Long-run labour demand

Long-run labour demand is the firm’s employment at each wage when it can change both its workforce $E$ and its capital $K$. The firm produces every output with the cheapest input mix, where $MP_E / w = MP_K / r$, and hires each input until the value of its marginal product equals its price. A wage change then works through a scale effect and a substitution effect, which makes long-run demand more elastic than short-run demand.

The intuition

A firm that can adjust everything responds to cheaper labour in two ways. Its costs fall, so it produces more and uses more of every normal input: the scale effect. Labour is also cheaper relative to machines, so it produces any given output with a more labour-intensive method: the substitution effect. A car wash facing lower wages opens another bay (scale) and replaces its automatic dryer with attendants who dry cars by hand (substitution).

Isoquants

An isoquant is the set of input combinations $(E, K)$ that produce the same output. Four properties follow from positive, diminishing marginal products:

1. Downward sloping. With both marginal products positive, using more labour requires less capital to keep output constant.
2. Non-intersecting. A crossing point would be a single input bundle with two different maximum outputs.
3. Higher is more. Isoquants further from the origin use more of both inputs and represent more output.
4. Convex to the origin. Moving down an isoquant, labour becomes less productive at the margin and capital more productive, so ever more labour is needed to replace each unit of capital.

Along an isoquant output does not change, so the output lost from $\Delta K$ fewer machines equals the output gained from $\Delta E$ more workers: $MP_K \, \Delta K + MP_E \, \Delta E = 0$. The absolute slope is the marginal rate of technical substitution:

$$MRTS = \left| \frac{\Delta K}{\Delta E} \right| = \frac{MP_E}{MP_K}.$$

Convexity means the MRTS falls as $E$ rises along the isoquant.

Isocost lines

An isocost line is the set of input combinations that cost the same total amount $C$:

$$C = wE + rK \quad \Longrightarrow \quad K = \frac{C}{r} - \frac{w}{r}E.$$

Its slope is $-w/r$, the relative price of labour. Parallel lines further from the origin cost more. A change in $w/r$ rotates the line.

The cost-minimizing mix

The cheapest way to produce output $q_0$ is the point where an isocost line is tangent to the isoquant. There the slopes are equal:

$$\frac{MP_E}{MP_K} = \frac{w}{r} \quad \Longleftrightarrow \quad \frac{MP_E}{w} = \frac{MP_K}{r}.$$

The second form says the last dollar spent on each input buys the same extra output. If it does not, reallocating spending lowers cost. Suppose $MP_E = 10$ at a wage of $20 (0.5 units per dollar) and $MP_K = 30$ at a price of capital of $40 (0.75 units per dollar). Spending $40 less on labour loses 20 units; spending that $40 on capital gains 30. Output rises at the same cost, so the original mix was not cost-minimizing.

Wage (w)

$20

Price of capital (r)

$20

Cost at P

$1,000

Cost at A

$1,408

MRTS at A (= K/E)

0.17

MRTS at P = w/r

1.00

Open in Graph Lab

Figure 1. The isoquant for 100 units and isocost lines with slope $-w/r$. P, where the isocost line $C_P$ is tangent to the isoquant and $MRTS = w/r$, is the cheapest way to produce 100 units. Any other point on the isoquant, such as A, lies on a higher isocost line $C_A$. A higher $w/r$ moves P up the isoquant to more capital and fewer workers.

Profit maximization implies cost minimization

Cost minimization says how to produce a given output, not how much to produce. The profit-maximizing firm hires each input until the value of its marginal product equals its price:

$$w = p \times MP_E \qquad \text{and} \qquad r = p \times MP_K.$$

Dividing one by the other gives $MP_E / MP_K = w / r$, so a profit-maximizing firm always minimizes cost. The converse fails: a firm can produce the wrong output at the lowest possible cost.

Key idea

Every profit-maximizing firm minimizes cost, but cost minimization alone does not pin down how many workers to hire: that also needs the profit-maximizing output.

Scale and substitution effects

Start at P, the profit-maximizing point at the old prices, producing $q_0$. When the wage falls, the marginal cost of output falls and the firm expands to $q_1$ at R. The move from P to R is split at a hypothetical point Q:

• Q produces the new output $q_1$ at the old input prices: the isocost line through Q is parallel to the old one. P to Q is the scale effect.
• R produces $q_1$ at the new prices: the flatter isocost line is tangent to $q_1$ at R. Q to R is the substitution effect, a move along a single isoquant.

After a wage cut the substitution effect always raises employment, and the scale effect does too as long as labour is a normal input (more output requires more labour). For capital the two pull in opposite directions: the scale effect raises $K$ and the substitution effect lowers it.

Price that changes: 0 = wage, 1 = price of capital

0

New value of that price (was $20)

$10

Output (was 100)

141.4

Scale effect on E (P to Q)

25.0

Substitution effect on E (Q to R)

20.7

Total change in E

45.7

Total change in K

10.4

Open in Graph Lab

Figure 2. Scale and substitution effects when one input price falls from $20 to $10. P to Q (scale effect) produces the new output $q_1 = 141$ at the old prices; Q to R (substitution effect) moves along $q_1$ to the new price ratio. A cheaper wage raises employment through both effects, by 25 and 20.7 workers. Cheaper capital raises employment through the scale effect (25 workers) but lowers it through the substitution effect (14.6 workers).

A change in the price of capital

A rise in the price of capital $r$ has two opposing effects on labour demand:

• Substitution effect: labour is now relatively cheaper, so at constant output the firm uses more workers and fewer machines. Employment rises.
• Scale effect: costs rise, so the firm produces less and uses less of both inputs. Employment falls.

The net effect on employment is ambiguous and depends on which effect is larger. In the technology of Figure 2, a rise in $r$ from $20 to $30 cuts output from 100 to 81.6; the scale effect removes 8.3 workers and the substitution effect adds back 3.7, so employment falls by 4.6. A fall in $r$ reverses every sign: the scale effect raises employment and the substitution effect lowers it.

On the exam

Draw three isocost lines: the original one tangent at P, a parallel one tangent to the new isoquant at Q, and one with the new slope tangent to the new isoquant at R. Mark $E_P$, $E_Q$ and $E_R$ on the horizontal axis and label the two gaps as the scale and substitution effects.

Perfect complements and perfect substitutes

The size of the substitution effect depends on how curved the isoquants are. Two extremes bound the possibilities.

Perfect complements are inputs used in fixed proportions, such as one worker per machine. Isoquants are right angles, and the cost-minimizing point is the corner whatever $w/r$ is, so there is no substitution effect. A fall in $r$ lowers costs, the firm expands, and it hires more workers and machines in the same ratio: labour demand rises through the scale effect alone. A wage increase likewise lowers employment only through the scale effect.

Perfect substitutes are inputs that replace each other at a constant rate, such as one machine for two workers. Isoquants are straight lines and the MRTS is constant, so there is no tangency. The firm uses only the input that is cheaper per unit of output, comparing $w / MP_E$ with $r / MP_K$. A fall in $r$ that leaves capital more expensive per unit of output changes nothing: the firm uses no capital, so its costs do not change. A fall large enough to make capital cheaper per unit of output switches the firm entirely to capital, and labour demand drops to zero. The scale effect raises output, but the substitution effect is total. A firm that already used only capital expands with more capital when $r$ falls and keeps hiring no labour.

Technology: 0 = perfect complements, 1 = perfect substitutes

0

Price of capital (r), was $50

$30

Output

245.0

Workers at R (were 25)

49.0

Scale effect on E (P to Q)

24.0

Substitution effect on E (Q to R)

0.0

Capital at R

49.0

Open in Graph Lab

Figure 3. A fall in the price of capital from $50 to $30 with a $20 wage. With perfect complements, the firm expands along the ray $K = E$ and employment rises from 25 to 49, a pure scale effect. With perfect substitutes (one machine replaces two workers), capital is cheaper per unit of output once $r$ is below $40, so the firm switches entirely to capital and employment falls from 25 to 0.

Why long-run demand is more elastic

In the short run capital is fixed, so there is no substitution effect, and output can respond only through labour against diminishing returns. In the long run the firm substitutes toward the cheaper input and adjusts its scale with every input free. Both add to the response, so the long-run elasticity of labour demand $\delta_{LR} = \%\Delta E_{LR} / \%\Delta w$ is larger in absolute value than $\delta_{SR}$.

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 4. 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 in the short run and by 20 percent in the long run.

Worked example

A firm produces with $q = E^{1/2} K^{1/2}$, so $MP_E = \tfrac{1}{2}\sqrt{K/E}$ and $MP_K = \tfrac{1}{2}\sqrt{E/K}$. It makes 40 units with a wage of $16 and a price of capital of $4.

1. Cost-minimization condition. $MRTS = MP_E / MP_K = K/E$. Setting it equal to $w/r = 16/4 = 4$ gives $K = 4E$.

2. Initial mix. Output is $\sqrt{E \cdot 4E} = 2E = 40$, so $E_P = 20$ and $K_P = 80$. Cost is $16(20) + 4(80) = 640$ dollars.

3. The wage falls to $4. Lower costs lead the firm to expand to 60 units. How far it expands depends on the demand for its product; here it is given.

4. Scale effect (P to Q). At the old prices the ratio stays $K = 4E$, so $2E = 60$: $E_Q = 30$ and $K_Q = 120$. Employment rises by 10 and capital by 40.

5. Substitution effect (Q to R). At the new prices $w/r = 1$, so $K = E$ and $\sqrt{E \cdot E} = E = 60$: $E_R = 60$ and $K_R = 60$. Employment rises by a further 30 and capital falls by 60.

6. Total. Employment rises from 20 to 60. Capital rises by 40 through the scale effect and falls by 60 through the substitution effect, a net fall from 80 to 60. At R, $MP_E / w = \tfrac{1}{2}(1)/4 = 0.125 = MP_K / r$.

Exercise: the price of capital rises instead

Start again from $E = 20$ and $K = 80$, with a wage of $16 and a price of capital of $4. The price of capital rises to $16 and output falls to 25 units. At the old prices ($K = 4E$), $2E = 25$ gives $E_Q = 12.5$: the scale effect cuts employment by 7.5. At the new prices $w/r = 1$, so $K = E = 25$: the substitution effect adds 12.5 workers. Employment rises from 20 to 25 even though output falls, because the substitution effect is larger. Capital falls from 80 to 25.

Evidence

Substitution effects are real but modest. Surveying the empirical literature, Hamermesh concluded that the long-run elasticity of demand for labour at constant output, which isolates the substitution effect, lies between $-0.15$ and $-0.5$.¹ A meta-analysis of more than 900 published estimates from 105 studies puts its preferred long-run, constant-output estimate at $-0.25$, inside that range.² The scale effect accounts for the rest of the long-run response; elasticity of labour demand shows how the two combine.

Common mistakes

Traps on exams

• Building Q at the new prices. Q produces the new output at the old prices; only R uses the new prices.
• Concluding that a higher interest rate must raise employment because labour is relatively cheaper. That is only the substitution effect; the scale effect lowers employment, and the net effect is ambiguous.
• Inverting the MRTS. With capital on the vertical axis, $MRTS = MP_E / MP_K$ and the isocost slope is $-w/r$.
• Looking for a tangency with perfect substitutes. The optimum is a corner: the firm uses only the input that is cheaper per unit of output.

Check yourself

1.

At a firm's current input mix, the marginal product of labour is 10 units and the wage is $20, while the marginal product of capital is 30 units and the price of capital is $40. To produce the same output at lower cost, the firm should:

1Use more capital and less labour2Use more labour and less capital3Keep its mix, because both inputs have positive marginal products4Use more of both inputs

Check answer

2.

The price of capital (the interest rate) rises. What happens to a profit-maximizing firm's long-run demand for labour?

1The substitution effect raises it and the scale effect lowers it, so the net effect is ambiguous2It rises, because labour is now relatively cheaper than capital3It falls, because the firm's costs rise and it produces less4It is unchanged, because the wage has not changed

Check answer

3.

One machine can do exactly the work of two workers (perfect substitutes). The wage is $20, and the firm currently uses only labour because a machine costs $50. The price of a machine falls to $30. What happens?

1The firm switches entirely to machines and employment falls to zero2Nothing, because machines are still more expensive than workers3The firm uses both inputs in a mix that sets $MP_E / MP_K = w / r$4Employment rises through the scale effect because costs fall

Check answer

Practise more questions on this topic

Sources

1. 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. 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). ↩