Compensating wage differentials and risky jobs

A compensating wage differential is the difference in pay between two jobs that offsets a difference in their non-wage characteristics, such as the risk of injury. In a competitive market, jobs with worse conditions pay more, and the size of the premium is set by the marginal worker and the marginal firm, not by the average worker. The differential can even be negative if enough workers like the condition.

The intuition

Adam Smith stated the idea in 1776: “The whole of the advantages and disadvantages of the different employments of labour and stock, must, in the same neighbourhood, be either perfectly equal, or continually tending to equality.”¹ If one job were better overall, workers would crowd into it and push its wage down; if one were worse, workers would leave until its wage rose. What tends to equality is the whole package, not the wage. Smith’s own example: the public executioner, holder of “the most detestable of all employments”, was better paid for the work done than any common trade.

Two kinds of differences drive the result. Workers differ in how much they mind a job characteristic. Firms differ in how costly it is to remove it. The market sorts workers who mind risk least into the jobs where safety is most expensive, and the wage gap that clears the market compensates the last worker it needs to attract.

Workers: the reservation price for risk

Suppose there are two kinds of jobs. A safe job has injury probability $\rho_0 = 0$ and pays $w_0$; a risky job has injury probability $\rho_1 > 0$ and pays $w_1$. Utility depends on the wage, a good, and on the risk of injury, a bad: $U = f(w, \rho)$ with $\partial U/\partial w > 0$ and $\partial U/\partial \rho < 0$.

In a graph with $\rho$ on the horizontal axis and $w$ on the vertical axis, indifference curves therefore slope upward: extra risk must be offset by extra pay to hold utility constant. Utility rises toward the north-west, where pay is higher and risk lower.

The indifference curve through the safe job, $U_0$, reaches the risky job’s injury rate at the wage $\hat w_1$. At $\hat w_1$ the worker is indifferent between the two jobs. Her reservation price for risk is

$$\Delta \hat w = \hat w_1 - w_0,$$

the smallest wage gap that persuades her to take the risky job. A risky job paying more than $\hat w_1$ puts her on a higher indifference curve than the safe job; one paying less leaves her in the safe job.

Wage offered in the risky job, w₁

$24.00

A's reservation price, ŵ_A − w₀

$6.00

B's reservation price, ŵ_B − w₀

$2.40

Differential offered, w₁ − w₀

$4.00

Workers who take the risky job

1

Open in Graph Lab

Figure 1. Indifference curves through a safe job paying $20 an hour. At the risky job's 5 percent injury rate the risk-averse worker A needs $26.00 and the risk-tolerant worker B $22.40, so their reservation prices are $6.00 and $2.40, and an offer of $24 attracts B but not A.

The slope of an indifference curve is the extra pay a worker needs per unit of extra risk. A risk-averse worker has steep curves and a large reservation price; a risk-tolerant worker has flat curves and a small one. Indifference curves of different workers can cross, because they describe different people.

Some workers like risk: a stunt performer or a mountain guide may value danger for its own sake. For a risk lover, $\partial U/\partial \rho > 0$, so indifference curves slope downward and the reservation price is negative: he would take the risky job at a lower wage than the safe job pays.

Injury rate in the risky job, ρ₁ (percent)

5.0

A's reservation price, ŵ_A − w₀

$6.00

C's reservation price, ŵ_C − w₀

−$2.40

Open in Graph Lab

Figure 2. Worker A dislikes risk, so her indifference curve slopes up; worker C likes risk, so his slopes down. At a 5 percent injury rate A's reservation price is $6.00 and C's is −$2.40, and both grow in size as the job becomes riskier.

Adding up workers gives the supply of labour to risky jobs. At a small differential only the most risk-tolerant workers take risky jobs; as the differential rises, workers with higher reservation prices join. Supply slopes upward against the differential $w_1 - w_0$.

Firms: safe or risky production

A firm’s output is $q = \alpha_0 E$ if its workplace is safe and $q = \alpha_1 E$ if it is risky, with $\alpha_1 > \alpha_0$: safety procedures and equipment take time that could go into production. The firm sells output at price $p$, so profit is $\pi_0 = p\alpha_0 E - w_0 E$ if safe and $\pi_1 = p\alpha_1 E - w_1 E$ if risky. Risky production is more profitable when

$$p\alpha_1 E - w_1 E > p\alpha_0 E - w_0 E \iff w_1 - w_0 < p(\alpha_1 - \alpha_0).$$

So the firm should:

• offer risky jobs if $w_1 - w_0 < p(\alpha_1 - \alpha_0)$,
• offer safe jobs if $w_1 - w_0 > p(\alpha_1 - \alpha_0)$.

The firm weighs the wage premium for risk against the value of the extra output risk allows. A higher output price or a larger productivity cost of safety pushes it toward risky jobs; a higher market premium for risk pushes it toward safe ones. Firms differ in how costly safety is, so $p(\alpha_1 - \alpha_0)$ differs across firms. As the differential rises, more firms switch to safety, and the demand for labour in risky jobs slopes downward against the differential. The firms that stay risky are those for which safety is most expensive.

Market equilibrium

Workers' risk tolerance

0.25

Largest gain to a firm from risky production, p(α₁ − α₀)

$6.00

Market compensating differential

$2.40

Workers in risky jobs

24.0k

Workers who would take a risky job at a zero premium

0.0k

Risky jobs firms would fill at a zero premium

40.0k

Open in Graph Lab

Figure 3. Supply of and demand for workers in risky jobs, with the wage gap $w_1 - w_0$ on the vertical axis. With risk tolerance at 0.25 the market differential is $2.40 and 24,000 workers hold risky jobs; at full risk tolerance 60,000 workers would take a risky job for no premium but firms want only 40,000 at that price, and the differential falls to −$1.20.

The market compensating differential is where supply meets demand. It equals the reservation price of the marginal worker and the productivity gain of the marginal firm. Three consequences follow.

1. Inframarginal workers earn rents. Every worker in a risky job other than the marginal one has a lower reservation price than the market pays. The shaded triangle in Figure 3 is their total gain.
2. The most risk-averse workers stay in safe jobs. No risky job pays enough to attract them.
3. The differential can be negative. If some workers like risk, supply starts below zero. When more workers would take risky jobs at a zero premium than firms want to fill at that premium, the market clears below zero, and risky jobs pay less than safe ones. With every worker averse to risk, the differential must be positive.

Key idea

The market compensating differential compensates the marginal worker, not the average one. It is negative only if risk lovers alone can fill the risky jobs firms want.

Worked example

Output sells for $10. A firm’s workers produce 4 units an hour in a safe workplace and 4.4 units in a risky one. In the market for risky jobs, supply is $w_1 - w_0 = 0.1E$ and demand is $w_1 - w_0 = 6 - 0.15E$, with $E$ in thousands of workers.

1. The firm’s threshold. $p(\alpha_1 - \alpha_0) = 10 \times (4.4 - 4) = 4$. The firm offers risky jobs if the differential is below $4.00 an hour.

2. Market equilibrium. $0.1E = 6 - 0.15E$ gives $E = 6/0.25 = 24$ thousand workers and a differential of $0.1 \times 24 = 2.40$ dollars an hour. Since $2.40 is below $4.00, the firm in step 1 is risky; a firm whose threshold is $1.50 is safe.

3. Rents. The 10,000th worker in the supply ranking has a reservation price of $0.1 \times 10 = 1.00$ dollar. She receives $2.40, a rent of $1.40 an hour. The 24,000th worker receives exactly his reservation price.

4. Risk lovers. Suppose instead supply is $w_1 - w_0 = -6 + 0.1E$: the first 60,000 workers would accept some pay cut to work in risky jobs. Then $-6 + 0.1E = 6 - 0.15E$ gives $E = 48$ thousand and a differential of $-6 + 4.8 = -1.20$ dollars. At a zero premium, 60,000 workers want risky jobs and firms want only $6/0.15 = 40$ thousand, so the premium must be negative.

Exercise: demand for risky labour falls

With the original supply, demand falls to $w_1 - w_0 = 4 - 0.15E$. Then $0.1E = 4 - 0.15E$ gives $E = 16$ thousand and a differential of $1.60. With fewer risky jobs to fill, the marginal worker is a more risk-tolerant one, so the premium falls.

Evidence

Explicit compensating differentials are common in pay rules. Federal public servants posted to designated isolated locations receive allowances on top of salary, including an environment allowance and a living cost differential.² Smith’s examples were the same kind: butchering, “a brutal and an odious business”, was “in most places more profitable than the greater part of common trades”.¹

For the risk of death, the evidence is extensive. A review of more than 60 studies of mortality risk across ten countries found risk premiums in developed countries typically between 1 and 2 percent of labour income, and noted that after the United States no country has been studied more than Canada.³ Evidence for other job characteristics is weaker and often has the wrong sign, for reasons covered in amenities, benefits and layoffs. How the wage-risk relationship is estimated, and turned into a value of a statistical life, is the subject of hedonic wages and the value of life.

Common mistakes

Traps on exams

• Putting the wage on the vertical axis of the market diagram. It is the differential $w_1 - w_0$. Supply and demand are for workers in risky jobs, not for all labour.
• Steep means small. A steep wage-risk indifference curve means the worker needs a large wage increase per unit of risk, so her reservation price is large.
• Downward-sloping indifference curves for everyone. Only risk lovers have them. For a worker who dislikes risk, wage-risk indifference curves slope up.
• The differential compensates everyone. It exactly compensates only the marginal worker; inframarginal workers earn rents.
• Higher productivity means risky. A firm with $\alpha_1 > \alpha_0$ still chooses safety if the differential exceeds $p(\alpha_1 - \alpha_0)$.

On the exam

For the worker’s problem, label the axes $\rho$ and $w$, mark the safe job at $(0, w_0)$, draw the vertical line at $\rho_1$, and label $\hat w_1$ and $\Delta \hat w$. For the market, label the vertical axis $w_1 - w_0$, draw supply up and demand down, and mark the equilibrium differential and the number of workers in risky jobs. If asked about a negative differential, start supply below zero and make demand small.

Check yourself

1.

True or false: a more risk-averse worker has steep wage-risk indifference curves because she needs only a small compensating differential to accept a risky job.

1True2False

Check answer

2.

A competitive firm sells output at price $p$. Each worker produces $\alpha_0$ units an hour in a safe workplace and $\alpha_1 > \alpha_0$ units in a risky one, because safety procedures take time. Safe jobs pay $w_0$ and risky jobs pay $w_1$. When does the firm choose risky production?

1When $w_1 - w_0 < p(\alpha_1 - \alpha_0)$2When $w_1 - w_0 > p(\alpha_1 - \alpha_0)$3When risky jobs pay less than safe jobs4Always, because $\alpha_1 > \alpha_0$

Check answer

3.

Which conditions can make the market compensating differential for risky jobs negative? Select all that apply.

Select all that apply.

1Some workers like risk, so their reservation prices are negative2Firms want few risky workers relative to the number who like risk3Every worker dislikes risk, but some dislike it only a little4Risky firms are far more productive than safe firms5Labour supply to risky jobs is perfectly elastic at a positive differential

Check answer

Practise more questions on this topic

Sources

1. Smith, A. (1776). An Inquiry into the Nature and Causes of the Wealth of Nations, Book I, chapter 10. Project Gutenberg edition. ↩ ↩²

2. National Joint Council of the Public Service of Canada, Isolated Posts and Government Housing Directive. ↩

3. Viscusi, W. K. and Aldy, J. E. (2003). The value of a statistical life: A critical review of market estimates throughout the world. Journal of Risk and Uncertainty, 27(1), 5 to 76. NBER working paper 9487. ↩