The simplest way to explain a function is to use its approximate affine function.
An affine function is characterized by a constant slope (noted a) and an intercept or reference (noted b).
Its simplicity is the basis of the most efficient and complex numerical modeling: finite element method. This method relies on a discretization to be able to define a value from one step to the next on the basis of a simple application of a law.
Assuming that on sufficiently small intervals, all variations can be expressed in an affine way (with two parameters). It allows to model, by cutting, extremely complex systems, which the theory could not describe. This method is used in fluid mechanics, on numerical models of aircrafts, in meteorology, …
Thus, the affine function is a good basis to explain any phenomenon, provided that we know the underlying laws and accept an approximation (bijective).
Let’s try to apply it to externalities to better understand how to define and measure externalities in a homogeneous context that is the balance of externalities.
Before evaluating the valuation function of externalities, let us try to identify the sources of externalities.
In our case, we will define an externality as the translation of a decrease/increase in the economic potential of a system (much like potential energy in physics), which is not translated into a direct transaction. Why is this?
Thus, the externality translates into a difference between a social cost and a private cost. This difference is equal to an enrichment / impoverishment of the system (environment): decrease of the potential.
In commercial exchanges, the transfer of goods takes place in equilibrium if we consider only the seller and the buyer (law of supply and demand). In this case, we are in a closed and isolated system. More realistically, and by extending this to third parties, this equilibrium does not exist: hence the presence of externalities. Hence the sources of externalities correspond to all the withdrawals from the environment, but must also include the intermediaries, unconstrained, who will seek to maintain an equilibrium within a group of people, not necessarily related to each other: in this case, it is of course the public authorities and the State, but also insurance.
Concerning the first method, it can be defined as any moment before a withdrawal is made: the potential of the environment. The t=0 can be set arbitrarily without affecting the value of the externality.
Thus the slope (a), which will define the externality, corresponds to the restitution of the resource withdrawn without further interactions, or in other words by using the recycling, or the (economic) effort necessary to restore the resource to its natural state.
Thus, if the removal of a resource has a private cost of 10, the social cost corresponds to the cost of recovering the resource and making it usable under the same conditions. This is the cost of recycling. If the cost of recycling 100% of the resource is 13, then the externality is 3.
For irreversible transformations (e.g. energy), we seek to replace the initial resource by an equivalent but renewable resource. Thus, if we consume 1MWh of fossil energy that cost 60€, then we will try to replace the energy consumed by an equivalent renewable resource, wind power for example, at 150€ per MWh: with an externality of 90€ as a result.
Note that in these substitution games, we must take comparable things and therefore in the previous case include in the calculation, the rates of charges and the duration of the underlying investments.
We have seen that the externality can be evaluated simply by taking an assumption: the natural, social and economic potential of the system. This equilibrium is guaranteed mainly by existing stocks, the State, and private economic actors whose business is mutualization (insurance).
It remains to solve the problem of the value chain. On top of this we advocate a progressive and levelled approach, this requirement and constraint being able to be summarized by a famous sentence of Nietzsche “Anyone who wants to act and speak only accurately ends up doing nothing at all.”