Wednesday, August 27, 2014

Fisher's proto-information transfer economics

Irving Fisher's 1892 thesis and an information equilibrium relationship with information transfer index k = 1.

One of Irving Fisher's thesis advisors was Willard Gibbs (of thermodynamics fame, which I mention because of the connection between information theory and thermodynamics). Here's a link to his 1892 thesis; I was struck by how close some of the equations are to the information transfer model.

Fisher looks at the exchange of some number of gallons of $A$ for some number of bushels of $B$ and states: "the last increment $dB$ is exchanged at the same rate for $dA$ as $A$ was exchanged for $B$". Fisher writes this as an equation on page 5 (see picture above, added 7/13/15):

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

The argument seems to have been introduced by both Jevons and Marshall. Of course it's generally false. Many goods exhibit economies of scale (i.e. buying in bulk) or other effects so that either the last increments of $dA$ and $dB$ are cheaper or more expensive than the first increments. A somewhat less restrictive assumption is that if we scale the total amount of $A$ and $B$ then the relationship between the rate $\alpha A$ was exchanged for $\alpha B$ and the rate the last increment $d(\alpha A) = \alpha dA$ is exchanged for the last increment $\alpha dB$ is unchanged. This property is called homogeneity of degree zero, and you can think of it as what would happen if we doubled the price of everything along with how much money we make: i.e. nothing.

Equation (1) is not the most general equation consistent with homogeneity of degree zero, but rather

\text{(2) } \frac{A}{B} = k \frac{dA}{dB}

This is identical to the result from this argument and the basis for the information transfer model. What is actually equilibrating in the market is the information the market is moving around when $A$ is exchanged for $B$.

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