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Microeconomics
ECON 100A
Problem Set 9
Due Dec 1, 2010 (Solutions Posted Dec 3, 2010)
1.
(This question is from a practice final)
Ralph consumes only two goods, potatoes (good 1)
and yams (good 2). He is not as particular with his portion sizes and has a utility function
given by
2
1
2
1
)
,
(
q
q
q
q
U
=
, where q
1
is the number of potatoes and q
2
is the number of yams
he consumes in a day. He wishes to maximize utility. The price of a potato is p
1
and the price
of a yam is p
2
. Ralph owns a farm that produces only potatoes and yams and selling these
crops are Ralph’s only source of income. Therefore his income, which depends on the prices
of potatoes and yams and on Ralph’s endowment of potatoes and yams, is given by Y =
p
1
q
1
0
+p
2
q
2
0
, where q
1
0
is Ralph’s endowment of potatoes (i.e. what he harvests) and q
2
0
is his
endowment of yams (i.e. what he harvests).
a.
(5) Solve for Ralph’s Marshallian demand function for potatoes and his Marshallian
demand function for yams. Your solution should be a function only of prices and
endowments. However, if you prefer, you
may
still use the notation “Y” in place of
p
1
q
1
0
+p
2
q
2
0
to keep the equations clean. If you do this, though, don’t forget that Y is still
a function of prices, that will matter!
Also solve for the indirect utility function.
Usual Cobb Douglas in the usual way.
q
2
/q
1
= p
1
/p
2
Y = p
1
q
1
+ p
2
q
2
q
1
= Y/2p
1
= (p
1
q
10
+ p
2
q
20
)/2p
1
q
2
= Y/2p
2
= (p
1
q
10
+ p
2
q
20
)/2p
2
V= (
Y/2p
1
)(
Y/2p
2
)
b.
(5) Solve for Ralph’s Hicksian demand function for potatoes and his Hicksian demand
function for yams.
usual cobb douglas hicksians
q
2
/q
1
= p
1
/p
2
q
1
q
2
= U
q
1
= (p
2
/p
1
)
0.5
* U
0.5
q
2
= (p
1
/p
2
)
0.5
* U
0.5
c.
(5) Verify the Slutsky equation holds for a change in the price of potatoes. That is,
calculate the left hand side of the equation and the right hand side of the equation directly
using the Marshallians and Hicksians above, and then show the left hand side equals the
right hand side. Identify the terms that represent 1) the ordinary income effect, 2) the
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View Full Documentendowment income effect, 3) the substitution effect, and 4) the total effect of the change
in the price in potatoes.
LHS. dq
1
/dp
1
= p
2
q
20
/2p
1
2
RHS SE. dq
1
h/dp
1
= 0.5p
1
1/2
* (p
2
*U)
1/2
= Y/4p
1
2
=  (p
1
q
10
+ p
2
q
20
)/4p
1
2
RHS OIE dq
1
/dY * q
1
= 1/(2p
1
) * Y/2p
1
= Y/4p
1
2
=  (p
1
q
10
+ p
2
q
20
)/4p
1
2
RHS EIE dq
1
/dY * q
10
= q
10
/(2p
1
)
RHS =  (p
1
q
10
+ p
2
q
20
)/4p
1
2
 (p
1
q
10
+ p
2
q
20
)/4p
1
2
+ q
10
/2p
1
=  (p
1
q
10
+ p
2
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 Fall '08
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 Microeconomics

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