Claudio Lucifora
email: claudio.lucifora@unicatt.it

Dominique Meurs
email: dominique.meurs@ined.fr

Elena Villar
email: elena.villar@unicatt.it

James Albrecht

Alex Bryson

Tommaso Colussi

Nabanita Datta Gupta

Francesco Fasani

Nicole Fortin

Andrea Ichino

Chinhui Juhn

Camille Landais

David Margolis

Pedro Martins

Martina Viarengo

Ulf Zölitz

Having a Child? Here is the Bill! Parenthood, Earnings and Careers in an Internal Labor Market

Keywords: Children, Motherhood penalty, Gender inequalities, Event study JEL Classification: J13, J16, J31

Using a unique 12-years panel of personnel records from a large French company, we find that becoming mother (extensive fertility margins) largely affects labor market outcomes. Instead, fatherhood does not significantly impact on men's wages or careers. An event study approach with the use of non-parents as control group enables us to show that, prior to childbirth, future mothers' earnings are in line with that of non-mothers. However, one year after birth, they start to fall, reaching -9% in total pay and -30% in individual bonuses. This drop is persistent: 8 years after childbirth there is no evidence of a catching-up trend. Mothers also have lower chances to climb-up the hierarchy of the firm and be promoted to managerial positions. A decomposition of the motherhood penalty shows that these "missed promotions", likely due to an increase in absenteeism during the child's pre-school age, are the main determinants of mothers' lower outcomes within the firm.

Introduction

Parenthood leads to huge changes in all spheres of life. Individuals who become parents for the first time need to reorganize their entire working and leisure lives, with relevant consequences for their economic outcomes. It is nothing new that women face most of the challenges that the arrival of the first child implies. They are less likely to stay active in the labor market. If employed, they tend to work fewer hours and earn lower wages not only than men, but also than childless women [START_REF] Sigle Rushton | Motherhood and Women' s Earnings in Anglo-American, Continental European, and Nordic Countries[END_REF]. It has been estimated that during her career, a typical working mother with a medium level of education loses roughly half the earnings she would have had without having a child [START_REF] Joshi | The Price of Parenthood and the Value of Children, chapter in Making Economic Policy in the 21st Century: Essays in Honour of Henry Neuberger[END_REF].

A growing body of literature is now focusing on the wage differential between women with children and otherwise comparable women, the so called "motherhood pay gap". Pay penalties for mothers have been found both in the US [START_REF] Waldfogel | The Effects of Children on Women's Wages[END_REF], [START_REF] Budig | The Wage Penalty for Motherhood[END_REF], [START_REF] Anderson | The Motherhood Wage Penalty Revisited: Experience, Heterogeneity, Work Effort, and Work-Schedule Flexibility[END_REF], [START_REF] Bertrand | New Perspectives on Gender, chapter in Handbook of Labor Economics[END_REF]), and Europe [START_REF] Simonsen | The Family Gap in Wages: what Wombmates Reveal[END_REF] for Denmark, [START_REF] Felfe | The Motherhood Wage Gap: What about Job Amenities?[END_REF] for Germany, [START_REF] Wilner | Worker-Firm Matching and the Family Pay Gap: Evidence from Linked Employer-Employee Data[END_REF] for France) 1 . These penalties range from 2% to 7% in hourly wages.

Traditionally, they have been explained within a human capital framework: mothers tend to earn less because they make lower investments in education, have more career breaks and work more part-time. Moreover, they are believed to bring less effort to the labor market because of the burden of childcare activities. But it can also simply be that women with children are different from childless women in relevant ways that are associated with their earnings potential.

For example, they may have higher preferences for family over career 2 .

Most of the literature on the motherhood pay gap focuses on the effect of having an additional child among women who already have children (intensive fertility margins). But very few works 1 The existence of penalties for mothers in all labor markets rises questions on the effectiveness of child-related policies. For example, there is no unique provision of maternity leave in the US, while in Germany mothers are allowed to take 3 years of leave after childbirth. But similar motherhood pay gaps are found in both countries. Norther European countries are usually praised for offering women good opportunities to pursue family and career. However, recent findings show that also within these institutional contexts the costs of motherhood are high and able to explain almost all the remaining gap in pay [START_REF] Kleven | Children and Gender Inequality: Evidence fom Denmark[END_REF], [START_REF] Lundborg | Can Women Have Children and a Career? IV Evidence from IVF Treatments[END_REF])

2 In order to address causality in interpretation of findings on the motherhood pay gap, these unobserved differences are usually assumed to be time-invariant [START_REF] Budig | The Wage Penalty for Motherhood[END_REF]. Other techniques used to disentangle the causal effect of childbirth from the association between children and earnings have been conditionalon-observables strategies such as OLS and propensity score matching (e.g. [START_REF] Simonsen | The Family Gap in Wages: what Wombmates Reveal[END_REF]Skipper (2006, 2008)), firm and worker fixed effects [START_REF] Budig | The Wage Penalty for Motherhood[END_REF], [START_REF] Anderson | The Motherhood Wage Penalty Revisited: Experience, Heterogeneity, Work Effort, and Work-Schedule Flexibility[END_REF], [START_REF] Wilner | Worker-Firm Matching and the Family Pay Gap: Evidence from Linked Employer-Employee Data[END_REF]), and quasi-natural experiments based on twin births [START_REF] Angrist | Children and Their Parents' Labor Supply: Evidence from Exogenous Variation in Family Size[END_REF], [START_REF] Simonsen | The Family Gap in Wages: what Wombmates Reveal[END_REF]) or IVF treatments [START_REF] Lundborg | Can Women Have Children and a Career? IV Evidence from IVF Treatments[END_REF] 1 have specifically looked at the labor market impact of turning into motherhood (extensive fertility margins)3 . The aim of this paper is contribute to this specific stream of literature by giving a more comprehensive picture of the economic consequences of having the first child. In particular, we want to investigate how becoming parent is related to adjustments in individual behaviors at the workplace and what is the impact of these adjustments on economic outcomes.

We are able to provide new evidence on these mechanisms thanks to the availability of a 12years panel (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016) on the population of employees of a large French company. In 2013 this company accounted for 71, 000 employees. Adding external collaborators, it reached a whole workforce of 129, 492 individuals. Given its dimension and rigid, hierarchical internal structure, it can be considered as a large internal labor market, thus mitigating concerns of external validity of our findings. To our knowledge, no previous studies have been done on the impact of parenthood on earnings and careers of men and women within a firm. Indeed, the records at our disposal allow us to focus on precise measures of individual productivity (bonuses linked to performance, absenteeism) not available in survey data. Moreover, we are among the few able to provide empirical findings on the consequences of becoming father on earnings and careers4 .

To study the child effect at the extensive margins we develop an event study around the timing of first childbirth. Our preferred specification includes a control group of non-parents in the estimation sample. This strategy allows us to add individual fixed effects to our estimations, solving the underidentification problem typical of event studies designs.

Our results suggest a clear penalization in total earnings for becoming mother. Instead, we do not find any significant effect of fatherhood. The motherhood penalty can be explained by three main factors: lower hours worked, "missed promotions", and "missed bonuses". During the child's pre-school age, mothers work around 4% less with respect to non-mothers. This reduction in working hours is due to a shift from full-time to part-time contracts in the years immediately after delivery. Only when children enroll school, 6 years after childbirth, mothers tend to come back to full-time, starting to catch up non-mothers. The majority of mothers likely opt for part-time contracts because of lack of discretion in adjusting working schedules. We find evidence that this lack of flexibility has a long and negative impact on their productivity, and, as a consequence, on their earnings. For example, among managers, who have the chance to adjust their working time more easily, the gap in earnings between mothers and non-mothers is lower and disappears by the time children start school. Lack of flexibility also means more requests for leaves, thus an increase in absenteeism. We find evidence of a rise in the number of hours of absence for mothers in the first years after delivery. This increase in absenteeism has two main consequences. First, a block in the progressions within the hierarchical ranks of the firm. These "missed promotions" are able to explain the largest part of the motherhood penalty, since they indicate a slowing down for mothers, as compared to childless women, in the career patterns within the firm. Second, they lead to lower bonuses. These bonuses are linked to individual productivity and are particularly relevant in the firm's division in which mothers represent the majority of the workforce: the commercial. In this division, productivity goals linked to specific incentive schemes play an important role, penalizing women who turn into motherhood, since they likely have lower chances to work long, and particular, hours [START_REF] Goldin | A Grand Gender Convergence: its Last Chapter[END_REF]. This paper is organized as follows. Section 2 describes the institutional background and data.

Section 3 presents the empirical strategies and the estimation samples. Section 4 shows the main results. Section 5 explains the underlying mechanisms of the child penalty for mothers. In Section 6 we present some heterogeneous effects. In Section 7 we propose a decomposition of the motherhood penalty. In Section 8 we perform two placebo tests. Section 9 concludes.

2 Parental leave and childcare services in France and within the firm Family policies have a long history in France, a country where fertility has traditionally been at the core of social and political agenda. This made France the European country with the highest fertility rate, 2 children per woman in 2015 5 , and among the public systems that offer good opportunities to balance working and family life. Despite reductions in public support for families after the 2008 economic crisis, in 2014 expenditures on childcare and early education services were relatively high: 1.1% of GDP , compared to the OECD average of 0.7% 6 .

5 World Bank 6 Family Policies, France (2014). Population Europe Resource Finder and Archive

3

A large network of local Family Allowances Funds (CAFs, Caisse d'Allocations Familiales) is responsible for the provision of job-protected child-related leaves and childcare services. Childrelated leaves include maternity and paternity leave. These can be further extended to parental leave.

All working women are eligible to maternity leave for a total of 16 weeks for the first and second child. The minimal mandatory leave is 8 weeks: at least 2 before childbirth and 6 after. Generally, 6 weeks are taken before childbirth and 10 after. 26 weeks can be taken for a third, or more, birth. During maternity leave, the employment contract is suspended and the woman is entitled Children have access to childcare services and preschool from a very young age, which is expected to help parents balancing work and family life. 48% of the country's children under age three are enrolled in some type of formal care 8 . These include publicly subsidized home-based care, accredited family daycare providers, and nursery (crèches). Allowances for crèches are guaranteed from CAFs according to family income and number of children, but a minimum of 15% of monthly payments is in charge of the family 9 .

A universal model of preschool education, the École maternelle, is available to all children aged 3-6. The program is fully founded and organized by the State. In many municipalities, enrollments can be made from the age of 2.

Our company can be defined as family friendly. Employees with dependent children receive specific pecuniary and non-pecuniary advantages. Maternity and paternity leave are paid at full salary. Additional 4 weeks of maternity leave are provided until the birth of the second child, while 2 starting from the third. Future mothers are allowed to take one hour off for each working day. Re-founding of pregnancy-related health costs are provided and both male and female employees get a childbirth premium. The premium consists in a full month salary. Only mothers are allowed to take 6 days off per-year for child-related reasons. They can take 2 additional days for "sick child". Contributions to childcare expenses are guaranteed to all employees with children aged less than 3. For children aged 3-6, additional monthly payments are provided.

In 2012 the firm signed the "Charte de la Parentalité", the Corporate Parenthood Charter. This is an agreement proposed at national level by the Observatory for the Balance of Time and Parenthood within the Company 10 and aims promoting working environments in which employees with children can easily reconcile professional and family lives. More than 500 firms in France have already signed it. The effort of our company especially regarded the possibility for parents to work from home through teleworking.

Data

Our analysis is based on unique personnel records on the universe of employees of a large multiutility French firm. In 2016 the company accounted for 61, 000 employees. As shown by Figure A1, which reports the share of employees by French Département, we observe individuals over the whole territory. Given its dimension and internal structure, this company can be considered as a large internal labor market.

Data cover a 12-years period, from 2005 to 2016. The dataset combines different registers linked 9 urlhttp://www.caf.fr/ma-caf/caf-du-bas-rhin/offre-de-service/petite-enfance/je-souhaite-placer-mon-enfanten-creche The main attribute of the firm is the rank level to which each employee is associated according to her occupational category, tenure, and level of responsibility. These rank levels describe the internal occupational hierarchy. Promotions are improvements within this hierarchical scale.

Ranks range from 30, associated to the lowest blue-collar position, to 370, associated to the highest managerial position.

Total annual earnings are made up by a base-pay, that depends on hours worked and includes job indemnities for overtime or night work, and a performance-related pay. This pay component is linked to individual productivity, evaluated by employees' direct managers at the end of each year.

We selected employees aged 20-50. Selection on age is standard in the literature studying the effect of childbirth and aims at avoiding not biological births. The final dataset consists of an unbalanced panel of 363, 125 employee-year observations.

Table 1 provides sample means and standard deviations for selected variables used in the empirical analysis. Descriptives are shown by gender, parenthood status (employees with children vs employees without children) and occupational category (managers vs non-managers).

Independently from the occupational category, the group that accounts for the lowest labor supply is women with children. They also have the highest number of hours of absence from work.

Interestingly, women without kids do not differ much from men in terms of hours supplied, although they tend to be more absent.

On average, mothers in a managerial position do better than non-mothers in terms of earnings. This is expected since, mainly because of seniority reasons, they tend to be at a higher rank level. However, despite being at a similar average rank, the differential in earnings with men with children is large.

On the other hand, mothers in a non-managerial positions perform slightly worse than non-

Empirical strategy

To study fertility effects at the extensive margin, we develop an event study. This methodology allows us to follow changes in employees' economic outcomes around the event "birth of the first child ".

The event study

Consider a panel of i = 1, .., N individuals for whom a specific outcome Y it is observed for t = 1, ..., T calendar times (e.g. years). In an ideal setting, each individual receives a treatment in some time periods r i (e.g. some specific years within the panel), and stays treated forever.

Let define S it = t -r i the relative event time: it indicates, for each individual, the relative distance to the event. D it = 1 • S it ≥ 0 is the indicator for being treated.

Within this framework, we estimate the dynamic effect of the event 11 on the observed outcome through the following equation:

Y it = S s γ s • 1{S it = s} + X it α + η i + ν t + ε it (1)
where γ s are coefficients on indicators for time relative to the event, which occurs at s = 0. For s < 0 they show pre-event trends. For s >= 0, they capture the effective treatment effects, the dynamic impact of the event on the outcome of interest. X it is a set of age cohort dummies allowing to control non-parametrically for life cycle trends. η i and ν t are individual and time fixed effects, ε it is a random noise.

γ s=-1 , the relative time prior to the occurrence of the event, is the omitted category.

Given this set up, we face a well-known problem of underidentification [START_REF] Dobkin | The Economic Consequences of Hospital Admissions[END_REF] and Borusyak and Jaravel ( 2017)). For each individual i, the calendar time t is simply the sum of the time in which the event happens, r i , and the relative event time, s it . There is a perfect linear relationship between these effects. Since individual fixed effects subsume event time fixed effects, they cannot be included and estimated along the full set of calendar time and relative event time fixed effects.

To estimate Equation ( 1) we include a control group in the estimation sample. The presence of a control group solves the underidentification issue because the control group can be used to estimate the year effects independently of the causal effect of treatment [START_REF] Borusyak | Revisiting Event Study Designs[END_REF]. Our control group is made up by all employees who are not parents.

In a robustness check, we drop the unit fixed effects and balance the sample of employees who experience the birth of the first child around the event time. Without individual fixed effects there is no underidentification problem.

In our main strategy, we focus on individuals who experience the birth of the first child within a specific time window: 2007-2009. This selection is driven by the need to have enough childbirths to develop the analysis and a relatively large time span before and after the event. Since we want to capture the child effect along the extensive fertility margins, we additionally drop all employees who will have other children after the first one12 . We have 7, 126 observations.

Our control group is made up by all the employees who, during the whole period, never receive the treatment, such as do not have any dependent children. We have 19, 319 childless employees, for a total of 91, 847 employee-year observations.

Our relative event time function, s, ranges from -3 years before the event up to 8 years after.

We estimate Equation (1) separately for men and women. Our identifying assumption is that, conditional on life-cycle patterns, time trends and time invariant characteristics between treated and control units, the timing of childbirth is as good as random. The individual fixed effects allow us to control for constant individual-specific components of the error term correlated with the timing of childbirth. However, our assumption would be violated if there were time-varying shocks that are correlated both with the timing of childbirth and Y it . For example, an unexpected promotion could impact on both the decision of having a child and earnings. In a robustness check, we will control for career progressions and show that results are not affected. We will additionally show that there is no correlation between the probability of having a child and present or past promotions.

Attrition, which in our setting is potentially due to exit from the firm because of the birth of the child, is another possible violation of our identification strategy. The share of employees who leave the firm is extremely low, both among parents and non-parents (Table A2). Most of those who leave the firm, both with and without having experienced the birth of the first child, move to another company. While no men who become fathers leave the company to take a period of parental leave, 23.20% of women who become mothers and leave (22 observations) do so. In Table A3 we additionally show that there is no correlation between the probability of leaving the firm at t conditional on having had a child at t -1, both for men and women.

Results on total annual earnings

Table A4 reports results on the static specification of the impact of first childbirth on total annual earnings, where we impose no pre-trends and assume that the effect is constant for all s. Results are positive for fathers, +2%, while negative for mothers, -6.4%. Before the birth of the child, the event time indicators are not significantly different from zero, suggesting the absence of pre-trends. Employees' earnings, despite gender or parental status, evolve in parallel until the event "birth of the first child". After the event, however, we find a clear penalization for mothers and no or slightly positive effects for fathers.

The penalization for mothers is not immediate: at event time 0 results are still not significant.

This may be due to the fact that our firm gives to new parents, both males and females, a childbirth premium corresponding to a full one month salary. It is just one year after birth that mothers' total pay starts to fall by around 6%. What is relevant is that it does not increase back.

Results show that 8 years after childbirth mothers are earning 9% less than non-mothers. 10

Findings on the event study show a penalty for mothers in terms of total annual earnings. In this section we investigate the channels through which this penalty arises: working hours, career paths, and absenteeism. We will also present results by earnings components (performancerelated pay, base-pay).

Working hours

The arrival of the first child leads to important changes in individuals' labor supply. Since our results are all conditional on staying employed in the firm, we can study the evolution of hours supplied around the timing of first childbirth. We thus replicate our event study using as dependent variable the log of annual hours actually worked by the employees.

Results are reported in Figure A2. A significant reduction in hours worked, around 5%, is found for women one year after delivery. The subsequent year the pattern starts to improve and stabilizes during children's pre-school age, with a gap of 4% with non-mothers. When the child enrolls school, 6 years after birth, a clear catch-up trend is found. 8 years after, the differential in working hours between mothers and non-mothers is 2%. No effect is found on fathers.

Given these findings, we deeper the study on working hours by analyzing changes in contractual working time before and after first childbirth on the subsample of mothers (Table A5). Before the birth of the child none of these women is working part-time. A sharp increase in the share of part-time contracts is observed between event time 0 and 1: +5%. During pre-school years, most of mothers opt for a working contract that accounts for 80% of full time hours. When the child starts primary school we observe that mothers start to come back a full-time contract.

These findings suggest that individual choices on hours worked after childbirth are a relevant mechanism that negatively impacts earnings. However, the event study on working hours shows a catching-up process that is not found in earnings' trends.

Career paths

According to [START_REF] Davies | The Origins of the Ideal Worker: the Separation of Work and Home in the United States From the Market Revolution to 1950[END_REF], the myth of the perfect worker began to emerge at the beginning of the 20 th century, when it was described as "a men completely devoted to his employer, his faithfulness rewarded by promotions" (p. 26). This way of conceiving work has developed over time, with the advent of the "24/7" working culture and the widespread idea that employers implicitly require that work schedule should take precedence over family. We thus want to study how career patterns diverge between employees who have children and employees who do not by looking at the trajectories of their rank level.

Table A6 shows that, before childbirth, there are no, or only slightly, significant differences within the hierarchical rank of the firm between employees who are going to have a child and employees who are not. After childbirth, however, mothers, constantly slow down with respect to non-mothers. 8 years after childbirth, they end up at 6 levels behind with respect to childless women. In Table A7 we additionally show that mothers see reduced their probability to enter the highest managerial positions. 8 years after birth the chances to become a manager for a woman with a child is more than 10% lower with respect to a childless woman.

Absenteeism

Two major arguments have been developed to justify promotions within a firm. The first is based on tournament theory [START_REF] Lazear | Rank-Order Tournaments as Optimal Labor Contracts[END_REF], [START_REF] Rosen | Prices and Incentives in Elimination Tournaments[END_REF]). A promotion is the prize allocated to the worker who ranks better than all other candidates. The winner will be moved to a position that involves higher responsibility and earnings. Her probability to win depends on her productivity and this probability is itself an incentive to exert higher effort. The second, based on job assignment models (Gibbons and Waldman, 1999a), views promotions as an instrument for the efficient allocation of the employees within the firm. The firm uses employment at lower hierarchical levels as a screening period during which it learns about employees' abilities and attitudes. According to [START_REF] Pfeifer | Determinants of Promotions in an Internal Labour Market[END_REF], both theories predict similar hypothesis on the determinants of promotions. Among these, absenteeism is particularly relevant. Indeed, absenteeism is an important proxy for productivity and work effort. According to the first theory on promotions, an absent employee cannot provide effort, and so her probability of winning the tournament declines. Instead, according to job assignment models, if an employee is absent the firm cannot learn about her productivity and can use absenteeism in the past as a proxy for productivity in the future.

We find confirmation of this fact in our firm. Absences are remunerated at full salary, so they do not directly impact on wages, but they do affect the career patterns of employees. A rise in the number of hours of absences has two main consequences. A stop in annual improvements along the hierarchical ranks and a reduction in individual bonuses.

To better investigate this issue, we replicate our event study on the log of annual hours of absence to see if there are changes in the absenteeism behavior of parents around childbirth as compared to non-parents (Figure A3). We exclude the hours of paternity and maternity leave. During the pre-school years of the child, mothers make around 20% more hours of absences with respect to non-mothers. These include 6 days per-year that are guaranteed to mother of small children for family reasons and additionally 2 days of leave for "sick child". Only when children enroll primary school, we do not observe significant differences between mothers and non-mothers in terms of absenteeism. No significant effects are found for fathers.

Earnings components

Table A8 reports results of the estimation of Equation ( 1) on base-pay and performance-related pay linked to individual productivity.

Again, before the birth of the child, there are no significant differences between our treatment and control units. Focusing on base-pay, we find some positive effect for fathers. This is likely due to an increase in overtime in the years immediately after the arrival of the child, a standard result for men [START_REF] Millimet | The Impact of Children on Wages, Job Tenure, and the Division of Household Labour[END_REF]. Mothers, instead, suffer a drop of 5.8% one year after delivery.

A gap with non-mothers that gets worse over time, reaching -7.7% 8 years after. But the largest penalization for mothers is in the performance-relate pay: -30% two years after the event, with no evidence of a catching-up. The individual productivity of fathers does not seem to be affected by the birth of the first child.

7 Heterogeneous effects

By occupational category

According to [START_REF] Blau | Work and Family[END_REF], women in manual occupations and service positions are those most penalized from childbirth. This type of jobs provides less flexibility in working time, as compared to professional jobs. Indeed, there is growing empirical evidence supporting the intuition that women value flexibility at work more than men do [START_REF] Bertrand | The Glass Ceiling[END_REF]. We know that, in our firm, employees in a managerial positions have the chance to arrange their working schedule more flexibly than other employees. We thus analyze the different impact of the birth of the first child on the subsamples of managers and non-managers (Table A9). Our results clearly point out that mothers in non-managerial positions are more penalized. For them, the gap with non-mothers is larger and more persistent (-8% 8 years after delivery). Mothers in managerial positions, instead, not only are less penalized, but also menage to catch-up non-mothers by the timing the child enrolls primary school. More flexible working time, allowing mothers to better reconcile working and family responsibilities, can help explaining these findings. Indeed, Table A9 shows that the reduction in working hours is much larger for women in non-managerial position, who likely face greater additional pressures on their time as they try to balance market and non-market work commitments. Moreover, the different incidence of the child penalty among managers and non-managers can be explained by the way employees in different occupational categories are evaluated and remunerated. It is more likely that managers are paid on the basis of their output, and so of their own productivity. Instead, non-managerial employees are likely paid according to their inputs, mainly how many hours they work and how many hours they are absent from work.

Among both categories, some positive or no effects are found for fathers.

By division

There is a relevant difference on how work is arranged and managed across the three main divisions of the firm: administrative, commerce, and production. In particular, in the commerce division, employees have to sell the product through direct contacts with clients. It is known that, in contrast to standard technology-based occupations, typical of the production division, business-type occupations tend to show larger elasticities of income with respect to hours worked. [START_REF] Goldin | A Grand Gender Convergence: its Last Chapter[END_REF] shows how these occupations involve work that is more structured, done under more time pressure, and that requires establishing, and maintaining, interpersonal relationships.

We know that only 10% of men in our firm are employed in the commerce division, while 35% of women are so (Table A1).

As Table A11 shows, a penalty for mothers in total earnings is found in all divisions, but the results are heterogeneous. In the administrative, we observe a penalty of 5% that disappears as soon as the child starts primary school. In the production, the penalty is persistent, but results have low power. The most penalized division is the commerce, where the motherhood penalty reaches 11.4% 5 years after childbirth and is still 5.6% 8 years after. In Table A12 we show that mothers working in the commerce division are also the most penalized in terms of performance-related pay. Indeed, the gap with non-mothers reaches more than 50% even 8 years after childbirth.

By workplace and place of residence

Having a high share of female colleagues at the workplace is not largely different, in terms of motherhood penalization, from having a low share of female colleagues, as shown in Table A13.

Fathers working with a majority of male colleagues, instead, tend to be rewarded from the birth of the first child.

The place where the individual lives seems to matter more: the motherhood penalty is higher in urban areas (Table A14). We can explain this result by assuming that in these areas the availability of childcare services is more expensive. Indeed, other studies have found that women living in areas where the outsourcing of home production activities is more affordable tend to have better economic outcomes [START_REF] Cortés | Low-Skilled Immigration and the Labor Supply of Highly Skilled Women[END_REF], Cortés and Pan (forthcoming)).

Decomposition of the motherhood penalty

We have shown that the main mechanisms that help us explaining the negative impact of the birth first child for female employees as compared to childless women are a reduction in working hours, lower chances to climb-up the hierarchy of the firm, and lower performance-related pay.

The last two mechanisms are linked to an increase in absenteeism observed during the child's pre-school age.

In this section we propose a decomposition of the motherhood penalty to understand the relative weight of these mechanisms. In particular, our main interest stands in explaining how much of the penalty observed between mothers and non-mothers can be decomposed into "less hours", "missed promotions" and "missed bonuses".

To do so, we rely on a standard decomposition of mean differences in log earnings based on a linear regression model [START_REF] Oaxaca | Male-Female Wage Differentials in Urban Labor Markets[END_REF]. In practice, we regress the log of total annual earnings on total annual hours worked, promotions and individual performance-related pay separately for parents and non-parents 13 . Then, we decompose the mean difference in log earnings into a component that accounts for differences in predictors between parents and non-parents (the explained part of the gap), and a component that accounts for differences in coefficients (the unexplained part of the gap).

We are aware that, in the right-side of the equation, we are including earnings determinants that directly respond to childbirth. But the aim of this exercise is to show how much each mechanism contributes to explain the gap, as if we were calculating an accounting identity.

Results of our exercise are presented in table A15. The average motherhood penalty is around 12%. Of this, almost 10% can be explained by our predictors. "Missed promotions", lower chances for mothers to improve their rank level within the firm, are able to explain the largest part of the gap. The reduction in working hours, due to the increase in part-time work, especially for mothers in non-managerial positions, during pre-school age of the child, is the second mechanism able to explain the penalty. The "missed bonuses" are able to explain a lower portion of the gap. However, we have seen that only mothers in the commerce division are largely penalized in terms of individual performance-related pay.

Results also show a positive gap for fathers, which cannot be explained by our predictors.

9 Results on the extended sample The patterns for fathers and mothers do not differ from our main results. For women we observe a first drop in total earnings 1 year after first childbirth, and a subsequent drop 4 years after.

This second drop is likely associated with the birth of another child. 8 years after the birth of the first child, mothers are earning 10% less than childless women.

Fathers, instead, show a slightly positive trend with respect to non-fathers.

13 Additional controls include age, education and year dummies 10 Robustness checks

In our first robustness check, we run an event study on a balanced sample of parents who who had their first child during the time window 2007 -2009. More specifically, for each parent, we set s = 0 for the years of the event (2007, 2008, or 2009) and relatively index all the other years.

Then, we balance the sample around the event time s = 0. As a consequence, our specification includes all parents that we are able to observe from -2 years before childbirth to +7 years after.

After having dropped the individual fixed effects, we run Equation ( 1) on the separate samples of fathers and mothers.

The main characteristic of this strategy is that each individual receives the treatment and is observed for the same number of years. Results are shown in Table A16 and confirm the patterns found in our main empirical strategy: negative, large and persistent effect for women, no effect for men.

In a second check we focus on the possibility that unexpected promotions influence both the decision of having a child and earnings. We start by additionally controlling for career progression.

We thus include in Equation ( 1) dummies for rank levels. Results, reported in Table A17, are robust. Pre-trends are not statistically significant. One year after childbirth mothers face a drop in earnings of around 4%, which persists 8 years after. Positive effects are found for fathers.

Then, we check for the correlation between past and current promotions and childbirth (Table A18). There is no association between being promoted and having a child, both for fathers and mothers. We find a slightly positive correlation between being promoted at t -1 and having a child at t for women. However, this has low power and is very low in magnitude, 0.2%.

In a third robustness, we replicate our event study using different birth windows: 2008 -2010 and 2009 -2011. Results are shown in Figure A5 and are highly robust.

In our last robustness we try a different kind of experiment. We focus only on employees who do not have children during the observation period, the control group in our main event study, but experience a medium length period of sick leave, between one and three months, during the years 2007-2009. We then follow their earnings and careers around this event. In this case, the control group is made up by employees who, in the window 2007-2009, experienced less than one month of sick leave. We remember that we are controlling for time-invariant unobserved characteristics between treatment and control units.

Results of this exercise are presented in Tables A19 andA20. Findings show a clear penalization for a medium-length sick leave period both for men and women, thus suggesting that the mechanism of absences has relevant economic consequences. What is interesting is that this effect is stronger for women, especially when analyzing their career patterns. 8 years after the sick leave period, women are falling 10 levels behind with respect to women who did not take it. The result for men is 7 levels.

Interpreting these results in light of the findings on the impact of the birth of the first child makes us thinking that while the sick-leave period is attached to the person and cannot be transferred, and because of this also men are penalized, all the burden of the consequences of childbirth stays with the mother and is transferred by the men to women. This translates in a clear penalization for mothers, but not for fathers.

Conclusions

Using a unique panel of personnel records from 2005 to 2016, we developed an event study with the use of a control group to capture the effect of becoming parent (extensive fertility margins) on the dynamic trajectories of earnings and careers among male and female employees of a large firm in France.

Despite the family-friendly institutional context, results show a clear penalty for women after first childbirth. While there are no significant differences between women who are becoming mothers and women who are not before the birth of the first child, 8 years after childbirth mothers are earning 9% less than childless female employees. Fathers' earnings, instead, are slightly positively or no significantly affected. The main channels underlying the motherhood penalty in total annual earnings pass through a reduction in working hours, a slowing down in the career tracks after childbirth and a large reduction in performance-related pay linked to individual productivity. The last two mechanisms are related to an increase in absenteeism during the child's pre-school age. A decomposition of the motherhood penalty shows that the "missed promotions" account for the largest part of the gap.

We can drive two main conclusions from our findings. First, even if family-friendly policies play a fundamental role in attracting and retaining women in the labor force, they will not reduce the penalty mothers face as long as they are negatively perceived and priced by employers. We show that the negative incidence of a reduction in working hours and an increase in absenteeism during the pre-school age of the child is strong and persists even when a clear catching-up trend in hours is found.

Second, it seems clear that a key driver of the motherhood penalty is still rooted in cultural norms that see the birth of a child as a female matter. The consequences of childbirth fully stay with the mother.

These results strengthen and give new evidence on what the current economic research is pointing out: the main explanation why many women's career is still largely penalized is childbearing and childcare. These are results of a twofold decomposition that uses non-mothers and non-fathers as norm.

Results are robust to a threefold decomposition

  to daily allowances paid by social security. Allowances are calculated taking into account the gross 3 wages earned before the date of interruption of work, divided by 91.25 (for monthly paid employees). A ceiling of 3, 218 Euros per month has been set in 2016 7 . Since 2002, paternity leave can be taken by all working fathers for up to 11 days for single birth and 18 days for multiple births. The leave must begin within 4 months after the birth of the child. Payment rules are equal to those of maternity leave. Maternity and paternity leave can be extended to parental leave. Parental leave is open to all employees with children who, at the timing of childbirth, have at least one year of tenure with the employer. The normal length of parental leave is one year, renewable two times. In case of multiple birth (at least three children), the leave can be renewed 5 times. Parental leave starts immediately after the end of maternity/paternity leave. During the leave, the employment contract is suspended. The employee perceives basic allowances for the provision of services for young children (PAJE, Prestation d'Accueil du Jeune Enfant) from the CAF. The amount of the allowance depends on family income and number of children.

  Figure 1, instead, shows the relative event time coefficients γ s estimated from the dynamic specification (Equation (1)). 95% confidence intervals are included. The vertical red lines indicate the timing of childbirth.

Figure 1 :

 1 Figure 1: Total Annual Earnings by Event Time

Figure

  FigureA4shows results on the sample of employees who had their first child within the time window 2007-2009 and then, possibly, had other kids.

Figure A2 :

 A2 Figure A2: Annual Working Hours by Event Time

Figure A3 :

 A3 Figure A3: Annual Hours of Absences by Event Time

Figure A4 :

 A4 Figure A4: Total Annual Earnings by Event Time, Extended Sample

  of medium-length (1-3 months) sick leave period Clustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

Table A2 :

 A2 Share of Employees who Leave the Firm and Reason for Exit

			Men				Women
		Child		No Child		Child		No Child
		%	Obs	%	Obs	%	Obs	%	Obs
	Exit	2.39	98	1.74	1132	3.31	94	2.38	639
	Change company	72.45	71	3.34	434	61.10	58	49.77	318
	Parental leave	0	0	0	0	23.30	22	0	0
	Pre-	0	0	8.04	91	5.32	5	9.10	58
	retirement/Invalidity
	Retirement	11.22	11	20.23	229	0	0	14.08	90
	Other	16.33	16	33.39	378	9.57	9	27.07	173

Table A3 :

 A3 Probability of Exit at t Conditional on Childbirth at t -1Clustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

		Exit t
		Fathers	Mothers
	Childbirth t-1	0.008	0.006
		(0.01)	(0.02)
	Individual FE
	Time FE
	Obs	3655	2647

Table A4 :

 A4 Static SpecificationClustered standard errors at the individual level in parentheses D the indicator variable for being treated D it = 1 • {S it ≥ 0} * p < 0.05, * * p < 0.01, * * * p < 0.001

		Total earnings
		Fathers	Mothers
	D	0.020 * *	-0.064 * * *
		(0.01)	(0.01)
	Individual FE
	Time FE
	Obs	69066	29907

Table A5 :

 A5 Share of Mothers by Contractual Working Time

		Full time 80% full time Part time
	-3 years	54.38	45.62	0
	-2 years	48.78	51.22	0
	-1 year	44.61	55.39	0
	0	41.49	58.21	0.30
	+1 year	39.73	54.88	5.39
	+2 years	35.84	59.86	4.30
	+3 years	36.17	59.57	4.26
	+4 years	39.93	55.59	4.48
	+5 years	43.80	52.32	3.88
	+6 years	50.62	44.44	4.94
	+7 years	55.93	39.41	4.66
	+8 years	61.00	35.50	3.50
	0: timing of first childbirth
	Full time= 100% of working hours
	80% full time= 80%-99% of working hours
	Part time= 50% or less of working hours

Table A6 :

 A6 Rank Level by Event TimeClustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

		Rank level
		Fathers	Mothers
	-3 years	-0.746	0.338
		(0.82)	(0.99)
	-2 years	-0.903 *	1.091 *
		(0.45)	(0.48)
	0	0.0698	0.272
		(0.37)	(0.44)
	+1 year	0.348	-2.043 * * *
		(0.44)	(0.51)
	+2 years	0.245	-3.641 * * *
		(0.51)	(0.58)
	+3 years	0.860	-3.154 * * *
		(0.63)	(0.70)
	+4 years	0.960	-3.256 * * *
		(0.71)	(0.78)
	+5 years	1.490 *	-3.557 * * *
		(0.72)	(0.87)
	+6 years	1.364	-3.734 * * *
		(0.78)	(0.95)
	+7 years	1.360	-4.645 * * *
		(0.88)	(1.11)
	+8 years	2.014	-6.085 * * *
		(1.14)	(1.34)
	Age cohorts
	Individual FE
	Time FE
	Obs	68807	29784
	0: timing of first childbirth

Table A7 :

 A7 Probability to Become Manager by Event Time

		Prob Manager
	Fathers	Mothers
	-3 years	0.016	-0.030
		(0.01)	(0.03)
	-2 years	-0.004	0.005
		(0.01)	(0.01)
	0	-0.001	-0.011
		(0.01)	(0.01)
	+1 year	-0.001	-0.022 *
		(0.01)	(0.01)
	+2 years	0.006	-0.035 * *
		(0.01)	(0.01)
	+3 years	0.009	-0.043 * *
		(0.02)	(0.01)
	+4 years	0.017	-0.059 * * *
		(0.02)	(0.02)
	+5 years	0.011	-0.052 *
		(0.02)	(0.02)
	+6 years	-0.002	-0.068 * *
		(0.02)	(0.02)
	+7 years	-0.012	-0.077 * *
		(0.02)	(0.02)
	+8 years	-0.036	-0.119 * * *
		(0.02)	(0.03)
	Age cohorts
	Individual FE
	Time FE
	Obs	68807	29784
	0: timing of first childbirth
	Clustered standard errors at the individual level in parentheses
	* p < 0.05, * * p < 0.01, * * * p < 0.001
		29

Table A8 :

 A8 Base-Pay and Performance-Related Pay by Event TimeClustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

		Base		Performance
		Fathers	Mothers	Fathers	Mothers
	-3 years	-0.007	-0.003	0.045	-0.121
		(0.01)	(0.01)	(0.08)	(0.09)
	-2 years	-0.0001	0.005	-0.008	0.005
		(0.01)	(0.01)	(0.06)	(0.07)
	0	0.002	-0.002	0.025	-0.039
		(0.01)	(0.01)	(0.06)	(0.07)
	+1 year	0.001	-0.058 * * *	0.084	-0.106
		(0.01)	(0.01)	(0.06)	(0.07)
	+2 years	0.010	-0.084 * * *	0.024	-0.303 * * *
		(0.01)	(0.01)	(0.06)	(0.08)
	+3 years	0.024 * *	-0.070 * * *	0.018	-0.289 * * *
		(0.01)	(0.01)	(0.06)	(0.08)
	+4 years	0.021 *	-0.075 * * *	0.010	-0.242 * *
		(0.01)	(0.01)	(0.06)	(0.08)
	+5 years	0.032 * *	-0.077 * * *	0.010	-0.308 * * *
		(0.01)	(0.01)	(0.06)	(0.08)
	+6 years	0.027 *	-0.086 * * *	0.002	-0.253 * *
		(0.01)	(0.01)	(0.06)	(0.08)
	+7 years	0.019	-0.083 * * *	-0.002	-0.393 * * *
		(0.01)	(0.01)	(0.07)	(0.08)
	+8 years	0.008	-0.077 * * *	-0.039	-0.314 * * *
		(0.01)	(0.02)	(0.08)	(0.09)
	Age cohorts
	Individual FE
	Time FE
	Obs	68807	29784	53810	21501
	0: timing of first childbirth

Table A9 :

 A9 Total Annual Earnings by Event Time and Occupational Category

		Non Managers		Managers
		Fathers	Mothers	Fathers	Mothers
	-3 years	-0.013	0.002	0.002	0.008
		(0.01)	(0.01)	(0.02)	(0.02)
	-2 years	0.002	0.003	-0.008	0.013
		(0.01)	(0.01)	(0.01)	(0.01)
	0	0.001	-0.006	0.010	-0.009
		(0.01)	(0.01)	(0.01)	(0.01)
	+1 year	-0.008	-0.072 * * *	0.034 * *	-0.025
		(0.01)	(0.01)	(0.01)	(0.01)
	+2 years	0.002	-0.10 * * *	0.032 *	-0.077 * *
		(0.01)	(0.01)	(0.02)	(0.02)
	+3 years	0.019	-0.086 * * *	0.031	-0.054 * *
		(0.01)	(0.01)	(0.02)	(0.02)
	+4 years	0.018	-0.090 * * *	0.027	-0.050 *
		(0.01)	(0.01)	(0.02)	(0.02)
	+5 years	0.023	-0.087 * * *	0.050 * *	-0.051 *
		(0.01)	(0.02)	(0.02)	(0.02)
	+6 years	0.028 *	-0.097 * * *	0.034	-0.041
		(0.01)	(0.02)	(0.02)	(0.02)
	+7 years	0.011	-0.086 * * *	0.034	-0.045 *
		(0.01)	(0.02)	(0.02)	(0.02)
	+8 years	0.005	-0.080 * * *	0.009	-0.025
		(0.02)	(0.02)	(0.02)	(0.03)
	Age cohorts
	Individual FE
	Time FE
	Obs	42481	16612	26326	13172
	0: timing of first childbirth
	Clustered standard errors at the individual level in parentheses
	* p < 0.05, * * p < 0.01, * * * p < 0.001

Table A10 :

 A10 Annual Working Hours by Event Time and Occupational Category

		Non Managers		Managers
		Fathers	Mothers	Fathers	Mothers
	-3 years	-0.004	-0.004	0.001	0.001
		(0.00)	(0.01)	(0.00)	(0.01)
	-2 years	-0.002	-0.003	0.001	0.001
		(0.00)	(0.00)	(0.00)	(0.00)
	0	-0.001	-0.002	-0.001	-0.001
		(0.00)	(0.00)	(0.00)	(0.00)
	+1 year	-0.001	-0.055 * * *	0.002	-0.026 *
		(0.00)	(0.01)	(0.00)	(0.01)
	+2 years	-0.001	-0.050 * * *	0.003	-0.021 *
		(0.00)	(0.01)	(0.00)	(0.01)
	+3 years	-0.002	-0.052 * * *	0.004	-0.027 *
		(0.00)	(0.01)	(0.00)	(0.01)
	+4 years	0.001	-0.051 * * *	0.004	-0.031 *
		(0.00)	(0.01)	(0.00)	(0.01)
	+5 years	0.004	-0.043 * * *	0.003	-0.028 *
		(0.00)	(0.01)	(0.00)	(0.01)
	+6 years	0.004	-0.041 * * *	0.004	-0.025 *
		(0.00)	(0.01)	(0.00)	(0.01)
	+7 years	0.004	-0.032 * *	0.005	-0.016
		(0.00)	(0.01)	(0.00)	(0.01)
	+8 years	0.001	-0.019 *	0.004	-0.009
		(0.01)	(0.01)	(0.00)	(0.01)
	Age cohorts
	Individual FE
	Time FE
	Obs	42481	16612	26326	13172
	0: timing of first childbirth
	Clustered standard errors at the individual level in parentheses
	* p < 0.05, * * p < 0.01, * * * p < 0.001

Table A11 :

 A11 Total Annual Earnings by Event Time and DivisionClustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

		Commerce		Production		Administrative
		Fathers	Mothers	Fathers	Mothers	Fathers	Mothers
	-3 years	-0.012	0.001	-0.002	0.001	0.000	-0.012
		(0.02)	(0.02)	(0.01)	(0.02)	(0.02)	(0.02)
	-2 years	-0.002	0.001	-0.001	0.004	0.002	0.012
		(0.01)	(0.01)	(0.01)	(0.02)	(0.01)	(0.01)
	0	0.002	-0.009	-0.001	-0.022	0.011	0.000
		(0.01)	(0.01)	(0.01)	(0.01)	(0.01)	(0.01)
	+1 year	0.012	-0.079 * * *	0.002	-0.053 * *	-0.003	-0.028
		(0.01)	(0.01)	(0.01)	(0.02)	(0.02)	(0.02)
	+2 years	0.014	-0.109 * * *	0.013	-0.076 * *	-0.001	-0.062 * * *
		(0.02)	(0.02)	(0.01)	(0.02)	(0.02)	(0.02)
	+3 years	0.009	-0.097 * * *	0.034 * *	-0.059 * *	0.023	-0.048 *
		(0.02)	(0.02)	(0.01)	(0.02)	(0.02)	(0.02)
	+4 years	0.005	-0.107 * * *	0.028 *	-0.050 *	0.053 *	-0.049 *
		(0.02)	(0.02)	(0.01)	(0.02)	(0.02)	(0.02)
	+5 years	0.007	-0.114 * * *	0.040 * *	-0.053 *	0.065 *	-0.045 *
		(0.02)	(0.02)	(0.01)	(0.03)	(0.03)	(0.02)
	+6 years	0.025	-0.109 * * *	0.027 *	-0.066 *	0.058 *	-0.046 *
		(0.02)	(0.02)	(0.01)	(0.03)	(0.03)	(0.02)
	+7 years	0.008	-0.103 * * *	0.020	-0.086 *	0.041	-0.031
		(0.03)	(0.02)	(0.01)	(0.04)	(0.03)	(0.02)
	+8 years	0.008	-0.056 *	0.010	-0.082 *	0.023	-0.019
		(0.04)	(0.02)	(0.02)	(0.04)	(0.02)	(0.03)
	Age cohorts
	Individual FE
	Time FE
	Obs	5657	7946	42185	8417	8743	8525
	0: timing of first childbirth

Table A12 :

 A12 Performance-Related Pay by Event Time and DivisionClustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

		Commerce		Production		Administrative
		Fathers	Mothers	Fathers	Mothers	Fathers	Mothers
	-3 years	0.010	-0.075	-0.196	-0.008	0.170	0.079
		(0.20)	(0.15)	(0.15)	(0.10)	(0.10)	(0.09)
	-2 years	-0.009	-0.046	0.001	-0.014	-0.030	0.143 *
		(0.13)	(0.10)	(0.14)	(0.08)	(0.10)	(0.07)
	0	0.106	0.002	-0.007	-0.010	0.063	0.043
		(0.12)	(0.10)	(0.14)	(0.07)	(0.11)	(0.07)
	+1 year	0.074	-0.096	0.006	0.058	0.088	0.0345
		(0.14)	(0.13)	(0.12)	(0.07)	(0.11)	(0.08)
	+2 years	0.218	-0.283	-0.238	-0.044	0.057	-0.210
		(0.14)	(0.17)	(0.14)	(0.08)	(0.13)	(0.11)
	+3 years	0.199	-0.323 *	-0.206	-0.064	0.173	-0.197
		(0.15)	(0.14)	(0.15)	(0.07)	(0.12)	(0.11)
	+4 years	0.190	-0.459 * *	-0.090	-0.068	0.220	-0.092
		(0.15)	(0.16)	(0.14)	(0.07)	(0.13)	(0.11)
	+5 years	0.199	-0.519 * * *	-0.127	-0.067	0.063	-0.263 *
		(0.15)	(0.13)	(0.13)	(0.07)	(0.13)	(0.11)
	+6 years	0.036	-0.555 * * *	-0.192	-0.099	0.007	-0.079
		(0.18)	(0.14)	(0.16)	(0.08)	(0.12)	(0.10)
	+7 years	0.233	-0.583 * * *	-0.175	-0.086	0.121	-0.202
		(0.25)	(0.14)	(0.15)	(0.08)	(0.14)	(0.12)
	+8 years	0.178	-0.564 * *	-0.105	-0.153	-0.094	-0.058
		(0.21)	(0.18)	(0.15)	(0.10)	(0.18)	(0.12)
	Age cohorts
	Individual FE
	Time FE
	Obs	2662	3340	6936	32405	7004	6484
	0: timing of first childbirth

Table A13 :

 A13 Total Annual Earnings by Event Time and Share of Male/Female Colleagues the share of male/female colleagues is below/above the median number of male/female employees at the work place Clustered standard errors at the individual level in parentheses

		<Median		>Median
		Fathers	Mothers	Fathers	Mothers
	-3 years	-0.018	-0.003	0.002	-0.010
		(0.01)	(0.02)	(0.01)	(0.02)
	-2 years	-0.010	-0.003	0.007	0.012
		(0.01)	(0.01)	(0.01)	(0.01)
	0	0.004	-0.001	0.005	-0.002
		(0.01)	(0.01)	(0.01)	(0.01)
	+1 year	-0.001	-0.068 * * *	0.010	-0.025
		(0.01)	(0.01)	(0.01)	(0.02)
	+2 years	0.001	-0.080 * * *	0.025 *	-0.097 * * *
		(0.01)	(0.01)	(0.01)	(0.02)
	+3 years	0.021	-0.069 * * *	0.036 * *	-0.072 * * *
		(0.01)	(0.01)	(0.01)	(0.02)
	+4 years	0.008	-0.076 * * *	0.045 * *	-0.064 * *
		(0.01)	(0.01)	(0.01)	(0.02)
	+5 years	0.020	-0.080 * * *	0.056 * * *	-0.059 * *
		(0.01)	(0.02)	(0.02)	(0.02)
	+6 years	0.015	-0.078 * * *	0.052 * *	-0.074 * * *
		(0.01)	(0.02)	(0.02)	(0.02)
	+7 years	0.003	-0.076 * * *	0.046 * *	-0.072 * * *
		(0.02)	(0.02)	(0.02)	(0.02)
	+8 years	-0.014	-0.074 * * *	0.031	-0.061 *
		(0.02)	(0.02)	(0.02)	(0.03)
	Age cohorts
	Individual FE
	Time FE
	Obs	34464	15087	34343	14697
	0: timing of first childbirth
	<Median, >Median:

* p < 0.05, * * p < 0.01, * * * p < 0.001

Table A14 :

 A14 Total Annual Earnings by Event Time and Place of Residence

		Urban		Rural
		Fathers	Mothers	Fathers	Mothers
	-3 years	-0.009	0.006	-0.005	-0.005
		(0.02)	(0.02)	(0.01)	(0.02)
	-2 years	-0.006	0.009	0.002	0.004
		(0.01)	(0.01)	(0.01)	(0.01)
	0	-0.000	-0.014	0.004	-0.001
		(0.01)	(0.01)	(0.01)	(0.01)
	+1 year	0.011	-0.070 * * *	0.003	-0.058 * * *
		(0.01)	(0.01)	(0.01)	(0.01)
	+2 years	0.025	-0.124 * * *	0.010	-0.081 * * *
		(0.01)	(0.02)	(0.01)	(0.01)
	+3 years	0.049 * *	-0.091 * * *	0.021 *	-0.075 * * *
		(0.02)	(0.02)	(0.01)	(0.01)
	+4 years	0.043 *	-0.092 * * *	0.020	-0.081 * * *
		(0.02)	(0.02)	(0.01)	(0.01)
	+5 years	0.058 * *	-0.102 * * *	0.028 *	-0.081 * * *
		(0.02)	(0.02)	(0.01)	(0.02)
	+6 years	0.061 * *	-0.110 * * *	0.022	-0.086 * * *
		(0.02)	(0.03)	(0.01)	(0.02)
	+7 years	0.048 *	-0.137 * * *	0.016	-0.075 * * *
		(0.02)	(0.03)	(0.01)	(0.02)
	+8 years	0.048 *	-0.124 * * *	0.001	-0.075 * * *
		(0.02)	(0.03)	(0.02)	(0.02)
	Age cohorts
	Individual FE
	Time FE
	Obs	42481	16612	26326	13172
	0: timing of first childbirth
	Urban: départements with the 10 most populated cities in France (Paris, Marseille, Lyon, Toulouse, Nice, Nantes, Strasbourg, Montpellier, Bordeaux, Lille)

Rural: the remaining départements.

Clustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

Table A15 :

 A15 Decomposition of the Parenthood Pay GapClustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001 Additional controls include age, year and education dummies

		Fathers	Mothers
	Pay gap	0.032 *	-0.121 * * *
		(0.02)	(0.02)
	Explained	0.001	-0.110 * * *
		(0.01)	(0.02)
	Promotions	0.029 *	-0.061 * * *
		(0.01)	(0.02)
	Working Hours	-0.002 *	-0.028 * * *
		(0.00)	(0.00)
	Performance Pay	0.003	-0.011 * * *
		(0.00)	(0.00)
	Unexplained	0.031 * * *	-0.011 *
		(0.01)	(0.01)
	Obs	68807	29784

Table A16 :

 A16 Total Annual Earnings by Event Time, Balanced Sample of Parents Robust standard errors in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001 40

		Total earnings
		Fathers	Mothers
	-2 years	0.004	0.036
		(0.03)	(0.03)
	0	-0.014	-0.022
		(0.03)	(0.03)
	+1 year	-0.021	-0.062
		(0.03)	(0.03)
	+2 years	-0.027	-0.136 * *
		(0.04)	(0.04)
	+3 years	-0.025	-0.158 * *
		(0.04)	(0.05)
	+4 years	-0.021	-0.170 * *
		(0.05)	(0.06)
	+5 years	-0.016	-0.176 * *
		(0.05)	(0.06)
	+6 years	-0.007	-0.180 * *
		(0.06)	(0.07)
	+7 years	-0.028	-0.190 *
		(0.06)	(0.08)
	Age cohorts
	Individual FE
	Time FE
	Obs	3539	2668
	0: timing of random assignment of childbirth

Table A17 :

 A17 Total Annual Earnings by Event Time controlling for Career ProgressionClustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001Clustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

			Total earnings
			Fathers	Mothers
		-3 years	0.007	0.002
	RL t		(0.01) 0.001	(0.01)	-0.002
		-2 years	0.015 * (0.00)	0.010	(0.00)
			(0.01)	(0.01)
	RL t-1	0	0.012 * 0.0004	0.004	0.002 *
			(0.01) (0.00)	(0.01)	(0.00)
	RL t-2	+1 year	0.010 (0.01) 0.001	-0.039 * * * (0.01)	0.001
		+2 years	0.018 * * (0.00)	-0.065 * * *	(0.00)
	Individual FE	(0.01)	(0.01)
	+3 years Time FE	0.031 * * *	-0.046 * * *
	Obs		(0.01) 3221	(0.01)	2278
		+4 years	0.031 * * *	-0.046 * * *
			(0.01)	(0.01)
		+5 years	0.039 * * *	-0.046 * * *
			(0.01)	(0.01)
		+6 years	0.037 * * *	-0.054 * * *
			(0.01)	(0.01)
		+7 years	0.028 * *	-0.051 * * *
			(0.01)	(0.01)
		+8 years	0.016	-0.039 * *
			(0.01)	(0.01)
		Rank levels
		Age cohorts
		Individual FE
		Time FE
		Obs	73250	29555
		0: timing of medium-length (1-3 months) sick leave period

Table A19 :

 A19 Sick Leave and Total Annual Earnings by Event Time

Table A20 :

 A20 Sick Leave and Rank Level by Event TimeClustered standard errors at the individual level in parentheses * p < 0.05, * * p < 0.01, * * * p < 0.001

		Rank level
		Men	Women
	-3 years	-0.069	-0.978
		(0.57)	(1.34)
	-2 years	0.150	-0.334
		(0.33)	(0.53)
	0	-0.442	-0.846 *
		(0.31)	(0.38)
	+1 year	-1.329 * *	-1.455 * *
		(0.42)	(0.47)
	+2 years	-1.912 * * *	-2.561 * * *
		(0.51)	(0.57)
	+3 years	-2.674 * * *	-3.417 * * *
		(0.61)	(0.71)
	+4 years	-3.163 * * *	-4.767 * * *
		(0.69)	(0.77)
	+5 years	-4.174 * * *	-4.973 * * *
		(0.77)	(0.90)
	+6 years	-4.817 * * *	-6.129 * * *
		(0.85)	(0.95)
	+7 years	-5.770 * * *	-8.674 * * *
		(1.04)	(1.12)
	+8 years	-7.211 * * *	-10.03 * * *
		(1.37)	(1.72)
	Age cohorts
	Individual FE
	Time FE
	Obs	73250	29555
	0: timing of medium-length (1-3 months) sick leave period

[START_REF] Lundborg | Can Women Have Children and a Career? IV Evidence from IVF Treatments[END_REF] develop a new IV strategy based on IVF induced fertility variation to estimate the causal effect of having children on women's career in Denmark

Results on the impact of fatherhood on earnings are scarce and puzzling. There is some evidence of a "fatherhood bonus" (4% -6% in Denmark[START_REF] Simonsen | The Family Gap in Wages: what Wombmates Reveal[END_REF]), but also evidence of no effect[START_REF] Wilner | Worker-Firm Matching and the Family Pay Gap: Evidence from Linked Employer-Employee Data[END_REF] in France, or even of a negative effect[START_REF] Cools | Parenthood Wage Penalties in a Double Income Society[END_REF] in Norway

https://www.service-public.fr/particuliers/vosdroits/F207

Population Europe Resource Finder and Archive

We will also estimate the static specification:Y it = ηi + νt + γ • D it + ε it .Compared to the fully dynamic specification, it imposes no pre-trends and constant treatment effects for all relative event time periods

In a second step of the analysis we will remove this restriction


Jin Yan
email: jin.yan@eurecom.fr

Jérôme Härri
email: jerome.haerri@eurecom.fr

On the Feasibility of URLLC for 5G-NR V2X Sidelink Communication at 5.9 GHz

Delivering ultra-reliable low-latency communication (URLLC) services on 5G New Radio (NR) is critical for extended Vehicle-to-everything (V2X) applications. Early studies focused either on infrastructure V2X communications, or exclusively evaluating performance on mmWave band, but never so far for V2X Sidelink communications. In this paper, we evaluate the feasibility of achieving URLLC critical requirements for 5G NR V2X Sidelink communication at the V2X 5.9 GHz band. We propose a higher 5G-NR numerology completed by a deterministic scheduler and propose an analytical V2X SL admission control algorithm to reach tight URLLC requirements.

I. INTRODUCTION

With the continuous rapid development of new innovative Vehicle-to-Everything (V2X) services requiring higher data volumes and reliability, 5G New Radio (NR) V2X has been designed to provide enhanced communication capabilities either on Uplink/Downlink (Uu) or on Slidelink (SL) 5G-NR interfaces. However, some promising services require V2X performance beyond the current 5G-NR V2X capabilities specified by 3GPP rel. 16 [START_REF] Garcia | A Tutorial on 5G NR V2X Communications[END_REF]. In particular, Ultra-Reliable Low Latency Communication (URLLC) is expected to be critical for highly innovative V2X services, such as tele-operated driving, platooning or industrial robotics. If specifications for 5G NR URLLC are due for rel.16, 5G NR V2X URLLC is not expected to be available before rel.18-19 at the earliest 1 .

A large number of studies are available on innovations and enablers for 5G NR URLLC as described and referenced by Ali et. al [START_REF] Ali | URLLC for 5G and Beyond: Requirements, Enabling Incumbent Technologies and Network Intelligence[END_REF], proposing various strategies to improve reliability. To address latency constraints, a typical approach in 5G NR development is to use a higher numerology in order to reduce the duration of transmission time intervals (TTI) (e.g. [START_REF] Segura | 5G Numerologies Assessment for URLLC in Industrial Communications[END_REF]), which may however interfere with other parts of the spectrum under different numerologies. Zambianco et al. [START_REF] Zambianco | Mixed-Numerology Interference-Aware Spectrum Allocation for eMBB and URLLC Network Slices[END_REF] investigated Inter-Numerology Interference (INI) between eMBB and URLLC slices and proposed using a deep reinforcement learning agent to mitigate INI.

Only few studies are available for 5G NR V2X. Feng et al. [START_REF] Feng | Joint Computation Offloading and URLLC Resource Allocation for Collaborative MEC Assisted Cellular-V2X Networks[END_REF] suggests multiple methods of centralised Cloud management, such as Mobile Edge Computing (MEC), or assignment by road-side units (RSUs). Another study by Ge et al. [START_REF] Ge | Ultra-Reliable Low-Latency Communications in Autonomous Vehicular Networks[END_REF] proposes a monte-carlo approach to tackle a combined 1 source: 3GPP WI NR SL enh2 Rel.18 and beyond. optimization of reliability and latency for URLLC V2X on the millimeter wave band (FR2). These studies do not, however, consider the use of Sidelink (SL) communication or the sub-6GHz band (FR1), including the C-ITS band at 5.9GHz.

In this paper, we evaluate the feasibility and network configurations enabling URLLC performance for 5G NR V2X SL at 5.9GHz. Our contributions are as follows: (i) we evaluate the impact of the 5G NR numerology-3 in FR1 providing 0.125ms TTI, (ii) we propose and evaluate the performance of a deterministic scheduler based on orthogonal optical codes (OOC) [START_REF] Gallo | Unsupervised Long-Term Evolution Device-to-Device: A Case Study for Safety-Critical V2X Communications[END_REF] to guarantee delay bounds, (iii) we introduce an admission control mechanism restricting the number of V2X UE (VUE) to guarantee URLLC requirements. This paper provides a holistic study to evaluate the conditions under which URLLC may be supported. These conditions may later be considered as the operational mechanisms required to operate a 5G NR V2X SL URLLC slice for future stringent enhanced V2X services.

The rest of the paper is organized as follows: Section II provides background on 5G NR V2X and URLLC. Section III describes the methodology used in this study, while Section IV provides evaluation results. Section V summarizes the key findings.

II. BACKGROUND & STATE-OF-ART

A. 5G NR Sidelink in V2X Communication

Sidelink (SL) is an extension of 5G NR communication [START_REF] Garcia | A Tutorial on 5G NR V2X Communications[END_REF] to support device-to-device (D2D) communication. Defined in 3GPP since LTE rel. 12 for Proximity Services (ProSe), SL has been specified for 5G NR in rel. 16 for V2X communication. Further releases (rel.17 and 18) define SL enhancements, aiming to support more stringent requirements and operation scenarios for V2X or ProSe, such as wider coverage, reliability improvement, latency reduction or power. More detailed information can be found on 5G NR V2X in Garcia et al. [START_REF] Garcia | A Tutorial on 5G NR V2X Communications[END_REF].

5G NR SL V2X physical resources are similar to 5G NR and span across time and frequency domains. In the frequency domain, the bandwidth is split into 15kHz Physical Resource Blocks (PRB). In the time domain, a 10 ms frame is divided into 10 sub-frames, which are further divided into numerous mini-slots.

Contrary to fixed sub-carrier spacing in C-V2X, 5G NR supports a flexible frame structure with reduced transmission The TTI can be reduced from 1 ms (under numerology-0) to 0.125 ms (under numerology-3). Keeping the number of resources per slot equal, mini-slots proportionally increase the required frequency resources as depicted in Fig. 1. Accordingly, the number of slots available per 5G NR sub-frame varies according to the applied numerology.

Two MAC layer scheduling modes -mode 1 (infrastructure supported) and mode 2 (ad-hoc) -are available for 5G NR V2X SL communication. In mode 2, four enhanced submodes are further defined (as depicted in Fig. 2): in 2(a), each vehicle can select its resources autonomously based on semipersistence sensing-based scheduling; 2(b) is a cooperative distributed scheduling approach, where UEs can assist each other in resource selection, for instance by sharing their selection window with neighboring VUEs and mitigating nearfar effects; in 2(c), a VUE assigns resources for others based on (pre-)configured transmission patterns; in 2(d), a VUE schedules the SL transmissions for its neighbouring VUEs. As can be seen from Fig. 2, mode 2(a), corresponding to the default scheduling mode for 5G-NR V2X SL, does not provide any delay bound and if mode 2(b) improves reliability, it cannot provide any delay bound neither. However, modes 2(c) and 2(d) correspond to deterministic scheduling options and accordingly are expected to be prime candidates for URLLC. URLLC is defined in two aspects : reliability/delay requirements and specific traffic patterns. 3GPP rel.17 [START_REF] Nr | Study on Scenarios and Requirements for Next Generation Access Technologies[END_REF] defines URLLC as providing a 10 -5 reception reliability with a 1 ms delay for 32-byte packets. 3GPP considers adapted requirements for 5G-NR V2X SL as 10 -5 reception reliability with 3-10 ms delay for 300-byte packets.

3GPP does not provide details on how the delay should be distributed between 3-10 ms. A more suitable formulation requires defining a reliability level of the expected delay. For example, a 4 ms delay with reliability 10 -5 (i.e. a prob. 10 -5 for the delay to exceed 4 ms). Accordingly, we define in this paper V2X URLLC in the following terms: reception reliability of 10 -x and a y ms delay with reliability of 10 -z . Evaluating 5G NR V2X capabilities to provide URLLC will require that we identify the x, y and z parameters. In particular, we will focus less on a given delay value but rather on the reliability of that delay value.

V2X traffic patterns for URLLC services are expected to be significantly different from regular (awareness) services, because they involve smaller data packets, a smaller transmit range and fewer users. Default 5G NR V2X SL parameters (numerology, sub-channels, MAC protocol, etc.) therefore need to be changed to adjust to such specific traffic patterns.

Finally, as previously described, most URLLC studies for 5G-NR V2X consider a FR2 spectrum (mmWAVE) and, to the best of our knowledge, none consider SL. As various extended V2X (eV2X) services with URLLC requirements focus on SL communication at 5.9GHz (FR1), we specifically focused on it in this paper. V2X SL URLLC at 5.9GHz is challenging to meet. First, only 10 or 20 MHz of bandwidth is reserved for V2X communication, which might be insufficient if a higher numerology is involved [START_REF] Popovski | Wireless Access in Ultra-Reliable Low-Latency Communication (URLLC)[END_REF].

Second, the eV2X services cannot rely on the full availability of a 5GS infrastructure, and the default 5G-NR V2X SL mode 2 scheduler does not support time-deterministic channel access time.

C. 5G NR V2X Admission Control for URLLC

URLLC services have specific traffic patterns and in exchange, V2X URLLC services need to meet extreme reliability and delay requirements. In the current rel. 16 specification, 5G-NR V2X SL mode 2 is designed to admit any VUE, although its performance quickly degrades as a function of the number of VUEs [START_REF] Garcia | A Tutorial on 5G NR V2X Communications[END_REF]. URLLC QoS cannot be met under unrestricted access policy, and a robust admission control strategy is required to accept the optimal number of VUEs for which stringent requirements can be met.

Admission control defines maximum users that can be admitted simultaneously to the system while efficiently using the available resources and satisfying QoS requirements. Admission control is therefore central to any 5G systems and more specifically to optimizing 5G NR slices (enhanced Massive Broadband massive Machine-Type Communication (mMTC) or URLLC).

Shashika et al. [START_REF] Shashika Manosha | Admission Control Algorithms for QoS-Constrained Multicell MISO Downlink Systems[END_REF] proposed an optimization algorithm to tackle admission control optimization in a Multiple-Input Single-Output system. In [START_REF] Ginige | Admission Control in 5G Networks for the Coexistence of eMBB-URLLC Users[END_REF], Ginige et al. proposed a coexistence support for both eMBB and URLLC users in 5G-NR, and pointed out the necessity of controlling the admissions of eMBB users to facilitate scheduling for all URLLC users. 3GPP so far does not describe admission control strategies for 5G-NR V2X SL mode 2 slices. In this paper, we evaluate the optimal number of users that can be admitted into a 5G-NR V2X SL URLLC slice, consisting of a NR numerology-3 and a dedicated mode 2(c) scheduler, and guaranteeing URLLC QoS requirements.

III. METHODOLOGY

In this section, we describe the three blocks required for 5G-NR V2X SL URLLC : higher physical layer numerology, a deterministic scheduler and URLLC admission control. A. Physical Resource Arrangement in 5G NR Since a higher numerology can effectively reduce the duration of a mini-slot, which is compensated for by its expansion of the frequency domain, we therefore apply numerology-3, which enables a unit duration as small as 0.125 ms. Considering the limited V2X bandwidth in 5.9 GHz frequency, in order to meet the 300 bytes packet size specified for URLLC SL from [START_REF] Nr | Study on Scenarios and Requirements for Next Generation Access Technologies[END_REF], and to avoid the half-duplexing problem, we consider a pure time-domain multiplexing. As depicted in the right-hand image in Fig 1, considering a 10 MHz assigned V2X bandwidth under numerology-3, all 6 available PRBs are uniformly assigned to one user 2 .

Two Modulation and Coding Scheme (MCS) values are investigated: MCS-8 (1/2 QPSK), which is the default V2X modulation to achieve a data rate of 6 Mbps; MCS-24 (64 QAM), which provides a better overall performance as described in [START_REF] Yan | MCS Analysis for 5G-NR V2X Sidelink Broadcast Communication[END_REF] for numerology-0. The maximum capacity S M supported per allocation unit (i.e one mini-slot) can be approximately evaluated based on Eq. 1.

2 Although 20 MHz is also baseline of bandwidth assigned for 5G NR V2X communication, this is left for future study.

S M = 1/8 × CR × Q m × N Sym × N P RB × N sub -S h (1)
We addressed two MCS impact factors : effective code rate value (CR) and modulation order (Q m ), which are described in Tables 5.1.3.1-2 in 3GPP standard [START_REF] Etsi | 5G NR; Physical layer procedures for data[END_REF]. N sym represents the available symbols per slot for V2X traffic load (we consider 8 out of 14 symbols), N P RB indicates the total available number of PRB in the frequency domain, N sub indicates the number of subcarriers per slot, which is fixed to 12, and finally S h is a 3-byte reservation for CRC Length.

The maximum capacity under different numerology values calculated based on Eq. 1 is listed in Table I. Focusing on numerology-3 and a mini-slot of 0.125 ms, MCS-24 has a capacity capable of supporting V2X standard 300-byte packets. Yet at a lower modulation MCS-8, 32-byte packets can still be handled. We conclude that although numerology-3 is not defined for FR1 (5.9GHz) by 3GPP, it can significantly improve latency while offering sufficient traffic capacity, even with a 10 MHz V2X bandwidth.

B. Deterministic Scheduler Design

According to Section II-A, mode 2(c) and 2(d) enable potential scheduling strategies suitable for URLLC services. In this paper, we propose an Optical Orthogonal Code (OOC)based deterministic mode-2(c) scheduler supporting channel access delay bounds. OOC has been adapted to a variety of channel access technologies, in particular by Gallo et al. [START_REF] Gallo | Short paper: A LTE-direct Broadcast Mechanism for Periodic Vehicular Safety Communications[END_REF] for C-V2X. This technique improves delivery reliability by restraining the maximum cross-correlation between binary code-words among any pairs of codewords v, u, as described in Eq. 2. The OOC-based deterministic scheduling procedure is listed below.

L j=1 vj • uj ≤ λ ∀u ̸ = v (2)
Figure 3: Example of OOC Access

• A L-bit long OOC codewords transmission pattern is pre-configured while satisfying the cross-correlation constraint condition. Each bit is associated with one mini-slot (duration of 0.125 ms).

• VUEs can transmit a packet in a mini-slot when the corresponding OOC indicator is 1-bit; VUEs switch to receiving mode while the OOC indicator is 0-bit. • The transmission status is illustrated in Fig. 3, where red 1-bits refer to transmission collisions while green 1-bits indicate successful, collision-free receptions. • w decides re-transmission rate, which corresponds to the total number of 1-bits within one code-word.

C. URLLC V2X SL Admission Control Mechanism

URLLC services require to restrict access to a limited number of users. To analyse such limit, the maximum capacity for the accessible VUE needs to be analyzed. We formulate the admission control problem into an optimization equation in Eq. ( 3). The binary array K(i) indicates whether SL link i meets both reliability and latency demands, as expressed in Eq. ( 4), therefore the optimal target is to maximize SL access link connections, indicating maximum system capacity.

Constraints for the optimization problem relate to both reliability and latency URLLC service requirements. We focus on a reception reliability P e of 10 -5 , and a T d ms delay with reliability of P min t . P e is expressed in Eq. ( 5), as a function of the probability of success reception P sw of one transmit packet over w transmission attempts.

In this study, λ is set to 1 so that every pair of codewords can have at most one overlapping '1'. µ represents transmission probability which is set to '1'. A uniform random access selection is considered, therefore Eq. ( 6) can be simplified as P * sw . Moreover P T d is defined in Eq. [START_REF] Nr | Study on Scenarios and Requirements for Next Generation Access Technologies[END_REF].

Pe = (1 -Ps w ) w (5)
Ps w = µ L-w w-j λ l=0 w l L-w w-l (6)
P * sw = (1 - w L ) N -1 (7)
PT d = P * sw + ⌈ T d ×8×W L ⌉ i=0 w/L × P * sw × (1 -P * sw ) i (8)
We can see that L and w have cost-benefit trade-offs on both P e and P T d performance: longer OOC codes L are capable of guaranteeing more admitted SL links, but are inclined to allocate re-transmission slots separately, which significantly increases delay. Moreover, longer OOC codes reduce the system efficiency, particularly for periodical connections, which is relevant to URLLC service. Higher values for w promise a higher probability of successful reception, however resource exhaustion is reached more quickly.

The process of solving this admission control optimization is described in Algorithm 1, which provides the maximum number of SL users that can be admitted under each L.

Algorithm 1 Admission Control Algorithm

1: while w < 4 do 2:

while N < N max do 3:

i ← N 4:

if Pe ≤ 10 -5 and P T d ≤ P min t then 5:

k(i) ← 1 6: else 7: k(i) ← 0 8: end if 9: N ← N + 1 10:
end while 11:

return

K * w (N ) = N i=1 k(i) 12:
w ← w + 1 13: end while

IV. EVALUATION A. Simulation Environment and KPI

The mechanisms described in Section III are evaluated by Matlab simulations according to parameter settings described in Table II. All vehicles are allocated on a one-lane road scenario sending messages in broadcast to one reception vehicle 3 . A configurable number of transmitters are randomly generated according to the target 1km-wide communication density. Analysis of the impact created by channel fading, phase shifting and other physical layer criteria is left for future studies. Data traffic is generated at 10 Hz (100 ms) over the total simulation time to allow an ergodic analysis.

Table II Three key performance indicators (KPI) are selected in this paper: Overall the PRR and delay reliability both decrease as the number of admitted users increases. The analytical model over-estimates the optimal number of admitted users, as the simulation results are subject to near-far issues, where additional collisions occur from out-of-coverage users, reducing the number of admitted users.

In a higher-density scenario, near-far characteristics results in a greater impact, consequently the difference between the analytical and simulation results widens. We therefore may consider the analytical model as an optimal benchmark. Nevertheless, we may observe correlated relationships between the analytical model and simulations under different retransmission time w values: firstly a higher w can effectively improve the overall reliability suggested in Fig. 5a, we can also verify that with lower w = 1, 2, it is extremely to achieve the target reliability demand; secondly in terms of latency depicted in Fig. 5b, as additional re-transmissions produce higher delays, the probability of fulfilling the subms latency requirement is considerably reduced when more re-transmissions are generated. We can therefore identify that w = 3 achieved the optimal number of users admitted under the target URLLC requirement.

Examining the absolute delay in more depth via simulations, Fig. 6 depicts the distribution of the absolute delay, as we are interested in estimating the probability of a user being admitted under a given delay threshold. Accordingly, we rely on a kernel density estimate providing an overall evaluation of various distribution tendencies. It can be observed that the number of peaks for certain delay values is correlated with w, each peak representing the average delay of a transmission event.

A comparison between different traffic densities is made between 5 transmitters on Fig. 6a and 20 transmitters on Fig. 6b. With fewer transmitters, the delay values are more closely packed at the beginning; however, with increasing transmitters, there is a higher probability of collision at the first transmission, and success reception events are spread sporadically over time, which leads to a higher delay. Higher w values also perform better under sparse scenario, but when traffic increases, there is a higher chance of collision at the first transmission.

We can conclude that if the proposed OOC mode 2(c) scheduler provides efficient URLLC performance in specific scenarios, codewords with additional dynamic correlation properties will need to be studied to better adapt to different scenarios. In this Section, we generalize this study using a wider range of codeword lengths, and summarize results in Table III. Note that lower w values can only support an extremely limited number of SL links under URLLC requirements. We therefore only focus on w = 3 and w = 4.

Without loss of generality, we conduct simulations with a codeword length of [8 × 60, 8 × 120] (60, 120 indicating length in ms) to evaluate if an optimal codeword length can be identified. Considering L, the lower bound is the minimal requirement to support at least 2 transmitters to meet the target reliability constraint, whilst the upper bound is designed to limit one transmission per link within one frame.

A -Result values are obtained from analytical model while S -Result values are produced from simulations from scenarios described in Section IV-A.

N * represents the maximum supported SL links to meet the target reliability P e of 10 -5 , while P T d ≤ 1ms gives the corresponding probability for a successful reception within 1 ms latency. Again, the analytical results generally overestimate the simulation results, due to simulation randomness and the aforementioned near-far problem.

With higher w both analytical and simulation results show correlated benefits maximizing the supported admission SL links. However, the relative delay increases, and the benefit is accordingly not significant enough to justify an increase from w = 3 to w = 4, or higher. Overall with w = 3 and an optimal code-word length ranging from [ It must be noted that this study is based on a worst-case scenario, only considering absolute collisions, and ignoring mobility, which has a favorable impact on fading and channel coding that could mitigate the number of packet losses. It is therefore possible to expect a better performance under more realistic conditions, which we plan for future work.

The reliability of a given delay to be met is also listed in Table III, with relative delay threshold P min t of 99%, 99.9% and 99.99% (i.e. 10 -1 , 10 -2 and 10 -3 failure for exceeding the target delay). Although some of the delay times are excessively long, this can still be considered a positive result, as it shows that 5G NR V2X SL at 5.9Ghz with 10Mhz bandwidth may provide a 10 -4 reception reliability and a 10 -1 delay reliability lower than 1 ms. Such reliability is better than what can be provided by the current 5G NR V2X SL communication systems.

V. CONCLUSION

This study proposed and evaluated a solution for 5G-NR V2X Sidelink (SL) to support URLLC at 5.9GHz. Our proposed concept consists of applying a 5G NR numerology-3, a 5G-NR V2X mode 2(c) deterministic scheduler and an analytical V2X SL URLLC admission control mechanism. We demonstrated that 5G NR V2X SL can achieve a reliability of 10 -4 at a latency lower than 1 ms with reliability of 10 -1 . Although such URLLC performance is limited to a group of fewer than 10 vehicles in 300-byte packets, it is a reasonable density for a 5G-NR V2X SL URLLC service. In future work, we plan to extend our analytical admission control algorithm to include more realistic mobility and channel conditions, and apply ML/AI mechanisms to better adapt the mode 2(c) scheduler to dynamic environments.

Figure 1 :

 1 Figure 1: 5G NR Sidelink Physical Numerology time intervals (TTI), which are equal to the mini-slot duration.The TTI can be reduced from 1 ms (under numerology-0) to 0.125 ms (under numerology-3). Keeping the number of resources per slot equal, mini-slots proportionally increase the required frequency resources as depicted in Fig.1. Accordingly, the number of slots available per 5G NR sub-frame varies according to the applied numerology.Two MAC layer scheduling modes -mode 1 (infrastructure supported) and mode 2 (ad-hoc) -are available for 5G NR V2X SL communication. In mode 2, four enhanced submodes are further defined (as depicted in Fig.2): in 2(a), each vehicle can select its resources autonomously based on semipersistence sensing-based scheduling; 2(b) is a cooperative distributed scheduling approach, where UEs can assist each other in resource selection, for instance by sharing their selection window with neighboring VUEs and mitigating nearfar effects; in 2(c), a VUE assigns resources for others based on (pre-)configured transmission patterns; in 2(d), a VUE schedules the SL transmissions for its neighbouring VUEs.As can be seen from Fig.2, mode 2(a), corresponding to the default scheduling mode for 5G-NR V2X SL, does not provide any delay bound and if mode 2(b) improves reliability, it cannot provide any delay bound neither. However, modes 2(c) and 2(d) correspond to deterministic scheduling options and accordingly are expected to be prime candidates for URLLC.

Figure 2 :

 2 Figure 2: Scheduling sub-modes for 5G NR V2X SL

  s.t. Pe ≤ 10 -5 PT d ≤ P min t L ∈ [0, 150], ∀L ∈ Z + (3) k(i) = 1 if Pe ≤ 10 -5, and PT d ≤ P min t 0 otherwise.

•

  Packet Reception Rate (PRR) : This parameter is expressed in a Complementary Cumulative Distribution Function (CCDF) in relation to the number of transmitters. • Delay : Delay is considered from two dimensions as depicted in Fig 4: the absolute delay measures the time interval from the generated data packet; the relative delay only considers the exact air time including channel access delay. Without re-transmission, this value is set to one mini-slot (0.125 ms), whereas with re-transmissions this value is calculated as the time between the first successful reception and the relative starting time. • Number of controlled admissions : This indicates the maximum number of admitted V2V (SL) links when both

Figure 4 :

 4 Figure 4: Delay Illustration

Figure 5 :

 5 Figure 5: PRR and Delay Distribution Graph of L = 90 × 8Nevertheless, we may observe correlated relationships between the analytical model and simulations under different retransmission time w values: firstly a higher w can effectively improve the overall reliability suggested in Fig.5a, we can also verify that with lower w = 1, 2, it is extremely to achieve the target reliability demand; secondly in terms of latency depicted in Fig.5b, as additional re-transmissions produce higher delays, the probability of fulfilling the subms latency requirement is considerably reduced when more re-transmissions are generated. We can therefore identify that w = 3 achieved the optimal number of users admitted under the target URLLC requirement.Examining the absolute delay in more depth via simulations, Fig.6depicts the distribution of the absolute delay, as we are interested in estimating the probability of a user being admitted under a given delay threshold. Accordingly, we rely on a kernel density estimate providing an overall evaluation of various distribution tendencies. It can be observed that the

Figure 6 :

 6 Figure 6: Absolute Delay of L = 90 × 8 C. General Result Analysis

Table I :

 I Max Capacity of one mini -slot

	µ mini-slot	MCS-8	MCS-24
		Size(ms)	N PRB	S M (bytes) N PRB	S M (bytes)
	0	1	53	736.2	53	2848.7
	1	0.5	26	361.1	26	1397.5
	2	0.25	13	180.6	13	698.8
	3	0.125	6	83.3	6	322.5

  : Simulation Parameters Setting

	Parameter	Value
	Bandwidth	10 MHz
	Applied Numerology	3
	Hamming Weight w	1,2,3,4
	Transmission Range	1000 m
	Packet Rate	100 ms
	Simulation time	30 ms
	Runs	300

Table III :

 III 8 × 90, 8 × 95], General Result under different Code-word Length the investigated 5G-NR V2X SL URLLC slice may support approximately 6 users.

	Index	Code Length L	60 × 8	70 × 8	80 × 8	85 × 8	90 × 8	95 × 8	100 × 8	110 × 8	120 × 8
		A-		N *	8	9	11	11	12	13	13	14	16
		Result	P Td ≤ 1ms	0.9591	0.9596	0.9557	0.9582	0.9565	0.9550	0.9571	0.9577	0.9552
				N *	2	3	4	4	6	6	5	7	8
	w = 3	S-Result	P Td ≤ 1ms P min t ≤ 0.9 P min t ≤ 0.99	0.92 ≤ 1ms ≤ 30ms	0.971 ≤ 1ms ≤ 30ms	0.964 ≤ 1ms ≤ 35ms	0.97 ≤ 1ms ≤ 35ms ≤ 35ms ≤ 40ms 0.947 0.962 ≤ 1ms ≤ 1ms	0.985 ≤ 1ms ≤ 10ms	0.956 ≤ 1ms ≤ 30ms	0.959 ≤ 1ms ≤ 50ms
			P min t	≤ 0.999 ≤ 40ms	≤ 45ms	≤ 70ms	≤ 60ms ≤ 80ms ≤ 70ms	≤ 40ms	≤ 90ms	≤ 95ms
		A-		N *	13	15	17	18	19	20	22	24	26
		Result	P Td ≤ 1ms	0.9102	0.9094	0.9088	0.9086	0.9084	0.9082	0.9036	0.9037	0.9037
				N *	5	7	8	8	7	6	12	13	14
	w = 4	S-Result	P Td ≤ 1ms P min t ≤ 0.9 P min t ≤ 0.99	0.932 ≤ 1ms ≤ 30ms	0.906 ≤ 1ms ≤ 40ms	0.915 ≤ 1ms ≤ 40ms	0.909 ≤ 1ms ≤ 30ms ≤ 40ms ≤ 50ms 0.934 0.922 ≤ 1ms ≤ 1ms	0.886 ≤ 5ms ≤ 45ms	0.909 ≤ 1ms ≤ 50ms	0.892 ≤ 5ms ≤ 55ms
			P min t	≤ 0.999 ≤ 40ms	≤ 55ms	≤ 60ms	≤ 45ms ≤ 65ms ≤ 70ms	≤ 75ms	≤ 80ms	≤ 90ms

The broadcast transmission is intended to all vehicles, but we only consider 1 receiving vehicle for analysis purpose.

ACKNOWLEDGMENT

This work has received funding from the European Union's Horizon 2020 R&D program under grant agreement No: 101015405 for Safe4RAIL-3 (https://safe4rail-3.eu/) and No: 957218 for Intelliot (https://intelliot.eu/).