1. Taking a model to the computer Martin Ellison University of Warwick and CEPR Bank of England, December 2005

2. Firms Baseline DSGE model HouseholdsMonetary authority

3. Households Two simplifying assumptions: CRRA utility function No capital

4. Dynamic IS curve Non-linear relationship Difficult for the computer to handle We need a simpler expression

5. Log-linear approximation Begin by taking logarithms of dynamic IS curve Problem is last term on right hand side

6. Properties of logarithms Taylor series expansion of logarithmic function To a first order (linear) approximation Applied to dynamic IS curve

7. Log-linearisation Log-linear expansion of dynamic IS curve Steady-state values (more later) (1) – (2) (1) (2)

8. Deviations from steady state What is ? In case of output, is output gap, percentage deviation of Z t from steady state Z

9. Log-linearised IS curve Slope = -σ

10. Advanced log-linearisation The dynamic IS curve was relatively easy to log-linearise For more complicated equations, need to apply following formula

11. Firms Previously solved for firm behaviour directly in log-linearised form. Original model is in Walsh (chapter 5).

12. Aggregate price level Original equationLog-linearised version

13. Optimal price setting Original equationLog-linearised version

14. Myopic price Original equationLog-linearised version

15. Marginal cost Original equationLog-linearised version

16. Wages Original equationLog-linearised version

17. Monetary authority We assumed Equivalent to Very similar to linear rule if i t small

18. Firms Log-linearised DSGE model HouseholdsMonetary authority

19. Assume for monetary authority From household Steady state Need to return to original equations to calculate steady-state

20. Steady state calculation From firm

21. Full DSGE model

22. Alternative representation

23. State-space form Generalised state-space form Models of this form (generalised linear rational expectations models) can be solved relatively easily by computer

24. Next steps Derive a solution for log-linearised models Blanchard-Kahn technique