Power laws and information equilibrium

One of the results of information equilibrium is a power law relationship between the information source and the information destination. If $A \rightleftarrows B$ (i.e. $I(A) = I(B)$) then

$$
\frac{dA}{dB} = k \frac{A}{B}
$$

which has solution

$$
\frac{A}{A_{ref}} = \left(\frac{B}{B_{ref}}\right)^{k}
$$

where $ref$ refers to the reference values for the integration (and become just some parameters in our model). I refer to this as the "general equilibrium" solution to make a connection with economics. Both $A$ and $B$ are adjusting to changes in the other. The "partial equilibrium" solutions follow from assuming $A$ or $B$ adjust slowly to changes in the other (are roughly constant) and result in supply and demand curves (see the paper). We can write this in a log-linear form:

$$
\log A = k \log B + c
$$

These are power law relationships -- not power law distributions. Therefore information equilibrium relationships should be between two aggregates, not rank orders or cumulative distributions. It seems that a lot of economics deals with the latter, but there are a few examples that come up that fit this mold. First, there is CEO compensation (C) versus firm size (S) from Xavier Gabaix (I borrowed a graph from this blog post about Gabaix recent paper). We have the model $S \rightleftarrows C$ so that

$$
\log S = k \log C + c
$$

And this works pretty well ...

... but not perfectly. What we have here is probably some measurement error, but also deviations from ideal information transfer (non-ideal information transfer) so that $I(C) < I(S)$ and therefore

$$
\log S \geq k \log C + c
$$

We'd say the information in the firm size isn't reaching the CEO's compensation for large and small firms. Since we expect ideal transfer to be a better approximation as our variables go to infinity, it seems likely that the high end represents more measurement error (fewer of the largest firms) than non-ideal information transfer. (I realized that after I drew on the figure.)

Here's another example relating GDP and population (P) from Nature:

This could be represented by the model $P/A \rightleftarrows GDP/A$ where $A$ is just the area unit to obtain population and GDP density. I drew in a schematic curve (again) that represents information equilibrium (ideal information transfer) and shows how non-ideal information transfer data would fall below the line. We can also see how allowing for non-ideal information transfer means you need to take a different view towards fitting the data -- we expect our results to fall below the line so a simple power law fit will be biased. I imagine new techniques will have to be developed to make this process more rigorous.

...

Update

In Gabaix's paper, footnote 7 (on page 194, regarding the CEO pay) is an information equilibrium model:

$$
\frac{dw}{dS} = k \frac{w}{S}
$$
with $k \equiv d \log w/dr$.