Unit 6 The firm and its employees

6.8 Counting the cost of job loss: Rents and reservation wages

In this section, we calculate the employment rent received by Maria, an employee earning $12 an hour for a 35-hour working week, when she exerts the amount of effort her employer requires. In the next section, we will show how her employer can use rents to motivate her to work hard.

To determine Maria’s rent, we need to think how she would evaluate two aspects of her job:

  • her pay: something she values
  • how hard she works: effort is costly for her.
utility
A numerical indicator of the value that one places on an outcome. Outcomes with higher utility will be chosen in preference to lower valued ones when both are feasible.

We can weigh these elements against each other using the concept of utility: her utility is increased by the goods and services she can buy with her wage, but reduced by the unpleasantness of going to work and working hard all day—the disutility of work.

Suppose that the required effort costs her the equivalent of $2 per hour. Then while she remains in her job she receives:

\[\begin{align*} \text{net utility per hour} &= \text{wage} − \text{disutility of effort per hour} \\ &= \$10 \end{align*}\]

To calculate her economic rent, we compare the value of staying in her job with the value of her next best alternative option, which is to be unemployed, and search for a new job.

unemployment benefit
A government transfer that is paid to an unemployed person while they are unemployed (or for part of the unemployment period). Also known as unemployment insurance.

People who lose their jobs can typically expect some help from others while they are out of work if their family and friends have jobs. And in many economies, they receive an unemployment benefit or financial assistance from the government. Suppose that for each hour Maria spends unemployed rather than working, her net utility—allowing for both income from these sources and the disutility of being unemployed—is $6.

It might be many weeks before she finds another job. The overall cost of job loss depends on how long she expects to be unemployed, and how much she expects to earn when she finds a new job. Maria estimates it will take her 44 weeks to find a new job, and that the average net utility she can expect in a new job is $9 (the wage minus effort costs).

To compare the value of her job with the next best option, we will suppose that Maria’s planning horizon is three years (156 weeks). In other words, what matters to her is how she will support herself and her family over the next three years. She cannot foresee what might happen after that. Figure 6.8 compares her current job with the next best alternative of becoming unemployed, over the planning period.

In this diagram, the horizontal axis shows time in weeks, and ranges between 0 and 156. The vertical axis shows different quantities, as follows. The hourly wage is $12. The wage minus cost of effort is $10 per hour which, multiplied by 35 hours and by 156 weeks, corresponds to a total value of current job of $54,600. The average net utility of other jobs is $9 per hour. The net utility in unemployment is $6 per hour. The total value of next best alternative is $6 times 35 hours times 44 weeks plus $9 times 35 hours times 112 weeks, totalling $44,520.
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Figure 6.8a Maria’s next best alternative and total employment rent.

The two lines in Figure 6.8a show Maria’s net hourly utility in her current job, and in her next best alternative, which is to become an unemployed job-searcher. The total value of the reservation option is $10,080 less than the value of her current job. Her total employment rent, $10,080, corresponds to the area between the two lines.

\[\text{total cost of job loss} = \text{total employment rent} = \text{\$10,080}\]

It is often more convenient to think about the value of different employment options, and hence employment rents, in hourly or weekly terms, rather than calculating the total value over a long period. That is easy for Maria’s current job: it is worth $10 per hour for the whole period.

But what about her reservation option of unemployment? To evaluate this, we need to take into account not only that she will receive $6 per hour while unemployed, but that it gives her the opportunity to search for a new job. On average, over the whole period, her reservation option is worth:

\[\frac{\text{\$44,520}}{156 \times 35 \text{ hours}} = $8.15 \text{ per hour}\]

Then we can say that Maria’s employment rent per hour is the difference between her net utility in the current job, and the average net utility of her reservation option:

\[\text{employment rent} = $10.00-$8.15 = $1.85 \text{ per hour}\]

Maria’s reservation wage

Using average values is also helpful because it allows us to think of the option ‘unemployment plus job search’ as equivalent, for Maria, to having a different job with net utility $8.15. She would be indifferent between an offer of a job worth $8.15 to her, and becoming unemployed and searching for a better job.

reservation wage
The reservation wage is the lowest wage a worker is willing to accept to take up a new job. It is the wage available in the worker’s next best job option (the reservation option). For workers whose next best option is unemployment, the reservation wage takes into account the wages they expect to receive when they find a new job as well as any income received while unemployed.

So we can say that $8.15 is Maria’s reservation wage. It is a measure of how she ‘values’ unemployment, her reservation option. Rather than being an unemployed job-searcher, she would accept any job at a wage (or net utility if effort was required) greater than $8.15. Figure 6.8b illustrates this way of thinking about unemployment.

In this diagram, the horizontal axis shows time in weeks, and ranges between 0 and 156. The vertical axis shows different quantities, as follows. The hourly wage is $12. The wage minus cost of effort is $10 per hour, and this is also the net utility of current employment. The net utility of other jobs is $9 per hour. The reservation wage (average value of reservation option) is $8.15 per hour. The difference between the net utility of current employment and the reservation wage is the employment rent ($1.85 per hour). The net utility of in employment is $6 per hour for the first 44 weeksm and $9 per hour thereafter.
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Figure 6.8b Maria’s reservation wage and employment rent per hour.

It is important to understand why Maria’s reservation wage is above the net utility of $6 she receives while unemployed. She would not accept a wage offer of $6, because she would do better to wait, and search for an offer closer to the average that other firms are offering. Her reservation wage, $8.15, represents the value to her of being unemployed and waiting for such an offer. While unemployed, she makes decisions as she would if she had a permanent job paying $8.15 per week.

The reservation wage depends both on her individual characteristics, which determine her own utility of unemployment, and on economy-wide things such as unemployment benefit and how easy it is to find a new job. To understand this more clearly, it is helpful to write a general expression for the reservation wage. Working in weeks rather than hours as above, suppose that:

  • Her planning horizon is h weeks.
  • Weekly unemployment benefit is b.
  • Her additional net utility of being unemployed is aM per week. We label it M for Maria as a reminder that it depends on things that are specific to her, such as her family responsibilities, and whether she has any savings she can use.
  • The average net utility in other jobs (the wage minus the cost of effort) is v per week.
  • She expects that it will take j weeks to find another job.

Then if Maria enters unemployment, she expects to receive: b + aM for j weeks, and net utility of v for the remaining hj weeks of her planning period. Maria’s reservation wage is the average value of her reservation option—that is, the total value divided by the number of weeks, h:

\[w_r=\frac{j(b+a^M)+(h−j)v}{h}\]

We can rearrange this equation to write it as:

\[w_r=τ(b+a^M)+(1−τ)v\]

In this expression, τ is equal to j/h. For an unemployed worker considering their planning horizon, τ is the proportion of time for which they can expect to remain unemployed. This will depend on the rate of unemployment in the economy. When there are many other unemployed people searching for jobs, the time taken to find a new job will be higher.

So Maria’s reservation wage is a weighted average of her utility while she is unemployed (b + aM), and the net utility v that she expects to receive when she finds a new job. When labour market conditions are bad for workers, finding a job takes a long time: she will put more weight on her utility while unemployed. But when she can find a job quickly, her reservation wage will be higher: it will be weighted towards the average value, v, of the job offers she expects to receive.

We have calculated Maria’s reservation wage using the ‘expected’ or ‘average’ length of unemployment. In practice, finding a job is uncertain: it may take less time, or more. Similarly, when she finds one the pay may be above or below average. Since she doesn’t know exactly what will happen, she bases the decision on the average values.

Exercise 6.5 Assumptions of the model

As in all economic models, our simplified representation of Maria’s employment rent has deliberately omitted some aspects of the problem that might be important. For example, we have assumed that:

  • Maria finds a job with a lower wage after her spell of unemployment.
  • Maria continues to receive unemployment benefits as long as she remains unemployed.

Redraw Figure 6.8b to show how relaxing each of these assumptions would alter the employment rent. Specifically, assume:

  • Maria finds a job with the same wage of $12 per hour after her spell of unemployment.
  • Maria’s eligibility for unemployment benefits lasts for only 13 weeks.

Question 6.10 Choose the correct answer(s)

Maria earns $12 per hour in her current job and works 35 hours a week. Her disutility of effort is equivalent to a cost of $2 per hour of work. If she loses her job, she will receive unemployment benefits equivalent to $4 per hour. Additionally, being unemployed has psychological and social costs equivalent to $1 per hour. Suppose that Maria’s planning horizon is 156 weeks, and that, if Maria were to become unemployed, she expects to find another job at the same wage and cost of effort after 44 weeks. Then:

  • The employment rent per hour is $7.
  • Maria’s reservation wage is $3 per hour.
  • If she can get another job with the same wage rate after 44 weeks, Maria’s total employment rent is $6,160.
  • Maria’s total employment rent if she can only get a job at a lower wage rate after 44 weeks of being unemployed is more than $10,780.
  • Maria’s net hourly benefit of being employed compared with unemployment is $7 for the first 44 weeks, but she receives a net hourly benefit of $0 for the remaining 112 weeks, so her employment rent will be less than $7 per hour. (Calculation: the total value of the next best alternative is ((10 − 3) × 35 × 44) + (10 × 35 × 112) = $43,820, and her employment rent per hour is 10 − (43,820 / (156 × 35)) = $1.97.)
  • From the reservation wage equation, Maria’s reservation wage does not just depend on the size of the unemployment benefits and the costs of being unemployed; it also depends on the net utility of being unemployed, her planning horizon, and how long she expects to wait to find another job. Maria’s reservation wage is the total value of the next best alternative divided by the number of hours = ((7 × 35 × 44) + (10 × 35 × 112)) / (156 × 35) = 43,820 / 5,460 = $8.03.
  • Maria’s employment rent = $7 (employment rent per hour) × 35 hours per week × 44 weeks = $10,780.
  • If she could get a job at the same wage after 44 weeks, Maria’s total employment rent = $7 (employment rent per hour) × 35 hours per week × 44 weeks = $10,780. If the new job were to have a lower wage, her total employment rent in the current job (cost of job loss) would be higher than $10,780.

Extension 6.8 From the reservation wage to the reservation wage curve

The equation we obtained for the firm’s reservation wage curve in Extension 6.5 looks rather different from the one we will use in the rest of this unit, which is based on the expression obtained in the main part of this section for Maria’s reservation wage. In this extension, we demonstrate that the two ways of writing the reservation wage curve are consistent with each other, and explain how they are linked.

If you wish, you can skip this extension, since we do not use the result anywhere else, and it is a little more difficult than the content of other extensions. It doesn’t use any advanced mathematics, but it requires you to think very carefully about the mathematical interpretation of algebraic equations and functions.

We have determined the reservation wage of a single worker, Maria:

\[w_r = \tau(b + \alpha^M) + (1 - \tau) \nu\]

It depends on \(b\) (unemployment benefit), \(ν\) (average net utility in other jobs), and \(τ\) (the proportion of time she expects to be unemployed). These three parameters are the same for all workers participating in the labour market. It also depends on the utility of being unemployed, \(α\), which differs between workers—\(α^M\) indicates Maria’s particular value of \(α\). Workers with higher levels of \(α\) have higher reservation wages.

Since workers have different unemployment utilities, \(α\), they have different reservation wages. This is the reason that the firm’s reservation wage curve slopes upward. By arranging all the firm’s potential employees in ascending order of their reservation wages, we can write its reservation wage curve as:

\[w = \tau(b + \alpha^N) + (1 - \tau) \nu \hspace{2cm} (1)\]

where \(α^N\) is the unemployment utility of the \(N^t{^h}\) employee. We will use this equation for the reservation wage curve in the following sections of this unit. But thinking about it carefully, you may realise that something is missing. If we choose a particular value of \(N\), how do we work out \(α^N\)? We need to know it in order to calculate the corresponding \(w\).

This equation has a completely different form to the one we obtained for the reservation wage curve in Extension 6.5, which can be written:

\[P(w) = \frac{qN}{m} \hspace{2cm} (2)\]

where \(P(w)\) captures the distribution of reservation wages among the population of workers. Specifically, it is the proportion of workers with reservation wages less than or equal to \(w\)—in other words, the proportion who will accept a wage offer \(w\). The parameters \(m\) and \(q\) are the firm’s meeting and quit rates. As we discussed in Section 6.5, equation (2) is an upward-sloping relationship between \(w\) and \(N\), and it tells us two things:

  1. the wage the firm must set if it wants to employ \(N\) workers in steady state
  2. for any \(N\), the reservation wage of the \(N^t{^h}\) employee.

In fact, these two equations are both valid ways of writing the reservation wage curve. They are written differently because some important information is hidden in each version—we haven’t explained what determines \(α^N\) in equation (1), or \(P(w)\) in equation (2). We will now explain where each of these comes from, and hence reconcile the two ways of writing the reservation wage curve.

What is \(P(w)\)?

In equation (2), \(P(w)\) is the proportion of workers whose reservation wage is less than or equal to a particular value, \(w\). Remember that the reason why workers have different reservation wages is because they have different unemployment utilities: these are fixed individual characteristics that remain constant whatever the state of the labour market. We will define \(P_α(α_0)\) as the proportion of workers whose unemployment utility, \(α\), is less than or equal to a particular value, \(α_0\).

Then we can work out the relationship between \(P\) and \(P_α\). A worker will accept a wage, \(w\), if their reservation wage satisfies:

\[w_r \leq w\]

Equivalently, using the expression we obtained for the reservation wage of a single worker (such as Maria), a worker with unemployment utility, \(α\), will accept a wage, \(w\), if:

\[\begin{align*} \tau(b + \alpha) + (1 - \tau) \nu &\leq w \\ \Rightarrow \alpha &\leq \frac{(w-\nu)}{\tau} + \nu - b \end{align*}\]

So the proportion of workers who will accept a wage, \(w\), is the proportion who have unemployment utility, \(α\), less than or equal to \((w-v)/τ + v - b\). That is:

\[P(w) = P_\alpha \bigl(\frac{(w-\nu)}{\tau} + \nu - b \bigr)\]

This shows what was hidden inside the term \(P(w)\): the proportion of workers who will accept a wage, \(w\), depends not only on the wage, but also on the distribution of unemployment utilities, \(P_α\), and the three parameters capturing the labour market conditions facing workers (\(b\), \(ν\), and \(τ\)).

What is \(α^N\)?

We defined \(α^N\) as the individual unemployment utility of the firm’s \(N^t{^h}\) potential worker.

  • From equation (2), we know the proportion of workers with reservation wage less than or equal to the reservation wage of the \(N^t{^h}\) worker is \(qN/m\).
  • So the proportion of workers with unemployment utility, \(α\), less than or equal to \(α^N\) is also equal to \(qN/m\).

Hence \(α^N\) is the solution of the equation:

\[P_\alpha(\alpha^N) = \frac{qN}{m}\]

We can think of this equation as determining \(α^N\) as a function of \(N\), \(q\), and \(m\). That is, for any set of values of \(N\), \(q\), and \(m\), there is a corresponding value of \(α^N\). It is an implicit function: to find this value of \(α^N\), we have to solve the equation. But even without solving it, the equation tells that \(α^N\) is all of the following:

  • an increasing function of \(N\)
  • an increasing function of \(q\)
  • a decreasing function of \(m\).

You can obtain these results more formally using the technique of implicit differentiation.

We know this because the right-hand side, \(qN/m\), increases with \(N\) and \(q\), and decreases with \(m\). And since \(P_α\) is an increasing function, \(α^N\) must increase or decrease in the same way.

Equations (1) and (2) are the same

We can now show that if we start from equation (2) for the reservation wage curve, we can rewrite it in the form of equation (1). Substituting the new expression for \(P(w)\) into (2), we obtain:

\[P_\alpha \bigl(\frac{(w-\nu)}{\tau} + \nu - b \bigr) = \frac{qN}{m}\]

We also know that \(\frac{qN}{m} = P_α(α^N)\), so:

\[P_\alpha \bigl( \frac{(w-\nu)}\tau+\nu-b \bigr)=P_\alpha(\alpha^N)\]

And since \(P_α~\) is an increasing function, the only way this can happen is if:

\[\begin{align*} \frac{(w-\nu)}{\tau} + \nu - b &= \alpha^N \\ \Rightarrow w &= \tau(b + \alpha^N) + (1 - \tau) \nu \end{align*}\]

—which is equation (1).

In summary, if we write the reservation wage curve as:

\[w = \tau(b + \alpha^N) + (1 - \tau) \nu \text{, where } \\ \alpha^N \text{ satisfies } P_\alpha(\alpha^N) = \frac{qN}{m} \hspace{2cm} (3)\]

we have a complete definition of the reservation wage curve that combines the information in equations (1) and (2). Equation (3) is an increasing relationship between \(w\) and \(N\), that depends on both the labour market conditions facing workers (\(b\), \(ν\), and \(τ\)) and the labour market conditions facing firms (\(q\) and \(m\)).

Note also that whether or not the reservation wage curve is a straight line depends on the distribution of unemployment utilities, \(P_α\). It will be a straight line if \(P_α\) increases linearly with \(α\). The particular distribution with this property is called the uniform distribution.

Exercise E6.2 Changes in the reservation wage curve

  1. For the uniform case, \(P_\alpha (\alpha) = \gamma(\alpha + \alpha_0)\), where \(\alpha_0\) is the lowest unemployment utility, derive the individual unemployment utility of the firm’s \(N^t{^h}\) potential worker (\(\alpha^N\)) and hence write down the full explicit equation for the reservation wage curve.
  2. By examining the expressions for the slope and position, interpret how these change with some key parameters.

Read more: Section 15.1 (on implicit differentiation) of Malcolm Pemberton and Nicholas Rau. Mathematics for economists: An introductory textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.