Econophysics and sunk costs

I highly recommend the work of Yakovenko and his collaborators on income and wealth distribution. His recent colloquium on "Statistical Mechanics of Money, Wealth, and Income" contains some of the most trenchant and clear science writing that I have seen in a while. It gives a very good background on the field of econophysics, something I imagined existed based on what I had read about Wall St. financial quants, yet I hadn't realized the depth and alternate focus on macroeconomics.

Econophysics distances itself from the verbose, narrative, and ideological style of political economy
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The econophysicist Joseph McCauley proclaimed that “Econophysics will displace economics in both the universities and boardrooms, simply because what is taught in economics classes doesn’t work” . (referencing P.Ball, 2006, “Econophysics: Culture Crash,” Nature 441, 686–688).

I consider the research very important but realize too that Yakovenko may have become lodged in a "no-man's land" of science. The economists don't completely appreciate him because he doesn't use their vaunted legacy of work; the statisticians don't recognize him because he deals with fat-tail models; and the physicists ignore him because they don't consider economics a science. Despite the lack of interest and threatened by the sunk costs of the status quo, Yakovenko makes a strong case by describing the history behind econophysics, reminding us that Boltzmann and company had suggested this field from the start, regrettably getting buried over the ensuing years.

“Today physicists regard the application of statistical mechanics to social phenomena as a new and risky venture. Few, it seems, recall how the process originated the other way around, in the days when physical science and social science were the twin siblings of a mechanistic philosophy and when it was not in the least disreputable to invoke the habits of people to explain the habits of inanimate particles.”

Curiously, the fractal and fat-tail proponents also seemed to have dismissed the field:

A long time ago, Benoit Mandelbrot (1960, p 83) observed: “There is a great temptation to consider the exchanges of money which occur in economic interaction as analogous to the exchanges of energy which occur in physical shocks between gas molecules.” He realized that this process should result in the exponential distribution, by analogy with the barometric distribution of density in the atmosphere. However, he discarded this idea, because it does not produce the Pareto power law, and proceeded to study the stable Levy distributions.

In spite of its early origins, econophysics-type approaches help to explain macroeconomics in a fresh way, separated from the conventional focus on supply/demand and whatever else that regular economists deem important. I looked at Yakovenko's work a couple of years ago when I used data from his paper on thermal and superthermal income classes. In the post, I provided a few of my own alternate math interpretations, which I think retain some validity, but I have since gained some valuable perspective. The big insight has to do with fast income growth and how that compares to adaptation of species and (of course) oil discovery. In all cases, we have very similar mathematics, which uses entropy to argue for large dispersive effects on measurable macro quantities.

Figure 1: Entroplet dispersion of species diversity. The relative abundance of species follows simple dispersive probability arguments. (top) Model histogram (bottom) Fit to North American bird data

A representative post on entroplets at TOD showed how the two channels of species adaptation work. One of the channels involves variations in time, in which species adapt via maximum entropy over slow periods. The other channel involves variations in adaptation levels themselves; this abstraction provides a kind of "short-circuit" to a slower evolution process in which small changes can provide faster adaptation. If the first case acts as a deltaTime and the second acts as a deltaX then deltaX/deltaT generates a velocity distribution leading to the observed relative abundance distribution as the entroplet or entropic dispersion function. That works very well in several other fat-tail power-law applications, including dispersive discovery of oil.

My intent on introducing that analysis was to show how this primitive adaptation model works in the context of human adaptation -- in particular, in the greedy sense of making as much money as possible.

What Yakovenko asserts, and I interpreted on my previous post on the subject, the adaptation channel of time remains a strong driver on the lower wage earners. If we only assume this as a variant, then according to maximum entropy, the income distribution drops off as exp(-v/v0) where v indicates income velocity, and ends up as proportional to a relative income if time drives the velocity, Income = v * time. The cumulative becomes, where t=time:

P(Income/t) = exp (-Income/t/v0))

This works for a portion of the income curve, primarily consisting of the low income classes. Yet it does not generate the observed Pareto power-law for the higher income part of the distribution. To get that we need the other fast channel for dispersive income growth. Humans can't mutate on command (and obviously don't have the diversity in geological formations) so they lack the same fast channels that exist for other fat-tail power laws.

To get the fast channel, we can hypothesize the possibility of income that builds on income, i.e. compound interest growth. This has some similarity to the ideas of preferential attachment, where a large volume or quantity will attract more material (or generate more mutations in species if a population gets large). Yakovenko calls it multiplicative diffusion:

This is known as the proportionality principle of Gibrat (1931), and the process is called the multiplicative diffusion (Silva and Yakovenko, 2005).
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Generally, the lower-class income comes from wages and salaries, where the additive process is appropriate, whereas the upper-class income comes from bonuses, investments, and capital gains, calculated in percentages, where the multiplicative process applies

The simplest variant of compound growth is the following equation:

dg(t)/dt = A*g(t) + 1

This has the solution

g(t) = 1/A*(exp(A*t)-1)

Figure 2: Compound growth starts immediately and will effect all income streams in a proportional amount.

Since we consider income growth as a relative or proportional process, the growth factor g(t) should fit directly into the income velocity expression, exp(-v/v0)

t = 1/A * ln(g(t)*A + 1)

If we plug this into the Income distribution we get

P(Income=g) = exp (-A / ln(g(t)*A+1)/v0))

This turns into a power law:

P(Income) = (Income*A+1) ^(-A/v0)

If the average growth of income v0 exactly compensates the compound growth term A, the Income cumulative has the form of an entroplet.

Figure 3: Income distribution in the USA. The measure income distribution fits between two-degree of freedom entropic dispersion (top curve, entroplet) and one-degree of freedom dispersion (bottom curve, exponential). The introduction of compound growth can magnify the dispersive effects

Unfortunately, this does not match the observed stratification of income classes (see Figure 3). The entroplet may work (at least conceptually) if all income classes continuously stored away part of their wages as compound savings, yet we know that lower income classes do not save much of their wages at all.

The growth equation which discounts low-incomes effectively looks like the following:

dg(t)/dt = A*( g(t)-t ) + 1

Early growth gets compensated by the linear growth term t, and the parametric deferred growth form looks like:

g(t) = t + B * exp(A*t)

Figure 4: Deferred growth has to overcome the linear term and then will grow according to the savings rate. Lower fractional savings (B) or lower compound interest (A) will defer the growth to the future.

As I discovered the last time I tried solving this problem, inverting the equation to obtain the time mapping requires a parametric substitution. I replace the running time scale with the substitution: t + B*exp(A*t), yet maintain the cumulative with a linear growth. This has the effect of compressing time during the intense compounding growth period.

See the red line in Figure 3 which assumes a deferred growth fit with the two parameters shown: B=0.9 and A=v0=0.03.

The exponential term, A, essentially borrows from the average income growth, v0, as a zero-sum game and builds up the compounding interest. I could have used an alternate exponential factor for A differing from v0, but in the steady-state economy money has to come from somewhere, so that having the interest rate track the average income growth dispersion makes sense (one less parameter to gain parsimony).

That leaves one other free parameter besides A; the term B refers to a proportional savings rate. If a person saves money relative to his non-compounded growth, it will provide the initial value for B. If B remains close to zero, it implies no savings; whereas a high value implies a large fraction goes into savings for compounding growth.

Figure 5: Variations of savings fraction, B, accounts for most of the yearly fat-tail fluctuations observed. (top) Plot from Yakovenko, recessions have lower investment savings than boom periods. (bottom) Model variations in B.

The combination of A and B generates the fast growth channel so that we can duplicate the fast growth needed for the observed income dispersion.

Modifying B moves the fat-tail up and down on the histogram. The meaning of B compares to the Gini factor or coefficient used to quantify the disparity between the income classes. Whereas modifying A, if needed, serves to flatten or steepen the fat-tail portion (according to a sensitivity analysis this has a larger effect the further right you go on the tail). If either of these disappears, it reduces to the exponential tail steep/narrow dispersion. The top curve to the right shows how effectively this procedure works on wide dynamic range data.

In the previous post, I also tried to separate the two segments by summing two distinct distributions, i.e. a low-income exponential distribution and a high-income continuous learning power-law. I have abandoned this approach because it appears too artificial. The compound growth seems to exhibit better understandability and we can also use that to accommodate that education or business experience can also lead to compound growth. By making the transition continuous, you begin to understand that people's behaviors in terms of money may show that same continuity.

Discussion

This approach explains everything you want to know in relation to how entropy drives income distributions and likely some part of wealth disparity.

The fact that income distribution consists of two distinct parts reveals the two-class structure of the American society.

About 3% of the population belonged to the upper class, and 97% belonged to the lower class.

... nations or groups of nations may have quite different Gini coefficients that persist over time due to specific historical, political, or social circumstances. The Nordic economies, with their famously redistributive welfare states, have G in the mid-20%, while many of the Latin American countries have G over 50%, reflecting entrenched social patterns inherited from the colonial era.

Money and the effects of compounded interest acts to disperse the wealth of individuals by artificially speeding up the evolution or adaptation of our species. Look at the case of Mexico; I recall noting that a few of the wealthiest people on the planet live there, which largely comes from oil income and whatever compounded interest their investments have gained. On the other hand, the Nordic countries tax the wealthy before they can tuck away their investments and the dispersion and disparity in incomes drops way down.

Which adaptation route should we follow? Using the proxy of compounded growth income does allow us to approach the natural relative abundance distribution of other biological species. Yet this comes at the expense of a rather artificial shortcut. The income distribution curve appears very precariously positioned and we will see large swings anytime we enter recessionary periods. The wealthy at the top appear to have enormous sensitivities to marginal rates of savings and the income growth at the bottom. The worker bees seem to show more stability. Propping up the wealthy through whatever debt-financing schemes one can find will likely keep the fat-tail from collapsing. In reality, a fat-tail can only exist in an infinite wealth (i.e. resource) world.

Do we need more data on income and wealth to really nail our understanding?

Despite imperfections (people may have accounts in different banks or not keep all their money in banks), the distribution of balances on bank accounts would give valuable information about the distribution of money. The data for a large enough bank would be representative of the distribution in the whole economy. Unfortunately, it has not been possible to obtain such data thus far, even though it would be completely anonymous and not compromise privacy of bank clients.

Like I said before, if we want to control our destiny, we have to understand the statistical dynamics. As long as people stay infatuated with deterministic outcomes and neglect to include entropic dispersion, the econophycists will have the field to themselves. The mainstream economists desperately trying to avoid sinking the costs of their intellectual investments apparently won't use these methods. Growth can't keep going unencumbered and we need to start paying attention to what the models can tell us.

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The Economic Undertow recommends the work of Steve Keen, another econophysicist.


A nice little utility called Winplot has a parametric mode and parameter sliders. The parameter D=a=v0 changes the low income rate and the parameter B modifies the savings fraction.
http://depositfiles.com/files/q76v4gqiu

The following Winplot figure provides a snapshot for one set of parameters. The blue curve is a non-compounded income growth. The gray curve is the entroplet form. The red curve shows the effects of compounding, B=0.32 and A=0.064 (with +/- 5% curves).