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Consumer theory: Demand functions
Maria Saez Marti
O
ﬃ
ce 210, IEW
tel. 044 634 37 13 email: [email protected]
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View Full Document Summary
x
∗
∈
B
such that
x
∗
%
x
for all
x
∈
B
(1)
max
x
∈
R
n
+
u
(
x
)s
.
t
.
p
·
x
≤
y.
(2)
•
Continuous preferences can be represented by utility functions
•
For such preferences a demand correspondence exists if all prices
are strictly positive
•
If, in addition preferences are strictly convex, demand function exists
•
If, in addition, utility functions are strictly monotone, the marginal
rate of substitution equals the price ratio.
1.5 Indirect Utility and Expenditure
The Indirect Utility Function
•
The relationship among prices, income, and the maximized value of
utility can be summarized by a realvalued function
v
:
R
n
+
×
R
+
→
R
de
f
ned by the indirect utility function:
v
(
p
,y
)= max
x
∈
R
n
+
u
(
x
)s
.
t
.
p
·
x
≤
y.
(3)
•
When
u
(
x
)i
scon
t
inuou
s
,
v
(
p
) is wellde
f
ned for all
p
>>
0
and
y
≥
0
•
If, in addition,
u
(
x
) is strictly quasiconcave, then the solution is
unique and we write it as
x
(
p
), the consumer’s demand function.
v
(
p
)=
u
(
x
(
p
))
.
(4)
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View Full Document Figure 1: Indirect utility at prices
p
and income
y
Theorem 6:
If
u
(
x
)
is continuous and strictly increasing on
R
n
+
,then
v
(
p
,y
)
de
f
ned
in (3) is
1.
Continuous on
R
n
++
×
R
+
2.
Homogeneous of degree zero in
(
p
)
3.
Strictly increasing in
y
4.
Decreasing in
p.
5.
Quasiconvex in
(
p
)
6.
Roy’s identity holds: If
v
(
p
)
is di
f
erentiable at
³
p
0
0
´
and
∂v
³
p
0
0
´
/∂y
6
=0
x
i
³
p
0
0
´
=
−
³
p
0
0
´
/∂p
i
³
p
0
0
´
,i
=1
,...,n.
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View Full Document CES Functions
Example 2
•
u
(
x
1
,x
2
)=
³
x
ρ
1
+
x
ρ
2
´
1
/ρ
,where0
6
=
p<
1
•
By Example 1, Marshallian demands are:
x
1
(
p
,y
p
r
−
1
1
y
p
r
1
+
p
r
2
,
(5)
x
2
(
p
p
r
−
1
2
y
p
r
1
+
p
r
2
for
r
≡
ρ/
(
ρ
−
1)
Calculation of indirect utility:
v
(
p
,y
)=[
(
x
1
(
p
))
ρ
+(
x
2
(
p
))
ρ
]
1
/ρ
(6)
=
⎡
⎣
⎛
⎝
p
r
−
1
1
y
p
r
1
+
p
r
2
⎞
⎠
ρ
+
⎛
⎝
p
r
−
1
2
y
p
r
1
+
p
r
2
⎞
⎠
ρ
⎤
⎦
1
/ρ
=
y
⎡
⎢
⎣
p
r
1
+
p
r
2
³
p
r
1
+
p
r
2
´
ρ
⎤
⎥
⎦
1
/ρ
=
y
(
p
r
1
+
p
r
2
)
−
1
/r
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View Full DocumentThe Expenditure Function
•
To construct the indirect utility function, we
f
xed prices and
in
come
, and sought the maximum utility the consumer could achieve.
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This note was uploaded on 06/20/2011 for the course OPR 201 taught by Professor Pp during the Spring '11 term at Thammasat University.
 Spring '11
 PP

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