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Recitation Notes 1

Xueting Wang

01/25/2016

1 firm’s labor demand

a Marginal Product of labor/capital: change in output produced by a change in the units of

labor/capital holding capital/labor constant

b Marginal Revenue Product: M RP = M P · M R. In perfectly competitive market M RP =

M P · p

c Marginal Expense of an Added labor: M E = w if labor market is perfectly competitive

Here we assume perfectly competitive product market and factor market.

Example

q = (K ρ + E ρ)1/ρ, 0 < ρ < 1.

M P E = 1

ρ(K ρ + E ρ)

1

ρ−1E ρ−1

M P K = 1

ρ(K ρ + E ρ)

1

ρ−1K ρ−1

MRTS = M P E /M P K = (K/E )1−ρ = W/r

In general, let production function be q=q(E,K), wage be w and price (rental rate) of

capital be r. Firm maximizes it profit.

Π = pq (E, K ) − wE − rK

1



Short Run vs. Long Run

In the short run, only labor is variable, and capital is fixed. So the maximization problem

becomes

maxE

Π = pq (E ) − wE − r K̄

So the FOC is

∂ Π

∂E = pq E − w = pM P E − w = 0

pMP E = M RP E = w

The labor demand curve with respect to nominal wage is the same as the marginal revenue

product curve.

Example

Suppose the production function is q = K 0.5E 0.5, what is the short run labor demand?

pMP E = p0.5(K

E )0.5 = w

E =K̄p

4w2

Labor demand is increase in the fixed level of capital because marginal product of labor

is higher with higher capital stock.

In the long run, all inputs can be varied. The maximization problem becomes

maxE,K

Π = pq (E, K ) − wE − rK

So the FOCs are

∂ Π

∂E = pq E −

w = pM P E −

w = 0

∂ Π

∂K = pq K − r = pM P K − r = 0

p = W/MP E (1)

2



p = r/MP K (2)

W/MP E = r/MP K (3)

W/r = M P E /MP K

How much to produce: (1) and (2) show that the added cost per unit of added output

must be the same as marginal revenue, which in perfect competition is price.

How to produce given quantity in a least cost way? (3) show that marginal cost per unit

of added output for labor must be equal to that of capital.

Example

What happens if wage falls?

If wage falls, (1) no long holds, even without adjusting capital the firm wants to hire

more worker. As the number of workers increases, M P K increases, to restore the equality of

(2) firm will employ more capital too. This is the scale effect. Additionally, the firm wants

to substitute labor for capital in the long run to restore the equality in (3). This is the

substitution effect.

3



100 1 2 3 4 5 6 7 8 9

10

0

1

2

3

4

5

6

7

8

9

Labor (hours)

C a p i t a l ( p h y s i c a l u n i t s )

Q*

Q**

MRTS=MP_L/MP_K=W/r

Lz Lz' Lz"

Substitution

EffectScale Effect

Z

Z'Z"

2 elasticity of labor demand

General definition of elasticity

η(x) = xf (x)

f (x) =

df (x)

dx

x

f (x) (4)

In the case of labor demand, f(x)=E and x=w

Example

If firm A has a labor demand function

E = 2

w.

4



What is the firm elasticity of labor demand?

η = −

2

w2

w

2/w = −1

5

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