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Lecture

# Solutions to Problem Set 2

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York University

Economics

ECON 3340

Andrea Podhorsky

Winter

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Chapter 12 #1
(a) + (b) See Figure below. Notice the aggregate marginal savings curve. Its derivation
and an explanation follow in part (c).
(c) It is straightforward to calculate aggregate marginal damage. Since pollution is non -
rival, aggregate dama ge is the vertical sum of the individual marginal damage functions:
MD =MD I+MD =e2 e=2e
Industry aggr egate marginal savings is slightly more complicated. How much is saved
between the two firms for a given level of total emissions? This will require a
horizontal summation of marginal savings curves, and the answer to this question
depends on how emissions are distributed among the firms. Efficiency would suggest that
this distribution take place as to maximize total savings. The necessary condition for thi s
is the equimarginal principle: marginal savings should be equated across firms to
maximize savings from pollution (minimize total costs of abatement). This is imposed
here in deriving the aggregate marginal savings function:
el= 5 –MS 1 2e 2=8 -MS2 then e 24 -(1/2) MS 2
Equimarginal principle: MS1 = MS 2 = MS
e +e = e = 5 -MS+4 - (1/2)MS
l 2 T
eT=9-(3/2)MS then MS =6- (2/3) e T
This is the effective aggregate level marginal savings function when emissions are distributed
cost effectively. The optimal level of emissions is found where this marginal savings function
equals aggregate marginal damage: 6- (2/3) e=2e then e*=2.25
The Pigouvian fee that supports this optimal level of emissions is equal to the level of marginal
savings (and marginal damage) at the opt imum: fee* =6-2/3(2.25) =2(2.25) =4.5
The last part of the solution is to determine the individual firms' levels of emissions at the
optimum. These can be found by setting the original marginal savings functions equal to the
Pigouvian fee: 5-e =4.5 then e = 0.5
1 1
1 8-2e 24.5 then e =27/4
Verify that these sum to the optimal total emissions: e +1 =224+7/4=2.25
#2. Consider the market for electricity. Suppose demand (in megawatt hours) is given by Q=50-
P and that the marginal private cost of generating electricity is $10 per megawatt hour (P is in the
same units). Suppose further that smoke is generated in the production of electricity in direct
proportion to the amount of electricity generated. The health damage from the smoke is $15 per
megawatt hour generated.
a) Suppose
the
electricity
is
produced
by
an
unregulated
monopolist.
What
price
will
be
charged,
and
how
much
electricity
will
be
produced?
b) In
part
(a),
what
is
the
consumer
surplus
from
the
electricity
generation?
What
is
the
net
surplus,
taking
into
account
the
pollution
damage?
(a) Depicted is a monopolist with linear demand and constant marginal private constant
marginal external cost. If this firm is unregulat ed, it would maximize profit by choosing a
quantity at which marginal revenue is equal to marginal private cost. For this problem:
P=50-Q= AND MR=50-2Q
MPC = 10
Profit max 50 -2Q=10 then Q`=20, and P' =50-20=30
(b) Refer to the Figure. Ignoring the externality for t he moment, the consumer surplus
associated with this market is represented by the shaded area under the demand curve, and
above the price:
CS = 1/2(50 - 30)20 = 200
Damages from pollution resulting from the generation of electricity are represented by the
stippled area: D= 15*20=300
Net consumer surplus from t his market is therefore negative: Net surplus = 200 - 300 = -100
We see that there is nothing that constrains a market from having a negative total effect on
consumers. How can this be, if consumers have the option to not participate in a free market?
The answer is that consumer surplus from the market itself [part (a)] is optional for any one
consumer: they can purchase electricity or not. But the damages from pollution are not optional
for any one consumer: if everyone else purchases electricity, I su ffer from the pollution
2 whether I buy electricity or not. The private benefit from purchasing electricity is greater than
the private cost of purchasing electricity. This outcome results because of the difference
between the private costs of purchasing ele ctricity (payment plus marginal pollution damage to
one consumer) and marginal social costs of purchasing electricity (payment plus marginal
pollution damage to all consumers). This is the nature of externalities.
Chapter 14, #3
a) The damage function given in the problem is written: D(S) = 0.01S
Where S is grams of sulfates deposited.
(b) Given in the problem is the transfer equation:
S = E E 3 E D
Where E Eepresents tons of sulfur emitted in England, and EDtons of sulfur fitted in Denmark.
The transfer coefficients areEa =l andDa =3
(d) Let E and qDrepresent emissions reductions in England and Denmark, respectively.
The goal at this point is to achieve
qE+ q D 12
At minimum emissions control costs between the two countries. We know that this is
accomplished where the marginal control costs are equal in the two countries:
q D 2 q E
The two equations in the two unknowns (q Dnd q )Eimply
qD=8 and q =4E
(e) Now we still want to reduce total emissions between the two countries by 12 tons, but to do
so in a way that accounts for the different transfer coefficients. The logic is that reductions in
Denmark are more valuable to the Swedes than reductions in England because Denmark is so
much closer. This would suggest equalizing the marginal costs of reducing sulfate deposits
in Sweden. As discussed in the text, this is the marginal co

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