System of Accounts for Global Entropy-Production (SAGE-P): The Accounting in the Topological Domain Space (TDS) of the Econosphere, Sociosphere, and the Ecosphere

Abstract

This chapter describes the accounts of the Econosphere, Sociosphere and Ecosphere governed by the Second Law of thermodynamics. Applied symmetries of ‘production’ (i.e., neg-entropy), ‘consumption’ (i.e., entropy) and capital accumulation (i.e., net-valued low entropy fund available for human consumption). These entropy production accounts permits holistic assessment of sustainability in the following state condition: (a) steady-state, (neg-entropy = entropy), (b) surplus-state, (neg-entropy > entropy) and (c) deficit-state (neg-entropy < entropy). Examined are the root of entropy accounting in the labour theory of values and the Ricardian (long-term) production equations of distribution of wages, profits and rent. This is contrasted to the neoclassical general equilibrium hypothesis of unlimited productivity per unit of consumption and where distributions are determined by state conditions of the supply and demand curve governed by consumer choice and/or preference functions. We introduce the equations of the entropic process described by G-R Flow-Fund Model whereby quantitative (material) production functions are transcribed into qualitative (immaterial) consumption functions, (i.e., enjoyment of Product). This powerful logic of entropy production as a limit function of Production, Consumption and Capital Accumulation, reinstates the Ricardian hypothesis of the longterm end result of Capitalism: Wages fall to subsistence, Profits fall to zero, and Rents rise to a maximum. The SAGE-P provides the accounts of entropy production essential for redirecting the trajectories of unsustainable growth, towards a sustainable economy reduced to maintaining the stock (i.e., wealth) of any well-defined Low Entropy Fund (LEF) available for human consumption, (see Fig. 6.4). The object of policy is thus redefined as the minimum socially acceptable rate of entropy production per unit of consumption. The Appendices provides a window on the formalism underpinning the accounting concepts of SAGE-P.

We have chosen the concept of Topological Domain Space (TDS) because of its generalisation of a mathematical object defined as any set of points that satisfy a set of postulates. These are postulates applied here are those expounded in the G-R Flow Fund Model, see Appendix I.

Appendices

Appendices

1.1 Appendix I

1.1.1 Georgescu-Roegen’s Flow-Fund Model

Factors of production are divided into two categories: the Fund elements, which represent the agents of the process, and the flow elements, which are used or acted upon. Each flow element is represented by one coordinate E _i (t). The fund element enters and leaves the process with its efficiency intact. Specifically, we can represent the participation of a Fund C _σ by a single function S _σ(t) showing the amount of services of C _σ up to the time t, where 0 ≤ t ≥ T.

The analytical presentation of a Process can thus be written [[E ^t _t0 (t), S ^t₀ (t)]], where the i subscript represents input or output and σ represents both input s and outputs.

As for the analytical coordinates of a partial process, analysis must renounce the idea of including in the description of a process, either inside or outside it, the problem associated with the happenings with a process reducing to recording only what crosses the boundary. For convenience, we may refer to any element crossing the boundary from the environment into the process as an input, and to any element crossing it from the opposite direction as an output. At this juncture, analysis must make some additional heroic steps all aimed at assuming away dialectical quality.

Discretely distinct qualities are still admitted into the picture as long as their number is finite and each one is cardinally measurable. If we denote the elements that may cross the t boundary of a given process by C ₁, C ₂, C ₃, … C _n, the analytical description is complete if, for every C _i , we have determined two non-decreasing functions F _i (t) and G _i (t), the first showing the cumulative input, the second, the cumulative output of C _i up to the time (t). Naturally, these functions must be defined over the entire duration of the process which may be always represented by a closed time interval such as [0, T]. The question of whether this analytical model is operational, outside paper-and-pencil operations, cannot be decided without an examination of the nature of the elements usually found in actual processes. Such an examination reveals that there exists numerous elements for which either F _i (t) or G _i (t) is identically null for the entire duration of the process. Solar energy is a typical example, which is only an input for any terrestrial process. The various materials ordinarily covered by the term ‘waste’ are clear examples of elements which are only outputs. In all these cases, we may simplify the analytical picture by representing each element by one coordinate only – namely, by:

$$ {E}_i(t)={G}_i(t)-{F}_i(t) $$

For an output element, E _i (t) = G _i (t) ≥ 0; for an input element, E _i (t) = -F _i (t) ≤ 0. The sign of the suffices indicate which is actually the case (Georgescu-Roegen 1971, p. 215).

Georgescu-Roegen further distinguishes E _i (t), which are (basic) elements necessary to maintain the production cycle at a steady-state (e.g., seeds → crops) and E _i (t), which are (non-basic) elements that are surplus available for consumption, E _i (t) = G _i (t) – F _i (t) ≥ 0.

1.2 Appendix II

1.2.1 Bayesian Statistical Methods

Thomas Bayes addressed both the case of discrete probability distribution of data and the more complicated case of continuous probability distribution . In the discrete case, Bayes’ theorem relates the conditional and marginal probabilities of events A and B, provided that the probability of B does not equal zero. Thus: P(A|B) = P(B|A) P(A)/P(B)

In Bayes’ theorem, each probability has a conventional name:

• P(A) is the prior probability (or ‘unconditional’ or ‘marginal’ probability) of A. It is ‘prior’ in the sense that it does not take into account any information about B; however, the event B need not occur after event A. In the nineteenth century, the unconditional probability P(A) in Bayes’ rule was called the ‘antecedent’ probability; in deductive logic, the antecedent set of propositions and the inference rule imply consequences. The unconditional probability P(A) is called ‘a priori’.

• P(A|B) is the conditional probability of A given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.

• P(B|A) is the conditional probability of B given A. It is also called the likelihood.

• P(B) is the prior or marginal probability of B, and acts as a normalising constant.

Bayes’ theorem, in this form, gives a mathematical representation of how the conditional probability of event A given B, is related to the converse conditional probability of B given A.

1.2.1.1 Bayes’ Theorem With Continuous Prior and Posterior Distribution s

Suppose a continuous probability distribution with probability density function ƒΘ is assigned to an uncertain quantity Θ. In the conventional language of mathematical probability theory, Θ would be a ‘random variable’. The probability that the event B will be the outcome of an experiment depends on Θ; it is P(B | Θ). As a function of Θ, this is the following likelihood function:

$$ L\left(\theta \right) = P\left(B\Big|\varTheta =\theta \right) $$

The posterior probability distribution of Θ (i.e., the conditional probability distribution of Θ, given the observed data B), has the probability density function:

$$ {f}_{\varTheta}\left(\theta \Big|B\right) = \mathrm{constant}\cdotp {f}_{\varTheta}\left(\theta \right)L\left(B\Big|\theta \right) $$

where the ‘constant’ is a normalising constant so chosen as to make the integral of the function equal to one, so that it is indeed a probability density function. This is the form of Bayes’ theorem actually considered by Thomas Bayes. In other words, Bayes’ theorem says: ‘To get the posterior probability distribution , multiply the prior probability distribution by the likelihood function and then normalise.’ More generally, still, the new data B maybe the value of an observed continuously distributed random variable X. The probability that it has any particular value is therefore zero. In such a case, the likelihood function is the value of a probability density function of X given Θ, rather than a probability of B given Θ:

$$ L\left(\theta \right) = {f}_x\left(x\Big|\varTheta =\theta \right) $$

1.2.1.2 Notation and Definitions

In the notation P(A|B), the symbol P is used only as a reference to the original probability. It should not be read as the probability P of some event A|B. Sometimes the more accurate notation P _B (A) is used, but the use of complex events as index of the symbol P is cumbersome. Notice that the line separating the two events A and B is a vertical line.

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read ‘the (conditional) probability of A, given B’ or ‘the probability of A under the condition B’. When in a random experiment, and the event B is known to have occurred, the possible outcomes of the experiment are reduced to B, and hence the probability of the occurrence of A is changed from the unconditional probability into the conditional probability given B.

Joint probability is the probability of two events in conjunction. That is, it is the probability of both events occurring together. The joint probability of A and B is written as P(A|B), P(AB), or P(A, B). Marginal probability is then the unconditional probability P(A) of the event A; that is, the probability of A, regardless of whether event B did or did not occur. If B can be thought of as the event of a random variable X having a given outcome, the marginal probability of A can be obtained by summing (or integrating, more generally) the joint probabilities over all outcomes for X.

The conditioning of probabilities (i.e., updating them to take account of (possibly new) information), may be achieved through Bayes’ theorem. In such conditioning, the probability of A given only initial information I, P(A|I), is known as the prior probability. The updated conditional probability of A, given I and the outcome of the event B, is known as the posterior probability, P(A|B, I).

A continuous probability distribution is a probability distribution which possesses a probability density function. Mathematicians also call such a distribution absolutely continuous, since its cumulative distribution is absolutely continuous with respect to the Lebesgue measure λ. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distribution s: normal, uniform, chi-squared, and others.

The probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region. The probability density function is non-negative everywhere, and its integral over the entire space is equal to one.

1.3 Appendix III

1.3.1 SAGE-P Datasets on the State and Change of State of the Econosphere, Sociosphere, and the Ecosphere

1.3.1.1 Propositions

Proposition I: An entropic Process in SAGE-P is an algorithm to map the transformation of (statistical) objects from lower into higher entropic states (i.e., consumption ), and its inverse from higher into lower entropic states (i.e., production).

The proposed methods to construct the algorithm are prescribed Bayesian Rules for scalar operators in any well-defined, hierarchical-structured, complex adaptive system in Topological Domain Space (TDS), defined as:

Econopshere (E ^’) ⊆ Sociosphere (E ^”) ⊆ 0 (E ^”’)

where ⊆ = subset.

Scalar operators of the ecological flow-fund are a slow moving, but much larger scale to the socio-demographic Fund, which, in turn, is larger, but slower moving to the economic Fund (i.e., E ^”’ > E ^” > E ^’).

• The elements of the E ^’ dataset belonging to the Econosphere TDS are characterised by fast moving state, and change of state, variables described by emergent properties of dissipative economic structures far from equilibrium: (i.e., domain properties of the low entropy economic fund);

• The elements of the E ^” dataset belonging to the Sociosphere TDS are characterised by slow moving state, and change of state, variables described by emergent properties of dissipative social-institution al structures near equilibrium (i.e., domain properties of the low entropy socio-demographic Fund);

• The elements of the E ^”’ dataset belonging to the Ecosphere TDS are characterised by very slow moving state, and change of state, variables described by emergent properties of dissipative ecosystem structures very near equilibrium (i.e., domain properties of the low entropy global ecosystem Fund).

Proposition II: The homomorphism of SAGE-P datasets are conserved value mappings of objects → objects; objects → functions; functions → objects; and functions → functions among any well-defined TDS:

• Econosphere: the homomorphism of economic objects/functions are the values conserved-in-exchange;

• Sociosphere: the homomorphism of socio-demographic objects/functions are the values conserved-in-use;

• Ecosphere: the homomorphism of socio-demographic objects/functions are the values conserved-in-themselves (i.e., intrinsic value).

Proposition III: SAGE-P functions are defined by the boundary conditions of processes unique to any well-defined TDS. Objects are defined by the boundary conditions of the objects-in-themselves, but change values with respect to function:

• Objects where values are conserved-in-exchange ∈ Econosphere

• Objects where values are conserved-in-use ∈ Sociosphere

• Objects where values are conserved-in-themselves ∈ Ecosphere

Proposition IV: SAGE-P qualitative properties of Objects change with Function:

• Social and ecological objects where qualities are values conserved-in-exchange ∈ Econosphere;

• Economic and ecological objects where qualities are values conserved-in-use ∈ Sociosphere;

• Economic and social objects where qualities are values conserved-in-themselves ∈ Ecosphere.

1.3.1.2 Structure of Datasets

SAGE-P material datasets are observed phenomena (i.e., statistical database); all other datasets are constructed from correspondence mappings of:

• (i) objects → objects;

• (ii) objects → functions;

• (iii) functions → objects;

• (iv) functions → functions.

1.3.1.3 Working Definitions

Objects are a collection of statistical elements of the dataset, (E _1-n = Θ), and represent numerical cardinal/ordinal values of the quantitative/qualitative properties of physical/abstract objects.

Functions are a collection of algorithmic operators where the elements of the set are the instructions, [f (E _1-n ) = π Θ], (i.e., vector mapping of objects on object, objects on functions, functions on objects, and function on functions).

SAGE-P algorithmic operators are formalisms expressed in terms of entailment properties of objects and functions. These may be classified to Aristotelean hierarchical structure of causes, viz: material cause (π¹) → efficient cause (π²) → formal cause (π³) ← final cause (π⁴). The reverse arrow of π⁴ reflects the (abstract) socio-cultural values mapped on sustainable development policy in terms of an intensity value measure of the socially acceptable rate of entropy production.

Homomorphism is a structure preserving algorithm permitting one-to-one correspondence mapping of the elements in the SAGE-P datasets – an example being the mapping of prices (values conserved-in-exchange) on economic objects to construct the System of National Accounting dataset (Note the symmetries conserved in linear datasets: inputs = outputs, and broken in nonlinear datasets, inputs ≠ outputs).

SAGE-P datasets are hierarchically-structured matrices of entropic processes representing the notion of:

• Production (P _e ) (i.e., negentropic processes);

• Consumption (C _e ) (i.e., entropic processes);

• Capital Accumulation (K _e ) (i.e., the low entropy Fund available for future consumption , or K _e(t)= P _e(t-n) – C _e(t-n).

1.3.1.4 Category Sets of the Statistical Database

SAGE-P datasets are divided into two separate and distinct categories, viz:

• Physical Objects/Function (Category I): material statistics subject to the Second Law of Thermodynamics, where Category I = (Θ ^p) and;

• Abstract Objects/Function (Category II): immaterial statistics not subject to the Second Law of Thermodynamic s, where Category II = (Θ ^a).

Category I: Quantitative values of inflow/outflow of Physical Objects in any well-defined entropic process and a parallel set of balance sheet accounts of the low entropy (physical) Fund:

• E ^’ I → Econosphere → Θ^p’

• E ^” I → Sociosphere → Θ^p”

• E ^”’ I → Ecosphere → Θ^p”’

Category II: Quantitative values of inflow/outflow of Abstract Objects in any well-defined entropic process and a parallel set of balance sheet accounts of the low entropy (abstract) Fund:

• E ^’ II → Econosphere → Θ^a’

• E ^” II → Sociosphere → Θ^a”

• E ^”’ II → Ecosphere → Θ^a”’

Category III: Mapping of Qualitative Values Algorithms (π) on Category I and II, π^’ = exchange, π^” = use, π^”’ = intrinsic

• E ^’ III → (EI, EII) → π^’ Θ

• E ^” III → (EI, EII) → π^” Θ

• E ^” III → (EI, EII) → π^”’ Θ

Category IV: Mapping of Spatial Co-ordinate Algorithms (π^s) on Category I (Note: Category II objects, by definition, have no geographical co-ordinates), π^s’ = economic co-ordinate space, π^s” = social coordinate space, π^s”’ = ecosystem coordinate space.

• E ^’ IV → Econosphere TDS → π^s Θ^p’

• E ^” IV → Sociosphere → π^s”’ Θ^p”

• E ^” IV → Ecosphere → π^s”’ Θ^p”’