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Microeconomics
Problem Set 4 Solutions
Due February 10
th
, 2010
Winter 2010 ECON 100A
Professor Michael Noel
University of California San Diego
1.
Now, let’s see if you can do the general perfect complements problem.
Matt gets utility from X and Z.
Matt always
consumes
α
X with
β
Z.
a.
What type of utility function represents Matt’s preferences?
Write down an expression for Matt’s utility.
U=min{(
1
/
α)
X, (
1
/
β)
Z}
b.
What is Matt’s utility maximizing combination of X*(P
x
, P
z
, Y) and Z*(P
x
, P
z
, Y)?
Use the facts that
(1)
that there will be no excess X or Z (so
1
/
α
X*=
1
/
β
Z*) and
(2)
the utility maximizing bundle must be on the budget constraint (PxX*+PzZ* =Y).
Substitute the expression for Z*=
β
/
α
X* from (1) into the budget constraint in (2) to get:
Px(X*)+ Pz(
β
/
α
X*)=Y
X*(Px+
β
/
α
Pz)
=
Y
and then solve for X*=
z
x
p
p
Y
α
β
+
Finally, substitute this expression for X* into Z*=
β
/
α
X* to solve for Z*:
Z*=
)
(
z
x
p
p
Y
+
c.
What is the Matt’s indirect utility function,
i.e., V(P
x
, P
z
, Y)=U(X*(P
x
, P
z
, Y), Z*(P
x
, P
z
, Y))?
U*(px, pz, Y)=min{
1
/
α
X*,
1
/
β
Z* }=min{
)
(
,
)
(
z
x
z
x
p
p
Y
p
p
Y
+
+
}
d.
What share of his budget does Matt spend on X*?
What share of his budget does Matt spend on Z*?
(Note:
check that these two shares sum to one.)
Share of Budget spent on X;
PxX*/Y=
z
x
x
z
x
x
p
p
p
p
p
Y
Y
p
+
=
+
Share of Budget spent on Z; PzZ*/Y=
z
x
z
z
x
z
p
p
p
p
p
Y
Y
p
+
=
+
)
(
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View Full Document Sum of the two budget shares?
1
=
+
+
=
+
+
+
z
x
z
x
z
x
z
z
x
x
p
p
p
p
p
p
p
p
p
p
α
β
2.
Now, let’s see if you can do the general perfect substitutes problem.
Toni gets utility from X and Z.
Toni thinks that
α
units of X are exactly as good as
β
units of Z.
a.
What type of utility function represents Toni’s preferences?
Write down an expression for Toni’s utility.
U=
β
X +
α
Z
b.
What are Toni’s ordinary demand functions, X*(P
x
, P
z
, Y) and Z*(P
x
, P
z
, Y)?
MRS=
β/α
X*=Y/Px
if
β/α>
Px/Pz
=0
if
β/α<
Px/Pz
Z*=0
if
β/α>
Px/Pz
= Y/Pz if
β/α<
Px/Pz
c.
What is the Toni’s indirect utility function,
i.e., V(P
x
, P
z
, Y)=U(X*(P
x
, P
z
, Y), Z*(P
x
, P
z
, Y))?
U*(px, pz, Y)=
β
Y/Px if
β/α>
Px/Pz
=
α
Y/Pz if
β/α<
Px/Pz
d.
What share of her budget does Toni spend on X*?
What share of her budget does Toni spend on Z*?
If
β/α>
Px/Pz then her budget share on X* is 1 and Z* is
0
If
β/α<
Px/Pz then her budget share on X* is 0 and Z* is
1
3.
Consider the utility maximization problem subject to a budget constraint with the following utility function:
z
x
z
x
U
z
x
2
)
,
(
max
,
+
=
a.
Are the commodities x and z goods?
Prove your answer.
What does this imply about the indifference curves
for U?
X is a good since utility increases as you get more X (holding Z constant),
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This note was uploaded on 11/27/2011 for the course ECON 100A taught by Professor Staff during the Winter '08 term at UCSD.
 Winter '08
 staff
 Microeconomics, Utility

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