Gregory Ponthiere
email: gregory.ponthiere@ens.fr

Long-Term Care, Altruism and Socialization

Keywords: long-term care, altruism, socialization, optimal policy, crowding out e¤ect. JEL codes: I12, I18, J14, Z13

The public provision of long-term care (LTC) can replace familyprovided LTC when adults are not su¢ ciently altruistic towards their elderly parents. But State intervention can also modify the transmission of values and reduce the long-run prevalence of family altruism in the population. That evolutionary e¤ect questions the desirability of the LTC public provision. To characterize the optimal LTC policy, we develop a three-period OLG model where the population is divided into altruistic and non-altruistic agents, and where the transmission of (non) altruism takes place through a socialization process à la Bisin and Verdier (2001). The optimal short-run and long-run LTC policies are shown to di¤er, to an extent varying with the particular socialization mechanism at work.

Introduction

The provision of long-term care (LTC) constitutes a major challenge for advanced economies. As a consequence of the ageing process, an increasingly large number of persons report having some type of functional limitation (e.g. sensory, physical or mental limitations), which prevent them from being autonomous. To give an idea, the European Union (2009) forecasts that the number of elderly dependents in the EU-27 will grow from about 21 millions of people in 2007 to more than 44 millions of people in 2060. Such a substantial rise in the number of elderly dependents will stimulate the aggregate demand for LTC services.

Whatever the demand for LTC services is satis…ed by informal care (provided by family or friends) or by formal care (either at home or in nursing homes), helping the dependent elderly is extremely costly. 1 This explains why the funding of LTC services has become a major research area.

A large number of articles have recently focused on what is called the "LTC insurance puzzle". 2 That puzzle can be stated as follows: although the probabilities to become dependent are large, and although LTC is costly, individuals do not, in general, purchase a private insurance against LTC. 3 Various explanations have been provided, such as excessive loading factors (Cutler 1993; Brown and Finkelstein 2004a), crowding out of private insurance by social insurance (Sloan and Norton 1997; Norton 2000; Brown and Finkelstein 2004b), and trust into family altruism (Hoerger et al 1996; Sloan et al 1997).

A related issue concerns the design of the optimal public intervention in the context of old-age dependency. In a pioneer paper, Jousten et al (2005) studied the optimal LTC policy in an economy where households are composed of one dependent parent and one child, under heterogeneity on the degree of altruism of children towards their parents. They found that, under asymmetric information about the altruism of children, the optimal policy consists of providing low quality of institutional care to the elderly in need, to satisfy the incentive compatibility constraint. More recently, Pestieau and Sato (2008) examined the optimal tax policy in an economy where a fraction of the elderly becomes dependent, and where young adults di¤er in their productivity at work.

Those articles, by paying attention to heterogeneity within families, and to the di¢ culties raised by informational asymmetries, cast an original light on the optimal State intervention in the context of LTC provision. However, those static frameworks cannot account for another key aspect of the problem: the impact of public intervention on the family's private provision of LTC, through its in ‡uence on the transmission of family values.

Suppose that we are in the economy studied by Jousten et al (2005), and that the government commits itself to helping elderly parents in need as a result of non-altruistic children. Is it then the case that altruism, as a value, will, in the long-run, survive in the population? Or will the State's intervention crowd out family altruism, by in ‡uencing the society's transmission of values? What kind of restrictions does the intergenerational transmission of values impose on the optimal LTC provision by governments?

Those questions can only be answered in a dynamic framework, where the composition of the population in terms of altruistic and non-altruistic individuals is a variable, which evolves over time as the output of a socialization process. The goal of the present paper is precisely to examine, within such a framework, the conditions under which the provision of LTC by the State crowds out the provision of LTC by families through its e¤ects on the intergenerational transmission of values.

For that purpose, we develop here a three-period overlapping generations model (OLG) where the adult population is divided into altruistic and nonaltruistic agents, and where the transmission of (non) altruism takes place through a socialization process à la Bisin and Verdier (2001). In that framework, parents, who are certain to be dependent at the old age, can in ‡uence the probability that their child will take some trait thanks to socialization e¤orts. Socialization is costly, but it exhibits also some potential gains at the old age: the parent of an altruistic child will, once dependent, receive the help of his child (whatever he was himself altruistic or not). The incentive to socialize the child depends thus on what the elderly receive when his child is not altruistic. This is where the State's intervention in ‡uences the socialization process, and, hence, the long-run composition of the population.

Anticipating our results, we …rstly study the dynamics of heterogeneity at the laissez-faire, and identify the determinants of the survival or disappearance of family altruism, under alternative assumptions regarding the transmission mechanisms of traits. Then, we introduce lump sum transfers towards the parents of non-altruistic children, and show how this transfer scheme may, under some conditions, a¤ect the long-run composition of the population. The conditions under which the State crowds out the family provision of LTC are iden-ti…ed. In a third stage, we characterize the optimal public intervention, both in the short-run (i.e. for a …xed partition of the population into altruistic and non-altruistic agents), and in the long-run (i.e. for a variable partition). We show that the existence of intergenerational composition e¤ects through socialization tends, under some conditions, to restrict the optimal public provision of LTC to a (positive) level that is lower than the optimal short-run provision.

The rest of the paper is organized as follows. Section 2 presents the model. The long-run population dynamics under the laissez-faire is studied in Section 3. Section 4 examines the impact of State-provided LTC on the population dynamics. The optimal short-run and long-run public interventions are compared in Section 5. Section 6 concludes.

The model

Environment

Let us consider a three-period OLG economy. Each cohort is a continuum of agents whose length is normalized to unity. First period is childhood, during which the child does not work, does not consume, and does not take any decision. Second period is young adulthood, during which agents work, consume, and make one child. All young adult agents earn an income y > 0 during that period. Third period is the old age; elderly persons are retired and dependent. 4 Thus every young adult has an elderly dependent parent. We assume also, like Jousten et al (2005), that there is no private saving for dependency. 5 All children are assumed to be identical. However, there exist two types of adult agents: altruistic agents (i.e. type a) care about their parents, whereas non-altruistic parents (i.e. type n) do not care about their parents. We denote by q t the fraction of altruistic adults within the young adult cohort at time t. The di¤erence between agents of types a and n lies in the fact that type-a agents derive, when being young, some welfare from the resources consumed by their parents, unlike type-n agents. Following Jousten et al (2005), that di¤erence can be modelled by assuming that the utility from consumption for a young adult agent of type i 2 fa; ng takes the form:

u(c i ) + i v(d i ) (1)
where c i denotes the young adult's own consumption when being young, d i is the consumption of his parent, while u( ) and v( ) are temporal utility functions for the young adult and the elderly adult. We assume that those functions satisfy standard properties: u 0 ( ) > 0, u 00 ( ) < 0, v 0 ( ) > 0 and v 00 ( ) < 0. The parameter i accounts for the altruism of children towards their elderly parent. Agents of types a and n di¤er as to the level of i :6

0 = n < a = 1 (2)

Socialization

The population follows an adaptation and imitation process of the type modelled by Bisin and Verdier (2001). The transmission of the parameter i re ‡ecting children's altruism towards their old dependent parents, is modelled as a mechanism where socialization inside the family and socialization outside the family interact. The …rst type of socialization is called the vertical transmission (from parents to children), whereas the second type is called the oblique transmission (from a "role model" in the society to the child).

Families are composed of one parent and one child. Children are born at time t without any cultural trait i 2 fa; ng. Direct vertical socialization to the parent's trait i 2 fa; ng occurs with a probability i t+1 . If the direct vertical socialization does not take place, which happens with a probability 1 i t+1 , the child then picks up the trait of a model chosen randomly in the population of reference, which is the population of young adults. Thus, the child will take the trait a with a probability q t , and the trait n with a probability 1 q t . Hence, if p aa t+1 and p an t+1 (resp. p nn t+1 and p na t+1 ) denote the probabilities that a child born at t in a family with trait a (resp. n) is socialized to, respectively, trait a and trait n (resp. n and a), the transition probabilities are:

p aa t+1 = a t+1 + 1 a t+1 q t p an t+1 = 1 a t+1 (1 q t ) p nn t+1 = n t+1 + 1 n t+1 (1 q t ) p na t+1 = 1 n t+1 (q t ) (3)
By the Law of Large Numbers, p ij t+1 is also equal to the proportion of children whose parents are of type i who have the cultural trait j. Hence, the proportion q t+1 of agents born at time t who become of type a follows the dynamic law:

q t+1 = a t+1 + 1 a t+1 q t q t + 1 n t+1 q t (1 q t ) (4)
The …rst term is the probability to be socialized to trait a when having a family of type a, multiplied by the probability to belong to a family of type a. The second term is the probability to acquire trait a when being born in a family of type n, multiplied by the probability to belong to a family of type n.

Following Bisin and Verdier (2001), we assume that parents of type i 2 fa; ng can socialize their children born at time t vertically, by educating them through a (purely physical) socialization e¤ort e i t (0 e i t 1). The socialization e¤ort e i t is an input in the cultural production of their children as adults: i t+1 = (e i t ): A welfare loss C(e i t ) is generated by a socialization e¤ort e i t . For simplicity, the disutility from socialization e¤orts takes a quadratic form:

C(e i t ) = e i t 2 2 (5)
where accounts for the disutility of socialization e¤orts ( > 0). Parents, when choosing e i t , weight the cost of socialization -C(e i t ) -against its expected gains, which depend on the in ‡uence of their e¤ort e i t on probabilities p ii t+1 and p ij t+1 , determined by the relation i t+1 = (e i t ). We assume that all parents would like their children to be altruistic towards them, in order to bene…t from their children's help. [START_REF] Bisin | The economics of cultural transmission and the dynamics of preferences[END_REF] This egoistic motivation invites a signi…cant re…nement of the standard Bisin-Verdier (2001) set up, which relies on parental imperfect altruism, parents wanting their children to be like them.

Regarding how parental e¤orts a¤ect the probabilities of direct transmissions of traits, there exist various possible transmission technologies. We will …rst consider a transmission technology called "It's the family" by Bisin and Verdier (2001). Under that technology, the probability of direct vertical socialization is independent from the composition of the population. We assume that, for a child in a family of type a, the probability of direct vertical socialization is:

a t+1 = e a t (6)
The probability of direct vertical socialization for a child in a family of type n is equal to 1 minus the parental e¤ort:

n t+1 = 1 e n t ( 7
)
The intuition is that egoistic parents, by their e¤orts, reduce the probability that their children are egoistic like them.

Alternatively, we assume that the probability of direct vertical socialization for a child depends on the parental socialization e¤ort, as well as on the current composition of the population. This consists of what Bisin and Verdier (2001) call the "It takes a village" technology. The probability of direct vertical socialization for a child in a family of type a is equal to the socialization e¤ort of the parent multiplied by the fraction of the population that is altruistic:

a t+1 = e a t q t (8)
Regarding the probability of direct vertical socialization for a child in a family of type n, this is assumed to be equal to 1 minus the socialization e¤ort of the parent, multiplied by the proportion of parents with the altruistic trait:

n t+1 = 1 q t e n t (9)

Preferences and decisions

Type-a's expected lifetime welfare is given by:

u(c a t ) + v(d a t ) (e a t ) 2 2 + p aa t+1 v(d a t+1 ) + p an t+1 v(d n t+1 ) (10)
Hence the problem faced by an altruistic child can be written as8 max

c a t ;d a t ;e a t u(c a t ) + v(d a t ) (e a t ) 2 2 + p aa t+1 v(d a t+1 ) + p an t+1 v(d n t+1 ) s.t. c a t + d a t y
First-order conditions for consumption and aid to the parent yield:

u 0 (c a t ) = v 0 (d a t ) (11)
This condition is independent from the socialization process, that is, from e a t , and from the expected gains from socialization p aa t+1 v(d a t+1 ) + p an t+1 v(d n t+1 ). Regarding the socialization e¤ort, we have, under "It's the family":9

e a t = (1 q t ) v(d a t+1 ) v(d n t+1 ) (12)
Thus the socialization e¤ort is decreasing in the cost parameter and increasing in the gains from having an altruistic child, equal to v(d a t+1 ) v(d n t+1 ). The e¤ort is decreasing in the proportion of the population being altruistic. That property is known as the "cultural substitution property" (see Bisin Verdier 2001).

The problem faced by a non-altruistic child can be written as:

max c n t ;d n t ;e n t u(c n t ) (e n t ) 2 2 + p na t+1 v(d a t+1 ) + p nn t+1 v(d n t+1 ) s.t. c n t + d n t y
First-order conditions for consumption and aid to the parent yield:

c n t = y (13) d n t = 0 (14)
Thus non-altruistic children consume their entire income, and do not give any help to their dependent parent.

Regarding socialization e¤orts, we have, under "It's the family":

e n t = q t v(d a t+1 ) v(d n t+1 ) (15)
e n t di¤ers from e a t , since e n t is increasing in the proportion of altruistic parents in the population q t . The reason is that, by socializing their children, non-altruistic parents reduce the probability of direct vertical socialization to their (egoistic) trait. That strategy makes sense only if the society includes a large number of altruistic role models. 10 Consider now the "It takes a village" transmission. We have:

e a t = q t (1 q t ) v(d a t+1 ) v(d n t+1 ) (16) e n t = q 2 t v(d a t+1 ) v(d n t+1 ) (17)
Given that q t < 1, a parent invests more, ceteris paribus, in socialization under "It's the family" than under "It takes a village", since the return on socialization is always lower under the latter technology. Hence the particular socialization mechanism at work is not neutral as far as the parental socialization decision is concerned. The dependency of the socialization e¤ort is observed both for altruistic and non-altruistic parents.

Having solved for consumptions, aid to parents, and socialization decisions, we can now characterize the temporary equilibrium of our economy at the laissez-faire, under the two distinct transmission technologies considered. 11

Proposition 1 Consider the laissez-faire economy.

Under the "It's the family" technology, we have:

c a < c n = y; d a = y > d n = 0 e a t = (1 q t ) v( y) v(0) ; e n t = q t v( y) v(0)
Under the "It takes a village" technology, we have:

c a < c n = y; d a = y > d n = 0 e a t = (1 q t ) q t v( y) v(0) ; e n t = q 2 t v( y) v(0)
Proof. See the above FOCs.

At the temporary equilibrium under the laissez-faire, there exists a large inequality between the resources enjoyed by elderly individuals, depending on whether their children are altruistic or not. Parents of altruistic children bene…t from a positive amount of resources given to them by their children, whereas parents of non-altruistic children are left with no resource at all, which is most problematic in front of LTC needs. Hence the laissez-faire involves large inequalities at the old age. Similar inequalities -but in the reverse directionoccurs at the young age, where non-altruistic young adults bene…t from higher consumptions than altruistic young adults. As a consequence, the proportion q t of altruistic adults in the population is a major determinant of the aggregate distribution of consumption at the di¤erent ages of life.

Long-run population dynamics

Whereas the previous section characterized the temporary equilibrium of the economy, that is, its equilibrium for a given partition of the population q t , let us now examine the dynamics of population. The proportion of altruistic adults in a cohort follows the dynamic law:

q t+1 = a t+1 + 1 a t+1 q t q t + 1 n t+1 q t (1 q t ) (18)
1 1 Given the constancy of income over time, we can rewrite d a t+1 as a constant fraction > 0 of the income (i.e. d a t+1 = y), and substitute for d n t+1 = 0 in the appropriate FOCs.

Consider …rst the "It's the family" transmission. Substituting for probabilities of vertical socialization and denoting [v( y) v(0)] by , we get:

q t+1 = [e a t + (1 e a t ) q t ] q t + [(1 (1 e n t )) q t ] (1 q t ) = q 2 t + q t (1 q t )
The existence, uniqueness and stability of a stationary equilibrium can be studied by analyzing the properties of the transition function F (q t ) q 2 t + q t (1 q t ) . The results are summarized in the following proposition.

Proposition 2 Consider the long-run dynamics of q t under the "It's the family" transmission technology:

If v( y) v(0) < 1, there exist only two stationary equilibria, q = 0 and q = 1. Only q = 0 is locally stable.

If v( y) v(0) = 1, there exists a continuum [0; 1] of stationary equilibria, which are all unstable.

If v( y) v(0) > 1, there exist only two stationary equilibria, q = 0 and q = 1. Only q = 1 is locally stable.

Proof. See the Appendix.

The dynamics of the population can take three forms. If the welfare gain from having altruistic children is low, any population including initially some non-altruistic individuals will turn out to be fully non-altruistic in the long-run. On the contrary, if the welfare gain from altruistic children is large, the longrun population will only include altruistic individuals. In the intermediate case, where v( y) v(0) = 1, any initial population will reproduce itself in the long-run.

Under the "It takes a village" technology, q t follows the law:

q t+1 = q 2 +
e a t q t q t (1 q t ) + e n t q t q t (1 q t ) = q 2 t + q 3 t (1 q t ) Proposition 3 summarizes the long-run dynamics of the population in that case.

Proposition 3 Consider the long-run dynamics of q t under the "It takes a village" transmission technology:

If v( y) v(0) < 1, there exist only two stationary equilibria, q = 0 and q = 1. Only q = 0 is locally stable.

If v( y) v(0) = 1, there exist only two stationary equilibria, q = 0 and q = 1. Only q = 0 is locally stable.

If v( y) v(0) > 1, there exist only three stationary equilibria, q = 0, q = 2 q v( y) v(0) and q = 1. Equilibria q = 0 and q = 1 are locally stable, whereas q = 2 q v( y) v(0) is unstable.

Proof. See the Appendix.

Under a low welfare gain from having altruistic children, the population dynamics is close to the one under "It's the family": altruism towards dependent parents will disappear in the long-run. However, when there is a large welfare gain from altruistic children, there exist now three -instead of two -stationary equilibria. The intermediate equilibrium is unstable, and acts as a threshold below which altruism will disappear, and beyond which altruism will be universal in the long-run. Such a threshold did not exist in the "It's the family" case, where, under v( y) v(0) > 1, the population would be fully altruistic in the long-run for any q 0 > 0. This result does not remain true under "It takes a village", where the generalization of altruism requires q 0 > 2 q v( y) v(0) . In sum, the proportion of the dependent who are helped by their children is likely to vary over time, in a direction that depends on how large the welfare gain from having altruistic children is in comparison to the welfare cost from socialization. In the long-run, altruism towards the dependent can either disappear or become general, depending on the relative welfare gains from altruistic children. That result is robust to the speci…cation of the transmission technology, even though the "It takes a village" technology involves an unstable intermediate stationary equilibrium that makes initial conditions relevant, unlike under the "It's the family" technology.

An economy with LTC policy

Let us now introduce a government, which taxes young adults in a non-distortionary way, in order to provide a help for the elderly parents who are in need of LTC. The government can observe whether individuals are altruistic or not, and helps the dependent elderly who are not helped by their children (and not the others). The government runs a balanced budget, and taxes only the non-altruistic young adults. Hence, denoting by g the resources spent by the government on each elderly dependent person in need, and by T the lump sum tax on non-altruistic children, we have:

(1 q t )T = (1 q t )g T = g (19)
Following Jousten et al (2005), we assume that the government's productivity in the provision of LTC services is less good than the one of children. Thus the elderly who is helped by the State bene…ts from an aid equal to a fraction (0 < < 1) of the dedicated resources g. Hence, if the government cannot tax more than what would be given by altruistic children, each dependent elderly parents with non-altruistic children receives an aid equal to g, with:

0 < g < T < y (20)

Socialization decisions

As a consequence of public intervention, we now have, under the "It's the family" technology, the following socialization e¤orts:

e a t = (1 q t ) v( y) v( g) and e n t = q t v( y) v( g) (21)
The welfare gain from having an altruistic child is now lower than under the laissez-faire, since the government, by providing g to the dependent in case of non-altruistic children, reduces the loss due to having non-altruistic children.

Consider now the "It takes a village" technology. We have:

e a t = (1 q t ) q t v( y) v( g) and e n t = q 2 t v( y) v( g) (22)
Here again, the socialization e¤orts chosen by parents are now lower, ceteris paribus, in comparison to the laissez-faire. The State aid, by raising the elderly's welfare in case of non-altruistic children, reduces welfare gains from having an altruistic child, as under the "It's the family" technology.

Proposition 4 Consider an economy with a lump sum transfer g > 0 to the elderly parents with non-altruistic children. Whatever we consider the "It's the family" or the "It takes a village" technologies, the optimal socialization e¤ orts for all parents are inferior to what these are at the laissez-faire.

Proof. The proof follows from the monotonicity of v( ).

The extent to which LTC public policy a¤ects parental socialization decisions depends on how large the governmental aid is with respect to the intrafamily aid provided by altruistic children. The impact of governmental aid depends also on the productivity of the government in the production of LTC services. The higher the productivity of the government is (i.e. the higher is), the lower the socialization e¤orts are ceteris paribus, yielding a larger di¤erential in terms of socialization e¤orts in comparison to the laissez-faire.

The impact of governmental aid depends also on the shape of the elderly's temporal utility function v( ). When v( ) is close to be ‡at on the [0; y] interval, the introduction of a governmental aid g 2 ]0; y] has little e¤ect on the welfare gain from altruistic children, and, thus, on socialization decisions. However, if v( ) is quite steeply on the [0; y] interval, the governmental aid reduces the welfare gain from altruistic children signi…cantly in comparison to the laissezfaire, which, in turn, a¤ects the socialization process strongly.

Long-run population dynamics

As shown above, the introduction of a governmental aid to the dependent elderly in need reduces the parental incentive to socialize their children. Given that this tendency a¤ects all parents -whatever these are altruistic or not -it is tempting to believe that the introduction of the governmental aid is neutral regarding long-run population dynamics. However, as we shall now see, that belief is misleading: the governmental intervention is not neutral for population dynamics.

Proposition 5 Consider the long-run dynamics of q t under the "It's the family" transmission technology. When the government provides a help g to the dependent elderly who have non-altruistic children, we have: If v( y) v( g) < 1, there exist only two stationary equilibria, q = 0 and q = 1. Only q = 0 is locally stable.

If v( y) v( g) = 1, there exists a continuum [0; 1] of stationary equilibria, which are all unstable.

If v( y) v( g) > 1, there exist only two stationary equilibria, q = 0 and q = 1. Only q = 1 is locally stable.

Proof. See the Appendix.

The governmental aid to the dependent in need, by reducing the level of v( y) v( g) , tends to make the …rst case more likely, and the third case less likely. Hence a public policy helping the dependent in need tends, in the longrun, to make the disappearance of altruism more likely. There is thus a kind of "crowding-out" of the family by the State. That crowding-out is larger the larger the State resources dedicated to LTC are (i.e. the larger g is), and the more e¤ective the State is in the production of LTC (i.e. the larger is).

That crowding-out e¤ect relies on socialization: as soon as the State provides a sizeable and e¢ cient help to the dependent elderly in need, young parents have little incentive to socialize their children, since making these altruistic is costly and has little welfare e¤ects. Hence, due to the State intervention, family altruism must disappear. Note, however, that although that crowding-out is possible, it is by no way necessary. It remains possible that v( y) v( g) > 1 despite a positive and e¢ cient public aid. In that case, the population will turn out to be fully altruistic in the long-run, despite the State aid. Hence the crowding-out e¤ect is by no way inevitable.

Let us now consider the "It takes a village" technology.

Proposition 6 Consider the long-run dynamics of q t under the "It takes a village" transmission technology. When the government provides a help g to the dependent elderly who have non-altruistic children, we have:

If v( y) v( g) < 1, there exist only two stationary equilibria, q = 0 and q = 1. Only q = 0 is locally stable.

If v( y) v( g) = 1, there exist only two stationary equilibria, q = 0 and q = 1. Only q = 0 is locally stable.

If v( y) v( g) > 1, there exist only three stationary equilibria, q = 0, q = 2 q v( y) v( g) and q = 1. Equilibria q = 0 and q = 1 are locally stable, whereas q = 2 q v( y) v( g) is unstable.

Proof. See the Appendix.

Here again, the introduction of governmental aid makes the long-run disappearance of altruism more likely than under the laissez-faire. But that "crowding out" is even more likely here than under the "It's the family" technology. Indeed, whereas the condition v( y) v( g) > 1 su¢ ces to avoid the disappearance of altruism under the "It's the family" transmission technology, the same is not true under the "It take a village" technology. Indeed, in that case, the level of the unstable intermediate steady-state equilibrium is now larger than at the laissez-faire, which suggests that, even if v( y) v( g) > 1, State aid may, depending on initial conditions, still make altruism disappear in the long-run.

In sum, the public provision of LTC may crowd out family-provided LTC, by making the disappearance of family altruism more likely. This suggests that the State and the family may be substitutes in the provision of LTC services. However, that section shows also that this crowding-out e¤ect varies with the size and e¢ ciency of the public provision of LTC, as well as with the transmission technology and with individual preferences.

The social optimum and optimal LTC policy

Governmental aid to the dependent in need may modify the composition of the population, allowing the disappearance of the -(possibly) otherwise survivingfamily altruism. Can we deduce from this that there should be no governmental policy from a long-run perspective? What would be the best long-run LTC policy? How would it di¤er from the optimal short-run LTC policy?

To address those issues, note that the lifetime welfares of altruistic agents with altruistic and non-altruistic children are given by:

u(c a ) + v(d a ) (e a t ) 2 2 + v(d a ) and u(c a ) + v(d a ) (e a t ) 2 2 + v( g) (23)
whereas the lifetime welfares of non-altruistic agents with altruistic and nonaltruistic children are given by:

u(c n ) (e n t ) 2 2 + v(d a ) and u(c n ) (e n t ) 2 2 + v( g) (24)
Those expressions di¤er by the additional term present in the utility function of the altruistic children, i.e. v(d a ).

As this is well-known, the construction of a social welfare function when the population is heterogeneous in altruism can raise serious di¢ culties, since altruism allows the double-counting of the interests of some agents, whereas the interests of others are counted only once (see Hammond 1987). Two possibilities arise regarding the treatment of altruism. On the one hand, the social planner can ignore altruism, and only consider the private part of individuals'welfare. On the other hand, the planner can take altruism into account, possibly by weighting individual utility functions di¤erently. Throughout this paper, we will choose the former solution, and consider only, in the social welfare function, the private part of agents'welfare, in order to avoid double-counting.

The optimal short-run policy

Let us …rst consider the …rst-best problem in the short-run, that is, for a given partition q of the population into altruistic and non-altruistic individuals. For that purpose, we will take, as a social objective, the maximization of social welfare ex post, that is, social welfare once the type of children has been revealed.

The social planner's problem can be rewritten as the following Lagrangian:

max c a ;d a ;c n ;g;e a ;e n qp aa h u(c a ) (e a ) 2 2 + v(d a ) i + qp an h u(c a ) (e a ) 2 2 + v( g) i +(1 q)p na h u(c n ) (e n ) 2 2 + v(d a ) i + (1 q)p nn h u(c n ) (e n ) 2 2 + v( g) i + [y qc a (1 q)c n qp aa d a (1 q)p na d a qp an g (1 q)p nn g]
where is the Lagrange multiplier associated with the resource constraint. The …rst term is the lifetime welfare of altruistic parents who had altruistic children; the second term is the lifetime welfare of altruistic parents who had non-altruistic children; the third term is the lifetime welfare of non-altruistic parents who had altruistic children, while the fourth term is the lifetime welfare of non-altruistic parents who had non-altruistic children. After simpli…cations, the …rst-order conditions for consumptions yield:

u 0 (c a ) = u 0 (c n ) = v 0 (d a ) = v 0 ( g) =
If we consider u( ) = v( ), we have:

c a = c n = d a > d n = g
Hence all …rst-period consumptions should be equal. Moreover, the consumption at the old age for the dependent elderly should be, if he has altruistic children, the same as what he had when being young. Only the dependent elderly with non-altruistic children should get fewer resources, due to the inef-…ciency of public LTC services. Under u( ) 6 = v( ), we still have an equalization of all consumptions at young adulthood (i.e. c a = c n ), but things are less clear regarding how these di¤er from consumption at the old age.

The FOCs for optimal e¤ort are: 12

e a = p aa0 " u(c a ) (e a t ) 2 2 + v(d a ) # + p an0 " u(c a ) (e a t ) 2 2 + v( g) # e n = p na0 " u(c n ) (e n t ) 2 2 + v(d a ) # + p nn0 " u(c n ) (e n t ) 2 2 + v( g)

#

Hence, after simpli…cation, we obtain:

e a = (1 q) [v(d a ) v( g )] e n = q v(d a ) v( g )
which coincide with the socialization e¤orts chosen in the decentralized economy when d a = d a and g = g .

Alternatively, under the "It takes a village" technology, we have: 13

e a = (1 q)q [v(d a ) v( g)] e n = q 2 [v(d a ) v( g)]
which coincide with the socialization e¤orts chosen in the decentralized economy when d a = d a and g = g .

Regarding the decentralization of the social optimum, note …rst that, at the laissez-faire, we have, under u( ) = v( ), that altruistic children dedicate 1/2 of their income to their elderly parent (i.e. = 1=2), so that:

c a = y 2 = d a c n = y d n = 0
Hence, we have, at the laissez-faire, the equality of c a and d a , in conformity with the social optimum. This is not the case for the consumptions of the nonaltruistic children and their parents. Hence the decentralization of the social 1 2 Remind that:

p aa0 = 1 q and p an0 = (1 q) p na0 = q and p nn0 = q 1 3 Remind that we have: p aa0 = q(1 q) and p an0 = q (1 q) p na0 = q 2 and p nn0 = q 2 optimum requires a lump sum transfer T from non-altruistic children towards their dependent parent, so insure that

c n = y T = y 2 = c a d n = T = g
Given that the decentralized levels of socialization coincide with the optimal ones under optimal consumptions, those transfers su¢ ce to decentralize the short-run social optimum. The following proposition summarizes our results.

Proposition 7 Consider a social planner maximizing the average ex post welfare under a …xed partition of the population q.

Under u( ) = v( ), the social optimum involves:

c a = c n = d a > d n = g
as well as socialization e¤ orts:

e a = (1 q) [v(d a ) v( g )] ; e n = q v(d a ) v( g )
under the "It's the family" technology and

e a = (1 q)q [v(d a ) v( g )] ; e n = q 2 v(d a ) v( g )
under the "It takes a village" technology.

Under u( ) = v( ), the social optimum can be decentralized by means of lump sum transfers g from young non-altruistic agents to the dependent elderly, with g = y 2 .

Proof. The proof follows from the comparison of FOCs of the laissez-faire and the social optimum. Thus, when the composition of the population is taken as …xed, the social optimum can be decentralized by means of lump sum transfers from non-altruistic children to their parents. Let us now compare this with the decentralization of the long-run optimum, where the composition of the population is a variable.

The optimal long-run policy

To examine the long-run social optimum and its decentralization, it is …rst necessary to consider the issue of the optimal composition of the population q. In the light of the analysis of the long-run dynamics, that question can be reduced to the comparison of two situations: one in which the population is entirely composed of altruistic persons (i.e. q = 1), and one where the population is Lemma 1). Hence, the optimum short-run policy may be at odds with the maximization of the long-run social welfare, because of the impact of that policy on the composition of the population through the socialization mechanism.

That tension between the short-run and the long-run social welfare maximization is not automatic, as this depends on various aspects of population dynamics. Several cases should be distinguished. For that purpose, we start by considering the economy with the "It's the family" technology.

Proposition 8 Consider the economy under the "It's the family" transmission technology. Suppose u( ) = v ( ) and 0 < q 0 < 1. The optimal provision of LTC by the government is determined as follows: If u( y 2 ) u( g) < 1 for any 0 g y 2 , the optimal (short-run and long-run) policy is a lump sum transfer g = y 2 from non-altruistic children towards their parents. If u( y 2 ) u( g) > 1 for any 0 g y 2 , the optimal (short-run) policy is a lump sum transfer g = y 2 from non-altruistic children towards their parents, whereas the laissez-faire is optimal in the long-run.

Otherwise, the convergence towards a stationary equilibrium with q = 1 requires that the (short-run) lump sum transfer from non-altruistic children towards their parents g satis…es the condition:

u( y 2 ) u( g ) > 1, implying g 2 0; y 2 .
The laissez-faire is optimal in the long-run.

Proof. See the Appendix.

The …rst two cases are situations where the State aid to the dependent has no e¤ect on the long-run composition of the population. As a consequence, the optimal policy coincides with the optimal short-run policy: a lump sum transfer from the non-altruistic children towards their dependent parents. In the …rst case, that policy will also hold in the long-run, where altruism has disappeared. In the second case, that policy will only be transitional, as the disappearance of non-altruism in the long-run will make that policy irrelevant.

In the third case, the long-run composition of the population is likely to vary with the level of the governmental aid to the dependent. The lump sum transfer g cannot take its optimal short-run level, because it has to be compatible with the inequality u( y Proof. See the Appendix.

In the …rst two cases, the long-run dynamics of the population will not vary with public intervention, and so there is no discrepancy between the optimal short-run policy and what maximizes social welfare at the stationary equilibrium. The same is not true for the third case, where the size of the transfer will a¤ect the dynamics of the population. In comparison to the "It's the family" technology, the lump sum transfer must satisfy here not one, but two conditions: u( y 2 ) u( g ) > 1 and q t > 2 q u( y 2 ) u( g ) . The second constraint, which depends on the current partition of the population, comes from the intermediate stationary equilibrium. This acts like a threshold, below which the population will be non-altruistic in the long-run. That condition restraints the interval of possible lump sum transfers more than under the "It's the family" technology. 1 6 Regarding this, note that, although it is tempting to recommend the highest transfer such that u( y We developed a three-period OLG model where the population is partitioned in two groups: altruistic persons, who care about their parents, and nonaltruistic persons, who do not. That partition is the outcome of a socialization process à la Bisin and Verdier (2001), where parental direct vertical socialization and oblique socialization interact. However, a signi…cant departure from the Bisin Verdier approach is that all parental e¤orts are here oriented towards making children altruistic, whatever parents are altruistic or not.

We showed that a public provision of LTC, by reducing the expected (net) welfare gains from having an altruistic child, lowers the parental socialization e¤orts. That reduction does not necessarily imply the disappearance of family altruism. Whether such a crowding out exists or not depends on (1) individual preferences;

(2) the characteristics of the socialization process (cost and transmission); (3) initial conditions (under the "It takes a village" technology).

We identi…ed conditions under which there is a con ‡ict between the optimal short-run LTC policy (i.e. under a …xed partition of the population) with the optimal long-run LTC policy (i.e. under a varying partition). Such a con ‡ict is not inevitable, but the public intervention, if …xed to its optimal short-run level, may, under general conditions, prevent the economy from reaching the long-run social optimum. It is then optimal to reduce the public aid to the highest level compatible with the long-run social optimum. That "compatible" public provision of LTC is not zero (i.e. unlike at the laissez-faire), but is generally inferior to the level decentralizing the short-run social optimum.

In sum, although the public provision of LTC can a¤ect the transmission of values, and may even make altruism disappear, such a crowding out is by no way automatic. The existence and size of such a crowding out e¤ect depends on individual preferences, and on the particular socialization process that governs the transmission of values. The latter point, although generally unnoticed, is worth being underlined. The optimal public intervention depends signi…cantly on the "response" induced by the socialization process at work. Given the complexity of socialization phenomena, there is little doubt that further attention will have to be paid, in the future, to the interactions between policy and socialization. is necessarily positive. We also have:

F 0 (0) = and F 0 (1) = 2
Hence, if > 1, we have: F 0 (0) > 1 and F 0 (1) < 1, so that the transition function lies above the 45 line in the neighborhood of 0, and above the 45 line in the neighborhood of 1. Taking the second-order derivative, we have:

F 00 (q t ) = 2 2
Given > 1, we have F 00 (q t ) < 0 for all levels of q t , so that F (q t ) is concave and admits no in ‡ection point, and remains above the 45 line for all levels of 0 < q t < 1. Hence, in that case, there is no intermediate stationary equilibrium. Moreover, only q = 1 is locally stable, as jF 0 (0)j > 1 and jF 0 (1)j < 1

Alternatively, if

< 1, we have: F 0 (0) < 1 and F 0 (1) > 1, so that F (q t ) lies below the 45 line in the neighborhood of 0, and below the 45 line in the neighborhood of 1. We have also:

F 00 (q t ) = 2 2
Given < 1, we have F 00 (q t ) > 0 for all levels of q t , so that F (q t ) is convex and admits no in ‡ection point, and remains below the 45 line for all levels of 0 < q t < 1. Hence, in that case, there is no intermediate stationary equilibrium. Moreover, only q = 0 is locally stable, as jF 0 (0)j < 1 and jF 0 (1)j > 1

Finally, if

= 1, we have:

F 0 (q t ) = 2q t + (1 2q t ) 1 = 1 F (q t
) is linear and of slope 1. Hence, given F (0) = 0, this coincides with the 45 line, so that any partition of the population is a stationary partition. There is a continuum of stationary equilibria, all of these being unstable.

Proof of Proposition 3

Consider now the "It takes a village" technology. We have: q t+1 G(q t ) = q 2 t + q 3 t (1 q t )

We have: G(0) = 0, G(1) = 1. Hence both q = 0 and q = 1, being …xed points of G( ), are stationary equilibria. Regarding the existence of an intermediate equilibrium, note that: G 0 (q t ) = 2q t + q 2 t (3 4q t )

We have G 0 (0) = 0 and G 0 (1) = 2

Hence G( ) lies below the 45 line in the neighborhood of 0, and lies above or below the 45 line in the neighborhood of 1, depending on: ? 1 Thus, if < 1, G( ) lies below the 45 line in the neighborhood of 1. On the contrary, if > 1, then G( ) lies above the 45 line in the neighborhood of 1. As a consequence, there must exist at least one intersection of G( ) with the 45 line for a level of q t that is larger than 0 and smaller than one: an intermediate steady-state must then exist.

Note also that the second-order derivative is:

G 00 (q t ) = 2 + 6q t (1 2q t )

Its sign is ambiguous, and may depend on q t . Note that:

G 00 (0) = 2 > 1 and G 00 (1) = 2 6

Thus the transition function is convex around q t = 0, and may turn out to be concave at q t = 1, provided > 1 3 . To study the stability of the stationary equilibria, note that jG 0 (0)j < 1 Hence q = 0 is a locally stable equilibrium. Moreover, we have, when < 1:

jG 0 (1)j > 1 so that q = 1 is not stable in that case. However, when > 1, we have: jG 0 (1)j < 1 so that q = 1 is then locally stable.

Regarding the intermediate stationary equilibrium, note that equalizing q t+1 and q t in the transition function yields:

q t = q 2 t + q 3 t (1 q t )
From which we get a unique value: q t = 2 q . Thus the interior equilibrium is unique. To study its stability, let us notice that:

G 0 2 r ! = 3 2 2 r
Given that < 1, we necessarily have G 02 q > 1, implying instability.

Finally, note that, if = 1, there exist only two stationary equilibria, q = 0 and q = 1. Moreover, we have: G 0 (0) = 0 < 1 and G 0 (1) = 1 Hence only the …rst equilibrium is locally stable.

Proof of Proposition 5

The dynamics of the economy is described by the equation:

q t+1 F (q t ) = [e a t +
(1 e a t ) q t ] q t + [(1 (1 e n t )) q t ] (1 q t ) = q 2 t + q t (1 q t ) v( a y) v( g)

The existence, uniqueness and stability of a stationary equilibrium can still be studied by analyzing the properties of the transition function F (q t ) q 2 t + q t (1 q t ) , where = v( a y) v( g) instead of v( a y) v(0) at the laissezfaire. The existence and stability analyses, which are close to the ones under the laissez-faire, are not reproduced here.

Proof of Proposition 6

The dynamics of the economy is described by the equation: q t+1 G(q t ) = q 2 + e a t q t q t (1 q t ) + e n t q t q t (1 q t ) = q 2 t + q 3 t (1 q t ) v( a y) v( g)

The existence, uniqueness and stability of a stationary equilibrium can thus be studied by analyzing the properties of the transition function G(q t ) q 2 t + q t (1 q t ) , where = v( a y) v( g) instead of v( a y) v(0) at the laissezfaire. Those analyses being close to the ones under the laissez-faire ones are not reproduced here.

Proof of Lemma 1

Let us compare social welfare under two extreme cases, where the population is composed exclusively of altruistic persons (i.e. q = 1) and where the population is composed exclusively of non-altruistic persons (i.e. q = 0). Those two cases coincide with the outcomes that will, under quite general conditions, prevail in the long-run, as shown in Section 4.

When the long-run population is fully altruistic, social welfare is equal to:

u(c a ) (e a )

For instance, the average cost of institutional LTC for old persons in France is about 35,000 euros per year (OECD

2006), whereas the yearly price of a nursing home in the U.S. ranges between 40,000 and 75,000 $ (see[START_REF] Taleyson | L'assurance dépendance privée: comparaisons internationales, Newsletters techniques SCOR[END_REF].2 See Pestieau and Ponthiere (2011) for a survey.3 According to[START_REF] Kemper | Lifetime use of nursing home care[END_REF], a man of age 65 has a 0.33 probability to enter a nursing home in the U.S., whereas that probability exceeds 0.50 for women.

We assume here no risk about the length of life, which is a signi…cant simpli…cation. See[START_REF] Ponthiere | Unequal longevities and lifestyles transmission[END_REF] on the interactions of life expectancy with the socialization process.

Although simplifying the picture, that assumption accounts for the fact that a substantial fraction of dependent people reach that status with totally insu¢ cient …nancial means (which are here approximated by zero for the sake of simplicity).

To simplify the presentation, we take altruistic agents as perfectly altruistic ( a = 1).

As such, we depart from[START_REF] Olivera | Old age support and demographic change in developing countries. A cultural transmission model[END_REF], who assumes that non-altruistic parents prefer their children to be non-altruistic.

As the agent takes d a t+1 as given, we abstract here from demonstration e¤ects. Note, however, that in a three-period OLG setting, demonstration e¤ects face an inconsistency. Indeed, a parent does not need to help his own parent to give the example to his child, since his child will have to help his parent to show the example to his own child. Hence demonstration cannot work here.

Throughout this paper, we assume the interiority of optimal socialization e¤orts, with 0 < e i t < 1. This imposes some (weak) restrictions on v( ) and on the parameter . 1 0 Otherwise, it is better not to socialize children, and to let them take their egoistic trait in a costless way.

) u( g ) > 1, which is necessary and su¢ cient for the convergence towards a fully altruistic society. That transfer will thus be smaller than the optimal short-run transfer g = y 2 , but will nonetheless be strictly positive.15 Note also that this transfer will only be transitory, since in the long-run, the population will be fully altruistic, making that transfer irrelevant. 1 5 The in ‡uence of intergenerational composition e¤ects on optimal policy in the context of socialization models was also highlighted in[START_REF] Ponthiere | Unequal longevities and lifestyles transmission[END_REF], but in the context of choices a¤ecting one's welfare rather than the welfare of others.

) u( g ) , implying g 2 0; y 2 . The laissez-faire is optimal in the long-run.

) u( g ) > 1 holds (in order to get as close as possible to what maximizes social welfare in the short-run), such a strategy may not maximize the intertemporal social welfare, since this may reduce the speed of convergence towards the steady-state.

+ v(d a )

entirely composed of non-altruistic persons (i.e. q = 0). 14 As stated in Lemma 1, social welfare is, under u( ) = v ( ), larger in the former case, even when the State intervenes optimally.

Lemma 1 Under u( ) = v ( ), social welfare is larger under a fully altruistic population (i.e. q = 1) than under a fully non-altruistic population (i.e. q = 0), even when the government intervenes optimally in the later case.

Proof. See the Appendix.

Whereas the proof of Lemma 1 is in the Appendix, we can give here a brief sketch, to provide the underlying intuitions. Under the laissez-faire, the indirect utilities under the situations where q = 1 and q = 0 are respectively:

Under u ( ) = v ( ), we know, by the concavity of u( ), that social welfare under q = 1 exceeds the one under q = 0. Moreover, when the government intervenes under q = 0, and taxes the non-altruistic children in order to fund some aid for their dependent parents, social welfare is, under u ( ) = v ( ):

However, comparing the maximum utility under q = 1 and q = 0 yields:

Therefore, the ine¢ ciency of the government in the provision of LTC services (i.e. < 1) implies that the maximum social welfare under a fully altruistic population always exceeds, ceteris paribus, the maximum social welfare under a fully non-altruistic population. Thus the optimal composition of the population consists of a population fully made of altruistic persons, since altruism leads to a more e¢ cient production of LTC in comparison to what a government could achieve. Hence, the social optimum involves q = 1.

Let us now turn back to the design of the optimum policy from a longrun perspective. In the light of Section 4, the optimal short-run policy g is not necessarily optimal from a long-run perspective. Indeed, if the parents of non-altruistic children receive a governmental aid g , this aid, by reducing the welfare gain from socializing a child, may induce too low socialization e¤orts, with the consequence that the long-run population will not be altruistic, in contradiction with what the maximization of social welfare recommends (see 1 4 Note that, in theory, under the "It's the family" technology, the case where v( y) v(0) = 1 coincides with a continuum of equilibria. However, we rule out that very special case here. Moreover, under the "It takes a village" transmission technology, the case where q = 2 q v( y) v(0) could also occur, but only under the special initial conditions q 0 = 2 q v( y) v(0) . But given that this third case is unlikely to occur, we will concentrate here on the two other stationary equilibria: q = 0 and q = 1.

Note that, in the latter case, we do not fully characterize the optimal long-run transfer, but only provide an interval for it. The reason is that, from a longrun perspective, what matters is that the population becomes fully altruistic.

Hence, as soon as g satis…es u( y 2 ) u( g ) > 1, social welfare in the long-run is maximized, whatever the precise level of g is. Of course, if one included, in the social objective, the welfare of all generations living during the transition towards the steady-state, then the choice of g would be more constrained. 16 Let us now consider the "It takes a village" technology.

Proposition 9 Consider the economy under the "It takes a village" transmission technology. Suppose u( ) = v ( ) and 0 < q 0 < 1. The optimal provision of LTC by the government is determined as follows:

1, for any 0 g y 2 , the optimal (short-run and longrun) policy is a lump sum transfer g = y 2 from non-altruistic children towards their parents.

If u( y

2 ) u( g) > 1 for any 0 g y 2 , the optimal (short-run) policy is a lump sum transfer g = y 2 from non-altruistic children towards their parents, whereas the laissez-faire is optimal in the long-run.

Otherwise, the convergence towards a stationary equilibrium with q = 1 requires that the (short-run) lump sum transfer from non-altruistic children towards their parents g satis…es the conditions:

Corollary 1 Take the case where neither u( y 2 ) u( g) 1, for any 0 g y 2 , nor u( y 2 ) u( g) > 1, for any 0 g y 2 . The maximum lump sum transfer g from non-altruistic children to their parents that is compatible with a convergence towards q = 1 is lower under the "It takes a village" technology than under the "It's the family" technology. The lower q t is, the larger is the gap between the two maximum transfers.

Proof. The two conditions from Proposition 9 can be rewritten as:

Given that 0 < q t < 1, the second term in the LHS of the …rst condition is smaller, ceteris paribus, than in the second condition. Hence the LHS of the …rst condition is larger than in the second condition. Thus the lump sum transfer g has a lower maximum value under the second condition.

The intuition is that, under the "It takes a village" technology, the convergence towards a fully altruistic population is conditional on initial conditions, unlike under "It's the family". Hence, the public aid towards the dependent is more constrained under the "It takes a village" technology, in order to prevent the non-convergence towards the equilibrium with a fully altruistic population.

To sum up, there exists a tension between the optimal short-run LTC policy, which reduces inequalities among the dependent, and the long-run social optimum. The former may, by a¤ecting parental socialization decisions, prevent the convergence towards the latter. Thus, if one wants to allow the -socially desirable -convergence towards a stationary equilibrium with generalized altruism, the LTC policy should be lower than its optimum short-run level. Evolutionary forces act thus here as a constraint for policy-makers.

Concluding remarks

As a consequence of the ageing process, the provision of LTC services to the elderly dependent is a major challenge for advanced economies. It is not straightforward to de…ne the optimal role to be played by governments in the provision of LTC. On the one hand, the large heterogeneity among families justi…es some intervention aimed at helping elderly persons in need. On the other hand, public intervention may crowd out intrafamily aid, and, hence, reduce social welfare.

The goal of this paper was to examine the optimal design of LTC policy when the public aid to the elderly dependent can crowd out intrafamily aid. We focused on a particular kind of crowding-out e¤ect: the government, by intervening, may a¤ect the transmission of values within and across families, and may make family altruism disappear.

Appendix

Proof of Proposition 2

The dynamics of the economy is described by the equation:

The existence, uniqueness and stability of a stationary equilibrium can be studied by analysing the properties of the transition function F (q t ).

Note that F (0) = 0 and F (1) = 1, so that both q = 0 and q = 1, being …xed points of F ( ), are stationary equilibria. Regarding the existence of an intermediate equilibrium, note that:

Hence maximum social welfare at the steady-state is obtained by solving:

First-order conditions yield, under u( ) = v( ):

Those consumptions prevail also in the long-run under the laissez-faire. Hence the indirect utility: u y 2 + v y 2 achieved at the steady-state under q = 1 is the same as the one that can be achieved at the social optimum.

When the long-run population is fully non-altruistic, social welfare is:

Hence maximum social welfare is obtained by solving the problem:

First-order conditions yield, under u( ) = v( ):

Consumptions do not coincide with the laissez-faire, at which we have: c n = y > d n = 0. If one introduces a system of lump sum transfers from the young to the old, we have, with T = y 2 ,

Hence, the indirect utility under the optimal policy when q = 0, which is equal to u y 2 + v y 2 , is inferior to the one obtained when q = 1, as a consequence of the ine¢ ciency of the State in the provision of LTC ( < 1):

Therefore the optimum consists of a population that is entirely made of altruistic persons, since altruism leads to a more e¢ cient production of LTC in comparison to what a government achieves. Hence, the social optimum involves q = 1.

Proof of Proposition 8

If u( y 2 ) u( g) < 1 for any 0 g y 2 , we have, in the long-run, q = 0. That result holds whatever g is. Hence, in that case where public intervention has no e¤ect on the population dynamics, the optimal policy in the long-run consists of the optimal policy in the short-run, that is, a lump sum transfer g = y 2 from non-altruistic children towards their parents. If u( y 2 ) u( g) > 1 for any 0 g y 2 , the long-run composition of the population is q = 1. That result holds whatever the public transfer g is. Hence, in that case, the optimal long-run policy is the laissez-faire. However, in the short-run, it is optimal to implement a lump sum transfer g = y 2 from nonaltruistic children towards their parents.

Otherwise, if we exclude those two cases, the level of the lump sum transfer g will determine the long-run composition of the population. Given that the optimal composition is q = 1, the optimal long-run transfer is limited from above, and is lower than the optimal short-run transfer, since g = y 2 would prevent the survival of altruism in the population in this case. The convergence towards the stationary equilibrium with q = 1 requires: u( y 2 ) u( g) > 1. Hence the (short-run) lump sum transfer g from non-altruistic children towards their parents. g must satisfy the condition: u( y 2 ) u( g ) > 1, implying g 2 0; y 2 . Here again, the laissez-faire is optimal in the long-run (when q = 1).

Proof of Proposition 9

Consider …rst the case where

1 for any 0 g y 2 . In that case, the long-run composition of the population is q = 0. That result holds whatever the level of g is. Hence, in that case where the public intervention has no e¤ect on the population dynamics, the optimal policy in the long-run consists of the optimal policy in the short-run, that is, a lump sum transfer g = y 2 from non-altruistic children towards their parents. If u( y 2 ) u( g) > 1 for any 0 g y 2 , the long-run composition of the population is q = 1. Here again, that result holds whatever the public transfer g is. Hence, in that case, the optimal long-run policy is the laissez-faire. However, in the short-run, it is optimal to implement a lump sum transfer g = y 2 from non-altruistic children towards their parents.

Otherwise, if we exclude those two cases, the level of the lump sum transfer g will determine the long-run composition of the population. Given that the optimal q equals 1, the optimal long-run transfer is limited from above, and is lower than the optimal short-run transfer, since g = y 2 would prevent the survival of altruism in the population in this case. The convergence towards the stationary equilibrium with q = 1 requires: u( y 2 ) u( g) > 1. implying g 2 0; y 2 . Moreover, unlike in the "It's the family" case, we need also q 0 to be strictly larger than the intermediate equilibrium, implying q 0 > 2 q u( y 2 ) u( g ) . Therefore the optimal short-run transfer must satisfy those two conditions. Here again, the laissez-faire is optimal in the long-run (when q = 1).


Laurent Najman

Ultrametric watersheds

We study hierachical segmentation in the framework of edgeweighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical edgesegmentations.

Introduction

This paper1 is a contribution to a theory of hierarchical image segmentation in the framework of edge-weighted graphs. Image segmentation is a process of decomposing an image into regions which are homogeneous according to some criteria. Intuitively, a hierarchical segmentation represents an image at different resolution levels.

In this paper, we introduce a subclass of edge-weighted graphs that we call ultrametric watersheds. Theorem 9 states that there exists a one-to-one correspondance, also called a bijection, between the set of indexed hierarchical edge-segmentations and the set of ultrametric watersheds. In other words, to any hierarchical edge-segmentation (whatever the way the hierarchy is built), it is possible to associate a representation of that hierarchy by an ultrametric watershed. Conversely, from any ultrametric watershed, one can infer a indexed hierarchical edge-segmentation.

Following [START_REF] Barthélemy | Combinatorial optimization and hierarchical classifications[END_REF], we can say that, independently of its theoretical interest, such a bijection theorem is useful in practice. Any hierarchical segmentation problem is a priori heterogeneous: assign to an edge-weighted graph an indexed hierarchy. Theorem 9 allows such classification problem to become homogeneous: assign to an edge-weighted graph a particular edge-weighted graph called ultrametric watershed. Thus, Theorem 9 gives a meaning to questions like: which hierarchy is the closest to a given edge-weighted graph with respect to a given measure or distance?

The paper is organised as follow. Related works are examined in section 1. We introduce segmentation on edges in section 2, and in section 3, we adapt the topological watershed framework from the framework of graphs with discrete weights on the nodes to the one of graphs with real-valued weights on the edges. We then define (section 4) hierarchies and ultrametric distances. The last part of the paper (section 5) introduces hierarchical edge-segmentations and ultrametric watersheds, the main result being the existence of a bijection between these two sets (theorem 9).

Apart from Theorems 2 and 3, and to the best of the author's knowledge, all the properties and theorems formally stated in this paper are new. Proofs of the various properties and theorems will be given in an extended version [START_REF] Najman | Ultrametric watersheds: a bijection theorem for hierarchical edgesegmentation[END_REF] to be published in a journal paper.

1 Related works

Hierarchical clustering

From its beginning in image processing, hierarchical segmentation is thought of as a particular instance of hierachical classification [START_REF] Benzécri | [END_REF]. One of the fundamental theorems for hierarchical clustering states that there exists a one-to-one correspondance between the set of indexed hierarchical classification and a particular subset of dissimilarity measures called ultrametric distances; This theorem is generally attributed to Johnson [START_REF] Johnson | Hierarchical clustering schemes[END_REF], Jardine et al. [START_REF] Jardine | Mathematical taxonomy[END_REF] and Benzécri [START_REF] Benzécri | [END_REF]. Since then, numerous generalisations of that bijection theorem have been proposed (see [START_REF] Barthélemy | Combinatorial optimization and hierarchical classifications[END_REF] for a recent review).

Our main theorem is an extension to hierarchical edge-segmentation of this fundamental hierachical clustering theorem.

Hierarchical segmentation

There exist many methods for building a hierachical segmentation [START_REF] Pavlidis | Hierarchies in structural pattern recognition[END_REF], which can be divided in three classes: bottom-up, top-down, and split-and-merge. A recent review of some of those approaches can be found in [START_REF] Soille | Constrained connectivity for hierarchical image decomposition and simplification[END_REF]. A useful representation of hierarchical segmentations was introduced in [START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF] under the name of saliency map. This representation has been used (under several names) by several authors, for example for visualisation purposes [START_REF] Guigues | Scale-sets image analysis[END_REF] or for comparing hierarchies [START_REF] Arbeláez | A metric approach to vector-valued image segmentation[END_REF].

In this paper, we show that any saliency map is an ultrametric watershed, and conversely.

Watersheds

For bottom-up approaches, a generic way to build a hierarchical segmentation is to start from an initial segmentation and progressively merge regions together [START_REF] Pavlidis | In: Structural Pattern Recognition[END_REF]. Often, this initial segmentation is obtained through a watershed [START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF][START_REF] Meyer | Morphological segmentation[END_REF][START_REF] Meyer | Morphological segmentation revisited[END_REF]. See [START_REF] Meyer | Segmentation, arbre de poids minimum et hiérarchies[END_REF] for a recent review of these notions in the context of mathematical morphology. Among many others [START_REF] Roerdink | The watershed transform: Definitions, algorithms and parallelization strategies[END_REF], topological watershed [START_REF] Bertrand | On topological watersheds[END_REF] is an original approach to watersheding that modifies a map (e.g., a grayscale image) while preserving the connectivity of each lower cross-section (see fig. 2). It as been proved [START_REF] Bertrand | On topological watersheds[END_REF][START_REF] Najman | Watersheds, mosaics and the emergence paradigm[END_REF] that this approach is the only one that preserves altitudes of the passes (named connection values in this paper) between regions of the segmentation.

Pass altitudes are fundamental for hierarchical schemes [START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF]. On the other hand, topological watersheds may be thick. A study of the properties of different kinds of graphs with respect to the thinness of watersheds can be found in [START_REF] Cousty | Fusion graphs: merging properties and watersheds[END_REF][START_REF] Cousty | Weighted fusion graphs: merging properties and watersheds[END_REF]. An interesting framework is that of edge-weighted graphs, where watersheds are naturally thin; furthermore, in that framework, a subclass of topological watersheds satisfies both the drop of water principle and a property of global optimality [START_REF] Cousty | Watershed cuts: minimum spanning forests, and the drop of water principle[END_REF].

In this paper, we translate topological watersheds from the framework of node-weigthed-graphs to the one of edge-weighted graphs, and we identify ultrametric watersheds, a subclass of topological watersheds that is interesting for hierarchical edge-segmentation.

Segmentation on edges

This paper is settled in the framework of edge-weighted graphs. Following the notations of [START_REF] Diestel | Graph Theory[END_REF], we present some basic definitions to handle such kind of graphs.

We define a graph as a pair X = (V, E) where V is a finite set and E is composed of unordered pairs of V , i.e., E is a subset of {{x, y} ⊆ V | x = y}. We denote by |V | the cardinal of V , i.e, the number of elements of V . Each element of V is called a vertex or a point (of X), and each element of E is called an edge (of X). If V = ∅, we say that X is non-empty. As several graphs are considered in this paper, whenever this is necessary, we denote by V (X) and by E(X) the vertex and edge set of a graph X.

A graph X is said complete if E = V (X) × V (X).
Let X be a graph. If u = {x, y} is an edge of X, we say that x and y are adjacent (for X). Let π = x 0 , . . . , x ℓ be an ordered sequence of vertices of X, π is a path from x 0 to x ℓ in X (or in V ) if for any i ∈ [1, ℓ], x i is adjacent to x i-1 . In this case, we say that x 0 and x ℓ are linked for X. We say that X is connected if any two vertices of X are linked for X. Let X and Y be two graphs. If V (Y ) ⊆ V (X) and E(Y ) ⊆ E(X), we say that Y is a subgraph of X and we write Y ⊆ X. We say that Y is a connected component of X, or simply a component of X, if Y is a connected subgraph of X which is maximal for this property, i.e., for any connected graph Z, Y ⊆ Z ⊆ X implies Z = Y . Let X be a graph, and let S ⊆ E(X). The graph induced by S is the graph whose edge set is S and whose vertex set is made of all points which belong to an edge in S, i.e., ({x ∈ V (X) | ∃u ∈ S, x ∈ u}, S).

Important remark. Throughout this paper G = (V, E) denotes a connected graph, and the letter V (resp. E) will always refer to the vertex set (resp. the edge set) of G. We will also assume that E = ∅. Let S ⊂ E. In the following, when no confusion may occur, the graph induced by S is also denoted by S.

Typically, in applications to image segmentation, V is the set of picture elements (pixels) and E is any of the usual adjacency relations, e.g., the 4-or 8-adjacency in 2D [START_REF] Kong | Digital topology: Introduction and survey[END_REF]. The previous definitions of cut and segmentation (illustrated on fig. 1) are not the usual ones. In particular, Prop. 1.(i) below states that there is no isolated point in an edge-segmentation. If we need an isolated point x, it is always possible to replace x with an edge {x ′ , y ′ }. Furthemore, isolated points are often noise in an image.

If S ⊂ E, we denote by S the complementary set of S in E, i.e., S = E \ S. A set C ⊂ E is an (edge-)cut (of G) if each edge of C is adjacent to two different nonempty connected components of C. A graph S is called an (edge-)segmentation (of G) if E(S) is a cut. Any connected component of a segmentation S is called a region (of S).
It is interesting to state the definition of a segmentation from the point of view of vertices of the graph. A graph X is said to be spanning (for V ) if V (X) = V . We denote by φ the map that associates, to any X ⊂ G, the graph φ(X) = {V (X), {{x, y} ∈ E|x ∈ V (X), y ∈ V (X)}}. We observe that φ(X) is maximal among all subgraphs of G that are spanning for V (X), it is thus a closing on the lattice of subgraphs of G [START_REF] Cousty | Some morphological operators in graph spaces[END_REF]. We call φ the edge-closing.

Property 1 A graph S ⊆ G = (V, E) is an edge-segmentation of G if and only if (i) The graph induced by E(S) is S; (ii) S is spanning for V ; (iii) for any connected component X of S, X = φ(X).
3 Topological watershed

Edge-weighted graphs

We denote by F the set of all mappings from E to R + and we say that any mapping in F weights the edges of G. For any F ∈ F, the pair (G, F ) is called an edge-weighted graph. Whenever no confusion can occur, we will denote the edge-weighted graph (G, F ) by F .

For applications to image segmentation, we will assume that the altitude of u, an edge between two pixels x and y, represents the dissimilarity between x and y (e.g., F (u) equals the absolute difference of intensity between x and y; see [START_REF] Cousty | Watershed cuts: thinnings, shortest-path forests and topological watersheds[END_REF] for a more complete discussion on different ways to set the mapping F for image segmentation). Thus, we suppose that the salient contours are located on the highest edges of (G, F ).

Let λ ∈ R + and F ∈ F, we define

F [λ] = {v ∈ E | F (v) ≤ λ}. The graph (induced by) F [λ] is called a (cross)-section of F . A connected component of a section F [λ] is called a (level λ) component of F .
We define C(F ) as the set composed of all the pairs [λ, C], where λ ∈ R + and C is a component of the graph F [λ]. We call altitude of [λ, C] the number λ. We note that one can reconstruct F from C(F ); more precisely, we have:

F (v) = min{λ | [λ, C] ∈ C(F ), v ∈ E(C)} For any component C of F , we set h(C) = min{λ | [λ, C] ∈ C(F )}. We define C ⋆ (F ) as the set composed by all [h(C), C] where C is a component of F . The set C ⋆ (F ), called the component tree of F [25, 26], is a finite subset of C(F ) that is widely used in practice for image filtering. A (regional) minimum of F is a component X of the graph F [λ] such that for all λ 1 < λ, F [λ 1 ] ∩ E(X) = ∅.
We remark that a minimum of F is a subgraph of G and not a subset of the points of G; we also remark that any minimum X of F is such that |V (X)| > 1.

We denote by M(F ) the graph whose vertex set and edge set are, respectively, the union of the vertex sets and edge sets of all minima of F .

Topological watersheds on edge-weighted graphs

Let X be a subgraph of G. An edge u ∈ E(X) is said to be W-simple (for X) (see [START_REF] Bertrand | On topological watersheds[END_REF]) if X has the same number of connected components as X + u = (V (X)∪u, E(X)∪{u}). An edge u such that F (u) = λ is said to be W-destructible (for F ) with lowest value λ 0 if there exists λ 0 such that, for all λ 1 , λ

0 < λ 1 ≤ λ, u is W-simple for F [λ 1 ] and if u is not W-simple for F [λ 0 ].
A topological watershed (on G) is a mapping that contains no W-destructible edges.

A mapping F ′ is a topological thinning (of F ) if:

-F ′ = F , or if there exists a mapping F ′′ which is a topological thinning of F and there exists an edge u W-destructible for

F ′′ with lowest value λ such that ∀v = u, F ′ (v) = F ′′ (v) and F ′ (v) = λ 0 , with λ ≤ λ 0 < F ′′ (v).
An illustration of a topological watershed can be found in fig. 2. The connection value between x ∈ V and y ∈ V is the number In other words, F (x, y) is the altitude of the lowest element [λ, C] of C(F ) such that x and y belong to C (rule of the least common ancestor). Two points x and y are separated (for

F (x, y) = min{λ | x ∈ V (C), y ∈ V (C), [λ, C] ∈ C(F )} (1)
F ) if F (x, y) > max{λ 1 , λ 2 }, where λ 1 (resp. λ 2 ) is the altitude of the lowest element [λ 1 , c 1 ] (resp. [λ 2 , c 2 ]) of C(F ) such that x ∈ c 1 (resp. y ∈ c 2 )
. The points x and y are λ-separated (for F ) if they are separated and λ = F (x, y).

The mapping F ′ is a separation of F if, whenever two points are λ-separated for F , they are λ-separated for F ′ .

If X and Y are two subgraphs of G, we set F (X, Y ) = min{F (x, y) | x ∈ X, y ∈ Y }.

Theorem 2 (Restriction to minima [START_REF] Bertrand | On topological watersheds[END_REF]). Let F ′ ≤ F be two elements of F. The mapping F ′ is a separation of F if and only if, for all distinct minima X and Y of M(F ), we have

F ′ (X, Y ) = F (X, Y ). A graph X is flat (for F ) if for all u, v ∈ E(X), F (u) = F (v). If X is flat, the altitude of X is the number F (X) such that F (X) = F (v) for any v ∈ E(X).
We say that F ′ is a strong separation of F if F ′ is a separation of F and if, for each X ′ ∈ M(F ′ ), there exists X ∈ M(F ) such that X ⊆ X ′ and F (X) = F (X ′ ).

Theorem 3 (strong separation [START_REF] Bertrand | On topological watersheds[END_REF]). Let F and

F ′ in F with F ′ ≤ F . Then F ′ is a topological thinning of F if and only if F ′ is a strong separation of F .
In other words, topological thinnings are the only way to obtain a watershed that preserves connection values.

In the framework of edge-weighted graphs, topological watersheds allows for a simple characterization.

Theorem 4. A mapping F is a topological watershed if and only if:

(i) M(F ) is a segmentation of G; (ii) for any edge v = {x, y}, if there exist X and Y in M(F ), X = Y , such that x ∈ V (X) and y ∈ V (Y ), then F (v) = F (X, Y ).
Note that if F is a topological watershed, then for any edge v = {x, y} such that there exists X ∈ M(F ) with x ∈ V (X) and y ∈ V (X), we have F (v) = F (X).

Hierarchies and ultrametric distances

Let Ω be a finite set. A hierarchy H on Ω is a set of parts of Ω such that

(i) Ω ∈ H (ii) for every ω ∈ Ω, {w} ∈ H (iii) for each pair (h, h ′ ) ∈ H 2 , h ∩ h ′ = ∅ =⇒ h ⊂ h ′ or h ′ ⊂ h.
An indexed hierarchy on Ω is a pair (H, µ), where H denotes a given hierarchy on Ω and µ is a positive function, defined on H and satisfying the following conditions:

(i) µ(h) = 0 if and only if h is reduced to a singleton of Ω; (ii) if h ⊂ h ′ , then µ(h) < µ(h ′ ).
A distance d, in general, obeys the triangular inequality

d(ω 1 , ω 2 ) ≤ d(ω 1 , ω 3 )+ d(ω 3 , ω 2 )
where ω 1 , ω 2 and ω 3 are any three points of the space. An ultrametric distance (on Ω) is a function

d from Ω × Ω to R + such that d(ω 1 , ω 2 ) = 0 if and only if ω 1 = ω 2 , such that d(ω 1 , ω 2 ) = d(ω 2 , ω 1
) and such that d obeys ultrametric inequality d(ω 1 , ω 2 ) ≤ max(d(ω 1 , ω 3 ), d(ω 2 , ω 3 )) for all ω 1 , ω 2 , ω 3 . The ultrametric inequality [START_REF] Krasner | Espaces ultramétrique[END_REF] is stronger than the triangular inequality.

A partition of Ω is a collection (Ω i ) of non-empty subsets of Ω such that any element of Ω is exactly in one of these subsets. Note that any given partition of the set Ω induces a large number of trivial ultrametric distances:

d(ω 1 , ω 1 ) = 0, d(ω 1 , ω 2 ) = 1 if ω 1 ∈ Ω i , ω 2 ∈ Ω j , i = j, and d(ω 1 , ω 2 ) = a if i = j, 0 < a < 1.
The general connection between indexed hierarchies and ultrametric distances was proved by Benzécri [START_REF] Benzécri | [END_REF] and Johnson [4]. This result states that there is a oneto-one correspondance between indexed hierarchies and ultrametric distances both defined on the same set. Indeed, associated with each indexed hierarchy (H, µ) on Ω is the following ultrametric distance:

d(ω 1 , ω 2 ) = min{µ(h) | ω 1 ∈ h, ω 2 ∈ h, h ∈ H}. (2)
In other words, the distance d(ω 1 , ω 2 ) between two elements ω 1 and ω 2 in Ω is given by the smallest element in H which contains both ω 1 and ω 2 . Conversely, each ultrametric distance d is associated with one and only one indexed hierarchy. Observe the similarity between eq. 2 and eq. 1. Indeed, connection value is an ultrametric distance on V whenever F > 0. More precisely, we have the following property.

Property 5 Let F ∈ F. Then F (X, Y ) is an ultrametric distance on M(F ). If furthemore, F > 0, then F (x, y) is an ultrametric distance on V .
Let Ψ be the mapping on F such that for any F ∈ F the map Ψ (F ) and for any edge {x, y} ∈ E, Ψ (F )({x, y}) = F (x, y). It is straightforward to see that

Ψ (F ) ≤ F , that Ψ (Ψ (F )) = Ψ (F ) and that if F ′ ≤ F , Ψ (F ′ ) ≤ Ψ (F ).
Thus Ψ is an opening on the lattice (F, ≤) [START_REF] Leclerc | Description combinatoire des ultramétriques[END_REF]. We remark that the subset of strictly positive maps that are defined on the complete graph (V, V × V ) and that are open with respect to Ψ is the set of ultrametric distances on V . The mapping Ψ is known under several names, in particular the one of subdominant ultrametric and the one of ultrametric opening. It is well known that Ψ is associated to the simplest method for hierarchical classification called single linkage clustering [START_REF] Jardine | Mathematical taxonomy[END_REF][START_REF] Gower | Minimum spanning tree and single linkage cluster analysis[END_REF], closely related to Kruskal's algorithm [START_REF] Kruskal | On the shortest spanning subtree of a graph and the traveling salesman problem[END_REF] for computing a minimum spanning tree.

Thanks to Th. 4, we observe that if F is a topological watershed, then Ψ (F ) = F . However, an ultrametric distance d may have plateaus, and thus the weighted complete graph (V, V × V, d) is not always a topological watershed. Nevertheless, those results underline that topological watersheds are related to hierarchical classification, but not yet to hierarchical edge-segmentation; the study of such relations is the subject of the rest of the paper.

Hierarchical edge-segmentations, saliency and ultrametric watersheds

Informally, a hierarchical segmentation is a hierarchy made of connected regions. However, in our framework, a segmentation is not a partition, and as the union of two disjoint connected subgraphs of G is not a connected subgraph of G, the formal definition is slightly more involved. A hierarchical (edge-)segmentation (on G) is an indexed hierarchy (H, µ) on the set of regions of a segmentation S of G, such that for any h ∈ H, φ(∪ X∈h X) is connected (φ being the edge-closing defined in section 2).

For any λ ≥ 0, we denote by H[λ] the graph induced by {φ(∪ X∈h X)|h ∈ H, µ(h) ≤ λ}. The following property is an easy consequence of the definition of a hierarchical segmentation.

Property 6 Let (H, µ) be a hierarchical segmentation. Then for any λ ≥ 0, the graph H[λ] is a segmentation of G.

Property 5 implies that the connection value defines a hierarchy on the set of minima of F . If F is a topological watershed, then by Th. 4, M(F ) is a segmentation of G, and thus from any topological watershed, one can infer a hierachical segmentation. However, F [λ] is not always a segmentation: if there exists a minimum X of F such that F (X) = λ 0 > 0, for any λ 1 < λ 0 , F [λ 1 ] contains at least two connected components X 1 and X 2 such that |V (X 1 )| = |V (X 2 )| = 1. Note that the value of F on the minima of F is not related to the position of the divide nor to the associated hierarchy of minima/segmentations. This leads us to introduce the following definition.

A map F ∈ F is an ultrametric watershed if F is a topological watershed, and if furthemore, for any X ∈ M(F ), F (X) = 0. This property is illustrated in fig. 3. By definition of a hierarchy, two elements of H are either disjoint or nested. If furthermore (H, µ) is a hierarchical segmentation, the graphs E(H[λ]) can be stacked to form a map. We call saliency map [START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF] the result of such a stacking, i.e. a saliency map is a map F such that there exists (H, µ) a hierarchical segmentation with F (v) = min{λ|v ∈ E(H[λ])}.

Property 8 A map F is a saliency map if and only if F is an ultrametric watershed.

A corrolary of property 8 states the equivalence between hierachical segmentations and ultrametric watersheds. The following theorem is the main result of this paper. Theorem 9. There exists a bijection between the set of hierachical edge-segmentations on G and the set of ultrametric watersheds on G.

As there exists a one-to-one correspondance between the set of indexed hierarchies and the set of ultrametric distances, it is interesting to search if there exists a similar property for the set of hierarchical segmentations. Let d be the ultrametric distance associated to a hierarchical segmentation (H, µ). We call ultrametric contour map (associated to (H, µ)) the map d E such that:

1. for any edge v ∈ E(H[0]), then d E (v) = 0; 2. for any edge v = {x, y} ∈ E(H[0]), d E (v) = d(X, Y ) where X (resp. Y ) is the connected component of H[0] that contains x (resp. y).
Property 10 A map F is an ultrametric watershed if and only if F is the ultrametric contour map associated to a hierarchical segmentation.

Fig. 4 is an illustration of the application of the framework developped in this paper to a classical hierarchical segmentation scheme based on attribute opening [START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF][START_REF] Meyer | Segmentation, arbre de poids minimum et hiérarchies[END_REF][START_REF] Salembier | Anti-extensive connected operators for image and sequence processing[END_REF]. Fig. 5 shows some of the differences between applying such scheme and applying a classical morphological segmentation scheme, e.g. attribute opening followed by a watershed [START_REF] Meyer | Morphological segmentation[END_REF]. As watershed algorithms generally place watershed lines in the middle of plateaus, the two schemes give quite different results.

It is important to note that most of the algorithms proposed in the litterature to compute saliency maps are not correct, often because they rely on wrong connection values or because they rely on thick watersheds where merging regions is difficult. Future papers will propose novel algorithms (based on the topological watershed algorithm [START_REF] Couprie | Quasi-linear algorithms for the topological watershed[END_REF]) to compute ultrametric watersheds, with proof of correctness.

On a more theoretical level, this work can be pursued in several directions.

-We will study lattices of watersheds [START_REF] Cousty | Raising in watershed lattices[END_REF] and will bring to that framework recent approaches like scale-sets [START_REF] Guigues | Scale-sets image analysis[END_REF] and other metric approaches to segmentation [START_REF] Arbeláez | A metric approach to vector-valued image segmentation[END_REF]. For example, scale-sets theory considers a rather general formulation of the partitioning problem which involves minimizing a two-term-based energy, of the form λC + D, where D is a goodness-of-fit term and C is a regularization term, and proposes an algorithm to compute the hierarchical segmentation we obtain by varying the λ parameter. We can hope that the topological watershed algorithm [START_REF] Couprie | Quasi-linear algorithms for the topological watershed[END_REF] can be used on a specific energy function to directly obtain the hierarchy. -Subdominant theory (mentionned at the end of section 4) links hierachical classification and optimisation. In particular, the subdominant ultrametric d ′ of a dissimilarity d is the solution to the following optimisation problem for p < ∞:

min{||d -d ′ || p p | d ′ is an ultrametric distance and d ′ ≤ d}
It is certainly of interest to search if topological watersheds can be solutions of similar optimisation problems. -Several generalisations of hierarchical clustering have been proposed in the literature [START_REF] Barthélemy | Combinatorial optimization and hierarchical classifications[END_REF]. An interesting direction of research is to see how to extend in the same way the topological watershed approach, for example for allowing regions to overlap.

Fig. 1 .

 1 Fig.1. Illustration of edge-segmentation and edge-cut. (a) A graph X. (b) An edgesegmentation of X; the set of dotted-lines edges is the associated edge-cut of X. (c) A subgraph of X which is not an edge-segmentation of X: the grey point is isolated.

Fig. 2 .

 2 Fig. 2. Illustration of topological watershed. (a) An edge-weighted graph F . (b) A topological watershed of F . The minima of (a) are ({m, i}), ({p, l}), ({g, h}, {c, d}, {g, c}, {h, d}) and are in bold in (a).

Fig. 3 .

 3 Fig.3. An example of an ultrametric watershed F and a cross-section of F .

Fig. 4 .Fig. 5 .

 45 Fig. 4. Example of ultrametric watershed.

This work was partially supported by ANR grant SURF-NT05-2 45825