Post on 23-Feb-2016
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Finance 30210: Managerial Economics
Consumer Demand Analysis
Suppose that you observed the following consumer behavior
P(Bananas) = $4/lb.P(Apples) = $2/Lb.
Q(Bananas) = 10lbsQ(Apples) = 20lbs
P(Bananas) = $3/lb.P(Apples) = $3/Lb.
Q(Bananas) = 15lbsQ(Apples) = 15lbs
What can you say about this consumer?
Is strictly preferred to
Choice A
Choice B
Choice B Choice A
How do we know this?
Consumers reveal their preferences through their observed choices!
P(Bananas) = $4/lb.P(Apples) = $2/Lb.
Q(Bananas) = 10lbsQ(Apples) = 20lbs
P(Bananas) = $3/lb.P(Apples) = $3/Lb.
Q(Bananas) = 15lbsQ(Apples) = 15lbs
Cost = $80 Cost = $90
Cost = $90 Cost = $90
B Was chosen even though A was the same price!
Choice A Choice B
What about this choice?
P(Bananas) = $2/lb.P(Apples) = $4/Lb.
Q(Bananas) = 25lbsQ(Apples) = 10lbs
Q(Bananas) = 10lbsQ(Apples) = 20lbs
Cost = $90
Q(Bananas) = 15lbsQ(Apples) = 15lbs
Cost = $90
Cost = $100
Is strictly preferred to Choice C Choice B
Choice C
Is choice C preferred to choice A?
Choice B
Choice A
Is strictly preferred to Choice B Choice A
Is strictly preferred to Choice C Choice B
Is strictly preferred to Choice C Choice A
Rational preferences exhibit transitivity
C > B > A
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
BAU :
A BU
Set of possible consumption choices “Utility Value”
Q(Bananas) = 25lbsQ(Apples) = 10lbs
Q(Bananas) = 10lbsQ(Apples) = 20lbs
Q(Bananas) = 15lbsQ(Apples) = 15lbs
Choice C
Choice A
Choice B
Using the previous example (Recall, C > B > A)
)20,10()15,15()10,25( UUU
We require that utility functions satisfy a few basic properties
20),( yxUx
y
A
B
C
)()()()( BUCUAUCU
There is a definite ranking of all choices
25),( yxU
20),( yxUx
y
A
B
C
More is always better!
)()( AUCU
We require that utility functions satisfy a few basic properties
20),( yxUx
y
A
B
C
People Prefer Moderation!
)()( AUCU 15
5 15
5
10
10
25),( yxU
We require that utility functions satisfy a few basic properties
x
y
*y
*x
20),( yxU
Suppose you are given a little extra of good X. How much Y is needed to return to the original indifference curve?
1x
?yMarginal Utility of Y
Marginal Utility of X
),(),(
**
**
yxUyxUMRS
xy
y
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
x
y
*y
*x
20),( yxU
)','(),( ** yxMRSyxMRS
'y
'x
If you have a lot of X relative to Y, then X is much less valuable than Y MRS is low!
The elasticity of substitution measures the curvature of the indifference curve
x
y'
xy
xy
MRSxy
%
%
Elasticity of substitution measures the degree to which your valuation of X depends on your holdings of X
y
x
small is
y
x
large is
The elasticity of substitution measures the curvature of the indifference curve
If the elasticity of substitution is small, then small changes in x and y cause large changes in the MRS
If the elasticity of substitution is large, then large changes in x and y cause small changes in the MRS
MRSxy
%
%
x
y
20),( yxU
40),( yxU30),( yxU
xy
We often assume that the marginal rate of substitution is dependant only on the ratio of X and Y – i.e. preferences are homogeneous
Consumers solve a constrained maximization – maximize utility subject to an income constraint.
),(max,
I ypx ptosubject
yxU
yx
yx
As before, set up the lagrangian…
)(),(),( ypxpIyxUyx yx
)(),(),( ypxpIyxUyx yx
First Order Necessary Conditions
0),( xx pyxU
ypxpI yx
0),( yy pyxU
y
x
y
x
PP
yxUyxU
),(),(
x
x
y
y
pyxU
pyxU ),(),(
y
x
),(max0,0
I ypx ptosubject
yxU
yx
yx
xpI
ypI
*y
*x
y
x
y
x
PP
yxUyxU
),(),(
ypxpI yx
y
x
Demand Curves present the same information in a different format – therefore, all the properties of preferences are present in the demand curve
x
xp
*x
xp
*x
ypID ,
Demand relationships are based off of the theory of consumer choice. We can characterize the average consumer by their utility function.
HLU ,
“Utility” is a function of lemonade and hot dogs
Consumers make choices on what to buy that satisfy two criteria:
L
L
H
H
PMU
PMU
ILPHP LH
Their decision on what to buy generates maximum utility
Their decision on what to buy generates is affordable
IPPDQ LHH ,,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
24
48
ILPHP LH
210,2,4 DQH
L
L
H
H
PMU
PMU
101224
This point satisfies both conditions and, hence, is one point of the demand curve
y
x x
y
x x
small is MRS
large is MRS
xp
xp
Willingness to pay is low
Willingness to pay is high
The marginal rate of substitution controls the height of the demand curve
$10
$2
ypID ,
ypID ,
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
24
68
ILPHP LH L
L
H
H
PMU
PMU
101226
Your decision at the margin has been affected. You need to buy less hot dogs and more lemonade (Substitution effect)
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
23
69
ILPHP LH L
L
H
H
PMU
PMU
102216
110,2,6 DQH
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
Quantity
Price
$4.00
2
10$ID
$6.00
1
%50100*2
21
%50100*4
46
15050
%%
PQ
D
y
x*x
The elasticity of substitution will control the slope of the demand curve
x
xp
'x *x
xp
xp'
D
MRSxy
%
%
xx p
x
%%
y
x x
xpsmall is small is x
y
x x
xp
large is large is x
Elasticity of Substitution vs. Price Elasticity
y
x x
xp0 0x
y
x x
xp
x
Perfect Complements vs. Perfect Substitutes
(Almost)
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
24
48
ILPHP LH L
L
H
H
PMU
PMU
201224
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
23
46
ILPHP LH L
L
H
H
PMU
PMU
202244
420,2,4 DQH
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
Quantity
Price
$4.00
2
10$ID
4
%100100*2
24%
Q
%100100*10
1020%
I
1100100
%%
IQ
D
20$ID
y
x x
xp
*x
xp x%
Ix
I
%%
Income elasticity measures the response of consumers to changes in income holding prices constant – the homogeneity of preferences will effect this
*x
Cross price elasticity refers to the impact on demand of another price changing
Quantity
Price
$4.00
2
2$LPD
6
%200100*2
26%
Q
%100100*2
24%
LP
2100200
%%
L
HL P
Q
4$LPD
Note: These numbers aren’t coming from the previous example!!
A positive cross price elasticity refers to a substitute while a negative cross price elasticity refers to a compliment
y
x*x
Cross price elasticity measures consumer response to changes in other prices – this is influenced by both homogeneity and elasticity of substitution
x*x
xp
xpy
y px
%%
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
)(),( 5.5. ypxpIyxyx yx
y
x
y
x
pp
yxyx
yxUyxU
5.5.
5.5.
5.5.
),(),(
xppy
y
x
An Example: Cobb-Douglas Utility
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
Ixpppxp
y
xyx
Iypxp yx
y
xpIx
2
ypIy
2
An Example: Cobb-Douglas Utility
An Example: Cobb-Douglas Utility
yxyxU ),(
yxyxU x1),(
1),( yxyxU y
xy
yxyx
yxUyxU
y
x
1
1
**
**
),(),(
With Cobb-Douglas Utility functions, your MRS is directly proportional to your relative consumption of the two goods.
xyMRS
MRSdxyd 1
xy
xy
Cobb-Douglas Utility functions have constant elasticity of substitution
MRSxy
An Example: Cobb-Douglas Utility yxyxU ),(
x
xp
*x
xp
xp
dpdx
px x
xxx
%%
xpIx
2
, 5.5. yxyxU
22 xx pI
dpdx
1
22 2
x
x
xx
pIp
pI
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
xpIx
2
ypIy
2
0%%
xp
dpdx
px y
yyy
Cobb-Douglas demands are independent of other prices!
max 5.5.
0,0
I ypx ptosubject
yx
yx
yx
xpIx
2
ypIy
2
xI
dIdx
Ix
I
%%
1
22
1
x
x
pII
p