EEP101/ECON125 Spring 00
Prof.: D. Zilberman
GSIs: Malick/McGregor/St-Pierre
Key to
PROBLEM SET 2
1.
There are three polluting firms in the economy with marginal benefit curves given by: MB
1
= 75 –
(5/4)q
1
for firm 1, MB
2
= 75 – q
2
for firm 2, and MB
3
= 75 – (3/4)q
3
where q
i
(i=1,2,3)
is the
amount of emissions produced by each firm
a)
The MB curve tracks the aggregate level of emissions. At any level of valuation (MB
1
= MB
2
= MB
3
= MB), aggregate emissions can be obtained by summing horizontally the three firms’
curves. Analytically, you need to find the inverse of the MB equations
q
1
= 60 - 4/5 MB;
q
2
= 75 - MB;
and q
3
= 100 - 4/3 MB, and sum quantities:
q
1
+ q
2
+ q
3
= Q = 235 – (47/15) MB; or inversely, MB = 75 – (15/47) Q = 75 – 0.32 Q
(decimals are rounded off) Your graph should look like the following:
235
100
75
64.1
30
3
MB (aggregate)
B
2
Q*/3
11.4
60
Q*
34.1
75
v* = t*
emissions
MSC
1
$
b)
Remember that “q
i
” does
not
represent the output of each firm, but instead the quantity of
pollution it emits. Firm 1 is more efficient in reducing that pollution. Intuitively, it is less
dependent on emitting pollution than others to create profits (think of emissions as inputs).
Alternatively, reducing emissions would cost it the least.
c)
The optimal level of emissions (Q*) can be obtained by equating the marginal social cost
curve with the aggregate MB curve: MSC = 30 + Q = MB
= 75 – (15/47) Q , which implies
Q* = 34.1 (and MB* = 64.1).
d)
Since marginal benefits of emissions are already net of marginal costs, the optimal tax
on
emissions
is simply equal to the aggregate MB at the optimum quantity, which is MB* = 64.1.
To find out the amount of emissions produced by each firm (q
1
*, q
2
*, and q
3
*) given the tax,
simply substitute MB* = tax = 64.1 into the equations you calculated in part a.
On the graph,
those q
i
* quantities (8.7, 10.9, and 14.5, respectively) are at the intersections of the horizontal
line running from the equilibrium point B to the left and each of the three MB
i
curves.

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