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Constructing and understanding stock-flow model

  1. Dec 15, 2012 #1
    Suppose that [itex]\textbf{x} = A\textbf{x} + B\dot{\textbf{x}}[/itex] where [itex]\textbf{x}[/itex] is vector of economic output level, [itex]A[/itex] is input-output matrix, [itex]B[/itex] is stock-flow matrix. The system represents closed and dynamic input-output system. [itex]\dot{\textbf{x}}[/itex] is time derivative of [itex]\textbf{x}[/itex]. Let [itex]C = B^{-1}(I-A)[/itex]. Then the system would be represented as [itex]\dot{\textbf{x}} = C\textbf{x}[/itex]. Then, let [itex]\textbf{p}[/itex] be a vector of price level. The equation would be written as the following: [itex]\textbf{p} = (1+\pi)(\textbf{p}A + \textbf{rp}B - \dot{\textbf{p}}B + wa_0)[/itex] where [itex]w[/itex] refers to wage rate and [itex]a_0[/itex] refers to a vector of labor requirement, [itex]\textbf{r}[/itex] refers to the interest rate, and [itex]\dot{\textbf{p}}[/itex] is first-time derivative of [itex]\textbf{p}[/itex]. In words, current cost: [itex]\textbf{p}A[/itex], interest charges: [itex]\textbf{rp}B[/itex], capital loss: [itex]\dot{\textbf{p}}B[/itex], wage costs: [itex]wa_0[/itex], uniform profit rate: [itex]\pi[/itex].

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

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

    Anyone wondering what I am referring to: https://docs.google.com/open?id=0B9NuS-9ksJO4ZFZSakhxRmE4T0k https://docs.google.com/open?id=0B9NuS-9ksJO4bE94U1VVUWV4Zjg https://docs.google.com/open?id=0B9NuS-9ksJO4SlV2cy0zYTFTbU0 https://docs.google.com/open?id=0B9NuS-9ksJO4QzhrRDJqbHBPMVk https://docs.google.com/open?id=0B9NuS-9ksJO4d284Snhnejd3aWM
  2. jcsd
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