A vector of information equilibrium relationships

This is a mathematical interlude that looks at some geometric interpretations of an ensemble of information equilibrium relationships. It represents some notes for some future work.

Let's start with a vector of information equilibrium relationships between output in a given sector $y_{i}$ and the money supply $p_{i} : y_{i} \rightleftarrows m$ so that

$$\frac{dy_{i}}{dm} = A_{ij}(m) y_{j}$$

The solution to this differential equation is

$$y_{i}(m) = \left[ \exp \int_{m_{ref}}^{m} dm' A_{ij}(m') \right] y_{j}(m_{ref})$$

if $A(m) = K/m$ (i.e. if $A(m_{1})A(m_{2}) = A(m_{2})A(m_{1})$ but not generally, see Magnus expansion) so that

$$y_{i}(m) = \left[ \exp \left( K_{ij} \log \frac{m}{m_{ref}} \right) \right] y_{j}(m_{ref})$$

The volume spanned by these vectors (spanning the economic output space) is

$$V = \det \exp \left( K \log \frac{m}{m_{ref}} \right) \approx 1 + \log \frac{m}{m_{ref}} \;\text{tr}\; K$$

So that the infinitesimal volume added to the economy is

$$dV = \left( \log \frac{m}{m_{ref}} \right) \;\text{tr}\; K$$

*  *  *

Update 30 November 2016

Let me continue this a bit, putting it in a more useful form. Starting with the expression for $V$ above:

$$V = \det \exp \left( K \log \frac{m}{m_{ref}} \right)$$

The $\log$ factor is a scalar and can be pulled through the determinant, gaining a factor of $n$ (the number of markets indexed with $i$ above ($p_{i} : y_{i} \rightleftarrows m$), giving us:

$$\begin{align} V & = \exp \left( n\; \log \frac{m}{m_{ref}}\right) \det \exp K \\ & = \exp \left(n\; \log \frac{m}{m_{ref}} \;\text{tr}\; K \right) \\ & = \left(\frac{m}{m_{ref}}\right)^{n} \; \exp  \text{tr}\; K \end{align}$$

If $m$ grows exponentially at some rate $\mu$ then $V$ will grow with rate $v$ where

$$v = n \; \mu \; \text{tr}\; K$$