1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Blanchard Kahn condition
Loading...