Marriage Surplus and Educational Endogamy in

This article deals with the fundamental question of educational homogamy in France. One interest in this interrogation finds its root in the fact that the global education level has highly increased during the last decades and particularly women'. These demographic changes may have influenced individuals well-being in some way. This article aims to highlight some of these eventual effects particularly within households in France.

Introduction

The family structure has undergone enormous changes in all the industrialized countries Over the last 50 years. The number of divorces is much more important than in the past while the number of marriages is declining. In married couples, the levels of education of husband and the wife are often very close, and that of the wife regularly exceeds that of the husband. Moreover, the wife's rate of participation in the labor market has increased significantly. And even if there is still a clear difference between men and women, the distribution of household chores within the household is much more egalitarian now than thirty or forty years ago. One of the important changes we can focus on is the homogamy more especially the educational homogamy. An increase of this would imply a reinforcement of the inequalities. Several studies have been done in different industrialized countries to investigate on the evolution of the homogamy, mainly in Sociology. In that way, we can mention some of those studies. [START_REF] Mare | Five Decades of Educational Assortative Mating[END_REF] has studied the variations of the educational homogamy in U.S. on the period 1940-1987 and found an increase until the 70s. [START_REF] Schwartz | Trends in Educational Assortative Marriage from 1940 to 2003[END_REF] using U.S data as well and covering a long period, confirmed a reinforcement of the educational homogamy until the 1990s and a decrease then. Similar results are found with Canadian data [START_REF] Hou | The Changing Role of Education in the Marriage Market : Assortative Marriage in Canada and the United States Since the 1970s[END_REF]. More generally, a study of [START_REF] Ultee | Educational Heterogamy and Father-to-Son Occupational Mobility in 23 Industrial Nations : General Societal Openness or Compensatory Strategies of Reproduction ?[END_REF] on 23 industrialized countries revealed a weakening of the educational over the time with a trend with a form of an inverted U, i.e an increase followed by a decrease over the time. That suggests that the educational homogamy has first increased with the increase of the education rate, and then has decreased. This is confirmed by Smits, Ultee and Lammers (1998). In Europe, we observe different evolutions of the educational homogamy: for instance, a decrease in U.K, Sweden, and Norway [START_REF] Halpin | Educational Homogamy in Ireland and Britain : Trends and Patterns[END_REF], Birkelund ans Heldal (2003), [START_REF] Henz | Who Marries Whom in Sweden ? dans[END_REF]) and a decrease in France and The Netherlands too [START_REF] Ultee | Educational Heterogamy and Father-to-Son Occupational Mobility in 23 Industrial Nations : General Societal Openness or Compensatory Strategies of Reproduction ?[END_REF] and in parallel, a global increase of the educational homogamy is observed in Italy [START_REF] Bernardi | Who Marries Whom in Italy[END_REF] and Spain [START_REF] Esteve | Changes in Educational Aassortative Mating in Contemporary Spain[END_REF]. In France, several sociological studies, have been carried out on this subject. The first study was carried out by Alain [START_REF] Girard | Le choix du conjoint. Une enquête psycho-sociologique en France[END_REF] for INED on the context and determinants of the choice of spouse. This study will be extended by Michel [START_REF] Bozon | La formation du couple. Textes essentiels pour la sociologie de la famille[END_REF]. [START_REF] Forsé | L'évolution de l'homogamie en France. Une méthode pour comparer les diagonalités de plusieurs tables[END_REF], Goux and Maurin (2003), [START_REF] Vanderschelden | Homogamie socioprofessionnelle et ressemblance en termes de niveau d'études : constat et évolution au fil des cohortes dunions[END_REF] and Bouchet-Valat (2011[START_REF] Bouchet-Valat | Les évolutions de l'homogamie de diplôme, de classe et d'origine sociales en France[END_REF] deal with the evolution of homogamy in France. [START_REF] Forsé | L'évolution de l'homogamie en France. Une méthode pour comparer les diagonalités de plusieurs tables[END_REF] used a measure derived from the odds ratio, the net diagonal index, and data from the survey "Emploi" 1989. They identified a decline in homogamy of social origin and stability of the homogamy of diploma in relative terms. A limit of this study is that the net diagonal index does not weight the odds ratios of the homogamy table in terms of the numbers they represent in the average population over the period. Using such a weighting, Bouchet-Valat (2011) observed a slight weakening of the degree of homogamy in the same survey. Using log-linear and log-multiplicative models, [START_REF] Vanderschelden | Homogamie socioprofessionnelle et ressemblance en termes de niveau d'études : constat et évolution au fil des cohortes dunions[END_REF] studied homogamy of education and socio-professional homogamy, using the survey "Etude de l'Histoire Familiale" 1999. She measured an irregular decline in educational homogamy, but no change in the level of socio-professional homogamy (both in relative terms). But regarding educational homogamy, only the first unions that had not experienced separation before the survey were taken into account in her study. Using different models based on odds ratios and using data from "Employment" surveys carried out between 1969 and 2011, [START_REF] Bouchet-Valat | Les évolutions de l'homogamie de diplôme, de classe et d'origine sociales en France[END_REF] shows that the homogamy of diploma and social class has clearly decreased, both in absolute and relative terms. Theses models are fundamentally statistical and do not have an economic approach of marriage; this will be the main point of difference of our study from theirs.

The interest of economists for marriage and its modeling is not new since it dates back at least to the work of [START_REF] Becker | A Theory of Marriage: Part I[END_REF][START_REF] Becker | A Theory of Marriage: Part II[END_REF]. Economic models of marriage that have been developed along this line are based on the idea that each person obtains some (positive or negative) utility from marriage. These models differ depending on the hypothesis made on the transferability of the utility. They rely on a concept of stable equilibrium defined so that: (a) there is no married person who would prefer to be single, (b) there are no pairs of people, married or not, who would prefer forming a couple together. These models are said "frictionless" (that is, matching instantly reach the equilibrium without the mechanism leading to this equilibrium being specified).

Recent research led to give empirical content to those models. Using a very simple model of marriage, known as with (perfectly) transferable utility, Choo and Siow (2006a, b) assume that the utility derived from marriage breaks down into two parts: the first is deterministic and takes a finite number of values based on observable characteristics (as, for example, the diploma, religion, income, expressed in the form of discrete variables) and the second consists of a continuous random term representing what is not observable by the analyst. They then show that the net gain (compared to the status of single) generated by marriage for a person of a given sex s having a certain characteristic c was equal, on average, to the logarithm of the ratio of the number of singles with sex s and characteristics c. By simply comparing the proportion of marriage of different groups of individuals, it becomes possible to measure the value for these groups of individuals to have certain intellectual, physiological or physical traits. It is also possible to assess the effect on the net gain of marriage of changes in the number of men or women with different characteristics. In this perspective, Choo and Siow (2006a,b) presented empirical results using data from the U.S Census and the Canadian Census.

The hypothesis of (perfectly) transferable utility means that the utility of an individual may be transferred to another individual at a constant rate. To fix ideas (but the interpretation of transferability is more general), individuals can exchange money or services that represent changes in the level of utility. Naturally, this hypothesis implies restrictions that are important. Whatever it may be, this model of marriage may provide answers to some important questions. For example, Chiappori and alii (2014) showed by using U.S Census data that, since education for women also had an effect on marriage opportunities, the global rate of return of education has proved to be larger for women than for men. These empirical models of marriage market are generalized by [START_REF] Galichon | Cupid's invisible hand: social surplus and identification in matching models[END_REF].

In this article, we will present these models and reproduce some empirical results presented with French data. It may be interesting to look at how the surplus generated by marriage has changed over time. The reduction in the number of marriages in France suggests that this surplus has declined. However, it is possible to go further and to identify groups of people and the types of marriage that have been most affected by changes. For example, Chiappori et alii (2014) argue that the increase in the correlation between spouses' income generally observed in recent years is not due to a amplification in the taste for assortative matchings. This increase is a mechanical consequence of the rise in the general level of education of women: a larger number of graduated men marry a larger number of graduated women because the latter are more numerous than before.

In this paper considering the Choo and Siow model, we empirically measure the marriage surplus and the educational endogamy on the global metropolitan french marriage market and on its different geographic regions. We use French data census from 1962 to 2011. In the first section, we describe the general theoretical model of Galichon and Salaniè (2012) inspired by the Choo and Siow model (2006). This is followed by the inference approaches: nonparametric and parametric. In the third part, we present the empirical results. We particularly observe that the joint surplus has increased on the period 1962-1982 and has decreased significantly on the period 1982-2011. More precisely, the intensity of increase of the joint surplus on the period 1962-1982 has been higher for very educated couples than for the less educated ; and the decrease of the joint surplus on the period 1982-2011 has been higher for less educated couples than for the very educated couples. About the endogamy, we remark that the endogamy presents a significant increase on the period 1962-1999 and a decrease on the period 1999-2011. On the period 1962-1982, the intensity of the increase of endogamy has been higher for very educated couples than for less educated couples. But on the period 1982-1990, the reinforcement of the endogamy within less educated couples became higher than within very educated couples. And finally, the decrease of endogamy on the period 1999-2011 has been higher for less educated couples than for the very educated ones. Our results are conform to the existence a trend with a form of an inverted U for the evolution of the educational homogamy in France, widely observed for The U.S and Canada. In that way we confirm the conclusion of studies of Schwart and [START_REF] Schwartz | Trends in Educational Assortative Marriage from 1940 to 2003[END_REF] and [START_REF] Hou | The Changing Role of Education in the Marriage Market : Assortative Marriage in Canada and the United States Since the 1970s[END_REF] on the evolution of the educational homogamy in U.S and Canada in the French case. But these results are in contrast with the global decrease observed by sociological studies in France. In our work, the educational homogamy has decreased only on the period 1999-2011 after it has increased on the period 1962-1999.

Theoretical Framework

At the heart of this theory is a two -sided assignment model with transferable utilities where agents on both sides of the market (men and women) are characterized by a set of attributes that is only partially observed by the econometrician. Each agent aims to match a member of the opposite sex in order to maximize his own utilities. This model is particularly interesting because under certain conditions (mainly restrictions on the form of the surplus function), it is possible to identify and estimate the characteristics of agent preferences. We suppose the set of observable attributes is discrete.

Marriage Market Description

We consider a bipartite matching market with transferable utility. We start with two sets H and F, whom we will refer to from now on as set of "men" and set of "women". Let P and Q be two measures defined respectively on H and F. Any individual m ∈ H can be matched with an individual w ∈ F or remain single, and conversely. Remaining single can be considered as a match to a null agent of the other side of the market. So, we add two dummy populations of null agents, denoted 0 to H and F respectively in order to take in to account single individuals. P represents the total population.

|H| + |F| = |P|

We suppose that there exists two random variables X and Y defined respectively from H to a discrete finite space X and from F to a discrete finite space Y.

We will adopt the Galichon and Salanié (2012) framework for its simplicity. Let M and N be the vectors of the numbers of men and women. M = (M x ) x∈X and N = (N y ) y∈Y where M x represents the number of men of type x and N y the number of women of type y. In the case the population is infinite we can consider the probabilities (m x ) x∈X and (n y ) y∈Y such that: ∀x ∈ X , ∀y ∈ Y, m x = P r(X = x) and n y = P r(Y = y)

In other terms, m x is the proportion of men of type x in the population of men and n y is the proportion of women of type y in the population. In the light of Galichon and Salanié (2012) we introduce the following notions:matching, feasible matching and aggregated matching matrix.

Definition 1 (Matching).

A matching is an application μ defined on the product space H × F to {0, 1} such that, for all (m, w) ∈ H × F, μ(m, w) = 1 if man m and woman w are matched and μ = 0 if they are not matched. Formally we have:

μ : H × F -→ {0, 1} (m, w) -→ 1 { m matched to w }
Intuitively, matching must satisfy some constraints to be feasible. A matching is feasible if and only if any individual is matched with at most one individual of the opposite side.

Definition 2 (Feasible Matching). A feasible matching is a matching such that:

∀(m, w) ∈ H × F, μ(m, w) = 1 ⇒ ∀m = m, ∀w = w, μ(m , w) = μ(m, w ) = 0
We introduce now the aggregated matching matrix. Definition 3 (Aggregated Matching Matrix). Let μ be a matching. The aggregated matching matrix µ associated to μ is defined as:

∀ (x, y) ∈ X × Y, µ xy = m,w 1(x m = x, y w = y)μ(m, w)
The number µ xy represents the number of realized marriages of type (x, y). In the case of feasible matching, we have:

∀x ∈ X , y∈Y µ xy ≤ M x ; ∀y ∈ Y, x∈X µ xy ≤ N y
For any type x of men, the number of singles men of this type is the difference between the total number of men of type x and the total number of matched men of type x.

∀x ∈ X , µ x0 = M x - y∈Y µ xy , ∀y ∈ Y, µ 0y = N y - x∈X µ xy
For notation, we define:

X 0 = X ∪ {0}, Y 0 = Y ∪ {0}

Matching Surplus

Matching Surplus And Heterogeneity The introduction of the concept of the matching surplus is essential to give an economic basis to matching. In fact, matching can be justified by the hypothesis of the existence of a surplus generated by any matching between two individuals. This surplus is a joint surplus for the two matched partners and they share it into two individual surpluses that represent their respective utilities from the current matching. The joint surplus is a primitive of the matching and its sharing is endogenous to the problem. So the main information we need to determinate equilibrium is the joint surpluses of any feasible matching.

To model the utility from marriage, we use the separability assumption suggested by [START_REF] Choo | Who marries whom and why[END_REF] and formalized by Galichon and Salanié (2012) and Chiappori, Salanié and Weiss (2014).

Assumption 1 (Separability Assumption). The joint surplus of the union of a man m of type x with a woman w of type y is:

s mw = S xy + ε my + η xw
where S xy is a deterministic function defined on the cartesian space X × Y

The term S xy represents the gross systematic joint surplus after transfers. By normalization, we assume respectively for a single man m of type x and for a single woman w of type y: S x0 = S 0y = 0, and

s m0 = ε m0 , s 0w = η 0w
We can normalize the surplus of singles to 0 because only the net surplus can identified. The following assumption will specify the distribution of the random terms.

Assumption 2 (Unobserved heterogeneity). The random terms ε my and η xw in the joint surplus of the union a man m of type x with a woman w of type y are supposed to have a Gumbel extreme values distribution conditional on X and Y .

∀(m, w, x, y) ∈ H×F ×X 0 ×Y 0 , ε my ∼ Gumbel(-γ, 1), and η xw ∼ Gumbel(-γ, 1).

As [START_REF] Galichon | Cupid's invisible hand: social surplus and identification in matching models[END_REF] showed it, this hypothesis of specification is not necessary. In fact, a more general hypothesis is to assume that the random terms ε my and η xw have respectively a distribution P X and Q Y conditional on Xand Y . But, for simplification, we stay in the logit framework used by [START_REF] Choo | Who marries whom and why[END_REF].

Equilibrium We assume that the equilibrium is defined in the sense of Gale and Shapley (1962).

Definition 4 (Stable Matching).

A matching is stable if: (i) no matched individual would rather be single, and (ii) no pair of individuals would rather be matched together than remain in their current situation.

This equilibrium is frictionless. All the agents are satisfied of their choices. The primitive of the problem is the joint surplus s mw . It is the generated surplus by the union of a man m and a woman w. This quantity is shared between the two partners into two parts u mw and v mw that represent respectively the individual utility of man m and the individual utility of woman w from their union. From Shapley and Shubik (1971), we know that the solution of the problem is given by a feasible matching μ that maximizes the social surplus that is the sum of the surplus of all of the individuals.

The utilities u m and v w are respectively the utility a man m obtains at the stable state of the market and the utility a woman obtains at the stable state of the market. Single individuals are self matched; a single man m gets ε 0 m and a single woman w gets η 0 w . At the equilibrium, partners are reciprocally the best choice of their spouse, and single individuals at the equilibrium haven't found any partner that brings to them a greater utility than their own utility. The following proposition showed by Galichon and Salanié (2012) characterizes the equilibrium.

Proposition 1 (Characterization of the sharing of the joint surplus). At the equilibrium, the respective individual utilities u m (x) and v w (y) of a man m of type x and of a woman w of type y matched to each other are such that:

u m (x) = max (ε m0 , U xy + ε my ) and v w (y) = max (η 0w , V xy + η xw )
where U xy and V xy are two functions such that U xy + V xy = S xy .

Proof. Let m be a man. We denote by u m his utility at the equilibrium.

u m ≥ ε m0
In the case m does not get matched, we have u m = ε m0 . For any woman w, with utility v w at the equilibrium, the utility of the man m from a union with w is:

s mw -v w
As u m is the utility of man m at the equilibrium, we have:

u m ≥ s mw -v w
In the case, m gets matched with w, we have:

u m = s mw -v w
We can then conclude:

u m = max ε m0 , max w (s mw -v w )
By exploiting the separability assumption, we have at the equilibrium:

u m = max u 0 m , max w (s mw -v w ) = max u 0 m , max w S x,y(w) + ε m,y(w) + η xw -v w
This leads to:

u m = max u 0 m , max y max w|yw=y (S xy + ε my -(-η xw + v w )) so u m = max u 0 m , max y S xy + ε my -min w|yw=y (v w -η xw )
We can denote by:

V xy = min w|yw=y (v w -η xw )
Then we get:

u m = max u 0 m , max y (S xy -V xy + ε my ) = max u 0 m , max y (U xy + ε my )
The proposition shows that the total surplus is shared into two parts between the partners. These parts are the deterministic parts of the individual utilities of the partners and sympathy shocks constitute the stochastic parts. The utility obtained at the equilibrium by an individual is the highest utility he can get from a union or of being single.

Identification of individual utilities

For any pair (m, w) in (H × F) of two individuals, their matching requires the existence of a joint surplus s mw that will be parted into two utilities u m and v w for each other. These utilities are the gains of the partners after transfers. Let μ be the optimal feasible matching. For any man m and woman w, the sum of their utilities u m and v w they can get on the market, is higher than or equal to their joint surplus s mw . In fact u m and v w are respectively higher than or equal to the individual parts of utility the man m and the woman w can get from the sharing of s mw . There will an equality u m + v w = s mw if and only if m and w are matched.

Under the assumption of the Gumbel distribution of the random terms ε my conditionally to x and of η xw conditionally to y, we can derive the distribution of the equilibrium utilities u m and v w . As shown by [START_REF] Galichon | Cupid's invisible hand: social surplus and identification in matching models[END_REF], the following proposition states the distribution of u m and of v w at the equilibrium. It is an important step in the identification of the individual surpluses.

Proposition 2. Assume a stable matching. Under the above assumptions, at the equilibrium, the utility u m (x) of a man m of type x and the utility v w (y) of a woman w of type y are such that:

u m (x) ∼ Gumbel   -γ + ln   1 + y∈Y e U (x,y)   , 1   and v w (y) ∼ Gumbel -γ + ln 1 + x∈X e V (x,y) , 1
where γ is the Euler-Mascherano constant.

Proof. Let's assume F um the distribution function of u m . We have: We adopt the notation of [START_REF] Galichon | Cupid's invisible hand: social surplus and identification in matching models[END_REF]. Let G x (U x. ) be the average utility of man m conditionally his type is x, where: U x. = (U xy ) y∈Y . We have:

∀ c ∈ R, F um (c) = P (U xy + ε my ≤ c, ∀y ∈ Y 0 ) = y∈Y 0 F εmy (c -U xy ) = exp   -exp   -   c + γ -ln   1 +
G x (U x. ) = E(u m |x m = x).
We deduce from the distribution of u m :

G x (U x. ) = ln(1 + y∈Y e Uxy )
Analogously, we denote by H y (V .y ) the average utility of woman with type y.

H y (V .y ) = ln(1 + y∈X e Vxy )
Following Galichon & Salanié (2015), we know that the optimal matching aims to maximize the total social surplus that is:

W = x∈X M x G x (U x. ) + y∈Y N y H y (V .y )
under the constraint U (x, y) + V (x, y) = S(x, y). This result is a consequence of the Gale-Shapley equilibrium. In fact at the equilibrium, any deviation from this position will diminish the total surplus.

The following proposition gives the identification formulas for the different systematic surpluses. Proposition 3. At the equilibrium, we have:

∀ (x, y) ∈ X × Y, U xy + V xy = S

Inference

In this section we will describe two approaches to estimate this model: nonparametric and parametric.

Nonparametric estimation

We estimate the systematic joint surplus by:

Ŝxy = ln μ2 xy μx0 .μ 0y
where μ is the observed matching matrix.

The support spaces X and Y are discrete. We smooth this estimator by the Nadaraya-Waston Kernel estimator on the interval of the variable age:

ŜNW (s, t) = x∈X y∈Y K s-x h 1 ; t-y h 2 * ln μ2 xy μx0 * μ 0y x∈X y∈Y K s-x h 1 ; t-y h 2
where the function K is the Epanechnikov Kernel.

Parametric Inference Maximum Likelihood estimation

We specify the systematic joint surplus S xy by a parametric function of the observable characteristics X and Y .

We can estimate the parameter vector by the maximum likelihood method. The likelihood is: where μxy is the observed number of unions of type (x, y). This is the product of the conditional probabilities of being married to a particular type of partner or to remain single with respect the type of the individuals. In fact the conditional probability a man of type x gets married to a woman of type y is:

L (θ) =
e U θ xy y∈Y 0 e U θ xy
and the probability he remains single is:

1 y∈Y 0 e U θ xy
The probability a woman of type y gets married to a man of type x is:

e V θ xy x∈X 0 e V θ xy
and the probability she remains single is:

1 x∈X 0 e V θ xy
The main problem is that we don't know the expression of U and V . We will express the likelihood in the function of the theoretical matching and of the observed matching and by an algorithm used by Galichon and Salanié (2012) we will compute it. At the equilibrium, we have: 1962,1968,1975,1982,1990,1999,2006 and 2011 to estimate the marriage surplus and its evolution over the time. Each observation in the data is individual. It precises the year of the interview, the household i.e the family unit in which the individual lives, her sex, her age, her relation with the head of the household, her matrimonial status, her educational attainment, her employment status, her work place, and her region of residence in the last census. We don't observe the current region of residence. We have also these information for her spouse in the case the individual is in union. But the spouse may not live the household. This information reveals the relationship between the various individuals of each household and allows us to reconstruct "couples". The data allow us to do spatial analysis because it contains the indication of the region of work; we will use this variable for the place of residence.

µ θ xy M x = e U θ

Variables

Our analysis variable is mainly Education. Educational attainment is measured from the respondent's reported highest level of education achieved. There are four levels:

1-No schooling or less then primary 2-Primary completed 3-Secondary completed 4-University completed

Identification of couples

We consider as a couple a man and a woman who live in the same household and reported as head of the household and spouse (partner) of the head. We construct four data-frames for each wave: one for married men, one for married women, one for single men and one for single women. We create in the initial data set, the variable "match" that indicates under our criteria if the individual is married or not. We keep only persons aged 17 or over. match=1 if the individual is married and 0 if not. We then, extract married men to construct the data for married men. We do so for married women, single men and single women. We verify that the number of married women and married men are same and they household identifiers match.

Empirical results

Nonparametric estimation Individual Surpluses and Joint Surplus from marriage in France

For each level of education i.e 5 years of education, 13 years of education, 16 years of education and 19 years of education, and for men and women, we estimate the individual surplus of men and women in function of the education of their partner. Note that we obtain continuous curves on the interval [START_REF] Bouchet-Valat | Mémoire de sociologie[END_REF][START_REF] Schwartz | Trends in Educational Assortative Marriage from 1940 to 2003[END_REF] of the variable age because the nonparametric estimator is a smoothing function defined on that interval. The nonparametric estimation of the marriage surpluses for men shows that for the different levels of education of men, the choice of partner is very linked to the educative distance between the partners. In fact, we remark that very educated men (level 4) maximize their individual surplus with very educated women. Their surplus curve is increasing. The less educated men (leve1) have a decreasing surplus curve: in the case the sympathy shock is dwarf, their utility is maximized with less educated women. For intermediary levels (level 2 and 3) the curves are parabolic and concave; they obtain a maximal surplus with women who have same education level as them. Their surplus curve is increasing until the maximum then is decreasing. Their highest surplus is brought by union with women who have same education level as them. We can also remark that the level of education seems to not influence the maximum of marriage surplus men can get from their optimal union. The maximum of surplus curves for the different levels of education are approximately the same.

The surplus curves of women are very similar to men' curves. High educated women have high surplus with high educated men. Women with intermediary level (level 2 and 3) have their maximum surplus from union with men who have same education level as them. One difference between men and women in the partner choice is particular in the case women are less educated (level 1). These women seem indifferent to men of level 2 and 3. Their surplus is maximal with men of level 1 and it significantly decreases with high educated men (level 4). Another difference we can notice is the level of the maximal marriage surpluses. We remark that very educated women, level 3 and 4, have maximal surplus higher than the maximal marriage surplus of less educated women, level 1 and 2.

The estimation of the joint surplus shows that it is maximal in the section plan x -y = 0. Individuals match with people who have the same education level as them. The joint surplus is negative. This can be explained by the fact that there is less married people than singles. The following graph shows the curve of the joint surplus in function of the education of the two partners.

Evolution of marriage surplus on the global french market

It can be interesting to look at the evolution of marriage surpluses over time in France. We compute the averaged values of U, V and S for each year from 1968 to 2011. The marriage surplus has been relatively stable on the period 1962-1968 with an increase on 1968-1982. Then it has decreased from 1982 to 2011. We can conclude that there has been a negative shock on the marriage surplus in 1982. Marriage in terms of cohabitation brings less surplus today than by the past. The drop of the men surplus is higher than the drop of the women surplus. We also notice the reduction of the surplus gap between men and women. Men take the greater part of the joint surplus but this difference have significantly decreased and the marriage tends to be more egalitarian now. The shock of marriage surplus has impacted more men then women.

Regional french markets

In this section, we focus on the regional marriage markets. Regions are considered as local local marriage markets. The different french regions we have studied are: Ile -de-France, Champagne, Picardie, Haute-Normandie, Centre, Basse Normandie, Bourgogne, Nord-Pas-Calais, Lorraine, Alsace, Franche Comté, Pays de la Loire, Bretagne, Poitou Charentes, Aquitaine, Midi-Pyrénées, Limousin, Rhône-Alpes, Auvergne, Languedoc-Roussillon, Provence Côte d'Azur, Corse. The estimation of the marriage surplus in these regions reveals that in term of evolution, the regional marriage surpluses for all the regions have had the same dynamic of evolution everywhere as the general dynamic marriage surplus has had on the global french market: that is , an increase on the period 1968-1982 and a decrease on the period 1982-2011. On the period 1962-1968 regional surpluses have decreased. Today the regional marriage surpluses are generally negative. In 1999, only Pays de la Loire (0.26), Haute Normandie (0.15), Basse Normandie (0.11), Poitou Charentes (0.11), Nord Pas de Calais (0.07), Bretagne (0.05) and Picardie (0.048) still had positive marriage surplus. The lowest marriage surpluses in 1999 were in Ile-deFrance (-0.68), Languedoc Roussillon (-0.39), Provence Cte d'Azur (-0.53) and Corse (-0.75).

Regional marriage surpluses have known two significant shocks: a positive shock in 1968 and a negative shock in 1982. These shocks have impacted all the regions but not with the same intensity. During the period 1962-1968, the highest impact of the positive shock is observed in Pays de la Loire and Aquitaine. In fact on that period, the marriage surplus has in increased in these regions by 0.4 points. The lowest impacts are observed in Basse Normandie, Alsace and Lorraine with an increase of the marriage surplus by 0.2 points. Particular fact is the decrease of the marriage surplus in this period in Languedoc-Roussillon and Provence Cte d'Azur; they are the only regions where the positive shock of marriage has not been observed.

The negative shock occurred in 1982 and its impact is observed everywhere. The highest losses of surplus are observed in Haute Normandie (-0.6), Ile-de-France (-0.61), Poitou Charentes (-0.61), Picardie (-0.66), Nord Pas de Calais (-0.71) and Languedoc Roussillon (-0.71);

Education and Endogamy

Effect of the education on the marriage surplus

We evaluate the averaged impact of x on U , V and S by α U , α V , α S on the population of married people. The averaged impact of y on U , V and S is evaluated by β U , β V and β S on the population of married people. We measure these effects and we analyze their evolution over the time.

α U = E X,Y ∆U XY ∆X , α V = E X,Y ∆V XY ∆X , α S = E X,Y ∆S XY ∆X
We can remark that:

α S = α U + α V
We have similar expressions for β.

β U = E X,Y U XY ∆Y , β V = E X,Y ∆V XY ∆Y , β S = E X,Y ∆S XY ∆Y
The averaged effect of the level of education of men and women on the marriage surpluses is negative. This can be explained by the fact the optimal choices of the partners on the section plane x = y; the education levels of the partners are very close. Any deviation from the section line intersection of the surplus curve with the plane of equation (x = y) of the surface of the surplus has a negative impact on the surplus. The sign of the impact of x and y is logical. The intensity of the impact of the men's education is higher on the women' surplus than on the men's surplus. The intensity of the impact of the men's education on the women's surplus has decreased over time where as it has increased on the men's surplus. The intensity of the impact of the men's education on the joint surplus has globally decreased. In others words, considering two stable couples of type (x, x) and (x + ∆x, x), the joint surplus of (x, x) is greater than the joint surplus of (x + ∆x, x), but this difference in surplus is weaker today than by the past. For women, things don't work necessarily same as men. The intensity of the impact of women' education on the joint surplus and on the men's surplus has significantly decreased over time where as this intensity has relatively been stable on their own surplus. The impact of women's education on the marriage surplus is lower today than by the past.

Endogamy

The endogamy is characterized by the assortative mating. We estimate it by the cross derivative of the joint surplus. In fact, as shown by [START_REF] Becker | A Theory of Marriage: Part II[END_REF], the positivity (resp. negativity) of the cross derivative of the joint surplus indicates that a positive (resp. negative) correlation between the characteristics of the partners is optimal.

s x,y := ∆ 2 S x,y ∆x∆y = ([S x+1,y+1 -S x+1,y ] -[S x,y+1 -S x,y ]) * 1 ∆x * ∆y
If s xy > 0 then there is a positive assortative mating and negative if s xy < 0. We then compute the averaged value of s. We can remark that in France, the endogamy is positive in France and has increased on the period 1962-1999 and then has decreased from 1999 to 2011. We have tried to analyze this evolution depending on the different geographic french regions, on the period 1962-1999.

E X,Y ∆ 2 S x,y ∆x∆y = x∈X y∈Y ∆ 2 S x,y ∆x∆y * µ xy x∈X y∈Y µ xy
The parisian region has the lowest level of endogamy and the lowest progression rate on the period 1962-1999. The endogamy has more increased in the North West of France than in the South. Pays de la Loire, Basse Normandie and Haute Normandie are the regions where endogamy has more increased on the period 1962-1999.

Parametric estimation

We complete the nonparametric analysis by a parametric analysis to prove the consistency of our results. To do this, we specify the joint surplus as follows:

S xy = λx α y β + 6 k=1 ρ k (x -y) k
where x and y represent respectively the numbers of education years of the men, and of the women. For partners who have the same degree, their joint surplus is explained only by their their numbers of education years x and y.

The parameter λ is simply the joint surplus of a couple of type (x = 1, y = 1).

λ = S 11
The parameters α and β represent respectively the elasticity of the joint surplus S xy with respect to x and the elasticity of the joint surplus S xy with respect to y, in the set of couples such that x = y, where the partners have the same level of education. In fact, if x = y, then S xy = λx So when we consider a couple of type x = y, an increase by 1 in x makes vary the joint surplus at a rate of sgn(λ) α

x and similarly, an increase by 1 in y makes vary the joint surplus at a rate of sgn(λ) α y . The effect of x and y are inversely proportional to the level of education of the partner. So the intensity of the impact of the education on the joint surplus is higher for couples less educated than for very educated couples. But this effect depends on the sign of the product λα for x and λβ for y.

An increase in x and y of a couple of type x = y from (x, x) to (x + 1, x + 1) impacts the joint surplus at a rate of sgn(λ) α+β x .

Education and Endogamy

Impact of the education on the marriage surplus The parameter λ is negative and has significantly decreased over time, so does S 11 . This is coherent to the general decrease of the joint surplus observed in the nonparametric estimation.

The parameters α has been positive on the period 1962-1975 whereas β has been negative. The fact that λ is negative induces that the conditional effect of men education on the joint surplus with respect to x = y has been negative on the period 1962-1975 and the conditional effect of women education on the joint surplus with respect to x = y has been positive. These two parameters change suddenly sign in 1982 and λ also so that the effects have preserved their sign. The intensity of these effects has decreased nearly to zero on the period 1982-2011. They are both positive today. The following tables show the estimated values of α, β, λ and the estimated conditional effects of x and y on the joint surplus with respect to x = y, and their evolution over time.

Year

λ α β ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 6
1962 -0.0018

(6.35×10 -7 )
2.9724

(2.47×10 -3 ) -0.6030

(3.9×10 -3 )
1.394

(1.19×10 -2 ) -1.7184 (2.59×10 -2 )
-0.0597 In the period 1962-1990, for a couple of type (x, x), an additional year of education of the man impact negatively the joint surplus. The consequent reduction can be very high, about 76% in 1975 for couples (x = 5, y = 5), whereas it is about 20% for couples (x = 19, y = 19). This effect of x turns positive by 1999 until 2011. Today, for couples of type (x = 16, y = 16), an increase in x by 1 brings 3.18% of the initial surplus as benefit. The change in the sign of the impact of the men' education on the joint surplus, negative by the past and positive today, can be explained by the reduction of the gap in education between men and women. By the past men were too much more educated than women so that an increase in their education increases that gap and affects more negatively the woman than it affects positively the man. This leads to a negative global effect on the joint surplus. Nowadays, the women education is close to the men education and even sometimes surpasses the men education. And increase in the men education reduces then actually the gap between him and his partner and they all both gain in surplus from this increase. For women, the effect has always been positive. Its highest intensity has been obtained in 1982. In this year, for a couple of type (x = 5, y = 5), an increase in y by 1 could bring 61% of the initial surplus as benefit. But today, the intensity is very few. In 2011, for a couple (x = 16, y = 16), when y by 1, the joint surplus increases 1.9%. Today the conditional effects of x and y on the joint surplus are very few but the impact of x is slightly higher than the impact of y.

As we have shown it, sgn(λ) α+β

x is the variation rate of the joint surplus from (x, x) to (x + 1, x + 1).It is the joint effect of x and y.

The sum α +β is positive on the period 1962-1990, so sgn(λ) α+β x is negative on this period. Less educated couples with same education level were happier then very educated couples. The conditional joint effect of x and y on the joint surplus turns positive by 1990 until 2011.

Endogamy We measure the endogamy that we denote by E x,y by the double cross derivative of S x,y with respect to x and y.

E x,y = ∂ 2 S x,y ∂x∂y = λαβx α-1 y β-1 - 6 k=2 ρ k k(k -1)(x -y) k-2
For simplification, we will consider couples of same education. The endogamy takes the form: Ẽx = λαβx α+β-2 .

The analysis shows that the endogamy has globally increased on the period 1962-1999 with a little depreciation of this dynamic in 1968, and a global decrease after 1999. The intensity of the evolution of the endogamy is evaluated by | d Ẽx dt |. Its analysis reveals that, on the period 1962-1982, the intensity of the increase of endogamy has been higher for very educated couples than for less educated couples. But on the period 1982-1990, the reinforcement of the endogamy within less educated couples became higher than within very educated couples. And finally, the decrease of endogamy on the period 1999-2011 has been higher for less educated couples than for the very educated. The parametric estimation of the averaged endogamy shows the positivity of the assortative mating in France, its increase on the period 1962-1999 and its decrease on the period 1999-2011. This trend is exactly what reveals the nonparametric estimation. As represented in the following graph, the two estimations are very near all over the time. dt | is increasing in x. We deduce that the increase of the joint surplus has been higher for very educated couples then for less educated couples.

On the period 1982-2011, we have x ≤ x s . So the surplus has decreased on the period 1982-2011. The function x → d Sx dt is increasing in x and negative on the period 1982-2011, so | d Sx dt | is decreasing in x. We deduce that the decrease of the joint surplus has been higher for less educated couples then for very educated couples.

The following table shows the estimation of the averaged joint surplus and its evolution over time in France. The parametric estimation of the averaged joint surplus in France shows an increase of the joint surplus on the period 1962-1982 and a decrease on the period 1982-2011 as noticed with the nonparametric estimation. In the nonparametric estimation, on the period 1962-1982, the joint surplus seems to be relatively constant before it decreases then by 1982.

Conclusion

Marriage generates a marital joint surplus. This joint surplus is parted into two individual utilities between the spouses. These surpluses depend on the levels of education of the spouses and particularly on the distance between their levels. Less educated men maximize their utility with less educated women and reversely, and high educated men maximize their utility with high educated women. This mechanism of partner selection is observed on the French Marriage Market all over the period of our study. The graphic representation of the joint surplus generated by marriage is a generally a concave surface with local maxima located on the section x = y. The nonparametric estimation shows that the joint surplus has decreased from 1962 to 1968, has increased on the period 1968-1982 and then has decreased on the period 1982-2011. This dynamic is also observed on the individual surpluses of the spouses. On the period 1962-1968, men utility and women utility have both decreased. On the period 1968-1982, men utility remains stable whereas women utility has increased. This is why the joint surplus has increased on this period. From 1982 to 2011 men utility and women utility have both decreased, with an intensity of decrease greater for men than for women. So men have lost more utility from marriage than women. This difference has had an important consequence mainly on the surplus gap between men and women. Estimations revealed that the sharing of marriage surplus is not egalitarian; men have the greater part of the joint surplus. But this difference has been reduced overtime. Today men remain happier in their union than women but this gap is very few now. We investigate on how the level of education of the spouses impact their surpluses. Nonparametric estimations shows that the averaged effect of the education level of the spouses on their surpluses (joint and individual) are negative on the population. The intensity of the impact of men education on their individual surplus has increased on the period 1968-1999 and decreased on the period 1999-2011. The intensity of the impact of men education on the women surplus has globally decreased over time, so today men education has less impact on women surplus than by the past. On the joint surplus the impact of men education has been relatively stable on the period 1968-1999 and then it decreased in intensity from 1999 to 2011. The intensity of the impact of men education on their own surplus is weaker than on women surplus. With women things work a bit differently. We remarked that the impact of women education on their own surplus, is negative but has remained relatively constant on the period 1968-1999 and then has decreased in intensity. The impact of women education on men surplus and on joint surplus has significantly decreased in intensity over time. The intensity of the impact of women education on their own surplus is weaker than on men surplus. Finally the nonparametric estimation of the endogamy reveals that the endogamy has been positive all over the time in France. This is equivalent with a positive assortative mating. People do not choose partners who are different from them in term of level of education. The nearer the partner is the higher is the attractiveness between the two. The analysis of the evolution of the endogamy shows a significant increase on the period 1962-1999 and a decrease on the period 1999-2011. It seems to be stable now. This is the general evolution of endogamy in France. Depending on the geographic regions, the level of endogamy vary. During the period 1962-1999, regions where we notice the greatest increase are the Center and the North-West except Bretagne. The parisian area has had the weakest increase on that period.

We refine the analysis with a parametric estimation. We have specified the joint surplus as the sum of a Cobb-Douglas function in x and y and a polynomial function in (x -y). The model fits perfectly the data. The parametric joint surplus has increased on the period 1962-1982 and has decreased significantly on the period 1982-2011. More precisely, the intensity of increase of the joint surplus on the period 1962-1982 has been higher for very educated couples than for the less educated; and the decrease of the joint surplus on the period 1982-2011 has been higher for less educated couples than for the very educated couples. There has been a negative shock on the marriage surplus in 1982 and this shock has more impacted the less educated couples than the very educated couples. When we focus on couples whose spouses have same education level, we derive the conditional effect of x and y on the joint surplus with respect to x = y. Men education level has had a negative conditional effect on the joint surplus with respect to x = y on the period 1962-1990. This effect turns positive after 1990 untill 2011. Women education level has had a positive conditional effect on the surplus with respect to x = y all over time. But the intensity of the effect of men education level on the joint surplus is higher than women effect today. The positivity of these two effects has a consequence: very educated couples where spouses have same education level are happier today than less educated couples where spouses have same education level. Finally the parametric of the endogamy shows a significant increase of the endogamy on the period 1962-1999 and a decrease on the period 1999-2011. On the period 1962-1982, the intensity of the increase of endogamy has been higher for very educated couples than for less educated couples. But on the period 1982-1990, the reinforcement of the endogamy within less educated couples became higher than within very educated couples. And finally, the decrease of endogamy on the period 1999-2011 has been higher for less educated couples than for the very educated.



  This is the cumulative distribution function of the Gumbel distribution stated in the proposition.

  V θ xy µ θ is the theoretical matching matrix. Therefore, we obtain: We use the Iterated Projection Fitting Procedure proposed by Galichon and Salanié (2012) to compute the likelihood for each θ chosen in sufficiently large range. The estimator estimator is given by: θMLE = Arg max θ log L(θ) 4 The Data In this paper, we use data from the French Census (Minnesota Population Center. Integrated Public Use Microdata Series, International: Version 7.1 [dataset]. Minneapolis, MN: IPUMS, 2018. https://doi.org/10.18128/D020.V7.1 ), in particular the waves

p

  xy S xy with p xy = µ xy x∈X y∈Y µ xy

Figure 1 :

 1 Figure 1: Evolution of Endogamy (Non-parametric)

α y β and

( 9 .57×10 - 3 )Figure 2 :

 932 Figure 2: Estimation

Figure 4 :

 4 Figure 4: Impact of women' education on the joint surplus

Figure 5 :

 5 Figure 5: Evolution of the parametric endogamy

Figure 7 :

 7 Figure 7: Evolution of the parametric joint surplus

Figure 8 :

 8 Figure 8: Estimation of the averaged joint surplus

Evolution of the joint surplus The evolution of the joint surplus is linked to the evolution of the different parameters on which it depends. To simplify the functional analysis, we will focus on couples in which partners have the same level of education. This is justified by the fact that the education levels of the partners are very close at the equilibrium and the education endogamy is positive in France. Let Sx be the joint surplus of a couple of type (x, x) takes the form.

The parameters λ and γ depend on the time t. λ = λ (t) and γ = γ (t)

We have:

The parameter λ is generally negative and γ has decreased over time so: dγ dt ≤ 0. So we have:

We denote x s by:

We have:

The data show that on the period 1962-1982, x ≥ x s . So the surplus has increased on the period 1962-1982.

Conflict of Interest: The author (Dr Ulysse Lawogni) declares that he has no conflict of interest.

Ethical approval : This article does not contain any studies with human participants or animals performed by any of the authors.


C Levard
email: levard@cerege.fr

M Le Bars

T Formentini

S Legros

E Doelsch

Organic waste-borne ZnS nanoparticles: the forgotten ones

Keywords: Incidental ZnS nanoparticles; organic wastes, environmental release, ecotoxicity

o Considerably more ZnS-NPs are released into the environment than most commercial NPs o The fate of incidental ZnS-NPs formed in organic waste has been completely overlooked o ZnS-NPs that form in organic waste are small and exhibit unique structural properties o The reactivity and ecotoxicity of ZnS-NPs may be enhanced due to their small size

The market boom in nanotechnology has led to unprecedented and well justified scientific interest in the risks posed by the release of manufactured nanoparticles (NPs) into the environment. This interest is reflected in the funding of a large number of modeling and experimental studies. The latter are mainly physico-chemical and ecotoxicity investigations that were initially performed in simplified systems, but are becoming increasingly complex and realistic.

Silver nanoparticles (Ag-NPs) are strong biocides, and in the last 15 years, have been among the most widely studied and publicized NPs. These studies have shown that the main pathway by which Ag-NPs enter the environment is via sewage sludge spread as fertilizer on agricultural soils. Concentrations of Ag-NPs in sewage sludge typically range between 1 and 10 mg/kg. Bare Ag-NPs can have toxic effects on a variety of organisms, particularly when soluble Ag is released. However, it is acknowledged that Ag-NPs in the environment are mostly transformed into poorly soluble and less toxic Ag 2 S [START_REF] Courtois | Ecotoxicology of silver nanoparticles and their derivatives introduced in soil with or without sewage sludge: A review of effects on microorganisms, plants and animals[END_REF][START_REF] Levard | Sulfidation of Silver Nanoparticles: Natural antidote to their toxicity[END_REF]. Considering the low concentrations of Ag-NPs released in all environmental sinks and the rapid transformation of Ag-NPs into chemically stable Ag 2 S, the actual environmental risk seems to be much lower than suspected 15 years ago.

Meanwhile, another class of nanomaterials formed unintentionally in organic wastes [START_REF] Kim | Integrated Approaches of X-Ray Absorption Spectroscopic and Electron Microscopic Techniques on Zinc Speciation and Characterization in a Final Sewage Sludge Product[END_REF][START_REF] Bars | Drastic change in zinc speciation during anaerobic digestion and composting: Instability of nanosized zinc sulfide[END_REF] has been vastly overlooked. Zinc sulfide nanoparticles (ZnS-NPs) are released in quantities that are several orders of magnitudes higher than those of Ag-NPs and other manufactured nanoparticles.

Zinc (Zn) is the fourth most abundant metal in use today, with 12 million tons produced in 2020 alone (International Lead and Zinc Study Group: https://ilzsg.org/static/statistics.aspx). Sources of Zn in sewage sludge include (i) runoff from Zn roofs, which can account for up to 50% of the Zn runoff load in urban areas [START_REF] Thévenot | Critical budget of metal sources and pathways in the Seine River basin (1994-2003) for Cd, Cr, Cu, Hg, Ni, Pb and Zn[END_REF][START_REF] Gromaire | Impact of zinc roofing on urban runoff pollutant loads: The case of Paris[END_REF], and (ii) the use of Zn as a pigment, catalyst, UV filter, luminescent, semiconductor (and many others) in electronics, paints, and care products (https://ilzsg.org/static/enduses.aspx). Furthermore, Zn is a ubiquitous supplement added to livestock feed to prevent disease and promote growth [START_REF] Romeo | Zinc fate in animal husbandry systems[END_REF]; maximum authorized levels of trace elements in animal feeds are regulated in the European Union according to the Commission Regulation (EC) 1334/2003). Most of the added Zn is not taken up by the animals and livestock wastes (manure, slurries) are consequently rich in Zn [START_REF] Romeo | Zinc fate in animal husbandry systems[END_REF]. Cropped soils are the most common destination of (agricultural) livestock waste and (urban) sewage sludge. Our calculations suggest that the Zn released into the environment every year by spreading on the soil corresponds to no less than 4.5% of annual Zn production. This overwhelming figure means about 553 000 t may enter the water-soilplant compartments every year.

Zn in livestock waste and sewage sludge is mostly found in the form of ZnS-NPs. For instance, 54% of the total Zn present in a wide range of organic wastes sampled from full-scale treatment plants were shown to consist of ZnS-NPs (Le [START_REF] Bars | Drastic change in zinc speciation during anaerobic digestion and composting: Instability of nanosized zinc sulfide[END_REF]. Extrapolation suggests that about 440 000 t/year of ZnS-NPs are released into the environment every year, i.e. 3 orders of magnitude more than the amount of Ag-NPs produced every year (400 t). If on the one hand, bulk ZnS has a very low solubility product constant (2 × 10 -25 ), the small diameter of ZnS-NPs in organic wastes (3-5 nm) gives them unique structural properties such as lattice strain, surface defects that may affect their solubility, surface reactivity and toxicity in the environment [START_REF] Gilbert | Nanoparticles: Strained and stiff[END_REF]. Paradoxically, only a few studies have addressed the fate and toxicity of ZnS-NPs in the environment (Le [START_REF] Bars | Contrasted fate of zinc sulfide nanoparticles in soil revealed by a combination of X-ray absorption spectroscopy, diffusive gradient in thin films and isotope tracing[END_REF]. In addition, most of these studies considered particle sizes > 10 nm, which are not representative of the ZnS-NPs that form in organic wastes [START_REF] Voegelin | Time-Dependent Changes of Zinc Speciation in Four Soils Contaminated with Zincite or Sphalerite[END_REF].

The pioneering work of Banfield's group on biofilms helped understand the formation and properties of ZnS-NPs in organic wastes (e.g. [START_REF] Labrenz | Formation of sphalerite (ZnS) deposits in natural biofilms of sulfate-reducing bacteria[END_REF]. A few recent studies point out that ZnS-NPs undergo (bio)chemical transformations after composting or stockpiling of organic waste [START_REF] Lombi | Fate of Zinc Oxide Nanoparticles during Anaerobic Digestion of Wastewater and Post-Treatment Processing of Sewage Sludge[END_REF][START_REF] Bars | Drastic change in zinc speciation during anaerobic digestion and composting: Instability of nanosized zinc sulfide[END_REF]. Only one study assessed the fate of organic waste-borne ZnS-NPs in soil, reporting complete transformation of ZnS in a clayey soil within 6 months, followed by rapid immobilization of released Zn due to sorption by the soil matrix [START_REF] Formentini | Radical change of Zn speciation in pig slurry amended soil: Key role of nano-sized sulfide particles[END_REF]. Similarly, studies of the fate of manufactured CdSe/ZnS core-shell quantum dots (3-14 nm, i.e., almost the same size as organic waste-borne ZnS-NPs) also suggest partial or total dissolution of the nano-ZnS shell in soil solution (Faucheur et al. 2018) or at the mineral/biofilm interface within hours [START_REF] Desmau | How microbial biofilms impact the interactions of Quantum Dots with mineral surfaces?[END_REF]. Although synthetic ZnS-NP (3-5 nm) were shown to dissolve much more slowly in a sandy soil than in a clayey soil, the released Zn was more freely available in the sandy soil, and consequently potentially more toxic for soil organisms (Le [START_REF] Bars | Contrasted fate of zinc sulfide nanoparticles in soil revealed by a combination of X-ray absorption spectroscopy, diffusive gradient in thin films and isotope tracing[END_REF].

Despite the potentially high reactivity of ZnS-NPs, to the best of our knowledge, their ecotoxicity to soil ecosystems for example has only been investigated in 19 studies, as opposed to ZnO-NPs and Ag-NPs, each of which has been the subject of more than more than 250 studies. No study on ZnS-NPs ecotoxicity has been performed in a realistic agricultural soil environment with organic waste born ZnS-NPs. Among the 19 studies, in the nine most relevant ones (of which only one included organic waste), ZnS-NPs were synthetized (or obtained by sulfidation of ZnO-NPs), added to the soil (sometimes artificial soil) system in pot experiments or in synthetic soil porewater, and the toxicity for model organisms (plants or soil organisms) was tested over a maximum period of a few weeks [START_REF] Adele | Soil Bacteria Override Speciation Effects on Zinc Phytotoxicity in Zinc-Contaminated Soils[END_REF][START_REF] Oleszczuk | Impact of ZnO and ZnS nanoparticles in sewage sludge-amended soil on bacteria, plant and invertebrates[END_REF][START_REF] Song | Metal(loid) oxides and metal sulfides nanomaterials reduced heavy metals uptake in soil cultivated cucumber plants[END_REF][START_REF] Starnes | Toxicogenomic responses of Caenorhabditis elegans to pristine and transformed zinc oxide nanoparticles[END_REF][START_REF] Bao | Opposite effects of the earthworm Eisenia fetida on the bioavailability of Zn in soils amended with ZnO and ZnS nanoparticles[END_REF][START_REF] Biruntha | Green Synthesis of Zinc Sulfide Nanoparticles Using Abrus precatorius and Its Effect on Coelomic Fluid Protein Profile and Enzymatic Activity of the Earthworm, Eudrilus eugeniae[END_REF][START_REF] Adele | Role of plant growth promoting bacteria in driving speciation gradients across soil-rhizosphere-plant interfaces in zinc-contaminated soils[END_REF][START_REF] Chen | Responses of soil bacteria and fungal communities to pristine and sulfidized zinc oxide nanoparticles relative to Zn ions[END_REF][START_REF] Imperiale | Interaction of hyperaccumulating plants with Zn and Cd nanoparticles[END_REF]. Generally, these studies showed that ZnS-NPs could be toxic for plants, earthworms, nematodes, arthropods, fungi, and bacteria. The toxic concentration varied among the studies as it depends on the target organism and the environment. For example, the mortality and reproduction of the soil springtail Folsomia candida was affected by exposure as low as 2.5 mg.kg-1 in soil [START_REF] Oleszczuk | Impact of ZnO and ZnS nanoparticles in sewage sludge-amended soil on bacteria, plant and invertebrates[END_REF] whereas no toxic effect was found against the earthworm Eisenia fetida at 400 mg.kg -1 in an artificial soil [START_REF] Bao | Opposite effects of the earthworm Eisenia fetida on the bioavailability of Zn in soils amended with ZnO and ZnS nanoparticles[END_REF]. Moreover, all the studies except three were performed using ZnS-NPs > 20 nm in size, that is, bigger than the ZnS-NPs found in organic wastes, even though the ecotoxicity of NPs is known to be highly dependent on the size of the nanoparticle [START_REF] Lowry | Transformations of Nanomaterials in the Environment[END_REF][START_REF] Spurgeon | Nanomaterial Transformations in the Environment: Effects of Changing Exposure Forms on Bioaccumulation and Toxicity[END_REF]. What is more, it is not clear whether ZnS-NPs are more or less toxic than ZnO-NPs and Zn 2+ , and whether longer exposure (i.e. more than few weeks) to ZnS-NPs alters toxicity. Extrapolating these results to more realistic conditions is risky because the small number of studies cannot account for the large number of variables that affect ecotoxicity. Indeed, the ecotoxicity of NPs depends on interactions between the nanoparticle and its environment (e.g. solubility, aggregation). In turn, this interaction depends on the quantity and quality of organic matter, pH, ionic strength, chemical composition of soil solution, and the presence of colloids, among others [START_REF] Lowry | Transformations of Nanomaterials in the Environment[END_REF][START_REF] Libralato | Toxicity Effects of Functionalized Quantum Dots, Gold and Polystyrene Nanoparticles on Target Aquatic Biological Models: A Review[END_REF][START_REF] Spurgeon | Nanomaterial Transformations in the Environment: Effects of Changing Exposure Forms on Bioaccumulation and Toxicity[END_REF]. A comprehensive range of experiments would be needed to assess the effect of exposure to organic waste-borne ZnS-NPs and associated hazards on a wider range of targeted organisms and of soil properties that may determine NP ecotoxicity.

We are convinced that the fate and ecotoxicity of ZnS-NPs formed in organic wastes deserve more attention in both modeling (currently not assessed, Avadi et al. 2022) and experimental studies, particularly in water-soil-plant systems, like other anthropogenic NPs such as Ag, ZnO, TiO 2 , CeO 2 , or carbon nanotubes. This is of utmost importance given the ever-increasing production of waste that calls on scientists, politicians, and civil society to transform waste into resources, while closing material loops in circular economies as a way to achieve sustainable development.

Acknowledgments

This study was part of the DIGESTATE project, funded by ANR (Agence Nationale de la Recherche, France) under Grant ANR-15-CE34-0003-01.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.