Tuesday, August 4, 2015

Information transfer models and communication channels

As part of dotting all the i's and crossing the t's in finishing up the draft of the paper, there is a piece I added to the discussion of the macroeconomic models that I've never made explicit. If you look at the information transfer models used in the IS-LM model and the Solow model, they are superficially similar:
ISLM:           N → M  and   N → S
Solow (1):     N → K   and   N → L
Solow (2):     K → I    and   K → D
i.e. "source → destination1" and "same source → destination2" (note, I've changed the Y to N because that's what I've done in the paper). However, you end up with completely different models. How can that happen? Obviously there are some unstated assumptions.

The ISLM model is a model of partial equilibrium (N varies slowly when M or S change, and the solution to the differential equation is N ~ log M) where both destinations (M and S) are receiving the same signal from the source (N). In a diagram it looks like this:



Solow (2) is similar, except it is general equilibrium. Changes in K immediately register in both I and D and vice versa (and the solution is log K ~ α log X). The diagram looks the same as the partial equilibrium one, though:



The third one is a bit different from the previous two. Solow (1) is a general equilibrium model where the information source is treated as a sum of two orthogonal signals (think two FM radio stations). It looks like this:



We have two separate "communication channels". This is important because it means that even if N is in information equilibrium with K and L, K and L are not necessarily in information equilibrium with each other (however, they will be "coupled" to each other by N).

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