detecting a pathogen by targeting the RNA Coronavirus (see Fig. 1) [START_REF] Ho | Practical application of ISO 15189 by accreditation bodies -: a comparison with ISO/IEC 17025[END_REF]. This partnership was authorized by the ordinance of the French government on 5 April 2020 [11] giving the possibility to ISO 17025 certified laboratories to perform biomedical testing by RT-PCR in order to detect COVID-19 infection.

Material and strategy

Validation process, sample processing and highthroughput COVID-19 detection by RT PCR Regarding this outbreak context, the Mobil'DNA lab of the French gendarmerie was adapted as a true extension of the fixed hospital laboratory by validating and following an analytical process.

The validation of the analytical process of the Mobil'DNA lab was carried out over a period of 10 days through comparative studies between the reference analytical technique used by the hospital and the analytical process performed by the Eur J Clin Microbiol Infect Dis IRCGN [START_REF] Huang | Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China[END_REF]. To this end, the analytical parameters of the processwerecleared with verificationof sensitivity, of repeatability and of reproducibility, carried out on more than 100 prospective clinical samples. The lack of crosscontamination was verified at all stages with the assays of high concentration RNA extracts.

The analytical process is performed in 5 steps:

1. Taking samples and recording them for traceability on the basis of the unique barcode identifier. All is done using a computerized system LIMS, which is connected to the hospital unit in order to transfer automatically the results of the essays. It provides a complete overview and control of analytical data to the microbiologists. 2. Sampling a fraction of the biological sample to be analysed. This operation is carried out under a type 2 microbiological safety cabinet (MSC). 3. Extraction of the RNA contained in the sample by lysis of the virus using the hospital's procedures [START_REF] Wölfel | Virological assessment of hospitalized patients with COVID-2019[END_REF]. 4. Quantification of purified DNA by real-time PCR (RTPCR) system.

Result interpretation and generation of reports, using

GendLink® an original gendarmerie software adapted to generate a report on the results of COVID analyses.

In order to ensure operator safety against COVID-19, each operatoriswearing a singleusepersonalprotectiveequipment including an integral Tyvek type suit, a double pair of gloves (external pair changed between each sample), a surgical mask or an FFP2 type mask for steps 1 and 2, a hood and protective goggles.

The analyses are done on an Applied™ 7500 real-time PCR system (software v2.3) according to the hospital procedure, using RNA extraction negative control, internal quality control (ICQ) composed by quantified virus RNA from patients diagnosed as COVID-19 infected, nontemplate control for RT-PCR and positive control from the GeneFirst® RUO COVID-19 Detection kit. In this configuration, the Mobil'DNA can provide a maximum of 1000 results per day involving five technical persons from IRCGN and 3 clinical microbiologists from the hospital to validate the results

Results

Since the validation ofthe technical procedure,on the 14April 2020 at Garches Hospital, around 4000 samples were analysed in 14 days with throughput between 100 and 700 samples per day depending on the delivery of samples. After reception of the samples, the results are given in less than 24 h. The Mobil'DNA lab and the forensic science laboratory of the French gendarmerie in the context of the COVID-19 pandemic can provide three short-term issues with a direct impact on this crisis:

1. The possibility within a short time of setting up an additional testing capacity to deal with a massive influx of individuals to be tested on a hospital site 2. To respond to a demand for test capacity in rural areas or overseas, for citizens distant from the test centres 3. To validate a global and efficient concept of mobile lab unit for rapid molecular diagnostic tests, easily transposable and deployable anywhere in the world during such outbreak scenario

Conclusion

Using this mixed concept mobile laboratory between the French National Gendarmerie and hospital centres has demonstrated the ability to pool skills and resources to cope with the current outbreak period by creating a single autonomous mobile solution, deployable in a very short time. This concept will provide an effective solution in the following cases:

-Mobile rapid diagnostic unit to be deployed on a remote site, isolated or contained contaminated areas, point of entry into the territory of a source of propagation of a pandemic (port, airport, border). -Extension of diagnostic test capacities to other pathogens (viruses such as Ebola and dengue or bacteria such as cholera and the tuberculosis bacillus). IRCGN is at the cutting edge of technology, as are collaborating medical microbiologists. This will make it possible to deploy the most efficient technologies depending on the situations in areas that cannot be equipped in such a short time. -The possibilities to extend the concept to other means of deployment (train, plane, etc.).

It shows that forensic police forces can collaborate with medical capabilities using the same technologies for the same purpose: health security and diagnostic assistance.

Fig. 1

 1 Fig. 1 The Mobil'DNA lab, the adaptation for genetic diagnosis (inside view)

Assistance Publique-Hôpitaux de Paris INSERM U1173, UFR Simone Veil, Versailles-Saint-Quentin University, 78180 Saint-Quentin en Yvelines, France

Acknowledgements The authors would like to thank Mr. Christophe Castaner, Minister of the Interior and General Christian RODRIGUEZ, Director General of the National Gendarmerie, who made it possible for the Gendarmerie Scientifique to work quickly with the AP-HP microbiologists to improve the detection capacity of COVID-19. We are also grateful to the Pr Philippe Berta, geneticis at Nîmes University and John Watson, expert for the European Commission, for their excellent technical assistance and reviewing.

Compliance with ethical standards

Conflict of interest

Hubac S. is the owner of the patent of Mobil'DNA.