Cyclicality of Wages and Union Power

Transcription

1 Cyliality of Wages and Union Power Annaïg Morin Booni University, Milan GATE, University of Lyon II Lumière, Lyon February 2011 Abstrat This paper proposes a dynami model of the labor market whih integrates two main features: mathing fritions and trade unions. To examine how trade unions shape the volatility of wages over the business yle, I deompose the volatility of union wages into two omponents: the volatility of the math surplus and the volatility of the union s effetive bargaining power. Formally, I define the effetive bargaining power of the union as the share of the total surplus alloated to the workers. Starting from the union s objetive funtion, I prove that its effetive bargaining power is endogenous and ounterylial. Intuitively, beause the union internalizes the relationship between the wage level and the job reation, it faes a trade-off between the wage rate and the employment rate. Therefore, the union s preferenes (wage-oriented or employmentoriented) flutuate along the yle and so does its effetive bargaining power. As a result, when the eonomy is hit by a produtivity shok, the dynamis of the union s effetive bargaining power partially ounterat the dynamis of the total surplus and this mehanism delivers wage rigidity. Relatedly, employment reats stronger when wages are olletively bargained, but its pattern features less persistene. Keywords: Searh and mathing; Trade unions; Cyles JEL lassifiation: J64; J51; E32 Eonomis Department, Booni university, Milan. annaig.morin@phd.unibooni.it. Web: I am grateful to Tito Boeri, Antonella Trigari and Alain Sand for their preious advie and support. Moreover, I thank Alberto Alesina, Christian Bauer, Mikael Burda, Vinenzo Cuiniello, Szabols Deák, Matthias Doepke, Salvador Ortigueira, Martin Shneider and seminar partiipants at Booni university, GATE-Lyon II University, the 2010 European Workshop in Maroeonomis, WIEM 2010, AFSE 2010, ISNE 2010 and ASSET 2010 for their very useful omments. 1

2 1 Introdution The role of trade unions is to protet the rights and interests of their members through representation within firms. It inludes negotiations with employers on behalf of the workers for better wages and working onditions. Therefore, through their diret partiipation in the wage determination proess, trade unions impat the wage rate level and shape its volatility. This paper integrates searh and mathing and trade unions to propose a dynami framework that aommodates two important soures of wage volatility: the volatility of the math surplus and the volatility of the union s effetive bargaining power, whih is formally defined as the share of the total surplus obtained by the workers. I then study how these two soures interat over the business yle to shed new light on the properties of the union wage volatility and thus on the properties of the union wage premium volatility. Speifially, the paper develops a tratable dynami model that distinguishes and ompares two wage bargaining proesses. The first one is the individual Nash bargaining traditionally used in the searh and mathing literature. The seond is haraterized by a olletive Nash bargaining, where a unique trade union negotiates with firms on behalf of its members, who are either employed or unemployed. Those firms reat in a seond stage by deiding unilaterally how many vaanies to post. I show that, when wages are olletively bargained, the maximization of the Nash produt leads to the following equilibrium ondition: the share of the total surplus alloated to the workers, I refer to this as the union s effetive bargaining power, is not onstant but rather flutuates with the degree of labor market tightness. This new variable enters the wage equation and its volatility gives rise to an additional soure of wage volatility. By studying the omponents of union wages, I make three ontributions to our understanding of wage and unemployment flutuations when wage negotiations take plae at a olletive level. First, I propose a theoretial framework whih allows me to analyze the union s behavior over the yle. In the present model, the union s preferenes (wage-oriented or employment-oriented) are endogenous and reat to the tightness of the labor market. Moreover, the flutuations of the union s preferenes are refleted in the flutuations of the union s effetive bargaining power. I demonstrate that this variable is in fat ounterylial, whih leads to the seond ontribution of my work. The union s effetive bargaining power enters naturally the wage equation. Therefore, wage flutuations are not only driven by the dynamis of the total surplus, but also by the dynamis of this additional variable, whose ylial properties are ruial: the ounteryliality of the union s effetive bargaining power partially ounterats the flutuations of the total surplus whih yields rigid wages. Colletively bargained wages are less responsive to the business yle than individually bargained ones, a result whih is onsistent with, beause diretly derived from, the union s utility funtion. Third, when the eonomy is hit by a positive produtivity shok, the moderate reation of the union wage translates into a stronger response of employment. Sine the union s effetive bargaining power inreases as the shok propagates, wages go bak slower towards their steady state values. In turn, the response of 2

3 employment in the union setor features less persistene. This paper builds on Mortensen and Pissarides (1994) searh and mathing model by introduing trade unions. Searh fritions ause math-speifi rents, as both the employer and the employee would be worse off if the math ended, due to the delays of filling a vaany and finding a new job. Suh a theoretial framework is promising as it allows to explore different rent sharing rules, i.e. different wage setting proesses. Most of the searh and mathing literature proposes an individual Nash bargaining solution, but alternatives have reently been investigated, often in order to improve the ability of the model to aount for the stylized fats of the business yle. 1 Despite this growing strand of the literature, few papers fous on olletive wage bargaining, even though olletive bargaining overage is still high, partiularly in Europe, 2 despite the overall deline in union density. In their role as a partiipant of wage negotiations, their behavior affets neessarily the level and the volatility of wages. Beause of this, it is important to take this institutional feature into onsideration when modeling wage formation. Moreover, to introdue trade unions into a searh and mathing framework is neessary in order to ompare olletive wage bargaining with individual wage bargaining and to isolate its speifiity. A first study integrating a trade union, in the form of a monopoly union, in a searh and mathing framework is the one of Pissarides (1986), extended subsequently by Delaroix (2006) and Garibaldi and Violante (2004). These authors model wage determination in the presene of trade unions and desribe the negative impat of unionism on employment due to the union wage premium. However, they propose models in steady state whih are inapable of giving any insight on the speifi volatility of olletively bargained wages. The model I develop aounts for a dynami struture of the role of trade union and allows a omparative analysis of the dynamis of both types of wages. In the present paper, I develop a bisetoral model eonomy: a non unionized setor where wages are negotiated at an individualized level oexists with a unionized setor where a union interats with firms in a olletive bargaining proess over wages. The ruial differene between the two wage setting proesses is as follows. Searh fritions are turnover osts whih give the workers or the union the power to bargain for a wage higher than the reservation wage, higher than the wage for whih the outsiders would be willing to work. In an individual wage bargaining, an employed worker, insider, makes full use of his bargaining power to push his wage to the highest possible level, without taking into aount the negative impat of this high wage on the firm s hiring deision and therefore on the unemployment rate. When wages are olletively bargained, the presene of unions inreases workers bargaining power as firms 1 Shimer (2005) points out that, at onventional parameters values, individually Nash bargained wages are exessively volatile, depressing vaany reation. As a result, employment is far less volatile than in the data. Wage rigidity has been explored as a way to improve the performane of the model. 2 For a reent study on wage bargaining institutions in most of European ountries, United States and Japan, see Caju, Gautier, Momferatou, and Ward-Warmedinger (2008). They provide among other data on olletive bargaining overage and establish that this rate exeeds 80% in most western European ountries. 3

4 would not be able to produe at all if both parties do not ome to an agreement. This lower fall bak utility of firms enables the union to apture a larger share of the total rent. The largely doumented 3 union wage premium reflets this differene in bargaining power. More interestingly, the union internalizes the impat of wages on the firms inentive to open vaanies and herewith gives onsideration to the outsiders who fae more diffiulty finding a job when wages are high. Indeed, the existene of unemployment and the threat of more unemployment is the major onstraint of union power, and this threat is materialized by the employer s labor demand urve (the job reation urve). The intuition works as follows. The onern of the trade union is the welfare of its members, who an be either employed or unemployed. Therefore the union s utility funtion inreases in both wages (though the insiders utility) and employment(though the outsiders utility) and the negative relationship between these two variables represents the trade-off faed by the union. The level of wages the union is demanding reflets its preferenes over these two variables. As a result, for a same bargaining power, the olletively bargained wage is lower than an individual one as employment enters the utility funtion of the union. The dynami model I present allows me to ompare how the two wage setting proesses affet the stati harateristis of the labor market as well as its dynami properties. Therefore, I am able to point out the role of trade unions in shaping the volatility of labor market variables. The argumentation onsists of three steps. As a start, I demonstrate that the preferenes of the union over wages and employment flutuate depending on the tightness of the labor market. Indeed, the union s preferenes reflet its members preferenes, whih are only based on the urrent status (insider or outsider) of eah member. Therefore, the omposition of the union s members, i.e. the proportion of (un)employed workers, translates into the orientation of the union. In good times, when the proportion of unemployed workers is low, the unemployment issue is played down and the union pushes for high wages as a large proportion of its members are employed and asking for a high wage rate. In bad times, when the stok of unemployed workers is large, the union moderates the wages in order to boost hirings. In addition, it is worth notiing that the union s preferenes translate diretly into the union s effetive bargaining power, whih is defined as the fration of the math surplus gotten by workers. Therefore, the aforementioned mehanism has an important impliation: the union s effetive bargaining power is endogenous and dereasing with unemployment. This finding is onsistent with the empirial study of Aronsson, Löfgren, and Wikström (1993) who test two models of wage determination, the first haraterized by a union s bargaining power whih is onstant over time, the seond where the bargaining power develops with unemployment and labor market harateristis, using time series data for the Swedish onstrution setor. They find evidene that unemployment tends to derease the bargaining power of the union. Two other papers (Campbell III (1997) and Fuess (2001)) onfirm empirially this result. Using U.S data, Campbell III (1997) finds that union wages are more 3 See for example Blanhflower and Bryson (2002). 4

5 sensitive to the unemployment rate than non union wages. Fuess (2001) obtains from Japanese data that the union power is positively orrelated with GDP. Seond, I spell out how the union s effetive bargaining power affets wages. In both setors, the bargained wage is a weighted average of two threat points delimiting the bargaining set, the weights being equal to the share of the math surplus obtained by the workers. This share is endogenous in the olletive bargaining setting. As a result, the dynamis of the union wage are not only driven by the flutuations of the math surplus, but also by the flutuations of the union s effetive bargaining power. Central to my analysis is thus the additional soure of wage volatility in the union setor arising from the volatility of the effetive bargaining power. In a third step, I fous on the ylial properties of this new variable, with the level of produtivity as the business yle indiator, and prove its ounteryliality. To get the intuition behind this result, onsider a positive produtivity shok. Both the union wage rate and the employment level shift upwards, but employment reats relatively smoothly. 4 In turn, the marginal rate of substitution of wage for employment inreases, and it beomes more valuable for the union to foster employment as eah worker benefits from a higher surplus. On impat, the union onverts the higher surplus into both higher wages and higher employment, but the shift in its preferenes reates a bias towards employment. Subsequently, as the shok propagates in the eonomy and raises the employment level, the union gives gradually more importane to the wage level. The opposite mehanism works in ase of an adverse produtivity shok. Given the higher reativity of wages ompared to employment, the union s first onern is to avoid a large derease in the wage rate and therefore prioritizes wages over employment. In the following periods, as unemployment inreases following the adverse shok, the union tends to be more employment-oriented. The behavior of the union is therefore totally explained by the hanges in valuation of wages and employment over the yle and hene in hanges in the marginal rate of substitution between these two elements. A similar behavior, unions being more aggressive in bad time, is desribed by Freeman and Medoff (1984) in their book What do unions do?, even though the underlying argumentation differs. Indeed, they explain that one of the main fators behind the ounteryliality of the union wage premium is the greater apaity of unionized workers to fight employers effort to redue wages when market onditions are unfavorable. This has been proposed by the authors as the reason of the widening of the union wage premium during the Depression of the deades 1920s, 1930 s. Blanhflower and Bryson (2004) onfirm this theory, based on an empirial study of the union wage premium in the US over the period The ounterylial properties of the union s effetive bargaining power is at the ore of the mehanism of wage rigidity. On impat, the dynamis of this variable partially ounterat the dynamis of the total surplus and make the union wage less responsive to the business yle. In the following periods, the 4 Note that firms take their hiring deision based on the differene between produtivity and wage rate. Employment flutuates therefore less that wages. This is exatly the starting point of Shimer s ritique. 5

6 response of the union wage rate to the shok is more persistent in the presene of a union. Symmetrially, employment reats stronger on impat and features less persistene. This model provides a onvenient framework to analyze the role of unions in shaping the volatility of wage and employment. In this sense, my paper is losely related to three papers: Mattesini and Rossi (2007), Zanetti (2007b) and Faia and Rossi (2009). These papers analyze how the response to produtivity shoks of employment and wage is affeted by the presene of the union and find that the union dampens wage dynamis and amplifies employment dynamis. However, none of these studies integrates trade unions into a searh and mathing framework, whih is yet ruial in order to understand the speifi power of unions and its volatility, as well as the union wage volatility. Moreover, a searh and mathing framework provides a natural way of modeling the union s objetive funtion, from where the endogenous union s effetive bargaining power is derived, and allows a lear omparison between individual and olletive wage bargaining. Finally, the seond soure of wage volatility, whih is at the ore of the origin of wage rigidity when wages are olletively bargained, is absent of their analysis. The paper proeeds as follows. The next setion lays out the dynami equations of the model. I introdue within a searh and mathing framework a setoral trade union negotiating wages with firms on behalf of its members and I ompare the wage formation proess with a standard individual Nash bargaining setting. Setion 3 fouses on how the unions effetive bargaining power evolves along the yle and on how this volatility impats the volatility of union wages. Setion 4 analyses quantitatively the dynami behavior of the model in ase of disturbanes (produtivity shoks). Setion 5 onludes. 2 The model 2.1 The environment A dual labor market Labor markets are generally haraterized by both individual and olletive wage formation. The present model reprodues this feature. Indeed, in the bisetoral model eonomy, two produing setors oexist: the non unionized setor denoted by the supersript N and the unionized setor denoted by the supersript U. The unique differene between these two setors stands in the wage bargaining proess. In the non unionized setor, wages are negotiated at an individualized level, i.e. employers and employees agree on wages through a bilateral bargaining proess. In the opposite, in the unionized setor, they are olletively bargained at the setoral level, meaning that a unique union negotiates wages with firms on behalf of its members. More preisely, I make two assumptions regarding the unionized setor. First, based on the observation that the bargaining proess over wages takes mainly plae at the setoral level 6

7 (OECD (1994), Caju, Gautier, Momferatou, and Ward-Warmedinger (2008)), I assume that all the firms belonging to the unionized setor are engaged in olletive bargaining over wages. Seond, the membership rule is suh that eah worker in the setor is unionized, regardless of his employment status. 5 The workers are unable to swith between setors beause eah oupation requires speifi skills and an extensive investment in training and qualifiations. 6 This segmentation an be brought together with the dual labor market desribed by Doeringer and Piore (1971) where a primary market with relatively high wages, high degree of onentration, and powerful unions, oexists with a seondary market with opposite harateristis. 7 As presented in MaDonald and Solow (1985), the existene of some sort of barriers prevents free movement aross setors. Some empirial studies (Nikell (1997), Nikell and Layard (1999)) show that a better measure of the unions size is provided by the olletive bargaining overage rate, given the presene of exess overage in several ountries. This rate is sluggish (Cahu and Zylberberg (2004)) whih justifies the assumption that the share of workers belonging to eah setor is fixed in the short run Timing At the beginning of the period, an exogenous shok ours. Existing mathes have a probability λ i (where i = N, U denotes respetively the non unionized and unionized setor) to end. Subsequently, firms post vaanies and searhing firms and unemployed workers meet. This mathing proess is time-onsuming meaning that only a ertain perentage of unemployed workers and firms atually form a math eah period. The number of new mathes in setor i at time t, m i t, is determined through a mathing funtion inreasing in both the size of the pool of unemployed workers searhing for a job u i t and the size of the pool of vaanies v i t. At the end of the period, prodution takes plae with a level of employment n i t and salaries are paid. Notie that in this framework, newly employed workers start to produe within the same period. This allows firms to reat immediately when faing a shok, in posting more or less vaanies. Conerning the point in time when wages are negotiated, the two setors differ. In the non unionized setor, by the very nature of individual wage bargaining, wages are negotiated one eah firm-worker pair has been formed. Firms rationally antiipate the outome of the wage negotiations when deiding the number of vaanies to post. In the unionized setor, I assume a right-to-manage timing, based on a sequene à la Stakelberg where the union first bargains with firms over wages and then firms respond by unilaterally determining employment. This timing ontrasts with the one adopted in the searh and mathing 5 In the more generalized ase where workers who have been laid off for more than a ertain number of periods loose membership, the qualitative results would remain valid. 6 The utilities setor makes a good example of the unionized setor, whereas the finanial setor an represent the non unionized one. 7 See also Silvestre (1971) who uses suh a labor market segmentation to desribe the Frenh industry sine

8 literature but is ommon pratie both in the trade unions literature and in papers integrating trade unions into searh and mathing frameworks (see Pissarides (1986), Mortensen and Pissarides (1999) and Delaroix (2006)) Stoks and flows The labor fore is homogeneous and normalized to one, out of whih α N (0, 1) belong to the non unionized setor and α U = 1 α N belong to the unionized setor. Employment in setor i = (N, U) at the end of period t, n i t, evolves aording to the following dynamis: n i t = α i u i t + m i t (1) whereas the number of unemployed workers at the beginning of period t evolves as: u i t = α i (1 λ i )n i t 1 (2) Plugging (2) into (1), I get: n i t = (1 λ i )n i t 1 + m i t (3) Notie that the law of motion of end-of-period unemployment, ū i t, is: ū i t = α i (1 λ i )n i t 1 m i t and that, by the normalization of the labor fore: ū N t + ū U t + n N t + n U t = 1 The mathing proess in setor i is summarized by the following funtion: where σ m is the mathing proess effiieny. m i t = σ m u i tσ uv i t 1 σ u (4) The vaany filling rate is qt(θ i t) i = mi t, where θ i vt i t = vi t represents the market u i t tightness, and the job finding rate is p i t(θt) i = mi t. The mathing proess presents u i t a ongestion externality in the sense that the slaker the labor market, the higher the probability for a vaany to be filled and the lower the probability for an unemployed worker to find a job. 2.2 Solving the model Two optimal deisions have to be derived: the optimal hiring deision of firms, in terms of number of vaanies to post for any antiipated (in the non unionized setor) or observed (in the unionized setor) wage rate, and the optimal wage. 8

9 2.2.1 Vaany posting deision Firms are assumed to be large enough so that by the law of large number the fration of vaanies filled in eah firm is equal to the vaany filling rate of the setor. Firms are similar whih allows me to fous on a representative firm in eah setor. The output in setor i is given by: yt i = z t n i t where z t is the produtivity level ommon to both setors. The prodution osts onsist of the wage ost and the ost of posting vaanies ( per vaany). In the non unionized setor, the employment level is determined before wages are negotiated. Therefore, in ase wages would depend on the sale of the firms, employers would have the possibility to manipulate the wage through their employment poliy. When hoosing the number of vaanies to post, wages would be taken into aount as a funtion of the employment level. For simpliity reason, I avoid this issue by assuming onstant returns to sale in the prodution funtion. As shown by Cahu and Wasmer (2001), the intra-firm bargaining element vanishes in this ase. In the unionized setor, given that firms take their hiring deision in a seond stage, one the wage rate has been negotiated, the wage shedule is observed and suh an intra-firm bargaining issue is therefore irrelevant. The number of posted vaanies results from the following profit s maximization proess: 8 max v i t The first order ondition is given by: F i t (n i t) = z t n i t w i tn i t v i t + E t βf i t+1(n i t+1) F i t v i t s.t. n i t = (1 λ i )n i t 1 + q i tv i t = z t qt i wtq i t i + E t β F t+1 i n i qt i = 0 (5) t Using the envelope ondition and equation (5), I obtain the expression of F i t : n i t 1 F i t n i t 1 = (1 λ) q i t (6) Plugging equation (6) into equation (5), I obtain the job reation equation (JC): q i t = z t w i t + E t β(1 λ i ) q i t+1 This result establishes that firms post vaanies up to the point where the ost of posting a vaany times the expeted duration of the vaany 1 equals qt i the ontribution of the worker to the flow of profit plus the vaany posting ost saved by the firm in t + 1 in ase the math does not end. (7) 8 Beause the intra-firm bargaining element, whih ould arise in the non unionized setor, vanishes due to the assumption of a onstant returns to sale prodution funtion, I do not write down the onstraint represented by the wage equation w N t = w N t (nn t ). 9

10 2.2.2 Wage negotiation Wage negotiation in the non unionized setor In this setor an individual bargaining proess takes plae over the wage between employers and employees one the mathing proess has taken plae. The Nash-bargained wage maximizes the produt of the net gains of agreement for both parties. At the onlusion of the bargaining, the marginal math provides a welfare Wt N to the worker, with Wt N being defined as: Wt N = wt N + E t β [ (1 λ N + λ N p N t+1)wt+1 N + λ N (1 p N t+1)ut+1 N ] (8) whereas the worker obtains a welfare of Ut N, with Ut N being defined as: Ut N = b + E t β [ p N t+1wt+1 N + (1 p N t+1)ut+1 N ] (9) if the bargaining fails. Notie that, due to the homogeneity of the labor fore and to the similarity of the firms, the value of employment for the marginal worker is the same for all the employees, regardless of the firm they are working in. As for the firms, the marginal math ontributes to the flow of profit up to the marginal value of employment F N t, without regards to the vaany posting osts whih n N t are sunk at the time of the wage negotiation. Therefore, the expression for the value of the marginal job is: F N t n N t v N t = v t N := J N t = z t w N t + E t β(1 λ N )J N t+1 (10) Hene, the Nash bargained wage results from the following maximization program: max [W N wt N t Ut N ] ηn [Jt N ] 1 ηn where η N [0, 1] is the workers bargaining power. The first order ondition states that the share of the total surplus alloated to the workers is equal to their bargaining power: W N t U N t W N t U N t + J N t = η N (11) Rearranging this first order ondition and using the expression of the values J N t, W N t and U N t, I obtain the equilibrium wage rate in setor N: w N t = η N [z t + E t β(1 λ N )θ N t+1] + (1 η N )b (12) The bargaining set is delimited by two threat points and ontains an infinity of equilibrium wage rates. The sharing rule alloates a onstant share of the bargaining set to the worker and the firm. Unlike in the walrasian model, the 10

11 wage does not equate the worker s produtivity. Indeed, the wage rate is a funtion of two labor market parameters: the ost of posting vaanies and the level of unemployment benefits. Moreover, and more importantly, the wage varies with the degree of the labor market tightness in the following period. To see this, onsider an inrease in the number of posted vaanies in t + 1. The duration of a vaany inreases, along with the real ost of posting a vaany q. t+1 This translates in period t into a bigger saving of hiring ost in ase of math and therefore into a higher value of employment for the firm and a higher total surplus. The workers surplus being a fixed part of the total surplus, the wage inreases. The same reasoning an be followed with an inrease of unemployment whih leads to a derease of the worker s surplus and finally of the wage rate. The effiieny ondition, established by Hosios (1990), entails that the bargaining power of the workers η N should equal the elastiity of the vaany filling rate to the degree of labor market tightness σ u. Indeed, if this ondition is respeted, the number of posted vaanies is equal to the one whih would prevail in an eonomy where firms take into aount that the vaany filling rate dereases with the number of posted vaanies. Wage negotiation in the unionized setor Unlike several reent papers studying trade unions and business yles (Zanetti (2007a), Mattesini and Rossi (2007), Faia and Rossi (2009)) who adopt a monopoly union model, I assume that, at the beginning of eah period, the union bargains with the firms over the wages. Subsequently, firms deide unilaterally the number of vaanies to post, based on the wage rate previously negotiated. This right-to-manage model, in line with Nikell (1982) and Nikell and Andrews (1983), allows me to keep the analysis as general as possible. 9 Net gain of agreement for the union. I assume that the unique onern of the union is the welfare of its members, who are either employed or unemployed. At the time the wage rate is bargained over, α U u U t members are employed and will attain with ertainty the level of utility Wt U at the end of the period, while u U t members are unemployed, out of whih p U t u U t will form a math and attain the level of utility Wt U at the end of the period, the remaining (1 p U t )u U t attaining the level of utility Ut U. The utility of the union is assumed to be the sum of the end-of-period utility levels of its members and takes onsequently the following form: Ω t = (α U u U t )W U t + u U t [p U t W U t + (1 p U t )U U t ] Ω t = n U t (W U t U U t ) + α U U U t (13) 9 Right-to-manage bargained wages are not Pareto-effiient. Effiient ontrats an be obtained if firms and unions would bargained simultaneously over wages and employment, as shown by Leontief (1946). However, as argued by Calmfors and Horn (1986) and Oswald (1993), negotiations do generally not inlude employment expliitly. 11

12 Beause all the workers of the setor would be unemployed in ase of failure of the wage negotiation, the fall bak welfare of the workers is α U Ut U. Therefore, the union s net value of agreement is n U t (Wt U Ut U ). As mentioned earlier, due to the assumptions of homogeneity of the labor fore and the similarity of the firms, the value of employment for the marginal worker is the same for all the employees. Moreover, due to the onstant returns to sale prodution funtion, the wage rate is independent from the level of employment and the welfare values Wt U and Ut U represent respetively the employment and unemployment values of both the marginal and the average worker. Beause of this, Wt U Ut U an be interpreted as the average worker s surplus and n U t (Wt U Ut U ) as the total surplus of the workers. This utilitarian speifiation presents several advantages. First, this speifiation, in keeping with MaDonald and Solow (1981) and Oswald (1982), is a ommon approah in the trade union literature (see Calmfors (1982), Sampson (1983), Kidd and Oswald (1987), Pissarides (1986) among others). Seond, it allows a immediate omparison of the workers objetives aross setors. Indeed, while eah worker in setor N seeks to maximize his own surplus W N t U N t, the union aims at maximizing the sum of the workers surpluses n U t (W U t U U t ). Both the workers and the union foresee that their demands will affet the hiring deision of the firms. However, the speifiity of the unionized setor is that the union internalizes the negative impat of wages on employment when negotiating wages, beause it seeks to maximize not only the individual surplus but also the proportion of workers reeiving this surplus. Third, the searh and mathing framework provides a natural way of modeling the workers fall bak utility U U t, whih is not fixed but flutuates with the expeted evolution of the degree of market tightness. Fourthly, the unions objetive is diretly derived from the members preferenes, whih is not the ase with Stone-Geary utility forms. Moreover, the unions speifiation, as desribed by equation (13), allows for politial onsiderations. Indeed, even if employment and wages have equal weight in the unions utility funtion, I will show later in this setion that the relative importane of these issues is endogenous and varies along the yle. Net gain of agreement for the firms. For the firms, the fall bak position is zero. Indeed, if the bargaining with the union fails and no wage agreement is found, the firms an not find any non unionized workers in the setor to be able to produe. If the bargaining is suessful, eah firm gets an end-of-period profit Ft U + vt U, beause the vaany filling osts, being paid at the middle of the period before the mathing proess takes plae, are sunk at the end of the period (even though they are not sunk at the time of the bargaining). 10 It is showed in Appendix A that, in both setors, the profit of the firms an be 10 This hoie of eliminating the vaany filling osts in the wage bargaining is done in order to ease the omparison with the non unionized setor. Otherwise, the differenes in the wage equations aross setors would stem from both the differene in the level at whih the wage bargaining takes plae and the differene in the timing (ex ante wage bargaining in the unionized setor, and ex post wage bargaining in the non unionized setor). 12

13 written as a funtion of the marginal value of employment: F i t = n i tj i t v i t This implies that the net gain for the firms is equal to n U t Jt U. It also implies that Jt U represents the firms ex post (i.e. one the vaany filling osts have been sunk) employment value of both the marginal and the average math. Wage equation. From equation (13), one an observe that the utility of the union depends both on the rate rate and on the employment level. A high wage rate inreases the individual worker s surplus, but, beause the job reation urve is downward sloping, lowers the inentive for the firms to post vaanies, and therefore lowers the employment rate. Ω t = Ω t (w U t, n U t (w U t )) with Ω w > 0, Ω n > 0 and n w < 0 The negative orrelation between the wage rate and the job finding rate embodies the trade-off faed by the union. This trade-off would disappear in ase the vaany posting deision were taken before the wage bargaining and both wage negotiation proesses would lead to the same equilibrium. In this sense, the timing assumption is ruial is this setor. 11 The union maximizes therefore its objetive funtion ontaining wage and employment, onstrained by the firms labor demand funtion 12, the job reation urve, and the maximization program of this right-to-manage model is written as the following: max[n U wt U t (Wt U Ut U )] ηu [n U t Jt U ] 1 ηu s.t JC urve: q U t where η U is the union s bargaining power. = z t w U t + E t β(1 λ U ) q U t+1 The FOC leads to the following equilibrium ondition whih states that the share of the total surplus alloated to the workers is (see Appendix B.1): where W U t U U t W U t U U t + J U t η t U η U σ u n U t = σ u n U t + (1 σ u )m U η U t = η U t (14) 11 To see this, onsider the alternative timing were vaanies are posted ex ante. With this timing, the hiring deision would be based on the expeted, or promised, wage level. But the union an not redibly announe that it will moderate the wage rate in order to promote hirings. Indeed, one the vaanies have been posted, nothing prevent the union to deviate from its announement and push the wage to the highest possible level. With this timing, the level of employment n U t would be taken as given. The maximization program would be redued to the ase of the wage bargaining of the average math, whih is equal to the wage bargaining of the marginal math due to the onstant returns to sale. 12 For an empirial test of this maximization program, see Dertouzos and Penavel (1981). 13

14 I define the onept of effetive bargaining power of a agent (workers or union) as the share of the total surplus alloated to the workers. In the non unionized setor, bargaining power and effetive bargaining power are two similar onepts: the workers bargaining power equals the share of the surplus alloated to the workers (see equation (11)). In the unionized setor, these two onepts are different: the union s bargaining power η U does not equal the unions effetive bargaining power η t U. It follows from equation (14) that the wage urve in the unionized setor is (see Appendix B.2): [ ] wt U = η t U z t + E t β(1 λ U )θt+1 U + (1 η t U )b Θ t (15) where Θ t = E t β(1 λ U ) q U t+1 (1 p U t+1) η t+1 η t 1 η t+1 Two omments are in order when onsidering the wage equation (15) and when omparing it with the orresponding equation for the non unionized setor, equation (12). First, the struture of the equation is the same as in the non unionized setor, with the two same extreme wage levels whih demarate the bargaining set, z t + E t β(1 λ U )θ U t+1 and b. Indeed, the third term in the right hand side of equation (15), Θ t, whih indiates how the expeted evolution of the union s effetive bargaining power impats the workers demand, beomes negligible as the persistene of the model inreases. Seond, unlike in the non unionized setor, the sharing rule does not alloate a onstant share of the bargaining set to the union and to the firm. Indeed, the weights, η t and (1 η t ), are not fixed but endogenous. These remark leads to an important result. The flutuations of the wage rate in the unionized setor stem from two different soures: the flutuations of the total surplus and the flutuations of the union s effetive bargaining power. Union s effetive bargaining power. In order to get the intuition behind the properties of the union s effetive bargaining power, I write it as an expression of the beginning-of-period stok of unemployed workers u U t and the out-of-unemployment probability p U t : η t = η σ u( 1 U u U t σ u ( 1 u U t for whih the following properties hold: 1 + p U t ) 1) + p U t (16) η t u U t < 0 The union s effetive bargaining power is equal to its bargaining power η U multiplied by a seond term smaller than one and dereasing in both the beginning-of-period unemployment level and the job finding rate. η t p U t < 0 14

15 To understand the negative orrelation between the union s effetive bargaining power and the beginning-of-period unemployment level, onsider the union s utility funtion, equation (13). The union s preferenes reflet its members preferenes, whih are only based on the urrent status (insider or outsider) of eah member. Indeed, the determination of the union s preferenes being renewed eah period, no ommitment fores the workers to take into aount their expeted status in the following periods. Therefore, the urrent omposition of the union s members, i.e. the proportion of (un)employed workers, translates into the orientation of the union. If the period starts with a large stok of unemployed workers, the union tends to moderate the pressure on the wage rate to give the firms the inentive to post vaanies. The union is naturally more willing to boost employment the more relevant the unemployment issue is, beause a relatively large proportion of its members are outsiders who benefit from a moderate wage rate. By the same token, if the employment level is high at the beginning of the period, the union is more wage oriented as the unemployment issue is played down and a relatively large share of its members are employed and laiming for a high wage rate. Consequently, I find the following salient result. The union s preferenes over wages and employment, driven by the proportion of (un)employed workers within the union, are endogenous and flutuating with the situation on the labor market. Said differently, the degree of market tightness reflets on the omposition of the unions and, through this hannel, shapes the unions preferenes over wages and employment. Notie that in the theoretial ase where there were no unemployment at the beginning of the period (or more realistially, if unemployed workers were not taken into onsideration by the union), the union s effetive bargaining power would be equal to its bargaining power η U. The result would be similar to an individual Nash bargaining. Indeed, in this ase, the union s members would all have the same preferenes over wages and employment (they would laim for a high wage rate), and the union would make full use of its bargaining power to push the wage to the highest possible level. The olletively bargained wage rate departs from the individually bargained one only beause the union is partly omposed by unemployed workers who value employment and wages differently ompared to employed workers. The heterogeneity of the union s members introdues politial onsiderations and reflets the internal onflit within the union. The higher the proportion of unemployed workers, the stronger the downwards pressure on the wage rate, the lower the union s power to drag a large part of the total surplus. As a result, the omposition of the union impats the outome of the wage negotiation. Next, onerning the negative orrelation between the union s effetive bargaining power and the job finding rate, two observations have to be made. First, notie that this job finding rate reflets the sale of the searh fritions haraterizing the labor market. The smaller these fritions, the faster the job finding. Seond, the math-speifi rents are generated by the searh fritions. Therefore, the smaller the searh fritions are, the loser the labor market is 15

16 from a ompetitive market, where the wage rate equals the reservation wage. Combining these two observations, the mehanism behind the negative orrelation between the union s effetive bargaining power and the job finding rate is the following. When the labor market onverges towards a ompetitive market (when the searh fritions beomes negligible), the probability to find a job tends towards infinity and the workers surplus fades. This indues the vanishing of the union s effetive bargaining power. In turns, the wage rate tends towards the ompetitive outome, i.e. the reservation wage. And vie versa. 3 Counterylial union power In this setion, I seek to illustrate the mehanism through whih wage rigidity arises when wages are olletively bargained. The degree of wage rigidity is assessed based on how the wage flutuates when the eonomy is hit by a produtivity shok. 3.1 Linearization In order to analyze the hannels through whih the produtivity shok affets the labor market, I onsider a log-linear approximation of the model. (End of period) employment dynamis: ˆn N t = λn p N ûn t + λ N ˆm N t ˆn U t = λu p U ûu t + λ U ˆm U t (Beginning of period) unemployment dynamis: û N t = pn λ N (1 λn )ˆn N t 1 û U t = pu λ U (1 λu )ˆn U t 1 Using these two equations and the Beveridge urve, I an write: ˆn N t = (1 λ)ˆn N t 1 + λ N ˆm N t ˆn U t = (1 λ)ˆn U t 1 + λ U ˆm U t Mathing proess: ˆm N t = σ u û N t + (1 σ u )ˆv N t ˆm U t = σ u û U t + (1 σ u )ˆv U t Vaany filling rate: Job finding rate: ˆq N t = ˆm N t ˆv N t ˆq U t = ˆm U t ˆv U t ˆp N t = ˆm N t û N t ˆp U t = ˆm U t û U t 16

17 Labor market tightness: ˆθ N t = ˆv N t û N t ˆθ U t = ˆv U t û U t JC urve: ˆθ N t = φ N ẑ t φ N w N ŵ N t + β(1 λ N )ˆθ N t+1 ˆθ U t = φ U ẑ t φ U w U ŵ U t + β(1 λ U )ˆθ U t+1 where w i = wi z, φi = 1 σ uc i and C i = zq i, i = N, U. Wage rate: ŵ N t = ηn w N ẑt + ηn w N [ β(1 λ N ) θ N]ˆθN t+1 (17) ŵt U = η w U ẑt + η w [ β(1 λ U U ) θ U ]ˆθU t+1 + η w [ (1 b) + β(1 λ U U ) θ U ]ˆ η t + ˆÕ t (18) where ˆÕ [ ] t = 1 β(1 λ U )C U (1 p U ) η w U 1 η (ˆ η t ˆ η t+1 ) and b = b z, = z and CU = zq U. Union s effetive bargaining power: ) ˆ η t = (γ U (1 λ U + λu p U ) û U t where γ U = (1 σu)λu (1 σ u)λ U +σ u. ( γ U (1 σ u )(1 λ U ))ˆθU t (19) 3.2 Counterylial effetive bargaining power and wage rigidity Equation (19) reveals that the union s effetive bargaining power is ounterylial. Indeed, u U t being the beginning-of-period unemployment, it does not shift on impat and the labor market tightness is obviously proylial. The intuition behind this property works as follows. Consider the utility funtion of the union, equation (13), whih is inreasing in both the wage rate and the level of employment. A produtivity shok impats both variables, but the wage rate reats relatively more than the employment level. Indeed, the vaany posting deision being motivated by the differene between the level of produtivity and the wage rate, the reation of the employment level is smoother than the one of the wage rate. This leads to the following observation: a produtivity shok modifies the marginal utility of both the employment level and the wage rate and hene modifies the marginal rate of substitution between these two elements. Said differently, a produtivity shok alters the relative 17

18 value that the union attahes to the employment level and the wage rate. To see this, onsider a positive produtivity shok. The wage rate inreases relatively more than the employment level. After the shok, an additional inrease in the employment level therefore provides a higher level of utility than previously, relative to an additional inrease in the wage rate. The marginal rate of substitution of wage for employment inreases. This leads to a shift in the union s preferenes towards employment. Intuitively, beause the wage rate and therefore the worker s surplus are relatively reative, it beomes more valuable for the union to foster employment when a positive shok ours as eah worker benefits now from a greater surplus. As a result, the orientation, or preferenes, of the union flutuates along the yle in the following way. By the aforementioned mehanism, when a positive produtivity shok ours, the union beomes, on impat, more employmentoriented. In the following periods, as employment goes up and the wage rate goes bak to its steady state value, the marginal utility of employment dereases while the marginal utility of the wage rate inreases. This leads to a derease of the marginal rate of substitution of wage for employment. The union gives therefore more and more importane to wages relative to employment. The opposite ours when the eonomy is hit by an adverse produtivity shok. The union beomes more aggressive on impat, and more populist as the shok propagates in the eonomy. Consequently, the wage rate is driven by two variables: the proylial surplus from the math and the ounterylial unions effetive bargaining power. The flutuations of the seond variable, whih is fixed in the non unionized setor, is at the ore of the wage rigidity mehanism. Indeed, the dynamis of the union s effetive bargaining power partially ounterats the dynamis of the total surplus and, as a result, dampens the wage rate flutuations. My omments are twofold. First, the urrent model provides a miro foundation for the wage rigidity. Colletively bargained wages are less responsive to the business yle not beause of any ad ho mehanism but beause of the ounterylial properties of a union-speifi variable whih arises diretly from the union s maximization program and enters the wage equation. Seond, it is worth noting that the nature of the wage rigidity arising here ontrasts with a part of the literature that models either onstant wages either wages with a bakward-looking omponent, often with the view to amplify the volatility of unemployment and vaanies and thus to respond to the Shimer ritique. For instane, Hall (2005) followed by Shimer (2004) assume that wages do not flutuate with the yle and are set at a onstant soially aeptable level. In Krause and Lubik (2007), the wage is a weighted average of the Nash bargained wage and a wage norm. Gertler and Trigari (2009) introdue a staggered wage setting, where only a ertain proportion of the ontrats are renegotiated eah period. In Blanhard and Gali (2007), Christoffel and Linzert (2006) and Shimer (2010), the wage is a weighted average of the past wage level and a urrent equi- 18

19 librium wage level 13. In the present model, the union wage rate flutuates with the urrent produtivity level and is not an expliit funtion of its past value, i.e. it is not a state variable. Notwithstanding, the union wage rate is funtion of the harateristis of the labor market at the beginning of the period, speifially of the level of employment whih determines the omposition of the union. Therefore, past labor market onditions impat the wage rate indiretly through the beginning-of-period employment level. This opens a seond hannel though whih shoks propagate. It is in sharp ontrast with the non unionized setor, where suh a propagation mehanism does not exist. Indeed, when wages are individually bargained, they are independent of the beginning-of-period employment level and hene of previous levels of produtivity. 4 Quantitative assessment of the model In this setion the quantitative properties of the model are investigated by studying the impulse response of the labor market to a positive produtivity shok in both setors. 4.1 Calibration The alibration of the model is desribed in table 1. These values are hosen to math the empirial regularities of U.S. I interpret a period as a month. The disount fator is set to /3 whih orresponds to a yearly interest rate of 4% ommonly used in the maro-rbc literature. The log produtivity level z t is assumed to follow an AR(1) proess: log(z t ) = ρ log(z t 1 )+ɛ t where ɛ N(0, σ 2 ). The persistene of the tehnology shok is set to ρ = 0.98 and the standard deviation to σ = 0, 008. This standard alibration is used by Rogerson and Shimer (2010) and is based on the estimations of Cooley and Presott (1995). The mean of z is normalized to one. I target the probability p N that an unemployed worker forms a math within the period to 45%, implying unemployment spells of around two months. This hoie is onsistent with Hall (2005) who estimates an monthly job finding rate of 0.48% and in line with the measure of this rate presented by Rogerson and Shimer (2010) for the US for the period Eah math has a probability to end λ N set to 0, 1/3. This value is omprised within the broadly aepted range of 8% 10% proposed by Hall (2005) and is similar to Shimer (2005) who measures this exit probability at 0, 1/3 in average in the US. I target the degree of labor market tightness θ N to 0.5, whih is onsistent with the estimate of obtained by Hall (2005). Considering the mathing proess, two parameters have to be disussed. First, the weight on unemployment σ u, whih represents the elastiity of the 13 This urrent equilibrium wage level is the Nash bargained wage level in Christoffel and Linzert (2006) and Shimer (2010) and the marginal rate of substitution between onsumption and leisure in Blanhard and Gali (2007). 19