Lecture 2.
LABOUR DEMAND
Theories for the demand for the factors of production
Labour demand theory sets to explain the demand for manpower
The demand for labour depends on the cost of labour, but also on the
cost of other factors, such as how effective its workforce are, and the
price they can sell their products.
The demand side of the labour
market
W
Supply
Demand
L
1
Share of total employment in some
industries in Norway
0,4
0,35
0,3
0,25
0,2
0,15
0,1
0,05
0
1970
1980
Industri
1990
Helse og sos
2000
Underv
2006
Offentlig
2
DISPOSITION
1. LABOUR DEMAND IN THE SHORT
RUN
2. THE SUBSTITUTION OF CAPITAL
FOR LABOUR
3. SCALE EFFECTS
4. BEYOND TWO INPUTS
5. MONOPSONY
3
1. LABOUR DEMAND IN THE SHORT RUN
Only labour is a variable input
Y(P):
Demand for particular good
P = P(Y) The inverse demand function
 PY 
P Y
Y P Elasticity of demand, assumed to be decreasing
 PY  0 Perfect competition
 PY  0 Im perfect competition
Y = F (L)
F’>0, F’’<0
The firm’s profit:
II(L) = P(Y)Y-WL
with Y=F(L)
Goal of the entrepreneur: To choose L so to maximise profits.
The first order condition:
II’(L) = F’(L)[P(Y)+P’(Y)Y] –W = F’(L)P(Y)(1+  PY ) –W = 0
When (1+  PY )>0 the labour demand is given by:
(1)
F ' ( L)  v
W
P
with v 
1
1   PY
4
Where v is the mark up, a measure of the firm’s market power.
v = 1: No market power
Derivating (1) with respect to w:
L
v

0
W
F ' F ' P ' pF ' '
Short run labour demand is a decreasing function of w
W
Supply
Demand
L
This represents a movement along the demand curve
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2. THE SUBSTITUTION OF CAPITAL FOR
LABOUR
In the longer run, the firm may replace part of its workforce with
machines, or conversely increase the number of workers and reduce
its stock of capital
Two inputs: Capital and labour
Y = F(K,L)
FK>0 FL>0
FKK<0 FLL<0
Minimization of total costs
Min (WL  RK )
( K , L)
The solutions:
subject to F ( K , L)  Y
L and K are
the conditional demand for labour and
capital.
The minimal value of total costs or WL  RK is a function of
the unit cost of each factor and the level of production.
This is called: The minimum cost function: C(W, R, Y)
6
The properties of the cost function:
i) It is increasing with respect to each of the arguments and is
homogenous of degree 1.
ii) It is concave in (W,R), which signifoes that Cww<0 and CRR<0
iii) It satisfises the Shepard’s lemma:
==> L = Cw(W,R,Y) and
K = CR(W,R,Y)
Where L and
K
are the optimal use of L and K.
The properties of the conditional factor prices:
Variation in factor prices:
L
 CW W  0
W
Where the sign follows from that the cost function is concave.
Cross elasticities:
The cross elasticities of the conditional demand for a factor with
respect to the price of the other factor are defined by:
 RL 
R L
L R
and
 RL 
W K
K W
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The elasticity of substitution
This is the elasticity of the variable ( K / L ) with respect to the relative
price W/R. The elasticity of substitution between capital and labour is
given by:

W / R ( K / L )
K / L ( w / R)
This formula says that the capital labour ratio increases by  % when
the ratio between the price of labour and the price of capital increases
by 1 %.
K
L
Put simple: The elasticity of substitution measures how many per cent
K/L has to change with when the isoquant become 1 % steeper
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3. SCALE EFFECTS
Now: The firm can choose the level of production
Unconditional factor demand
The profit function:
(14) II (W, R,Y) = P(Y)Y – C(W,R,Y)
Maximising (14) with respect to Y
The optimal level of production is given by:
(15) P(Y) = vCY(W,R,Y) with
v
1
1   PY
The firm sets its price by applying the markup v to its marginal costs
CY
The optimality condition takes the form:
W
P
R
FK ( K , L)  v
P
FL ( K , L)  v
and
At optimum, the marginal productivity of each factor equal to its
costs, multiplied by the mark up==>The long run demand for capital
and for labour
When v = 1: perfect competition
v>1 : imperfect competition
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The relationship between the demand for a factor and its cost:
The profit function is equal to the maximal value of profit for given
values of the inputs. It is defined by:
II(W, R)  Max II(W, R, Y)  P(Y )Y  C (W , R, Y )
Y
Differentiating the profit function with respect to W, we get:
II W (W, R)  [P(Y*)(1  η PY ) - C Y (W, R, Y*)]
Y *
 C W (W, R, Y*)
W
where Y* be the optimal level of production:
According to (15) the term in brackets are zero.
Moreover: Shepard’s lemma (6) states that the CW = L*.
Then we arrive at the following relation
IIW = -L* and
IIR = -K*
The profit function II(W,R) being convex, we have IIWW  0 and
IIRR  0
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L *
  II W W  0 and
W
L *
  II RR  0
R
Thus, unconditional demand for a factor is a decreasing function of
the cost of this factor.
Labour demand elasticities:
It is possible to be more exact about the unconditional labour demand.
We have L* = CW(W,R,Y*)
Differentiating this equality with respect to W, we get:
L *
Y *
 CW W  CW Y
W
W
Multiply with W/L*. We then get:
L * W
W
Y * W
 CW W
 CW Y
W L *
L*
W L *
WL 
Y * CW Y
W
Y
CW W  (
)W
L
L*
W
CW W and
Since L* =CW(W,R,Y*), the terms:
L
(
Y * CW Y
) designate respectively the elasticity  WL
L*
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of the conditional labour demand, and the elasticity of this demand
with respect to the level of output at point Y=Y* This last elasticity
can be denoted:  YL
We finally obtain:
(19)
Y
WL  WL   YLW
L

Substitution effect: W
Always negative
Scale effect:  Y  W
Always negative
L
Y
The substitution effect: For a given output; a rise in labour costs
always leads to reduced utilization of this factor.
The scale effect measures on the profit maximating production level
from increased W.
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4. BEYOND TWO INPUTS
Now, more than two inputs
1.Conditional demand
Minimization of total costs for a given level of output:
The product function
Y = F(X1,...,Xn)
where Xi is the quantity of factor i.
The conditional demands are obtained by minimizing total costs of a
given quantity Y:
n
i
i
Min
W
X

1
n
X ,..., X
st
F(X1 ,..., X n )  Y
i 1
The Lagrangian of this problem is:
n
  W i X i
  (F(X 1 ,..., X n ) - Y)
i 1
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Differentiating the Lagrangian wrt Xi gives us the first order condition

i

W
 Fi  0
i
X
The conditional factor demand
X i are
defined by the following
equation:
(23)
F ( X ..., X )  Y
1,
n
and
Fi( X 1, ..., X n ) W i

Fj( X 1, ..., X n ) W j
Eq (23) says that the technical rate of substitution (Fi/Fj)between
factor i and j is equal to the relative cost (Wi/Wj).
The cost function
The minimum value of total costs, or
W
i
X
j
is also called the
cost function of the firm. It depends on the output and its price:
C(W1,...,Wn,Y)
As earlier: It satisfies Shepard’s lemma:
(24)
X i  Ci (W1 ,..., Wn , Y )
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P-substitute and P-complements
Differentiating (24) with respect to W gives
X i
 C ii  0
W i
The conditional demand for input is always decreasing with the price
of input. General result. Does not depend on the number of inputs
The cross-price effect
Contrary to the two-factor case, the variation in the conditional
demand for a factor resulting from a change in the cost of another
factor does not always have the same sign.
When Wi rises the firm reduces their demand for i, and they
must increase their demand for at least one other factor to keep output
constant.
Without any further information about technology, it is not
possible to know either which factor will be utilized more.
Still, the symmetry condition holds:
X i X j
(26)

 Cij
j
i
W
W
Eq (26) says that the effect of a variation in the price of factor j on
conditional demand for factor j on conditional demand for factor i is
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the same as that of a variation in the price of factor i on the
conditional demand for factor j.
X i
 0 Substitutes
W j
X i
 0 complements
W j
Factor i and j are substitutes if, to keep a given level of Y, the demand
for one of the factors increases when the price of the other factor rises.
Elasticity of substitution
The cross elasticity of the conditional demand for factor i with respct
to price j:
W j X i W j
(27)

Cij
X i W j X i
The partial elasticity of substitution is given by:
( 28)
 ij
CCij
Ci C j
This is analogous to the two-factor situation.
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2. Unconditional demands
Now the goal is to maximise profits ==> Unconditional demand
Maximise profits
The previous cost function C(W,R,Y) is replaced by C(W1,...,Wn, Y)
The optimal level of output is given by:
(30) P(Y) = vCY(W1,...,Wn, Y)
i.e., the firm set its price by applying the markup to the marginal costs.
We have:
II(W1,...,Wn)
This is the maximal value of profit for given factor
costs
The unconditional demand for factor i is given by:
Xi = - IIi(W1,...,Wn)
IIi is the partial derivative of the profit function with respect to W
The profit function is convex, i.e., IIii >0
Then we can present the relationship between demand for a factor and
its price:
X i
  II ii  0
i
W
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The demand for a factor diminishes with the price of this factor
applies irrespective of the number of inputs. Often called the “law” of
demand.
W
Supply
Long run
Short run
L
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Gross substitutes and gross complements
From earlier we have the equation for the optimal use of factor i:
X i  Ci (W 1 ,...,W n , Y )
Differentiating this with respect to Wj we get:
X i
Y

C

C
ij
iY
W j
W j
Multiplying by Wj/Xi :
Wj
YC W Y
  i Cij  ( iYi ) j
X
X
Y W j
i
j
This is a measure of the elasticity of the demand for factor i with
respect to price Wj
The term
Wj
Cij represents
Xi
the the elasticity of the conditional demand
taken at profit optimum.
From Shepard’s lemmaXi=Ci, the term
YC iY
Xi
measures the output
elasticity of the conditional demand for input i.
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Finally, then we have:
(31)  ij   ji  Yi jY
This measures the effects of a rise in Wj of factor j on the demand for
factor i.
 ij is the substitution effect
Yi jY is the scale effect
When i  j,  j , has no determinate sign. It is positive (negative) if the
i
factor i and j are substitutes (complements).
The second term is the scale effect, which also has an indeterminate
sign.
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MONOPSONY
Imperfect competition, barriers to entry
Enables the firm to discriminate
The basic model
The firm faces a labour supply Ls(w)=G(W)
which is an increasing function of w
The problem of the monopolist is given by:
II ( w)  Ls ( w)( y  w)
Max
w
The first order condition
 wL ( w M )
wM 
y
1   wL ( w M )
where
wLs ' ( w)
 (w )  s
0
L ( w)
L
w
M
The term
 wL ( w M )
1   wL ( w M )
lies between 0 and 1.
The wage chosen by the monopsonist is below the competitive wage,
except were labour supply is infinite.
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