Normann Rion
email: normann.rion@psemail.eu

CHAPTER 2. WAITING FOR THE PRINCE CHARMING

Keywords: JEL Classification: J41, J64, J38, J68 Fixed-term Contracts, Unemployment, Employment Protection, Policy, Dynamics Chapter 3 JEL Classification: J42, J64, E31, E32, D81 Fixed-term contracts, Employment protection, New Keynesian model, Inflation dynamics, Uncertainty

In this paper, I build a simple Mortensen-Pissarides model embedding a dual labor market. I derive conditions for the existence of an equilibrium with both protected open-ended contracts and exogenously short fixed-term contracts. I also study dynamics after a change in firing costs. Fixed-term contracts play the role of fillers while open-ended contracts are used to lock up most productive matches. High firing costs favor the emergence of a dual equilibrium and encourage the resort to fixed-term employment in job creation. This substitution scheme is intertwined with a generalequilibrium effect. Open-ended contracts represent the bulk of employed workers; higher firing costs reduces aggregate job destruction, which pushes down unemployment and in turn reduces job creation flows through fixed-term contracts. I calibrate the model on the French labor market.

Policy experiments demonstrate that there is no joint gain in employment and social welfare through changes on firing costs around the baseline economy. Welfare improving policies consist in large cuts in firing costs, where fixed-term employment eventually disappears. Increases in firing costs within a dual labor market lead to a sluggish adjustment, while large cuts in firing costs lead to a quicker one. The adjustment time is highly non-monotonous between these two extremes. Uncertainty over firing costs significantly strengthens fixed-term employment on behalf of open-ended employment.

I have received a great deal of support throughout the writing of my dissertation.

I am grateful to the Ecole Normale Supérieure, the Paris School of economics, the doctoral school 465 Économie Panthéon Sorbonne and the laboratory Paris-Jourdan Sciences Économiques for hosting my research in comfortable conditions. I would especially like to thank Véronique Guillotin for her everyday administrative and human support.

I would like to thank my supervisor Gilles Saint-Paul for his insightful guidance and availability. I could not have imagined a better mentor for my Ph.D., leaving me utterly free to explore ideas and providing enlightening advice whenever solicited.

I would like to acknowledge the members of my thesis committee Axelle Ferrière and François Langot for their useful comments and patience. I really appreciated the endeavor you involved in providing feedback as well as the great quality of our conversations.

I would like to express gratitude to the members of the jury Hélène Turon, Claudio Michelacci and Jean-Baptiste Michau for carefully reviewing my work despite the tight schedule.

I would like to extend my sincere thanks to Pablo Winant for his encouragements and guidance. Our collaboration on dolo taught me a lot. I would like to thank my colleagues of the R4-64 office Yassine Kirat, Mehdi Bartal, Can Askan Mavi, Emanuele Franceschi, Jaime Levya, Leonard Bocquet, Yagan Hazard and Simon Löwe for bearing my politically incorrect jokes from day to day. At the end of the 1980s, the economic downturn following the oil shocks lead firms to ask for looser firing rules. Consequently, many European countries enforced policies to make labor markets more flexible. Fixed-term contracts were thus introduced in France in 1979.

Labor contracts and the law

In France, the law regulates many aspects of the relationship between employers and workers. Ray (2020) describes its highways and byways.

Open-ended contracts cover most employees. Open-ended contracts include a probationaryemployment period of up to 4 months. Both the employer and the worker may costlessly dissolve the agreement during this time. Once the probationary period has come to an end, the employer may invoke economic or personal motives for redundancies, which they should be able to justify on court if the involved worker challenges the employer's decision. Economic redundancies compel the employer to demonstrate that reducing employment is the only available solution, while personal redundancies impose showing that the worker cannot fulfill the job he was hired for. Blatant misconducts excepted, the firm has to notify the worker before laying him off. The worker carries on in his position until the notification delay expires. For example, a worker with a seniority between 6 months and 2 years has to be notified at least 1 month before being dismissed. The notification delay amounts to 2 months if seniority exceeds 2 years.

Employer-initiated terminations and retirements confer severance payments to the involved workers.

Severance payments are difficult to assess as they depend on agreements reached through sectorlevel collective bargaining, firm-level collective bargaining or individual bargaining. Labor law sets a minimum level rule for severance payment, though. The severance payment cannot be lower than 1/4th of the monthly reference wage per year for the first 10 years of seniority and 1/3rd of the monthly reference wage per year beyond these first 10 years. A generous formula pinpoints the reference wage, which is the highest amount between last year's average monthly wage, or one third of last quarter's total wages with 1/12th of bonuses if any. Note that collective and individual redundancies do not abide by the same rules. Firms with more than 50 employees that dismiss more than 10 workers in a month have to formulate a specific plan, which sets a limit in the number of redundancies and offers specific training or job proposals to ease the fired workers' reclassification. As workers or unions can question their dismissal in court, firing costs extend far beyond the minimum severance payment labor law spells out. Abowd and [START_REF] John | The costs of hiring and separations[END_REF] and [START_REF] Kramarz | The shape of hiring and separation costs in france[END_REF] notably estimate firing costs for French firms. Their results confirm that the actual cost of terminations is much larger than the minimal one labor law delineates as severance payments. [START_REF] Yazidi | Le traitement des litiges en droit du travail : constats et perspective économiques[END_REF] and [START_REF] Fraisse | Labor disputes and job flows[END_REF] quantify the extent of legal actions in front of labor courts in France. In 2013, 60% of disputes are challenged in appeal, which is far higher than in other types of courts.

On the workers' side, an employee can quit its open-ended job at the expense of an advance notice delay. Quitting workers cannot get unemployment benefits or severance payments in contrast with dismissed workers. An employer-worker pair can also bargain on termination conditions if both parties want to end the contractual relationship. In this case, the employer has to compensate the worker for at least the minimum legal severance payment. The worker is better off than quitting as he can get unemployment benefits as well as a severance payment once laid off. The employer may avoid time-consuming and expensive legal actions classic redundancies entail. Note that the procedure must be validated by an administrative authority. After its approval, the worker can challenge the reached agreement for one year. Thus, negotiated terminations do not utterly prevent labor court judges from taking action into redundancies.

Nowadays, quantitatively speaking, quits represent roughly 40% of terminations. 20% of terminations occur at the end of the probationary-employment period. Terminations involving personal motives amount to 16% of all redundancies, whereas economic terminations represent only 3% of terminations. Negotiated terminations cover 10% of redundancies1 .

In legal terms, open-ended contracts are considered the normal form of any labor relationship 2 .

Thus, employers must justify the resort to fixed-term contracts. Legitimate reasons include replacing an ill worker, workload fluctuations and dealing with seasonal activities 3 . Fixed-term contracts can generally last up to 18 months, including up to two renewals. Beyond this maximum duration, both parties need to sign an open-ended contract to carry on the labor relationship. [START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF] reports that the average duration of fixed-term contracts was 46 days in 2017. The distribution of duration is strongly skewed to the right: 25% of fixed-term contracts last less than 5 days, 75% less than 27 days. At the end of the contract, the firm pays the worker a precariousness severance payment amounting to 10% of the total gross wages paid during the contractual relationship. Theoretically, the law punishes the abusive resort to fixed-term contracts, which may may lead the involved employer to redefine the contract as open-ended and pay damages to the worker.

In general, fixed-term contracts cannot end before their stipulated termination date at zero cost. The requester exposes himself to paying a compensation to the other party. Labor courts decide on each party's potential misconducts. A few exceptions arise, though. Beyond exceptional situations, the firm and the worker can costlessly repeal the contract by mutual agreement. Moreover, a fixed-term worker can freely leave his current position if he signs an open-ended contract. In this case, the worker has to notice his employer at most 2 weeks before leaving. Fixed-term contracts scarcely end before their stipulated termination date. According to [START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF], only 2% of fixed-term contracts terminated prematurely in 2017.

Nevertheless, the term "fixed-term contract" covers a myriad of different types of contracts. For example, some fixed-term contracts called contrats d'usage significantly contribute to the shortening of fixed-term contracts, especially in the service sector. A few industries may use these contracts, which include no constraints on renewals or precariousness severance payment. As stated above, the law chastises the abusive resort to contrats d'usage. However, the worker has little incentive to prosecute a malicious employer. Such a legal action negatively impacts the worker's reputation towards other employers in the same industry, which in turn harms the worker's future employment prospects.

Interim work may also be considered as fixed-term employment as it is legally constrained as well.

At the blurry frontier of fixed-term employment, self-employed workers who ensure temporary missions for very few customers can also be considered. These workers do not benefit from the same legal protection as employees, which may encourage their use. Expanding digital platforms resort a lot to self-employed workers. Note that this specific form of temporary employment also covers countries where fixed-term contracts do not legally exist.

Employment protection and the macro-economy 1.2.1 Stylized facts

As I have shown above, French labor law supposedly constrains the resort to fixed-term contracts.

However, it has not prevented fixed-term employment from expanding. Figure 1.1 shows the proportion of fixed-term contracts among employees. The share of fixed-term workers has steeply increased, from 6% of employees in 1982 to nearly 13% in 2019. Hires essentially occur through fixed-term contracts.

Figure 1.2 shows the share of each type of contract in job creation, distinguishing fixed-term contracts of more than one month and less than one month. Fixed-term contracts are shorter and shorter; the share of fixed-term contracts of less than one month in job creation has increased from nearly 50% in 2000 up to 70% now. [START_REF] Fontaine | CDD vs CDI: les effets d'un dualisme contractuel[END_REF] contends that it is specifically occurring in France and not necessarily in other European countries. The share of fixed-term contracts of more than one month has decreased from 30% in 2000 down to 17% now. Nowadays, fixed-term contracts represent 85% of hires. [START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF] provides further details about the resort to fixed-term contracts. Some sectors particularly contribute to the expansion of fixed-term contracts lasting less than one month such as residential care activities, the audio-visual industry and catering.

Insee (2019) reports that fixed-term contracts notably affect young workers. In 2019, fixed-term and interim employment cover 35% of workers aged 15 to 24. This share falls to 10% for workers aged 24 to 49 and to 6% for workers aged at least 50. Using cross-country data, OECD (2006) finds that stringent employment protection reduces youth employment whereas it advantages older workers. The role of fixed-term contracts as an entry point towards open-ended employment has been extensively discussed in the literature4 . Fixed-term employment does not clearly facilitate transition towards open-ended employment when compared to unemployment. In France, [START_REF] Givord | When does the stepping-stone work? fixed-term contracts versus temporary agency work in changing economic conditions[END_REF] shows that 1/10th of fixed-term contracts quarterly transit into open-ended contracts.

In countries with segmented labor markets, fixed-term workers benefit less from on-the-job training than open-ended contracts5 . Firms do not have incentives to invest in training for short-term workers. [START_REF] Hagen | Do temporary workers receive risk premiums? assessing the wage effects of fixedterm contracts in west germany by a matching estimator compared with parametric approaches[END_REF], [START_REF] Booth | Temporary Jobs: Stepping Stones or Dead Ends?[END_REF] and [START_REF] Pfeifer | Fixed-term contracts and wages revisited using linked employer-employee data[END_REF][START_REF] Pfeifer | A note on dual internal labor markets and wages of temporary workers: Evidence from linked-employer-employee data[END_REF] estimate the contractual wage gap using British, Spanish and German panel data. All else equal, fixed-term workers earn a few per cent less compared to open-ended workers.

Fixed-term employment vary in prominence across Europe. Figure 1.3 shows the share of fixed-term contracts in dependent employment in various European countries. In 2019, fixed-term employment is particularly strong in Spain, Poland, Portugal and Netherlands, where it represents more than 20% of dependent employment. At the other end of the spectrum, fixed-term employment represent less than 5% of dependent employment in Great Britain, Bulgaria and Baltic countries.

European countries vary much in their employment protection, which makes international comparisons difficult 6 . The OECD built a widely used indicator to sum up the stringency of employment protection. As OECD (2020b) describes, the indicator takes into account four dimensions: (i) procedural requirements before notice to the open-ended employee; (ii) notice period and severance 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 2 0 1 7 2 0 1 8 2 0 1 9

Employment protection and the labor market

The impact of employment protection on labor markets, especially unemployment, has been extensively discussed in the literature. The raise of a high and persistent unemployment in Western Europe stirred up the interest for this issue. [START_REF] Michael | Perspectives on lobor market flexibility[END_REF] tection legislations to the situation in some European countires and contends that both are rigid in different ways. Still, many academics point at high severance payments as responsible for the sticky European unemployment rates. [START_REF] Olivier | Hysteresis and the european unemployment problem[END_REF] describe the perverse role of stringent employment protection legislation in this hysteresis phenomenon. Companies are reluctant to hire, out of fear for future firing costs. The important severance payments provide the in-place employees with a stronger bargaining position. In-place employees decide wages such that it is not profitable to fire them. The consequent wage-setting rules are persistent and so is the unemployment rate. [START_REF] Galí | Hysteresis and the european unemployment problem revisited[END_REF] estimate that this hypothesis alone could not account for persistent unemployment rates.

Employment protection legislation also has macroeconomic consequences in terms of employment volatility. [START_REF] Blanchard | The Perverse Effects of Partial Labour Market Reform: FixedTerm Contracts in France[END_REF] builds up a matching model with exogenous firing costs.

New matches deliver a low productivity and come with a low firing cost. Firms can either change the expired entry-level jobs into regular jobs with an higher expected productivity and enlarged firing costs, or fire the worker and reopen a vacancy. Entry-level jobs embody fixed-term contracts, which are prominent among the young. Firms assesses the quality of matches before signing regular contracts and do not entrust new employees with many critical tasks, which explains why entry-level jobs are not as productive as regular jobs. A trade-off appears when entry-level jobs expire: transformations into regular contracts may deliver a higher productivity, but the nature of the contract prevents from looking for matches with a higher quality. As the firing costs associated with regular contracts increase, firms are more reluctant to keep entry-level workers into regular jobs as the regular wages are higher. The demanded productivity of a regular job rises as it is cheaper to try out new matches. Firms resort to entry-level jobs more frequently, which lowers the average productivity and output in the economy. Introducing temporary contracts may thus increase unemployment and decrease the outsiders' welfare. [START_REF] Edward P Lazear | Job security provisions and employment[END_REF] estimates the impact of severance payments on aggregate labor market variables using cross-country differences and finds that higher severance payments reduce employment. Bentolila et al. (2012b) calibrates a matching model of dual labor markets to both France and Spain. It finds that Spanish looser constraints on the resort to fixed-term employment as well as stronger employment protection of regular contracts account for the post-2008-crisis dramatic increase in unemployment compared to the French situation.

Such clear-cut conclusions about the impact of employment protection on employment is not wide-spread, though. Employment protection both reduces job creation and job destruction flows, which ambiguously impacts unemployment. [START_REF] Boeri | Institutional Reforms and Dualism in European Labor Markets[END_REF] reviews the literature and confirms the inconclusiveness of empirical work in that matter. Theoretical work is on the same page8 . This conclusion has already been advertised in the literature before segmented labor markets were considered. Many papers have modeled labor demand when firms face labor adjustment costs9 .

Overall, these models conclude that firing costs decrease both job creation in good times and job destruction in bad times. The impact of firing costs on unemployment is thus unclear.

The literature has also considered the impact of employment protection on productivity. OECD (2008) finds a small negative impact of employment protection legislation on long-run productivity growth using cross-country data. Employment protection prevents workers from moving into more productive activities. Overall, OECD (2010) shows that employment protection has a negative effect on labor reallocation. Variety in employment protection explains between 20 and 30% of cross-country differences in workers reallocation rates. Using cross-country data, [START_REF] Bassanini | Job protection legislation and productivity growth in OECD countries[END_REF] shows that employment protection reduces productivity growth in industries with many openended contracts. Employment protection does not impact the productivity of fixed-term workers, though. As for the impact of growth on fixed-term employment, [START_REF] Wasmer | Competition for jobs in a growing economy and the emergence of dualism[END_REF] builds a matching model and shows that a slowdown in growth increases the share of fixed-term employment in total employment.

In terms of labor market dynamics, [START_REF] Bentolila | The macroeconomic impact of flexible labor contracts, with an application to spain[END_REF] builds a partial-equilibrium dynamic model to account for the effects of temporary contracts introduction. It compares labor market dynamics in a models involving open-ended contracts with and without fixed-term contracts as an alternative. If finds that aggregate fluctuations in employment are larger with fixed-term contracts while less persistent. Open-ended employees are costly to fire in opposition to fixed-term ones. Firms just need to wait for the expiry of fixed-term contracts to get rid of them in bad times.

Thus, firms have an incentive to hire open-ended workers who will never be fired and face all demand fluctuations with fixed-term contracts. When a negative shock occurs, the firm separates from all temporary workers before considering laying off permanent workers. The on-shock response of employment is thus stronger. When the situation improves, the firm can hire fixed-term workers to meet its orders: employment gets back faster to its initial value. The insider-outsider divide lies in this dynamic scheme. Permanent workers are never laid off and benefit from a strong bargaining position whereas temporary workers face successive periods of unemployment separated by short-time jobs. [START_REF] Saint-Paul | Dual Labor Markets: A Macroeconomic Perspective[END_REF] studies in depth this phenomenon from a theoretical point of view.

The fact that employment protection stabilizes open-ended employment and exacerbates fluctuations of fixed-term employment is a well-known result in the literature. [START_REF] Sala | Flexibility at the margin and labor market volatility in oecd countries*[END_REF] builds a discrete time model to study business-cycle features of dual labor markets and finds that introducing fixed-term contracts increases unemployment volatility. [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF] also highlights the excessive turnover on the fixed-term side of the labor market.

The contractual choice at the hiring stage

Job creation flows substantially contribute to unemployment fluctuations10 . As we have seen, flows highly differ across contract types. Thus, in segmented labor markets, the choice between hiring through fixed-term or open-ended contracts matters a lot to unemployment fluctuations.

In empirical terms, [START_REF] Rémy | Pourquoi les employeurs choisissent-ils d'embaucher en cdd plutôt qu'en cdi ? Dares analyses[END_REF] surveys French employers' motives for hiring fixed-term contracts in 2017: 69% of employers state that they hire through fixed-term contracts because of limited-time needs; 65% refer to a way of assessing workers' skills; 56% cite the habit of hiring fixed-term contracts for specific jobs as well as avoiding risks associated with future downturns; 45% mention avoiding employment protection rules associated with open-ended employment.

Some opportunities of production may be too short to be exploited through permanent contracts but adapted to the durations of temporary ones. [START_REF] Dräger | Do firms demand temporary workers when they face workload fluctuation? cross-country firm-level evidence[END_REF] carries out empirical analyses over a firm-level data including 20 countries. They demonstrate that workload fluctuations explain a significant share of the probability to hire a temporary worker. Interestingly, this result limits to rigid labor markets. The introduction of temporary contracts changes the way firms deal with workload fluctuations. [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF] builds a Mortensen-Pissarides model with heterogeneous arrival rates in adverse productivity shocks. When adverse productivity shocks are frequent, fixed-term contracts are preferable. On the contrary, open-ended contracts cover matches with low arrival rates of adverse productivity shocks.

Faccini (2014) explore the screening motive for hiring fixed-term workers 11 . It builds a randomsearch model where matches have either a bad or high productivity. Matches perceive the productivity of the match with some noise. Fixed-term contracts provide some time to discover the productivity of the job and may costlessly split if the quality of the match is low. An open-ended contract incurs firing costs with such a termination. Thus, the screening motive encourages the use of fixed-term contracts. Screening cannot account for most job creations in France. Indeed, 75% of fixed-term contracts last less than 27 days, whereas the probationary employment of open-ended contracts lasts up to 4 months. Fixed-term employment as a screening device is credible for highly complex jobs; a few months of probationary employment may not be enough to accurately assess the workers' skills.

Financial constraints may also impact the contractual composition of hires. [START_REF] Caggese | Financing constraints and fixed-term employment contracts*[END_REF] Heterogeneous job-finding and job-filling rates could also be a criterion to rank contracts in vacancy posting and job search activities. [START_REF] Berton | Workers and firms sorting into temporary jobs*[END_REF] models a segmented labor market with directed search. Open-ended contracts stipulate firing costs, whereas fixed-term matches can split any time at zero cost. Workers are heterogeneous in their unemployment utility flow.

Workers with the lowest unemployment utility flows are ready to accept contracts that yield a lower surplus, which are fixed-term contracts in this case. On the contrary, fixed-term contracts do not inspire workers with high unemployment utility flows, who prefer open-ended contracts. At the equilibrium, workers searching on the fixed-term side are compensated for the lower surplus with a high job-finding rate, whereas workers looking for open-ended jobs experience a low job-finding rate. Overall, workers that experience heterogeneous outside options rank contracts with respect to the associated job-finding rates, which enables segmented labor markets to emerge as equilibria.

Fixed-term contracts may also serve as a waiting device for firms before they find a suitable worker. [START_REF] Smith | Limited duration employment[END_REF] builds a matching model with separate pools of workers. Firms pay a fee to review candidates in a given pool of worker. If a great worker is in the pool, a contract with a long duration is chosen to lock up the match. On the contrary, if the quality of matches is too low, the firm may have an interest to opt for a short contract. It may then review the workers' pool once the contract is over. New workers may have regenerated the candidates' pool.

Reforming dual labor markets

Academics have proposed many reforms to improve the functioning of segmented labor markets. I follow the review of [START_REF] Fontaine | CDD vs CDI: les effets d'un dualisme contractuel[END_REF] in that regard. I do not consider 2013's French reform and Italian Jobs act it is beyond the scope of the issues my dissertation deals with. I discuss some ingredients of these reforms below.

The social cost of expanding fixed-term employment arouses great interest in terms of policy action.

Some academics call for targeted policies for the low-skilled and the young, who are more prone to experience fixed-term contracts. For example, [START_REF] Dolado | A positive analysis of targeted employment protection legislation. The B.E[END_REF] Some academics also call for a transition towards a unique contract. Such an approach would make labor law simpler. However, as [START_REF] Lepage-Saucier | 14. le contrat de travail unique: quid pro quo ou simple quiproquo? Regards croises sur l'economie[END_REF] and [START_REF] Barthelemy | Le contrat unique[END_REF] state, the unique contract is a holdall term that gathers a variety of policies. According to a first definition, the unique contract would gradually provide for higher severance payments with seniority. Under a second definition, the probationary employment time would be longer. [START_REF] Siassi | Moving Towards a Single Labor Contract: Transition vs[END_REF] analyzes the optimal way of implementing such a unique contract under the two scenarii above. It also considers transitional and steady-state winners and losers of such a reform.

Complementary proposals cover the funding of unemployment benefits. In France, firms fund unemployment benefits through payroll taxes, which do not depend on each firm's use of fixed-term contracts. However, fixed-term workers frequently oscillate between short employment periods and unemployment. Thus, the funding of unemployment benefits indirectly encourages firms to resort to fixed-term employment. In the United States, an experience rating system was adopted;

the federal government changes contribution rates considering each firm's balance with respect to the unemployment benefit system. In this manner, it encourages firms to ensure long-lasting jobs.13

Contributions

As far as labor is concerned, policymakers' activism has been mind-boggling over the last decades. Open-ended contracts stipulate a red-tape firing cost, whereas fixed-term contracts have a settled average duration. New firm-worker pairs choose the contract that maximizes their joint surplus depending on their productivity.

In this framework, I find that most productive matches end up in open-ended contracts. Most productive matches are ready to pay for future firing costs to make the most out of their current good productivity draw; fixed-term contracts may be too short to do so. A firm-worker pair with a mediocre productivity is better off producing for a while and go back hunting for high productivity into open-ended contracts, which is counter-factual. In models with heterogeneous processes for workload fluctuations, the latter counter-factual result arises as well if the productivity of new matches is no longer fixed. Thus, a successful model of job creation in a dual labor market would craftily navigate between the pros and cons of the flexible and rigid approaches.

The three chapters detailed above follow. Then, I conclude.

Chapter 2

Waiting for the Prince Charming:

Fixed-Term Contracts as Stopgaps

Introduction

Since the end of the 1980s and the introduction of fixed-term contracts, the segmentation of European labor markets, opposing strongly protected open-ended contracts and short fixed-term contracts, has worsened. In France, fixed-term contracts represented more than 80 % of created jobs in 2018. Academics often point at firing costs as encouraging the resort to fixed-term contracts.

Should a dual labor market be reformed? Which post-reform dynamic behavior should we expect?

To address these questions, I extend the classic model of [START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF] to include

both open-ended and fixed-term contracts. My contribution includes theoretical and numerical insights. I design the model to be the most tractable possible allowing an endogenous choice between open-ended contracts and fixed-term contracts at the hiring stage, which facilitates its integration into more complex frameworks, as is done in my third chapter. In numerical terms, I analyze the dynamic response of the dual labor market following reforms on employment protection legislation, which, as far as I know, has not been done yet. Overall, the resulting model provides rich theoretical insights about the economic schemes at stake in a dual labor market. It also provides results on the desirability of employment protection legislation reforms in terms of welfare and employment. The numerical results I present must be taken with a grain of salt considering the assumptions made to ensure tractability: productivity shocks are i.i.d, agents are risk neutral and there is no on-the-job search.

At the equilibrium, when a firm-worker pair meets and draws a high productivity, the best way to maximize benefits consists in locking up the pair through an open-ended contract. Otherwise, agents can commit to a short productive relationship through a fixed-term contract or go back to searching for a better match on the labor market. The meeting process being costly, agents opt for a fixed-term 2.1. INTRODUCTION contract if the productivity draw is not too disappointing. Thus, fixed-term contracts provide a productive interlude, while enabling to go back to seeking a high-productivity match before long:

fixed-term contracts act as fillers before meeting the Prince Charming2 .

I find that two opposed effects mix up to shape the behavior of the labor market with respect to firing costs. The first effect is a substitution effect towards fixed-term employment as firing costs increase.

Higher firing costs make open-ended contracts more rigid and fosters the resort to less productive fixed-term contracts for the sake of flexibility. Thus, higher firing costs provide an incentive to hire Secondly, it delineates the mechanisms underpinning the choice between fixed-term and open-ended contracts4 . [START_REF] Smith | Limited duration employment[END_REF] has first highlighted the role of fixed-term contracts as stopgaps. In his stock-flow matching model, firms hire poorly productive workers on limited duration to be ready to hire highly productive workers when they appear. Its framework is very different, though; search is directed and labor market institutions are not explicitly modeled.

Thirdly, my paper contributes to the literature assessing policy measures using structural models of the labor market. The closest papers in that regard are [START_REF] Jose Dolado | From dual to unified employment protection: Transition and steady state[END_REF] and [START_REF] Cahuc | Taxation of Temporary Jobs: Good Intentions with Bad Outcomes?[END_REF]. [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF][START_REF] Cahuc | Taxation of Temporary Jobs: Good Intentions with Bad Outcomes?[END_REF] [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF][START_REF] Cahuc | Taxation of Temporary Jobs: Good Intentions with Bad Outcomes?[END_REF] in several aspects; new matches are heterogeneous in productivity and not in productivity shock arrival rate. Moreover, I consider transitions between states while [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF][START_REF] Cahuc | Taxation of Temporary Jobs: Good Intentions with Bad Outcomes?[END_REF] limit to steady-state analyses.

2.1. INTRODUCTION [START_REF] Siassi | Moving Towards a Single Labor Contract: Transition vs[END_REF] is the closest paper in terms of ambitions when it comes to studying post-reform dynamics, but it is rather a complement than a substitute. [START_REF] Siassi | Moving Towards a Single Labor Contract: Transition vs[END_REF] builds a discrete-time stochastic life-cycle model where workers are risk-averse. One-job firms and workers meet according to a standard matching function. When matches split, firms pay severance payments to the workers that enables them to smooth consumption while unemployed. The severance payment is a function of the worker's tenure. The model is calibrated on Spanish data with the 2008's severance payment scheme. The authors derive the optimal linear severance scheme with respect new entrants' utility.

They find that the actual severance payment scheme is too generous and increases unemployment.

The authors then derive transitions from the actual severance payment scheme towards the optimal one.

I depart from [START_REF] Siassi | Moving Towards a Single Labor Contract: Transition vs[END_REF] in tree important dimensions. First, I model labor both fixed-term and open-ended contracts explicitly. In [START_REF] Siassi | Moving Towards a Single Labor Contract: Transition vs[END_REF], the workers with low-tenure workers stand for fixed-term contracts while high-tenure workers embody open-ended contracts. This approach overlooks the rigidity of fixed-term contracts before their stipulated end date. Secondly, I focus more on theoretical mechanisms than numerical results. [START_REF] Siassi | Moving Towards a Single Labor Contract: Transition vs[END_REF] uses computerintensive methods to derive equilibria and transitions. The transition paths I derive have an analytic expression, which provides a transparent appraisal of the phenomena at stake. Thirdly, I consider redtape firing costs instead of severance payments. Severance payments are a pure transfer between the firm and the worker. Thus, they do not impact jointly efficient job destruction decisions5 . Severance payments indirectly impact job creation through the stronger workers' bargaining position and, thus, higher wages. Note that the impact of severance payments can be offset through a bargaining process with full commitment6 . On the contrary, red-tape firing costs impact directly the surplus of open-ended matches and distort job destruction decisions. They also push up wages, which alters job creation incentives. Unlike severance payments, a full-commitment bargaining of wages before the hiring decision cannot offset the distortion red-tape firing costs induce. Beyond the fact that red-tape firing costs have an allocative role in contrast with severance payments, red-tape firing costs are substantial. [START_REF] Kramarz | The shape of hiring and separation costs in france[END_REF] shows that the sole law-prescribed severance payment understates the extent of termination costs.

THE MODEL CHAPTER 2. WAITING FOR THE PRINCE CHARMING

The paper is structured as follows. Section 2 describes the model and derives the main theoretical results. In Section 3, I calibrate the model on French data, study employment protection legislation reforms, the associated post-reform dynamics and the impact of regulatory uncertainty. Section 4 concludes.

The model

In this model a-la [START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF] The matching function is standard, with constant returns to scale over the number of vacancies v and the number of job seekers e. The number of matches per unit of time is denoted m(e, v), while the job market tightness θ is classically set as the number of vacancies over the number of job seekers θ = v/e. The constant-return-to-scale feature of the matching function entails that the job-meeting probability p = m(e, v)/e and the seeker-meeting probability q = m(e, v)/v only depend on the labor market tightness. q is a non-increasing function, whereas p is a non-decreasing one. Both q and p are convex. When paired with a worker, firms can hire through a fixed-term contract or hire through a open-ended contract. They are also able to resume searching for a worker if they are not satisfied with the productivity of the match.

As in the classic Mortensen-Pissarides model with Nash bargaining, the surplus of matches entirely determine hiring and firing decisions. Joint surpluses are defined as the sum of workers' and firms'

surpluses. The surplus of a firm with a continuing open-ended contract with productivity z is

J p (z) -(V -F ) = J p (z) -V + F ,
S f (z) = J f (z) -V + W f (z) -U S p (z) = (J p (z) -[V -F ]) + (W p (z) -U ) S p 0 (z) = (J p 0 (z) -V ) + (W p 0 (z) -U )
Wages are determined following a Nash bargaining. I mark down the workers' bargaining power as η. It is common to both open-ended and fixed-term contracts. Indeed, analyzing the role of employment protection legislation implies shutting down all differences between open-ended and fixed-term contracts, with the exception of genuinely legal ones.

J p (z) -(V -F ) = (1 -η) S p (z) (2.1) J p 0 (z) -V = (1 -η) S p 0 (z) (2.2) J f (z) -V = (1 -η) S f (z) (2.3)
Nash-bargaining makes endogenous separations as well as hiring decisions jointly efficient. In other words, there is no conflict over the hiring and firing choices between firms and workers. The formulas above enable the computation of the workers' rents.

W p (z) = U + η 1 -η (J p (z) -V ) + η 1 -η F W p 0 (z) = U + η 1 -η (J p 0 (z) -V ) W f (z) = U + η 1 -η J f (z) -V
Workers appropriate a fraction of the firms' surplus because of the sunk hiring costs firms pawn. The matching procedure can be considered as a production function with two inputs : job seekers and vacancies. The involvement of firms through recruiting costs in this process before any production takes place generates a hold-up situation favoring workers at the moment of wage bargaining

The agents' value functions

Vacancies and unemployed workers A firm-worker contact occurs with probability q (θ) per unit of time. The cost of a vacancy is γ per unit of time regardless of the contract type. When the idiosyncratic productivity of the match reveals, the firm chooses between hiring the worker through an open-ended contract or a fixed-term contract and letting the worker go back into the unemployed's pool.

rV = -γ + q (θ) max J p 0 (z) -V, J f (z) -V, 0 dG(z) (2.4)
Firing costs only apply if the match is validated in the first place. The potentially ephemeral constitution of a match does not boil down to the payment of firing costs if immediate separation is preferable. Thus, the role of firing costs is not unrealistically exacerbated in job creation.

The unemployed workers' value function embeds the unemployment benefit b and the various possibilities stemming from the eventual contact with a firm, which occurs with probability p(θ) per unit of time.

rU = b + p(θ) max W p 0 z -U, W f z -U, 0 dG z (2.5)
Open-ended contracts The firm's capital value of a continuing open-ended match consists in an immediate profit from production z net of the worker's wage w p (z). When the match exogenously separates with probability s, the firm goes back to searching a worker without paying the firing cost. In real life, this probability is associated with events such as quits, retirements or probationary employment terminations. In this case, firms do not have to pay firing costs. The firm may also face a productivity shock with probability λ and assess whether it keeps or lays off the worker regarding the new idiosyncratic productivity of the match. Firms laying off workers because of adverse productivity shocks have to pay firing costs.

rJ p (z) = z -w p (z) + s (V -J p (z)) + λ max J p z , V -F -J p (z) dG z (2.6)
J p being increasing in the idiosyncratic productivity of the match, there exists a job destruction

margin for continuing open-ended contracts z p defined as

J p (z p ) = V -F
The subsequent probability of separation for open-ended contracts is

ξ = s + λG (z p ) (2.7)
The workers' value function when he is under a continuing open-ended contract firstly consists in the wage w p . The different alternatives mentioned above apply.

rW p (z) = w p (z) + λ max W p z , U -W p (z) dG z + s (U -W p (z)) (2.8)
The

rJ p 0 (z) = z -w p 0 (z) + λ max J p z , V -F -J p 0 (z) dG z + s (V -J p 0 (z)) (2.9) rW p 0 (z) = w p 0 (z) + λ max W p z , U -W p 0 (z) dG z + s (U -W p 0 (z)) (2.10)
Note that new open-ended matches become continuing open-ended matches after the first productivity shock. It is counter-factual as the probationary employment period is legally prescribed. In France, as far as open-ended contracts are concerned, probationary employment lasts at most 4 months, 6 months and 8 months for employees, supervisors and executives respectively. Fixed-term contracts that last more than 6 months generally have a 1-month probationary employment. As for fixed-term contracts that last less than 6 months, probationary employment lasts at most 2 weeks. Probationary employment enables both the firm and the worker to terminate the contractual relationship at zero cost. Theoretically speaking, the end of a contract during the probationary employment needs not to be justified. However, it is subject to an advance notice and a worker may legally challenge it and claim for compensation if he suspects his employer's decision is not bound to his lack of skill. Economic reasons cannot justify an end of contract during probationary employment, for example.

A match in probationary employment learning about its disappointing productivity may choose to split. In my framework, new firm-worker pairs immediately learn about their productivity.

The probationary employment period is skipped in this case. I include probationary employment terminations in the costless separation rate s. Introducing uncertainty over the productivity of the match and some learning effects over it would go beyond the scope of this paper. [START_REF] Bucher | Hiring Practices, Employment Protection and Temporary Jobs[END_REF] and [START_REF] Faccini | Reassessing labour market reforms: Temporary contracts as a screening device[END_REF] do so in a dual labor market.

Fixed-term contracts

Similarly to open-ended contracts, the firm employing a fixed-term worker gets an immediate profit from a match with productivity z. It also pays a wage w f (z). The match may split if the stipulated termination date is reached, which happens with probability δ per unit of time. In this case, the firm gets back to the labor market and earns the net return V -J f (z). A productivity shock may also take place with probability λ per unit of time.

rJ f (z) = z -w f (z) + λ J f z -J f (z) dG(z ) + δ V -J f (z) (2.11)
The fixed-term worker's value function embeds the wage w f (z) and the expectations about nextperiod outcomes. If the match is not separated, the worker expects to get the average worker's value from a fixed-term contract. If a split occurs, the worker returns searching for a job.

rW f (z) = w f (z) + λ W f z -W f (z) dG z + δ U -W f (z) (2.12)
Note that fixed-term matches split because of expiry shocks only. The probability of resignation, retirement or probationary employment termination s does not apply here as it is not empirically relevant. According to [START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF], only 2 % of fixed-term contracts terminated before their stipulated date in 2017.

Two serious discrepancies between the model and the actual French labor code need to be highlighted.

The desire for simplicity guides these two strong assumptions.

First, the duration of fixed-term contracts is exogenous. Endogenous separations for fixed-term contracts are not possible. At first view, a certain lack of realism must be pointed at here. Indeed, according to the French legislation, a firm may fire a fixed-term worker before the stipulated date through the payment of a firing cost. Nevertheless, this makes the analysis vainly cumbersome. The

Joint surpluses and wages

At the equilibrium, we assume that there is free-entry in the vacancy posting activity. The present discounted value of a vacancy is zero. Competition between firms depletes the profit opportunities from new jobs. In other words, the rent provided by the posting of vacancies attracts new entrants until its disappearance at the equilibrium.

V = 0

The free-entry condition leads to a condition for job creation, which states that the firm's expected cost for a firm-worker contact γ/q(θ) equals the expected profit it yields.

γ q(θ) = max J p 0 (z) , J f (z) , 0 dG(z) (2.13)
Using the Nash-bargaining rules (2.1)-( 2.3) and the job creation condition (2.13), I get

γ (1 -η)q(θ) = max S p 0 (z) , S f (z) , 0 dG(z) (2.14)
Deriving the expression for surpluses using the proper value functions along with (2.14), I get (r + s + λ)S p (z) =z -rU + (r + s) F + λ max S p z , 0 dG z (2.15)

S p 0 (z) =S p (z) -F (2.16) (r + δ + λ)S f (z) =z -rU + λ S f z dG z (2.17) rU =b + ηγθ 1 -η (2.18)
The unemployed value the benefit b and the rent they obtain from finding a job on the next-period.

A common rent for all ex-post insiders is a share of the firms' recruitment cost. γ is lost for the firm at the very moment of agreement over the formation of a match and the subsequent wage bargaining. The asset value of being unemployed is also the employed's outside option.

Nash-sharing rules as well as workers' and joint surpluses pinpoint wages. they enhance the workers' outside option. Similarly, a higher labor market tightness encompasses greater job-finding opportunities for unemployed workers, which raises their outside option: wages increase.

w p (z) = η (z + (r + s)F + γθ) + (1 -η)b (2.19) w p 0 (z) = η (z -λF + γθ) + (1 -η)b (2.20) w f (z) = η (z + γθ) + (1 -η)b (2.

Job creation and job destruction

Joint surpluses of matches are enough to pinpoint hiring and firing decisions. Joint surpluses are linear and increasing in match productivities. Consequently, I define the job destruction margin for continuing open-ended contracts z p as

S p (z p ) = 0 (2.22)
An open-ended match separates when the pair prefers paying the firing cost instead of continuing.

The productivity must be sufficiently low to fill this requirement ; the worker and the firm benefit from a separation if z < z p . Using an integration by part in the definition of z p (2.22) jointly with the definition of S p (2.15), the job destruction condition for open-ended contracts is

z p -b + (r + s)F + λ r + s + λ +∞ z p (1 -G(z)) dz = ηγθ 1 -η (2.23)
The above equation defines a positive relationship between θ and z p . The intuition behind this result is classic in the literature. A looser market tightness implies a higher job-finding probability for the unemployed, which makes their outside option stronger at the moment of wage bargaining: wages raise. It encourages firms to be more demanding in terms of match productivities and to increase the job destruction margin. An enhanced firing cost diminishes the job destruction margin; firms are more reluctant to pay firing costs and accept to maintain matches with worse productivities than before. Interestingly, a continuing open-ended match behaves as if it held a bond with face value F and yield (r + s) and used it to pay firing costs in case of an endogenous separation.

In the same manner, I define the threshold z f as follows

S f z f = 0 (2.24)
The interpretation of this threshold is twofold. As far as a contact between a vacancy and a jobseeker is concerned, it states whether a fixed-term contract is profitable. A fixed-term contract is profitable as soon as z ≥ z f8 . Conversely, when z < z f , an existing match would have interest into splitting. The firm and the worker would prefer to get back into searching. (2.24) jointly with the Nash-bargaining rule (2.3) entail

z f = 1 + λ r + δ b + ηγθ 1 -η - λ r + δ Ez (2.25)
where Ez = zdG(z). As previously, a higher labor market tightness enlarges wages because of stronger workers' outside option. Fixed-term contracts are profitable on a thinner range and z f increases. A higher average productivity E z encourages the resort to fixed-term contracts: the profitability margin decreases as fixed-term contracts become more often beneficial.

An analogous threshold z c can be defined for the desirability of new open-ended contracts

S p 0 (z c ) = 0 (2.26)
Meanwhile, since S p (z p ) = 0 and ∂S p /∂z = 1/(r + s + λ), I can rewrite S p as

S p (z) = z -z p r + s + λ (2.27)
(2.26) and (2.27) enable a convenient writing for z c .

z c = z p + (r + s + λ)F (2.28)
Similarly, the joint surpluses of fixed-term matches and new open-ended matches verify

S f (z) = z -z f r + δ + λ (2.29) S p 0 (z) = z -z c r + s + λ (2.30)
Note that ∂S p 0 /∂z = 1/(r + s + λ) > 1/(r + δ + λ) = ∂S f /∂z if and only if s < δ. I assume the validity of the latter condition, which states that the destruction rate of fixed-term contract is higher than the exogenous separation rate of open-ended contracts. It is undoubtedly the case in the data as I show in the calibration section. As a result, S p 0 and S f being increasing and linear in z, there exists z * such that

S p 0 (z * ) = S f (z * ) (2.31)
For all z ≥ z * , S p 0 (z) ≥ S f (z). I use (2.29), (2.30) and (2.31) to derive z * .

1 r + s + λ - 1 r + δ + λ z * = z c r + s + λ - z f r + δ + λ (2.32)
The behavior of the thresholds is characterized by the following proposition Proposition 1. These assertions are equivalent

1. z * > z f 2. z * > z c 3. z c > z f Proof. See Appendix 2.B
In the same manner, the equality of two of the thresholds z f , z c or z * is equivalent to the equality between all of them z f = z c = z * .

One ambition of this paper is to describe the endogenous choice between fixed-term contracts and open-ended contracts in job creation. With the thresholds previously defined and the increasing character of S p 0 and S f , the job creation condition (2.14) becomes

γ (1 -η)q(θ) = +∞ max[z c ,z * ] S p 0 (z) dG(z) + max z f ,z * z f S f (z) dG(z) (2.33)
Using two integrations by parts, the definitions of thresholds and proposition 1, the above equation writes

γ (1 -η)q(θ) = 1 r + s + λ +∞ max[z c ,z * ] (1 -G(z)) dz + 1 r + δ + λ max z f ,z * z f (1 -G(z)) dz (2.34)
Now, I spell out the formal definition of a steady-state equilibrium in this model. Definition 1. A steady-state equilibrium in this economy is characterized by the tuple θ, z p , z c , z f , z * verifying equations (2.23), (2.25), (2.28), (2.32) and (2.34).

This job creation condition (2.34) heavily depends on the distribution of idiosyncratic shocks and the subsequent value of thresholds. It is possible to have no fixed-term contracts as well as both contracts at the hiring stage. Thus, a formal definition of dual job creation is useful. Definition 2. Job creation is said to be dual if one kind of contracts is not systematically preferred to the other at the hiring stage.

In this model, job creation obeys the following proposition Proposition 2. Considering an equilibrium θ, z p , z c , z f , z * .

Job creation only occurs through open-ended contracts if and only if z

* ≤ z f ≤ z c . Open-ended contracts are hired when z ∈ (max [0, z c ] , +∞) as figure 2.1 displays. z * z f max [0, z c ] +∞
Back to search Open-ended contract A new firm-worker pair drawing a high idiosyncratic productivity may want to take advantage of this opportunity in full. To this extent, the best way to make the most of the situation consists in

hiring through an open-ended contract, which lasts longer and thus provides a higher surplus than a fixed-term contract. The match willingly locks itself through an open-ended contract in order to maximize the expected surplus. Otherwise, when the firm-worker pair draws a low productivity, both the worker and the firm go back searching. Fixed-term contracts serve as a compromise between a rigid contract or no productive relationship at all. Hiring through a fixed-term contract appears as an intermediate action: it generates a surplus through production while enabling to go back to searching for a better match before long. Fixed-term contracts are expedients to still take advantage of some positive surplus while waiting for better days to come. When, finally, a high productivity shock arises in a firm-worker meeting, an open-ended contract is signed.

In the current model, the exogenous separation rate s accounts for quits in open-ended matches among other costless termination channels.9 . However, most quits are job-to-job transitions. What would happen if we explicitly modeled on-the-job search in the current framework? Assume workers review offers at the same rate as the unemployed. A workers quits his current position if and only if he finds a match that delivers a higher surplus. Note that, according to the law, fixed-term contracts can leave their position before the stipulated expiry date if they are hired under an open-ended contract. Thus, fixed-term workers can search on the job too. If there is no search cost, every worker searches on the job. In this case, on-the-job search simply increase the workers' outside option compared to the baseline case. Now, assume that workers search at a cost. The higher the workers' surplus, the lower the eventual gain from on-the-job search. As continuing open-ended contracts deliver higher wages than fixed-term jobs all else equal, fixed-term workers would tend to search on the job more than open-ended workers do. A highly productive new firm-worker pair would sign an open-ended contract to lock up the match and avoid losing the worker through on-the-job search.

Thus, a priori, on-the-job search would not change the ranking of contracts in job creation. [START_REF] Cao | Fixed-Term and Permanent Employment Contracts: Theory and Evidence[END_REF] studies in depth this question and builds a dual labor market with on-the-job search.

Aggregate flows and stocks on a dual labor market

The share of open-ended workers that moves into unemployment falls into exogenous separations, which occur with probability s, and endogenous separations, which occur when an adverse produc-

= p (θ) (1 -G (max [z * , z c ])).
On the fixed-term side of the labor market, matches come to their stipulated end date with probability δ.

An unemployed worker finds a fixed-term job if he contacts a firm and the productivity of the resulting match is in the proper region. It occurs with probability

µ f = p (θ) G max z * , z f -G z f .
Denoting u the measure of non-employed workers, n p the measure of open-ended workers and n f the measure of fixed-term workers, fixed-term and open-ended employments evolve according to the following system.

.

n p = -ξn p + µ p u (2.

35)

.

n f = -δn f + µ f u (2.36)
Normalizing the workers' population to 1 leads to the following expressions for steady-state openended employment, fixed-term employment and non-employment.

n p = µ p δ µ p δ + ξδ + µ f ξ (2.37) n f = µ f ξ µ p δ + ξδ + µ f ξ (2.38) u = ξδ µ p δ + ξδ + µ f ξ (2.39)
As previously, an increase in θ has ambiguous consequences. On one hand, it strengthens the workers' outside option, which in turn raises wages, encourages the destruction of open-ended jobs and makes firms more demanding at the hiring stage in terms of productivity. On the other hand, job creation is bolstered by the enlarged probability of contact. Therefore, the impact on labor market stocks is ambiguous.

Comparative statics

In this subsection, I carry out comparative-statics exercises to describe the steady-state behavior of the model. The plus + and minussubscripts respectively denote the equilibrium values before and after the change in firing costs.

θ + θ - z p + z p - z f + z f - z * - z * + H p H f (H p ) H f θ z • JD p (JD p ) JC (JC) z f (z * ) (z * )
H p and (H p ) respectively are the hiring regions under open-ended contracts before and after the change in firing costs.

H f and H f are their counterparts for fixed-term contracts. Overall, the evolution of open-ended employment is ambiguous : job creation reduces as well as job destruction. As for fixed-term employment, job creation probability increases but the evolution of unemployment is unclear. Therefore, the response of the fixed-term job creation flow to an increase in firing costs is ambiguous.

An important alternative situation where the reasoning above is no longer valid consists in the insensibility of the steady-state equilibrium to firing costs. This happens if there is no endogenous destruction of open-ended jobs. Proposition 3 mathematically recapitulates the previous results.

Proposition 3. At the steady-state equilibrium, with endogenous job destruction of open-ended contracts

∂θ ∂F < 0, ∂z p ∂F < 0, ∂z c ∂F > 0, ∂z f ∂F < 0, ∂z * ∂F > 0 otherwise ∂θ ∂F = ∂z c ∂F = ∂z f i ∂F = ∂z * ∂F = 0 Proof. See Appendix 2.B
An increase in firing costs entails a substitution towards fixed-term employment on behalf of open-ended employment, while the classic result of an ambiguous response of unemployment remains.

The enhanced relative flexibility of fixed-term contracts makes them significantly more attractive than open-ended contracts, which are progressively replaced by fixed-term contracts at the margin.

When there is no endogenous destruction of open-ended jobs, the equilibrium becomes insensitive to firing costs.

The above exercise shows that the relative desirability of each contract at the hiring stage heavily depends on firing costs. Intuitively, sufficiently low firing costs lead to a complete shutdown of fixed-term employment, the corner case being firing costs such that z c = z f = z * as propositions The result above demonstrates the adaptability of our approach. The productivity-flexibility trade-off may deliver a classic labor market with a rigid side only, a dual labor market or a fixed-term-oriented labor market. This is ideal to study transitions and steady-state outcomes related large-scale policies, such as the ban of fixed-term contracts or large cuts in firing costs. The insensitive equilibrium associated with F ≥ F is interesting too. The data suggests that only 30% of open-ended job destruction is currently endogenous, which is not so far from the 0% of our extreme case. To this extent, the economic schemes at stake in the region close to F might be insightful.

Numerical analyses

As we saw in the comparative statics section, quantitative analyses are necessary to circumscribe the behavior of employments and unemployment in response to changes in firing costs.

Calibration

In this section, I calibrate a model to mimic the behavior of the French labor market. Availability of data concerning dualism on the labor market guided this choice. Moreover, the scope of the following analyses can be widened to many Western Europe countries, which share quite similar labor market institutions even though the strength of employment protection legislation may differ.

One may quote Germany, Italy, Spain or Portugal among others10 .

I set the unit time period to one month in accordance with the average duration of 1.5 months for temporary contracts in 2017 documented by [START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF] 11 . Interestingly, the median duration associated with temporary contracts is much lower and amounts to 5 days. The interest rate is set to 5 % annually, which boils down to 0.4 % on a monthly basis. The matching function is specified as

a Cobb-Douglas function m(u, v) = mu σ v 1-σ .
The elasticity of the matching function with respect to unemployment σ is set to 0.6, which stands in the middle of the range 0.5-0.7 estimated as reasonable by [START_REF] Burda | Gross worker and job flows in europe[END_REF] for Western Europe economies. In order to avoid the effect of congestion externalities and the complexity of their interlacing with the labor market institutions, we set the bargaining power η to 0.612 . As I mention in the theoretical discussion above, one important question is the distribution of idiosyncratic shocks. Uniform distributions are used in seminal papers 13 and include important benefits in terms of tractability. However, [START_REF] Tejada | Dual labor markets and labor protection in an estimated search and matching model[END_REF] shows that log-normal distributions replicate well the distribution of wages in dual labor markets.

Results are qualitatively robust to changes in distributions of idiosyncratic shocks.

F/w p µ f / µ p + µ f n f /n s/ξ u q(θ)
1.33 0.83 0.12 0.70 0.26 0.33 in the determination of (σ z , F, b, s, m, γ), whose values are detailed in table 2.3.

Steady-state employment and welfare

Employment Left-hand side of 16 To be accurate, [START_REF] Kramarz | The shape of hiring and separation costs in france[END_REF] assesses that firms with more than 50 employees face a marginal cost of 97,727 FFr (Table 1b), which represents 14 months of the workers' median wage. Consequently, the associated median wage of fired workers is 6980 FFr. Thus, Table 2 shows that individual terminations cost 27,389 FFr, which amounts to 4 months of the fired workers' median wage, while the termination within a collective firing plan marginally costs 81,850 FFr, which equals 12 months of the median wage. Employment On the x-axis, F is the threshold beyond which firing costs generate an equilibrium with dual job creation and is defined in Proposition 4. F denotes the value of firing costs as specified in the baseline calibration (see Table 2.3).

n p -Dual n f -Dual u -Dual n p -Classic u -Classic
option is bolstered by the possibility to find a job through fixed-term contracts for a given labor market tightness.

Accordingly, the open-ended job destruction margin is higher in the dual than in the classic case and so is the open-ended job destruction probability.

The right-hand side of Figure 2.5 displays the evolution of employments in a classic and a dual Mortensen-Pissarides model. On the left-hand part of the graph, firing costs are lower than F .

Job creation only happens through open-ended contracts and there is no fixed-term employment.

The dual and the classic Mortensen-Pissarides models coincide when firing costs are in 0, F .

Theoretically, firing costs have a relatively ambiguous effect on employment. On one hand, firing costs discourage lay-offs because of a magnified separation cost. On the other hand, the workers' enhanced threat point in the wage-bargaining process discourages hires. As firing costs exceed F , an equilibrium with dual job creation becomes possible. The classic model delivers lower and lower employment levels. At one point, the increase in firing costs ends up jeopardizing hiring incentives more than it discourages lay-offs: unemployment reaches a minimum level and begins increasing.

In the dual labor market, the same phenomenon takes place regarding overall employment but the outcome differs. The difference between the classic and dual cases stem from the emergence of an additional mechanism in the dual case. When dual job creation is an equilibrium outcome, higher firing costs encourage the substitution of open-ended employment towards fixed-term employment.

Starting from F , an increase in firing costs widens the spectrum of situations where fixed-term contracts are profitable. As fixed-term workers operate a larger share of new jobs, the job destruction flow enlarges. In my calibration, the widened possibilities of hires brought in by fixed-term contracts decrease less unemployment than magnified job destruction pushes it up : unemployment increases up to a maximum value of 0.28. This substitution effect is quantitatively substantial. Open-ended employment drops to a minimum of 0.62, while fixed-term employment increases from 0 to a maximum of 0.10.

When firing takes even higher values, the behavior of employments reverts. Open-ended employment increases from 0.62 to 0.65, while fixed-term employment decreases from 0.10 to 0.09. At first view, however, fixed-term employment should still increase according to the substitution effect we Welfare Beyond employment, important considerations include the impact of firing costs on welfare. The comparative statics carried out earlier as well as proposition 3 provide preliminary results. The unemployed would prefer the highest possible value for θ, which is associated with F = 0. Indeed, the unemployed workers value high job-finding probabilities. As for the fixedterm and open-ended matches, the results are theoretically ambiguous and a numerical analysis is necessary. Social welfare SW is defined as the sum of production from open-ended matches, production from fixed-term matches, the unemployed's home production net of the vacancy costs.

Therefore, the steady-state social welfare SW verifies

rSW = n p z p + n f z f + bu -γv -λG (z p ) n p F
where z p and z f are the average productivities of open-ended and fixed-term matches respectively. Productivity On the fixed-term side of the labor market, an increase in firing costs has two opposite effects.

z p -Dual z f -Dual z -Dual z -Classic
On the one hand, the higher the firing cost, the lower the labor market tightness and the lower the profitability threshold z f , which reduces fixed-term jobs' productivity. On the other hand, A benevolent social planner willing to reduce unemployment and improving welfare faces a dilemma if it wants to keep the dual structure of the labor market at minimum political cost. Indeed, starting from the baseline firing costs, there is no free lunch for small changes in firing costs. In the neighborhood of baseline firing costs F , increasing firing costs decreases unemployment and welfare, while decreasing firing costs increases unemployment and welfare. A reform improving both unemployment and welfare needs a large cut in firing costs. The post-reform firing costs need to be close to or below F . A large welfare gain and a small unemployment decrease come at a high political cost since the optimal reform consists in strongly cutting firing costs, which leads to a persistent increase in unemployment and a persistent decrease in open-ended employment.

Interestingly, the shape of the latter reform roughly resembles a unique-contract reform as there are no longer fixed-term contracts with firing costs below F .

Dynamics

In this section, we study dynamics after a change in firing costs as well as transitions between dual and classic labor markets. The mathematical aspect of these dynamics is discussed in Appendix 2.C.

I also study the consequences of regulatory uncertainty. In the classic framework, adjustment time reflects the interaction between the impact reaction and its absorption through bigger job creation and destruction flows. On the right-hand side of F, F , as post-reform firing costs move father and farther from the baseline, the impact reaction brings employment farther and farther from the new steady-state and transition rates do not increase enough to quickly absorb the impact reaction: adjustment time increases and reaches a maximum.

Post-reform adjustment speed

As post-reform firing costs decrease, transition rates become high enough to quickly absorb the initial response of employment and adjustment time decreases. Overall, adjustment takes up to 1.5 years in the classic framework. Note that if, by a happy coincidence, the impact response of employment sets it close to its new steady-state, adjustment would be much shorter. It does not take place in the current calibration, but may occur with other specifications. Overall, the policy maker faces a high uncertainty with respect to adjustment time when it implements a reform. In the following paragraph, we describe the post-reform dynamics in two specific cases. to adjust to each of the reforms on the acceptance rate.

Post-reform dynamics

In both examples of reforms, the impact response of employments and the post-reform transition probabilities shape the dynamics. As far as I know, there is no exploitable data to verify my findings.

F = F 1 . When F = F 1
, matches fear a shock magnifying firing costs, which encourages substitution towards fixed-term contracts at the hiring step.

To measure the numerical impact of regulatory uncertainty, I consider different ranges for firing costs between states 1 and 2 and find the parameters that match the baseline moments in Table | using the model with regulatory uncertainty. A simulated method of moments with 10,000 paths of policy shocks with a 1,000-month duration each is carried. The average and 95-per-cent confidence interval appear as plain and dashed lines.

2.2 using a Simulated Method of Moments. More specifically, I simulate 10,000 paths of policy shocks over 1,000 months randomly starting from one of the steady-states with F = F 1 or F = F 2 . I minimize the wedge between the simulated and actual moments.

Then, I compute the steady state when the shock arrival rate on firing costs is set to zero and firing costs are back to their baseline value. Figure 2.10 displays the percentage deviation in employments with respect to the baseline steady-state. It shows what happens when the uncertainty over firing costs vanishes. Starting from an equilibrium with 2.5 % changes of firing costs with respect to their baseline value, shutting down uncertainty provides a 15% increase in open-ended employment and 30% decrease in fixed-term employment. The impact of firing cost uncertainty becomes significant at the 5% level when firing costs vary of more than 1.5% around their baseline value. Policy uncertainty significantly fuels substitution towards fixed-term employment, which increases unemployment.

Robustness As previously mentioned, there is a diversity of views about the value of the productivity shock arrival rate λ. In Appendix 2.A21 , I plot the same graphs for calibrations with significantly higher and lower productivity shock arrival rates. I carry out three calibrations with three different average durations between productivity shocks: a month, a year and the mid-length of a typical business cycle, namely 3 years. In the same manner, the red-tape costs associated with an endogenous job destruction are difficult to assess. I calibrate the model with a zero share of exogenous splits of open-ended contracts to assess the sensitivity of my results to this parameter.

Assuming that all inactive workers search for a job may seem a pretty polar case despite the blurry frontier between employment, unemployment and inactivity in the fixed-term contracts. Thus, I also carry out a calibration targeting a 10% unemployment rate.

A higher arrival rate of productivity shocks and a zero share of costless separations of open-ended jobs magnify the curvature of transition probabilities, employments, average productivities, welfare and adjustment time with respect to firing costs. Open-ended job destruction flows are thicker and firing costs are more frequently paid. The differences between the dual and classic labor markets enlarge. In quantitative terms, changes in firing cost with a similar size have a stronger impact in terms of employment and welfare. The productivity shock arrival rate significantly impacts the steady-state outcome of a change in firing costs. If the value of the productivity shock arrival rate is not well known, a cut in firing costs leads to pretty unpredictable steady-state employment outcomes.

Higher productivity shock arrival rate lead to longer adjustment durations up to 5 years.

Conclusion

In this paper, I have built a simple matching model with both fixed-term and open-ended contracts.

The model provides a theoretical rationale to explain the contractual choice at the hiring step: fixed-term contracts act as stopgaps, offering both production and a possibility to return quickly on the labor market to fall onto a high-productivity match. In terms of policy, the removal of fixed-term contracts and a strong cut in firing costs leads to a gain in both welfare and employment within 18 months. Frequent marginal changes in firing costs push up unemployment and fixed-term

employment and weakens open-ended employment.

Tractability comes at a high cost: productivity shocks are i.i.d and there is no on-the-job search.

However, understanding the mechanisms underpinning dynamics in a dual labor market requires a few sacrifices. The next model should involve fewer restrictive assumptions to make numerical analyses more trustworthy. Employment Employment Employment

Firing costs

n p -Dual n f -Dual u -Dual n p -Classic u -Classic
n p -Dual n f -Dual u -Dual n p -Classic u -Classic
n p -Dual n f -Dual u -Dual n p -Classic u -Classic

2.B Proofs

Proposition 1 I denote ρ p = 1/(r + s + λ) and ρ f = 1/(r + δ + λ). As mentioned above,

ρ p > ρ f . Assume that z * > z f . (2.32) implies that ρ p z * = ρ p -ρ f z * + ρ f z * = ρ p z c + ρ f z * -z f .
Since z * -z f > 0, the latter equality implies z * > z c .

Assume that z * > z c . Again, jointly with algebraic manipulations, (2.32) implies that

ρ f z c = -ρ p -ρ f z c + (ρ p -ρ f )z * + ρ f z f > -ρ p -ρ f z c + (ρ p -ρ f )z c + ρ f z f > ρ f z f , which entails that z c > z f .
Assume that z c > z f . Algebraic manipulations and (2.32) imply that

ρ p -ρ f z * = ρ p z c -z f + ρ p -ρ f z f > ρ p -ρ f z f , which implies z * > z f . Proposition 2 Referring to (2.34),
If open-ended workers are the only ones hired, then max z f , z * ≤ z f , implying that z * ≤ z f .

Referring to proposition 2, the latter inequality entails

z f ≤ z c . As a result, z * ≤ z f ≤ z c . If job creation is dual, then      0 < max z f , z * z f < z *
Using proposition 2, the latter system of inequalities boils down to max 0, z f < z * .

For each case, the converse propositions are straightforward using (2.34).

Proposition 3

Considering the case where job creation is dual and there is endogenous destruction of open-ended jobs, We differentiate the different equations.

r + ξ r + s + λ dz p = ηγ 1 -η dθ -(r + s)dF r + δ r + δ + λ dz f = ηγ 1 -η dθ dz c = dz p + (r + s + λ)dF 1 r + s + λ - 1 r + δ + λ dz * = dz c r + s + λ - dz f r + δ + λ - γq (1 -η)q 2 (θ) dθ = -(1 -G (z * )) 1 r + s + λ - 1 r + δ + λ dz * -1 -G z f dz f r + δ + λ
Substituting the expression of dz p into the definition of dz c , we get

dz c r + s + λ = 1 r + ξ ηγ 1 -η dθ + λG (z p ) dF
In turn, this expression for dz c can be substituted into the definition of dz * .

1 r + s + λ - 1 r + δ + λ dz * = 1 r + ξ ηγ 1 -η dθ + λG (z p ) dF - 1 r + δ + λ dz f
Reintroducing the expression of dz f , the differentiated job creation condition becomes,

- γq (1 -η)q 2 (θ) dθ = - 1 -G (z * ) r + ξ ηγ 1 -η dθ + λG (z p ) dF - G (z * ) -G z f r + δ ηγ 1 -η dθ
As a result,

∂θ ∂F = - 1-G(z * ) r+ξ λG (z p ) -γq (1-η)q 2 (θ) + ηγ 1-η 1-G(z * ) r+ξ + G(z * )-G z f r+δ < 0
This entails ∂z f ∂F < 0 and ∂z p ∂F < 0.

In addition,

∂z c ∂F ∝ ηγ 1 -η ∂θ ∂F + λG (z p ) ∂z c ∂F ∝ - γq (1 -η)q 2 (θ) + G (z * ) -G z f r + δ ηγ 1 -η > 0
Jointly with the fact that ∂z f ∂F < 0, We get that ∂z * ∂F > 0.

When z p ≤ 0, we have that ∂θ ∂F = 0 and, consequently, ∂z f ∂F = 0 and ∂z c ∂F = 0. This leads to

∂z * ∂F = 0.
Similar computations can be carried out for the other case, where job creation occurs through open-ended contracts only. The only change lies in the job creation condition.

-

γq (1 -η)q 2 (θ) dθ = - 1 -G (z c ) r + s + λ dz c
Introducing the expression of dz c , the differentiated job creation condition becomes,

- γq (θ) (1 -η)q 2 (θ) dθ = - 1 -G (z c ) r + ξ ηγ 1 -η dθ + λG (z p ) dF As a result, ∂θ ∂F = - 1-G(z c ) r+ξ λG (z p ) -γq (1-η)q 2 (θ) + ηγ 1-η 1-G(z c ) r+ξ < 0 and ∂z c ∂F ∝ ηγ 1 -η ∂θ ∂F + λG (z p ) ∂z c ∂F ∝ - γq (1 -η)q 2 (θ) > 0 ∂z p ∂F < 0 is still true.
Proposition 4 To prove this result, we will rely on the two corner equilibria that are implicitly present in proposition 2.

In the first case, consider the case where z f = z c = z * = z, which is the knife-edge case associated with proposition 1. In this case, the equilibrium is summed up by equations (2.34) and (2.25), where θ and z f are respectively replaced by θ and z. Taking into account the fact that z f = z c = z * in the job creation equation, the two latter equations boil down to

z = 1 + λ r + δ b + ηγ θ 1 -η - λ r + δ E z γ (1 -η)q θ = +∞ z (1 -G(z)) dz
Given the equilibrium values θ, z , z c = z along with (2.23) and (2.28) entail that

F 1 verifies z -b -λ F + λ r + s + λ +∞ z-(r+s+λ)F 1 (1 -G(z))dz = ηγ θ 1 -η
Proposition 2 ensures that job creation only occurs through open-ended contracts. Replacing µ f by zero in (2.38) demonstrates that n f = 0 at the equilibrium.

A marginal increase in firing costs immediately entails an increase in z c while z f diminishes according to proposition 3. The initial situation being z f = z c = z, we now face z f < z < z c , which is equivalent to z f < z * as Proposition 1 states. Job creation is now dual, as asserts Proposition 2. Conversely, a marginal decrease in firing costs implies a cut in z * and z c as well as an increase in z f , which results in z c < z f . The latter is equivalent to z * < z c under proposition 2. The resulting equilibrium only involves open-ended contracts.

In the second case, consider the case where z p = 0. There is no endogenous destruction of open-ended matches. The equilibrium can be summed up by F , θ, z p , z c , z f , z * verifying equations (2.23), (2.25), (2.28), (2.32) and ( 2.34) under the additional constraint that z p = 0.

2.C Dynamics

In this section, I describe the dynamics of the equilibrium after a shock to firing costs.

2.C.1 Dual-to-dual and classic-to-dual transitions

The comments made here also apply to the model that includes policy uncertainty.

Let us consider the model starting from the equilibrium defined by θ 0 , z p 0 , z c 0 , z f 0 , z * 0 with firing costs F 0 at time t = 0 -and the associated open-ended and fixed-term employment values n p 0 and n f 0 . The former tuple solely consists in forward looking variables. Therefore, assuming that firing costs jump from F 0 to F on time t = 0, the equilibrium tuple defined in definition 1 immediately jumps to its new value θ, z p , z c , z f , z * . As for employment, notice that if z p > z p 0 -which occurs if .

F < F 0 -open-ended matches such that z p 0 < z < z p split
n p t = -ξn p t + µ p 1 -n p t -n f t - G(max z p 0 ,z p )-G z p 0 1-G z p 0 δ 0 n p t . n f t = -δn f t + µ f 1 -n p t -n f t
where ξ, µ p and µ f are computed as described in section 2.2.4 using the equilibrium tuple

θ, z p , z c , z f , z * .
Laplace transforms are particularly useful when it comes to solving systems of differential equations embedding Dirac functions. Denoting N p and N f the Laplace transforms of n p and n f , the latter system becomes

     sN p (s) -n p 0 = µ p s -α 0 n p 0 -(ξ + µ p ) N p (s) -µ p N f (s) sN f (s) -n f 0 = µ f s -δ + µ f N f (s) -µ f N p (s)
where

α 0 = G(max z p 0 ,z p )-G z p 0 1-G z p 0 .
The system above being linear in N p (s) and N f (s), one may isolate the latter to obtain

N p (s) = (1 -α 0 )n p 0 s 2 + µ p + δ + µ f (1 -α 0 )n p 0 -µ p n f 0 s + δµ p s (s 2 + (ξ + δ + µ p + µ f ) s + ξδ + ξµ f + µ p δ) N f (s) = n f 0 s 2 + µ f + (ξ + µ p ) n f 0 -µ f (1 -α 0 )n p 0 s + ξµ f s (s 2 + (ξ + δ + µ p + µ f ) s + ξδ + ξµ f + µ p δ)
The common denominator of these two expressions is a third-degree polynomial with three real roots: 0, α 1 and α 2 defined as

α 1 = 1 2 -ξ + δ + µ p + µ f + (ξ + δ + µ p + µ f ) 2 -4 (ξδ + ξµ f + µ p δ) α 2 = 1 2 -ξ + δ + µ p + µ f -(ξ + δ + µ p + µ f ) 2 -4 (ξδ + ξµ f + µ p δ)
Thus, N p (s) and N f (s) verify

N p (s) = (1 -α 0 )n p 0 s 2 + µ p + δ + µ f (1 -α 0 )n p 0 -µ p n f 0 s + δµ p s (s -α 1 ) (s -α 2 ) N f (s) = n f 0 s 2 + µ f + (ξ + µ p ) n f 0 -µ f (1 -α 0 )n p 0 s + ξµ f s (s -α 1 ) (s -α 2 )
For i ∈ {p, f }, a partial fraction decomposition yields

N i (s) = n i ∞ s + ρ i 1 s -α 1 + ρ i 2 s -α 2
where n i ∞ , ρ i 1 and ρ i 2 verify

n p ∞ = µ p δ µ p δ + ξδ + µ f ξ n f ∞ = µ f ξ µ p δ + ξδ + µ f ξ ρ i 1 = - α 1 A i + B i α 2 -α 1 ∀i ∈ {p, f } ρ i 2 = α 2 A i + B i α 2 -α 1 ∀i ∈ {p, f }
where

A p = (1 -α 0 )n p 0 -n p ∞ A f = n f 0 -n f ∞ B p = µ p + δ + µ f (1 -α 0 )n p 0 -µ p n f 0 + (α 1 + α 2 ) n p ∞ B f = µ f + (ξ + µ p ) n f 0 -µ f (1 -α 0 )n p 0 + (α 1 + α 2 ) n f ∞
For all i ∈ {p, f }, using the inverse Laplace transform, we get

n i t = n i ∞ + ρ i 1 exp {-α 1 t} + ρ i 2 exp {-α 2 t}
The classic-to-dual transition obeys the same equations with the only additional condition n f 0 = 0.

2.C.2 Dual-to-classic transition

Keeping the same notations as above, with the exception that the possibility to hire through fixedterm contracts is shut down after the reform. The resulting equilibrium is defined by the tuple (z p , z c , θ) which verifies equations (2.23), (2.28) and the modified job creation condition

γ (1 -η)q(θ) = 1 r + s + λ +∞ z c (1 -G(z)) dz (2.C.1)
The only remaining transition probabilities on the labor market are the job-finding probability

µ = p(θ) (1 -G (z c
)) and the job-destruction probability ξ, which is left unchanged. Thus, the system of equation describing dynamics after the reform on time t = 0 is .

n p t = -ξn p t + µ 1 -n p t -n f t -α 0 δ 0 (t)n p t (2.C.2)
.

n f t = -δn f t (2.C.3)
where α 0 is defined as in the dual-to-dual-reform paragraph.

(2.C.3) can be solved right away

n f t = n f 0 exp {-δt}
As for (2.C.3), using Laplace transforms yield

sN p (s) -n p 0 = µ s -(ξ + µ)N p (s) - µ s + δ n f 0 -α 0 n p 0 id est N p (s) = (1 -α 0 )n p 0 s 2 + µ + (1 -α 0 )δn p 0 -µn f 0 s + µδ s(s + δ)(s + ξ + µ) A partial fraction decomposition yields N (s) = n p ∞ s + ρ p 1 s + δ + ρ p 2 s + ξ + µ
where

n p ∞ = µ ξ + µ ρ p 1 = µ δ -ξ -µ n f 0 ρ p 2 = ((1 -α 0 )n p 0 -n p ∞ ) - µ δ -ξ -µ n f 0
Using the inverse Laplace transform, n p t verifies

n p t = n p ∞ + ρ p 1 exp {-δt} + ρ p 2 exp {-(ξ + µ)t}

2.D The model with regulatory uncertainty

The Bellman equations defining the firms' and workers' programs now include the shock in firing costs. Firing costs become a new state variable. When a shock hits the latter, which occurs with probability , firing costs jumps to either F 1 or F 2 with an equal probability 1/2 and the present discounted values adjust accordingly. Whenever possible, I simplify notations and substitute dependence in F i with the corresponding subscripts i ∈ {1, 2}.

rV (F ) = -γ + q (θ(F )) max J p 0 (z, F ) -V (F ), J f (F ) -V (F ), 0 dG(z) + V 1 + V 2 2 -V (F ) rU (F ) = b + p(θ(F )) max W p 0 z , F -U (F ), W f z , F -U (F ), 0 dG z + U 1 + U 2 2 -U (F ) rJ p (z, F ) = z -w p (z, F ) + s (V -J p (z, F )) + λ max J p z , F , V (F ) -F -J p (z, F ) dG z + 1 2 max [J p (z, F 1 ) , V 1 -F 1 ] + 1 2 max [J p (z, F 2 ) , V 2 -F 2 ] -J p (z, F ) rW p (z, F ) = w p (z, F ) + λ max W p z , F , U (F ) -W p (z, F ) dG z + s (U (F ) -W p (z, F )) + 1 2 max [W p (z, F 1 ) , U 1 ] + 1 2 max [W p (z, F 2 ) , U 2 ] -W p (z, F ) rJ p 0 (z, F ) = z -w p 0 (z, F ) + s (V (F ) -J p 0 (z)) + λ max J p z , F , V (F ) -F -J p 0 (z, F ) dG z + 1 2 max [J p (z, F 1 ) , V 1 -F 1 ] + 1 2 max [J p (z, F 2 ) , V 2 -F 2 ] -J p 0 (z, F ) rW p 0 (z, F ) = w p 0 (z, F ) + λ max W p z , F , U (F ) -W p 0 (z, F ) dG z + s (U (F ) -W p 0 (z, F )) + 1 2 max [W p (z, F 1 ) , U 1 ] + 1 2 max [W p (z, F 2 ) , U 2 ] -W p 0 (z, F ) rJ f (z, F ) = z -w f (z, F ) + λ J f z , F -J f (z, F ) dG(z ) + δ V (F ) -J f (z, F ) + 1 2 J f (z, F 1 ) + 1 2 J f (z, F 2 ) -J f (z, F ) rW f (z, F ) = w f (z, F ) + λ W f z , F -W f (z, F ) dG z + δ U (F ) -W f (z, F ) + 1 2 W f (z, F 1 ) + 1 2 W f (z, F 2 ) -W f (z, F ) Free entry implies V 1 = V 2 = 0.
Taking into account the Nash-bargaining rules (2.1)-( 2.3), the job creation condition is

γ (1 -η)q (θ(F )) = max S p 0 z , F , S f z , F , 0 dG z
Considering the unemployed's value function, the previous equation and Nash-bargaining rules

(2.1)-(2.3) imply (r + )U i = 1 + r b + ηγ 1 -η θ i + r θ 1 + θ 2 2
Algebraic manipulations along with the Nash-bargaining assumption deliver the following expressions for surpluses

(r + s + λ + )S p (z, F ) = z -(r + )U (F ) + (r + s + )F + λ max S p z , F , 0 dG z + 2 (max [S p (z, F 1 ) , 0] + max [S p (z, F 2 ) , 0]) + U 1 + U 2 2 - F 1 + F 2 2 S p 0 (z, F ) = S p (z, F ) -F (r + δ + λ + )S f (z, F ) = z -(r + )U (F ) + λ S f z , F dG z + S f (z, F 1 ) + S f (z, F 2 ) 2 + U 1 + U 2 2
S p is increasing in both z and F and has its values in R. Thus, there exists z p 1 and z p 2 such that S p (z p 1 , F 1 ) = 0 and S p (z p 2 , F 2 ) = 0. Notice that ∂S p /∂z does not depend on F and verifies

∂S p ∂z =            1 r+s+λ+ if z ≤ z p 2 1 r+s+λ+ 2 if z p 2 ≤ z ≤ z p 1 1 r+s+λ if z ≥ z p 1
Using an integration by parts, we know that z p 1 and z p 2 verify

z p 2 -(r + )U 2 + (r + s + )F 2 + U 1 + U 2 2 - F 1 + F 2 2 + λ r + s + λ + 2 z p 1 z p 2 (1 -G(x)) dx + λ r + s + λ +∞ z p 1 (1 -G(x)) = 0 z p 1 -(r + )U 1 + (r + s + )F 1 + U 1 + U 2 2 - F 1 + F 2 2 + λ r + s + λ +∞ z p 1 (1 -G(x)) dx + 2 z p 1 -z p 2 r + s + λ + 2 = 0
The present discounted value of unemployment U depends on the firing cost through the labor market tightness, which accounts for the i subscript, with i ∈ {1, 2}. Their expressions still verify (2.18).

The derivative above yields another expression for S p .

S p (z, F 2 ) =            z-z p 2 r+s+λ+ if z ≤ z p 2 z-z p 2 r+s+λ+ 2 if z p 2 < z < z p 1 z-z p 1 r+s+λ + z p 1 -z p 2 r+s+λ+ 2 if z ≥ z p 1 S p (z, F 1 ) =            z-z p 2 r+s+λ+ + z p 2 -z p 1 r+s+λ+ 2 if z ≤ z p 2 z-z p 1 r+s+λ+ 2 if z p 2 < z < z p 1 z-z p 1 r+s+λ if z ≥ z p 1
Considering the surplus of fixed-term contracts, ∂S f /∂z equals 1/(r + δ + λ). One may define z f i such that S f z f i , F i = 0. Thus, another definition of S f is

S f (z, F i ) = z -z f i r + δ + λ
where thresholds z f i are such that

z f i = r + δ + λ r + δ + (r + )U i - δ r + δ U 1 + U 2 2 - λ r + δ Ez
In the same manner, one may define z c i as

S p 0 (z c i , F i ) = 0. Since for all z ≥ z p 2 , ∂S p /∂z = 1/(r + s + λ) > 1/(r + δ + λ) = ∂S f /∂z, there still exists z * i such that S f (z * i , F i ) = S p 0 (z * i , F i ), for
i ∈ {1, 2}. I assume that and the distance between F 1 and F 2 is small enough so that z * i > z p 2 for all i ∈ {1, 2}. In this case, z * i verify

1 r + s + λ - 1 r + λ + δ z * i = z p 1 r + s + λ - z p 1 -z p 2 r + s + λ + 2 1 {i = 2} + F i - z f i r + λ + δ
As previously, assume that and the distance between F 1 and F 2 is small enough so that z c i > z p 2 for all i ∈ {1, 2}, we can rewrite the job creation conditions (2.14), which pinpoint θ i with i ∈ {1, 2}.

γ (1 -η)q (θ i ) = 1 r + s + λ +∞ max z c i ,z * i (1 -G(z)) dz + 1 r + δ + λ max z f i ,z * i z f i (1 -G(z)) dz
Consequently, the equilibrium can be summed up by a continuous time Markov process over space

θ i , z p i , z c i , z f i , z * i i=1,2
with switching probability and 2x2 transition matrix filled with 1/2. See Appendix 2.C for the study of the corresponding dynamic behavior.

Introduction

In Firms resort to fixed-term contracts to cope with uncertainty and demand fluctuations [START_REF] Rémy | Pourquoi les employeurs choisissent-ils d'embaucher en cdd plutôt qu'en cdi ? Dares analyses[END_REF].

On the one hand, [START_REF] Saint-Paul | Dual Labor Markets: A Macroeconomic Perspective[END_REF] Secondly, my paper is close to a few papers in the DSGE literature4 that assess the impact of firing costs on macroeconomic variables. The closest paper is [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF], which builds a model with a Mortensen-Pissarides framework into a traditional New Keynesian framework. Firing costs protect employed workers. The model is estimated on Euro area data. [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF] finds that changes in firing costs do not impact inflation volatility. My findings corroborate this result: while a 5-percent cut in firing costs significantly alters the cyclical properties of labor market variables, inflation volatility remain the same. However, the study of inflation components shows that their behavior changes much in response to a change in firing costs. The neutrality of firing costs with respect to inflation volatility rests on the small value of the ex ante contractual productivity wedge. Using similar tools as [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF], [START_REF] Zanetti | Labor market institutions and aggregate fluctuations in a search and matching model[END_REF] finds that firing costs decrease the volatility of output, employment and job flows. Thus, the higher firing costs, the more rigid is the labor force and the more firms adjust prices in response to shocks. I depart from the approach of [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF] and [START_REF] Zanetti | Labor market institutions and aggregate fluctuations in a search and matching model[END_REF] and introduce fixed-term contracts.

My intuition is that fixed-term contracts mitigate the need for stronger price adjustments firing costs alone may encourage.

Thirdly, my paper also relates to the literature explaining the macroeconomic impact of risk. [START_REF] Bloom | The impact of uncertainty shocks[END_REF] shows that uncertainty shocks encourage firms to postpone hiring and investment decisions when they face adjustment costs. A difficulty in this literature is identifying uncertainty shocks. For example, [START_REF] Leduc | Uncertainty shocks are aggregate demand shocks[END_REF] asserts that uncertainty shocks have the same effect as negative aggregate demand shocks: both increase unemployment and decrease inflation5 . In my model, the different behavior of fixed-term employment in front of negative demand shocks and uncertainty shocks may provide a way to disentangle both types of shocks.

The paper is organized as follows. Section 2 presents the model. Section 3 exposes the calibration and estimation procedure. Section 4 displays the main experiments. Section 5 concludes.

Model

The model follows a discrete timing and embeds 3 types of agents; the households, the firms and the

Households

Households are identical and constitute a continuum represented by the interval (0, 1). They can be employed under a fixed-term or an open-ended contract, or unemployed. They earn wages or unemployment benefits accordingly. Households also hold firms, consume the homogeneous good produced by final good firms, save using one-period nominal bonds, earn interests on their savings and pay lump-sum taxes. Hence, I assume that taxes do not distort the choice of households over consumption and investment. I do not consider the interplay between payroll taxes and labor market dualism.

If households are identical ex ante, their different employment histories make them heterogeneous ex post. How labor market dualism impacts the consumption and saving behavior of households is beyond the scope of this paper. As [START_REF] Merz | Search in the labor market and the real business cycle[END_REF] and [START_REF] Andolfatto | Business cycles and labor-market search[END_REF] first did, I assume that households pool revenues and that capital markets are perfect. Thereby, I rule out the complication heterogeneity brings on. Households share equal consumption and investments. This assumption is not innocuous: unemployed, open-ended and fixed-term workers have different borrowing constraints and [START_REF] Lise | On-the-job search and precautionary savings[END_REF] shows that assets shape the search behavior. I leave these issues for future research. The first order conditions with respect to c t and B t+1 lead to the following Euler equation

u (c t ) = βE t R t P t P t+1 u (c t+1 ) (3.2.1)
As households own firms, firms discount profits of date s from date t with the factor β t,s = β s-t u (c s ) /u (c t ).

Final good firms

Final good firms are identical and compete to produce the good consumed by households. They put together retail goods through a Dixit-Stiglitz technology and produce Y t .

Y t = 1 0 Y t -1 t i,t di t t -1 (3.2.2)
where t is the elasticity of substitution between retail goods.

The firm takes as given the price of the retail goods P i,t and the price of the final good P t and maximizes its profits with respect to the components of its input {Y i,t } i∈(0,1) under the constraint (3.2.2). The program of the final good firm boils down to

max {Y i,t } i∈[0,1] P t Y t - 1 0 P i,t Y i,t di subject to (3.2.2)
The subsequent first order condition provides an expression for the demand of retail good i.

Y i,t = P i,t P t -t Y t (3.2.3)

Retailers

Retailers buy goods from intermediate firms and sell the obtained production to final good producer.

They are in monopolistic competition and lie on the interval (0, 1). Retailers accomplish the one-toone transformation of the intermediate good into a retail good. Denoting X i,t retailer i's input in the intermediate good, the production technology writes

Y i,t = X i,t
As a result, retailers face a real marginal cost that equals the relative price of the intermediate good φ t . I assume that retailers adjust prices as [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF] describes.

P i,t =      P i,t-1 with probability ψ P * i,t
with probability 1 -ψ

A fraction ψ of retailers is able to adjust its prices to the optimal value P * i,t , whereas the remaining retailers stick to their former prices. There is no indexation of non-adjusted prices on inflation in this model. The price-setting retailer i at period t has the following program.

max

P * i,t E t +∞ T =t β t,T ψ T -t P * i,t P T -φ T Y i,T subject to Y i,T = P * i,t P T -T Y T
It leads to the following first order condition.

E t +∞ T =t β t,T ψ T -t P T T Y T P * i,t P T -µ T φ T = 0 (3.2.4) µ t = t /( t -1) is a mark-up shock such that µ t = /( -1)e µ t , with µ t iid ∼ N 0, σ 2 µ .

Intermediate good firms and the labor market

Intermediate-good firms use labor as sole input. They can employ one worker or maintain one vacancy. Workers can be unemployed or employed under a fixed-term or an open-ended contract.

They are identical. There is no on-the-job search, which implies that only unemployed workers search for a job. When a firm and a worker meet, the idiosyncratic productivity of the match z is revealed. I assume that the idiosyncratic productivity of a match is i.i.d across time and drawn from a distribution with cumulative distribution function G. I reluctantly make this assumption for the sake of simplicity. With persistent idiosyncratic productivity, the computation of the equilibrium requires keeping track of the productivity distribution of matches as a state variable. Since the literature considering cycles and dual labor market is in early stages, I prefer to leave the distributional issues for future research. Matches also face an aggregate productivity shock A t with process

log (A t ) = ρ A log (A t-1 ) + A t , where A t iid ∼ N 0, σ 2 A .
The number of firm-worker contacts per period is m(e, v), where e is the number of job-seekers and v is the number of vacancies. A classic measure of the matching activity is the labor market tightness θ = v/e. The matching function m has constant returns to scale, which enables the definition of the firm-worker meeting probability p(θ) on the job seeker's side and its counterpart q(θ) on the firm's side.

q = m (e, v) v = m θ -1 , 1 p = m (e, v) e = m (1, θ) = θq(θ)
p is increasing in labor market tightness, whereas q is decreasing in labor market tightness. Note that the meeting probabilities are not the classic job-finding and vacancy-filling probabilities. A firm-worker meeting does not lead to production if the idiosyncratic productivity is too low.

The timing in the economy is summed up in Figure 3.2.4. At the beginning of the period, agents learn the value of shocks and firms manage their workforce accordingly. They lay off poorly productive workers and post vacancies. Next, new matches are revealed. Workers fired in the current period are able to participate to the present meeting round. Hence, I avoid understating labor market flows as most fixed-term contracts last less than a quarter: fixed-term jobs last 1.5 months on average in France [START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF]. Finally, production is carried out, firms pay for wages and firing costs, households consume, and the period ends. Vacancies When paired with a worker, firms hire through a fixed-term contract or an open-ended contract. They can also return searching for a worker and get the chance to be matched with another one on the next period. Job creation is unconstrained; firms choose the actions that maximize their surpluses. The present discounted value of a vacancy V t bears witness of these different options:

V t = -γ + q (θ t ) max J 0,p t (z) , J f t (z) , E t β t,t+1 V t+1 dG(z) + (1 -q (θ t )) E t β t,t+1 V t+1 (3.2.5)
where γ is the cost of a vacancy, J 0,p t is the firm's surplus with a new open-ended match. I denote J p t its counterpart for continuing open-ended matches. J f t is the firm's surplus with a fixed-term match.

The workers' surpluses are denoted W , while the total surpluses are denoted S.

Surplus

W p t (z t ) -U t = ηS p t (z t ) (3.2.6) W 0,p t (z t ) -U t = ηS 0,p t (z t ) (3.2.7) W f t (z t ) -U t = ηS f t (z t ) (3.2.8)
Total surpluses verify

S p t (z t ) = J p t (z t ) -(V t -F t ) + W p t (z t ) -U t (3.2.9) S 0,p t (z t ) = J 0,p t (z t ) -V t + W 0,p t (z t ) -U t (3.2.10) S f t (z t ) = J f t (z t ) -V t + W f t (z t ) -U t (3.2.11)
where U t is the value of unemployment at the beginning of period t. The outside option of workers includes taking part in current period's market trades and earning the unemployment benefit if no beneficial meeting occurred. U 0 t is the present discounted utility of workers who have not found a job on period t.

U t = p (θ t ) max W 0,p t (z), W f t (z), U 0 t dG(z) + (1 -p (θ t )) U 0 t (3.2.12) U 0 t = b + E t β t,t+1 U t+1 (3.
J p t (z t ) = φ t A t z t -w p t (z t ) + E t β t,t+1 (1 -s) max J p t+1 (z) , V t+1 -F dG(z) + sV t+1 (3.2.14)
An incumbent open-ended worker earns a wage and may leave with probability s. If he does not, he faces a productivity shock and the match may split if the latter shock is adverse enough. Laid-off or quitting workers go back to the job seekers' pool immediately and are therefore eligible to participate to firm-worker contacts in the current period. Otherwise, the match goes on.

W p t (z t ) = w p t (z t ) + E t β t,t+1 (1 -s) max W p t+1 (z), U t+1 dG(z) + sU t+1 (3.2.15)
New open-ended workers have a different wage function w 0,p t : their outside option does not include the payment of the firing cost in case of disagreement during the wage bargaining and the possibility to search for a new job in case of disagreement. After one-period, if there is no separation, a wage renegotiation occurs because idiosyncratic productivity changed. The firm pays firing costs if an endogenous split occurs. Otherwise, the firm's continuation value is the expected surplus of an incumbent open-ended contract.

J 0,p t (z t ) = φ t A t z t -w 0,p t (z t ) + E t β t,t+1 (1 -s) +∞ z p t+1 J p t+1 (z) dG(z) -G z p t+1 F + sV t+1 (3.2.16) W 0,p t (z t ) = w 0,p t (z t ) + E t β t,t+1 (1 -s) max W p t+1 (z), U t+1 dG(z) + sU t+1 (3.2.17)
In this paper, I consider firing costs instead of severance payments. Indeed, severance payments are pure transfers between firms and workers. They do not impact the joint surplus of a match. As a result, severance payments do not directly shape job destruction decisions. They increase wages, though; on each period, severance payments are part of the workers' outside option in case of a disagreement over negotiated wages. The fact that firms and workers renegotiate wages each period explains why severance payments raise workers' wages. The impact of severance payments can be utterly offset in the bargaining process if there is enough commitment6 . Without renegotiations, the pre-hiring bargaining of wages can utterly wipe out the impact of severance payments to reach the efficient employment level. On the contrary, firing costs, alter the joint surplus of the match and is .

still push up wages. They increase wages through the stronger workers' outside option and directly alter job destruction decisions. Firing costs alter labor market outcomes through more channels than severance payments do.

Fixed-term contracts A fixed-term contract stipulates the wage function w f t . Fixed-term matches split with the exogenous probability δ. I assume that fixed-term contracts yield a lower productivity than open-ended matches with a factor ρ < 1. I discuss this assumption in detail below. Firms value the immediate production net of the wage. As for the continuation value, it simply embeds the possibility of an exogenous separation shock.

J f t (z t ) = ρA t z t φ t -w f t (z t ) + E t β t,t+1 (1 -δ) J f t+1 (z) dG(z) + δV t+1 (3.2.18)
Fixed-term workers immediately value their wages. In case of a separation, they can immediately indulge in labor market trades.

W f t (z t ) = w f t (z t ) + E t β t,t+1 (1 -δ) W f t+1 (z) dG(z) + δU t+1 (3.2.19)
Using the different definitions of the firms' and the workers' surpluses, the total surpluses write 7 3.2. MODEL Using the firms' surpluses, joint surpluses and the surplus sharing rules, wages verify

S p t (z t ) = A t z t φ t -b - ηγθ t 1 -η + (1 -E t β t,t+1 (1 -s)) F + E t β t,t+1 (1 -s) max S p t+1 (z) , 0 dG(z) (3.2.20) S 0,p t (z t ) = S p t (z t ) -F (3.2.21) S f t (z t ) = ρA t z t φ t -b - ηγθ t 1 -η + E t β t,t+1 (1 -δ) S f t+1 (z)dG(z) (3.
w p t (z t ) = η (A t z t φ t + (1 -E t β t,t+1 (1 -s)) F + γθ t ) + (1 -η)b (3.2.23) w 0,p t (z t ) = η (A t z t φ t -E t β t,t+1 (1 -s)F + γθ t ) + (1 -η)b (3.2.24) w f t (z t ) = η (ρA t z t φ t + γθ t ) + (1 -η)b (3.2.25) (3.2.26)
The new open-ended worker is penalized with higher firing costs to compensate the future wage gains in case of continuation. Moreover, labor market tightness increases the outside option of workers through the higher probability of finding a job, which increases wages.

Job creation and job destruction

I assume there is free entry on vacancy posting.

V t = 0 (3.2.27)
The job creation condition stems from the free-entry condition (3.

A t z p t φ t + (1 -E t β t,t+1 (1 -s)) F + E t β t,t+1 (1 -s)A t+1 φ t+1 +∞ z p t+1 (1 -G(z)) dG(z) = b + ηγθ t 1 -η (3.2.36) A t φ t z c t = A t φ t z p t + F (3.2.37) ρA t z f t φ t + ρE t β t,t+1 (1 -δ)A t+1 φ t+1 Ez -z f t+1 = b + ηγθ t 1 -η (3.2.38)
Using these thresholds and integrations by part, I rewrite the job creation condition (3.2.28).

γ (1 -η)q (θ t ) A t φ t = +∞ max[z c t ,z * t ] (1 -G(z)) dz + ρ max[z c t ,z * t ] z f t (1 -G(z)) dz (3.2.39)
The following proposition sums up job creation. A new match with a high productivity chooses a contract to last as long as possible: it would be a pity to split before any productivity shock hits and to lose the advantage of a high productivity draw. Thus, on average, an open-ended contract lasts longer than a fixed-term contract and enables to lock up highly productive matches. A new match with a lower initial productivity may not find it optimal to face the risk of paying firing costs in the doldrums to secure current gains. In worst cases, going back to search is the best option. In the middle ground, though, fixed-term contracts are relevant; they enable both production and a quick return to search for a better match. When meeting, a firm and a worker know that the current productivity draw will last one period.

Proposition 5. Given z p t , z c t , z f t , z * t , if z * t ≤ z f t ≤ z c t ,
z f z c z * 0 A t φ t ρA t φ t z S S p 0 S f
They do not hope a high productivity draw to last forever. Thereby, fixed-term contracts lose their role of median solution between producing and searching for more productive matches. One contract would be systematically preferred to the other without the contractual productivity gap because productivities of matches are not persistent. Still, the assumption that fixed-term contracts are per se less productive than open-ended ones is not far-fetched. Fixed-term positions are mainly filled by low-skilled or inexperienced workers [START_REF] Fontaine | CDD vs CDI: les effets d'un dualisme contractuel[END_REF]) and benefit less from on-the-job training [START_REF] Arulampalam | Training and labour market flexibility: Is there a trade-off[END_REF][START_REF] Arulampalam | Training in Europe[END_REF][START_REF] Albert | Firm-provided training and temporary contracts[END_REF][START_REF] Cutuli | Fixed-Term Contracts, Economic Conjuncture, and Training Opportunities: A Comparative Analysis Across European Labour Markets[END_REF]. Aguirregabiria and Alonso-Borrego The displayed pdf belongs to a log-Normal low with zero mean and standard deviation 0.2.

z p z f z * 0.
(2014) even estimates that fixed-term workers are 20 % less productive than open-ended workers.

The main departure of my approach with respect to [START_REF] Sala | Flexibility at the margin and labour market volatility: The case of Spain[END_REF] and [START_REF] Sala | Flexibility at the margin and labor market volatility in oecd countries*[END_REF] is the appearance of the threshold z * t . The movements in thresholds z p t , z c t , z f t and z * t shape the behavior of the labor market and its fluctuations. If job creation involves both fixed-term and open-ended contracts -as will be the case in my calibration -then Figure 3.2.6 sums up the hiring and firing policies and the position of the thresholds if idiosyncratic productivity shocks are drawn from a log-Normal distribution.

Fiscal and monetary policy

The government taxes households to provide for unemployment benefits. The revenues from the firing tax get back to the government. Thus, the budget constraint of the government is

τ t + (1 -s)G (z p t ) n p t F = bu t + g t (3.2.40)
where g t is the government expenditure and follows the AR(1) log-process log (g t ) = (1 -ρ g ) log (g)+

ρ g log (g t-1 ) + g t , g t ∼ N 0, σ 2 g .
The monetary policy is decided in accordance with the interest rate rule

log (R t /R) = ρ R log (R t-1 /R) + (1 -ρ R ) ρ π E t log P t+1 P t + ρ y log y t y + m t (3.2.41)
where y is the steady-state output and m t iid

∼ N 0, σ 2 m .
Fixed-term contracts are known to behave as buffers in front of workload fluctuations, as Saint-Paul (1996) documented. Thus, the case of indeterminacy in the Taylor rule and the subsequent appearance of sunspot equilibria may prove interesting. For the sake of simplicity, though, I shall restrain the present analysis to the determinate case with ρ π > 1 and ρ y > 0 and leave indeterminacy and its consequences on dual labor markets for future research. is .

Aggregate values and the equilibrium

This paragraph sums up the different conditions enabling an utter closing of the model. The employment values sum to the measure of households, namely 1.

n p t + n f t + u t = 1 (3.2.42)
The aggregate stock of job-seekers includes the formerly unemployed households and the new entrants in the unemployment pool from the current period.

e t = u t-1 + δn f t-1 + ξ t n p t-1 (3.2.43)
The employment variables drop by the job destruction flow and augment by the job creation flow.

n p t = (1 -ξ t ) n p t-1 + µ p t v t (3.2.44
)

n f t = (1 -δ) n f t-1 + µ f t v t (3.2.45) µ p t = q (θ t ) (1 -G (z * t )) and µ f t = q (θ t ) G (z * t ) -G z f
t are the open-ended and fixed-term job-filling rates.

ξ t = s + (1 -s)G (z p t )
0 Y i,t di = A t E z [z | z ≥ z p t ] (1 -ξ t ) n p t-1 + µ p t v t A t E z [z | z ≥ z * t ] + ρA t E z [z] (1 -δ) n f t-1 + ρµ f t v t A t E z z | z * t ≥ z ≥ z f t
Using the first order condition from the final good firm's program (3.2.3), I get

Y t ∆ t = A t E z [z | z ≥ z p t ] (1 -ξ t ) n p t-1 + µ p t v t A t E z [z | z ≥ z * t ] + ρA t E z [z] (1 -δ) n f t-1 + ρµ f t v t A t E z z | z * t ≥ z ≥ z f t (3.2.47) with ∆ t = 1 0 P i,t Pt -t
di measures price dispersion. [START_REF] Yun | Nominal price rigidity, money supply endogeneity, and business cycles[END_REF] demonstrated that the associated law of motion is

∆ t = (1 -ψ) P * t P t -t + ψ P t P t-1 t ∆ t-1 (3.2.48)
while the price level follows

P t = ψP 1-t t-1 + (1 -ψ) (P * t ) 1-t 1 1-t (3.2.49)
As detailed in Appendix 3.B.3, the log-linearization of the price-level equation (3.2.49) and the price-setting equation (3.2.4) leads to the New-Keynesian Phillips curve. π t = βE t π t+1 + κ φ t + µ t where κ = (1 -βψ)(1 -ψ)/ψ and µ t is an i.i.d mark-up cost that follows a normal law N 0, σ 2 µ .

The current specification of the New-Keynesian Phillips curve makes identification difficult in the estimation. As a result, I define σ µ = κ σ µ , which enables to rewrite the New-Keynesian Phillips curve.

π t = βE t π t+1 + κ φ t + µ t
where µ t is i.i.d and follows a normal law N 0, σ 2 µ .

Given the path of exogenous shocks

Calibration and estimation

I bridge the model to the data through two steps. First, I calibrate some parameters calibrated to a typical dual European labor markets. Then, parameters driving shock processes and the nominal rigidity are estimated through a Bayesian technique. Time is measured quarterly.

Calibration

The intended calibration exercise needs to meet two requirements to be relevant heuristically speaking. A natural objective is the faithful reproduction of labor market dualism in the Euro area.

Comparability with existing research is another one; labor market dualism has not been combined with a New Keynesian framework already, as far as I know. I shall rely on modeling choices Thomas and Zanetti ( 2009) make, as it looks at a classic labor market embedding firing costs within the Euro Area.

The central role of the distribution for idiosyncratic productivity shocks in the shaping of hiring and separation decisions falls reluctantly within the need for comparability, in absence of a proper estimation procedure. Consequently, I assume that the standard deviation for these shocks amounts to 0.2 as in [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF]. The discount factor is similarly set to 0.99. The matching function is assumed to be in a Cobb-Douglas form m (e, v) = me σ v 1-σ with σ set to 0.6, which stands in the middle of the 0.5-0.7 range [START_REF] Burda | Gross worker and job flows in europe[END_REF] estimated for some European countries. I set the workers' bargaining power to 0.6. Note that the Hosios condition is not verified as unemployment benefits are financed through a tax. The elasticity of demand curves is set to 6. I also assume that the government-spending-to-gdp ratio is 20 % as in [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF]. I also set the quarterly separation probability of open-ended matches ξ to 3 % consistently with the magnitude of data from Eurostat. This leads to a steady-state separation probability of fixed-term matches of 20 %, which is equivalent to an average duration of 15 months. 15 months lies above the average duration of fixed-term contracts in the Euro area. In this manner, the volatile character of fixed-term employment is not exaggerated on the job destruction side.

Among separations involving open-ended contracts, an essential factor is the probability of paying firing costs. [START_REF] Jolivet | The empirical content of the job search model: Labor mobility and wage distributions in Europe and the US[END_REF] explain that more sclerotic European markets lead to a higher share of voluntary quits, whereas lay-offs tend to be more significant in countries with more flexible labor markets. The French case is described as the most representative of the former phenomenon, with 70 % of separations involving open-ended matches happening through voluntary channels according to [START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF]. A reasonable value for the Euro Area is thus inferior. As a result, I target a rate of 60 % for the ratio of exogenous separations among total separations for open-ended matches. The resulting value of s is 2.1 %.

The calibration of firing costs constitutes a challenge. The data is scarce, which makes reasonable proposals difficult to spell. It is all the more true since heterogeneity between countries is high, whether it be from a legal or an economic point of view. Moreover, labor court decisions often meddle with firing decisions, bringing their share of uncertainty and hidden costs as Bentolila et al.

(

Estimation

Labor market parameters being calibrated on the Euro Area, I now estimate the unknown parameters related to shocks and nominal rigidities. I use a Sequential Monte Carlo procedure9 . Following [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF], I choose the same time period extending from 1997-Q1 to 2007-Q4 and a similar set of quarterly time series as observables (Y t , π t , R t , n t ), namely GDP at constant prices, employment, EONIA rates in annual terms and the GDP deflator 10 . All involve the 19-country Euro area and are obtained from the ECB Data Warehouse. The demeaned growth rate of the GDP deflator is plugged to inflation π t . Since R t corresponds to the nominal interest rates, they correspond to the demeaned quarterly equivalent of EONIA rates. Real GDP and employment are logged and reluctantly linearly detrended. As a matter of fact, the use of any detrending method or filter as well as the comprehension of adequate observable equations to match growth rates in the data is mainly arbitrary and may deliver heterogeneous and mis-specified estimation results. [START_REF] Ferroni | Trend agnostic one-step estimation of dsge models[END_REF] and [START_REF] Canova | Bridging dsge models and the raw data[END_REF] roughly propose to simultaneously estimate the parameters of flexible specifications for trends along deep parameters so as to let the model explain the part of the data it is able to account for. However, as [START_REF] Canova | Back to square one: Identification issues in dsge models[END_REF] and Iskrev (2010);Iskrev et al. (2010) testify, identification issues are real in the estimation of DSGE models. The resort to the methods advocated by [START_REF] Ferroni | Trend agnostic one-step estimation of dsge models[END_REF] and [START_REF] Canova | Bridging dsge models and the raw data[END_REF] would magnify these problems through the introduction of new parameters. For this reason, I stick to the detrending method employed by [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF], which remains simple and enables comparisons between my model and their own. In Appendix 3.C.3 Table 3.C.3, I report the estimation results using different filtering methods and observables ranging from 1995Q3 to 2019Q2. Overall, the estimates are pretty robust.

Most chosen priors are chosen to be diffuse, reflecting the void of the DSGE literature with respect to dual labor markets. Thus, the priors for autocorrelations of the shock processes are uniform laws on [0, 1], while standard deviations follow inverse gamma distributions with mean 0.5 and standard deviation 411 . As for the Taylor parameters r π and r y , the prior needs to be vague and embed the fact that r π > 1 and r y > 0. As a result, truncated normal laws with large standard deviations are employed. [START_REF] Druant | Firms' price and wage adjustment in europe: Survey evidence on nominal stickiness[END_REF] assess the average duration of firms' prices in the Euro Area to 10 months, which corresponds to a value of ψ around 0.7. A tight normal law around this value is subsequently chosen.

I run the Sequential Monte Carlo procedure with 500 likelihood-tempering steps, which involve a swarm of 30,000 particles and one Random-Walk-Metropolis-Hastings step each. Table 3 Government spending shocks are more volatile and more auto-correlated in the estimated classic model than in its dual counterpart. Government spending is a pure aggregate demand shock that does not impact the production function. Moreover, given the absence of capital, note that the shock on government spending g t also includes the missing values aggregate demand need to match fluctuations in GDP. Productivity shocks impact production, inflation and employment.

Experiments

Labor market moments

In this paragraph, I assess the ability of the model with respect to the reproduction of labor market moments. The absence of proper series for the computation of the latter in the Euro area represents a significant obstacle. As a result, I use time series from French data as a proxy to account for the main features of the Euro area labor market. I detail the sources in Table 3.C.2. on the open-ended side of the labor market with respect to its fixed-term counterparts. Compared to [START_REF] Sala | Flexibility at the margin and labour market volatility: The case of Spain[END_REF] and [START_REF] Sala | Flexibility at the margin and labor market volatility in oecd countries*[END_REF], my main contribution is replicating the volatility and counter-cyclicality of the share of fixed-term workers in job creation jc f /jc. [START_REF] Sala | Flexibility at the margin and labour market volatility: The case of Spain[END_REF] and [START_REF] Sala | Flexibility at the margin and labor market volatility in oecd countries*[END_REF] assume that it is exogenously set by law. [START_REF] Cole | Can the mortensen-pissarides matching model match the business-cycle facts?[END_REF] showed that replicating the Beveridge curve in real business cycles models with a frictional labor market is challenging. The actual negative correlation between unemployment

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Impulse Response Functions

The computation of impulse response functions highlights several interesting features of the model.

First of all, a substitution effect between fixed-term and open-ended contracts emerges, with longrun consequences on employment whether it be fixed-term or open-ended and a strong influence on impact. Secondly, a general-equilibrium effect appears. Indeed, shocks imply changes in the size of the job seekers' pool, which, in turn, has important implication for job creation flows. Finally, the common source of both effects being the evolution of z p , z f , z * , θ , the two effects interact through time depending on the persistence of the considered shock. As a result, viewed from the labor market, the monetary policy shock represents the negative of the government spending shock. The economic schemes at stake are the same. There is however a nuance. The effects of the monetary policy shock on the interest rate vanish much faster and the oscillations between the substitution effect and the general-equilibrium effect develop in a rougher way. This observation is also relevant when one considers the cost-push shock. The IRF of the latter will not be described, as it involves the same mechanisms as the other IRFs.

Volatility shocks As surveys report, uncertainty in demand makes firms willing to resort to fixedterm employment [START_REF] Rémy | Pourquoi les employeurs choisissent-ils d'embaucher en cdd plutôt qu'en cdi ? Dares analyses[END_REF]. What is the impact of uncertainty in my model? Solving the model with a first-order perturbation method 13 is convenient to estimate the model within a reasonable time span. However, it rules out risk as a driving force in policy functions.

To catch a glimpse of the impact of uncertainty, I strike a bargain between time feasibility and the introduction of risk. I keep the estimates obtained from the first-order perturbation method, introduce stochastic volatility and solve the model using a third-order perturbation method. The third order is the lowest one where policy functions involve innovations in stochastic volatility. This approach is far from ideal as it implicitly assumes that introducing stochastic volatility does not change the parameters estimates. It is not the case. The third-order perturbation method along the introduction of stochastic volatilities includes new terms in the response of states to shocks, which, i(scarce) n turn, should change the estimates. Ideally, I should estimate again the model with the third-order perturbation method. The seminal paper [START_REF] Bloom | The impact of uncertainty shocks[END_REF] highlights the impact of uncertainty on investment and hiring. A larger uncertainty motivates a pause in the hiring and investment decisions that involve fixed costs.

In my framework, according to this mechanism, increasing uncertainty leads to postponing openended job destruction and open-ended job creation. It encourages substitution towards fixed-term hires and the share of fixed-term contracts in job creation increases. Following [START_REF] Bloom | The impact of uncertainty shocks[END_REF], I estimate a VAR to corroborate this intuition with CAC40 and its volatility, the share of fixed-term contracts in job creation and total employment. The underpinning assumption is that shocks firstly hit stock markets, then the composition of hires and finally total employment. Including prices and other labor market variables would be better, but the short length of my quarterly time series do not allow me to do so. Figure 3.C.5 shows the impulse response functions to a one standard-deviation volatility shock. I find that the share of fixed-term contracts in hires increases while total employment

decreases. An adverse demand shock -embodied by a negative government spending shock -13 Perturbation methods consist in approaching the behavior of variables around the steady-state using Taylor expansions along a correction parameter. A third-order perturbation method uses monomials up to degree three. Judd (1998) describes it in more detail. As a result, in a dual labor market, an uncertainty shock and an adverse demand shock have different implications. It is a relevant result as the literature over macroeconomic risk argues that uncertainty shocks are difficult to distinguish from negative aggregate demand shocks [START_REF] Leduc | Uncertainty shocks are aggregate demand shocks[END_REF].

Fluctuations in fixed-term employment may help to identify uncertainty shocks and demand shocks separately. My model is a naive representation of a dual labor market and cannot pretend to do it seriously, but it should be considered in future research.

Reforms in employment protection legislation

In this paragraph, I examine the macroeconomic effects of a reform on employment protection legislation, which is summed up in firing costs here.

Steady-state analysis

The steady-state labor market can be summed up by the thresholds and the labor market tightness z p , z c , z f , z the open-ended job destruction threshold z p and the labor market tightness θ. Indeed, as labor market tightness increases, wages increase and agents demand a higher productivity when hiring.

The expected surplus of a vacancy must increase to compensate firms for higher wages. What happens to inflation dynamics? Using the job creation condition (3.2.39) to isolate φ t , the New-Keynesian Phillips curve becomes

z p (z p ) z f z f (z * ) z * (H p ) H f H p H f θ z • JD p (JD p ) JC (JC) (JP ) (JC p ) (JC p )
π t = hc t + Π f t + subs p t - κ 1 -βρ A A t + µ t
The hiring cost component hc is bound to the cost of vacancy posting.

hc t = κσ +∞ T =t β T -t E t θ T
The fixed-term hiring component Π f represents the profitability of hiring fixed-term contracts only.

Π f t = (1 -η) γ -1 κ µ f + µ p ρz f +∞ T =t β T -t E t z f T
The substitution component subs p relates to the profitability of substituting fixed-term contracts towards open-ended hiring.

subs p t = (1 -η) γ -1 κµ p z * +∞ T =t β T -t E t z * T
The variance of inflation verifies Table 3.4.3: Firing costs and inflation volatility Table 3.4.3 reports the variances of inflation and its relevant components in annual terms. As the small value of var (subs p ) bears witness of, the substitution motive towards open-ended contracts plays a minor role. It is all the more true as the contractual productivity wedge (1 -ρ) is tiny. Being able to substitute contracts without facing strong differences in production abilities ex ante matters to inflation dynamics. In the present case, the contractual composition of flows and employment can be altered in depth without influencing inflation dynamics. Meanwhile, reduced firing costs push up the profitability threshold z f to a region where it affects fewer new matches, as Figure 3.2.6 suggests. Movements in the substitution threshold z * do not alter much the expected surplus of a vacancy as ρ is close to one. Thus, the same percentage fluctuations in labor market tightness and, thus, hiring costs, require larger adjustments in the profitability threshold z f . The steady-state value z f µ p + µ f does not move much. As a result, reduced firing costs push up the variance of the fixed-term hiring component Π f . For the same reasons, the same percentage fluctuations in the profitability threshold z f require larger adjustments in hiring costs ; the volatility of labor market tightness increases and so does the volatility of the hiring cost component hc. The higher volatility of the hiring cost component and the fixed-term profitability component is compensated by the stronger negative co-variance of the labor market tightness and the profitability threshold.

var ( π) = var (hc) + var Π f + var (subs p ) + κ 2 (1 -βρ A ) 2 σ 2 A 1 -ρ 2 A + σ 2 µ +
When the labor market tightness increases after a transitory shock, the expected cost of a vacancy increases. Consequently, the expected value of a vacancy must increase. A first possible response is to increase the profitability threshold to have more productive new hires, but fewer of them. A second solution is to decrease the profitability threshold to have less productive new hires, but more of them. In my calibration, the second solution prevails, which makes the labor market tightness and the profitability threshold negatively correlated. The enhanced volatility in the hiring cost component and the fixed-term hiring component mechanically increase the magnitude of their co-variance. Overall, in theoretical terms, the dual character of the labor market does not impact inflation dynamics as long as fixed-term contracts and open-ended contracts do not embed wide productivity wedges. Numerically speaking, I tried to check robustness with respect to ρ. As I highlight above, this parameter enable dualism to arise as an equilibrium. Small changes to ρ are enough to lose dualism. Introducing some persistence in the idiosyncratic productivity of matches would help reducing the sensitivity of the model with respect to this parameter.

Conclusion

In this paper, I have described and estimated a tractable New-Keynesian model with a dual labor market. The estimated model replicates well the main moments of a typical European dual labor market and shows that fixed-term employment could help identify uncertainty shocks from negative aggregate demand shocks. Moreover, changes in firing costs do not impact inflation volatility as long as the ex ante contractual productivity wedge remains small.

A few points merit further discussion, though. Tractability comes at a high cost. The assumption that fixed-term jobs produce less than open-ended ones all else equal enable a dual equilibrium to arise, but the resulting model is very sensitive to the parameter embodying this productivity wedge. Assuming that the quality of matches undergoes i.i.d shocks every period explains this lack of robustness. Moreover, ruling out heterogeneity enables to easily delineate the mechanisms at stake, but it leads to an incomplete investigation of inflation dynamics. Inflation dynamics seem unaffected by change in firing costs as long as fixed-term and open-ended workers substitute well. Accounting 123 for firm-worker heterogeneity and rigidity in wages would make the model more realistic. I plan to work out these extensions in future research.

Appendix 3.A Proofs

Proof Proposition 5 The behavior of the thresholds is characterized by the following proposition Lemma 1. These assertions are equivalent

1. z * t > z f t 2. z * t > z c t 3. z c t > z f t Proof. Assume that z * t > z f t . The definition of z * t (3.2.35) implies that z * t = (1 -ρ) z * t + ρz * t = z c t + ρ z * t -z f t . Since z * t -z f t > 0, the latter equality implies z * t > z c t .
Assume that z * t > z c t . Again, jointly with algebraic manipulations, (3.2.35) implies that

ρz c t = -(1 -ρ) z c t + (1 -ρ)z * t + ρz f t > -(1 -ρ) z c t + (1 -ρ)z c t + ρz f t > ρz f t , which entails that z c t > z f t .
Assume that z c t > z f t . Algebraic manipulations and (3.2.35) imply that

(1 -ρ) z * t = z c t -z f t + (1 -ρ) z f t > (1 -ρ) z f t , which implies z * t > z f t .
Referring to the job creation condition (3.2.39),

If open-ended workers are the only ones hired, then max z f , z * ≤ z f , implying that z * ≤ z f .

Referring to Lemma 1, the latter inequality entails

z f ≤ z c . As a result, z * ≤ z f ≤ z c . If job creation is dual, then      0 < max z f , z * z f < z *
Using Lemma 1, the latter system of inequalities boils down to max 0, z f < z * .

For each case, the converse propositions are straightforward using (3.2.39).

Proof Proposition 6 Differentiating (3.2.36)-(3.2.37), I get.

(

1 -β(1 -ξ)) φ ∂z p ∂F = -(1 -β(1 -s)) + ηγ 1 -η ∂θ ∂F φ ∂z c ∂F = φ ∂z p ∂F + 1
Combining both equation leads to

φ ∂z c ∂F = β(1 -s)G (z p ) 1 -β(1 -ξ) + 1 1 -β(1 -ξ) ηγ 1 -η ∂θ ∂F Differentiating (3.2.35) and (3.2.38) leads to (1 -ρ)φ ∂z * ∂F = φ ∂z c ∂F -ρφ ∂z f ∂F ρ (1 -β(1 -δ)) φ ∂z f ∂F = ηγ 1 -η ∂θ ∂F
Using these equations in the differentiated (3.2.39), I get

- γq (1 -η)q 2 ∂θ ∂F = -(1 -G (z * )) (1 -ρ)φ ∂z * ∂F -1 -G z f ρφ ∂z f ∂F = -(1 -G (z * )) φ ∂z c ∂F -ρ G (z * ) -G z f φ ∂z f ∂F = -(1 -G (z * )) β(1 -s)G (z p ) 1 -β(1 -ξ) - 1 -G (z * ) 1 -β(1 -ξ) + G (z * ) -G z f 1 -β(1 -δ) ηγ 1 -η ∂θ ∂F Isolating ∂θ ∂F yields ∂θ ∂F = - (1 -G (z * )) β(1-s)G(z p ) 1-β(1-ξ) -γq (1-η)q 2 + 1-G(z * ) 1-β(1-ξ) + G(z * )-G z f 1-β(1-δ) ηγ 1-η < 0
Knowing the sign of ∂θ ∂F leads to

∂z p ∂F < 0 ∂z f ∂F < 0 ∂z c ∂F ∝ - γq (1 -η)q 2 + G (z * ) -G z f 1 -β(1 -δ) ηγ 1 -η > 0 ∂z * ∂F > 0

3.B Detailed Calculations

3.B.1 Nash-bargaining joint surpluses

Using the different definitions of surpluses with the free-entry condition -namely equations (3.2.14), (3.2.15), (3.2.27) and (3.2.9) -I get

S p t (z t ) = z t -U t + E t β t,t+1 U t+1 + (1 -E t β t,t+1 (1 -s)) F + E t β t,t+1 (1 -s) max S p t+1 (z), 0 dG(z) (3.B.1)
Meanwhile, the definition of U t (3.2.12) yields

U t = p (θ t ) max W 0,p t (z) -U 0 t , W f t (z) -U 0 t , 0 dG(z) + U 0 t (3.B.2)
Nash-sharing rules (3.2.7) imply that

U t = p (θ t ) η 1 -η max J 0,p t (z), J f t (z), E t β t,t+1 V t+1 dG(z) + (1 -p (θ t )) U 0 t (3.B.3)
But the definition of V t (3.2.5) and the free entry condition (3.2.27) imply that

max J 0,p t (z), J f t (z), E t β t,t+1 V t+1 dG(z) = γ q (θ t ) (3.B.4)
Since p (θ t ) = θ t q (θ t ), the definition of U t boils down to

U t = ηγθ t 1 -η + U 0 t (3.B.5)
Consequently, using (3.2.13) the outside value of an incumbent worker is

U t = b + ηγθ t 1 -η + E t β t,t+1 U t+1 (3.B.6)
Reintroducing (3.B.6) into (3.2.20) leads to the following expression for the surplus of continuing permanent contracts

S p t (z t ) = A t z t φ t -b - ηγθ t 1 -η + (1 -E t β t,t+1 (1 -s)) F + E t β t,t+1 (1 -s) max S p t+1 (z), 0 dG(z) (3.B.7)
Following the same steps, I find that

S 0,p t (z t ) = S p t (z t ) -F (3.B.8)
As for temporary contracts, equations (3.2.18), (3.2.19) and (3.2.27) boil down to

S f t (z t ) = ρA t z t φ t -U t + E t β t,t+1 (1 -δ) S f t+1 (z)dG(z) + U t+1 (3.B.9)
Using (3.B.6), the surplus of incumbent fixed-term contracts is

S f t (z t ) = ρA t z t φ t -b - ηγθ t 1 -η + E t β t,t+1 (1 -δ) S f t+1 (z)dG(z) (3.B.10)

3.B.2 Steady-state equations

Given parameters β, σ, η, σ z , , F, b, δ, ρ, m, γ, we need to derive the steady-state values of R, c, Y , n p , n f , u, ∆, z p , z c , z f , z * , θ, φ, v, π, A, g. Some of them can be directly computed through the following equations

R = 1/β φ = -1 P = P * = ∆ = 1 A = 1
Given (z p , θ), many variables can be derived in a tractable way.

q = mθ -σ p = θq U = b + ηγθ 1 -η z c = z p + F φ z f = U -ρφ(1 -δ)βEz ρφ (1 -β(1 -δ)) z * = z c -ρz f 1 -ρ ξ = s + (1 -s)G (z p ) µ p = p (1 -G (z * )) µ f = p G (z * ) -G z f
Then, one can circumscribe (z p , θ) by solving the following system numerically.

φz p + (1 -β(1 -s)) F + (1 -s)βφ +∞ z p (1 -G(z)) dz = U γ (1 -η)φq = +∞ z * (1 -G(z)) dz + ρ z * t z f (1 -G(z)) dz
Solving the following linear system yields n p , n f , u, e .

n p + n f + u = 1 e = u + δn f + ξn p n p = (1 -ξ) n p + µ p e n f = (1 -δ) n f + µ f e
The resulting expressions are

n p = δµ p ξ(1 -δ)µ f + δ(1 -ξ)µ p + ξδ n f = ξµ f ξ(1 -δ)µ f + δ(1 -ξ)µ p + ξδ u = 1 -n p -n f e = u + ξn p + δn f
Now, the steady-state values v, Y , g and c can be obtained.

v = θe Y = (1 -ξ) E z [z | z ≥ z p ] n p + µ p E z [z | z ≥ z * ] e + ρ (1 -δ) E z [z] n f + ρµ f E z z | z * ≥ z ≥ z f e g = Y g Y c = Y -g -γv

3.B.3 Log-linearization

The Euler equation can be log-linearized starting from 3.B.11) where π t = P t /P t-1 denotes inflation.

c t = E t c t+1 -- cu u -1 R t -E t π t+1 (
The definition of price level dynamics (3.2.49) can be log-linearized as

π t = (1 -ψ) P * t -P t-1 (3.B.12)
The retailers' price-setting equation (3.2.4) becomes

P * t = (1 -βψ)E t +∞ k=0 (βψ) k P t+k + µ t+k + φ t+k (3.B.13)
Substracting P t-1 on each side of this equation, we get

P * t -P t-1 = (1 -βψ)E t +∞ k=0 (βψ) k P t+k -P t-1 + µ t+k + φ t+k (3.B.14) = π t + (1 -βψ) (µ t + φ t ) + (1 -βψ)E t +∞ k=1 (βψ) k P t+k -P t + µ t+k + φ t+k (3.B.15) = π t + (1 -βψ) φ t + µ t + βψ E t P * t+1 -P t (3.B.16)
Using the definition of mark-up shocks µ t and using (3.B.12), I get

π t = βE t π t+1 + κ φ t + µ t (3.B.17) where κ = (1 -βψ)(1 -ψ)/ψ.
The log-linearization of exogenous processes yields

A t = ρ A A t-1 + A t g t = ρ g g t-1 + g t
The other log-linearizations of equations ( 3

Aφz p A t + z p t + φ t + β(1 -s) Aφ +∞ z p (1 -G(z)) dz -F (c t -E t c t+1 ) + β(1 -s)Aφ +∞ z p (1 -G(z)) dz E t A t+1 + E t φ t+1 -(1 -G (z p )) z p E t z p t+1 - ηγθ 1 -η θ t = 0 ρAφz f A t + z f t + φ t + ρβ(1 -δ)Aφ Ez -z f E t A t+1 + E t φ t+1 + c t -E t c t+1 -ρβ(1 -δ)Aφz f E t z f t+1 - ηγθ 1 -η θ t = 0 (1 -ρ)z * z * t -z p z p t + F Aφ A t + φ t + ρz f z f t = 0 γ (1 -η)φq(θ) -A t -φ t + σ θ t + (1 -ρ) (1 -G (z * )) z * z * t + ρ 1 -G z f z f z f t = 0 R t = ρ R R t-1 + (1 -ρ R ) ρ π E t π t+1 + ρ y Y t + m t v t = θ t - (1 -ξ p ) θn p v n p t-1 - (1 -δ) θn f v n f t-1 + (1 -s)θg (z p ) z p n p v z p t n p t = (1 -ξ) n p t-1 -(1 -s)z p g (z p ) z p t + (1 -G (z * )) q(θ)v n p v t -σ θ t - z * g (z * ) q(θ)v n p z * t n f t = (1 -δ) n f t-1 + G (z * ) -G z f q(θ)v n f v t -σ θ t + q(θ)v n f z * g (z * ) z * t -z f g z f z f t Y t = c Y c t + g Y g t + γv Y v t Y t = A t + (1 -s) +∞ z p zg(z)dz n p Y n p t-1 + ρ (1 -δ) Ezn f Y n f t-1 - (1 -s)n p (z p ) 2 g (z p ) Y z p t -(1 -ρ) (z * ) 2 g (z * ) vq(θ) Y z * t -ρ z f 2 g z f vq(θ) Y z f t + +∞ z * zg(z)dz + ρ z * z f zg(z)dz vq(θ) Y v t -σ θ t

3.C Estimation

3.C.1 Calibration

I carry out the calibration using known parameters and steady-state targets. A first block of steady-state values and parameters is tractable.

φ = -1 u = 1 -n n f = n n f n n p = n -n f s = ξ s ξ δ = 1 -n f n µ f µ p +µ f n f n 1 - µ f µ p +µ f ξ e = u + δn f + ξn p µ p = ξn p e µ f = δn f e
Knowing ξ, s and σ z , it is possible to numerically solve for z p using the fact that ξ = s + (1 -s)G (z p ).

Then, the following system in (w p , U, F ) -where U = b + ηγθ 1-η -is tractable.

w p = ηφE [z | z ≥ z p ] + η(1 -β(1 -s))F + (1 -η)U γ = (1 -η)φq +∞ z * (1 -G(z)) dz + ρ z * z f (1 -G(z)) dz b = U - ηγθ 1 -η m = qθ σ
The parameters (F, b, s, δ, ρ, m, γ) are all determined at this point

3.C.2 Data

t t + h 0 1 2 3 4 5 -0.2 -0.1 0.0 0.1 0.2 t Rt+h 0 1 2 3 4 5 -0.2 -0.1 0.0 0.1 0.2 0.3 t nt+h 0 1 2 3 4 5 -0.2 -0.1 0.0 0.1 0.2 0.3 t jc p t + h 0 1 2 3 4 5 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 t jc f t + h 0 1 2 3 4 5 -0.3 -0.2 -0.1 0.0 0.1 0.2 t (jc f /jc)t+h 0 1 2 3 4 5 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 t n f t + h 0 1 2 3 4 5 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 t jd p t + h 0 1 2 3 4 5 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 t jd f t + h 0 1 2 3 4 -0.4 -0.2 0.0 0.2
jc f t n f t + h 0 1 2 3 4 5 -1.00 -0.75 -0.50 -0.25 0.00 jc f t jd p t + h 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 jc f t jd f t + h 0 1 2 3 4 0.0 0.3 0.6 0.9 jc f t vt+h 0 1 2 3 4 5 -0.6 -0.4 -0.2 0.0 (jc f /jc)t Yt+h 0 1 2 3 4 5 -0.3 -0.2 -0.1 0.0 0.1 0.2 (jc f /jc)t t + h 0 1 2 3 4 5 -0.4 -0.2 0.0 0.2 (jc f /jc)t Rt+h 0 1 2 3 4 5 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 (jc f /jc)t nt+h 0 1 2 3 4 5 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 (jc f /jc)t jc p t + h 0 1 2 3 4 5 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 (jc f /jc)t jc f t + h 0 1 2 3 4 5 0.0 0.3 0.6 0.9
(jc f /jc)t (jc f /jc)t+h 0 1 2 3 4 5 -0.50 -0.25 0.00 0.25 0.50 The x-axis is the lag h and the y-axis is the correlation between the variable xt and the variable xt+h. The solid line is the simulated value, the dashed line is the 95 % confidence interval around the latter and the dotted line is the value from the data. Yt nt+h 0 1 2 3 4 5 -2.5 0.0 2.5 5.0 7.5

(jc f /jc)t n f t + h 0 1 2 3 4 5 -0.5 0.0 0.5 1.0 (jc f /jc)t jd p t + h

Yt jc p

t + h 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 Yt jc f t + h 0 1 2 3 4 5 -2 -1 0 1 Yt (jc f /jc)t+h 0 1 2 3 4 5 -2 -1 0 1 Yt n f t + h 0 1 2 3 4 5 -4 -2 0 2 4 Yt jd p t + h 0 1 2 3 4 5 -2 -1 0 1 Yt jd f t + h 0 1 2 3 4 5 -2 0 2 4
Yt vt+h (jc f /jc)t jc f t + h 0 1 2 3 4 5 0 5 10 15

nt jc p t + h 0 1 2 3 4 5 -2 -1 0 1 2 3 nt jc f t + h 0 1 2 3 4 5 -2 -1 0 1 nt (jc f /jc)t+h 0 1 2 3 4 5 -2 -1 0 1 2 nt n f t + h 0 1 2 3 4 5 -6 -4 -2 0 2 nt jd p t + h 0 1 2 3 4 5 -2 -1 0 1 2 nt jd f t + h
(jc f /jc)t (jc f /jc)t+h 0 1 2 3 4 5 -4 -2 0 2 (jc f /jc)t n f t + h 0 1 2 3 4 5 0 10 20 30 (jc f /jc)t jd p t + h 0 1 2 3 4 5 -4 -2 0 2 (jc f /jc)t jd f t + h
A t z p t φ t + (1 -E t β t,t+1 (1 -s)) F + E t β t,t+1 (1 -s)A t+1 φ t+1 +∞ z p t+1 (1 -G(z)) dG(z) = b + ηγθ t 1 -η z c t = z p t + F t A t φ t γ (1 -η)A t φ t q (θ t ) = +∞ z c t [1 -G(z)] dz n p t = (1 -ξ t ) n p t-1 + µ p t e t v t = θ t 1 -(1 -ξ t ) n p t-1 Y t = c t + g t + γv t Y t ∆ t = A t E z [z | z ≥ z p t ] (1 -ξ t ) n p t-1 + (1 -G (z c t )) q (θ t ) v t A t E z [z | z ≥ z c t ] E t +∞ T =t β t,T ψ T -t P T Y T P * i,t P T -µφ T = 0 log (R t /R) = ρ R log (R t-1 /R) + (1 -ρ R ) ρ π E t log P t+1 P t + ρ y log y t y + m t log (A t /A) = ρ A log (A t-1 /A) + A t log (g t /g) = ρ g log (g t-1 /g) + g t 3.D.2 Log-linearization c t = E t c t+1 -- cu u -1 R t -E t π t+1 Aφz p A t + z p t + φ t + β(1 -s) Aφ +∞ z p (1 -G(z)) dz -F (c t -E t c t+1 ) + β(1 -s)Aφ +∞ z p (1 -G(z)) dz E t A t+1 + E t φ t+1 -(1 -G (z p )) z p E t z p t+1 - ηγθ 1 -η θ t = 0 z c z c t = z p z p t - F Aφ A t + φ t γ (1 -η)φq(θ) -A t -φ t + σ θ t + (1 -G (z c )) z c z c t = 0 n p t = (1 -ξ) n p t-1 -(1 -s)z p g (z p ) z p t + (1 -G (z c )) q(θ)v n p v t -σ θ t - z c g (z c ) q(θ)v n p z c t v t = θ t - (1 -ξ) θn p v n p t-1 + (1 -s)θg (z p ) z p n p v z p t Y t = c Y c t + g Y g t + γv Y v t Y t = A t + (1 -s) +∞ z p zg(z)dz n p Y n p t-1 - (1 -s)n p (z p ) 2 g (z p ) Y z p t -(z c ) 2 g (z c ) vq(θ) Y z c t + +∞ z c zg(z)dz vq(θ) Y v t -σ θ t π t = βE t π t+1 + (1 -βψ)(1 -ψ) ψ φ t + µ t R t = ρ R R t-1 + (1 -ρ R ) ρ π E t π t+1 + ρ y Y t + m t

Introduction

In Western Europe, fixed-term employment has considerably expanded over the last decades. Fixedterm contracts are now pervasive in job creation flows. As the celebrated paper [START_REF] Shimer | Reassessing the ins and outs of unemployment[END_REF] shows, job creation flows mostly account for unemployment fluctuations. Understanding why agents prefer hiring through one type of contract is thus essential. A few papers tried to theorize the contractual choice at the hiring stage using matching models. Surprisingly, though, if papers have reviewed this literature, none has assessed the commonly used assumptions and their consequences over job creation. I try to bridge that gap.

I find that two main modeling strategies arise. Some papers model fixed-term contracts as highly flexible; firm-worker matches may split at zero cost any time. On the contrary, other papers assume that fixed-term contracts are rigid; separations cannot occur before reaching the stipulated end date.

Matching models with flexible fixed-term contracts tend to predict job creation flows with fixed-term contracts only, which is counter-factual. As a result, some papers introduce ad hoc hiring rules with a settled share of open-ended contracts in job creation. Matching models with rigid fixed-term contracts, on the contrary, generally leave room for both open-ended and fixed-term contracts in job creation. However, I show that models with rigid fixed-term contracts lack robustness when it comes to the ranking of contracts in job creation. Overall, my results suggest that a matching model with realistic implications would strike a balance between flexible and rigid fixed-term contracts, avoiding both ad hoc hiring rules and robustness issues rigid fixed-term contracts bring along.

Why do flexible and rigid fixed-term contracts shape job creation differently? As the modeling of open-ended contracts displays little variability across papers, the differences stem from the modeling of fixed-term contracts. To be more specific, assumptions impact the surplus of fixed-term matches, which in turn shapes the way job creation resorts to open-ended and fixed-term contracts. Flexible fixed-term contracts deliver a higher surplus than rigid fixed-term contracts, as rigid fixed-term contracts may force some matches to stay together after an adverse productivity shock, for example.

Thus, in uncertain environments, job creation would naturally favor flexible fixed-term contracts to rigid fixed-term contracts.

I propose a constructivist thought experiment to delineate the consequences of both rigid and This new ranking of contracts in job creation leads to a counter-factual result if any new match can maximize the joint surplus of a fixed-term match with respect to its average duration. As the optimal average duration increases with the productivity of the match, the least productive fixed-term matches are the most likely to be converted into open-ended contracts.

Many papers review the literature. The closest paper is [START_REF] Dolado | Drawing Lessons from the Boom of Temporary Jobs in Spain[END_REF], which presents models and stylized facts of labor market dualism. Beyond including new papers, I focus on matching models, whereas the scope of [START_REF] Dolado | Drawing Lessons from the Boom of Temporary Jobs in Spain[END_REF] is wider. [START_REF] Boeri | Institutional Reforms and Dualism in European Labor Markets[END_REF] describes the evolution of labor market institutions over time and reviews the empirical literature assessing the impact of employment protection on unemployment. It also builds a theoretical model to assess the impact of labor market institutions on unemployment. [START_REF] Bentolila | Dual Labour Markets Revisited[END_REF] depicts theoretical and empirical insights about the impact of employment protection on dual labor markets.

The first section reviews the literature of matching models of dual labor markets. The second section builds a variety of matching models with rigid fixed-term contracts and highlights some robustness issues. The third section concludes.

Literature review

In this section, I review the literature dealing with labor market dualism using matching models.

Note that there are also models describing labor demand when firms face adjustment costs2 . I first delineate two polar categories of papers with respect to the modeling of fixed-term contracts. Then, I review the papers and their approaches towards job creation in a dual labor market.

Flexible or rigid fixed-term contracts?

Considering papers with both fixed-term and open-ended contracts, the chosen specifications to model open-ended contracts look alike.

In general, open-ended contracts stipulate a firing cost that needs to be paid in case of a split. Some academics also include a cost-less separation channel in their models. Papers mostly differ on the fixed-term side of the labor market. In that regard, most models can be classified in two main categories: models with rigid fixed-term contracts and models with flexible fixed-term contracts.

Rigid fixed-term contracts cannot split before the stipulated expiry date is reached, or it is expensive to do so. Flexible fixed-term contracts, on the contrary, can be terminated any time at no cost. What does the actual law say about it? In France, a fixed-term worker may leave his position for an open-ended contract without compensating the firm. Otherwise, regardless gross misconducts or exceptional situations, the unilateral end of a fixed-term contract before its due date is costly to the requester. Concerted terminations are possible and entail no transactions, though.

Both the rigid and flexible approaches to the modeling of fixed-term contracts are relevant. On the one hand, the fact that most fixed-term contracts reach their expiry date in the data supports the idea that fixed-term contracts are rigid; concerted terminations seldom occur. In France, Milin At the other end of the spectrum, rigid fixed-term contracts blur the ranking of contracts in job creation. In a model with random search and stochastic productivity, rigid fixed-term contracts are no longer systematically delivering a higher total surplus than open-ended contracts. After an adverse productivity shock, a fixed-term match cannot split and must endure a low surplus for a while. Thus, in very bad times, open-ended contracts could be better; splitting at some cost may be better than bearing a fixed-term match with a depressed productivity. As [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF][START_REF] Cahuc | Taxation of Temporary Jobs: Good Intentions with Bad Outcomes?[END_REF] Other models focus on heterogeneity in workload fluctuations. [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF][START_REF] Cahuc | Taxation of Temporary Jobs: Good Intentions with Bad Outcomes?[END_REF] build a continuous-time model where firms and workers randomly meet according to a matching function.

Matches face adverse productivity shocks that occur at heterogeneous rates. Open-ended contracts Once it has arrived, the firm is bound to pay a firing cost. Between the notification and the firing permission issuance, the match delivers the lowest possible productivity and the worker is paid the average wage in the economy. Job creation is constrained; new-matches are directed towards each type of contract with a given probability. If job creation was unconstrained, since new matches deliver all the same productivity, new firm-worker pairs would simply opt for the type of contract that delivers the highest joint surplus. As a result, job creation would always occur through either open-ended or fixed-term contracts.

Rigid fixed-term contracts: a constructivist approach

The following sections build assumption by assumption various models with rigid fixed-term contracts to delineate robustness issues and the assumptions responsible for them. The first subsection describes the common core of these models. The second subsection focuses on models with heterogeneous processes of workload fluctuations. The last subsection describes models with heterogeneous productivities and the robustness issues that arise in models with rigid fixed-term contracts.

Initial assumptions

In this subsection, I build a basic model as a starting point. I shall extend this model over various dimensions and see the way job creation changes in the next subsections. I am particularly interested in the contractual composition of created jobs and the ranking of contract types in job creation.

Time is continuous. Firms as well as workers are identical ex ante. Firms post vacancies to attract workers and may employ one worker. I assume that the number of new matches per unit of time follows a function with constant returns to scale and the number of vacancies and job seekers as inputs. Meeting rates only depend on the labor market tightness θ, which is the ratio of the number of vacancies and job seekers. Firms meet unemployed workers with probability q(θ), while unemployed workers face a probability p(θ) ≡ θq(θ) of finding a vacancy. A firm-worker pair produces y per unit of time, whether it be under an open-ended or a fixed-term contract. Matched firms and workers face a shock that makes their associations unproductive with probability λ per unit of time. This sudden drop in productivity may stem from a demand drop as well as a failure from the firm, the worker or the match itself.

Newly matched firms and workers maximize their expected joint surplus using either fixed-term or open-ended contracts.

Open-ended jobs stipulate a wage and may end through two channels: one involving firing costs and one which does not. The adverse productivity shock involves a split with firing costs F . The firing cost is a red-tape, dead-weight cost. It is not a firm-worker transfer such as a severance payment. In

real life, open-ended matches may also end without entailing the payment of a firing cost (quits and retirement for example). I denote s the probability of cost-less separations per unit of time.

Fixed-term contracts stipulate a duration and a wage. For now, I assume that the duration of fixed-term contracts is fixed in expectation. I model it through a memory-less job destruction rate δ.

In other words, the duration of fixed-term contracts is δ -1 in expectations and follows an exponential law with parameter δ. One may think it is unrealistic, as fixed-term contracts specify a duration and not a probabilistic job destruction rate. A Poisson process for fixed-term job destruction does not change conclusions qualitatively speaking, while it makes expressions more tractable and elegant than definite durations. Note that a productivity shock does not end a fixed-term contract in contrast with open-ended contracts. The firm has to pay the initially negotiated wage up to the end of the contract.

At this point, the trade-off is shaped. The new firm-worker pairs choose between a long open-ended contract with a costly separation in case of an adverse productivity shock, and a short fixed-term contract with a cost-less separation and potentially unproductive times.

I assume that the hiring decisions are jointly efficient to rule out incentive compatibility constraints and other theoretical difficulties related to firm-worker asymmetry. Newly matched firms and workers choose the contract that maximizes their expected joint surplus; the wage is then pinned down according to a Nash-bargaining rule. Note that the wage is set once for all and does not change until the match splits. I shall relax this assumption later in the paper.

I also assume that the vacancy-posting activity brings no profit to firms. Otherwise, as there are no barrier to entry, new comers would compete until profits are washed out. This so-called free-entry condition is classic in the literature [START_REF] Mortensen | Job creation and job destruction in the theory of unemployment[END_REF]. I denote V the present discounted value of a vacancy.

V = 0 (4.2.1) I denote J p and W p the firm's and worker's expected surpluses from an open-ended contract with wage w. J f and W f are their fixed-term counterparts. I denote S p and S f the joint surpluses from an open-ended contract and a fixed-term contract. The joint surpluses do not depend on the wage as it is a simple transfer between the firm and the worker. U is the present discounted utility from unemployment. The joint surpluses verify

S p = J p -V + W p -U S f = J f -V + W f -U
The joint surplus associated with open-ended contracts does not include firing costs. A disagreement over the negotiation of wages does not incur the payment of firing costs for the firm. Firing costs do not belong to the firms' outside option while agents negotiate wages.

A firm employing a worker under an open-ended contract with wage w earns (y -w) per unit of time.

These flows stop if the match splits, whether it be with or without firing costs. An open-ended match splits without firing costs with probability s. An adverse productivity shock strikes with probability λ and involves the payment of firing costs F . As for the worker, he earns w per unit of time until the match ends.

rJ p = y -w + s(V -J p ) + λ(V -F -J p ) rW p = w + (s + λ)(U -W p )
I assumed that open-ended matches split when an adverse productivity shock hits. It has to be consistent to do so, though; separations need to be jointly efficient. It implies that the firm-worker pair renegotiates a wage w after the adverse productivity shock. The joint surplus associated with continuing the match must be negative for the separation to be jointly efficient. Under which conditions is it the case? I denote J c and W c the surpluses of a firm and a worker carrying on the match after an adverse productivity shock.

rJ c = -w + s(V -J c ) rW c = w + s(U -W c )
The associated joint surplus S c is defined by

S c = J c -(V -F ) + W c -U
In contrast with the surplus of a new open-ended match, the surplus of a continuing match includes the firing cost. If the firm and the worker cannot agree on a new wage, splitting now involves paying the firing cost; the firm-worker pair formed a while ago at that point.

A little algebra provides an expression for S c .

S c = F - rU r + s
The following assumption needs to hold for the surplus of a continuing open-ended match to be negative.

Assumption 1. F < rU/(r + s)

If the firing cost is too high, it may be better to carry the burden of a zero productivity until a cost-free separation shock hits instead of immediately paying the firing cost.

A fixed-term contract reaches its end with probability δ per unit of time. The match faces an adverse productivity shock with probability λ, in which case the productivity of the match reaches zero. It bears no consequence on the worker, who earns the initially negotiated wage up to a job destruction shock.

rJ f (y, w) = y -w + λ(J f (0, w) -J f (y, w)) + δ(V -J f (y, w)) rW f (w) = w + δ(U -W f (w))
Using the free-entry condition (4.2.1) and the expressions of surpluses above, a little algebra leads to the following joint surpluses.

S p = y -λF -rU r + s + λ S f = y r + δ + λ - rU r + δ

Heterogeneous workload fluctuations

In this section, I extend the basic framework with heterogeneous arrival rates of adverse productivity shocks λ. I also assume that new firm-worker pairs optimize the fixed-term job destruction rate δ. Then, I broaden the model to make possible conversion of expired fixed-term contracts into open-ended ones. Appendix 4.A.1 includes all the proofs of the following propositions.

Basic framework

As in [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF] and [START_REF] Cahuc | Taxation of Temporary Jobs: Good Intentions with Bad Outcomes?[END_REF], I assume first that jobs are heterogeneous in the arrival rate of adverse productivity shocks. [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF] assumes that firms pay a sunk cost to draw a shock arrival rate λ from a distribution G and then maintain a vacancy. Once the match splits, the firm has to pay to draw a new value of the shock arrival rate λ. For the sake of simplicity, I assume that a job-specific arrival rate of productivity shocks λ is drawn from distribution G when a firm-worker pair forms. Another difference with [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF] is that durations follow an exponential law instead of being fixed. In the current model, agents optimize the expected duration of fixed-term jobs instead of the duration itself. Qualitatively speaking, it keeps job creation as in [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF] and eases the following exposition of robustness issues. With these assumptions, the value of a vacancy verifies

rV = -γ + q(θ)(1 -η) max S p (λ), S f o (λ), 0 dG(λ)
where γ is the maintaining cost of the vacancy, η is the worker's share of the joint surplus and S f o (λ)

is the optimized joint surplus of fixed-term contracts with shock arrival rate λ.

S f o (λ) = max δ≥0 S f (λ, δ)
Joint surpluses now depend on the arrival rate of productivity shocks λ.

Considering open-ended jobs, I assume that there are no cost-less separations to stick to the framework of [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF]: for now, s = 0. I consider the case s > 0 later in this section.

Thus, the joint surplus of an open-ended job with an arrival rate of adverse productivity shocks λ boils down to

S p (λ) = y -λF -rU r + λ
Proposition 7 describes the behavior of S p .

Proposition 7. S p is continuous and decreasing. Moreover, S p (0) = y/r-U > 0 and lim λ→+∞ S p (λ) = -F < 0. Thus, there exists a unique λ p such that S p (λ p ) = 0.

The higher the probability of adverse productivity shocks, the less relevant open-ended contracts are. As λ increases, open-ended contracts shorten, which brings forward firing costs.

In addition to the arrival rate of productivity shocks, the surplus associated with fixed-term contracts also depends on the fixed-term job destruction rate δ.

Proposition 8. The function δ → S f (δ, λ) reaches a maximum at δ * (λ) such that

δ * (λ) = λ y rU -1 -1 -r
Moreover, δ * (0) = -r < 0 and lim λ→+∞ δ * (λ) = +∞. δ * being continuous, there exists a unique λ such that δ * (λ) = 0.

δ * increases with the shock arrival rate λ. The more frequently adverse productivity shocks strike, the less the optimal fixed-term contract lasts in expectation. The optimal duration finds a middle ground between too short contracts that are still productive when they expire and too long contracts that lock down the agents in unproductive matches for some time.

S p (λ) = y -λF -rU r + λ -c
The following proposition describes its behavior.

Proposition 11. S p is continuous and strictly decreasing in λ. Moreover, S p (0) = y r -U -c and lim λ→+∞ S p (λ) = -F -c. Hence, there exists a unique λ p such that S p (λ p ) = 0.

As for fixed-term contracts, they are worth converting if no adverse productivity shock has struck before the termination shock. The conversion of fixed-term contracts into open-ended contracts entails the renegotiation of the wage according to a Nash-sharing rule. Thus, the conversion decision is efficient for both the firm and the worker. The joint surplus from a fixed-term contract with destruction probability δ and an adverse shock probability λ is

S f (λ, δ) = y + δS p (λ) + r + δ + λ - rU r + δ -c (4.2.2) Proposition 12. The function δ → S f (δ, λ) reaches a maximum at δ * (λ) such that δ * (λ) =      λ λ(F +c)+r(U +c) rU -1 -1 -r if λ ≤ λ p λ y rU -1 -1
-r otherwise δ * is increasing in λ and there exists a unique λ such that δ * (λ) = 0. The solid line is the curve of the optimal fixed-term job destruction rate δ * when fixed-term contracts are convertible; the dashed line when they are not.

The following proposition details the behavior of the optimized joint surplus of fixed-term jobs The next section shows that the model with heterogeneous arrival rates of adverse productivity shocks may lead to counter-factual results if the productivity of new matches is no longer fixed.

S f o (λ) = max δ≥0 S f (λ, δ).
More specifically, regardless the arrival rate of productivity shocks, the shortest fixed-term contracts are the most likely to be converted into open-ended contracts when the productivity of matches is heterogeneous.

rV = -γ + q(θ)(1 -η) max S p (y), S f (y), 0 dG(y) (4.2.3)
where η is the worker's share of the joint surplus and joint surpluses now depend on the productivity of the match y. Using the free-entry condition (4.2.1), the definition of V (4.2.3) and integrations by parts, the job creation condition arises. In this framework, open-ended contracts serve to lock up highly productive matches. The match must be productive enough for immediate gains to exceed the expected firing costs. Open-ended contracts make the best out of a high productivity draw, as they last longer than fixed-term contracts.

γ (1 -η)q(θ) = 1 r + s + λ +∞ max(y c ,y * ) (1 -G(y))dy + 1 r + δ + λ max y f ,y * y f (1 -G(y))dy (4.
Fixed-term contracts provide a proper option when productivity is not that high; they strike a balance between going back to searching for a high productivity draw and producing for a while. contracts entails the renegotiation of the wage according to a Nash-sharing rule. Thus, the conversion decision is efficient for both the firm and the worker. I denote w and w the negotiated wages when fixed-term jobs are created and converted. The firm's and worker's surpluses from a fixed-term contract with productivity y write

rJ f (y, w) = y -w + λ(J f (0, w) -J f (y, w)) + δ max J p (y, w ), 0 -J f (y, w) rW f (y, w) = w + λ(W f (0, w) -W f (y, w)) + δ max W p (y, w ), U -W f (y, w)
The joint surplus of a fixed-term contract with productivity y boils down to

S f (y) = y + δS p (y) + r + δ + λ - rU r + δ
The definition of y c and y f remain the same. The following proposition describes job creation.

Optimized duration and i.i.d productivity shocks

The following analysis shows that the models with heterogeneous arrival rate in adverse productivity shocks as well as heterogeneous productivities of matches lead to counter-factual results when some assumptions are relaxed: the shortest fixed-term contracts are the most likely to be converted into open-ended ones and the most productive matches are hired under fixed-term contracts.

I relax the assumption that productivity shocks are necessarily adverse. Now, I assume that they are independent and identically distributed and drawn from a law with cumulative distribution function

S c (z) = (J c (z) -[V -F ]) + (W c (z) -U )
In contrast with the surplus of new open-ended contracts, firing costs belong to the outside option of firms with continuing open-ended workers. The associated firm's and worker's surpluses are

rJ c (y) = y -w c (y) + s (V -J c (y)) + λ max J c y , V -F -J c (y) dG y rW c (y) = w c (y) + λ max W c y , U -W c (y) dG y + s (U -W c (y))
where w c (y) is the negotiated wage through Nash bargaining. Therefore, the joint surplus boils The duration of fixed-term contracts is endogenously chosen to maximize the joint surplus when a worker-firm pair forms and conversions into open-ended contracts become possible when fixed-term contracts end. The surplus associated with a job creation through a z-productivity fixed-term contract denoted as S f o (z) now includes the choice of the instantaneous expiration probability δ and verifies

S f o (y) = sup δ≥0 S f (y, δ) -c
where S f (y, δ) denotes the joint surplus of a fixed-term contract with productivity y and expiration probability δ.

Firm's and worker's surpluses of a fixed-term contract become

rJ f (y, δ) = y -w f (y) + λ J f y , δ -J f (y, δ) dG(y ) + δ max [J p 0 (y), V ] -J f (y, δ) rW f (y, δ) = w f (y) + λ W f y , δ -W f (y, δ) dG y + δ max [W p 0 (y), U ] -W f (y, δ)
The resulting joint surplus of a fixed-term contract with productivity y and job destruction rate δ write (r + λ + δ)S f (y, δ) = y -rU + δS p 0 (y) + + λ S f y , δ dG y (4.2.6)

The job creation condition is now

γ (1 -η)q(θ) = max S p (y), S f o (y), 0 dG(y)
The following proposition characterizes the optimal duration of fixed-term contracts.

Proposition 18. Let y be in the support of G.

If Ey/r < α ≡ U + S p (y ) + dG (y ), the optimal expiration rate δ * verifies

δ * (y) =            +∞ if x(y) ≤ 0 λ 1+ 1+x(y) x(y) -r if 0 < x(y) < λ r 2 + λ r 0 otherwise where x(y) = y -rα -(r + λ) S p (y) + -S p (y ) + dG (y ) rα -Ey Otherwise, δ * (y) ∈ {0, +∞}
Ey/r -α is the expected surplus of a fixed-term contract with zero probability of destruction after a productivity shock. The gain is the present discounted value of expected production Ey/r, while the losses α are the present discounted values of a return into unemployment U and the expected value of a conversion into a open-ended contract S p (y ) + dG (y ). These two events do not occur under a zero probability of fixed-term job destruction, which explains why they appear as losses.

Conversely, a productivity shock has a probability one to hit such an eternal fixed-term contract, which explains the role of the expected surplus as a hiring criterion.

If Ey/r ≥ α, a no-end fixed-term contract is expected to be profitable considering the impact of productivity shocks throughout its existence. Therefore, if the immediate surplus of a new match -namely the surplus before any productivity shock hits -is not too low, a zero probability of destruction is optimal. Otherwise, an immediate destruction is preferable and an infinite probability of destruction is chosen.

Conversely, if Ey/r < α, a no-end fixed-term contract has an expected negative surplus after a productivity shock. This encourages firms to shorten the stipulated durations of fixed-term contracts in order to avoid productivity shocks. Intuitively, if the productivity of the match is neither too low nor too high to opt for an infinite or zero probability of job destruction, there is room for optimization in terms of durations. The contract must be long enough to benefit from the current level of productivity but short enough to avoid losses associated with a productivity shock, which is expected to be detrimental. In that regard, an increase in the probability of shock occurrence pushes up the destruction probability for a given productivity y.

The probability of destruction decreases with the productivity of the firm-worker pair. Importantly, proposition 18 and its ramifications are still valid if the match chooses the duration instead of the destruction probability of a fixed-term contract as is the case in [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF].

According to proposition 18, the next assumption is necessary to rule out labor markets with polar durations of fixed-term contracts.

Assumption 2. Ey/r < α

The following proposition states the optimal choice between fixed-term and open-ended contracts in function of the productivity of the match. The only caveat of fixed-term contracts with a finite duration is a superior administrative costs. In case of conversion, the contracting cost is paid twice. Firms and workers no longer strike a balance between productivity and flexibility as in the previous model. The sole compromise takes place between flexibility and hiring costs.

When productivity is moderate, the match can opt for a short fixed-term contract. On one hand, this provides a flexibility gain: if an adverse productivity shock occurs, the contract will end up quickly.

The shorter the contract, the thinner this advantage. On the other hand, if an open-ended contract is beneficial, the contracting cost is paid twice. The shorter the contract, the heavier this drawback.

Consequently, open-ended contracts tend to be preferred to very short fixed-term contracts whenever the former are beneficial. Conversely, when productivity is high, the hiring costs become small compared to the flexibility gains a longer fixed-term contract provides. As productivity converges towards infinity, a no-term fixed-term contract is even better than a open-ended contract. Somehow, the rigid open-ended contract is more flexible than a no-end fixed-term contract, which constitutes a better device to lock a firm-worker pair with a high productivity as there is a zero probability of separation.

When the hiring cost is high, the scope for short fixed-term contracts is reduced and open-ended contracts turn out to be more attractive. Finite fixed-term contracts are irrelevant as they impose paying twice hiring costs. Job creation only occurs through no-end fixed-term contracts and openended contracts. The limit point is the expected difference in surplus a finite fixed-term contract provides when a productivity shock occurs compared to a no-end fixed-term contract, namely -λ (Ey/r -α) /r. This situation is not realistic. Therefore, a high contracting cost is incompatible with a proper fit of the data, hence the following additional assumption.

Assumption 3. 0 < c < λ r α -Ey r

The necessary theoretical foundations are elaborated enough to demonstrate the inadequacy of such a model to match data. Under assumptions 2 and 3, if job creation is dual, the resulting equilibrium is such that the shortest fixed-term contracts are the most likely to be converted into open-ended contracts, which is at odds with the data. The only way to remedy this problem is to consider equilibria with job creation through fixed-term contracts exclusively. Open-ended job creation solely occurs through conversion of fixed-term contracts. This is not relevant considering the ridiculous empirical probabilities of the latter event. Interestingly, notice that introducing heterogeneous firms in terms of shock arrival rate λ does not change these findings. To this extent, the model of [START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF] critically relies on the hypothesis that new jobs have a fixed productivity. In this manner, a one-to-one link can be established between the distribution of the productivity shock arrival rate λ and the distribution of fixed-term matches' durations. Assuming heterogeneous productivities for new matches leads to the same counter-factual conclusions for each value of the shock arrival rate λ.

Conclusion

In this paper, I have reviewed random-matching models accounting for labor market dualism. I have shown that the contractual layout of job creation crucially depends on the modeling of fixed-term contracts.

When fixed-term matches can costlessly split in response to shocks, fixed-term contracts are flexible;

job creation tends to favor fixed-term contracts. One way leave room for open-ended jobs in the labor market equilibrium is to make expiring fixed-term contracts convertible into open-ended contracts, or constrain job creation. In that regard, many papers direct a settled share of new matches to each type of contract, which prevents studying contractual substitution effects in job creation.

On the contrary, when fixed-term matches cannot split before their stipulated end date regardless undergone adverse shocks, fixed-term contracts are rigid. In that case, job creation may leave room for both contracts. Job creation does not only occur through fixed-term contracts and there is no need for ad hoc job creation rules to get both types of contracts involved. Contractual substitution effects are not shut down at the hiring stage.

The nice features of models with rigid fixed-term contracts come at a high price; the resulting models lack robustness. In models with heterogeneous productivities, the ranking of contracts Thus, S f (λ) > S p (λ) for all λ ≤ λ.

If λ > λ, ∆ can be expressed as

∆(λ) = F λ 2 -2 √ rU √ y - √ rU λ + r √ y - √ rU 2 λ(r + λ)
The sign of ∆ boils down to the sign of the numerator, which is a second-degree polynomial function.

It has two positive roots λ * 1 and λ * 2 .

λ * 1 = √ rU √ y - √ rU F 1 -1 - F U λ * 2 = √ rU √ y - √ rU F 1 + 1 - F U
It is possible to locate these roots with respect to the thresholds λ and λ p .

λ * 1 = √ rU √ y - √ rU F 1 -1 - F U < √ rU √ y - √ rU F 1 -1 - F U = λ λ = √ rU √ y - √ rU F 1 -1 - F U < √ rU √ y - √ rU F 1 + 1 - F U = λ * 2 λ * 2 = √ rU √ y - √ rU F 1 + 1 - F U < √ rU √ y - √ rU F 1 + y rU = λ p
The ordering λ * 1 < λ < λ * ≡ λ * 2 < λ p implies that Note also that S p (0) = y r -U -c and lim λ→+∞ S p (λ) = -F -c. Hence, there exists a unique λ p such that S p (λ p ) = 0. As a result, δ * is increasing in λ. Moreover, δ * is continuous and δ * (0) = -r, lim λ→∞ δ * (λ) = +∞.

           S p (λ) > S f (λ) > 0 if λ ≤ λ * S f (λ) > S p (λ) > 0 if λ * < λ < λ p
Thus, there exists a unique λ such that δ * (λ) = 0.

Proposition 13

Proof. For a given λ, 2 cases arise in the program S f o (λ) = max δ≥0 S f (λ, δ)

If δ * (λ) > 0, id est if λ > λ then δ * (λ) is the unique maximizer of δ → S f (δ, λ) and S f o = S f (δ * (λ), λ).

Otherwise, if δ * (λ) ≤ 0, id est if λ ≤ λ, δ * (λ) does not comply with the non-negativity constraint over the optimal δ. Note that δ → S f (δ, λ) is continuous and decreasing over the Proof. I denote ρ p = 1/(r + s + λ) and ρ f = 1/(r + δ + λ). As mentioned above, ρ p > ρ f . Assume that y * > y f . (4.2.4) implies that ρ p y * = ρ p -ρ f y * + ρ f y * = ρ p y c + ρ f y * -y f . Since y * -y f > 0, the latter equality implies y * > y c . Assume that y * > y c . Again, jointly with algebraic manipulations, (4.2.4) implies that ρ f y c = -ρ p -ρ f y c + (ρ p -ρ f )y * + ρ f y f > -ρ p -ρ f y c + (ρ p -ρ f )y c + ρ f y f > ρ f y f , which entails that y c > y f . Assume that y c > y f . Algebraic manipulations and (4.2.4) imply that ρ p -ρ f y * = ρ p y c -y f + ρ p -ρ f y f > ρ p -ρ f y f , which implies y * > y f .

Proposition 16

Proof.

If F ≥ rU/(r + δ), y c ≥ y f . Referring to proposition 15, the latter inequality entails The first inequality inequality stems from the fact that s < δ, which ensures that (r +s+λ) -1 > (r + δ + λ) -1 .

As a result, the only hires with a productivity lower than y c occur through fixed-term contracts with a minimal productivity y f . As for productivities y > y c , using the linearity of S f and S p , one may show that fixed-term contracts still dominate open-ended ones. -If x(y) ≤ -1, ∂S f /∂δ ≥ 0 and δ * (y) = +∞ -If -1 < x(y) < 0, P y has two negative roots and, thus, is positive on (0, +∞). Consequently, S f is increasing in δ over the latter interval and δ * (y) = +∞ -If x(y) = 0, P y is linear and positive over (0, +∞) and δ * (y) = +∞.

-If x(y) > 0, P y has one negative and one positive root, the latter verifying r + δ 0 = λ 1+ 1+x(y)

x(y)

. Either δ 0 < 0, in which case S f decreases over (0, ∞) and δ * (y) = 0, or δ 0 ≥ 0, in which case S f attains a maximum at δ * (y) = δ 0 ∈ (0, ∞).

When do we have δ 0 ≥ 0 ? The latter condition is equivalent to r(1 + x(y)) -λ 1 + x(y) -(r + λ) ≤ 0, which is -again -a second-degree polynomial in 1 + x(y) with roots -1 and (λ + r)/r.

Consequently, δ 0 ≥ 0 if and only if -1 ≤ 1 + x(y) ≤ (λ + r)/r. Since x(y) > 0, this is true whenever x(y) ≤ λ r 2 + λ r .

If rα < Ey, sign ∂S f ∂δ = -sign (P y (r + δ)). One may revisit the cases tackled above.

-If x(y) ≤ 0, S f is decreasing in δ over (0, +∞) and δ * (y) = 0 -If x(y) > 0, P y has one negative and one positive root, the latter verifying r + δ 0 = λ 1+ 1+x(y)

x(y)

. Either δ 0 < 0, in which case S f increases over (0, ∞) and δ * (y) = +∞, or δ 0 ≥ 0, in which case S f attains a minimum at δ 0 ∈ (0, ∞) and δ * (y) ∈ {0, +∞}.

Proposition 19

Proof. The choice between a fixed-term and a open-ended contract at the hiring stage can be summed up in the sign of the function ∆ defined as ∆(y) = S f 0 (y) -S p 0 (y) + . The latter can be rewritten as Conversely, if λ r + c(r+λ) rα-Ey = λ r + β 2 1 + r λ ≤ λ r 2 + λ r , algebraic manipulations directly entail that β 2 ≤ λ r 2 , which proves that β < λ r .

We now have all the necessary information to circumscribe the equilibria with dual job creation.

If β < λ r , ∆(y) ≥ 0 if and only if x(y) ≥ (2 + β) β. In other words, fixed-term contracts are hired whenever x(y) ≥ (2 + β) β. x being increasing in y, there is room for job creation through open-ended contracts if and only if x (y c ) < (2 + β) β.

If β ≥ λ r , ∆(y) ≥ 0 if and only if x(y) ≥ λ r + β 2 1 + r λ . Job creation is dual if and only if x (y c ) < λ r + β 2 1 + r λ . In this case, all fixed-term contracts have a zero probability of job destruction.

To end the proof, notice that β < λ r is equivalent to c < λ r α -Ey r . Moreover, when c = 0, revisiting each point above ensures that hiring through a fixed-term contract is weakly preferable to

hiring through a open-ended contract.

4.B.2 Additional graphs

I

  am intellectually indebted to Marc Audran, my mathematics teacher in Mathématiques Spéciales, who taught me rigor and passed on his passion for mathematics to me. I would like to thank my in-laws Clotilde, Vincent, Pierre, Marie, Éloi, Maximilien, Élizabeth, Nathanaël, Silouane, Paul-Étienne and Marthe for their practical assistance, especially during the lock-down. I would like to express my gratitude to my parents Catherine and Max for their wholehearted support throughout all these years. iii Writing down a Ph.D. dissertation comes close to asceticism sometimes. I would like to thank my friends for their emotional support and the relaxing moments we shared. I would like to offer my special thanks to my spouse Marguerite-Marie and my daughter Irène for their patience and unwavering support. iv 3.C.1 Prior and posterior distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.C.2 Simulated and data cross-correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.C.3 Simulated and data cross-covariances . . . . . . . . . . . . . . . . . . . . . . . . . . 3.C.4 Historical decomposition of observable variables with respect to shocks . . . . . . . 3.C.5 IRF to a one-standard deviation volatility shock -VAR estimation. . . . . . . . . . . 4.2.1 Optimal duration and arrival rate of adverse productivity shocks . . . . . . . . . . . 4.2.2 Arrival rate of productivity shocks and job creation . . . . . . . . . . . . . . . . . . . 4.2.3 Surpluses and arrival rate of adverse productivity shocks . . . . . . . . . . . . . . . 4.2.4 Optimal duration and arrival rate of adverse productivity shocks . . . . . . . . . . . 4.2.5 Arrival rate of productivity shocks and job creation: the case S f o (λ p ) ≤ 0 . . . . . . . 4.2.6 Arrival rate of productivity shocks and job creation: the case S f o (λ p ) > 0 . . . . . . . 4.2.7 Hiring open-ended contracts only . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.8 Dual job creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.9 Hiring fixed-term contracts only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.10 Dual job creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.11 Dual job creation when 0 < c < λ r α -Ey r . . . . . . . . . . . . . . . . . . . . . . 4.2.12 Dual job creation when c ≥ λ r α -Ey r . . . . . . . . . . . . . . . . . . . . . . . . . 4.A.1 Arrival rate of productivity shocks and job creation: dual case . . . . . . . . . . . . . 4.A.2 Arrival rate of productivity shocks and job creation: open-ended jobs only . . . . . . 4.B.1 Job creation with open-ended jobs only . . . . . . . . . . . . . . . . . . . . . . . . . 4.B.2 Dual job creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

  Figure 1.1: A sharp increase in the proportion of fixed-term contracts over the last decades POPULATION: Employees who are between 15 year-old and 64 year-old and live in France, apprentices excluded. SOURCE: Insee, enquête Emploi Fixed-term employees include fixed-term workers (Contrat à Durée Déterminée in French) and interim workers.

Figure 1 . 2 :

 12 Figure 1.2: Contractual composition of hires POPULATION: Employees who are between 15 year-old and 64 year-old and live in France, interim workers and apprentices excluded SOURCE: Acoss-Urssaf, Déclarations Préalables à l'Embauche

  settles a discrete-time model where firms experience decreasing returns to scale and use labor as sole input. Investment in labor occurs one period before production and is financially constrained. Both contracts are perfect substitutes, but fixed-term contracts are less productive than open-ended contracts. Fixed-term contracts last one-period and reducing open-ended employment requires firing costs. Financial constraints encourage firms to use fixed-term employment rather than open-ended ones out of fear of paying for future firing costs. It is even more the case for expanding firms, which suffer more from financial frictions. Cao et al. (2010) highlights how on-the-job search may impact the contractual choice at the hiring stage. Firm-worker matches are heterogeneous in productivity. All else equal, open-ended contracts offer higher wages than fixed-term contracts because of firing costs. Firms tend to offer open-ended contracts to most productive matches to avoid losing the worker through on-the-job search. As for lower productivities, losing the worker through on-the-job search does not incur a big loss and a fixed-term contract fits the bill.

  makes a case for reducing firing costs for workers with volatile and lower productivity. It would favor employment of workers usually trapped in streams of unemployment and short temporary jobs. More generally, a large body of literature deals with active labor market policies including vocational training, subsidized employment and wages or job search assistance 12 . Despite the involvement of important fiscal resources, results are rather controversial.

  draws. The model is flexible; when firing costs are low, only open-ended contracts emerge in equilibrium. If firing costs are high enough, both open-ended and fixed-term contracts intervene in job creation. The model being entirely tractable, it is possible to analytically derive transitions after changes in firing costs. I calibrate the model to French data and find that a contractual substitution effect and a general-equilibrium effect shape steady-state. I notably study the transition from a benchmark dual equilibrium towards a labor market with open-ended contracts only and a reduced firing cost. I find that it is adverse to open-ended employment and takes up to several years to complete. I also find that frequent stochastic changes in firing costs tend to increase average fixed-term employment.Employment protection affects the way firms optimize their labor force to maximize profits. It is all the more true with a segmented labor market, where labor reallocation essentially takes place on the fixed-term side of the labor market. Fixed-term contracts being a short-term tool, firms are likely to use them a lot to face shocks, especially if they face price rigidity. Thus, employment protection impacts the way firms adjust prices in response to shocks on costs and demand. Fluctuations in fixed-term employment could reflects firms' expectations over demand and costs. I tackle these issues in my third chapter, where I build a New-Keynesian model with a dual labor market. The New-Keynesian framework is standard and includes price rigidity, aggregate productivity shocks, cost-push shocks, monetary policy shocks and government-spending shocks. The model embeds a classic discrete-time Mortensen-Pissarides model: firms and workers negotiate wages using Nash bargaining and face i.i.d productivity shocks. Open-ended contracts are protected by a firing tax and fixed-term contracts have a given average duration. In contrast with the second chapter, I assume that fixed-term contracts are less productive than open-ended contracts all else equal. New firm-worker pairs trade off between flexibility and productivity at the hiring stage: the agents choose the contract yielding the highest joint surplus at job creation. It is worth hiring most productive matches using open-ended contracts as expected firing costs are worth less than the productivity gain such contracts yield compared to fixed-term contracts. As for matches with an intermediate productivity, fixed-term contracts are better. I calibrate some parameters using Euro area labor market moments. I estimate the parameters associated with nominal rigidities and shocks with Euro Area quarterly time series. using a Sequential Monte Carlo method. I do not use time series that are specific to a dual labor market for estimation; I use inflation, interest rate, GDP and employment time series. Still, the model replicates well labor market moments. Ideally, I would need Euro area data about labor market dualism to assess the fit of the model. Unfortunately, I do not dispose of such data. I use French labor market data data to assess the fit of the model as it is available on a quarterly basis and seems a good middle ground in terms of employment protection among Western European countries. The model replicates well the counter-cyclical share of fixed-term jobs in job creation and the Beveridge curve, which is notably difficult to obtain in real business cycle models with matching frictions. I find that a change in firing costs does not change inflation volatility but alters dynamics of inflation components. Using a third-order perturbation method and introducing stochastic volatility in shock processes, I find that uncertainty shocks increase fixed-term employment, whereas a negative demand shock decreases fixed-term employment. Fixed-term employment could provide a way of disentangling negative demand shocks and uncertainty shocks. It is all the more relevant as telling apart uncertainty shocks and negative demand shocks is a notably difficult task in macroeconomics with standard models.Fluctuations in job creation flows mostly explain unemployment fluctuations. Yet, few models account for the choice between fixed-term and open-ended contracts in job creation. Many models use ad hoc hiring rules that specify the share of new matches directed to each type of contract. My fourth chapter reviews the literature dealing with segmented labor markets using matching models from the perspective of job creation. I delineate a distinction based on the modeling of fixed-term contracts. Some papers model fixed-term contracts as flexible; fixed-term jobs can be terminated anytime at no cost. Others assume that fixed-term matches cannot split before reaching their stipulated end. Assuming fixed-term contracts as flexible makes them highly desirable compared to open-ended contract and the associated firing costs. If job creation were to occur without constraints, new matches would tend to hire through fixed-term contracts only. It is counter-factual as job creation involves open-ended contracts as well. As a result, some papers resort to hiring rules which assign each new match to a given contract type with a given probability. On the contrary, modeling fixed-term contracts as rigid leaves room for both contracts in job creation. Rigid fixedterm contracts are less profitable than flexible ones; a match may have to bear a depressed joint surplus for a while after an adverse shock as it cannot split before the stipulate end date. This enables open-ended contracts to be involved in job creation along with fixed-term contracts. Models with rigid fixed-term contracts lack robustness, though. In particular, the ranking of contracts in job creation chapters 2 and 3 display changes if expiring fixed-term contracts are convertible into open-ended contracts. Most productive matches end up in fixed-term contracts, whereas open-ended contracts are relegated to mediocre productivities. If new firm-worker pairs can optimize the average duration of fixed-term contracts, I show that the shortest contracts are the most likely to be converted

  more fixed-term contracts and favor the emergence of an equilibrium with coexisting fixed-term and open-ended contracts. The second effect is a general-equilibrium effect. Higher firing costs reduce open-ended job destruction. Since most workers operate under open-ended contracts, aggregate job destruction decreases and so does unemployment. In turn, the depressed unemployment rate reduces fixed-term job creation flows.As for the social welfare and employment, I find that there is no free lunch when the dual nature of the labor market is preserved: a benevolent planner cannot both decrease unemployment and improve welfare around the baseline economy. The optimal policy consists in a large cut in firing costs, which both increases welfare and decreases unemployment. If the cut in firing costs is large enough, fixed-term employment is no longer relevant and only open-ended employment remains at the equilibrium. Consequently, this model supports a transition towards a unique open-ended contract, a reform the literature has extensively tackled 3 . In terms of dynamics, an increase in firing costs with respect to the baseline calibration leads to a several-year long adjustment. Indeed, flows are reduced on the open-ended side of the labor market, which tends to slow down transitions. Conversely, a strong decrease in firing costs enhances open-ended job creation and destruction flows, which reduces the adjustment duration. Between these two endpoints, the behavior of the adjustment time after a moderate cut in firing posts is non-monotonous and intricate. Transitions toward a unique-contract equilibrium with a cut in firing costs end by 2 years. Taking account of frequent marginal employment protection legislation reforms 2.1. INTRODUCTION CHAPTER 2. WAITING FOR THE PRINCE CHARMING -one every 7 months on average in France -I extend the model to include uncertainty with respect to firing costs. This policy risk significantly bolsters fixed-term employment on behalf of open-ended employment and pushes up unemployment. My paper relates to three strands in the literature. First, it contributes to explain the dynamic interaction between fixed-term and open-ended contracts as Saint-Paul (1996) and Dolado et al. (2002) do.

  build a matching model where firm-worker pairs face heterogeneous arrival rates in adverse productivity shocks. Firms-worker pairs opt for fixed-term contracts when adverse productivity shocks are frequent, whereas open-ended contracts are beneficial when adverse productivity shocks are unusual. Expiring fixed-term contracts can be converted into open-ended contracts. In addition, firm-worker pairs optimize hired fixed-term contracts' duration. Cahuc et al. (2016) calibrate the model on French data and find that higher firing costs do not change aggregate employment much but dramatically increase turnover on the fixed-term side of the market . Cahuc et al. (2019) extend the model of Cahuc et al. (2016) and assume that the resort to fixed-term contracts is taxed depending on the duration of contracts. Taxes may be refunded if the fixed-term contract is converted into an open-ended contract one at expiry. The authors estimate the model on French data and find a detrimental effect of taxes on the duration of fixed-term contracts. My paper departs from

  7 and enables them to extract a rent. Moreover, continuing open-ended workers benefit from a supplementary rent ηF/(1 -η) when compared with fixed-term workers and new open-ended workers. The firing cost pushes up the open-ended workers' bargaining position by enhancing the threat of a costly separation. The firm will reward the worker through the wage for avoiding the separation and its associated cost F . Thus, the firing cost influences labor market outcomes through two channels: as a pure firing tax and through wages. The firing cost is not involved in Nash bargaining for a new open-ended match since the worker is not yet an insider.

  main trade-off between fixed-term and open-ended contracts is set between long-lasting rigid openended contracts and very short fixed-term contracts. Consequently, the advantage of immediately paying the firing cost when fixed-term contracts are very short is thin when the firm can wait for the stipulated termination date to get rid of the worker. For the sake of simplicity, I assume that firms always prefer to wait for the end of fixed-term contracts instead of paying firing costs. The results barely change when this subtlety is accounted for. Here, the separation probability δ implicitly models the average duration of fixed-term contracts. Secondly, the model does not embed the possibility of conversion into an open-ended contract when the stipulated duration of the fixed-term contract is reached, whereas the convertibility option actually exists. French data shows that fixed-term contracts seldom convert into open-ended contracts. The quarterly probability of transition from fixed-term to open-ended employment constitutes an upper bound of the conversion probability and amounts to 7.4 % according to Hairaultet al. (2015). Meanwhile, the distribution of the duration of fixed-term contracts has its mean at 1.5 months and a median located around 5 days as[START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF] mentions. Thus, these 7.4 % more likely include multiple round trips between fixed-term employment and unemployment somehow ending into open-ended employment rather than unique and direct fixed-term-to-open-ended-employment trajectories. Consequently, the conversion rate probably lies far below transition rates from fixedterm to open-ended employment. Our calibration being carried out on a monthly basis, the value of the conversion probability becomes insignificant. The no-convertibility assumption is not innocuous in theoretical terms, as I demonstrate in chapter 4.

Figure 2 . 1 :

 21 Figure 2.1: Hiring open-ended contracts only

Figure 2 Figure 2 . 2 :

 222 Figure 2.2: Dual job creation

Figure 2 . 3 :

 23 Figure 2.3: An increase in firing costs in the dual Mortensen-Pissarides model

Figure 2 .

 2 Figure 2.3 diagrammatically sums up the movements of the different loci equations (JD p ) (2.23), z f (2.25), (z * ) (2.32) and (JC) (2.34) after an increase in firing costs.

1 and 2 Figure 2 . 4 :

 224 Figure 2.4 sums up the different sort of equilibria in terms of contractual composition depending on firing costs.

  of fixed-term contracts in job creation of 83 % 15 . I consider that exogenous splits of open-ended matches constitute 70 % of separations. As for the calibration of the French firing costs, I rely on[START_REF] Kramarz | The shape of hiring and separation costs in france[END_REF]. Individual lay-offs marginally cost 4 months of the median wage, while the marginal cost of lay-off within a collective-termination plan represents 12 months of the median wage16 . The former being the most frequent case, I reckon that total firing costs represent 4months of the open-ended workers' average wage. As in[START_REF] Bentolila | The macroeconomic impact of flexible labor contracts, with an application to spain[END_REF] and[START_REF] Cahuc | Explaining the spread of temporary jobs and its impact on labor turnover[END_REF], I assume that red-tape costs actually embodied by firing costs only represent one third of total firing costs. Thus, I target a ratio of 4/3 for firing costs F with respect to the monthly open-ended workers' average wage. I also target the quarterly vacancy filling rate to 70 %, which is equivalent to a monthly rate of 33 %. As for the productivity shock arrival λ, there is no consensus stemming from the empirical literature. I assume a yearly average frequency. The calibration results

Figure 2 .

 2 5 displays the steady-state transition probabilities and their evolution with respect to firing costs. As firing costs increase, the separation probability of open-ended contracts decreases. In the same manner, the open-ended-contract job-finding rate decreases with firing costs while the fixed-term-contract job-finding rate increases. Magnified firing costs encourage the substitution towards fixed-term contracts. When firing costs are higher than F , the substitution effect is responsible for the lower open-ended job finding rate in a dual labor market compared with a classic labor market with open-ended contracts only. Employees' outside 14 Source: Recensement de la population 2016 -Insee 15 Average from Q1-2000 to Q2-2019 computed with data from Acoss -Urssaf (Déclarations préalables à l'embauche)

Figure 2 .

 2 Figure 2.5: Steady-state transition probabilities, employments and firing costs

  previously described. A general-equilibrium effect is responsible for this behavior. Since open-ended employment represents the bulk of workers, unemployment tends to decrease as the open-ended job destruction probability shrinks. Even though higher firing costs push up the fixed-term jobs' hiring probability, the latter reduction in unemployment is strong enough to push down the job creation flow into fixed-term employment. Since fixed-term job destruction rates are constant, fixed-term employment shrinks. In the French data, a minor share of open-ended matches' separations induce the payment of a firing cost. The calibrated economy is close to the corner equilibrium with no endogenous open-ended job destruction. The French economy is set in the region where the generalequilibrium effect of an increase in firing costs outreaches the substitution effect. In graphic terms, open-ended employment increases with firing costs in the neighborhood of the baseline firing costs materialized by the dashed vertical line F in Figure 2.5.

Figure 2 .

 2 Figure 2.6 displays steady-state social welfare on the left panel and average productivities on the right panel as firing costs vary.

Figure 2 . 6 :

 26 Figure 2.6: Evolution of social welfare and productivities with respect to firing costs

  higher firing costs encourage the substitution towards fixed-term contracts on behalf of open-ended contracts in the neighborhood of z * because z * increases with firing costs. This makes fixed-term contracts more productive on average. The right-hand side graph of figure2.6 shows that those two effects cancel out over the chosen interval of firing costs.The average productivity is higher in the dual framework than in the classic framework. The substitution account for this phenomenon. In the dual framework, most job creations occur through fixed-term contracts, and these new jobs are necessarily productive enough to generate a positive joint surplus. Moreover, fixed-term jobs are short enough to remain productive throughout their whole life span. In a classic framework, new open-ended jobs are more and more productive as well, but job creation flows are smaller because of firing costs and the continuing open-ended jobs are less and less productive as job destruction flows tighten. The higher the firing costs, the more the average productivity in the dual framework dominates the average productivity in the classic framework.The overall behavior of welfare results from the intertwined productivity-related and employmentrelated mechanisms. As open-ended contracts cover most workers, the negative impact of firing costs on the average productivity of open-ended jobs dominates and the total average productivity decreases with firing costs. Since employment moves between 0.7 and 0.8 in response to changes in firing cost, the decreasing average productivity drives the welfare as the left panel of Figure2.6.Why is the steady-state welfare higher in a dual labor market than in a classic labor market? The unemployment rate and the average productivity are higher in the dual framework. Theses two opposite effects lead to an ambiguous result in terms of social welfare. The difference between both frameworks is to be found on the open-ended job destruction side and the welfare loss firing costs induce. Open-ended job destruction rates are comparable between the dual and the classic framework. However, open-ended employment is higher in the classic framework. The welfare loss associated with open-ended job destruction is thus higher in the classic framework than in the dual framework. The latter mechanism tilts the balance in favor of the dual labor market, which dominates the classic labor market in terms of welfare.

  The speed of adjustment of an economy after a reform constitutes an important factor from the policy maker's point of view. It depends both on employment values and employment transition probabilities. Employment transition probabilities are forward looking because they are functions of the labor market tightness and the thresholds pinpointing job destruction and job creation are forward-looking variables. Therefore, they jump immediately to the new steady-state values after an unexpected change in the parameters 17 . As for employment values, they are stock variables. These results are valid for both the classic and dual Mortensen-Pissarides model. Figure 2.7 displays the time at which 99 % of the adjustment is completed for each type of employment. Starting from the baseline value, a cut in firing costs immediately increases the open-ended job creation and destruction probabilities. Flows thicken in the open-ended side of the labor market. Meanwhile, on impact, the least productive continuing open-ended matches now deliver a negative surplus and split; open-ended employment dives and unemployment soars.

Figure 2 .

 2 Figure 2.7: τ 99% of different reforms with respect to the post-reform firing costs in months

Figure 2 .

 2 A.6 in Appendix 2.A shows such a case when the average duration between productivity shocks is 3 years. In the classic framework, employment immediately gets close to its new steady-state value when post-reform firing costs are half way between the baseline firing costs and F . In the dual framework, a similar balance between the new steady-state employment values, the new transition probabilities and the employment impact responses shape dynamics. A novel force emerges in contrast with the classic framework; the fixed-term job creation rate decreases as firing costs decrease. Starting from the baseline economy, as post-reform firing costs decrease, the impact response sets open-ended employment further and further from its new steady-state value. The higher open-ended job transition rates cannot overcome this mechanism when firing costs are close to the baseline. The presence of fixed-term contracts that capture a decreasing but still important share of job creations extends the adjustment time. Adjustment times reach a maximum. As postreform firing costs further decrease, the prominence of fixed-term contracts reduces and open-ended transition rates increase enough to overcome the impact response of open-ended employment. Adjustment takes up to 3 years. In the same manner as in the classic framework, if by chance the impact response brings employment close to its new steady state value, transitions are much shorter. As figure 2.A.3 in Appendix 2.A shows, when the average duration between two productivity shocks is one month, the impact response of open-ended employment brings it close to its new steady state value when post-reform firing costs are a bit higher than F . Starting from the baseline value, an increase in firing costs reduces the transition probabilities on the open-ended side of the labor market. Since open-ended matches represent the bulk of employees in the baseline calibration, the new unemployment value depends on open-ended employment to reach its new steady-state value. The sluggish motion of open-ended employment slows down the adjustment of unemployment. The higher fixed-term job creation rate is not high enough to overcome the low transition rates on the open-ended side of the labor market. It reflects on the adjustment of fixed-term employment, which essentially relies on job creation. A 3-per-cent increase in firing costs needs an adjustment of 30 months, which represents 2.5 years.

Figure 2

 2 Figure 2.8: Post-reform dynamics

Figure 2 .

 2 Figure 2.10: Extend of regulatory uncertainty and employment Steady-state employments when = 0 and the value of F is the baseline. Parameters are estimated for different values of | F1-F F |=| F2-F F

  Figure 2.A.1: Steady-state employment values and firing costs

Figure 2 .

 2 Figure 2.A.3: τ 99% of different reforms with respect to the post-reform firing costs in months

Figure 2 .Figure 2 .

 22 Figure 2.A.4: Steady-state employment values and firing costs

Figure 2 .

 2 Figure 2.A.6: τ 99% of different reforms with respect to the post-reform firing costs in months

Figure 2 .

 2 Figure 2.A.7: Steady-state employment values and firing costs

Figure 2 .Figure 2 .

 22 Figure 2.A.8: Social welfare and firing costs

Figure 2 .Figure 2 .

 22 Figure 2.A.10: Steady-state employment values and firing costs

Figure 2 .

 2 Figure 2.A.12: τ 99% of different reforms with respect to the post-reform firing costs in months

  right away. Denoting δ 0 Dirac function in 0, open-ended and fixed-term contracts, equations (2.35)-(2.36) are amended as follows

  the 1980s, European governments introduced fixed-term contracts in labor markets otherwise characterized by stringent employment protection legislations. Employment protection changes the way firms compose their staff. In this manner, employment protection impacts the firms' pricing decisions in response to shocks on demand and costs and, thus, impact inflation dynamics. Fixedterm contracts being more flexible than regular contracts, they should be one of firms' preferred tools to deal with shocks, especially if pricing decisions are constrained. As a result, fixed-term employment fluctuations should reflect well firms' changes in expected costs and demand. Fixedterm employment could thus play the role of a short-term macroeconomic indicator if its fluctuations are better understood. The literature has seldom considered fluctuations in dual labor markets and its response to shocks. I try to bridge that gap. How does a dual labor market respond to most common types of shocks? How does dualism impact inflation dynamics? To address these questions, I estimate a New-Keynesian dynamic stochastic general-equilibrium model with a dual labor market using Euro area data. The New-Keynesian block of the economy remains classic. As for the labor market, I extend the traditional Diamond-Mortensen-Pissarides framework to embed fixed-term and open-ended contracts. Firms post vacancies, unemployed workers search for a job and firm-worker pairs arise following a matching function. New firm-worker pairs differ in quality and optimally choose between going back to search, or carrying on either with an open-ended or a fixed-term contract. While laying off an unprofitable open-ended contract entails the payment of a firing cost, fixed-term contracts split at zero cost and are less productive all else equal. Thus, a trade-off between productivity and flexibility drives the choice between a fixed-term and an open-ended hire. When the quality of the match is high enough, the productivity gains overcome the future payment of firing costs in bad times; the best way to make the most out of the match is hiring through an open-ended contract. Otherwise, the fear of firing costs prevails and a fixed-term contract is preferred. I estimate the log-linearized model on quarterly data from the Euro area. Overall, the model is able to fit the cyclical features of a dual labor market despite being estimated on classic general macroeconomic time series -Output, nominal interest rates, inflation and employment. One main contribution is the model ability to account for the fluctuations in contractual composition of hires.It is not the case in the few references studying business cycles and dualism. The model fits well the counter-cyclicality of the share of fixed-term contracts in job creation. Solving the estimated model with a third-order perturbation method, I find that fixed-term employment responds oppositely to a negative aggregate demand shock and to a volatility shock on aggregate productivity. This result suggests that fixed-term employment could be used to tell apart uncertainty shocks and negative aggregate demand shocks, which is a well-known difficulty in the literature. My last main contribution involves inflation dynamics. I find that dualism does not impact inflation volatility as long as the ex ante productivity wedge between open-ended and fixed-term contracts remains small.The assumption that fixed-term workers produce less than open-ended workers ceteris paribus is fundamental to obtain a dual labor market as equilibrium. If fixed-term and open-ended workers with the same idiosyncratic quality produced the same, hires would involve either fixed-term workers or open-workers only. While fundamental, my assumption substantiates with empirical work. Fixed-term workers are likely to undergo successions of short employment periods and long unemployment spans[START_REF] Fontaine | CDD vs CDI: les effets d'un dualisme contractuel[END_REF].[START_REF] Christopher | Loss of skill during unemployment and the persistence of employment shocks[END_REF] shows that the latter reduce concerned workers' skills. Moreover, fixed-term workers benefit less from on-the-job training 2 . The productivity-flexibility trade-off is highlighted in several papers related to labor market dualism.[START_REF] Caggese | Financing constraints and fixed-term employment contracts*[END_REF] introduces fixed-term contracts, which are less productive than openended contracts by assumption, in firms with a decreasing-return-to-scale technology. Firms hire open-ended contracts until the productivity gains are offset by the expected losses from costly separations, and hire fixed-term contracts beyond that point. Explaining business cycles in a dual labor market requires relevant labor market data. As far as I know, it does not exist on a quarterly basis concerning the Euro area. Thus, I assess the fit of the model using French data from the Dares. Beyond the counter-cyclical share of fixed-term hires mentioned above, open-ended job creation and destruction flows are more volatile than their fixedterm counterparts. While open-ended job destruction is counter-cyclical, fixed-term job destructionis pro-cyclical. The short duration of fixed-term contracts explains this counter-intuitive result; as most fixed-term contracts last less than a quarter, demand boosts fuel fixed-term job creation that expire within the same quarter. My model also replicates the Beveridge curve. Real business cycles with frictional labor markets a-la Mortensen Pissarides mimic the negative correlation between unemployment and vacancies with difficulty. Talking about dynamics, a substitution effect and general-equilibrium effect shape the labor market. Shocks change the contractual composition of job creation -substitution effect -, which in turn impacts the size of the job seekers' pool and influences job creation -general-equilibrium effect.

  explains that fixed-term employment acts as a securing buffer to open-ended workers. In the doldrums, firms lay off fixed-term workers and leave open-ended employment as unchanged as possible to avoid firing costs. Firms restore fixed-term employment during booms. Thus, a negative aggregate demand shock leads to a decrease in fixed-term employment. On the other hand, bolstered uncertainty encourages firms to postpone open-ended hires and opt for fixed-term hires instead3 . Fixed-term employment should increase following an increase in uncertainty. My model is consistent with both views ; fixed-term employment decreases after an aggregate demand shock and increases after an uncertainty shock. It suggests that fixed-term employment could be used in future research as a way to identify uncertainty shocks.My paper relates to three branches of the literature. First, a few papers have tried to measure the cyclical properties of dual labor markets and account for the higher volatility on its fixed-term side.[START_REF] Sala | Flexibility at the margin and labour market volatility: The case of Spain[END_REF],[START_REF] Costain | Employment fluctuations in a dual labour market[END_REF] and[START_REF] Sala | Flexibility at the margin and labor market volatility in oecd countries*[END_REF] are the closest papers in that regard. These papers either assume that job creation only occurs through fixed-term contracts, or that the share of fixed-term contracts in job creation is exogenous. I depart from this approach as I do not rule out contractual substitution effects in job creation, which involves both open-ended and fixed-term contracts. In that regard, I can study the fluctuations in the contractual composition of hires.

Figure 3 .

 3 Figure 3.2.1: An overview of the structure of the model

  sharing I assume that wages are set each period by Nash bargaining. It is not realistic considering the evidence supporting rigidity in wages. In addition, wage flexibility seems inconsistent with the stickiness of retailers' prices. Still, I leave rigid wages for future research. Tractability motivates the use of Nash bargaining. It makes hiring and firing decisions jointly efficient and only dependent on the total surplus of a match. Denoting η the worker's share of the match surplus, the sharing rules write

  11), note that the surplus of incumbent open-ended matches includes firing costs, while the one of new open-ended workers does not. In case of disagreement during wage bargaining, a newly paired worker goes back to the pool of job seekers and the firm does not pay firing costs. That way, I do not overstate the role of firing costs. Now, I describe the open-ended and the fixed-term contracts as well as the associated surpluses. Open-ended contracts A continuing open-ended contract delivers the wage w p t and stipulates a firing tax F . An open-ended match may separate with the exogenous probability s. In this case, the separation bears no cost. Otherwise, the firm chooses whether it keeps or lays off the worker regarding the idiosyncratic productivity of the match. An endogenous separation entails the payment of the firing cost and the opening of a new vacancy. Hence, a firm with an incumbent open-ended worker has the following surplus

  2.22) As intended, the surplus of continuing open-ended workers includes firing costs in case of endogenous separations. It increases the continuing open-ended worker's threat point in Nash bargaining and pushes up his wage. The new open-ended workers does not benefit from this effect since a failure in the bargaining process does not entail the payment of F . The total surplus of fixed-term workers shows the productivity gap ρ in the flows and the exogenous termination of the contract in the continuation value. Fixed-term contracts struck by a job destruction shock split regardless the productivity of the match.

Figure 3 . 2 . 4 :

 324 Figure 3.2.4: Dual job creation

Figure 3 .

 3 Figure 3.2.5: Joint surpluses and idiosyncratic productivity of matches

Figure 3 .

 3 2.5 shows the joint surplus of fixed-term and new open-ended jobs in function of the idiosyncratic productivity and illustrates the importance of the productivity wedge for the shaping of job creation.

  1

  [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF] in several dimensions. I assume that the vacancy-worker meeting probability from the firm's point of view is 0.7 instead of 0.9. The latter figure replicates flows on the US labor market, which are known to be bigger than the European ones. One important feature of the labor market is its size, since it influences labor market tightness and job-finding rates. Should I consider ILO-defined unemployment stricto sensu or include the inactive population? Elsby et al. (2015) and[START_REF] Fontaine | CDD vs CDI: les effets d'un dualisme contractuel[END_REF] demonstrate the importance of the participation margin to explain unemployment fluctuations respectively in the United States and in France. This is all the more true with precarious employment, which involves people at the blurred frontier between unemployment and inactivity. According to data from Eurostat extending between 2002 and 2017, the participation rate in the Euro Area rate is around 67 %. Thus, I set steady-state employment to 0.67. In the same manner, I target a ratio of fixed-term contracts over total employment of 13 % in accordance with Eurostat estimates from 2006 to 2017.The contractual composition of creation flows matters much, as it impacts the turnover of the labor force. Targeting the share of fixed-term contracts among new jobs enables to control for the contractual composition of hires. However, an issue arises: the model implicitly assumes that fixedterm and open-ended contracts are substitutes for all tasks in the economy. It is a strong assumption as fixed-term workers could be better suited to some specific tasks, such as one-time tasks. Some industries frequently cope with one-time demand. Thus, targeting the share of fixed-term contracts in job creation should be done with caution, as it mixes fixed-term contracts with diverse degrees of substitutability with respect to open-ended contracts. Ruling out very short fixed-term contracts in the calibration may help to consider fixed-term contracts and open-ended contracts that may substitute. In France, data from Acoss -Dares witness that 83 % of job creations occur through fixed-term contracts and 30 % of these new fixed-term contracts last less than one day. Excluding these contracts leads to a share of fixed-term contracts in jobs creation of 77 %. Unfortunately, I do not have data about the proportion of very short contracts in job creation in other countries.In Spain,[START_REF] Felgueroso | Recent trends in the use of temporary contracts in Spain[END_REF] documents that 90 % of job creations occur through fixed-term contracts. At the other end of the spectrum, Addison et al. (2019) report a 45 % share of fixed-term contracts in German job creation. I target a share of fixed-term contracts in job creation of 70 % to reach a middle ground in the Euro area.

Figure 3 .

 3 4.1 shows the logged and filtered data along the lines of[START_REF] James D Hamilton | Why you should never use the hodrick-prescott filter[END_REF]. For this graph only, I do not use linear detrending because the 2008's crisis belongs to the time interval of data; a break of trend occurs for most variables 12 . It is not a problem as the qualitative cyclical features I describe below remain when using linear detrending.

Figure 3 .

 3 Figure 3.4.1: Fluctuations in labor market values.

  Is the model able to reproduce cross-correlations and volatility of these labor market time series? I compute the model-induced cross-correlations using simulations. For each of the 30,000 particles of the SMC algorithm, I compute the correlations over 200-period chains of shocks with a 100-period burn-in time. Then, I derive a weighted average of cross covariances using the particle-specific weights generated by the SMC procedure. Then, I compare the simulated moments to those of the time series displayed in Figure3.4.1, which are linearly detrended between 1997-Q1 and 2007-Q4. I also consider vacancies v, inflation π and EONIA interest rates R. Table 3.4.1 compounds the results. Overall, the model reproduces well the standard deviations and correlation with output of the considered time series. The model retrieves the pro-cyclicality of job creation flows. On the job destruction side, the model reproduces the pro-cyclicality of the fixed-term job destruction rate and the counter-cyclicality of the open-ended job destruction. It also mimics the higher volatility of flows

  and vacancies is difficult to replicate without making productivity implausibly volatile. In the original Diamond-Mortensen-Pissarides model, a positive productivity shock increases the surplus of matches and encourages the posting of vacancies. It pushes up labor market tightness and workers benefit from a higher job-finding probability that bolsters their threat point in Nash-bargaining. Wages end up increasing and fuel job destruction. Therefore, the positive effect of the productivity shock on wages significantly weakens the movements of job creation and vacancies, making the Beveridge curve flatter or even positively sloped.[START_REF] Ljungqvist | The fundamental surplus[END_REF] explain that the endeavors to obtain sufficiently strong responses of unemployment to productivity shocks consist in reducing the fundamental surplus of the match. The so-called fundamental surplus is the share of the surplus that can be allocated to job creation. The literature teem with implementations of wage rigidity, high utility of unemployment or firing costs; all boil down to reducing the fundamental surplus of new matches. What are the dimensions impacting the fundamental surplus in my model? The utility flow associated with unemployment is calibrated at a high value. Moreover, firing costs discourage open-ended job destruction. Both reduce the fundamental surplus. Fixed-term contracts ambiguously impact the fundamental surplus. On the one hand, it makes production possible for low values of idiosyncratic productivity for which firms and workers would get back to searching if only open-ended contracts were available. On the other hand, for a given idiosyncratic productivity, the new fixed-term contracts produce less. Numerically speaking, the simulated correlation between unemployment and vacancies amounts to -0.4, which fits the data well. My model suffers from a widespread problem in the DSGE literature: it is unable to generate enough persistence in cross-correlations and covariances. Figures 3.C.2 and 3.C.2 bear witness of this issue. Introducing wage rigidity and persistent idiosyncratic shocks could help. I leave it for future research.

Figure 3 Figure 3 . 4 . 3 :Figure 3

 33433 Figure 3.4.2: IRF of main variables to a one-standard-deviation productivity shock

Figure 3 .

 3 Figure 3.4.5 shows the impulse response functions of macroeconomic and labor market variables to a one-standard-deviation volatility shock to productivity starting from the stochastic steady state.

Figure 3

 3 Figure 3.4.5: IRF of main variables to a one-standard deviation volatility shock to productivity.

  The higher labor market tightness pushes up the profitability threshold z f ; the hiring region tightens and the expected gain from a vacancy decreases. The only manner to enhance the expected benefit from vacancies is to encourage substitution towards open-ended workers that are more productive; the substitution threshold z * must decrease, and so does the open-ended profitability threshold z c and the open-ended job destruction threshold z p . I denote (JC) the associated locus. I consider a steady-state decrease in firing costs. Let me first keep the labor market tightness θ constant as a thought experiment. A cut in firing costs encourages job destruction; the threshold for open-ended job destruction z p increases and the JD p curve moves up to (JD p ) . Are open-ended jobs more profitable overall? On the one hand, reduced firing costs directly encourage job creation as separations are cheaper. On the other hand, enhanced job destruction rate pushes up the probability of paying the firing cost in the future, which indirectly depletes open-ended job profitability. The former effect prevails and the profitability threshold z c decreases. Reduced firing costs leave the profitability of fixed-term contracts unaffected: z f does not change and the JP locus does not move.θ (θ)

Figure 3 .

 3 Figure 3.4.6: A decrease in F and the steady-state equilibrium. The prime index the equilibrium values after the change in firing costs. H p denote the hiring region of open-ended contracts.

  about volatility, open-ended job destruction is more volatile as the cut in firing costs hinders its movements less. Note that the volatility of flows on the fixed-term side soars, whereas the volatility of job creation decreased on the open-ended side of the labor market. The tightened hiring region of fixed-term workers strengthens the sensitivity of the related labor market moments. Surprisingly, the change in the composition of flows and employment does not impact inflation and employment volatility and cyclicality.

Fixed

Figure 3

 3 Figure 3.C.1: Prior and posterior distributions

Figure 3 .C. 2 :

 32 Figure 3.C.2: Simulated and data cross-correlations

Figure 3 .Figure 3 .

 33 Figure 3.C.3: Simulated and data cross-covariances

Figure 3 .

 3 Figure 3.C.5: IRF to a one-standard deviation volatility shock -VAR estimation.

  productivities. It is better to pay firing costs and split immediately than bearing with a depressed productivity up to the end date of the contract. As a result, making expired fixed-term matches convertible into open-ended ones reverses the ranking of contracts in job creation. Most productive matches end up in fixed-term contracts, while open-ended contracts cover less productive matches.

(

  2018) mentions that only 2 % of fixed-term contracts splitted before their stipulated termination date in 2017. On the other hand, firms and workers negotiate wages through Nash bargaining in most models. It makes hiring and firing decisions efficient for both the firm and the worker. In this case, as all separations are mutually beneficial, assuming that fixed-term matches may split at zero cost any time is consistent with the legal possibility of concerted termination. The flexible and rigid approaches towards fixed-term-contracts lead to different trade-offs at the hiring stage. In a basic environment with stochastic productivity and random search, flexible fixed-term contracts dominate open-ended contracts in terms of surplus. Indeed, a fixed-term match facing an adverse productivity shock may costlessly split if the contract is flexible, whereas splitting an open-ended match incurs a firing cost. Thus, in a framework with flexible fixed-term contracts, firm-worker pairs always opt for fixed-term contracts at the hiring stage if they are free to do so. Without additional constraints, job creation only occurs through fixed-term contracts. Costain et al. (2010) is a polar example of unconstrained job creation with flexible fixed-term contracts. In what follows, I also review papers with constrained job creation.

  and Rion (2019, 2020) show, rigid fixed-term contracts lead to coexisting fixed-term and open-ended contracts at the hiring stage. While flexible fixed-term contracts lead to a counter-factual unconstrained job creation through fixed-term contracts only, rigid fixed-term contracts enable job creation to occur through both fixed-term and open-ended contracts. Rigid fixed-term contracts enable the study of substitution effects between hiring fixed-term contracts and open-ended contracts. This nice feature of models with rigid fixed-term contracts comes at the cost of robustness issues as I demonstrate below. More specifically, in models with heterogeneous productivity, the ranking of contracts in job creation dramatically changes when expiring fixed-term matches can be converted into open-ended matches. 4.1.2 Flexible fixed-term contracts I now review models with flexible fixed-term contracts. I distinguish what I call unconstrained and constrained job creation. Unconstrained job creation consists in choosing the contract that maximizes the joint surplus of the new match. Constrained job creation, on the contrary, makes the choice of the contract inconsistent with the highest joint surplus. Costain et al. (2010) also builds a continuous-time framework with random search and matches face aggregate and idiosyncratic productivity shocks. Aggregate shocks follow a discrete Markov process while idiosyncratic productivity shocks are i.i.d. Open-ended contracts stipulate a firing cost, whereas fixed-term ones can be terminated any time. Fixed-term contracts reach their stipulated end date at a given a exogenous rate and may be converted into open-ended contracts at that moment if much if expiring fixed-term matches were convertible.

  stipulate a firing cost. Firm-worker pair can optimize the duration of the contract and cannot split before its stipulated end. Fixed-term matches can be converted into an open-ended contract once expired. The optimal duration of fixed-term contracts decreases with the arrival rate of adverse productivity shocks; the more frequent the shocks, the shorter fixed-term contracts. The riskiest contracts end up in fixed-term contracts. On the contrary, open-ended contracts cover low-risk matches. The least risky fixed-term matches spared by adverse productivity shocks are worth converting into open-ended matches.In some papers with rigid fixed-term contracts, job creation is constrained.Bentolila et al. (2012a) builds a continuous-time random-matching model. Matches undergo i.i.d productivity shocks that occur at a given rate and start at a settled high productivity. Open-ended matches renegotiate wages through Nash-bargaining each time a productivity shock occurs, whereas fixed-term matches keep the initially negotiated wages up to expiry, which occurs at a given rate. Expiring fixed-term matches can be converted into open-ended matches. The paper focuses on the role labor courts play in open-ended job destruction and its impact on the labor market. If an open-ended match faces an adverse enough shock, the firm sends an advance notice to the worker and the match splits after the firing permission is issued, which takes time. Firing permission is issued at a given Poisson rate.

FigureFigure 4 . 2 . 1 :Figure 4 . 2 . 2 :Figure 4 . 2 . 3 :

 421422423 Figure 4.2.1: Optimal duration and arrival rate of adverse productivity shocks

Figure 4 . 2 .Figure 4

 424 Figure 4.2.2 displays the optimal job destruction rate δ * . For all λ > λ p , δ * is the same as in the previous case without convertible fixed-term contracts. It makes sense as the associated fixed term jobs will not be converted into open-ended contracts; it is not profitable to do so. On the contrary, if λ ≤ λ p , the conversion to an open-ended contract takes place if no adverse productivity shock occurs before the fixed-term contract expires. In this case, the optimal job destruction rate exceeds its no-convertibility counterpart, the dashed line on the graph. Why are short fixed-term contracts less attractive when they are convertible? Short contracts loose flexibility with respect to the no-convertibility case. They are more likely converted into open-ended contracts, which brings in contracting and firing costs. Long fixed-term contract, on the contrary, postpone these costs, which makes them more appealing.

  Figure 4.2.5: Arrival rate of productivity shocks and job creation: the case S f o (λ p ) ≤ 0

I

  shall define a few useful thresholds to describe job creation in this framework. I define profitability thresholds y c and y f for open-ended and fixed-term contracts. S p (y c ) = 0 S f y f = 0 I assume that splits of fixed-term jobs occur more frequently than cost-less separations of open-ended jobs. Mathematically speaking, it boils down to s < δ. This assumption is in line with the data in Western European countries. Under this assumption, the surplus of open-ended jobs has a higher slope than the surplus of fixed-term jobs. , there exists a threshold y * that breaks even the surplus of open-ended jobs and the surplus of fixed-term jobs. S p (y * ) = S f (y * ) (4.2.4) Open-ended jobs dominate fixed-term ones as productivity exceeds y * . The ranking of the thresholds y c , y f and y * shapes job creation. Proposition 15 makes it clearer. Proposition 15. These assertions are equivalent 1. y * > y f 2. y * > y c 3. y c > y f

  Figure 4.2.7: Hiring open-ended contracts only

FigureFigure 4

 4 Figure 4.2.8: Dual job creation

Proposition 17 .

 17 Figure 4.2.9: Hiring fixed-term contracts only

  Figure 4.2.10: Dual job creation

  verify (r + s + λ)S c (y) =y -rU + (r + s) F + λ max S c y , 0 dG y In this framework, as we shall demonstrate later, the possibility to convert expiring fixed-term contracts into open-ended ones makes open-ended hires irrelevant. To this extent, I introduce a fixed hiring and transformation cost c, which corresponds to the administrative cost of hiring a worker or transforming an expired fixed-term contract into an open-ended one. Consequently, the surplus of a new open-ended contract with productivity y becomes S p (y) = S c (y) -F -c

Proposition 19 .Figure 4

 194 Figure 4.2.11: Dual job creation when 0 < c < λ r α -Ey r

-

  in job creation dramatically changes if fixed-term matches become convertible into open-ended ones. If firm-worker pairs optimize the duration of fixed-term contracts, the shortest fixed-term contracts are the most likely to be converted into open-ended contracts at expiry. As for models with heterogeneous processes of workload fluctuations, they critically depend on the assumption that the productivity of new matches is settled. Relaxing this assumption leads to the aforementioned counter-factual conclusions.Overall, future models have to consider carefully the modeling of fixed-term contracts and navigate through the pros and cons of both modeling strategies. Building a robust model of job creation in a Proposition 8Proof. Consider the derivative of S f with respect to δ. r ≡ δ * (λ) for all λ ≥ λ p Note that δ * is linear in λ with a positive slope and, thus, is continuous and increasing in λ. In addition, δ * (0) = -r < 0 and lim λ→+∞ δ * (λ) = +∞. Thus, there exists a unique λ such that δ * (λ) = 0. For a given λ, 2 cases arise in the program S f o (λ) = max δ≥0 S f (λ, δ)If δ * (λ) > 0, id est if λ > λ then δ * (λ) is the unique maximizer of δ → S f (δ, λ) and S f o = S f (δ * (λ), λ).Otherwise, if δ * (λ) ≤ 0, id est if λ ≤ λ, δ * (λ) does not comply with the non-negativity constraint over the optimal δ. Note that δ → S f (δ, λ) is continuous and decreasing over the interval [δ * (λ), +∞]. Hence, δ = 0 leads to the highest possible value of the surplus and respects the constraint δ ≥ 0. Therefore, the optimized surplus of fixed-term jobs S f o (λ) = S f (λ, δ * (λ)) verifies decreases with λ. Moreover, S f o (0) = y/r -U > 0 and lim λ→+∞ S f o (λ) = 0 + . Thus, S f o > 0. Proposition 10 Proof. Now, I consider the function ∆ = S f -S p . If λ ≤ λ, Assumption 1 entails that ∆(λ) = λ(F -U ) r + λ < 0

S

  f (λ) > 0 > S p (λ) otherwise Job creation takes place accordingly. Proposition 11 Proof. I rule out trivial cases where open-ended contracts are never relevant and assume open-ended contracts are profitable when there are no adverse productivity shocks. Assumption 5. y > r(U + c) S p is strictly decreasing in λ.∂Sp ∂λ = -y + r(U -F ) (r + λ) 2 < -(y -rU ) (r + λ) 2 < 0Assumption 5 ensures that y > r(U + c) > y > rU .

1 - 1 -- 2 √ 2 +

 1122 Consider 2 cases for a given λ:If λ < λ p , then S p (λ) > 0 and∂S f ∂δ (λ, δ) = -λ(F + c) + r(U + c) (r + δ + λ) 2 + rU (r + δ) 2 > 0 i.e. δ < λ λ(F + c) + r(U + c) rU -r ≡ δ * (λ) for all λ < λ p If λ ≥ λ p , then S p (λ) r ≡ δ * (λ) for all λ ≥ λ pNow, I compute the derivative of δ * with respect to λ to study its variations.If λ ≥ λ p , S p (λ) ≤ 0 and δ * is linearly increasing in λ.Otherwise, S p (λ) < 0, and δ * writesδ * (λ) = λ λ(F +c)+r(U +c) rU -1 -rThe derivative of δ * verifies(δ * ) (λ) ∝ λ(F + c) + r(U + c) -√ rU -λ(F + c) 2 λ(F + c) + r(U + c) ∝ 2r(U + c) + λ(F + c) -rU λ(F + c) + r(U + c) ∝ √ rU -λ(F + c) + r(U + c) rc > 0

-- 2 rUFigure 4 2 :

 242 Figure 4.A.1: Arrival rate of productivity shocks and job creation: dual case

y

  * ≤ y f ≤ y c . (4.2.5) entails that only open-ended contracts weigh in job creation for productivities y > y c . Otherwise, y c < y f and proposition 15 ensures that y f < y c < y * . (4.2.5) entails that job creation is dual. For productivities y f < y < y * , job creation occurs through fixed-term contracts. Productivities y such that y > y * involve open-ended contracts.Proposition 17Proof.If F ≥ rU/(r + δ), then S f (y c ) ≥ 0 = S p (y c ). Since S f is increasing in y and S f y f = 0, y f and y c verify y f ≤ y c .For all y ≤ y c , S p and S f are linear in y andS p (y) = y -y c r + s + λ S f (y) = y -y f r + δ + λUsing these expressions, one may demonstrate that fixed-term contracts deliver a higher jointsurplus than their open-ended counterparts for productivities y ≤ y c .S f (y) = y c -y f r + δ + λ + y -y c r + δ + λ > S f (y c ) ≥0 + y -y c r + s + λ =S p (y)≥ S p (y)

S

  f (y) = 1 + s r + δ + λ >1 y -y c r + s + λ + S f (y c ) > S p (y) + S f (y c ) ≥0 > S p (y)Otherwise, S f (y c ) < 0 = S p (y c ). S f being increasing in y and S f y f = 0, y f and y c verify y f > y c . Moreover, S p and S f are in linear in y and for all y > y c exists y * > y f such that S p (y * ) = S f (y * ). Consequently, the ranking of jointsurpluses verify y) > 0 > S f (y) ∀y c < y < y f S p (y) > S f (y) > 0 ∀y f < y < y * S f (y) > S p (y) > 0 ∀y > y *Job creation takes place accordingly.Proposition 18Proof. Let me denote z(y) = y-rα-(r+λ) S p 0 (y) + -S p 0 (y ) + dG (y ) . Notice that y is increasing in y. Algebraic manipulations deliver another expression of S f as followsS f (y, δ) δ) 2 (rα -Ey)Consequently, provided that r + δ > 0,∂S f ∂δ ≥ 0 ⇔ z(y)(r + δ) 2 -2λ(rα -Ey)(r + δ) -λ 2 (rα -Ey) ≤ 0Studying the variations of S f boils down to assessing the sign of a second-degree polynomial in (r + δ).If rα = Ey, sign ∂S f ∂δ = sign (-z(y)). In this case, δ * (y) ∈ {0, +∞} if z(y) = 0. Otherwise, any non-negative δ maximizes S f .If rα > Ey, sign ∂S f ∂δ = sign (P y (r + δ)), where P y (X) = -x(y)X 2 + 2λX + λ 2 . Several subcases arise.

∆

  (y) = S f (y, δ * (y)) -c -S p 0 (y) + Using (4.B.1), the previous equation becomes ∆(y) = rα -Ey r + δ * (y) + λ x(y) -λ r + δ * (y) -c Several cases arise when c > 0 If x(y) ≤ 0, δ * (y) = +∞ and ∆(y) = -c < 0. Hiring only takes place through open-ended contracts under the constraint that y ≥ y c . If 0 < x(y) < λ r 2 + λ r , using the definition of δ * spelled in proposition 18, one may rewrite ∆Algebraic manipulations entail that sign (∆(y)) = sign x(y) + 1 -β 1 + x(y) -(1 + β) where β = cλ rα -EyThe right-hand side of the equation above is a second-degree polynomial in 1 + x(y).Therefore, since x(y) > 0, ∆(y) ≥ 0 if and only if x(y) ≥ (2 + β) β.If x(y) ≥ λ r 2 + λ r , δ * (y) = 0 and ∆(y) = rα-Ey r+λ x(y) -λ r -c. Thus, ∆(y) ≥ 0 if and only if x(y) ≥ λ r + c(r+λ) rα-Ey .Notice that the latter condition is always fulfilled in this specific case if and only if β ≤ λ r

2 :

 2 Figure 4.B.1: Job creation with open-ended jobs only

  , there are two continua of risk neutral firms and households and time is continuous. The interest rate is denoted r. Firms are numerous and can

either employ one worker or maintain a vacancy opened. They face i.i.d idiosyncratic productivity shocks that occur with a probability λ per unit of time and are drawn from a log-Normal distribution log N 0, σ 2 z with cumulative distribution function G and support (0, ∞). Our model mainly departs from the classic Mortensen-Pissarides framework by assuming that workers are either employed through open-ended contracts or fixed-term contracts.

  where J p (z) is the firm's surplus from a continuing open-ended contract with productivity z, V the firm's surplus from an unfilled vacancy and F is the firing cost.

	J p 0 (z) is the firm's surplus from a new open-ended contract with productivity z. Indeed, when a
	contact occurs between a firm and a worker, the firm's outside option does not include the payment
	of a firing cost if no contract is signed. The open-ended and fixed-term workers' surpluses are
	standard. We denote W p (z) the worker's surplus from a continuing open-ended contract with
	productivity z, W p 0 (z) the worker's surplus from a new open-ended contract with productivity z,
	W f (z) the worker's surplus from a fixed-term contract with a productivity z and U the unemployed's
	surplus.
	Interestingly, the firm's surplus for a new open-ended match with productivity z is J p 0 (z) -V , where

  shock takes place. Thus, the associated probability is ξ = s + λG (z p ). Unemployed workers' probability to become employed under open-ended contracts is µ p

	2.2. THE MODEL	CHAPTER 2. WAITING FOR THE PRINCE CHARMING

tivity

Table 2

 2

	r	σ	η	λ	δ
	0.4 % 0.6 0.6 0.083 0.67

.1 sums up the choice of parameters.

Table 2

 2

.1: Parameters

Table 2 .

 2 2: Targets for a calibration of the French labor market

	F	b	s	σ z	m	γ
	1.30 1.00 0.013 0.05 0.53 3.78 •10 -3

Table 2 .

 2

	3: Calibrated parameters

Table 2 .

 2 2 outlines the targeted labor markets moments.[START_REF] Fontaine | CDD vs CDI: les effets d'un dualisme contractuel[END_REF] suggests that inactive workers significantly contribute to job creations through fixed-term contracts. As a result, I target an unemployment rate of 26 %, which corresponds to the 2016's French inactivity rate 14 . I set the steady state share of fixed-term contracts in employment to 12 %. I target an average share

  Let me assume that firing costs undergo i.i.d shocks with probability per unit of time. Firing costs follow a uniform distribution over {F 1 , F 2 } such that F 1 < F 2 . I describe the derivation details of the model in Appendix 2.D. I set to match the French and Portuguese score of 17 reforms in 9 years, which seems to be a lower bound in Western and Southern European countries. Consequently, our results will understate the actual impact of regulatory uncertainty in the mentioned countries. A steady-state no longer exists: the system is now a Markov Jump Non-Linear System 20 Figure2.9shows the paths of fixed-term and open-ended employment for a given realization of policy shocks.

			F 2
	F
			F 1	0	100	200	300	400
	n p	0.59 0.60 0.61 0.63 0.62	0	100	200	300	400
		0.1050
	n f	0.1000 0.1025
		0.0975
		0.0950	0	100	200 Time (months) 300	400
	reforms are actually frequent and often marginal in Western and Southern Europe. Between 2005
	and 2013, they count 17 employment protection legislation reforms in France, 49 in Italy, 38 in
	Spain, 23 in Greece and 17 in Portugal. Thus, a natural question to address concerns the impact of
	regulatory uncertainty on the labor market equilibrium.

Earliest time series of open-ended and fixed-term employents are annual and start on 1982, while fixed-term contracts were introduced in 1979. As for in-depth reforms, there were virtually none about employment protection in France in the last few years. The biggest reform was Accord national interprofessionnel concluded on 2013 January 11 and was about introducing a pinch of experience rating in payroll taxes. Relevant data may stem from 2014's Italian Jobs Act, which imposes simpler termination procedures and caps on compensations

19

.

Regulatory uncertainty If studying one-and-for-all in-depth reforms is relevant from a policy point of view, it remains a purely theoretical view. According to

[START_REF] Fontaine | CDD vs CDI: les effets d'un dualisme contractuel[END_REF]

, Figure 2.9: Employment dynamics with a given path of policy shocks Intuitively, uncertainty in firing costs should discourage hires through open-ended contracts. When F = F 2 , matches with a productivity just over the job destruction cap split if policy shock leads to

  government. Households can be unemployed or employed through a fixed-term or an open-ended contract. Three types of firms coexist. Perfectly competitive firms produce the final good valued by households for consumption and investment. Final good producers aggregate the differentiated 3.2. MODEL CHAPTER 3. FLUCTUATIONS IN A DLM goods produced by the retailers. Retailers are in monopolistic competition and transform the homogeneous intermediate good into a differentiated retail good. Intermediate-good firms produce the associated intermediate good and experience perfect competition. Intermediate firms use labor as their only input. Figure 3.2.1 sums up the structure of the model. I now describe the behavior of the different types of agents in more detail.

		Government:
		Interest-rate rule
		Government spending shock
		Monetary policy shock
	Taxes
	Unemployment benefits
	Households: Consumption Savings	Homegeneous consumption good supply	Final-good firms: Dixit-Stiglitz aggregation
		Firing tax
				Retail good supply
	Dual labor supply
		Intermediate
			good	Retailers:
	Intermediate-good firms	supply	Monopolistic competition
	Aggregate productivity shocks		Calvo pricing
				Mark-up shocks

  {A t , g t } +∞ t=0 and initial values for the state variables R -1 , n p -1 , n f -1 , ∆ -1 , P -1 , the equilibrium sums up into the set of endogenous variables R t , c t , Y t , n p t , n f t , u t , ∆ t , z p t , z c t , z f t , z * t , θ t , φ t , v t , P t , P *

			+∞
		t	t=0
	pinned down by equations (3.2.1), (3.2.4), (3.2.35)-(3.2.38) and (3.2.41)-(3.2.49).
	A t , µ t , g t , m t	+∞ t=0 , laws of motions of exogenous shocks

Table 3 .

 3 3.2: Targets for a calibration of the labor market in the Euro areaThe last parameters to be pinpointed are F , b, m, ρ and γ. The empirical evidence concerning b is thin. Indeed, the flow value of non-employment is a highly debated issue in labor economics.

	2012a) documents. The 0-to-6 OECD indicator for employment protection legislation enables a
	cursory comparison between Euro area countries.
	While Portugal and Italy display the highest employment protection indicators around 2.9, German
	and Spanish values amount to 2.4. France is close to 2.7 and thus constitutes a middle ground
	regarding employment protection legislation in the biggest Euro zone countries. Fortunately, Kramarz

and Michaud (

2010

) is one of the scarce studies that estimates firing costs and it considers the French case. The latter assess that individual lay-offs marginally cost 4 months of the median wage, while the marginal cost lay-offs within a collective termination plan represents 12 months

Table 3 .

 3

3.3: Calibrated parameters

  .3.4 compound results. I similarly estimate a classic labor market without fixed-term contracts to compare results. All details are available in Appendix 3.D. The results are displayed in Table 3.D.3. Overall, the estimates are pretty similar across both models and estimates are not significantly different. The log marginal density do not significantly differ either, as Table 3.D.2 shows.

Table 3 .

 3

	Variables	Data		Model
		Std. Dev. Cor (Y, .) Std. Dev. Cor (Y, .)
	Y	1.16	1.0	0.79	1.0
	π	0.32	0.15	0.31	0.01
	R	0.33	0.29	0.18	0.38
	n	0.73	0.93	0.87	0.85
	jc p	7.19	0.65	12.16	0.45
	jc f	4.97	0.52	3.89	0.21
	jc f /jc	1.29	-0.4	2.89	-0.47
	n f	1.87	0.18	2.0	0.1
	jd p	5.7	-0.43	7.05	-0.54
	jd f	2.84	0.39	2.0	0.24
	v	10.32	0.61	7.29	0.41

4.1: Actual and simulated correlations with output of different variables

  , θ . Equations(3.2.35)-(3.2.39) pinpoint the steady-state equilibrium in the labor market. Figure 3.4.6 illustrates it. (3.2.36) defines job destruction for open-ended matches. It defines a positive relationship between the open-ended job destruction threshold z p and labor market tightness θ; a higher labor market tightness increases wages and encourages job destruction. I denote JD p the corresponding locus. For the same reason, a higher labor market tightness reduces the surplus of new matches. As a result, the profitability threshold z f and substitution threshold z

* * increase with labor market tightness. I denote JP and JC p the corresponding loci. The job creation condition (3.2.39) defines a negative relationship between

Table 3

 3

	Variables	Data		Baseline	Reduced firing costs
	Y	1.16	1.0	0.79	1.0	0.79	1.0
	π	0.32	0.15	0.31	0.01	0.31	0.03
	R	0.33	0.29	0.18	0.38	0.18	0.37
	n	0.73	0.93	0.87	0.85	0.9	0.85
	jc p	7.19	0.65	12.16	0.45	9.61	0.34
	jc f	4.97	0.52	3.89	0.21	15.64	0.14
	µ f / µ f + µ p	1.29	-0.4	2.89	-0.47	9.95	-0.04
	n f	1.87	0.18	2.0	0.1	8.89	0.03
	jd p	5.7	-0.43	7.05	-0.54	10.18	-0.51
	jd f	2.84	0.39	2.0	0.24	8.89	0.14
	v	10.32	0.61	7.29	0.41	12.08	0.35
		Table 3.4.2: Labor market moments and institutions.

Std. Dev. Cor (Y, .) Std. Dev. Cor (Y, .) Std. Dev. Cor (Y, .) .4.2 shows that the cyclicality of variables has not changed much. The only exception is the share of fixed-term contracts in job creation, which was counter-cyclical and is now acyclical. Talking

Table 3 .

 3 C.1: Times series used for estimation All these time series are drawn from the ECB Data Warehouse

	Observable	Time range	Unique Identifier
	GDP at constant prices 1995:Q1 -2019:Q3 MNA.Q.Y.I8.W2.S1.S1.B.B1GQ._Z._Z._Z.EUR.LR.N
	GDP deflator	1995:Q2 -2019:Q2 MNA.Q.Y.I8.W2.S1.S1.B.B1GQ._Z._Z._Z.IX.D.N
	Nominal interest rates 1994:Q1 -2019:Q3 FM.M.U2.EUR.4F.MM.EONIA.HSTA
	Employment	1995:Q1 -2019:Q3 ENA.Q.Y.I8.W2.S1.S1._Z.EMP._Z._T._Z.PS._Z.N

Table 3 .

 3 Dev. Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. C.3: Estimations with Hamilton, linear-trend, first-difference and Hodrick-Prescott filters

	Parameters	Hamilton	Linear trend	First difference	Hodrick-Prescott
	Mean Std. ρ A 0.52 0.11	0.69	0.08	0.29	0.27	0.62	0.09
	ρ g	0.86	0.05	0.88	0.04	0.78	0.08	0.86	0.05
	ρ R	0.79	0.04	0.77	0.05	0.79	0.08	0.7	0.05
	ρ π	2.02	0.59	2.06	0.54	1.87	0.74	1.99	0.51
	ρ y	0.25	0.05	0.29	0.06	0.46	0.12	0.43	0.06
	ψ	0.84	0.03	0.83	0.03	0.71	0.06	0.8	0.03
	σ A	0.79	0.11	0.31	0.03	0.32	0.04	0.29	0.03
	σ µ	0.23	0.03	0.3	0.03	0.28	0.04	0.3	0.03
	σ g	7.67	1.74	5.61	1.36	3.04	0.87	5.13	1.17
	σ m	0.06	0.01	0.08	0.01	0.09	0.02	0.06	0.01

3.D Classic Model 3.D.1 Equilibrium equations

	u (c t ) = βR t E t	P t P t+1	u (c t+1 )

  [START_REF] Rion | Waiting for the Prince Charming: Fixed-Term Contracts as Stopgaps[END_REF] models job creation in the same manner. The only difference is that productivity shocks are i.i.d and job destruction is endogenous. Productivity shocks are not necessarily adverse and do not systematically lead to a split of open-ended contracts.

Convertible fixed-term contracts

Now, I assume that fixed-term contracts are converted into open-ended ones if it is profitable to do so. Fixed-term contracts are worth converting if no adverse productivity shock has struck before the termination shock. I also assume that the conversion of fixed-term contracts into open-ended

  G. Consequently, Assumption 1 is no longer needed to ensure that separating open-ended matches be efficient after a productivity shock. Continuing open-ended contracts now display non-trivial surpluses. The joint surpluses writes

I take these figures from the DMMO-EMMO surveys (Dares,

2020) 2 article L1221-2, Code du travail 3 article L1242-2, Code du travail

See Blasco and Givord (2010),[START_REF] Ichino | From temporary help jobs to permanent employment: what can we learn from matching estimators and their sensitivity[END_REF],[START_REF] Güell | How binding are legal limits? transitions from temporary to permanent work in spain[END_REF],[START_REF] Berton | Are temporary jobs a port of entry into permanent employment[END_REF][START_REF] De Graaf-Zijl | Stepping stones for the unemployed: the effect of temporary jobs on the duration until (regular) work[END_REF]), OECD (2014) and[START_REF] Givord | When does the stepping-stone work? fixed-term contracts versus temporary agency work in changing economic conditions[END_REF]

See Arulampalam and Booth (1998);[START_REF] Arulampalam | Training in Europe[END_REF];[START_REF] Albert | Firm-provided training and temporary contracts[END_REF];[START_REF] Cutuli | Fixed-Term Contracts, Economic Conjuncture, and Training Opportunities: A Comparative Analysis Across European Labour Markets[END_REF];OECD (2014)

In a pioneering work, Piore (1986) compares employment protection in France, Spain, Italy and West Germany and finds substantial differences.

See https://eplex.ilo.org/

See among others[START_REF] Cahuc | Temporary jobs, employment protection and labor market performance[END_REF];[START_REF] Ljungqvist | How do lay-off costs affect employment?*[END_REF]

See among others[START_REF] Nickell | Fixed costs, employment and labour demand over the cycle[END_REF][START_REF] Nickell | Unemployment and the structure of labor costs[END_REF][START_REF] Nickell | Chapter 9 dynamic models of labour demand[END_REF];[START_REF] Bentolila | Firing costs and labour demand: How bad is eurosclerosis?[END_REF];[START_REF] Bertola | Labor turnover costs and average labor demand[END_REF];[START_REF] Hopenhayn | Job turnover and policy evaluation: A general equilibrium analysis[END_REF];[START_REF] Saint-Paul | Dual Labor Markets: A Macroeconomic Perspective[END_REF];[START_REF] Risager | On the effects of firing costs when investment is endogenous: An extension of a model by bertola[END_REF];[START_REF] Dolado | Drawing Lessons from the Boom of Temporary Jobs in Spain[END_REF]

See Shimer (2012);[START_REF] Petrongolo | The ins and outs of european unemployment[END_REF];[START_REF] Fujita | The cyclicality of separation and job finding rates[END_REF];[START_REF] Michael | Unemployment dynamics in the oecd[END_REF].

See also[START_REF] Bucher | Hiring Practices, Employment Protection and Temporary Jobs[END_REF];[START_REF] Portugal | Why Do Firms Use Fixed-Term Contracts[END_REF]

See Caliendo and Schmidl (2015) for a recent review

See[START_REF] Patricia | The effects of the unemployment insurance payroll tax on wages, employment, claims and denials[END_REF] for a natural-experiment assessment of the experience rating system in the US.

In France, Collectif (1996) portrays 40 years of labor policy and the hords of acronym-designated targeted contracts that go with it.

[START_REF] Bertola | Crossing the river: a comparative perspective on Italian employment dynamics[END_REF] and[START_REF] Siassi | Moving Towards a Single Labor Contract: Transition vs[END_REF] are notable exceptions

This all the more true if on-the-job search is an option.[START_REF] Polivka | Contingent and alternative work arrangements, defined[END_REF] finds that most contingent workers do not search for a perennial job. The model thus delineates a lower bound in terms of fixed-term contracts' desirability at the hiring stage

See[START_REF] Lepage-Saucier | 14. le contrat de travail unique: quid pro quo ou simple quiproquo? Regards croises sur l'economie[END_REF] and[START_REF] Amable | Who wants the contrat de travail unique? social support for labor market flexibilization in france[END_REF] for reviews

See Cahuc et al. (2016) and the general introduction for a review

Hiring and firing decisions are jointly efficient if firms and workers negotiate wages using Nash bargaining, which is the case in both the current chapter and[START_REF] Siassi | Moving Towards a Single Labor Contract: Transition vs[END_REF].

[START_REF] Edward P Lazear | Job security provisions and employment[END_REF] detail this idea.

See Grout (1984) for a thorough development of this question

Of course, if a fixed-term contract is profitable, but still less beneficial than a open-ended contract, then the hire takes place through a open-ended contract.

Quits are marginal in fixed-term matches: only 2% of fixed-term contracts terminate before their stipulated end[START_REF] Milin | Cdd, cdi : comment évoluent les embauches et les ruptures depuis 25 ans ? Dares analyses[END_REF].

OECD (2013) and ILO detail the employment protection legislations of many countries.

I essentially rely on this note to target moments characterizing the French labor market. It is based on the Enquête sur les mouvements de main d'oeuvre carried out by Dares

The Hosios condition is still valid in our framework.

Among others,Mortensen and Pissarides (1994, 1999);[START_REF] Cahuc | Temporary jobs, employment protection and labor market performance[END_REF]

This result is extensively developed in Pissarides et al. (2000) page 59-63.

[START_REF] Cahuc | Temporary jobs, employment protection and labor market performance[END_REF];[START_REF] Sala | Flexibility at the margin and labour market volatility: The case of Spain[END_REF];Bentolila et al. (2012a) and[START_REF] Sala | Flexibility at the margin and labor market volatility in oecd countries*[END_REF] make this assumption.

See Boeri and Garibaldi (2019) for an assessment of the Italian Jobs Act

[START_REF] Luiz Do Valle | Continuous-time Markov jump linear systems[END_REF] is an approachable introduction to the linear case.

The interested reader can visualize the impact of different probabilities of productivity-shock occurrence on MyBinder[START_REF] Jupyter | Binder 2.0 -Reproducible, interactive, sharable environments for science at scale[END_REF]

See Arulampalam and Booth (1998);[START_REF] Arulampalam | Training in Europe[END_REF];[START_REF] Albert | Firm-provided training and temporary contracts[END_REF];[START_REF] Cutuli | Fixed-Term Contracts, Economic Conjuncture, and Training Opportunities: A Comparative Analysis Across European Labour Markets[END_REF]

[START_REF] Bloom | Fluctuations in uncertainty[END_REF] highlights that firm are more reluctant to go for investments requiring fixed costs in uncertain times.

A few seminal papers in the DSGE literature include[START_REF] Smets | Shocks and frictions in us business cycles: A bayesian dsge approach[END_REF],[START_REF] Christiano | Nominal rigidities and the dynamic effects of a shock to monetary policy[END_REF],[START_REF] Gertler | An estimated monetary dsge model with unemployment and staggered nominal wage bargaining[END_REF][START_REF] Trigari | Equilibrium unemployment, job flows, and inflation dynamics[END_REF],[START_REF] Blanchard | Labor markets and monetary policy: A new keynesian model with unemployment[END_REF] and[START_REF] Lawrence J Christiano | Unemployment and business cycles[END_REF]

In the same vein, see also[START_REF] Jurado | Measuring uncertainty[END_REF] and[START_REF] Fasani | Are uncertainty shocks aggregate demand shocks?[END_REF]

[START_REF] Edward P Lazear | Job security provisions and employment[END_REF] lays out this idea in more detail.

Detailed calculations are available in Appendix 3.B.1

To be accurate,[START_REF] Kramarz | The shape of hiring and separation costs in france[END_REF] assess that firms with more than 50 employees face a marginal cost of

97,727 FFr (Table 1b), which represents 14 months of the workers' median wage. Consequently, the associated median wage of fired workers is 6,980 FFr. Thus, Table2shows that individual terminations cost 27,389 FFr, which amounts to 4 months of the fired workers' median wage, while the termination within a collective firing plan marginally costs 81,850 FFr, which equals 12 months of the median wage.

See Herbst and Schorfheide (2016) pp

100-129 for an extensive treatment of this method. 10 Table3.C.2 details the series used for estimation

Standard deviations are expressed in percentage.

See Gorodnichenko and Ng (2010),[START_REF] Ferroni | Trend agnostic one-step estimation of dsge models[END_REF], Lafourcade and de Wind (2012),[START_REF] Canova | Bridging dsge models and the raw data[END_REF] and[START_REF] James D Hamilton | Why you should never use the hodrick-prescott filter[END_REF] for thorough developments about the consequences of filtering data

See Nickell (1978[START_REF] Nickell | Unemployment and the structure of labor costs[END_REF][START_REF] Nickell | Chapter 9 dynamic models of labour demand[END_REF];[START_REF] Bentolila | Firing costs and labour demand: How bad is eurosclerosis?[END_REF];[START_REF] Bertola | Labor turnover costs and average labor demand[END_REF];[START_REF] Saint-Paul | Dual Labor Markets: A Macroeconomic Perspective[END_REF];[START_REF] Risager | On the effects of firing costs when investment is endogenous: An extension of a model by bertola[END_REF];[START_REF] Wasmer | Competition for jobs in a growing economy and the emergence of dualism[END_REF];[START_REF] Dolado | Drawing Lessons from the Boom of Temporary Jobs in Spain[END_REF];[START_REF] Caggese | Financing constraints and fixed-term employment contracts*[END_REF]

Acknowledgments

Para(1) and Para (2) correspond to mean and standard deviation of the prior distribution if the latter is Normal or Inverse Gamma. Para(1) and Para (2) correspond to lower and upper bound of the prior distribution when the latter is uniform Thus, productivity shocks alone cannot account for the whole extent of fluctuations in GDP without producing crazy results in terms of interest rate, inflation and employment. Government spending shocks capture the fluctuations in GDP that productivity shocks cannot fully account for. Table 3.D.2

reports that both models have comparable logged marginal data densities. Thus, roughly speaking, the more volatile and persistent government spending shocks are, the more mis-specified aggregate demand is to fit the GDP time series. To this extent, the dual model fits observed fluctuations in GDP better than the classic model. The volatility and persistence of government spending shocks remains high in the dual model, though. The model lacks essential features -capital and investment for example -to properly fit fluctuations in GDP. The estimates of the government spending shock in [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF] exhibit a similar behavior.

The mark-up shock µ t is the counterpart of the government spending shock for inflation. Its volatility amounts to one standard deviation of the inflation time series. The auto correlation parameter of productivity shocks is 0.7, lower than usual estimates in the literature. The parameters driving the interest rate rule (3.2.41) are similar to [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF], with the exception of the weight of output. The Calvo pricing parameter amounts to 0.83, a bit lower than the estimate of [START_REF] Thomas | Labor market reform and price stability: An application to the euro area[END_REF] around 0.9.

Combining these three expressions, I get

A non-linear system in z f , z * , ρ can then be solved numerically. whereas an open-ended match may be able to do so at some cost.

My approach also shows how robustness issues arise in models with rigid fixed-term contracts.

Assume first that fixed-term contracts cannot be converted into open-ended contracts and that their average duration is fixed. New firm-worker pairs that draw a high productivity want to make the most out of it and a fixed-term contracts may be too short in that regard; the contract may reach its stipulated end before the production opportunity has come to an end. In this case, open-ended contracts are more relevant. The productivity draw needs to be high enough to compensate the match for future firing costs. If the drawn productivity is not high enough to compensate future firing costs, a fixed-term contract is a good option as it enables quickly going back to searching for high productivity draws and generates some positive surplus for a while. Open-ended contracts lock up highly productive matches, whereas fixed-term contracts are good a compromise for intermediate productivities.

Rigid fixed-term contracts

In this section, I review papers with rigid fixed-term contracts. Again, I distinguish constrained and unconstrained job creation.

The closest paper to the thought experiment I propose in introduction is the model of chapter 2 [START_REF] Rion | Waiting for the Prince Charming: Fixed-Term Contracts as Stopgaps[END_REF]. Firms and workers meet according to matching function in a continuous-time framework.

Firm-worker pairs are heterogeneous in productivity and receive i.i.d shocks in productivity at a

given rate. Open-ended contracts stipulate a firing cost while fixed-term contracts split following an exogenous termination rate. A fixed-term match cannot split before a termination shock hits regardless its productivity. Note that fixed-term matches cannot be converted into open-ended ones when expiring. Job creation is unconstrained; a new-firm worker pair draw a productivity from a given distribution and optimizes the choice of the contract accordingly. Most productive draws

end up in open-ended contracts; fixed-term contracts may be too short to take full advantage of the initial high productivity draw. When the productivity is not high enough to consider paying firing costs in the future, fixed-term contract are a good fit. Firms and workers can go back to search for good matches before long and benefit from a positive surplus. As I show below, Rion (2019) is not robust. Making fixed-term matches convertible into open-ended matches dramatically changes the contractual layout of job creation.

In the same vein of models with heterogeneous productivities, the model of chapter 3 [START_REF] Rion | Fluctuations in a Dual Labor Market[END_REF] is a discrete-time model and is very similar to [START_REF] Rion | Waiting for the Prince Charming: Fixed-Term Contracts as Stopgaps[END_REF]. A few departures arise as productivities of matches are i.i.d across periods and fixed-term matches are less productive than open-ended ones all else equal. Agents strike a balance between productivity and flexibility. Most productive matches

end up in open-ended matches; the agents are ready to pay firing costs in order to avoid the ad hoc productivity loss a fixed-term contract would incur. When the initial productivity draw is lower, matches favor fixed-term contracts. Again, the contractual framework of job creation would change

Heterogeneous productivities: the rise of robustness issues

In this section, I introduce heterogeneous productivities starting for the baseline model described

Basic framework

Starting from the basic framework delineated above, what happens with job creation if the productivity of jobs is heterogeneous? I assume that new firm-worker pairs draw their productivity y job from a distribution with cdf G.

The match chooses the contract that maximizes the joint surplus, going back to searching being also an available option. Nash bargaining ensures that the chosen contract also maximizes the firm's and worker's surpluses. If firms pay γ per unit of time to maintain a vacancy, the present discounted value of a vacancy verifies Assumption 4. y > rU S p is continuous and decreases with λ. Indeed, the derivative of S p with respect to λ goes by

Moreover, S p (0) = y/r -U and lim λ→+∞ S p (λ) = -F < 0. As a result, there exists a unique λ p such that

General Conclusion

fluctuations. However, only a few papers have tried to model the choice between fixed-term and open-ended contracts in job creation. I have reviewed these papers in the fourth chapter. Two polar assumptions about the modeling of fixed-term contracts drive the results in the literature. Some papers assume that fixed-term contracts are flexible and can end any time at zero cost. Others assume that fixed-term contracts are rigid and cannot split before their stipulated termination date.

Modeling fixed-term contracts as flexible leads to job creation flows with fixed-term contracts only.

On the contrary, rigid fixed-term contracts enable both open-ended and fixed-term contracts to naturally intervene in job creation. Being able to account for contractual substitution effects using rigid fixed-term contracts comes at some cost. The ranking of contracts in job creation is not robust when the productivity of new matches is heterogeneous. This observation notably covers the models of chapters 2 and 3.

In

KEYWORDS

Firing costs, Dualism, Fixed-term contracts, Macroeconomics