# Energy-Based Economic Sustainability Protocols
## Appendix B Pigouvian Relativism
### One Transaction  
To introduce a Pigouvian applicative case, the following assumptions are needed. First, a hypothetical relation between price and quantity that considers the price to be a 1-1 monetary form of benefit is represented by a monotonically decreasing line. This assumption, together with the monotone constant increase to the marginal social cost \(MS_{i}\) and the marginal production cost \(MC_{i}\), is arbitrarily set for the sake of understanding through a simple visual explanation. This demand is intrinsically related to the Utility (U), defined in Equation (1), and represents the stakeholder's perceived benefit for the activity considered. The demand curve for the hypothetical considered good is described by the following arbitrary relation:  
\[Y=\alpha-\beta C_{i}\ \ \alpha=4\ \ \beta=0.5 \tag{1}\]  
\(Y\) is the demand and \(C_{i}\) is the quantity; let us proceed to identify the marginal production and social costs (MC and MS) stemming from the previous Equation (2). Let us proceed to substitute the value in (3) into the term \(E\) and consider that the activity does not require raw materials (e.g., the supplier does not have incoming transactions). Let us proceed to assume the marginal cost of the producer \(i\) (\(MC_{i}\), blue) for the production of the activity (the same can be done for a set of activities, with its resultant unit being a weighted average of the single entities) considered. The result will be the following:  
\[MC_{i}=E_{i}=\mu\phi\psi C_{i}\]  
where \(C_{i}\) is the accounting unit in I.S.U. (International System of Units), and \(\mu\), \(\phi\), and \(\psi\) are, respectively, the efficiency of the energy processes of the producer, the compound exergy (or retrievable amount of work from the source material), and the energy price. Now let us consider the marginal cost of production as:  
\[MC_{i}=\mu\phi\psi C_{i}=0.5C_{i} \tag{2}\]  
We can then assume and derive a hypothetical value for the \(\mu\), \(\phi\), and \(\psi\) (\(MC_{i}\)):  
\[\mu= 0.5 \tag{3}\] \[\phi= \frac{1}{\psi}\] \[\psi= 0.25\text{ EUR per kWh}\]Then, we assume the value for the price of carbon dioxide (\(\tau\)) and its specific emission per quantity of output (\(p_{i}\)) to be, respectively, EUR 28 per ton of CO\({}_{\text{2 eq}}\) and 0.2/28 ton CO\({}_{\text{2 eq}}\) per Kg. It follows that the impact cost per output of activity i is dependent on the specific materials required for the production of the energy necessary to produce the activity:  
\[P_{i}=\tau p_{i}C_{i}=0.2C_{i} \tag{10}\]  
We can derive the resultant marginal social total cost by adding the cost of pollutants to the energetic cost of the activity. The marginal social cost plus the private cost curve (social total) are shown in the following equation (\(MST_{i}\)):  
\[\begin{split} MST_{i}=E_{i}+\tau p_{i}C_{i}=0.7C_{i}\\ \Downarrow\\ MST_{i}=MC_{i}+MS_{i}\end{split} \tag{11}\]  
Now, let us introduce the criterion that the producer would tend to maximize:  
\[\max_{C}\int MB\;dC-\int MC\;dC \tag{12}\]  
which is the chosen amount of production \(C\) that defines the maximum of the revenue. This optimum will be a private (local) optimum because it does not yet include the externalities. By knowing also the criterion of the social authority (representing the whole set of stakeholders affected by this activity):  
\[\max_{C}\int MB-MST\;dC \tag{13}\]  
Solving graphically Equations (12) and (13), we get the respective private (local) and social (global) optimum, where the social optimum corresponds to a complete internalisation of the impact cost (MS). Particularly, its values are those of the following tuples of (C,Y) coordinates:  
\[\begin{split}\text{MC equilibrium: }(4,2)\\ \text{MST equilibrium: }(3.\overline{3},2.\overline{3})\end{split} \tag{14}\]  
and a graphical representation is shown in the following plot: