THE GENERALIZED COASE THEOREM FOR NEGATIVE EXTERNALITIES IN THE CONTEXT OF SURVIVAL PROBABILITY MAXIMIZATION
Let us now assume that both producers (the polluter and the injured party) maximize their probability of survival.
We use the following notation (with j = 1 denoting the polluter and j = 2 the injured party):
x is the amount of pollution,
dj(x) is the j-th party's income (including any compensation from the other party) at the polluter's level of activity causing pollution x, bj is the boundary of certain extinction of the j-th party due to low income, is logically an increasing function, since a greater reduction in pollution is possible only at the cost of a greater decrease in income. Conversely d?(x) is logically decreasing, since higher pollution affects the injured party more. As it is reasonable to assume a decreasing marginal effect of a reduction in costs on the maintenance of pollution below level x, we can also assume that d1(x) is strictly concave.
We distinguish two variants for the legal regime:
A. The injured party cannot stop the polluter producing
In this variant, the injured party must pay the polluter compensation if it wants it to cut production. The final decision is up to the polluter. We use the following notation:
τ is the compensation paid by the injured party,
x1(τ) is the injured party's variant offer, i.e. the polluter's output level at which the injured party is willing to pay it compensation r,
x2(τ) is the injured party's output level at the pollution level caused by the polluter's output level x1(τ),
p1 is the price of the polluter's output,
p2 is the price of the injured party's output,
φ1(τ) is the polluter's income [including compensation from the injured party) in relation to the amount of compensation r.
It is reasonable to assume that x1(τ) is a decreasing function, because a larger [smaller) reduction in output is accompanied by higher [lower) compensation. As the utility of the injured party is assumed to be the Pareto probability of survival, the level of output and the amount of compensation are bound by the condition
where x20 = x2(0). This implies
The polluter's probability of survival when the polluter is not restricted by law, pays no regard to its effect on the injured party, and is offered no compensation (τ = 0), is
where The polluter's probability of survival if it accepts a subsidy of τ > 0 is
where the denominator is given by the relation (*).
The optimal amount of compensation (from the polluter's point of view) τ* > 0 must satisfy the condition p1 (τ) = 0, i.e.
Substituting from relation (*):
So, if equation (*) has a solution τ* > 0, the maximum sum of the output of the two parties at this optimum is . We can regard the follow ing as the overall (“social”) criterion, since (except for the constant - b1 - b2) it is the sum of revenues multiplied by the probability of survival for the relevant party:
So, in variant A, where the polluter decides the amount of compensation based on an accommodating offer made by the injured party, these compensation negotiations produce the overall ("social") optimum.
In variant A, an environmental investment costing I and having a given environmental effect which is socially considered worth implementing (for example, through voting in a referendum), is of course financed by the injured party, whose probability of survival is thereby reduced from
B. The injured party can stop the polluter producing
In this variant, the polluter (the party with index j = 1) must pay the injured party (the party with index j = 2) compensation for its loss. The final decision is up to the injured party and the polluter makes the variant offer. So, the injured party decides the amount of compensation and thus also the level of pollution based on the polluter's accommodating offer. We use the following notation: a is the compensation paid by the polluter, x1(σ) is the polluter's variant offer, i.e. the permitted output level requested by the polluter in return for compensation a, x2(τ) is the injured party's output level at the pollution level caused by the polluter's output level x1(τ), p1 is the price of the polluter's output, p2 is the price of the injured party's output, φ1(σ) is the polluter's income (net of compensation paid to the injured party) in relation to the amount of compensation σ.
Here we assume that the variant offer x1(σ) is an increasing function, because higher compensation means a higher permitted level of pollution. Maximization of the probability of survival by the polluter leads to the condition
where. For the polluter's offer this implies
The injured party's probability of survival if it accepts no compensation (σ = 0) is
where . The injured party's probability of survival if it accepts a subsidy of is
where the denominator is given by relation (**).
The optimal amount of compensation (from the polluter's point of view) must satisfy the condition . After substituting in the same way as in variant A we get
Substituting from relation (**):
If an optimum exists (i.e. equation (**) has a solution σ* > 0), the value of the "social” criterion C(σ) here, as in variant A, is at a maximum.
For both variant A and variant B, the decision-taker's optimum is identical to the overall ("social") optimum.
The same cannot be said, however, for environmental investment. In the standard Coase theorem it does not matter "socially" what the legal regime is (and who, therefore, finances this investment), but in generalized microeconomics it does matter.
The value of the "social” criterion after the implementation of investment I in variant A is
where as the value of the criterion after the implementation of investment I in variant B is
where the following certainlyapplies (except in the totally unrealistic and exceptional case :
On the basis of these conclusions we can now for mulate the following theorem: Generalized Coase theorem for negative externalities
From the point of view of generalized economics, where agents maximize their own Pareto probability of survival, it holds that:
• allowing two parties to negotiate compensation for environmental damage will lead to optimization of the utility of both parties and of social utility regardless of the legal regime. In this respect, there is no difference from the Coase theorem in standard microeconomics with the homo economicus paradigm. By contrast,
• in generalized economics (as distinct from the Coase theorem in standard microeconomics) the legal regime will influence the individual and social "costs” (compared in the criteria of individuals and of society as a whole under consideration) of investment in environmental cleanup. If the legal regime disadvantages the party that is more threatened, these "costs" are higher than in the opposite case.
THE COASE THEOREM FOR THE CASE WHERE A PRODUCER HARMS A CONSUMER
In this case it makes sense to consider the restrictive regime only, with consumers being willing to allow the producer to pollute in return forcertain compensation. We will assume that there is an authority — for example the state — that defends the interests of all injured parties simultaneously. How much does this change the conclusions derived above?
The difference here is that consumers do not have a profit criterion. Their decision-making can of course be modelled using a function representing the amount that consumers would regard as sufficient compensation for their losses. In that case we can proceed in the same way as in the previous section. The problem, however, lies in establishing the shape of this compensation function.
We do not consider sociological surveys to be very useful for solving this problem. Answers to questions such as "What proportion of your income are you willing to for go in compensation for a reduction in the pollution level by one ton per square mile?” are always hypothetical. The payment is not real and the assessment is very subjective and can differ significantly from person to person in terms of the amount and of the credibility of the figure. It is hard to imagine constructing a reliable compensation function in such a way.
One possible way of establishing the shape of the compensation function is provided by the Tiebout model. It assumes that individuals will (if allowed to) for m communities with similar preferences. People "vote with their feet". Those who prefer higher taxes and lower pollution will move to "green havens". These havens, however, will be unaffordable for the poor, who, conversely, will move to high-pollution locations. This will clearly not be the overall (globally) optimal solution. The benefit of this model, however, is that it reveals preferences, which can be used to help us establish the shape of the compensation function.
We can, however, view the problem symmetrically as an ef for t to achieve the maximum utility from a positive externality. This allows us to use the methodology set out in the next section.
-  See Tiebout, C.: The Pure Theory of Local Expenditure. Journal of Political Economy 64, 5(1956): 416–24.