Thursday, March 19, 2015

The slowly variying information transfer index approximation


Previously, I had looked at the changing information transfer index as an approximation (see e.g. here). What if we look at the exact result assuming the functional form of k

k(N,M)=logN/clogM/c


So that

dNdM=k(N,M)NM=logN/clogM/cNM


Re-arranging and integrating the differential equation

dNNlogN/c=dMMlogM/c


Nn0dNNlogN/c=Mm0dMMlogM/c


loglogN/cloglogn0/c=loglogM/cloglogm0/c


loglogN/clogn0/c=loglogM/clogm0/c


logN/clogn0/c=logM/clogm0/c


logN/c=logn0/clogm0/clogM/c


If we define the constant

k0logn0/clogm0/c


We have

N=c(Mc)k0


And the price level is

P=αk0(Mc)k01


where α represents the freedom to define the price level to be P=100 for any given year. This is basically the same result where we take k to be constant, which means the approximation where we take

Nn0dNNk(N,M)Mm0dMM


for slowly varying k(N,M) represents simply moving to a local fit rather than a global fit.

[Assuming my math is right.]

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