Taking a model to the computer

1
Taking a modelto the computer

• Martin Ellison
• University of Warwick and CEPR
• Bank of England, December 2005

2
Baseline DSGE model
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
(1)
Steady-state values (more later)
(2)
(1) (2)
8
Deviations from steady state
?

• What is

9
Log-linearised IS curve
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 equation Log-linearised version

13
Optimal price setting
Original equation Log-linearised version

14
Myopic price
Original equation Log-linearised version

15
Marginal cost
Original equation Log-linearised version

16
Wages
Original equation Log-linearised version

17
Monetary authority
We assumed
Equivalent to
Very similar to linear rule if it small
18
Log-linearised DSGE model
19
Steady state

• Need to return to original equations to calculate
  steady-state

Assume for monetary authority
From household
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