49
Demand Functions
Examples of Problems
Cobb-Douglas Utility Function U=X
1
a
X
2
b
or alnX
1
+ blnX
2
1.
Suppose a consumer has the utility function
U(X
1
,X
2
) = lnX
1
+ 2lnX
2
a.
Find the MRS
b.
Find the demand functions for X
1
and X
2
c.
Find the optimal choices and optimal utility if I=900,
P
1
=10 and P
2
=30.

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50
2.
Suppose a consumer has the utility function
(quasi-linear U = aX
1
+ bf(X
2
))
U(X
1
,X
2
) = X
1
+ 5lnX
2
a.
Find the MRS
b.
Find the demand functions for X
1
and X
2
c.
Find the optimal choices and optimal utility if I=100,
P
1
=4 and P
2
=1.

51
3.
Suppose a consumer has the utility function (perfect
substitutes U = aX
1
+ bX
2
)
U(X
1
,X
2
) = 4X
1
+ 2X
2
a.
Find the MRS
b.
What should the consumer do if I=60, P
1
=3 and P
2
=4?
Why?
c.
Find the optimal choices and optimal utility when
P
1
<2P
2
, for any I.

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