Economics 310 – Microeconomic Theory: A Mathematical
Approach
Fall 2008
Solution to Problem Set 3
Question 1:
For each of the following utility functions over two goods, find the Marshallian
demand, the Hicksian demand and expenditure function
The Marshallian demand is a result of utility maximization subject to the
budget constraint. The Hicksian demand can be found by minimizing the ex
penditure subject to the utility being equal to some prespecified level
u.
From
duality of the utility maximization problem and the expenditure minimization
problem we know that the optimal consumption levels of
x
and
y
have to satisfy
the same FOCs for both problems.
This can be seen graphically.
No matter
whether we want to maximize the utility or minimize the expenditure, we have
to find a tangency point between an indifference curve and a budget line. After
finding the relationship that
x
and
y
have to satisfy at the tangency point, we
need to plug this formula into the budget constraint for a given income (if we
are utility maximizing) or into the utility formula for a given utility level (if we
are expenditure minimizing).
(a)
U
(
x, y
) =
x
Note: we can’t use Lagragian method here. There will be a corner solution.
The prices are given, and you are not supposed to solve them.
Assume
p
x
,
p
y
>
0.
The consumer receives utility only from
x
; therefore, she will spend all her
money on it, whether she is utility maximizing or expenditure minimizing.
Hence, the Marshallian demand is
(
x
m
, y
m
) =
m
p
x
,
0
.
To achieve the utility level
u,
the consumer has to buy
x
=
u
units of good
x,
which implies that the Hicksian demand is
(
x
h
, y
h
) = (
u,
0)
.
The expenditure function is
e
(
p
x
, p
y
, u
) =
p
x
u
(b)
U
(
x, y
) =
xy
x
+
y
Lagrangian for the utility maximization problem is
L
=
xy
x
+
y
+
λ
(
m

p
x
x

p
y
y
)
.
1
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The FOC
L
x
=
y
2
(
x
+
y
)
2

λp
x
= 0
,
L
y
=
x
2
(
x
+
y
)
2

λp
y
= 0
,
imply
y
2
(
x
+
y
)
2
p
x
=
x
2
(
x
+
y
)
2
p
y
⇒
y
2
p
y
=
x
2
p
x
.
(1)
Plugging this into the budget constraint, we get
p
x
x
+
p
y
p
x
p
y
x
=
m
⇒
x
m
=
m
p
x
+
√
p
y
p
x
, y
m
=
m
√
p
x
p
y
+
p
y
Note:
The solution below is based on the duality between utility maxi
mization and cost minimization. However, it is good to directly solve the cost
minimization problem. Make sure you understand the equivalence between the
two sets of first order conditions.
To obtain the Hicksian demand we need to plug the optimality condition (1)
into the utility constraint:
u
=
xy
x
+
y
⇒
u
=
x
p
x
p
y
x
x
+
p
x
p
y
x
=
x
√
p
x
√
p
y
+
√
p
x
,
x
h
=
u
√
p
y
+
√
p
x
√
p
x
, y
h
=
u
√
p
y
+
√
p
x
√
p
y
.
The expenditure function is
e
(
p
x
, p
y
, u
) =
p
x
x
h
+
p
y
y
h
=
u
(
√
p
y
+
√
p
x
)
2
.
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 '08
 StephenE.Morris
 Utility, Roy, Px, Della

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