Introduction to Real Business Cyclesihome.ust.hk/~dxie/OnlineMacro/lecture16.pdf · Lecture 16. Introduction to Real Business Cycle Theory (RBC) 1. Business Cycle Facts 2. Detrending

Lecture 16. Introduction to Real Business Cycle Theory (RBC)

1. Business Cycle Facts 2. Detrending Procedures 3. Basic Model (Long and Plosser 1983 JPE) 4. Stability of a system of difference equations 5. Log Linearize a system of equations 6. King and Rebelo (RCER 2000) 7. Farmer and Guo (JET 1994) 1. Business Cycle Facts: Comovement; Persistence; Recurrent but not period; Most

variables are procyclical except unemployment (countercyclical) and real interest rate (acyclical)

2. Leading variables: stock prices; residential investment etc. 3. Lagging variables: inflation; nominal interest rate 4. Detrending I: linear detrending (log linear) 5. Detrending II: Piecewise linear detrending (log linear) 6. Detrending III: Hodrick-Prescott filter (HP filter)

( )

( ) ( )

( ) ( )

1

2

1

1 2

1 1

2

1 22

1 1

1 2

min ( )

subject to:

( )

FOCs are:

0

NT

t

t N

T

t t

Y t

t N

T T T T

t t t t

t

t N t N

T T T T T

t t t t t t

t t

T

t

Y Y

Y Y Y Y

L Y Y Y Y Y Y

L

Y

µ

! µ

=

=

= "

+ "=

= = "

+ "= =

"

# $" " " %& '

( )# $= " " " " " "* +& ', -

.=

.

/

/

/ /

7. ! is a function of µ . Selecting either ! or µ is equivalent. Is '( ) 0 or '( ) 0?! µ ! µ" #

8. Think about the following: If µ = +! If 0µ = If 0 µ< < +! For quarterly data, ! is recommended to be set to 1600.

9. The idea of RBC is to model business fluctuation as responses of rational individuals to productivity shocks. The simple model of one-sector Long and Plosser (1983 JPE) demonstrate this possibility.

10. In Long and Plosser, the representative agent is assumed to have log utility function:

[ ]0

ln (1 ) lnt t tE c l

!! !

!

" # #$

+ +

=

+ %&

11. The resource constraint is given by, 1

1t t t t tK z K n c

! !

" " " " "

#

+ + + + + += # In other words, 100 percent depreciation is assumed.

12. With these assumptions, the model has an explicit solution. To proceed, write down the Lagrangian and derive the first order conditions,

( )1

1

0

ln (1 ) lnt t t t t t t t t

L E c l z K n c K! " "

! ! ! ! ! ! ! !!

# $ $ %&

'+ + + + + + + + +

=

( )* += + ' + ' ', -. /0 12

13. FOCs are (for all ! ):

( )1 1

1 1 1 1

0

1(1 ) 0

1

0

t t

t

t t t t t

t

t t t t t t

Ec

E z K nn

E z K n

!

!

" "! ! ! !

!

" "! ! ! ! !

#$

#$ "

$ %$ "

+

+

&+ + + +

+

& &+ + + + + + + + +

' (& =) *

+ ,

' (&& & =) *

&+ ,

& =

14. Thus, the same has to be true for 0! = , which yields:

( )1 1

1 1 1 1

1(1 )

1

t

t

t t t t

t

t t t t t t

c

z K nn

E z K n

! !

! !

"#

"# !

# !$ #

%

% %

+ + + +

=

%= %

%

=

and 1

1t t t t tK z K n c

! !"

+ = " plus the transversality condition:

1TVC: lim 0

t t tE K

!

! !!

" # + + +$%

=

15. The solution method is the same as we used for the case without uncertainty, namely, guess and check:

1

1

1

1

1 1

(1 )

1

(1 )

1 and 1

t t t t

t t t t

t

t

t t t

t

t t

c az K n

K a z K n

YE

c c K

EaY a a Y

a a

! !

! !

" "!#

" "!#

!# !#

$

$+

+

+ +

=

= $

% &= ' (

) *

% &= ' (

$) *

$ = = $

16. Need to check the labor-leisure equation is also satisfied: 1

(1 )1

(1 )

(1 )

Thus, constant:

(1 )

(1 ) (1 )(1 )

t

t

t t

t

t t

t

t

Y

n n

Y

c n

an

n n

n

!" #

!#

! #

! #

! # ! #$

%= %

%

= %

%=

= =

%=

% + % %

17. Also need to verify the transversality condition. 18. Final result:

( )

1

1

1

2

1 1

1

ln constant + ln ln

Thus, if ln , , which is i.i.d, then,

ln (1) stationary

If ln is itself an (1), namely,

ln ln , is i.i.d.

Then ln is (2), so ar

t t t

t t t

t z

t

t

t t t t

t

K z K n

K K z

z

K AR

z AR

z z

K AR

! !!"

!

µ #

$ % %

&

+

+

+ +

+

=

= +

= +

e ln and lnt tc Y

19. This example shows that a simple general equilibrium model with productivity shocks may general business cycle phenomena.

1. Linear Difference Equations. 2. Simplest different equation:

1

0Solution:

t t

t

t

x x

x x

!

!

+ =

=

3. Slightly more difficult one: 1t t

x x b!+ = + 4. Find the steady state first (to economize on notation, denote the steady state by x)

, ( 1)1

bx x b x! !

!= + " = #

$

5. Let t

t

x xx

x

!=% denote the percentage deviation from the steady state, then

1 0

t

t t tx x x x! !+ = " =% % % %

6. Higher dimension case: 1 0

t

t t tx Ax x A x+ = ! =

7. But how to compute tA and its properties?

8. Look at easy case first. If 1

1

0

.

.

.

Then

.

.

.

n

t

t

t

n

A

x x

!

!

!

!

" #$ %$ %

= $ %$ %$ %$ %& '

" #$ %$ %$ %=$ %$ %$ %& '

9. A little bit more complicated case. If matrix A has n distinct eigenvalues. Let i!

be the thi eigenvalue. Namely,

[ ]det 0, 1,2,...iI A i n! " = =

10. Then there exists a matrix Q such that 1

1

.

.

.

n

Q AQ

!

!

"

# $% &% &

= % &% &% &% &' (

11. Redefine 1ˆt tx Q x

!= , then

1 1 1 1

1

1

1

1

1

0

1

.

ˆ ˆ.

.

.

ˆ = .

.

t t t

t t

n

t

t

n

Q x Q Ax Q AQQ x

x x

x

!

!

!

!

" " " "+

+

+

+

= =

# $% &% &

= % &% &% &% &' (

# $% &% &% &% &% &% &' (

12. Then we can obtain ˆt tx Qx=

13. Stability Issues. 14. Case 1.

1t tx x!+ =% %

0

0

| | 1 unstable diverges to unless 0

| | 1 stable: converges to zero for any

| | 1 periodic cycles

t

t

x x

x x

!

!

!

> ±" =

<

=

% %

% %

15. Case 2. 1t t

x Ax+ =% % 16. Then: 17. Case of 1 2 33 and | | 1,| | 1,| | 1n ! ! != > < < . Saddle point stable. Need one linear

constraint on 0x% .

18. What if 1 2 33 and | | 1,| | 1,| | 1n ! ! != > > < ? 19. Suppose that 1 22, and | | 1 and | | 1n ! != < < , then what happens? (Draw

diagrams)

How to log linearize an equation

1. Again define (thus, (1 ) )t

t t t

x xx x x x

x

!= = +% %

2. Why is it called log linearize? 3. If

t t ty x z= , then

t t ty x z= +% % %

4. If t ty x

!= , then

t ty x!=% %

5. If '( )( ), then

( )t t t t

f x xy f x y x

f x

! "= = # $

% &% %

6. If , then t t t t t t

x zy x z y x z

y y= + = +% % %

7. Here is a practice question: how to log linearize 1t t t

k Ak c!

+ = " ?