Unit 6 The firm and its employees
6.5 Managing hiring and quitting: The reservation wage curve
How do firms and workers meet each other in the labour market, and how do they decide whether to form a match? Typically, firms advertise vacancies, and workers apply; the firm then offers an employment contract to those who fulfil its requirements, which the worker may accept or reject.
An important factor in their decision is the wage. For some workers—such as senior managers—the salary is negotiated. In some types of work in some countries, a rate for the job may be agreed with a trade union. But for many workers the wage is set by the employer, and is the same for everyone doing the same work within the firm. So employers must think about how high the wage needs to be to recruit and retain the number of workers required.
Reservation wages
In the models in this unit, we assume that firms recruit workers who are currently unemployed. In practice, of course, some workers move directly from job to job, but focusing on workers moving in and out of employment helps to simplify our analysis, and in particular gives us insight into one of the most important questions about the labour market: What determines the overall levels of employment and unemployment?
Consider the situation of Françoise, who has recently graduated from a university in Paris. She has not yet decided what career to pursue, so she decides to search for work teaching French as a foreign language in the meantime. Given her qualifications and what she knows about teaching opportunities, she thinks she might be able to find a job paying €700 a week.
But one of the language schools is advertising vacancies now at €580 per week. Should she apply?
- reservation option
- When someone makes a choice amongst the available options in a particular transaction, the reservation option is their next best alternative option. Also known as: fallback option. See also: reservation price.
- 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.
That depends on whether the option to take the job makes her better off than her reservation option, or next best alternative, which is to remain unemployed and search for a job at a higher wage. The value of the reservation option for people in her position depends on three things:
- her income while unemployed, including an unemployment benefit (if she is entitled to it) plus any support her family can provide
- other things that affect her utility while she is unemployed, such as whether she can use her time for further study or helping her family, or feelings of boredom, isolation, and anxiety about finding a job
- how long she thinks it would take to find a better opportunity.
- 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.
Having considered these factors, she decides that she should not accept a wage of less than €600. This is her reservation wage. Effectively, she thinks of being unemployed as equivalent to being in a job that pays €600 per week. So she would not accept a job at €580. She decides not to apply.
Hiring and quitting
Now think about the situation of a language school in Paris, offering intensive French courses to visiting students. It employs language tutors, most of whom, like Françoise, are recent graduates likely to stay only six months or a year before moving on to the next stage in their careers.
Suppose that the number of tutors employed is N, and on average 4% of them leave per week. So the number leaving will be 0.04N. If the school wishes to have a workforce of N = 50 tutors, it can expect two tutors (0.04 × 50) to leave each week, and it will need to hire two tutors per week to replace them.
Workers will not accept job offers if the wage is lower than their reservation wage—and reservation wages differ among potential workers, depending on their individual circumstances. So when the school advertises vacancies, the number of tutors it can hire depends on how many are searching for work and see the advert, and how many of them have reservation wages below the wage that the school offers.
The upward-sloping line in Figure 6.5 shows how many tutors the school can hire per week, depending on its wage. The lowest reservation wage among all potential applicants is €550. If the school offers little more than €550, very few workers will accept a job offer. If it raises the wage, workers with higher reservation wages will accept and it can hire more workers per week. At any wage w, the hiring line tells us how many of the potential applicants each week have a reservation wage below w.
To maintain a workforce of 50, the school needs to hire two workers per week, so it should set a wage of €675. Work through the steps to understand what happens if it wants more or fewer employees.
Figure 6.5 Setting the wage so that hiring equals quitting.
The hiring line
A workforce of 50
N = 70 and N = 20
Figure 6.5 shows that the wage that equalizes hiring and quitting increases with the number, N, of workers employed. We have used this to plot the wage required for employment of N workers in Figure 6.6. The hiring and quitting lines in the top panel show that if N = 50, an average of two workers per week will leave, and a wage of €675 is required to replace them. In the bottom panel, we have plotted the point (N = 50, w = 675)—and similarly for other levels of employment.
Figure 6.6 The wage required to maintain employment of N workers.
The upward-sloping line in Figure 6.6 tells us the wage required to employ N workers. But it also tells us the reservation wages of the tutors who could be employed at this wage. If the school sets a wage of €725 and employs 70 tutors, its employees will have reservation wages between €550 and €725. And 50 of them will have reservation wages below €675. A firm’s required-wage line is also its reservation wage curve (although it is a straight line in this example), showing the reservation wages of its employees.
When the language school decides which wage to set (Section 6.10) the reservation wage curve will be an important factor in the decision. But first we need to consider another aspect of the employment relationship that will affect its profits: work effort.
Question 6.6 Choose the correct answer(s)
Read the following statements about Figure 6.6 and choose the correct option(s).
- The vertical axis intercept represents the smallest reservation wage; all other points on the hiring line also represent reservation wages.
- The hiring line shows the wage the firm needs to set to replace the number of workers that quit per week, on average.
- €725 is the largest reservation wage of all employed workers (those who have lower reservation wages are willing to be hired; those with higher reservation wages are not willing to be hired); €550 is the smallest reservation wage.
- The firm now needs to hire five workers per week instead of two; if the hiring line and reservation wage curves stay the same, the wage the firm needs to set is higher than initially.
Extension 6.5 The hiring and quitting model
In the main part of this section we showed that there is an upward-sloping relationship between \(w\) and \(N\), which determines the wage \(w\) that a firm needs to pay if it wants to have a workforce of size \(N\). The relationship reflects the reservation wages of potential employees. In this extension we derive the algebraic equation of this reservation wage curve, which depends on the parameters representing the labour market conditions facing the firm. We use calculus (differentiation) to determine the properties of the curve.
Consider a firm that wants to employ \(N\) workers. It knows that—on average—a proportion, \(q\), of its employees will leave each week: \(q\) is the expected quit rate. So to maintain a workforce of size \(N\), it will need to hire \(qN\) workers each week.
Finding workers takes time. Firms search for workers with the skills and qualities required for the job, and workers likewise search for jobs that will suit them. Suppose that the firm encounters an average of \(m\) suitable matches per week: these workers will accept a job offer if the wage is at least as high as their reservation wage.
But workers have different reservation wages, while the firm must set the same wage for all workers doing the same job. Suppose that the firm offers a wage of \(w\). We will write \(P(w)\) for the proportion of workers who would accept the offer—in other words, the acceptance probability. The higher \(w\) is, the greater the number of workers who will accept will be. Therefore, \(P(w)\) is an increasing function.
Then the number of workers the firm can recruit per week is \(mP(w)\). To keep the level of employment in the firm at \(N\), the wage should be set high enough so that the number of hires is equal to the number of workers who quit:
\[\underbrace{mP(w)}_{hires} = \underbrace{qN}_{quits}\]When this equation is satisfied, the firm is in a ‘steady state’: the wage is such that the number of employees, \(N\), remains constant (steady) over time.
Figure E6.1 (taken from Figure 6.5) shows the particular case of this model that we used for the language school in Section 6.5. We assumed that the firm wanted to employ \(N = 50\) workers, and that the quit rate, \(q\), was 0.04. The vertical line represents the average quitting per week, \(qN = 2\), and the upward-sloping line is hires per week, \(mP(w)\); the number of hires per week increases with the wage, \(w\).
The wage that satisfies the equation, \(mP(w)=qN\), is the wage required for steady-state employment, \(N\). In the figure, the solution is \(w = 675\), where the two lines cross.
Figure E6.1 The hiring and quitting model for the language school, when \(N = 50\), and \(q = 0.04\).
In the language school example we have drawn, the hiring line as a straight line to keep the model simple. More generally, we can assume that it will be upward-sloping—\(P(w)\) is an increasing function—but its shape depends on the distribution of reservation wages in the population of workers.
The firm’s reservation wage curve
The equation, \(mP(w)=qN\), is an important relationship between \(w\) and \(N\). We can describe it as the equation of the reservation wage curve, provided that we can be sure that it gives us a unique value of \(w\) for each value of \(N\). In other words, we need to be sure that the hiring line and quitting line in Figure E6.1 cross at just one point.
The exception would be if the quit rate (\(q\)) were so high and the number of suitable matches per week (\(m\)) were so low, that even with a wage high enough that all workers would accept—that is, \(P(w) = 1\)—the firm could not find the number of workers it needed.
But this equation is what we call an implicit equation for \(w\)—without knowing the function \(P(w)\), we cannot rearrange it to get \(w\) on one side and everything else on the other (and we might not be able to do that even if we knew \(P\)). However, it is straightforward to reason that we can expect it to have a unique solution for \(w\): since the right-hand side is an increasing function of \(w\) and the left-hand side is constant, they can only be equal at one point.
So we can think of the equation, \(mP(w)=qN\), as implicitly determining \(w\) as a function of \(N\). Whatever the value of \(N\), it tells us the corresponding required wage, \(w\). Equally, for any value of \(w\), it tells us what level of steady-state employment the firm could have.
Secondly, it tells us that this is a positive relationship: \(w\) is an increasing function of \(N\). In the main part of this section, we deduced that by reasoning from the diagram; in Figure E6.1, an increase in desired employment, \(N\), would shift the vertical quitting line to the right, so the lines would cross at a higher wage.
We can deduce it algebraically by differentiating. Since the equation can be rearranged to give \(N\) as an explicit function of \(w\), an easy way to do this is to work out \(dN/dw\):
\[N= \frac{mP(w)}{q} \Rightarrow \frac{dN}{dw}= \frac{mP'(w)}{q}\]Since \(P'(w)>0\), and \(m\) and \(q\) are positive too, this confirms that \(dN/dw>0\); \(N\) is an increasing function of \(w\). Equivalently, \(w\) is an increasing function of \(N\). We can apply the inverse function rule to invert the derivative:
\[\frac{dw}{dN}= \frac{1}{\frac{dN}{dw}}=\frac{q}{mP'(w)}>0\]This is the relationship shown for the language school in Figure E6.2, taken from Figure 6.6. Since we plot it with \(w\) on the vertical axis, the slope of the line in the figure is \(dw/dN\), and the expression above tells us that the line is quite flat if raising or lowering the wage makes a big difference to hiring (\(P'(w)\) is high), and steep if hiring is not very sensitive to the wage.
Figure E6.2 The wage required for employment of \(N\) workers (the reservation wage curve).
Remember from Section 6.5, that since the relationship in Figure E6.2 is derived from the distribution of reservation wages among workers, it also tells us the distribution of reservation wages among the firm’s employees. For any value of \(w\), it tells us how many of the employees have reservation wages at or below that level. That is why we call it ‘the reservation wage curve’.
Changes in matching and quitting rates
Returning to the equation underlying the reservation wage curve, we can work out how the line shifts if either of the parameters, \(m\) or \(q\), changes. \(N\) is an increasing function of \(w\), but it also depends on \(m\) and \(q\), so we can use partial differentiation to determine the effect:
\[N= \frac{mP(w)}{q} \Rightarrow \frac{\partial N}{\partial m}= \frac{P(w)}{q}>0 \text{ and }\frac{\partial N}{\partial q}= -\frac{mP(w)}{q^2}<0\]Thus if the rate at which the firm can find suitable workers rises (but the quit rate remains the same) the firm can employ more workers at any given wage, so the reservation wage curve shifts down. This happens because it can find the workers who will accept that wage more quickly. An increase in the quit rate (ceteris paribus) has the opposite effect. If more workers quit each week, the curve shifts up. At a given wage, hiring hasn’t changed. For quitting to be equal to hiring, the number of employees in steady state must be lower.
Exercise E6.1 Deriving the reservation wage curve
Suppose the acceptance probability for wage, \(w\), is the function, \(P(w) = k(w \ – \ r_0)\), where \(r_0\) is the lowest reservation wage in the population (and the highest reservation is high enough that the firm will never be able to recruit every worker).
Derive the reservation wage curve for this function and draw an appropriate sketch of it on a diagram similar to Figure E6.2. Explain, using mathematics and intuition, how it changes when the quit rate rises.
Read more: Section 7.4 (on the inverse function rule) of Malcolm Pemberton and Nicholas Rau. Mathematics for economists: An introductory textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.
