The labour supply curve and its elasticity

The labour supply curve shows the hours a person chooses to work at each wage, holding non-labour income and preferences fixed. It starts at the reservation wage, slopes upward while the substitution effect of a raise dominates, and bends backward if the income effect takes over. The labour supply elasticity, $\sigma$, is the percentage change in hours divided by the percentage change in the wage.

From indifference curves to the supply curve

A change in the wage rotates the budget line $C = w(T - L) + V$ around the endowment point $E = (T, V)$. Each wage produces one optimal bundle: the tangency $MU_L/MU_C = w$, or the corner at $E$ when the wage is below the reservation wage. Recording the hours $h = T - L^*$ at each wage gives the supply curve. Every point on the supply curve is a tangency in the consumption-leisure diagram.

A raise has two effects on someone who already works (income and substitution effects has the full decomposition):

• The substitution effect: leisure now costs more forgone consumption, so the person works more.
• The income effect: at current hours the person is richer by about $h \times \Delta w$, and with leisure a normal good they buy more leisure, so they work less.

The income effect is proportional to hours worked. Just above the reservation wage hours are few, the income effect is small and the substitution effect dominates: the curve slopes upward. At high wages and long hours the income effect is large and can dominate: hours fall as the wage rises and the curve becomes a backward-bending labour supply curve.

Wage per hour (w)

$12.00

Leisure L

73.4

Hours worked h = T − L

36.6

Consumption C

$1,239

Open in Graph Lab

Figure 1. A worker with 110 hours a week and $800 of non-labour income, with preferences in which the income effect eventually dominates. Each wage gives a tangency $P$; the purple curve is the path of optimal points for every wage between the $5 reservation wage and the wage shown. At $12 she works 36.6 hours. The path moves left (more work) until the wage reaches $20 and turns back toward more leisure above it.

For these preferences ($U = -1/C + L/128{,}000$), hours at any wage above $5 are $h = (\sqrt{128{,}000\,w} - 800)/w$, which peaks at 40 hours when $w = 20$.

Wage per hour (w)

$12.00

Hours worked h

36.6

Labour supply elasticity at this wage

0.41

Open in Graph Lab

Figure 2. The same worker's labour supply curve. Hours are zero below the $5 reservation wage, rise steeply while the substitution effect dominates, peak at 40 hours at $20, and fall to 32.9 hours at $60 as the income effect takes over. The elasticity is 0.41 at a wage of $12 and −0.25 at $45.

Key idea

Each point of a labour supply curve is a tangency between an indifference curve and a budget line rotated around the endowment point. The curve slopes up where the substitution effect dominates and bends back where the income effect, which grows with hours worked, dominates.

Market labour supply

The market labour supply curve is the horizontal sum of individual curves: at each wage, add the hours of every person, counting zero for anyone below their reservation wage. Adding wages at a given number of hours has no meaning.

Market supply responds at both margins. A raise changes the hours of people already working (intensive margin) and draws in people whose reservation wages it passes (extensive margin). Each entry adds a kink at which the market curve becomes flatter. The market curve can slope upward even where some individuals’ curves bend backward.

Wage per hour (w)

$16.00

Ana's hours

22.0

Ben's hours

6.0

Market hours

28.0

Open in Graph Lab

Figure 3. Ana supplies $h = 2(w - 5)$ hours above her $5 reservation wage and Ben supplies $h = 1.5(w - 12)$ above his $12 reservation wage. The market curve is $2w - 10$ below $12 and $3.5w - 28$ above, with a kink where Ben starts work. At $16 the market supplies 22 + 6 = 28 hours.

The labour supply elasticity

$$\sigma = \frac{\Delta h / h}{\Delta w / w} = \frac{\%\Delta h}{\%\Delta w}.$$

The sign records which effect dominates: $\sigma > 0$ when the substitution effect dominates, $\sigma < 0$ when the income effect does. The size records responsiveness: supply is inelastic when $|\sigma| < 1$ and elastic when $|\sigma| > 1$. An elasticity of 0.1 means a doubling of the wage (a 100 percent rise) raises hours by 10 percent.

Borjas’s reading of the evidence puts the elasticity for men’s annual hours at about $-0.1$: hours barely respond, and the income effect slightly dominates. Estimates for women centre around 0.2. Most of women’s response is at the extensive margin; women who already work change their hours about as little as men do.¹ The model explains the gap. Historically many women were near their reservation wages, so a raise drew some into work, while nearly all prime-age men worked already and could respond only in hours. Women who worked fewer hours also had a smaller income effect to offset the substitution effect.

Wage per hour (w)

$30.00

Change in the wage from $25

20.0%

Change in men's hours

−1.8%

Change in women's hours

3.7%

Open in Graph Lab

Figure 4. Constant-elasticity supply curves with elasticities of −0.1 for men and 0.2 for women, through 40 and 30 hours at a wage of $25. A 20 percent raise to $30 lowers men's hours by 1.8 percent and raises women's by 3.7 percent. Both curves are steep: labour supply is inelastic for both groups.

The gap has narrowed as women’s attachment to the labour force has grown. In US data, Blau and Kahn find that married women’s own-wage elasticity fell by 50 to 56 percent between 1980 and 2000, and their response to husbands’ wages also fell.² With fewer women near the participation margin, their supply behaves more like men’s.

Why the elasticity is hard to estimate

Estimates come from regressions such as $h_i = \beta w_i + \gamma V_i + \ldots$ across workers, with $\sigma = \beta \times w/h$. Published estimates range widely, from negative to positive, because of four problems:

1. Measurement error. Hours and wages are reported with error, and the wage is often computed as earnings divided by hours. An over-reported hours figure then lowers the computed wage, creating a spurious negative relation between hours and wages. This division bias pushes estimated elasticities down.³
2. Selection. Wages are observed only for people who work. Non-workers also respond to wages, at the extensive margin, and workers are not a random sample: they tend to have higher wage offers or lower reservation wages. Ignoring this biases estimates.⁴
3. Endogenous non-labour income. Much non-labour income is the return on past saving, and people with a strong taste for work both work more and save more. The estimated income effect, and with it the wage effect, is contaminated by tastes.
4. The time horizon. Hours can be measured per day, week or year. Responses are larger over longer horizons, when people can also change weeks worked and whether they work at all, so the elasticity depends on the horizon chosen.

Worked example

1. Elasticity from two points. A worker’s wage rises from $20 to $24 and her hours fall from 40 to 39 a week. The wage rises by $4/20 = 20\%$ and hours change by $-1/40 = -2.5\%$, so $\sigma = -2.5/20 = -0.125$. The income effect dominates, and supply is inelastic.

2. The midpoint version. Measured from the averages, hours change by $-1/39.5 = -2.53\%$ and the wage by $4/22 = 18.2\%$, so $\sigma = -0.139$. State which method you use; the two agree only for small changes.

3. Market supply. With Ana and Ben from Figure 3, market hours at $16 are $2(11) + 1.5(4) = 28$ and at $20 are $2(15) + 1.5(8) = 42$.

4. Market elasticity. From $16 to $20 the wage rises 25 percent and market hours rise $14/28 = 50$ percent, so the market elasticity is $50/25 = 2.0$. Ana’s elasticity is $(8/22)/0.25 = 1.45$ and Ben’s is $(6/6)/0.25 = 4.0$, high because he is close to his reservation wage. The market elasticity is their average weighted by initial hours: $\tfrac{22}{28}(1.45) + \tfrac{6}{28}(4.0) = 2.0$.

Common mistakes

Traps on exams

• Calling a backward bend an inferior good. A negative elasticity requires leisure to be normal: the income effect, toward more leisure, beats the substitution effect.
• Adding vertically. Market supply adds hours at a given wage. Count zero, not a negative number, for anyone below their reservation wage.
• Confusing a shift with a movement. A wage change is a movement along the supply curve. A change in non-labour income or preferences shifts it.
• Using levels instead of percentages. $\Delta h/\Delta w$ is a slope, not an elasticity.

On the exam

Draw the consumption-leisure diagram with three budget lines rotated around $E$ and their three tangencies. Then draw the hours-wage diagram and plot each tangency’s hours, $T - L^*$, against its wage. Label the reservation wage where the curve meets the vertical axis and mark where the substitution and income effects dominate.

Check yourself

1.

Why can an individual's labour supply curve bend backward at high wages?

1The income effect of a raise grows with hours worked, so at high wages and hours it can outweigh the substitution effect2At high wages leisure becomes an inferior good3The substitution effect reverses sign once the wage is high enough4Employers offer fewer hours at high wages

Check answer

2.

A worker's wage rises from $20 to $24 an hour and her hours fall from 40 to 39 a week. Using percentage changes measured from the initial values, what is her labour supply elasticity?

Your answer

Check answer

3.

Borjas summarizes the evidence as an elasticity of about $-0.1$ for men and about $0.2$ for women. Which explanation of the difference is most consistent with the model?

1Many women were close to their reservation wages, so a raise drew some into work, while nearly all prime-age men already worked2Women's indifference curves are tangent to the budget line while men's are not3Leisure is a normal good for men and an inferior good for women4Men's wages are measured with less error, which biases their elasticity toward zero

Check answer

Practise more questions on this topic

Sources

1. Borjas, G. J. Labor Economics, 8th edition. McGraw-Hill Education, chapter 2. ↩

2. Blau, F. D. and Kahn, L. M. (2007). Changes in the labor supply behavior of married women: 1980 to 2000. Journal of Labor Economics, 25(3), 393 to 438. NBER working paper 11230. ↩

3. Borjas, G. J. (1980). The relationship between wages and weekly hours of work: The role of division bias. Journal of Human Resources, 15(3), 409 to 423. IDEAS record. ↩

4. Heckman, J. J. (1979). Sample selection bias as a specification error. Econometrica, 47(1), 153 to 161. IDEAS record. ↩