THE COASE THEOREM FOR POSITIVE EXTERNALITIES
The fundamental principle of the Coase theorem applies to positive externalities as well as to negative ones. If the output chosen by an externality provider when maximizing its own income is low from the perspective of the recipient of the externality, the recipient can financially stimulate the provider. In this case we focus on the survival probability maximization model.
The Coase theorem states that maximization of the private utility of the parties will generate the "socially” optimal outcome. Any external directive or regulation will inevitably spoil this outcome.
The application of the Coase theorem to positive externalities has some specific features. It involves redistribution rather than compensation for damage caused to another party — one agent provides support to another in whose survival and increased activity it has an economic interest. Rather than altruism (which we will discuss in the final chapter of this book), it is therefore a case of action motivated by the donor's own economic benefit.
THE GENERALIZED COASE THEOREM FOR POSITIVE EXTERNALITIES AND AGENTS MAXIMIZING THEIR OWN SURVIVAL PROBABILITY
We take as our example the relationship between an apple grower (orchard owner) and a beekeeper. There are two possibilities: either we can assume that the number of beehives is insufficient to ensure cross-pollination of the apple grower's trees and so the apple grower is willing, in his own interest, to subsidize the beekeeper; or, conversely, we can assume that the number of trees is too low to provide enough pollen for the hives and so the beekeeper — also in his own interest — will donate funds to the apple grower so that he can expand his orchard.
Suppose, for example, that the bees are the "bottleneck", i.e. that the orchard owner is the recipient of the positive externality. From the purely modelling perspective, the opposite case merely involves renumbering the parties.
Furthermore, let us assume that the survival of both parties, namely the provider and the recipient of the positive externality, depends exclusively on their income.
The sales revenue es of the recipient of the positive externality [the apple grower) comes exclusively from the sale of apples of quantity qs at unit price πs, i.e. the sales revenue of the apple grower is
Let us assume for simplicity that qs = 0 if the provider of the positive externality (the beekeeper) goes out of business. The income of the apple grower, which provides the beekeeper with a subsidy of σ, is 
.
The income dv of the provider of the positive externality (the beekeeper) comes partly from the sale of honey of quantity qv at unit price nv, and partly from the subsidy provided by the orchard owner σ, i.e.
, where
Let us assume that in both cases these are the agents' sole sources of income. In line with the other chapters of this book, we assume that the risk of extinction of an agent with income d and subsistence level b is determined by his relative margin vis-a-vis the subsistence level
, which is consistent with a first-order Pareto distribution.[1] Let us denote the subsistence level b (the extinction zone boundary) of the orchard owner (the recipient of the positive externality) by b0 and that of the beekeeper (the provider of the positive externality) by b1.
SINGLE POSITIVE EXTERNALITY PROVIDER MODEL
Let us assume that the apple grower's output qs is a smooth increasing function of honey output qv:
The apple price πs (like the honey price πv) is given, so the apple grower's sales revenue function is also a smooth increasing function:
Let us furthermore assume that the beekeeper's output qv is also given and that there is a risk that the beekeeper will go out of business (owing to insufficient income) and his output will fall to zero. His income is
and his survival probability is
By contrast, the apple grower is in a situation of two threats: first due to low income of his own, and second due to the beekeeper going out of business. The apple grower's income for his own needs (net of the subsidy he provides to the beekeeper) is
and his probability of survival is given by the product of the probability of survival of the beekeeper and the probability of the apple grower avoiding extinction due to low own income:
Let us denote the first factor by C1 and the second one by C0:
The condition for a maximum is
Because the derived functions on either side of the equation are [given the assumed smoothness of function qs(qv)) simple, strictly monotonic, smooth and positive, and because 
, there exists a single optimum subsidy level σ- . It is clear that for this level σ* > 0, i.e. if there is a single beekeeper in the locality it is economically advantageous for the orchard owner to subsidize him regardless of the parameters of the model.
MULTIPLE POSITIVE EXTERNALITY PROVIDERS MODEL
We will start by analysing the situation with two beekeepers. Will the apple grower choose one of them, or will it support both? Will it prefer the stronger one (providing him with a greater degree of security than the weaker one) or the weaker one (the existence of two beekeepers may suit the apple grower more)? Will the fact that there are two beekeepers affect the amount of the subsidy? Our approach, as we will demonstrate, allows us to analyse such nontrivial problems.
We assume that the apple grower will go out of business only if both beekeepers do likewise. His survival probability is therefore given by the relation:
and honey output will be divided (at the same overall volume) into the output of the two beekeepers:
Let us start by assuming that the two beekeepers are in an equal economic position, i.e. 
and are therefore equally at risk of extinction. In this case, of course, the apple grower has no reason to prefer one beekeeper over the other (from his perspective they are identical), so we can assume that 
. The apple grower's probability of survival is then:
If the beekeepers are in an unequal economic position, the apple grower will prefer the weaker one. If the probability of extinction of the weaker one will remain higher even after receiving the entire subsidy σ, it will get the entire subsidy. If not, the subsidy will be split as follows:
where σ0 is the subsidy that balances the threats to both, i.e.
For n beekeepers having the same economic position, the subsidy will be split equally, i.e. σj = σ/n, and the following will hold for the apple grower's probability of survival, similarly as in the n = 2 case:
By providing the same overall subsidy σ to multiple beekeepers, the apple grower will increase his probability of survival. The higher the number of beekeepers n, the lower the increase in his survival probability. We have proved this by conducting computer experiments for the case where there are two beekeepers in an equal economic position and the orchard owner's sales revenues are 10 times as high as the beekeepers' combined sales revenues. The results of these experiments are shown in Figure 55.
Figure 55: The increase in the apple grower's probability of survival (on the vertical axis) as a result of his subsidy to n beekeepers in % of own income (on the horizontal axis)
The effect of the subsidy (of around 10% of the apple grower's income overall at the optimum point) is therefore the highest for the single beekeeper case.
This conclusion is logical: the smaller the number of beekeepers, the larger the risk posed to the apple grower by a threat to any one of them, and hence ceteris paribus the greater the incentive to provide a subsidy. With a higher number of beekeepers, the apple grower will reduce the subsidy [because by providing the subsidy he is causing a threat to himself by reducing his income). If the number of agents providing the positive externality is very high, the orchard owner will stop providing the subsidy.
These conclusions are obviously conditional on the highly restrictive and simplifying assumptions of the model. Nevertheless, we can derive from them some rules that apply in reality. The same is true for a model constructed for a specific positive externality, namely for information.
- [1] See Chapter 1.
