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In this piece I will expand upon my transactional economics thought experiment (model if you like). The supposition of this model is that transactions are fundamental in an economy, and are the building blocks for everything else. The behaviour of one particular person (node) is not necessarily important but rather the aggregate effect of many people is important. Naturally, different people will have different levels of influence depending upon their wealth as illustrated in my previous article but essentially what we have is an network of people (nodes) that ought to have some modelable behaviour as well as some randomness that we may not be able to model.

Networks and flow

With this model I mainly wish to figure out how wealth (capital / money / goods / etc) flows around the network; are there "attractors" that encourage growth in certain regions or perhaps barriers that inhibit flow. Appealing to induction and limits I want to build a model contains elements (ideas) that mimic a real economy. My impetus for this came from my brief reading on networking as well as a general interest in graph theory and topology, and now I try to apply this to economics with some input from real financial markets as well as virtual markets. The reason I look at the latter is my assumption that as they are populated by people then they ought to provide an insight into special cases of real economies, as well as provide some illustration into markets that are not so easily observable in a real market.

In the previous piece I considered transactions between a limited number of nodes and implicitly used symmetry to suggest how things would look on aggregate. I suggested four particular types of transactional economy that we can call close or open, and with appeal to simplified special cases I provided an insight into how these 4 cases behave. A real economy, even with a fixed supply of money, ought to behave like an open system in physics.

Energy conservation

That is to say that "energy" is not conserved, or by similar analogy "wealth" is not conserved. The reason I believe this to be the case is partly from the high degree of unpredictable in human actions but also due to the fluctuating number of humans at any one time. The rate of births and deaths are not static numbers, nor is the distribution across the age ranges flat or static. Add to this that humans interact with each other in "strange" ways that are likely to be poorly modelled by closed-system models, thus it is very hard to know how many participants you have in an economy at any given time. We all spend and save money at different rates which is dynamic rather than static, also it is likely to be non-linear and probably not like a unitary transformation (re: conserved flow). However, it is likely to be easier to assume unitarity as a starting point and then consider how to add in non-unitary elements. Note that fluids are generally modelled as mass conserving (divergenceless), as are wavefunctions that obey Schroedinger's wave equation. On the latter point we can confer the equivalence of the conservation of probability density in quantum mechanics with that of mass conservation in classical fluids.

From my PhD work I am drawing parallels between fluid flow which is modelled on a computer and that of real life. In my particular topic I was looking at gravity and the movement of dark matter. This isn't as fancy as it sounds but as with all simulations we have to appreciate that approximations are made in order to gain any kind of insight into a complex system. In a computer model we deal with lattice-like network (the nodes are arranged in a cube) and the flow evolves over forward in discrete units of time and over discrete units of space. This is analogous to an economy where humans are discrete (separate) entities interacting with each other. Transactions are not instantaneous so this too could be viewed as discrete steps of time. I am aware that electronic transactions can essentially be instantaneous when we compare the timescale of an electronic transactions versus that of two people trading physical items face-to-face.

Gaussians and power laws

After reading the Black Swan by Taleb I became more acutely aware of the problem of assuming the ubiquity of Gaussian distributions in everything we do. We often assume that something starts as a Gaussian and remains a Gaussian through out its life. Interestingly, as an initial condition for simulating the universe we assume that distribution of mass is Gaussian (in practical terms: put the mass into bins and plot a histogram to see a Gaussian). The observational data backs this up to a degree but I don't wish to get into that here. The main point I wish to make here is that by the end of the simulation the mass distribution is not Gaussian but tends to log-normal. How much it changes depends on far you run the simulation. A log-normal has some similar looking features to a Gaussian but they are not the same. The Gaussian is far more appropriate in a regime where the dynamics of the universe are linear. In our current Universe we see non-linearities everywhere, although to fair approximation we can say that on scales beyond 8 MPc (Mega-Parsec) the universe is linear. This example shows that Gaussians can change when the underlying process is not amenable to preserving the nature of the Gaussian (not a surprise).

One wonders how the mass distribution of the Universe compares to the wealth distribution on Earth. We will see a power law although the two phenomena will have different powers. I know Taleb lists a figure for the distribution of populations for cities but I don't have that to hand; I also know that Philip Ball lists many such numbers in his book Critical Mass. It would be interesting to know what the indices of these power laws are and how they have changed over time. Furthermore it would be possible to model such phenomena in a similar way to gravitational simulations of the universe. The exact initial conditions are not important but rather the overall picture, the statistical measures, is what is important. In simulations of the universe it is not important to closely monitor nor define the exact position or velocity of each particle but rather to make sure that on aggregate that the positions and velocities of each particle fit with the statistical distribution that you expect.

Conclusion