Elasticity of labour demand

The elasticity of labour demand is the percentage change in employment caused by a 1 percent change in the wage, $\delta = \%\Delta E / \%\Delta w$. It is negative and larger in absolute value in the long run than in the short run. It is larger the more easily other inputs replace labour and the more elastic the demand for the product labour makes.

Why the size matters

Policies that raise the cost of labour, such as a minimum wage or a payroll tax, cost few jobs when labour demand is inelastic and many when it is elastic. The elasticity also decides what a wage increase does to total labour earnings. With $|\delta|$ below 1, a 10 percent raise cuts employment by less than 10 percent, so the wage bill $wE$ rises; with $|\delta|$ above 1, the wage bill falls.

Measuring the response

Demand is elastic when $\delta$ is less than $-1$, inelastic when it lies between $-1$ and $0$, and unit elastic at $-1$. The short-run elasticity $\delta_{SR}$ holds capital fixed; the long-run elasticity $\delta_{LR}$ lets it adjust. Because the long run adds a substitution effect and a fuller scale response (see long-run labour demand), $|\delta_{LR}| > |\delta_{SR}|$.

Wage (w), was $20

$24.00

Change in the wage

20.0%

Change in employment, short run

−8.0%

Change in employment, long run

−20.0%

Open in Graph Lab

Figure 1. Short-run and long-run labour demand through the same point A, with elasticities of $-0.4$ and $-1$ at A. A 20 percent wage increase cuts employment by 8 percent along the short-run curve and by 20 percent along the long-run curve.

What the estimates show

The course textbook summarizes the evidence as a short-run elasticity of about $-0.4$ to $-0.5$ and a long-run elasticity of about $-1$, with roughly one third of the long-run response coming from the substitution effect and two thirds from the scale effect.¹ Two surveys frame those numbers:

• Hamermesh concluded that the long-run elasticity at constant output, which isolates the substitution effect, lies between $-0.15$ and $-0.5$.²
• A meta-analysis of more than 900 published estimates from 105 studies found a mean of $-0.51$ and a median of $-0.39$, with more than 80 percent of estimates between 0 and $-1$. Demand was more elastic in the intermediate and long run than in the short run, and the authors found evidence that publication bias inflates reported elasticities.³

The elasticity of substitution

How much the input mix responds to relative prices is measured by the elasticity of substitution,

$$\sigma = \frac{\%\Delta (K/E)}{\%\Delta (w/r)},$$

the percentage change in the capital-labour ratio caused by a 1 percent change in the price of labour relative to capital, with output held constant. With $\sigma = 0.5$, a 10 percent rise in $w/r$ raises $K/E$ by about 5 percent. For large changes the percentage formula is only approximate: with a constant $\sigma = 0.5$, a 50 percent rise in $w/r$ raises $K/E$ by 22 percent, not 25.

$\sigma$ reflects the curvature of the isoquant:

• Perfect complements, $\sigma = 0$: right-angled isoquants; the input ratio never changes.
• Perfect substitutes, $\sigma \to \infty$: straight-line isoquants; a small price change can switch the firm entirely from one input to the other.
• Any convex isoquant has $\sigma > 0$. The Cobb-Douglas technology $q = E^{\alpha}K^{1-\alpha}$ has $\sigma = 1$.

Elasticity of substitution (σ)

0.5

Relative wage (w/r), was 1

1.50

Capital-labour ratio K/E (was 1)

1.22

Change in w/r

50%

Change in K/E

22%

Open in Graph Lab

Figure 2. An isoquant through $E = K = 40$ and the cost-minimizing mix after $w/r$ rises from 1 to 1.5. With $\sigma = 0.5$ the capital-labour ratio rises by 22 percent; with $\sigma = 4$ it rises about fivefold. As $\sigma$ falls toward 0 the isoquant approaches a right angle (perfect complements); as $\sigma$ rises it approaches a straight line (perfect substitutes).

For capital and labour in aggregate, Chirinko’s review of the estimates concludes that $\sigma$ most likely lies between 0.4 and 0.6, and finds little support for the Cobb-Douglas value of 1.⁴

The Hicks-Marshall rules of derived demand

Four conditions, known as the Hicks-Marshall rules of derived demand (Marshall’s rules), determine how elastic the demand for an input is. Labour demand is more elastic when:

1. The elasticity of substitution is larger. Firms can replace workers with machines more easily.
2. The demand for the output is more elastic. A wage increase raises marginal cost and the product price; with elastic product demand, sales and employment fall further.
3. Labour’s share of total costs is larger. A raise for workers who are a small share of costs barely moves the product price (the “importance of being unimportant”). This rule holds only when the product demand elasticity exceeds the elasticity of substitution.
4. The supply of other inputs is more elastic. If buying more machines drives up their price, substitution away from labour is choked off.

With two inputs, constant returns to scale and perfectly elastic capital supply, rules 1 to 3 combine into one formula. With $s$ labour’s share of costs and $\eta$ the absolute elasticity of product demand,

$$\delta = -\left[(1 - s)\,\sigma + s\,\eta\right],$$

where $-(1 - s)\sigma$ is the substitution effect and $-s\eta$ the scale effect. Raising $s$ changes $|\delta|$ by $\eta - \sigma$, which is why rule 3 needs $\eta > \sigma$.

Elasticity of substitution (σ)

0.5

Elasticity of product demand (η)

1.5

Labour's share of costs (s)

0.70

Elasticity of labour demand (δ)

−1.20

Substitution part, −(1 − s)σ

−0.15

Scale part, −sη

−1.05

Open in Graph Lab

Figure 3. The absolute elasticity of labour demand, $(1 - s)\sigma + s\eta$, against labour's share of costs, with $\sigma = 0.5$ and $\eta = 1.5$. The lower band is the substitution effect and the upper band the scale effect. At $s = 0.7$, $\delta = -1.2$, of which $-1.05$ comes from scale. The line slopes up because $\eta$ exceeds $\sigma$; with $\sigma$ above $\eta$ it slopes down.

Key idea

A group of workers can win a raise at little cost in jobs when machines are poor substitutes for them, demand for their product is inelastic, they are a small share of costs (when $\eta > \sigma$) and other inputs are inelastically supplied.

Several inputs and cross-elasticities

With many inputs (skilled and unskilled labour, young and old workers, old and new machines), each input $i$ is hired until $w_i = p \times MP_i$, and every earlier result carries over: demand curves slope down, long-run demand is more elastic, and price changes have scale and substitution effects. What is new is the cross-elasticity of factor demand,

$$\delta_{ij} = \frac{\%\Delta x_i}{\%\Delta w_j},$$

the percentage change in the quantity of input $i$ from a 1 percent change in the price of input $j$. If $\delta_{ij} > 0$ the inputs are substitutes: a higher price of $j$ shifts the demand for $i$ up. If $\delta_{ij} < 0$ they are complements: a higher price of $j$ shifts the demand for $i$ down.

Price of capital (index, was 100)

70

Change in the price of capital

−30%

Unskilled employment at a $15 wage (was 100k)

83.7k

Skilled employment at a $35 wage (was 100k)

119.5k

Open in Graph Lab

Figure 4. Demand for unskilled workers (substitutes for capital, cross-elasticity 0.5) and skilled workers (complements, cross-elasticity $-0.5$) after the price of capital falls 30 percent. At unchanged wages, unskilled employment falls from 100 to 84 thousand and skilled employment rises from 100 to 120 thousand.

Capital-skill complementarity

The capital-skill complementarity hypothesis says that capital substitutes for unskilled labour but complements skilled labour. Griliches proposed it from evidence that capital was more substitutable for unskilled than for skilled workers in US manufacturing.⁵ Hamermesh’s survey reached the same conclusion: skilled labour and physical capital are complements.² The hypothesis implies that cheaper capital raises the demand for skilled workers and lowers it for unskilled workers. Krusell, Ohanian, Ríos-Rull and Violante built this technology into a model of the US economy and found that the growth of capital equipment, together with changes in the relative supply of skilled workers, accounts for nearly all of the movement in the skill premium over the three decades they studied.⁶

U.S. college wage premium

Median weekly earnings, bachelor's degree or higher ÷ high school only, full-time workers aged 25 and over

Figure 5. US full-time workers with a bachelor's degree or more have earned between about 1.7 and 1.9 times the median weekly earnings of high-school graduates in every quarter since 2000. A persistently large premium is consistent with capital-skill complementarity, but the ratio alone cannot separate that cause from changes in the supply of graduates.Source: FRED, Employed full time: Median usual weekly nominal earnings (second quartile): Wage and salary workers, 25 years and over: Bachelor's degree and higher (LEU0252918500Q); High School graduates, no college (LEU0252917300Q), U.S. BLS Current Population Survey. Retrieved 2026-10-23.

Show the data and notes

Ratio of the two published medians (quarterly, not seasonally adjusted). High school only = high school graduates with no college. Raw ratio: it does not control for age, experience or occupation. Not published: 2025-Q4 (October 2025 Current Population Survey data were not collected during the federal government shutdown).

Period	Bachelor's or higher ÷ high school
2000 Q1	1.761
2000 Q2	1.753
2000 Q3	1.771
2000 Q4	1.767
2001 Q1	1.789
2001 Q2	1.757
2001 Q3	1.778
2001 Q4	1.746
2002 Q1	1.764
2002 Q2	1.765
2002 Q3	1.769
2002 Q4	1.740
2003 Q1	1.750
2003 Q2	1.771
2003 Q3	1.702
2003 Q4	1.739
2004 Q1	1.772
2004 Q2	1.689
2004 Q3	1.714
2004 Q4	1.714
2005 Q1	1.750
2005 Q2	1.705
2005 Q3	1.739
2005 Q4	1.759
2006 Q1	1.721
2006 Q2	1.772
2006 Q3	1.739
2006 Q4	1.752
2007 Q1	1.711
2007 Q2	1.829
2007 Q3	1.784
2007 Q4	1.780
2008 Q1	1.802
2008 Q2	1.782
2008 Q3	1.830
2008 Q4	1.801
2009 Q1	1.835
2009 Q2	1.810
2009 Q3	1.844
2009 Q4	1.757
2010 Q1	1.827
2010 Q2	1.809
2010 Q3	1.862
2010 Q4	1.799
2011 Q1	1.817
2011 Q2	1.774
2011 Q3	1.811
2011 Q4	1.807
2012 Q1	1.773
2012 Q2	1.766
2012 Q3	1.806
2012 Q4	1.805
2013 Q1	1.826
2013 Q2	1.844
2013 Q3	1.781
2013 Q4	1.881
2014 Q1	1.817
2014 Q2	1.782
2014 Q3	1.718
2014 Q4	1.843
2015 Q1	1.834
2015 Q2	1.785
2015 Q3	1.836
2015 Q4	1.804
2016 Q1	1.841
2016 Q2	1.810
2016 Q3	1.809
2016 Q4	1.819
2017 Q1	1.823
2017 Q2	1.797
2017 Q3	1.780
2017 Q4	1.790
2018 Q1	1.804
2018 Q2	1.804
2018 Q3	1.818
2018 Q4	1.796
2019 Q1	1.827
2019 Q2	1.807
2019 Q3	1.846
2019 Q4	1.850
2020 Q1	1.805
2020 Q2	1.797
2020 Q3	1.837
2020 Q4	1.819
2021 Q1	1.801
2021 Q2	1.813
2021 Q3	1.804
2021 Q4	1.765
2022 Q1	1.844
2022 Q2	1.846
2022 Q3	1.797
2022 Q4	1.768
2023 Q1	1.834
2023 Q2	1.786
2023 Q3	1.790
2023 Q4	1.754
2024 Q1	1.865
2024 Q2	1.838
2024 Q3	1.794
2024 Q4	1.745
2025 Q1	1.841
2025 Q2	1.804
2025 Q3	1.783
2026 Q1	1.805
2026 Q2	1.779
2026 Q3	1.764

Worked example

1. Labour demand elasticity. The wage rises from $20 to $22 (10 percent) and employment falls from 50,000 to 47,000 (6 percent). Then $\delta = -6 / 10 = -0.6$: inelastic. The hourly wage bill rises from $20 \times 50{,}000 = 1{,}000{,}000$ dollars to $22 \times 47{,}000 = 1{,}034{,}000$ dollars.

2. Elasticity of substitution. The ratio $w/r$ rises from 2.0 to 2.2 (10 percent) and the firm’s $K/E$ rises from 0.50 to 0.53 (6 percent). Then $\sigma = 6 / 10 = 0.6$.

3. Hicks-Marshall decomposition. With $s = 0.7$, $\sigma = 1$ and $\eta = 1$, the substitution effect is $-(1 - 0.7)(1) = -0.3$ and the scale effect is $-(0.7)(1) = -0.7$, so $\delta = -1.0$. The scale effect supplies 70 percent of the response, close to the textbook’s two thirds.

4. Cross-elasticity. The price of capital rises 20 percent and employment of unskilled workers rises 5 percent. Then $\delta_{ij} = 5 / 20 = 0.25$: positive, so unskilled labour and capital are substitutes.

Exercise: which change raises the elasticity more?

With $s = 0.7$, $\sigma = 0.5$ and $\eta = 1$, $|\delta| = 0.3(0.5) + 0.7(1) = 0.85$. Doubling $\sigma$ to 1 gives $0.3(1) + 0.7(1) = 1.0$, an increase of 0.15. Raising $\eta$ to 1.5 instead gives $0.3(0.5) + 0.7(1.5) = 1.2$, an increase of 0.35. When labour is a large share of costs, product demand matters more than substitution.

Common mistakes

Traps on exams

• Sign and size. $\delta$ is negative; “more elastic” means larger in absolute value, so $-1.2$ is more elastic than $-0.4$.
• Confusing $\sigma$ with $\delta$. $\sigma$ is the response of the input ratio $K/E$ to $w/r$ at constant output; $\delta$ is the response of employment to the wage, including the scale effect.
• Applying the importance of being unimportant everywhere. A larger labour share raises the elasticity only when $\eta > \sigma$.
• Reversing the cross-elasticity sign. Positive means substitutes; negative means complements.

Check yourself

1.

The wage in an industry rises from $25.00 to $27.00 an hour and employment falls from 80,000 to 72,000 workers. Using the initial values as the base, what is the elasticity of labour demand?

Your answer

Check answer

2.

What are the elasticities of substitution for perfect complements and perfect substitutes?

1Zero for perfect complements; it approaches infinity for perfect substitutes2Infinity for perfect complements; zero for perfect substitutes3One for both, because both have constant MRTS4Negative for perfect complements; positive for perfect substitutes

Check answer

3.

When the price of computers rises, firms hire fewer software developers. What are the cross-elasticity and the relationship between the two inputs?

1The cross-elasticity is negative, so computers and developers are complements2The cross-elasticity is positive, so computers and developers are substitutes3The cross-elasticity is negative, so computers and developers are substitutes4The cross-elasticity is zero, because developers are labour and computers are capital

Check answer

Practise more questions on this topic

Sources

1. Borjas, G. J. Labor Economics, 8th edition, chapter 3. McGraw-Hill. ↩

2. Hamermesh, D. S. (1986). The demand for labor in the long run. In O. Ashenfelter and R. Layard (eds.), Handbook of Labor Economics. North-Holland. NBER working paper 1297. ↩ ↩²

3. Lichter, A., Peichl, A. and Siegloch, S. (2014). The own-wage elasticity of labor demand: A meta-regression analysis. IZA Discussion Paper 7958. A revised version appeared in European Economic Review, 80 (2015). ↩

4. Chirinko, R. S. (2008). σ: The long and short of it. Journal of Macroeconomics, 30(2), 671 to 686. CESifo working paper 2234. ↩

5. Griliches, Z. (1969). Capital-skill complementarity. Review of Economics and Statistics, 51(4), 465 to 468. ↩

6. Krusell, P., Ohanian, L. E., Ríos-Rull, J.-V. and Violante, G. L. (2000). Capital-skill complementarity and inequality: A macroeconomic analysis. Econometrica, 68(5), 1029 to 1053. Federal Reserve Bank of Minneapolis Staff Report 239. ↩