Stationary Rational Bubbles in Non-Linear Business Cycle Models - - PowerPoint PPT Presentation

SLIDE 1

Stationary Rational Bubbles in Non-Linear Business Cycle Models

Robert Kollmann Université Libre de Bruxelles & CEPR

June 17, 2019

SLIDE 2

Main result:

non-linear DSGE models have more stationary equilibria than you think! Blanchard & Kahn (1980): conditions for existence of unique stable solution of linear(ized) models are IRRELEVANT for non- linear models This paper shows: standard non-linear DSGE models have MULTIPLE stable equilibria, even when the linearized versions

• f these models have unique solution

⇒ DSGE models may have multiple sunspot equilibria, if non-linearity taken into account

SLIDE 3

Sunspot equilibria look like ‘bubbles’:

Economy temporarily diverges from no-sunspots trajectory, before reverting abruptly towards no-sunspots trajectory Key ingredient: heteroskedastic sunspots The further the system has diverged from ‘fundamental’ (stable) solution, the bigger the subsequent ‘correction’ ⇒ Conditional variance of future state is greater, the farther system has deviated from ‘fundamental’ solution. Heteroskedasticity stabilizes the system!

SLIDE 4

• “Divergent behavior”: similar to ‘rational’

bubbles in linear models (Blanchard (1979))

• Big difference compared to Blanchard

bubbles: bubbles here are STATIONARY.

SLIDE 5

1

,

t t t

E y y λ

+ = ⋅

1 λ> ;

t

y : scalar jump variable

Unique stable solution:

t

y =

Blanchard (1979), Blanchard & Watson (1982)

Bubble:

1 ( /(1

))

t t

y y λ π

+ =

− ⋅

with probability 1 π

−

1 0 t

y + = with probability π lims

t t s

E y

→∞ + = ±∞ if t

y ≠

expected path of bubble diverges to ±∞ Big influence on financial economics BUT little influence on structural (DSGE) macro Expected path of bubbles in non-linear DSGE described here do NOT diverge to ±∞

SLIDE 6

Note: Can construct DSGE models whose

linearized versions have stable sunspots:

1 t t t

E y y λ

+ = ⋅ need | | 1

λ≤ . ⇒

1 1 t t t

y y λ ε

+ +

= ⋅ +

is stationary solution for any

1

{ }

t

ε +

with

1 0 t t

E ε + =

Needed ingredients:

• Increasing returns, externalities (e.g., Schmitt-Grohé (1997),

Benhabib and Farmer (1999))

• Financial frictions (e.g., Martin and Ventura (2018))
• Overlapping generations (e.g., Galí (2018))

Specific assumptions & calibrations that deliver | | 1 λ < can be debatable & fragile (e.g. in standard OLG model: need dynamic inefficiency, r≤g) By contrast, paper here argues that very standard DSGE models with | | 1

λ> can deliver stationary

sunspot equilibria, if non-linearities are considered.

SLIDE 7

Basic intuition I:

Consider non-linear model with just 1 non- predetermined variable (no exogenous driver)

1

( , )

t t t

E G Y Y

+

=

Linearization (around steady state) gives:

1 t t t

E y y λ

+ =

⋅ ,

SS t t

y Y Y ≡ − Linearized model has unique non-explosive solution iff | | 1 λ > . That unique solution is:

t

y = (Blanchard & Kahn (1980), Prop. 1)

SLIDE 8

I show: even when |

| 1 λ > , the non-linear model can have stationary sunspot equilibrium

1

( , )

t t t

E G Y Y

+

= ⇔

1 1

( , )

t t t

G Y Y ε

+ +

=

with

1 0 t t

E ε + =

⇒

1 1

( , )

t t t

Y Y ε

+ +

=Λ

.

1 t

ε +

: “sunspot shock”

Even if |

| 1,

Y

Λ >

there may exist process

1

{ }

t

ε + with

1 0 t t

E ε + =

such that

1

{ }

t

Y +

is stationary.

Note: when white noise

1

{ }

t

ε + is fed into

1 1

( , )

t t t

Y Y ε

+ +

=Λ , then

1

{ }

t

Y +

diverges if |

| 1.

Y

Λ >

SLIDE 9

Key requirements for stationary solution:

• 1

1

( , )

t t t

Y Y ε

+ +

=Λ has to be NON-LINEAR in

1 t

ε +

• Distribution of

1 t

ε +

has to depend on

t

Y

2 1 1 1 1 2

( ,0) ( ,0) ( ,0)( )

t t t t t t

Y Y Y Y

ε εε

ε ε

+ + +

≅Λ +Λ ⋅ + Λ ⋅

2 1 1 1 2

( ,0) ( ,0) ( )

t t t t t t

E Y Y Y E

εε

ε

+ +

≅Λ + Λ ⋅ Let

2 1

( ) ( ) 0.

t t t

E f Y ε + = ≥

If

( ,0)

t

Y

εε

Λ ≠

then can set

2 1

( ) ( )

t t t

E f Y ε + =

such that

1

| / | 1

t t t

dE Y dY

+

< : “MEAN REVERSION” Example: ( ,0) 1,

Y t

Y Λ > ( ,0) 0.

t

Y

εε

Λ <

Then need

'( ) 0

t

f Y >

for mean reversion:

2 1

( )

t t

E ε +

must be

increasing in .

t

Y

SLIDE 10

Basic intuition II: RBC model

1

;

t t t

C K Y

+

+ = ( ),

t t

Y F K = ' 0, F > '' 0 F <

1 1

{ [ '( )]/ '( )} '( ) 1

t t t t

E u C u C F K β

+ +

⋅ = ; assume ''' 0

u > (CRRA)

Sunspot: assume

1 t

K + ↑

t

C ⇒ ↓ '( ) ,

t

u C ↑

1

'( )

t

F K + ↓ Euler eqn requires:

1 1 2

'( ) '( ( ) )

t t t t t

E u C E u F K K

+ + +

= −

↑

• In deterministic economy: need

1 t

C + ↓ &

2 t

K +

↑

2 t

K + has to rise more than

1 t

K + ! ⇒ K diverges

• With stochastic sunspot:

2 t

K +

random.

1

'( )

t

u C +

is convex in

2 t

K +

⇒ if

2

( )

t t

Var K + rises,

1

'( )

t t

E u C +

↑ ⇒

2 t t

E K +

can rise less than

1 t

K +

!

⇒ possibility of mean reversion

SLIDE 11

Bursting bubble: sudden, unexpected drop

• f K towards ‘fundamental’ (no-sunspots)

level. The further K has diverged from ‘no-sunspot path’, the sharper the ‘correction’

1 t

K + ↑ ⇒

2

( )

t t

Var K + ↑

Heteroskedasticity can stabilize the system!

SLIDE 12

Several of the models considered below are usually presented

as outcome of decision problem of an infinitely-lived representative agent.

Bubbles violate the transversality condition (TVC) of infinitely lived household:

1

lim '( )

t t t

E u C K

τ τ τ τ

β

→∞ + + + >

This paper disregards TVC: 1) Lansing (2010) disregards the TVC in a Lucas- style asset pricing models with bubbles, arguing that “agents are forward-looking but not to the extreme degree implied by the transversality condition”

SLIDE 13

2) In richer models with heterogeneous agents and

distortions: equilibrium is not solution of decision problem of representative agent. Detection of TVC violations in stochastic economies: virtually impossible, even with very long simulation runs (billions of periods): States with very low consumption might only occur with extremely small probabilities. 3) Assume OLG population structure with agents who live N<∞ periods: then the TVC (infinite horizon) does not hold

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Novel result about OLG economy:

(i) if there is a complete financial market that allows all generations alive at both dates t and t+1 (ii) if the generation born at date t receives a wealth endowment that is a constant share 1/N of aggregate date t wealth (across all generations) THEN an ‘aggregate’ Euler equation holds that is identical to the Euler equation of a representative infinitely lived household:

1 1

{ '( )/ '( )} 1

t t t t

E u C u C MPK β

+ + =

BUT: there is no TVC in the OLG economy!

SLIDE 15

OLG structure with efficient

intergenerational risk sharing: justification for considering macro models that lack a TVC, but whose other equilibrium conditions are identical to those of standard business cycle models (that assume infinitely lived agents)

SLIDE 16

Detailed Example I:

Long-Plosser RBC model with sunspots ( ) ln( ) u C C = ;

1

;

t t t

C K Y

+

+ = ( ) ( )

t t t

Y F K K

α

= ≡ , 0 1 α < < Euler equation:

1 1

{ '( )/ '( )} '( ) 1

t t t t

E u C u C F K β

+ +

⋅ = ⇒

1 1 1

{ / } / 1

t t t t t

E C C Y K β α

+ + +

⋅ = ⇒

1 1 2 1 1

{( )/( )} / 1

t t t t t t t

E Y K Y K Y K β α

+ + + + +

− − ⋅ = ⇒

1 2 1 1

{(1 / )/(1 / )} / 1

t t t t t t t

E K Y K Y Y K αβ

+ + + +

⋅ − − ⋅ = ⇒

1

{(1 )/(1 )}/ 1

t t t t

E Z Z Z αβ

+

⋅ − − = ,

1/ t t t

Z K Y

+

≡

: investment/output ratio Textbook solution:

t

Z αβ =

SLIDE 17

1

{(1 )/(1 )}/ 1

t t t t

E Z Z Z αβ

+

⋅ − − =

Linearization around Z αβ

=

:

1

,

t t t

E z z λ

+ = ⋅

;

t t

z Z Z ≡ − 1/( ) 1 λ αβ ≡ > .

⇒

t

z =

is unique non-explosive solution of

linearized model. But: non-linear model has other stationary solutions.

1 1

{(1 )/(1 )}/ 1

t t t t

Z Z Z αβ ε

+ +

⋅ − − = +

,

1 0 t t

E ε + =

⇒

1 1 1

( , ) 1 (1/ 1)/(1 ).

t t t t t

Z Z Z ε αβ ε

+ + +

= Λ ≡ − − +

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Fig.1. Long & Plosser model: investment/output ratio at t+1,

1, t

Z + as function of

t

Z for

1

{ 0.5;0;0.5}

t

ε + ∈ −

1 1 1

( , ) 1 (1/ 1)/(1 )

t t t t t

Z Z Z ε αβ ε

+ + +

=Λ ≡ − − +

;

0.35, 0.99. α β = =

SLIDE 19

1 1 1

( , ) 1 (1/ 1)/(1 )

t t t t t

Z Z Z ε αβ ε

+ + +

=Λ ≡ − − +

• When

t

Z αβ < , the model can hit zero-capital corner solution in later periods ⇒ restrict attention to solutions with [ ,1) Zτ αβ τ ∈ ∀

• Support of

1 t

ε + has to be bounded below:

1

1 [ /(1 )] [1/ 1]

t t

Z ε αβ αβ

+ ≥ − +

− ⋅ − ⇒ distribution of

1 t

ε + must depend on

t

Z !

• Let

1 t

ε + only takes two values:

t

ε − and /(1 )

t t t

ε π π ⋅ − with probabilities

t

π and 1 ,

t

π − respectively, [0,1)

t

ε ∈

⇒

1 t

Z +

takes two values:

1

( , )

L t t t

Z Z ε

+ ≡Λ

− &

1

( , /(1 ))

H t t t t t

Z Z ε π π

+ ≡Λ

− with

1 1 1. L H t t

Z Z

+ +

≤ ≤

• Postulate

1

( )

L t t

Z f Z

+ =

, with ( ) ( ,0)

t t

f Z Z αβ≤ ≤Λ

for

[ ,1)

t

Z αβ ∈ . Solve

1

( , )

L t t t

Z Z ε

+ ≡Λ

− for

t

ε

& substitute into

1

( , /(1 ))

H t t t t t

Z Z ε π π

+ ≡Λ

−

SLIDE 20

Two degrees of freedom in modeling sunspot:

• bust investment/GDP ratio,

1 L t

Z +

• conditional probability of bust,

t

π

SLIDE 21

Specification I:

1

,

L t

Z αβ

+ =

+Δ 0.01, Δ= 0.5 π=

(When Δ= , then

0.346 Z αβ = =

is absorbing state; thus set

0) Δ>

SLIDE 22

Simulated series with constant probability:

0.5

t

π =

Simulated output (Y), consumption (C) and investment (I) normalized by steady state output

SLIDE 23

Lower volatility if probability of investment bust

rises once investment/output ration

t

Z crosses

threshold.

Simulated series with state-contingent probability of bust:

0.5

t

π =

for

0.356 0.36

t

Z αβ+Δ = ≤ ≤

&

100

1 10

t

π

−

= −

for

0.36

t

Z >

SLIDE 24

Specification II: gradual contractions

1

0.95( ),

L t t

Z Z αβ αβ

+ =

+ ⋅ − 0.5 π=

SLIDE 25

Simulated series with constant probability of bust:

0.5

t

π =

Simulated series with state- contingent probability of bust:

0.5

t

π = for 0.356 0.36

t

Z αβ+Δ= ≤ ≤ & 1

t

π for 0.36

t

Z >

SLIDE 26

Table 1. Long-Plosser model with bubbles: predicted business cycle statistics Standard dev. % Corr. with Y Autocorr. Mean (% deviation from SS) Y C I C I Y C I Y C I Z

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

(a) Specification I: Zt

L =αβ+∆

πt=0.5 11.72 100.19 33.48

• 0.42

0.62 0.62 0.47 0.62 13.49 -7.62 53.31 31.15 πt≅1 for zt>0.36 1.33 3.51 3.82 0.77

• 0.26
• 0.26 -0.66 -0.26

3.27 -0.13 9.71 6.25 (b) Specification II: Zt

L =αβ+0.95×(zt –αβ)

πt=0.5 1.73 210.28 4.94

• 0.31

0.68 0.68 0.46 0.68 73.09 -89.68 380.11 177.21 πt≅1 for zt>0.36 1.40 1.30 4.00 0.14 0.85 0.85 0.28 0.85 4.45 -0.26 13.34 8.46 (c) US Data (from King and Rebelo (1999)) 1.81 1.35 5.30 0.88 0.80 0.88 0.80 0.87

SLIDE 27

Example II: RBC model with incomplete capital

depreciation & endogenous labor

( , )

s t t t

Max E u C L β

∞ =

∑

subject to resource constraint

t t t

C I Y + = with

1 (1

)

t t t

I K K δ

+

= − − and

1

( ) ( )

t t t

Y K L

α α −

= .

1 1

( , ) (1 ) ( ( )) ,

t t t t

u C L C L

σ

σ ν

− −

= − −

1 1/ 1 1/

( ) ( /(1 1/ )){ },

t t

L L L

η η

ν η

+ +

= Ψ + −

1/

(1 ) / ( )

t t t

Y L L

η

α − =Ψ⋅

⇒

1/( 1/ )

( ) ((1 )( )/ )

t t t

L n K K

α α η

α

+

= ≡ − Ψ

1 1 1 1

{( ( )) /( ( )) }( / 1 ) 1

t t t t t t t

E C L C L Y K

σ σ

β ν ν α δ

− − + + + +

− − + − =

1 1 1 1 1

{( ( )) /( ( )) }( / 1 ) 1 ,

t t t t t t t

C L C L Y K

σ σ

β ν ν α δ ε

− − + + + + +

− − + − = +

1 t

ε +

: sunspot with

1 0 t t

E ε + =

⇒

2 1 1

( , , )

t t t t

K K K κ ε

+ + +

=

SLIDE 28

Intuition about stationary sunspot equilibrium:

• 2

1

( )

L t t

K K λ

+ +

=

: stationary no-sunspot decision rule

• Gap between K with & without sunspot:

1 2 1

( ).

t t t

g K K λ

+ + +

≡ − In linearized system: gap explodes

1 1 3 1

( ') ;

t t t

g g κ λ κ ε

+ +

= − ⋅ + ⋅

1

( ') 1 κ λ − >

In non-linear system: gap can be stationary Non-sunspot solution is ‘attractor’

SLIDE 29

2 1 1

( , , )

t t t t

K K K κ ε

+ + +

=

: solution with sunspots

■ Assume

1 t

ε + takes only two values,

t

ε −

and

/(1 )

t

ε π π −

, with probabilities π & 1

; π −

[0,1].

t

ε ∈

• ‘Bust’:

2 1

( )

L t t

K K λ

+ +

=

2 1

( , , )

L t t t t

K K K κ ε

+ +

= −

⇒ can solve this for

t

ε

• This pins down ‘Boom’ capital stock:

2 1

( , , /(1 ))

H t t t t

K K K κ ε π π

+ +

= −

Parameters: 0.99, 0.35, 0.025 β α δ = = =

(depreciation rate)

1 σ η = = (risk aversion, labor supply elasticity

0.5 π =

(bust probability)

SLIDE 30

Non-linear RBC model (incomplete capital depreciation, variable labor) with bubbles. Simulated paths of GDP (Y), consumption (C), investment (I) are normalized by steady state GDP. Capital (K) and hours series (L) are normalized by their respective steady states.

SLIDE 31

Table 2. RBC model (incomplete capital depreciation) with bubbles: predicted business cycle statistics Standard dev. % Corr. with Y

• Autocorr. Mean (% deviation from SS)

Y C I L C I L Y C I L Y C I L K

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

πt=0.5 0.46 9.67 11.12 0.23 0.13 -0.21 1.00 0.91 0.54 0.52 0.91 18.6 11.8 39.0 8.90 39.0

SLIDE 32

Example III: Small Open Economy

ln( )

s t t

Max E C β

∞ =

∑

s.t.

1 t t t t

C A A R Y

+

+ = ⋅ +

:

t

C consumption; : Y output

1: t

A +

net foreign assets, NFA;

:

t

R gross interest rate;

• Euler equation:

1 1 1

( / ) 1

t t t t

E C C R β

− + + =

• ( ),

t t

R R A =

with '

R <

■ No-sunspots solution: policy function

1

( )

t t

A A λ

+ =

SLIDE 33

■ How to construct stationary sunspot equilibrium:

• Euler equation ⇒

1 1 1 1

( / ) ( ) 1 ;

t t t t

C C R A β ε

− + + +

= +

1 t

ε + : sunspot with

1 0. t t

E ε + =

⇒

1 1 1

( )/(1 )

t t t t

C C R A β ε

+ + +

= ⋅ ⋅ +

• Budget constraint:

1

( )

t t t t

C A R A Y A+ = ⋅ + −

• Substitute budget constraint into Euler equation:

⇒

1/ 2 1 1 1 1

( ) ( ( ) ) { ( )/(1 )}

t t t t t t t t

A A R A Y A R A Y A R A

σ

β ε

+ + + + +

= + − + − ⋅ +

2 1 1

( , , )

t t t t

A A A κ ε

+ + +

=

: NFA law of motion with sunspot

0,

ε

κ >

εε

κ > : non-linearity permits stationary sunspot equil.

SLIDE 34

2 1 1

( , , )

t t t t

A A A κ ε

+ + +

=

: NFA law of motion with sunspot

2 1

( ):

t t

A A λ

+ +

= stationary no-sunspots decision rule ■ Structure of stationary sunspot equilibrium:

• Gap between NFA with & without sunspot:

1 2 1

( ).

t t t

g A A λ

+ + +

≡ −

In linearized system: gap explodes

1 1 3 1

( ') ;

t t t

g g κ λ κ ε

+ +

= − ⋅ + ⋅

1

( ') 1 κ λ − >

In non-linear model: gap can be stationary

SLIDE 35

2 1 1

( , , )

t t t t

A A A κ ε

+ + +

=

: NFA law of motion with sunspot

►Assume

1 t

ε + takes only two values,

t

ε −

and

/(1 )

t t t

ε π π −

, with probabilities

t

π & 1

;

t

π −

[0,1].

t

ε ∈

• ‘Bust’:

2 1

( ) .

L t t

A A λ

+ +

= +Δ

No-sunspot decision rule.

Δ>

2 1

( , , )

L t t t t

A A A κ ε

+ +

= −

: this pins down

t

ε

• ‘Boom’:

2 1

( , , /(1 ))

H t t t t

A A A κ ε π π

+ +

= −

Parameters: 0.99; β=

( ) exp(

/ )/

t t

R A a A Y β = − ⋅

,

0.01. a =

0.5

t

π =

(bust probability)

SLIDE 36

(a) Constant bust probability:

0.5

t

π =

(b) State-contingent bust probability:

0.5

t

π =

for

1/

0.25;

t

A Y

+

≤ 1

t

π for

1/

0.25

t

A Y

+

>

Non-linear Small Open Economy model with bubbles

Simulated paths of net foreign assets (A), GDP (Y), consumption (C) and net exports (NX). All series normalized by steady state GDP.

SLIDE 37

Table 3. πt=0.5 for

1/ t

A y

+

>

Small Ope St A

(1)

11.4

0.25 >

2.0 en Economy tandard de C

(2)

43 9.07 07 1.30 y model (en

• ev. %

NX

(3)

4.95 1.27 ndowments)

• Corr. wi

C N

(4)

• ) with bub

th Y NX

(5)

• bles: predi

Autocorr A C

(6) (7)

0.90 0.53 0.81 0.26 icted busine

• r. M

NX

(8) (

0.52 24 0.26 16 ess cycle sta Mean (% de A Y

(9) (10)

4.44 0.00 6.70 0.00 atistics eviation fro C

(11)

0.09

• 0.13
• m SS)

NX

(12)

• 0.09
• 0.13

SLIDE 38

CONCLUSIONS

• Stationary sunspot equilibria exist in standard

non-linear DSGE models, even when the linearized versions of those models have unique solutions.

• In the sunspot equilibria considered here, the

economy temporarily diverges from the no- sunspots trajectory, before abruptly reverting towards that trajectory.

• In contrast to rational bubbles in linear models

(Blanchard (1979)), the bubbles considered here are stationary--their expected path does not explode to infinity.

SLIDE 39

ADDITIONAL MATERIAL

Blanchard (1979):

1

,

t t t

E y y λ

+ = ⋅

1 λ> ⇒

1 1, t t t

y y λ ε

+ +

= ⋅ +

1 0 t t

E ε + =

• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

How non-linearity may generate stationary bubble: Assume:

1

exp( )

t t t

E z z a λ

+ −

= ,

1, a λ > >

1 1

exp( )

t t t

z z a λ η

+ +

= ⇒ − +

with

1 0 t t

Eη + =

1 1

log( ).

t t t

z z a λ η

+ +

= + ⇒ +

Let

1 1

ln( )/( 1), /

t t t t

y z a a λ ε η

+ +

≡ + − ≡

⇒

1 1

ln(1 ),

t t t

y y λ ε

+ +

= ⋅ + +

1 0 t t

E ε + =

1 t

y + is concave in

1 t

ε +

⇒

1 t t t

E y y λ

+ <

⋅ Let

1

{ ; /(1 )}

t t t

ε ε ε π π

+ ∈ −

−

with prob. , 1

. π π −

t

ε >

Set

[0,1)

t

ε ∈

so that

1

ln(1 )

t t t

y y λ ε

+ = ⋅ +

− = Δ <

1 1

ln{ 1 [1 exp( )] /(1 )}

H t t t t

y y y y λ λ π π

+ +

= ≡ ⋅ + + − Δ− ⋅ ⋅ −

with prob. 1 π

−

1 t

y + = Δ with probability π