E
CONOMICS
11,
W
INTER
2010:
HW
A
SSIGNMENT
#3

S
OLUTIONS
Question 1
The Lagrangian can be written as follows:
ℒ
(
𝑥𝑥
,
𝑦𝑦
,
𝜆𝜆
) =
𝑥𝑥
+
𝑦𝑦
1
2
�
+
𝜆𝜆
(
𝑀𝑀 − 𝑥𝑥𝑝𝑝
𝑥𝑥
− 𝑦𝑦𝑝𝑝
𝑦𝑦
)
The first order conditions are:
1
− 𝜆𝜆𝑝𝑝
𝑥𝑥
= 0
1
2
𝑦𝑦
−
1
2
�
− 𝜆𝜆𝑝𝑝
𝑦𝑦
= 0
𝑀𝑀 − 𝑥𝑥𝑝𝑝
𝑥𝑥
− 𝑦𝑦𝑝𝑝
𝑦𝑦
= 0
Solving the first two equations for
λ
and setting them equal yields:
1
𝑝𝑝
𝑥𝑥
=
1
2
𝑦𝑦
1
2
�
𝑝𝑝
𝑦𝑦
Solve for y to get:
𝑦𝑦
=
�
𝑝𝑝
𝑥𝑥
2
𝑝𝑝
𝑦𝑦
�
2
Plugging this expression into the first order condition for
λ
and solving for x yields:
𝑀𝑀 − 𝑥𝑥𝑝𝑝
𝑥𝑥
− �
𝑝𝑝
𝑥𝑥
2
𝑝𝑝
𝑦𝑦
�
2
𝑝𝑝
𝑦𝑦
= 0
𝑥𝑥
=
𝑀𝑀
𝑝𝑝
𝑥𝑥
−
𝑝𝑝
𝑥𝑥
4
𝑝𝑝
𝑦𝑦
Note that x must be greater than or equal to zero.
If x > 0, then we have an interior solution
and the demand functions are given by the solutions to the maximization problem above.
If, on
the other hand, x = 0, then we have a corner solution where the individual spends all of their
income on y.
Hence, the demand functions can be expressed as follows:
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𝑋𝑋�𝑝𝑝
𝑥𝑥
,
𝑝𝑝
𝑦𝑦
,
𝑀𝑀�
=
⎩
⎪
⎨
⎪
⎧
𝑀𝑀
𝑝𝑝
𝑥𝑥
−
𝑝𝑝
𝑥𝑥
4
𝑝𝑝
𝑦𝑦
𝑖𝑖𝑖𝑖
𝑀𝑀
𝑝𝑝
𝑥𝑥
−
𝑝𝑝
𝑥𝑥
4
𝑝𝑝
𝑦𝑦
> 0
0
𝑖𝑖𝑖𝑖
𝑀𝑀
𝑝𝑝
𝑥𝑥
−
𝑝𝑝
𝑥𝑥
4
𝑝𝑝
𝑦𝑦
≤
0
𝑌𝑌�𝑝𝑝
𝑥𝑥
,
𝑝𝑝
𝑦𝑦
,
𝑀𝑀�
=
⎩
⎪
⎨
⎪
⎧
�
𝑝𝑝
𝑥𝑥
2
𝑝𝑝
𝑦𝑦
�
2
𝑖𝑖𝑖𝑖
𝑀𝑀
𝑝𝑝
𝑥𝑥
−
𝑝𝑝
𝑥𝑥
4
𝑝𝑝
𝑦𝑦
> 0
𝑀𝑀
𝑝𝑝
𝑦𝑦
𝑖𝑖𝑖𝑖
𝑀𝑀
𝑝𝑝
𝑥𝑥
−
𝑝𝑝
𝑥𝑥
4
𝑝𝑝
𝑦𝑦
≤
0
Question 2
Part (a) – The Lagrangian can be written as follows:
ℒ
(
𝑥𝑥
,
𝑦𝑦
,
𝜆𝜆
) =
𝑥𝑥𝑦𝑦
3
+
𝜆𝜆
(
𝑀𝑀 − 𝑥𝑥𝑝𝑝
𝑥𝑥
− 𝑦𝑦𝑝𝑝
𝑦𝑦
)
The first order conditions are:
𝑦𝑦
3
− 𝜆𝜆𝑝𝑝
𝑥𝑥
= 0
3
𝑥𝑥𝑦𝑦
2
− 𝜆𝜆𝑝𝑝
𝑦𝑦
= 0
𝑀𝑀 − 𝑥𝑥𝑝𝑝
𝑥𝑥
− 𝑦𝑦𝑝𝑝
𝑦𝑦
= 0
Solving the first two equations for
λ
and setting them equal yields:
𝑦𝑦
3
𝑝𝑝
𝑥𝑥
=
3
𝑥𝑥𝑦𝑦
2
𝑝𝑝
𝑦𝑦
Note that this equation is identical to that which you would get if you used the marginal utility
per dollar method.
Solving this equation for y as a function of x yields:
𝑦𝑦
=
3
𝑥𝑥𝑝𝑝
𝑥𝑥
𝑝𝑝
𝑦𝑦
Plugging this expression into the first order condition for
λ
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 Winter '09
 McDevitt
 Supply And Demand, Hicks, Hicksian demand function, income expansion path

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