Department of Economics
W3211
Columbia University
Fall 2011
SO L UTIONS T O
Probl
e
m S
e
t 7
Int
e
rm
e
diat
e
Mi
c
ro
ec
onomi
c
s
Prof
.
S
e
yhan E Arkona
c
1
.
Suppose that the minimum wage covers all sectors of the economy; however, for unionized
laborers, the minimum wage is ineffective. That is, the union wage is already above the
minimum wage. Analyze the impact of an increase in the minimum wage on both the
unionized and nonunionized labor markets. (Assume that the higher minimum wage is still
ineffective in the unionized sector and that union and nonunion labor are substitutable.)
Answer:
See the above figure. The higher minimum wage from min1 to min2 reduces the quantity
demanded of nonunion labor from L
1
to L
2
. This causes a rightward shift of the demand curve
for union labor from D
1
to D
2
. The equilibrium union wage rises from W
1
to W
2
. This explains
why unions typically support increases in the minimum wage even though it does not directly
affect their members.
2
.
Suppose there are only two goods
–
Food (
F
) and Shelter (
S
).
The demand equations for
these two goods depend on their prices,
p
and
as follows:
D
(
,
) = 10
–
2
–
(
) = 10
–
–
2p
The supply curves depend only on their own prices:
(
) = p
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(
p
) = 5
Determine the equilibrium price and quantity of these goods.
Answer:
The two equations describing the equilibrium in each market are given by:
F
=
10
–
2
–
5
10
–
–
2p
Using the first equation, we can solve for
in terms of
:
10
–
3
Substituting this into the second equilibrium condition for
yields:
7(10
–
3
) = 10
–
Solving for
3.0.
Then
=
1.0.
The quantities are
Q
= 3
and
= 5
.
Notice that while the
demand equations are symmetrical, the higher supply of Shelter results in a lower price and
higher quantity in equilibrium.
3
.
Consider trade between two consumers (1 and 2) and two goods, X and Y.
Suppose the total
quantities of each good are 100 units.
Each consumer has CobbDouglas preferences given
by:
U
(X,Y) = XY
What is the shape of the contract curve, i.e. derive the equation?
How does the contract curve
change if consumer one has the utility function
(X,Y) = X
2
Y
while the other consumer’s preferences are as before? Again, derive the equat
ion for the contract
curve.
Answer: In both cases, we find the contract curve by maximizing one of the two consumer’s
utility subject to the other consumer’s utility remaining constant. When both have the same
utility function, the Lagrangian is:
L = X
1
Y
1
+ λ ((100 –
X
1
)(100
–
Y
1
)
–
U
0
)
The first order conditions yield the result:
MRS
1
= MRS
2
In other words, the contract curve is describing all points at which the indifference curves are
tangent.
The equation is thus:
Y
1
/X
1
= (100
–
Y
1
)/(100
–
X
1
)
Simplifying, Y
1
= X
1
.
The contract curve consists of all points at which the quantities of each
good are equal.
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 Fall '09
 Elmes
 Economics, Microeconomics, Supply And Demand, NC, contract curve

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