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Winter 2011
Answers for Problem Set IX
1.a. The Marshallian demand functions are
*
(,, )
2
M
xp
sM
p
and
*
2
M
yp
s
.
The indirect utility
function is
2
*
4
M
Up
ps
.
1.b The Hicksian demand functions are
1/2 1/2
1/2
(,,)
H
x
psu
p s u
and
1/2
1/2
1/2
H
y
.
The
expenditure function is
*
1/2 1/2 1/2
(,,) 2
M psu
.
1.c.i.
Substitute
4
p
,
9
s
, and
216
M
into the indirect utility function to see that the individual's
initial utility index is
2
*
(216)
46656
( , ,
)
324
4(4)(9)
144
.
Alternatively, you can find the initial
utility index by substituting
4
p
,
9
s
, and
216
M
into the Marshallian demand functions to see
that the individual consumes
27
x
and
12
y
, and then substituting these values into the utility
function to get
324
U
.
Now, substitute
324
u
,
9
p
, and
9
s
into the expenditure function to
see that the individual needs
1/2
1/2
1/2
2(9) (9) (324)
2(3)(3)(18)
324
M
to stay on the original
indifference curve after the price change.
(The numbers happen to work out so that the utility index
and the required expenditure both equal 324; this does not happen in general.)
1.c.ii.
The compensating variation is the expenditure in the original situation minus the expenditure required
to stay on the original indifference curve after the price change:
1/2
1/2
1/2
216 2(9) (9) (324)
216 2(3)(3)(18)
216 324
108
CV
.
This means that the individual
would need an additional $108 to be as well off after the price increase as before.
1.c.iii. Substitute
9
p
,
9
s
, and
216
M
into the indirect utility function to see that the individual's
initial utility index would be
2
*
(216)
46656
( , ,
)
144
4(9)(9)
324
if the price increase occurs.
Alternatively, you can find this initial utility index by substituting
9
p
,
9
s
, and
216
M
into the
Marshallian demand functions to see that the individual would consume
12
x