Pascal Belan
email: pascal.belan@univ-nantes.fr

Pierre-Jean Messe
email: pjmesse@gmail.com

François-Charles Wol
email: wol@sc-eco.univ-nantes.fr

Michael Kaganovich

André Masson

Lemna

Postponing retirement age and labor force participation: The role of family transfers *

Keywords: Retirement, domestic production, family transfers JEL Classication: J26

In this paper, we examine the consequences of delaying retirement in an overlapping generations model with domestic production and parental transfers in the form of grandchild care. We show that a change in age at retirement inuences the employment rates of both the young and the old. This interdependency stems from the provision of family transfers. Postponing retirement may increase time devoted to grandchild care transfers, which allows the young to work more on the labor market. We then study the conditions under which this positive family externality holds. Finally, using numerical simulations, we assess the consequences of delaying retirement on labor participation and account for public policy implications.

1 Introduction Many European countries have been characterized by a trend towards early retirement over the last three decades. This seems problematic in a setting where fertility rates tend to fall and life expectancy is steadily increasing, thereby putting a strong pressure on the nancing of pension schemes.

1 Given the expected growth of the demographic dependency ratio, solutions have to be found to make pay-as-you-go social security sustainable. While raising taxes or increasing the social security debt are clearly implausible solutions, several authors have suggested to increase the activity rates of older workers, ideally along with a reduction in social security benets.

It has been shown that postponing retirement may lead to a so-called double dividend [START_REF] Cremer | The double dividend of postponing retirement[END_REF]. First, delaying retirement is expected to restore at least partially the nancial balance of the pension system. Second, it may lead to more income equality among the retirees, at least if the system operates redistribution within generations. A good understanding of the consequences of postponing retirement is then needed if the challenge of aging in developed countries may only be solved through a reform aimed at increasing age of retirement. Such a policy would itself have an impact on economic growth [START_REF] Echevarria | Life expectancy, retirement and endogenous growth[END_REF].

Assuming that delaying retirement will be helpful to avoid a nancial crisis of social security means that this reform is indeed eective in terms of worker's employment.

Economists certainly agree that a reform of labor market rules is needed to prolong activity. However, even if we put this argument related to the functioning of the labor market aside, we argue that postponing retirement may strongly inuence the pattern of employment not only of older workers, but also of younger workers. The idea is simply to account for intergenerational relationships. In many families, at a given date, two generations take part in the labor market. The labor force participation of these two generations cannot be disconnected as long as one observes substantial ows of family transfers.

If a change in the labor supply of old workers aects their own transfer decisions, this may in turn lead to a change in the employment rate of younger workers. That family matters in the context of retirement issues is not a new idea. For instance, it has been shown that there is a strong tendency of husbands and wives to retire together and that each spouse values retirement more once their spouse has retired [START_REF] Gustman | Retirement in dual-career families: A structural model[END_REF]. Instead of focusing on spousal relationships, we account here for intergenerational links and family transfers to investigate how a public policy aimed at postponing retirement inuence employment rates.

We investigate in this paper the consequences of postponing retirement on the labor 1 Delaying retirement is subject to an implicit tax: prolonging activity implies paying additional payroll taxes and it also leads to a reduction in pension rights. Casamatta et alii (2006) show that this implicit tax on continued activity may result from some political process. participation of both young and old workers when grandchild care matters. For that purpose, we consider an overlapping generations model with domestic production and intergenerational transfers. Both the old and young generations take part in the labor market. Young workers have children and have to care for them. They personally devote time to raise their children. During their period of activity, they may either pay for formal child care or benet from grandchild care from their parents. In this setting, we study the relationship between the provision of parental transfers and retirement age.

Our results are twofold. First, we show that the children are more likely to work when older workers provide more grandchild care. Second, we prove that an increase in the length of the working period for older workers has an ambiguous eect on the labor participation of the younger workers. If the latter benet from less time-related resources, then they will devote more time to their children and thus will reduce their labor participation. However, in some cases, we evidence the reverse result. Older people will both work longer and care more for the grandchildren, which will in turn improve the employment rate of the children. Here, the main mechanism rests on the trade o of the old between time devoted to grandchild care and time devoted to their own domestic production. If later retirement brings higher income to grandparent and allows them to dramatically reduce time devoted to domestic production, time transfer to the young parent may increase.

A close look at this family externality matters in terms of public policy. Increasing the labor rates of older workers is most often viewed as a way for a government to spend less resources on pensions through two dierent channels. First, by postponing age of retirement, this will delay the receipt of the pension for all the workers who have to work longer. Second, in the case of a pay-as-you-go pension system, a government will receive additional taxes from the workers still involved in the labor market. Once private transfers are taken into account, a dierent picture emerges since expected changes in the labor participation of younger workers with children also aect payroll taxes.

The rest of the paper is organized as follows. We briey review the existing literature in Section 2. In Section 3, we present a theoretical model with two generations, domestic production and grandchild care. We investigate in Section 4 the consequences of postponing age of retirement on the pattern of parental transfers and labor participation of the young workers. In Section 5, we perform a numerical analysis to illustrate the fact that delaying retirement may increase the labor force participation of both the young and old workers. Finally, Section 6 concludes.

Previous literature

Several microeconomic studies have focused on the interplay between private transfers and labor supply decisions. 2 In the upward direction, individuals who have to care for their elderly parents may be induced to increase their number of worked hours if they have for instance to pay for formal services and housing retirement. Conversely, their labor participation may be reduced if they provide time transfers in the form of caregiving activities (Ettner, 1996, Wolf andSoldo,1994). Family transfers are also likely to aect the behavior of recipients. [START_REF] Joulfaian | Inheritance and labor supply[END_REF] nd that inheritances lead to a small decrease in the labor supply of women. Using samples of teenagers, [START_REF] Dustmann | Intra-household transfers and the part-time work of children[END_REF] and [START_REF] Wol | Parental transfers and the labor supply of children[END_REF] examine whether the receipt of nancial transfers reduces the labor participation of children still enrolled in school.

Parents may also rely on non-nancial transfers to help their children. [START_REF] Ermisch | Family structure, home time demands and the employment patterns of Japanese married women[END_REF] and [START_REF] Sasaki | The causal eect of family structure on labor force participation among Japanese married women[END_REF] account for intergenerational co-residence and nd that the labor supply of young women is much higher when they live with their parents. [START_REF] Dimova | Do downward private transfers enhance maternal labor supply ? Evidence from around Europe[END_REF] focus on the role of time transfers in the form of grandchild care. Using the SHARE European data, they show that the receipt of grandchild care has a positive impact on the labor force participation of young mothers. Conversely, the receipt of monetary transfers does not aect the decision of young mothers to have a paid job. Using a unique sample on ageing rst generation immigrants in France, Dimova and Wol (2007) show that grandchild care is spread unequally across siblings and has a strong positive impact on the labor supply of care-receiving mothers.

3

In what follows, we argue that such transfers in the form of grandchild care matter when studying the consequences of postponing retirement age. It is thus important to shed light on the magnitude of these time transfers. In the U.S., [START_REF] Soldo | Family structure and transfer measures in the Health and Retirement study[END_REF] and [START_REF] Cardia | Intergenerational time transfers and childcare[END_REF] nd that more than 40% of married women spend more than 100 hours per year caring for grandchildren. [START_REF] Cardia | Intergenerational time transfers and childcare[END_REF] provide additional evidence that intergenerational transfers of time are substantial both in the U.S. and Canada, grandparents playing an important role in child care. The Share data conrm that transfers in the form of grandchild care are frequent in all European countries. Regular grandchild care, dened as care provided on a daily or weekly basis, is received by exactly one third of mothers in working age with at least one child less than 10. South European countries are marked by the highest incidence of regular grandchild care.

4

Our attempt to account for private transfers in the context of retirement decisions using an overlapping generations model is innovative with respect to the previous liter-ature. Despite of their importance, transfers in the form of grandchild care has been widely neglected by economists. As it stands, our contribution is more closely related to the work of [START_REF] Cardia | Intergenerational time transfers and childcare[END_REF], who also consider an overlapping generations model with domestic production and both time and monetary transfers. However, contrary to these authors who calibrate this model to the US economy, we do not focus on capital accumulation and macroeconomic eects of child care policies. Our primary interest lies in the impact of postponing retirement and the labor participation consequences.

The theoretical model

We consider an overlapping generations model with two adult generations, the young (children) and the old (their parents). We assume that the young adults have themselves kids, so that there is implicitly three cohorts of agents in the model. However, as grandchildren are young kids, they do not make any economic decision, so that we focus in what follows on transfer and labor participation choices within a two-generations framework.

5

Following [START_REF] Cardia | Intergenerational time transfers and childcare[END_REF], we introduce domestic production and time family transfers in the OLG model. The utility of each agent depends on a composite good, which is made of a consumed market good and a home produced good. Market goods and time are the two inputs necessary to home production. We assume that young adults will essentially devote their time to raise their children. The home-produced goods may thus be seen as child care for the young generation, educational expenditures and child care services being examples of market goods. Conversely, the domestic time of their parents is related to personal activities like cooking, shopping, housework and odd jobs among others. A crucial distinction between the two generations is related to the denition of the time inputs, since we allow for time transfers from the old to the young generation.

The setting is as follows. During the rst period, denoted by 1 as upscript, the young adult allocates her time between a paid activity and home production, i.e. child care. Let h t be the labor supply and k 1 t the number of hours devoted to the kids. Endowment of time for the young is normalized to one, so that

h t + k 1 t = 1 (1)
The second period, denoted by 2 and also normalized to one, is made up of both working time, retirement and domestic production. We denote by θ t+1 the fraction of time devoted to labor activities, so that (1 -θ t+1 ) is the length of the retirement period. In our model, the retirement age is exogenous. Let e 2 t+1 be time spent in pure domestic production by the parent and let T t+1 be a transfer in the form of grandchild care. The time constraint for the parent may be expressed as

θ t+1 + e 2 t+1 + T t+1 = 1 (2)
Two remarks are in order. First, we only consider transfers in the form of grandchild care provided by parents. In so doing, we neglect the possibility of home-sharing arrangements and nancial transfers. This restriction is in fact mainly empirically driven, as grandchild care transfers are much more frequent than cash gifts for young adults and represent a substantial number of hours per week [START_REF] Cardia | Intergenerational time transfers and childcare[END_REF][START_REF] Dimova | Do downward private transfers enhance maternal labor supply ? Evidence from around Europe[END_REF][START_REF] Dimova | Grandchild care transfers by ageing immigrants in France: Intra-household allocation and labour market implications[END_REF].

6 Secondly, we clearly need a motivation for the provision of grandchild care: why would parents give time to their children since these transfers are presumably costly owing to transportation costs for instance ? Several motives for private transfers have been suggested in the literature on intergenerational relationships (see [START_REF] Laferrère | Microeconomic models of family transfers[END_REF].

A rst possibility is that the parent is inuenced by the utility level of the child, which is the spirit of the altruistic model [START_REF] Becker | A Treatise on the Family[END_REF]. While the basic altruistic model usually only includes nancial transfers, it can be easily extended to the case of services from parents to children (Sloan et alii, 2002). A second model relies on exchange considerations [START_REF] Cox | Motives for private income transfers[END_REF]. This would lead to a framework where parents help their children through grandchild care, but expect a transfer of money in exchange of their services (for instance to pay for formal old-age support). A third possibility is the demonstration eect theory [START_REF] Cox | On the demand for grandchildren: Tied transfers and the demonstration eect[END_REF], according to which the child's propensity to care for parents is conditioned by parental example.

In this paper, we choose an alternative strategy and rely on an impure form of the altruistic motive, i.e. the so-called warm-glow motive described in [START_REF] Andreoni | Impure altruism and donations to public goods: A theory of warm-glow giving[END_REF]. The underlying idea is that parents obtain satisfaction not from the well-being of their children, but instead from the act of giving time per se. In so doing, we thus suppose that grandparents enjoy spending time with young kids. The donor's utility is furthermore rising with the amount given. It should be noted that we depart here from the specication of [START_REF] Cardia | Intergenerational time transfers and childcare[END_REF]. Following the Beckerian altruistic model, these authors consider instead that grandparents take the well-being of their children when maximizing their own utility.

6 Furthermore, the labor supply decisions of young adults seem to be quite insensitive to the provision of nancial transfers from parents. See for instance [START_REF] Wol | Parental transfers and the labor supply of children[END_REF].

Consumer problem

We now describe the budget constraint for each generation. In the rst period, the child's resources are devoted to market good purchases c t and savings s t . Denoting by w the wage rate and considering a pay-as-you-go pension scheme with payroll tax rate of τ t , the budget constraint for the young is given by:

c t + s t = (1 -τ t ) h t w (3)
We consider a small open economy, so that the real interest rate r is exogenous. Let us denote by R = 1 + r, the gross interest rate. During the second period of life, resources consist of wage income θ t+1 (1 -τ t+1 ) w during the working period and pension benets (1 -θ t+1 ) bw once being retired, where b represents the replacement rate. Accounting for the returns on rst-period savings Rs t , the market good purchased by the household is

d t+1 such that d t+1 = ω t+1 (θ t+1 , τ t+1 ) + Rs t (4)
where

ω (θ t+1 , τ t+1 ) = [(1 -θ t+1 ) b + θ t+1 (1 -τ t+1 )] w is the second period income.
For the young, we dene ct as the composite of market good c t -z 1 t and domestically produced good q 1 t . The latter comprises child care. Let g 1 be the family production function, whose arguments are purchased inputs z 1 t (like child care services), personal time to children k 1 t and grandchild care transfers T t . We consider a production function of the form q 1 t = g 1 (z 1 t , k 1 t + T t ). We thus assume that time values k 1 t and T t are perfect substitutes. What matters for instance for the young kid is to be with an adult, which is either a parent or a grandparent. Conversely, we make no assumption a priori concerning the complementarity or substitutability between z 1 t and k 1 t + T t . The composite good ct is itself the result of a function f

1 such that ct = f 1 c t -z 1 t , g 1 z 1 t , k 1 t + T t (5)
In a similar way, the second-period composite good dt+1 is obtained by combining the market good d t+1 -z 2 t+1 and the produced good q 2 t+1 . Family production is achieved through the production function q 2 t+1 = g 2 z 2 t+1 , e 2 t+1 where z 2 t+1 and e 2 t+1 denote respec- tively purchased inputs and time inputs. Then, dt+1 is given by

dt+1 = f 2 d t+1 -z 2 t+1 , g 2 z 2 t+1 , e 2 t+1 (6)
Functions f 1 , g 1 , f 2 and g 2 are linear homogeneous. Marginal productivities are positive and decreasing, so that

f i 1 > 0, f i 11 < 0, f i 2 > 0, f i 22 < 0, f i 12 > 0 and g i 1 > 0, g i 11 < 0, g i 2 > 0, g i 22 < 0, g i 12 > 0 (i = 1, 2).
The main dierence between the parent and the child specications for the extended consumption is related to the receipt of grandchild care, which is an additional input in the production function of the young. Given the paternalistic altruism, each individual seeks to maximize the intertemporal utility function U t which is dened in the following way:

U t = u (c t ) + δ v dt+1 + γΦ (T t+1 ) (7)
where δ represents subjective discount factor. Utility functions u(.) and v(.) are twice dierentiable, increasing and strictly concave. The term γΦ (T t+1 ) picks up the warm-glow motive for providing grandchild care, the parameter γ being strictly positive [START_REF] Andreoni | Impure altruism and donations to public goods: A theory of warm-glow giving[END_REF].

7

The problem for an agent is thus to maximize the intertemporal utility function ( 7)

with respect to ct , dt+1 , c t , d t+1 , s t , h t , z 1 t , k 1 t , z 2
t+1 , e 2 t+1 and T t+1 subject to time constraints (1) and ( 2), nancial constraints (3) and ( 4), and denitions of ct and dt+1 , ( 5) and ( 6).

All variables must be positive. The maximization program can be expressed as

max k 1 t ,z 1 t ,z 2 t+1 ,T t+1 ,st u f 1 1 -k 1 t (1 -τ t ) w -s t -z 1 t , g 1 z 1 t , k 1 t + T t +δv f 2 ω (θ t+1 , τ t+1 ) + Rs t -z 2 t+1 , g 2 z 2 t+1 , 1 -θ t+1 -T t+1 +δγΦ (T t+1 )
The ve rst-order conditions for an interior solution, i.e. ∂U t /∂k 1 t = 0, ∂U t /∂z 1 t = 0, ∂U t /∂z 2 t+1 = 0, ∂U t /∂T t+1 = 0 and ∂U t /∂s t = 0, lead to 8

-(1 -τ t )wf 1 1 (.) + g 1 2 (.)f 1 2 (.) = 0 (8) -f 1 1 (.) + g 1 1 (.)f 1 2 (.) = 0 (9) -f 2 1 (.) + g 2 1 (.)f 2 2 (.) = 0 (10) -g 2 2 (.)f 2 2 (.)v 1 (.) + γΦ 1 (.) = 0 (11) -f 1 1 (.)u 1 (.) + δRf 2 1 (.)v 1 (.) = 0 (12)
The interpretation of the rst-order conditions is as follows. From (8), the marginal benet of a rise in leisure g 1 2 f 1 2 is equal to its marginal cost resulting from the loss of income (1 -τ t ) wf 1 1 . From ( 9), the marginal disutility f 1 1 of the young adult involved by an increase in purchased inputs z 1 t used to produce the non-market good is equal to its marginal benet g 1 1 f 1 2 , while (10) has the same interpretation for the old parent.

According to (11), the marginal cost of giving time to the young g 2 2 f 2 2 v 1 is equal to its marginal benet γΦ 1 resulting from the joy-of-giving motivation. Finally, (12) represents the trade-o between the two periods of life. 7 The satisfaction function Φ is also supposed to be twice dierentiable, strictly concave and increasing. 8 For any function F in the following, F i stands for the rst-order derivative with respect to the i th argument.

Dynamics

Consumer's choice may be expressed as a function of four variables : the net wage when young (1 -τ t ) w, the time-transfer received from his parent T t , the second-period income ω (θ t+1 , τ t+1 ), and the retirement age θ t+1 . In particular, the time-transfer of the consumer born in t to his child writes

T t+1 = T ((1 -τ t ) w, T t , ω (θ t+1 , τ t+1 ) , θ t+1 )
Similarly, labor supply when young is

h t = 1 -k 1 t = h ((1 -τ t ) w, T t , ω (θ t+1 , τ t+1 ) , θ t+1 )
In order to disentangle the multiple eects of postponing retirement and isolate the relevant ones for our purpose, we shall focus on an increase in the current period retirement age under two assumptions. First, we keep the retirement age in future periods at its initial value. Second, the contribution rate of the pay-as-you-go pension system is constant and set at its initial value.

The second assumption means that we do not take into account the government budget constraint. Normalizing the size of each generation to unity, the gap between contributions receipts and pension benets

D t = (1 -θ t ) b (θ t ) w -τ h ((1 -τ ) w, T t , ω (θ t+1 , τ ) , θ t+1 ) + θ t w
is added to the public decit, that is, we consider the case of a country with a large public decit (as in almost all European countries) and assume that part of the decit is due to the pension system 9 .

With a constant retirement age θ, dynamic evolution of time transfers is backward and writes

T t+1 = Γ ((1 -τ ) w, T t , ω (θ, τ ) , θ)
Starting the economy at period 0, time-transfer T 0 of the old born in period -1 results from the maximization program max

z 2 0 ,T 0 v f 2 ω (θ, τ ) + Rs -1 -z 2 0 , g 2 z 2 0 , 1 -θ -T 0 + γΦ (T 0 )
where saving s -1 is given.

We then study the consequences of postponing the current retirement age, i.e. an increase in θ 0 , as an economic policy aimed at reducing the public budget decit, keeping both the contribution rate τ and the future retirement age θ 1 set at their initial value before the reform. A particular attention is put on labor supply within the family h 0 + θ 0 . 9 Note that there is no diculty to nance the decit since we consider a small open economy.

A convenient interpretation of our framework is thus the case of a government that postpones retirement in order to get additional resources for the public pension system.

Nevertheless, it could be argued that a temporary rise in retirement age is not totally convincing in a context where policy aging is expected to be steady. If the government raises the retirement age for the nancial aspect of the pay-as-you-go program facing population aging, then the retirement age is expected to raise in all periods. We shall give some insights on this matter in Section 5.

4

The optimal pattern of labor supply and transfers

Let us now investigate the consequences of postponing retirement on the pattern of family transfers and labor participation. We proceed in the following way. First, we study how a change in parental transfers aects the labor supply of the young. Then, we focus on the relationship between an increase in the retirement age and the provision of time transfers by the grandparents. Finally, we analyze the overall eect of prolonged activity on the labor participation of both generations.

Grandchild care and child's labor supply

When the young makes her decisions, she takes the grandchild care transfer T t from her parents as given. For convenience, we omit time indices in this section and denote by T -, time transfer received by the consumer from his parent. We focus on the relationship between the parental transfer and the child's labor participation. We deduce from ( 8) and ( 9) that:

g 1 2 (z 1 , k 1 + T -) g 1 1 (z 1 , k 1 + T -) = (1 -τ ) w
which means that the marginal rate of substitution between child care k 1 and purchased input z 1 is equal to the net wage (1 -τ ) w, i.e. the marginal cost of reallocating time from labor to child care. Let η 1 = z 1 k 1 +T -be the ratio between the purchased and time inputs devoted to domestic production. Since g 1 is linear homogeneous, η 1 is the solution of g

1 2 (η 1 , 1) /g 1 1 (η 1 , 1) = (1 -τ ) w. Now, let ζ 1 = c
k 1 +T -be the consumption per unit of time devoted to domestic pro- duction. Using the rst-order condition (9) and linear homogeneity of f 1 and g 1 , ζ 1 is the solution of

f 1 1 (ζ 1 -η 1 , g 1 (η 1 , 1)) f 1 2 (ζ 1 -η 1 , g 1 (η 1 , 1)) = g 1 1 η 1 , 1 Moreover, since ζ 1 (k 1 + T -) = c = (1 -k 1 )(1 -τ )w -s and h + k 1 = 1, one obtains
that labor supply of the young satises

h = 1 -k 1 = s + ζ 1 (1 + T -) (1 -τ ) w + ζ 1 (13)
From the linear homogeneity of f 1 and g 1 , we get c = f

1 (ζ 1 -η 1 , g 1 (η 1 , 1)) (k 1 + T -).
The augmented consumption of the young adult may then be expressed as follows

c = c (1 -τ ) w 1 + T --s (14) with c ≡ f 1 (ζ 1 -η 1 ,g 1 (η 1 ,1))
(1-τ )w+ζ 1 .

Proposition 1 Savings increase with the intensity of grandchild care, i.e. ∂s ∂T -> 0.

Proof. From equation ( 12) and using the expression ( 14) of c, we get

0 = -f 1 1 ζ 1 -η 1 , g 1 η 1 , 1 u 1 c (1 -τ ) w 1 + T --s +δRf 2 1 ω + Rs -z 2 , g 2 z 2 , 1 -θ -T v 1 (f 2 ω + Rs -z 2 , g 2 z 2 , 1 -θ -T )
By concavity of the consumer problem, the right-hand side is decreasing with respect to s. Thus sgn

ds dT -= sgn -f 1 1 (•) c (1 -τ ) wu 11 (•)
so that the derivative ds dT -is positive.

Proposition 2 The child's labor supply is an increasing function of grandchild care, i.e.

∂h

∂T -> 0.

Proof: Using (13), we get the following derivative

∂h ∂T -= 1 (1 -τ ) w + ζ 1 ∂s ∂T -+ ζ 1 Since ∂s ∂T -+ ζ 1 > 0 from Proposition 1, this implies ∂h ∂T -> 0.
A rise in the provision of grandchild care leads to a decrease in parental child care k 1 and then to a rise in the child's labor participation h.

Not surprisingly, with more grandparental services devoted to the care of young babies, young adults are able to spend more time on paid employment. Much simpler models of transfers with endogenous labor supply lead to similar conclusions. Interestingly, this prediction receives empirical support in the literature. In Europe, estimates from simultaneous equation models indicate that the coecient of the endogenously treated grandchild care variable has a positive impact on the labor force participation of young mothers [START_REF] Dimova | Do downward private transfers enhance maternal labor supply ? Evidence from around Europe[END_REF].

The parental decision of transfer

We now study the parental decision of giving time to the young generation. As mentioned before, we consider an increase in the retirement age in period 0 that was not expected by generation -1 in the preceding period. Savings s -1 accumulated during period -1 is given. Omitting time indices, let Ω = ω + Rs be the second-period income of the old of period 0, and L = 1 -θ the length of their retirement period. The problem for the parent may be expressed as

max T,z 2 Ψ z 2 , T, Ω, L = v f 2 Ω -z 2 , g 2 z 2 , L -T + γΦ (T )
We seek to understand how an increase in retirement age inuences the provision of grandchild care. It is straightforward to see that there are two eects:

• a time constraint eect: with an increase in labor participation, the parent has less time to devote to grandchild care, which will reduce T ;

• an income eect: when the government increases the value of θ, the second-period parental income also increases since dΩ dθ > 0. As a consequence, the old can devote more nancial resources to purchase input z 2 related to domestic production. The consequence on time e 2 is ambiguous and depends on the substitutability of time and market good in domestic production. Then, as we shall see, the income eect leads grandparent to spend more or less time with grandchildren.

Accounting for time constraint eect and income eect implies that the impact of θ on T cannot be signed in the general case. Considering the parental maximization program, the rst-order conditions for an interior solution are equations ( 10) and ( 11) and may be expressed as

Ψ z z 2 , T, Ω, L = 0 Ψ T z 2 , T, Ω, L = 0 with Ψ z (z 2 , T, Ω, L) = (-f 2 1 + f 2 2 g 2 1 ) v 1 and Ψ T (z 2 , T, Ω, L) = -v 1 f 2 2 g 2 2 + γΦ 1 .
By concavity of the consumer problem, Ψ zz < 0, Ψ T T < 0, and the determinant ∆ = Ψ zz Ψ T T -(Ψ zT ) 2 of the Hessian matrix of Ψ is positive. Dierentiating the rst-order conditions with respect to z 2 , T , Ω and L, we deduce that

dT dΩ = Ψ zT Ψ zΩ -Ψ T Ω Ψ zz ∆ dT dL = Ψ zT Ψ zL -Ψ T L Ψ zz ∆
Proposition 3 A decrease in the length of the retirement period reduces time devoted to grandchild care, i.e.

dT dL > 0. A rise in parental resources during the second period has an ambiguous impact on time transfers, i.e.

dT dΩ ≥ 0 or dT dΩ ≤ 0.

Proof: We denote by σ 2 g and σ 2

f the elasticity of substitution respectively for the production function g 2 and f 2 . In the Appendix, we calculate the second-order derivatives Ψ zz , Ψ zT , Ψ zΩ , Ψ zL , Ψ T T , Ψ T Ω and Ψ T L . It is then straightforward to obtain dT dL and dT dΩ .

After some manipulations, we get

∆ σ 2 g σ 2 f Ω -z 2 + g 2 g 2 1 v 1 f 2 12 g 2 12 2 dT dL = - v 11 v 1 f 2 1 Ω -z 2 g 2 g 2 1   σ 2 g + σ 2 f 1 + 1 Ω-z 2 g 2 g 2 1 (L -T ) g 2 2 z 2 g 2 1   + 1 + 1 Ω -z 2 g 2 g 2 1 z 2 g 2 1 (L -T ) g 2 2   1 + 1 1 + 1 Ω-z 2 g 2 g 2 1 (L -T ) g 2 2 z 2 g 2 1   2 that implies dT dL > 0.
In a similar way, calculation of dT dΩ leads to:

∆ σ 2 f (δv 1 ) 2 (f 2 12 ) 2 g 2 12 Ω -z 2 + g 2 g 2 1 dT dΩ = - v 11 v 1 f 2 σ 2 g g 2 Ω -z 2 + σ 2 f g 2 z 2 + σ 2 g -σ 2 f g 2 1 - g 2 Ω -z 2 + g 2 z 2
so that dT dΩ can be either positive or negative depending on the magnitude of the two terms on the right-hand side of the previous equality.

To study the consequences of Proposition Both the time constraint and income eects may be more closely related to the possibility of substitution between inputs of the dierent production functions. As evidenced in the proof of Proposition 3, the sign of dT dΩ depends on the values of σ 2 g and σ 2

f .

Specically, dT dΩ will be positive only if σ 2 g is suciently high. When the purchased input z 2 and time devoted to domestic activities 1 -θ -T are strongly substitutable, postponing retirement will lead to a rise in private downward transfers. With the rise in θ, grandparents have now more nancial resources (income eect). Since z 2 is a normal good and owing to the substitutability, grandparents will essentially devote this supplement of revenue to purchase additional units of z 2 and lessen time devoted to domestic tasks.

They will therefore spend more time with their grandchildren. Conversely, in situations where z 2 and leisure 1 -θ are not very substitutable, then the reverse conclusion holds.

Parents have less time to devote to their own consumption and not enough additional resources to buy purchased domestic inputs instead of providing domestic time. Children will then benet from less time transfers related to grandchild care.

The labor participation of young and old workers

We now investigate the overall impact on the labor market of a rise in retirement age θ.

Since the child's labor participation depends on the family transfer and since postponing retirement inuences grandchild care, this means that delaying retirement will aect the labor supply decision of both the parent and the child. Labor supply is given by h for the young and θ for the old. The impact of a change in θ is then given by:

∂ (h + θ) ∂θ = 1 + ∂h ∂T - ∂T ∂Ω ω θ - ∂T ∂L
Combining results from Propositions 1, 2 and 3 sheds light on the interplay between the receipt of family transfers and employment on the labor market. Our main result is that postponing retirement may either increase or decrease the labor participation of the young parents.

Since ∂h ∂T -> 0, if the time constraint eect is dominant, postponing retirement age decreases time transfer and therefore labor supply of the young, i.e.

∂(h+θ) ∂θ

< 1. But, when the income eect dominates the time constraint eect, grandparents are expected to devote more time to grandchild care when retirement age is postponed. Specically, the derivative ∂(h+θ) ∂θ will be higher than 1 when the following condition holds:

∂T ∂Ω ω θ > ∂T ∂L
that is, z 2 and e 2 are strongly substitutable and the additional income eect due to the postponed retirement has a large impact on grandchild care.

With respect to the existing literature [START_REF] Cremer | The double dividend of postponing retirement[END_REF], our model exhibits the potential for a new dividend of a public policy aimed at postponing retirement. The provision of grandchild care transfers gives rise to an interdependency within the family.

A change in the labor participation of the old generation will have a direct impact on the labor supply of the young generation. In many cases, grandparents will reduce time devoted to their adult children in response to a rise in θ, as they have less time for nonworking activities ( ∂T ∂Ω ω θ < ∂T

∂L

). This will in turn reduce the participation of children on the labor market, since the latter have now to care for their own children instead of relying on the parental support.

However, postponing retirement may also increase the provision of time transfers

( ∂T ∂Ω ω θ > ∂T ∂L
). By working longer, grandparents are in a position to buy more purchased inputs related to domestic production. If these purchased inputs and their own domestic time are strongly substitutable, they will spend more time with the grandchildren. This is then the reverse story. Adults with young children have now more time to devote to non-domestic activities, and they will presumably spend part of this extra-time in paid activities. A modication in θ has now an unintended, albeit benecial, consequence.

Delaying retirement increases employment of both the old and young generations.

Although we do not focus on the sustainability and the nancial balance of the pension scheme in our model, it is clear that delaying retirement will be helpful for the pension system in terms of additional payroll taxes. A question worth is then to assess the magnitude of the `crowding-out' or `crowding-in' eects occurring through intergenerational linkages. If postponing retirement improves the labor participation of the old, but reduces at the same time employment for young adults, then the expected increase in payroll taxes will be much lower than initially expected. Knowing the overall eect of a change in θ on (h+θ) deserves further attention and we perform in the next section numerical simulations to assess the possibility that postponing retirement may improve the labor participation of both the young and old generations.

Numerical illustration

We rely on the following functional forms for utility and production functions. First, we consider Cobb-Douglas functions for the extended levels of consumption of the young and the old, i.e. ct and dt+1 . This implies that in both periods, the elasticities of substitution σ 1 f and σ 2 f between the market good and the domestically-produced good are equal to one.

Second, we assume CES forms for the domestic production functions g 1 and g 2 . Hence, ct can now be expressed as

ct = (1 -τ t ) w 1 -k 1 t -s t -z 1 t α 1 a 11 z 1 t ρ 1 + a 12 k 1 t + T t ρ 1 1-α 1 ρ 1
where a 11 and a 12 measure the respective contributions of input and time devoted to child care. In a similar way, we get dt+1

dt+1 = ω (θ t+1 , τ t+1 ) + Rs t -z 2 t+1 α 2 a 21 z 2 t+1 ρ 2 + a 22 (1 -θ t+1 -T t+1 ) ρ 2 1-α 2 ρ 2
with a 21 and a 22 the respective weights of purchased input and time devoted to domestic tasks in the second-period production function. At last, we rely on a logarithmic utility function ln ct + δ ln dt+1 + γ ln T t+1

With these specications and for an interior solution of the consumer problem in each period, the evolution of time transfers is characterized by the following recurrence equation

(1 -α 2 ) T t+1 1 -θ t+1 -T t+1 -γ 1-ρ 2 - a 21 γ a 22 R (1 -τ t ) w (1 + T t ) + ω (θ t+1 , τ t+1 ) 1 δ + 1 + γ T t+1 -γ (1 -θ t+1 ) ρ 2 = 0 (15)
where the interval of admissible values for T t+1 is I = γ(1-θ t+1 ) 1-α 2 +γ ; 1 -θ t+1 . Since ρ 2 < 1, straightforward calculus shows that the left-hand side is an increasing function of T t+1 and grows from 0 to +∞ when T t+1 goes from the lower to the upper bound of I. Thus, for any time transfer T t received by an individual born in t, there exists a unique time transfer T t+1 left to his ospring.

Our numerical illustration draws on the parameter values summarized in Table 1.

10

We suppose that domestic production functions g 1 and g 2 are the same in both periods.

Recalling that the elasticity of substitution between the purchased input and time devoted to domestic activities is a crucial parameter, we do not set its value for the moment.

In the benchmark case, we set retirement age θ to one-third. Assuming, for instance, economic life starts at twenty and period length is thirty years, this means that people enter the labor force at twenty and leave it at sixty. Then, we assess the consequences of an increase in retirement age at period 0, from one-third to one-half, i.e. from sixty to sixty-ve. As in the preceding section, we consider a one-shot policy, i.e. an increase in θ 0 , holding θ t , t ≥ 1, constant. Retirement age is postponed in order to reduce the decit of the pay-as-you-go pension system for given contribution rate and replacement rate.

10 Following de la [START_REF] De La Croix | A theory of economic growth[END_REF], we choose δ in the line of the RBC literature, with a quarterly psychological discount factor of 0.99. With period length of thirty years, the parameter δ is thus set equal to 0.99 120 = 0.3. We compute the steady-state time transfer T , 11 solution of the following equation 12

(1 -α 2 ) T 1 -θ -T -γ 1-ρ 2 = a 21 γ a 22 R (1 -τ ) w (1 + T ) + ω (θ, τ ) 1 δ + 1 + γ T -γ (1 -θ) ρ 2
Then, we compute T 0 and h 0 that result from a one-shot increase in the retirement age from 1/3 to 1/2 in period 0. Time transfer of the old in period 0 is determined by 13

(1

-α 2 ) T 0 1 -θ 0 -T 0 -γ 1-ρ 2 = a 21 γ a 22 Rs -1 + ω (θ 0 , τ ) (1 + γ) T 0 -γ (1 -θ 0 ) ρ 2
As shown in Proposition 3, an increase in θ 0 may have two osetting eects on time transfer T 0 . Indeed, the eect of θ 0 associated with retirement duration (L) appears on the left-hand side, while the eect of θ 0 through the second-period income (Ω) appears on the right-hand side. From Proposition 3, we know that the former eect (time constraint eect) is negative on time transfer, while the latter one (income eect) is ambiguous.

Nevertheless, from the expression of dT dΩ in the proof of Proposition 3, we deduce that the specications of the utility function that we have chosen imply that the sign of the income eect is the same as the sign of ρ 2 . This is conrmed by the equation that characterizes T 0 in this example.

11 The resulting decit of the pay-as-you-go pension system represents between 4.2% and 10.9% of total labor income according as the value of the elasticity of substitution.

12 Straightforward calculus show that this equation admits a unique solution in the interval γ(1-θ)

1-α2+γ ; 1 -θ . 13 Here again, the left-hand side is increasing with respect to T 0 from 0 to +∞ when T 0 goes from γ(1-θ0)

1-α2+γ to 1 -θ 0 . Thus, the equation allows for a unique solution in the set of admissible values for T 0 .

From equation ( 13), the resulting labor supply of the young born in 0 is

h 0 = s 0 + ζ 1 0 (1 + T 0 ) (1 -τ 0 ) w + ζ 1 0 (16) where ζ 1 0 = η 1 0 1 -α 1 1 + α 1 a 12 a 11 (η 1 0 ) ρ 1 and η 1 0 = a 11 (1 -τ 0 ) w a 12 1 1-ρ 1 and savings write s 0 = δ 1 + γ -γ 1-θ 1 T 1 (1 -τ 0 ) w (1 + T 0 ) -R -1 ω (θ 1 , τ 1 ) 1 + δ 1 + γ -γ 1-θ 1 T 1 (17)
In the last expression, since we consider a one-shot increase, we have θ 1 = 1/3 and T 1 results from equation (15). When considering values of the elasticity of substitution (σ 2 = 0.1), we nd that grandparents reduce their time transfers of 0.12 in response to an increase in the retirement age of 0.16. This leads in turn to a signicative fall in labor participation of young adults.

Nevertheless, the overall labor participation is higher since the rise in θ (+0.16) is higher than the fall in h 0 (-0.1). These eects are attenuated for higher values of the elasticity of substitution. In fact, the income eect becomes positive for σ 2 higher than 1, i.e. time transfer becomes a superior good. But, the substitution eect is still dominant and leads to a fall in time transfer and labor supply of the young.

A positive eect of the retirement age on time transfer is obtained when the three following conditions are simultaneously satised : (i) high values of the ratio a 21 /a 22 , (ii) high elasticity σ 2 and (iii) low degree of altruism. For instance, as plotted on Figure 2, if a 21 = 5, a 22 = 1, σ 2 ≥ 9, γ = 0.02, and keeping the values of Table 1 for the other parameters, the retirement age has a small positive eect on time transfer. Since the weight of the purchased input in domestic production is high, the income eect of the rise in the retirement age allows the grandparent to sharply reduce time input (e 2 = 1-θ -T ).

Grandparent chooses to reduce time input e 2 of an amount higher than the fall in time endowment 1 -θ. This results in a rise in time transfer to the young, and consequently in higher labor supply of the young in period 0. Let us now discuss the eect of a permanent increase in retirement age on labor force participation. The dierence with respect to the one-shot increase is that young workers, born in period 0, expect an increase in their second-period income. As equations ( 16) and ( 17) show, labor supply of the young in period 0 depend on θ 1 through two channels.

First, a direct eect appears in equation ( 17): a permanent increase in the retirement age raises the second-period income ω (θ 1 , τ ). Since both extended consumption ct and dt+1 are normal goods in our example, rst-period consumption of the young increases and savings s 0 is reduced. Through this channel, the increase in θ 1 implies a rise in h 0 . Nevertheless, there also exists an indirect eect that passes through time transfer T 1 and that may be ambiguous.

14 It is then clear that permanent postponing of retirement has also an amboguous eect on labor participation of young parents.

Conclusion

The purpose of this paper was to study the consequences of prolonging activity in a setting where family transfers matter. We show that owing to intergenerational linkages, a change in the labor participation of one generation is expected to aect the employment rate of the other generation. Interestingly, we nd that this family externality may be either positive or negative. In some circumstances, postponing retirement may have a boosting eect on the labor market, in that it increases the labor force participation of both the young and the old workers. This nding, which is innovative with respect to the previous literature, is of importance with respect to the nancing of pension scheme, as delaying retirement will impact the amount of expected additional payroll taxes.

As our primary aim was to show that intergenerational relationships and family support have to be taken into account when studying the functioning of the labor market, we have restricted our attention to the consequence in the retirement age. An interesting extension would be to explore the consequence of alternatives change in the pay-as-yougo pension system as increasing contribution rate or decreasing replacement rate and compare the results with those from changing the retirement age. Moreover, we have considered a simple framework with only one type of transfer, i.e. grandchild care. Several extensions of this model may come to mind. First, it would be useful to account for nancial gifts made by the parents to their children. Second, transfers may ow in the reverse direction and it could be that children pay for the services and time transfers provided by the parent. Third, older workers may themselves have alive parents, which would give rise to a trade-o between caring for elders and helping children. This suggests that the consequences and magnitude of the underlying family externality have to be examined within an extended framework, and we leave this issue for future research.

14 Indeed, it is straightforward to see that s 0 is increasing with T 1 , but T 1 is aected by θ 1 and T 0 .

Figure 1

 1 Figure 1 plots the impact of delaying retirement on the time transfer and labor supply, as a function of the elasticity of substitution between purchased input and time in the domestic production.

Figure 1 :

 1 Figure 1: Increasing retirement age in period 0 (a 21 = 1, γ = 0.2)

Figure 2 :

 2 Figure 2: Increasing retirement age in period 0 (a 21 = 5, γ = 0.02)

Table 1 .

 1 ParametersFirst period of life weight of the market good in the consumption function

	α 1	0.5

For an overview of the consequences of endogenous labor supply in models of family transfers, see the survey of[START_REF] Laferrère | Microeconomic models of family transfers[END_REF].

The unequal distribution of grandchild care is driven to a higher extent by better labor market potential than weaker nancial status of the recipient.

In Europe, grandchild care is also much more frequent for women than the receipt of nancial gift.

The presence of grandchildren is in fact only necessary for grandchild care to take place. Another interpretation of our model is to assume that parents provide time to their children, even if the latter do not have themselves children. A very dierent framework would be to consider a three-generational framework, with specic investment decisions in grandchildren (related to human capital). On this issue, see for instance[START_REF] Duo | Grandmothers and granddaughters: Old age pension and intrahousehold allocation in South Africa[END_REF].

Appendix

We calculate the second-order derivatives of Ψ. From the following denition of σ 2 g :


« Se passer des imams formés à l'étranger suppose d'être capable de répondre aux attentes des musulmans de France, ce qui est loin d'être le cas » Le politiste et islamologue Franck Frégosi liste, dans une tribune au « Monde », les limites de la fin du recours aux imams payés et formés hors de l'Hexagone. Ce « n'est qu'une réponse partielle à un défi bien plus vaste », met-il en garde.

La fin des imams fonctionnaires étrangers détachés en France marque une nouvelle étape dans la gouvernance de l'islam hexagonal. Cette information n'a surpris personne : elle figurait déjà, en février 2020, dans le discours de Mulhouse (Haut-Rhin) d'Emmanuel Macron, qui énonçait les grandes lignes de la politique de lutte contre le séparatisme islamique.

L'arrêt des filières de détachement d'imams fonctionnaires étrangers en France, entré en vigueur au 1 er janvier, est censé mettre un terme à l'islam consulaire, c'est-à-dire à une politique qui, depuis les années 1990, consistait, sur la base d'accords bilatéraux, à délivrer des visas de séjour pour une durée de quatre ans à des contingents d'imams. Ceux-ci étaient non seulement salariés par des Etats musulmans mais aussi formés hors de France, dans des sociétés où l'islam est la religion dominante de la population, la religion officielle des pouvoirs publics et où des pans de législation sont marqués par des emprunts, à des degrés divers, à la législation islamique.

Cette décision d'y mettre fin s'inscrit dans une politique gouvernementale plus large : tout en faisant de la laïcité en valeurs (plus qu'en droit) un horizon indépassable opposable à tous les cultes, le ministre de l'intérieur, Gérald Darmanin, et le chef de l'Etat ne s'interdisent pas de vouloir peser sur les contours organisationnels et même religieux de l'islam, afin de l'ajuster à leurs attentes politiques. Certains Etats, dont les dirigeants sont entrés en délicatesse avec nos responsables, sont ainsi invités à cesser d'envoyer dans l'Hexagone des professionnels du culte qui étaient rétribués par leurs soins. Face aux aspirations de ces Etats à exercer un contrôle religieux sur leurs diasporas respectives, le gouvernement laïque français entend ainsi opposer un souverainisme religieux.

Lire aussi l'analyse (2021) | Article réservé à nos abonnés Le double pari d'Emmanuel

Macron sur l'islam consulaire Cette « gallicanisation » [en référence à la doctrine politico-religieuse affirmant la spécificité de l'Eglise de France face à l'autorité du pape] forcée de l'islam en France conduit à considérer que les musulmans de France devraient pouvoir disposer de leurs propres imams, formés sur le territoire national, des clercs à la fois religieusement compétents, et familiers des usages et de la spécificité de l'histoire française de cette religion comme de son cadre juridique d'exercice.

Les autorités françaises ont longtemps privilégié les tenants de l'islam policé des chancelleries étrangères, réputé plus fiable politiquement parce que contrôlé par des régimes autoritaires partenaires des gouvernements français successifs. Mais, depuis des décennies, certains imams et de nombreux musulmans parmi les jeunes générations appelaient de leurs voeux cet impératif de gallicanisation. Cette intention, louable, ne doit cependant pas faire illusion. Abondamment relayée par la presse, cette décision concerne tout au plus trois cents fonctionnaires étrangers, en majorité turcs, algériens et marocains, soit environ 10 % des imams officiant en France au quotidien, selon diverses estimations.

Connaissances profanes

Vouloir se passer de la contribution des imams formés hors de l'Hexagone suppose d'être en capacité de répondre, de France, aux attentes religieuses des fidèles musulmans, ce qui est encore loin d'être le cas. Depuis le discours de Mulhouse, des institutions comme la Grande Mosquée de Paris se sont empressées de tisser des partenariats avec d'autres fédérations, ouvrant diverses antennes de l'institut parisien Al-Ghazali (à Marseille, à Istres, aux Mureaux…), afin de former un personnel religieux qui fait toujours défaut à l'islam de France. D'autres institutions, comme la Grande Mosquée de Lyon, avaient pris une longueur d'avance en la matière, en mettant sur pied, depuis plusieurs années, des formations destinées à des imams et à des aumôniers, sans parler de la fédération Musulmans de France (anciennement Union des organisations islamiques de France) qui fut sans doute l'une des premières à se prononcer pour des formations depuis la France durant la décennie 1990. Et à avoir agi en conséquence. Certaines mosquées régionales ont également mis en place des cycles de cours ouverts aux imams en poste ou aux futurs impétrants.

Lire aussi | Article réservé à nos abonnés Pourquoi la fin des « imams détachés » est un mirage

Encore faudrait-il s'assurer que ces formations religieuses font appel à des personnalités à la fois détentrices d'un savoir religieux avéré et de connaissances plus profanes, tels que les savoirs pratiques en lien avec les transformations qui affectent la société. Si l'on se prive d'un imam formé à l'étranger, ce n'est pas pour y substituer un imam formé sur place mais étranger aux réalités françaises contemporaines.

La question du coût

Depuis plus de dix ans, il existe des formations dûment reconnues, dites « civiles et civiques », dispensées par des universités françaises qui délivrent des enseignements aussi bien en droit de la laïcité et en droit des cultes qu'en histoire des rapports Etat-religions, ou en sciences sociales du religieux et de la sécularisation. Nombre d'imams indépendants les fréquentent et y côtoient leurs homologues des autres cultes (catholique, protestant, juif, etc.) ainsi que des personnes concernées par la pluralité religieuse en régime de laïcité. Il suffirait donc peut-être d'inviter tous les autres à les suivre sans réserve.

Reste cependant une ultime question, encore non tranchée à ce jour : ne plus vouloir que des imams soient financés par des Etats étrangers mais par des associations de droit français suppose que celles-ci soient en mesure d'assurer ce coût financier. Or, aujourd'hui, seuls les imams détachés [qui perdront définitivement leur statut en avril] jouissent d'une bonne rémunération et souvent d'un logement affecté. Le statut de la plupart des autres reste précaire : ils sont rarement salariés et souvent assujettis au bon vouloir des présidents d'association.

Lire aussi la tribune (2020) | Article réservé à nos abonnés « Il est temps d'accorder aux imams déjà en activité la considération à laquelle ils ont droit » Ajouter à vos sélections Ajouter à vos sélections La fin des imams détachés n'est donc qu'une réponse partielle à un défi bien plus vaste. Vouloir poser les bases d'un imamat de France devrait commencer par chercher à recueillir les besoins et les attentes de ces acteurs de premier plan de l'islam au quotidien que sont les imams, plutôt que les réduire à de simples « sujets » de politiques publiques.

Franck Frégosi est directeur de recherche au Groupe sociétés, religions, laïcités (GSRL) du CNRS. Il enseigne à l'institut d'études politiques d'Aix-en-Provence et a codirigé, entre autres, l'ouvrage « L'Islam et la cité. Engagements musulmans dans les quartiers populaires » (Presses universitaires du Septentrion, 2017).