Finance 30210: Managerial
Economics
Consumer Demand Analysis
Suppose that you observed the following consumer behavior
P(Bananas) = $4/lb.
Q(Bananas) = 10lbs
P(Apples) = $2/Lb.
Q(Apples) = 20lbs
P(Bananas) = $3/lb.
Q(Bananas) = 15lbs
P(Apples) = $3/Lb.
Q(Apples) = 15lbs
Choice A
Choice B
What can you say about this consumer?
Choice B
Is strictly
preferred to
Choice A
How do we know this?
Consumers reveal their preferences through their observed choices!
Choice A
Choice B
Q(Bananas) = 10lbs
Q(Bananas) = 15lbs
Q(Apples) = 20lbs
Q(Apples) = 15lbs
P(Bananas) = $4/lb.
P(Apples) = $2/Lb.
P(Bananas) = $3/lb.
Cost = $80
Cost = $90
Cost = $90
Cost = $90
P(Apples) = $3/Lb.
B Was chosen even though A was the same price!
What about this choice?
Choice C
P(Bananas) = $2/lb.
Q(Bananas) = 25lbs
P(Apples) = $4/Lb.
Q(Apples) = 10lbs
Q(Bananas) = 15lbs
Choice B
Cost = $90
Cost = $90
Q(Apples) = 15lbs
Q(Bananas) = 10lbs
Choice A
Choice C
Cost = $100
Q(Apples) = 20lbs
Is strictly
preferred to
Choice B
Is choice C
preferred to choice
A?
Choice B
Choice C
Is strictly
preferred to
Choice A
Is strictly
preferred to
Choice B
C>B>A
Choice C
Is strictly
preferred to
Choice A
Rational preferences exhibit transitivity
Consumer theory begins with the assumption that every
consumer has preferences over various combinations of
consumer goods. Its usually convenient to represent
these preferences with a utility function
U : AB
U
A
Set of possible
consumption choices
B
“Utility Value”
Using the previous example (Recall, C > B > A)
Choice A
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
Choice B
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
Choice C
Q(Bananas) = 25lbs
Q(Apples) = 10lbs
U (25,10)  U (15,15)  U (10,20)
We require that utility functions satisfy a few basic
properties
There is a definite ranking of all
choices
y
U (C )  U ( A)  U (C )  U ( B)
A
C
B
U ( x, y )  25
U ( x, y )  20
x
We require that utility functions satisfy a few basic
properties
More is always better!
y
U (C )  U ( A)
C
A
B
U ( x, y )  20
x
We require that utility functions satisfy a few basic
properties
y
People Prefer Moderation!
U (C )  U ( A)
15
A
C
10
5
B
5
10
15
U ( x, y )  25
U ( x, y )  20
x
Suppose you are given a little extra of good X. How much Y is needed
to return to the original indifference curve?
Marginal Utility of X
y
U x ( x* , y * )
y
 MRS  
x
U y ( x* , y * )
x  1
y*
y  ?
Marginal Utility of Y
U ( x, y )  20
x
x*
The marginal rate of substitution (MRS) measures the amount of Y
you are willing to give up in order to acquire a little more of X
The marginal rate of substitution (MRS) measures the amount of Y
you are willing to give up in order to acquire a little more of X
MRS ( x* , y* )  MRS ( x' , y' )
y
If you have a lot of X relative to Y, then X is
much less valuable than Y MRS is low!
y*
U ( x, y )  20
y'
x
*
x'
x
The elasticity of substitution measures the curvature of the
indifference curve
y
 y
 
x
 y
%  
x

%MRS
'
 y
 
x
x
Elasticity of
substitution
measures the
degree to which
your valuation of X
depends on your
holdings of X
The elasticity of substitution measures the curvature of the
indifference curve
y
 is small
If the elasticity of substitution
is small, then small changes in
x and y cause large changes in
the MRS
x
y
 y
%  
x

%MRS
 is large
If the elasticity of substitution
is large, then large changes in
x and y cause small changes
in the MRS
x
We often assume that the marginal rate of substitution is dependant
only on the ratio of X and Y – i.e. preferences are homogeneous
y
 y
 
x
U ( x, y )  40
U ( x, y )  30
U ( x, y )  20
x
Consumers solve a constrained maximization – maximize
utility subject to an income constraint.
max U ( x, y )
x, y
subject to
px x  p y y  I
As before, set up the lagrangian…
 ( x, y )  U ( x, y )   ( I  p x x  p y y )
 ( x, y )  U ( x, y )   ( I  p x x  p y y )
First Order Necessary Conditions
U x ( x, y)  px  0
U y ( x, y )   p y  0

U y ( x, y )
py
U x ( x, y )

px
I  px x  p y y
U x ( x, y ) Px

U y ( x, y ) Py
max U ( x, y )
x0, y 0
subject to
px x  p y y  I
y
U x ( x, y ) Px

U y ( x, y ) Py
I
py
y
I  px x  p y y
*
x
*
I
px
x
Demand Curves present the same information in a different
format – therefore, all the properties of preferences are present in
the demand curve
px
y
px
x
x
*
DI , p y 
x*
x
Demand relationships are based off of the theory of consumer choice. We can
characterize the average consumer by their utility function.
“Utility” is a function of lemonade and hot dogs
U L, H 
Consumers make choices on what to buy that satisfy two criteria:
Their decision on what to
buy generates maximum
utility
Their decision on what to
buy generates is affordable
PH H  PL L  I
MU H MU L

PH
PL
QH  DPH , PL , I 
These decisions can be represented
by a demand curve
Example: Suppose that you have $10 to spend. Hot Dogs cost $4 apiece and glasses of
lemonade cost $2 apiece.
# Hot Dogs
MU
(Hot Dogs)
# Lemonade
MU
(Lemonade)
1
9
1
4
2
8
2
3
3
7
3
1.5
4
6
4
1
5
5
5
.5
MU H MU L

PH
PL
8 4

4 2
PH H  PL L  I
QH  D4,2,10  2
42  21  10
This point satisfies both
conditions and, hence, is one
point of the demand curve
The marginal rate of substitution controls the height of the demand
curve
px
y
Willingness to pay
is low
MRS is small
$2
x
y
px
$10
MRS is large
x
DI , p y 
x
Willingness
to pay is
high
DI , p y 
x
Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still
have $10 to spend)
# Hot Dogs
MU
(Hot Dogs)
# Lemonade
MU
(Lemonade)
1
9
1
4
2
8
2
3
3
7
3
1.5
4
6
4
1
5
5
5
.5
MU H MU L

PH
PL
8 4

6 2
Your decision at the margin
has been affected. You
need to buy less hot dogs
and more lemonade
(Substitution effect)
PH H  PL L  I
62  21  10
You can’t afford what you
used to be able to afford –
you need to buy less of
something! (Income effect)
Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still
have $10 to spend)
# Hot Dogs
MU
(Hot Dogs)
# Lemonade
MU
(Lemonade)
1
9
1
4
2
8
2
3
3
7
3
1.5
4
6
4
1
5
5
5
.5
MU H MU L

PH
PL
9 3

6 2
PH H  PL L  I
QH  D6,2,10  1
61  22  10
This point satisfies both
conditions and, hence, is one
point of the demand curve
Demand curves slope downwards – this reflects the negative relationship between price
and quantity. Elasticity of Demand measures this effect quantitatively
%Q  50
D 

 1
%P
50
Price
64

 *100  50%
 4 
$6.00
$4.00
DI  $10
Quantity
1
2
1 2 

 *100  50%
2


The elasticity of substitution will control the slope of the demand
curve
y
px
 y
%  
x

%MRS
% x
x 
% p x
p' x
px
x*
x
D
x'
x*
x
Elasticity of Substitution vs. Price Elasticity
y
px
 is small
 x is small
x
x
px
y
 is large
x
 x is large
x
Perfect Complements vs. Perfect Substitutes
y
px
 0
x  0
(Almost)
x
x
px
y
 
x
x  
x
Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to
spend.
# Hot Dogs
MU
(Hot Dogs)
# Lemonade
MU
(Lemonade)
1
9
1
4
2
8
2
3
3
7
3
1.5
4
6
4
1
5
5
5
.5
MU H MU L

PH
PL
8 4

4 2
PH H  PL L  I
42  21  20
Your decision at the margin is unaffected, but you have some
income left over (this is a pure income effect)
Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to
spend.
# Hot Dogs
MU
(Hot Dogs)
# Lemonade
MU
(Lemonade)
1
9
1
4
2
8
2
3
3
7
3
1.5
4
6
4
1
5
5
5
.5
MU H MU L

PH
PL
6 3

4 2
PH H  PL L  I
QH  D4,2,20  4
44  22  20
This point satisfies both
conditions and, hence, is one
point of the demand curve
For any fixed price, demand (typically) responds positively to increases in income. Income
Elasticity measures this effect quantitatively
%Q 100
D 

1
%I 100
Price
 20  10 
%I  
 *100  100%
 10 
$4.00
DI  $20
DI  $10
Quantity
2
4
42
%Q  
 *100  100%
2


Income elasticity measures the response of consumers to changes
in income holding prices constant – the homogeneity of
preferences will effect this
y
px
% x
I 
% I
%x
px
x
*
x
x*
x
Cross price elasticity refers to the impact on demand of another price changing
Note: These numbers aren’t coming
from the previous example!!
Price
%QH 200
L 

2
%PL 100
42
%PL  
 *100  100%
 2 
$4.00
DPL  $4
DPL  $2
Quantity
2
6
62
%Q  
 *100  200%
2


A positive cross price elasticity refers to a substitute while a negative cross
price elasticity refers to a compliment
Cross price elasticity measures consumer response to changes
in other prices – this is influenced by both homogeneity and
elasticity of substitution
y
px
%x
y 
%p y
px
x*
x
x*
x
.5
max x y
An Example:
Cobb-Douglas
Utility
.5
x 0, y 0
subject to
px x  p y y  I
( x, y)  x y   ( I  px x  p y y)
.5
.5
U x ( x , y ) .5 x . 5 y .5 p x


.5 .5
U y ( x , y ) .5 x y
py
 px 
y   x
p 
 y
An Example:
Cobb-Douglas
Utility
px x  p y y  I
 px 
px x  p y   x  I
p 
 y
y
.5
max x y
.5
x 0, y 0
subject to
px x  p y y  I
I
x
2 px
I
y
2 py
An Example: Cobb-Douglas Utility

U ( x, y)  x y
U x ( x , y )  x
 1

U y ( x, y)  x y
y

 1

U x ( x* , y * ) x 1 y    y 
   1   
*
*
U y ( x , y ) x y
 x
With Cobb-Douglas Utility functions, your MRS is directly
proportional to your relative consumption of the two goods.
An Example: Cobb-Douglas Utility

MRS 

 y
 
x
 y
d 
x  
d MRS  

U ( x, y)  x y
 y 
 
 x  


 MRS

   y 
  
      x 
  
1
   y 
 
x
Cobb-Douglas Utility functions have constant elasticity of substitution
U  x, y   x .5 y .5
px
I
x
2 px
%x
dx p x
x 

% p x dp x x
dx
I
 2
dp x
2 px
px
x*
x
px
I
x   2
 1
2 px  I 


 2 px 
.5
max x y
.5
x 0, y 0
subject to
I
x
2 px
I
y
2 py
px x  p y y  I
%x
dx p y
y 

0
%p y dp y x
Cobb-Douglas demands are
independent of other prices!
.5
max x y
.5
x 0, y 0
subject to
I
x
2 px
I
y
2 py
px x  p y y  I
%x dx I
I 

%I dI x
1
I

1
2 px  I 


 2 px 