Collective bargaining, hours vs jobs and the fiscal multiplier. EC-IILS JOINT DISCUSSION PAPER SERIES No. 9. International Labour Organization

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1 International Labour Organization European Union International Institute for Labour Studies Collective bargaining, hours vs jobs and the fiscal multiplier EC-IILS JOINT DISCUSSION PAPER SERIES No. 9

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3 COLLECTIVE BARGAINING, HOURS VS JOBS AND THE FISCAL MULTIPLIER

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5 COLLECTIVE BARGAINING, HOURS VS JOBS AND THE FISCAL MULTIPLIER INTERNATIONAL LABOUR ORGANIZATION INTERNATIONAL INSTITUTE FOR LABOUR STUDIES

6 Abstract This paper is part of a series of discussion papers that have been prepared by the International Institute for Labour Studies IILS within the framework of the joint project Addressing European labour market and social challenges for a sustainable globalization, which has been carried out by the European Commission EC and the International Labour Organization ILO. The discussion paper series provides background information and in-depth analysis for two concluding synthesis reports that summarize the main findings of the project. This paper relates to first part of the project Addressing the short- and medium-term labour market and social challenges of the current economic and financial crisis and the concluding synthesis report Building a sustainable job-rich recovery. The current economic crisis has led several governments to conduct discretionary fiscal expansion to foster aggregate demand. The question of the effectiveness of fiscal policy has attracted an increasing interest within academic circles. A new body of literature on fiscal policy, making use of dynamic stochastic general equilibrium DSGE models, has recently emerged. In this paper, we aim at contributing to this new literature with a particular focus on the effects of expansionary fiscal policy on the labour market. In particular, we focus on the following three questions: i What are the effects of an increase in government spending on economic activity? ii How do changes in the labour input, induced by the rise in government spending, adjust along its two margins hours of work per employed worker and employment? iii Is the choice of the bargaining scheme important for the propagation of government spending shocks to key labour market variables?

7 TABLE OF CONTENTS Main findings 1 1 Introduction 1 2 The Related Literature and Modeling Approach 3 Is the wealth effect relevant according to the data? Modeling strategy The Prototype Model 8 The Labor Market Households Firms Two bargaining problems in the labor market Government and Monetary Authority Aggregation and Market Clearing Equilibrium Definitions Calibration 29 5 Quantitative Analysis 31 6 Conclusion 35 A Appendix 37 How to Reduce the Negative Wealth Effect on Labor Supply: an Overview Household s optimization problem Household and Firms marginal values of a match Equilibrium equations Additional Figures i

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9 COLLECTIVE BARGAINING, HOURS VS JOBS AND THE FISCAL MULTIPLIER Main findings In this paper we assess to what extent labor market institutions affect the effectiveness of fiscal policy. The ability of fiscal policy to reduce unemployment depends on the adjustment of labor input along the intensive and extensive margin. The motivation is to reflect on the attempts by some governments to save jobs by reducing hours. For that purpose, this paper considers two types of labor market institutions. First, workers and firms bargain over wages and hours efficient bargaining. Second, bargaining is restricted to wages and hours are set unilaterally by firms right to manage. We show that fiscal policy is more effective in reducing unemployment in the case of efficient bargaining. We also show that efficient bargaining generates a larger fiscal multiplier. This result underlines that collective bargaining may contribute to the design of successful economic policy. 1 Introduction The current economic crisis has led several governments to conduct discretionary fiscal expansion to foster aggregate demand. The question of the effectiveness of fiscal policy has attracted an 1

10 increasing interest within academic circles. A new body of literature on fiscal policy, making use of dynamic stochastic general equilibrium DSGE hereafter models, has recently emerged. In this paper, we aim at contributing to this new literature with a particular focus on the effects of expansionary fiscal policy on the labor market. In particular, we focus on the following three questions: i What are the effects of an increase in government spendings on economic activity? ii How do changes in the labor input, induced by the rise in government spendings, adjust along its two margins hours of work per employed worker and employment? iii Is the choice of the bargaining scheme important for the propagation of government spendings shocks to key labor market variables? These questions are central to policy makers who aim at improving the economic activity and the level of employment. The main goal of fiscal policy is to increase employment by stimulating aggregate demand. It follows that the key aspect is that firms adjust their labor input along the extensive margin employment rather than along the intensive margin hours of work per employed worker. It is also central to understand the implications of different bargaining schemes for the dynamic of wages, hours, and employment. In particular, it is worth understanding whether the terms of negociations between firms and workers matter for the adjustment of labor input along the two margins or not. In return, the type of bargaining might have implications for the size of the fiscal multiplier. In this paper, we develop a DSGE model with labor market search and matching frictions à la Mortensen and Pissarides 1994 and Pissarides 2000 in order to analyse the effects of a positive change in government spendings on economic activity and in particular on the labor market variables employment, hours worked per employed worker, and real wage. There has been few attempts to study the effects of government spendings on labor market characterized by search and matching frictions within a DSGE framework. Among the few who do, Monacelli et al MPT hereafter conduct a similar analysis but consider only the intensive margin of labor input. In contrast, we build our model so that it displays the following features: 1. the extensive and intensive margin of labor input are explicitly identified; 2. positive government spending shocks increase output, consumption, total hours worked, and real wage, in line with standard empirical findings see Monacelli and Perotti 2008; 3. alternative assumptions on the determination of real wage and hours of work per employed worker are considered. In a standard neoclassical growth model an increase of government spendings leads to a rise in the expected net present value of taxes that households have to pay: households become poorer. It follows that households reduce their consumption and postpone their leisure. This intertemporal substitution effect is usually called negative wealth effect in the literature for instance see Barro and King The increase in labor supply leads the marginal product of labor to decline. In the context of a Walrasian labor market, this implies that the real wage falls too. Hence, in order to build our model consistently with feature ii, we follow Monacelli and Perotti 2008 MP hereafter. 2

11 These authors propose to shut down the negative wealth effect on the labor supply by assuming that households preferences are described by the utility function introduced by Greenwood et al GHH hereafter in the literature. There is no clear agreement on how changes in the labor input, caused by a government spending shock, are adjusted along both the intensive and the extensive margins. As a consequence, the model is not forced to deliver any specific impulse responses of employment and hours following a positive shock to the government spendings. Instead, we consider two different assumptions with regard to the setting of the real wage and the hours worked per employed worker, as indicated in feature iii. First, following Trigari 2006, we assume that the real wage and hours of work per employed individual are jointly negotiated by both firms and households through a Nash bargaining process. Second, we assume that while the real wage is Nash bargained by firms and households, the number of hours is determined unilaterally by firms. The first bargaining process is called efficient bargaining while the second one is called right-to-manage bargaining. The results of our quantitative analysis can be summarized as follows. The increase in government spendings leads output, consumption, total hours worked, and the real wage to shift upward under both efficient bargaining and right-to-manage bargaining. We show that our model generates an output multiplier that is significantly larger than unity on impact under either bargaining scheme. The choice of bargaining specification affects the size of the impact responses of several variables. In particular, efficient bargaining scheme offers a better amplification mechanism to the government spending shocks in terms of expansions in output and employment than right to manage bargaining scheme does. Regarding output, the difference between the two alternative bargaining schemes is significant but small. The adjustment along the intensive margin is prevailing in both cases. The rest of the paper is organized as follows. A discussion on the related literature and further detail of our modeling approach is proposed in Section. The model is formally outlined in Section. The calibration of the model is presented and discussed in Section. The quantitative analysis of the model is reported in Section. Conclusions are provided in Section. 2 The Related Literature and Modeling Approach Is the wealth effect relevant according to the data? In a standard neoclassical growth model an increase of government spendings leads to a rise in the expected net present value of taxes that households have to pay: households become poorer households permanent incomes decline. It follows that households reduce their consumption and postpone their leisure increase their supply of labor. These effects of the increase in government spendings on consumption and labor supply are called negative wealth/income effects in the literature Barro and King Hence, output increases in response to the positive government 3

12 spending shock but by less than the increase of the latter. Firms labor demand curve is not affected by the government spending shock only the labor supply schedule is shifted by the shock. As a consequence, the labor market being competitive, the real wage declines. The theoretical results concerning the dynamics of consumption and real wage are not always empirically verified. Indeed, some economists such as Fatás and Mihov 2001, Blanchard and Perotti 2002, Galí et al. 2007b, and Perotti , using U.S. data, find that consumption and real wage increase following a positive government spending shock. Other authors, such as Ramey and Shapiro 1998, Edelberg et al. 1999, Burnside et al. 2004, and Ramey 2009, using also U.S. data, show that positive government spending shocks have instead negative effects on consumption and real wage as in a standard neoclassical growth model. MP, using a different U.S. data set from those used by the authors quoted above notably, they add a constructed data on the price markup apply the identification scheme of government spending shocks advocated by Blanchard and Perotti 2002 and the one developed by Ramey and Shapiro Under both identification strategies, they find that the increase in government spendings lead consumption and real wage to shift upward. The size of the output fiscal multiplier whether output increases by more or less than the increase of government spendings is also debated in the empirical literature. Some economists have estimated a U.S. output fiscal multiplier smaller than unity as in a standard neoclassical growth model such as Barro while others have estimated a U.S. output fiscal multiplier larger than unity such as Ramey and MPT. Regarding the labour market impact, some empirical studies report that labor input rises in response to an increase in government spendings see for instance Galí et al. 2007b. As mentioned above, according to the neoclassical growth model, the increase of labor input measured as the number of total hours worked is due to the negative wealth effect inherent to the government spending shock. However, if one is interested in understanding the effect of fiscal policies on the employment/unemployment level, this finding is not fully informative. Indeed, changes in the labor input total hours worked can be done along its extensive margin the number of employed worker and/or its intensive margin the number of hours of work per employed worker. To our knowledge, little empirical analysis has been carried out to understand the effects of government spending shocks on employment and average hours worked. In the case of the U.S., Yuan and Li 2000 find that an increase in government spendings leads average hours worked to rise and employment to fall. In addition, average hours worked react faster to a positive government spending shock than employment. However, the change in the latter happens to be more persistent than the change in the former. In contrast, MPT, also on the basis of U.S. data, report that an increase in government spendings has strong and positive effects on both employment and total hours worked. Their empirical study shows that the dynamics of employment and total hours worked are very similar. They deduce that the average hours worked increases in response to a governement spending shock at 1 Perotti 2007 extends also his analysis to three other countries from OECD: Australia, Canada, and United Kingdom. 2 Barro 1981calculates a multiplier of about Ramey 2009 claims a multiplier of 1.2 4

13 least on impact but very mildly. Hence, there is not a clear consensus on the effect of a government spending shock on the two labor input margins. On the theoretical side, few studies investigate which margin contributes the most to the changes in the labor input induced by the government spending shocks 4. Yuan and Li 2000 develop a business cycle model with a government spending shock, featuring unemployment through the search and matching mechanism à la Pissarides 2000 and Mortensen and Pissarides 1994 and identifying explicitly the two margins of the labor input. They show that an increase in government spendings shifts upward average hours worked and reduces the employment level. The decline of the number of employees is slow and relatively small. As a consequence, total hours worked increases in response to the positive government spending shock. Hence, the simulation results of the model match the empirical results reported by those same authors. Modeling strategy In this paper, we develop a DSGE model with labor market search and matching frictions that allow us to analyse the effects of a positive change in government spendings on economic activity and in particular on the labor market variables employment, average hours worked, and the real wage. We build the model so that it displays the following features: 1. the extensive and intensive margins of labor input are explicitly identified; 2. positive government spending shocks increase output, consumption, total hours worked, and the real wage; and 3. alternative assumptions on the determination of real wage and hours of work per employed worker are considered. We impose the model to have the feature 2. in order to match MP s empirical findings. In contrast, since there is no clear agreement on how changes in the labor input are adjusted along both margins, we do not restrict the model to deliver some specific impulse responses of employment and average hours worked with respect to government spending shocks. Instead, following Trigari 2006, we consider different assumptions regarding the setting of the real wage and average hours worked, as indicated in the feature 3.. First, we assume that the real wage and average hours worked are jointly negotiated by both firms and households through a Nash bargaining mechanism efficient bargaining, see Trigari Indeed, the outcome of this Nash bargaining process happens to be privately efficient. Note that this determination rule of average hours worked and the real wage is commonly used in models with labor market search and matching frictions in which both labor input margins are time-varying. Second, we assume that only the real wage is Nash-bargained by 4 The role played by the number of hours worked per employed worker and the number of employed individuals in the changes of the labor input are mainly analysed in the context of monetary policy shocks as in Trigari 2006 and neutral/investment specific technology shocks as in Canova et al and Krause and Lubik

14 firms and households whereas average hours worked are unilaterlly determined by firms right-tomanage bargaining, see Trigari The right-to-manage bargaining rule has also been used by other authors such as Christoffel et al. 2009, and Christoffel and Linzert Note that our aim is different from Yuan and Li While they are particularly interested on the dynamics of employment and hours of work per employed worker, we focus on a larger number of variables. In Yuan and Li 2000, because of the negative wealth effect, consumption and real wage decline following an increase of government spendings, which contradicts the empirical findings of MP. Note that in Yuan and Li 2000, the determination of average hours worked and the real wage does not depend on whether bargaining is efficient or right-to-mange. The intensive margin of labor input is chosen unilaterally by households and the real wage is set according to an ad-hoc rule: the real wage is proportional to the marginal product of labor input. Specifically, the larger the households bargaining power, the closer is the rel wage to the marginal product of labor. We follow the strategy of MP in building our model to ensure that the latter displays feature 2.. MP develop a theoretical model which is able to match their empirical findings with regard to consumption, real wage, and the real price markup. According to MP, the government spending shock would impact positively consumption and the real wage if the negative wealth effect of government spendings on labor supply is weakened enough. Those authors choose to shut down the negative wealth effect by assuming that households preferences are described by the utility function introduced by GHH. Those authors specify a utility function which implies a marginal rate of substitution of consumption for leisure that depends only on labor put differently, the intertemporal consumption/saving behavior does not alter the labor supply decision. As a consequence, the GHH preferences imply that the wealth effect on labor supply is nil 6. MP stress that lowering the negative wealth effect on labor supply is not a sufficient condition to have a positive response of consumption to government spending shocks. The model should also include a monopolistically competitive intermediate goods market with nominal price rigidities. MP give the following intuition to support their strategy. The baseline model that MP use to support their intuition is a business cycle model with monopolistic competition, flexible price, and a frictionless labor market. An increase of government spendings does not affect the households labor supply curve if the preferences are GHH since the wealth effect on labor supply is inexistent in this case. Hence, the real wage cannot be changed through the supply side of the labor market. MP show that if the real wage is fixed, an increase in government spendings does not affect the labor input but reduces consumption under GHH preferences. Hence, in order to have a positive response of consumption to the government spending shock the real wage needs to increase. As pointed by MP, 5 As mentioned by Trigari 2006, the right to manage term was previously used in a different kind of model in the labor market literature. In those models for instance Andrews and Nickell 1983, the frictions in the labor market are due to the existence of unions rather than the search and matching mechanism. The real wage is Nash bargained by firms and unions while only firms can decide the number of workers to be employed. The intensive margin of the labor input is generally assumed to be fixed. 6 As noted by MP, the strategy of undermining the negative wealth effect on labor supply is rather counterintuitive because it also implies a large negative wealth effect of government spending shocks on consumption. 6

15 if the real wage shifts upward, households would, via a substitution effect, supply more labor and henceforth consume more 7. Under GHH preferences, the increase of the real wage can be triggered off only from the demand side of the labor market through an upward shift of the labor demand curve. Price stickiness à la Calvo-Yun 8 is an additional necessary ingredient in order to ensure that a positive government spending shock shifts upward the firms labor demand curve. Under the Calvo- Yun price stickiness specification, each period only a fraction of firms, selected randomly, are allowed to set optimally their prices. Hence, the increase of government spendings leads the aggregate demand schedule to shift upward. Assuming that a large fraction of firms cannot change their prices in the current period, the aggregate supply curve shifts upward in response to the government spending shock. Indeed, firms that cannot set optimally their prices have no other choice than adjusting the quantities of goods they produce by increasing them to face the shift in the aggregate demand schedule. Firms which increase their production will, as a consequence, increase their labor demand. Thus, if the number of non-optimizing firms is large enough, the aggregate labor demand schedule moves upward along the aggregate labor supply curve, implying an increase in both labor input and real wage. The literature proposes other strategies to overcome the negative wealth effect on labor supply of government spending shocks. These alternative mechanisms are presented briefly in Appendix. We depart from MP in modeling the labor market. In particular, we consider labor market search and matching frictions, which allows us to analyse the determinants of the equilibrium unemployment. In addition, labour input can be modified both alongside hours worked and the number of jobs. This will help us in understanding how an increase in government spendings affects both hours worked individually and employment and want to analyse the consequences of these margin adjustements on the level of unemployment. We are not the first to study the effects of government spending shocks on a labor market characterized by search and matching frictions within a DSGE framework. Yuan and Li 2000 and MPT also focus on the effects of changes in government spendings on the labor market with search and matching frictions. However, as mentioned above, the former authors do not overcome the negative wealth effect on labor supply. MPT explore the negative wealth effect of government spending shocks on labor supply on the labor market with search and matching frictions. Those authors use a specification of preferences introduced by Shimer With this utility function, being employed is costly in terms of satisfaction for households. MPT show formally how the negative wealth effect of government spending shock on labor supply interacts with the firms dynamic decision on hiring new workers in the equilibrium. Specifically, the channel through which the wealth effect affects the hiring rate is given by the marginal value of non-work activity. The latter resemble the marginal rate of substitution of consumption for leisure, being defined as the ratio of the marginal utility of employment 7 As emphasized by MP, the increase of the real wage implies no wealth effect which would reduce the labor supply because, in the context of monopolistic competition, the rise in households total income due to the increase in real wage is cancelled out by the induced decline in real profits. 8 See Calvo 1983 and Yun

16 to the marginal utility of wealth. In order to reduce the negative wealth effect on labor supply, MPT increase the complementarity between consumption and employment. Indeed, as shown by MPT, the effect of the negative wealth effect on the hiring rate becomes weaker as the degree of complementarity between consumption and employment increases. However, MPT assume that only the extensive margin of the labor input is time-varying The Prototype Model In this section we outline the model and discuss its properties. The Labor Market The labor market is characterized by search and matching frictions. The workers are assumed to be ex ante all identical. In the beginning of each period t there is a continuum of workers of mass 1 among which n t 1 are employed workers and u t are workers seeking for a job in the beginning of period t. Hence, we have u t = 1 n t 1. Following Mortensen and Pissarides 1994, the search and matching frictions in the labor market are modeled by a standard Cobb-Douglas matching function: m t mu t,v t = γ m u γ t v 1 γ t 1 where v t denotes the number of vacancies that firms post in the beginning of the period t in order to hire new workers. The efficiency of the matching process is measured by parameter γ m. It is useful to define the probability for a firm to fill a vacancy, q t, as well as the probability for a worker to find a job, p t : q t = m t v t q t = γ m θ γ t qθ t 2 p t = m t u t p t = γ m θ 1 γ t pθ t 3 where θ t denotes the labor market tightness and is defined as follows θ t v t v t = 4 u t 1 n t 1 9 Several authors have incorporated the labor market search and matching frictions in DSGE models. Some of them focus on the effects of productivity shocks such as Merz 1995 and Andolafatto 1996, while others analyze the effects of monetary policy shocks such as Trigari The identification of both labor input margins does not require to model the labor market with search and matching frictions. Yuan and Li 2000 show that a model with a Walrasian labor market and a time-varying labor effort, such as the one developed by Burnside et al. 1993, can be reinterpreted as a model with a Walrasian labor market in which changes in the labor input can occur on both margins the intensive margin corresponds to the labor effort. 8

17 Households The economy is inhabited by a large representative household which consists of a continuum of individuals of mass one as in Merz The preferences of that representative } household are described by the following function: E 0 t=0 β t {n t c e,t ψh ζ t 1 σ 1 σ σ + 1 n t c1 σ u,t where n t denotes the number of the household s members that are employed that is the current level of employment, 1 n t denotes the number of the household s members that do not have a job different from the number of workers seeking a job, u t, c e,t is the consumption level of the employed members of the household, c e,t represents the consumption level of the unemployed members of the household, h t denotes the hours worked by each employed member of the household, σ > 0 is a parameter that measures the degree of substitutability between consumption and leisure, ψ is a preference parameter for leisure, ζ is a parameter that governs the elasticity of labor supply in terms of hours per employed worker, and β is the discount factor. Hence, the employed members of the household have GHH preferences over consumption and hours of work while the unemployed members have constant relative risk aversion preferences over consumption. The intratemporal budget constraint of the representative household is given by c t + b t w t h t n t + rn t 1 π t b t 1 τ t + D t 5 where c t n t c e,t + 1 n t c u,t represents the aggregate consumption level of the household, w t is the real wage, D t denotes the profits received from firms the household is assumed to own all firms, τ t is a lump-sum tax that the household has to pay to the government, r n t is the gross nominal interest rate, b t is the holding of one period state contingent real bonds, and π t is the gross inflation rate. The representative household is aware that its employment situation in the labor market depends on the flows of its members in and out of employment, accordingly to n t = ρn t 1 + p t 1 n t 1 6 By computing the first order conditions of the household s optimization problem with respect to c e,t and c u,t, one can demonstrate that, in equilibrium, the household behaves as if its preferences are described by 11 E 0 t=0 β t c t n t ψht ζ 1 σ 1 1 σ 7 In other words, the household chooses its optimal sequences of c t and b t by maximizing its modified 11 See Appendix for a formal proof. 9

18 expected discounted sum of utilities 7 subject to its employment accumulation law 6, and the budget constraint 5. We restate the household s problem with a Bellman equation H n t 1,b t 1 = max c t,b t c t n t ψht ζ 1 σ 1 1 σ + βe t H n t,b t 8 subject to the budget constraint 5, and the employment accumulation law 6. The first order conditions of the household s problem with respect to c t and b t, in that order, are = c t n t ψht ζ σ 9 1 = βe t λt+1 The first order condition 9 states that the marginal utility of wealth represented by the Lagrange multiplier associated the household s budget constraint, is equal to the marginal utility of consumption in equilibrium. r n t π t+1 10 In turn, the first order condition 10 corresponds to the standard Euler equation stating that the expected gross growth rate of the marginal utility of consumption depends on the gross real interest rate, E t r n t π t+1. For the purpose of the bargaining problem in the labor market, we derive the representative household s marginal value of having one of its member hired in the labor market rather than unemployed. We follow MPT to get a recursive expression of the household s marginal value of having one of its member hired 12 : where H n,t Hn t 1,b t 1 n t. H n,t = w t h t ψh ζ t + βe t H n,t+1 ρ p t+1 11 Firms There are two kinds of firms in the economy: wholesale firms and retail firms. Wholesale firms produce homogenous intermediate goods using labor as production input. They operate in a competitive market. However, wholesale producers have to face search and matching frictions when they express their labor demand. In turn, retail firms produce differentiated final goods using the wholesale firms intermediate goods as production inputs. They operate in a monopolistically competitive market. The differentiated final goods market is characterized by the existence of menue cost: we incorporate price rigidities in the differentiated final goods market using Calvo-Yun-type price stick- 12 See Appendix for the details of the computations. 10

19 iness. Specifically, each period only a fraction of retailers, selected randomly, are allowed to set optimally their prices. Wholesale firms The wholesale firms are all identical and have a small size, meaning that each intermediate goods producer can offer only one job. The production function of each firm is given by f h t = aht α, 0 < α 1 where a is the productivity factor common to all wholesale firms. For convenience, the wholesale firms problem will be solved at the aggregate level. Thus, the quantity of homegenous intermediate goods produced by all wholesale firms, denoted by yt w, is given by y w t = n t ah α t 12 The timing of events in the labor market is the same as in MPT. In the beginning of the period t, there are n t 1 employed workers in the intermediate goods market. All wholesale firms post v t vacancies in the beginning of the period in order to increase the stock of employees, n t. We assume that it is costly for a firm to post a vacancy. Specifically, the vacancy cost function, C v t, is assumed to be linear in the number of posted vacancies: C v t = κv t, where κ > 0 is the unit cost of vacancy posting. In the other side of the labor market, there are u t = 1 n t 1 unemployed workers that are looking for jobs in the beginning of the period in all wholesale firms. m t new matches come out of the searching process. Afterward in the current period, a fraction 1 ρ of the workers that were employed in wholesale firms in the previous period exogenously exits from those firms. The workers that have lost their jobs have to wait until the next period to look for new jobs. Then, a firm can make use of the new match to produce. Note that the new matches cannot be broken until the next period. Thus, formally, in the wholesale market, the employment evolves over time as follows n t = ρn t 1 + q t v t 13 The period-t real profits profits in terms the price index of the differentiated final goods, P t of all wholesale firms are defined as follows Φ t = e t y w t w t h t n t κv t 14 where e t Pw t P t denotes the real price of the homogenous intermediate goods. wholesale producers choose the quantities of employment, and vacancies, n t and v t, respectively, to maximize their expected discounted profits E t s=0 +s Φ t+s 15 subject to the production function 12, the employment accumulation law 13, and the expression 11

20 of the real profits 14. The wholesale firms problem can be restated with a Bellman equation as follows { } F n t 1 = max e t an t h α λt+1 n t,v t w t h t n t κv t + βe t F n t t subject to the employment accumulation law 13. The first order conditions of the wholesale firms optimization problem with respect to v t, and n t, in that order, are 16 ψ t = κ q t 17 ψ t = e t ah α t w t h t βe t +1 ψ t+1 ρ 18 where ψ t is the Lagrange multiplier associated to the employment accumulation law 13. Combining the first order conditions 17 and 18 one gets the job creation condition for a wholesale firm: κ = e t ah α +1 κ t w t h t + βρe t 19 q t q t+1 The job creation equation states that the current cost of posting a vacancy unit cost of opening a vacancy, κ, times the average duration of that vacancy, q 1 t is equal to the current wholesale firm s revenues net of its current labor cost, e t aht α w t h t, plus the present expected gain from not having to seek for a worker in the next period conditionally that the current match survives the following period. As for households, following MPT, we derive the wholesale firms marginal value of hiring a new worker once the vacancy posting costs are sunk, for the purpose of the bargaining process. It can be easily shown that, F n t 1 n t = κ q t 20 Hence, combining 19 and 20, one gets the wholesale firms marginal value of hiring a new worker 13 : where F n,t Fn t 1 n t. 13 See Appendix for the details of the computations. F n,t = e t ah α t w t h t + βρe t +1 F n,t

21 Retail firms The economy is also populated by a continuum of differentiated final goods producers, indexed by j on the unit interval, that operate in a monopolistically competitive market. Retailers purchase the homogenous intermediate goods from wholesale firms. Then, they use a technology that converts one unit of intermediate goods into one unit of a variety j of the differentiated final goods. Finally, retail firms sell their differentiated final goods to the household. The technology of production used by retailers implies that their real marginal cost coincides with the real price of the intermediate goods. Formally, we have mc j,t = e t = mc t, where mc t denotes the real marginal cost for retail firms. The optimal demand function for a variety of final goods of type j is given by Pj,t y j,t = d j,y t P t where P j,t and P t denote, respectively, the price of the variety of final goods j and the price index of final goods, in turn y j,t and y t are, respectively, the output of the retail firm j and the index of final goods. The nominal price index P t and the final goods index, in that order, ar defined as follows P t = y t = ˆ 1 0 ˆ 1 0 Pj,t 1 ε 1 ε 1 d j 1 y 1 ε 1 1 ε 1 j,t d j ε > where the parameter ε represents the intratemporal elasticity of substitution across the different varieties of final goods. The demand function for each variety of final goods, d j Pj,t P t,y t, results from the production cost minimization problem of the differentiated final goods producers, 1 subject to y t = 1 0 y1 ε j,t d j 1 min y j,t 0 P j,ty j,t ε 1 The resolution of this optimization problem leads to the following demand function for y j,t : y j,t = Pj,t P t ε Pj,t y t d j,y t P t We incorporate price rigidities in the differentiated final goods market using Calvo-Yun-type price stickiness. Specifically, only a fraction 1 ω of all retail firms, selected randomly, are allowed to set 25 13

22 optimally their prices. The remaining ω retailers set their prices accordingly to the the following rule of thumb 14 : P j,t = P j,t 1. Here we assume that retail firms which are allowed to set optimally their prices choose the same optimal price: P j,t = P t for all j. In order to derive the equilibrium condition on the optimal price, P t, we focus on retail firms which set optimally their prices in period t, assuming that in the subsequent periods, t + k, those prices remain unchanged with a probability of ω. Specifically, we solve the following optimization problem max P j,t k=0 βω k E t λt+k [ Pj,t subject to y j,t+k = P t+k mc t+k Pj,t P t+k ]y j,t+k η y t+k Thus, retail firm j chooses its optimal price, P t, that maximizes its expected discounted sum of profits, taking into account the demand function for its variety of the final goods and that its price can remain unchanged with a probability of ω. Note that the t-period profit of firm j is defined as follows: P j,t P t y j,t e t y w j,t, where yw j,t denotes the quantity of the homogenous intermediate goods that firm j purchases from wholesale firms. Since the production function of a retailer is given by y j,t = y w j,t and e t = mc t, the t-period profit of firm j can restated as P j,t P t y j,t mc t y j,t. The first order condition of the retail firms problem can be stated as follows where ft 1 = ε 1 ft 2 26 ε ft 1 = p t 1 ε ft 2 = p t ε k=0 βω k λt+k E t y t+k βω k +k E t y t+k k=0 Pt+k Pt+k P t P t ε mc t+k ε 1 with p t = P t P t. It is convenient to rewrite the expressions of f 1 t and f 2 t in recursive forms, as in?: ft 1 = p t 1 ε +1 y t mc t + βωe t π ε pt t+1 p t+1 ft 2 = p t ε λ ε t+1 y t + βωe t πt+1 ε 1 pt ft+1 2 p t+1 1 ε f 1 t where we recall that π t = P t P t 1 is the current period gross inflation rate. 14 For simplicity, we assume that there is no indexation of non-optimized prices on the gross inflation rate that prevailed in the previous period. 14

23 Two bargaining problems in the labor market Here we outline two different bargaining specifications that are commonly used in the literature: efficient bargaining and right to manage bargaining. Below, we stand close to Trigari 2006 and MPT in explaining those two bargaining schemes. Efficient bargaining Each period, the real wage and the hours of work per employee in the labor market are determined through a generalized Nash-bargaining process between wholesale firms and workers. The assumption that the hours of work are also negotiated is conventional in search and matching models where the labor input can be changed at both the extensive and intensive margin. Trigari 2006 labels this Nash-bargaining mechanism the efficient bargaining model. Wage negotiation Each period, the real wage in the labor market is determined through a generalized Nash-bargaining process between wholesale firms and the representative household: { w t argmax H n,t η F n,t 1 η}, 0 < η < 1 29 where η denotes the bargaining power of workers and where the expressions of H n,t and F n,t are given by 11 and 21, respectively. The first order condition of the Nash-bargaining process is given by ηf n,t = 1 η H n,t 30 where H n,t represents the household s marginal value of an additional employed worker expressed in units of consumption goods. The total surplus from a marginal match in the labor market or surplus for short, denoted by S n,t, is defined as the sum of the wholesale firms marginal value of hiring an additional worker and the household s marginal value of an additional employed worker defined in units of consumption goods: S n,t F n,t + H n,t. Straightforwardly, using equation 30, one can show that the Nash-bargaining process leads the household and wholesale firms to share that surplus as follows F n,t = 1 ηs n,t and H n,t = ηs n,t. The household s reservation wage, w t, is defined as the real wage from which the household is willing to work in the labor market. In turn, the wholesale firms reservation wage, w t, is defined as the maximum value of the real wage that wholesale firms are willing to pay a worker. Note that if the real wage is set equal to the household s reservation wage, w t = w t, then the household s marginal value of an additional worker expressed in units of consumption goods becomes nil: H n,t = 0. Hence, in this case, equation 11 becomes 15

24 w t h t = ψh ζ t βe t λt+1 H n,t+1 ρ p t Note that the household s reservation wage, w t, increases with the hours of work, h t. Hence, if the household is willing to supply more hours of work, it would ask for a higher reservation wage. In turn, w t decreases with the household s expected future continuation value of the match, H n,t+1 +1 λ βe t+1 t its expected future continuation value of the match rises. ρ p t+1. This means that the household accepts a lower reservation wage as long as If the real wage is rather set equal to the wholesale firms reservation wage, w t = w t, then the wholesale firms marginal value of an additional hiring of a worker is zero: F n,t = 0. Hence, in this case, equation 21 becomes w t h t = mc t aht α λt+1 + βρe t F n,t+1 32 Note that the wholesale firms reservation wage, w t, increases with the wholesale firms expected future continuation value of the match, βρe t λt+1 F n,t+1. This means that wholesale firms accept to raise their reservation wage as long as their expected future continuation value of the match increases. In turn, w t declines when the retailers real price markup, 1 mct, increases. Hence, as the real price of the wholesale intermediate goods decreases, wholesale firms are less willing to raise their reservation wage. The bargained real wage, w t, is then obtained by taking the average sum of the two reservation wages, the weights being given by the bargaining powers of both wholesale firms and the household. Indeed, combining equations 21 and 32 we get F n,t = h t w t w t. As well, combining equations 11 and 31, we get H n,t obtain = h t w t w t. Plugging the two previous equations into equation 30 we Note that by combining equations 2 and 20 with 30, we get w t = η w t + 1 ηw t 33 F n,t = κ γ m θ γ t 34 H n,t = η κ θt γ 35 1 η γ m By making use of equations 3, 31, 32, 34, and 35, we can restate the expression of the real wage, 33, as follows: 16

25 w t h t = ηmc t aht α + 1 ηψht ζ λt+1 + ηβe t κθ t+1 36 We can also compute a recursive expression for the surplus, S n,t. First note that S n,t = w t h t w t h t 37 Then, plug 31 and 32 into 37 and use the relations between the surplus, S n,t, and the marginal values of an additional employed worker, H n,t and F n,t : S n,t = mc t aht α ψht ζ λt+1 + βe t S n,t+1 [ρ η p t+1 ] 38 Recall that a fraction 1 η of the surplus goes to wholesale firms: F n,t = 1 ηs n,t. Combining the latter with equations 34 and 2, one gets κ γ m θ γ t = 1 ηs n,t 39 Thus, higher the size of the surplus, S n,t, is, higher the wholesale firms hiring rate, κ γm θ γ t, is too. In turn, the wholesale firms hiring rate depends positively on the labor market tightness, θ t. By making use of equations 38 and 39, one can get a recursive equation reflecting the dynamic of employment: κ [ ] θt γ = 1 η mc t aht α ψht ζ λt+1 κ + βe t θ γ t+1 γ m γ [ρ η p t+1] m 40 Hours of work negotiation Each period, the number of hours of work in the labor market is determined through a generalized Nash-bargaining process between wholesale firms and the household: The first order condition of the Nash-bargaining process is given by { h t argmax H n,t η F n,t 1 η} 41 η H n,t F n,t + 1 η F n,t H n,t = 0 h t h t where H n,t h t = w t ψζ ht ζ 1 and F n,t h t = αmc t aht α 1 w t. Using these two derivatives and equation 30, one can restate the first order condition of the hours of work Nash-bargaining problem as follows 17

26 h t = αmct a ψζ 1 ζ α 42 Hence, as noted by Trigari 2006, the Nash condition 42 resembles the standard equilibrium condition on labor in models with a Walrasian labor market and a monopolistically competitive intermediate goods market. In particular, the hours of work decrease with the real price markup, 1 mct. However, contrary to what happens in a frictionless labor market, here the real wage plays no role in the determination of the optimal condition on hours of work, 42. As stressed by Trigari 2006, under efficient bargaining, wholesale firms take the marginal disutility of supplying hours of work in terms of consumption goods as a measure of labor costs instead of the real wage. As a consequence, the optimal condition on hours of work is the same as in a Walrasian labor market even though the real wage is not allocational. Right to manage bargaining Here we consider the case where the hours of work are not negotiated but are rather chosen unilaterally by wholesale firms. Hence, the first order condition of the wholesale firms problem with respect to the hours of work, h t, is given by w t = αmc t a t h α 1 t 43 From equation 43 we can derive the demand function of hours of work h t = αmct a t w t 1 1 α hwt 44 Wholesale firms and the household bargain the real wage before the former set the hours as in 44. Nonetheless, when wholesale firms and the household bargain the real wage they take as given how the real wage affects the hours of work decision. Hence, the real wage bargaining problem can be expressed as follows { w t argmax H n,t η F n,t 1 η} subject to The first order condition of the Nash-bargaining process is given by η H n,t w t F n,t + 1 η F n,t w t H n,t = 0 18

27 where F n,t = mc t ahw t α +1 w t hw t + βρe t F n,t+1 H n,t = w t hw t ψhw t ζ + βe t H n,t+1 ρ p t+1 So H n,t w t = δ W t and F n,t w t = δ F t where δ W t = h t 1 α ψζ hζ 1 t w t α 46 δ F t = h t 47 δt W denotes the household s net marginal benefits from increasing the real wage and δt F represents the wholesale firms net marginal benefits from increasing the real wage. Hence the first order condition of the Nash-bargaining process can be restated as follows δ W t ηf n,t = δt F 1 η H n,t 48 As Trigari 2006 pointed out, condition 48 states that the real wage is set so that the household s net marginal benefit weighted by both its bargaining power and the firms marginal value of hiring an additional worker is equal to the firms net marginal benefit weighted by both its bargaining power and the household s marginal value of an additional employed worker. Let us define the relative weighted household s net marginal benefits from increasing the real wage as follows Γ t ηδ W t ηδ W t + 1 ηδ F t 49 Hence, the condition 48 can be restated as follows H n,t = H n,t = ηδ W t 1 ηδ F t F n,t Γ t 1 Γ t F n,t 50 Thus, the household s marginal value of an additional employed worker rises with its net marginal benefits of increasing the real wage. Using equation 50 and the definition of the surplus, S n,t, 19

28 one can show how the latter is shared between the households and firms under right to manage bargaining F n,t = 1 Γ t S n,t 51 H n,t = Γ t S n,t 52 As in the efficient bargaining case, we can also derive a real wage equation for the right to manage case. Recall that F n,t = h t w t w t and H n,t equation 50 we obtain = h t w t w t. Plugging the two previous equations into w t = Γ t w t + 1 Γ t w t 53 Using the expressions of the reservation wages 31 and 32, the optimal condition on real wage 50, the optimal condition 34, and the probability for a worker to find a job 3, the wage equation can be restated as follows w t h t = Γ t mc t aht α + 1 Γ t ψht ζ λt+1 1 Γ t Γ t+1 + Γ t βe t κθ t+1 Γ t 1 Γ t+1 λt+1 + Γ t βρe t κ θ γ t Γ t Γ t+1 γ m Γ t 1 Γ t The right to manage wage equation, 54, presents some differences with the efficient wage equation, 36. In equation 54 the real wage depends on a weighted sum of the wholesale firms revenues, household s desutility of supplying hours of work, and the expected future labor market tightness as in 36. However, the weights, Γ t, are time-varying under right to manage bargaining. As noted by Trigari 2006, the weights are not only given by the relative bargaining power, η, but also by the real wage relative allocational effect, measured by δt W and δt F. A dynamic equation for the surplus can also be derived under right to manage bargaining from equations 37, 31, 32, 51, and 52 S n,t = mc t aht α ψht ζ λt+1 + βe t S n,t+1 [ρ Γ t+1 p t+1 ] 55 Note that the right to manage dynamic equation for the surplus, 55, resembles the efficient dynamic equation for the surplus, 40. The main difference is that the household s future surplus from a match in case of separation conditional on finding a new job in the following period is weighted by the relative weighted household s net marginal benefits from increasing the real wage, Γ t, rather than 20