## Interpretation of the Conditions

There are several useful interpretations of the conditions for Pareto efficiency derived above. Each of these interpretations suggests a scheme to correct the efficiency loss created by the production externality.

The first interpretation is that the steel firm faces the wrong price for pollution. As far as the steel firm is concerned, its production of pollution costs it nothing. But that neglects the costs that the pollution imposes on the fishery. According to this view, the situation can be rectified by making sure that the polluter faces the correct social cost of its actions.

One way to do this is to place a tax on the pollution generated by the steel firm. Suppose that we put a tax of t dollars per unit of pollution generated by the steel firm. Then the profit-maximization problem of the steel firm becomes max pss — cs(s, x) — tx.

The profit-maximization conditions for this problem will be

Comparing these conditions to equation (34.3), we see that setting t_Acf(f,x) Ax will make these conditions the same as the conditions characterizing the Pareto efficient level of pollution.

This kind of a tax is known as a Pigouvian tax.3 The problem with Pigouvian taxes is that we need to know the optimal level of pollution in order to impose the tax. But if we knew the optimal level of pollution we could just tell the steel firm to produce exactly that much and not have to mess with this taxation scheme at all.

Another interpretation of the problem is that there is a missing market— the market for the pollutant. The externality problem arises because the polluter faces a zero price for an output good that it produces, even though people would be willing to pay money to have that output level reduced. Prom a social point of view, the output of pollution should have a negative price.

We could imagine a world where the fishery had the right to clean water, but could sell the right to allow pollution. Let q be the price per unit of pollution, and let x be the amount of pollution that the steel mill produces. Then the steel mill's profit-maximization problem is max pss — qx — cs(s7 x), s,x and the fishery's profit-maximization problem is max pff + qx - c/(/,x).

The term qx enters with a negative sign in the profit expression for the steel firm since it represents a cost—the steel firm must buy the right to generate x units of pollution. But it enters with a positive sign in the expression for the profits of the fishery, since the fishery gets revenue from selling this right.

The profit-maximization conditions are

ih Ax

3 Arthur Pigou (1877-1959), an economist at Cambridge University, suggested such taxes in his influential book The Economics of Welfare.

Thus each firm is facing the social marginal cost of each of its actions when it chooses how much pollution to buy or sell. If the price of pollution is adjusted until the demand for pollution equals the supply of pollution, we will have an efficient equilibrium, just as with any other good.

Note that at the optimal solution, equations (34.5) and (34.7) imply that

Ax Ax

This says that the marginal cost to the steel firm of reducing pollution should equal the marginal benefit to the fishery of that pollution reduction. If this condition were not satisfied, we couldn't have the optimal level of pollution. This is, of course, the same condition we encountered in equation (34.3).

In analyzing this problem we have stated that the fishery had a right to clean water and that the steel mill had to purchase the right to pollute. But we could have assigned the property rights in the opposite way: the steel mill could have the right to pollute and the fishery would have to pay to induce the steel mill to pollute less. Just as in the case of the smoker and nonsmoker, this would also give an efficient outcome. In fact, it would give precisely the same outcome, since exactly the same equations would have to be satisfied.

To see this, we now suppose that the steel mill has the right to pollute up to some amount x, say, but the fishery is willing to pay it to reduce its pollution. The profit-maximization problem for the steel mill is then max ps,s -f- q(x — x) — cs(s, x).

Now the steel mill has two sources of income: it can sell steel, and it can sell pollution relief. The price equals marginal cost conditions become

The fishery's maximization problem is now max pff - q{x - x) - cf{f,x), which has optimality conditions

Ax v

Now observe: the four equations (34.8)-(34.11) are precisely the same as the four equations (34.4)-(34.7), In the case of production externalities, the optimal pattern of production is independent of the assignment of property rights. Of course, the distribution of profits will generally depend on the assignment of property rights. Even though the social outcome will be independent of the distribution of property rights, the owners of the firms in question may have strong views about what is an appropriate distribution. 