Theories of identities are nonsensical; information equilibrium conditions are better

In reading David Glasner's two posts (so far) on accounting identities:

• Why Theories of National Income Based on Accounting Identities Are Nonsensical and Error-Ridden, Part I
• Why Theories of National Income Based on Accounting Identities Are Nonsensical and Error-Ridden, Part II

I suddenly realized I was looking at an argument that these so-called accounting identities represent information equilibrium (~) conditions. Let's posit something called aggregate demand N (for NGDP) that is an information source for national aggregate variables. Additionally, let's say aggregate demand is in information equilibrium with income (Y) and expenditure (E):

N ~ Y
N ~ E

Now if we have ideal information transfer ('economic equilibrium'), information equilibrium is an equivalence relation, so that we can immediately say:

E ~ Y

log E = k log Y + m

In order to allow E = Y at some point, we must have k = 1  and m = 0, which means that E = Y in equilibrium (ideal information transfer). Now away from economic equilibrium (non-ideal information transfer), we have

a log Y + b < log N
a log E + b < log N

so in general E ≠ Y. (The coefficients of the logs, i.e. a, and the intercepts, i.e. b, must be equal in order to allow the possibility that E = Y.) The information transfer model doesn't tell you how far Y or E fall; it just says there should be a trend where Y ~ E if markets are typically in equilibrium.

This general argument would apply to any national income identity, such as savings and investment (S ≡ I), that isn't based on a definition (e.g. per Glasner purchases equal sales). More interestingly, it applies to another definition: the equation of exchange.

I got in an argument with Scott Sumner on his post that says MV = PY just means V ≡ PY/M for saying that, sure, in the economics profession it's just a definition, but I think the equation of exchange can be usefully restated as an information equilibrium condition.

If we look at the market P:N→M [1] where the price level P is a detector, N = PY is aggregate demand (NGDP) is an information source and M is the money supply (we'll say base money minus reserves), we can write down the equations (in economic equilibrium, i.e. ideal information transfer):

N ~ M

log N = k log M + c1

log P = (k - 1) log M + c2

And we can show:

PY/M = N/M = (1/k) P ≡ V

How does this compare with empirical base velocity? Well, if we take the expected value <P/k> in 1000 random markets with random k values between 0 and 2, we get a pretty good fit (for such a simple model):

This is the Monte Carlo result with 10 different random sets of 1000 markets, hence the 10 gray lines. The blue points are the data (from FRED). Again, it's the trend we're capturing here, and the fluctuations represent non-ideal information transfer and/or shocks.

Footnotes:

[1] The notation A:X→Y means that X is an information source, Y is an information destination and A is a detector per the definitions in the original information transfer model paper.