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1
EEP101/ECON125 Spring 00
Prof.: D. Zilberman
GSIs: Malick/McGregor/StPierre
Key to
PROBLEM SET 1
1.
The graph below illustrates the case where a monopolist supplies the market. (Note that the graph is not exactly
to scale, so coordinates should be calculated as the intersection of curves, or by substituting values in the relevant
functions. Ex: at point “g” MB = MSC; at point “k”, MC = 40 +2*100 = 240)
40
50
300
200
100
100
200
300
400
k
MB
MSC
g
MC
MC (with subsidy)
d
b
h
90
220
240
310
l
e
c
j
i
275
MR
m
a
f
0
a)
Social Optimum
is attained at a point (g on the graph) where the marginal social cost (sum of private and
external marginal costs) is just equal to the marginal social benefit (simply the inverse demand in this case).
MSC = MC + MEC = (40 + 2Q) + (10 + 0.5Q) = 50 + 2.5Q
Social Equilibrium => MSC = MB => Point g => 50 + 2.5Q = 400 – Q
=>
=> Q* = 100
Total External Cost , TEC,
at Q
*
=>
Area
ekgl
=> 100 (10 + 60)(1/2)
= 3,500
=> TEC* = 3,500
CS* (varies, but assuming quota) => area
agh
=> (1/2)(400300)(100) = 5,000
=> CS* = 5,000
PS* (varies, but assuming quota) => area
ekgh
=> (1/2)(260+60)(100) = 16,000
=> PS* = 16,000
Total Welfare, W* = CS* + PS*  TEC* = 5,000 + 16,000 – 3,500 = 17,500
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but you should also see that W* => area lga
=> (1/2)(350)(100) = 17,500
=> W* = 17,500
A common mistake was “PS* = 12,500” (perhaps representing the area ghl).
Recall that producer surplus
is the difference between price received and marginal cost ,
over the actual quantity sold
. It is often a triangle, but
not always.
b)
The
monopolist
produces at a level where the private marginal cost equals private marginal revenue:
Total Revenues
TR = P(Q)Q = (400  Q)Q = 400Q  Q
2
.
Marginal Revenue MR =
ƒ
ƒ
TR
Q
= 400  2Q
Equilibrium Quantity => MR = MC =>
Point d
=> 400  2Q = 40 + 2Q
=> Q
m
= 90.
Equilibrium Price => 400  Q
m
=> 400  90 => 310
=> P
m
= 310
c)
Remember that the
deadweight loss
is the welfare deficit, relative to the socially optimal level.
Consumer's Surplus, CS
m
=> Area
abc
=>
(400  310)(90)(1/2) = 4,050
=> CS
m
= 4,050
Producer's
Surplus, PS
m
=> Area
bcde
=>
(90)(270 + 90)(1/2) = 16,200
=> PS
m
= 16,200
Total external cost, TEC
m
=> Area
elfd => (90) (10+55)(1/2) = 2,925
=> TEC
m
=
2,925
Total Welfare, W
m
= CS
m
+ PS
m
 TEC
m
= 4,050 + 16,200 – 2,925 = 17,325
also see that W
m
=> area acde minus elfd is alcf => (1/2)(350+35)(90) = 17,325
=> W
m
= 17,325
Deadweight Loss => DWL
m
= W* W
m
= 17,500 – 17,325 = 175
also see that DWL
m
=> area agl minus alcf
is cgf => (1/2)(35*10) = 175
=> DWL
m
= 175
d)
Correction
of
the externality
The government plans to intervene with a tax or a subsidy to shift the private MC curve so that it intersects the MR
at the desired optimal level of output Q*.
Let's call "x" the amount by which the MC curve should change.
The
way we have written the equation, a negative value for x implies a subsidy (reducing MC), while a positive value
implies a tax (increasing MC).
MC(Q*) + x
=
MR(Q*)
(40 + 2 Q*) + x =
(400  2 Q*)
We know from a) that Q* = 100, so x is the only unknown.
Therefore
40 + 2(100) + x =
400  2 (100)
240 + x = 200
x* =  40.
The government should thus grant the monopolist a unit subsidy of $40 per unit produced.
This seems strange
subsidizing a polluter—but the intuition is simple. Relative to the social optimum, the market power of a
monopolist make her produce “not enough” (and charge too high a price), but the unregulated externality makes her
produce “too much”, so the two effects work in opposite directions.
If the externality is relatively severe, the
government should encourage the monopolist to restrict even further its production by charging a tax. But if the
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 Spring '09

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