Sunday, March 1, 2015

Information equilibrium is an equivalence relation


Something for the math nerds. I've said it a couple times, but haven't actually shown the proof. However, it is true that information equilibrium is an equivalence relation. If we define the statement $A$ to be in information equilibrium with $B$ (which we'll denote $A \cong B$) by the relationship (i.e. ideal information transfer from $A$ to $B$):

$$
\text{(1) }\;\; \frac{dA}{dB} = k \frac{A}{B}
$$

for some value of $k$, then, first we can show that $A \cong A$ because

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

$$
1 = k \cdot 1
$$

and we can take $k = 1$. Second we can show that $A \cong B$ implies $B \cong A$ by re-deriving the relationship (1), except moving the variables to the opposite side:

$$
\frac{dB}{dA} = \frac{1}{k}\;\; \frac{B}{A} = k' \; \frac{B}{A}
$$

for some $k'$ (i.e. $k' = 1/k$). Lastly we can show that $A \cong B$ and $B \cong C$ implies $A \cong C$ via the chain rule:

$$
\text{(2a) }\;\; \frac{dA}{dB} = a \frac{A}{B}
$$

$$
\text{(2b) }\;\; \frac{dB}{dC} = b \frac{B}{C}
$$

such that

$$
\frac{dA}{dC} = \frac{dA}{dB}\; \frac{dB}{dC} = a b \; \frac{A}{B} \frac{B}{C}
$$

$$
\frac{dA}{dC} = k \; \frac{A}{C}
$$

with information transfer index $k = a b$. That gives us the three properties of an equivalence relation: reflexivity, symmetry and transitivity.


3 comments:

  1. Jason, how does it work in the (very) long run?

    http://www.bankofengland.co.uk/research/Pages/onebank/datasets.aspx#5

    ReplyDelete
    Replies
    1. It works fairly well on that dataset:

      http://informationtransfereconomics.blogspot.com/2013/11/the-long-run-in-uk.html

      In looking at that dataset, there were some interesting insights on how the labor market works:

      http://informationtransfereconomics.blogspot.com/2013/11/the-labour-supply-part-1.html

      http://informationtransfereconomics.blogspot.com/2013/11/the-labour-supply-part-2.html

      However, I hadn't quite worked out how the monetary base reserves worked in the model at the time, so I should probably re-do the results.

      It also works reasonably well on a long run US dataset:

      http://informationtransfereconomics.blogspot.com/2014/09/the-us-economy-1798-to-present.html

      The separation into monetary regimes still seems somewhat ad hoc to me, but remains important.

      Delete

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