El presente estudio muestra que el camino más eficiente para mejorar el medio ambiente en cualquier área y en cualquier dimensión es con el uso de una versión

This study shows that the most efficient way to improve the natural environment, is by using a "soft" version of the Principal-Agent methodology by means of the emission of improvement certificates that embrace large entities and therefore stimulate the transference of technologies. We compare two cases of optimal solutions, collusive optima and fusion optima. "Fusion" means that one agent-owner of a certificate- can make improvements in another agent's domain, (which we'll call "land"). We analyze specific and relatively simple models for which explicit or almost explicit solutions can be reached. We use these models as reference models only because their practical justification, including the calibration of parameters, is its major flaw when applied to environmental phenomena. Although the methodology could, in theory, be applied to several environmental problems, this study deals basically with pollution. We work with difusion processes, introducing the cooperation factor.

There have been many attempts to use financial markets to combat or diminish environmental deterioration. Among the most significant are the following:

Emission trading (known as cap-and-trade) related to permits to pollute.

Economic evaluation based on "willingness to pay” known as full "cost-benefit” analysis.

Establishment of property of rights, i.e. the privatization of nature.

Valuing the environment through contingent valuation. (

However, none of the above mentioned approaches have rendered the desired effects.

The main deficiency of the "permits to pollute” approach is that it does not stimulate any cooperation. If someone finds a new method to capture carbon (just to give one example), there is no reason to believe that he or she would share this invention with others instead of profiting by selling permits. Besides, it leads to a wild market with strong governmental intervention, for example, assigning initial quota, this is called the

Full economical analysis needs very precise models but natural phenomena are far too complex and depend on too many processes to be fully understood or measured. M. Sagoff rightly stressed that "(...) the immense effort economists have invested over decades in trying to measure the benefits of environmental resources and services has resulted and can result only in confusion”, (Sagoff, 2004).

Establishing property rights that require institutional arrangements and procedures that are difficult to accomplish, should not be proposed as a solution to the "tragedy of the commons”. Natural resources are hard to privatize. Even dealing with deforestation (this being a problem that mostly affects developing countries), the attempt to set property rights encounters increasing social problems rather hard to solve.

The Contingent valuation method (CVM) is used to estimate economic values of all kinds of ecosystems and environmental goods by asking how much one would be willing to pay for a specific good. Unfortunately the answers were closely related to the educational level of people involved and the kind of questions asked.

Although the comprehensive conservation of the biological diversity requires a strategy that goes beyond cost-benefit analysis -the monetary valuation can play a supportive role in environmental policy, but its multiple practical and normative problems have to be considered when using such a method, above ah in developing countries where people are too poor to think about environmental degradation. Philip E. Graves wrote: "To the extent that we value public goods, we also realize that getting extra income to buy them will accomplish nothing”, (Graves, 2003). It was A. Fitzsimmons who, in his controversial book

In this article we assume that an environmental fund has already been created and could be used by two parties. To describe these entities we’ll use the word

Presently, we analyze different aspects of cooperation in three mathematical models:

Elementary deterministic model.

Squared Bessel processes with linear improvements only. We show how state dependent agents’ actions can reduce the cooperation factor needed to make fusion worthy. We also explain how to analyze and value certificates in this particular setting.

General processes with external factor independent of agents’ actions, (

We use the

In all cases, we consider that the payment would increase when pollution levels diminish. Principal problem-optimality of certificates, needs precise estimation of social costs of pollution, and these estimations seem to be

The approach presented here, pretends to open the path toward practical solutions to prevent environmental destruction.

To explain what we understand by

These examples with general parameters have been studied by Ray, who compared competitive Nash and collusive equilibria, (

The real advantage of fusion could be appreciated in more complicated stochastic models.

Let each "land” emit pollution level and loss function are given by

where the second term represents its social cost that is unknown, and set just for illustration purpose, and the first term is the cost for abatements.

In this example we assume that pollution in one land doesn’t affect the counterpart. Now optimal

Consider fusion and assume that joint loss function is given by:

Where Y= X_{1}+ X_{2}.

If we assume that

We will call

On the other hand, when we consider neighboring lands (pollution in one affects the other in a straightforward way), and social loss function

In the case of fusion the total loss being

and once again the total level of pollution is smaller when

Therefore, emissions of certificates of improvement in the form

will lead directly to fusion if

In this part we would like to show how to value different certificates and explain how to do it for state dependent actions (linear with fixed parameters) the cooperation factor needed to make fusion worthy would be smaller than 2. (Similar conclusions can be obtained in more general models).

To begin with, we would like to formulate some basic facts about these processes and comment the modeling in this setting.

(We will set X (0) = 1).

We assume that δ ≥ 0, but at this moment do not specify the sign of β. If β = 0 then

Pollution levels are increasing with time, but at some random moments actions are taken to combat it. See (Szatzschneider &

In our example we will compare agents actions for certificates of the form

For X(s), Y(s) independent

There are well known formulas for E(X(s)), Var (x(s)),

Therefore, one can design many reasonable certificates with explicit valuations.

Now, if agents actions are limited to making δ smaller, then the cooperation factor

Coming back to our example:

Assume that an agent can make improvements in his (her) land, changing

Assume now that one agent can make. improvements changing

Let ^{
2
}

The cost of improvements is individually

Therefore, the joint cost in the collusive case is:

and in the fusion:

and clearly even for

The exact calculation of will be performed in the next section for a different model.

In this case we work with different lands and borrow the general idea from (

We will deal with a more general case. However we solve exclusively the Agent Problem.

The pollution level is modeled as

S(0) > 0, □, and ^{1}

Assume also that there will be a unique strong solution to this equation for our choice of

This model will produce a very simple optimal solution for in the case of certificates:

Let us rewrite

The Agent's problem is to maximize

being

Let

Now

Because neither ⊂, _{t} depends on u_{t} and assuming a quadratic cost of improvements

We would like to compare fusion versus collusive optimality.

Let

W_{1}, W_{2} being independent.

Joint expectation (assuming that stochastic integrals are true martingales) is:

while fusion solution for _{
1
}
_{
2
} ) would give

The choice of certificates could be the subject of separate studies and matched in some sense to the social costs of pollution. However, it is very difficult to estimate them correctly. Therefore, for the first two applications the approach can be any of the proposed. We leave to the market the exact costs of improvements.

It is frequent in quantitative finance that the design of financial products anticipates their valuation. We propose first to apply and later to discuss and analyze the performance of the certificates of improvements. For one agent's problem see (

Los símbolos ⊂ y

Let

set Y(0) = 1 (to simplify).

We will formulate here the result that will be useful to prove Theorem 2.

Let σ > 0

Where φ is the solution of

This theorem is a particular case of one presented in (

Clearly it is very easy to calculate φ:

with

This result has been obtained with the use of exponential martingales, Girsanov, and integration by parts.

Assume

^{2} - 2σ > 0

In this case φ(s) has the same form as in theorem 1, with

Now

Apply the same martingale method as in theorem 1.