Nicolas Jacquemet
email: knicolas.jacquemet@univ-paris1.fr

Jean-Marc Robin
email: kjeanmarc.robin@sciences-po.fr

Elizabeth Caucutt

Pierre-André Chiappori

Hélène Couprie

Annick Dauphin

Olivier Donni

Bernard Fortin

Guy Lacroix

Costas Meghir

Rob Shimer

Aloysius Siow

Marriage with Labor Supply *

Keywords: Marriage search model, Collective labor supply, Structural estimation. JEL classification: C78, D83, J12, J22. Résumé Marché du mariage, Modèle collectif d'offre de travail, Estimation structurelle

 to allow for labor supply. We characterize the steady-state equilibrium when exogenous divorce is the only source of risk. The estimated matching probabilities that can be derived from the steady-state flow conditions are strongly increasing in both male and female wages. We estimate that the share of marriage surplus appropriated by the man increases with his wage and that the share appropriated by the woman decreases with her wage. We find that leisure is an inferior good for men and a normal good for women.

Introduction

One of the key issues in understanding how tax policies affect labor supply is the intra-household allocation of time and consumption. This is in particular the case of welfare benefits, such as the Working Family Tax Credit program in the UK and the Earned Income Tax Credit in the US, aimed at providing work incentives and a safety net against poverty at the same time. The models used to address these issues typically take the household as a unit with unitary preferences (e.g. [START_REF] Eissa | Taxes and the labor market participation of married couples: the earned income tax credit[END_REF]. The Collective models of the family [START_REF] Chiappori | Collective Labor Supply and Welfare[END_REF][START_REF] Chiappori | Collective Labor Supply and Welfare[END_REF] offer a solution for improvement by modeling intrahousehold resource allocation, but the interest of this framework for policy evaluation is in turn limited by its inability to predict the impact of welfare policies on the sharing rule. Yet, the factors influencing the sharing rule, such as sex ratios or rules about divorce, are now well understood. 1 They are distribution factors conditioning the process of match formation and implicitly determining the threats that each household member can summon in the strategic negotiation for sharing resources. The ultimate model to evaluate welfare policies for the family must therefore be an equilibrium model of both match formation and intra-household resource allocation. The present paper offers one attempt at constructing such a model.

The literature on the family can be broadly classified into three main strands. A first series of papers address the issue of intra-household resource allocation for a given population of families, without worrying about their formation. The oldest papers used bargaining models. 2 Becker's (1981) theory of the benevolent dictator is a special theory of efficient resource sharing that does not use Nash bargaining. Chiappori's Collective model is another one, in many ways the least restrictive of all. 3 Then, in the last ten years a large variety of search-matching models of marriage markets have been proposed to explain certain stylized facts or trends such as declining marriage rates or increasing female college graduation rates, and to analyze various policies affecting the family. 4 At the same time, the general theory of search and matching in marriage markets (existence of equilibrium, multiplicity, conditions of positive assortative mating, PAM, etc.) was further developed, 5 but generated almost no empirical microeconomic applications. [START_REF] Wong | Structural Estimation of Marriage Models[END_REF] stands like one rare exception. 1 See [START_REF] Grossbard-Shechtman | A Theory of Allocation of Time in Markets for Labour and Marriage[END_REF], [START_REF] Brien | Racial Differences in Marriage and the Role of Marriage Markets[END_REF], [START_REF] Lundberg | Do Husbands and Wives Pool Their Resources? Evidence from the United Kingdom Child Benefit[END_REF], [START_REF] Chiappori | Marriage market, divorce legislation, and household labor supply[END_REF], [START_REF] Delboca | Household Time Allocation and Modes of Behavior: A Theory of Sorts[END_REF], Amuedo-Dorantes and Grossbard (2007), Seitz (2009).

2 See [START_REF] Manser | Marriage and Household Decision-Making: A Bargaining Analysis[END_REF][START_REF] Mcelroy | Nash-Bargained Household Decisions: Toward a Generalization of the Theory of Demand[END_REF][START_REF] Lundberg | Separate Spheres Bargaining and the Marriage Market[END_REF], 1996). 3 See [START_REF] Chiappori | Non-unitary Models of Household Behavior: A Survey of the Literature[END_REF] for a recent survey of non-unitary models of the household. Chiappori's seminal contributions generated a long list of papers building on the model of the family as a Pareto equilibrium. We can only cite a few of them: [START_REF] Browning | Income and Outcomes: A Structural Model of Intrahousehold Allocation[END_REF], [START_REF] Fortin | A Test of the Unitary and Collective Models of Household Labour Supply[END_REF], [START_REF] Browning | Efficient Intra-Household Allocations: A General Characterization and Empirical Tests[END_REF], [START_REF] Chiappori | Marriage market, divorce legislation, and household labor supply[END_REF], [START_REF] Mazzocco | Saving, Risk Sharing, and Preferences for Risk[END_REF]Mazzocco ( , 2007)), [START_REF] Blundell | Collective Labour Supply: Heterogeneity and Non-Participation[END_REF], etc. Note also that the assumption of efficient allocations within the family has been disputed, in particular, by [START_REF] Delboca | Household Time Allocation and Modes of Behavior: A Theory of Sorts[END_REF][START_REF] Waldvogel | Fast Construction of the Fejér and Clenshaw-Curtis Quadrature Rules[END_REF], 2009). 4 See e.g. [START_REF] Aiyagari | On the State of the Union[END_REF], [START_REF] Greenwood | Women on Welfare: A Macroeconomic Analysis[END_REF], [START_REF] Caucutt | Why Do Women Wait? Matching, Wage Inequality, and the Incentives for Fertility Delay[END_REF], [START_REF] Gould | Waiting for Mr. Right: rising inequality and declining marriage rates[END_REF], [START_REF] Fernandez | Love and Money: A Theoretical and Empirical Analysis of Household Sorting and Inequality[END_REF], [START_REF] Chiappori | Collective Labor Supply and Welfare[END_REF]Weiss (2006, 2007), [START_REF] Chiappori | Birth Control and Female Empowerment: An Equilibrium Analysis[END_REF] 5 See e.g. [START_REF] Sattinger | Search and the Efficient Assignment of Workers to Jobs[END_REF], [START_REF] Lu | Matching and Expectations in a Market with Heterogeneous Agents[END_REF], [START_REF] Burdett | Marriage and Class[END_REF], [START_REF] Shimer | Assortative Matching and Search[END_REF], [START_REF] Sattinger | Search and the Efficient Assignment of Workers to Jobs[END_REF], [START_REF] Eeckhout | Bilateral Search and Vertical Heterogeneity[END_REF], Eeckhout and Kircher (2010b,a).

2 Documents de Travail du Centre d'Economie de la Sorbonne -2011.50 Recently, the perfect-information assignment framework of [START_REF] Shapley | The assignment game I: The core[END_REF] was revived in order to integrate the collective model within a matching framework. 6 In this paper, we aim at a similar target. However, instead of assuming a frictionless environment, we assume that single individuals randomly search for a partner and that they can date only one person at a time. We design a search-matching model of the marriage market with labor supply by building on the seminal works of [START_REF] Sattinger | Search and the Efficient Assignment of Workers to Jobs[END_REF], [START_REF] Lu | Matching and Expectations in a Market with Heterogeneous Agents[END_REF] and [START_REF] Shimer | Assortative Matching and Search[END_REF]. The equilibrium model of this paper thus defines the outside option as the value of remaining single, which is equal to the instantaneous utility of the wage plus the option value of an eventual future marriage. Couples are formed when enough public goods are produced in the association. The surplus is divided via Nash bargaining.

There are many reason that explain our preference for search models. First, casual experience seems to suggest that it takes time and trial to find the right partner in life. Second, sequential search naturally yields mismatch. Third, it is easy with a search model to deal with continuous individual characteristics and difficult in a perfect-assignment model. Fourth and finally, forward looking behavior and risk are natural ingredients of search-matching models.

Despite a rather complicated structure the model remains tractable thanks to the steady state assumption. Although an important application of the matching framework is understanding long-term demographic changes such as increasing divorce and remarriage rates, we assume that these changes are slow enough for a steady state to hold at least approximately at all times. In other words, flows can vary over time in a trended or cyclical way, but net flows must remain small compared with gross flows. We show that steady-state flow conditions deliver important identifying restrictions on matching probabilities, and indirectly on the relationship between transfers-or the sharing rule defined as the share of male transfers in total transfers-and wages.

The steady-state flow conditions relate the matching probability to wage distributions in a simple way. Despite a small correlation between spouses' wages (25%), we find that the matching probability increases in both wages exponentially. So its shape is rather flat for most wages but very steep when both wages cross the median. We take this result as a strong indication of positive assortative matching, albeit unconventional.

Under the steady-state restriction, a lot of information can be drawn from cross-section data. Using SIPP data we observe that married men earn more than anybody else, and that the distribution of wages is about the same for married women, single women and single men. The model's estimates let us conclude from these facts that a man who is more productive in the labor market is also more attractive in the marriage market both by the higher income that he brings to the family and also by the larger public good value that he contributes to. Women contribute to family income and marriage surplus in the same way, but female wage is a less efficient input of household production than male wage. This makes a high wage a relatively less attractive trait in a woman than in a man. Consequently, high-wage men are over-represented among married couples and they are in a favorable position to bargain a higher share of the rent.

Hours worked do not bring any additional information on matching but allow to identify preferences for consumption and leisure. Married men, at all wages, work more that anybody else and married women less. We show that intrahousehold transfers cannot explain this fact.

High-wage men take a bigger share of the rent and high-wage women a smaller share. If men, receiving bigger transfers, work more, it has to be that leisure is an inferior good for them. For women, the opposite is true: more non labor income implies more specialization into household production. These results should not be taken at face value. The model is overly simplistic. First there are many dimensions of individual heterogeneity that matter in marriage decisions. Second, the reasons that make marriage attractive are certainly more sophisticated and diverse than household production, the way we model it in this paper. Children, for example, are certainly an important component of the marriage externality, which deserves to be properly modeled. Third and lastly, but the list could be longer, individuals and couples are subject to shocks that we rule out completely. Future versions of the model will see to that. This paper's main achievement is to show that search-matching models, with realistic distributions of individual heterogeneity, are more than theoretical objects of interest, and that they can be used efficaciously in empirical work.

The layout of the paper is as follows. First we construct the model. Second we study identification. Third we estimate the model non parametrically. The last section concludes. An appendix details the numerical techniques used to perform the nonparametric analyses.

The Model

The model builds on [START_REF] Shimer | Assortative Matching and Search[END_REF], which we extend to allow for labor supply decisions, non symmetric equilibria, and a general matching function, not necessarily quadratic (Shimer and Smith) or linear [START_REF] Tröger | Matching Heterogeneous Agents with a Linear Search Technology[END_REF].

Basic Setup

We consider a marriage market with L m males and L f females. The number of married couples is denoted by N and the respective numbers of single males and single females are

U m = L m -N and U f = L f -N.
Individuals differ in labor productivity, x ∈ [x, x] for males and y ∈ [y, y] for females. Let ℓ m (x) and ℓ f (y) denote the density functions of the measures of males of type x and females of type y, with L m = ´ℓm (x) dx and L f = ´ℓ f (y) dy. The wage densities for the sub-populations of singles are u m (x) and u f (y), with U m = ´um (x) dx and U f = ´u f (y) dy. The measure density of couples of type (x, y) is n (x, y), with N = ˜n(x, y) dx dy and ℓ m (x) = ˆn(x, y) dy + u m (x) ,

(1)

ℓ f (y) = ˆn(x, y) dx + u f (y). (2)
In this paper, labor productivity is the only permanent source of individual heterogeneity. This is also the first dimension to consider as we are interested in modeling family labor supply.

However, we shall later introduce a match-specific heterogeneity component. We assume that only singles search for a partner, ruling out "on-the-marriage" search. The number of meetings per period is a function of the numbers of male and female singles,

M(U m ,U f ), and λ i = M(U m ,U f ) U i
is the instantaneous probability that a searching individual of gender i = m, f meets with a single of the other sex. We also denote λ = M(U m ,U f ) U m U f . All datings do not end up in wedlock. We assume that there exists a function α(x, y) ∈ [0, 1] indicating the probability that a match (x, y) be consummated. The matching probability is an equilibrium outcome that will be later derived from fundamentals. The matching set is the support of α. Lastly, matches are exogenously dissolved with instantaneous probability δ .

Equilibrium Flows

In steady state, flows in and out of the stocks of married couples of each type must exactly balance each other out. This means that, for all (x, y),

δ n (x, y) = u m (x) λ m u f (y) U f α(x, y) = λ u m (x) u f (y) α(x, y). ( 3
)
The left-hand side is the flow of divorces. The right-hand side measures the flow of new (x, y)marriages. It has three components: a single male of type x, out of the u m (x) ones, meets a single female with probability λ m ; this woman is of type y with probability u f (y)/U f ; the knot is tied with probability α(x, y).

Integrating equation (3) over y, we obtain the steady-state flow condition for the stock of all married males of type x: δ ˆn(x, y) dy = λ u m (x) ˆu f (y)α(x, y) dy.

Using equation (1) to substitute ´n(x, y) dy out of this equation yields

δ [ℓ m (x) -u m (x)] = λ u m (x) ˆu f (y)α(x, y) dy, or, equivalently, u m (x) = δ ℓ m (x) δ + λ ´u f (y)α(x, y) dy . ( 4
)
By symmetry, the equation defining the equilibrium distribution of wages in the population of single females is

u f (y) = δ ℓ f (y) δ + λ ´um (x)α(x, y) dx .
(5)

Optimal Labor Supply

Individuals draw utility from consumption and leisure. Let

v m (x, xT + t) = xT + t -A m (x) B m (x) , (6)
denote the indirect utility for a man of wage x and non-labor income t. T denotes total time endowment, B m (x) is an aggregate price index, with B m (0) = 1, and A m (x) is the minimum expenditure to attain a positive utility, with A m (0) = 0.7 The linearity of utility with respect to non labor income will produce a simple rent sharing mechanism. But other specifications are possible.

For tractability, we rule out labor market non participation. So, hours worked straightforwardly follow from indirect utility by application of Roy's identity:

h m (x, xT + t) = T -- ∂ 1 ∂ 2 v m (x, xT + t) (7) = T -A ′ m (x) -b ′ m (x)[xT + t -A m (x)],
where ∂ 1 and ∂ 2 denote partial derivatives, a prime (such as in b ′ m and A ′ m ) denotes a derivative, and b m (x) = log B m (x). A standard specification is A m (x) = xa m and B m (x) = x b m , yielding the linear expenditure system:

xh m = xT -xa m -b m [xT -xa m + t].
We use symmetric definitions for females. In particular,

h f (y, yT + t) = T -A ′ f (y) -b ′ f (y)[yT + t -A f (y)], (8)

Optimal Rent Sharing Between Spouses

Let W m (v, x) denote the present value of marriage for a married male of type x receiving a flow utility v, and let W m (x) denote the value of singlehood (derived in the next section). The flow value of a marriage contract delivering v utils is

rW m (v, x) = v + δ [W m (x) -W m (v, x)] ,
where r is the discount rate and the second term of the right-hand side is the option value of divorce. We define marriage surplus for males as

S m (v, x) = W m (v, x) -W m (x) = v -rW m (x) r + δ . ( 9
)
Spouses share resources through monetary transfers t m and t f respectively for husband and wife. These transfers divide the value of the public goods produced in the household. Although transfers can be positive or negative, both should be positive in equilibrium, otherwise one is better off remaining single. We assume that spouses share resources cooperatively using Generalized Nash Bargaining with bargaining coefficient β , whereby transfers t m and t f solve max

t m ,t f S m (v m (x, xT + t m ), x) β S f v f (y, yT + t f ), y 1-β subject to the condition t m + t f ≤ C(x, y) + z.
We thus assume that marriage is made attractive by the production of public goods, whose value is the sum of two components. The first one is a deterministic function of wages, C(x, y), while the other, z, is a match-specific component that is drawn independently of wages from a zero-mean distribution denoted G. Public goods may take various forms like home production or other household services making certain purchases redundant. For example, the possibility of talking to each other is a source of entertainment that makes purchasing books or theatre tickets less imperative to married couples. In the same way as one imputes housing services to home owners using hedonic prices, one can think of C(x, y) + z as an imputed provision for all the public goods and services produced in the marriage. Children is a well-known example of such "public goods" (??). This analogy might be disputed on the ground that single individuals can adopt children.

It is yet more difficult for singles to find children to adopt and to rear them. Marriage (or cohabitation) can be seen as a more efficient technology to produce children, the value of which could be imputed by looking at the cost of adoption.

Of course, this framework is overly simplistic. Individuals differ in many other ways but labor productivity. For example, the child rearing technology is of lesser value for divorced individuals with children from first marriage. Indeed, the main reason why we have introduced the match-specific component of the public good, z, which allows some couples with wages (x, y) to match and others not to, is that there are multiple personal characteristics that enter the definition of attractiveness, which we do not take into account.8 Designing empirically tractable multidimensional matching models with random search is definitely a promising area for further research.9

With quasi-linear utility functions, the solution is trivially found to be such that

t m (x, y, z) -s m (x) = β [C(x, y) + z -s m (x) -s f (y)],
(10)

t f (x, y, z) -s f (y) = (1 -β )[C(x, y) + z -s m (x) -s f (y)], (11)
where we denote

s m (x) = B m (x)rW m (x) -xT + A m (x), (12)
s f (y) = B f (y)rW f (y) -yT + A f (y). ( 13
)
Functions s m and s f are the non labor incomes that exactly compensate singles for not being offered the marriage option. Hence, a configuration of types (x, y, z) induces marriage only if the negotiated transfers t m ,t f exceed thresholds s m , s f .

Two dating bachelors decide to match if the overall surplus is positive, i.e. s(x, y)

+ z > 0 with s(x, y) = C(x, y) -s m (x) -s f (y). The matching probability then follows as α(x, y) = Pr{s(x, y) + z > 0|x, y} = 1 -G[-s(x, y)].
(14)

The Value of Singlehood

The (flow) value of being single for males is

rW m (x) = v m (x, xT ) + λ m E (y,z) max {S m [v m [x, xT + t m (x, y, z)], x] , 0} = v m (x, xT ) + λ m ¨max {S m [v m [x, xT + t m (x, y, z)], x], 0} dG(z) u f (y) U f dy.
Using the expression for marriage surplus ( 9), and substituting s m (x) for rW m (x) using ( 12), we obtain:

s m (x) = λ β r + δ ¨max{C(x, y) + z -s m (x) -s f (y), 0} dG(z) u f (y) dy. ( 15
)
A similar expression can be derived for females: 16)

s f (y) = λ (1 -β ) r + δ ¨max{C(x, y) + z -s m (x) -s f (y), 0} dG(z) u m (x) dx. (

Equilibrium

An equilibrium is a fixed point (u m , u f , s m , s f ) of the following system of equations, where the first two equations determine equilibrium wage distributions for singles (derived from ( 4) and ( 5)), and the last two equations determine equilibrium values of singlehood:

u m (x) = ℓ m (x) 1 + λ δ ´u f (y)α(x, y) dy , ( 17
)
u f (y) = ℓ f (y) 1 + λ δ ´um (x)α(x, y) dx , ( 18
)
s m (x) = λ β r+δ ˜max{z +C(x, y) -s f (y), s m (x)} dG(z) u f (y) dy 1 + λ β r+δ U f , ( 19
)
s f (y) = λ (1-β ) r+δ ˜max{z +C(x, y) -s m (x), s f (y)} dG(z) u m (x) dx 1 + λ (1-β ) r+δ U m , (20)
with

U m = ´um (x) dx, U f = ´u f (y) dy, λ = M(U m ,U f ) U m U f , and α(x, y) = 1 -G[-C(x, y) + s m (x) + s f (y)].
We write equations ( 19), (20) in that form so that s m and s f become fixed points of contracting operators given u m and u f (see [START_REF] Shimer | Assortative Matching and Search[END_REF]. [START_REF] Shimer | Assortative Matching and Search[END_REF] prove the existence of an equilibrium for a simpler version of the model. They consider a symmetric equilibrium with a quadratic matching function (i.e. λ constant). The common distribution of singles (u = u m = u f ) is the solution to an equation similar to equations (4) or (5) :

u(x) = ℓ(x) 1 + λ δ ´u(y)α(x, y) dy , (21)
that can be shown to be contracting once u is reparameterized as v = log(u). However, the general equilibrium fixed-point operator that involves α as well as u is not globally contracting. Shimer and Smith show that an equilibrium exists but it is not necessarily unique. [START_REF] Tröger | Matching Heterogeneous Agents with a Linear Search Technology[END_REF] prove the existence of an equilibrium in u for all α (the first step of Shimer and Smith's proof) for the linear matching case (λ = 1/ √ U).

Data

In this section we present the data used in estimation, and we emphasize a few salient facts on wage and hour distributions that the model is challenged to replicate.

Demography of Marriages and Divorces

In the US in 2001, 30.1% of men (24.6% of women), 15 years and plus, were not married and 21 % (23.1%) were divorcees [START_REF] Kreider | Number, Timing, and Duration of Marriage and Divorces: 2001[END_REF]. The median age at first marriage was 24 for men and 21.8 for women. Table 1 displays the percents of men and women of various cohorts who had not married at different ages. People are generally getting married later, but women persistently earlier than men.

In 2001, the median duration of first marriages was 8.2 and 7.9 years, respectively, for men and women. The median duration between first divorce and remarriage, for those married two times, was 3.3 years and 3.5, and second marriages lasted 9.2 and 8.1 years on average. About 75-80% of first marriages, depending on cohorts, reached 10 years, 60-65% 20 years, 50-60% 30 years. This indicates a separation rate of around 2.5% per year. For second marriages, 70-80% reached 10 years, 55% 15 years, and 50% 20 years, consistently with a slightly higher separation rate, around 3% annual.

According to survival data the median marriage duration should therefore be of 23-28 years instead of 8-9 years. The Poisson assumption is at odds with the data because a large proportion of marriages never end, and those who end in divorce do it relatively fast, in the first two years.

One way of making divorce rates non stationary in the model is to permit z to change rapidly. Thus, marriages resulting from a very large z would end fast if new, likely lower values are drawn soon.

Wage and Labor Supply Data

We use the US Survey of Income and Program Participation (SIPP) from 1996-1999. For every quarter that an individual is in the panel we collect information on the labor market state at the time of the survey, wages if employed, the number of hours worked, gender, and the corresponding information for the respondent's spouse if married. Our sample is restricted to individuals who are not self-employed or in the military, between the ages of 21 and 65.

We assume the environment stationary and calculate individual mean wages over employment spells, and mean hours worked over all quarters including non-employment spells. By this way, we somewhat reduce the transitory noise in wages and hours, and we reduce the number of labor-market non-participation spells (with declared hours equal to zero). Then we drop all observations with zero hours worked (individuals and individuals' spouses never employed in the 4-year period). This is definitely not a satisfactory procedure but the model cannot deal (for the moment) with both the extensive and the intensive margins of labor market participation. We also trim the 1% top and bottom wages to clean the data. In our sample, 2N/(2N + U m + U f ) = 50.3% of the population is married, out of L m = 13, 223 males and L f = 13, 925 females. There is a slight deficit of single males vis-a-vis single females: U m /U f = 0.9 (N = 6, 827,U m = 6, 386,U f = 7, 098).

Wage Distributions

Figure 1a shows the Gaussian kernel density estimates of wage distributions by gender and marriage status. Married males have higher and more dispersed wages than single males. Single males, and single and married females exhibit strikingly similar wage distributions. Panel (b) displays the corresponding CDFs. The wage scale is in logs so as to emphasize the non normality of the distributions: both tails are fatter than for a normal distribution.

The figure in Panel (c) plots the joint distribution of wages among married couples, also estimated using a Gaussian kernel density. The most salient feature of this distribution is its very large support. Virtually all wage configurations, like a low male wage and a high female wage or vice versa, exist in the sample. Spouses' wages are only weakly correlated (25%), but the wage density is clearly oriented along the dominant diagonal (see the flat projection in Panel (d)). These patterns (wide support, low correlation) justify the introduction of the matchspecific externality component (z): it allows for imperfect sorting due to unobserved matching characteristics and explains both the low correlation and the large support.

At this stage, the data seem like an impossible challenge for the theory. Such a low correlation between x and y tends to indicate a very little amount of sorting based on wages. However, the estimation of the model has some interesting surprises in store. There is some evidence of complementarity: male hours are higher and female hours lower for high wage men married to low wage women, and male hours are lower and female hours higher for high wage women married to low wage men.

Steady State and Matching Probabilities

In this section, we use the steady-state restriction (3) to estimate the shape of the matching probability α(x, y) as a function of wages. Before showing what it is, we first argue that the steady-state assumption is an acceptable first-order approximation, even in a changing macroenvironment.

Divorce Rates

The steady-state equation (3), by relating marriages to divorces, makes it possible to use data on first marriage ages to learn about marriage duration and divorce frequency. In addition it can help to tell us which estimate in the bracket [2.5, 8] percent per year to choose for δ in the estimation of other structural parameters.

The average probability for a single man of type x to randomly meeting a single woman and marry her is equal to

µ m (x) ≡ λ m ˆu f (y) U f α(x, y) dy.
At the steady-state equilibrium described by equation ( 3), we have

µ m (x) = δ ˆn(x, y) u m (x) dy = δ ℓ m (x) -u m (x) u m (x) ,
with a similar formula for single women. The average marriage rate among single men is the expectation of this quantity:

µ m ≡ ˆµm (x) u m (x) U m dx = δ L m -U m U m .
Using the data displayed in Table 1 9.0 years for women (mean of 13.0). Figure 3b plots the implied average search durations by gender and wage (i.e. log(2)/µ i (x), i = m, f ). Low wage individuals have to wait for a very long time, and women more than men. The waiting time decreases with the wage. So, women get married before men but start searching earlier and take longer.

We obtain this result because there are more female singles (7,098) than male (6,386) in the sample. Given its simplicity, the model can only explain this difference from different wage distributions and different effects of wages on preferences and marriage externality. Of course, many other factors can explain the relative male scarcity in the marriage market, such as a higher mortality rate.

In the end, we conclude that the steady-state assumption is a reasonable approximation because estimates of marriage duration from data on wedlock age are consistent with direct observation. However, Table 1 seems to indicate that marriage and divorce habits do change over time, younger cohorts marrying both later and divorcing more often. This also happens with unemployment rates, for example, which do fluctuate over time (between 4% and 10% for the US). Still the steady-state approximation-job destruction rate divided by job destruction rate plus job finding rate-is a very good approximation. This is because inflows and outflows may fluctuate over time, yet they do so in tandem, so that net flows remain small compared to gross flows in all circumstances, which is what the steady-state restriction effectively means.

The Matching Probability

The equilibrium flow condition (3) implies that So the matching probability is identified up to the multiplicative factor λ (or λ /δ ) from the observed distributions of wages among singles and couples.

λ α(x, y) = δ n(x, y) u m (x)u f (y) . ( 22
In absence of (good) data on datings, it is extremely difficult to separate the meeting probability from the probability of matching given meeting. In order to show the implied shape of α(x, y), we thus arbitrarily chose λ so that the meeting rate would be twice a year for men (λ m = 1/6). The shape of the implied matching probability, as a function of wages, is unaffected by this choice. Note that it is likely that no wage combination (x, y) can induce marriage for sure: α(x, y) ≤ 1 for all (x, y). This condition imposes the lower bound max x,y n(x,y) u m (x)u f (y) = 1.37e -03 on λ /δ , or a minimum meeting rate for men of λ m = .065, or again a maximum of 15 datings per year.

Figure 4 displays the estimated matching probability function obtained by replacing wage densities in equation ( 22) by the estimates plotted in Figure 1. It is unambiguously increasing in both wages. More precisely, the matching probability increases exponentially with both wages and is rather flat for wages below the median with high-wage women having a very low probability of matching with anybody else but a high-wage man. This pattern indicates that positive assortative mating is definitely at work in the marriage market albeit mostly in the upper tail of the wage distribution. This explains why we find such a low overall correlation between spouses' wages.

Transfers and Sharing Rule

We now turn to the identification of transfers t m ,t f , and the sharing rule in particular, defined by the proportion t m /(t m + t f ). We first consider this problem using data on wage distributions only. We shall address the issue of the information that is provided by labor hours in the next section.

Let σ be the standard deviation of the distribution of the match-specific component z, and define G 0 as the distribution of z/σ , that is G(z) = G 0 (z/σ ).

Average Transfers

By inverting the equilibrium relationship between α(x, y) and s(x, y) in equation ( 14), we obtain

s(x, y) = -σ G -1 0 (1 -α(x, y)). ( 23
)
Then, noting that

ˆmax{z + s, 0} dG(z) = s[1 -G(-s)] + ˆ+∞ -s z dG(z),
for all s, equation ( 15) relates s m (x) to total surplus s(x, y) as

s m (x) = β σ λ r + δ ˆµG 0 (α(x, y ′ ))u f (y ′ ) dy ′ , ( 24
) with µ G 0 (α) = -αG -1 0 (1 -α) + ˆ+∞ G -1 0 (1-α) v dG 0 (v).
And by symmetry,

s f (y) = (1 -β )σ λ r + δ ˆµG 0 (α(x ′ , y))u m (x ′ ) dx ′ . ( 25
)
It follows that

s(x,y) σ , s m (x)
β σ and s f (y)

(1-β )σ are identified given G 0 and λ . Actual transfers depend on the realized value of the match specific component of the public good, z, which we never observe. Define instead expected transfers as

t m (x, y) ≡ E[t m (x, y, z)|x, y, s(x, y) + z > 0], t f (x, y) ≡ E[t f (x, y, z)|x, y, s(x, y) + z > 0],
where the expectation is of course conditional on matching, s(x, y) + z > 0. Equations ( 10) and ( 11) imply that

t m (x, y) = s m (x) + β E[s(x, y) + z|x, y, s(x, y) + z > 0] = β σ λ r + δ ˆµG 0 (α(x, y ′ ))u f (y ′ ) dy ′ + µ G 0 (α(x, y)) α(x, y) (26)
and

t f (x, y) = s f (y) + (1 -β )E[s(x, y) + z|x, y, s(x, y) + z > 0] = (1 -β )σ λ r + δ ˆµG 0 (α(x ′ , y))u m (x ′ ) dx ′ + µ G 0 (α(x, y)) α(x, y) . ( 27
)
Hence,

t m (x,y)
β σ and t f (x,y)

(1-β )σ are in turn also identified given G 0 and λ on the support of α(x, y). Mean transfers are proportional to the bargaining power coefficient (β for men, 1 -β for women). Clearly enough, the same transfer can be obtained with a better outside option and a lower β . Collective models do not separate these two sources of bargaining power within the family. Indeed, there is a one-to-one relationship between the minimal utility that the Pareto program assigns to household members and the equivalent utility weight (or the Kuhn-Tucker multiplier). In a bargaining model, however, the weight of each individual (log) surplus in the Nash program (β and 1 -β ) is structurally independent of the minimal utility levels (or outside options). This superior flexibility calls for more data, as identifying β effectively requires separate data on the size of the cake to be shared between parties and the shares themselves. 10With data only on wages and matching, it is not possible to identify σ either. This is because the only information that is used to identify transfers is the frequency of marriage for any particular wage configuration. Marriage occurs in the model when

t m (x, y, z) + t f (x, y, z) = C(x, y) + z > 0.
In absence of any additional information on the component C(x, y) of household production, allowing to anchor it at some known level, we can divide all terms of this equation by any positive number and the inequality remains true for all x, y. In the next section, we shall ask whether labor supply data can help to improve the inference on β and σ given our model. The answer is no.

Average Total Transfers

Average total transfers

t m (x, y) + t f (x, y) = C(x, y) + E[z|x, y, s(x, y) + z > 0]
are identified up to the scale factor σ given β , λ and G 0 . To use equations ( 26) and ( 27) for estimation, we set β = 0.5 and σ = 1000, the order of magnitude of monthly earnings. Moreover, we set G 0 equal to the CDF of a standard normal distribution. We shall later address the issue of the separate identification of G 0 . For the time being, let us assume that it is not too Figure 5a shows an estimate of the average total transfer function, t m (x, y) + t f (x, y), and Panel (b) shows total household earnings (wage times hours worked) for comparison. The two functions have very different shapes. Household earnings are more or less linearly increasing in both wages (the mapping is approximately a linear plane). Total transfers, however, seem largely independent of female wages for all but the highest male wages. Thus, in the marriage market, a higher wage makes a woman more attractive only to high-wage men (say above the median wage). By contrast, a higher wage always makes a man more attractive.

The informational source of this inference is two-fold. First, single men do not have very different wages as single women, whereas married men earn more. This can only reflect a greater demand for high-wage men in the labor market. Second, this general effect being factored out, the matching probability strongly increases in both wages above the median. The joint density of married couples' wages is thus much fatter in that region than the product of wage densities for singles, signaling another selection effect: the configuration of two high wages produces a lot of public goods and is therefore highly demanded.

Of course there may exist alternative explanations, possibly along the line: married men are older, have accumulated more human capital and are thus better paid. A better model would indeed be non stationarity, but this is for the moment, analytically and empirically, out of reach. In addition, it is striking that the marriage gap in wages applies to men and not to women. Now, wage differentials may reflect other differences, like differences in education, and it may be that women value men's education more than most men value women's education, with the exception of highly educated men. This explanation calls for another type of extension, with multidimensional heterogeneity. This one seems easier to grasp.

t m (x,y) t m (x,y)+t f (x,y)

The Sharing Rule

The sharing rule t m (x,y) t m (x,y)+t f (x,y) is identified given β , λ and G 0 , irrespective of σ . Figure 6a plots its estimate using the same calibrations of β , λ and G 0 as in the preceding sections. The median share of total transfers that goes to a married man with a median wage (2.6 in logs) is about one half, with some variance depending on his wife's wage. The same is true for a married woman with a median wage (around 2.4 in logs). This is expected given the arbitrary choice of 1/2 for β . However, it is most remarkable that the variance of transfer shares given one's own wage is much larger for females than for males. This can be seen most clearly in Panels (b) and (c) which plot flat projections in (x, z) and (y, z) planes. Looking at the picture more closely, we see that a linear function of x and y would not be a bad approximation of the sharing rule, with a positive and steep slope in the x-direction (male wages), and a negative and relatively flatter slope in the y-direction (female wages).

Summing up, a man who is more productive in the labor market is also more attractive in the marriage market both by the higher income that he brings to the family, and also by the larger public good value that he generates. Women contribute to family income and marriage surplus in the same way, although female wage is a less efficient input of household production than male wage. This makes a high wage a less attractive trait in a woman than in a man.

Consequently, high-wage men are over-represented among married couples and they are in a better position to bargain a higher share of the rent.

Inference from Hours

We now consider data on hours worked. Let h 0 m (x) ≡ h m (x, xT ) and h 0 f (y) ≡ h f (y, yT ) denote hours for single males and females, respectively. Let h 1 m (x, y, z) ≡ h m (x, xT + t m (x, y, z)) and h 1 f (x, y, z) ≡ h f (y, yT + t f (x, y, z)) denote hours for married individuals. First, note from equation ( 7) that, for any two men with the same wage x, one being married and the other single,

h 1 m (x, y, z) -h 0 m (x) = -b ′ m (x)t m (x, y, z), ( 28
)
where y and z are the other characteristics of the married man (i.e. his wife's wage and the match-specific unobserved component). The match component z being not observed, let us average it out and define

∆ m (x, y) ≡ E(h 1 m |x, y) -h 0 m (x) = -b ′ m (x)β σ t m (x, y) β σ . ( 29
)
The normalized male transfer t m (x,y)

do not help to identify β and σ . We thus obtain a different result from [START_REF] Chiappori | Collective Labor Supply and Welfare[END_REF] first because standard collective models do not have unobserved heterogeneity in the sharing rule (here z), and second because the Pareto equilibrium concept does not allow to define bargaining power and outside options independently-a difference that we have already emphasized.

Partial Identification of Individual Preferences

For a given calibration of β and σ , Chiappori's fundamental result holds: one private good (labor) is enough to identify individual preferences. Indeed, B m (x), B f (y) follow as

B m (x) = exp ˆx 0 b ′ m (x ′ ) dx ′ , (30)
B f (y) = exp ˆy 0 b ′ f (y ′ ) dy ′ . (31)
Moreover, the two remaining unknowns A m (x), A f (y), can easily be recovered from hours worked by single individuals. Setting the transfer equal to 0 in equations ( 7) and ( 8), these equations can be rewritten as linear ordinary differential equations:

d[xT -A m (x)] dx -b ′ m (x)[xT -A m (x)] = h 0 m (x), d[yT -A f (y)] dy -b ′ f (y)[yT -A f (y)] = h 0 f (y).
The solution is

xT -A m (x) = B m (x) ˆx 0 h 0 m (x ′ ) B m (x ′ ) dx ′ , ( 32
) yT -A f (y) = B f (y) ˆy 0 h 0 f (y ′ ) B f (y ′ ) dy ′ , (33)
using initial conditions A m (0) = A f (0) = 0 and B m (0) = B f (0) = 1.

Estimation of Preference Parameters

We estimate income effects, b ′ m (x), by regressing ∆ m (x, y) on t m (x, y). The corresponding population parameter is b ′ m (x) = ´∆m (x, y)t m (x, y)n(x, y) dy ´tm (x, y) 2 n(x, y) dy , and similarly for women:

b ′ f (x) = ´∆ f (x, y)t f (x, y)n(x, y) dx ´t f (x, y) 2 n(x, y) dx .
Then preference parameters A and B can be calculated using equations ( 32), ( 33), ( 30) and (31).11 Figure 7a shows the estimated income effects, -b ′ m (x) and -b ′ f (y), obtained with β = 0.5 and σ = 1000, together with B m (x) and B f (y) (see appendix for estimation details). Panel (b) shows estimates of price effects T -A ′ m (x) and T -A ′ f (y), as well as A m (x) and A f (y). A low-order polynomial approximation is shown (dashed curves) for comparison.

For men, leisure (household production) is an inferior good-higher transfers increase hours worked-whereas for women, it is a normal good. This seems necessary to explain why married men work more than singles and married women less. Note that this is not an uninteresting economic result (as it looks like an explanation by exogenous differences in preferences). We could have found that household transfers are unfavorable to married men, forcing them to work more to compensate. This is not the way it works. Married men have higher wages than other categories of individuals because they are more attractive (selection effect). Better outside options let them draw a bigger share of the surplus. Receiving more transfers, if they also work more, it must be because they like it. The specialization story where men work more because they have a comparative advantage in wage-work-indeed they are better paid-leaving women to perform more household tasks, does not seem to provide a satisfactory description of both wage and hour distributions when all equilibrium restrictions are respected.

Identification of G 0

The identification of the standardized distribution of the match specific component of the public good, G 0 , comes from the residuals of the labor supply equations for couples. Thus, equation ( 28) and ( 29) imply that

h 1 m (x, y, z) -E(h 1 m |x, y) -b ′ m (x)β σ = t m (x, y, z) -t m (x, y) β σ = z σ -E z σ x, y, z σ > - s(x, y) σ , (34)
and similarly for females:

h 1 f (x, y, z) -E(h 1 f |x, y) -b ′ f (y)(1 -β )σ = z σ -E z σ x, y, z σ > - s(x, y) σ . (35)
Suppose for a moment that b ′ m (x)β σ and s(x, y)/σ can be identified without knowledge of G 0 , then the first equation identifies the distribution of z/σ given x, y and conditional on matching:

z σ > - s(x,y)
σ . It follows that the unconditional distribution of z/σ , alias G 0 , is identified if α(x, y) tends to one for some limiting value of (x, y). However, it is unlikely that the decision to marry could be entirely determined by wages, so α(x, y) should be less than one for all x, y. Moreover, the estimators of b ′ m (x)β σ and s(x, y)/σ that we have previously designed depend on G 0 . So, equation ( 34) identifies G 0 as the solution to a complicated inverse problem. It is thus difficult, if not impossible, to construct a nonparametric estimator for the distribution of z/σ . For this reason, we did not attempt to estimate G 0 and postulated a standard normal distribution instead; in which case,

µ G 0 (α) = -αΦ -1 (1 -α) + φ • Φ -1 (1 -α),
where φ is the PDF of the standard normal distribution.12 Figure 8 shows the estimated distribution density of

h 1 m (x, y, z) -E(h 1 m |x, y) -b ′ m (x)β σ + E z σ x, y, z σ > - s(x, y) σ = h 1 m (x, y, z) -h 0 m (x) -b ′ m (x)β σ - s m (x) β σ - s(x, y) σ
for males (labeled "predicted males" in the plot) and the symmetric PDF for females (labeled "predicted females"):

h 1 f (x, y, z) -E(h 1 f |x, y) -b ′ f (y)(1 -β )σ + E z σ x, y, z σ > - s(x, y) σ = h 1 f (x, y, z) -h 0 f (y) -b ′ f (y)(1 -β )σ - s f (y) (1 -β )σ - s(x, y) σ
The solid line plots the truncated PDF of z/σ for married couples (i.e. given z σ > -s(x,y) σ ), assuming z/σ drawn from a normal distribution at meeting time.

Two remarks are in order. First the distributions of the thus standardized hours of married men and women, h 1 m and h 1 f , are remarkably aligned. Second, the truncated distribution of z/σ has a much smaller variance (two orders of magnitude smaller!). This discrepancy reveals the presence of two classical measurement errors with independent and identical distribution.

Equilibrium Computation and Low Order Approximation

In this section we want to verify that we can go backwards, that is, calculate the equilibrium wage distributions and labor supply functions from the previous nonparametric estimates of the structural parameters, namely, the marriage externality function C(x, y) and the preference parameters. To calculate the equilibrium (equations ( 17), ( 18), ( 19), ( 20)) we postulated a Cobb-Douglas meeting function

M(U m ,U f ) = M 0 U 1/2 m U 1/2 f , and estimated M 0 as M 0 = λU 1/2 m U 1/2 f
for the calibrated value of λ . Despite the lack of a global contraction mapping property, we found that the standard fixed-point iteration algorithm, x n+1 = T x n , worked well in practice, even starting far from the equilibrium (like with s m (x) = 0 and u m (x) = ℓ m (x)). Details about the numerical computation of the equilibrium (discretization plus standard fixed point algorithm) can be found in the appendix.

The just-identified nonparametric estimates unsurprisingly deliver a perfect fit of wage distributions and own wage-hour supply functions. However the ability of the model to fit the conditional mean hours given both spouses' wages is limited by the form of the transfer functions, and the model fails to some extent to fit hours for high wage men and low wage women (see Figure 9). Simulating the model using the full nonparametric estimates of the structural parameters yields, as expected, a perfect fit. What is more challenging is to use a small-order polynomial approximation of the functional parameters (technical details in appendix). Figure 10 shows the fit of a 4th order polynomial approximation. It is quite good, despite unwanted undulations of hours at higher wages.

Conclusion

In this paper, we have developed a prototypical version of a search-matching model of the marriage market with labor supply, extending the model in [START_REF] Shimer | Assortative Matching and Search[END_REF] to allow for labor supply. We study its identification and estimation from cross-section data.

The model is rich of interesting lessons. We first show that wage distributions provide useful information on matching patterns. Despite a low correlation between spouses' wages, we estimate a matching probability function that is strongly increasing in both wages. However, if a high wage is always increasing male attractiveness, a high wage makes a female attractive only to high wage males. In consequence, a higher wage provides men with a greater share of the marriage surplus but not women. Finally, with only one private good (labour supply), hours give no additional information on matching and rent sharing. Differences in hours between single and married individuals suggest that labor is a normal good for men, but an inferior good for women.

Many possible extensions of the model easily come to mind, like endogenizing divorce, either through shocks to z or via on-the-marriage search, or like allowing for other dimensions of heterogeneity but wages, or introducing children.

It was sometimes necessary to "shrink" steps by using iterations of the form u p+1 = u p +θ (Tu pu p ) with θ ∈ (0, 1]. A stepsize θ < 1 may help if T is not everywhere strictly contracting.

Fifth, the fact that CC quadrature relies on Chebyshev polynomials of the first kind also allows to interpolate functions very easily between points y 0 = f (x 0 ), ..., y n = f (x n ) using Discrete Cosine Transform (DCT): The fact that the grid (x 0 , ..., x n ) is not uniform and is denser towards the edges of the support interval allows to minimize the interpolation error and thus avoids the standard problem of strong oscillations at the edges of the interpolation interval (Runge's phenomenon).

f (x) = n ∑ k=0 Y k • T k (x
Another advantage of DCT is that, having calculated Y 0 , ...,Y n , then polynomial projections of y = (y 0 , ..., y n ) of any order p ≤ n are obtained by stopping the summation in (A.1) at k = p. Finally, it is easy to approximate the derivative f ′ or the primitive ´f simply by differentiating or integrating Chebyshev polynomials using if k ≥ 2.

In calculating an approximation of the derivative, it is useful to smoothen the function by summing over only a few polynomials. Derivatives are otherwise badly calculated near the boundary. Moreover, our experience is that the approximation:

ˆx x 1{t ≤ x} f (x) dx ≃ n ∑ k=0 w k 1{t ≤ x k } f (x k )
gave similar results as integrating the interpolated function.

We implemented these procedures with numbers of grid points such as n = 50, 100, 500 on a laptop without running into any memory or computing time difficulty.

  Joint wage density for married couples (b) Flat projection on the (x, y) plane

FigureFigure

  Figure 1: Wage distributions

  Figure 2: Mean hours

  -marks show 1955-59 cohort data.

Figure 3 :

 3 Figure 3: Duration of singlehood

Figure 5 :

 5 Figure 5: Total transfers and earnings

Figure 6 :

 6 Figure 6: Sharing rule. Share of total transfers that goes to the man,

Figure 7 :Figure 8 :

 78 Figure 7: Preference parameters (the dotted line correspond to 4th order approximations)

Figure 9 :

 9 Figure 9: Fit of conditional mean hours, h 1 i (x, y)/E(h 1 i |x, y) -1, i = m, f

  are the OLS estimates of the regression of y = (y 0 , ..., y n ) on Chebishev polynomials T k (x) = cos k arccos xeffectively calculated using FFT. A MATLAB code for DCT is, with y = (y 0 , ..., y n ): Y = y([1:n+1 n:-1:2],:); Y = real(fft(Y/2/n)); Y = [Y(1,:); Y(2:n,:)+Y(2*n:-1:n+2,:); Y(n+1,:)]; f = @(x) cos(acos((2*x-(xmin+xmax))/(xmax-xmin)) *(0:n))*Y(1:n+1); :); Y(2:n,:)+Y(2*n:-1:n+2,:); Y(n+1,:)]'; f=@(x,y) cos(acos((2*x-(xmin+xmax))/(xmax-xmin))*(0:n))... *Y(1:n+1,1:n+1)... *cos((0:n)'*acos((2*y'-(ymin+ymax))/(ymax-ymin)));

  cos(k arccos x) ′ = k sin(k arccos x) sin(arccos x) , and ˆcos(k arccos x) dx = ) -cos(k-1)x 2(k-1)

Table 1 :

 1 Percent never married by age Note: The Table reports the percentage of men and women who had never married, by age (in row) and cohort (in column). Source; Kreider (2005), US Census Bureau, SIPP, 2001 Panel, Wave 2 Topical Module.

			Men				Women
		1945 to 1950 to 1955 to 1960 to 1945 to 1950 to 1955 to 1960 to
	Age	1949	1954	1959	1964	1949	1954	1959	1964
	20	20.4	23.0	17.6	15.8	44.8	40.5	36.6	30.2
	25	66.6	59.2	49.9	45.0	78.7	70.1	66.0	59.5
	30	79.7	74.0	68.8	65.6	85.4	80.7	78.1	74.4
	35	86.2	81.7	78.5	76.6	88.3	86.2	84.5	83.0
	40	89.6	85.9	83.6		90.9	89.1	87.7
	45	91.5	88.2			92.1	90.6
	50	93.1				93.0

  on the distribution of first marriage age of the 1955-1959 cohort we estimate both an age at which individuals start searching for a partner (age 0m and age 0 f ) and δ by running jointly the regressions of log survival probabilities on search durations:

		(a) Percent never married by age	(b) Median search duration before marriage
						males
	1
	0.8
	0.6
	0.4
	0.2
	20 0	22	24	26	28	30	32	34	36	38	40
	20	22	24	26	28	30	32	34	36	38	40

log S m (t) = -δ (tage 0m ), log S f (t) = -δ (tage 0 f ),

where S m (t), S f (t) denotes the probability of not being married by age t respectively for men and women. We estimate starting ages age 0m = 17.3 years and age 0 f = 11.8 years. The estimated divorce rate is δ = 8.0% annual. Figure

3a

shows the fit of this simple model, which is good. The implied median first marriage duration is 8.7 years (mean of 12.6 years), which is remarkably similar to the value that can be directly estimated from divorce data.

The median waiting time before marriage is estimated 8.1 years for men (mean of 11.7) and

See Choo and Siow (2006),Choo, Seitz, and Siow (2008b,a),[START_REF] Chiappori | An Assignment Model with Divorce and Remarriage[END_REF],[START_REF] Siow | Testing Becker's Theory of Positive Assortative Matching[END_REF],[START_REF] Chiappori | Investment in Schooling and the Marriage Market[END_REF],[START_REF] Chiappori | Partner Choice and the Marital College Premium[END_REF].

Documents de Travail du Centre d'Economie de la Sorbonne -2011.50

The indirect utility function maximizes the utility of consumption c and labor supply h, for a given wage x and non labor income t, subject to the budget constraint c = xh + t and the participation constraints c > 0 and h < T . 6 Documents de Travail du Centre d'Economie de la Sorbonne -2011.50

From a technical point of view, it also allows to smooth out the discontinuity at the boundary of the matching set.

[START_REF] Wong | Structural Estimation of Marriage Models[END_REF] aggregates individual characteristics into one single index in a Shimer-Smith model with positive assortative mating. An obvious extension of our model would add to z a function of individual characteristics, using single indexes to model complementarity in a simple way. 8 Documents de Travail du Centre d'Economie de la Sorbonne -2011.50

For example,[START_REF] Cahuc | Wage Bargaining with On-the-Job Search: Theory and Evidence[END_REF] use data on firm value-added and wages to identify the bargaining power of workers in an equilibrium search-matching model.

β σ being already identified, this equation identifies b ′ m (x)β σ . As there is no obvious normalization that can be applied to the income effect -b ′ m (x), hours

We set the maximal number of hours T equal to the upper bound of hours in the sample, i.e. T = 667 hours per month (28 full 24-hour days!). 23 Documents de Travail du Centre d'Economie de la Sorbonne -2011.50

We also experimented with mixtures of two normals with varying kurtosis, and obtained similar results.

It can be shown that the error associated to a polynomial approximation (of any order) of an unknown function at any point x is proportional to ∏ n j=0 (xx j ). The Chebyshev points are the {x j } j=0,...n minimizing this quantity.29Documents de Travail du Centre d'Economie de la Sorbonne -2011.50

gratefully acknowledge financial support from ANR-GINDHILA. Robin gratefully acknowledges financial support from the Economic and Social Research Council through the ESRC Centre for Microdata Methods and Practice grant RES-589-28-0001, and from the European Research Council (ERC) grant ERC-2010-AdG-269693-WASP.

Appendix

Computational Details

This appendix shortly describes the numerical tools used for estimation. First, we discretize continuous functions on a compact domain using Chebyshev grids. 13 For example, let [x, x] denote the support of male wages, we construct a grid of n + 1 points as

Second, to estimate wage densities n(x, y)/N, u m (x)/U m and u f (y)/U f on those grids we use kernel density estimators with twice the usual bandwidth to smooth the density functions in the tails. This is important as, for instance, we divide n by u m u f to calculate α according to (22). Additional smoothing is thus required.

Third, many equations involve integrals. Given Chebyshev grids, it is natural to use Clenshaw-Curtis quadrature to approximate these integrals:

where the weights w j can be easily computed using Fast Fourier Transform (FFT). The following MATLAB code can be used to implement CC quadrature [START_REF] Waldvogel | Fast Construction of the Fejér and Clenshaw-Curtis Quadrature Rules[END_REF]:

N=[1:2:n-1]'; l=length(N); m=n-l; v0=[2./N./(N-2); 1/N(end); zeros(m,1)]; v2=-v0(1:end-1)-v0(end:-1:2); g0=-ones(n,1); g0(1+l)=g0(1+l)+n; g0(1+m)=g0(1+m)+n; g=g0/(n^2-1+mod(n,2)); wcc=real(ifft(v2+g)); wcc=[wcc;wcc(1)];

Note that, although Gaussian quadrature provides exact evaluations of integrals for higher order polynomials than CC, in practice CC works as well as Gaussian. On the other hand, quadrature weights are much more difficult to calculate for Gaussian quadrature. See [START_REF] Trefethen | Is Gauss Quadrature Better than Clenshaw-Curtis?[END_REF]. Fourth, we need to solve functional fixed point equations. The standard algorithm to calculate the fixed point u(x) = T [u](x) is to iterate u p+1 (x) = Tu p (x) on a grid. If the fixed point operator T involves integrals, we simply iterate the finite dimensional operator T obtained by replacing the integrals by their approximations at grid points. For example, an equation like


These paper is issued from Biodivalloc programme (ANR05 BDIV02) titled: "From localised products to geographical indications: which tools to conserve biodiversity in mega-biodiverse countries?" and a symposium organised in Dakar in November 2009. ** Main processed fish are : gej (putrified, salted, dried fish), sali (salted-dried), kecax (braiseddried), yeet (putrified, salted, dried volute), pañe (boiled-dried arks), tuffë (boiled-dried murex), yoxos (grilled-dried oyster), tambajang (salted-dried pelagic fish), metora (smoked). *** Other Imrâgen products -tishtar, oil, etc-have the same trajectories than cockles.There is room for collective brands -like Park label-to furnish national markets and West African markets.

In the context of Ocean resource depletion and marine biodiversity erosion, most of initiatives focus on value chain promotion in addition and in combination with all activities related to fisheries co-management, eco-labelling and fair trade models to create synergies and maximize benefits and long term sustainability of the fishery. While industrial fisheries will never be able to cope with ecofriendly criteria, sharps argument favor small-scale or artisanal fisheries to developpe schemes rending better fish quality. Thanks to higher quality, the fishermen have access to premium market, get higher earnings and social provisions.

Illustration of two main trajectories

In developping countries, and especially in West-african sea countries such as Senegal and Mauritania, two main trajectories could be highlighted: in one hand, valorisation of origin-based products (promotion of products as origin), which are more or less « traditional », export oriented and supposed to integrate world market, to improve value chain thanks to international norms, labels and even, ecocertification; in the other hand, local traditional products, with a well established reputation and distributed through diversified commercial chains (local, national and interreregional) controled by women producers, not candidate to ecocertification or official labellisation, but rather to quality signs and collective brand.

The first one, very few and limited (in exchanged volume, extension of production area, involved actors along the sea food chain) could be illustrated through the study case of mullet fishery ecocertification in Mauritania [29, 30 ,31, 32]. Mullet fishery system is the basis of Imrâgen livelihood. Mullet (Mugil cephalus Linné) and processed products are the main (unique) source of income : dried fish (tishtâr), salted and crushed fish (khlî), poutargue, gonads pressed, salted and dried (beydh ezôl) and oil from fish head and internal organs (dhen) are traditionnaly processed and sold by Imrâgen fishermen wifes. With the creation of PNBA (National Park of Banc d'Arguin) in 1976, Imrâgen population is divided into two communities : inside the Marine Protected Areas, strong constraints in terms of resources access and uses are imposed, in favor of « traditional » and « ecological » fishing methods (only « shoulders » net, canoë with veil called « lanches », etc.) ; outside, motorized small-scale fishing with various gillnets develops. Since mid-1990s, processing activities controled by women are in crisis, due to several factors, among which: growing fishing effort linked to increasing fishermen populations (from Mauritania and Senegal) and less selective gears outside the Park ; increasing demand of Mugil gonads sold fresh by wholesalers to Nouakchott companies and exported to European market.

In 2001, IUCN (International Union for Conservation of Nature) and FIBA (International Fund for Banc d'Arguin) decide to support Imrâgen fishery system inside the Park in promoting mullet products as origin. This project, which aims at mid-term an eco-certification of Mugil fishery, has three goals : Mugil stocks preservation, traditional knowledge conservation and Imrâgen economic development. Innovations concern landing fish access (microcredit), production and processing conditions (training in hygienic norms and Western criteria of quality ; distribution of working riel: buckets, pots, knifes, balances, cupboards, etc., new), packaging and sales. During the fishing campaign 2006-2007, a test was proposed to the women of two villages of the Park. It concerns above all poutargue product, which has a high value (around 200 $/kg on world market) and is destinated to European consumers, living in Nouakchott. Poutargue is putting under vacuum to improve the appearance and the preservation. The product is bought to the women between 20 and 30 $M/kg according to its quality, a little more than the price offered by the local wholesaler, and sold around 60 $/kg in Nouakchott in supermarkets as well as in the headoffice of the PNBA. At the end of season, benefices are supposed to be redistributed to the women.

Contacts are established between IUCN and Marine Stewardship Council (MSC), leading international organisation for fisheries eco-certification, whose principles are consistent with FAO-Guidelines for Eco-labelling of marine capture fisheries [START_REF] Fao | Guidelines for the Eco-labelling of Fish and Fishery Products from Marine Capture Fisheries[END_REF]. At the end of November, 2007, MSC declares officially PNBA mullet fishery ecocertification is undergoing full assessment procedures.

Another initiative in favor of « poutargue » promotion is conducted by a Mauritanian NGO, called « Mauritanie 2000 » supported by Slow Fish [START_REF] Food | La sentinelle de la poutargue de mullet des femmes imraguen. Manuel de production[END_REF][START_REF] Boulay | La valorisation des produits imrâgen : une voie durable au service de la diversité biologique et culturelle du parc national du Banc d'Arguin, Mauritanie. Contribution au colloque « localiser les produits[END_REF]. It concerns all the Imrâgen women of the coast (inside and ouside the Park) involved in poutargue production and aims to improve processing techniques and products packaging and so, give opportunities to enter new markets. Today, the poutargue of Imrâgen is one of the ten "sentinels " products, qualified by Slow-Food in Africa. This concept « sentinel » designates local product and the attached know-how, badly known and jeopardised of disappearance.

The second trajectory is well illustrated by Saloum or Casamance cockles exploitation and value-adding initiatives that aim to develop alternatives sustainable activities in a context of rural crisis (drought, population pressure, renewable resource overexploitaiton, decline of cereal culture, migration of young people, etc.) [START_REF] Sarr | Shell's Valorisation Policy in Saloum (Senegal)[END_REF][START_REF] Cormier-Salem | Managing the coastal land-water interface in the tropical delta systems[END_REF]. Most of them focus on women, who play a main role in fish and cockles collect, processing and trade. Initiatives are either exogenous like Enda Graf Sahel project called « Women and Cockles », either endogenous, in the hand of women entrepreners like Alimatou Sarr from Dionewar or supported by Santa Yalla project of Casamance. Women replicate the traditional organisation of « age group » into operational groups and co-operatives (Economic Interest Group) officially recognised and supported by funding agencies. The process of labelling mainly consists of guaranteeing the safety of the product (flushing and washing several times with water, addition of chlorine, three successive cookings) and improving its presentation and its traceability (vacuum bagging, labelling). Innovations also include exploitation techniques in favor of mangrove ecosystem and biodiversity conservation. For example, the Japanese cooperation has introduced oyster seedings in shallow waters. Traditional and sustainable techniques of shellfish collect are also encouraged, such as the "Moundé" baskets, which ensures that the smallest shellfish are not harvested.

Terriorial qualification of seafood: new challenges

Certification tools are not the panacea. As a matter of fact, West African artisanal or processed seafood products are very few to be certified, and even less to candidate to be certified. According to a GTZ-feasibility study on eco-labelling in Senegal [START_REF] Blueyou | Feasability study. Eco-labelling of artisanal coastal fisheries in Senegal. Bueyou & ENDA/REPAO[END_REF], only 4 fish could be candidated : shrimp, lobster, octopus and cuttlefish. Mismatches beetween certification requirement and the reality of many small-scale fisheries are obvious. Also, only few developing countries' fisheries have been applying for certification by the MSC. Main reasons are the predominant small-scale multi-gear and multi-species fisheries, the general lack of data and organisational structure, the lack of fishery management and regulation and also insufficient capacities and capabilities for efficient enforcement [START_REF] Wessells | Product certification and ecolabelling for fisheries sustainability[END_REF][START_REF] Cormier-Salem | Des produits de terroir pour conserver la diversité biologique et culturelle au Sud. Enjeux, acteurs, instruments[END_REF].

In the case of Fair-fish in Senegal, according to his director Billo Heinzpeter Studer [START_REF] Cormier-Salem | Biodiversité littorale et projets de valorisation des productions localisées en Afrique de l'Ouest. Dakar, Sénégal[END_REF], project stops because :1) there was a too long period till first success ; 2) there was a huge gap between demands of Swiss supermarket Migros and conditions of small-scale fisheries in Senegal and so, continuous needs of increasing project funds ; 3) Swiss project management was optimistic : regarding the gap between demands and conditions, there was a too long period of remuneration of the local team by time instead of achievement ; 4) local staff was overstrained. There was a mutual frustration by not fulfilled tasks that leads to cumulated cases of neglect of duties ; 5) one-track orientation and one retail chain lead to dependancy and heteronomy.

Labels and official signs to guaranty products quality « theoritically » allow consumers to make informed choice and producers to secure their market acces (stability of furniture and quality, credibility), increase their income (higher price of local products) and improve their farming and fishing system (sustainable and equitable share of benefices). However, as it is pointed out by many studies on eco-labelling in developping countries [START_REF] Wessells | Product certification and ecolabelling for fisheries sustainability[END_REF][START_REF] Blueyou | Feasability study. Eco-labelling of artisanal coastal fisheries in Senegal. Bueyou & ENDA/REPAO[END_REF][START_REF] Boulay | La valorisation des produits imrâgen : une voie durable au service de la diversité biologique et culturelle du parc national du Banc d'Arguin, Mauritanie. Contribution au colloque « localiser les produits[END_REF][START_REF] Cormier-Salem | Managing the coastal land-water interface in the tropical delta systems[END_REF], tensions are significant between markets, norms and actors involved in these processes. Also, international market induces more complex networks with new middlemen (wholesalers, retailors) and longer chain between producers and consumers. Moreover, local norms are not compliant with increasingly stringent food safety, hygiene and product traceability norms. So, there are sharp tensions beween origin-products typicity and standardisation. Basis of quality criteria often lack of transparency and certificate organisms are not enough independant. Role of government in voluntary labelling and certificat and implication of actors along the entire supply chain are also in question: degrees of acceptance vary from one stakeholder to another. As tools are heavy, expensive and binding, some actors are de facto excluded from this valuechain [START_REF] Cormier-Salem | Managing the coastal land-water interface in the tropical delta systems[END_REF].

Lessons from African eco-labelling initiatives in fishery

African ecolabelling in fishery studies and Biodivalloc program analysis [START_REF] Cormier-Salem | Biodiversité littorale et projets de valorisation des productions localisées en Afrique de l'Ouest. Dakar, Sénégal[END_REF][START_REF] Blueyou | Feasability study. Eco-labelling of artisanal coastal fisheries in Senegal. Bueyou & ENDA/REPAO[END_REF][START_REF] Queffelec | Analyse juridique des initiatives et potentialités de valorisation de produits de la mer en Afrique de l'ouest[END_REF][START_REF] Cormier-Salem | Managing the coastal land-water interface in the tropical delta systems[END_REF] demonstrate labelling tools have to be choosen according to the market orientation and adapted to the local context.

Labelling tools are free and volontary. They encompass various procedures and steps, from « simple » labels and norms to certifications (which need guaranty and third external organism to certificate), from informal to official signs of quality. Each of them have threats and opportunities [START_REF] Queffelec | Analyse juridique des initiatives et potentialités de valorisation de produits de la mer en Afrique de l'ouest[END_REF]. Also, official signs of quality allow public recognition but they cost a lot ; they allow immediate credibility but also possible sanctions ; price is higher but multiple diversified networks are unadequate and local specificities not preserved. On the opposite, informal signs or collective brand are more adaptable, more flexible to local context but not recognised ; reputation is based on confident and self control, which is not enough to construct credibility and justify higher price. A major stake concerns « quality », currently focused on two aspects, sanitation and sustainability [START_REF] Mariojouls | Introduction to quality: quality concepts, quality perception by producers, clients and consumers; quality signs (geographic origin, ecolabelling, etc.); translation of quality concepts into products procedures and services[END_REF]. In fact, two kinds of quality has to be distinguised [START_REF] Charles | Exploitation d'une ressource naturelle et politique de valorisation par des signes de qualité. La pêcherie du bar de ligne de Bretagne[END_REF]: intrinsic depends on the products themselves -gustatory, nutritional or medical value (see for instance in France Red Label and AOC) ; "extrinsic" quality is conferred by the conditions of exploitation, processing and marketing -respect for the environment, responsible and sustainable uses, poverty reduction, equitable sharing of benefits generated by Protected Areas, etc. (see for instance Fairtrade, IGP, ecocertification). Some of these criteria are compatible, but very often they are contradictory -for instance hygiene criteria often conflict with organoleptic criteria, normalisation does not fit with specification (or typicity) [START_REF] Cormier-Salem | Biodiversité littorale et projets de valorisation des productions localisées en Afrique de l'Ouest. Dakar, Sénégal[END_REF][START_REF] Cormier-Salem | Les "produits de terroir" dans les Suds : des liens incontournables entre qualité et durabilité ?[END_REF].

In the case of mullet fishery and cockles exploitation, the two trajectories have diverse promises and threats : poutargue, high-valued product, export-oriented, is not really an Imrâgen product ; innovations, imposed by international norms and eco-certification via MSC are not always well accepted by local actors, but they ensure access to « market niches». On the contrary, cockles are traditional identary products iii . The « old » food chain for domestic consumers and the improved food chain for urban or foreign, richer and warned consumers, could co-exist and respond to high and diversified demand on the local, national and African market. Nevertheless, cockles cannot be exported towards European or Asian markets because of sanitary constraints. Elaborate a collective brand could be a better tool than ecocertification to protect biodiversity and promote local products on national markets [START_REF] Sarr | Shell's Valorisation Policy in Saloum (Senegal)[END_REF][START_REF] Queffelec | Analyse juridique des initiatives et potentialités de valorisation de produits de la mer en Afrique de l'ouest[END_REF].

Conclusion : Issues and recommandations

Most of the eco-labelling processes aim first to improve local products value chains. They are oriented to market niches and are per se limited in terms of exchanged volumes. Beyond legal and commercial goals, ethical and ecological preoccupations have growing place. Quality economy, solidarity, equity and responsability are becoming key words. However, as illustrated by Senegalese and Mauritanian studies, these tools suffer from inner tensions. Innovative processes lead to various changes, especially in terms of : resource and territory access, social organisation (with complex recomposition of actors'power, from producers to consumers), value-chain and quality criteria. These tools, which are supposed to alleviate poverty, on the contrary could stress disparities and aggravate inequities (beteen men and women, young and elder, résident and migrant, etc.). In the context of African food crisis, another major stake is to guaranty for every body the access to a secure, heathy diet and quality (intrinsic and extrinsic) products.

Finally, the success depends on 1) the capacity of the producers to get organized and identify the relevant markets for their products, 2) consumers awareness for sustainable issues related to production, food quality and trade and responsiveness (militant market and consumers reliability) [START_REF] Cormier-Salem | Des produits de terroir pour conserver la diversité biologique et culturelle au Sud. Enjeux, acteurs, instruments[END_REF][START_REF] Jacquet | The rise of seafood awareness campaigns in an era of collapsing fisheries[END_REF].

To prevent the mismatches, following recommandations are suggested : 1) Strenghten the links between eco-labelling incentives and coastal management policies ; incorportate seafood products promotion in the ICZM ; considere ecolabelling as a tool of marine biodiversity conservation. There is a need of public incentives and local actors' capacity-building support. Particuliar attention has to be paid to property rules (access and uses right, benefices sharing, actors interactions) ; 2) conduct markets analysis at various scales (especially at West African scales) and study complexity of value-chain and market segmentation and/or competition and substitution ; 3) promote informations sharing and initiatives exchanges among all stakeholders ; make Localised Seafood Systems more recognised ; create a meeting and advertising place like a « house of seafood products ».