Finance 30210: Managerial
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  .1512   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  .1512   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