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100A:
Intermediate Microeconomics, Spring 2010, Dr. Famulari
Problem Set #3
Question 1:
Matt gets utility from candy bars (C) and movies (M).
Matt ALWAYS eats 2 candy
bars at each movie he attends.
Extra candy without seeing another movie is of no value to
Matt, just as going to the movies is of no value without the candy bars.
Matt maximizes his
utility from movies and candy bars subject to a budget constraint.
1A.
What type of utility function represents Matt’s preferences?
Write down an expression for
Matt’s utility.
U=min{1/2C, M}
A hint on how to check your specification:
Let’s look at the indifference curve for U=1.
When there is no excess M or C, we will have U = 1 = 1/2C = M.
Solving for C and M
we have that C=2 and M=1 – just what we wanted.
(Note:
What do I mean by no
“excess” C or M?
How would utility change if C increases but M stays at 1?
Alternatively, how would utility change if M increases but C stays at 2?
Do you see how
there is no excess C or M when C=2 and M=1?)
1B.
Draw a picture of Matt’s utility maximizing problem. (See 1F below)
1C.
Will you be able to use the Lagrange multiplier technique to solve for Matt’s utility
maximizing combinations of movies and candy bars?
Why or why not?
No, from the picture you can see that the solution to Matt’s utility maximizing problem
will be at a kink point where there is no excess C or B.
The utility function is not
differentiable at this point.
1D.
What is Matt’s utility maximizing combination of movies, M*(P
M
, P
C
, I), and candy bars,
C*(P
M
, P
C
, I)?
Use the facts that
(1)
that the solution will be a t a kink point, with no excess C or M (so 1/2C*=M*) and
(2)
the utility maximizing bundle must be on the budget constraint (PcC*+ PmM* =I).
Substitute the expression for M* from (1) into the budget constraint in (2) to get:
Pc(C*)+ Pm(1/2C*)=I and then solve for C*=
)
2
1
(
m
c
p
P
I
+
Since M*=1/2C*, M*=
)
2
1
(
*
2
m
c
p
P
I
+
1E.
What is the Matt’s indirect utility function, i.e., V(P
M
, P
C
, I)=U(M*(P
M
, P
C
, I),C*(P
M
, P
C
,I))?
Plug in the values of M* and C* into the utility function.
This gives you the indirect
utility function (utility as a function of prices and income).
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)
2
1
(
*
2
m
c
p
P
I
+
,
)
2
1
(
*
2
m
c
p
P
I
+
}
You have solved for general expressions for C*, M* and U*.
Now, suppose the price of movies
is $4, the price of candy bars is $2, and the consumer’s income is $24.
1F.
What are the numerical values of C*, M* and U*?
Put these values on your graph,
C*=(24)/(2 + .5*4)=24/4=6
M*=24/((2*(2+1/2*4)=24/8=3
U*=3
M
U* = 3
3
6
C
Question 2:
Now, let’s see if you can do the general perfect complements problem.
Matt gets
utility from X and Y.
Matt always consumes
α
X with
β
Y.
2A.
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 Spring '07
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