Rice University
Economics 370
Professor Diamond
Microeconomic Theory
SOLUTION TO PROBLEM SET 3: UTILITY AND DEMAND AND INCOME
AND SUBSTITUTION EFFECTS
I.
Suppose that an individual’s indifference curves are described by straight lines with a slope
equal to
–b
, and that, as usual, prices and income,
p
1
,
p
2
,
m
are determined exogenously.
1.
What is the demand function for
x
1
?
The budget line is
x
2
m/p
2
slope=p
1
/p
2
So,
x
1
=
m/p
1
if
p
p
1
<
b
p
2
m/p
1
x
1
0
if
1
>
b
p
2
p
p
1
0
≤
x
1
≤
m/p
1
if
=
b
2
2.
What is the demand function for
x
2
?
x
2
=
m/p
2
if
p
p
1
>
b
p
2
0
if
1
<
b
p
2
p
p
1
0
≤
x
2
≤
m/p
2
if
=
b
2
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2
2
II.
Suppose
an
individual’s
utility
function (defined
over
two
goods)
is
expressed
as
u
(
x
1
,
x
2
) =
x
1
x
2
.
1.
Derive the demand function for
x
1
.
2
2
2
1
2
1
2
2
1
1
2
2
2
2
1 2
1
1
2
1
2
1 1
2
2
1
1 1
2
1
2
1
1
,
2
2
2
2
2
3
u
u
mu
x
mu
xx
x
x
mu
p
x
x
mu
p
xx
x
p
x
x
p
px
p x
m
p
px
p
x
m
p
m
x
p
∂
∂
=
=
=
=
∂
∂
=
=
=
=
+
=
⎛
⎞
+
=
⎜
⎟
⎝
⎠
=
1
2.
Derive the demand function for
1
1
2
1
2
2
1
2
2
2
3
3
p
p
m
x
x
p
p
p
=
=
=
x
2
.
2
m
p
III.
Suppose now that the utility function is
u
(
x
1
,
x
2
) = ln(
x
1
) + 2 ln(
x
2
) .
x
1
and
x
2
.
1.
Again, derive the demand functions for both
4
1
2
1
1
2
1
1
2
1
2
1
2
2
1
2
1
1 1
2
1
2
1
1
1
1
2
1
2
2
1
2
1
2
,
2
2
2
3
2
2
2
3
3
u
u
mu
mu
2
x
x
x
mu
p
x
p
x
x
x
mu
p
x
p
p
px
p
x
m
p
m
x
p
p
p
m
m
x
x
p
p
p
p
∂
∂
=
=
=
=
∂
∂
=
=
⇒
=
⎛
⎞
+
=
⎜
⎟
⎝
⎠
=
=
=
=
2.
Are your answers for II and III the same? Why or why not?
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 Spring '08
 DIAMOND
 Economics, Utility, X1, m Engel

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