Loyal Déborah

Anne-Laure Sutter

Rascle Nicole

Validity of the Measure of Intensive Mothering Ideology

Keywords: Intensive Mothering Ideology, Motherhood, Scale Development, Validity

Objective. The aim of this study was to test the validity of the Measure of Intensive Mothering Ideology (MIMI) a French scale assessing beliefs about mothering and childcare. Method. The MIMI was submitted online to Mothers / mothers-to-be (N = 249) and Childless women (N = 231). To test structural validity, confirmatory factor analyses were conducted in both groups. Then, to test knowngroups validity, means comparisons were conducted according to parity (mothers / mothers-to-be and childless women) and employment status (full-time, part-time and housewives). It was also hypothesized that MIMI scores would be negatively correlated with education. Results. Model fit was satisfactory for Mothers / mothers-to-be (X2/df = 2.52, AGFI = .957, NFI = .937 RMR = .087) and, to a lesser extent, for Childless women (X2/df = 3.31, AGFI = .948, NFI = .907, RMR= .104). In both groups, most dimensions were moderately correlated (.22-.70). As hypothesized, Mothers / mothersto-be and Housewives had higher score than Childless women and Employed women. MIMI scores were also negatively and moderately correlated with education. Conclusion. The MIMI shows good structural validity and known-groups validity. This scale offers interesting research avenues notably regarding perinatal parental adaptation.

Introduction

Our ideas about family, motherhood and childcare do not stem from nature but are mainly socially built [START_REF] Ambert | An international perspective on parenting: Social change and social Constructs[END_REF][START_REF] Arendell | Conceiving and investigating motherhood: The decade's scholarship[END_REF][START_REF] Hays | The cultural contradictions of motherhood[END_REF][START_REF] Harkness | Culture and Parenting[END_REF][START_REF] Bornstein | Cultural Approaches to Parenting[END_REF]. According to Hays [START_REF] Hays | The cultural contradictions of motherhood[END_REF], in developed nations, contemporary dominant discourse about motherhood constitute an ideology, a collection interrelated of beliefs, that she named Intensive Mothering Ideology (IMI). According to Hays [START_REF] Hays | The cultural contradictions of motherhood[END_REF], most people think that mothers are intrinsically the best caregivers for children. Childcare is described as a consuming expert-guided and child-centered activity. Child rearing is seen as the hardest job but also as the most meaningful and rewarding one.

Finally, children are revered owing to their purity and genuineness.

According to the attachment theory, women deep involvement in mothering is important for the child development and mothers satisfaction and efficacy [START_REF] Bowlby | Maternal care and mental health[END_REF][START_REF] Bradley | Parents' socioemotional investment in Children[END_REF][START_REF] Wolff | Sensitivity and Attachment: A Meta-Analysis on Parental Antecedents of Infant Attachment[END_REF]. However normative aspects of motherhood might also be significant contributors to distress in mothers through excessive expectations or idealization [START_REF] Tummala-Narra | Contemporary impingements on mothering[END_REF][START_REF] Liss | Maternal Guilt and Shame: The Role of Self-discrepancy and Fear of Negative Evaluation[END_REF][START_REF] Beck | Postpartum depression: A metasynthesis[END_REF][START_REF] Haga | A qualitative study of depressive symptoms and wellbeing among first-time mothers: The transition to motherhood[END_REF][START_REF] Miller | Is This What Motherhood is All About?": Weaving Experiences and Discourse through Transition to First-Time Motherhood[END_REF]. Moreover, IMI might also threaten gender equality notably through the belief that the mother is the natural and best caregiver which places an undue burden on mothers [START_REF] Arendell | Conceiving and investigating motherhood: The decade's scholarship[END_REF][START_REF] Hays | The cultural contradictions of motherhood[END_REF][START_REF] Macdonald | What's Culture Got to Do with It? Mothering Ideologies as Barriers to Gender Equity[END_REF][START_REF] Dillaway | Locating Mothers: How Cultural Debates About Stay-at-Home Versus Working Mothers Define Women and Home[END_REF][START_REF] Johnston | Cognitive acrobatics in the construction of worker-mother identity[END_REF][START_REF] Brugeilles | Partage des activités parentales: les inégalités perdurent[END_REF][START_REF] Meda | Le temps des femmes : pour un nouveau partage des rôles[END_REF][START_REF] Garner | Conciliation entre vie professionnelle et vie familiale, les leçons des enquêtes auprès des ménages[END_REF]. Finally, IMI might also impact work-family balance because of the view that mothering is all-encompassing and much more important than any professional success [START_REF] Hays | The cultural contradictions of motherhood[END_REF][START_REF] Dillaway | Locating Mothers: How Cultural Debates About Stay-at-Home Versus Working Mothers Define Women and Home[END_REF][START_REF] Johnston | Cognitive acrobatics in the construction of worker-mother identity[END_REF][START_REF] Loyal | Mothering Ideology and Work Involvement in Late Pregnancy: A Clustering Approach[END_REF].

However, few quantitative studies focus on beliefs about motherhood. According to Liss et al., the main barrier to understand IMI was the lack of a quantitative measure to assess it [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. To address this problem, the Intensive Parenting Attitudes Questionaire (IPAQ) has been constructed in the United States [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. The IPAQ measures 5 dimensions: Essentialism (mothers are the best caregivers for children), Fulfillment (being a parent is fulfilling), Child-centrism (child-rearing should be focused on the child's needs), Challenge (child-rearing is the hardest job) and Stimulation (children should be properly stimulated). All dimensions were moderately correlated, highlighting that IMI is a set of interconnect beliefs. Various dimensions of IMI have been linked with mothers' anxiety about leaving the child, perception of maternal employment as detrimental to children and more participation in household duties and childcare [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. It was also found that Essentialism and Challenge were more prevalent in mothers than in childless women. However, Stimulation and Child centrism were more prevalent in childless women [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. In the same vein, it was found that Essentialism and Challenge were stronger in housewives than in employed mothers However, Stimulation was lower in housewives. [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. Incidentally, one should note than numerous studies have showed that gender ideology was more traditional in less educated women [START_REF] Davis | Gender ideology: components, predictors, and consequences[END_REF]. Our guess is that it might hold true for IMI. Finally, in a cross-sectional study with mothers of preschoolers, Essentialism was linked with less life satisfaction and Challenge was linked with more depressive symptoms and stress, while controlling for social support [START_REF] Rizzo | Insight into the parenthood paradox: Mental health outcomes of intensive mothering[END_REF].

In France the Measure of Intensive Mothering Ideology (MIMI) has been built [START_REF] Loyal | Intensive mothering ideology in France: A pilot study[END_REF]. This scale is made up of IPAQ items and new items generated from French mothers and mothers-to-be speech.

All the items have been submitted jointly to exploratory factor analysis. This procedure was undertaken to increase the cultural validity of the scale in the French context. Indeed, parenting is highly culture-bound [START_REF] Harkness | Culture and Parenting[END_REF][START_REF] Bornstein | Cultural Approaches to Parenting[END_REF]. American and French women might share some ideas about childcare and motherhood but they might also exhibit differences [START_REF] Bornstein | Cultural Approaches to Parenting[END_REF][START_REF] Suizzo | French and European American child-mother play: Culture and gender considerations[END_REF]. For instance, French women have been found to place less value on bonding, breastfeeding, sacrificing their own needs and responding quickly to baby's ones, and more value on equal task-sharing between parents and child autonomy than American mothers [START_REF] Suizzo | French and American Mothers' Childrearing Beliefs: Stimulating, Responding, and Long-Term Goals[END_REF]. Moreover, United States and France have very different child and family policies [START_REF] Thevenon | Family policies in OECD countries: a comparative analysis[END_REF]. French parents benefit from childcare provision, paid parental leave and family allowance [START_REF] Thevenon | Family Policies: France[END_REF]. On the other hand, United States "has no explicit family policy, nor does it have a coherent package of social policies that are targeted on children and their families" [START_REF] Kamerman | Child and Family Policies in the United States at the Opening of the Twenty-first Century[END_REF]. For instance, in the United States, except for a limited number of exceptions, there is no maternity or paternity paid leave [START_REF]Maternity and paternity at work: Law and practice across the world[END_REF]. Nearly 1 in 4 American women go back to work within two weeks of having a child [START_REF] Lerner | The Real War on Families: Why the U.S. Needs Paid Leave Now[END_REF].

The MIMI assesses 6 dimensions of IMI: Essentialism, Consuming Fulfillment, Childcentrism, Challenge, Sacrifice and Stimulation. The MIMI shows some differences with the IPAQ.

First, a new "Sacrifice" dimension was found and describe how parents should set aside many things (i.e. "When becoming a parent, one must be prepared to put on hold one's personal life.").

Interestingly, the idea of sacrifice had been described in Hays seminal work [START_REF] Hays | The cultural contradictions of motherhood[END_REF]. Secondly, the Essentialism dimension was more nuanced and depicts fathers as uncomfortable rather than incompetent with children matters (i.e. "Mothers are often more comfortable with babies than fathers"). Moreover, the original Fulfillment dimension has been turned into a "Consuming Fulfillment" dimension about how parenting is fulfilling ("Being a parent brings a person the greatest joy he or she can possibly experience") but also consuming and all encompassing (i.e. "A parent always fears that something might happen to his children"). Finally, the Stimulation dimension was more about providing nurturing one-to-one interactions (i.e. "One must stimulate children, play with them, talk to them, sing songs to them, read stories to them") rather than intensive cognitive stimulation. Most dimensions were moderately correlated, highlighting that IMI is a set of interconnect beliefs. IMI, assessed with the MIMI, in conjunction with worries about maternal employment effects on children, has been associated with more depressive symptoms and preoccupied attachment in a cross-sectional study with pregnant women [START_REF] Loyal | Mothering Ideology and Work Involvement in Late Pregnancy: A Clustering Approach[END_REF].

The MIMI has been created and validated in French women recently [START_REF] Loyal | Intensive mothering ideology in France: A pilot study[END_REF] and its validity should be explored more deeply [START_REF] Boateng | Best Practices for Developing and Validating Scales for Health, Social, and Behavioral Research: A Primer[END_REF]. Thus, this study was designed to assess the structural and known-groups validity of this scale. It was hypothesized that the MIMI would show good structural validity through confirmatory factor analysis. It was also hypothesized that the MIMI would discriminate women according to parity and employment status [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. Finally, it was hypothesized that IMI would be negatively associated with education [START_REF] Davis | Gender ideology: components, predictors, and consequences[END_REF].

Method

Participants

Mothers / mothers to be (N = 249) and Childless women (N = 231) were recruited online.

Participants characteristics are presented on Table 1. The questionnaire was hosted with LimeSurvey (https://www.limesurvey.org/) and administrated on French forums and Facebook groups (generalist, pregnancy, motherhood, childcare…) using a snowball sampling method.

TABLE 1

Research about online recruitment and data collection is still scarce but it offers very promising avenue [START_REF] King | Social media recruitment and online data collection: A beginner's guide and best practices for accessing low-prevalence and hard-to-reach populations[END_REF][START_REF] Skitka | The internet as psychological laboratory[END_REF][START_REF] Birnbaum | Psychological Experiments on the Internet[END_REF]. First, internet and social media offer a cost effective way to recruit quickly a large amount of participants. Moreover, participants are responding in their natural context and might be less influenced by social desirability bias. However, some potential disadvantages should be noted. First, it is impossible to control how participants are completing the study (alone or not, noisy surroundings…). It is also impossible to ensure that the person has read thoughtfully and understood the consent form before signing it. Moreover, because of anonymity, online participants might be more prone to conform to group norms and expectations. More importantly, issues have been raised regarding the representativeness of samples recruited online. However, some researchers have observed that online recruitment might in fact lead to more diverse samples than other recruitment methods [START_REF] King | Social media recruitment and online data collection: A beginner's guide and best practices for accessing low-prevalence and hard-to-reach populations[END_REF][START_REF] Gosling | Should we trust web-based studies? A comparative analysis of six preconceptions about internet questionnaires[END_REF].

This study was approved by a national informatics right committee (https://www.cnil.fr/) and a regional ethic committee (http://www.cpp-soom3.u-bordeaux2.fr/). All procedures were conducted in accordance with the declaration of Helsinki.

Measures

The Measure of Intensive Mothering Ideology (MIMI) was administered [START_REF] Loyal | Intensive mothering ideology in France: A pilot study[END_REF]. The MIMI contains 21 items assessing six dimensions of IMI: Essentialism (seven items), Consuming Fulfillment (five items), Child-centrism (two items), Challenge (two items), Sacrifice (three items) and Stimulation (two items). Participants were asked to answer each item on a Likert scale ranging from 1 (disagree very strongly) to 6 (agree very strongly). A higher score is indicative of a stronger endorsement of the IMI component. Internal consistencies assessed with Cronbach's alpha have previously been computed in a sample of mothers and mothers-to-be for Essentialism (.83), Consuming Fulfillment (.74), Child-centrism (.75), Challenge (61), Sacrifice (.61) and Stimulation (.64) [START_REF] Loyal | Intensive mothering ideology in France: A pilot study[END_REF].

Statistical Analysis

Statistical analyses were carried out with SPSS 20 and SPSS Amos 20. First, confirmatory factor analyses (Weighted Least Squares) were conducted for Mothers / mothers-to-be and Childless women. Our sample size was adequate to compute such analyses [START_REF] Worthington | Scale Development Research: A Content Analysis and Recommendations for Best Practices[END_REF]. Model fit is usually assessed with various indicators such as : χ 2 /ddl < 2-5, AGFI and NFI >.90 and low RMR [START_REF] Worthington | Scale Development Research: A Content Analysis and Recommendations for Best Practices[END_REF][START_REF] Hooper | Structural Equation Modelling: Guidelines for Determining Model Fit[END_REF]. Factor loadings (>.40) and correlations between variables were examined. Finally, internal consistencies have been computed. Then, IMI scores were compared between Mothers / mothers-to-be and Childless women (Student T test). Then, in Mothers / mothers-to-be, IMI scores were compared between women who were housewives or were employed full-time or part time (Anova). Because most childless women were students (60%) comparison according to their employment status was not computed.

Results

Structural Validity.

According to current recommendations [START_REF] Worthington | Scale Development Research: A Content Analysis and Recommendations for Best Practices[END_REF][START_REF] Hooper | Structural Equation Modelling: Guidelines for Determining Model Fit[END_REF], model fit was regarded as satisfactory in Mothers / mothers-to-be (X2/df = 2.52, AGFI = .957, NFI = .937 RMR = .087) and, to a lesser extent, in Childless women (X2/df = 3.31, AGFI = .948, NFI = .907, RMR= .104). First, as expected, IMI (except Stimulation) was stronger in Mothers / mothers-to-be than in Childless women (Table 2). Moreover, IMI (except Stimulation) was stronger in Mothers / mothers-tobe who were housewives than in those who were employed either part time or full time. However, Consuming Fulfilment was lower in women who were employed part time than in those who were either housewives or employed full time (Table 3).

TABLE 2 TABLE 3

In Mothers / mothers to be, education was associated with Consuming Fulfillment, Child centrism, Essentialism, Stimulation (r = -.34, -.25, -.24, -.17, p<.01) and Sacrifice (r =-.14, p<.05). This was also observed, to a lesser extent, in Childless women regarding Consuming Fulfillment (r = -.15, p<.05) and Essentialism (r = -.14, p<.05).

Discussion

As hypothesized, the MIMI showed good structural validity observed through confirmatory factor analysis in Mother / mothers-to-be and, to a lesser extent, in Childless women. Essentialism, Consuming Fulfilment and Child Centrism had good internal consistencies. However, Challenge, Sacrifice and Stimulation dimensions had lower internal consistency but it is well known that Cronbach alpha is highly influenced by the number of items [START_REF] Streiner | Starting at the Beginning: an Introduction to Coefficient Alpha and Internal Consistency[END_REF][START_REF] Lance | The Sources of Four Commonly Reported Cutoff Criteria: What Did They Really Say?[END_REF][START_REF] Rammstedt | Can't We Make It Any Shorter?: The Limits of Personality Assessment and Ways to Overcome Them[END_REF][START_REF] Ziegler | Short Scales -Five Misunderstandings and Ways to Overcome Them[END_REF]. Since those dimensions have few items (2 or 3), getting values above .60 might be seen as satisfactory. Two-item factors are usually considered less reliable [START_REF] Worthington | Scale Development Research: A Content Analysis and Recommendations for Best Practices[END_REF] but some authors raise questions about this misgiving [START_REF] Rammstedt | Can't We Make It Any Shorter?: The Limits of Personality Assessment and Ways to Overcome Them[END_REF][START_REF] Ziegler | Short Scales -Five Misunderstandings and Ways to Overcome Them[END_REF]. However, internal consistency of those dimensions in childless women was troublesome (<.50). Thus, using the MIMI in childless women requires some caution. On another note, in both groups, dimensions were correlated, showing that IMI is a collection of interconnected beliefs. Only Stimulation was not associated with Sacrifice and Essentialism. Thus, this dimension might be regarded as less central to the very definition of IMI. However, it was clearly correlated to Consuming Fulfillment, Child Centrism and Challenge.

As hypothesized, IMI (except Stimulation) was higher in Mothers / mothers-to-be than in Childless women. Moreover, for Mothers / mothers-to-be, IMI (except Stimulation and Consuming Fulfillment) was higher in those who were housewives than in those who were employed. Finally, IMI was negatively correlated with education in Mothers / mothers-to-be, and, to a lesser extent, in Childless women. Thus, our hypotheses are confirmed: IMI endorsement is higher in mothers / mothers-to-be, housewives and less educated women. Our guess is that a very interesting research avenue will be to explore if women who are choosing to have children have higher IMI endorsement (self-selection hypothesis) or if women who become mothers exhibit a growth in IMI endorsement (role transition hypothesis). The MIMI might be used to conduct such research. Moreover, IMI was higher in Housewives. One could conclude that IMI might inhibit work engagement in women. Indeed, Liss et al. found that IMI was associated with fear to leave the child and perception that maternal employment is detrimental to children [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. In the same vein, IMI associated with perception that maternal employment is detrimental to children has been associated with a delayed planned return to work after the birth [START_REF] Loyal | Mothering Ideology and Work Involvement in Late Pregnancy: A Clustering Approach[END_REF]. However, IMI might also change to accommodate employment status. Indeed, it has been observed that gender ideology is modified by changes in the division of housework [START_REF] Carlson | Housework: Cause and consequence of gender ideology?[END_REF]. Further research might contemplate using the MIMI to explore the presumably reciprocal influences between work and IMI. Finally, one unexpected result was that Consuming Fulfillment was lower in women who were employed part-time. This interesting result might be explained by the fact that those women are not engaged in either full-time childcare or full-time employment. Those two situations might be associated with the perception that motherhood is a demanding and consuming activity because of the time spend (housewifery) or the difficulty in work and family life balance (fulltime employment).

On the whole, our results are far more univocal than Liss et al. ones [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. They found that only Essentialism and Challenge were higher in Mothers whereas Stimulation and Child-Centrism were higher in Childless women. They found no differences for Fulfillment. Regarding employment, they found that housewives had higher Essentialism and Challenge but lower Stimulation. They found no differences regarding Fulfilment and Child-Centrism. The explanation for those differences between Liss et al. results [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF] and ours probably lay in the different measures used. Our guess is that the IPAQ probably assess a more virulent version of IMI. For instance, mothers in Liss et al. study might have been unable to mentally reconcile Stimulation and Child-centrism with their everyday life. This

[

  Validité de la Mesure de l'Idéologie du Maternage Intensif] Résumé Objectif. Le but de cette étude était de tester la validité de la Mesure de l'Idéologie du Maternage Intensif (MIMI), une échelle française évaluant les croyances sur le maternage et les soins aux enfants. Méthode. La MIMI a été soumise en ligne a des mères / futures mères (N = 249) et à des femmes sans enfants (N = 231). Cette échelle comprend 21 items évaluant 6 dimensions de l'IMI. Pour vérifier la validité structurelle, des analyses factorielles confirmatoires (moindres carrés pondérés) ont été effectuées dans les deux groupes. Puis, pour tester la validité de groupes connus, des comparaisons de moyennes ont été effectuées selon la parité (mères / futures mères et femmes sans enfant) et l'emploi (temps plein, temps partiel et au foyer). Nous avons également fait l'hypothèse que les scores d'IMI seraient corrélés négativement avec l'éducation. Résultats. L'ajustement du modèle était satisfaisant pour les mères et les futures mères (X2/df = 2.52, AGFI = 0.957, NFI = 0.937 RMR = 0.087) et, dans une moindre mesure, pour les femmes sans enfant (X2/df = 3.31, AGFI = 0.948, NFI = 0.907, RMR = 0.104). Dans les deux groupes, la plupart des dimensions étaient modérément corrélées (.22-.70). Conformément à nos hypothèses, les mères / futures mères et les femmes au foyer présentent des scores plus élevés que les femmes sans enfant et les femmes ayant un emploi. Les scores d'IMI étaient également corrélés négativement et modérément avec le niveau d'éducation. Conclusion. La MIMI présente une bonne validité structurelle et de groupes connus. Cette échelle offre des pistes de recherche intéressantes notamment en matière d'adaptation au rôle parental dans la période périnatale. Mots-clefs : Idéologie du Maternage Intensif ; Maternité ; Développent d'Echelle ; Validité

Figure 1

 1 Figure 1 displays factor loadings and correlations between variables. Most factor loadings are

FIGURE 1 Known

 1 FIGURE 1

Figure 1 .Figure 1 .

 11 Figure 1. Validity of the Measure of Intensive Mothering Ideology

Table 1 .

 1 Descriptive statistics 381

		Mothers
		Mothers-to-be	Childless
		N = 249	N=231
	Age (years)
	Min-max	20 -47	-48
	Mean (SD)	30.49 (5.00)	23.95 (4.29)
	Marital Status (%)
	Marriage	55
	Couple	41
	Single	3
	Education (%)
	≥ 2 years of college	62
	≤ High school diploma	22
	Employment (%)
	Full-time	38
	Part time	25
	Housewives	23
	Student	1
	Pregnant (%)	14
	Not pregnant women
	Number of child (%)
	0	0
	1	52
	2	30
	3	14
	4 or more	4
	Youngest child age (months)
	Min-Max	0-60	---
	Mean (SD)	15.43 (12.92)
	Pregnant women
	Number of child (%)
	0	42
	1	28	---
	2	25
	3 or more	6
	Youngest child age (months)
	Min-Max	6 -84	---
	Mean (SD)	33.10 (20.10)
	382
	383
	384
	385
	386

Table 2 .

 2 Mean Comparisons between Mothers / Mothers-to-be and Childless Women.

	Dimensions	Student T Test	Mothers / Mother-to-be (N = 249)	Childless Women (N = 231)
	Essentialism	t(478) = 3.68, p<.01	3.41 (0.93)	3.09 (0.91)
	C. Fulfilment	t(478) = 4.04, p<.01	5.09 (0.79)	4.78 (0.87)
	Child centrism	t(478) = 3.31, p<.01	4.68 (1.03)	4.34 (1.22)
	Challenge	t(478) = 2.11, p<.05	4.91 (1.01)	4.70 (1.11)
	Sacrifice	t(478) = 5.42, p<.01	3.38 (1.05)	2.89 (0.93)
	Stimulation	t(478) = -0.33, p =.75	5.64 (0.56)	5.65 (0.45)

Table 3 .

 3 Mean Comparisons between Mothers / Mothers-to-be according to their employment status.

	Dimensions	Anova	Full-Time (n = 95)	Part-Time (n = 62)	Housewive (n = 58)
	Essentialism	F(2.212) = 21.43, p<.01	3.12 (0.89) a	3.13 (0.82) a	4.01(0.94)
	C. Fulfilment	F(2.212) = 8.59, p<.01	5.12 (0.73) a	4.76 (0.99)	5.34 (0.58) a
	Child centrism F(2.212) = 9.27, p<.01	4.66 (1.01) a	4.30 (1.13) a	5.10 (0.92)
	Challenge	F(2.212) = 5.36, p<.01	4.73(1.05) a	4.77(1.07) a	5.25 (0.84)
	Sacrifice	F(2.212) = 12.68, p<.01	3.23(0.95) a	3.06 (1.02) a	3.92(1.06)
	Stimulation	F(2.212) = 0.623, p=.54	5.68(0.47)	5.59(0.69)	5.59(0.57)

situation of cognitive dissonance might lead them to reject those ideas [START_REF] Festinger | A theory of cognitive dissonance[END_REF]. It might also be explained by strong differences between French and American contexts. United States and France have very different child and family policies [START_REF] Thevenon | Family policies in OECD countries: a comparative analysis[END_REF]. French parents benefit from childcare provision, paid parental leave and family allowance [START_REF] Thevenon | Family Policies: France[END_REF]. Thus, American and French housewives might be very different. For instance, French mothers have been found to place more value on child autonomy [START_REF] Suizzo | French and American Mothers' Childrearing Beliefs: Stimulating, Responding, and Long-Term Goals[END_REF], so they might hold a softer understanding of what Stimulation and Child-centrism is about.

Thus study suffers some limitation. First, our participants were rather privileged women, mainly well-educated, young and married or living as a couple, especially for mothers / mothers-to-be.

Further research should contemplate using the MIMI in more deprived populations. Moreover, the housewife status might encompass various situations ranging from women choosing housewifery as a lifestyle to women forced to stay at home for various reasons. Getting information about motives for housewifery and employment could provide interesting perspective. For instance, women who are housewives but would like to work are prone to distress [START_REF] Kroska | Exploring the consequences of gender ideology -work discrepancies[END_REF][START_REF] Des Rivières-Pigeon | Travailler ou ne pas travailler… Le désir d'emploi des nouvelles mères et la dépression postnatale[END_REF]. In the same vein, even if employment have generally been found to be beneficial to mothers mental health [START_REF] Barnett | Women and multiple roles: Myths and reality[END_REF], those who are employed but belief that men should be the sole bread-winner have been found to be more distressed [START_REF] Helms | Provider role attitudes, marital satisfaction, role overload, and housework: A dyadic approach[END_REF][START_REF] Perry-Jenkins | Linkages Between Women's Provider-Role Attitudes, Psychological Well-Being, and Family Relationships[END_REF] Finally, further studies should investigate the test-retest reliability of the MIMI [START_REF] Boateng | Best Practices for Developing and Validating Scales for Health, Social, and Behavioral Research: A Primer[END_REF] and its association with various related concepts [START_REF] Boateng | Best Practices for Developing and Validating Scales for Health, Social, and Behavioral Research: A Primer[END_REF] such as beliefs about the consequences of maternal employment on children [START_REF] Loyal | Mothering Ideology and Work Involvement in Late Pregnancy: A Clustering Approach[END_REF][START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF] or household and parental duties [START_REF] Liss | Development and validation of a quantitative measure of intensive parenting attitudes[END_REF]. Attitude toward highly normative childcare practices such as breastfeeding [START_REF] Lee | Breastfeeding advocacy, risk society and health moralism: A decade's scholarship[END_REF] might also be of particular interest

Disclosure of interest.

The authors declare that they have no competing interest.


Fig 3) Impact of the Mix4 on the immune response

PCA summarizing the patterns of 66 immune gene expression levels at 11 days of age (i.e. 4 days post infection) in isolator 1 and 2. The levels of gene expression were assessed using a medium-throughput qPCR approach (Biomark). The patterns were slightly different (p=0.046, between group analysis, Monte Carlo permutation test). The same result was found at 14 days of age (p=0.044; not shown).

Fig 2) Impact of the Mix4 on Salmonella colonization

Levels of Salmonella shedding in isolator 1 and 2 (cf. Fig1) after Salmonella infection, assessed using bacteriologic counts. At 14, 20 and 27 days of age, the difference between Isolator 1 and 2 was significant. In isolator 1, 35 chickens were bred until 27 days of age and inoculated with Salmonella at 7 days of age. A subset of 14 chickens was kept for the metabarcoding characterization of their gut microbiota.

In isolator 2, 35 chickens were bred until 27 days of age, inoculated at 1 day of age with the Mix4, and infected with Salmonella at 7 days of age. A subset of 15 chickens was kept for the metabarcoding characterization of their gut microbiota.

We observed significant differences between the Unifrac β-diversities measured in isolator 1 and 2 at every age category except on day 4 (Permanova tests).

Conclusion

Inoculation of four commensal bacteria (namely: Mix4) after hatching durably modified gut microbiota composition and Salmonella colonization. The inoculation of these commensal bacteria also induces a slighly different immune response against Salmonella infection. These results pave the way of the development of bacterial consortia able to increase the resistance of chicks to pathogens.

Introduction

Salmonella infections are diseases of particular medical and economic interest in developed countries. Salmonella-infected host animals may present highly heterogeneous shedding levels, depending on the outcome of infection. Several studies have investigated the host-gut microbiota-pathogen interactions and their role in the occurrence of Salmonella super-shedding. They suggested the possibility to manipulate the relationships between Salmonella and the GM to reduce the fecal shedding. In this project, we analysed the impact of four commensal bacteria inoculated before infection on: 1-the faecal and caecal microbiota composition ; 2-Salmonella colonization 3-the immune status of chicks measured in kinetic in blood.

Mix4

• 1x E. faecium

Fig 1 )

 1 Fig 1) Impact of the Mix4 on gut microbiota β-diversities Principal coordinate analysis summarizing Unifrac distances among the chicken bred in isolator 1 and 2, at every age category.

→

  Isolator 1: only Salmonella inoculation at 7 days of age → Isolator 2: Mix4 exposure prior Salmonella inoculation at 7 days of age INOCULATION OF FOUR COMMENSAL BACTERIA IN CHICKEN HAS AN IMPACT ON THE GUT MICROBIOTA COMPOSITION, SALMONELLA COLONISATION AND THE IMMUNE RESPONSE KEMPF F. 1 , DRUMO R. 1 , CHAUSSE A.-M. 1 , KUBASOVA T. 2 , MENANTEAU P. 1 , GUITTON E. 3 , RYCHLIK I. 2 , VELGE P. 1

ISP, INRAE, Université Tours, UMR 1282, 37380, Nouzilly, France

Veterinary Research Institute, Hudcova 70, Brno, 621 00, Czech Republic

Plateforme d'Infectiologie Expérimentale, INRAE, 37380, Nouzilly, France


Michel Lubrano
email: lubrano@univmed.fr

Serge-Christophe Kolm

David Later

De La Croix

The Redistributive Aspects of ELIE: a simulation approach

Keywords: JEL Classification : C15, D63, D78, H21

. Il étudie successivement l'influence de l'incertitude dans la connaisance des revenus indivduels, l'impact des échelles d'équivalence et finalement les conséquences de l'accumulation du capital. Si l'incertitude ne modifie pas fondamentalement les propriétés d'équité de ELIE, les échelles d'équivalences peuvent avoir des cosnéquences non triviales en fonction de la relation entre fertilité et revenu. Enfin, l'accumulation du capital introduit de fortes inégalités dans la taxation. Le papier repose sur des simulations calibrées de la distribution des revenus en France et sur l'utilisation des divers indices de taxation.

Introduction

Macrojustice, as discussed in [START_REF] Kolm | The optimal production of social justice[END_REF] concerns the rules for distributing the benefits of the main social resources. Because initial dotations are unequally distributed including human capacities, macrojustice implies some kind of redistribution which depends on the socially accepted degree of solidarity or altruism. [START_REF] Kolm | The optimal production of social justice[END_REF] considers that the main resources are labour capacities so that redistribution should imply an equal labour income equalisation or ELIE. "Practically, each individual receives or yields in proportion to the difference between her wage rate and the average, the proportion being the equalisation labour. Hence, a difference in productivity is compensated by a proportional difference in income transfer." As capital was initially produced by labour, the real means of production at the steady state are labour and natural resources. The contribution of labour to total resources is accounted for 97%. Consequently, [START_REF] Kolm | The optimal production of social justice[END_REF] assumes that the question of taxation can be centred on labour only: Labour is the main source of income in society.

This being said, let us turn to usual public finance theory in order to situate the possible impact of ELIE. Public finance has different aims and functions that can be summarised as follows.

-A redistributive function which is based essentially on progressive income tax and various policies of assistance such as the RMI (minimum income for insertion) in France or allowances for old people. Obviously an implementation of ELIE should replace these existing taxes and transfers.

-An incentive function. For instance, a carbon tax aiming at reducing CO 2 emissions, or family allowances that aim at promoting a greater number of children per family. ELIE cannot be a full substitute for these taxes and transfers.

-The production of public goods. This aspect of public finance aims at organising and improving economic growth and capital accumulation. Here again, we are outside the role and attributions of ELIE.

These functions cannot of course be totaly independent one from another. But we would like to be able to separate as much as possible these different effects. We would not like for instance that a redistributive policy might generate disincentives or that an incentive policy could have regressive distributional features. This is the reason why ELIE is based on capacities and not on effective labour supply. As its basis for taxation is inelastic, there should be no inefficiency disincentives for labour supply with ELIE. [START_REF] Kolm | The optimal production of social justice[END_REF] assumes that the distribution of talents is perfectly known. We shall detail a number of reasons why this might not be the case. It is thus reasonable to investigate the sensitivity of the ELIE scheme to an uncertainty in the knowledge of the distribution of talents. Are the basic properties greatly modified or is uncertainty only a minor problem? This question is essential for judging the redistributive performances of ELIE. In other words, from a purely redistributive point of view, is it worth changing the existing system for ELIE?

Modern societies and particular France are concerned with the behaviour of households as regards to fertility decisions. Family allowances have been implemented as an incentive to fertility. They have been quite successful in France for maintaining a decent rate of birth, compared to other European countries. But family allowances, on the other side, might have some unwanted redistributive effects. So what is the exact trade-off between family incentives and redistribution? Because of their incentive motives, they cannot be a substitute to ELIE, but have to be a complement. How can the two systems cohabitate? How will family allowances modify the redistributive properties of ELIE? This is also an important question. [START_REF] Kolm | The optimal production of social justice[END_REF] does not consider capital income by arguing that physical capital is itself produced by labour, so that for macrojustice (not for microjustice) capital can be neglected. This can be true in a steady state, but it ignores the question of accumulation and economic growth and so does not give any hint on how to investigate the trade-off between growth and redistribution. On one hand, individual saving rates rise with the level of income. Any redistributive scheme, by transferring resources from rich to poor tends to lower the aggregate rate of saving and by consequence the rate of capital accumulation. On the other hand, capital income modifies the total income distribution and has a strong influence on inequality. In most countries and certainly in France, wealth inequality is much stronger than income inequality.

To answer the above questions, we shall have to formalise the ELIE scheme in various contexts. The basic ELIE formula is simple and linear. But this attractive simplicity hides in fact many non trivial properties. For instance, the degree of redistribution depends on the characteristics and shape of the initial gross income distribution. When mixed with other public finance mechanisms, the initial model is also much complexified. For these two reasons (shape of the distribution, and more complex models), we have to resort to simulation techniques. We have clearly two ways of proceeding. We could start from a microeconomic sample of French wages and simulate from this empirical distribution. The alternative solution is to simulate a given density and calibrate its parameters so as to match some inequality or redistributive indices computed from the empirical distribution of gross or net incomes in France. de la [START_REF] De La Croix | The Tradeoff Between Growth and Equality: ELIE in an Overlapping Generations Model[END_REF] adopt the same type of calibrated simulation as well as many authors do in the literature. This approach is much simpler to implement because it is easier to get the figures for some indexes than the figures of individual incomes collected from household surveys. Moreover, calibrated simulation is more adapted when one want to study variants of the original model, because in this case, the income distribution can be simulated in different way across the variants of the initial model.

A fiscal system has been characterised in the econometric literature by a number of indices that we recall in the appendix. They are, apart from the well-known Gini index, the progressivity index of [START_REF] Kakwani | Measurement of Tax Progressivity: An International Comparison[END_REF], the vertical equity index of [START_REF] Reynolds | Public Expenditure, Taxes and the Distribution of Income[END_REF] and the horizontal inequity index of Atkinson (1980) and [START_REF] Plotnick | A Measure of Horizontal Inequity[END_REF]. We detail these indices in an appendix, because it is rather difficult to find a comprehensive presentation of the related literature which might not be well known to all the readers of this book.

The paper is organised as follows. In section 2, we give a mathematical presentation of the ELIE scheme. In section 3, we simulate this model and verify its redistributive properties. We show that a small value of the fundamental redistributive parameter is needed in order to obtain the same degree of redistribution that exists in France. With section 4, we enter the core of the debate, examining the consequences of uncertainty in the knowledge of the distribution of talents. In section 5, we introduce family allowances while section 6 details a growth model à la [START_REF] Solow | A Contribution to the Theory of Economic Growth[END_REF] to investigate the relation between capital accumulation and income distribution. Section 7 concludes.

The initial formulation of the model

The economy we consider is composed of n individuals with a labour supply ℓ i and a potential wage rate of w i corresponding to individual capacities or productivities. Taxation bears on w i , independently of ℓ i , the effective quantity of supplied labour. That, with the assumption that there is a total freedom for the amount of labour supplied, means that there is no interaction between labour supply and taxation. Consequently, the question of optimal taxation can be treated in a first best framework provided that wage rates or productivities are perfectly known. Kolm imagines a self-financing distributive system where taxes and subsidies, both noted t i , balance with t i = 0. As labour is the main resource, taxes and subsidies are measured in terms of a quantity of labour. An equal quantity of labour k is taken from each individual and measured at his productivity so the tax is equal to kw i . Meantime, an equal amount k w, to be determined, is redistributed to each individual so that the net transfer for individual i is equal to

t i = k( w -w i ). ( 1
)
The value of k has to be chosen by society while w is determined by the assumption that transfers are totaly financed by taxes. As a matter of fact, the system must verify

t i = 0 ⇒ k(n w - w i ) = 0. ( 2
)
This equation has only two solutions. Either k = 0 or w = w i /n = w. Thus w is equal to the mean wage w. It plays a central role in the system and we shall call it here the pivot.

Remark : 1 w determines the position of richer and poorer individuals. In the EU and in the UK as well, any individual with an income below a fraction of the mean income in society is considered as poor. In France, the same definition uses instead the median. When the income distribution is not symmetric, this makes a difference. So the choice of w as a pivot is not neutral.

When accounting for taxes and transfers, disposable income or net total income is given by

y i = w i ℓ i + k( w -w i ) = k w + w i (ℓ i -k).
(3)

In this framework, taxation is based on capacities and not on actual income. So that an individual determines his labour supply independently of k. But he is implicitly constrained to work at least for k days. Everybody receives the same basic income k w plus a fraction ℓ i -k of his labour productivity. This is the Equal Labour Income Equalisation or ELIE. ELIE is thus a kind of universal basic income.1 Everybody receives k w. At the difference of a usual basic income, the financing mechanism is already contained in the definition of ELIE as everybody also pays kw i for financing the system.

Characterising the initial model

We start from an hypothetical population of n = 10 000 individuals. For the time being, we restrict our attention to individuals, leaving the question of household composition to section 5. We suppose also that wages are for the while the sole income. These two hypothesis characterise the initial simple model of Kolm. We are here interested in exploring the consequences of wage heterogeneity on redistribution, or in other words to show that ELIE properties heavily depends on the shape of the gross income distribution. We shall thus simulate a calibrated distribution of gross income and analyse the resulting distribution of net income (after taxes and transfers) when k varies. In the original model of Kolm, labour supply is exogenous. So for the ease of simulation, we shall simply suppose that ℓ i = 1. Consequently, the net total disposable income (3) is transformed into

y i = w i + k( w -w i ) (4)
This assumption is a non trivial one. In the original model, w i is the hourly wage rate which has to be multiplied by ℓ i or order to get the gross income ℓ i w i . Assuming ℓ i exogenous and equal to 1 means that w i now represents gross labour income. In a way the first best framework is lost, in another way is it kept because ℓ i is supposed to be exogenous. The model also supposes that w i is perfectly known. We shall relax this last assumption in section 3.

We suppose that gross wages are distributed according to a Gamma distribution:

f (w|ν, s) = 1 Γ(ν)
s ν w ν-1 e -w s .

We recall that the mean of this distribution is equal to ν/s and its variance to ν/s2 . We then impose the restriction s = ν in order to get a normalised mean of 1. 2 . We have then to calibrate this distribution by adjusting its remaining parameter ν. For a normalised mean, the empirical distribution of gross wages can be characterised in a simple way by its Gini coefficient. This coefficient was equal to 0.327 in France in 1998. 3 To obtain the same Gini coefficient for our simulated sample, we had to choose ν = 2.75. Note however that these figures are in a way not exactly comparable because here we have individuals and earnings for the while when real data are concerned with households and total incomes. Note also that with the Gamma distribution, very high incomes are not so well represented. In our sample, the maximum income is 6.27 times the mean. We could have chosen another usual distribution such as the logNormal or the Pareto without changing much the results. We choose the Gamma for the sake of algebraic simplicity. All moments exist, contrary to the Pareto. 4Let us now apply the ELIE scheme to this sample, taking three different values for k: 0.2, 0.3, 0.4, as suggested in [START_REF] Kolm | The optimal production of social justice[END_REF] so as to obtain the distribution of net incomes (after taxes and transfers) as given by (4). A coefficient k = 0.2 corresponds to a tax rate of one working day in a five working day week. And k = 0.4 corresponds to two working days.

Figure 1 displays a non-parametric estimate of the different income distributions obtained when varying k. We immediately notice three facts: firstly, ELIE introduces a concentration of net incomes around the mean; secondly it raises the minimum disposable income as k grows; thirdly, if poverty is defined an income lower than half the mean, a k greater than 0.4 is needed to totaly eliminate it (See [START_REF] Leroux | Macrojustice : A pluridisciplinary appraisal of Kolm's theorychap. ELIE-minating poverty? Limits of the mechanism and potential improvements[END_REF] as an illustration).

The decrease of the Gini coefficients given in Table 1 shows that ELIE strongly reduces inequality. But the chosen values for k entail Gini coefficients which are much lower than that computed on French data for net incomes and which is equal to 0.27 in 2000 and 2001. We should have chosen k = 0.17 to recover such a value for the Gini coefficient, which would corresponds to slightly less of one working day. This is a relatively low value when compared to the importance of social contributions and payroll taxes in France. However, we are studying here only redistribution and redistribution, when properly measured, is rather weak in France. For instance, income taxes are only a small part of GDP in France. In this simple model, a rather low value for k is finally realistic. When examining various variants of the basic model, we shall see that the French Gini coefficient for net incomes is reproduced for increasing values of k as we depart from the original model.

It is rather easy to verify that ELIE complies with the Pigou-Dalton requirement, which means that it does not change the mean of the distribution after taxes and transfers and that it entails Lorenz curves that get closer to the main diagonal without crossing. A a consequence, all concentration curves, computed on the sole basis of positive taxes are identical. 5Let us now detail various indices which characterise taxation, leaving aside transfers. Taxes, noted tt i , concern only individuals who are above the mean: Those below the mean receive only transfers. The marginal tax rate is constant and equal to k for those paying taxes. However, despite this linear structure, ELIE is a progressive taxation scheme. The mean taxation rate, obtained by dividing taxes by gross wages, is

tt i = 0 if w i ≤ w k(w i -w) otherwise.
tx i = tt i /w i = k(1 - w w i ). (5)
This rate grows with w i , but less than proportionally as illustrated on Figure 3. The shape of this graph, which illustrates the bearing of taxation on our particular population, depends heavily on the shape of the distribution of income. On the contrary, the marginal rate of taxation is constant and equal to k for individuals above the mean, and of course equal to zero for those below the mean.

The usual index of progressivity of [START_REF] Kakwani | Measurement of Tax Progressivity: An International Comparison[END_REF] is strong with 0.45 compared to its value computed on actual French data which is only 0.31. It is not a function of k because of the linear structure of ELIE (constant marginal rate). It is only a function of the shape of the distribution and of the chosen pivot for ELIE.6 A smaller index could be obtained with a lower ν entailing stronger asymmetry and stronger initial inequality. The average fiscal pressure, computed as the ratio between the sum of total taxes and the sum of gross incomes, is very low, because only incomes which are above the mean are taxed. But even when the fiscal pressure is computed for only those actual paying taxes, it is not very high. The amount of inequality removed as indicated by the Reynolds index as well as the redistribution index grows with k.

Horizontal equity, as defined in the appendix is perfect, taxes and transfers imply no rank permutation. The Atkinson-Plotnick horizontal inequity index is always zero. This is due to the fact that there is no effect due to family composition or to the combination of different taxes.

We can conclude that the pure ELIE transfer scheme is a rather powerful mechanism of redistribution. Its implementation in France would require a rather low value for k to reproduce the same Gini coefficient for net income. However, ELIE is much more progressive. That would mean putting the burden of redistribution more on richer people than what it is now.

Uncertainty in the knowledge of wages

Wages and income are not in general perfectly known by the tax authority. The basic wage rate can be public, but there are bonuses and extra hours for instance that can be more difficult to know for the tax authority. These can be important for some categories. We have then tax evasion and tax avoidance. The former is illegal but some professions are famous for practicing it. The later is perfectly legal and results from an optimisation behaviour as described for instance in [START_REF] Stiglitz | The General Theory of Tax Avoidance[END_REF]. Of course in this game of tax evasion and tax avoidance, richer individuals have more opportunities than poorer ones.

We are now going to modify the initial model so as to take into account this aspect which is important for practical implementation. We have to know how much ELIE looses of its nice redistributive properties when it becomes difficult to have a precise knowledge of the individual talents. We now modify the taxation scheme so that w i would be lowered by a random effect e i :

t i = k( w -w i -e i ).
(6)

We want to implement the idea that uncertainty in the knowledge of wages increases with its level and that the importance of dissimulation is random. For that purpose, e i is a strictly positive process and once again we chose the Gamma distribution for convenience. Its mean will depend on the level of w i and on its rank in the distribution. The maximum rate of tax avoidance (expressed as a fraction of gross income) is supposed to be 30%. More precisely, if the w i are sorted in increasing order, we shall have

e i ∼ 0.3 i n w i G(ν, ν), (7)
where G(ν, ν) means the Gamma distribution. This formula implies that e i is positive with a mean equal to 0.3 * w i * i/n and a variance equal to (0.3 * w i * i/n) 2 /ν. The richer people will on average manage to hide 30% of their income while the poorer people will manage to hide a negligible part of it. For the system to be balanced, the pivot has to be taken equal to the mean of the declared wages w i -e i and no longer equal the mean of the exact wages w i . Introducing a random dissimulation entails little changes when compared to the initial 1. Redistribution is of course weaker as well as the fiscal pressure on richer people. A weak horizontal inequity appears due to the random character of the modeled tax avoidance.

In order to get a Gini of 0.27, we must chose a k = 0.24, a value slightly higher than in the original case where a value of k = 0.17 was sufficient. Despite the fact that we have introduced a rather strong possibility of tax avoidance and evasion, the final influence is weak. We just have to increase k in order to get the same amount of inequality. Redistribution is simply slightly less important. Taxing capacities instead of incomes seems a crucial hypothesis because incomes are rather easy to measure whereas capacities are not. The results of this section show that finally this hypothesis is not so crucial because when relaxing it, the main properties of the model are not so much altered. We must however keep in mind that that we have supposed that labour supply is exogenous and thus does not depend on the value of k.

Family allowances and family composition

ELIE is a fair and powerful redistributive scheme with nice labour incentives properties. However, incentives are not limited to the labour market, as underlined in the introduction. In this section, we want to investigate how the pure redistributive ELIE mechanism can interact with the incentive mechanism of family allowances.

Family allowances exists in many European countries and are very important in France. They have been successfully designed to increase natality. With family allowances, we have a mechanism which mixes redistribution and incentives. Clearly, ELIE cannot be a substitute for family allowances, simply because it is not a targeted mechanism. If we want to experiment the feasibility of implementing ELIE in France, we have to study the properties of a mix system combining ELIE with a complementary mechanism of redistribution based on family composition and not solely on the position of gross income versus mean income.7

ELIE and household composition

The basic unit i is no longer an individual but a household. We suppose that a household is composed of two adults earning the same amount each and a random number of children noted ch i that can be zero. 8 The relationship between income and fertility is rather controversial. [START_REF] Becker | A Treatise on the Family[END_REF] argues for a positive relationship on the ground that children are consumer goods competing with alternative goods. But he also put forward a trade-off between quality of education and quantity, so that the relation can be negative (see also [START_REF] De La Croix | Inequality and Growth: Why Differential Fertility Matters[END_REF] for an alternative point of view). How could we decide between these alternative views? We had access to survey data collected from interviews in Marseilles in 2006 at the occasion of an exercise in experimental economics concerning willingness to pay for a better air quality. The sample is made of 549 individuals, coming from distinct households for which we have the number of children, the total net income of the household and the marital status of the responder. We can thus run a regression which links the number of children to family income and marital status. Cel is a dummy variable indicating if the responder is single or not. Expressed in logarithms, this regression has a low R 2 = 0.205, but clearly indicates a slightly positive relation between fertility and family income as we have:

ln(1 + ch i ) = 0.034 [0.16] + 0.061 [2.43] ln(1 + w i ) -0.40 [-9.01] cel.
Following this regression9 , we shall suppose that the number of children follows a Poisson distribution with parameter θ (which represents both the mean and the variance) and we note ch i ∼ P (θ). The average number of children in France is two per family. We suppose that part of the fertility decision is independent of income, say for the first child, but that the decision for an extra child depends on income. We have thus chosen θ = 1+w i / w. The realised number of children in a family for the 10 000 households of the sample lies in the interval [0,9]. In Figure 4, we compare the distribution of the number of children in a household when fertility depends on gross income to a distribution that would suppose that fertility is totaly independent of income, while keeping the same mean. Clearly, the assumption that fertility depends on income has a strong influence on the decision to have a second child, but lowers the probability to have more than 3 children for the given gross income distribution.

As we now define the income distribution of a household and no longer the income distribution of an individual, comparisons between households become more difficult. Clearly a couple with many children has not the same needs as a couple without any children. We have to introduce an equivalence scale in order make household incomes comparable. We have chosen a rather mild equivalence scale which gives a weight of 1 to the first adult, 0.7 to the second adult and 0.3 to each child. This is a mix between the Oxford equivalence scale and the OECD equivalence scale. Let us call sn i the number thus obtained for household i. If we divide w i by this number, we have an income expressed in terms of family needs. The basic ELIE formula is changed into:

y i sn i = 2 w i sn i + 2 k ws - w i sn i (8)
where ws = n -1 i w i /sn i .

The influence of equivalence scales

Might ELIE gain some incentive properties for fertility decisions just by using an equivalence scale without introducing a specific family allowance mechanism? The answer might depend on the assumption made for fertility which determines in which type of family children are located. It is thus wise to consider two variants: one where fertility is independent of income and one where it is income dependent. Let us first suppose that the number of children per household is drawn from a Poisson distribution with θ = 2. We have simulated an income distribution for 10 000 households composed of two adults earning each the same w i and having a random number of children, independent of their income. Results are displayed in In Table 3. In column U , we have divided the total gross household income by 2 (the number of adults in the household) while in column S, total income is divided by an equivalence scale depending on the number of children. We then apply the usual ELIE scheme on the resulting scaled income, using (8). Considering an equivalence scale slightly increases inequality, fiscal pressure and decreases redistribution. The order of magnitude of these changes is small, but significant. It would be stronger with a different equivalence scale. Column U is identical to the initial case of section 3. With an equivalence scale, the French Gini coefficient is recovered for k = 0.19 instead of k = 0.17 when there is no equivalence scale.

Let us now consider the income distribution conditional on the number of children. What is the more favourable system for large families? In Table 4, we give the mean income of the 10 000 households having zero, one, two, three and more children. We give the total income after ELIE taxes and transfers, the exact income that the household will have in its pocket, once transfers are operated. This figure is obtained by re-multiplying the net income obtained in (8) by sn i . In column k = 0 of Table 4, there is of course no difference between U and S. For k > 0, the U columns are simply an estimate of the conditional mean of the income distribution. The computed marginal mean income is equal to 2.025. Let us concentrate on the columns marked S. When k is positive, ELIE seems to redistribute money as a function of the number of children. Households with less than two children receive less that the mean income. Household with more than two children receive more and transfers increase with the number of children.

As a conclusion, an equivalence scale seems to introduce a trade-off between fertility incentives and fairness.

Equivalence scales and income dependent fertility

If we believe in the incentive effect of family allowances, fertility should be assumed to depend on income. This assumption is going to change dramatically the previous results. We have redone the same simulation exercise as in subsection 5.2, but this time with ch i ∼ P (θ) where θ = 1 + w i / w instead of θ = 2. Table 5 shows that the use of an equivalence scale now decreases inequality as shown by the Gini indices computed on net income after ELIE transfers, instead of increasing it as in the case of random fertility. The fiscal pressure is lowered while progressivity is increased. Thus the dependence between fertility and income has a drastic influence on the redistribution properties of ELIE when an equivalence scale is used. This influence is now positive in term of fairness. A value of k = 0.13 is now sufficient to get the French Gini coefficient on net income instead of k = 0.17 in the original model without equivalence scales. Let us now examine the money which actually enter the household pocket. Table 6 shows first that before any redistribution (k = 0), the fertility model implies that households with children have more money than households without children, while the computed average income remains the same at 2.025. This is just a consequence of our assumption made on fertility. When k > 0, the mean income per household type is not significantly changed. Families with more than two children get more money with the equivalence scales and family with less than two children get less money as in the previous configuration. So the incentive mechanism introduced by the equivalence scales is unchanged, while the fairness properties of the model were increased.

ELIE and family allowances

ELIE is a mechanism designed for macro justice and not for incentives, because it is not a targeted mechanism. Its sole aim is to reduce the distance between average income and actual income, independently of any other factors. We have here introduced a new dimension, which is family composition and its corollary equivalence scales. Depending on the modeling of fertility, the use of an equivalence scale can confer to ELIE some incentive properties, but at the expense of a lower degree of fairness measured here as progressivity and inequality. When fertility increases with income, the incentive effect disappear and fairness is restored.

Family allowances is a sensitive topic in France and also in many other European countries. A government who would like to implement the redistributive ELIE mechanism cannot politically suppress family allowances, even if we can truly suspect this mechanism to distort the nice properties of ELIE. Let us try to conceive a complementary redistribution scheme based solely on family composition and independent of income. We need extra resources in order to finance explicit family allowances, which means distributing an amount of money only on the basis of family composition, independently of income. The redistributive ELIE system balances because the pivot is equal to the mean ws . If we choose a pivot lower than the mean, say wq < ws , we get the needed extra amount to redistribute:

T r = 2 n i=1 k( wq - w i sn i ) > 0. ( 9
)
So that net income is now defined as

y i sn i = 2 w i sn i + 2 k wq - w i sn i + T r ch i /n ch , ( 10
)
where n ch is the total number of children in the complete sample. How can we calibrate wq ? In the ELIE scheme, the Kakwani progressivity index is independent of k, but varies with the size of the pivot. In the previous sections, the Kakwani index was much higher than its French value of 0.31. We can decide to chose wq so as to match the value of 0.31 for the Kakwani index. Compared to Table 5, inequality has slightly increased, but mainly progressivity was calibrated to be much lower. This is translated in a huge increase in the fiscal pressure because now much more households are taxed. The Reynolds index is smaller indicating that taxes are less successful in reducing inequality.

Table 8 show that large families get much more money. But we have seen that this extra amount of money is not obtained in a fair way. If ELIE had to be applied, explicit family allowances should be avoided. Equivalence scales are enough.

6 Introducing capital stock [START_REF] Kolm | The optimal production of social justice[END_REF] claims that since capital is by definition produced, the other primary resources finally account for about 97.5% for labour and 2.5% for the non-human natural resources. So that the problem of overall distribution in macrojustice is the allocation of the rights in the value of productive capacities, i.e. labour capacities. 10 The idea is attractive from a normative point of view and in a steady state, but it is not adapted if the objective is to confront the ELIE scheme to an empirical reality. Kolm's point of view implicitly assumes that capital and credit markets are efficient so that there is no problem of capital reallocation in the long term.

Stylised facts

We observe a fundamental inequality in the sharing of the fruits of capital and of labour which sets forward the question of the efficiency of any redistributive policy. At the macro level, the share of capital in total value added can remain relatively constant despite of large wage rises as in the US, or experience a marked increase as in Europe and especially in France as shown in Figure 5 and as noted in [START_REF] Piketty | Inégalités et redistribution[END_REF]. The modification of the capital share in France implied a reallocation of more than 10% of total GNP on a rather short period of 10 years. This is more than what any fiscal redistributive policy could have operated. At the micro level, we can notice that capital allocation remains extraordinary stable in the long term between individuals. This is due to several reasons. There is first the question of initial conditions in the dynamic process of wealth transmission: death duty does not manage to reallocate efficiently ownership because rates of taxation are too low for that purpose. Secondly, the rate of saving is different between poor and rich individuals. A high rate of saving is necessary for accumulation. Moreover, as documented in [START_REF] Direr | Modéliser la distribution des richesses en France[END_REF], mostly rich individuals have access to really profitable investments. Thirdly, credit market is rationed, banks avoid financing risky projects initiated by lower income holders as exemplified for instance in [START_REF] Piketty | Inégalités et redistribution[END_REF]. Finally, wages and capital gains are not taxed in the same way. In France, for the same level of income, wage earners are more taxed than holders of capital gains. All these features put together contribute to inequality in the income and wealth distributions. As a matter of fact, in every country, there is a stronger inequality in wealth distribution, than in income distribution. For instance, the Gini index for net income distribution was reported to be 0.27 in France. It rises up to 0.65 for wealth inequality in 2004 (see for instance [START_REF] Cordier | Les inégalités de patrimoine des ménages entre 1992 et 2004[END_REF]).

Regulating the distribution of income between capital and labour can become thus one of the major goals of income distribution policies. This justifies introducing capital accumulation in the ELIE framework and see what is the performance of ELIE for regulating the inequality in income distribution generated by capital accumulation.

General features

The model we shall use for capital accumulation11 is inspired from [START_REF] Solow | A Contribution to the Theory of Economic Growth[END_REF] where the rate of saving determines the rate of accumulation. In the original growth model of Solow, population grows at a fixed rate while there is no depreciation for capital. Here we want to keep the labour force constant with the same wage distribution for ease of comparison with our previous results. We shall thus suppose that the rate of growth of the labour supply is zero while capital depreciates at a fixed rate. The fundamental properties of the model should not be affected. We consider that there is one active generation with an economic activity lasting for 30 periods. In this framework, we can leave aside the question of inheritance taxes (death duty). We just have to precise initial conditions, saving behaviour and transfers.

The first key assumption concerns the rate of saving. As we have an heterogenous population (contrary to the model of Solow), we shall suppose that the rate of saving varies across the population. Our key assumption is that rich people save more than less rich people. More precisely, the rate of saving will be a positive function of income. This assumption summarises the stylised facts detailed above and is central to obtaining a distribution of wealth more concentrated than the distribution of income. The second key assumption follows [START_REF] Direr | Modéliser la distribution des richesses en France[END_REF] where only rich people have access to really profitable investments. Capital profitability will also be dependent on income.

With this model, we want to compare two types of taxation-redistribution in the presence of capital accumulation. In a first scenario, we introduce ELIE only for labour income. There is a flat tax on capital income, very much like what happens in France. The fruit of these taxes is redistributed uniformly among the individuals as would be a basic income. This scenario aims at reproducing more or less the French situation or a minimal adaptation of ELIE to France. The second scenario is a kind of full application of ELIE when there is capital accumulation. Capital income is added to labour income and serves a basic to the ELIE formula. Capital and income are equally treated.

A dynamic model of capital accumulation

Initial conditions are crucial for any non-stationary processes, because they have a direct influence on their trajectories. We have calibrated the wage distribution on the French Gini coefficient and normalised it with a mean equal to one. We have to do the same for the initial wealth distribution. We suppose that the initial capital stock is a function of labour income and that it is not equally distributed in such a way so as to reproduce the French Gini coefficient computed for wealth. Our proposed formula is

s 0,i = α 0 w i µ r 1I(w i > w) + (1 -α 0 ) w i µ r 1I(w i < w), (11)
where µ r is the mean rate of return on capital and 1I is the indicator function. With α 0 = 0.75, we get a Gini coefficient of 0.651 for the initial capital stock. These initial conditions are interpreted as a bequeath of the previous generation. With an initial mean rate of return of 3%, the income generated by the initial capital stock will represent roughly 39% of total income at t = 0. The rate of saving determines the rate of accumulation. Capital stock available at time t, s i,t comes from capital at time t -1, s i,t-1 , diminished of a fixed depreciation rate δ, but augmented of savings. It is commonly admitted that the saving rate increases with income, but less than proportionally (see e.g. [START_REF] Dynan | Do the Rich Save More?[END_REF]). We propose modeling the saving rate as

γ i = µ γ √ w i , (12)
where µ γ is a scaling parameter. The mean saving rate is equal to 15% in 2005 in France. This rate is reproduced for µ γ = 0.156. For individuals with a labour income lower than w, this rate is on average equal to 12%; for individuals with a labour income greater than w, this rate is equal to 19% on average, given our wage distribution. Following [START_REF] Direr | Modéliser la distribution des richesses en France[END_REF], profitability of capital must be a function of income to illustrate the fact that only rich individuals have access to really profitable investments. We shall suppose the rate of return on capital, r i,t , is a random variable with a mean which is a linear function of the square root of w i and a fixed variance:

r i,t ∼ G(µ r √ w i , σ 2 r ) (13)
where G(., .) is a Gaussian random variable. At each period, we will have a new draw for the returns. We have previously chosen µ r = 0.03 and now we chose σ r = 0.01. This formulation does not exclude temporary negative returns. Capital income at time t, sr i,t , is given by:

sr i,t = s i,t r i,t , (14)
while capital stock is defined by

s i,t = (1 -δ)s i,t-1 + γ i y i,t . (15)
The growth rate of the capital stock depends on the balance between the proportion of depreciation δ and the share of net income devoted to savings. As savings depends on net income, the growth rate of capital is a function of k, the redistribution parameter. There is a clear trade-off between growth and redistribution. Net income is given by the combination of labour income, capital income and the implementation of a given redistribution scheme. In our first scenario, ELIE is applied only to labour income, while capital income is taxed at a fixed rate τ of 20% (a realistic value in the French context). The product of capital taxation is used to finance a basic income. We have:

y i,t = w i -k(w i -w) + (1 -τ )sr i,t + al t /n, (16)
where al t is the fruit of taxation on capital income and is given by

al t = τ i sr i,t . (17)
We can now calibrate δ. As there is a trade-off between capital growth and redistribution, we chose δ so as to maintain over the 30 periods a constant ratio between capital income and total income for a given value of k. For k = 0.3, the corresponding value of δ is found to be 0.0175. In our second scenario, capital and labour incomes are treated in the same way (which they are not in reality). We add capital and labour income and then apply the ELIE formula to total income;

y i,t = w i + sr i,t -k( wcap -w i -sr i,t ), (18)
with wcap = n -1 (w i + sr i,t ). Simulating these two variants for our 10 000 individuals will give us clues on the properties of ELIE in the presence of capital accumulation.

A trade-off between growth and inequality

We first analyse the trade-off between growth and inequality in our simple calibrated growth model. Table 10 clearly illustrates the trade-off resulting from ELIE. Increasing k deceases the net total income but also inequality both for net income, capital income and capital stock.

There is still more inequality on capital income than on labour income. The two systems of taxation-redistribution, that is to say, on one side ELIE applied only to labour income but coupled with a flat tax on capital income, and a basic income and on the other side the full ELIE applied to the sum of labour and capital income, are strictly equivalent for k = 0.2. But for greater values of k, the full ELIE is much more powerful to reduce inequality. For k = 0.2, the share of capital income increases slightly in both cases, compared to the initial situation.

For k = 0.3, it diminished with the full ELIE while it remains slightly constant in the first scenario. The redistributive power of ELIE is made even more apparent if we examine Figure 6 which displays for both scenarios the income distribution. With the full ELIE, the right 2001) on the disappearance of large fortunes at the beginning of the XX th century with the introduction of a progressive taxation on income. However, the redistributive scheme of ELIE should be combined with a careful study for tuning death duties. Initial conditions are of a considerable importance, even after 30 periods, to determine the shape of the capital distribution.

Analysing taxation

The previous picture can be completed by analysing some taxation indices. For k = 0.20, we have seen that the macroeconomic situation is the same for both scenarios. However, we see now that the full ELIE system implies a lower fiscal pressure and a stronger progressivity of taxation in order to achieve at the same redistribution. Fiscal pressure starts to become greater with the full ELIE only for k = 0.40. Finally, we note that with capital accumulation, our calibrated model needs a k = 0.40 together with the full ELIE in order to reach the French value of 0.270 for the Gini coefficient on net income. We can thus conclude that introducing capital accumulation leads to a very strong modification of the income distribution and of the ELIE principle of equity based on the sole labour factor. Clearly, ELIE has to be completed by a taxation redistribution of bequeath.

Conclusion

The ELIE scheme is a principle of taxation and transfers that has many attractive properties in terms of macrojustice and equity. In this paper, we have tried to address the question of its implementation. We examined that nice principle of taxation-redistribution and tried to see how its properties would be transformed when it has to be applied in a specific context with its own characteristics where everything cannot be erased. Taxation systems cannot be changed completely in one year. Reformation is always discussed at the parliament which means that some categories will pay or receive more or less than what the original idea would imply.

In France family allowances are historically very important and certainly been successful in maintaining a decent rate of natality, even if we have seen that their impact in term of equity is dubious. Since capital is much more mobile than labour, the impact of capital taxation is always discussed in terms of incentives and international fiscal competition more than in terms of equity.

We have tried to propose options which make the basic ELIE model more realistic for the French case. We have found that uncertainty on wages does not modify fundamentally the properties of the model and this is a nice result. However it might lower greatly its efficiency. Family allowances represents a greater challenge. The incentive properties of family allowances can be recovered simply by introducing equivalence scales. Their impact is very sensitive to the relation between income and fertility. If fertility is increasing with income, an equivalence scale can increase the redistributive efficiency of the system in the sense that the same degree of inequality can be obtained with a lower k. But we can have exactly the reverse property with a different model for fertility. The incentive effect can be greatly increased if we introduce a specific family allowance mechanism, but at cost of a huge increase in inequality. Finally, capital accumulation has the strongest distortion impact and changes many of the ELIE properties. In the presence of capital accumulation, ELIE should be completed by a mechanism of taxation-redistribution on bequeath which is outside the scope of this paper.

Our model was calibrated in such a way so as to reproduce most of the inequality and taxation indexes obtained on French data. Various values for k were considered. They are directly linked to the desired degree of aversion for inequality in society. In the pure ELIE model, k had to be rather low to fit French data. But in this basic model, the income distribution does not reproduce correctly the importance of high incomes. A model with capital accumulation is necessary to obtain an distribution representing correctly high incomes. In this case k has to be rather high in order to reproduce the value of the various taxation indexes. The determination of k thus heavily depends on the model considered. The more we depart from the original model, the greater k has to be. Further investigation is needed so as to estimate the desired degree of aversion for inequality in society. Many papers have been written on this topic. Building on the seminal work of [START_REF] Kolm | The optimal production of social justice[END_REF] and [START_REF] Atkinson | On the measurement of inequality[END_REF], we can quote [START_REF] Gevers | Professed Inequality Aversion and Its Error Component[END_REF] followed by Van Praag, Goedhart, and Kapteyn (1980) or [START_REF] Amiel | Measuring attitudes toward inequality[END_REF], all using experiments and questionnaires. This is planned for future work.

It will be used extensively in the next definitions. The concentration curve C T (p) indicates the proportion of total taxes that the population holding the proportion p of the total income pays: p = F (q) ⇒ C T (p) = 1 g x q 0 t(x)f (x) dx.

Let us now order the observations t i according to the ascending order of x i . The concentration curve can be estimated by

C T (p = i/n) = 1 n g x i j=0 t j = 1 n t i j=0 t j .
A concentration index CI is defined in the same way as the Gini index as a measure of the surface between the concentration curve and the diagonal

CI = 1 -2 1 0 C T (p) dp.
Let us go from gross income x i to net income y i by subtracting taxes t i . The concentration curve of net incomes C Y (p) gives the proportion of total net income that is held by the the population holding the proportion p of the total gross income: There exists a deterministic relationship between the three curves:

p = F (q) ⇒ C Y (p) = 1 (1 -g)x q 0 [x -t(x)]f (x) dx,
L X (p) = gC T (p) + (1 -g)C Y (p).
The various indexes that characterise a taxation and redistribution scheme are defined in the same way as the Gini index. They correspond to surfaces between two Lorenz and/or concentration curves.

The progressivity index of [START_REF] Kakwani | Measurement of Tax Progressivity: An International Comparison[END_REF] measures the surface between the Lorenz curve associated to gross incomes and the concentration curve of taxes. It characterises progressivity as a departure from proportionality:

K = 2 0 1 [L X (p) -C T (p)]dp = CI T -G X .
This index is positive if a tax is progressive, equal to zero if the tax is proportional and negative if the tax is regressive. The negative bound is -(1 + G X ) and the positive bound (1 -G X ) where G X is the Gini index for gross income x i .

The index of vertical equity of [START_REF] Reynolds | Public Expenditure, Taxes and the Distribution of Income[END_REF] measures the reduction in inequality brought in by taxes as the surface between the concentration curve of net income and the Lorenz curve of gross income

RS = 2 0 1 [C Y (p) -L X (p)]dp = G X -CI Y .
It equivalently measures the reduction operated by taxation on the Gini index of gross income. Because of the deterministic relationship existing between Lorenz and concentrations curves, these two indices are related by RS = g 1 -g K.

The index of horizontal inequity of Atkinson (1980) and [START_REF] Plotnick | A Measure of Horizontal Inequity[END_REF] is concerned by the unequal treatment of equals or re-ranking operated by taxation. It measures the surface between the concentration curve and the Lorenz curve of net income:

AP = 2 0 1 [C Y (p) -L Y (p)]dp = G Y -CI Y .
If taxes do not induce a rank permutation, then the concentration and Lorenz curves of net income are identical and this index is zero. Or the Gini and concentration index of net income are identical.

The redistributive index measures the surface between the Lorenz curve of net and gross incomes:

IR = 2 0 1 [L Y (p) -L X (p)]dp = G X -G Y .
This index is at value between 0 and 1. It increases with the degree of redistribution. It is simple to discover that this index is equal to the difference between the index of vertical equity and the index of horizontal inequity: IR = RS -AP.

For more details on the topic, see [START_REF] Lambert | Distribution and Redistribution of Income[END_REF], [START_REF] Duclos | Inégalité et redistribution revenu avec une application au Canada[END_REF] or [START_REF] Essama-Nssah | Inégalité, Pauvreté et Bien-Être Social[END_REF].

Figure 1 :

 1 Figure 1: Income distribution before and after ELIE transfers Multiply by 28 935 euros (the annual mean disposable net income in France in 2004) to have an idea of the range of net incomes in France.

Figure 2 :

 2 Figure 2: Lorenz and concentration curves associated with ELIE

Figure 3 :

 3 Figure 3: Taxation rates implied by ELIE (in percentage)

Figure 4 :

 4 Figure 4: Distribution of the number of children for the two scenarios

Figure 5 :

 5 Figure 5: Capital share in some OECD countries

Figure 6 :Figure 7 :

 67 Figure 6: Income distribution with capital accumulation

Table 1 :

 1 The original ELIE model

	k Mean Gini Fiscal	Partial Kakwani Reynolds Atkinson Redis-
	wages	pressure fisc. pres.				tribution
	0 1.013 0.327 0.00	0.00	0.000	0.000	0.00	0.000
	0.2 1.013 0.262 4.70	7.23	0.454	0.065	0.00	0.065
	0.3 1.013 0.229 7.05	10.84	0.454	0.098	0.00	0.098
	0.4 1.013 0.196 9.39	14.45	0.454	0.131	0.00	0.131

Table 2 :

 2 ELIE and uncertainty in the knowledge of wages

	k Mean Gini Fiscal Kakwani Reynolds Atkinson Redistribution
	wages	pressure
	0.0 1.013 0.327 0.00	0.000	0.000	0.0000	0.000
	0.2 1.013 0.282 3.16	0.437	0.044	0.0000	0.044
	0.3 1.013 0.261 4.73	0.437	0.066	0.0002	0.066
	0.4 1.013 0.239 6.31	0.437	0.089	0.0003	0.088
	Table

Table 3 :

 3 Equivalence scales and ELIE with a random fertility

		k = 0.0	k = 0.2	k = 0.3	k = 0.4
		U	S	U	S	U	S	U	S
	Gini	0.327 0.342 0.262 0.273 0.229 0.239 0.196 0.205
	Fisc. pres. 0.000 0.000 4.697 4.911 7.045 7.366 9.393 9.821
	Kakwani 0.000 0.000 0.454 0.448 0.454 0.448 0.454 0.448
	Reynolds 0.000 0.000 0.065 0.068 0.098 0.103 0.131 0.137
	In columns U, household income is divided by 2, while in columns
	S it is divided by an equivalence scale.

Table 4 :

 4 Total net income per family with a uniform fertility

	k = 0.0	k = 0.2	k = 0.3	k = 0.4
	Children U	S	U	S	U	S	U	S
	0 2.062 2.062 2.054 1.959 2.051 1.907 2.047 1.856
	1 2.025 2.025 2.025 1.984 2.025 1.963 2.025 1.943
	2 2.025 2.025 2.025 2.039 2.025 2.046 2.025 2.053
	3 2.018 2.018 2.020 2.088 2.020 2.123 2.021 2.158
	> 3 2.003 2.003 2.008 2.160 2.010 2.239 2.012 2.317

Table 5 :

 5 Equivalence scales and ELIE with income dependent fertility

		k = 0.0	k = 0.2	k = 0.3	k = 0.4
		U	S	U	S	U	S	U	S
	Gini	0.327 0.307 0.262 0.245 0.229 0.215 0.196 0.184
	Fisc. pres. 0.000 0.000 4.697 4.393 7.045 6.589 9.393 8.753
	Kakwani 0.000 0.000 0.454 0.465 0.454 0.465 0.454 0.465
	Reynolds 0.000 0.000 0.065 0.061 0.098 0.092 0.131 0.123

Table 6 :

 6 Equivalence scales and ELIE with income dependent fertility (continued)

	k = 0.0	k = 0.2	k = 0.3	k = 0.4
	Children U	S	U	S	U	S	U	S
	0 1.502 1.502 1.607 1.500 1.659 1.499 1.711 1.498
	1 1.712 1.712 1.774 1.720 1.806 1.724 1.837 1.729
	2 1.997 1.997 2.002 2.001 2.005 2.003 2.008 2.005
	3 2.271 2.271 2.222 2.273 2.197 2.273 2.173 2.275
	> 3 2.892 2.892 2.719 2.860 2.632 2.843 2.546 2.827

Table 7 :

 7 Family allowances and ELIE with income dependent fertility

		k = 0.0 k = 0.2 k = 0.3 k = 0.4
	Gini	0.306 0.249 0.222 0.197
	FiscPres 0.000 8.154 12.231 16.308
	Kakwani	-	0.310 0.310 0.310
	Reynolds 0.000 0.059 0.089 0.119

Table 8 :

 8 Equivalence scales and ELIE with income dependent fertility (continued)

	k = 0.0 k = 0.2 k = 0.3 k = 0.4
	Children
	0 1.488 1.387 1.337 1.286
	1 1.731 1.676 1.649 1.621
	2 1.955 1.967 1.974 1.980
	3 2.344 2.409 2.441 2.474
	> 3 2.857 3.090 3.206 3.322

Table 9 :

 9 The trade off between growth and inequality k Mean net Mean capital Mean capital Gini net Gini capital Gini capital

		total inc.	share	stock	income	income	stock
				Initial situation
	0.000 1.659	0.389	16.540	0.417	0.706	0.651
				ELIE on labour income only
	0.00	1.670	0.393	17.004	0.410	0.680	0.601
	0.20	1.660	0.390	16.842	0.367	0.673	0.590
	0.30	1.655	0.388	16.761	0.346	0.670	0.584
	0.40	1.650	0.386	16.680	0.324	0.666	0.578
				ELIE on total income
	0.000 1.684	0.399	17.240	0.470	0.692	0.619
	0.200 1.660	0.390	16.842	0.367	0.673	0.590
	0.300 1.648	0.385	16.650	0.318	0.664	0.575
	0.400 1.636	0.381	16.462	0.269	0.654	0.561

Table 10 :

 10 The trade off between growth and inequality everybody can save and accumulate in our model, even if all are not equal in their opportunities and saving rates. But the capital distribution is still bimodal. The full ELIE has a larger impact on holders of capital stock greater than 60 times the mean wage. The full ELIE is able to greatly diminish their importance. This corroborates the remarks made in Piketty (

	k Mean net	Gini	Fiscal Kakwani Reynolds
		total inc. total inc. pressure
			ELIE on labour income only
	0.00	1.670	0.410	7.868	0.214	0.053
	0.20	1.660	0.367	10.663	0.238	0.092
	0.30	1.655	0.346	12.072	0.247	0.111
	0.40	1.650	0.324	13.490	0.254	0.131
			ELIE on total income
	0.000 1.684	0.470	0.000	-	0.000
	0.200 1.660	0.367	6.945	0.350	0.092
	0.300 1.648	0.318	10.298	0.352	0.136
	0.400 1.636	0.269	13.574	0.355	0.179
	periods, because

The interpretation of ELIE as a basic income mechanism is uncontroversial when everybody is working. It can lead to controversies when there are inactive individuals.

The annual mean disposable net income in France was 28 935 euros in 2004 and the median 24 599 euros. As we have chosen a normalized mean, the horizontal axis can be read as x times the mean. It is then easy to interpret the right tail of the distribution. Actual income figures can be obtained just by multiplying the figures of the horizontal axis by the empirical mean given in this footnote.

Some of these data can be found for instance on the Web site of the World Institute for Development and Economic research, UNU-WIDER World Income Inequality Database, Version 2.0a, June 2005. http://www.wider.unu.edu/wiid/wiid.htm .

 4 A real alternative would have been to choose a richer distribution with much more parameters in order to get a better tail behaviour. We can quote the Generalised Beta II distribution which has four parameters. However, the calibration of the parameters is far from trivial. We should have used at least the deciles of the income distribution for this purpose and not just a simple Gini coefficient. The gain for the subsequent analysis is not clear.

For instance, as wi ∼ G(ν, ν), then by properties of the Gamma distribution, yi = (1 -k)wi + k w is also Gamma with yi ∼ G(ν, ν/(1 -k)) + k. Its mean is equal to 1 and its variance to (1 -k) 2 /ν. So ELIE does not change the mean but reduces the variance.

The pivot is chosen here equal to the mean because the system is balanced in this case. Choosing for instance the median would have led to an unbalanced system. The pivot can be chosen lower than the mean if extra resources are to be collected for paying for instance civil servants and collective equipments. But in this case the original ELIE is distorted.

In France, family allowances are independent of income, except for the complement familial which is proportional to wages.

We exclude for simplicity mono-parental households. This is certainly a limitation of our analysis. We are concerned with the whole French income distribution, of which households with a single adult are a significant fraction. We also do not introduce the possibility of income disparities between the two parents and their consequences on taxation.

A non-parametric regression, not reported here, indicates also a positive and quasi linear relation.

See also section 3.3 of[START_REF] Kolm | The optimal production of social justice[END_REF], this book.

A complete model would have to consider overlapping generations such as in[START_REF] Direr | Modéliser la distribution des richesses en France[END_REF]. We would like, at least in a first attempt, to leave this question aside.

APPENDIX : About the instruments

Most of the indices used in the literature devoted to taxation and transfer analysis are based on the geometry of the Gini index. Let us start from a series of gross income X with density f (x) and distribution F (q). The Lorenz curve is defined as

Let us now suppose that the observations have been ordered in ascending order. The Lorenz curve can be estimated using

The Lorenz curve indicates the proportion L X (p) of total income that is hold by the proportion p of the total population. Perfect equity corresponds to the main diagonal L X (p) = p. The Gini index G is defined as the surface between that diagonal and the Lorenz curve:

which can be approximated by, using ascending order

Let us now define a tax function t(x) and the amount of taxes t i = t(x i ). The mean tax rate is obtained as the ratio of total taxes by total gross income:

x .