14.03 Fall 2000 Problem Set 2
Due
in Class #7
Theory:
1) Nicholson 3.2 (Note that:
∂
log
base a
x
1
x
=
∂
(
x
ln
a
) , 3.4, 3.7 (Hint: observe how these three
functions are related), 4.2, 4.5 (Note: use low-tech math and common sense here), 5.2 and
5.9.
2) Let
U
(
x
,
y
)
= −
1 x
−
1
y
. Suppose that prices are
P
x
and
P
y
, and income is
I
.
(A) Calculate the utility-maximizing choices of x and y, that is, the Marshallian demand
functions
d
x
(
P
x
,
P
y
,
I
)
and
d
y
(
P
x
,
P
y
,
I
)
.
(B) Calculate “indirect utility,” i.e., the utility at the optimal choices,
V
(
P
x
,
P
y
,
I
).
(C) For a given utility level
U
0
, solve the dual expenditure-minimization problem, and
compute the optimal choices of
X
and
Y
, i.e., the “compensated demand functions”
h
x
(
P
x
,
P
y
,
U
o
)
and
h
y
(
P
x
,
P
y
,
U
o
)
.
(D) Calculate the minimum expenditure function
E
(
P
x
,
P
y
,
U
0
)
. Show that the expenditure
and indirect utility functions you have calculated are inverses of one another, i.e., show that