Listening to workers: The overtime versus hiring dilemma

Journal of Business Research 66 (2013) 1771–1779

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Journal of Business Research

Listening to workers: The overtime versus hiring dilemma☆ Marcos Singer ⁎, Juan José Obach Pontificia Universidad Católica de Chile, Chile

a r t i c l e

i n f o

Article history: Received 1 August 2012 Received in revised form 1 December 2012 Accepted 1 January 2013 Available online 12 February 2013 Keywords: Overtime hours Headcount Information asymmetry

a b s t r a c t To reach a certain production level, firms sometimes allow overtime and/or adjust the number of their personnel. Under some circumstances, workers can decide to work overtime to gain additional compensation, even though the firm might not need that time. This type of overtime exists because of an information asymmetry that favors workers: they know better than management the everyday routines, the temporary bottlenecks, and the malfunctions in the workplace. This study models this situation as an infinitely repeated game. In each stage-game the workers decide whether to work overtime, and the firm decides whether to adjust the amount of personnel. The game characterizes the conditions of the Nash equilibriums, some of which might lead to collaborative communication between the workers and the firm. The study empirically tests two propositions with data from a Chilean smelting plant. The results identify under which circumstances the firm should “listen” to the workers (i.e., take into account how much overtime they incur) when making personnel decisions. © 2013 Elsevier Inc. All rights reserved.

1. Introduction To reach a certain production level, the firm must decide the average working hours of its employees and the amount of personnel needed (Holt, Modigliani, & Simon, 1955; Rosen, 1968). If the hours exceed a standard working day, the firm must pay an overtime charge. In many Latin American countries, firms pay a surcharge of 50% to 100% for overtime during weekends. In the United States, the surcharge is 50% for any hour that exceeds 40 h of work within seven days. In Japan, the surcharge is 25% above the agreed-upon wage for an ordinary working day. Extensive research exists about finding the right combination between overtime and personnel adjustments. Overtime might cause the exhaustion of the workers (Lester, 1939) and a negative feeling toward the firm (Babbar & Aspelin, 1998). That is why, according to Stamas (1979), the overtime surcharge is a type of reparation for the workers. In contrast, Hart and Ma (2010) find that incorporating overtime helps to improve contract efficiency, and therefore, both employers and employees prefer the overtime. Accordingly, Trejo (1993) and Bell and Hart (1998) empirically show that a reduction in contractual hours often translates into overtime increments, suggesting that overtime is a useful alternative when available. Adjusting personnel poses at least two disadvantages for the firm. First, fixed hiring and layoff costs exist; and second, once hired the workers receive a full wage, even if the workload decreases. Therefore, a number of issues require firms to adjust the number of personnel. ☆ We thank two anonymous reviewers and a co-editor of this special issue for helpful comments. ⁎ Corresponding author at: Pontificia Universidad Católica de Chile, Escuela de Administración, Vicuña Mackenna 4860, Macul, Santiago, Chile. E-mail addresses: [email protected] (M. Singer), [email protected] (J.J. Obach). 0148-2963/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jbusres.2013.01.009

Bentolila and Bertola (1990) propose a dynamic model of optimal employment under linear adjustment costs. Goux, Maurin, and Pauchet (2001) study the difference between indefinite-term contracts (ITC) and fixed-term contracts (FTC) when adjusting personnel. The authors also consider the asymmetry between hiring and layoff costs. Abowd and Kramarz (2003) propose an optimization model that considers past hiring costs, past training costs (both of which are sunken), termination costs, total compensation, and marginal productivity. Addison, Bellmann, Schank, and Teixeira (2008) distinguish between highly skilled, skilled, and unskilled workers to estimate a personnel policy. In summary, a robust line of research exists that explores what overtime policies and personnel adjustments can optimize the firm's production. Unfortunately, in complex systems, managers have difficulty in detecting the actual workload needed. Many tasks do not correlate with sales or other observable indicators. Only workers and their direct supervisors, who are usually senior workers, have enough information about the work to be done, and therefore, the workers themselves define the need for overtime. Management is left with the limitation of observing overtime and then making decisions on whether to hire additional personnel. Only occasionally are managers able to audit whether or not the overtime is in the best interest of the firm. Such an information asymmetry might harm the firm, because supervisors and workers can conspire to work more hours than necessary. In the face of this moral hazard problem, management might distrust the overtime as a credible signal for the need of more hours. Then, if a workload peak indeed occurs, the firm will find its personnel below the optimal level and, as a result, the firm might lose its competitiveness. The available literature does not consider the strategic point of view of the overtime versus hiring personnel dilemma. To fill this gap, this study proposes an infinitely repeated game. In each stage

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of the game, a team of workers has the possibility of working overtime. The firm observes any overtime and then decides whether or not to hire an additional worker for the next period. Once the amount of personnel increases, the firm reduces the maximum overtime in the next period. By equilibrium analysis, the study finds that under certain conditions the firm might adopt two extreme strategies: removing or permanently keeping all of the additional personnel. To depart from the first strategy is rather difficult: the firm must increase the punishment to deceitful workers or increase wages above the labor productivity. To depart from the second strategy is also difficult: the additional payroll must be very productive or the firm must reduce wages. The study also finds conditions that align worker incentives to the firm's objectives. In this case, observed overtime becomes a credible signal of the workload that the firm should “listen”. Such conditions prescribe that the firm should reduce the hiring cost when a high probability of an extra workload exists and should reduce layoff costs when a low probability of an excessive workload exists. The organization of the paper is as follows. Section 2 describes a stage-game that models the interaction between a team of workers and the firm within one month and then extends this stage-game as an infinite repetition. Section 3, by means of a number of propositions and their proofs, describes the different strategies and conditions for collaborative and non-collaborative equilibriums. Section 4 empirically tests two of these propositions. Section 5 summarizes the results. 2. Overtime versus hiring as a repeated game 2.1. The team-firm relationship as a stage-game (within one month) This study models the interaction between a team of workers and a firm during one month as a dynamic stage-game with incomplete information. In the stage-game, events happen according to the following logic (Harsanyi, 1967). First, fate chooses how much of an excessive workload the firm faces; only the team has this information. The assumption is that this workload can be either high or low; the extra workload is high with probability θ and is low with probability (1− θ). The value of θ is common knowledge for the managers and the team because θ depends on sales and other public information. Nevertheless, factors exist, such as temporary bottlenecks or malfunctions, where the team has exclusive knowledge. Therefore, only the team knows how much extra workload the firm really has. Next in the stage-game, the team makes a decision on whether to work overtime or not. If a high workload exists, then overtime is necessary. If a low workload exists, then overtime is unnecessary. Thus, the team that works the overtime is deceiving the firm with

Fate

Work Team

Incur overtime high extra workload

θ

Firm

4

2 Not incur overtime

the purpose of increasing the team's income. The firm only observes whether workers take overtime or not, and then the firm decides whether to keep additional personnel or not, thus setting the initial conditions for the next stage-game. Fig. 1 shows a stage-game that has the following order of events: θ is realized, the team works overtime or not, and the firm retains support personnel or not. If a team works overtime, then the firm does not know whether the overtime corresponds to node 4 or node 6; if the team does not work overtime, the firm does not know whether the overtime corresponds to nodes 5 or 7. Therefore, imperfect information exists because the firm does not know in which branch of the tree the firm's decision lies. For simplicity, from now on the assumption is that each stagegame spans one month. The study also assumes that the extra workload can take two values only: 16 and 0. Overtime can be 16 or 8 h, depending on whether an extra worker is available or not. The firm pays a surcharge of 50% above the regular wage for the overtime. A team comprises eight workers, and a working day consists of 8 h per worker. The study defines the term s as the total regular wage per month that a team of eight workers receives. The payment per month is 1.5 ⋅ (2/8) for a team that earns overtime in the months in which the support personnel is zero. In these months each member of the team works 2 h of overtime per day. The payment per month is 1.5 s ⋅ (1/8) for a team that earns overtime in the months in which one support person is available; that is, months in which one additional worker joins the team. During these months, the additional person works 8 out of the 16 overtime hours and each of the 8 members of the team takes the other 8 h. As workers extend their working day, they are substituting leisure hours for work that entails a cost (Babbar & Aspelin, 1998). The study defines the terms for cost as c, the cost per month for a team if each member works 2 h per day of required overtime (when the extra workload is high). If each of the eight members work 1 h of the required overtime, then the cost is c/2; if each member works 2 h per day of unnecessary overtime, then the cost per month for a team is b. If each of the eight members works 1 h of unnecessary overtime, then the cost is b/2. Since the required overtime hours involve greater effort, c > b. The firm imposes punishment on a team if the firm detects deceitful overtime. This detection is possible only if the firm does not keep support personnel. In contrast, if the firm increases its personnel, the firm refrains from inquiring whether the overtime is unnecessary to avoid the awkward suggestion that workers are idle when the firm is hiring new personnel. No certainty exists that the firm detects deception, and so punishment only occurs with a given probability. The study defines r as the expected value of the punishment imposed on the team if the firm detects that the team is deceiving the firm with unnecessary overtime. The punishment is

Support Personnel in the Next Period

Hold support personnel

1

Not hold support personnel

0

Hold support personnel

5

Not hold support personnel

1 Incur overtime

1–θ low extra workload

6

7

0

Hold support personnel

1

Not hold support personnel

0

Hold support personnel

1

3 Not incur overtime

1

Not hold support personnel

Fig. 1. Relationship between the firm and a work team within a month.

0

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Table 1 Payoffs for scenarios without initial support personnel.

High workload

Team works overtime Team does not work overtime

Low workload

Team works overtime Team does not work overtime

Firm keeps support personnel in next period

Firm does not keep support personnel in next period

Team: 1.5 s/4 − c + ρw Team: ρw Team: 1.5 s/4 − b+ ρw Team: ρw

Team: 1.5 s/4 − c + ρw Team: ρw Team: 1.5 s/4 − b− r + ρw Team: ρw

not transferrable to the firm (as with a fine); the punishment is a cost as in the case of a verbal reprimand. Recall that the firm can keep support personnel or not for the next period. The attitude toward keeping personnel translates into certain actions. In months that begin without support personnel, the firm hires one worker; in months that begin with one support worker, the firm retains the worker for the following month. The action of hiring or laying off a worker entails costs. The definitions of the costs are the terms h for the cost of hiring one additional worker and d for the cost of laying off a worker. If an extra workload indeed exists, and the firm has the proper manpower, then the production increases. The following terms are definitions for the production conditions. The term p is the increase in the monthly production due to a high workload that needs eight additional hours per day. One hour of overtime per day per team member, or a regular working day of one support worker can supply such hours. The term 1.5p is the increase in monthly production due to a high workload that requires 16 h of overtime per day. The production increase due to the second 8 h is only 0.5p because of exhaustion (Hutchins and Harrison, 1926). In such a scenario, if the firm decides to hire a support worker for the next period, the worker recovers the 0.5p loss. The term 2p is the increase in monthly production due to an extra workload that requires 16 additional hours per day. One hour per day of overtime per team member provides the first 8 h, and one regular working day of one support worker supplies the other 8 h. These parameters are the labor and contractual conditions represented by B = (θ, s, c, b, r, p, h, d). 2.2. Long-term relationship modeled as a repeated game The study models the long-term relationship between a team and the firm as an infinite sequence of consecutive stage-games. The following terms define the relationship. The term w0 is the expected payoff for the work team from playing an infinite repetition of the stage-game without initial support personnel. The term w1 is the expected payoff for the work team from playing an infinite repetition of the stage-game with one initial support person. The term f0 is the expected payoff for the firm from playing an infinite repetition of the stage-game without the initial support personnel. The term f1 is the expected payoff for the firm from playing an infinite repetition of the stage-game with one initial support person. The term ρ is the monthly discount factor for the work team, with 0 b ρ b 1. And μ is

Firm: 1.5p− 1.5 s/4 − h + μ(0.5p) + μf Firm: −h + μ(0.5p) + μf Firm: −1.5 s/4 − h + μf Firm: -h + μf

Firm: 1.5p− 1.5 s/4 + μf Firm: μf Firm: −1.5 s/4 + μf Firm: μf

the monthly discount factor for the firm, with 0 b μ b 1. The parameters ρ and μ represent how much both players value future payoffs. The low values indicate that the players strongly prefer immediate rather than remote benefits. Table 1 shows the payoffs of a game without initial support personnel, and Table 2 exhibits the payoffs with initial support personnel. For each scenario, the corresponding discount factor weighs the expected payoff in the next period. For example, if a high workload exists, then the team works overtime and the firm keeps support personnel. In this scenario, the firm earns 1.5p − 1.5 s / 4 − h + μ(0.5p) + μf . This expression can be understood as follows. The production increment is 1.5p because the overtime can handle all of the workload. The payment for the team is 1.5 s/4 and a hiring cost h exists. The productivity loss of 0.5p is recovered in the next month so it is discounted by μ. Also in the next period, the firm earns f1 that is discounted by μ. 2.3. Repeated game strategies and payoffs The study defines the following strategies for each work team: Honest occurs when the team works overtime only if an extra workload exists. The Dodger occurs when the team does not work overtime if an extra workload exists and works overtime if no extra workload exists. The Workaholic occurs when the team works overtime regardless of the extra workload. And Lazy occurs when the team does not work overtime under any circumstances. Apart from these strategies, a team could randomize the choice. This paper uses the following definitions that are similar to Singer, Donoso, and Konstantinidis (2009) for this set of decision variables. The term α is the probability that the team works overtime when an extra workload exists. The expression 1 − α is the probability that the team does not work overtime when an extra workload exists. The term β is the probability that the team works overtime when a low extra workload exists. The expression 1 − β is the probability that the team does not work overtime when no extra workload exists. For example, if α equals 0.7 and β equals 0.4; then with a high workload the team works overtime with a probability of 70%, and with little workload the team takes overtime with a probability of 40%. Depending upon the actions of the work team, that is, whether the team works overtime or not; the firm has the following strategies. If the firm is Trustful, then the firm hires or keeps a support worker only if the team works overtime. If the firm is Distrustful, then the

Table 2 Payoffs for scenarios with initial support personnel.

High workload

Team works overtime Team does not work overtime

Low workload

Team works overtime Team does not work overtime

Firm keeps support personnel in next period

Firm does not keep support personnel in next period

Team: (1.5 s/8) − c/2 + ρw Team: ρw Team: (1.5 s/8) − b/2 + ρw Team: ρw

Team: (1.5 s/8) − c/2 + ρw Team: ρw Team: (1.5 s/8) − b/ 2 − r + ρw Team: ρw

Firm: 2p − (1.5 s/8) − s/8 + μf Firm: p − s/8 + μf Firm: −s/8 − (1.5 s/8) + μf Firm: −s/8 + μf

Firm: 2p − (1.5 s/8) − s/8 − d + μf Firm: p− s/8 − d + μf Firm: −(1.5 s/8) − s/8 − d + μf Firm: −s/8 − d + μf

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firm refrains from hiring or keeping a support worker when the team works overtime, but the firm hires a support worker when a team does not work overtime. The firm can be Expansive if the firm hires or keeps the support worker regardless of whether the team works overtime or not. The firm is Restrictive when the firm does not hire or keep support personnel. The firm could again randomize the decision to hire or keep the support worker. Thus, the study uses the following definitions for this set of decision variables. The term γ is the probability that the firm hires or keeps a support worker when the team works overtime. The expression 1 − γ is the probability that the firm does not hire or keep the support worker when the team works overtime. The term δ is the probability that the firm hires or keeps the support worker when the team does not work overtime. The expression 1 − δ is the probability that the firm does not hire or keep the support worker when the team does not work overtime. Fig. 2 shows the strategy spaces for each player. The aforementioned strategies are called corner strategies. Players can choose any of these strategies for a stage-game, which vector S = (α, β, γ, δ) can represent. In what follows, the assumption is that S is the same for all stage-games. This assumption implies that the team and the firm choose their strategies simultaneously at the beginning of the first stage-game and keep the strategies invariable in the future (Baron & Bezanko, 1984; Fudenberg & Tirole, 1992, pp. 299–301). This invariance expresses the sequential game as a static game that one can solve using the Nash equilibrium criteria. The validity of the assumption might be in question because of new information that might become available over time that one could use to revise past decisions. Nevertheless, the decisions of a team are influenced by group practices and culture that do not change in the short run. As for firms, they do not revise their human resource policies often because of the costs of gathering and processing new information (Lundgren & Schneider, 1971). w0 is the expected payoff of a team for playing an infinite game without initial support personnel. In Table 1, w0 is the aggregation of the following payoffs. 1. There is a high workload, the team works overtime, and the firm hires or keeps additional personnel: θ ⋅ α ⋅ γ ⋅ (1.5 s / 4 − c + ρw ). 2. There is a high workload, the team works overtime, and the firm does not hire or keep additional personnel: θ ⋅ α ⋅ (1 − γ) ⋅ (1.5 s / 4 − c + ρw0). 3. There is a high workload, the team does not work overtime, and the firm hires or keeps additional personnel: θ ⋅ (1 − α) ⋅ δ ⋅ (ρw ). 4. There is a high workload, the team does not work overtime, and the firm does not hire or keep additional personnel: θ ⋅ (1 − α) − (1 − δ) ⋅ (ρw0). 5. There is a low workload, the team works overtime, and the firm hires or keeps additional personnel: (1− θ) ⋅ β ⋅ γ ⋅ (1.5 s / 4 − c + ρw ).

6. There is a low workload, the team works overtime, and the firm does not hire or keep additional personnel: (1− θ) ⋅ β ⋅ (1− γ) ⋅ (1.5 s/4 − b− r + ρw0). 7. There is a low workload, the team does not work overtime, and the firm hires or keeps additional personnel: (1 − θ) ⋅ (1 − β) ⋅ δ ⋅ (ρw ). 8. There is a low workload, the team does not work overtime, and the firm does not hire or keep additional personnel: (1 − θ) ⋅ (1 − β) ⋅ (1 − δ) ⋅ (ρw0). These eight payoffs generate an equation for w0 that depends on labor and contractual conditions B, the discount factor ρ, and the strategy vector S (see Appendix A). The expected payoff w0 recursively depends on w0 and w1 (both discounted by ρ), that is, w0 = w0(B, w0, w1, ρ, S). Analogously, Table 2 garners a recursive equation for w1 = w1(B, w0, w1, ρ, S). Combining the equations for w0 and w1 creates the value of w0(B, ρ, S) and w1(B, ρ, S), that is, w0 and w1 are functions of B, ρ , and S only. Analogously, the calculations for f0 and f1 from Table 1 are f0(B, f0, f1, μ, S) and from Table 2 f1(B, f0, f1, μ, S). Combining both equations creates the values of f0(B, μ, S) and f1(B, μ, S).

3. Player strategies in equilibrium 3.1. Propositions for the players' behavior Proposition 1 states that in equilibrium the decision of a team to work overtime is not arbitrary where arbitrary means that the possibility of working overtime is equally likely to occur, regardless of the workload. The proof is via contradiction. If the decision of whether to work overtime does not depend on the workload, that is, α = β = 0.5, then the necessary condition for a Nash equilibrium is that a team does not have unilateral incentives to deviate. If the repeated game starts without support personnel, the following conditions should exist. First, ∂w0(B, ρ, S)/∂α evaluated at S* = (0.5, 0.5, 0.5, 0.5) equals zero. And, second, ∂w0(B, ρ, S)/∂β evaluated at S* = (0.5, 0.5, 0.5, 0.5) equals zero. The first condition shows that θ = (1 − ρ) / (0.25ρ(ρ − 2)) is negative for any value of ρ. If the repeated game starts with support personnel, then the result is similar. Proposition 2 states that if a team is a dodger and support workers never exist, then the expected payoff for the firm is always negative. If a team is a dodger and one support worker always exists, then the firm earns a profit only if its productivity in overtime is above a certain threshold. The proof is the following. If support workers never exist, then the firm is restrictive. Given that, if a team is a dodger, S* = (0, 1, 0, 0). The game begins without workers, so the expected payoff for the firm is f (B, ρ, S*) = (6 s(θ − 1)) / (16(1 − μ)), which is always negative. If one support worker always exists, then the firm is expansive. Given that, if the team is a dodger, S* = (0, 1, 1, 1). The game begins with one support worker, so the expected payoff for

Fig. 2. Player strategy spaces.

M. Singer, J.J. Obach / Journal of Business Research 66 (2013) 1771–1779

the firm is f1(B, ρ, S*) = (16pθ − s(5 − 3θ)) / (16(1 − μ)) that is positive only if p > (5 s − 3sθ) / (16θ). 3.2. Propositions for non-collaborative game equilibriums Proposition 3 states that if the expected value of the punishment that the firm imposes against a deceitful team is higher than a threshold, then the outcome (workaholic, restrictive) is never an equilibrium. The proof is that a required condition for this outcome (workaholic, restrictive) to be an equilibrium is that ∂w0 (B, ρ, S) / ∂β evaluated at S* = (1, 1, 0, 0) must be nonnegative. Solving the inequality for r creates r ≤ 3 s/8-b. The expression 3s / 8 − b corresponds to the utility that a team obtains by working two daily hours of unnecessary overtime. If the expected punishment r is higher than this value, the outcome (workaholic, restrictive) is not an equilibrium. Proposition 4 states that if the production increment because of overtime is high enough, then a high wage makes the outcome (workaholic, restrictive) unlikely to be an equilibrium. The proof is that if the outcome (workaholic, restrictive) is an equilibrium, then ∂f0 (B, ρ, S)/∂γ evaluated at S*=(1, 1, 0, 0) must be non-positive. Solving the inequality for p creates p ≤(1/(16θμ))(16 h + 16dμ − sμ). Taking the derivative at the right-hand side of the inequality with respect to s creates a negative value: −1 /(16θ). This result implies that as the wage grows, the inequality becomes more restrictive and therefore less likely to hold. Proposition 5 states that if the firm decides to keep a support worker, then the firm can avoid workers systematically working overtime, which is the outcome (workaholic, expansive), by setting the wage s such that s b (8/3)c. The proof is that the following conditions must hold for the outcome (workaholic, expansive) to be an equilibrium: ∂w0 (B, ρ, S)/∂α evaluated at S* = (1, 1, 1, 1), ∂w0 (B, ρ, S)/∂β evaluated at S* = (1, 1, 1, 1), and ∂f0 (B, ρ, S)/∂γ evaluated at S*= (1, 1, 1, 1) must all be nonnegative. The value s from the first two conditions is s≥(8/3)c and s≥(8/3)b. As c>b, the second condition is redundant. From the third condition, θ≥(16 h(1−2 μ)−16dμ−sμ)/(16pμ(1−μ)). Assuming a discount factor μ>1/2 (which means that $2 next month are preferable to $1 now), the right-hand side of the inequality is always negative, and so the condition is also redundant. Therefore, if the firm does not want to converge to the outcome (workaholic, expansive), s can be set such that sb (8/3)c. 3.3. Propositions for collaborative equilibriums Proposition 6 states that if the cost for a team of the necessary overtime is higher than the payment the team receives for such overtime (at 150% rate), then a higher value of θ makes the outcome (honest, trustful) less probable for being an equilibrium. The proof is that the outcome (honest, trustful) is equivalent to S*=(1, 0, 1, 0). A condition required for such an outcome to be an equilibrium is that ∂w0 (B, ρ, S)/∂β evaluated at S*=(1, 0, 1, 0) is non-positive. This condition requires b≥(3 s+θρ(4c−3 s))/(8−4θρ). Taking the derivative of the righthand side of the inequality (that is, the lower limit of b) with respect to θ generates (1/4)⋅(ρ(8c−3 s)/(θρ−2)2). If c is higher than 3 s/8 (the 150% payment for 2 h), such an expression is positive. A higher value of θ indicates a higher value for the expression, and so the range of b for which the outcome (honest, trustful) is an equilibrium decreases. If c is lower than 3 s/8, the expression (1/4)∙(ρ(8c−3 s)/(θρ−2)2) is negative, so the effect of θ is negative. Proposition 7 states that if the hiring costs are high, then the outcome (honest, trustful) is less likely to be an equilibrium during high-workload seasons than in low-workload seasons. The term season means a number of months with the same θ. The proof is that the necessary condition for the outcome (honest, trustful) to be an equilibrium is that ∂f0 (B, ρ, S)/∂γ evaluated at S* = (1, 0, 1, 0) is nonnegative. This value requires h ≤ (8pμ − 16dμ − 2sμ + 32dθμ + 8pθμ + 3sθμ − 16pθμ 2) / (16 − 32θμ). Taking the derivative of the right-hand side of the inequality with respect to θ generates

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(1/ 16)(μ / ((2θμ − 1)2))(8p + s(3− 4 μ) + 32d(1− μ)). Assuming that μ > 3/4 (which means that $4 next month are preferable to $3 now) and that h is close enough to its upper limit, an increment in θ can cause the value of h to be both higher than its upper limit and exceed its feasible range. Therefore, for high values of h, a higher value of θ reduces the probability that the outcome (honest, trustful) is an equilibrium. Proposition 8 states that, if θ b 1/2, then an increase in the layoff costs makes the outcome (honest, trustful) less likely to be an equilibrium. The proof is that the necessary conditions for the outcome (honest, trustful) to be an equilibrium are the following: ∂f0 (B, ρ, S)/∂γ evaluated at S* = (1, 0, 1, 0) must be nonnegative, and ∂f0 (B, ρ, S)/∂δ evaluated at S*=(1, 0, 1, 0) must be non-positive. Solving these two inequalities for p generates two expression: p≥((16 h(1−2θμ)+ 16dμ(1−2θ)+sμ(2−3θ))/(8 μ+8θμ−16θμ2)) and p≥((16 h(1− 2θμ)+16dμ(1−2θ)+sμ(2−3θ))/(8θμ(1−μ)). Given that 1≥θμ, the second expression has the higher lower limit for p. Taking the derivative to the right-hand side of this expression with respect to d generates (2(1−2θ))/(θ(1−μ)). If θb 1/2, then the lower limit of p increases as d increases. Thus, this condition becomes more restrictive. 4. Empirical tests for two propositions This section empirically evaluates two of the propositions in Section 3 with data from a copper smelting and refining plant in Chile. The database consists of monthly information for the year 2008, with 31 of the teams working on the production line and 63 in charge of support activities such as maintenance or quality control. The data available comprise production, activities, personnel, contracted hours, overtime hours, absenteeism hours, wages, and worker seniority. According to Proposition 4, if the increment in production p because of overtime is important, then a higher wage makes the outcome (workaholic, restrictive) less likely to be an equilibrium. Fig. 3 shows the average overtime rate, measured as the number of hours after the number of regular working hours, for the 31 teams that work on the production line. The assumption is that these teams have a higher p than the 63 work teams in charge of support activities. The basis for this assumption is the direct effect that any additional work time has on the production line as compared to a support activity that only has an indirect effect on production. The figure also shows the personnel coefficient of variation (standard deviation over average) for each team over the course of 12 months, which is a measure of how much the personnel adapts to the workload (in general, the personnel correlates by 0.40 to the workload). If the outcome (workaholic, restrictive) was the equilibrium, the personnel coefficient of variation would be close to zero and the overtime rate would be high. In contrast, the figure shows that as the average wage increases, the personnel coefficient of variation increases significantly and that the overtime rate decreases (and therefore the unnecessary overtime rate, as θ is unbiased). Regressions 1 and 2 in Table 3 confirm these tendencies. For example, if the average wage in a team increases from Ch$500.000 (US$962) to Ch$600.000 (US$1,154), then the overtime rate decreases from 10% to 7.4% and the personnel coefficient of variation increases from 12% to 19%. Fig. 4 shows the average overtime rate and the personnel coefficient of variation for the 63 supporting teams. Here, the assumption is that the teams have a low p. As the average wage increases, no trend is observable in the personnel coefficient of variation, which the regressions 3 and 4 in Table 3 confirm. These results suggest that for high productivity teams, higher salaries imply a higher probability of a non-cooperative equilibrium, as Proposition 4 states. On the other hand, low productivity teams do not appear to have any effect in the observed type of equilibrium. As mentioned earlier, Proposition 8 states that if θ b 1/2, then higher layoff costs make the outcome (honest, trustful) less likely to be an equilibrium. Following a comparative statics analysis, this

0,6

35%

0,5

30% 25%

0,4

20% 0,3 15% 0,2

10%

0,1

Overtime Rate

M. Singer, J.J. Obach / Journal of Business Research 66 (2013) 1771–1779

Headcount Coefficient of Variation

1776

5%

0,0

0% $ 460

$ 531

$ 570

$ 594

$ 600

$ 623

$ 704

$ 1.047

Average Wage (US$) Headcount Coefficient of Variation Tendency of Headcount C. of V.

Overtime Rate Tendency of Overtime R.

Fig. 3. Wages for production line teams (high hourly overtime productivity).

study calculates the monthly average production in order to associate θ > 1/2 with months that have higher than average production, and vice versa. This criterion is valid for teams on the production line only. The study approximates the layoff costs by using the average seniority of a work team, because Chilean law mandates a dismissal payment of one month of wages for each year under contract. Fig. 5 shows the personnel coefficient of variation and the average overtime rate for months with θ b 1/2 for the 31 teams that work on the production line. As seniority increases, the personnel coefficient of variation decreases, while the overtime rate remains constant. Such trends, according to the regressions in Table 4, show that in low-seniority teams, the variation in personnel increases. The trustful strategy implies that the personnel varies depending on observed overtime. Therefore, when θ b 1/2, high layoff costs make the outcome (honest, trustful) less likely to be an equilibrium. This result does not occur when θ > 1/2. These results empirically support Proposition 8. When the layoff cost is high and the work load is low, the cooperative equilibrium is less probable. In other words, when business is slow, if the firm faces high layoff costs (because of unionization, for instance); the firm might not want to reduce its head count, making the cooperative equilibrium improbable. The above analysis makes an additional assumption that workers and managers have common knowledge about the value of θ when this probability is high and when it is low. This common knowledge can occur in industries with seasonality such as Coca-Cola, where workers and managers well know that θ is around 0.7 during the summer and around 0.4 during the winter. Also, the theoretical threshold is 1/2 due to a number of assumptions and simplifications. In reality, such a threshold could be different, so the arbitrary assumption is that half of the months are above the actual threshold.

5. Summary of results and conclusions When a team of workers is not honest, the firm should dismiss overtime as a signal of the actual workload. Without proper information, the firm might adopt two extreme strategies: removing all of its support personnel or keeping the personnel permanently. As described below, both strategies can be harmful under fairly general conditions. In the scenario according to Proposition 2 in which the firm removes all of the support personnel, if the work team is a dodger (cheater), the firm always incurs a negative utility. In this case the firm is being permissive with the overtime by not keeping the support personnel. On the other hand, if the firm decides to permanently keep the support personnel and to, therefore, restrict the overtime hours, then the firm earns money only if the overtime productivity is sufficiently high. Reasonably, if a team is a dodger, then the only way the firm can earn money is to hire an additional worker. And this worker has to be very productive to compensate for the loss of production because of the necessary overtime the team does not incur. Without support personnel, Proposition 3 sets a threshold of punishment for deceitful teams above which a team does not incur unnecessary overtime. This threshold requires an increment in monitoring the actual workload on the floor plant, or a rise in the punishment for the team caught working unnecessary overtime. These measures can be very costly for the firm or counterproductive if they appear abusive. Therefore, the calculated threshold of punishment might not be attainable in practice. Proposition 4, as empirically validated, shows that the firm might use other measures to leave this non-collaborative equilibrium (workaholic, restrictive). This outcome is less likely if a team receives a high wage, as long as the overtime is sufficiently productive. Reasonably, in high productivity plants, if

Table 3 Regressions for the overtime rate and personnel coefficient of variation vs. wages.

Dep. variable. Ind. variable. Coefficient t-statistics P-value

Regression 1 (teams on production line)

Regression 2 (teams on production line)

Regression 3 (support teams)

Regression 4 (support teams)

Overtime rate Log(Wage) −0.13 −4.06 0.000

Coeff. variation personnel Log(Wage) 0.33 3.726 0.001

Overtime rate Log(Wage) −0.10 −3.43 0.001

Coeff. variation personnel Log(Wage) 0.005 0.19 0.85

1777

30%

0,3

25% 20%

0,2

15% 10%

0,1

Overtime Rate

Headcount Coefficient of Variaton

M. Singer, J.J. Obach / Journal of Business Research 66 (2013) 1771–1779

5% 0,0 $ 969

0% $ 1.190

$ 1.339

$ 1.470

$ 1.727

$ 2.102

Average Wage (US$) Headcount Coefficient of Variation

Overtime Rate

Tendency of Headcount C. of V.

Fig. 4. Wages for supporting teams (low hourly overtime productivity).

beyond the normal hours. Understandably, the difficulty for the firm is to quantify these costs; but if the firm is facing a season with an extra workload, the firm has to be aware of these costs in order to keep a collaborative equilibrium. Second, the hiring costs are high and the firm is in a high-workload season (Proposition 7). And third, the layoff costs are high and the workload is relatively low (Proposition 8, tested empirically). These two last results imply that turnover costs are relevant to reach a collaborative equilibrium. If the firm faces elevated hiring costs (highly skilled jobs, shortages in the labor supply); then in high workload seasons, the firm might find difficulty in trusting an honest team when that team works the necessary overtime. Notice that being trustful in this scenario means holding/hiring support personnel. On the other hand, if the firm faces high layoff costs (seniority of the workers, unionization); then in low workload seasons, the firm might also distrust the workers. If workers are honest and are not working overtime, the firm should not keep extra personnel. If these costs are high, then the firm is likely to deviate from a collaborative equilibrium. Even though the collaborative scenario makes the most efficient use of overtime hours, this study does not suggest that firms should adjust their personnel period-by-period as a function of the observed overtime. According to Lilien and Hall (1986), variations in the workforce should respond to structural changes in the demand for labor, whereas changes in work hours might respond to short-term fluctuations. The results in this study suggest that if the conditions are in place for collaboration, then firms should take overtime as a credible signal of the actual workload faced by the firm.

0,5

35% 30%

0,4

25% 0,3

20% 15%

0,2

10% 0,1 5% 0,0

0% 14

17

20

22

22

24

26

Seniority (years) Headcount Coefficient of Variation

Overtime Rate

Tendency of Headcount C. of V.

Tendency of Overtime R.

Fig. 5. Layoff costs for production line teams in months in which θ b 1/2.

30

Overtime Rate

Headcount Coefficient of Variation

the cost of the overtime is high because of high wages; then the firm has incentives to not only hire additional workers to capture this high productivity but to restrict the actual overtime. Combining Propositions 3 and 4, if the firm suspects that unnecessary overtime exists, then the firm should increase punishment. If increasing the punishment is unfeasible, then the firm must hire new personnel as long as the firm is productive enough. In the scenario according to Proposition 2 in which the firm keeps support personnel permanently, the firm makes a profit only if the overtime productivity is sufficiently high. In that case, Proposition 5 points out that the firm can avoid workers systematically working overtime by setting a wage below a certain maximum in relation to the cost of the necessary overtime. Reasonably, when the personnel are systematically high, and therefore the firm restricts the overtime, then the overtime must be very productive to be profitable for the firm. If the overtime is not very productive, the firm must reduce the wages in relation to the effort of the workers that work the overtime. Fortunately, many firms achieve collaborative equilibriums in which workers truthfully communicate their workload through overtime, and the firm takes this information into account when hiring support personnel. However, the following circumstances make this mutually beneficial situation less likely. First, the cost to the team of the required overtime is higher than a certain threshold, and an extra workload exists (Proposition 6). If an extra workload exists and the cost for the team of working the necessary overtime is high, then the team could deviate from a collaborative equilibrium. These overtime costs can be interpreted as exhaustion from working

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M. Singer, J.J. Obach / Journal of Business Research 66 (2013) 1771–1779

Table 4 Regressions for the overtime rate and coefficient of variation of personnel vs. seniority. Months in which θ b 1/2

Dep. variable. Ind. variable Coefficient t-statistic P-value

Months in which θ >1/2

Regression 1

Regression 2

Regression 3

Regression 4

Overtime rate Log(Seniority) −0.03 −0.68 0.502

Coeff. variation personnel Log(Seniority) −0.22 −2.06 0.05

Overtime rate Log(Seniority) −0.02 −0.4 0.691

Coeff. variation personnel Log(Seniority) −0.15 −1.31 0.2

Firm (starting the game with support personnel)

Appendix A. Expected payoffs for the players Work Team (starting the game without support personnel)

  2 f 1 ¼ ð1=ð16ð1  μ ÞÞÞγð8pθμ θαβ þ αμ þ 2θα  2θβ þ 2β  αβ   θα 2

þsμðαθðð3θα  2Þ  6βðθ  1ÞÞ þ βð2ðθ  1Þ þ 3β 1 þ θ

w0 ¼ ðθα ðρδ  2Þð3s  8cÞ þ βð1  θÞðð3s  8bÞðρδ  2Þ þ 16r ð1  γ ÞÞ þρβ2 ð3sð2θδ þ γ  2θγ  δÞ þ 3s2 ðθγ  θδÞ

2

β þ 2θαÞÞ þ δð8pθμðαβ þ θα  α  2β  θαβ þ 2 þ 2μ þ θβ 2θα  αμÞ þ sμð5θα  5θβ  3θ2 β2 þ 6θ2 αβ  6θαβ þ 6θβ 2

8bðδ  γÞð1  θÞðb  2ÞÞ þ α 2 θ2 ρðγ  δÞð3 s  8cÞ þ2θραβðδ  γÞð1  θÞð4b þ 4c  3sÞ=ð32θραð1  γ Þðρδ  1Þ  2

2 2

6θβÞÞ þ 16dðθα  θβ  2βμ  2θαμ þ 2θβμ Þ þ 16hμ ðθβ  θα

2

2

þ32θ α ρ ð1  γÞðγ  δÞ þ 32θαβρ ð1  θÞ γ  δ γ þ γδ

16ð1  ρÞÞÞ

2  3β2 þ 5β  3θ2 α 2 Þ þ ð16d  16hμ Þð1  β þ θβ–θα Þ þ16dμ ð2θα  2θβ  2 þ 2βÞ þ 32hμβÞ þ γ2 μð8pθðθαβ2 αβ2 θα 2 Þþ 16d β2 þ θ2 α 2 þ 2θαβ  2θ2 αβ þ θ2 β 2θβ 2

þ16h θ2 β2 2θβ2 þ 2θαβ  2θ2 αβ þ β2 þ θ2 α2 Þ þμδ2 ð8pθμ α þ θαβ  αβ þ β  θα  θα 2  θβ

þð16d þ 16hÞð1 þ β2 þ θ2 α 2 þ θ2 β2  2β þ 2θαβ  2θ2 αβ 2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

θ α  θ β Þ þ 32hð2θβ  θ α  θ β  θβ  2θαβ þ 2θ αβ 2θαβ  θβ  β2 þ 2θ2 αβ þ 2θβ 2  θ2 α 2  θ2 β2 Þ þ 32hð2θβ 2

w1 ¼ ðθα ð1 þ ρ  δρÞð8c  3sÞ þ ð1  θÞβðð3s  8bÞðρδ  ρ  1Þ þβ ρð3s  8bÞ γ  δ  2θγ þ 2θδ þ θ γ  θ δ

2

2

θα  2θαβÞ þ 32dðβ þ θα  2θαβ  θβ  β þ 2θ αβ þ 2θβ 2

þ16rð1  γ ÞÞ þ θ2 α 2 ρð3s  8cÞð3γ  2  δÞ

2

2θα þ 2θβ  2θβ ÞÞ þ γμδð8pθμðθβ þ 2αβ þ 2θ α  α  β

Work Team (starting the game with support personnel)

θ2 α 2  θ2 β2  θβ  2θαβ þ 2θ2 αβ  β2 ÞÞ þ 8pðθαμ þ 2θα þ 2θ



2θμÞ  2sð1  μ Þ  3sαθðμ þ 1Þ  16dð1  μ Þ þ βð3sðθ  μ  1 þθμÞ þ 16dγÞ

þ2αβρθð3sðθ þ θδ  δ þ 2γ  2θγ  1Þ þ4bð3θγ  þ 2  2θ  3γ þ δ  θδÞ þ4cðθγ  θδ  γ þ δÞ

þ16r 1  θ þ γ2 þ 2θγ  θγ2  2γ Þ=ð32θρα ð1  γ Þðρδ  1Þ   2 2 2 2 2 þ32θ α ρ ð1  γ Þðγ  δÞ þ 32θαβρ ð1  θÞ γ  δ γ þ γδ 16ð1  ρÞÞÞ

Expected payoffs for the players by corner strategies (starting the game without support personnel) 1. Work Team Honest strategy:

Firm (starting the game without support personnel) ∧

w0 fhonest g ¼ ðθð3s  8cÞ ðθγρ þ ð1  θþÞδρ  2ÞÞ= ð32θðγ þ γρðθ  θγ  δÞ þ δρð1  θ þ θγÞ  1Þ  16ð1  ρÞÞ

f 0 ¼ ð1=ð16  16μ ÞÞðαθð24p  6sÞ þ β6sðθ  1Þ þ γðμθ8pðαð2θ  þ1   þθβ  β  θαÞ þ βð2  2θÞÞ þ sμðα 3θ2 α  2θ þ βð3β 1 þ θ2 þ2θð1  3βÞ  2Þ þ θð1  θÞαβ6Þ  ð16h þ 16dμ Þðθα–θβ þ βÞÞ

Dodger strategy:

þδðθ8pμ þ 3Þ  ðα ðβ þ αθ  2θ 2Þ þ βð2θ  2  θα Þ  þsμð6θ θαβ þ β2 αβ þ 5ðθα  θβ þ βÞ  3 θ2 α 2 þ θ2 β2 þ β2 2

2Þ þ ð16h þ 16dμ Þ β  1  2β γμ þ θα  θβ þ 2βγμ Þ 2

2

2

2

2

2

þγ μðθ8pμðθαβ  αβ  θα Þ þ ð16d þ 16hÞðβ þ θ α þ θ β 2 2 2 2θ αβ þ 2θαβ  2θβ ÞÞ þ δ μðθ8pμðα þ β þ θðα  β  θα Þ

2

αβð1  θÞ  1Þ  ð16d þ 16hÞð2β  β2 þ 2θα  2θβ  2θαβ þ2θ2 αβ  θ2 α 2  θ2 β2 þ 2θβ 2  1Þ þ μγδθð8pμð2θα 2  α  β θα þ θβ þ 2αβ  2θαβÞ  ð32d þ 32hÞðθα 2  α  2β2 þ θβ 2 þβ þ 2αβ  2θαβÞÞÞ

  ` U 2 w0 fdodgerg ¼ ðð3s  8bÞ ρð1  θÞðθδ þ 2γ Þ þ θ γρ þ 2θ  2 16rðθ  1 þ γ  θγÞÞ=ð16ρ  16Þ ` U w0 fworkaholicg ¼ ðð3s  θ8c  8bÞð1  θÞÞðγρ  2Þ 2

16r ðθ  1 þ γ  θγÞÞ=ð32θρ γρ  1  γ ρ þ γ 16ð1  ρÞÞ

U`

w0 flazyg ¼ 0



M. Singer, J.J. Obach / Journal of Business Research 66 (2013) 1771–1779

References

2. Firm Trustful strategy: ∧

2 2

2

f 0 ftrustfulg ¼ ððθ8pð3α þ αμ  θα μ  αβμ þ θαβμ  αβμ þθαβμ2 þ 2θαμ  θα 2 μ  2θβμ þ 2βμÞ sðθα ð6 þ 2μ þ 6θβμ  6βμ Þ þ βð6  6θ  2θμ  2μ Þ   3θ2 μ α 2 þ β 2 þ 3β2 μ ð2θ  1ÞÞ þ 16hðβ2 μ ð1  2θÞ   þθ2 μ β 2 þ α 2 þ 2θαβμ ð1–θÞ–ðθα  βð1  θÞÞÞ   þ16dμðβ2 ð1  2θÞ þ θ2 α 2 þ β2 þ θðβ–α Þ þ2θαβð1–θÞ  βÞÞ=ð16ð1  μ Þ

Distrustful strategy f 0 ∧ fdistrustfulg ¼ ðθ8pð3α þ μð3 þ θα 2 þ β ð1  θÞðα  2Þ 2

2α ð1 þ θÞÞ þ μ ðα ð1 þ θÞ þ ðβ  αβÞð1  θÞ θα 2  1Þ  sðθ α ð6  5 μ Þ þ βð1  θÞð6  5μ þ 6θαμ Þ   þ3θ2 μ α 2 þ β2 þ 3β 2 μ ð1  2θÞ þ 2μÞ þ16hðθα ð1  2μ Þ þ β ð1  θ  2μ þ 2θμ Þ   þ2θαβμ ð1  θÞ þ θ2 μ α 2 þ β 2 þ β 2 μ ð1  2θÞ þμ  1Þ þ 16dðμβðθ  1Þ  θαμ þ 2θαβμ ð1  θÞ    þμ θ2 α 2  β2 2θ þ θ2 þ 1 ÞÞ=ð16ð1  μ ÞÞ

Expansive   ∧ 2 f 0 fexpansiveg ¼ ðð8pθ μ þ αμ  3α  3μ  sðβð3μ  6Þð1  θÞ þ3θα ðμ  2Þ  2μÞ  16hðμ  1ÞÞ=ð16μ  16Þ Restrictive ∧

1779

f 0 frestrictiveg ¼ ð24pθα  6sðβ þ θα  θβÞÞ=ð16ð1  μ ÞÞ

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