05

Feb

The Lotka-Volterra equations describe the evolution of an interacting prey population and predator population.

Mathematically, this gives a cyclic function (if the two populations manage to cohabit), one period of which is decomposed into the following 4 phases:

- the prey population is expanding, and where the predators are at a low level,
- the prey population is at a maximum, and where predators are expanding,
- the prey population is declining, and where predators are at a maximum,
- the prey population is at a minimum, and where the predator population is declining,

Quite simple to pose, the problem nevertheless has no simple algebraic solution (even when there are only two factors). Nevertheless, they allow us to know the equilibrium between two populations (mean of the solutions and stability of the fixed points) and are useful to understand the dynamics of systems (hysteresis). The initial values make it possible to predict whether a species will periclite or thrive by following cycles.

It can be transposed to economics, when we try to explain the link between two variables that are both self-regulating: unemployment and inflation.

This modeling has been widely used in the World3 project of Dennis Meadows (Nobel Prize in Economics).

Dennis MEADOWS (and his team) exploited this system, but by including many factors (not only 2), involving feedback loops that are more complex and less obvious than those mentioned above.

In this study, several scenarios were established based on entry points set in the simulations.

Without these initial values, there is no simulation possible: this is called system initialization.

Small variations in these values can lead to chaotic behavior (especially as the number of input variables increases).

In the framework of the balance of externalities, a simple transposition exists between social cost, prey, and private cost, predation (similar mechanics), the equilibrium point being when the social cost is equal to the private cost. This can be done because the environment has the capacity to absorb a defined volume of nuisance, and therefore a social cost, which, thanks to self-regulation, returns to its initial value. Zero nuisance is not conceivable, as it would be tantamount to stopping all economic activity. This point of balance between the absorption capacity of the environment and economic activity is much more interesting. Indeed, it can be defined by nuisances, but can also be extended to other sources of externalities.

We shall therefore propose a formalism for determining the critical points of this function, so that the evaluation of externalities works, … in all cases.

In the construction of a unified reporting, two points are crucial:

- Define the concept you want to measure,
- Define the function that gives the value of the measurement,

When trying to define the function that gives the measurement, the simplest way is to start with an affine function (with one or more input variables). On this type of function, it is necessary to be able to define:

- A marginal progression,
- A specific point (usually ordered at the origin, or any other characteristic point),

In a measure of externality, three things are important:

- The true function of externalities in a finite world, with feedbacks, threshold effects, and which are non-linear,
- The simplest critical point to generalize (across the whole range of possible interactions) is when the externality cancels itself out,
- Around the critical point, the function is stable (can be represented by an affine function), because the function is assumed to be continuous,

It should be noted that taking the y-intercept as the critical point does not necessarily make sense, because whatever the input variables, it corresponds to a null activity – whereas the externality, by definition, is not definable in the absence of economic activity).

Let us therefore try to define this critical point a little more precisely!

This point can be defined in three ways:

- Point of equilibrium allowing an economic activity in which all nuisances can be absorbed by the environment (absorbed within an acceptable time) (also works for resource withdrawals),
- Point of balance between well-being and economic development (of general interest),
- Point of balance between resource extraction and environmental rhythm,

The first definition applies to ecosystems, the second to social ties.

For the ecosystem, the equilibrium point is relatively simple because the social cost consists in evaluating the cost of returning to an initial state, or the cost of the status quo. The equilibrium point is therefore the status quo. It has a disadvantage that concerns irreversible impacts (or reversible impacts on time scales too long for a realistic assessment to be made), but we deal with the point of marginal cost assessment in another article.

This (status quo) notion is more difficult to transpose when we talk about well-being.

What is well-being? How is it defined? Are we talking about the same way of well-being from one culture to another?

Well-being is subjective, and when it varies it is the consequence of an externality.

Nevertheless, the general interest is well known (at least at the level of a developed country). It corresponds to the public policies implemented, laws, taxes, … that populations decide (by voting) and act (by political action).

A pragmatic approach therefore consists in taking as a point of balance the deviation from the standards (set by public action) that companies allow themselves. This deviation from standards varies from one country to another, with advantages and disadvantages (discussed in another article).

Finally, there remains the balance point to be defined for natural resources. This point exists, as in the Lotka-Volterra equations, and for biological resources we are in exactly the same case. Man is a predator for the rest of the environment. We know a second thing: man dominates demographically all habitats, and in the dynamics of Lotka-Volterra, the predator takes enough to upset the balance and kill the ecosystem. So what can this point of balance be? A simple way is to take any removal as a source of externality. A restoration of the resource to compensate for the removal.

We have defined arbitrary equilibrium points, which in the end are all defined to maintain a point of equilibrium. The definition of the equilibrium point requires a major effort of adaptation in the social and economic sphere, but an effort consistent with our need for adaptation.