ECON 383
Practice Problems from Chapter 10
1, 2, 3, 5
H. K. Chen (SFU)
ECON 383
1/5
Chapter 10 — Ex.1
1. Two sellers, a and b, each selling one house. Two buyers x and y,
whose valuations of houses are as follows:
Buyer j
x
y
va,j
2
3
vb,j
4
6
Suppose pa = 0 and pb = 1. Is this set of prices market-clearing?
Construct each buyer’s preferred-seller
graph
There is no perfect matching for the
given set of prices, because...
the set {x, y} is constricted
The set of prices is not market-clearing
pi
Seller
Buyer
va,j , vb,j
0
a
x
2, 4
1
b
y
3, 6
vx,a − pa = 2
vx,b − pb = 3
)
vy,a − pa = 3
vy,b − pb = 5
H. K. Chen (SFU)
ECON 383
)
⇒
x prefers b
the most
⇒
y prefers b
the most
2/5
Chapter 10 — Ex.2
2. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{x, y, z}, with the following valuations:
Buyer j
x
y
z
vb,j
7
3
4
va,j
5
2
5
vc,j
1
1
4
Suppose pa = pb = 2 and pc = 1. Is this set of prices market clearing?
Construct the preferred-seller graph
x prefers b the most
y prefers b the most
z prefers a and c the most
Again, the set {x, y} is constricted.
Therefore, the set of prices fails to clear
the market.
H. K. Chen (SFU)
ECON 383
pi
Seller
Buyer
va,j , vb,j , vc,j
2
a
x
5, 7, 1
2
b
y
2, 3, 1
1
c
z
5, 4, 4
3/5
Chapter 10 — Ex.3
3. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{x, y, z}, with the following valuations:
Buyer j
x
y
z
vb,j
4
5
7
va,j
2
3
4
vc,j
6
1
5
Suppose pa = pc = 1 and pb = 3. Is this set of prices market clearing?
Construct the preferred-seller graph
x prefers c the most
y prefers a and b the most
z prefers b and c the most
With some coordination, a perfect
matching can be found. Therefore, the
set of prices clears the market.
H. K. Chen (SFU)
ECON 383
pi
Seller
Buyer
va,j , vb,j , vc,j
1
a
x
2, 4, 6
3
b
y
3, 5, 1
1
c
z
4, 7, 5
4/5
Chapter 10 — Ex.5
5. Three sellers, {a, b, c}, each with one house to sell. Three buyers,
{x, y, z}, with the following valuations:
Buyer j
x
y
z
vb,j
7
6
4
va,j
7
7
5
vc,j
4
3
3
Suppose pa = 4, pb = 3 and pc = 1. Is this set of prices market clearing?
Construct the preferred-seller graph
x prefers b the most
y prefers a and b the most
z prefers c the most
With some coordination, a perfect
matching can be found. Therefore, the
set of prices clears the market.
H. K. Chen (SFU)
ECON 383
pi
Seller
Buyer
va,j , vb,j , vc,j
4
a
x
7, 7, 4
3
b
y
7, 6, 3
1
c
z
5, 4, 3
5/5