Constructing and understanding stock-flow model

1. Dec 15, 2012

mackenzies

Suppose that $\textbf{x} = A\textbf{x} + B\dot{\textbf{x}}$ where $\textbf{x}$ is vector of economic output level, $A$ is input-output matrix, $B$ is stock-flow matrix. The system represents closed and dynamic input-output system. $\dot{\textbf{x}}$ is time derivative of $\textbf{x}$. Let $C = B^{-1}(I-A)$. Then the system would be represented as $\dot{\textbf{x}} = C\textbf{x}$. Then, let $\textbf{p}$ be a vector of price level. The equation would be written as the following: $\textbf{p} = (1+\pi)(\textbf{p}A + \textbf{rp}B - \dot{\textbf{p}}B + wa_0)$ where $w$ refers to wage rate and $a_0$ refers to a vector of labor requirement, $\textbf{r}$ refers to the interest rate, and $\dot{\textbf{p}}$ is first-time derivative of $\textbf{p}$. In words, current cost: $\textbf{p}A$, interest charges: $\textbf{rp}B$, capital loss: $\dot{\textbf{p}}B$, wage costs: $wa_0$, uniform profit rate: $\pi$.

I don't get what this means. OK, the economic output at one point, $\textbf{x}$, equals to input-output matrix times itself, $A\textbf{x}$ and add this to $B\dot{\textbf{x}}$. Then that means that on economy where output is greater than input, $B$ is acting as negative one. What is stock-flow matrix, first of all, in this case? Can anyone explain this? Furthermore, why is $\textbf{rp}B$ interest charges and why is $\dot{\textbf{p}}B$ capital loss?

Also, suppose that we simplify things to $\textbf{p} = \textbf{p}A - \dot{\textbf{p}}B = \textbf{p}D$, why does $-D^{-1}$ become Frobenius matrix?