EEP101/ECON125 Spring 99
Prof D. Zilberman
TA's: Malick/Marceau
Suggested Solutions to Problem Set 2
*
1.
All answers referring to graphical details are to be found in figure 1.
a)
To find the aggregate marginal benefit curve, solve each of the individual
marginal benefit curves for q
i
and then add them together.
This gives q
1
= 50 
MB for firm 1 and q
2
= 150  1.5
×
MB for firm 2.
Adding these together, we get Q
= q
1
+ q
2
= 200  2
×
MB and MB = 100  0.5
×
Q.
b)
Firm 1 is more efficient at reducing pollution.
It derives less additional benefit
from a given increase in pollution.
Therefore, its opportunity costs of not polluting
are lower.
At a given level of pollution, it can reduce its emissions by one unit
more cheaply than firm 2.
For example, if the two firms were emitting 30 units of
pollution, firm 1 has a marginal abatement cost of 40 while firm 2 has a marginal
abatement cost of 80.
Notice that what matters for efficiency in this context is not
so much the slope of the curves, but their relative heights.
c)
The optimal level of emissions is where marginal social costs equal marginal
social benefits.
Equating the marginal benefits found above with marginal social
costs leads to the condition: MB = 100  0.5
×
Q* = 40 + Q* = MSC.
Solving for
Q gives Q* = 40.
The intersection of these two curves is point
e
in figure 1.
d)
The optimal tax, t*, will be the one that induces firms to choose to emit the
optimal amount of pollution.
Firms will choose to pollute and pay the tax as long
as that costs less than abating.
They will choose to reduce their pollution if the
cost of abating one more unit is cheaper than the cost of paying the tax on one
more unit.
Thus, firms will abate up to the point where MB = t.
Marginal benefits
at Q* = 40 are MB(Q*) = 100  0.5
×
40 = 80.
The optimal level of the tax is T* =
80.
We may also find the tax as the marginal social cost at Q*  since marginal
benefits and marginal costs are the same at that point.
This gives t* = MSC(Q*) =
40 + 40 = 80, as before.
Firm 1 will pollute up to the point where MB = 100  2q
1
= 80, implying q
1
* = 10.
And firm 2 will pollute up to the point where MB
2
= 100  (2/3)q
2
= 80, implying
q
2
* = 30.
Notice that q
1
* + q
2
* = 10 + 30 = 40 = Q*.
e)
Because the firms’ marginal benefit curves are also their marginal abatement
cost curves, the total cost of reducing emissions up to q
1
* and q
2
* is the area under
*
Solution to question 1 provided by S. Marceau. Solutions to questions 2 and 3 provided by
G. Malick.
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View Full DocumentEEP101/ECON125 Spring 99
Prof D. Zilberman
TA's: Malick/Marceau
the marginal benefit curves of each firm over the units of pollution they are no
longer producing.
That is, we measure the costs starting from the level of
pollution before regulation up to the level of pollution that results from the
environmental policy.
For firm 1, total abatement costs are TC
1
= 0.5
×
(50 
10)
×
80 = 1600.
For firm 2, total abatement costs are TC
2
= 0.5
×
(150  30)
×
80 =
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 Spring '09
 Economics, Prof D. Zilberman, D. Zilberman TA

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