An empirical validation of k - (k - 1) = 1

The equations for the price level (P) and output (N) in the information equilibrium model have the forms:

N ~ (M/m0)ᵏ
P ~ k (M/m0)ᵏ⁻¹

Which means that

log N ~ k log M

log P ~ (k-1) log M

log N ~ k₁ log M

log P ~ k₂ log M

So that the difference in the slopes in log-log space between the first relationship and the second relationship is k₁ - k₂ = k - (k - 1) = 1 (if measured over short time periods -- since k is changing). In the real world there are shocks that come from the changes in the labor supply (E) so that we should actually add in the slope of 

log E ~ s log M

So that 

k₁ + s - k₂ ≈ 1

And guess what: it works! Non-ideal information transfer during recessions mean there should be deviations, and in general I(N) > I(M) so P < P* (the ideal price level) so we fall below the ideal value: