Unit 6 The firm and its employees

6.9 Getting employees to work hard: The labour discipline model

Before you start

To understand the model in this section, you will need to know about game theory and Nash equilibrium. If you are not familiar with these concepts, read Sections 4.2 and 4.3 before beginning work on it.

When the cost of job loss (the employment rent) is large, workers will be willing to work hard in order to reduce the likelihood of losing their jobs. And the employment rent depends on the wage. So the firm owners and managers can give their workers an incentive for effort by raising wages.

labour discipline problem, labour discipline model
Employers face a labour discipline problem when they need to give employees an incentive to ensure that they work hard and well. In the labour discipline model, they do this by setting wages that include an economic rent (employment rent), which will be lost if the job is terminated. See also: employment rent.

We will represent this social interaction within the firm as a game played by the owners and the employees, which we call the labour discipline model. As with other models, we simplify the interaction to focus on what is important, following the principle that sometimes we see more by looking at less. So consider again the case of Maria, whose employment rent we calculated in the previous section. What wage should the employer set to ensure that she will work hard?

The players are Maria and her employer. The game is sequential (one of them chooses first). Here is the order of play:

  1. Based on his knowledge of how employees like Maria respond to higher or lower wages, the employer chooses a wage and informs her that she will be employed in subsequent weeks at the same wage—as long as she works at the required level of effort.
  2. In response to the wage offered and how likely it is that the employer will find out if she is not working up to the required level, Maria chooses a level of work effort, taking into account the costs of losing her job if she does not provide enough effort.

The pay-off for the employer is the profit he will make if Maria works hard producing more goods or services that will be his to sell. Maria’s pay-off is her net utility in the job, taking into account the wage and the effort she has provided.

Nash equilibrium
A Nash equilibrium is an economic outcome where none of the individuals involved could bring about an outcome they prefer by unilaterally changing their own action. More formally, in game theory it is defined as a set of strategies, one for each player in the game, such that each player’s strategy is a best response to the strategies chosen by everyone else. See also Game theory.

If she chooses her work effort as a best response to the employer’s offer, and the employer chooses the wage that maximizes his profit given that Maria responds the way she does, their two strategies are a Nash equilibrium.

Employers typically hire work supervisors and may install surveillance equipment to keep watch on their employees, increasing the likelihood that the management will find out if a worker is not working hard and well. Here we will ignore these extra costs and just assume that the employer occasionally gets some information on whether or not an employee is working as required—not enough information to implement a piece rate contract, but enough to fire a worker if the news is not good.

To choose the wage, the employer needs to know how the employee’s work effort will respond to higher wages. So we consider Maria’s decision first.

Maria chooses her effort

Suppose that the employer sets a weekly wage w, and Maria’s reservation wage is wr. If she exerts the required effort this week:

  • Her cost of effort (the disutility of work) is c, and the net utility of working is wc.
  • She will remain in the job next week.

If she decides to exert no effort:

  • She will not experience any disutility of work, so her net utility of working is w.
  • She may be caught shirking, and be fired.

If Maria thought only about getting the highest pay-off this week, she would make no effort. But she needs to consider the implications of shirking for her future: what are her chances of being caught and fired? If that happens, she will be left with her reservation option, being an unemployed job-searcher, which is equivalent to having a job with net utility of wr.

Suppose that, given the employer’s ability to monitor her work, the length of time that she can expect to hang on to her job while shirking is s weeks. The full consequences of her options are:

  • Exert no effort: w for s weeks, then become an unemployed job-searcher, with an average pay-off of wr.
  • Exert the required effort: wc for the foreseeable future.

Figure 6.9 shows how to compare the pay-offs for her two options.

There are two diagrams. Diagram 1 refers to working. The horizontal axis shows the time in weeks, and the vertical axis shows different measures of wage w. If a worker works for h weeks, they receive a wage minus cost of effort equal to w minus c, which is also their payoff. This worker has a reservation wage w_res, which is lower than w minus c. The payoff w minus c times h weeks of working is the worker’s total payoff. Diagram 2 refers to shirking. The horizontal axis shows the time in weeks, and the vertical axis shows different measures of wage w. If a worker works for s weeks, they receive a wage w, which is also their payoff. If after s weeks, they shirks for h-s weeks thereafter, they receive their reservation wage w_res, which is also their payoff. The reservation wage is lower than the wage w. The sum of payoff w times s weeks of working, and payoff w_res times h-s weeks of shirking is the worker’s total payoff.
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Figure 6.9 Comparing the pay-offs from working and shirking.

Pay-off from working: This diagram refers to working. The horizontal axis shows the time in weeks, and the vertical axis shows different measures of wage w. If a worker works for h weeks, they receive a wage minus cost of effort equal to w minus c, which is also their payoff. This worker has a reservation wage w_res, which is lower than w minus c. The payoff w minus c times h weeks of working is the worker’s total payoff.
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Pay-off from working

If Maria exerts the required effort, she receives a pay-off of wc for h weeks. Her total pay-off is h(wc).

Pay-off from shirking: This diagram refers to shirking. The horizontal axis shows the time in weeks, and the vertical axis shows different measures of wage w. If a worker works for s weeks, they receive a wage w, which is also their payoff. If after s weeks, they shirks for h-s weeks thereafter, they receive their reservation wage w_res, which is also their payoff. The reservation wage is lower than the wage w. The sum of payoff w times s weeks of working, and payoff w_res times h-s weeks of shirking is the worker’s total payoff.
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Pay-off from shirking

If she exerts no effort, she receives w for each of the s weeks she expects to remain in the job. But then she will become unemployed and search for a new job—equivalent to receiving wr per week thereafter. Her total pay-off is sw + (hs)wr.

Net pay-off from working compared with shirking: There are two diagrams. Diagram 1 refers to working. The horizontal axis shows the time in weeks, and the vertical axis shows different measures of wage w. If a worker works for h weeks, they receive a wage minus cost of effort equal to w minus c, which is also their payoff. This worker has a reservation wage w_res, which is lower than w minus c. The payoff w minus c times h weeks of working is the worker’s total payoff. Diagram 2 refers to shirking. The horizontal axis shows the time in weeks, and the vertical axis shows different measures of wage w. If a worker works for s weeks, they receive a wage w, which is also their payoff. If after s weeks, they shirks for h-s weeks thereafter, they receive their reservation wage w_res, which is also their payoff. The reservation wage is lower than the wage w. The sum of payoff w times s weeks of working, and payoff w_res times h-s weeks of shirking is the worker’s total payoff.
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Net pay-off from working compared with shirking

The shaded areas show the total pay-offs from each option. The net pay-off from working is the difference between them: h(wc) – (sw + (hs)wr).

Figure 6.9 shows that Maria’s best response to the wage chosen by the employer is:

  • Work at required level of effort if \(\begin{align*} \left({ \text{pay-off from} \atop \text{working hard for } h \text{ weeks} }\right)& \geq \left({ \text{pay-off from} \atop \text{shirking (for } s \text{ weeks until she is caught) }}\right) \\ h(w - c) &\geq sw + (h - s) w_r \end{align*}\)
  • Otherwise make no effort.
no-shirking condition
The condition that must be satisfied by the wage to ensure that the worker’s pay-off from exerting the level of effort required by the employer is greater than or equal to the pay-off from shirking. See also: no-shirking wage

This mathematical inequality is called the no-shirking condition. If the wage satisfies the no-shirking condition, Maria will work hard.

Just as in the calculation of the reservation wage, we have assumed that Maria uses the ‘expected’ length of time before the employer catches her shirking when she makes her decision. In practice, being caught shirking is a matter of chance. But the average number of weeks, s, gives her the best estimate she can make of the pay-off from shirking.

The employer chooses the wage

The firm’s profit from employing Maria is the difference between the amount of output she produces, and the cost of employing her (the wage). Suppose that she produces output of y per week if she works at the required level, but zero if she does not. The firm’s weekly profit is:

  • yw if she works at the required level
  • w if she shirks and has not been caught
  • w if she is caught shirking and fired. (She still has to be paid for her time on the job until she is fired.)

So the only way the employer can make a positive amount of profit is to set the wage high enough to ensure that Maria doesn’t shirk, but below her output, y.

no-shirking wage
The wage that is just sufficient to motivate a worker to provide effort at the level specified by their employer. See also: no-shirking condition

The employer can do the same calculation as Maria, to work out her response to the wage he sets. To maximize profit, he should choose the lowest wage that gives her an incentive to work hard. By rearranging the no-shirking condition, we can find Maria’s no-shirking wage:

\[w=w_r+c+(\frac{s}{h − s})c\]

For simplicity, we assume that if working at the required level has the same pay-off to her as not working, Maria will not shirk. Requiring the employer to pay a little more than the no-shirking wage would complicate the analysis without adding anything of importance.

The employer will choose this wage, provided that it is below her productivity, y. (If it isn’t, there is no wage at which the employer can make a profit, and he would be better off not employing her.)

Equilibrium

In the Nash equilibrium of the game, the employer sets a wage:

\[w=w_r+c+(\frac{s}{h − s})c\]

and Maria chooses to exert effort.

If the required effort is not costly for Maria (c = 0), the employer will just pay her reservation wage, wr. But otherwise he has to pay more, for two reasons:

  • to cover the cost of the effort she is required to make, c
  • to give her enough rent, \((\frac{s}{h − s})c\) so that she does not want to risk losing her job by shirking.

The rent depends on how long Maria can get away with shirking. If monitoring her is difficult, the average number of weeks before she is caught, s, will be high. In this case, the temptation to shirk will be strong, so the employer will have to increase the rent to deter her. But if monitoring is very easy, so that she would lose her job instantly if she shirked (s = 0), the rent would not be needed: she would never choose to shirk. A simpler way to remember the no-shirking wage is:

\[w=w_r+c+\text{rent}(s,c)\]

(remembering also that the rent increases with both c and s).

What has the labour discipline model told us?

  • Equilibrium: In the owner–employee game, the employer offers a wage and Maria provides a level of effort in response. Their strategies are a Nash equilibrium.
  • Shirking is deterred by employment rent: Maria provides effort because her net utility is high compared with the utility of unemployment (her reservation wage).
  • Power: Because Maria fears losing her job, the employer is able to exercise power over her, getting her to act in ways that she would not do without this threat of job loss. This contributes to the profits of the employer.

Exercise 6.6 The no-shirking wage

This section explained why the no-shirking wage satisfies the mathematical equality

\[w=w_r+c+(\frac{s}{h − s})c\]

where wr is the worker’s reservation wage, c is the cost of effort, s is the number of weeks the worker can get away with shirking before being fired, and h is the worker’s planning horizon (in weeks).

Using the numbers from Section 6.8 (c = $2, h = 156):

  1. Find the no-shirking wage for s = 10, 20, 30, …, 100 weeks. (Hint: Use statistical software like Excel.) Plot your answers on a chart and explain intuitively how the no-shirking wage varies with s.
  2. Assume that s = 20 (that is, it takes 20 weeks to catch shirking workers). Find the no-shirking wage for h = 0, 20, …, 200 weeks. Plot your answers on a chart and explain intuitively how the no-shirking wage varies with h.

Exercise 6.7 Equilibrium employment rents

For each of the following scenarios, draw a diagram like Figure 6.9 to illustrate how (if at all) Maria’s employment rent changes. Explain your answers.

  1. The government decides to increase childcare subsidies for working parents, but not for those who are unemployed. Assume Maria has a child and is eligible for the subsidy, and that if she were unemployed she would still keep her child in childcare (at the non-subsidized rate).
  2. Demand for the firm’s output rises as celebrities endorse the good.
  3. Improved technology makes Maria’s job easier.

Exercise 6.8 Employee monitoring technologies

Technology is making it easier for firms to track workers, both at home and on-site. For example:

  1. Read the introduction and conclusion of both studies linked in the bullet points, and summarize how the introduction of the monitoring technology (on-board computers, point-of-sale IT systems) affected workers’ behaviour, and any broader consequences for firms in that sector.
  2. Discuss the conditions under which employee monitoring is likely to backfire. (Hint: You may find this Harvard Business Review article helpful).