Pierre Pestieau

Grégory Ponthière

Gregory Ponthiere
email: gregory.ponthiere@ens.fr

WORKING PAPER N° 2011 -28 Childbearing Age, Family Allowances and Social Security

Keywords: endogenous fertility, childbearing age, family benefits endogenous fertility, childbearing age, pensions, family bene…ts. JEL classi…cation codes: J13, D10, H21, H55

Although the optimal policy under endogenous fertility has been widely studied, the optimal public intervention under endogenous childbearing age has remained largely unexplored. This paper examines the optimal family policy in a context where the number and the timing of births are chosen by individuals who di¤er as to how early fertility can weaken future earnings growth. We analyze the design of a policy of family allowances and of public pensions in such a setting, under distinct informational environments. We show how endogenous childbearing ages a¤ect the optimal policy, through the redistribution across the earnings dimension and the internalization of fertility externalities. It is also shown that, contrary to common practice, children bene…ts di¤erentiated according to the age of parents can, under some conditions, be part of the optimal family policy.

Introduction

As this is well-known among demographers, there has been a continuous postponement of fertility in European economies since the late 1970s. That strong demographic trend is well illustrated by the case of France. As shown on Figure 1, the average age at motherhood has raised from about 26.9 years in 1977 to about 29.7 years in 2005. [START_REF] Balestrino | Endogenous fertility and the design of family taxation[END_REF] The postponement of fertility appears even more strongly when we look at the mode of the distribution of the age at motherhood: the mode age has grown from 25 years in 1977 to 29 years today. With the development of assisted reproductive technologies, that evolution is likely to be sustained -if not reinforced -in the future. The causes of the postponement of fertility have been widely explored. [START_REF] Blackorby | Population issues in social choice theory, welfare economics, and ethics[END_REF] Empirical studies highlighted that women's earnings opportunities and educational achievements are a major determinant of fertility behavior. The rise in women's wage, by implying a higher opportunity cost of motherhood, appears to be a major factor of fertility decline, which coincides also with later motherhood. [START_REF] Cigno | Fertility and the tax-bene…t system: a reconsideration of the theory of family taxation[END_REF] Besides those empirical studies, theoretical models have also been developed to explain the choice of a particular fertility age-pattern over the lifecycle. Those models, such as Happel et al. (1984), Cigno and Ermisch (1989) and Cigno (1991), highlighted the central in ‡uence of lifecycle e¤ects on the tim-ing of births. Whereas Happel et al. (1984) emphasized that the consumption smoothing induced by the maximization of lifetime welfare may lead to delaying births, Cigno and Ermisch (1989) focused on the shape of the earnings pro…le induced by human capital investment, and argued that steeper earnings pro…les lead to the postponement of births later in life. [START_REF] Cigno | Economics of the family[END_REF] The various in ‡uences of governments on the timing of births have also been studied. While the income tax rate and parental leave bene…ts reduce the opportunity cost of early children, children allowances reduce the net direct expenditures, and the income tax rate reduces also the forgone returns to forgone human capital investment. Those in ‡uences of governments have been con…rmed by various empirical studies. [START_REF] Cigno | A microeconomic analysis of the timing of births[END_REF] Macroeconometric studies, such as Ermisch (1988) on UK and Walker (1995) on Sweden, identi…ed a positive impact of child allowances on early motherhood. That e¤ect was con…rmed by microeconometric studies, such as Laroque and Salanié (2004) on France, who found that cash bene…ts increase the probability of having a …rst child.

In the light of those in ‡uences of the government on fertility behavior, a natural question to raise concerns the optimal design of family policies. What should government do in front of the postponement of fertility? Should governments implement a transfer policy in such a way as to reinforce (or weaken) that demographic trend? Should governments subsidize births di¤erently depending on the age of parents? By which channels are the optimal fertility policy and the optimal pension policy related to each other?

The literature on the optimal policy under endogenous fertility has examined various aspects of the problem, but without paying attention to the optimal timing of births. Cigno (1986) showed that family allowances aimed at reducing child poverty may, by raising fertility, have quite the opposite e¤ects, so that the optimal policy may consist in taxing -instead of subsidizing -the number of children. [START_REF] Cigno | Optimal family policy in the presence of moral hazard, when the quantity and the quality of children are stochastic[END_REF] More recently, Cremer et al. (2006) studied the design of the optimal pension system under endogenous and stochastic fertility, and showed that, under positive fertility externalities, one should grant parents who have more children with larger pension bene…ts. [START_REF] Cremer | Pensions with endogenous and stochastic fertility[END_REF] The goal of this paper is to complement that literature, by examining the design of the optimal public intervention in a context where not only fertility, but, also, the childbearing age, are endogenous. For that purpose, we develop a three-period model. In the …rst two periods, individuals are active, and they can have children. In the third period, individuals retire. In order to account for the widely observed e¤ect of heterogeneity in education and in the timing of births (Cigno and Ermisch 1989), we assume that some agents bene…t from a rise in productivity and earnings (to an extent that is decreasing in the number of early children), whereas other agents do not bene…t from such a process, and face a ‡at earnings pro…le. In that framework, having a child in the …rst period relative to the second is cheaper, but it may impose a cost in terms of educational and professional achievement, so that some trade-o¤ exists between lower fertility cost and higher future wages. Moreover, when parents retire, and count on some help from their children, they will, if earnings increase with age, receive fewer resources per child under late childbearing than under early childbearing. Retired agents will also bene…t from the contributions of fewer individuals under late childbearing than under early childbearing (where they bene…t from the contributions of their working children and grandchildren).

In the following, we made several simplifying assumptions. First, we abstract here from intrafamily decision making issues, and treat a couple as a single individual capable of working, saving and having children. Second, we assume away the childhood period, i.e. the period during which children live as dependent of their parents. Third, we focus on a static overlapping-generations model, and do not explore here the e¤ects of childbearing ages on the long-run dynamics. [START_REF] Cremer | Pensions with heterogeneous individuals and endogenous fertility[END_REF] Fourth, we assume here that the number of children is deterministic, and thus rule out uncertainty about the number of children. [START_REF] D'albis | Demographiceconomic equilibria when the age at motherhood is endogenous[END_REF] Anticipating our results, we show that, under the laissez-faire, agents who face ‡at earnings pro…le tend to make children earlier than agents with an increasing earnings pro…le. We characterize the …rst-best utilitarian social optimum, and show that this can be decentralized by means of lump sum transfers from increasing-productivity agents to constant-productivity agents, in order to equalize the consumption per period and the total number of children for all agents (despite di¤erent timings for births). Then, we consider the secondbest optimal policy under linear taxation instruments, and show that the optimal family allowances are positive only if subsidizing children favours early childbearing. The study of the second-best problem under non-linear instruments shows that children allowances should concern children from constantproductivity parents only. Finally, we examine the optimal family policy under the presence of a Pay-As-You-Go (PAYG) pension system, and show that the social planner faces, under endogenous childbearing age, a trade-o¤ between e¢ ciency (favouring late childbearing) and equity (helping early childbearing).

The rest of the paper is organized as follows. Section 2 presents the general framework. The …rst-best utilitarian optimum is derived in Section 3, where its decentralization is also examined. In Section 4, we look at the optimal child bene…ts when there is a perfect capital market. In Section 5, we introduce the idea that pensions can be of the PAYG type, which may give an incentive to both the government and individuals to have early children. Section 6 concludes. [START_REF] Cremer | Pensions with heterogeneous individuals and endogenous fertility[END_REF] Those e¤ects are discussed in the companion paper Pestieau and Ponthiere (2011). It is shown there that, compared to the 2-period OLG model, the conditions for optimal capital accumulation and optimal fertility di¤er quite a lot under varying childbearing ages. However, Samuelson's (1975) Serendipity Theorem still holds in that broader demographic environment. [START_REF] D'albis | Demographiceconomic equilibria when the age at motherhood is endogenous[END_REF] See Cigno and Luporini (2009) on the optimal family policy when the number and the future earnings of children are random. The absence of risk rules out a case for early childbearing: the possibility to insure oneself against a total number of children lower than desired.

Basic model 2.1 Environment

We consider a three period model. [START_REF] Ermisch | The econometric analysis of birth rate dynamics in Britain[END_REF] The periods are labeled a, `and o (i.e. advanced, late, old). In the …rst two periods, individuals work, consume, save and have children. It is assumed that a perfect capital market exists. In the last period, individuals are retired and consume the proceeds of their savings.

To do justice to the observed heterogeneity in terms of education, career and childbearing ages, we assume that individuals di¤er in the shape of the earnings lifecycle pro…le. For simplicity, we consider the following two types:

Type-1 agents have a constant productivity and earnings along the lifecycle (periods 1 and 2); [START_REF] Ermisch | Age at motherhood in Japan[END_REF] Type-2 agents have, at period 2, a productivity that is larger than in period 1, to an extent that is decreasing in the number of children they had in period 1.

The lifetime welfare of an individual of type i 2 f1; 2g is expressed as:

U i = u (c i ) + u (d i ) + 2 u (b i ) + v (n ai + n `i) (1)
where c i , d i and b i are consumption, u ( ) is strictly concave, is a time preference factor (0 1) and v ( ) is the utility for early and late children, which is also strictly concave. [START_REF] Ermisch | Early motherhood and later partnerships[END_REF] In the …rst period, consumption is equal to wage, w, minus saving and minus the cost of raising children, n ai e a : c i = w s ai n ai e a :

(2)

In the second period, an individual of type i 2 f1; 2g earns wh i (n ai ), with h 2 (n a2 ) > h 1 (n a1 ) 1 and h 0 i (n ai ) < 0: This re ‡ects the idea that, for some individuals with a broad career potential, early childbearing has a cost on earning in period `. Indeed, various empirical studies, such as Joshi (1990, 1998) on Great Britain and Dankmeyer (1996) on the Netherlands, showed that having children in an early stage of a career slows down human capital accumulation and professional promotion. [START_REF] Gahvari | Pensions and fertility: in search of a link[END_REF] In the second period, the individual earns also savings income Rs ai , where R is the interest factor. He saves also s `i for the old age, and spends e `n`i as cost of raising his n `i late children. Throughout this paper, we assume that the direct cost of children is larger for late children than for early children:

e `> e a
That assumption is compatible with the medical literature showing the larger costs of late motherhood in comparison to the ones under early motherhood. Those additional costs are of various kinds, and concern both the mother and the child (see Gilbert et al. 1999, Gustafsson 2001). [START_REF] Gauthier | The impact of family policies on fertility in industrialized countries: a review of the literature[END_REF] We assume, for simplicity, that the parents face the entire additional cost from late childbearing. Hence, second-period consumption for an agent of type i 2 f1; 2g is:

d i = wh i (n ai ) + Rs ai s `i n `ie `: (3)
In the third period, an agent of type i 2 f1; 2g consumes the proceeds of his savings:

b i = Rs `i: (4)

Laissez-faire

Let us characterize the laissez-faire equilibrium, where each agent of type i 2 f1; 2g chooses …rst-, second-and third-period consumptions (i.e. c i , d i and b i ), as well as …rst-and second-period children (i.e. n ai and n `i) in such a way as to maximize his lifetime welfare, subject to his budget constraint. For an agent of type i 2 f1; 2g, the …rst-order conditions are:

u 0 (c i ) = Ru 0 (d i ) = 2 Ru 0 (b i ) u 0 (c i ) e a + u 0 (d i ) wh 0 i (n ai ) + v 0 (n ai + n `i) = 0 u 0 (d i ) e `+ v 0 (n ai + n `i) = 0:
The …rst equalities, which describe the optimal consumption pro…le, are quite standard. However, the second and third conditions, which characterize the optimal early and late number of children, are not straightforward.

To interpret those conditions, let us consider the case where h 1 (n a1 ) = 1 and assume further that = R = 1: Then we have, at the laissez-faire equilibrium, a perfect consumption smoothing for all agents:

c i = d i = b i
with higher consumption levels for type-2 agents, who bene…t from an increasing earnings pro…le.

We also have that type-1 agents have no children in the second period:

n `1 = 0
Actually, given that late children are more costly than early children under e `> e a , and given that their productivity will remain the same whatever the number of early children is, type-1 agents have no reason to postpone fertility. The number of early children for type-1 agents is given by:

u 0 (c 1 ) e a = v 0 (n a1 )
The number of early children should be at a level where the marginal welfare cost from early fertility (the LHS) is exactly equal to the marginal welfare gain from early fertility (the RHS).

Regarding agents of type-2, three possible cases can arise, depending on the size of the cost di¤erential between early and late children, and on the impact of early parenthood on future productivity. The three cases are: (1)

n a2 > 0 and n `2 > 0; (2) n a2 = 0 and n `2 > 0; (3) n a2 > 0 and n `2 = 0.
If there exist some n a2 > 0 and n `2 > 0 such that

u 0 (c 2 ) e a + u 0 (d 2 ) wh 0 2 (n a2 ) + v 0 (n a2 + n `2) = 0 u 0 (d 2 ) e `+ v 0 (n a2 + n `2) = 0
then we have an interior solution for fertility in the two ages. [START_REF] Gilbert | Childbearing beyond age 40: pregnancy outcome in 24032 cases[END_REF] Simplifying those two conditions, we obtain:

e ` e a = wh 0 2 (n a2 )

Hence the marginal gain from early childbearing (the LHS), i.e. lower children costs, equals the marginal loss from early children (the RHS).

Alternatively, when we have, for any n a2 > 0, the strict inequality:

e ` e a < wh 0 2 (n a2 )

then the marginal gain from early childbearing is necessarily inferior to the marginal loss from early children, so that the optimal number of early children is a corner solution, i.e. n a2 = 0. In that case, n `2 is given by:

u 0 (d 2 ) e `= v 0 (n `2)
That conditions equalizes the marginal loss from late childbearing (the LHS) with the marginal gain from late childbearing (the RHS). Hence, in that case, we have n a2 = 0 and n `2 > 0.

Finally, when we have, for any n a2 > 0, the strict inequality:

e ` e a > wh 0 2 (n a2 )

then the marginal gain from early childbearing is necessarily superior to the marginal loss from early children, so that the optimal number of late children is a corner solution, i.e. n `2 = 0. Then, n a2 is given by:

u 0 (c 2 ) e a + u 0 (d 2 ) wh 0 2 (n a2 ) + v 0 (n a2 ) = 0
Hence, in that case, we have n a2 > 0 and n `2 = 0.

Regarding the total number of children, the number of early children for type-1 agents is given by:

u 0 (c 1 ) e a = v 0 (n a1 )
whereas, under n a2 > 0 and n `2 > 0, we have

u 0 (c 2 ) e a u 0 (d 2 ) wh 0 2 (n a2 ) = v 0 (n a2 + n `2)
If jh 0 2 (n a2 )j is su¢ ciently large, type-2 agents have fewer children than type-1 agents: n a2 + n `2 < n a1 .

When n a2 = 0, the optimal number of late children n `2 is given:

u 0 (d 2 ) e `= v 0 (n `2)
With a strongly concave u( ), it must be the case that n `2 < n a1 . Finally, when n `2 = 0, the optimal number of early children n a2 satis…es:

u 0 (c 2 ) e a u 0 (d 2 ) wh 0 2 (n a2 ) = v 0 (n a2 )
Here again, if jh 0 2 (n a2 )j is large, we have, despite c 2 > c 1 , that n a2 < n a1 . In sum, whereas type-1 agents do not have late children, but only early children, three possible cases can arise for type-2 agents. If the cost di¤erential between late and early parenthood is extremely large, as it used to be the case before the massive development of assisted reproductive technologies, then type-2 agents also opt for early children only. If, on the contrary, the cost di¤erential between late and early is low, then type-2 agents have no early children, and only late children, since this will preserve their future productivity at little additional child costs. In the intermediate case, there is an interior solution with n a2 > 0 and n `2 > 0, with a shape of fertility pro…le that depends on how sensitive the earnings pro…le is to early parenthood. Finally, the total number of children for type-1 agents tends to exceed the number of children for type-2 agents when the earnings pro…le of the latter is strongly sensitive to early births.

Therefore, although simple, the model developed in this section can explain the observed postponement of fertility as a result of the development of assisted reproductive technologies, which reduced the gap e ` e a . Our model allows us also to rationalize the observed heterogeneity: in conformity with empirical studies (Cigno and Ermisch 1989, Ermisch and Ogawa 1995, Joshi 2002), adults with lower career opportunities tend to make children earlier than adults with (potentially) steeper earnings pro…les. Finally, our framework can also, under general conditions, explain why the tendency towards steeper potential earnings pro…les over time tends to reduce total fertility.

The …rst-best problem

Let us now characterize the social optimum of our economy. For that purpose, we will focus on a classical utilitarian social objective, whose goal coincides with the maximization of aggregate welfare. We also focus here on the aggregate welfare of a single cohort of individuals, which includes a …xed fraction 1 of agents of type 1, and a …xed fraction 2 of agents of type 2. Whereas focusing on a single cohort of given size is an obvious simpli…cation, this allows us to escape from well-known di¢ culties raised by population ethics. [START_REF] Gustafsson | Optimal age at motherhood. Theoretical and empirical considerations on postponement of maternity in Europe[END_REF] The utilitarian …rst-best optimum is obtained by maximizing the following Lagrangian:

X i (U i [c i + d i + b i + n ai e a + n `ie ` w (1 + h i (e ai ))])
where denotes the Lagrange multiplier associated with the resource constraint of the economy.

Assuming interior solutions, we obtain the following conditions:

u 0 (c i ) = u 0 (d i ) = u 0 (b i ) = v 0 (n ai + n `i) = [e a wh 0 i (n ai )] v 0 (n ai + n `i) = e
The utilitarian social optimum involves an equalization of all consumptions across individuals of all types, and across periods. Thus, whatever agents have an increasing productivity pro…le (i.e. type-2 agents) or a ‡at productivity pro…le (i.e. type-1 agents), they should all enjoy the same consumptions at all periods of their life.

Regarding optimal fertility, we have, for type-1 agents, that the FOCs for n a1 and n `1 cannot be both satis…ed. Given e `> e a , the social optimum involves, as in the laissez-faire, n a1 > 0 and n `1 = 0, that is, it is not socially optimal that type-1 agents have late children. As far as type-2 agents are concerned, three cases can arise, like at the laissez-faire.

If there exists some n a2 > 0 such that:

e ` e a = wh 0 2 (n a2 )

then the FOCs for n a2 and n `2 can be simultaneously valid. As a consequence, we have: n a2 > 0 and n `2 > 0 at the social optimum.

On the contrary, if, for any n a2 > 0, we have e ` e a < wh 0 2 (n a2 ) then we have n a2 = 0 and n `2 > 0 at the social optimum.

Finally, if, for any n a2 > 0, we have e ` e a > wh 0 2 (n a2 ) then we have n a2 > 0 and n `2 = 0 at the social optimum.

Regarding the total number of children, we have, for agents of types 1:

v 0 (n a1 ) = e a
whereas, type-2 agents's fertility satis…es one of the following conditions:

v 0 (n a2 + n `2) = e `if n `2 > 0 v 0 (n a2 ) = [e a wh 0 2 (n a2 )] if n `2 = 0
Thus, provided the optimal number of late children for type-2 agents is positive, we obtain, given e `> e a , that the optimal number of children for type-2 agents is lower than the optimal number of children for type-1 agents:

n a1 > n a2 + n `2
The same result holds if the optimal number of late children for type-2 agents is zero. Indeed, in that case, we obtain, given wh 0 2 (n a2 ) < 0, that the optimal number of children for type-2 agents is lower than the optimal number of children for type-1 agents:

n a1 > n a2
Thus utilitarianism recommends the equalization of consumptions across individuals and time periods, as well as a larger total fertility for type-1 agents than for type-2 agents. The social optimum involves only early children for type-1 agents, and possibly early and/or late children for type-2 agents, depending on the cost gap between late and early children, and on the sensitivity of productivity to early parenthood.

Hence, the utilitarian social optimum di¤ers signi…cantly from the laissezfaire. Whereas type-1 agents have lower consumptions, and may even have fewer children than type-2 agents at the laissez-faire, the utilitarian optimum recommends the same consumptions for all agents, as well as more children for type-1 agents than for type-2 agents.

Regarding the decentralization of the social optimum, this can be achieved by means of lump-sum transfers from type-2 agents towards type-1 agents, in such a way as to equalize all consumptions. Note that such transfers, by reducing the marginal utility of consumption for type-1 agents, will raise early fertility n a1 above its laissez-faire level. Inversely, by increasing the marginal utility of consumption for type-2 agents, those transfers will tend to reduce the fertility of type-2 agents, but without a¤ecting the timing of births. [START_REF] Happel | An economic analysis of the timing of childbirth[END_REF]

Family allowances

The previous section showed how the …rst-best optimum can be decentralized by means of lump sum transfers from individuals with high earnings potential towards individuals with low earnings potential. However, such an intervention requires both that lump sum transfers are available policy instruments, and that the government can observe the types of individuals. These are strong assumptions. In this section, we depart from such a …rst-best problem, and consider instead the characterization of the second-best social optimum.

For that purpose, we will proceed in two stages. We will …rst focus on the optimal policy when the only available …scal instruments are uniform payroll taxes, demogrants and children allowances. Then, we will consider the optimal policy under asymmetric information, under non-linear …scal instruments.

Linear case

The government has, as available policy instruments, a uniform payroll tax, , a uniform demogrant, T , and a uniform subsidy on the number of children, . The instrument can be understood as a family allowance.

We assume, as above, that the two types of agents di¤er regarding the slope of their (potential) earning pro…le: h 2 (n a2 ) > h 1 (n a1 ) 1, with h 0 i (n ai ) < 0. However, given that we are here mainly concerned with the timing of fertility rather than with the total fertility, we assume that, for each individual, the total fertility is …xed to its replacement level, i.e. n ai + n `i = 1. To simplify notations, we denote early fertility n ai by n i = 1 n `i, and we normalize the utility from children, in such a way that v (1) = 0.

In the laissez-faire, each agent of type i 2 f1; 2g maximizes: 1) ((e a e `) n i + e `)] :

U i = u i (c i ) + u (d i ) + u (b i ) [c i + d i + b i T (1 ) w (1 + h i (n i )) + (
where is the Lagrange multiplier associated with the agent's budget constraint. The FOCs are:

u 0 i (c i ) = u 0 (d i ) = u 0 (b i ) = (1
) wh 0 i (n i ) + ( 1) (e a e `) = 0

Thus agents tend, here again, to smooth consumption across periods, and organize the timing of births in such a way as to equalize the marginal welfare losses and marginal welfare gains from early parenthood. As to the government, its objective is to maximize the sum of individual utilities subject to its revenue constraint. Namely, it maximizes the following Lagrangian:

L = X i 3u (1 ) w (1 + h i (n i )) + T (1 ) ((e a e `) n i + e `) 3 + [ w (1 + h i (n i )) T (e a e `) n i + e `]g
where is the Lagrange multiplier associated with the revenue constraint.

Using the envelope theorem, we obtain the following FOCs: 18

@L @ = P i [u 0 (x i ) w (1 + h i (n i )) w (1 + h i (n i )) wh 0 i (n i ) @ni @
(e a e `) @ni @ = 0

@L @ = P i u 0 (x i ) [(e a e `) n i + e `] + wh 0 i (n i ) @ni @
(e a e `) @ni @

[(e a e `) n i + e `] = 0 @L @T =

P i u 0 (x i ) (1 + ) (e a e `) @ni @T + wh 0 i (n i ) @ni @T = 0 where x i (1
)w(1+hi(ni))+T ( 1)((ea e `)ni+e `) 3

is the argument of the agent i's temporal utility u( ).

Those conditions characterize the optimal values for our instruments , and T . Note, however, that the simultaneous study of the optimal levels of the three taxation tools would be quite laborious, as their values are related to each others through the government's budget constraint. Hence, to keep the analysis simple, we will proceed as follows. To interpret those optimality conditions, we will consider alternative pairs of instruments, holding the other instrument equal to 0. Hence, we will focus on the pairs ( ; T ) and ( ; T ), while keeping, each time, the other …scal tool set to 0. This will allow us to derive, in …ne, closed-form solutions for the optimal levels of instruments.

Let us start with the pair ( ; T ), composed of a payroll tax on labor earnings and a …rst-period demogrant. The …rst FOC from above does not su¢ ce, on its own, to characterize the optimal level of , as a rise in must, under the government's budget constraint, imply a change in the demogrant T , in such a way as to maintain the budget equilibrium. Therefore, in order to characterize the optimal , we will use a compensated Lagrangian expression, whose derivative with respect to the policy instrument gives us the e¤ect of a variation of on the Lagrangian when that change is compensated by a variation of T that keeps the government's budget equilibrium. Using the optimality conditions, the derivative of the compensated Lagrangian can be de…ned as:

@ L @ @L @ + @L @T @T @ = @L @ + @L @T E [w (1 + h (n))]
where L denotes the compensated Lagrangian, and where the second term of the RHS accounts for the e¤ect of a change in the tax rate on the …rst-period demogrant T , under the government's budget equilibrium constraint. The operator E( ) denotes the average value of its argument among the population.

Substituting for the FOCs for optimal and T and equalizing to zero yields:

@ L @ = @L @ + @L @T E [w (1 + h (n))] = E [u 0 (x) w (1 + h (n))] + E [u 0 (x) w] E [(1 + h (n))] E A @ ñ @ = 0
where A wh 0 (n)+ (e ` e a ) is the e¤ect of n on the revenue constraint (the …rst term is negative and the second positive) and @ ñ @ @n @ + @n @T @T @ denotes the e¤ect of a change in on early fertility, when that change is compensated by a change in T so as to maintain the budget equilibrium.

Regarding the pair ( ; T ), one can proceed in the same way as with the pair ( ; T ), and de…ne the derivative of the compensated Lagrangian as follows:

@ L @ @L @ + @L @T @T @ = @L @ @L @T [(e a e `) E (n) e `]
where the second term is the e¤ect of a change in on the demogrant, under the government's budget equilibrium. Substituting for the FOCs for optimal and T and equalizing to zero yields:

@ L @ = @L @ @L @T [(e a e `) E (n) e `]
= (e a e `) [E (u 0 (x) n) E (u 0 (x)) E (n)] + E A @ ñ @ = 0

where @ ñ @ @n @ + @n @T @T @ denotes the e¤ect of a change in on early fertility when that change is compensated by a change in T in such a way as to maintain the budget equilibrium.

The two compensated Lagrangian conditions can be rewritten as:

cov (u 0 (x) ; (1 + h (n))) E A @ ñ @ = 0 cov (u 0 (x) ; n) (e a e `) + E A @ ñ @ = 0
To interpret these tax formulae, we look at pairs of instruments : and T and and T . Then, we obtain the following expressions for the optimal payroll tax and the optimal child subsidy:

= cov (u 0 (x) ; w (1 + h (n))) E @ ñ @ h 0 (n) (5) = cov (u 0 (x) ; n) E @ ñ @ (6)
Regarding the optimal payroll tax, note …rst that the numerator of ( 5) is positive, as the covariance is negative, the correlation between x and h (n) being positive, given h 0 (n) < 0: This is the standard equity term of the tax formula. However, the sign of depends also on that of the denominator, whose sign depends on the one of @ ñ=@ : If the (aggregate) compensated e¤ect of the tax on n is negative, then the tax is desirable. This is the e¢ ciency term of the tax formula. On the contrary, if the (aggregate) compensated e¤ect of on n is positive, the denominator is negative, implying a negative optimal .

As far as optimal family allowances are concerned, the numerator of ( 6) is positive (low income agents have early children). The equity term of the …scal formula pushes towards the subsidization of children. We thus have positive child bene…ts if the denominator is also positive, that is, if those children bene…ts have a positive (aggregate) e¤ect on early childbearing in compensated terms.

In sum, the optimal policy consists, in the absence of lump sum transfers, into a subsidization of children, to the extent that such family allowances increase early parenthood. However, if subsidizing children reduces early parenthood, then family allowances are no longer justi…ed, and a taxation of children is required instead. The intuition is the following. If subsidizing children fosters early parenthood, this means, given that early parenthood concerns generally the individuals with low career opportunities, that family allowances are a way to redistribute resources in the right direction (i.e. individuals with low career opportunities), by subsidizing a good that those agents consume more than others. Inversely, if subsidizing children reduces early parenthood and raises late parenthood, family allowances are regressive, since these amount to subsidize a good that is mostly consumed by individuals with high career opportunities.

Finally, note that, as an alternative to this presentation, we could replace the general children subsidy by a subsidy a on early childbearing only. [START_REF]Evolution de la fertilité[END_REF] Under such a re…ned instrument, children would be subsidized di¤erently, according to the age of their parents. In that case, the optimal child allowance would be:

a = cov (u 0 (x) ; n) E @ ñ @ a
where @ ñ=@ a is the e¤ect, on early fertility, of a change in a compensated by a change in T in such a way as to maintain the budget equilibrium. The compensated e¤ect @ ñ=@ a is positive, leading to a subsidy on early children.

Hence, the availability of children allowances di¤erentiated according to the age of parents would be an indirect way to redistribute resources towards agents with low career opportunities, who are usually the ones who have early children.

Non linear case: 2 types

Let us now turn to an alternative formulation of the second-best problem, where the available …scal instruments are not restricted to linear (uniform) instruments, but where the government cannot observe the types of agents. Here again, we assume, as above, a perfect capital market with zero rate of interest and no time preference.

The problem for an individual of type i is to maximize:

u (c i ) + u (d i ) + u (b i ) [c i + d i + b i w (1 + h i (n i )) + n i (e a e `) + e `]:
This implies c i = d i = b i , so that the utility can be reduced to

' [w (1 + h i (n i )) n i (e a e `) e `+ T i ]
where ' ( ) denotes the agent's lifetime welfare (' 0 ( ) > 0, ' 00 ( ) < 0), whereas T i denotes a lump sum transfer such that P i T i = 0: The central planner cannot observe the types of agents. We assume that

h 2 (n) > h 1 (n) > 1 with h 0 i (n) < 0:
If the di¤erence between h 2 and h 1 is big enough, type-1 agents will choose a very high value of n 1 , and type-2 agents will postpone childbearing. In the …rst-best, there will be some redistribution from type 2 to type 1, namely T 1 > 0 > T 2 :

In the second-best, the central planner cannot observe the types of agents. Hence, it is tempting, for an agent of type 2, that is, with high career opportunities, to pretend to be of type 1, in such a way as to bene…t from public transfers. Therefore, the social planner must make sure that type-2 agents do not mimic type-1 agents. This leads us to the self-selection constraint:

' [w (1 + h 2 (n 2 )) + n 2 (e ` e a ) e `+ T 2 ] > ' [w (1 + h 2 (n 1 )) + n 1 (e ` e a ) e `+ T 1 ] : (7)
It is clear that, if we had assumed h 1 (n 1 ) = 1 implying n 1 = 1; then

T 1 = 2 (wh 2 (n 2 ) + n 2 (e a e `))
which is the …rst-best solution.

Formally, the second-best problem can be written by means of the Lagrangian:

L = X i f' (w (1 + h i (n i )) + n i (e ` e a ) e `+ T i ) T i g + ' (w (1 + h 2 (n 2 )) + n 2 (e ` e a ) e `+ T 2 ) ' (w (1 + h 2 (n 1 )) + n 1 (e ` e a ) e `+ T 1 )
where is the Lagrange multiplier associated with the constraint on lump sum transfers, whereas is the Lagrange multiplier associated with the self-selection constraint.

The FOCs are:

-

n 1 : 1 ' 0 (x 1 ) [wh 0 1 (n 1 ) + (e ` e a )] ' 0 (x 2 ) [wh 0 2 (n 1 ) + (e ` e a )] = 0 -n 2 : [' 0 (x 2 ) 2 + ] [wh 0 2 (n 2 ) + (e ` e a )] = 0 -T 1 : 1 (' 0 (x 1 ) ) ' 0 (x 2 ) = 0
system can be an individualized system, where parents internalize the e¤ect of their fertility choices on retirement bene…ts, or, alternatively, a collective system, where parents can free-ride on the system (either PAYG or familial), to get large pensions without supporting the cost of fertility.

As we shall see, relaxing the perfect capital market assumption and replacing it by a pension system is not neutral at all for the design of the optimal family allowances in the context of endogenous childbearing ages. The reason is that, once a (formal or informal) pension system is introduced, there might be, for parents, an incentive in having early children. The reasons are twofold. First, children, if born a longer time ago, can, under increasing earnings pro…les, provide higher pensions to their parents. Second, and more importantly, when parents with early children are retired, they can count not on one, but on two generations of workers: their own children and their grandchildren.

To illustrate this, we will …rst, for the sake of presentation, study fertility choices under an individualized pension system, when there is only one type of agents. We will then reintroduce heterogeneity later on in this section.

Laissez-faire and optimal policy with one type

In that context, the problem faced by an individual amounts to choosing savings s a , early children n a , late children n `and contribution rate to maximize: 20 1) w e a n a s a ] + u [( 1)

U = u [(
wh (n a ) e `n`+ s a ] +u [ (wn `+ wh ( n a ) n a + wn a n a )] + v (n a + n `)
where is the fraction of earnings that is paid to the elderly, while n a means that the individual expects his early children to have made the same number of early children as he made himself (i.e. n a ), without having any control on it. Note that the earnings'basis available for the funding of the elderly's pensions includes three terms: the …rst two terms (i.e. wn `and wh ( n a ) n a ) concern the incomes from the late and the early children, whereas the third term, i.e. wn a n a , concerns the income of grandchildren, whose number is n a n a (as each of the n a early child had also n a early children once adult).

We can thus obtain the following FOCs:

s a : u 0 (c) = u 0 (d) (8)
:

u 0 (c) = u 0 (b) n `+ h ( n a ) n a + n a n a 1 + h (n a ) (9)
n `: u 0 (c) e `= u 0 (b) w + v 0 (n a + n `) (10) n a : u 0 (c) [e a + (1 ) wh 0 (n a )] = u 0 (b) (wh ( n a ) + w n a ) + v 0 (n a + n `) (11)
In a world of identical individuals, these 4 conditions lead to the optimal decision concerning s a ; n a ; n `and as long as each individual considers that the pension scheme operates at the level of the family. It is interesting to observe that such a PAYG scheme creates a distortion in favor of early childbearing, relative to a setting of a fully funded pension system. The reason is that early children have higher earnings when their parents retire, and that they have themselves some working children, unlike the parents'late children.

If, instead of that individualized pension system, the PAYG scheme were collective, each agent would not, when making his fertility choices, perceive his impact on pension bene…ts. Hence, agents would choose a lower level of fertility. To see this, note that, under collective pensions, conditions ( 8) and ( 9) would be unchanged, but conditions ( 10) and ( 11) would become:

u 0 (c) e `= v 0 (n a + n `) (10') u 0 (c) [e a (1 ) wh 0 (n a )] = v 0 (n a + n `) (11')
Given that the RHS of ( 10) includes an additional positive term in comparison to (10'), agents tend to make, under a collective pensions system, fewer late children n `than under an individualized system. Similarly, the RHS of ( 11) includes an additional positive term absent in (11'), so that the collective pensions system induces fewer early children than under individualized pensions. Therefore, in order to induce the optimal fertility behavior under a collective system, one would need Pigouvian subsidies a and `on early and late children, such that

a = w (h ( n a ) + n a ) u 0 (b) u 0 (c) `= w u 0 (b) u 0 (c)
This is a standard problem with PAYG and fertility (see Gahavari 2009): individuals do not internalize the e¤ect that their fertility choice can have on pension bene…ts. We have the following inequality between Pigouvian subsidies: a > The children subsidy should be higher on early than on late childbearing, even though one can have n `> n a : There are two reasons why the external e¤ects of early fertility exceed the ones of late fertility. First, under h ( n a ) > 1, the levels of earnings of contributors born from young parents are larger than the earnings of contributors born from older parents, since only the former can bene…t from increasing productivity. Second, the number of contributors is strongly a¤ected by early fertility decisions, through the number of grandchildren. On the contrary, in the case of late parenthood, grandchildren cannot fund the pension system, and so the fertility externality is less sizeable.

Those arguments tend to justify a di¤erentiated …scal treatment of early and late parenthoods, under the form of children allowances that depend on the age of parents. Thus, even if one assumes a perfect homogeneity of the population in terms of career opportunity and earnings pro…le, there is already a strong argument not only for subsidizing children, but also for subsidizing them di¤erently depending on the age of their parents. The justi…cation is that fertility externalities induced by early and late fertility are not of the same magnitudes, inviting distinct Pigouvian subsidies.

5.2 Laissez-faire and optimal policy under two types Whereas the above policy analysis was carried out for a unique type of agents, let us now turn back to the case when individuals di¤er in their future career and earnings opportunities (i.e. the future earnings function h i (n ai )). For that purpose, we assume, here again, h 2 (n a2 ) > h 1 (n a1 ) 1, with h 0 i (n ai ) < 0. For simplicity of presentation, we suppose also that n ai = n i = 1 n `i, and that v (1) = 0: We also assume saving between …rst and second period.

Given that children allowances di¤erentiated according to the age of parents have been already studied above, we will abstract here from these, and assume that the available …scal instruments are: a uniform child bene…t of rate , a payroll tax and ‡at rate PAYG pension p. Given those assumptions, we have:

c i = d i
and thus

U i = 2u (1 ) w (1 + h i (n i )) (1 ) (n i (e a e `) e `)

+ u (p) :

We also have:

p = X i w 1 n i + h i (n i ) n i + n 2 i (e a n i + e `(1 n i )) :
Each individual i 2 f1; 2g chooses n i without seeing the e¤ect his choice has on his future pension bene…ts. His choice of n i is determined by the FOC:

u 0 (c i ) [w (1 ) h 0 i (n i ) (1
) (e a e `)] = 0:

That condition does not yield the socially optimal level of early children, since this neglects the impact of early fertility on old-age pensions through its impact on the budget constraint of the economy.

To see this, let us now turn to the social planner's problem. We can rewrite it by means of the following Lagrangian:

L = X i U i p + (e a n i + e `(1 n i )) w 1 n i + h i (n i ) n i + n 2 i :
where is the Lagrange multiplier associated with the government's budget constraint.

Using again the operator E to lighten the presentation, we have the FOCs:

@L @p = E [u 0 (p) ] = 0 @L @ = E (u 0 (c) w (1 + h (n))) + E w 1 n + h (n) n + n 2 E B @n @ = 0 @L @ = E [u 0 (c) [n (e a e `) e `]] E ((e a n + e `(1 + n))) E B @n @ = 0
where B (e a e `) w ( 1 + h (n) + nh 0 (n) + 2n) denotes the e¤ect of n on the budget constraint. A marginal increase in early fertility n has 4 revenue e¤ects: (1) a revenue increase thanks to lower child cost (i.e. as e a < e `);

(2) a revenue loss as h 0 (n) < 0 (due to the productivity loss induced by early childbearing); (3) another revenue gain as h > 1 (thanks to the larger contributions of the children once older and more productive); (4) and another revenue gain through grandchildren (which is linear in the number of early children). A su¢ cient condition for B < 0 is:

nh 0 (n) h < 1 + 2n 1
h . Those FOCs can be rewritten as:

@L @ = cov (u 0 (x) ; 1 + h) + E [u 0 (p) v 0 (x)] E (1 + h) +u 0 (p) E 1 n + h (n) n + n 2 E (u 0 (p)) w E B @n @ @L @ = (e a e `) cov [u 0 (x)] E (u 0 (b) u 0 (x)) E (n) E (u 0 (b)) E B @n @ :
To better interpret these expressions, let us consider the case where only pairs of …scal instruments are available: on the one hand, the pair ( ; p), that is, only a tax rate and a pension are available instruments; on the other hand, the pair ( ; p), that is, only a child allowance and a pension are available instruments.

Let us …rst consider the pair ( ; p), being …xed to 0. Equalizing the above FOCs to 0 and isolating yields:

= cov(u 0 (x);1+h)+E(u 0 (p) u 0 (c))E(1+h)+u 0 (p)E(1 n+h(n)+n 2 ) E(u 0 (p)) w E( @n @ ( 1+nh 0 +h+2n))
That formula de…nes the optimal level of the payroll tax when the …scal revenues are used to fund a pension system only. The numerator consists of 3 terms that are all expected to be positive (assuming that E (u 0 (p) u 0 (c)) > 0). Hence those three terms push toward more redistribution through the income tax. The denominator gives the e¤ect of the tax on public revenue. The sign of this e¢ ciency e¤ect depends on the impact of earnings taxation on aggregate early fertility (i.e. E (@n=@ )), and on the impact of aggregate early fertility on the government's budget constraint. If the impact of on the aggregate early fertility is negative (i.e. E (@n=@ ) < 0), and if the impact of early fertility on …scal revenues is also negative, (i.e. if nh 0 (n) h < 1 2n 1 h ), the e¢ ciency e¤ect is also positive. As a consequence, the optimal tax is positive: > 0.

Turning now to the pair ( ; p), under the assumption = 0. Equalizing the above FOCs to 0 and isolating yields:

= (e ` e a ) cov (u 0 (x) ; n) E (u 0 (p) u 0 (c))
(e ` e a ) Eu 0 (p) E @n @ : Assume that the poor (low h i ) tend to have more early children than the rich (i.e. dn=dh < 0). Assume further that child allowances have a positive e¤ect on (total) early fertility (i.e. E (@n=@ ) > 0). Then the sign of the optimal child allowance rate depends on the relative strength of the two terms of the numerator. On the one hand, the positive covariance term pushes for a positive and a low pension. Indeed, if the poor tend to make more children, the equity term pushes towards the subsidization of children, i.e. > 0. That e¤ect is increasing in the cost di¤erential between late and early childbearing, i.e. (e ` e a ). On the other hand, the positive term Eu 0 (p) Eu 0 (c) pushes towards a negative aimed at …nancing the pension scheme. Hence, the sign of the optimal child allowance depends on the strengths of those two terms.

In sum, once we reintroduce heterogeneity in terms of earnings pro…le, the optimal family policy requires an arbitrage between equity and e¢ ciency. Given that agents with lower future career opportunities tend to make children earlier, the subsidization of fertility is supported by equity concerns. The cost gap between late and early fertility tends to reinforce that equity case for children allowances. However, a higher cost gap between late and early childbearing pushes, from an e¢ ciency point of view, towards lower family allowances. As a consequence, the observed falling cost gap between late and early childbearing has a somewhat ambiguous e¤ect on the optimal family allowances.

Conclusion

As this is widely acknowledged among demographers, European economies have been characterized, during the last three decades, by a tendency towards the postponement of fertility. Through its consequences on the economy as a whole, that demographic trend raises various challenges to policy-makers.

The present paper aimed at examining the design of the optimal family policy in an economy where not only total fertility, but also the timing of births, are chosen by individuals. For that purpose, we developed a simple three-period model, where the adult population can have children either in young adulthood, or in older adulthood, and where late childbearing, although possible, is more costly than early childbearing. Moreover, the adult population was assumed to be partitioned in two groups, some individuals having promising earnings opportunities in the future (conditionally on the number of children at the young age), whereas other individuals face a ‡at earnings pro…le.

That simple model explains the observed postponement of fertility as a result of the development of assisted reproductive technologies, which reduced the cost gap between late and early childbearing. Our model allows us also to rationalize the observed heterogeneity in terms of childbearing age, that is, the stylized fact according to which adults with lower career opportunities tend to make children earlier than adults with steeper earnings pro…les.

Regarding the optimal public intervention, we …rstly considered the …rstbest utilitarian optimum, and showed that this can be decentralized by means of lump-sum transfers from increasing-productivity agents towards constant-productivity agents, in such a way as to equalize consumptions at all periods for all agents. Such transfers were shown to raise constant-productivity individuals'early fertility above its laissez-faire level, and to reduce the increasingproductivity individuals'late fertility, but without a¤ecting the timing of births for the two types.

Then, considering an economy where only linear uniform policy instruments are available, we showed that the optimal second-best policy involves a subsidization of children, to the extent that it fosters early parenthood. Indeed, given that early parenthood concerns generally parents with low career opportunities, family allowances are justi…ed on redistributive grounds. Children allowances di¤erentiated according to the parent's age, would, if available, achieve that redistribution even better. Turning then to the optimal non-linear policy under asymmetric information, we showed that only children from parents with ‡at earnings pro…le should be subsidized, to solve the self-selection problem.

Finally, we investigated the impact of di¤erential childbearing ages on the funding of social security, by introducing, in our set-up, a pension system. At the laissez-faire, one of the most striking e¤ects of endogenous childbearing ages is that, under normal saving, or, alternatively, under a fully-funded pension system, individuals tend to adopt late childbearing with more earnings, whereas, under a PAYG system, there is a bias for early childbearing with overall lower earnings. We also showed, in a framework without heterogeneity, that, under a collective pension system, the fertility externalities related to early and late parenthoods di¤er signi…cantly, inviting a larger Pigouvian subsidy for early children. The underlying reason is both qualitative (older children, under increasing productivity, can contribute more than younger, late children) and quantitative (early parents can, once retired, bene…t from the contributions of their grandchildren). Turning back to the 2-type case, we then highlighted that the social planner faces, under endogenous childbearing age, a trade-o¤ between e¢ ciency (favoring late childbearing) and equity (helping early childbearing).

In sum, the present study, although it has no pretension to completeness, highlights some major challenges faced by policy-makers in the context of endogenous childbearing ages. Endogenous childbearing age a¤ects many fundamental aspects of the optimal policy design. As far as redistribution is concerned, the observed correlation between the heterogeneity in terms of fertility age-pattern and in terms of career opportunities makes children allowances di¤erentiated in terms of the age of parent socially desirable. Moreover, the internalization of fertility externalities in the context of pensions funding would also require children allowances di¤erentiated according to the age of parents. Hence, even though policy debates are usually concerned with changes in the total fertility rate, the timing of births is far from a detail for the design of the optimal family policy, and, as such, will require more attention in the future.

Figure 1 :

 1 Figure 1: Age at motherhood in France

Sources: INED (2011).

See[START_REF] Gustafsson | Optimal age at motherhood. Theoretical and empirical considerations on postponement of maternity in Europe[END_REF] for a survey on the literature on childbearing age.

Empirical studies include:[START_REF] Schultz | Changing world prices, women's wages and the fertility transition: Sweden 1860-1910[END_REF],[START_REF] Heckman | The relationship between wages and income and the timing and spacing of births: evidence from Swedish longitudinal data[END_REF] and[START_REF] Tasiran | Fertility dynamics. Spacing and timing of births in Sweden and the United States[END_REF] on Sweden, Ermisch and Ogawa (1994) on Japan, Merrigan and St Pierre (1998) on Canada, and Joshi (2002) on Great Britain.

[START_REF] Cigno | A microeconomic analysis of the timing of births[END_REF] also found, on the basis of UK data, empirical support for that explanation of the observed heterogeneity in terms of fertility patterns.

On this, see the survey by[START_REF] Gauthier | The impact of family policies on fertility in industrialized countries: a review of the literature[END_REF].

This was con…rmed by[START_REF] Balestrino | Endogenous fertility and the design of family taxation[END_REF] in a model where households di¤er in their productivities and in their ability to raise children.

That result is also obtained in[START_REF] Cremer | Pensions with heterogeneous individuals and endogenous fertility[END_REF], who examined the optimal pensions scheme when fertility is endogenous and parents di¤er in their ability to raise children.

0 Actually, we consider a model that corresponds to the steady-state of an OLG model wherein the longitudinal view coincides with crossesectional one.1 1 In some cases below, we allow type-1 agents to have a higher and endogenous productivity in the second period, but always lower than that of type-2 agents.1

We assume here a perfect substituability, in welfare terms, between early and late children.That assumption, which is made for simplicity, amounts to assume that children are not durable consumption goods. As stressed in[START_REF] Gustafsson | Optimal age at motherhood. Theoretical and empirical considerations on postponement of maternity in Europe[END_REF], taking children as durable goods would favour early childbearing ceteris paribus.1

Note that the precise causes of the negative e¤ect of early childbearing remain largely unknown. For instance,[START_REF] Ermisch | Early motherhood and later partnerships[END_REF] showed that very early motherhood (teen births) worsens later outcomes on the marriage market.

[START_REF] Cigno | Economics of the family[END_REF] According to[START_REF] Gustafsson | Optimal age at motherhood. Theoretical and empirical considerations on postponement of maternity in Europe[END_REF], late births imply, on average, more pregnancy complications, more ceasareans and more breast cancer for mothers. Moreover, late children are also more subject to somatic and learning problems.

[START_REF] Cigno | A microeconomic analysis of the timing of births[END_REF] Obviously, given ea < e `, those two FOCs would not be all valid if h 0

(n a2 ) = 0.

[START_REF] Cigno | Optimal family policy in the presence of moral hazard, when the quantity and the quality of children are stochastic[END_REF] On this, see[START_REF] Blackorby | Population issues in social choice theory, welfare economics, and ethics[END_REF].

[START_REF] Cremer | Pensions with endogenous and stochastic fertility[END_REF] The conditions describing the type of optimum (interior or corner) are independent from consumption levels, and depend only on ea, e `and h 0

( ).

[START_REF] Cremer | Pensions with heterogeneous individuals and endogenous fertility[END_REF] For simplicity we assume interior solutions, which excludes the case where h 1 = 1.

[START_REF] D'albis | Demographiceconomic equilibria when the age at motherhood is endogenous[END_REF] In that case, there is no tax nor subsidy on late childbearing.

0 To keep things simple, we assume that savings between the …rst and the second periods is possible.

-T 2 : 2 (' 0 (x 2 ) ) + ' 0 (x 2 ) = 0 First of all, we see that, when the self-selection constraint is not binding ( = 0), we have the …rst-best outcome:

e ` e a = wh 0 i (n i ) ' 0 (x i ) = On the contrary, when > 0, we have:

In other words, in the choice of n 2 , one has the standard "non distortion at the top" result. As to redistribution, it is only partial, since we have: x 2 > x 1 . Thus, the presence of asymmetric information prevents the equalization of lifetime welfare across types, unlike in the …rst-best optimum. Finally, we have

This expression means that one should have a subsidy on n 1 to relax the selfselection constraint. Indeed, given that early children are more important for agents of type 1 in comparison to agents of type 2 (for whom early childbearing has worse e¤ects on the earnings pro…le), a simple way to prevent type-2 agents from pretending to be of type 1 to get transfers consists of subsidizing children.

In sum, the optimal policy under asymmetric information involves children allowances that are di¤erentiated according to the parents'(potential) earnings pro…les. Whereas early children from parents with (potentially) steep earnings pro…les are not subsidized, early children from parents with ‡at earnings pro…le should be subsidized, as a way to solve the self-selection problem. Thus, even if the government cannot observe the type of agents, it can nonetheless insure the self-selection of types by proposing di¤erent children allowances, which make, by construction, any mimicking suboptimal for individuals.

Childbearing and PAYG social security

Up to now, we studied the optimal family policy under a perfect capital market. We will now relax that assumption. For that purpose, we will assume that the only way individuals can provide resources for their old age is through a contract such that when retired they get a fraction of their children's earnings.

Such a contract can take various forms. It can be a standard Pay-As-You-Go pension system (PAYG), which provides a pension to the elderly thanks to the contributions of the young active individuals. It can also take the form of intergenerational trade within the (extended) family, each active child giving some fraction of his income to the inactive elderly in his family. Moreover, the


; F Genelle

M Dabas

C Sirieix

J Riss

S Renié

B Dubearnes

P Bégassat

On the geoelectrical characterization of an old landfill cover

Two geoelectrical methods were used to characterize the state of an old French landfill cover. The objectives were to locate different materials used for the covering such as clay material or geomembrane and to identify heterogeneities that could be linked to possible defects in the cover due to fractures or cracks. These damages can induce preferential water pathways and unusual increase of leachate within the waste mass. The geoelectrical methods used were the electrical resistivity cartography with an Automatic Resistivity Profiling (ARP©) and the Self Potential method (SP). Results have put in evidence three distinct zones with different geoelectrical signatures that are correlated with three different phases of landfill covering. ARP also seems to have detected the presence of geomembrane installed all around the old landfill to improve its stability and geodrains over two closed alveoli. Local differences in apparent resistivity and self-potential signals have also been identified and could result in default in the clay cover due to thickness variations but also in different lithology, compaction and water content.

Introduction

To minimize infiltration of precipitation and the accumulation of contaminant groundwater within a landfill, impermeable barriers such as compacted clay or geocomposite are covering the overall waste storage. Although ideal barriers are composed of a continuous impermeable clay layer and a geocomposite, real covers are often fractured and eroded due to mechanical, climatic and hydraulic stresses acting on its surface or even be damaged during its laying. These damages can induce an escape of landfill gases and creation of leachate through infiltration of surface water. Geophysical methods would be a good way to detect anomalies due to preferential water pathways and unusual increase of leachate within the waste mass. Currently, as leachate is highly electrically conductive, geophysical methods are mainly used to trace the migration of leachate inside and away from the landfill [START_REF] Naudet | Ground water redox conditions and conductivity in a contaminant plume from geoelectrical investigations[END_REF][START_REF] Guérin | Leachate recirculation: moisture content assessment by means of geophysical technique[END_REF]. Little attention and few published studies have used geophysical methods to evaluate water recharge through the landfill cap [START_REF] Carpenter | Assessing a fractured landfill cover using electrical resisitivty and seismic refraction techniques[END_REF][START_REF] White | Electrical leak detection system for landfill liners: a case history[END_REF][START_REF] Guyonnet | Heterogeneity detection in an experimental clay liner[END_REF][START_REF] Cassiani | Vertical radar profiling for the assessment of landfill capping effectiveness[END_REF]. Our goal was to test the feasibility of the electrical resistivity cartography with an Automatic Resistivity Profiling (ARP©) and the Self Potential method (SP) to improve the knowledge of an old French landfill cover.

Methods

The self-potential (SP) method is a passive electrical method measuring the natural electrical potential at the ground surface with non-polarisable electrodes. It aims at detecting potential changes linked to a natural overload of underground electric charges. These charges are likely to be linked to in-body fluids circulation through the electrokinetic coupling effect, electrochemical coupling like diffusion of ionic species, and oxide-reduction reactions (i.e. [START_REF] Jouniaux | Review of self-potential methods in hydrogeophysics[END_REF]. In particular, due to the general inhomogeneity of the subsoil, the SP sources can be distinguished between primary and secondary charge polarizations. The primary sources are associated with the electrokinetic fluid flow through rock pores and fissures, while the second sources are correlated to the distribution of electric charges, induced by the primary sources, along resistivity discontinuities. As concerns the landfill covers, we expect to detect sources mainly due to electrokinetic effect and electrical contrasts separating different kinds of cells or to locate fractures or cracks in the cover that facilitate surface water infiltration. The ARP technique uses a patented multi-electrode device in which wheel-based electrodes are inserted in the ground and rolled along the surface [START_REF] Dabas | Theroy and practice of the new fast electrical imaging system ARP©[END_REF]. The device is a Vshaped multipole system with one transmitting dipole and three receiving dipoles. The length of the current dipole is fixed and equal to 1 m, while the length of the receiving dipoles increased from 0.5 to 1.00 and 1.70 m respectively with their distance from the transmitter. The system is attached to a quad bike, which facilitates rapid data acquisition. Use of differential GPS navigation within the system enables accurate surveying. Continuous electrical soil mapping is carried out at three different theoretical depths below ground surface level. Apparent electrical resistivity data are then presented in the form of three maps corresponding to the length of the different measuring dipoles. These maps represent the cumulative volume contribution of the soil from top to the three increasing pseudodepths of investigation.

The studied site

The studied site has been exploited from 1979 to 1998 with around 600 000 tons of municipal waste stored without compaction during this period and composed of oversized waste, raw wastes, crushed and compostable household waste. These wastes form a dome of 10 m height. From February to April 2002, wastes from the eastern part have been removed and replaced over the western part of this dome (from zone 3 to zone 1 and 2, fig. 1

Figure 1 a. Presentation of the studied municipal landfill with location of the three zones in the old waste dome covered at different dates and the two closed alveoli (AS2 and AS3). ARP measurements

Results

The apparent electrical resistivity data obtained with the ARP technique are presented in the form of horizontal pseudo-depth slices obtained at three different measuring dipole lengths (0.5, 1 and 1.70 m) on figure 2. The maps obtained on the old landfill outline the three zones covered at different dates.

From zone 3 (more recent) to zone 1 (the oldest), the apparent electrical resistivity decreases. This could be due to a decrease of the clayey cover layer thickness but also differences in the type of clay used, its compaction and also in water content. The more conductive zones (<5 Ohm.m) could reflect the influence of the very conductive waste on the measurement and therefore reflect a less important cover layer thickness. The three maps show more resistive zones (>50 Ohm.m) all around the landfill and mainly in zone 3. These zones could be attributed to the presence of geomembrane as it is electrically insulating, which were installed in the earth banks and anchored at the top of the slopes under the clayey layer for stability reasons (fig. 1.b). Note also that the electrical signature of this geomembrane, which tends to decrease with the investigation depth, is more important for the shallower measuring dipole length (0.7 m). This could indicate that the topsoil plus clayey cover layer thickness could be less than 1.5 m as initially installed (fig. 1.b). ARP measurements performed on the two adjacent alveoli outline distinguished more resistive lines with around 5 m width, which are detected since the first investigation depth. These lines seem to correspond to geodrains installed above 0.5 m of topsoil.

The self-potential map presented on figure 3 also shows a contrast between zone 1 characterized by more elevated SP values and zone 3. This difference could be due to the electrical contrast between these zones due to water content and thickness variations of the clayey cover layer. The SP map also identified negative anomalies (>8mV in absolute value) in the resistive zones identified by the ARP method on the edge of the landfill and that could be attributed to the geomembrane. The Automatic Resistivity Profiling (ARP©) and the Self Potential method (SP) have been applied on an old French landfill in order to better characterize its covering in terms of different material used and in terms of heterogeneities that could be linked to possible defects in the cover due to fractures or cracks. Knowledge of these defects location is important because they can induce preferential water pathways and unusual increase of leachate within the waste mass. Results have shown that the ARP method can clearly locate geodrains as resistive bodies. Both the ARP and SP methods have also outlined three zones that have been covered at different dates. Local differences in apparent resistivity and self-potential signals have also been identified and could result in default in the clay cover due to thickness variations but also in different lithology, compaction and water content. To discriminate these different causes, intrusive geotechnical measurements or geophysical experiments on controlled parcels must be performed

  .a). Contrary to the first storage, these wastes were intensely compacted by layers of 50 cm. The cells of the northern part of this zone were dedicated to exclusively compacted wastes stored since 2002 (zone 3, fig.1.a). Therefore, the landfill site comprises three main zones: 1) a zone mainly composed of non-compacted wastes covered before 2002, 2) a zone composed of around 7 m of non-compacted wastes overlain by 6 m of compacted waste covered after 2002, and 3) a zone of only compacted wastes covered at the end (fig. 1.a). Except for zone 3, the covering layer was composed of a 1 m clayed material overlain by a 0.50 m topsoil layer. For stability reasons, in the earth banks, the clay has been replaced by a geomembrane overlain by topsoil and anchored at the top of the slopes under the clayey layer (fig. 1.b). The geophysical measurements were performed in June 2009. Around 440 self-potential data were obtained each 10 m with Pb/PbCl 2 non-polarizable electrodes. The ARP measurements covered the overall site with a point every 10 cm along profiles separated by 2 m. Two other closed alveoli covered more recently with geomembrane bands were also prospected with a profile spacing of 1 m. (fig.1.a).

Figure 2 Figure 3 .

 23 Figure 2 Apparent electrical resistivity maps obtained with the ARP technique on the old landfill and the two adjacent alveoli (receiving dipole length of 0.5 m (a.), 1 m (b.) and 1.7 m (c.)

Acknowledgements

The CALITOM public service union is thanked for assess to the site. Financial supports by ADEME, HydroInvest and the Association Nationale Recherche Technologie (ANRT) through a PhD grant to F. Genelle are acknowledged.