Unit 6 The firm and its employees

6.10 Combining recruitment and labour discipline: The wage-setting model

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.

The wage affects both the number of workers a firm can employ and how hard its employees will work (the labour discipline problem). How can the firm set the wage to address both of these concerns? In general, the wage required to hire enough workers to achieve a particular level of employment will not be high enough to motivate them to work hard.

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.

We will consider the example of the Parisian language school (introduced in Section 6.5), which employs young graduates to teach short courses in French for visiting students. Figure 6.10 shows the school’s reservation wage curve, which tells us the wage required to employ N workers. Potential tutors only accept a job offer if the wage is above their reservation wage, and have different reservation wages according to their individual utility of unemployment. So to recruit and retain more employees, the school has to increase the wage.

For example, if the wage is €650, its potential recruits are limited to those with reservation wages less than or equal to €650, and the maximum number of tutors it can employ is 40. If it raises the wage to €700, it can also attract tutors with reservation wages between €650 and €700, increasing employment to 60.

In this diagram, the horizontal axis shows the average number of workers arriving or leaving per week. The vertical axis shows the weekly wage w. Coordinates are (average number of workers, weekly wage). An upward-sloping, straight line is labelled ‘Hires per week’ and starts from (0, 550).
Fullscreen

Figure 6.10 The school’s reservation wage curve.

If the school sets a wage €650 and employs 40 tutors, its employees will have reservation wages between €550 and €650. But it also wants them to work hard—effective teaching requires careful preparation—and it is impossible to monitor and assess the quality of every lesson.

employment rent
The economic rent a worker receives when the net value of their job exceeds the net value of their next best alternative (that is, being unemployed). See also: economic rent.

Consider the case of Marc, a tutor whose reservation wage is equal to €650. He will not choose to work hard if the wage is €650, because he will be indifferent between being employed while exerting no effort, and his next best option, unemployment. To give an employee an incentive to work hard, the wage must be above their reservation wage: firstly to compensate them for the cost of effort, and secondly to ensure that losing the job is costly for them. In other words, they must receive an employment rent, so that they prefer to work rather than risk being caught shirking and fired.

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 rent required to deter shirking depends on two things: the cost of effort, c, and the number of weeks, s, that a shirker could expect to remain in the job before being caught. For a tutor with reservation wage wr, the no-shirking wage, just sufficient to deter shirking is:

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

Section 6.9 explains how to derive this expression for the no-shirking wage.

We will suppose that for each tutor, the cost of effort amounts to €25 per week, and the required rent is equal to €35 per week. Then the no-shirking wage is:

\[\begin{align*} w&=w_r+25+35 \\ &=w_r+60 \end{align*}\]

In Figure 6.11, we have plotted the no-shirking wage curve €60 above the reservation wage curve. If we think of potential tutors as lined up in order of their reservation wages, Marc, with reservation wage €650, is the 40th, and Françoise, whose reservation wage is €600, is the 20th. In each case, their no-shirking wage is €60 above their reservation wage.

There are two diagrams. In diagram 1, the horizontal axis shows employment N, and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 500 and 850. Coordinates are (employment, wage). An upward-sloping, straight line is labelled reservation wage curve and passes through points (0, 550) and (40, 650). A parallel, upward-sloping, straight line is labelled no-shirking wage curve and passes through points (0, 610) and (40, 710). The vertical distance between the two lines is equal to €60. A segment labelled Marc connects points (40, 0) and (40, 650). The cost of effort plus the rent to deter shirking is equal to €60. In diagram 2, the horizontal axis shows employment N, and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 500 and 850. Coordinates are (employment, wage). An upward-sloping, straight line is labelled reservation wage curve and passes through points (0, 550) and (20, 600). A parallel, upward-sloping, straight line is labelled no-shirking wage curve and passes through points (0, 610) and (20, 660). The vertical distance between the two lines is equal to €60. A segment labelled Françoise connects points (20, 0) and (40, 600). The cost of effort plus the rent to deter shirking is equal to €60.
Fullscreen

Figure 6.11 The no-shirking wage curve.

What can the school do if it wants to employ 40 tutors? A wage of €650 is enough to recruit them, but then Marc, Françoise, and more than half of the others have no incentive to work because the wage is below their no-shirking wage.

Figure 6.11 suggests that to ensure that none of them shirk, the school should set the wage at €710.

But there is one more problem: if the school offered €710, some workers with reservation wages higher than Marc would accept the offer. But they would shirk, because their no-shirking wage is above €710. To overcome this problem, the firm will need to interview its applicants to find out more about them, and make offers only to those it expects to work hard—that is, the ones with reservation wages below €650.

The importance for individual firms and workers of learning more about each other before they commit themselves to an employment contract is one of the reasons that matching in the labour market takes time and effort. In practice, firms hold interviews for most jobs. Assessing who is likely to work hard may be difficult, but in this simple model we will assume that our firm can screen applicants perfectly.

Then, the no-shirking line in Figure 6.11 tells us the lowest wage the school can set if it wants to employ a given number of workers and ensure that they work hard. To employ 40 non-shirking workers, it could set a wage of €710, and screen applicants to recruit only those who will work hard at this wage.

feasible set
All of the combinations of goods or outcomes that a decision-maker could choose, given the economic, physical, or other constraints that they face. See also: feasible frontier.

But it could set a higher wage. If the school chose a wage of €730, there would be more applicants for each vacancy, but it could choose only to hire enough to maintain employment at 40. Figure 6.12 shows the school’s feasible set of wages and employment. All the points above the no-shirking wage curve are feasible.

In this diagram, the horizontal axis shows employment N, and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 500 and 850. Coordinates are (employment, wage). An upward-sloping, straight line is labelled reservation wage curve and passes through points (0, 550) and (40, 650). A parallel, upward-sloping, straight line is labelled no-shirking wage curve and passes through points (0, 610), (40, 710) and (50, 735). The vertical distance between the two lines is equal to €60. The area above the no-shirking wage curve is the feasible set. Point (40, 735) is in the feasible set.
Fullscreen

Figure 6.12 The feasible set.

If the school’s owners want to make as much profit as possible, what wage should they set? To answer this question they need to consider how profit depends on N and w.

Profit and isoprofit curves

A firm’s profits are equal to sales revenue minus input costs.

Suppose that each tutor generates revenue of y = €800 per week for the school in student fees. To keep the model simple, assume that wages are the school’s only input cost. Then if N tutors are employed at wage w, the net profit from employing each tutor is 800 – w, and the school’s total profit is:

\[\begin{align*} \text{profit per week}&=(y-w) \times N\\ &=(800-w)N \end{align*}\]

As long as the wage is below €800, it will make a profit. The lower the wage, w, and the larger the number, N, of tutors employed, the more profit it will make.

isoprofit curve
A curve that joins together the combinations of prices and quantities of a good that provide equal profits to a firm.

We can represent profit in a diagram by finding different combinations of w and N that give the same amount of profit. For example, with N = 10 and w = €650, profit = €1,500. Other ways to obtain profit of €1,500 would be N = 40 and w = €762.50, or N = 75 and w = €780. In Figure 6.13, we have drawn a curve joining these three points with all the other combinations of N and w that give €1,500 of profit. This is called an isoprofit curve (‘iso’ means ‘same’ in Greek). Work through Figure 6.13 to understand how other isoprofit curves can be drawn.

In this diagram the horizontal axis shows employment, N and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 400 and 850. Coordinates are (employment, wage). An upward-sloping, concave curve passes through points A (10, 650), B (40, 762.50) and C (75, 780). At all points on this curve, profit is €1500. Other upward-sloping, concave curves lie at all points below the isoprofit curve for 1500, and these are the isoprofit curves for 3000, 5000, 8000, 12000. Isoprofit curves for lower level of profits lie at all points below isoprofit curves for higher levels of profits. The zero-profit line consists of a vertical segment between (0, 0) and (0, 800), and a horizontal line starting from (0, 800).
Fullscreen

Figure 6.13 Isoprofit curves when profit = (800 – w) × N.

Different ways of making €1,500 of profit:
Fullscreen

Different ways of making €1,500 of profit

In this diagram, the horizontal axis shows employment, N and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 400 and 850. Coordinates are (employment, wage). An upward-sloping, concave curve passes through points A (10, 650), B (40, 762.50) and C (75, 780). At all points on this curve, profit is 1,500.

The zero-profit line: In this diagram the horizontal axis shows employment, N and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 400 and 850. Coordinates are (employment, wage). An upward-sloping, concave curve passes through points A (10, 650), B (40, 762.50) and C (75, 780). At all points on this curve, profit is 1500. The zero-profit line consists of a vertical segment between (0, 0) and (0, 800), and a horizontal line starting from (0, 800).
Fullscreen

The zero-profit line

Each tutor produces revenue of €800. If the wage is €800, profit per tutor is zero, so for all values of N, total profit is zero. Also, profit is zero when N = 0. The zero-profit line is horizontal at 800 and vertical at N = 0.

Another isoprofit curve: In this diagram the horizontal axis shows employment, N and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 400 and 850. Coordinates are (employment, wage). An upward-sloping, concave curve passes through points A (10, 650), B (40, 762.50) and C (75, 780). At all points on this curve, profit is 1500. Another upward-sloping, concave curve lies at all points below the isoprofit curve for 1500, and is the isoprofit for 3000. The zero-profit line consists of a vertical segment between (0, 0) and (0, 800), and a horizontal line starting from (0, 800).
Fullscreen

Another isoprofit curve

We have added the curve that joins all the (N, w) combinations where profit is €3,000.

Higher isoprofit curves: In this diagram the horizontal axis shows employment, N and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 400 and 850. Coordinates are (employment, wage). An upward-sloping, concave curve passes through points A (10, 650), B (40, 762.50) and C (75, 780). At all points on this curve, profit is 1500. Other upward-sloping, concave curves lie at all points below the isoprofit curve for 1500, and these are the isoprofit curves for 3000, 5000, 8000, 12000. Isoprofit curves for lower level of profits lie at all points below isoprofit curves for higher levels of profits. The zero-profit line consists of a vertical segment between (0, 0) and (0, 800), and a horizontal line starting from (0, 800).
Fullscreen

Higher isoprofit curves

Isoprofit curves with higher levels of profit are closer to the bottom right of the diagram, where N is higher, and w is lower.

indifference curve
A curve that joins together all the combinations of goods that provide a given level of utility to the individual.

We could describe the isoprofit curves as the firm’s indifference curves: the firm is indifferent between combinations of w and N giving the same level of profit. Isoprofit curves have the following characteristics:

  • They slope upward: If you start at point A, for example, and then increase the number of workers, you do not need as much profit per worker to keep total profit constant. Therefore you can raise the wage.
  • They have higher levels of profit towards the bottom right of the diagram, where N is high and w is low.
  • They all have a similar curved shape: They are steep when N and w are both low, and quite flat when N and w are both high.

To understand the last property, consider what happens when you increase the number of workers, N, by one. How much do you have to change the wage to stay on the same isoprofit curve—that is, to make the same profit as before? Figure 6.14 shows this calculation for two points on the 1,500 isoprofit curve. When w and N are low, you make a lot of profit on the additional worker, and you have to increase w a lot to offset this extra profit: the slope of the curve is high. When w and N are high, profit on the extra worker is low and you don’t have to adjust the wage as much.

Compare two points on the 1,500 isoprofit curve If you increase N by 1, how much do you have to raise the wage to stay on the same curve?
N w Profit Profit per worker N + 1 Profit goes up by: To keep profit constant, raise the wage by: Slope
Low w and N 5 500 1,500 300 6 300 300/6 = 50 High
High w and N 75 780 1,500 20 76 20 20/76 = 0.26 Low

Figure 6.14 Calculating the slope at two points on an isoprofit curve.

Maximizing profit

Profit is high when N is high and w is low. But not all combinations of N and w are feasible. The best the school can do is to find the most profitable combination in the feasible set. Figure 6.15 brings together the isoprofit curves and the feasible set of (N, w) combinations—the points on or above the no-shirking wage curve—so that we can find the one that gives maximum profit.

In this diagram the horizontal axis shows employment, N and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 400 and 850. Coordinates are (employment, wage). An upward-sloping, concave curve passes through points A (10, 650), B (40, 762.50) and C (75, 780). At all points on this curve, profit is 1500. Other upward-sloping, concave curves lie at all points below the isoprofit curve for 1500, and these are the isoprofit curves for 3000, 5000 3610, 8000, 12000. Isoprofit curves for lower level of profits lie at all points below isoprofit curves for higher levels of profits. The zero-profit line consists of a vertical segment between (0, 0) and (0, 800), and a horizontal line starting from (0, 800). An upward-sloping, convex line labelled no-shirking wage starts from point (0, 610) and intersects the isoprofits for 1500 and 3000 at two points.
Fullscreen

Figure 6.15 Where is the highest profit in the feasible set?

There are no feasible points with profit as high as €5,000: the firm cannot reach the €5,000 isoprofit curve. But there are feasible points with profit of €3,000, and other points with higher profit than that.

Profit is always higher at a point on the no-shirking wage curve than at any point vertically above it. So the school will always choose a point on this curve: it tells us the wage that the school will set for any given level of employment.

In Figure 6.16, we have drawn the highest isoprofit curve that the firm can reach—the one where profit is €3,610, which just touches the no-shirking wage curve.

In this diagram the horizontal axis shows employment, N and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 400 and 850. Coordinates are (employment, wage). An upward-sloping, concave curve passes through points A (10, 650), B (40, 762.50) and C (75, 780). At all points on this curve, profit is 1500. Other upward-sloping, concave curves lie at all points below the isoprofit curve for 1500, and these are the isoprofit curves for 3000, 3610, 5000, 8000, 12000. Isoprofit curves for lower level of profits lie at all points below isoprofit curves for higher levels of profits. The zero-profit line consists of a vertical segment between (0, 0) and (0, 800), and a horizontal line starting from (0, 800). An upward-sloping, convex live labelled no-shirking wage starts from point (0, 610) and is tangential to the isoproft curve for 3610 at point E (38, 705).
Fullscreen

Figure 6.16 Maximum profit of €3,610 is achieved at point E, where w = €705 and N = 38.

The firm makes the maximum feasible amount of profit at point E, employing 38 tutors at a wage of €705.

Profit is maximized at the tangency point of the reservation curve and the isoprofit curves—just as, in Unit 3, utility is maximized where the budget constraint is tangent to an indifference curve.

To understand why, imagine yourself moving along the no-shirking wage curve. Start where N is small, near the vertical axis: profit there will be low. As you move along the line, you cross the €1,500 isoprofit curve, then the €3,000 one—profit is rising until you reach point E, where profit is €3,610. But if you keep going, profit starts to fall; you hit the €3,000 curve again, then €1,500. Point E is the best you can do.

Our model of wage setting tells us the following:

  • The number of workers a firm can employ depends on its wage, and the reservation wages of potential employees (the reservation wage curve). To increase employment it needs to raise the wage, in order to recruit employees with higher reservation wages.
  • The firm chooses a wage on the no-shirking wage curve, which lies above the reservation wage curve. The difference between the two is the cost of effort, and the employment rent required to deter shirking.
  • It chooses the point where the no-shirking wage curve touches the highest possible isoprofit curve.
labour market power
A firm has labour market power (sometimes called monopsony power) if it can reduce the wage it needs to pay its workers by lowering the number of workers that it employs.

In this model, the firm faces a trade-off: to employ more workers it has to raise wages. As long as the wage is less than €800, the language school makes a profit on each tutor employed—so it would like to employ more. But then it would have to raise the wage, reducing the profit on every tutor. Restraining employment keeps the wage down, enabling it to make a high profit on each employee. The firm’s ability to control wages in this way is called labour market power.

Our model of wage setting demonstrates how firms can set wages both to recruit and retain workers, and to provide them with an incentive to work hard. In the next section, we will examine the implications: how wages and employment are affected when economic conditions change.

Question 6.11 Choose the correct answer(s)

Read the following statements about Figure 6.16 and choose the correct option(s).

  • (30, €700) is a feasible employment and wage choice for the firm.
  • The (employment, wage) combinations of (19, €610) and (16, €550) are both on the same isoprofit as the profit-maximizing point E (38, €705).
  • If the no-shirking wage curve became steeper while keeping the vertical axis intercept unchanged, the firm would choose to hire fewer workers.
  • If the firm had labour market power, then the profit-maximizing point may not be on the no-shirking curve.
  • (30, €700) is above the no-shirking wage curve, so it is a feasible choice for the firm (though it gives lower profits of €3,000).
  • (19, €610) is on the same isoprofit (€3,610) as point E, but (16, €550) gives a higher profit of (800 – 550) × 16 = €4,000.
  • The firm’s profit is always maximized at the tangency point of the no-shirking wage curve and the isoprofit curves. The isoprofit curves have steeper slopes at lower levels of employment, so the tangency point would be at a lower level of employment than point E.
  • The firm’s profit is always maximized at the tangency point of the no-shirking wage curve and the isoprofit curves, so the profit-maximizing point will be on the no-shirking wage curve even if the firm has labour market power.

Exercise 6.9 Competition and profits

Suppose that the language school in Figure 6.16 now faces more competition from other schools. Explain, using a diagram like Figure 6.16, how increased competition would affect:

  • the no-shirking wage curve
  • the firm’s profit maximizing choice
  • the firm’s profits.

Extension 6.10 Setting the wage to maximize profit

In the main part of this section we used diagrams to show how a firm (the language school) would set its wage, \(w\), and a corresponding level of employment, \(N\). Its profit-maximizing combination of \(w\) and \(N\) lies at the point where the no-shirking wage curve is tangent to an isoprofit curve. In this extension, we show how to do this by using calculus to solve a constrained choice problem. The method we use is explained in Extension 3.5, which you may need to re-read before reading this extension.

The symbol Π is the Greek capital letter ‘pi’ , and is often used in economics to represent profit.

The firm’s total profit is equal to the number of workers, \(N\), multiplied by the net profit on each worker, (\(y \ – \ w\)), where \(y\) is the firm’s revenue (or income) per worker employed. We write profit as a function, that we call Π, of wages and employment:

\[\Pi(w, N) = (y - w)N\]

Figure 6.13, reproduced as Figure E6.3, shows the isoprofit curves—curves joining combinations of \(w\) and \(N\) that give equal amounts of profit—for the case \(y = 800\).

In this diagram the horizontal axis shows employment, N and ranges between 0 and 80. The vertical axis shows the wage w, and ranges between 400 and 850. Coordinates are (employment, wage). An upward-sloping, concave curve passes through points A (10, 650), B (40, 762.50) and C (75, 780). At all points on this curve, profit is 1500. Other upward-sloping, concave curves lie at all points below the isoprofit curve for 1500, and these are the isoprofit curves for 3000, 5000, 8000, and 12000. Isoprofit curves for lower level of profits lie at all points below isoprofit curves for higher levels of profits. The zero-profit line consists of a vertical segment between (0, 0) and (0, 800), and a horizontal line starting from (0, 800).
Fullscreen

Figure E6.3 Isoprofit curves when \(\text{profit} = (800 \ – \ w) × N\).

The properties of isoprofit curves

The equation of the isoprofit curve corresponding to a particular level of profit, \(Π_0\), is:

\[(y - w)N = \Pi_0\]

We can work out the shape of the isoprofit curves by examining the algebraic properties of this equation. Since the graph is drawn with \(w\) on the vertical axis, it is helpful to rearrange the equation to obtain:

\[w = y - \frac{\Pi_0}{N}\]

This equation shows that \(w\) increases with \(N\) (because \(Π_0/N\) gets smaller). We can verify this by differentiating:

\[\frac{dw}{dN} = \frac{\Pi_0}{N^2} > 0\]

This tells us that isoprofit curves are upward-sloping. It is useful to note that the slope can be written in terms of w and N only by substituting back the original expression for profit, \(Π_0 = (y-w)N\):

\[\frac{dw}{dN} = \frac{(y - w)N}{N^2} = \frac{(y - w)}{N}\]

This second way of writing the slope shows us how it varies in different parts of the diagram without having to work out the level of profit at different points: the curves are flatter towards the top right, where \(N\) and \(w\) are both high.

Returning to the previous expression, we can determine the second derivative, which shows that as we move along an isoprofit curve the slope, \(dw/dN\) decreases—the curve gets flatter—as \(N\) increases:

\[\frac{d^2 w}{d N^2} = -\frac{\Pi_0}{N^3} < 0\]

In summary, these isoprofit curves are increasing and concave, in contrast to indifference curves which are typically decreasing and convex (as explained in Extension 5.4).

Lastly, the isoprofit curve equation tells us that if we draw a set of isoprofit curves for different values of \(Π_0\), higher levels of profit correspond to lower levels of the wage at each value of \(N\). In other words, the curves lower down in the figure have higher profit. To verify this algebraically, we can partially differentiate \(w\) with respect to \(Π_0\), holding \(N\) constant:

\[w = y - \frac{\Pi_0}{N} \Rightarrow \frac{\partial w}{\partial \Pi_0} = -\frac{1}{N} < 0\]

—that is to say, higher profit requires a lower wage at each level of employment.

Solving the firm’s constrained choice problem

The owners of the firm want to choose \(w\) and \(N\) from the firm’s feasible set to make the highest possible profit. The feasible set is the set of combinations of \(w\) and \(N\) that lie on or above the no-shirking wage curve.

In the model we have developed in this unit, the no-shirking wage is greater than the worker’s reservation wage because the wage has to allow for the cost of effort, and give the worker enough rent to ensure that it is better to work than shirk. So the no-shirking wage curve lies above and parallel to the reservation wage curve, which is an upward-sloping relationship between \(w\) and \(N\). To keep the analysis of profit maximization more general than in the main section above, we will not assume that it is a straight line. We will simply assume that \(w\) is an increasing function of \(N\):

\[w = W(N) \text{, where } W'(N) > 0\]

The parameters are: \(m\) and \(q\), which determine the firm’s recruitment and retention (Extension 6.5); and \(b\), \(ν\), and \(τ\) which determine workers’ reservation wages (Section 6.8). Extension 6.8 explains how all of these fit together to form the reservation wage curve. The no-shirking wage also depends on \(c\), \(h\), and \(s\) which affect the incentives for working hard (Section 6.9).

We know that the position and slope of the no-shirking wage curve depend on a number of different labour market parameters. But in this extension we will keep things simple by not including these parameters explicitly when we write the function, \(W\)(\(N\)); we focus only on the relationship between \(w\) and \(N\).

The firm faces a constrained choice problem of a similar form to the problem for a worker choosing consumption and free time to maximize utility (Extensions 3.5 and 5.5). There is one difference, though: consumption and free time are both ‘goods’ for workers, so they want both to be as large as possible, and the indifference curves slope upward. But the firm maximizes profit when \(N\) is as large as possible, and \(w\) is as small as possible, so its isoprofit curves slope upward.

Feasible combinations of \(w\) and \(N\) satisfy \(w ≥ W(N)\), but since the firm wants \(w\) to be as small as possible, we know it will choose a combination where \(w = W(N)\). So, as we did for Karim in Extension 3.5, we can write the constraint as an equality, rather than an inequality.

The employer’s constrained choice problem

Choose \(w\) and \(N\) to maximize \(\Pi(w, N)\) subject to the constraint \(w=W(N)\).

Since the profit function has a particularly simple form, we can solve this problem easily using the method of substitution. Profit is:

\[\Pi = (y - w)N\]

and substituting for w using the constraint gives profit as a function of N only:

\[\Pi = (y - W(N))N\]

To find the value of N that maximizes profit, we differentiate with respect to N (using the product rule) and set the derivative to zero:

\[\frac{d \Pi}{dN} = (y - W(N)) - W'(N)N = 0\]

Rearranging this equation, we get the first-order condition:

\[W'(N) = \frac{y - W(N)}{N}\]

which tells us that profit is maximized at the point on the no-shirking wage curve \(w = W(N)\) where its slope is equal to the slope of the isoprofit curve. (We showed above that the slope of an isoprofit curve at any point is given by \((y \ – \ w)/N\).)

If we know the explicit equation of the no-shirking wage curve, we can solve the pair of simultaneous equations (the first-order condition and the no-shirking wage equation) to find the particular values of \(w\) and \(N\) that the employer will choose. In Exercise E6.3, you can try this for the case of a linear no-shirking wage curve.

The second-order condition

We should check the second-order condition to be sure that the solution we have found is indeed a maximum point. Differentiating the expression for profit again gives:

\[\frac{d^2 \Pi}{dN^2} = -2W'(N) - W''(N)\]

For a maximum point, the second derivative must be negative. We know that \(W\)(\(N\)) is an increasing function, so the first term is certainly negative. However, if it is highly concave (that is, if \(W’’(N)\) is a large negative number) it is possible that the second term could be sufficiently positive to make the second derivative positive overall. In that case, we would have found a point of minimum profit. So to solve this constrained choice problem for any particular no-shirking wage curve, we ought to check that the second-order condition holds.

If you like, you can try sketching a diagram to show what happens when the no-shirking wage curve is highly concave. If it is sufficiently curved, the no-shirking wage curve lies below the isoprofit curve either side of the tangency point (rather than above it, as in Figure 6.16). Then, imagine yourself approaching the tangency point along the no-shirking wage curve from the left. As you cross the isoprofit curves, profit gets lower and lower—until you pass the tangency point, when it starts to rise again. The tangency point is a point of minimum profit.

Exercise E6.3 A linear no-shirking wage

Consider the linear case, where the no-shirking wage curve is given by the equation \(W = W_0 + W_1 N\) (where \(W_0 \text { and } W_1\) are positive constants).

  1. Find the profit-maximizing wage and employment in terms of \(W_0\) and \(W_1\). (Make sure to check that the second-order condition is satisfied.)
  2. Explain how the profit-maximizing wage and employment change if the no-shirking wage curve shifts upwards.
  3. Explain what happens if the slope of the no-shirking wage curve increases.

Read more: Chapter 8 (on curve-sketching, and finding maxima and minima) of Malcolm Pemberton and Nicholas Rau. Mathematics for economists: An introductory textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.