Microeconomics
Problem Set 6 Questions
Due February 24th, 2010
Winter 2010 ECON 100A
1.
Rene also consumes kangaroo burgers and vegemite but she is picky about the ratio in which she eats
them. Her preferences can be represented by the utility function U = min(V,B). Rene receives fixed
money income Y.
a.
Compute her Marshallian demand functions x
B
*
(p
B
, p
V
, Y) and x
V
*
(p
B
, p
V
, Y). Find her indirect utility
function v(p
B
, p
V
, Y) and show Roy’s identity holds.
V
B
V
B
p
p
Y
x
x
+
=
=
*
*
V
B
V
B
p
p
Y
Y
p
p
V
+
=
)
,
,
(
Roy’s identity:
V
B
p
p
dY
dV
+
=
1
/
2
)
(
/
/
V
B
V
B
p
p
Y
dp
dV
dp
dV
+
=
=
)
(
/
/
*
V
B
V
B
p
p
Y
dY
dV
dp
dV
x
+
=

=
b.
Compute her Hicksian demand functions x
B
h
(p
B
, p
V
, U) and x
V
h
(p
B
, p
V
, U). (Hint: think about it
before plunging in: take a specific indifference curve, what is the cheapest combo of B and V that will
get Rene on it.) Find her expenditure function E(p
B
, p
V
, U) = p
B
x
B
h
+ p
V
x
V
h
and
show that
∂
E/
∂
pi =
x
i
h
for each i.
U
x
x
h
V
h
B
=
=
U
p
p
U
p
p
E
V
B
V
B
)
(
)
,
,
(
+
=
Shepard’s Lemma (
∂
E/
∂
pi = x
i
h
) falls out of this easily.
c.
Imagine a change in the price of kangaroo burgers. Show the Slutsky equation holds in this case.
(Hint: remember to differentiate each piece first, plug in for U after.)
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View Full Document∂
x
B
/
∂
p
B
= Y/(p
B
+ p
V
)
2
∂
x
B
h
/
∂
p
B
= 0
∂
x
B
/
∂
E * x
B
= 1/(p
B
+ p
V
) * Y/(p
B
+ p
V
) = Y/(p
B
+ p
V
)
2
.
The first is equal to the second plus the negative of the third.
Note: the SE is zero! Makes sense, though. Rene does not substitute in any way when prices
change…only bundles on the kink are possible.
2.
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 Winter '08
 staff
 Microeconomics, Utility, Hicksian demand function, Hicksian

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