Update: x is now defined! See below.
Warning, very wonkish. This will be the first in a series laying out the math of complexity systems, and how they relate to the economy. Please bear with me, as I won’t be able to complete the entire article in one post, and thus, it may get confusing.
In economics, like much of science, a lot of attention is paid to that which exists. All of our measurements of stock and flow within the system measure production and consumption which actually took place in a given period. Economists also measure output trends, and when a recession happens, measure real output against trend output to come up with an “output gap“. There has been a lot of ink spilled about the causes of output gaps. Indeed, a robust, unified theory as to the causes of the output gap elude us to this day.
What is Not
In this blog post, I’m going to take a look at the monetary system through the lens of biological transfer. That is, the flow of biomass (carbon) through an ecosystem. I believe that this concept transfers well to both information in a network, and money in an economy. The framework starts with Boltzmann’s famous definition of surprisal, given as:
s = –k log(p)
Where s is one’s surprisal at seeing an event that occurs with the probability of p, and k is the appropriate (positive) scalar constant. This framework is fitting, considering that no one plans to have a recession, and our society is not built around the assumption that recession is the inertial frame of reference in which we operate (which may have been true during the Great Depression!). Because the probability, p, is normalized into a fraction between zero and one, there arises the view that the negative sign is placed there as a mathematical convenience to make s work out positive. I would like to to take the logisticians standpoint, and state that the equation defines s as gauging what p is not. Thus, if p is the weight we give to an occurrence, s is the measure of that occurrence’s absence.
This measure of the interaction between presence and absence plays a very important role in whether a system survives or disappears.
The Importance of Indeterminacy
The product of the measure of presence of an event, i, (Pi) by the magnitude of its absence (si) yields a quantity that represents the indeterminacy (hi) of that event. Given by:
hi = –k(pi) log(pi).
When pi ≈ 1, the event is almost certain, and hi ≈ 0. When pi ≈ 0, the event is surely absent, so that again hi ≈ 0. It is only for the less determinate values of pi that hi remains appreciable, achieving its maximum at pi = (1/e). It is only when pi is intermediate that the event is both present frequently enough and has sufficient potential for change. Thus, hi represents the capacity for event i to be a significant causal factor in system change or evolution. In order to gain some perspective as to the aggregate system’s indeterminacy, we write H as:
This is a metric of the total capacity of the system to undergo change. Whether the change will be coordinated or stochastic depends on the interrelations of events i to eachother and by how much. In order for change to meaningful and directional, constraints must exist among possible events (Atlan, 1974). In order to view the relationship between events, we write Boltzmann’s equation as:
si j = –k log(pi j)
Where pi j is the joint probability that events i and j co-occur. Whenever i and j are totally independent, p = pipj. We define that maximum as s*i j. The difference by which s*i j exceeds si j in any instance is the measure of the constraint i exerts on j, xi|j:
As you can tell, this equation also shows the constraint that j exerts upon i. Thus, it is a statement of the mutual constraint i and j exert upon eachother.
In order to calculate the mutual constraint (x) extant in the whole system, we will weight each xi|j by the joint probability that i and j occur, and sums over all combinations of i and j:
Here is where the advantage of Boltzmann’s equation as the formal estimate of lacunae becomes apparent, because the convexity of the logarithmic function guarantees that:
This says that the aggregate indeterminacy is an upper bound on how much order (constraint) can appear in the system. Most of the time H > x, so that the difference:
…is greater than (>) or equal to (=) zero as well. φ is called “conditional entropy”. The previous equation can be rewritten as:
This states that the capacity for evolution or self-organization (H) can be decomposed into two components. The first (x) quantifies all that is regular, orderly, coherent and efficient. The concerns of conventional economics! φ represents irregularity, disorder, incoherent, and inefficient behaviors. φ also represents the reserve that allows the system to persist (Conrad).
— — — —
In the next section, we will define a few more terms, look at how this mathematical framework fits into biological systems, and then (maybe!) adapt those concepts to look at economic transactions and our money system. Stay tuned!
Probability of i for any possible j: pi. = ∑ipi j. Probability of j for any possible i: p.j = ∑ipi j
 See Newton’s Third Law of Motion.