Taking a modelto the computer

Martin Ellison

University of Warwick and CEPR

Bank of England, December 2005

Firms

Baseline DSGE model

Households

Monetaryauthority

ttt vi ˆˆ

11

1)1

(')('t

tt

itCUE

tCU

)ˆˆ(

)1)(1(ˆ 1

tttt Ex

Households

Two simplifying assumptions:

CRRA utility function

1)(

1t

t

CCU tt CCU )('

tt YC No capital

Dynamic IS curve

11 1

1

t

tttt

iYEY

Non-linear relationship

Difficult for the computer to handle

We need a simpler expression

Log-linear approximation

Begin by taking logarithms of dynamic IS curve

11 1

1lnlnln

t

tttt

iYEY

Problem is last term on right hand side

Properties of logarithms

...432

)1ln(432

zzz

zz 11 z

)1ln()()1(ln zEzEzE

11

11 1

1ln

1

1ln

t

ttt

t

ttt

iYE

iYE

Taylor series expansion of logarithmic function

To a first order (linear) approximation

Applied to dynamic IS curve

Log-linearisation

)1ln()1ln(lnlnln 11 tttttt EiYEY

ln)1ln(lnlnln iYY

))1(ln()1ln()ln(ln)ln(ln 11 tttttt EiYYEYY

Log-linear expansion of dynamic IS curve

Steady-state values (more later)

(1) – (2)

(1)

(2)

Deviations from steady state

What is YYt lnln ?

tttt

t ZZ

ZZ

Z

Z

Z

ZZZ ˆ1lnlnln

In case of output, is output gap, YYt lnln tx̂

percentage deviation of Zt from steady state Z

Log-linearised IS curve

)ˆˆ(ˆˆ 11

1

tttttt EixEx 1ˆˆ ttt Ei

1ˆ tt xE

tx̂

Slope = -σ

Advanced log-linearisation

The dynamic IS curve was relatively easy to log-linearise

For more complicated equations, need to apply following formula

)(),(

),()(

),(

),(),(ln),(ln ,, yy

yxf

yxfxx

yxf

yxfyxfyxf yxyyxx

Firms

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

Aggregate price level

Original equation Log-linearised version

11

11 )1( titt ppp 1ˆˆ)1(ˆ titt ppp

Optimal price setting

Original equation Log-linearised version

0

1

1

0

*1

1

i t

itit

iit

i t

itit

iit

t

it

PP

CE

PP

pCE

P

P it

1* ˆˆ)1(ˆ itttit pEpp

Myopic price

Original equation Log-linearised version

ttt mcpp

1*

ttt cmpp ˆˆˆ *

Marginal cost

Original equation Log-linearised version

tt

t mcP

W

tt wcm ˆˆ

Wages

Original equation Log-linearised version

tt

t

t

t

NC

C

N

P

W

t

tt xw ˆ1

ˆ

Monetary authority

ttt vi ˆˆWe assumed

ttt vii lnlnlnln 1Equivalent to

Very similar to linear rule if it small

Firms

Log-linearised DSGE model

Households

Monetaryauthority

ttt vi ˆˆ

)ˆˆ(ˆˆ 11

1

tttttt EixEx )ˆˆ()1)(1(

ˆ 1

tttt Ex

Assume for monetary authority

From household

Steady state

Need to return to original equations to calculate steady-state

01

i

0

1

1

iYY

Steady state calculation

From firm

1

*

1

1

Y

P

Wmc

PPP i

Full DSGE model

)1)(1(

ˆˆ

ˆˆˆ

)ˆˆ(ˆˆ

1

11

1

ttt

tttt

tttttt

vi

xE

EixEx

Alternative representation

tttt

ttttttt

xE

vxExE

ˆˆˆ

ˆˆˆˆ

1

111

11

tt

t

tt

tt vx

E

xE

0ˆ

ˆ

1

1ˆ

ˆ

0

1 11

1

11

State-space form

10110 tttt vBXAXEA

Generalised state-space form

Models of this form (generalised linear rational expectations models) can be solved relatively easily by computer

Next steps

Derive a solution for log-linearised models

Blanchard-Kahn technique