David K.H. Begg, Gianluigi Vernasca-Economics-McGraw Hill Higher Education (2011)

CHAPTER 13 Welfare economics
0 Firm A chooses the optimal quantity to produce (QA) at which the marginal revenue is equal to the marginal
cost of producing that quantity: p = iJTCA!dQA"
Similarly, the optimal quantity of pollution that maximizes profits is where marginal revenue of pollution
(in this case, zero) is equal to the marginal cost of pollution:
_ arcA O =
(}pA
The firm chooses a level of pollution such that the cost of an extra unit of pollution is zero. Since higher is the
pollution, lower is the total cost of the firm, we should expect that the level of pollution that solves that
condition to be quite high.
Firm B has the following total cost function: TCB = TCs( QB, PA) with the following properties: iJTCB!dQB > 0
and iJTC8/dPA > 0.
This means that the total cost of firm B increases with output produced and with the pollution made by
firm A. The externality problem is the following: firm A in deciding how much to pollute does not take into
account the effects that its decision has on firm B. Denote by f the market price of fish for firm B. The profit
of firm B is given by: nB = fO.s - TCs( QB, PA).
Suppose we give the right to pollute to firm A. Firm A can sell its right to firm B.
The profit function of firm A becomes nA = pQA - TCA(QA, PA) + qPA, where q is now the price that firm A can
get by selling its right to pollute to firm B.
For firm B, the profit function is now 1t8 = fQ8 - TC8(Q8, PA ) - qPA.
For firm A, the optimal level of pollution that maximizes profits is given by the condition:
arc A arc A
--
+q =O = q
()pA
()pA
That condition simply says marginal cost is equal to marginal revenue from polluting (now equal to q).
For firm B, the quantity of pollution that maximizes its profit is given by the condition:
(1)
arcB - q =O-arcB = q
i}pA
()pA
(2)
Equations (1) and (2) imply that the price q should satisfy the following: -dTCA/dPA = iJTC8/dPA.
In deciding the optimal level of PA , firm A now takes into account the effect that its decision has on firm B.
In particular, it must set a level of pollution such that the marginal private cost of polluting (-dTCA/dPA) is
equal to the marginal social cost of polluting (dTCB/dPA). So, by assigning the property rights we can obtain
the efficient level of pollution. You can try to work out the case in which firm B has the right not to be polluted
and can sell this right to firm A; does the result above still hold?
Environmental issues and the economics of climate change
When there is no implicit market for pollution, pollutants are overproduced. Private producers ignore the
costs they impose on others. In equilibrium, social marginal cost exceeds social marginal benefit.
The most topical environmental externality we are currently facing is global warming, or climate change;
that is, the rise in global temperature due to human activity. In particular, global warming is an externality
in two main dimensions:
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