1. Solution techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

2. State-space form Generalised state-space form Many techniques available to solve this class of models We use industry standard: Blanchard-Kahn

3. Alternative state-space form

4. Partitioning of model backward-looking variables predetermined variables forward-looking variables control variables

5. Jordan decomposition of A eigenvectorsdiagonal matrix of eigenvalues

6. Blanchard-Kahn condition The solution of the rational expectations model is unique if the number of unstable eigenvectors of the system is exactly equal to the number of forward-looking (control) variables. i.e., number of eigenvalues in Λ greater than 1 in magnitude must be equal to number of forward-looking variables

7. Too many stable roots multiple solutions equilibrium path not unique need alternative techniques

8. Too many unstable roots no solution all paths are explosive transversality conditions violated

9. Blanchard-Kahn satisfied one solution equilibrium path is unique system has saddle path stability

10. Rearrangement of Jordan form

11. Partition of model stable unstable

12. Transformed problem

13. Decoupled equations Decoupled equations can be solved separately stable unstable

14. Solution strategy Solve unstable transformed equation Translate back into original problem Solve stable transformed equation

15. Solution of unstable equation As, only stable solution is Solve unstable equation forward to time t+j Forward-looking (control) variables are function of backward-looking (predetermined) variables

16. Solution of stable equation Solve stable equation forward to time t+j As, no problems with instability

17. Solution of stable equation Future backward-looking (predetermined) variables are function of current backward- looking (predetermined) variables

18. Full solution All variables are function of backward-looking (predetermined) variables: recursive structure

19. Baseline DSGE model State space form To make model more interesting, assume policy shocks v t follow an AR(1) process

20. New state-space form One backward-looking variable Two forward-looking variables

21. Blanchard-Khan conditions Require one stable root and two unstable roots Partition model according to

22. Next steps Exercise to check Blanchard-Kahn conditions numerically in MATLAB Numerical solution of model Simulation techniques