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!

+ = " ?