# elasticity

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```					Example of linear demand with different measures
Numbers       Wage (\$)       Hours      Wage (Cents)
1             24             8           2400
2             22            16           2200
3             20            24           2000
4             18            32           1800
5             16            40           1600
6             14            48           1400
7             12            56           1200
8             10            64           1000
9             8             72            800
10            6             80            600
11            4             88            400
12            2             96            200

Same relationship
Labor Demand in Dollars and Numbers
W = 26 - 2*N

30
25
20                           Slope = -2 =(ΔW)/(ΔN)
\$ 15
10
5
0
0   1   2   3   4   5    6   7    8   9   10   11   12   13
Numbers
Labor Demand in Hours and Cents
C = 2600 - 25H

3000
2500
2000                              Slope = -25 =(ΔW)/(ΔN)
Cents 1500
1000
500
0
0       20         40        60         80     100         120
Hours

Slopes are sensitive to the units
Need a unit free measure of labor demand sensitivity
Computing the elasticity
Own wage elasticity of demand for labor:
Percentage change in labor demand caused by
a 1% change in the wage
• N: labor
• W: wage
N
% change in labor =
N
W
% change in wage =
W
Computing the elasticity
Own wage elasticity of demand for labor:
Percentage change in labor demand caused by
a 1% change in the wage

 N   W 
Own wage elasticity =     /    
 N   W 
 N   W 
Units cancel            =     / 
 W   N 
Example of linear demand with different measures
Numbers        Wage (\$)       Hours       Wage (Cents)
1              24            8            2400
2              22           16            2200
3              20           24            2000
4              18           32            1800
5              16           40            1600
6              14           48            1400
7              12           56            1200
8              10           64            1000

9               8           72            800
>9.5      7<
10               6           80            600
11              4            88             400
12              2            96             200
Labor Demand in Dollars and Numbers
W = 26 - 2*N

30
25
Slope = -2
20
\$ 15
ΔW=2   10
W 7 5
0
0   1   2   3   4   5    6    7      8   9   10    11   12   13
Numbers          ΔN=1
N  9.5

 N   W 
    /     =(1/9.5) / (2/7) = |-.368|
  N   W 
Example of linear demand with different measures
Numbers        Wage (\$)       Hours       Wage (Cents)
1             24             8            2400
2             22            16            2200
3             20            24            2000
4             18            32            1800
5             16            40            1600
6             14            48            1400
7             12            56            1200
8             10            64            1000

9              8            72       800
>76 700<
10              6            80       600
11              4            88             400
12              2            96             200
Labor Demand in Hours and Cents
C = 2600 - 25H

3000
2500
2000                           Slope = -25
Cents 1500
ΔW=200 1000
W  700 500
0
0      20      40         60          80    100   120
Hours     ΔN=8
N  76
 N   W 
    /            =(8/76) / (200/700) = |-.368|
  N   W 
Relationship between demand slope and
elasticity

 N   W 
Own wage elasticity =     /     
 N   W 
Slope of demand curve is
(ΔW)/(ΔN)                  N   W 
=    / 
 W   N 
Relationship between demand slope and
elasticity

 N   W 
Own wage elasticity =           /     
 N   W 
Elasticity = |(1/slope)*(W/N)|
 N   W 
=    / 
 W   N 
Relationship between demand slope and
elasticity
As the demand slope
Elasticity = |(1/slope)*(W/N)| =>   gets bigger , the
demand elasticity
W                       4          gets smaller

3

2

1
N
Relationship between demand slope and
elasticity
Extremes: 3: slope = 0
Elasticity = |(1/slope)*(W/N)|       ηNN
W                       4

3

2

1
N
Relationship between demand slope and
elasticity
Extremes: 3: slope = 0
Elasticity = |(1/slope)*(W/N)|         ηNN
W                       4

Perfectly E          lastic

3

2

1
N
Relationship between demand slope and
elasticity

Elasticity = |(1/slope)*(W/N)|
W                       4

Extremes: 4: slope = -
ηNN = 0

3

2

1
N
Relationship between demand slope and
elasticity

Elasticity = |(1/slope)*(W/N)|       Extremes: 4: slope = -
ηNN = 0
W                       4

Perfectly nelastic
3

2

1
N
Relationship between demand slope and
elasticity

Elasticity = |(1/slope)*(W/N)|
W                       4               Relatively Inelastic
Demand

3      Relatively
Elastic Demand
2

1
N
If you are a union representative, which
demand curve would you want?

W                 4

Aim: Maximize the wage bill = W*N

3

2

1
N
Labor demand elasticity and the wage bill

Labor demand: N: number of workers; W: Wage

W

W1

W0

Demand

N1    N0               N

Wage Bill = W*N; Change in wage bill = W1N1 – W0N0
Labor demand elasticity and the wage bill

W
Relatively Inelastic
Demand

W1

W0
Relatively
Elastic Demand

N1 N2 N0                    N

Change in wage bill
Relatively Inelastic demand, Δ(W*N) = W1N2 – W0N0
Relatively Elastic demand, Δ(W*N) = W1N1 – W0N0
Labor demand elasticity and the wage bill

W
Relatively Inelastic
Demand

W1

W0
Relatively
Elastic Demand

N1 N2 N0                    N

Change in wage bill
Relatively Inelastic demand, Δ(W*N) = W1N2 – W0N0         Bigger
Relatively Elastic demand, Δ(W*N) = W1N1 – W0N0
Precise relationship between demand
elasticity and the wage bill
ED = Elasticity of demand =
% change in employment
% change in wage

0 < ED < 1: inelastic demand
ED = 1: unitary elastic demand
ED > 1: elastic demand

Wage increase with inelastic demand will raise the
wage bill
Wage increase with elastic demand will lower the
wage bill
EXAMPLE
ED = Elasticity of demand = 0.3 < 1, inelastic
% change in employment = 3%
% change in wage = 10%

W1 = W0 (1.10)
N1 = N0 (0.97)
Change in wage bill = W1N1 – W0N0
= W0 (1.10)* N0 (0.97) - W0N0
= 0.067*W0N0
So wage bill rises when wage rises when the elasticity of demand is
below 1.
.
Demand Schedule Estimated as N = 10 - 1*W

12

10

8
.           (ΔN)/(ΔW) = -1
N

6                             .

(W = 6; N = 4)
4

2

0
0   2              4               6             8      10
W

Point Elasticity: [(ΔN)/(ΔW)]*(W/N) = | (-1)*(6/4) |
= 1.5
Cross price elasticity of demand

Cross-price elasticity of demand for labor: Percentage
change in labor demand caused by a 1% change in the
price of another input

Two inputs N and K are gross substitutes if as the price of
K rises, the quantity of N demanded rises

ηNK = ΔN           Δr
>0
N           r
Cross price elasticity of demand

Cross-price elasticity of demand for labor: Percentage
change in labor demand caused by a 1% change in the
price of another input

Two inputs N and K are gross complements if as the price
of K rises, the quantity of N demanded falls

ηNK = ΔN          Δr
<0
N          r
Price of IT
Indexes of Computer Price and Business Capital Stock, 1960-1996
Source: Ruttan, Technology, Growth and Development: An Induced Innovation Perspective . 2001

Index   400

350

300

250

200

Capital Stock
150

100

50

Price
0

1955   1960            1965           1970            1975           1980            1985            1990        1995   2000
Year
Estimated own and cross price elasticities between capital,
labor and human capital per worker
Price of
Human
Physical            Numbers of Capital per
Demand for              Capital             Workers    Worker
Physical Capital             -0.45             1.07              -0.11
Numbers of
0.66              -1.44              0.15
Workers
Human Capital
-0.15             0.35              -0.13
per Worker

Red: Complements;     Blue: Substitutes

Note: Based on share-weighted elasticities of substitution reported in Table 6 of Huang.
Hallam, Orazem and Paterno, "Empirical Tests of Efficiency Wage Models."Economica
65 (February 1998):125-143.
Laws of Derived Demand:
Relating the size of the scale and the substitution
effects to the own wage elasticity of demand

1) The more elastic is the demand for
the product, the more elastic is the
demand for labor.
Union affiliation of employed wage and salary workers by industry,
2002
Members       Covered
Private wage and salary
workers                                           8.5          9.3
Mining                                            8.5         10.0
Construction                                     17.2         17.8
Manufacturing                                    14.3         15.1
Transportation and public
utilities.                                       23.0         24.3
Wholesale and retail trade                        4.5          4.9
Finance, insurance, real
estate                                            1.9          2.5
Services                                          5.7          6.7
Government workers                               37.5           42
Source: Bureau of Labor Statistics
Source: OECD, Employment Outlook, 2004.
Laws of Derived Demand:
Relating the size of the scale and the substitution
effects to the own wage elasticity of demand

2) The more substitutable are other inputs
for labor, the more elastic is the demand
for labor
3) The more readily available are
substitutes for labor, the more elastic is
the demand for labor
Laws of Derived Demand:
Relating the size of the scale and the substitution
effects to the own wage elasticity of demand

4) ‘The importance of being unimportant’
The greater is labor’s share of total cost,
the greater is the elasticity of demand for
labor

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