Pierre Pestieau

Gregory Ponthiere
email: gregory.ponthiere@ens.fr

Long-term care and births timing

Keywords: long term care, birth timing, childbearing age, family policy, OLG models Long term care, birth timing, childbearing age, family policy, OLG models. JEL classi…cation codes: E13, J13, J14

Due to the ageing process, the provision of long-term care (LTC) to the dependent elderly has become a major challenge of our epoch. But at the same time, our societies are characterized, since the 1970s, by a signi…cant postponement of births. This paper aims at examining the impact of those demographic trends on the optimal family policy. We develop a four-period OLG model where individuals, who receive children's informal LTC at the old age, must choose, when being young, how to allocate births along their lifecycle. It is shown that early children provide more LTC to their elderly parents than late children, because of the lower opportunity cost of providing LTC when being retired. In comparison with the social optimum, individuals have, at the laissez-faire, too few children early in their life, and too many later on in their life. The decentralization of the …rst-best optimum requires thus to subsidize early births. We study also the design of the optimal subsidy on early births in a second-best setting. Its level depends on e¢ ciency and equity issues, as well as on its incidence on the long-run population composition and on LTC provision.

Introduction

The beginning of the 21st century is characterized by two fundamental demographic trends, which constitute the most recent corollaries of the demographic transition started two centuries ago [START_REF] Lee | The demographic transition: three centuries of fundamental change[END_REF].

On the one hand, the ageing process raises the proportion of elderly persons in need of long-term care (LTC), i.e. persons who can no longer carry out simple daily activities such as eating, washing, etc. The number of dependent elderly at the European level (EU-27) is expected to grow from about 38 million people in 2010 to about 57 million in 2060 (EU 2012). As shown on Figure 1, the rise in the number of dependent elderly persons is expected to be substantial in most large European economies. The rise of LTC constitutes a major challenge for families, since two thirds of LTC is provided informally [START_REF] Norton | Long term care[END_REF]. Forecasts from the EU suggest that a signi…cant part of LTC provision will remain informal in the future. Given the limited development of private LTC insurance markets, this constitutes also a major challenge for policy-makers.1 On the other hand, our societies witness also, since the 1970s, a signi…cant postponement of births (see Gustafsson 2001). Because of various reasons, such as the rise in education, medical advances or cultural norms, individuals tend to have their children later on in their life. 2 To illustrate that trend, Figure 2 shows the rise in the mean age at birth in di¤erent European countries. That rise is substantial: whereas the mean age at birth was below 27 years in France in the late 1970s, it is about 30 years today. The postponement of births has also a signi…cant impact on the dynamics of the age-structure of the economy, and, hence, may a¤ect the …nancial sustainability of systems of intergenerational trade, such as PAYG pensions schemes. Although those two demographic trends may seem, at …rst glance, to be unrelated, those two phenomena are linked through various channels. First of all, as already mentioned, a large volume of LTC services are informal, and provided by the family. Hence the rise of the demand for LTC imposes some pressure on the time constraints faced by other members of the family, such as spouses and children. Given that the time dedicated to LTC cannot be spent on other activities, such as working or educating one's children, LTC makes time even more scarce for informal care givers. Secondly, when the caregivers are children of the dependent elderly, the age at which caregivers are born has a major impact on the opportunity cost of helping their elderly parents. Clearly, the oldest child of the family is, when his parent becomes dependent, quite old already, and close to retirement. Hence, for the …rst children born in a family, the opportunity cost of helping an elderly parent is lower. On the contrary, younger children are, when their parent becomes dependent, quite young, and are not close to retire, implying a higher opportunity cost of providing LTC.

Whereas those links may appear purely theoretical, empirical studies on informal LTC suggest that informal LTC is far from a marginal phenomenon.

According to recent …gures of the SCAN Foundation (2012), informal caregivers spend, on average, 20.4 hours per week providing care. Moreover, the period during which informal care is given is, on average, equal to 4.6 years. Thus informal LTC really a¤ects time constraints in real life. Regarding the impact of the age of children, recent studies, such as [START_REF] Fontaine | Les con…gurations d'aide familiale mobilisées autour des personnes âgées dépendantes en Europe[END_REF] showed, on the basis of SHARE data, that the behavior of the children of a dependent parent without spouse is far from symmetric, but, rather, varies with the age of the child and his/her involvement in the labor market. In particular, if younger children are working full time, older children tend to be more involved in the provision of LTC to their elderly parent, even though younger children also participate (in particular when their older brothers and sisters are strongly involved). Thus the age (and, hence, working status) of children providing informal LTC to an elderly parent seems to be an important variable, and this variable is directly related to the fertility choices made by their parents when they were young.

The goal of this paper is precisely to reexamine the formal relations between, on the one hand, the provision of informal LTC by children, and, on the other hand, the timing of births, by paying a particular attention to the impact of LTC on time constraints, and also to the unequal opportunity costs of providing LTC among children of di¤erent ages. More precisely, we would like to examine the conditions under which the timing of birth of children can be used by parents as a kind of insurance device for LTC provision at the old age in an economy where there exist no social insurance system, and where LTC is only provided informally through children.

For that purpose, we develop a 4-period OLG model where individuals become dependent at the old age (period 4), and require some help from their children. In order to discuss the fertility timing, we consider a model of lifecycle fertility, where parents can choose to have children in two fertility periods (i.e. periods 2 and 3). We thus consider the joint of decisions of the timing of births and of the provision of LTC to the elderly parents, within a dynamic model of capital accumulation.

Our analysis proceeds in two stages. Our model is …rst used to examine, at the laissez-faire, the relation between birth timing and LTC provision. The question raised is the following: under which conditions do older children provide more or less LTC to their elderly parents, in comparison with younger children? We also characterize the stationary equilibrium of the economy, to examine to what extent our conclusions are robust to the economic development process, which a¤ects the dynamics of the wages and interest rates, and, hence, the opportunity cost of providing informal LTC for children. Then, in a second stage, we propose to characterize the long-run social optimum, and we examine the decentralization of that long-run social optimum by means of appropriate policy instruments. Our question are: what does the optimal family policy look like? Is the birth timing chosen by parents socially optimal in the long-run? Is the postponement of births desirable in times of a rise in LTC?

Anticipating on our results, we …rst show that, at the laissez-faire, early children provide, under mild conditions, more LTC to their elderly parents in comparison to late children, because of a lower opportunity cost of helping their parents. That tendency is shown to be persistent in the long-run. Then, we characterize the long-run social optimum. We show that individuals have, at the laissez-faire, too few children early in their life, and too many children later on in their life in comparison with the social optimum. The intuition behind that result lies in the fact that, in our model, parents do not perfectly anticipate, when they plan the births of their children, that the chosen birth timing will a¤ect the opportunity costs faced by their children when they will provide LTC 50 or 60 years later, once they will be dependent. As a consequence of this, the decentralization of the social optimum requires subsidies on early births, in such a way as to encourage parents to have children early, so as to minimize the opportunity cost of providing LTC in the future. When considering the secondbest problem (and focusing, for simplicity, on an open economy where factor prices are taken as given), we show that the optimal (uniform) subsidy on early births depends not only on standard equity and e¢ ciency concerns, but aims also at internalizing population composition e¤ects, as well as at encouraging the provision of LTC to the dependent parents.

As such, this paper pertains to several branches of the literature. First of all, it complements the existing literature on LTC and family games. That literature already emphasized the crucial role played by the age of children in LTC provision, in particular when the child decides his geographical location (see [START_REF] Konrad | Geography of the family[END_REF][START_REF] Wakabayashi | Is the eldest son di¤erent? The residential choice of siblings in Japan[END_REF]. The present paper is not about geographical location choices, but focuses on the di¤erences in the time cost of providing LTC due to the proximity with the retirement period. We also complement other papers on family games and LTC, such as [START_REF] Pezzin | E¢ ciency in family bargaining: living arrangements and caregiving decisions of adult children and disabled elderly parents[END_REF][START_REF] Pezzin | Long term care of the disabled elderly: do children increase caregiving by spouses?[END_REF], which focus more on strategic motives for LTC provision, without considering endogenous fertility choices. Secondly, our paper is also in line with the literature focusing on fertility as insurance device in case of LTC risk at the old ages (see [START_REF] Cremer | Uncertain altruism and the provision of long term care[END_REF]. Our contribution with respect to that literature is to consider a particular aspect of fertility choices: the timing of births. Thirdly, we also contribute to the literature on optimal public policy under LTC (see [START_REF] Jousten | Long-term care insurance and optimal taxation for altruistic children[END_REF]Pestieau andSato 2006, 2008;[START_REF] Cremer | Securing Long-term Care in the EU: Some Key Issues[END_REF][START_REF] Cremer | Long term care policy, myopia and redistribution[END_REF]. Those papers study various aspects of the design of optimal LTC public insurance, but do not consider endogenous fertility. Finally, we also complement the existing literature on childbearing ages and the choice of optimal fertility pro…le in OLG models, such as d 'Albis et al (2012), and Pestieau andPonthiere (2014, 2015). Our contribution with respect to those papers is to highlight the e¤ects of birth timing decisions on the informal provision of LTC in the long-run.

The rest of the paper is organized as follows. The model is presented in Section 2. The laissez-faire (temporary equilibrium and intertemporal equilibrium) is studied in Section 3. Section 4 characterizes the long-run social optimum. The optimal family policy is derived in Section 5. The second-best optimal policy is studied in Section 6. Section 7 concludes.

The model

We consider a four-period OLG model. The duration of each period is normalized to 1. Each cohort is a continuum of individuals of size 1. Fertility is at the replacement level, that is, one child per agent.

Period one is childhood. Period 2 is young adulthood, during which individuals work, consume, save resources and can have n t children. Period 3 is older adulthood, during which individuals work during a fraction of time z 2 ]0; 1[, and enjoy a period of retirement of duration 1 z. During that period, individuals consume, save and can have 1 n t children. Period 4 is a period of old-age dependency, during which individuals need long term care.

There exist two types of agents, depending on the age of their parent:

Type-E agents: children born from young parents (i.e. "early" children)

Type-L agents: children born from older parents (i.e."late" children).

Within the population of young adults at time t, the proportion of type E agents t is denoted by q t (we have 0 q t 1). The proportion of agents of type L is 1 q t . Figure 3 illustrate the coexistence of agents of types E and L; those agents di¤er regarding the age of their parents: while type-E individuals have younger parents (i.e. the age-gap between them and their parents is low), type-L individuals have older parents (i.e. the age-gap between them and their parents is higher). As a consequence of this, those agents will also di¤er in terms of age at the time of providing informal help to their elderly dependent parents. Type-E agents will be older when their parents will fall into dependency and will need LTC. On the contrary, type-L agents will be younger and fully active when their parents will become dependent.

Preferences

The preferences of a young adult of type i 2 fE; Lg are represented by the following utility function:

u(c i t ) + v n i t + u(d i t+1 ) + v(1 n i t ) + (a i ) + '(b ie t+2 ) (1)
where c i t is consumption at the young age, d i t+1 denotes consumption in third period. a i is the informal temporal aid given to the parent. It is equal to a E t+1 for type E and to a L t for type L. Finally, b ie t+2 is the expected temporal aid received from children at the old age. As usual, we assume u 0 ( ) > 0; u 00 ( ) < 0; v 0 ( ) > 0; v 00 ( ) < 0; 0 ( ) > 0; 00 ( ) < 0; ' 0 ( ) > 0 and ' 00 ( ) < 0: The parameter 2 [0; 1] captures the extent to which young individuals take into account the impact of their fertility choices on future LTC received at the old age. Full myopia corresponds to the case where = 0, whereas = 1 coincides with the absence of myopia. The expected total help received at the old age is the sum of the aid given by each child:

b ie t+2 = n i t a Ee t+2 + (1 n i t )a Le t+2 (2)
We assume here a perfect substitutability between the aid received from the early children and from the late children. The expected levels of temporal aid received from each early child and from each late child, i.e. a Ee t+2 and a Le t+2 , are likely to di¤er. Actually, since the two types of children will, because of the birth spacing, help their elderly parents at di¤erent points in their life, it is plausible to believe that they will not face the same opportunity cost of helping, and, thus, that they may provide di¤erent amounts of help. Note, however, that the expectations of parents may not fully re ‡ect this, and therefore we will consider various kinds of assumptions regarding a Ee t+2 and a Le t+2 when characterizing the laissez-faire equilibrium.

Budget constraints

When an agent is of type E, his parent becomes dependent when he is in the third period of his life. Hence, he provides aid to his dependent parent when being an old adult, that is, we have a E t = 0 and a E t+1 > 0. Hence, assuming that a child costs a fraction in terms of time, the resource constraints are:

w t 1 n E t = c E t + s E t (3) w e t+1 z(1 (1 n E t ) a E t+1 ) + R e t+1 s E t = d E t+1 (4)
where w t is the hourly wage earned at time t, w e t+1 is the expected wage rate at time t + 1, while R e t+1 is equal to one plus the expected interest rate prevailing at time t + 1. The associated intertemporal budget constraint is:

w t 1 n E t + w e t+1 z(1 (1 n E t ) a E t+1 ) R e t+1 = c E t + d E t+1 R e t+1 (5)
When an agent is of type L, his parent becomes dependent when he is in the second period of his life. Hence, he provides aid to his dependent parent when being a young adult, that is, we have a L t > 0 and a L t+1 = 0. Hence, the resource constraints are:

w t (1 n L t a L t ) = c L t + s L t (6) w e t+1 z 1 (1 n L t ) + R e t+1 s L t = d L t+1 (7)
The unique di¤erence between the two sets of budget constraints lies in the moment of life at which children help their elderly parents. In the hypothetical case where there was no help (i.e. a E t+1 = a L t = 0), the two sets of budget constraints would be the same across types E and L. The associated intertemporal budget constraint is:

w t (1 n L t a L t ) + w e t+1 z 1 (1 n L t ) R e t+1 = c L t + d L t+1 R e t+1 (8)
From those constraints, it is obvious that the two types of children do not face the same opportunity cost of helping their dependent parents. Given that early children are close to retirement at the time of helping their dependent parents, they face a lower opportunity cost; on the contrary, late children are at the beginning of their career at the time of helping their parents. Hence, the labor earnings that they must give up to help their parents are larger.

Production

The production process involves capital K t and labour L t , and exhibits constant returns to scale:

Y t = F (K t ; L t ) (9)
The total labor force is made of young and old adults of types E and L:

L t = q t 1 n E t + (1 q t ) 1 n L t a L t +q t 1 z(1 a E t (1 n E t 1 )) + (1 q t 1 )z 1 (1 n L t 1 ) (10)
Hence the production process can be written as:

Y t = F (K t ; L t ) (11)
For simplicity, it is assumed that there is full depreciation of capital after one period of use. Hence capital accumulates according to the law:

K t+1 = q t s E t + (1 q t )s L t (12)
As usual, factors are paid at their marginal productivity:

w t = F L (K t ; L t ) (13) R t = F K (K t ; L t ) (14)
where w t is the wage rate and R t equals 1 plus the interest rate.

The resource constraint of the economy is:

F (K t ; L t ) = q t c E t + (1 q t )c L t + q t 1 d E t + (1 q t 1 )d L t + K t+1
3 The laissez-faire

Let us now consider how individuals of the two types will make their decisions in terms of consumption, fertility and parental help. For that purpose, we will …rst consider the temporary equilibrium, which is a situation where each agent maximizes his utility, conditionally on his beliefs regarding future production factor prices (i.e. w e t+1 and R e t+1 ) and conditionally on his beliefs regarding the future help received from the children born early and late (i.e. a Ee t+2 and a Le t+2 ). For the sake of simpli…cation, we will assume, in the rest of this section, that all agents, whatever these are of type E or L, formulate exactly the same beliefs regarding future factor prices and future help received at the old age.

Temporary equilibrium

Conditionally on their anticipations, individuals of type E solve the following problem:

max c E t ;d E t+1 n E t ;a E t+1 u(c E t ) + v n E t + u(d E t+1 ) + v 1 n E t + (a E t+1 ) + '(n E t a Ee t+2 + (1 n E t )a Le t+2 ) s.t. w t 1 n E t + w e t+1 z(1 (1 n E t ) a E t+1 ) R e t+1 = c E t + d E t+1 R e t+1
First-order conditions (FOCs) yield:

u 0 (c E t ) = R e t+1 u 0 (d E t+1 ) (15) v 0 n E t v 0 1 n E t + ' 0 (b Ee t+2 ) a Ee t+2 a Le t+2 = u 0 (c E t ) w t w e t+1 z R e t+1 (16) 0 (a E t+1 ) = u 0 (c E t ) w e t+1 z R e t+1 (17)
The …rst condition in those FOCs is the standard Euler equation, relating present and future consumption. The second condition characterizes the optimal birth timing. It says that the birth timing is optimally chosen when the marginal welfare gain from increasing the number of early births (LHS of the condition, which is decreasing in n E t ) is exactly equal to the marginal welfare loss from increasing early births (RHS of the condition). Note that the LHS of that condition depends on the di¤erence in terms of the expected LTC received at the old age, depending on the type of child (third term of the LHS). When parents anticipate equal help from early and late children, i.e. a Ee t+2 = a Le t+2 , that term vanishes. If, on the contrary, parents expect to receive more help from their early children, i.e. a Ee t+2 > a Le t+2 , this increases the marginal welfare gain from having early children ceteris paribus. Hence, for an equal RHS, this leads to a rise in the optimal level of early children n E t . Inversely, the beliefs a Ee t+2 < a Le t+2 reduces the marginal welfare from from having early children, inviting, ceteris paribus, later births (i.e. a smaller n E t ). The last FOC characterizes the optimal level of aid provided to the parent. In case of early children, the marginal welfare loss of helping their elderly parent (RHS) is increasing with the retirement age z. Later retirement raises the opportunity cost of helping the elderly.

Individuals of type L solve the following problem:

max c L t ;d L t+1 n L t ;a L t u(c L t ) + v n L t + u(d L t+1 ) + v 1 n L t + (a L t ) + '(n L t a Ee t+2 + (1 n L t )a Le t+2 ) s.t. w t (1 n L t a L t ) + w e t+1 z 1 (1 n L t ) R e t+1 = c L t + d L t+1 R e t+1
FOCs are:

u 0 (c L t ) = R e t+1 u 0 (d L t+1 ) (18) v 0 n L t v 0 1 n L t + ' 0 (b Le t+2 ) a Ee t+2 a Le t+2 = u 0 (c L t ) w t w e t+1 z R e t+1 (19) 0 (a L t ) = u 0 (c L t )w t (20)
The comparison of the FOCs for the two types of young individuals reveals several things. First, the slope of the consumption pro…le is the same across agents; but the levels of the consumption pro…les can vary across types. Second, the condition for optimal fertility timing is the same across agents. Hence, given that those agents face the same market prices, the unique di¤erence in birth timing will come from di¤erences in the consumption pro…les. Individuals with larger consumption at the young age exhibit a lower marginal utility of consumption, which reduces the marginal welfare loss associated with early births (RHS), which encourages advancing births. On the contrary, individuals with a lower consumption at the young age exhibit a larger marginal welfare loss of early births, which encourages birth postponement. Thirdly, the RHS of the condition for optimal help to the elderly parent is also di¤erent from its counterpart for type-E agents. Under the assumption w e t+1 z R e t+1 < w t , late children face, ceteris paribus (i.e. for a given consumption), a higher marginal welfare loss from helping their parents. As shown below, the lower opportunity cost of helping the elderly explains why early children provide, at the laissez-faire, more LTC than late children.

Proposition 1 Given the anticipated levels of future production factor prices w e t+1 and R e t+1 , as well as the anticipated levels of future LTC received from the children a Ee t+2 and a Le t+2 , the temporary equilibrium is a vector c E t ; d

u 0 (c E t ) = R e t+1 u 0 (d E t+1 ) v 0 n E t v 0 1 n E t + ' 0 (b Ee t+2 ) a Ee t+2 a Le t+2 = u 0 (c E t ) w t w e t+1 z R e t+1 0 (a E t+1 ) = u 0 (c E t ) w e t+1 z R e t+1 u 0 (c L t ) = R e t+1 u 0 (d L t+1 ) v 0 n L t v 0 1 n L t + ' 0 (b Le t+2 ) a Ee t+2 a Le t+2 = u 0 (c L t ) w t w e t+1 z R e t+1 0 (a L t ) = u 0 (c L t )w t w t = F L (K t ; L t )
Under the assumption

w e t+1 z R e t+1
< w t , individuals of type E provide, in comparison with type-L individuals, a larger amount of LTC to their elderly parents, they consume also more and have more early children than type-L individuals:

a E t+1 > a L t c E t > c L t d E t+1 > d L t+1 n E t > n L t
Proof. The …rst part of the proposition is obtained from the FOCs of the problems faced by individuals of types E and L. Regarding the second part of the proposition, this can be obtained by considering the following three cases:

Case 1:

c E t = c L t = c t . Then d E t+1 = d L t+1 = d t+1
from the FOCs for optimal savings and n E t = n L t from the FOCs for optimal fertility. Hence, from the FOC for aid, we have:

0 (a E t+1 ) = u 0 (c t ) w e t+1 z R e t+1 0 (a L t ) = u 0 (c t )w t
Substituting for u 0 (c t ), we have:

0 (a E t+1 ) = 0 (a L t ) w e t+1 z w t R e t+1 =) a E t+1 = 0 1 0 (a L t ) w e t+1 z w t R e t+1 Under c E t = c L t = c t , d E t+1 = d L t+1 = d t+1
and n E t = n L t , we obtain, from the comparison of the budget constraints of the two types, that:

a E t+1 = R e t+1 w t w e t+1 z a L t
Collecting those things, we obtain, by transitivity of equality:

0 1 0 (a L t ) w e t+1 z w t R e t+1 = a L t R e t+1 w t w e t+1 z
which is a contradiction. Indeed, taking the transform 0 ( ) on both sides, we obtain:

0 (a L t ) w e t+1 z w t R e t+1 = 0 a L t R e t+1 w t w e t+1 z
which is false except when R e t+1 wt w e t+1 z = 1. Hence this case cannot arise. Case 2:

c E t < c L t . Then d E t+1 < d L t+1 and n E t < n L t .
From the FOC for aid, it seems that two subcases can arise: either a E t+1 > a L t or a E t+1 a L t . But further analysis shows that only the …rst subcase is possible. To show this, rewrite the two budget constraints as follows:

w t + w e t+1 z R e t+1 w t n E t + w e t+1 z( (1 n E t )) R e t+1 w e t+1 z a E t+1 R e t+1 = c E t + d E t+1 R e t+1 w t + w e t+1 z R e t+1 w t n L t + w e t+1 z (1 n L t ) R e t+1 ! w t a L t = c L t + d L t+1 R e t+1
Suppose that c E t < c L t and a E t+1 a L t . Then the RHS of the second condition is higher than the RHS of the …rst condition. On the LHS, the …rst two terms are identical across conditions. Given

w e t+1 z R e t+1 < w t
and n E t < n L t , the term in brackets term is higher, in absolute value, in the second condition, implying that more is subtracted there. But if a E t+1 a L t , the last term of the LHS is also larger in absolute value, implying also that more is subtracted there. Hence, the LHS of the second condition is unambiguously lower than the LHS of the …rst condition. But since the RHS must be, under our assumptions, unambiguously larger in the second condition than in the …rst one, a contradiction is reached. Thus the case a E t+1

a L t cannot arise. Case 3: c E t > c L t . Then d E t+1 > d L t+1 and n E t > n L t .
Hence, from the FOC for aid, we obtain that a E t+1 > a L t . Only that case can arise.

Proposition 1 states that, provided

w e t+1 z R e t+1
< w t , individuals born from young parents provide, at the laissez-faire, more LTC to the dependent parents, in comparison with individuals born from older parents. This result is obtained from the simple fact that early children are older at the time their parents become dependent, and, hence, face a lower opportunity cost of helping them, in comparison to late children. Hence they provide a larger amount of informal LTC.

That result can be related to the existing empirical literature on LTC informal provision. Clearly, if we compare the aid provided by two children of the same parent, i.e. a E t and a L t , we obtain that, under wtz Rt < w t or z Rt < 1, the elderly dependent parent will receive more help from his older children. That theoretical prediction is consistent with the empirical literature on informal LTC provision, which shows that the amount of informal care provided by older children tends to exceed the amount of care provided by younger children when younger children are still working full time whereas older children are not [START_REF] Fontaine | Les con…gurations d'aide familiale mobilisées autour des personnes âgées dépendantes en Europe[END_REF].

Note, however, that the size of the inequality in informal help depends on the level of the interest rate, and, more importantly, on the retirement age. A lower retirement age (a lower z) tends to raise inequalities in informal LTC provided between early and late children. Indeed, in that case, the opportunity cost of providing LTC to the dependent parent is much lower for early children in comparison with late children. On the contrary, if z is close to 1, the inequality becomes less sizeable.

Finally, it should be stressed that the di¤erences in the behavior of the two types of agents concern not only the LTC provision, but, also the fertility timing. As stated in Proposition 1, early children tend to have a larger proportion of children born early in life, whereas late children tend to have a larger proportion of children born later on in life. Those results come merely from the di¤erent time constraints faced by the two types of agents. Type-E children must provide LTC to their elderly parents when they are mature, and this leaves them plenty of time to have more children when they are younger. On the contrary, type-L children must provide LTC to their parents when they are young adults, and this encourages them to postpone births later on in their life.

Intertemporal equilibrium

Up to now, we focused on the temporary equilibrium of the economy. Note that whether early children tend to provide a larger informal help than late children depends signi…cantly on the relative levels of wages and interest rates. However, those factor prices are likely to evolve over time. Hence it makes sense to study the intertemporal equilibrium of the economy, to see how the relative levels of factor prices will evolve over time. In this section, we will characterize the intertemporal equilibrium with perfect foresight.

The intertemporal equilibrium with perfect foresight is a sequence of temporary equilibria for a given initial capital stock K 0 , sequence satisfying the following capital accumulation equation:

K t+1 = q t s E t + (1 q t )s L t ( 21
)
as well as the conditions:

w e t+1 = w t+1 = F L (K t+1 ; L t+1 ) (22) R e t+1 = R t+1 = F K (K t+1 ; L t+1 ) (23)
a Ee t+2 = a E t+2 (24)

a Le t+2 = a L t+2 (25)
At intertemporal equilibrium, the labor market clears:

L t = q t 1 n E t + (1 q t ) 1 n L t a L t +q t 1 z(1 a E t (1 n E t 1 )) + (1 q t 1 )z 1 (1 n L t 1 ) (26)
as well as the market for goods:

F (K; L t ) = q t c E t + (1 q t )c L t + q t 1 d E t + (1 q t 1 )d L t + K t+1 (27)
At the intertemporal equilibrium, the FOCs relative to individuals'utility maximization problems are also satis…ed:

u 0 (c E t ) = R t+1 u 0 (d E t+1 ) v 0 n E t v 0 1 n E t + ' 0 (b E t+2 ) a E t+2 a L t+2 = u 0 (c E t ) w t w t+1 z R t+1 0 (a E t+1 ) = u 0 (c E t ) w t+1 z R t+1 u 0 (c L t ) = R t+1 u 0 (d L t+1 ) v 0 n L t v 0 1 n L t + ' 0 (b L t+2 ) a E t+2 a L t+2 = u 0 (c L t ) w t w t+1 z R t+1 0 (a L t ) = u 0 (c L t )w t
Similarly, …rms maximize their pro…ts, so that the following conditions hold:

w t = F L (K t ; ) R t = F K (K t ; )
Finally, the proportion of individuals of type E in each cohort follows the dynamic law:

q t = q t 1 n E t 1 + (1 q t 1 )n L t 1 q t 1 n E t 1 + (1 q t 1 )n L t 1 + q t 2 (1 n E t 2 ) + (1 q t 2 )(1 n L t 2 ) (28)

Stationary equilibrium

Let us now consider a stationary equilibrium. At the stationary equilibrium, the economy perfectly reproduces itself over time. The capital stock is constant, the composition of the population in the two types is constant, consumptions are constant over time, and informal care to the dependent parents is also constant over time, meaning that what a child of a given type gives to his dependent parent is exactly equal to the aid that person will receive from a child of the same type as himself at the old age. At steady-state, the following conditions are satis…ed (time indexes are dropped):

K = qs E + (1 q)s L L = q 1 n E + (1 q) 1 n L a L +qz(1 a E (1 n E )) + (1 q)z 1 (1 n L )
That condition can be rewritten as:

L = q n L ( z ) n E ( z ) + a L za E + 1 n L a L + z z + zn L
The resource constraint of the economy is thus:

F (K; L) = q c E + d E + (1 q) c L + d L + K
Factors are paid at their marginal productivity:

w = F L K; q n L ( z ) n E ( z ) + a L za E + 1 n L a L + z z + zn L R = F K K; q n L ( z ) n E ( z ) + a L za E + 1 n L a L + z z + zn L
Both types of agents maximize their lifetime welfare subject to their intertemporal budget constraints:

u 0 (c E ) = Ru 0 (d E ) v 0 n E v 0 1 n E + ' 0 (n E a E + (1 n E )a L ) a E a L = u 0 (c E ) h w wz R i 0 (a E ) = u 0 (c E ) wz R u 0 (c L ) = Ru 0 (d L ) v 0 n L v 0 1 n L + ' 0 (n L a E + (1 n L )a L ) a E a L = u 0 (c L ) h w wz R i 0 (a L ) = u 0 (c L )w
Regarding the long-run composition of the population, imposing q t = q t 1 = q t 2 in the condition:

q t = q t 1 n E + (1 q t 1 )n L (q t 1 n E + (1 q t 1 )n L ) + q t 2 (1 n E ) + (1 q t 2 )(1 n L ) yields: q = n L 1 n E + n L
The following proposition summarizes key features of the stationary equilibrium in our economy.

Proposition 2 The stationary equilibrium is a vector c E ; d E ; n E ; a E ; c L ; d L ; n L ; a L ; K; q satisfying the conditions:

u 0 (c E ) = Ru 0 (d E ) v 0 n E v 0 1 n E + ' 0 (n E a E + (1 n E )a L ) a E a L = u 0 (c E )w h 1 z R i 0 (a E ) = u 0 (c E ) wz R u 0 (c L ) = Ru 0 (d L ) v 0 n L v 0 1 n L + ' 0 (n L a E + (1 n L )a L ) a E a L = u 0 (c L )w h 1 z R i 0 (a L ) = u 0 (c L )w K = q w 1 n E c E +(1 q) w 1 n L a L c L q = n L 1 n E + n L
At the stationary equilibrium, and assuming z R < 1, individuals of type E provide a larger amount of LTC to their elderly parents, in comparison with type-L individuals. They also consume more and have a larger proportion of early births:

a E > a L c E > c L d E > d L n E > n L
Proof. The …rst part of the proposition is obtained from the FOCs of the problems faced by individuals of types E and L. Regarding the second part of the proposition, this can be obtained by the same kind of argument as at the temporary equilibrium. Consider the following three cases: Case 1: c E = c L . Then d E = d L and n E = n L . Hence, from the FOC for aid, we obtain that a E > a L . Hence, from the FOC for aid, we have:

0 (a E ) = u 0 (c) wz R 0 (a L t ) = u 0 (c t )w t
Substituting for u 0 (c t ), we have:

0 (a E ) = 0 (a L ) wz wR =) a E = 0 1 0 (a L ) wz wR Under c E = c L = c, d E = d L = d
and n E = n L , we obtain, from the comparison of the budget constraints of the two types, that:

a E = Rw t wz a L
Collecting those things, we obtain, by transitivity of equality:

0 1 0 (a L ) z R = a L t R z
which is a contradiction. Indeed, taking the transform 0 ( ) on both sides, we obtain:

0 (a L t ) z R = 0 a L t R z
which is false except when R z = 1. Hence this case cannot arise. Case 2: c E < c L . Then d E < d L and n E < n L . From the FOC for aid, it seems that two subcases can arise: either a E > a L or a E a L . But further analysis shows that only the …rst subcase is possible. To show this, rewrite the two budget constraints as follows:

w 1 + z R w n E + z(1 n E )) R wz a E R = c E + d E R w 1 + z R w n L + z(1 n L ) R wa L = c L + d L R
Suppose that c E < c L and a E a L . Then the RHS of the second condition is higher than the RHS of the …rst condition. On the LHS, the …rst two terms are identical across conditions. Given z R < 1 and n E t < n L t , the term in brackets term is higher, in absolute value, in the second condition, implying that more is subtracted there. But if a E a L , the last term of the LHS is also larger in absolute value, implying also that more is substracted there. Hence, the LHS of the second condition is unambiguously lower than the LHS of the …rst condition. But since the RHS must be, under our assumptions, unambiguously larger in the second condition than in the …rst one, a contradiction is reached. Thus the case a E a L cannot arise.

Case 3: c E > c L . Then d E > d L and n E > n L . Hence, from the FOC for aid, we obtain also that a E > a L . Proposition 2 states that the tendency of early children to provide more LTC to their elderly parent in comparison to late children is not a short-run phenomenon. Actually, in the long-run, we …nd, under the general condition z R < 1, the same tendency. Note that, since at the stationary equilibrium the wage rate is constant over time, the relative gap in the opportunity cost of providing LTC to the elderly parents is even larger here that at the temporary equilibrium, where, under capital accumulation, we have w t+1 > w t , which raises the cost of helping the elderly for early children. Thus, the tendency of early children to provide more LTC to their elderly dependent parents is not a temporary tendency, but is even reinforced in the long-run.

Long-run social optimum

Let us now characterize the long-run social optimum. For that purpose, we will consider the standard utilitarian social welfare function. The goal of the social planner is here to select consumption, fertility and informal LTC variables, as well as some stock of capital, to maximize the lifetime welfare prevailing at the stationary equilibrium.

The social planning problem of the utilitarian planner di¤ers from the problems faced by agents in several important dimensions. First, the social planner considers the maximization of social welfare in the long run (i.e. at the stationary equilibrium), and considers thus a time horizon quite di¤erent from the ones faced by individuals. Second, the social planner does not su¤er from myopia, so that is set to 1 in the social planning problem. Besides those classical di¤erences, another, more original di¤erence concerns the fact that the social planner takes into account the impact of fertility choices on the composition of the population in the long-run. In other words, the social planner internalizes compositional externalities related to individual fertility choices.

The problem of the social planner can be written by means of the following Lagrangian:

max c E ;d E ;a E ;;n E c L ;d L ;a L ;n L ;K n L 1 n E +n L u(c E ) + v(n E ) + u(d E ) + v(1 n E ) + ' n E a E + (1 n E )a L + (a E ) + 1 n E 1 n E +n L u(c L ) + v(n L ) + u(d L ) + v(1 n L ) + ' n L a E + (1 n L )a L + (a L ) + " F K; h n L 1 n E +n L n L ( z ) n E ( z ) + a L za E + 1 n L a L + z z + zn L i n L 1 n E +n L c E + d E 1 n E 1 n E +n L c L + d L K #
FOCs for consumptions are: :

u 0 (c E ) = u 0 (c L ) = u 0 (d E ) = u 0 (d L ) = (29)
The social optimum involves thus an equalization of consumption levels across time periods and across types.

The FOC for K is:

F K K; " n L n L ( z ) n E ( z ) + a L za E 1 n E + n L + 1 n L a L + z z + zn L #! = 1
(30) This condition is the standard Golden Rule of capital accumulation [START_REF] Phelps | The golden rule of accumulation: a fable for growthmen[END_REF], in the case where there is no cohort growth and where there is full depreciation of capital.

The FOCs for a E and a L are:

n L 1 n E + n L ' 0 ( )n E + 0 (a E ) + 1 n E 1 n E + n L ' 0 ( ) n L = F L (K; ) zn L 1 n E + n L (31) 1 n E 1 n E + n L ' 0 ( ) (1 n L ) + 0 (a L ) + 1 n E 1 n E + n L ' 0 ( ) n L = F L (K; ) 1 n E 1 n E + n L (32)
Further simpli…cations yield:

' 0 ( )n E + 0 (a E ) + ' 0 ( ) 1 n E = u 0 (c)F L (K; ) z (33) ' 0 ( ) n L + 0 (a L ) + ' 0 ( ) (1 n L ) = u 0 (c)F L (K; ) ( 34
)
The LHS is the marginal social welfare gain from helping the elderly parents, whereas the RHS is the marginal social welfare loss from providing LTC. Obviously, that loss depends on how large the marginal productivity of labour is. Moreover, the marginal social welfare loss from providing LTC is larger for individuals of type L than for individuals of type E.

The FOCs for n E and n L are:

n L (1 n E + n L ) 2 u(c E ) + v(n E ) + u(d E ) + v(1 n E ) + ' n E a E + (1 n E )a L + (a E ) u(c L ) + v(n L ) + u(d L ) + v(1 n L ) + ' n L a E + (1 n L )a L + (a L ) + n L 1 n E + n L v 0 (n E ) v 0 (1 n E ) + ' 0 ( ) a E a L +u 0 (c)F L (K; ) " n L n L ( z ) n E ( z ) + a L za E (1 n E + n L ) 2 + n L ( z ) 1 n E + n L # +u 0 (c) n L (1 n E + n L ) 2 c E + d E n L (1 n E + n L ) 2 c L + d L = 0 (35)
and

1 n E (1 n E + n L ) 2 u(c E ) + v(n E ) + u(d E ) + v(1 n E ) + ' n E a E + (1 n E )a L + (a E ) u(c L ) + v(n L ) + u(d L ) + v(1 n L ) + ' n L a E + (1 n L )a L + (a L ) + 1 n E 1 n E + n L v 0 (n L ) v 0 (1 n L ) + ' 0 ( ) a E a L +u 0 (c)F L (K; ) " (1 n E ) n L ( z ) n E ( z ) + a L za E (1 n E + n L ) 2 + n L ( z ) 1 n E + n L + z # +u 0 (c) " 1 n E (1 n E + n L ) 2 c E + d E (1 n E ) (1 n E + n L ) 2 c L + d L # = 0 (36)
Further simpli…cations yield:

v 0 (n E ) v 0 (1 n E ) + ' 0 ( ) a E a L = u 0 (c)F L (K; ) " a L za E ( z ) (1 n E + n L ) # u(c E ) + v(n E ) + u(d E ) + v(1 n E ) + ' n E a E + (1 n E )a L + (a E ) u(c L ) + v(n L ) + u(d L ) + v(1 n L ) + ' n L a E + (1 n L )a L + (a L ) (1 n E + n L ) (37)
while the FOC for n L becomes:

v 0 (n L ) v 0 (1 n L ) + ' 0 ( ) a E a L = u 0 (c)F L (K; ) " a L za E ( z) (1 n E + n L ) # u(c E ) + v(n E ) + u(d E ) + v(1 n E ) + ' n E a E + (1 n E )a L + (a E ) u(c L ) + v(n L ) + u(d L ) + v(1 n L ) + ' n L a E + (1 n L )a L + (a L ) (1 n E + n L ) (38)
Note that the two conditions have the same RHS. Hence we have:

v 0 (n E ) v 0 (1 n E ) + ' 0 n E a E + (1 n E )a L a E a L = v 0 (n L ) v 0 (1 n L ) + ' 0 n L a E + (1 n L )a L a E a L
Denote the function present in the LHS and RHS by (x) v 0 (x) v 0 (1 x) + ' 0 xa E + (1 x)a L a E a L . We have 0 (x) < 0 on the interval [0; 1] for a E 7 a L . Given that the function (x) is strictly monotone, and that it appears on both sides of the above condition, it must be the case that its argument must take the same value on both sides. From which it follows that, at the social optimum, we have equal fertility across types:

n E = n L = n (39)
Hence, the optimal long-run proportion of early children is:

q = n 1 n + n = n (40)
In the light of this, the FOC for n becomes:

v 0 (n) v 0 (1 n) + ' 0 na E + (1 n)a L a E a L = u 0 (c)F L (K; ) (1 z) a L za E (a E ) (a L ) (41)
The LHS of that condition is similar to the LHS of the condition for fertility at the laissez-faire. However, the RHS di¤ers …rst by the fact that R = 1 at the social optimum, and by the addition of two terms: u 0 (c)F L (K; ) a L za E and (a E ) (a L ) . Therefore, the FOCs for informal LTC become:

0 (a E ) = u 0 (c)F L (K; ) z ' 0 na E + (1 n)a L (42) 0 (a L ) = u 0 (c)F L (K; ) ' 0 na E + (1 n)a L (43)
The last term of RHS is the same. First term identical up to z. Hence 0 (a E ) < 0 (a L ), implying a E > a L . Hence, at the long-run social optimum, early children should provide more help to their elderly parents than late children. The following proposition summarizes our results.

Proposition 3 The long-run social optimum is a vector c E ; c L ; d E ; d L ; a E ; a L ; n E ; n L ; K ; q such that:

c E = c L = d E = d L = c n E = n L = n v 0 (n ) v 0 (1 n ) + ' 0 ( ) a E a L = u 0 (c )F L (K ; ) (1 z) a L za E (a E ) (a L ) F K (K ; ) = 1 0 (a E ) = u 0 (c )F L (K ; ) z ' 0 ( ) 0 (a L ) = u 0 (c )F L (K ; ) ' 0 ( ) 0 (a L ) 0 (a E ) = u 0 (c )F L (K ; ) (1 z) > 0 =) a E > a L q = n Proof. See above.
At the utilitarian social optimum, early and late children should have the same life in terms of consumption patterns and fertility patterns. The unique di¤erence is that early children should provide a larger LTC than late children. The intuition goes as follows: from the perspective of social welfare, it is better to ask early children to provide more LTC, since the social opportunity cost of helping their elderly parent is lower for them, as long as z < 1. Note that the gap between a E and a L depends on how large the retirement age is. When z is low, the aid gap is large, whereas, when z is close to 1 (i.e. late retirement), the aid gap becomes tiny.

Having characterized the social optimum, let us now compare it with the stationary equilibrium achieved at the laissez-faire. Proposition 3 summarizes our …ndings.

Proposition 4 Comparing the laissez-faire (i) with the social optimum (i ), we obtain: K i < K i when R > 1 prevails at the laissez-faire, whereas K i > K i when R < 1 prevails at the laissez-faire.

c i = d i , whereas c i < d i (under R > 1) or c i > d i (under R < 1).
Under R > 1 at the laissez-faire, we have:

a E > a E a L > a L Provided u 0 (c i )F L (K; ) 1 z R ' 0 n i a E + (1 n i )a L a E a L ? u 0 (c )F L (K ; ) (1 z) a L za E (a E ) (a L ) ' 0 n a E + (1 n )a L a E a L
we have:

n E ? n E n L ? n L
Proof. The result regarding the capital stock is derived from the Golden Rule condition, in comparison with R ? 1 at the laissez-faire. The di¤erence concerning the shape of consumption pro…les follows from comparing the FOCs at the social optimum and at the laissez-faire, under R ? 1 at the laissez-faire.

Regarding the amount of informal LTC provided to the dependent parent, the FOCs are, at the laissez faire:

0 (a E ) = u 0 (c E )F L (K; ) z F K (K; ) 0 (a L ) = u 0 (c L )F L (K; )
whereas these are, at the optimum:

0 (a E ) = u 0 (c )F L (K ; ) z ' 0 n a E + (1 n )a L 0 (a L ) = u 0 (c )F L (K ; ) ' 0 n a E + (1 n )a L
Provided the conditions:

u 0 (c E )F L (K; ) z F K (K; ) > u 0 (c )F L (K ; ) z ' 0 n a E + (1 n )a L u 0 (c L )F L (K; ) > u 0 (c )F L (K ; ) ' 0 n a E + (1 n )a L
hold, we obtain a E > a E and a L > a L . If we are in underaccumulation at the laissez-faire (R > 1), we obtain that the LHS exceeds the …rst term of the RHS (since c > c i under R > 1). Given that the second term of the RHS is negative, it is clear that, under underaccumulation, the two strict inequalities are satis…ed, yielding: a E > a E and a L > a L .

Let us now consider fertility. At the laissez-faire, the FOC for n i was:

v 0 n i v 0 1 n i = u 0 (c i )F L (K; ) h 1 z R i ' 0 n i a E + (1 n i )a L a E a L
whereas the FOC for the socially optimal n i is:

v 0 (n ) v 0 (1 n ) = u 0 (c )F L (K ; ) (1 z) a L za E (a E ) (a L ) ' 0 n a E + (1 n )a L a E a L
The LHS is the same. But the RHS di¤ers. Hence, provided

u 0 (c i )F L (K; ) 1 z R ' 0 n i a E + (1 n i )a L a E a L ? u 0 (c )F L (K ; ) (1 z) a L za E (a E ) (a L ) ' 0 n a E + (1 n )a L a E a L
we have: n E ? n E and n L ? n L . The social optimum exhibits several important di¤erences in comparison with the laissez-faire. First, it exhibits a di¤erent level of capital, as well as ‡at consumption pro…les, unlike at the laissez-faire. Secondly, and more importantly, the long-run social optimum exhibits a larger proportion of early births in comparison with the laissez-faire, as well as a lower proportion of late births. The underlying intuition goes as follows. The marginal social welfare loss associated to the informal provision of LTC is larger among late children than among early children. But at the laissez-faire, parents do not take this into account when deciding the timing of births. This explains why they tend to make too many late births in comparison with what is socially optimal. Thirdly, another important di¤erence between the laissez-faire and the social optimum concerns the amount of LTC provision. Individuals of both types tend to provide too little LTC to their elderly parents, in comparison with the social optimum. The intuition is here that young individuals do not, when choosing the amount of help, take into account the welfare gains this generates among the dependent elderly. On the contrary, the social planner takes those gains into account, and this leads to a larger amount of informal aid.

Decentralization and optimal policy

Let us now examine how the long-run social optimum can be decentralized.

For that purpose, we design policy instruments in such a way that individual decisions at the laissez-faire would, under those policy instrument, lead to the social optimum. The following proposition summarizes our results.

Proposition 5 The long-run social optimum can be decentralized by means of the following instruments:

A system of intergenerational transfers allowing the capital stock to reach its optimal level K .

Intra-generational lump-sum transfers allowing the equalization of consumption levels across types E and L.

Finally, regarding the aid to the parents, the FOCs would become, under a subsidy i :

0 (a E ) = u 0 (c E ) F L (K; ) z F K (K; ) u 0 (c E ) E 0 (a L ) = u 0 (c L )F L (K; ) u 0 (c L ) L
Given that the optimal a E and a L satisfy:

0 (a E ) = u 0 (c )F L (K ; ) z ' 0 ( ) 0 (a L ) = u 0 (c )F L (K ; ) ' 0 ( )
It is straightforward to deduce, given F K (K; ) = 1:

E = L = ' 0 ( ) u 0 (c )
Proposition 4 states that the decentralization of the long-run social optimum requires to subsidize early births, in such a way as to induce parents to advance births in a way compatible with long-run social welfare maximization. The decentralization of the long-run social optimum also requires some subsidies on the help provided to the elderly parents. The intuition goes as follows. Individuals help their parents at the laissez-faire, but their help is purely driven by their own interests. However, the dependent parents bene…t a lot from the help provided by their children. The social planner takes into account those welfare gains, and subsidize children's help to an extent that depends on those welfare gains. Note that the health status of the dependent may strongly a¤ect the level of the derivative ' 0 ( ). If the health status is so bad that children's help does not make them better o¤, i.e. ' 0 ( ) 0, then the subsidy should be very low. Thus the public subsidy fully internalizes the welfare externalities induced by children's provision of LTC.

The second-best problem

As shown above, the decentralization of the …rst-best social optimum requires various kinds of policy instruments, some of these being hardly available in real life. In this section, we propose to explore the decentralization of the secondbest social optimum, that is, under a limited set of available policy instruments. For that purpose, we will consider an economy at the stationary equilibrium, at which q = n L

1 n E +n L and 1 q = 1 n E 1 n E +n L . Moreover, given that our interest lies mainly on the design of family policy, we will consider here, for the sake of simplicity, that the economy is open, so that wages and interest rates are …xed. 3We will consider three policy instruments: a tax on labour earnings , a demogrant T and a subsidy on early children . The last instrument can be used by the government to reduce the cost of early children. For the sake of analytical convenience, the cost of children is here de…ned in terms of goods (and not in terms of time, as in the previous sections).

Laissez faire

Under those policy instruments, parents of types E and L choose savings s i , informal LTC a i and fertility n i and 1 n i conditionally on policy instruments , T and . In what follows, we assume complete myopia of agents, namely = 0. This makes the analysis simpler, but it does not change the qualitative nature of results compared to a case of partial myopia (0 < < 1).

Type E's problem is:

max s E ;n E ;a E u w(1 ) s E n E (1 ) + T +u zw(1 )(1 a E ) + s E (1 n E ) +v(n E ) + v(1 n E ) + (a E )
FOCs are:

u 0 (c E ) = u 0 (d E ) (44) u 0 (c E ) (1 ) = v 0 (n E ) v 0 (1 n E ) (45) u 0 (d E )zw(1 ) = 0 (a E ) (46)
Consumption is smoothed along the life cycle. The subsidy on early births tends to encourage earlier births. The last FOC concern the informal LTC provided to the elderly. Type L's problem is:

max s L ;n L ;a L u w(1 )(1 a L ) s L n L (1 ) + T +u zw(1 ) + s L (1 n L ) +v(n L ) + v(1 n L ) + (a L )
FOCs are:

u 0 (c L ) = u 0 (d L ) (47) u 0 (c L ) (1 ) = v 0 (n L ) v 0 (1 n L ) (48) u 0 (c L )w(1 ) = 0 (a L ) (49)
Those conditions are similar to the ones of agents of type E, except regarding the marginal cost of providing informal LTC to the dependent parents, which is here larger (given z < 1).

Second best solution

Let us now consider the second-best social planning problem. Under the presence of instruments , and T , the social planner chooses the optimal levels of savings, fertility and long term care for both types of agents.

The demand functions obtained from the laissez-faire are, for i 2 fE; Lg:

s i = s i ( ; ; T ) n i = n i ( ; ; T ) a i = a i ( ; ; T )
The second-best planning problem can be written as the following Lagrangian:

$ = q[u w(1 ) s E n E (1 ) + T + u wz(1 )(1 a E ) + s E (1 n E ) +v(n E ) + v(1 n E ) + (a E ) + '( bE )] +(1 q)[u w(1 )(1 a L ) s L n L (1 ) + T + u wz(1 ) + s L (1 n L ) +v(n L ) + v(1 n L ) + (a L ) + '( bL )] + q w + (1 a E )zw + (1 q)(w(1 a L ) + zw qn E + (1 q)(n L ) T where bi = a E n i + a L (1 n) i and q = n L
1+n L n E . Using the envelope theorem and the laissez-faire FOCs, we obtain:

@$ @s i = 0 (50) @$ @a E = q' 0 ( bE )n E + (1 q) ' 0 ( bL )n L qzw (51) @$ @a L = (1 q) ' 0 ( bL ) 1 n L + q' 0 ( bE )(1 n E ) (1 q)w (52) @$ @n E = U E U L + w a L za E @q @n E + q' 0 ( bE )(a E a L ) q + (n E n L ) @q @n E (53) @$ @n L = U E U L + w a L + za E @q @n L + (1 q) ' 0 ( bL )(a E a L )
1 q + (n E n L ) @q @n L (54)

Earning tax

Let us introduce the following notations (using the expectation operator E):

y = w (q + (1 q) (1 a L )) + wz q(1 a E ) + (1 q) Eu 0 = q u 0 (c E ) + u 0 (d E ) + (1 q) u 0 (c L ) + u 0 (d L ) Eu 0 y = q u 0 (c E )w + u 0 (d E )zw(1 a E ) + (1 q) u 0 (c L )w(1 a L ) + u 0 (d L )zw
In order to derive the formula for the optimal tax rate , we consider the derivatives of the compensated Lagrangian with respect to that …scal instrument. That derivative amounts to the e¤ect of a marginal change of on the Lagrangian when the change in is compensated by a variation in T , in such a way as to maintain the government's budget balanced.

Assuming that cross derivatives in compensated terms are negligible (i.e.

@ ñE @ = @ ñL @ = @ã E @ = @ã L @ = 0), that derivative can be written as:

@ $ @ = @$ @ + @$ @T y = cov(u 0 ; y) + A w qz @ã E @ + (1 q) @ã L @ (55)
where

A @ã E @ h q' 0 ( bE )n E + (1 q) ' 0 ( bL )n L i + @ã L @ h q' 0 ( bE )(1 n E ) + (1 q) ' 0 ( bL )(1 n L )
i and @ã i @ @a i @ + @a i @T @T @ = @a i @ + @a i @T y

is the e¤ect of a change of on the amount of LTC provided by type-i agents to their elderly parents when that change in is compensated by a variation in the demogrant T in such a way as to maintain the government's budget balanced. Equalizing @ $ @ to 0 and isolating yields the following tax formula:

= cov(u 0 ; y) + A w h qz @ã E @ + (1 q) @ã L @ i (56)
To interpret that earning tax formula, we can distinguish between three terms. The …rst term of the numerator is an equity term. Given that individuals with higher incomes bene…t from a lower marginal utility of consumption, we have cov(u 0 ; y) < 0, so that this …rst term is positive, and pushes towards a higher earning tax. The second term of the numerator, A, captures the incidence of the earning tax on the provision of informal LTC by children. By taxing labor earnings, the opportunity cost of providing informal LTC to elderly parents is reduced, which is likely to raise the amount of LTC provided. That term pushes towards more taxation of labor. Finally, the denominator of the tax formula is a standard e¢ ciency term, which captures the incidence of the tax on the tax base.

Family allowances

Let us now derive the formula for the subsidy on early children . For that purpose, we will use the following notations:

n E = qn E + (1 q)n L Eu 0 E = qu 0 (c E ) + (1 q)u 0 (c L ) Eu 0 E n E = qn E u 0 (c E ) + (1 q)n L u 0 (c L )
In order to derive the formula for the optimal child allowance , we consider, as above, the derivatives of the compensated Lagrangian with respect to that subsidy. That derivative amounts to the e¤ect of a marginal change of on the Lagrangian when the change in is compensated by a variation in T , in such a way as to maintain the government's budget balanced.

Assuming that cross derivatives in compensated terms are negligible (i.e.

@ ñE @ = @ ñL @ = @ã E @ = @ã L @ = 0), and using the FOCs of the laissez-faire, that derivative can be written as:

@ $ @ = @$ @ + @$ @T n E = Eu 0 E n E n E Eu 0 E + B + C D (57)
where

B U E U L + (wa L zwa E ) @q @n E @ ñE @ + @q @n L @ ñL @ C (a E a L ) q' 0 ( bE ) @ ñE @ + (1 q) ' 0 ( bL ) @ ñL @ D @ ñE @ q + n E n L @q @n E + @ ñL @ 1 q + n E n L @q @n L
and where:

@ ñE @ @n E @ + @n E @T @T @ = @n E @ + @n E @T n E @ ñL @ @n L @ + @n L @T @T @ = @n L @ + @n L @T n E
Hence the tax formula for the early births subsidy is:

= cov(u 0 E ; n E ) + B + C D (58)
That formula includes four terms. The …rst term of the numerator is the equity term. The intuition behind that term goes as follows. A family allowance on early births is socially desirable if …rst-period consumption and …rst-period births are negatively correlated. Actually, if individuals with lower …rst-period consumption also have more early children, it makes sense, on equity grounds, to subsidize early births, as a way to redistribute resources towards the disadvantaged individuals. 4 The term B at the numerator gives the e¤ect of composition (q) on overall utility and on earning tax revenue. That term re ‡ects that the optimal subsidy on early births depends on its incidence on the long-run composition of the population. If the decentralized population involves a too large proportion of late children from a social perspective, then this justi…es the subsidization of early births, as a way to make the population closer to its optimal structure in the long-run. The term C re ‡ects the incidence of family allowances on the amount of LTC provided by children. In a hypothetical case where the amounts of informal LTC provided by early and late children would be equal, that term would vanish, since a¤ecting the timing of births would then be neutral for the provision of LTC. But as early children face a lower opportunity cost of helping their parents, we have a E a L > 0, implying that this term is likely to be positive. Finally, the term D is the standard e¢ ciency term. It is also expected to be positive. Our results are summarized in the following proposition.

Proposition 6 Assuming a small open economy with R = 1, and assuming that the only available instruments are a tax on earnings , a subsidy on early births and a demogrant T , we obtain that, under negligible cross derivatives, the optimal and the optimal are given by:

= cov(u 0 ; y) + A w h qz @ã E @ + (1 q) @ã L @ i = cov(u 0 E ; n E ) + B + C D where A @ã E @ h q' 0 ( bE )n E + (1 q) ' 0 ( bL )n L i + @ã L @ h q' 0 ( bE )(1 n E ) + (1 q) ' 0 ( bL )(1 n L ) i B U E U L + (wa L zwa E ) @q @n E @ ñE @ + @q @n L @ ñL @ C (a E a L ) q' 0 ( bE ) @ ñE @ + (1 q) ' 0 ( bL ) @ ñL @ D @ ñE @ q + n E n L @q @n E + @ ñL @ 1 q + n E n L @q @n L Proof. See above.
In sum, our analysis of the second-best problem identi…es the various determinants of the optimal subsidy on early births. Besides standard equity and e¢ ciency terms, the level of the optimal subsidy on early births depends also on the incidence of that allowance on the composition of the population, as well as on its incidence on the amount of LTC provided to the dependent parents.

Conclusions

The rise of LTC and the postponement of births are two key demographic trends of the last 40 years. This paper proposed to analyze, within a dynamic OLG model with endogenous birth timing and old-age dependency, the formal relation between those two phenomena.

Our results suggest that the timing of birth de…nitely matters for informal LTC provision. Because of their proximity with retirement, early children face a lower opportunity cost of providing LTC to their elderly dependent parents, in comparison with the opportunity cost faced by late children. Thus, even if all children have exactly the same preferences, and the same interest for the health and the well-being of their parent, they will provide di¤erent amounts of informal LTC, simply because they are not at the same moment in their life, and, hence, do not face the same time constraints.

When studying the stationary equilibrium of the economy, it appeared that the gap in terms of opportunity costs of providing informal LTC to the elderly does not vanish over time, but, rather, is a persistent phenomenon. Thus, the development process will not erase that gap: early children still face, in the long-run, lower opportunity costs, and, hence, will still provide more help to their dependent parents.

On the normative side, we showed that, at the utilitarian optimum, it is also the case that early children should provide more LTC than late children. More importantly, the long-run social optimum involves a di¤erent birth timing than the laissez-faire: it is socially desirable, for the sake of long-run social welfare maximization, to increase the proportion of early births, and to reduce the proportion of late births, in comparison with the laissez-faire. Thus, in the context of LTC provision, decentralized fertility choices do not necessarily coincide with what maximizes social interests in the long-run.

From a policy perspective, this study recommends a di¤erentiated …scal treatment of births, depending on the age of parents. Early births should, to some extent, be encouraged, since these allow the society to bene…t from cheap informal LTC provision from children of dependents who are already retired or close to retirement. Thus the optimal …rst-best subsidy on early births is likely to be positive, and its size depends on the extent to which the decentralized composition of the population in the long-run di¤ers from the socially optimal one, and on how much the decentralized provision of LTC di¤ers from the …rst-best one. We also showed that, once we restrict the set of available …scal instruments to a uniform subsidy on early births, a linear earning tax and a demogrant, the optimal subsidy on early births still depends on its incidence in terms of long run population composition and in terms of LTC provision, but re ‡ects also the trade-o¤ between e¢ ciency and equity concerns.

Obviously, this paper, by focusing on the link between fertility choices and informal LTC provision, left aside some important determinants of fertility choices, in particular regarding the timing of births. As this is well-known in the literature, fertility decisions and education decisions are strongly related, in the sense that more educated persons have, on average children later on in their life. Hence, the present paper does not have the pretension to cover all dimensions of fertility timing choices. On the contrary, it aimed at exploring the link between fertility timing and informal LTC provision at the old age, and showed that having children earlier can be regarded as some form of preventive behavior or insurance for receiving more informal LTC once dependent at the old age.

Another interesting feature of this paper is to show some mechanisms by which children tend to reproduce the behaviors of their parents. As we showed, early children tend to have more children early, whereas late children tend to have more children later on in their life. Whereas this may look like a main social reproduction process, the reproduction that we obtain is not based on social or family norms. On the contrary, this reproduction is dictated by rationality. Parents, by choosing the age at which they have their children, determine the age gap between them and their descendants, which will constraint these in all their choices (through the time constraints), including fertility choices. Thus our simple model of rational choices can mimic social reproduction behaviors.

Those few observations su¢ ce to underline the various ways by which LTC provision and birth timing choices are related to each other. In the light of this paper, it can be concluded that the recent demographic trends observed -the rise of LTC and the postponement of births -raise deep challenges to policymakers, but are also strongly connected, and, as such, invite a consistent family policy aimed at dealing with these both issues in an adequate way. The strong link between those two demographic trends of the 21st century -rise of LTC and birth postponement -invites a joint design of fertility policies and LTC policies.

Figure 1 :

 1 Figure 1: Number of dependent persons (total / relying on informal care and cash bene…ts), in thousands (source EU 2012)

Figure 2 :

 2 Figure 2: Mean age at birth (source: Human Fertility Database).

Figure 3 :

 3 Figure 3: Type-E and Type-L agents in the OLG economy

On this, seeCremer et al (2012).

On the various causes of birth postponement, see[START_REF] Cigno | A microeconomic analysis of the timing of births[END_REF] Ermisch (1991) and Happel et al (1984).

We also suppose, for the sake of simplicity, that R = 1.

On that equity e¤ect, see[START_REF] Pestieau | Childbearing age, Family Allowances and Social Security[END_REF], who study, in a static model with endogenous birth timing, the design of optimal family allowances when parents di¤er in wage levels.

Subsidies on early births E and L equal to:

Subsidies on informal aid to the elderly parents equal to:

Proof. Once the Golden Rule capital level is reached, we have F K (K; ) = R = 1, which automatically makes individual consumption pro…les ‡at:

The lump-sum transfers also guarantee that all consumption levels are equal across types, i.e. c E = c L and

Regarding the subsidies on early births, remind that the FOC for fertility choices becomes, under such a subsidy:

Given that the goal is to have n i satisfying:

it is easy to see that the decentralization requires:

Given that F K (K; ) = 1 thanks to intergenerational transfers, the formula can be simpli…ed to:

If some subsidies to LTC provision bring a E = a E and a L = a L (see below), this formula can be reduced to:


Brunilde Girardet
email: girardet@recherche.enac.frllapasse@capgemini.com

Laurent Lapasset

Daniel Delahaye
email: delahaye@recherche.enac.fr

Christophe Rabut
email: christophe.rabut@insa-toulouse.fr

Wind-Optimal Path Planning: Application to Aircraft Trajectories

Keywords: Obstacle avoidance, path planning, wind, aircraft trajectory

In this paper, an algorithm to plan a continuous wind-optimal path is proposed, and simulations are made for aircraft trajectories. We consider a mobile which can move in a two dimensional space. The mobile is controlled only by the heading direction, the speed of the mobile is assumed to be constant. The objective is to plan the optimal path avoiding obstacles and taking into account wind currents. The algorithm is based on Ordered Upwind Method which gives an optimality proof for the solution. The algorithm is then extended to spherical coordinates in order to be able to handle long paths.

I. INTRODUCTION

Planning optimal trajectories is a rich and dynamic research domain with many application areas like robotics, space or aviation. Depending on the problems needs, the issues are different in nature and so are the techniques used to solve them. Here, we are interested in finding the global optimal path in presence of currents in a two dimensional space. Several methods, as Dijkstra algorithm [START_REF] Dijkstra | A note on two problems in connexion with graphs[END_REF] or A* algorithm [START_REF] Hart | A Formal Basis for the Heuristic Determination of Minimum Cost Paths[END_REF], discretize the domain and work on the generated network to find the optimal path. These algorithms are very efficient but the computed solution is restricted to the network. Thus, the quality of the solution depends on it. With the proposed algorithm, the solution is not constrained to follow a network and is computed on the continuous space. The algorithm is based on front propagation methods as Level Set methods, Fast Marching methods and Ordered Upwind methods. These different algorithms are developed by Sethian in [START_REF] Sethian | Level Set Methods and Fast Marching Methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science[END_REF].

In [START_REF] Pêtrès | Path Planning for Autonomous Underwater Vehicles[END_REF], Pêtrès adapts the Fast Marching Method to path planning for Autonomous Underwater Vehicles taking into account underwater currents. However, his algorithm cannot be applied to vehicles featuring behaviours more complex than a linear reaction to currents. In [START_REF] Alton | Dijkstra-like Ordered Upwind Methods for Solving Static Hamilton-Jacobi Equations[END_REF], Alton uses the Ordered Upwind algorithm with the Semi-Lagrangian method to generate optimal trajectories. The drawback of this approach is the need of a local minimization at each mesh point used to discretize the space, which increases the computation time.

Here, we base our algorithm on the Ordered Upwind algorithm with the Eulerian method.

The main contribution of this paper is, first, to model the optimal path planning in order to use the Eulerian discretization in the Ordered Upwind Method. Usually, the drawback of this discretization is the need to find the roots of a nonlinear equation. In our case, we will show that the resolution is easy since the equation is reduced to a quadratic equation. The second contribution of the paper is to adapt the algorithm to the spherical space to be able to handle long paths around the Earth.

The paper is organized as follows. Section II presents the modelling of the problem. In Section III, we present the algorithm to plan wind-optimal path. Simulations and experiments are performed to validate the proposed approach in Section IV. Finally, conclusions of the work are given in Section V.

II. MODELLING PATH PLANNING PROBLEM

In this section, we present the mathematical formulation of the problem. The goal is to drive the mobile to reach a goal position (x f , y f ) from a start position (x 0 , y 0 ) in a minimum time while avoiding obstacles. The speed of the mobile is constant. The research area is a two dimensional space with the presence of currents. In this paper, we talk about wind currents since we are interested in aircraft trajectories but it can be also valid for other types of currents such as underwater currents.

A. Equation of Motion

Considering wind currents, the motion of the mobile is defined as follows:

ẋ(t) = V a sin(θ (t)) +W x (x, y) ẏ(t) = V a cos(θ (t)) +W y (x, y)
where (x, y) is the mobile position, θ is the heading angle relative to North direction, V a is the constant speed of the mobile, W x (x, y) is the east component of the wind, and W y (x, y) is the north component of the wind.

B. Optimization Problem

The control parameter of the optimal path is the heading angle θ . The optimization problem is then to find the best control θ over time to minimize the total travel time. The optimization problem can be written as:

               min θ (t f -t 0 ) s.t. ẋ(t) = V a sin(θ (t)) +W x ẏ(t) = V a cos(θ (t)) +W y (x(t 0 ), y(t 0 )) = (x 0 , y 0 ) (x(t f ), y(t f )) = (x f , y f )
It is convenient to let the control variable be a(t) = (sin(θ (t)), cos(θ (t))), a unit vector in the direction of motion of the mobile. In this case, the optimization problem becomes:

     min a∈A (t f -t 0 ) s.t. Ẋ = f (X(t), a(t)) X(t 0 ) = X 0 and X(t f ) = X f , (1)
where X is the mobile position, and f (X(t), a(t)) is the real speed of the mobile taking into account wind currents at time t.

III. PATH PLANNING BASED ON ORDERED UPWIND METHOD

The problem (1) is a standard formulation of optimal control problems. One resolution method is to write this problem as an Hamilton-Jacobi equation and to solve it. The form of the Hamilton-Jacobi equation is based on Bellman's optimality principle in discrete case [START_REF] Bellman | Dynamic Programming[END_REF]. The details of the calculations can be found in [START_REF] Bertsekas | Dynamic Programming and Optimal Control[END_REF].

A. The Hamilton-Jacobi Equation

In our problem, the Hamilton-Jacobi equation corresponding to the optimal control problem (1) is given by:

max a∈A {-∇u(X). f (X, a)} = 1 (2)
The resolution of this Partial Differential Equation (PDE) on the whole space gives a necessary and sufficient condition for optimality. The principle of the resolution is to decompose recursively the problem into linked sub-problems as in Dynamic Programming for a discrete problem. The resolution can be seen as a front expansion problem where the wavefront represents the minimum time to reach the arrival point. The wavefront is computed gradually from the previous front positions. It is based on the Huygen's principle which that states every point reached by a wavefront becomes a source of a spherical wavefront; the sum of these secondary waves determines the form of the wavefront at any subsequent time.

The evolution of the wavefront is described by (3):

∇u(X) F(X, ∇u(X) ∇u(X) ) = 1 (3)
where: F(X, n) is the front speed in the direction n, with n the outward unit vector normal to the front at point X, and u(X) represents the minimum time to reach the end point starting from the point X.

With [START_REF] Sethian | Level Set Methods and Fast Marching Methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science[END_REF] and by rewriting (2) as ( 4), we obtain a relation between the speed of the wavefront, F(X, n), and the speed of the mobile, f (X, a).

∇u(X) max a∈A - ∇u(X) ∇u(X) . f (X, a) = 1 (4)
The optimal path problem can then be written as a front expansion problem where the speed of the wavefront, F(X, n), depends on the speed of the mobile, f (X, a).

F(X, n) = max a∈A {-n. f (X, a)} (5)
To design the optimal path between the departure point and the arrival point, we have to follow the characteristics of the Hamilton-Jacobi PDE from the departure point until we reach the arrival point since we know that these characteristics are the optimal trajectories of the corresponding control problem.

B. Ordered Upwind Algorithm

The Ordered Upwind algorithm was developed by Sethian and Valdirminsky to approximate the solution of the Hamilton-Jacobi equations in [START_REF] Sethian | Ordered Upwind Methods for Static Hamilton-Jacobi Equations[END_REF]. In [START_REF]Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Algorithms[END_REF], Sethian proved that the algorithm converges to the viscosity solution of the Hamilton-Jacobi equations, a weak solution of the PDE (3).

The principle of the Ordered Upwind algorithm is to avoid iterations thanks to a careful use of the information about the characteristic directions of the PDE. It uses the fact that the value function, u, is strictly increasing along the characteristics and then verifies the causality property. The value function u(X) is then constructed gradually using the previous smaller values u(X) along the characteristic. This principle makes the algorithm highly efficient. In general, characteristic directions of the PDE are not known in advance. The strength of the Ordered Upwind method is the ability to compute information about the characteristics as the solution is constructed. In our problem, the characteristics of (2) represent the optimal trajectories.

To compute the value function, u, we consider a 2D nonregular triangular mesh. Let (X, X j , X k ) be a simplex, the value of u(X) is computed from u(X j ) and u(X k ) if the characteristic direction at mesh point X lies inside the simplex (X, X j , X k ). Let h be the maximum distance between two adjacent mesh points (i.e. if the mesh points X j and X k are adjacent, then X j -X k h). All mesh points belong to one of the following classes (see Fig. 1):

• Accepted is the set of mesh points where the function u has been computed. Such points are considered frozen. • Considered is the set of mesh points where an estimate v of u has been computed but not frozen. • Far is the set of all other mesh points where an estimate v of u has not been computed yet.

Two other sets are also defined:

• AcceptedFront is the subset of Accepted mesh points that are adjacent to some not-yet-accepted (i.e. Considered) mesh points.

Considered Far AF

Fig. 1. All the mesh points are assigned to three different sets: Accepted, Considered, and Far. AcceptedFront is a subset of the Accepted set. The AF set describes the front.

• AF is the set of line segments [X j , X k ], such that X j and X k are adjacent mesh points on the AcceptedFront, and X j and X k are adjacent to a Considered mesh point X. For each Considered mesh point X, we define a new set called NearFront (NF). It is the subset of AF segments that are close to the Considered mesh point X:

NF(x) = (x j , x k ) ∈ AF|∃x on (x j , x k )s.t. x -x h f 2 f 1 ,
where f 1 and f 2 are respectively the minimum and the maximum speed of the mobile taking wind into account.

Ordered Upwind Algorithm 1) Start with all the mesh points in Far (u = + ∞);

2) Move the initial point X 0 to Accepted (u(X 0 ) = 0); 3) Move all the mesh points X adjacent to the initial point into Considered and evaluate the trial value v(X) as:

v(X) := min

x i ∈NF(x) v x i (X)
4) Find the mesh point X with the smallest value of v among all Considered; 5) Move X to Accepted (u( X) = v( X)), and update the AcceptedFront; 6) Move the Far mesh points X adjacent to X in Considered, and compute their trial values by: v(X) := min

x j x k ∈NF(x) v x j x k (X) (6)
7) Recompute the values for all the other Considered X such that XX i ∈ NF(X) by:

v(X) = min v(X), min xx i ∈NF(x) v x,x i (X)
8) If Considered is not empty, then go to step 4.

Two equivalent methods are presented in [START_REF]Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Algorithms[END_REF] to compute the trial value v(X) from a simplex (X, X j , X k ). The Semi-Lagrangian method requires performing a local minimization at each mesh point, whereas the finite-differences upwind update formula (Eulerian discretization) requires finding the roots of a non-linear equation. In our case, the speed of the wavefront F has a closed form, and the value v x j x k (X) in ( 6) can be computed using a finite-differences upwind formula of the Hamilton-Jacobi equation ( 4). The discretized equation is then a quadratic equation. By modelling the mobile speed as f (X, a) = V a a + W in (5), the speed of the wavefront is equal to:

F(X, n) = V a -n,W (7)
More details about the calculus of the speed of the wavefront are given in [START_REF] Girardet | Generating Optimal Aircraft Trajectories with respect to Weather Conditions[END_REF]. From this definition of the wavefront speed, (3) is discretized using an upwind finite-difference discretization on the simplex (X, X j , X k ) to compute the value v X j X k (X). Equation ( 3) becomes:

P -1 w(X) 2 V 2 a = 1 + P -1 w(X),W 2 (8)
The vector P -1 w(X) is the discretization of ∇u(X) from the directional derivatives of u in the directions defined by the edges of the simplex (x, x j , x k ) (see [START_REF] Girardet | Generating Optimal Aircraft Trajectories with respect to Weather Conditions[END_REF] for the details). We note that ( 8) is a quadratic equation of the form:

Av 2 x j x k (X) + Bv x j x k (X) +C = 0 (9)
with :

A = V 2 a P -1 α, P -1 α -P -1 α,W 2 B = 2V 2 a P -1 α, P -1 β -2 P -1 α,W P -1 β ,W + 1 C = V 2 a P -1 β , P -1 β -P -1 β ,W + 1 2
The value v x j x k is then computed by solving [START_REF]Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Algorithms[END_REF]. To ensure that v x j x k is a good approximation of the value function, u, at the point X, the characteristic direction for the mesh point X needs to lie inside the simplex (X, X j , X k ). It is shown in [START_REF] Girardet | Generating Optimal Aircraft Trajectories with respect to Weather Conditions[END_REF] how to check this property.

The optimal trajectory is built by moving from the initial point to the destination point along the characteristic direction determined by:

dX dt = -V a ∇u(X) ∇u(X) + W (X)
The computational complexity of this algorithm is O(ϒN log N), where N is the number of mesh points and ϒ the anisotropy ratio: the ratio of the fastest to slowest propagation speed for each point (see [START_REF]Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Algorithms[END_REF] for more details).

C. Obstacles Avoidance

We propose to modify the speed of propagation. We slow down the wavefront in parts of the environment that have to be avoided. Thus, the value function u is increased, penalizing the passage through these areas. We create a map of values ξ as a function of the obstacles. The values ξ are between 0 and a maximum value ξ max . The scaled values ξ are then used to slow down the wavefront speed as follows: (1ξ )F(X, n). The maximum value ξ max needs to be less than 1 to keep the wavefront speed strictly positive.

D. Adaptation to Spherical Coordinates

The algorithm presented above works on a Cartesian grid; it therefore relies on the assumption that geodesics are straight lines. Recall that geodesics are shortest paths in terms of distance between two points in a given space. In order to work with long paths for which the wind-optimal trajectories offer the most benefits, we have to take into account the sphericity of Earth where geodesics are then segments of great circles. We approximate the form of Earth by a sphere, and we adapt the algorithm to spherical coordinates. The idea is to project the spherical coordinates on a plane with Cartesian coordinates via a map projection.

The map projection, ψ, used in this work is a transformation of latitudes, φ , and longitudes, θ , of points on the surface of a sphere into points on a plane:

ψ : [-180; 180] × [-90; 90] → R × R (θ , φ ) → (x, y)
However, projecting points on a sphere into a plane is not sufficient to use the Ordered Upwind Method since all map projections cause distortion in the shapes, areas, or distances. Different map projections exist in order to preserve some of these properties. Earth being represented by a sphere, any flat representation generates distortions such that angles and areas cannot both be maintained simultaneously, and distances can never be preserved. This last point explains why the Ordered Upwind Method needs adaptations to be used on a sphere.

The principle of the Ordered Upwind Method is to compute the minimum time to reach a point of the space from the propagation of the front by solving (3). To keep the equation on the map, the projection needs to preserve the relation between vectors. The projection of a vector -→ v on a sphere in spherical coordinates into a vector -→ w on a plane in Cartesian coordinates by an application ψ can be done as follows.

Let γ be a path on the sphere and x be a point on this path: γ :]ε, ε[→ S ⊂ R 3 and γ(0) = x. The speed vector in spherical coordinates -→ v at the point x is defined by γ (0). By the application ψ, the speed vector -→ w on the map projection is equal to:

- → w = (ψ • γ) (0) = ψ (γ(0)) J ψ (x) .γ (0) = J ψ (x). - → v (10)
Thus, the inner product of two projected vectors w 1 and w 2 is equal to:

J ψ (θ , φ ) v 1 , J ψ (θ , φ ) v 2 = J T ψ (θ , φ ) J ψ (θ , φ ) v 1 , v 2
For the Ordered Upwind algorithm, the inner product has to be preserved up to a positive multiplicative constant. To meet this requirement, the projection has to be conformal meaning that the matrix J ψ (θ , φ ) is a positive multiple of a rotation matrix. Equivalently, the matrix J T ψ (θ , φ ) J ψ (θ , φ ) is a positive multiple of the identity. Under this assumption, the distortion induced by the projection is isotropic and can be corrected by scaling. However, note that the scaling factor depends on the point where the correction has to be applied.

In the presented algorithm, we first project the points into the plane, and then adapt the Ordered Upwind Method by correcting the distance distortions at the neighbourhood of each point on the plane. The optimal trajectory will be computed on the plane and then projected back on the sphere by the inverse projection.

1) Lambert Conformal Conic Projection: The map projection, ψ, used here is a Lambert conformal conic projection. It is a conformal map. The general principle of the projection construction is as follows (see [START_REF] Snyder | Map Projections-A Working Manual[END_REF] for more details on different map projections). The idea is to seat a cone over the sphere of Earth and projects conformally onto the cone. The cone is unrolled to give a plane, and the parallel touching the sphere is assigned unitary scale and is called the reference parallel or standard parallel. The transformation is based on two key points: first, as the projection is conform, the angles are preserved; secondly, the distance between points on the standard parallel is preserved on the plane. The scale factor m is then equal to 1 for the latitude of the reference parallel. The equations to project a point on the surface of the sphere into a point on a plane by the Lambert conformal conic projection are presented in [START_REF] Snyder | Map Projections-A Working Manual[END_REF] .

2) Scale Factor: The scale factor m gives the distortion of the projection in the area of interest. It is the ratio of the scale at a given point to the true scale. Scale distortions exist at locations where the scale factor is smaller or larger than 1. Distortions increase as the distance from the standard point (tangent plane) increases. The scale factor is not constant, changing with direction and location. It depends on the properties of the projection, and is defined at a given point by: m = Distance on the map projection Distance on the sphere

The scale factor is a key point in the adaptation of the Ordered Upwind Method since we will used it to compute the value function u(X). As we saw in Section III-B, the time at each point is determined using two neighbouring points. We use the distance between points and the speed of the propagation to compute a time. Thus, if the distance between projected points is not equal to the real distance on the sphere, the computed time in the flat map will not be correct. The distance between projected points has to be corrected using the scale factor. This correction is only valid as long as points are not too far from each other.

As we consider two neighbouring points X j and X k to compute the time at one point X, the distance between these three points is not large. Thus, we assume that on the triangle described by X, X j and X k , a straight line approximates the geodesic. With this reasonable hypothesis, the distance between X, X j and X k can be corrected using the computed scale factor. And based on this critical distance correction on the triangle X, X j and X k , we can use the same steps of the Ordered Upwind algorithm to compute the minimum time at the point X.

As in [START_REF] Bertsekas | Dynamic Programming and Optimal Control[END_REF], the speed of propagation F depends on the aircraft speed, the wind speed and the normal vector of the propagation front, all of these parameters have to be in the same coordinates systems. The normal of the propagation front is computed within the algorithm; this vector is already in Cartesian coordinates and takes distortions into account. Thus, instead of correcting the projected distance, we work with the projected speed of propagation F on the map using the scale factor. As the normal vector of the propagation front is already computed on the map, we only have to modify the aircraft speed and to project the wind speed on the map. The aircraft speed is multiplied by the scale factor to offset the distortions, and the wind speed is projected on the map. Finally, the propagation speed on the map is equal to:

F(X, n) = m V a -n,W P
where W P is the wind vector projected on the map and m is the scale factor at the point X.

3) Algorithm: Let us summarize the different steps of the adaptation of the algorithm.

1) Discretization of the workspace and transformation of the grid points in longitude/latitude coordinates into the plane by a Lambert conformal conic projection; 2) Transformation of winds data into the plane; 3) Application of the Ordered Upwind algorithm on the projected map using scale factor to correct distortions; 4) Computation of the trajectory. This step is detailed below.

5) Inverse transformation of the trajectory points from

Cartesian coordinates into longitude/latitude coordinates. The optimal trajectory is built by moving from the initial point to the destination point along the characteristic direction by solving the following equation:

dX dt = -m V a ∇u(X) ∇u(X) + W P (X)
To compute the gradient ∇u(X), we compute the gradient ∇u(θ , φ ) using finite differences on the regular grid on the sphere. The gradient ∇u(θ , φ ) can be written as a function of the gradient ∇u(x, y) and of the map projection ψ. Equation [START_REF] Snyder | Map Projections-A Working Manual[END_REF] gives the relation between the gradient in spherical coordinates and the gradient in Cartesian coordinates (with J ψ the Jacobian matrix of ψ):

∇u(x, y), - → w = ∇u(x, y), J ψ (θ , φ ) - → v = J T ψ (θ , φ ) ∇u(x, y) ∇u(θ ,φ ) , - → v = ∇u(θ , φ ), - → v (11)
The gradient ∇u(X) used to construct the optimal trajectory is then obtained in (12) from the gradient ∇u(θ , φ ):

∇u(X) = (J T ψ (θ , φ )) -1 ∇u(θ , φ ) (12)
IV. SIMULATIONS AND RESULTS ANALYSIS

A. Trajectory Simulation with Obstacles

Our first test problem aims at emphasizing the impact of wind on trajectories while avoiding obstacles. From the point of view of airlines' operations, these obstacles could model some adverse weather conditions such as storms or turbulences. In Fig. 2, the optimal trajectories without wind (1.) and with wind (2.) have been computed using the Ordered Upwind algorithm with the Eulerian discretization. The obstacles are described by coloured iso-contours, red represents impassable obstacles and the colour goes toward the blue as the importance of the obstacles decreases. In the north part, wind is in the east direction and increases northwards. In the south part, wind is in the west direction and increases southwards.

The trajectory taking into account the wind takes advantage of favourable winds. The two trajectories do not behave in the same way to get around the obstacles. This is attractive to take into account the wind to plan the aircraft trajectory. The wind-optimal trajectory can be far from the optimal trajectory computed without wind.

B. Trajectory Simulation in Spherical Coordinates

1) Traffic Data for Simulations: The data used comes from a simulation of one day of traffic in Europe. The traffic sample contains 28,278 flights with different types of aircraft flying at different speeds. We only select a part of the flights: the cruise phase. We assume that during this phase, the altitude remains constant. In the simulation, we first deal with all the flights. Then, we concentrate on the longest trajectories (¿800km) to study the benefit of wind-optimal routes. An analysis of the traffic data can be found in Table I.

2) Wind Data for Simulations: Three days of wind forecast over Europe are considered: September 30th, October 2nd, and October 4th 2013. An analysis of the wind data for the three days is displayed in Table II. Figure 3b shows the wind forecast on September 30th 2013. Wind vectors (arrows) represent the direction and speed of the wind. Colours also show the different speeds, red colour symbolizes the maximum wind speed and blue colour denotes the minimum wind speed. The optimal route of each flight is computed with the three different wind conditions, and is compared with the direct route.

Table III presents the benefit in time if aircraft follow optimal routes rather than direct routes. Q1, Q2 and Q3 denote respectively the first, second and third quartile of the time benefit. Depending on the day, the average time benefit is different. For half of the flights, the direct route is the best solution; the time benefit is thereby null and the algorithm finds the direct route. In the three cases, it is interesting to note that optimizing the longest trajectories is most beneficial. About three-quarters of these flights get a time benefit.

The direct and optimal routes computed for one particular flight, JWD020, have been depicted in Fig. 3a in which the blue trajectory (aircraft 1) is the direct route and the red trajectory (aircraft 2) is the wind-optimal one. Table IV illustrates the difference between the optimal and the direct routes for the flight JWD020. The optimal route is deviated from the direct route to take advantage of tail winds. The travelled distance is then longer but the flight time is shorter. The obtained benefit is around 6 minutes.

V. CONCLUSION

We presented in this paper a new model for the optimal path planning problem of a single trajectory. We proposed an analytical solution of the discretized equation for the Ordered Upwind algorithm. Thus, this model has the advantage to avoid an iterative algorithm for the resolution of the equation. The results show the benefits of taking into account currents while avoiding obstacles to plan optimal path especially for long paths or for strong currents.

Fig. 2 .

 2 Fig. 2. Optimal trajectory with obstacles: 1. Without wind (green), 2. With wind (red)

(a) Trajectories: 1 .Fig. 3 .

 13 Fig. 3. Simulation results for one flight: JWD020.