Solution techniques

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Solution techniques

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

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State-space form
Generalised state-space form
Many techniques available to solve this class of
models We use industry standard Blanchard-Kahn
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Alternative state-space form
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Partitioning of model
backward-looking variables predetermined variables
forward-looking variables control variables
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Jordan decomposition of A
eigenvectors
diagonal matrix of eigenvalues
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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 ? greater than 1
in magnitude must be equal to number of
forward-looking variables
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Too many stable roots
multiple solutions equilibrium path not
unique need alternative techniques
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Too many unstable roots
no solution all paths are explosive transversali
ty conditions violated
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Blanchard-Kahn satisfied
one solution equilibrium path is
unique system has saddle path stability
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Rearrangement of Jordan form
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Partition of model
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Transformed problem
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Decoupled equations
stable unstable
Decoupled equations can be solved separately
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Solution strategy

• Solve unstable transformed equation

Solve stable transformed equation
Translate back into original problem
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Solution of unstable equation
Solve unstable equation forward to time tj
Forward-looking (control) variables are function
of backward-looking (predetermined) variables
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Solution of stable equation
Solve stable equation forward to time tj
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Solution of stable equation
Future backward-looking (predetermined) variables
are function of current backward-looking
(predetermined) variables
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Full solution
All variables are function of backward-looking
(predetermined) variables recursive structure
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Baseline DSGE model
State space form
To make model more interesting, assume policy
shocks vt follow an AR(1) process
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New state-space form
One backward-looking variable
Two forward-looking variables
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Blanchard-Khan conditions
Require one stable root and two unstable roots
Partition model according to
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Next steps

• Exercise to check Blanchard-Kahn conditions
  numerically in MATLAB
• Numerical solution of model
• Simulation techniques