Other than the solutions to the ODE (see Eqs. 8 and 9 here) for constant information source/destination, there is the solution for "floating" source/destination where:

$$Q^d = (Q^s)^{1/\kappa}

$$

and

$$

P = \frac{1}{\kappa} (Q^s)^{1/\kappa - 1}

$$

Let's assume $Q^s(t) \sim \exp r_0 t $ so that we have

$$

Q^d \sim \exp r_0 t/\kappa

$$

and

$$

P \sim \exp r_0 t (1/\kappa - 1)

$$

I plotted these functions for $\kappa = 0.5, 1.0 and 2.0$ (Green, blue and red in the picture -- I orignally used a Wiener process with drift in place of the $r_0 t$, but then turned down the variance so it would be easier to see).

The dashed lines show $P$ and the solid lines show $Q^d$. The black dashed line (coinciding with the solid blue line) is $Q^s$. We basically get the story that when demand outpaces supply ($\kappa \lt 1$), the price level goes up. The opposite happens when $\kappa \gt 1$. My next thought, based on the idea that no one knows where economic growth comes from (i.e. total factor productivity), was to ask: what if $\kappa$ controls the fluctuations of the economy from recessions to growth rates? So I fixed $Q^s(t) \sim \exp r_0 t $ and let $\kappa$ be a function of time (this time an autoregressive process; I'm all over the stochastic map):

Here we have the demand (blue solid) outpacing the supply (gray dashed) since $\kappa< 1$ on average and the price level rising (blue dashed). Here is $\kappa$

Now $\kappa = K_{\sigma}^{Q^s}/K_{\sigma}^{Q^d}$ where $K_\sigma \sim\log \sigma$ where $\sigma$ is the number of symbols used to encode information in the source/destination. This allows us venture a few hypotheses:

- "Inflation" is when $\langle \kappa \rangle < 1$, i.e. $\langle \sigma^s \rangle < \langle \sigma^d \rangle$, or the number of symbols used in the demand information source is on average greater than the number in the supply information destination. (This mechanism could still be involved.)
- "Recessions" occur when $\sigma^d$ increases and/or $\sigma^s$ decreases such that $\kappa$ falls below its mean.
- The selection rate of symbols must be lower for higher $\sigma$ in order for information transfer to remain "ideal" $I_{Q^d} = I_{Q^s}$; a recession in this sense is a slowdown in the selection rate of an increasing number of demand symbols (or an increase in the selection rate of a decreasing number of supply symbols).
- For small amounts of inflation in a normal economy, this would imply the selection rate for supply symbols is typically slightly faster than the selection rate for demand symbols.

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