Chapter 32
NAME
Externalities
Introduction.
When there are externalities, the outcome from indepen
dently chosen actions is typically not Pareto eﬃcient. In these exercises,
you explore the consequences of alternative mechanisms and institutional
arrangements for dealing with externalities.
Example:
A large factory pumps its waste into a nearby lake. The lake
is also used for recreation by 1,000 people. Let
X
be the amount of waste
that the firm pumps into the lake. Let
Y
i
be the number of hours per day
that person
i
spends swimming and boating in the lake, and let
C
i
be the
number of dollars that person
i
spends on consumption goods. If the firm
pumps
X
units of waste into the lake, its profits will be 1
,
200
X
−
100
X
2
.
Consumers have identical utility functions,
U
(
Y
i
, C
i
, X
) =
C
i
+ 9
Y
i
−
Y
2
i
−
XY
i
, and identical incomes. Suppose that there are no restrictions
on pumping waste into the lake and there is no charge to consumers for
using the lake. Also, suppose that the factory and the consumers make
their decisions independently.
The factory will maximize its profits by
choosing
X
= 6. (Set the derivative of profits with respect to
X
equal
to zero.)
When
X
= 6, each consumer maximizes utility by choosing
Y
i
= 1
.
5. (Set the derivative of utility with respect to
Y
i
equal to zero.)
Notice from the utility functions that when each person is spending 1.5
hours a day in the lake, she will be willing to pay 1.5 dollars to reduce
X
by 1 unit. Since there are 1,000 people, the total amount that people
will be willing to pay to reduce the amount of waste by 1 unit is $1,500.
If the amount of waste is reduced from 6 to 5 units, the factory’s profits
will fall from $3,600 to $3,500. Evidently the consumers could afford to
bribe the factory to reduce its waste production by 1 unit.
32.1 (2)
The picturesque village of Horsehead, Massachusetts, lies on a
bay that is inhabited by the delectable crustacean,
homarus americanus
,
also known as the lobster. The town council of Horsehead issues permits
to trap lobsters and is trying to determine how many permits to issue.
The economics of the situation is this:
1. It costs $2,000 dollars a month to operate a lobster boat.
2. If there are
x
boats operating in Horsehead Bay, the total revenue
from the lobster catch per month will be
f
(
x
) = $1
,
000(10
x
−
x
2
).
(a)
In the graph below, plot the curves for the average product,
AP
(
x
) =
f
(
x
)
/x
, and the marginal product,
MP
(
x
) = 10
,
000
−
2
,
000
x
.
In the
same graph, plot the line indicating the cost of operating a boat.
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394
EXTERNALITIES
(Ch.
32)
2
4
6
8
10
12
2
4
6
8
10
12
x
AP, MP
0
AP
Cost
MP
(b)
If the permits are free of charge, how many boats will trap lobsters
in Horsehead, Massachusetts? (Hint: How many boats must enter before
there are zero profits?)
8 boats.
(c)
What number of boats maximizes total profits?
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 Spring '10
 JOSEPHPETRY
 Utility, developer, total proﬁts, Middle Potato

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