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
∫Nn0dN′N′logN′/c=∫Mm0dM′M′logM′/c
loglogN/c−loglogn0/c=loglogM/c−loglogm0/c
loglogN/clogn0/c=loglogM/clogm0/c
logN/clogn0/c=logM/clogm0/c
logN/c=logn0/clogm0/clogM/c
If we define the constant
k0≡logn0/clogm0/c
We have
N=c(Mc)k0
And the price level is
P=αk0(Mc)k0−1
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
∫Nn0dN′N′≈k(N,M)∫Mm0dM′M′
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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