1
ECON 520: Intermediate Microeconomics
Problem Set 2
Solutions
Professor D. Weisman
2
1. For Bob, we are given that
a) MRS P  B  
MRSP B  2
B
 2  B  2P
P
Let B  1 , then P  
Hence, Bob would give up
1
2
1
pizza for additional beer.
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b) Let P  1 , then B  2 . Hence Bob would give up 2 beers for 1 additional pizza.
c) For Bob
MRSP B  2 ; and for Carol MRSP B  4 .
This implies that Bob is willing to give up 2 beers for 1 additional pizza, but Carol is willing to give up 4
beers for 1 additional pizza. Because 4  2 , Carol must value pizza more than Bob. [Note: similar
reasoning implies that Bob values beer more than Carol]
2. Write down equation for the utility function and indifference map for each of the cases given
a) The
MRS P  B or MRS B  P is constant. This implies the utility function is linear in Beers(B) and
Pizza(P). That is, we have a case of perfect substitutes.
(1)
u  2B  1P or more generally (1’) u   B   P , where   2
Notice that we can write (1) in the following form:
(2) B 
u 1
 P
2 2
The coefficient of 
1
1
on P implies that Carol is willing to give up
Beer for 1 additional Pizza, or 1
2
2
Beer for 2 additional Pizzas.
Indifference Map
B
U2
U1
U0
-1/2
-1/2
-1/2
P
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b) This is a case of perfect complements because Bob consumes Beer and Pizza in fixed proportions:
2 Beers with every Pizza. His utility function is given by
1

B, P   is an unknown.
2

(3) u   Min 
Set
1
B  P and obtain B  2 p ; that is for every Pizza Bob consumes, he also consumes 2Beers.
2
We are also told that when P  4 and
(4)
B  10 , u  32 . Hence
32   Min 5, 4    4    8 . Hence, Bob’s utility function is given by
1

B, P 
2

(5) u  8Min 
B
Efficient consumption locus: Neither excess Beer
nor excess pizza
U1
U0
2
P
c) We are told that Kathy loves Pizza and is neutral toward Beer. This means that Kathy derives no
positive (or negative) utility from Beer. Also, she derives 4 units of satisfaction for each Pizza she
consumes [Note: This value of 4 is a constant.]
(6)
u  4P
B
U0
U1
U2
P
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3. Given
pP  4, pB  2 and I  100
a) Determine equilibrium number of Beers and Pizza
(1) u  2 P  4 B or B 
Budget constraint:
u 1
 P
4 2
2B  4P  100 or B  50  2P
Observe that the slope of budget constraint
 2
 1
 are not
 2
and the slope of indifference curve  
equal. Hence, we will have a corner solution (i.e., allocate entirety of income to Beers or entirety of
income to Pizza). We need to determine which outcome generates higher utility.
If purchase only Pizzas, P 
If purchase only Beers, B 
100
 25 and u  2  25  50 .
4
100
 50 and u  4  50   200 .
2
Since 200>50, purchase only Beers.
Equilibrium outcome: B
0
 50; P0  0; u 0  200 .
 1 
B  Solve following 2 equations simultaneously:
 3 
B  3P
(Efficient consumption locus) 
  3P  50  2 P  5P  50  P  10 .
B  50  2 P (Budget constraint)

(2) u  2 Min  P,
When P  10, B  30 and
u  2Min 10,10  20 .
Hence, Equilibrium outcome: B
(3)
u  4BP
 Given
0
 30; P0  10; u 0  20 .
MU B  4P and MU P  4B  .
Consumer equilibrium (Interior solution) requires that
MU B MU P
4P 4B



 B  2P .
pB
pP
2
4
B  50  2P . Solving these 2 equations
simultaneously yields 4P  50 or P  12.5 and B  25 . This consumption bundle generates
Also, the budget constraint must be satisfied. Hence,
utility of
u  4  2512.5  1250 . Hence, the equilibrium outcome is given by:
Equilibrium outcome: B
0
 25; P0  12.5; u 0  1250 .
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b) Graphical illustration of results
(1)
B
Equilibrium
(Corner Slution)
50
Indifference
Curve
Budget
Constrain
t
U=200
25
100
P
(2)
Indifference
Curve
B
50
Equilibrium
30
U=20
Budget
Constraint
10
25
P
(3)
B
Indifference
Curve
50
Equilibrium
25
U=1250
Budget
Constraint
12.5
25
P
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4. Compute marginal rates of substitution for each of the utility functions in question 3 when
B  20 .
and
(1)
P  10
u  2P  4B and B 


(2) u  2 Min  P,
u 1
1
 P  MRS P  B  ; MRS B  P  2 [These are constants]
4 2
2
1 
B
3 
Efficient consumption requires
B  3P . At P  10 and B  20 , we have too many Pizzas and not
enough Beers. Hence
MRS P  B  0 and MRS B  P   .
(3) u  4BP [Recall: MU B  4 p and MU P  4 B ]
MRS P  B 
MU P 4 B B 20

 
2
MU B 4 P P 10
MRS B  P 
1
1

MRS P  B 2
5. Utility functions from question 3 on problem set 2.
(1)
 1 
u  2P  4B ; (2) u  2Min  P, B  ; (3) u  4BP
 3 
1. Utility function (1) is perfect substitutes. Three possibilities for consumer equilibrium [Normalize
pP  1 ]
I

 p
P

 I  pB  B
Demand functions P  
 pP

0


pB  2 pP
pB  2 pP
pB  2 pP


0

 I  pP  P
B
 pB

I

 pB
pB  2 pP
pB  2 pP
pB  2 pP
(2) Perfect complements.
The efficient consumption locus is given by (i)
B
p
I
 P  P . Solve (i) and (ii) simultaneously
pB pB
3P 
or
B  3P ; The budget constraint is given by (ii)

p
p  I
I
 P  P  P 3  P  
 P  3 pB  p P   I
pB pB
p
p

B 
B
P
I
3I
and B 
(Demand functions)
3 pB  pP 
3 pB  pP 
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(3) “Typical preferences”
Equilibrium condition:
Budget constraint: B 
 Given
MU B  4P and MU P  4B 
MU B MU P
p
4B 4P



 B  P P
pP
pB
pP pB
pB
p
I
 P P
pB pB
Solve simultaneously,
pP
p
I
I
I
P 
 P  P  P  2 pP   I  P 
and B 
pB
pB pB
2 pP
2 pB
(Demand functions)