Friday, July 28, 2017

Dynamic equilibrium and ensembles (and collected results)


I previously worked out that ensembles of information equilibrium relationships have a formal resemblance to a single aggregate information equilibrium relationship involving the ensemble averages:

$$
\frac{d \langle A \rangle}{dB} = \langle k \rangle \frac{\langle A \rangle}{B}
$$

I wanted to point out that this means ensemble ratios and abstract prices will exhibit a dynamic equilibrium just like individual information equilibrium relationships if $\langle k \rangle$ changes slowly (with respect to both $B$ and now time $t$):

$$
\frac{d}{dt} \log  \frac{\langle A \rangle}{B} \approx (\langle k \rangle - 1) \beta
$$

plus terms $\sim d\langle k \rangle /dt$ where we assume (really, empirically observe) $B \sim e^{\beta t}$ with growth rate $\beta$. The ensemble average version allows for the possibility that $\langle k \rangle$ can change over time (if it changes too quickly, additional terms become important in the solution to the differential equation as well as the last dynamic equilibrium equation).

Generally, considering the first equation above with a slowly changing $\langle k \rangle$, we can apply nearly all of the results collected in the tour of information equilibrium chart package to ensembles of information equilibrium relationships. These have been described in three blog posts:
1. Self-similarity of macro and micro 
Derives the original ensemble information equilibrium relationship 
2. Macro ensembles and factors of production 
Lists the result for two or more factors of production (same result gives matching models) 
3. Dynamic equilibrium and ensembles 
The present post arguing the extension of the dynamic equilibrium approach to ensemble averages

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