Economics
Consumer Demand Analysis
We can begin our representation of the consumer with a demand function…
Is a function of…
Qd D P, I , Ps , Pc
Quantity
demanded
Price
Income
Prices of
Substitutes
Price
40 D 20,100,40,10
$20
D I 100, Ps 40, Pc 10
40
Quantity
Prices of
Compliments
Given a demand function we can characterize the behavior of demand with elasticity.. p
Price
%Qd
% P
Price elasticity will always be a negative number
p
$20
50
2
25
$15
%P 25%
D I , Ps , Pc
40
60
%Qd 50%
Quantity
Given a demand function we can characterize the behavior of demand with elasticity.. I
Income elasticity will generally be a positive number 40
I 4
10
Price
$20
D I 110
D I 100
40
56
%Qd 40%
%Qd
% I
Quantity
%I 10%
Given a demand function we can characterize the behavior of demand with elasticity.. Ps
Cross price elasticity will be a positive number for substitutes 10
Ps .5
20
Price
$20
D Ps 48
D Ps 40
40
44
%Qd 10%
%Qd
%Ps
Quantity
%Ps 20%
Given a demand function we can characterize the behavior of demand with elasticity.. Pc
20
I
. 5
40
Price
Cross price elasticity will be a negative number for compliments $20
D Ps 10
D Pc 14
32
40
%Qd 20%
Quantity
%Qd
%Pc
%Ps 40%
Note that if the demand relationship is linear, elasticity is not constant
Qd a bP cI dPs ePc
%Qd
p
%P
Q P
p
P Q
P
p b
Q
Ps
%Qd
d p
Ps
%I
Q
Q I
p
I Q
I
I c
Q
P
Pc e c
Q
For example….
Qd 60 2 P .0025 I
30
p 2
.6
100
40,000
I .0025
1
100
Price
$80
$30
D I 40,000
100
Quantity
Let’s try something a little more complicated…a non-linear demand relationship 1
2
Qd 100 2 P .02 I
2
1
2
1
2
P
400 .29
P
Q P
p
P
Q
68
P Q
Q
1
2
Price
$2916
2
I
.04 I 2
.04 20
Q I
I
.24
.04 I
Q
68
I Q
Q
$400
D I 20
Quantity
68
Sometimes we use demand functions that are linear in logs…
LN Qd LN 150e .15 P .012 I
Qd 150e
.15 P .012 I
LN Qd LN 150 LN e .15 P .012 I
LN Qd 5 .15 P 012 I
Price
Price
$12
$12
D I 40
40
D I 40
Quantity
3.7
LN(Quantity)
P
150e .15 P .012 I P
Q P
.15 P .012 I
.15 (.15)
p
.15 P .1512 1.8
150e
.15 P .012 I
Q
150 e P Q
I
150e .15 P .012 I I
Q I
.15 P .012 I
I
.012 (.012)
.012 I .012 40 .48
150e
.15 P .012 I
Q
150 e P Q
Price
Qd 150e
.15 P .012 I
$12
D I 40
40
Quantity
A little math trick…recall the derivative of the natural log
ln x 1
x x ln x
x x This just says that the difference in logs is a percentage change
Therefore, if we start with elasticity
%Q ln Q ln Q
p
P
%P %P P
Sometimes we use demand functions that are linear in logs…
LN Qd 5 .15P 012 I
%Q ln Q
p
P .15 P .1512 1.8
%
P
P
Price
%Q ln Q
I
I .012 I .012 40 .48
%I I
$12
D I 40
3.7
LN(Quantity)
Sometimes we use demand functions that are linear in logs…
LN e Qd LN 16 P .65e.045 I
e
Qd
16 P
Qd LN 16 LN P .65 LN e.045 I
.65 .045 I
e
Qd 2.7 .65LN P .045 I
Price
LN(Price)
$10
2.3
D I 40
3
D I 40
Quantity
3
Quantity
Sometimes we use demand functions that are linear in logs…
Qd 2.7 .65 LN P .045I
1
.65
%Q Q 1
p
.22
.65
Q
3
%P ln P Q
LN(Price)
I
%Q Q I
40
I
.045 .045