In a recent paper, Bick et al. (2022) show the presence of a hump-shaped relationship between hours and hourly wages with a maximum around 50 hours worked. We show that a model with fixed labor costs where workers and firms bargain in

wages and hours can help explain this non-linear relationship. Also, a quantitative version of the model is able to match the empirical hourly-wage to hours worked relationship estimated by those authors for the US

Del Rey, Elena

Naval, Joaquín

Silva, José I.

Munich RePEc Personal Archive

Munich Personal RePEc ArchiveHours and Wages: A BargainingApproachDel Rey, Elena and Naval, Joaquín and Silva, José I.11 March 2022Online at https://mpra.ub.uni-muenchen.de/112349/MPRA Paper No. 112349, posted 11 Mar 2022 11:14 UTCHours and Wages: A Bargaining Approach ✯Elena Del Rey1, Joaqúın Naval1, and José I. Silva21Universitat de Girona2Universitat de Girona and University of KentMarch 11, 2022AbstractIn a recent paper, Bick et al. (2022) show the presence of a hump-shaped rela-tionship between hours and hourly wages with a maximum around 50 hours worked.We show that a model with fixed labor costs where workers and firms bargain inwages and hours can help explain this non-linear relationship. Also, a quantitativeversion of the model is able to match the empirical hourly-wage to hours workedrelationship estimated by those authors for the US.JEL Classifications : C78, E24, J31.Keywords : Fixed labor costs, wage-hours relationship, bargaining.✯We would like to thank the financial support of the Ministerio de Ciencia e Innovación PID2020-113452RB-I00 (Naval and Silva) and PID2019-106642GB-I00 (Del Rey).Corresponding author José I. Silva, Universitat de Girona, Campus de Montilivi, C/Universitat 10,FCEE, 17003, Girona, Spain, email: jose.silva@udg.edu1 IntroductionMost of the empirical literature assumes a constant elasticity between earnings and hours(See, for example, Goldin (2014), Cortés and Pan (2019) and Denning et al. (2021)).The assumption of a constant elasticity, however, may hide important non-linearitiesin the data. In a recent paper, Bick et al. (2022) show that hourly wages in the USincrease for workers below 50 hours, but decrease when individuals work more hours.These authors show that this non-monotonic relationship is robust to different datasets,different individuals and it is happening in different countries like US and Denmark.To reproduce this relationship between wages and hours, Bick et al. (2022) assumeincreasing returns to hours worked when workers work short hours (below 40 hours perweek) and decreasing returns for greater numbers of hours worked. However, previousworks such as Pencavel (2015) find that returns are decreasing for more than 48 hoursper week but constant for shorter hours worked.The aim of this paper is to show that simultaneous bargaining over wages and hoursbetween workers and firms with fixed labor costs can also explain the aforementioned non-linear relationship between hours and hourly wages. In contrast to Bick et al. (2022), wedepart from the competitive market assumption where wages equal the marginal productof labor and assume decreasing returns in hours worked.Similar to Pissarides (2000) (Chapter 7.3) and Kudoh et al. (2019) among others, inour model workers and firms bargain over wage and hours. If an agreement is reached,firms incur a fixed training cost. If an agreement is not reached, workers remain joblessand suffer a fixed cost that can be interpreted as the cost of continuing the job search.Since search is costly in terms of time, this cost can be increasing in the value of leisure.Fixed costs reduce wages because, on the one hand, firms share part of the training costswith workers, and, on the other, worker search costs increase the value of reaching anagreement for a given wage.We account for worker heterogeneity in terms of preferences for leisure.1 Higherpreferences for leisure improve the outside option and the bargaining position of workers,and wages increase. Fixed costs, in contrast, reduce wages and, for sufficiently highpreferences for leisure (low hours), hourly fixed costs become too large and wages start tofall. This generates an inverted-U shape relationship between equilibrium hourly wagesand hours.We perform a simple numerical exercise and find that the model can reproduce thehump-shaped relationship observed in the data. We also show that removing either thefixed firm’s training costs or worker’s search cost from the model generate a negativerelationship between hourly wages and hours worked. Starting with Cogan (1981), fixedcosts have been introduced in several labor supply models (see, for example, French (2005)and Erosa et al. (2016)). However, we are the first to analyze the role of these type ofcosts to generate the hump-shaped relationship between hours and wages.1The value of the alternative use of time needs not only be interpreted in terms of leisure, it may alsobe motivated by the existence of home production. Having to perform these other activities makes workmore costly.22 The modelWe consider the simultaneous bargaining over wages and hours of work by firms andworkers at a given point in time. If an agreement is reached, workers obtainW = hw − xhµ (1)where h stand for hours, w for hourly wage and xhµ is the cost of working h hours withµ > 1 and x an idiosyncratic component of the cost that measures preferences for leisureand characterises workers. If an agreement is not reached, workers continue the job searchincurring a fixed cost F . Hence, upon reaching an agreement, workers obtain W plus F,the cost they do not incur. In turn, if an agreement is reached, the firm obtainsJ = ahλ − hw − T (2)where ahλ is the output generated in h hours of work, with λ < 1. T stands for a fixedhiring cost incurred by firms. It includes training and administrative costs of signing thecontract.Hours and wages are determined simultaneously by Nash bargaining:maxh,w(W + F )β (J)1−β .The first order condition that determines wages isβJ∂W∂w= −(1− β) (W + F )∂J∂w,since ∂J∂w= −∂W∂w= h, this impliesβJ = (1− β) (W + F ) (3)Substituting (1) and (2) in (3), and manipulating, we obtainw = βahλ−1 + (1− β)xhµ−1 −(1− β)F + βTh(4)Similarly, the first order condition that determines hours of work isβJ∂J∂h= −(1− β) (W + F )∂W∂h, (5)which can be simplified using (3) to obtain∂J∂h= −∂W∂h.From (1) and (2)∂W∂h= w − µxhµ−1∂J∂h= −(w − λahλ−1),Then, (5) impliesh∗x =(aλµx)1µ−λ. (6)3Substituting (6) into (4) we obtain the equilibrium wagew∗x = βa (h∗x)λ−1 + (1− β) (h∗x)µ−1 x−(1− β)F + βTh∗x(7)Hence, at equilibrium, each individual x agrees with the firm on a number of hours ofwork and a wage that are given by (6) and (7). The equilibrium hourly wage dependspositively on output produced and costs incurred by worker and negatively, on firms’scosts of hiring and worker’ search costs, i.e. fixed costs.We now study how hours of work and the hourly wage change with the individualpreferences for leisure x. Clearly, from (6):dh∗xdx= −1µ− λh∗xx< 0Higher preferences for leisure unambiguoulsy lead to lower hours of agreed work. Theeffect of individual preferences for leisure on hourly wages is more involved:dw∗xdx= βa(λ− 1) (h∗x)λ−2 dh∗xdx+ (1− β) (h∗x)µ−1+(1− β)x(µ− 1) (h∗x)µ−2 dh∗xdx+(1− β)F + βT(h∗x)2dh∗xdx(8)The first two terms are positive. First, higher preferences for leisure reduce hours of workand, since returns are decreasing, increase hourly output. This has a positive effect onthe hourly wage. Second, individuals demand a higher compensation if their preferencesfor leisure are higher. This also has a positive effect on the hourly wage.In contrast, the last two terms in (8) are negative: since higher preferences for leisurereduce hours of work, this reduces the compensation workers receive for x and increasesthe size of fixed costs per hour. The effect of these two terms on the hourly wage isnegative.Further manipulation of (8) allows us to identify the range of x for which this derivativeis positive or negative. In particular, we can show thatdw∗xdx> 0 ↔ x̂ = λ(aµ)µλ[(1− λ)(βµ+ (1− β)λ)(1− β)F + βT]µ−λλ> x. (9)Therefore, the equilibrium hourly wage first increases in the individual preference forleisure x and, after x̂, it decreases. While as working costs increase from zero workersare increasingly compensated for these costs, and hours fall, after a certain point the lowhours make working costs too small and fixed costs too large. Then wages start goingdown. In Fig. 1, we represent the equilibrium wage as a function of x (Quadrant II),the equilibrium hours as a function of x (Quadrant IV) and the resulting relationshipbetween w∗x and h∗x (Quadrant I).4Figure 1: The relationship between wages and hours at the bargaining equilibrium3 Numerical exerciseWe have seen that a bargaining model with fixed labor costs is able to generate aninverted U-shape relationship between hours worked and hourly wages. In this sectionwe set the model’s parameters to be consistent with reasonable labor costs. Table 1 showsthe obtained parameter values. The goal is to see if a quantitative version of the modelis able to match the empirical hourly-wage relationship estimated by Bick et al. (2022)for the US.We start by setting the workers bargaining power parameter to β = 0.5 and assumethe presence of quadratic costs of working µ = 2. We parameterize the model in annualbasis and consider an individual working 40 weekly hours when setting the differentparameters. Thus, we normalize to one both her weekly hours worked (h40 = 1) as wellas her annual wage (wa40= w40×h40×52=1). We also assume that there are 300 differentx ranged between 0.001 and 0.025.Following Silva and Toledo (2009), the firm’s on-the-job training fixed costs T = 0.14are set to match 14% of annual wages (55% of quarterly wages). To match the data, itis convenient to assume that worker search costs F are linearly increasing in the value ofleisure: F (x) = χ× x. That is, individuals with higher working costs per hour also showhigher fixed costs of looking for jobs since leisure is relatively more important for them.From a theoretical point of view, note that this assumption adds an additional negativeterm to (8) but does not change the main result on the hump-shaped relationship betweenhourly-wages and hours worked.5Figure 2: Hourly-wage relationshipNext, χ = 2 and λ = 0.06 are set to match the log of wages observed in the data for 30and 65 weekly hours worked, respectively. Notice that this implies that the fixed costs oflooking for a job F (x) range between 0.002 and 0.05 for individuals with the lowest andhighest working costs x, respectively. In the case of a worker working 40 hours per week,we obtain an annual job searching cost of 1% of her wage (F (x40) = 2 × 0.005 = 0.01).Finally, the productivity parameter a = 0.1682 can be obtained by substituting ourobtained parameters in the normalized annual wage equation for an individual working40 weekly hours. That is, we multiply 52 by equation (4) and obtain:a = (1/52 + β × T + (1− β)× F40)/(β + (1− β)× λ/µ).Finally, we keep the parameters unchanged and depart from our benchmark 40 weeklyhours scenario by calculating the optimal hourly wage w∗x and weekly hours worked h∗xfor each x by using equations (6)-(7).Table 1: Calculated parameter valuesParameter Value RationaleParameter of worker’s job searching costs, χ 2 Log wages at 30 hours workedFixed firm’s cost of training a worker, T 0.14 Silva and Toledo (2009)Worker’s bargaining power, β 0.5 Own assumptionParameter of working variable cost, µ 2 Quadratic costsParameter of the production function, λ 0.06 Log wages at 65 hours workedAggregate productivity, a 0.1682 Wage equation6Figure 3: Hourly-wage relationship with non Fixed costsFigure 2 compares the log hourly wage of our simulated scenarios with the estimateddata in Bick et al. (2022). It matches quite well the inverted U-shape relationship betweenhours and wages. However, it reaches the maximum log wages around 45 hours insteadof 50. To show the importance of having fixed costs in the model, we next simulatetwo alternative scenarios where we eliminate either the training costs T = 0 or the jobsearching costs F = 0. The rest of parameters remain unchanged. Figures 3 (a)-(b) showa monotonically decreasing relationship between hours and wages in both cases withinthe range of 30 to 65 hours.4 Final commentIntroducing fixed labor costs in a scenario of simultaneous bargaining in hours and wages,we have created a non-linear relationship between hours and hourly wages. Further, wehave been able to match the empirical hourly-wage relationship estimated by Bick et al.(2022). This paper suggests that further empirical research should evaluate the role offixed costs behind the inverted-U relationship between hourly wages and hours worked.7ReferencesBick, Alexander, Adam Blandin, and Richard Rogerson, “Hours and Wages,”The Quarterly Journal of Economics, 01 2022. qjac005.Cogan, John F., “Fixed Costs and Labor Supply,” Econometrica, 1981, 49 (4), 945–963.Cortés, Patricia and Jessica Pan, “When Time Binds: Substitutes for HouseholdProduction, Returns to Working Long Hours, and the Skilled Gender Wage Gap,”Journal of Labor Economics, 2019, 37 (2), 351–398.Denning, Jeffrey T., Brian A. Jacob, Lars Lefgren, and Christian vom Lehn,“The Return to Hours Worked within and across Occupations: Implications for theGender Wage Gap,” IZA Discussion Papers 14325, Institute of Labor Economics (IZA)April 2021.Erosa, Andrés, Luisa Fuster, and Gueorgui Kambourov, “Towards a Micro-Founded Theory of Aggregate Labour Supply,” Review of Economic Studies, 2016,83 (3), 1001–1039.French, Eric, “The Effects of Health, Wealth, and Wages on Labour Supply and Re-tirement Behaviour,” Review of Economic Studies, 2005, 72 (2), 395–427.Goldin, Claudia, “A Grand Gender Convergence: Its Last Chapter,” American Eco-nomic Review, April 2014, 104 (4), 1091–1119.Kudoh, Noritaka, Hiroaki Miyamoto, and Masaru Sasaki, “Employment andhours over the business cycle in a model with search frictions,” Review of EconomicDynamics, 2019, 31, 436–461.Pencavel, John, “The Productivity of Working Hours,” The Economic Journal, 2015,125 (589), 2052–2076.Pissarides, Christopher A., Equilibrium Unemployment Theory, 2nd Edition, Vol. 1of MIT Press Books, The MIT Press, December 2000.Silva, José Ignacio and Manuel Toledo, “Labor Turnover Costs and the CyclicalBehavior of Vacancies and Unemployment,” Macroeconomic Dynamics, May 2009, 13(S1), 76–96.8

Hours and Wages: A Bargaining Approach

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Hours and Wages: A Bargaining Approach

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