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I have understood the point with the Blanchard Kahn conditio

Paper help Economics I have understood the point with the Blanchard Kahn conditio

Economics

I have understood the point with the Blanchard Kahn conditio

I have understood the point with the Blanchard Kahn condition, my problem is to find the explicit so… Show more 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. The original problem is as follows: (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}) 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 (Lambda) be the diagonal matrix of A, we then have . (A=QLambda Q^{-1}) (begin{pmatrix} beta & gamma/lambda \ 0 & 1/gamma \ end{pmatrix} = begin{pmatrix} 1 & 1 \ 0 & (1-lambdabeta)/gamma \ end{pmatrix} begin{pmatrix} beta & 0 \ 0 & 1/lambda \ end{pmatrix} begin{pmatrix} 1 & -gamma/(1-lambdabeta) \ 0 & gamma/(1-lambdabeta) \ end{pmatrix}) I know that beta is stable and 1/lamba is unstable, and there is one predetermined variable and one non-predetermined one, thus there exists one unique solution. I am then supposed to find this solution. I have followed the lecture notes where the lecturer starts by taking expectations such that: (begin{pmatrix} p_{t} \ m_{t} \ end{pmatrix} = A E_{t} begin{pmatrix} p_{t+1} \ m_{t+1} \ end{pmatrix}) Premultiply by Q^(-1) and define: (Q^{-1} begin{pmatrix} p_{t} \ m_{t} \ end{pmatrix} = begin{pmatrix} z_{t}^{1} \ z_{t}^{2} \ end{pmatrix}) We then get: (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}) So far so good. However, I do not manage to go from this to the solution which is given by: (p_{t} = frac{-gamma}{1-lambdabeta}m_{t}) Anyone can help?? • Show less

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