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Homework Help: Blanchard Kahn condition

  1. May 4, 2013 #1
    1. The problem statement, all variables and given/known data
    I have understood the point with the Blanchard Kahn condition, my problem is to find the explicit solution when I know there exists one unique solution to the problem. The problem comes from a DSGE model.

    2. Relevant equations
    [itex]\begin{pmatrix} p_{t} \\ m_{t} \\ \end{pmatrix} = \begin{pmatrix} \beta & \gamma/\lambda \\ 0 & 1/\gamma \\ \end{pmatrix} \begin{pmatrix} p_{t+1} \\ m_{t+1} \\ \end{pmatrix} + \begin{pmatrix} \gamma/\lambda & \beta \\ 1/\lambda & 0 \\ \end{pmatrix} \begin{pmatrix} u_{t+1} \\ w_{t+1} \\ \end{pmatrix} [/itex]

    I have found the diagonal matrix of the matrix in front of p(t+1) and m(t+1). Let Q be the matrix of eigenvectors from A and [itex]\Lambda[/itex] be the diagonal matrix of A, we then have [itex]A=Q\Lambda Q^{-1}[/itex]:

    [itex] \begin{pmatrix} \beta & \gamma/\lambda \\ 0 & 1/\gamma \\ \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & (1-\lambda\beta)/\gamma \\ \end{pmatrix} \begin{pmatrix} \beta & 0 \\ 0 & 1/\lambda \\ \end{pmatrix} \begin{pmatrix} 1 & -\gamma/(1-\lambda\beta) \\ 0 & \gamma/(1-\lambda\beta) \\ \end{pmatrix} [/itex].

    3. The attempt at a solution
    I have followed the lecture notes where the lecturer starts by taking expectations such that:

    [itex] \begin{pmatrix} p_{t} \\ m_{t} \\ \end{pmatrix} = A E_{t} \begin{pmatrix} p_{t+1} \\ m_{t+1} \\ \end{pmatrix} [/itex]

    Then we premultiply by [itex]Q^{-1}[/itex] and define [itex] Q^{-1} \begin{pmatrix} p_{t} \\ m_{t} \\ \end{pmatrix} = \begin{pmatrix} z_{t}^{1} \\ z_{t}^{2} \\ \end{pmatrix} [/itex]. We then get:

    [itex] \begin{pmatrix} z_{t}^{1} \\ z_{t}^{2} \\ \end{pmatrix} = \begin{pmatrix} \beta & 0 \\ 0 & 1/\lambda \\ \end{pmatrix} E_{t} \begin{pmatrix} z_{t+1}^{1} \\ z_{t+1}^{2} \\ \end{pmatrix} [/itex]

    The stable eigenvalue in the diagonal matrix of A is [itex] \beta [/itex] and the unstable one [itex] 1/\lambda [/itex]. I have tried to follow theoretical papers I have found online but I just do not manage to do it.

    The solution is supposed to be:
    [itex] p_{t} = \frac{-\gamma}{1-\lambda\beta}m_{t} [/itex]
  2. jcsd
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