# Energy-Based Economic Sustainability Protocols
## 4 Theory
### Energy Theory of Value
#### 4.1.3 Measuring the Exchange Value  
Based on the work of [PERSON] and [PERSON] [37], we use the CExC to be able to account for raw materials and energy and for the perfection of production processes leading to the activity (or supply chain) under consideration. The value of the activity can be expressed as the energy consumed (\(E_{i,j}\)) plus the value of the materials processed by the stakeholder to create a certain output of its activity (\(C^{H(N)}\)), where \(H\) represents the set of extracted flows from the common reservoirs for a degree of depth of the analysis \(N\).  
\[O_{i,j}=C^{H(N)}_{i,j}+E_{i,j} \tag{2}\]  
The wealth \(O_{i,j}\) in this case represents the exchange value of the produced output, which is an objective and quantitative reference of value. However, in addition to this value, we need to consider the subjective difference in cost or benefit, as defined in Equation (7), which relates to the value of the common as defined in Equation (1). The customers' willingness to pay will also depend on the environmental externalities generated by each activity, which, in our first approximation, are captured through a custom utility function that is dependent on socio-economic aspects. The internalization of these social costs will be discretised in a time unit and referred to as the _Stakeholder's Responsibility Paradigm_(see Section 5.1), considering that stakeholders as well as industrial policymakers fully internalize them as a cost upstream from their supply chains. For now, let us continue deriving the dependence of activities on each other and on the raw materials and resources stakeholders process and extract. Using Equation (2) and knowing that the value of the commodities in input \(C^{H(N)}\) is the weighted composition of the value of commodities purchased by the supplier plus the energy consumed to process them and proceeding recursively until extraction from the reservoir of the considered resource (e.g., \(N=\infty\)), we can define the energetic cost as:  
\[E_{i,j}=\mu_{i,j}\phi_{i,j}\psi_{i,j}C_{i,j} \tag{3}\]  
where:  
\[\mu_{i,j} =\text{efficiency}[\%] \tag{4}\] \[\phi_{i,j} =\text{compound energy}[\frac{kWh}{Kg_{C_{ij}}}]\] \[\psi_{i,j} =\text{energy price}[\frac{\text{EUR}}{kWh}]\]  
The term \(Kg_{C_{ij}}\) in Equation (4) is the weight of the resource used by \(j\) to produce activity \(i\). Equation (2) can be rewritten as the summation over all users belonging to the tree of the transaction (\(p\)) considered; we drop the index \(j\) for simplicity:  
\[O_{i}=\sum_{p}^{\pi(N=\infty)}\mu_{p}\phi_{p}\psi_{p}R_{p} \tag{5}\]  
where:  
\[R_{p}=R_{p}^{*}[\frac{kg}{m^{2}\cdot yr}]\cdot A_{p}[m^{2}]\cdot\Delta t_{i}[ yr] \tag{6}\]  
The term \(C_{i,j}\) in (4) is now substituted by the term \(R_{p}\), which represents the intake from common resources \(R\) as specified in (7). Equation (5) represents the relation between the exchange value of an activity to all the activities it depends on and the raw resources dispersed over an area \(A\), whose depletion occurs during a time \(\Delta t\). The term \(\pi\) stands for the whole set of transactions limited by a degree of depth, or accuracy, \(N\); while \(p\) stands for the current level of the transaction tree being accounted for. Both sizes will be further described in Section 5.1. The density of the extraction capabilities of the raw materials over a considered area is defined via Equation (6), where the surface and time units have been expressed in \(m^{2}\) and \(yr\), respectively. It is worth pointing out that this cumulative expression of energy exchanges, represented in the form of embodied wealth, is already integrated along time units through the term \(\Delta t\). This formulation will be re-contextualised in (11).