The rest of the Solow model

Here, I mostly referred to the Cobb-Douglas production function piece, not the piece of the Solow model responsible for creating the equilibrium level of capital. That part is relatively straight-forward. Here we go ...

Let's assume two additional information equilibrium relationships with capital $K$ being the information source and investment $I$ and depreciation $D$ (include population growth in here if you'd like) being information destinations. In the notation I've been using: $K \rightarrow I$ and $K \rightarrow D$.

This immediately leads to the solutions of the differential equations:

$$
\frac{K}{K_{0}} = \left( \frac{D}{D_{0}}\right)^{\delta}
$$

$$
\frac{K}{K_{0}} = \left( \frac{I}{I_{0}}\right)^{\sigma}
$$

Therefore we have (the first relationship coming from the Cobb-Douglas production function)

$$
Y \sim K^{\alpha} \text{ , }\;\;\;\; I \sim K^{1/\sigma} \text{ and }\;\;\;\; D \sim K^{1/\delta}
$$

If $\sigma = 1/\alpha$ and $\delta = 1$ we recover the original Solow model, but in general $\sigma > \delta$ allows there to be an equilibrium. Here is a generic plot:

Assuming the relationships $K \rightarrow I$ and $K \rightarrow D$ hold simultaneously gives us the equilibrium value of $K = K^{*}$:

$$
K^{*} = K_{0} \exp \left( \frac{\sigma \delta \log I_{0}/D_{0}}{\sigma - \delta} \right)
$$

As a side note, I left the small $K$ region off on purpose. The information equilibrium model is not valid for small values of $K$ (or any variable). That allows one to choose parameters for investment and depreciation that could be e.g. greater than output for small $K$ -- a nonsense result in the Solow model, but just an invalid region of the model in the information equilibrium framework.

An interesting add-on is that $Y$ and $I$ have a supply and demand relationship in partial equilibrium with capital being demand and investment being supply (since $Y \rightarrow K$, by transitivity they are in information equilibrium). If $s$ is the savings rate (the price in the market $Y \rightarrow I = Y \rightarrow K \rightarrow I$), we should be able to work out how it changes depending on shocks to demand. There should be a direct connection to the IS-LM model as well.