More on the information transfer index

The information transfer index is defined as 

$$ \kappa = \frac{K_0 \log \sigma^s}{K_0 \log \sigma^d} $$ 

where we are measuring information in the same units (defined by $K_0$). Now we take the floating information source solution 

$$ \frac{Q^{d}}{Q^{d}_{ref}} = \left(\frac{Q^{d}}{Q^{d}_{ref}}\right)^{1/\kappa} $$ 

and solve for $\kappa$ 

$$ \kappa = \frac{\log Q^{s}/Q^{s}_{ref}}{\log Q^{d}/Q^{d}_{ref}} $$ 

I believe we can make the identification 

$$ \sigma^x \sim Q^{x}/Q^{x}_{ref} $$ 

i.e. the "number of demand symbols" is basically proportional to the size of the demand which makes intuitive sense (well, to me). We next plot how $\kappa$ behaves vs $\sigma^s$ and $\sigma^d$. The colors indicate high magnitude (in red) or low magnitude (meaning zero, in blue) of the gradient. The line across the figure show where $\log \kappa = 0$ i.e. $\kappa = 1$.

One interesting thing that appears is that as both $\sigma^s$ and $\sigma^d$ become large, $\kappa \rightarrow 1$. Random thought at this moment is that as aggregate demand and aggregate supply become large, we should see both growth rates converge across international data as economies become large and, from a monetarist perspective, growth rates approach the monetary base growth rate (see $r_0$ here). Maybe. I will in a future post look at this information in this light.

I decided to use some empirical data to play around with these concepts. For example, if we say that $Q^d$ is nominal GDP (aggregate demand) and $Q^s$ is the (St. Louis Adjusted) Monetary Base, we can measure $\kappa$

This is somewhat close to $\kappa = 1$ as we might expect for a large economy (although any value of $\kappa$ can be acheived given any $\sigma^s$). It seems we should be approaching 1 more monotonically as the scale of the economy grows.

If we use

$$
P \sim \frac{1}{\kappa} \left( Q^s\right)^{1/\kappa - 1}
$$

With the empirically defined $\kappa$ we get the picture above for the equilibrium price level $P$ (in an AD/AS model).

Do these results make sense? I would say no. But I'm going to think about it some more.