Michel-Pierre Chelini
email: mpchelini@gmail.com

Georges Prat
email: georges.prat@parisnanterre.fr

Understanding the long run dynamics of French unemployment and wages

Keywords: Classification J.E.L.: E24, J2, J30 equilibrium unemployment, wages, France

A standard specification of the WS-PS model based on wage bargaining between unions and firms makes it possible to understand the main features of long-term dynamics of unemployment and wages in France at the macroeconomic level. This result is conditional on auxiliary hypotheses made on the representations of the degree of rigidity in the labour market (depicted by a stochastic state variable), of the reservation wage (depending on the legal minimum wage), and on the nature of "other factors" of wages and prices that are not a priori specified in the WS-PS theoretical framework (summarized by the output gap). We find that the observed unemployment adjusts gradually to its equilibrium value, which is composed of three components: a "chronic" component due to the repartition in the added-value (real reservation wage, social contributions, productivity, profit margins of companies), a "cyclical" component depending on the output gap, and a "frictional" component due to the imperfect mobility of labour factor and technical progress. The observed wage also adjusts gradually to its negotiated value, the latter depending on the reservation wage, the social contributions, the price level, the labor productivity, the profit margins of companies, the unionization rate and on the unemployment rate whose influence is time-varying. Our results suggest that, in the average, the power of firms dominates that of unions during the negotiations, while, as predicted by the theory, change in employment intervenes effectively in the adjustment between wage desired by employees and wage offered by employers to achieve equilibrium.

Introduction

This article aims to portray the macroeconomic evolution in the unemployment rate and wage rate in the long run in France, in accordance both with economic theory and historical and institutional facts. 1 Our study is based on the thought that unemployment and wages must be modelled simultaneously. Indeed, from the point of view of the employee for whom the wage is an income, the requested wages are weakened by unemployment. In contrast, from the point of view of the employer for which the wage constitutes a cost of production, the wage determines the amount of labour input it wishes to use, and thus the extent of underemployment.

Our work does not propose a breakthrough in pure theory, but its objective imposes of course the choice of a general theoretical framework. As will be explained later, our framework refers to a simple specification of the WS-PS model (Wage Setting -Price Setting), which seems relevant to describe the behaviour of actors in the French labour market. In this model, the behaviours of employees and employers are supposed to be rational, but the labor market is characterized by an imperfection of the competitive mechanisms since companies fix their workforces according to wage agreements concluded with the trade unions. This theoretical framework makes it possible to estimate simultaneously at any time the value of the equilibrium unemployment rate and the value of the corresponding wage agreement bargained by the two social partners. To this end, our modelling introduces auxiliary hypotheses, covering the representation of the reservation wage, the other factors likely to influence wages and prices but not directly specified by theory, and the gradual adjustment of unemployment and wages to their equilibrium values. Moreover, we propose an econometric state-space modelling of unemployment and wages using the Kalman filter methodology, which allows a joint representation of the evolutions over time of the degree of labour market rigidity and of the sensibilities to unemployment of wages claimed by unions and those offered by companies. Our results show that the proposed specification of the WS-PS model reproduces the main features of the dynamics of the unemployment rate and the wage rate in France over the period 1955-2008. The remainder of this article is organized as follows. Part 2 gives an overview of the literature, especially that devoted to the applications of the WS-PS model in France, and justifies the choice of this model compared to other approaches in the literature. Part 3 discusses the auxiliary assumptions made to achieve the econometric specification. Part 4 presents the data, the estimation method and empirical results. The conclusion draws the main lessons from this study.

The literature and the choice of the WS-PS model

The economic literature on the relationship between wage rates and unemployment rates has developed along three lines 2 : the Phillips curve with its many extensions, the WS-PS negotiation model, and finally, the matching model pertaining to job supply and demand. Following the seminal article by [START_REF] Phillips | The Relation Between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957[END_REF], which highlighted a negative relationship between the nominal wage rate of change and the unemployment rate for the United Kingdom, the theoretical and empirical developments since then have led to the concept of "Non-accelerating inflation rate of unemployment" (NAIRU) proposed by Tobin, reflecting the theoretical value of the unemployment rate when real wages grow at the same rate as productivity. The debates in the literature highlight that this approach is characterized by theoretical and empirical limitations. At the theoretical level, the NAIRU was considered as representing more a reference value reflecting an economic equilibrium condition in the labour market than an approach describing the actual behaviour of the agents, even if, under certain restrictive conditions, the Phillips curve may join the WS-PS model or can be combined with it [START_REF] Blanchard | Wage Dynamic: Reconciling Theory and Evidence[END_REF], [START_REF] Heyer | Variables observables et inobservables dans la théorie du taux de chômage d'équilibre. Une comparaison France/Etats-Unis[END_REF][START_REF] Heyer | Structural and reduced approaches of the equilibrium rate of unemployment, a comparison between France and the United States[END_REF]). On the empirical side, studies have ascertained that, while these approaches explain wage changes rather well (see, for example, [START_REF] Heyer | Relation de Phillips, boucle prix -salaire : une estimation par la méthode de Johansen[END_REF]), the NAIRU often does not adequately represent the trend in the unemployment rate [START_REF] L'horty | Le NAIRU en France : les insuffisances d'une courbe de Phillips[END_REF][START_REF] Richardson | The Concept, Policy use and Measurement of Structural Unemployment: Estimating a Time Varying NAIRU across 21 OECD countries[END_REF], except by admitting a priori exogenous hypotheses, including the hysteresis hypothesis suggested by [START_REF] Blanchard | Hysteresis And The European Unemployment Problem, NBER Chapters[END_REF] [START_REF] Mc Morrow | Time-varying Nairu/Nawru estimates for the EU's member states[END_REF]).

The WS-PS model [START_REF] Layard | The cause of British unemployment[END_REF], 1986], [START_REF] Layard | Unemployment[END_REF]) explicitly outlines the behaviours of employees and firms in an imperfect competition framework and leads to the definition of an equilibrium rate of unemployment (ERU) that characterizes a situation wherein both employees and employers have arrived at a wage agreement; the ERU may be both higher and more fluctuating than the so-called "natural rate of unemployment". In fact, the specifications of this model in the literature are as numerous as the number of contributions, not to mention the variety of quantitative methods used to confront this model with observation data. In this regard, it is worth noting that two modes of empirical verification have been followed in the literature. The first is to estimate the structural wage (WS) and price (PS) equations in order to deduce the equilibrium value of the unemployment rate, which requires the introduction of constraints to identify the parameters of the model. The second mode consists of directly estimating a reduced-form of the equilibrium rate of unemployment equation, which is deduced from the equality between the desired and offered values of the real wages given by the two structural equations; this approach assumes that the spread between the observed unemployment and the ERU is a white noise, or that unemployment progressively adjusts to its equilibrium value according to a process to be specified. By adopting this second approach and assuming an adaptive adjustment of observed unemployment to its equilibrium value, Layard et al. [1991, p. 435] show that the WS-PS model fairly describes changes in the unemployment rate in 19 OECD countries including France; the specification used by the authors explains on average 86% of the variance of unemployment rates over the period 1956-88. These results were confirmed for France with the first kind of approach [START_REF] Bonnet | The Influence of different Specifications of Wages-Prices Spirals on the Measure of the NAIRU: the Case of France », Insee, document de travail G9612 de la Direction des Etudes et Synthèses Economiques[END_REF], L'Horty and Sobczak [1997], Cotis and al. [1998], L'Horty and Thibault [1998], Cahuc and[START_REF] Cahuc | Economie du travail : la formation des salaires et les déterminants du chômage[END_REF], l'Horty and Rault [2003], Gérard-Prenveille [2003]). 3 However, empirical analyses have shown that the introduction of numerous institutional variables capable of reflecting the rigidity of the labour market does not lead to very conclusive results [START_REF] Chagny | Le taux de chômage d'équilibrediscussion théorique et évaluation empirique[END_REF], [START_REF] Heyer | Structural and reduced approaches of the equilibrium rate of unemployment, a comparison between France and the United States[END_REF]). Interestingly, in order to capture the evolution in the institutional environment of the labour market, [START_REF] Heyer | Structural and reduced approaches of the equilibrium rate of unemployment, a comparison between France and the United States[END_REF] employed the Kalman filter method to analyse the time variability of the structural parameters of the WS and PS equations over the period in United States and France. The results show that if parameters stability can be accepted for the United States, some parameters vary significantly for France due to the increasing flexibility of the labour market subsequent to 1982, in particular regarding the deindexation of wages on the price level.

At about the same time as WS-PS, a matching model was proposed to capture the frictions between the flows of job supply and job demand [START_REF] Pissarides | Short-run equilibrium dynamics of unemployment vacancies and real wages[END_REF][START_REF] Pissarides | Equilibrium unemployment theory, 2 nd Ed[END_REF]). This approach then led to many developments that have been rewarded by the 2010 Nobel Prize in economy assigned to Diamond, Mortensen and Pissarides. This "DMP model" is indeed notable because it helps to determine not only an equilibrium level of unemployment and wage (as WS-PS), but also the job creation and job destruction flows at a disaggregated level. However, [START_REF] Diamond | Wage Determination and Efficiency in Search Equilibrium[END_REF] has shown that these frictions can give rise to several possible equilibria, so that agents must be able to coordinate in order to achieve a "good" equilibrium.

Moreover, the DMP model ability to describe the dynamics of the labour market was subsequently challenged by [START_REF] Shimer | The cyclical behavior of equilibrium unemployment and vacancies[END_REF] who showed that this model generates excessive volatility in real wages (and thus a too-low volatility for employment). Moreover, unlike the WS-PS model in which corporations face a reversed demand curve in the goods market (decreasing as a function of price), the DMP model does not incorporate a Keynesian mechanism involving the level of aggregate demand.

Consistent with the objective of this article, we have chosen to carry out our work within the framework of the WS-PS model. In contrast to the Phillips curve approaches, the agents' behaviours are explicitly outlined and seem rather well adapted to the French context.

While the desired values of wages and prices are respectively the results of rational behaviours on the part of employees and firms, the two parties negotiate between themselves to arrive at a wage agreement, ultimately leading the employers to set the level of employment. As for the DMP model, it is inherently more complex than WS-PS because it must describe the frictional forces associated with the inadequacy of job offers with job demand and draw all the consequences concerning the flows of supply and demand of employment. The WS-PS model provides a simple tool well-suited to the limited objective of our study, which deals with the historical dynamics in unemployment and wages in France at the macroeconomic level. Nevertheless, as will be explained below, the choice of this model should not hide the fact that the many specifications encountered in the literature raises difficulties, most of them being related to measurement problems of the variables.

The proposed model

The WS and PS theoretical equations

We consider a static specification of the WS-PS model where agents are representative and the capital exogenous, the aim being to deduce at each date the equilibrium values of the unemployment rate and the corresponding wage rate (one thus determines a time series of short-term equilibria of the labour market). As usual in the literature, the PS and WS equations are presented in a linear form by considering the logarithms of the variables, except for those expressed as a rate. 4 By relaxing the assumption of constant sensibilities of prices and wages to unemployment (i.e. according to the date, unemployment plays more or less the role of an adjustment variable between prices and wages), the general equations WS and PS can be expressed as follows 5 : PS :

p t t t p t t t c t t c t ps t f m u k ce p p p s          ) ( t k p t   0
(1) In a monopolistic competition situation where the marginal productivity of labour is decreasing, the PS equation is obtained under the assumption that firms set their prices in a way that maximizes their profits. The variable ps t s thus corresponds to the wage offered by the representative firm in view of the value of the price level t p which it considers optimal to set. In this framework of analysis, the "right to manage" principle leads companies to set their workforce according to the wage agreements reached with the unions. It follows that the unemployment rate can act as a variable of adjustment between the aspirations of both parties. The positive influence of the unemployment rate reflects the fact that, in the general case, the marginal productivity of labour increases when the number of workers decreases, thus encouraging the firm to offer a higher real wage, all other things being equal. As a direct result of the model, it can be seen that firms integrate in their selling prices the differential between the general price level and the consumer prices, which captures the influence of the terms of trade (price of exports / import prices). As is usual, Eq.( 1) is obtained by considering at the outset that firms integrate in the prices they should set the margin rate t m 6 and the rate of compulsory social contributions t ce . Regarding the WS equation, it determines the wage rate desired by employees under the assumption that trade unions aim to maximize the utility function of employees; doing so, trade unions take into account the likelihood that wage earners could be unemployed, receiving in this case an "equivalent income" equal to the monetary values of leisure benefits augmented by the disutility of labour and domestic labour. In Eq.( 2), the fact that unemployment weakens the level of wage required by of employees is a very intuitive result. As for the PS equation, it can be seen that the rate of social contributions t cs paid by employees belongs at the outset to the vector of variables conditioning the gross wage before taxes required by employees. The "other factors" p t f and s t f in the PS and WS equations represent variables that are a priori unspecified by the theory but that may also influence prices and wages.

The equilibrium unemployment rate

Equalizing the right-hand sides of equations ( 1) and ( 2 the theoretical content of the latter is complex and there is no compelling empirical indicator in the literature. Studies show that using unemployment indemnities as an indicator do not lead to very convincing results [START_REF] Manning | Wage bargaining and the Phillips Curve: The identification and specification of aggregate wage equations[END_REF], Cotis et al. [1998]). One explanation could be that the share of the unemployed at the end of entitlement is often both large (up to 50%) and fluctuating. This is the reason why some contributions assume that the real reservation wage follows a linear trend (L'Horty and Sobczak [1996], Cotis et al [1996], L'Horty and Sobczak [1997] 2001), or that it can be represented by the wage of the previous period [START_REF] Manning | Wage bargaining and the Phillips Curve: The identification and specification of aggregate wage equations[END_REF], [START_REF] Blanchard | Wage Dynamic: Reconciling Theory and Evidence[END_REF]). Even if these choices are understandable because of the unobservable nature of the reservation wage, they remain arbitrary. Another hypothesis is to index the reservation wage on labour productivity (Blanchard and Katz [1997], Cotis et al., 1998). In fact, if we introduce this hypothesis back into Eq.(3), the equilibrium unemployment no longer depends on the reservation wage and on productivity, which is of course not intuitive.

To circumvent these difficulties, we have followed an alternative direction, that of using the existence in France since 1950 of a legal minimum wage. 7 Indeed, [START_REF] Rioux | Salaire de réserve, allocation chômage dégressive et revenu minimum d'insertion[END_REF] analysed the results of four surveys carried out by the French National Statistic Office (INSEE) between 1994 and 1998, by interviewing the unemployed persons on the minimum hourly wage that they required to accept a job, that is to say the value of their reservation wage. It is important to emphasize here that the questions posed did not mention any reference to the SMIC: they are not therefore a priori incited to respond towards the value of the legal minimum. The author finds that two-thirds of the unemployed benefiting from the Minimum Income of (Social) Insertion (RMI, program supported by the French government between 1988 and 2009) and therefore not benefiting from unemployment allowances, ask at most the national minimum wage (SMIC) in order to accept an offer of employment.

Conversely, two-thirds of the unemployed benefiting from allowances require at least this minimum wage (SMIC). Moreover, for this latter group, it appears that the reservation wage increases with the wage level of their last job, while it reduces as the unemployment experience continues. In spite of the heterogeneity of the reservation wage according to individuals 8 , these results suggest that the average reservation wage can be linked to the national minimum wage (SMIC), the latter being seen as a social reference value of the labour price, over which minimum wage requirements will be positioned by default or by excess. This leads us to represent the average hourly reservation wage R t s as a function of the trend of the (official) net hourly minimum wage. Consideration of the trend is justified by the fact that we try to represent at each date the average in individual reservation wages, so that we can expect a smoother evolution than that of the SMIC whose revisions are discontinuous. The econometric results obtained support this choice insofar as they have been improved by considering the trend in place of the observed values of the SMIC. We have thus 9 :  will be all the greater because rigidity in prices (e.g. price control policy), in wages 10 (including social contributions) or in employment (e.g. presence of severance compensation) are strong during the period of negotiation. On the whole, the socalled "rigidity of the labor market" results from these three kinds of rigidities. In this regard, it should be noted that the contributions of the literature show that the introduction of numerous institutional variables capable of translating these rigidities does not lead to very conclusive results. Therefore, the multidimensional, latent and not directly quantifiable character of the degree of rigidity in the labour market has led us to represent the dynamics in the coefficient t  by a recursive stochastic process estimated using the Kalman filter methodology, which allows the estimation of unobservable variables 11 :

) log( o t o R t s SMIC s    0  o  0   o t s
u t t u u t         1 1 0 1 0 1   u  u t  = N(0, ) ( 2 u t   ) (5)
Finally, it is necessary to formulate a hypothesis to represent the "other factors" t f involved in Eq.(3) of equilibrium unemployment. As we have already noted, the empirical contributions of the literature often introduce many explanatory variables of unemployment as "proxies" of institutional or cyclical phenomena that can be more or less suggested by theory 12 , to the point where it is sometimes difficult to know if it is the theory that is validated or if there occurs simply correlations between unemployment and other macroeconomic variables. Such approaches also involve a multiplication of arbitrary parameters to be estimated. Since we wanted to avoid these drawbacks, and because factors related to the repartition of added value are represented by the variables specified by the WS-PS model, we assumed that t f reflects the influence of the underlying factors of the level of real added value. Accordingly, the factors that determine t f are summarized by the sum of a cyclical element proportional to the output gap t q  and of a constant o f representing structural factors that may generate some uncompressible unemployment :

o t t f q b f    0  b 0  o f t t t q q q    (6)
t q  : output gap, measured by the deviation (%) between the log of potential GDP t q and the log of real observed GDP t q t q : HP trend ( hp t q ) of t q plus a constant of 4.46% ensuring a positive or zero value of the output gap t q  over the whole period (this constant corresponds to the minimum negative value of hp t t q q  observed in 1973) 13 At the theoretical level, the output gap t q  -which of course depends on variables it X -can be seen as conditioning t f through the expected growth rate. Indeed, because of decreasing returns, the bargaining agents will anticipate in principle a rate of growth of production increasingly smaller as the production nears its maximum (i.e. as t q  is close to zero). In these circumstances, an increase in t q  will reflect the expectation of higher output growth, leading the unions to higher wage demands and better acceptance of these demands by employers, who may compensate by setting higher prices to maintain their profit margins. These behaviours will thus lead to simultaneous increases in s t f and p t f , hence to their sum t f , and in fine to an increase of equilibrium unemployment (Eq.( 3)). Eq.( 6) offers the advantage of being very simple 14 and parsimonious about the number of parameters to be estimated. However, it is important to question the identity of the factors it X underlying t q  and therefore t f . 15 On this question, Chélini and Prat [2016, Annex 2] show that a linear combination of variables such as the real interest rate, international competitiveness, corporate profit margins, money supply growth, inflation and unionization rates, provide a fairly good representation of t q  .

The negotiated wage rate

The wage rates ps t s and ws t s (equations ( 1) and ( 2)) being the solutions of maximization programs of firms and unions before negotiation, these notional rates are interpreted respectively as offered and desired values. Because the negotiated value

     1 0    (7)
The mechanism leading to the value * t s can be described as follows. If, at the beginning of the negotiation, the wage desired by employees ws t s exceeds that offered by firms ps t s by x%, these latter will be induced to reduce the labour factor in the production process (using the principle of "right to manage"), thus leading to an increase t du % (in points) of the unemployment rate. According to the WS equation ( 2), this increase in unemployment leads to a decrease of t s t du k % of the wage desired by employees and, according to the PS equation ( 1), to an increase of t p t du k % of the wage offered by the firm.

The rise in unemployment towards its equilibrium level thus allows both parties to reach an agreement on an intermediate value * t s of wages, thanks to a reduction in the claims of the employees and an acceptance by the firm of an upward revision of wages offered; as a result, unemployment appears as the key mechanism reducing the gap between ws t s and ps t s (see in particular Layard and Nickell (1986)). At the end of the bargaining process, the wage agreement leads to the equilibrium condition

ps t s = ws t s = * t s .
As in any situation of bilateral monopoly, the value of * t s depends on the power and the cleverness of both parties. These qualities are a priori unknown, but Nash's bargaining theory -a situation in which no player has an interest in deviating individually from his strategy -makes it possible to circumvent this indeterminacy by considering that negotiation maximizes the weighted average of additional utility obtained by the two parties in comparison to a fall-back situation. The overall optimal solution thus corresponds to a negotiated value of the wage * t s which necessarily lies between the desired values ws t s and ps t s prevailing at the beginning of the negotiation. The parameter  of Eq. ( 7) then reflects the balance of the bargaining power between the union and the firm. A value  = 0 means that the company is all powerful and can therefore impose employment and wage levels, whereas the value  = 1 means on the contrary that the union has complete power in the negotiation. The values of  ranging from 0 to 1 thus allow to describe the curve of the possible values of the employment contract (d'Autume [2001], p.6). By including equations ( 1) and ( 2) in ( 7), we obtain the theoretical expression of the negotiated wage:

p t s t t t t t t t t R t c t t f f u m ce p cs sr p s ) 1 ( ) ( ) 1 ( ) ( *                   (8) with ] ) 1 [( s t p t t k k       In the relationship (8), the time-varying coefficient ] ) 1 [( s t p t t k k       represents
the sensibility of the wage rate to unemployment rate. This sensibility is negative when the weight attributed to unemployment by the employees during the negotiations is greater than the weight attributed to this variable by the employers (i.e.

s t p t k k     ) 1 (
), and positive in the opposite case. The condition 0  t  expresses a situation where the two weights are balanced. Eq.( 8) shows that the estimation of * t s requires auxiliary assumptions about the representation of the real reservation wage R t sr , on that of the coefficient t  , and finally on the "other factors" of prices and wages that are captured by the pseudo-difference

p t s t f f ) 1 (     .
The real reservation wage in Eq.( 8) is defined as :

 R t sr c t R t P s log  (9)
where the nominal reservation wage R t s is given by Eq.( 4), while, in line with determination of the latter, c t P is the HP trend of the consumer price index. As with t  , the coefficient t  is determined by a stochastic recursive process estimated using the Kalman filter methodology:

s t t s s t         1 1 0 1 0 1   s  s t  = N(0, ) ( 2 s t   ) (10)
Finally, since s t f and p t f are assumed to depend linearly on variables it X , we can deduce that the influence of the "other factors" results in a linear combination of the it X ,

where the influence of each variable on wage can a priori be positive, negative or null 17 (in the latter case, the effects of it X compensate each other):

o i i i p t s t c X f f         ) 1 ( (11)
One can notice that equations ( 9), ( 10) and ( 11) associated with the determination of the negotiated wage rate are directly coherent with the auxiliary equations ( 4), ( 5) and ( 6) associated with the equilibrium unemployment rate, so that they do not really add any arbitrariness to the whole model.

The gradual adjustments toward the equilibrium values

The empirical contributions of the literature suggest that the assumption of a gradual adjustment of observed unemployment to equilibrium unemployment * t u may be relevant. 18 Many causes are put forth by the authors to explain such an adjustment occurring after a macroeconomic shock, known as "partial" or "gradual" hysteresis: the costs of collecting information concerning vacancies and mobility costs, the social costs associated with unemployment that include redundancy payments, the duration of unemployment (implying a degradation of human capital), a shortage of physical capital that follows a durable shortfall of demand in the goods market, … This type of adjustment between two successive periods adds a form of rigidity different from that represented by the alpha coefficient which reflects the degree of rigidity during the negotiation period. This persistence is characterized by an adaptive process 19 : * 1 ) 1 (

t u t u t u u u       1 0   u  (12)
Concerning wages, because of the cost and duration of negotiations [START_REF] Layard | Unemployment[END_REF], p. 425) and the delays due to the establishment employment contracts and to the first payment after negotiations [START_REF] Avouyi-Dovi | Les négociations salariales en France : une analyse à partir de données d'entreprises (1994 -2005)[END_REF], the average wage observed in the statistics only adjusts progressively over its negotiated value, which is also represented by an adaptive process:

* 1 ) 1 ( t s t s t s s s       1 0   s  (13)
4 The joint estimate of unemployment and wage rates : data, method and empirical results

Data

The statistical time series chosen are annual data because wages are governed by collective agreements giving rise to specific agreements (branches or companies) spread over the full year [START_REF] Avouyi-Dovi | Les négociations salariales en France : une analyse à partir de données d'entreprises (1994 -2005)[END_REF]). These latter are negotiated every year or every two years, not every quarter. This choice obviously reduces the number of observations, but the length of the analysis period, which exceeds half a century partially compensates for this limit. The time series cover the period 1950-2008, the sources being INSEE, the Ministry of Labour and the Banque de France. 20 The objective being to identify long-term structural relationships, we preferred to exclude years of turbulence corresponding to the global financial crisis that began in mid-2007. Although there was a second phase of bank failures in the autumn of 2008, it can be assumed that macroeconomic effects only became significant in France in 2009 (we observe a continuous decline in the unemployment rate between 2006 and 2008, with unemployment rising appreciably only in 2009). For similar reasons, the years 1950-54, which are characterized by very high wage changes (reconstruction, inflation of the Korean War, post-inflationary catch-up after 1952), were excluded from the estimation period, which will therefore cover the years 1955 through 2008 included.

4.2

The state-space system to be estimated By reporting Eq.( 6) specifying the influence of "other factors" t f in the ERU's Eq.(3), and then including the resulting equation in the adjustment process (12), we get the observed unemployment rate equation (Eq. ( 14) below). Similarly, by reporting Eq.( 11) specifying the influence of the "other factors"

p t s t f f ) 1 (    
in the negotiated wage Eq. ( 8), and then including the resulting equation in the adjustment process (13), we arrive at the observed wage rate equation (Eq. ( 15) below). The stochastic coefficients t  and t  being described by Eq.( 5) and Eq.( 10), the set of hypotheses described above leads to the spacestate system composed of the four following equations 21 :

The two measurement equations :

u t o t t t t u t u t f q b m spread u u              ) ( ) 1 ( 1 (14) 1 0   u  t t t t R t t ce cs p s spread       s t o i i i t t t t t t t R t c t s t s t c X u m ce p cs sr p s s                                   ) ( ) 1 ( ) ( ) 1 ( 1 (15) 1 0   s  ] ) 1 [( s t p t t k k       1 0   
The two stochastic state equations :

u t t u u t         1 1 0 1 0 1   u  (5) s t t s s t         1 1 0 1 0 1   s  (10)
Given equations ( 4) and ( 9) of the reservation wage R t s and its real value R t sr , equations ( 14) and ( 15

N(0, ) ( 2 u t   ) and N(0, ) ( 2 s t   ).

Estimation method and empirical results

The above four-equation system does not pose problems of identification and was estimated over the period 1955-2008 (54 years) using the maximum likelihood method [START_REF] Harvey | Forecasting, structural time series models and the Kalman filter[END_REF], [START_REF] Hamilton | Time series analysis[END_REF]). 22 The initial values of the state variables t  and t  were determined by a grid search to minimize the information criteria (Akaike (AIC), Schwarz (SC) and Hannan-Quinn (HQ)). 23 It should be noted that the Kalman filter method can give rise to difficulties in estimating the variances of the residuals ("innovations") of the measurement equations (also called "signals") and the variances of the residuals ("noises") of the state equations, but it is possible to minimize this risk by initializing these variances with sufficiently high values (Stock and Watson, 1998;[START_REF] Durbin | Time Series Analysis by Space State Methods[END_REF], as we done. In case of persistent difficulty, it is the practice of econometricians to impose constraints on the variances, in particular by arbitrarily fixing the value of the ratio between the variance of innovations and the variance of noises ("Signal-to-Noise Ratio"). This constraint was not necessary in our estimates, the variances being estimated by imposing a positivity constraint (see note of Table 1).

According to the Kalman filter, the successive values of the state variables t  and t  are revised each year on the basis of new information assumed to reflect a positive, negative or null variation in the degree of the labour market rigidity. This new information takes the form of additive normally distributed white noises, whose variances belong to the vector of the hyperparameters to be estimated. Since the effects of these informational shocks are supposed to spread gradually over the labour market, the dynamics of these two state variables are characterized by AR(1) processes with drift. An AR(2) process or the introduction of observable variables in the state equations have not proved to be empirically relevant. Note that this nonlinear method does not impose stationarity conditions on the observable variables of the system. However, it imposes other conditions which may appear more severe on the error terms u t  and s t  of the measurement equations ( 14) and ( 15): absence of autocorrelation, homoscedasticity and normality (see in particular Durbin and Koopman [2001, pp. 33-34]). Therefore, diagnostic tests on standardized residuals were carried out using respectively the tests Q, hH and HK, which were developed by [START_REF] Harvey | Forecasting, structural time series models and the Kalman filter[END_REF] specifically for the Kalman filter.

Table 1 gives the estimated values of the parameters 24 as well as the statistics associated with the various diagnostic tests carried out on the standardized values of residuals and . For these two residuals, results lead to the conclusion of an absence of autocorrelation (Q statistic) and of heteroscedasticity (hH statistic) at the threshold of 5%, while the normality hypothesis can be accepted at the 1% level of significance (HK statistic), hence indicating that there is no abnormal frequency of extreme points. These properties of residuals thus show that the conditions of application of the Kalman filter are realized and that the specification of the model is statistically globally acceptable. Note -The space-state system represented by Eqs. ( 14), ( 15), ( 5) and ( 10



, with 18 = T/3), one accept the null (no heteroscedasticity). HK is the pvalues of the normality test proposed by Harvey-Koopman specified for the Kalman filter: the values obtained allow to accept the normality hypothesis at the 1% level for unemployment and at the 5% level for wages. AIC, SC and HQ stand for Akaike, Schwarz and Hannan-Quinn information criteria for the system; AICs, SCs et HQs give the values of the same statistics under the constancy hypothesis for  is the level of significance), allowing to test the null Ho (

o t o t       , ) in the system; for % 5  
, the calculated value LR=58.2 being higher than the critical value 9.49, Ho is rejected.

The unemployment rate estimated equation

Examination of the estimated unemployment equation ( 14 4)). These outcomes position the reservation wage at around 15% above the national minimum wage SMIC on average over the entire period, with values between the latter and the median wage, approaching the median wage at the end of the period. Given that the national minimum wage SMIC is about 60% of the median wage, our results mean that the reserve wage would be on the order of 70% of the median wage on average, which seems plausible. On the other hand, the constant and March 1983 also contributed to this movement: a policy of progressive price liberalization and competitive disinflation, progressive deindexation of wages on inflation, a decrease in social benefits. Numerous facts were subsequently linked to maintain this movement of flexibility: lower unemployment benefits and the introduction of a new system making them dependent on the contribution period, decentralization of collective agreements, abolition of the administrative authorization for economic dismissal, development of individualized differentiation of wages, the increase in fixed-term contracts, the development of part-time and temporary work, the easing of traineeship conditions, the degressivity of allowances and the increased inspection of job searches following the financial crisis of UNEDIC in 1992, and so on.

Within this general movement for flexibility, the creation of the French social minimum income RMI in 1988 under the Rocard government can be seen as reflecting the desire for a minimal state support for the growing number of unemployed people, who had lost their rights. This general downward trend was nevertheless at points counteracted by positive shocks of rigidity. In particular, over the period 1991-94 following the reunification of Germany (October 1989 to October 1990), there was an increase in t  because this event had given rise to fears of structural economic difficulties for the France's leading trading partner. French firms were thus more cautious about hiring (Germany was down by 2% of GDP in 1993). The increase in t  during this period was also presumably exacerbated by the adoption of the Maastricht Treaty in September 1992, which arguably gave rise to fears of the emergence of new constraints imposed by Europe (EU). Similarly, it can be seen that the implementation of the 35-hour law in 2000 was followed for the three ensuing years by an increase in the rigidity coefficient before its declining trend re-established itself. This temporary upward in t  was probably reinforced by the introduction of the single European currency in 2002, because the uncertainty associated with this major institutional change created a period of wait-and-see for companies in the labour market and a downward revision of production plans (a decline in the level of activity was observed during this period). On the whole, this fairly good correspondence between the evolution of the coefficient of rigidity t  and the historical reality is likely to give credibility to the economic significance of the estimated values of this coefficient.

It may at this point be interesting to distinguish three components in the estimated equilibrium unemployment rate, namely u will be all the more important as the real reservation wage (including the social contributions) will be higher than the labour productivity. The variable t spread can be positive, negative or zero, depending on whether the wage requirements increased by the compulsory levies are higher, lower or equal to the productivity. 27 Since the values of t  are positive, it follows that the chronic component of unemployment * , 1 t u can also take positive, negative or zero values, even with a positive profit margin. 28 In other words, the behaviours underlying the distribution of the value added can be generally net creators or destructors of employment, thus leading to a reduction or increase in the equilibrium unemployment rate in relation to a situation where this component would not exist. Secondly, the

t t t q b u    * , 2
rate can be regarded as the cyclical component (positive or null) of equilibrium unemployment, resulting from a level of production that is too low to ensure the full use of the production factors. Finally, the values of t  and o f being positive, the rate

o t t f u   * , 3
can be regarded as reflecting a frictional component directly related to the degree of rigidity of the labour market. This component may reflect the effects of the necessary duration between the start of the job search and beginning the new job, as well as the effects of technical progress, which may create fewer jobs in the innovative activities than are destroyed in the traditional low-innovation activities, while extending the time required for inter-activity mobility of the workforce, thus accentuating the inadequacy of job supply to job demand. 29  and structural parameters are of course with a margin or error. Although the specific factors to these three components remain broadly dominant, these three components are linked by a common factor represented by the stochastic coefficient t  , which means that the degree of labour market rigidity can condition the level of unemployment by these three different ways. Over the period as a whole, the chronic, cyclical and frictional components of the unemployment rate averaged 2.8%, 2.1% and 2.5%, respectively. On the 1975-2008 subperiod, these values are respectively 3.5%, 2.4% and 2.9%. 30 The negative values of the chronic component * , 1 t u observed up to 1968 indicate that this latter could have contributed positively to employment by creating a very favourable situation for employers to hire: * , 1 t u would have made it possible to lower the unemployment rate between 1% and 2% compared with unemployment which would have prevailed if only the other two components had existed. From the early 1970s onwards, and especially after the oil shock of 1973, the chronic unemployment grew strongly, reaching a first level of 4% in 1985, followed by a ceiling of about 5% in 1993-95, then decreasing to a value close to 3% at the end of the period. 31 This evolution is in line with the observation often made in the economic literature that unemployment attributable to the sharing of value added in favour of employees has developed in France from the 1970s to the end of the 1980s [START_REF] Laroque | Labour market institutions and employment in France[END_REF], [START_REF] Baron | Le ralentissement de la productivité du travail au cours des années 1990[END_REF], Lannes and Paris [2010]). The cyclical component * , 2 t u experienced fluctuations of significant amplitude ranging between 0 and 4.5%. 32 There was also an upward trend in this type of unemployment from the time of the 1973 oil shock through the end of the 1990s. As for the frictional component * , 3 t u , it oscillates between a minimum of 1% in 1958 and a maximum around 4% during the years 1983-86, this component exhibiting a smoother evolution than the other two. As a whole, these results suggest that neither chronic factors nor cyclical factors, nor indeed frictional factors, can be neglected. In particular, after the oil price shock of 1973, the increases in these three categories of unemployment have accumulated to explain the evolution of the unemployment rate at a two-digit level.

To understand what has happened since 2008, it is interesting to refer to the results issued from the surveys conducted at the micro-level by the Bank of France with a large sample of 1,150 French firms between 2010 and 2013 [START_REF] Jadeau | The Behavior of French Firms During the Crisis: Evidence From the Wage Dynamics Network Survey[END_REF]), which clearly joins our analytical framework. Despite the slowdown in both economic growth and productivity during this period, wages continued to rise for most firms, showing persistent downward wage rigidities that could lead to a rise in the chronic component of unemployment. The other phenomena mentioned by a majority of companies as constituting a brake on employment are related to future economic growth, changes in labour legislation, changes in social contributions and in dismissal costs, all of which are taken into account in our modelling with the output gap, the rigidity coefficient, the rate of social charges, and the gradual adjustment of the unemployment rate towards its equilibrium value, respectively. Likewise, starting from the fact that unemployment has long been structurally higher in France than in comparable countries, Tirole (2014) draws up a report showing the dysfunction of the French labor market which is in accordance with our results. Indeed, he emphasizes the existence of excessive social charges on the labor factor, the mismatch between labor supply and demand, the rigidity induced by the Smic, the importance of fixed-term contracts which induces tensions between employers and employees, and the cost of dismissal. About this last point, the author finds that companies that do not lay off pay for those who lay off and proposes to tax the dismissal to restore a more equitable functioning of the labor market.

The wage rate estimated equation

Regarding the estimated wage rate (Eq. ( 15)), only the unionization rate t txs was found to be significant among the "other factors" it X considered. The positive value of the estimated coefficient 1  means that t txs is a factor favourable to wages, which seems quite intuitive. The value is within the theoretical range [0,1] and indicates a balance of power ) 1 (    on the order of 1 to 4 on average between the employees and the employers, in favour of the latter. A ratio approaching 1 to 2 would nevertheless remain statistically acceptable. The slope

The overall properties of the unemployment-wage model

The estimated values of the state variables k are worth respectively 13.7 and 3.1 over the entire period, and 10.2 and 2.4 over the period 1982-2008. We can illustrate how unemployment can play the role of adjustment variable by considering the median values over the second period. For example, if at the beginning of a negotiation, the wage desired by the employees exceeds that offered by firms by 12.6%, a reduction in employment decided by the firms, which would imply a 1% increase (in points) of the unemployment rate, would lead to a 10.2% decrease in the wage desired by the employees (WS equation) and to a 2.4% increase in the wage offered by firms (equation PS); all other things being equal, the rise in unemployment would allow the two parties to reach a wage agreement. In addition to the historical analysis of institutional rigidities in the labour market (presented above) in relation with the dynamics of coefficient

          t t t t t p t s t dsmic m dp j u k k 785 . 0 2  R 29 . 1  DW (17a) ) 7 . 2 ( ) 2 . 8 ( 1 ) 9 . 4 ( ) 5 . 3 ( ) 4 . 3 ( ) 1 . 3 ( 1 05 . 0 13 . 0 23 . 0 15 . 0 12 . 0 06 . 0 ˆ           t t t t t s t p t dsmic m dp j u k k 756 . 0 2  R 49 . 1  DW (17b)
Although the autocorrelation of residuals suggests missing factors (DW), non-reported ADF tests (using McKinnon's critical values) lead us to conclude at the 5% level that Eqs (17a) and (17b) are cointegrating relations. Overall, the signs of the estimated coefficients are rather intuitive. Considering the one-period lagged sensibilities of the other agent, the estimates suggest a form of inertia in the negotiations, since everything happens as if labour union and firm began the wage bargaining at year t by keeping some memory of the sensibility to unemployment of the other partner, as revealed during the negotiation at t-1. We also note that higher unemployment is accompanied on average by lower sensibilities of the desired and offered wages. An interpretation could refer to the Beveridge curve, which reflects the negative relationship between the number of unemployed and the number of job vacancies. Indeed, according to this curve, job vacancies are relatively high in times of low unemployment, because the tension in the labour market is accompanied by an unsuitability of job offers to job demand that is greater than in times of significant underemployment. In this context, the competition between firms to attract a workforce adapted to their needs will lead to a wage offered that is all the more (positively) sensitive to unemployment the lower it is, this behaviour being reflected in the PS equation by an increase of p t k when unemployment decreases. At the same time, the worsening of the mismatch of job offers with job demands may weaken the sensibility of union claims to unemployment, a behaviour reflected in the WS equation by a value of s t k which increases when unemployment decreases.

It can be seen that the long-term interest rate also has the effect of lowering the two sensitivities. On the firm side, for a given level of unemployment, an increase in the cost of capital may encourage them to offer a lower salary to maintain their profit margins (decline of p t k ). On the employee side, a decrease in the ratio labour price / capital price could provide unions an argument against firms, which would have the effect of weakening the (negative) influence of the unemployment rate on wage demands (decline of s t k ); but this last effect appears to be insignificant. The results also show that both sensibilities tend to be larger when inflation is high. For a given level of unemployment, inflation allows firms to offer higher wages without lowering their margin rates (increase of p t k ). Concerning employees, the increase in the sensibility of the desired wage to unemployment (negative influence) could be understood by the fact that inflation has a negative effect on the "index of consumer sentiment" 34 , thus weakening -for a given unemployment rate -wage claims of unions (increase of s t k ). The two sensibilities are also all the stronger when the one-period lagged margin rate of firms is high. For a given level of employment, it seems intuitive that employers could consider offering a wage today that is even higher than their last rate of margin was large (increase of p t k ). On the other hand, everything happens as if a higher margin rate in the previous period had the effect of weakening the union wage demands, which results in an increase in the sensitivity s t k of desired wages for a given unemployment rate (negative influence). This result could be interpreted by the conjecture that, when underemployment is important (as observed on average over our period), a rise in profit margins may lead unions to encourage companies to hire rather than apply pressure on wages. Finally, for a given level of unemployment, a strong increase in the SMIC (minimum wage) over the previous year tends to moderate both the wage claims of the unions and the level of wages offered by the firms; this result seems intuitive and is manifested by a joint decline of We conclude our analysis of the factors of the two sensibilities by calculating the correlation between the values of the state variable t  issued from the Kalman filter (Figure 1) and the values recomposed as

) ( 1 ˆp t s t t k k   
, where and are from the tworegression system (17a) and (17b); in the same manner, we calculate the correlation between the state variable t  (Figure 3) and the recomposed values . We found and , which suggests that the degree of rigidity in the labor market and the sensibility of the wage rate to unemployment are in part conditioned by some observable factors that themselves likely embed some stickiness (inertia from the preceding bargaining, unemployment, cost of capital, change in price level, desired margins, SMIC,…). Overall, despite their arbitrary character, these ex-post results appear to be rather intuitive and thus enhance the credibility of the estimated dynamics of t  and t  using the state-space modelling. However, a more in-depth theoretical and empirical research about the underlying factors of the two sensibilities would be necessary to consider them ex-ante in the structural equations. Indeed, in the measurement equations, when t  and t  are replaced with the values of and issued from equations ( 17a) and (17b), the fitted values of unemployment and wages deteriorate notably.

Our model also makes it possible to assess at each date the relative difference between wages desired by employees and wages offered by employers, namely ) (

ps t ws t t s s dif   .
Indeed, according to the theoretical equations ( 1) and ( 2) and the auxiliary hypotheses retained for empirical estimations, this difference can be calculated by the relation ( 18) where (Eq.( 9)) with (Eq.( 4)), and (Eq. ( 6)), while the values of s t k and p t k issued from the state equations ( 5) and ( 10)) are those represented in Figure 4 (for estimates, see Table 1). The average value over the whole period found for is 5.12%, meaning that employees generally require a wage greater than the one offered by firms, which seems not very surprising. However, as shown by Figure 5, is strongly time-varying (the values being in the range of + 20% to -10%), excluding the extreme negative points in 1980 and 2001. Considering the gap between real notional wages and labour productivity (the latter being viewed as the classical equilibrium value of real wage), these outcomes seem plausible (notwithstanding uncertainty affecting the estimated values of ); for example, if unions claim a wage only 5% greater (lower) than productivity while firms offer a wage only 5% lower (greater) than productivity, the difference between the two notional wages would be +10% (-10%). 35 Moreover, it should be noted that the extreme negative values of that occurred in 1980 coincides with the highest growth rate over the whole period in real GNP (about 7% per year) -this rate being about 4 % for the second extreme negative point in at 2001. This suggests that when the growth in production is very strong, wages offered by firms tend to exceed wages claimed by unions, which is not very surprising since firms are of course informed earlier than unions regarding production figures. Anyway, the dynamics of suggests one should examine if this gap is negatively (positively) linked to change in employment (or unemployment), as implied by the WS-PS model according to which the labour factor is an adjustment variable to achieve equilibrium. Using the Newey-West method providing estimators whose variances are robust to autocorrelation and heteroscedasticity, we find that, beyond some inertia between two successive periods, the difference between desired and offered wages are correlated with two (uncorrelated) variables that can be considered as representing employment' adjustment. We found s (see above for the description of the mechanism). However, it is somewhat astonishing that the optimal one period lag found in the change in the unemployment rate; concerning this, we can but conjecture that firms believe that subsequent negotiations will partly validate their strategic plans, thus leading them to make early adjustments in the labor factor. This hypothesis is in accordance with the estimated value of coefficient  , which suggests that the bargaining power of employers clearly dominates that of employees. We finally examine the capacity of our model to represent the historical dynamics of unemployment and wages. Despite sometimes substantial differences, Figure 6 and Figure 7 show that the values of the unemployment rate and of the wage rate estimated from equations ( 14) and ( 15) describe the fluctuations and the major turning points in observed values. 36 As indicated in Table 1, the quality of adjustments is assessed both with the conventional coefficient 2 R and the coefficient proposed by [START_REF] Harvey | Forecasting, structural time series models and the Kalman filter[END_REF]. The latter is more stringent since it makes it possible to evaluate the relevance of the structural model with respect to a benchmark characterized by a random walk with drift. 37 The values of 2 D R of the measurement equations being 0.67 and 0.94 respectively for unemployment and wages, we conclude that the model leads to significantly better quality adjustments than those corresponding to the benchmark. The good statistical properties of the standardised values of residuals and show that the specification of the model is statistically acceptable; this implies especially that the theoretical constraints that are assumed a priori in the basic WS and PS equations -resulting in the unit values of the coefficients of price level, productivity, profit margin and contribution ratios -are a posteriori admissible. We finally estimated a system composed of equations ( 14) and ( 15) by imposing the joint constraint

o t    and o t    t  .
The results strongly reject this constraint, hence confirming the relevance of our model with time-varying coefficients (see Table 1, the value of the log-likelihood ratio test LR).

Conclusions

We conclude this work with the feeling that economic theory allows to shed light on the main features of the historical changes in unemployment and wages in France at the macroeconomic level. The main contribution of this paper is probably to have exhibited over a long period the main economic factors of changes in unemployment and wages with a basic version of the WS-PS bargaining model, provided that the degree of rigidity in the labour market is time-varying. The multidimensional and not directly quantifiable character of this degree of rigidity has led us to represent its dynamics by a stochastic process estimated using the Kalman filter methodology. Concerning this, we note a fairly good concordance between, on the one hand, the evolution of the degree of rigidity estimated and, on the other hand, institutional changes and historical facts. Moreover, we attempted to identify factors of the sensibilities of the desired and offered wages to unemployment by a brief econometric analysis and found that these sensibilities are conditioned by some macroeconomic variables which themselves likely embed some stickiness. However, although these results constitute a rudimentary economic explanation of the degree of rigidity in the labour market, factors are not yet sufficiently specified to consider them directly in our structural equations. Using other auxiliary assumptions concerning the representation of the reservation wage (depending on the legal minimum wage) and "other factors" complementing those specified by the WS and PS equations (these factors being summarized by the output gap), we estimated the equilibrium unemployment rate and the corresponding negotiated wage rate between unions and firms.

Overall, the values of the unemployment rate and of the wage rate estimated from our modelling describe the main fluctuations and the major turning points in observed values. The observed unemployment rate adjusts gradually to its equilibrium value (here we can mention the many social costs associated with unemployment). Three significant components can be distinguished in the equilibrium unemployment rate, the relative importance of which varies significantly over time: a "chronic" component resulting from the factors underlying the distribution of added value (real reservation wage, profit margin, social contributions, productivity), a "cyclical" component resulting from a level of production below its potential value, and a "frictional" component attributable to an insufficient labour mobility and to technology progress. These results are in line with the idea that the fight against French unemployment must be carried out simultaneously on several fronts : by acting on the factors that determine the sharing of added value (minimum wages, social contributions, taxation, training, and innovations designed to increase productivity, etc.), by pursuing a macroeconomic policy of reviving activity, or by setting up a structural policy aimed at improving labour mobility or increasing the flexibility of the labour force (unemployment insurance rules, dismissal rules and practices, characteristics of employment contracts, etc.). However, our results are of course conditional on the auxiliary hypotheses representing the variables not directly measurable or not identified by the theory (reservation wage, degree of rigidity, "other factors").

Regarding wages, the negotiated value is represented by a weighted average of the wage desired by employees (WS equation) and the wage offered by firms (PS equation), the weighting coefficient reflecting the bargaining power between the two parties. Moreover, due to the duration taken to set up employment contracts, the observed wage adjusts with delay to its negotiated value, which depends on the reservation wage, social contributions, price level, productivity, margin rate, rate of unionization, and finally on the rate of unemployment, the influence of which varies over time. Overall, results about the wage rate highlighted five general teachings. The first is that, on average, employers take the lead on employees during negotiations. The second lesson is that the reservation wage would be on average about 15% above the SMIC, with this gap tending to increase over the period. The third lesson is that the sensibilities of the desired (WS) and offered (PS) wages to unemployment vary over time. This phenomenon may provide an explanation for the instability of the empirical wageunemployment relationship found in the literature. The fourth lesson is that the difference we assessed between wage desired by employees and wage offered by employers allows us to show that, as predicted by the theory, employment intervenes effectively in the adjustment between the two notional wages to achieve equilibrium. A final lesson is the confirmation of the relevance of the distinction made between five concepts of wage : the wage desired by employees whose value depends on the reservation wage (WS side), the wage offered by the employers (PS side), the negotiated wage resulting from the confrontation between the wage claimed by unions and the one offered by firms, and finally the observed wage which is the result of the implementation of the negotiated wage contract.

2 Our analysis focuses on dynamics of unemployment and wages and so leaves aside contributions based on the wage curve, which show a negative correlation between real wages and unemployment rates across regions on average over a given period (Blanchflower andOswald [1995], [2005]). 3 It should be noted that, contrary to the approaches issued from the Phillips curve, the WS-PS model is applicable to the analysis of disaggregated data (see in particular [START_REF] Cahuc | Equalizing wage difference and bargaining power: evidence from a panel of French firms[END_REF], [START_REF] Doisy | Allègement de charges au voisinage du SMIC et chômage d'équilibre : une modélisation à partir d'une maquette désagrégée du marché du travail[END_REF]). 4 For a description of the passage between the raw solutions of the maximization programs of the two agents and the two log-linear equations, see in particular [START_REF] Chélini | Cliométrie du chômage et des salaires en France[END_REF]. 5 Equations ( 1) and ( 2) are notably inspired by the presentations made by Blanchard & Katz [1997], [START_REF] Bonnet | Peut-on mettre en évidence des rigidités à la baisse des salaires nominaux ? Une étude sur quelques grands pays de l'OCDE[END_REF], Le [START_REF] Le Bihan | Courbe de Phillips et modèle WS-PS. Quelques réflexions[END_REF] and [START_REF] Simonnet | Asymptotically Median Unbiased Estimation of coefficient Variance in a Time Varying Parameter Model[END_REF]. We note that agents are supposed to be rational and so are not subject to the monetary illusion. 6 The inclusion of the margin rate as an argument in the price equation is justified by the fact that, if the real wage is equal to or greater than labour productivity (as, for example, supposed by the theory of efficiency wage), this should not imply that the firm accepts a zero or negative profit. Indeed, in the situation of monopolistic competition situation, the firm can set a price higher than the marginal cost in order to obtain a positive margin. 7 The French Guaranteed Minimum Interprofessional Wage (SMIG) was created in February 1950 and replaced in January 1970 by the Minimum Interprofessional Growth Wage (SMIC) and the Guaranteed Minimum Income used to calculate certain social benefits. While the SMIG was indexed to inflation, the SMIC is revalued by adding to the inflation rate half the increase in the purchasing power of the basic hourly wage of workers (it can also benefit from additional increases from government). Under certain conditions, it could be compared to the US national minimum wage according to the Fair Labor Standards Act (1938). 8 For the unemployed benefiting from the Minimum Income of Social Insertion (RMI), the coefficient of variation of the responses (standard deviation across respondents / average) is 26%, this value reaching 45% for the unemployed persons who are compensated. 9 The parameters o s and o  each influence in a different manner both the level and the slope of the reservation wage; the latter may be more or less above or below the trend of the SMIC depending on the dates. In order to capture the possible effects of missing variables, we tried to add a linear trend to the right of equation ( 4): this trend proved to be insignificant. 10 The microeconomic theories of the labour market describe the many endogenous sources of wage rigidity, in particular the efficiency wage theory [START_REF] Shapiro | Equilibrium unemployment as a worker discipline device[END_REF]), the insider-outsider model [START_REF] Lindbeck | The Insider-Outsider Theory of Employment and Unemployment[END_REF]) and the implicit contract theory [START_REF] Azariadis | Implicit contracts and underemployment equilibria[END_REF]). The degree of flexibility of wages and employment may therefore vary over time not only because of legal institutional constraints (types of labour contracts, social minima, redundancy payments, collective agreement, wage indexation, etc.), but also because of the behaviour of the actors themselves. Empirically, [START_REF] Bonnet | Peut-on mettre en évidence des rigidités à la baisse des salaires nominaux ? Une étude sur quelques grands pays de l'OCDE[END_REF] shows the importance of downward wage rigidities in OECD countries. We can also mention the analysis of [START_REF] Laroque | Labour market institutions and employment in France[END_REF] which shows, on the basis of a survey carried out in 1997 among married French women, that there is unemployment called "classic", which is attributable to a wage rigidity induced by the existence of a minimum wage.

11 It is interesting to note that, by comparing the European countries to the United States, [START_REF] Nickell | Unemployment and labour market rigidities: Europe versus north America[END_REF] shows that all institutional rigidities have not an adverse effect on employment (see also [START_REF] Blanchard | The Role of Shocks and Institutions in the Rise of European Unemployment: The Aggregate Evidence[END_REF]). This is likely to reinforce an approach with stochastic parameters, which can be seen as capturing the overall effects of institutional changes.

12 These include the real interest rate, replacement rate, the ratio of minimum wage to average wage, unemployment benefits, the trade balance, duration of unemployment, the price level as a mark-up on wages, terms of trade, unionization rate, job destruction rate, labour force growth rate, share of profits in value added, inflation rate, structure of unemployment,.... 13 Although arbitrary, this approach seems to moderately condition the evaluation of t q  . Indeed, the different methods envisaged in the literature to estimate the output gap -including production function based approachesoften lead to similar results (see [START_REF] Bonnet | Ecart d'activité ('Output gap') et diagnostic conjoncturel[END_REF]. 14 When included in the equation of equilibrium unemployment (3), Eq.( 6) establishes a link between unemployment and production, thus joining the Okun's empirical law [1962].

15 This identification will be useful for estimating the wage equation (see Eq. ( 11)). 16 Even if trade unions take into account the price level during the negotiations, the employment contracts are fixed in terms of nominal wages. That is why we considered the latter by transferring the consumer price level 18 See specifically Layard-Nickell-Jackman [1991], Eq. ( 16), p. 431. 19 We have alternatively considered an error correction model: because the results was not improved, the (simpler) adaptive process has been retained as did [START_REF] Layard | Unemployment[END_REF]. 20 For details about the data, see [START_REF] Chélini | Cliométrie du chômage et des salaires en France[END_REF], Annex 3. 21 The variables in the form of rates are expressed in % in the equation of the unemployment rate and in decimal value in the equation of the wage rate. 22 In a first step, we have estimated among the hyperparameters the (two) covariances between the residuals of the measurement equations and the auxiliary residuals of the corresponding state equations. As these covariances were not significant, the system was estimated by excluding them. 23 Estimates were made using the Eviews7 software. The state variables are represented in the "one-step-ahead predicted states" mode, which implies that the degree of rigidity at date t results from the cumulated effects of the past stochastic shocks reflecting the successive changes in this degree. 24 In Eq.( 14) of the wage rate, the average unemployment rate

) ( 2 1 1    t t t u u u
was considered because this specification allowed an improvement in the results, suggesting that the influence of the level of employment on wages is exerting more slowly than the influence of prices and other variables. 25 The breakdown of 1982-83 is linked to changes in collective agreements -including the progressive deindexation of wages on the price level -and had the effect of significantly increasing the flexibility of the labour market [START_REF] Blanchard | L'indexation des salaires: quelle rupture en 1982 ?[END_REF][START_REF] Ralle | Indexation des salaires : la rupture de 1983[END_REF], [START_REF] Desplatz | La modération salariale en France depuis le début des années 1980[END_REF]). [START_REF] Heyer | Structural and reduced approaches of the equilibrium rate of unemployment, a comparison between France and the United States[END_REF] confirm these results with the WS-PS model using the Kalman filter method. The decline in the share of wages in the added value since these years illustrates this change in trend [START_REF] Meurs | Salaires réels et emploi dans les pays de l'OCDE[END_REF], [START_REF] Gérard-Prenveille | Déterminants du taux de chômage d'équilibre et ajustements sur le marché du travail : une analyse sur données françaises[END_REF]). 26 Indeed, referring to the theoretical exposition presented in Chapter V of his Treatise on Pure Economy [1994, 1943 1st ed.], [START_REF] Allais | Les facteurs déterminants du chômage français, 1952-1978[END_REF][START_REF] Allais | Le chômage et les charges salariales globales[END_REF] proposed an explanation of the unemployment rate in France over the period 1952-1978 by a linear combination of the "chronic", "cyclical" and "technological" components (see [START_REF] Prat | Rueff, Allais et le chômage d'équilibre[END_REF]). [START_REF] Allais | La mondialisation, la destruction des emplois et de la croissance : l'évidence empirique[END_REF] proposed later a partition of unemployment into five categories: chronic unemployment, cyclical unemployment, unemployment due to the free trade of the world, unemployment due to immigration, and finally technological unemployment. Let us concede that the empirical approach proposed by the author to measure these five types of unemployment did not dampen our conviction. 27 The composite variable t spread is a priori defined to more or less a constant, since, unlike the net hourly minimum wage expressed in current euros, the series of the price level and labour productivity are in the form of indices. In order to "calibrate" these series together, it was assumed that, at the time of the creation of the minimum wage in 1950, the institutional variables determining the real cost of labour (i.e. minimum wage including contributions) balance with labour productivity (see Chelini and Prat [2016], footnote (54)). The estimated values of the unemployment rate and the wage rate do not depend on this calibration, but the latter determines the relative importance of the estimated chronic and frictional components of unemployment. 28 Between 1950 and 2008, the profit margin rate oscillates around an average level of 28.4% with a moderate amplitude (the standard deviation equals 2.6%) 29 Frictional unemployment may be regarded as referring to the "Job Search" (Stigler [1962]) and "matching" models [START_REF] Pissarides | Equilibrium unemployment theory, 2 nd Ed[END_REF]). 30 By comparison with our frictional component, the value of the "natural rate of unemployment" as defined by Milton Friedman is generally estimated between 3% and 5% (see for example [START_REF] Weiner | New estimates of the natural rate of unemployment[END_REF], and especially for France, [START_REF] Heyer | Le chômage structurel à 5% en France ?[END_REF]). 31 Note that, after 2008 and until 2014, there have been a widening gap between real wages and productivity accompanied by an increase in the unemployment rate, which suggests that the chronic unemployment has likely increased. After 2014 the gap and unemployment both decreased, suggesting that the chronic component likely lessened. 32 In particular, one observe in 1990 the coexistence between a low cyclical unemployment (1%) and a consequent chronic unemployment (4%). This result is in line with those of [START_REF] Salanié | Une maquette analytique de long terme du marché du travail[END_REF], which shows that even during good economic times, unemployment can remain high in France, as this phenomenon is attributed to excessive labour costs, notably because of the minimum wage. Note that, after 2008, the sluggish growth in French production suggests an increase in the cyclical component. 33 Intercepts was removed from the final regressions since not significant. 34 The "Index of consumer confidence" of French households provided by Insee' surveys appears to be negatively correlated with inflation. It is the same for USA with the well-known « Index of Consumer Sentiment » built by the University of Michigan. 35 In the same vein and to give an empirical comparison, we note that the gap between observed gross wage and labor productivity fluctuates between +10% and -10% over the period 1950-2008 (supposing a zero gap in 1950). 36 In order to better illustrate the differences, Figure 6 compares the rate of change of the observed wage with the rate of change of the values estimated in level according to (14). 37 We have:

     T t



  ) is estimated over the period 1955-2008 (54 years) using the maximum likelihood method. Numbers in brackets at the right of estimates give the statistic z, which is interpreted as a t-value ratio. To ensure positivity, the variances of i t  ˆand (i = u , s) are estimated as and . The initial values of t  and t  are those minimizing the information criteria (we obtain 0.053 and -0.795, respectively). 2 D R is the Harvey [1992] modified coefficient of determination: the positive values obtained indicate that the structural model is more performing than a random walk with drift. (*) The intercept was found to be insignificant and then dropped from the final estimation. (**) Q and hH are the statistics of the Ljung-Box autocorrelation test (number of lags: heteroscedasticity test that are applied to the standardised values of residuals i t ˆ, as proposed by[START_REF] Harvey | Forecasting, structural time series models and the Kalman filter[END_REF] for the Kalman filter. At the 5% level of significance, the values obtained for Q being less than the critical tabulated (wage) (indicated in brackets, in italics after the statistics), one accepts the null (no autocorrelation) ; values of the statistic hH being less than the tabulated values for these two equations (

t

   and t  (estimation using theSeemingly Unrelated Regression method). (***) LR is the log-likelihood ratio (distributed as a

  are common to the unemployment and wage measurement equations -are very significant (see Eq. (

  Figure 1 -Coefficient alpha in the unemployment rate equation (*) (degree of rigidity of the labor market)

  Figure 2 -The three components in the equilibrium unemployment rate %

  Figure 3 -Coefficient omega in the wage rate equation (*) (sensibility of the wage rate to the unemployment rate)

Figure 4

 4 Figure4exhibits the time patterns of the two calculated sensibilities; in accordance with the WS and PS equations, the estimated values of s t k and p t k are always positive over the period and show a sensibility to unemployment significantly higher for employees than for employers. This finding suggests that economic and institutional constraints faced by firms in the exercise of their "right to manage" have a smaller effect than those represented by the employees' aversion to the risk of being unemployed. The medians for the values of s t k and

  Figure 4 -Sensibilities of employees and employersto the unemployment rate

  Figure 5 -Differences between the wage desired by employees and the wage offered by employers : "observed" and fitted values values

Figure 7 -

 7 Figure 6 -Observed values and estimated values of the unemployment rate

  represents the negotiated value of the wage rate, we can express the value of the ERU which makes compatible the aspirations of both agents :

																											), that is ps t s = ws t s = * t s where * t s
	u	* t			t	[	s	R t		cs	t		t ce		p	t			t		t m		f	t	]	(3)
								with			t		1	(	s t k 	k	t	p	)	and	f	t		f	t	s		f	t	p
	The relationship (3) shows that the empirical estimate of	* t u requires auxiliary
	assumptions to represent the reservation wage	R t s , the coefficient t  and the "other
	factors" t f of prices and wages. Concerning the representation of the reservation wage R t s ,

  ) depends on the trend of the national minimum wage SMIC and

	involves parameters o  and o s which are of course estimated simultaneously with other
	parameters of the system. The error terms u t  and s t  (called "innovations") of the
	measurement equations are assumed to be N(0,		( 2 u t 	)	) and N(0,		( 2 s t 	)	), as well as the
	errors terms (called "noises") u t  and s

t

 of the two state variables are assumed to be

Table 1 -Estimating of the unemployment rate and the wage rate using the Kalman filter method

 1

			unemployment rate t u	gross hourly wage rate t s
			state equation (5) ( u i  )	state equation (10) ( s i  )
		i 1	0.88 (16.2)	0.94 (21.0)
		i o	0.07 (1.8)	(*)

  ) show that these two coefficients are characterized by autocorrelation and that their dynamics are non-stationary, which would lead one to subsequently conduct a search for factors. Because this empirical exploration is made without a pre-requisite theoretical, we considered many macroeconomic variables to be potential candidates, such as: unemployment rate, growth rates in the wage rate and in the SMIC, inflation rate, production growth rate, interest rate, growth rate in money supply, margin rate, unionization rate, … Because the rules and in the state of the economythese sensibilities depend on common factors (see Figure4), we thus estimate a two-equations system using the Seemingly Unrelated Regression method which avoids biases due to contemporary correlation between the two residuals and to heteroscedasticity. For each sensibility, outcomes bring into high six significant factors 33 that are the one-period lagged sensibility of the other agent, the

	comparison of the dynamics of s t k and p t k suggests that -likely as a result of a common
	knowledge in institutional unemployment rate t u (%), the long term interest rate t j (government bonds, % per
	annum), the inflation rate for the previous year	t dp (deflator of the added value, % per
	annum), the one-period lagged margin rate t m (%), and the growth rate of the net hourly
	SMIC (	t dsmic ) :
	5 . 3 ˆ ( ) 5 . 8 ( 1 ) 1 . 4 ( ) 2 . 1 ) 4 . 6 ( ) 0 . 3 ( ( 1 31 . 0 56 . 0 84 . 0 22 . 0 86 . 0 06 . 1	)
					t		1	(	s t k 	k	p t	)	, we now
	conduct a brief econometric analysis that aims to straightforwardly identify some underlying
	observable factors of s t k and p

t k . Confirming the visual of Figure 4, preliminary tests (non-reported

  factors. One can see that coefficients i

		f	t	s		1 (			)	f	t	p		i  	i	X	i		c	o	with		i				s i a		1 (  	)	p i a	and	o c			f	s o		1 (  	)	f	p o	, where
	s i a and p i a represent the sensitivities of		s f and p t t f to the it						
	and o c can eventually cancel each other.
	c t p to the right-hand side of equations (1) and (2).
	17 with	Consider   i p t f	p i a	  t q  it f X	 i p o ,	i c	X t f	it s		and i   i a  it s i f X a	b s o	i c ,	.	a	i		Eq. (6) allows: s i p i a a  and o f	t  f	 o p f f	p  t	 o s f f	 , which leads to  o it i s t f X a  i

X

  SSR is the sum of the squares of the structural model residuals. A positive (negative) value of 2 D R means that the structural model gives a better (less good) representation than a simple random walk with drift.

	R	2	1	SSR	/	1	(	y	t	y	)	2	and	R	D 2		1		SSR	/	  t 2 T	(	 y	t		 y	)	2	, (	t u y  , t t s ) and	dy	t		y	t		y	t	1 	, where

The present paper is a modified English version of an article written in French[START_REF] Chélini | Cliométrie du chômage et des salaires en France[END_REF]), especially adding novel contributions on two important points. First, it brings up new results about factors of the time-varying sensibilities to unemployment of wages claimed by employees and wages offered by employers, these two sensibilities playing a major role in the WS-PS model on which our approach is based. Second, the present paper assesses the difference between the level of wage claimed by employees and that offered by employers, and show that, as supposed by the theory based on negotiation between the two parties, change in employment intervenes indeed in the adjustment between these two notional wages to achieve equilibrium.