UCLA
Economics 11
–
Fall 2009
Professor Mazzocco
Problem Set 3
Due by October 22 before 9:00am in Room 8271
1)
Suppose that an individual with income I cares about two goods, X and Y. The price of
the two goods is P
x
and P
Y
. The individual has the following utility function:
U(X,Y) = X (1 + Y)
a)
Find the Marshallian (uncompensated) demand for X and Y. Are X and Y normal or
inferior goods?
b)
Find the Hicksian (compensated) demand for X and Y.
c)
What is the minimum expenditure necessary to achieve a utility level of U= 72 with
Px=4 and
P
Y
=2?
a)
L(x,y,
λ
) = X(1+Y) +
λ
[I P
x
x
–
P
z
y]
L
X
= 1 + Y 
λP
x
= 0
(1)
L
Y
= X
–
λP
y
= 0
(2)
L
λ
= I P
x
x
–
P
y
y = 0
(3)
With equations (1) and (2) we get:
P
X
/P
y
= (1+Y)/X
(4)
X
*
=(I+P
Y
)/ 2p
x
, Y
*
=(IP
Y
)/ 2p
y
,
a)
L(x,y,
λ
) = P
x
x + P
z
y +
λ
[U X
–
XY]
L
X
= Px
–
λ(1+Y)
= 0
(1)
L
Y
= Py
–
λ(X)
= 0
(2)
L
λ
= U X
–
XY
= 0
(3)
With equations (1) and (2) we get:
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P
X
/P
y
= (1+Y)/X
(4)
Replacing back in (3) we get:
Y=(UPx/Py)
1/2
1
X=U(Py/Px)
1/2
c) We can obtain the indirect utility function replacing the demand functions in U=X+XY
V=((I+Py)
2
)/4PxPy
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 Fall '07
 McDevitt
 Economics, Px, Britney, py

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