## Wednesday, July 13, 2016

### "A statistical equilibrium approach to the distribution of profit rates"

That's the title of a paper [pdf] by Gregor Semieniuk (and his co-author) who tried to get a hold of me at the BPE meeting (which I was unfortunately unable to attend, but did make some slides). The paper notes that the distributions of the rates of profit appear to have invariant distributions suggestive of statistical equilibrium; here's a diagram:

This diagram is reminiscent of the diagrams I've used when talking about the distribution of growth states in an economy, most colorfully illustrated at this link (and discussed here and in my paper; note that Gregor's diagram has a log scale for the y-axis).

In fact, it might be directly related. If we have a series of markets (or firms) where income $I_{k}$ is in information equilibrium with expenses $E_{k}$ with "price" $p_{k}$, i.e. $p : I \rightleftarrows E$, in general equilibrium we have

$$\frac{I_{k}}{I_{k, ref}} = \left( \frac{E_{k}}{E_{k, ref}}\right)^{p_{k}}$$

so that for $E_{k} \approx E_{k, ref}$

$$I_{k} \approx I_{k, ref} + p_{k} \frac{I_{k, ref}}{E_{k, ref}} (E_{k}-E_{k, ref})$$

Therefore $p_{k}$ determines the rate of profit (difference between income and expenses). It determines how much bigger $I$ is given $E$. If $p = 1$, then profit is zero because income equals expenses. If $p > 1$, the firm is profitable because $I > E$; vice versa for $p < 1$. In the partition function approach (also discussed in the paper and the link above), the "prices" represent growth states; here, the "prices" represent profit states. The distribution represents an equilibrium "temperature".

We'd also expect the profit states to have some distribution around a mean value, but have negative profit states to be over-represented due to non-ideal information transfer. This is as observed in the diagram from Gregor's paper. It is also observed in nominal growth data over time** (figure from my paper linked above; also see here):

** This implies a macroeconomy is ergodic: the distribution of temporal states are the same as distributions of ensembles of states.

...

Update 01 August 2016

Michael Williams sent me links to two of his (and co-authors') papers (here and here) where they look at the distribution of economic profit rates and firm growth rates. These appear to have well-defined distributions as in the paper from Semieniuk et al above. The difference is that Williams et al find that the distributions appear to be Cauchy distributions rather than Laplace distributions. I noted previously that the distribution of wage changes also appears (by eye -- I did no rigorous testing) to be Cauchy (plus perturbations).

Laplace distributions are maximum entropy distributions subject to the constraint on the absolute deviation |Δ| = |x - μ| of the variable (profit rate in this case) from the mean. Semieniuk et al also suggest the asymmetric Laplace distribution which is the maximum entropy distribution subject to a constraint on the average deviation (Δ = x - μ).

Cauchy distributions are "heavy-tailed" and are the maximum entropy distributions for random variables subject to a constraint on log(1+Δ²) where Δ = x - x₀ (the Cauchy distribution has no mean, so there is a parameter x₀ that represents the "center").

In my view the Laplace distribution represents a "first order approximation" which is improved upon with the Cauchy distribution. However, I'd take a more agnostic view that what we have is some form of stable distribution (of which the Cauchy distribution is a particular example). Note that Mandelbrot showed that stable distributions appeared to fit various financial data. Here's a graph with a Cauchy (blue), stable (yellow), and skewed stable (green) distributions along with a Laplace distribution (gray dashed):

Stable distributions are stable in the sense that they have their own (more general) central limit theorem that doesn't require a finite variance.

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