Sarah Le Duigou
email: sarah.leduigou@univ-smb.fr

Bérangère Legendre

Mareva Sabatier

Financial incentives to postpone retirement: Estimating the impacts of the 2003 French pension reform on the employment and wages of older workers by using a lifetime job search model

Keywords: financial retirement incentives, employment, endogenous wages, equilibrium search model JEL Codes: J26, J31, C21

This article analyses whether financial incentives to delay retirement affect the employment and wages of older workers, not just their retirement behavior. We focus on the outcomes of France's 2003 reform that offered pension bonus to older workers who postponed their retirement until the legal age. An equilibrium search model is constructed in which both employment and wage are endogenous, allowing us to analyze the role of both labor supply and demand sides in the retirement decision. Simulations of the model show that some firms choose to post higher wages to induce their workers to work longer. Thanks to a higher retention, firms save recruitment and training costs. Finally, demand-side considerations can attenuate the impacts of financial incentives that cause wage increases and few new jobs.

Introduction

In France, the recent pension reform project establishing a universal system in 2025 has put the issue of financial incentives to work beyond the legal retirement age back at the center of the debate. This debate is not new and echoes that of 2003 when the Fillon government had already introduced a bonus/malus system 1 . In this article, we return to this landmark episode of pension reform in France to try to understand how the functioning of the labor market may have been modified by the increase in the labor supply of older people wishing to delay their retirement. This is all the more important since the literature has focused mainly on the effect of financial incentives on individual retirement decisions and thus on the supply side of the labour market, but has been almost silent on the potential effects of the demand side on the labour market. Indeed, several empirical studies, spanning various nations, suggest that these financial incentives increase employment rates among people older than the legal retirement age [START_REF] Gruber | Social Security Programs and Retirement Around the World: Micro-Estimation[END_REF] [START_REF] Gruber | Social Security Programs and Retirement Around the World: Micro-Estimation[END_REF]; Hanel, 2010 [START_REF] Hanel | Financial incentives to postpone retirement and further effects on employment ? evidence from a natural experiment[END_REF]; Ferrari, 2019 [START_REF] Ferrari | The effectiveness of incentives to postpone retirement: evidence from italy[END_REF]; Belloni et Alessie, 2013 [START_REF] Belloni | Retirement choices in italy: What an option value model tells us[END_REF], Benallah, 2011[5]; Engels et al., 2017[11]). In frictional labour markets, the return on jobs is determined by their expected duration (Seater, 1977 [27]), Bettendorf and Broer, 2003[6], Hairault et al.,2010 [17]). By extending the time horizon during which workers remain in the labor market, these incentives also might alter the preferences that firms exhibit prior to the legal retirement age (Ilmakunnas and Ilmakunnas , 2015 [START_REF] Ilmakunnas | Hiring Older Employees: Do the Age Limits of Early Retirement and the Contribution Rates of Firms Matter?[END_REF]), namely, by changing their job creation efforts, wages, training policies and promotion rules. Firms might react by adjusting either the quantity of jobs available (job creation) or the wage they offer (the prices). The former adjustment benefits outsiders, whereas the latter benefits insiders.

To better understand the effects of financial incentives to delay retirement on both the supply and demand for labour, we develop a job-search model with age groups able to account for the effect of the policy on both employment and wages. This model must also make the wage distribution explicit to show the possible heterogenous effect according to wage. The equilibrium search model (Mortensen, 1998 [START_REF] Dt | Equilibrium unemployment with wage posting: Burdett-mortensen meet pissarides[END_REF]) is particularly suitable for this purpose. We extend it by making retirement endogenous and therefore potentially affected by financial incentives.

We contribute to the literature on lifetime job search model applied to retirement behavior by making the contact rate and the productivity endogenous and thus by analysing both labor demand and supply. Doing so, we include the firm behavior in the complex process of retirement.

So far, only the labor supply was taken into account [START_REF] Hairault | Distance to retirement and older workers' employment: The case for delaying the retirement age[END_REF] [START_REF] Hairault | Distance to retirement and older workers' employment: The case for delaying the retirement age[END_REF]. In our model, we go further: firms can react to the change in workers' retirement decisions (due to incentives) by adjusting their job openings and wage offers. Cross-country variability in older workers' labor market participation highlights the impact of institutional differences (Nickell, 1997[23]; Abowd et al. [START_REF] Abowd | High wage workers and high wage firms[END_REF]; Blanchard and Wolfers, 2000 [START_REF] Blanchard | The role of shocks and institutions in the rise of european unemployment : the aggregate evidence[END_REF]; Saint-Paul, 2009 [START_REF] Saint-Paul | Does welfare state make older workers unemployable? IZA Discussion Papers[END_REF]). Prior empirical studies also note the heterogenous impacts of financial incentives across countries (Albert et al., 2008[2]; Hanel, 2010 [START_REF] Hanel | Financial incentives to postpone retirement and further effects on employment ? evidence from a natural experiment[END_REF]) and in relation to individual preferences and productivity (Van Soest and Vonkova, 2014 [START_REF] Van Soest | How sensitive are retirement decisions to financial incentives? a stated preference analysis[END_REF] and Reichlin, 2018 [START_REF] Reichlin | Social Security and Old Age Unemployment Risk with Efficient Wage Bargaining[END_REF]). Therefore, to model workers' behavior, we augment Mortensen's (1998 [START_REF] Dt | Equilibrium unemployment with wage posting: Burdett-mortensen meet pissarides[END_REF]) model with unemployment benefits indexed on past earnings.

We capture heterogeneity in leisure preferences and health conditions (Gielen, 2009[14]; Currie et Madrian, 1999 [START_REF] Currie | Handbook of Labor Economics, chapter Health, health insurance and the labor market[END_REF]; Cai et Kalb, 2006 [9]; Garcia-Gomez, 2011 [START_REF] Garcia-Gomez | Institutions, health shocks and labour market outcomes across europe[END_REF]) by assuming a disutility at work that is heterogenous among workers. With this theoretical approach, we identify the transmission channel of the policy on the 50 to 59 years old labour market and simulate the effect of the policy without being exposed to the Lucas' critique (1976 [START_REF] Lucas | Econometric policy evalation: A critique[END_REF]). Implementing this type of model makes indeed it possible to highlight the mechanisms underlying the implementation of financial incentives.

We conclude that postponing workers' retirement horizon with an incentive policy affects wages in the previous age classe and particularly for earners at the top of the wage distribution, but it does not influence employment. Financial incentives also do not affect workers homogeneously; the presence of disutility at work and unemployment insurance causes only highly paid workers to lengthen their career. That is, the main effect of incentive policies is a deformation of the offered wage distribution, such that the policy only benefits insiders already employed with a high wage.

We study the anticipation effects of both senior workers and the firms that employ them. Furthermore, we model firms' anticipatory behaviours in terms of both job creation and the wage posting strategy. Simulations based on this structural model provide clear guidance with regards to using financial incentives to delay workers' retirement. This article brings a new contribution to the literature. Financial incentives to postpone retirement create a high inertia of income and status inequalities. Since high-paid workers will have greater interest in benefiting from financial incentives, inherited inequalities will therefore continue through these increased pensions. These inequalities existed just before retirement, this period being characterized by more accidents at work, health issues and a high risk of long-term unemployment.

Section 2 presents the equilibrium model and its properties. Section 3 discusses the results from the simulations. Section 4 concludes.

A lifetime job search model

To understand how financial incentives affect wages and employment of seniors, we use the framework developed in Mortensen's (1998 [START_REF] Dt | Equilibrium unemployment with wage posting: Burdett-mortensen meet pissarides[END_REF]). This framework has the advantage to model a frictional labour market in which the hiring process is costly in terms of time for firms. In this context, the working horizon of workers affect firms' behavior (Seater, 1977 [27]), Bettendorf and Broer, 2003[6], Hairault et al.,2010 [17]). Besides, this framework models this behavior in terms of both job creation, by making the contact rate endogenous and wage strategy, by making job productivity endogenous. In this model, time is continuous and the model is in steady state2 .

Model Assumptions

Population dynamics

To study the effect of financial incentives on wages and employment of seniors, we augment Mortensen's (1998 [START_REF] Dt | Equilibrium unemployment with wage posting: Burdett-mortensen meet pissarides[END_REF]) model with life cycle and endogenous retirement variables. We divide the life cycle into four parts. All variables that depend on workers' age class are indexed by i, which can take values of i = 0 for young workers, i = 1 for young seniors who cannot retire (they have not reached the legal retirement age), i = 2 for older seniors who can retire (they are beyond the legal retirement age), and i = 3 for retired workers. The mass of the population in each age class is denoted m i . We assume the economy is in a steady state. At each period, the probability of ageing equals δ i , and the arrival of new agents replaces an equal number of retired workers, so there is no labor force growth. The probability δ 3 is the probability of dying. Masses m i therefore solve:

δ 0 m 0 = δ 1 m 1 = δ 2 m 2 = δ 3 m 3
Depending on their age, workers can achieve three different status levels: Employed, unemployed or retired. We denote, by u i and e i , the mass of unemployed and employed workers of age class i, and r ji indicates the mass of retired workers of age class i who retire in age class j. We differentiate retired workers according to their retirement age, because the bonus only applies to those who retire later. We then can summarize the composition of workers in each age class as follows:

m 0 = e 0 + u 0 m 1 = e 1 + u 1 m 2 = e 2 + u 2 + r 22 m 3 = r 32 + r 33

Employment opportunity

Workers search for a job when they are unemployed or employed, there is on the job search.

Firms and workers meet according to the following matching process:

M i = v η i (φ 0 u i + φe i ) 1-η
where η is the matching function elasticity, v i indicates the number of job vacancies, and φ 0 and φ refer to the search effectiveness of unemployed and employed workers, respectively.

We set θ i = vi φ 0 ui+φei to reflect labor market tightness on each market. The meeting frequencies between workers and firms are given by:

λ i = φθ 1-η i and λ 0 i = φ 0 θ 1-η i
for employed and unemployed workers, respectively; and

q i = φθ -η i and q 0 i = φ 0 θ -η i
for firms to contact an employed or an unemployed worker, respectively.

The cumulative distribution of wages offered by firms on each market, regardless of the worker's status, is endogenous and denoted by F i (.), as detailed in subsection 2.3.2.

Workers' Behavior

Bellman Equations

To capture the effect of the reforms on the labour market, we also account for the health conditions of these older workers. Employment decisions by workers at the end of their careers often are meaningfully informed by their health conditions (Currie and Madrian, 1999 [START_REF] Currie | Handbook of Labor Economics, chapter Health, health insurance and the labor market[END_REF],

Cai and Kalb, 2006 [START_REF] Cai | Health Economics, chapter Health status and labour force participation : evidence from Australia[END_REF], Garcia-Gomez, 2011 [START_REF] Garcia-Gomez | Institutions, health shocks and labour market outcomes across europe[END_REF]). Therefore we introduce workers' disutility at work, according to their age class (Gielen 2009, [14]). This disutility is attached to each worker and is heterogenous across workers. We denote it with d. According to data from the SHARE database, 60-64 year-old seniors have 40% more chronic deceases on average than those between the ages of 50 and 59 years. Health tends to deteriorate between these two age class. To represent this deterioration, we introduce a parameter α (with α > 1 )that increases workers' disutility.

Let V e i (.) denote the value in each age class of the optimization problem of an employed worker according to the wage, and the disutility at work if his age class is 1 or 2:

rV e 0 (w) = w -τ + λ 0 w w (V e 0 (x) -V e 0 (w))dF 0 (x) -s(V e 0 (w) -V u 0 (b(w))) -δ 0 (V e 0 (w) -M ax(V e 1 (w), V u 1 )) (1
)

rV e 1 (w, d) = w -τ -d + λ 1 w w (V e 1 (x, d) -V e 1 (w, d))dF 1 (x) -s(V e 1 (w, d) -V u 1 (b(w), d)) -δ 1 (V e 1 (w, d) -M ax(V e 2 (w, d), V u 2 (b(w), d), V r 22 )) (2
)

rV e 2 (w, d) = w -τ -αd + λ 2 w w (V e 2 (x, d) -V e 2 (w, d))dF 2 (x) -s(V e 2 (w, d) -M ax(V u 2 (b(w), d), V r 22 )) -δ 2 (V e 2 (w, d) -V r 33 ) (3)
The value of being unemployed, according to unemployment benefits and the worker's disutility, is denoted by V u i (., .). The value of retiring while in age class i is denoted by V r ii . In each age class, an employed worker can receive a better job proposal at the arrival rate λ i (1 -F i (w)), and his job is destroyed at a rate s. Retirement is endogenous, because workers can choose to retire once they move from age class 1 to 2, or once they lose their jobs while in age class 2. Workers are forced to retire at the end of age class 2. Unemployed workers receive unemployment benefits denoted by b that depend on their previous wage with a replacement rate of ρ, as follows:

b(w) = ρw
The pensions are founded by a lump tax τ that is equal whatever the status of the worker and therefore that does not affect any workers' trade-offs in the economy.

The value of being unemployed in each age class V u i (.), according to unemployment benefits and disutility at work, thus can be determined as follows:

rV u 0 (b) = b -τ + λ 0 0 w w M ax(V e 0 (x) -V u 0 (b), 0)dF 0 (x) -δ 0 (V u 0 (b) -V u 1 (b)) (4) rV u 1 (b, d) = b -τ + λ 0 1 w w M ax(V e 1 (x, d) -V u 1 (b, d), 0)dF 1 (x) -δ 1 (V u 1 (b, d) -M ax(V u 2 (b, d), V r 22 )) (5) rV u 2 (b, d) = b -τ + λ 0 2 w w M ax(V e 2 (x, d) -V u 2 (b, d), 0)dF 2 (x) -δ 2 (V u 2 (b, d) -V r 33 ) (6)
The disutility at work also affects V u i for i = 1, 2 by decreasing the value of job opportunities V e i (x, d). Similar to employed workers, unemployed workers can choose to retire when they move from age class 1 to 2.

The value of being retired is V r ij , where i is the current age class and j is the age class in which the worker retires. The value of being retired in age class 2 is:

rV r 22 = p 2 -τ -δ 2 (V r 22 -V r 32 ) (7)
In age class 3, the asset value of these workers is given by:

rV r 32 = p 2 -τ -δ 3 V r 32 ( 8
)
If workers only retire in age class 3, their asset value is given by:

rV r 33 = p 3 -τ -δ 3 V r 33 (9)
In age class 3, early retirees keep receiving the pension p 2 , and all others receive a pension p 3 .

As in Hairault et al. (2010[17]), we assume the pensions are lump sums and not indexed on the last wage. Indeed, in France the pensions are computed based on the 20 best years and not only on the last wage. This last wage can be quite heterogenous among workers, yet, as the agents in the model are ex-ante homogenous (same individual productivity, same layoff and hiring hazard, same wage offers lottery), the mean wage of their 20 best years is necessarily very homogenous. A pension that takes the form of a lump sum is therefore more accurate to model the workers retirement behavior than a wage indexation.

Pension scheme: The incentive policy and the workers' choices

The incentive policy suggests that pensions rise when workers work longer, so p 3 > p 2 . Both an increase in p 3 and a decrease in p 2 widens the gap between being active or retiring early.

An increase in p 3 raises the value of being active in age class 2, V e 2 and V u 2 . A decrease in p 2 reduces the value of retiring early.

Workers make several decisions relative to the labor market: When to retire, whether they are employed or unemployed, when to resign if employed, and when to accept a job if unemployed. These arbitrages therefore involve three states: unemployment and employment, employment and retirement, and unemployment and retirement. Using equations 1, 2, 3, 4, 5, 6, 7, 8, we define a decision matrix that summarize workers' arbitrage as follows:

• I eu 0 such that I eu 0 (w, b) = 1 if V e 0 (w) > V u 0 (b) and I eu 0 (w, b) = 0 if V e 0 (w) < V u 0 (b)
. Each point of the matrix I eu 0 (w, b) takes the value 1 if and only if the value of being employed at w is higher than the value of being unemployed at b.

• I eu i for i = 1, 2, such that I eu i (w, b, d) = 1 if V e i (w, d) > V u i (b, d) and I eu i (w, b, d) = 0 if V e i (w, d) < V u i (b, d).
Each point of the matrix I eu i (w, b, d) takes the value 1 if and only if in the age class i, the value of being employed at w with a desutility at work d is higher than the value of being unemployed at b.

• I ur such that I ur (b, d) = 1 if V u 2 (b, d) > V r 22 and I ur (b, d) = 0 if V u 2 (b, d) < V r 22 .
Each point of the matrix I ur takes the value 1 if and only if in the age class 2, the value of being unemployed at b with a desutility at work d is higher than the value of being retired.

• I er such that

I er (w, d) = 1 if V e 2 (w, d) > V r 22 and I er (w, d) = 0 if V e 2 (w, d) < V r
22 for all w. Each point of the matrix I er takes the value 1 if and only if in the age class 2, the value of being employed at w with a desutility at work d is higher than the value of being retired.

The incentive policy affects I ur and I er directly by increasing the value of staying active. It also affects I eu indirectly, because employment is more valuable if the horizon is more distant. Through these decisions, the incentive policy affects the mass of workers, according to their status, wages, unemployment benefits, and disutility. The mass of employed workers according to their wage and disutility e i (., .) and the mass of unemployed workers according to their unemployment benefits and disutility u i (., .) can be deduced from the decision matrix, as presented in Appendix A.

Firms' Behaviour

By changing the condition of workers according to their status, wages, and unemployment benefits, the incentive policy also indirectly affects firms' behavior, both in relation to job creation, and in wage decisions. We assume firms can define their searches according to workers' age classes 3 such that there are three labor markets: i = 0, 1, 2. We assume firms cannot observe disutility at work, the status, or the reservation wage of workers, yet they are aware that a distribution of workers exists according to these characteristics.

Job Creation Decision

Firms enter each market as long as this equiprofit is superior to the vacancy cost, denoted as c. We use Π i (θ i ) to denote the profit of firms that target age class i, according to the tightness of the labor market. At an equilibrium, labor market tightness can reveal the following free 3 They can discriminate among workers by setting experience requirements. When a firm enters one of the three markets, the production generated by employing a worker from the two other markets is null. Therefore, workers do not cheat. entry condition:

Π i (θ i ) = c (10)
Labor market tightness also depends on firms' profit and drives the meeting frequencies between firms and workers. As indicated in subsection 2.1.2, the higher the labor market tightness, the higher the exit rate from unemployment.

Wage Posting Decision

The intuition behind the wage game of firms (Burdett and Mortensen, 1998 [START_REF] Burdett | Wage differencials, employer size, and unemployment[END_REF]) starts with the assumption that firms successively enter each market. When only one firm appears in the market, its maximum instantaneous profit occurs at the lowest wage possible (here, minimum wage). A second firm entering the market might then offer a wage slightly superior to the first firm, to poach the employed workers of the first firm, and so on for other firms entering the market. At equilibrium, when firms have equiprofit, this wage game generates a wage distribution on an interval [w;w] Therefore, the offered wage distribution F i solves the following equiprofit condition:

Π i (w i ) = Π i (w) (11)
With w i the minimum wage proposed in age class i. 4 The incentive policy raises the expected profit in age classes 1 and 2 by inducing a lower separation probability. In the presence of generous unemployment benefits, and given the disutility of workers, this increase should be particularly notable among firms offering rather high wages. In this case, this policy is likely to affect the offered wage distribution in age classes 1 and 2 by inducing more firms to offer higher wages.

Effects on Firms' Behavior

By noting h i (.), the hiring frequency of firms according to the offered wage, and J i (.), the surplus of the firms according to the wage,we can calculate firms' expected profit according to the wage:

Π i (w) = h i (w)J i (w)
4 Without any minimum wage regulations, the lowest wage offered by firms on each market is the wage that maximizes the profit when F i (w) = 0. By definition, w such that F i (w) = 0 is the lowest wage in the economy since no offered wage is below it. The shape of the profit differs from one market to another, so these minimum wages also should be different. If the institutional minimum wage w is above these wages, the market minimum wages equal this institutional minimum wage. Actual minimum wages can therefore be computed as follows:

w i = max{argmax w Π i (w), w}
where Πy, Πa, and Πs refer to the profit of firms offering the lowest wage on each market (i.e, when F i (w) = 0)

Hiring frequency depends on the mass of workers ready to accept a given wage. Firms can recruit both unemployed and employed workers, so this frequency depends on the distribution of workers according to their current wage, unemployment benefits, and disutility. Firms hire workers at the following frequencies according to the offered wage:

h 0 (w) = q 0 0 b b u 0 (b)I eu 0 (w, b)db + q 0 w w e 0 (x)dx h i (w) = q 0 i b b d d u i (b, d)I eu i (w, b, d)dd.db + q i d d w w e i (x, d)dddx
The policy increases the mass of workers in age class 2 that remain in the labor force (e 2 + u 2 ) and therefore increases hiring frequency on this market.

Firm Surplus

With this frequency, firms generate a surplus. We use k to denote the match-specific investment that firms devote to each match, and β i is the cost of this investment per worker at the job creation date according to age class. This parameter allows the investment cost to differ over age classes. It for instance accounts for the fact that a worker with a larger work experience have already accumulated some human capital that makes this investment less costly.

Mortensen (1998 [START_REF] Dt | Equilibrium unemployment with wage posting: Burdett-mortensen meet pissarides[END_REF]) shows that the endogenous productivity resulting from this investment when it contributes at a deacreasing rate to productivity generates a realistic wage distribution. Production also depends on an individual productivity component that reflects the age class y i .

The matches' productivity according to the match-specific investment is therefore given by:

y i (k) = y i + q γ k γ
Where the exogenous parameter q > 0 is the share of productivity that results from the firms' human capital investment and where the exogenous parameter 0 < γ < 1 accounts for the decreasing return of the production function according to human capital. The value of the firms' expected surplus is given by:

J 0 (w) = y(k) -w + δ 0 d d I eu 1 (w, d)h(d)dd M ax(J 1 (w), 0) r + s + δ 0 + λ 0 (1 -F 0 (w)) -β 0 k (12) J 1 (w) = y(k) -w + δ 1 d d I er (w, d)I eu 2 (w, d) e1(w,d) e1(w) dd M ax(J 2 (w), 0) r + s + δ 1 + λ 1 (1 -F 1 (w)) -β 1 k (13) J 2 (w) = y(k) -w r + s + δ 2 + λ 2 (1 -F 2 (w)) -β 2 k (14)
Maximizing equations 12, 13, and 14, subject to k provides the optimal level of this investment chosen by firms. It fully depends on wages. The probability that a firm keeps an employee hired in age class 0 during age class 1 depends on the wage and disutility at work that the worker has in this age class. If the wage is enough to compensate for the disutility, the worker remains employed. In age class 2, workers remain employed if they choose not to resign and not to retire early, I eu 2 = I er = 1. This decision also depends on the wage and disutility at work. The repartition of workers according to their disutility is not homogenous according to wages though, because workers with high disutility reject more job offers. Intuitively, workers with a high disutility are less often employed. Yet when employed, they enjoy high wages. The probability that a firm employ a worker at wage w when that worker has disutility d is e1(w,d) e1(w) . With e 1 (w), we denote the density of workers employed at wage w.

Across age classes 0, 1, and 2, the probability that the firm keeps its employee in the next age class increases with wages, because the higher the wage, the lower the poaching risk. Also, in age class 1, the higher the wage, the less likely workers are to resign or retire. The incentive policy reduces the risk of workers retirement from firms offering high wages. The expected surplus mostly increases for firms offering high wages.

Equilibrium Conditions

The equilibrium distributions e i (., .), u i (., .), and F i (.); the decision vectors; and the equilibrium value of θ i can be achieved when four conditions are fulfilled, in each market:

• Decision vectors are such that conditions of section 2.2.2 are fulfilled.

• Firms post wages so that equiprofit is guaranteed (equation 11).

• Firms choose the optimal level of human capital investment by maximizing equations 12, 13, and 14, subject to k.

• Firms enter the labor market until all expected profit is exhausted (equation 10).

• Workers' flows in and out of the market for each status and wage level are equal (Appendix A).

The incentive policy raises the profit in age classes 1 and 2. Whether the profit of firms increases only if they offer high wages or for any firm is a crucial question. If the policy raises the level of equiprofit in a market, labor market tightness increases and firms create more jobs.

If instead the policy only affects the profit of firms offering high wages, equiprofit might not be affected such that only the wage distribution would change.

We proceed to numerical simulations to compute these equilibrium results. To take into account a potential increase of labour disutility with age, we choose a distribution of disutility that has the same shape as the distribution of the health conditions of older workers. We use the number of chronic diseases as a proxy for health condition and use the SHARE database to compute the number of chronic diseases. Appendix B details the number of chronic diseases among men 50 to 59 years we use as a proxy for health condition. The shape of this distribution leads us to anticipate that workers' disutility follows a log-normal distribution, with µ as the mean and σ as the standard deviation. The values of the model parameters are in table 1.

We distinguish two subsets of parameters, according to the calibration strategy. We use the same calibration strategy as Le Duigou (2020 [START_REF] Duigou | Endogenous unemployment benefits in an equilibrium job search model over the life-cycle[END_REF]). The first subset {r, δ i , s, p 2 , p 3 } is based on external information. We use values from Postel-Vinay and Robin (2004) [START_REF] Postel-Vinay | To match or not to match? optimal wage policy with endogenous worker search intensity[END_REF] to set the job destruction rate. The age class transition rates depend on the length of the age class: Age class 0 lasts 30 years, so δ 0 = 1/30; age class 1 lasts 10 years, so δ 1 = 1/10; and age class 2 lasts 5 years so δ 2 = 1/5. The value of the pensions p 2 and p 3 represent the French policy before 2003. At retirement age, we set the mean retirement ratio to 0.85, similar to Hairault et al. (2012[16]).

The second subset, {φ, φ 0 , q, y i , β i , ρ, γ, σ, µ, α}, is designed to reproduce different moments in the economy. The parameters {φ, φ 0 } respectively reproduce the contact rate of employed workers (Postel-Vinay and Robin, 2004 [START_REF] Postel-Vinay | To match or not to match? optimal wage policy with endogenous worker search intensity[END_REF]) and the unemployment rates of workers who are 20-49 years old. Without disutility, considering the observed levels of wages and unemployment, the employment rate of workers between 50 and 64 should be higher. We therefore use µ and α to fit correctly the employment rate of the 50-59-year and 60-64-year age groups. The value of the parameter α indicates that the disutility of the 60-64-year age group is 50% higher than the disutility of the younger group, assuming 40% more chronic diseases. The parameter σ reproduces the shape of the distribution of the number of chronic diseases among men between 50 to 59 years old, for the 5th to 9th decile ratio of the distribution. Appendix B depicts the theoretical distribution of disutility for the 50-59-year age group, with calibrated values of µ and σ.

The parameters {y i , β i , γ, q} are set to reproduce the level and shape of wage distributions over the age classes. The parameters y 0 is set to reproduce the mean wage of 20-49 years old and β 0 is normalized to one. The parameters y 1 , y 2 and β 1 , β 2 capture the wage growth with age due to human capital accumulation. We assume y 1 = y 2 and β 1 = β 2 , meaning that we assume that there is no significant human capital accumulation between 50-59 years old and 60-64 years old. These two parameters are set to reproduce the 50-59 years old and 60-64 years old mean wage. The two remaining parameters of the production function γ and q allow to create wage dispersion (Mortensen (1998 [START_REF] Dt | Equilibrium unemployment with wage posting: Burdett-mortensen meet pissarides[END_REF])) in the model. We use these two parameters to reproduce the 6th and 7th decile of the 20-49 wage distribution. The parameter ρ is set to reproduce the 50-59 mean unemployment benefits 6 . We assume here that unemployment benefits never end in the model. This assumption is consistant with the long elegibility period for senior workers in France (3 years and then some special programs for seniors exist). Note that we do not address the unemployment rate for the 60 to 64 years old group, yet the model, given wages, the unemployment benefits, and the disutility distribution, can reproduce its value of 0%.

Results

Table 2 lists statistics pertaining to employment, unemployment, retirement rate, mean wage, offered wages and unemployment benefits according to age classes in the benchmark economy.

The incentive policy of 0.75% per quarter corresponds, over five years, to a bonus of 15%.

Consequently, before the policy p 3 = p 2 , and after p 3 = p 2 * 1.15. Table 3 presents the results of a simulation that introduces the bonus. By comparing the benchmark in table 2 with the economy after the reform in table 3, we observe that the policy increased the employment rate among workers aged 60-64 years. Some workers responded to the reform by delaying their retirement. Thus, active workers in this age group become more numerous in our theoretical model: from 11.5% to 48%. This result is consistent with research that shows that an incentive 6 Calibration is precise at 2% on wage levels and half a percentage point for employment rate. policy can raise the employment rate of workers who are older than the legal retirement age (Gruber and Wise, 2004[15], Hanel, 2010 [START_REF] Hanel | Financial incentives to postpone retirement and further effects on employment ? evidence from a natural experiment[END_REF]; Ferrari, 2019 [START_REF] Ferrari | The effectiveness of incentives to postpone retirement: evidence from italy[END_REF], Belloni and Alessie, 2009 [START_REF] Belloni | The importance of financial incentives on retirement choices: New evidence for italy[END_REF],

Benallah, 2011 [START_REF] Benallah | Comportements de départ en retraite et réforme de 2003. les effets de la surcote[END_REF]). We also assess the theoretical effects on received and offered wages. Before the reform, only very well paid workers were active between 60 and 64 years, as reflected in the wage increase in table 2 between the 50-59-year and 60-64-year age classes. This increase results from a mere composition effect; the offered wages decrease between the two age classes.

In table 3, after the reform, the selection effect diminishes, because active workers between 60 and 64 of age grow more numerous. The mean wage and offered wage for these workers decrease compared with before the reform.

The spillover effect on the 50-59 years old wages. Beyond this effect on workers aged 60 years or older, we note the effect of the reform on employment and wage of younger workers Firms have at first interest to pay the lowest wage possible, yet as increasing the wage offer implies to raise the probability to hire a worker (speed up the hiring process) and to raise the job horizon, firms can raise their wage as long as their profit does not become below the profit of paying the minimum wage (the equiprofit). Before the incentive policy, very few workers were working after 60. Firms anticipated that they would very likely loose their employee as soon as he or she reaches 60. After the policy is implemented, more than a third still work after 60. Firms can anticipate an extended horizon. Yet because of desutility at work, those who remain employed after 60 necessarily have a rather high wage, the other retire early or resign themselves to remaining unemployed. Given this possible extended horizon, new wages, at the top of the wage distribution becomes profitable, some firms therefore decide to raise again their wage offer. Yet, this spreading of wage does not affect the equiprofit of opening a vacancy on the 50-59 years-old market: it is not more profitable to employ someone aged 50-59.

It is only more profitable to offer high wages to these workers, firms start offering high wages that used to induce too low a profit before the implementation of the policy. The job creation equation 10 remains unchanged. However, even if we observe no change in employment among workers aged 50 to 59 years, the employment rate for those 50-64 years of age increases overall from 54% to 61% due to the reform.Beyond this effect on workers aged 60 years or older, we note the effect of the reform on younger workers too. In the immediately younger age class, contrary to Hairault et al. (2012, [16]), we find no effect on unemployment.

Firms do not create more jobs, yet they choose to raise the wage of existing jobs. We obtain this new result as the model accounts for endogenous job creation and wage distribution.

Conclusion

We provide some initial evidence that incentive policies designed to delay retirement affects the wage policies adopted by firms. Using a theoretical approach, we show that financial incentives might imply a wage increase for older workers, especially highly paid ones.

We focus on the 2003 French pension reform (implemented in 2004) that gave older workers who decided to postpone their retirement a pension bonus of 0.75% per additional quarter worked. The theoretical analysis details the various mechanisms that can explain these results and identify how policy exerts an impact on older workers. The equilibrium search model à la

Mortensen (Mortensen, 2008 [START_REF] Dt | Equilibrium unemployment with wage posting: Burdett-mortensen meet pissarides[END_REF]), with endogenous retirement and heterogenous work disutility among workers shows that after the reform, the median wage for workers between 50 and 59 years of age increased by 1%, and they rose even more among the highest paid workers.

The wage increase therefore appears due to a change in the firms' wage posting behavior. As the horizon grows longer, firms anticipate an extended employment duration for their employees, and offering higher wages becomes more interesting as a means to retain workers. This adjustment occurs at the expense of an adjustment to the quantity of jobs, and therefore at the expense of increased employment. Insiders benefit more from these incentive policies than outsiders. Therefore postponing workers' retirement horizon implies a selection effect: only the best paid are encouraged to remain active.

To improve the effect of financial incentives on the employment of older workers in general

and not only at retirement, it would therefore be necessary to accompany them with policies aimed at reducing the inequalities between older workers even before retirement, for example through specific job-search aids for this category of workers and trainings aimed at increasing the employability of older workers in skilled jobs.

Following this research, several extensions could be proposed. The optimal incentive policy could be analysed, taking into account the welfare costs for workers who suffer from high disutility at work at the end of their career but continue to work anyway.

A Workers' flows

We denote by u 0 (b) and e 0 (w), the mass of young unemployed and employed workers receiving Given these flows equation, the distribution of disutility among the unemployed, the employed, and the retired is different in steady state. Workers with high disutility are more likely to retire earlier, yet some workers with high disutility can also remain employed if they are employed at a high wage. Both wage and disutility distribution are endogenous.

B Chronic disease and disutility

3

  Incentive Policy Simulations: An application to the 2003 French pension reform 3.1 Model Calibration We use parameter values consistent with the French economy before the reform in 2003. We use data from the 2003 French Labor Force Survey, and retain only men in the sample. We use data for the three age classes, as defined in the model. That is, workers can retire at the end of the age class 1, and the legal retirement age in 2003 was 60 years, so we set the end of the age class 1 to 59. We set the beginning of this age class at 50 years. For the numerical simulation, we assume the three age classes to correspond to the following ages: Age class 0 from 20 to 49 years, age class 1 from 50 to 59 years, and age class 2 from 60 to 64 years 5 . Workers younger than 50 years can be considered unaffected by the reform. Workers older than 60 years are necessarily affected; they are the target of the reform. Workers between 50-59 years can be affected by the reform through an anticipation effect.

1 2 δ 1 w w e 1 ( 1 - 1 w w e 1 s b b e 2

 1211112 respectively the unemployment benefit b and the wage w and by u i (b, d) and e i (w, d), the mass of unemployed and employed workers with a disutility d and receiving respectively the unemployment benefit b and the wage w.We assume that workers' disutility at work is heterogenous. The cumulative distribution of d in the population is denoted by H(.), its associated density function by h(.). In steady state, u 0 (b) solves the following flows equation:b)f 0 (x) dx + δ 0 u 0 (b) = δ 3 m 3 + se 0 (b -1 )For unemployment benefits larger than b, u 0 (b) solves:λ w w I eu 0 (x, b)f 0 (x) dx + δ 0 u 0 (b) = se 0 (b -1 ) For i = 1, 2, u i (b, d) solves: , b, d)f 1 (x) dx + δ 1 u 1 (b, d) = δ 0 u 0 (b)h(d) + se 1 (b -1 , d) + δ 0 (1 -I eu 1 (b -1 , b, d))e 0 (b -1 )h(d) , b, d)f 2 (x) dx + δ 2 u 2 (b, d) = δ 1 I ur (b, d)u 1 (b, d) + sI ur (b, d)e 2 (b -1 , d) + δ 1 I ur (b, d)(1 -I 2 eu (b -1 , b, d))e 1 (b -1 , d)The cumulative distribution functions of wage earned by employed workers of age class 2 with disutility d is noted G 2 (w, d) and solve in steady state:(δ 0 + s + λ(1 -F 0 (w))) b)I eu 1 (b, x)f 0 (x)dxdb (δ 1 + s + λ(1 -F 1 (w))) (b, d)I eu 1 (d, b, x)f 1 (x)dxdb + δ 0 w w e 0 (x)h(d)I eu 1 (x, b(x), d)dx (δ 2 + s + λ(1 -F 2 (w))) (b, d)I eu 2 (d, b, x)f 2 (x)dxdb + (x, d)I er (x, d)I eu (x, b(x), d)dxThe mass of early retirees of age class 2 and 3 according to disutility solves:δ 2 r 22 (d) = δ 1 b b I ur (b, d))u 1 (b, d)db + δ (x, d)(1 -I ur (b(x), d))(1 -I er (x, b(x), d))dx + (x, d)(1 -I ur (b(x), d))dbδ 3 r 23 (d) = δ 2 r 22 (d) The mass of late retirees of age class 3 according to disutility solves: δ 3 r 33 (d) = δ 2 (h(d)m 2 -r 22 (d))

Figure 1 :

 1 Figure 1: Distribution of the number of chronic disease among men between 50 and 59 years old

Figure 2 :

 2 Figure 2: Theoretical distribution of disutility among the 50 to 59 years old

Table 1 :

 1 Model parameter values

		Parameters based on external information
	Parameter	Value	Moment targeted
	w	1	Normalised
	βy	1	Normalised
	r	0.04	Discount rate
	δ i	1/n	Age class
	s	0.1	Postel-Vinay and Robin (2004) [24]
	p 2	0.85*50-59 mean wage	Hairault et al.(2012)[16]
	p 3	p 2	No incentive policy
		Parameters based on internal information
	φ	3.2	Contact rate of the employed (0.7)
	φ 0	4.8	20-49 Unemployment rate (9%)
	q	0.29	20-49 7th decile (1.92)
	y 0	1.6	20-49 Mean Wage (1.79)
	y 1 = y 2	2.1	50-59 Mean Wage (2.17)
	β 1 = β 2	0.85	60-64 Mean Wage (2.27)
	γ	0.78	20-49 6th decile (1.73)
	ρ	0.5	50-59 Mean unemployment benefits (1)
	µ	0.9	50-59 Employment rate (75%)
	σ	0.95	Median to the 9th decile of disutility distribution (0.33)
	α	1.45	60-64 Employment rate (11%)

Table 2 :

 2 Benchmark economy before the reform (p 3 = p 2 )

		Employment	Unemployment	Inactivity Mean Wage Mean offered wage Mean unemployment
							benefits
	20-49	90.9%	9.1%	-	1.779	1.554	0.791
	50-59	74.5%	25.5%	-	2.176	2.084	1.012
	60-64	11.6%	0%	88.4%	2.303	1.99	-

Table 3 :

 3 Simulation of the economy after the reform (p 3 = p 2 * 1.15)

		Employment	Unemployment	Inactivity Mean Wage Mean offered wage Mean unemployment
							benefits
	20-49	90.7%	9.3%	-	1.786	1.564	0.794
	50-59	74.5%	25.5%	-	2.222	2.106	1.034
	60-64	35.9%	11.7%	52.4%	2.224	1.791	1.265
	too. To understand what happens in this age class, we need to go back to the wage game:

Table 4 :

 4 As workers undergo a growing desutility at work with age, those who earn low wages retire early or resign themselves to remaining unemployed. Firms recognize that the extended horizon comes only at the cost of a rather high wage. The lower paying jobs do not exhibit a lengthened time horizon. Nor does the equiprofit of firms (equation 11); only the wage distribution changes. However, even if we observe no change in employment among workers aged 50 to 59 years, the employment rate for those 50-64 years of age increases overall from 54% to 61% due to the reform. Wage increases after the reform (p 3 = p 2 * 1.15), 50-59-year age class Only the best paid workers remain active. Taking into account the firms' behaviors allows us to show that such policies affect wages among the immediately younger age class but not on their employment. With this model, we demonstrate that the positive effect of the reform is limited to the highest paid workers. Theoretically, the main effect of the policy thus is the deformation of the wage distribution, rightwards.

	According to the simulation, wages increase by 2% (from 2.176 to 2.222). Besides, the model
	shows that this raise does not occur homogenously over the wage distribution. It is driven
	mostly by high paying jobs (see table 4) such that a median wage increases by 1% (from 2.200
	to 2.222), but the highest quartile increases by 3.7% (from 2.329 to 2.415). This increase in
	wages results from an increase in the wage offer (from 2.084 to 2.106) and therefore reflects a
	change in firms' wage policy.

This bonus consisted of a pension increase, by 0.75% per additional quarter worked. Between

and 2006, more than 83,000 retirees (equal to approximately 5% of the retiree population) benefited from this pension bonus. The average bonus pension was about 20 euros per month.In 2008, the pension bonus was increased to 1.25% per additional quarter worked. In 2016, 14.1% of the retirees perceive a bonus for postponing their retirement.

Note that a model close to the one presented below (with life cycle yet without endogenous retirement) is applied to unemployment insurance issues in Le Duigou (2020)[START_REF] Duigou | Endogenous unemployment benefits in an equilibrium job search model over the life-cycle[END_REF].

Because we assume that workers between 20 and 59 years of age cannot retire, we exclude inactivity from the data for these workers. After 50 years of age, many unemployed workers are registered as inactive, because in 2003, job search was not compulsory for unemployed senior workers. We assume the inactive are registered as unemployed rather than retired.