EXCERCISES:
EXTERNALITIES
Here are the solutions to the first exercise.
1.
Negative externality.
Demand for Q is given by:
P
=
120Q
The total cost (TC) of producing Q is given by:
TC =
½Q
2
Finally, there is a negative externality associated with the production of Q.
The value of
the total damage (TD) done by the pollution is give by:
TD =
¼Q
2
Total surplus is equal to consumer surplus (CS) plus producer surplus (PS) less the total
damage to the environment (TD).
Consumer surplus
Consumer surplus is equal to the area under the demand curve up to the amount of Q
consumed (Q) minus the amount paid for Q.
The area under the demand curve is given
by integrating the demand curve, from zero to Q:
∫
P(Q)dQ = 120Q  ½Q
2
The amount paid for Q equals P*Q
=
(120Q)*Q
=
120QQ
2
So consumer surplus is given by:
CS =
120Q ½Q
2
– (120QQ
2
)
=
½Q
2
Producer surplus
Producer surplus equals total revenues, less the cost of production.
Total revenues of
producing Q equals P*Q, or:
TR = (120Q)*Q
1
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View Full DocumentThe total cost is given by TC, above. So producer surplus is given by:
PS
=
120Q – Q
2
– ½Q
2
=
120Q  1½Q
2
Total damage
Total damage is given by TD, above:
TD = ¼Q
2
Total surplus
Total surplus equals consumer surplus + producer surplus – total damage:
TS =
½Q
2
+ 120Q – 1½Q
2
– ¼Q
2
=
120Q – 1¼Q
2
1.a.
What is the optimal outcome?
The optimal outcome maximizes total surplus. To maximize total surplus, we take the
derivative of TS with respect to Q, and set it equal to zero:
dTS/dQ =
120 – 2½Q = 0
Q =
48
Thus, the optimal quantity is equal to 48. Price is equal to 120Q = 12048 = 72.
Plugging 48 into the equation for TS above gives us the total welfare at the optimal
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 Fall '07
 SandraBlack
 Economics, Microeconomics, Externalities, Monopoly, Supply And Demand, Externality, total surplus

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