Economics 101, Problem Set 4 and 5
Alan C. Marco
Solutions.
25.
Suppose demand is given by
p
= 100
°
2
Q
and supply is given by
p
=
1
2
Q:
Give the equilibria for the following
situations:
(a)
The free market equilibrium
100
°
2
Q
=
Q
2
Q
°
=
40
P
°
=
20
(b)
A tax of $15.
100
°
2
Q
°
15
=
Q
2
Q
°
=
34
P
s
=
17
P
d
=
32
The DWL is given by
DWL
b
=
°
Q
±
°
P
±
1
2
=
(40
°
34)(32
°
17)
1
2
=
45
Note that
°
P
is just the di/erence between demand and supply along the price axis at the equilibrium
quantity. It°s not really a ±change²in price. We just use it to measure the height of the DWL triangle.
(c)
A subsidy of $20.
100
°
2
Q
+ 20
=
Q
2
Q
°
=
48
P
s
=
24
P
d
=
4
The DWL is given by
DWL
c
=
°
Q
±
°
P
±
1
2
=
(48
°
40)(24
°
4)
1
2
=
80
(d)
A price °oor at $40.
The price ³oor will be binding, so
Q
°
will be given by the quanity demanded.
100
°
2
Q
=
40
Q
°
=
30
P
=
40
P
s
=
15
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P
s
= 15
is the ±price²on the supply curve; we need this just to calculate the DWL. The DWL is given by
DWL
d
=
°
Q
±
°
P
±
1
2
=
(40
°
30)(40
°
15)
1
2
=
125
(e)
A price ceiling at $10.
The price ceiling will be binding, so
Q
°
will be given by the quanity supplied.
Q
2
=
10
Q
°
=
20
P
=
10
P
d
=
40
P
d
= 40
is the ±price²on the demand curve; we need this just to calculate the DWL. The DWL is given
by
DWL
e
=
°
Q
±
°
P
±
1
2
=
(40
°
20)(40
°
10)
1
2
=
300
(f)
For (b) through (e), show that the deadweight loss is related to the quantity distortion (at least in rank
order). [You must calculate the actual DWL for each case.] Is DWL related to the price distortion?
Part
DWL
Q distortion
Price distortion
(e)
300
20
20-10=10
(d)
125
10
40-20=20
(c)
80
8
24-20=4 or 20-4=16
(b)
45
6
32-20=12 or 20-17=3
You can see that the quantity distortion is in line with the DWL. Price ±distortion²(the distance between
the equilibrium price and the free market price) is actually a bit hard to de´ne, esp. in the case of taxes
and subsidies since two prices really exist. If one de´nes the distortion as the largest of the price distortions
(and one de´nes another ±price² in the case of price ³oors and ceilings), one can actually get a similar
ranking. But, it°s really much easier to do this with quantity, since that°s always observable.
26.
Suppose the (perfectly competitive) market for wheat is initially at a short-run and long-run equilibrium. Show
(using a two-panel graph of the ±rm and the market) the short-run and long-run impact on
p; Q
(market
quantity), and
q
(±rm quantity) of the following events:
(a)
A tax increases per-unit costs by $1 per unit.
P
q
AC1
MC1
AC2
MC2
q1
q0
P2
P1
P0
P
Q
D
S3
SL2
Q1
Q2
P2
P1
P0
SL1
S2
S1
Q0
Short run:
2