The Ohio State University
Department of Economics
Econ 501.02—Prof. James Peck
Homework #1 Answers
1.
Suppose we have the utility function,
U
=
x
2
y
3
.
(a)
Find the function for the marginal rate of substitution.
(b)
Show that this utility function satis
fi
es the
fi
rst 3 axioms of consumer
preference.
Answer:
(a)
The MRS is given by
∂u
∂x
∂u
∂y
=
2
xy
3
3
x
2
y
2
=
2
y
3
x
.
(b) Suppose preferences are based on this utility function, and denote bun
dle A by
(
x
A
, y
A
)
, bundle B by
(
x
B
, y
B
)
, and bundle C by
(
x
C
, y
C
)
.
Then
preferences are complete, because either
(
x
A
)
2
(
y
A
)
3
>
(
x
B
)
2
(
y
B
)
3
(in which
case
A
P
B
), or
(
x
A
)
2
(
y
A
)
3
<
(
x
B
)
2
(
y
B
)
3
(in which case
B
P
A
), or
(
x
A
)
2
(
y
A
)
3
=
(
x
B
)
2
(
y
B
)
3
(in which case
A
I
B
). Preferences are re
fl
exive, because
(
x
A
)
2
(
y
A
)
3
=
(
x
A
)
2
(
y
A
)
3
. Preferences are transitive, because if
(
x
A
)
2
(
y
A
)
3
>
(
x
B
)
2
(
y
B
)
3
and
(
x
B
)
2
(
y
B
)
3
>
(
x
C
)
2
(
y
C
)
3
hold, then
(
x
A
)
2
(
y
A
)
3
>
(
x
C
)
2
(
y
C
)
3
must hold. In
other words, the "
>
" relation on real numbers is complete, re
fl
exive, and tran
sitive.
2.
Suppose we have the utility function,
U
= 2
xy
+
x
+
y.
(a)
Find the function for the marginal rate of substitution.
(b)
If prices are
p
x
= $2
and
p
y
= $4
, and if income is
M
= $17
,
fi
nd the
utility maximizing consumption bundle.
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 Spring '08
 YANG
 Microeconomics, Utility, $2, Ohio State University Department of Economics Econ

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