Answers to Selected Questions in Practice Problem Set 2
ECON 11
Summer Session C 2010
Microeconomic Theory
Yong Yang
1. [Utility maximization]
(a) A line conneting (0,30) and (45,0). A ﬁgure is omitted.
(b) No because the MRS is not the same with the price ratio.
(c) The optimal consumption is
X
= 30 and
Y
= 10.
(d)
X
= 20 and
Y
= 10.
(e) A curve passing through points (c) and (d). A ﬁgure is omitted.
2. [Marginal rate of substitution]
(a)
MU
X
=
Y
2
and
MU
Y
= 2
XY
.
(b)
MU
X
does not depend on
X
and is constant given
Y
. So it is not decreasing in
X
.
MU
Y
is
increasing in
X
, but not decreasing.
(c) The MRS is
MU
X
MU
Y
=
Y
2
X
. It gets smaller as
X
increases, so the MRS is decreasing in
X
.
(d) False. In the above example, the MRS is diminishing although the marginal utilies are not
decreasing.
3. [Cardinality of utility function]
(a)
X
= 50 and
Y
= 30.
(b)
X
= 50 and
Y
= 30, which is the same as above.
(c)
X
= 50 and
Y
= 30, which is the same again.
(d) Note that Ted, Emily, and Alex have the same MRS=
Y
X
, so the condition for optimal
consumption is the same when the price ratio is the same. If even the income is the same,
they would choose the same consumption. The statement is true.
(e) Jessica’s marginal utilities are
MU
X
=
f
0
(
XY
)
Y
MU
Y
=
f
0
(
XY
)
X
where
f
0
(
·
) denotes the ﬁrst order derivative of
f
with respect to its argument. Then her
MRS is given by
MRS =
f
0
(
XY
)
Y
f
0
(
XY
)
X
=
Y
X
which is the same as Ted’s. So she would choose the same consumption given the same price
ratio and the same income as Ted faces.
4. [Derivation of demand curve] The MRS is
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 Summer '08
 cunningham
 Supply And Demand, Utility, TED, Px

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