Monetary Policy and Bubbles in a New Keynesian Model with Overlapping Generations

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Monetary Policy and Bubbles in a New Keynesian Model with Overlapping Generations

Jordi Galí

CREI, UPF and Barcelona GSE

August 2016

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Motivation

Asset price bubbles: present in the policy debate...

- key source of macro instability - monetary policy as cause and cure ...but absent in modern monetary models

- no room for bubbles in the New Keynesian model - no discussion of possible role of monetary policy

Present paper: modi…cation of the NK model to allow for bubbles Key ingredients:

(i) …nitely-lived consumers (Blanchard (1984), Yaari (1965)) (ii) stochastic retirement (Gertler (1996))

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Related Literature

Real models of rational bubbles: Tirole (1985),..., Martín-Ventura (2012) Monetary models with bubbles: Samuelson (1958),..., Asriyan et al.

(2016)

) ‡exible prices

New Keynesian models with overlapping-generations à la Blanchard-Yaari:

Piergallini (2018), Nisticò (2012), Del Negro et al. (2015) ) no discussion of bubbles

Monetary policy, sticky prices and bubbles:

- Bernanke and Gertler (1999,2001): ad-hoc bubble speci…cation - Galí (2014). Main di¤erences here:

- variable employment and output

- many-period, stochastic lifetimes (Blanchard-Yaari) - nests standard NK model as a limiting case

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A New Keynesian Model with Overlapping Generations

Survival probability: γ

Size of cohort born in periods: (1 γ)γ^{t s} Total population size: 1

Two types of individuals:

- "Active": manages own …rm, works for others.

- "Retired": consume …nancial wealth Probability of remaining active: υ

Labor force (and measure of …rms): α ₁¹ _υγ^γ 2(0,1]

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Consumers

Consumer’s problem:

maxE₀

∑

∞ t=0

(βγ)^tlogC_t_j_s 1

Pt

Z _α

0

P_t(i)C_t_j_s(i)di+E_tfΛt,t+1Z_t₊₁_j_sg=A_t_j_s+W_tN_t_j_s A_t_j_s =Z_t_j_s/γ

Optimality conditions:

C_t_j_s(i) = ¹ α

P_t(i) Pt

e

C_t_j_s Λt,t+1 = β

C_t_j_s C_t₊₁_j_s

Tlim!∞γ^TEt Λt,t+TA_t₊_T_j_s =0

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Firms (I)

Technology

Yt(i) =Γ^tNt(i) whereΓ 1+g

Calvo price setting: a fractionυγθ of …rms keeps prices unchanged Law of motion for the price level

p_t =υγθp_t ₁+ (1 υγθ)p_t Optimal price setting

p_t = µ+ (1 ΛΓυγθ)

∑

∞ k=0

(ΛΓυγθ)^kE_tf^p^t⁺^k+w_t₊_kg wherew_t log(W_t/Γ^t) andΛ ₁₊¹_r. Assumption: ΛΓυγθ<1.

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Firms (II)

Implied in‡ation equation

πt =ΛΓE_tf^π^t⁺¹g+λ(w_t w) whereλ ⁽¹ ^υγθ⁾⁽_υγθ¹ ^ΛΓυγθ⁾.

Remark: in the standard NK model,ΛΓ= β (i.e. r = (1+ρ)(1+g) 1' ^ρ+g).

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Asset Markets (I)

Nominally riskless bond 1

1+i_t =Et Λt,t+1

Pt

P_t+1

Valuation of individual stocks Q_t^F(i) =

∑

∞ k=0

(υγ)^kE_tf^Λ^t^,t⁺^k^D^t⁺^k(i)g whereDt(i) Yt(i) ^P_P^t⁽ⁱ⁾

t Wt

Aggregate stock market Q_t^F

Z _α

0

Q_t^F(i)di

=

∑

∞ k=0

(υγ)^kEtf^Λ^t,t⁺^k^D^t⁺^kg

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Asset Markets (II)

Bubbly asset

Q_t^B(j) =Etf^Λ^t,t⁺¹^Qt^B+1(j)g with Q_t^B(j) 0 for allt. Recursively:

Q_t^B(j) =EtfΛt,t+TQ_t^B₊_T(j)g forT =1,2,3, ...

Remark: in the standard NK model 0= lim

T!∞E_tf^Λ^t,t⁺^T^A^t⁺^Tg _T^lim

!∞E_tn

Λt,t+TQ_t^B₊_T(j)^o=Q_t^B(j) implyingQ_t^B(j) =0.

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Asset Markets (III)

Aggregate bubble:

Q_t^B =Ut +Q_t^B_j_t ₁ whereQ_t^B

j^{t k}

R

j2Bt kQ_t^B(j)dj Equilibrium condition:

Q_t^B =E_tfΛt,t+1Q_t^B₊₁

j^tg Financial wealth "at birth":

A_t_j_t =Q_t^F_j_t +Ut/(1 γ) Remark: in the absence of bubble creation

Q_t^B =EtfΛt,t+1Q_t^B₊₁g A_t_j_t =Q_t^F_j_t sinceU_t =0 and Q_t^B₊₁

j^t =Q_t^B for all t

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Labor Markets and Monetary Policy

Wage equation:

W^t = ^N^t α

ϕ

whereW^t ^W^t^/Γ^t andN_t Rα

0 N_t(i)di. Natural level of output

Y_tⁿ = _Γ^tαM ¹^ϕ Γ^tY New Keynesian Phillips curve

πt =ΛΓE_tf^π^t⁺¹g+κby_t whereκ λϕ, and by_t log(Y_t/Y_tⁿ).

Monetary Policy

bi_t =φ_ππt+φ_qbq_t^B wherebi_t log¹₁⁺₊ⁱ_r^t,q_t^B _Γ^Qt^B^tY

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Market Clearing

Goods market

Y_t(i) = (1 γ)

∑

t s= _∞

γ^{t s}C_t_j_s(i) for all i 2[0,α], implying

Y_t = (1 γ)

∑

t s= _∞

γ^{t s}C_t_j_s =C_t Labor market

Nt =

Z _α

0

Nt(i)di =_∆^p_tY^t ' Y^t whereY^t ^Y^t/Γ^t

Asset markets

(1 γ)

∑

t s= _∞

γ^{t s}A_t_j_s =Q_t^F +Q_t^B

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Balanced Growth Paths

Consumption function (agej, normalized by productivity) (i) active individuals:

C^j = (1 βγ) A^aj + ¹

1 ΛΓυγ W^N

α

(ii) retired individuals

C^j = (1 βγ)A^rj

Aggregate consumption function

C = (1 βγ) Q^F +Q^B+ W^N

1 ΛΓυγ

= (1 βγ) Q^B+ Y

1 ΛΓυγ

using Q^F =D^/(1 ΛΓυγ)andY =W^N+D^.

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Balanced Growth Paths

Bubbleless BGP (Q^B =0)

ΛΓυ=β or, equivalently,

r = (1+ρ)(1+g)υ 1

Remark #1: υ=1 )^r = (1+ρ)(1+g) 1>g Remark #2: υ<β ,^r <g

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Balanced Growth Paths

Recall:

Q_t^B =Etf^Λ^t^,t⁺¹^Qt^B+1j^tg or, lettingq_t^B _Γ^Qt^t^BY and u_t _Γ^Ut^tY

q_t^B = E_tfΛt,t+1Γq_t^B₊₁_j_tg

= EtfΛt,t+1Γ(q_t^B₊₁ ut+1)g Bubbly BGP with no bubble creation (Q^B >0, u_t =0 all t):

ΛΓ=1 or, equivalently,

r =g Implied bubble size:

q^B = ^γ(β υ)

(1 βγ)(1 υγ) ^q

B

Remark: necessary and su¢ cient condition for existence: υ< β

Jordi Galí (CREI, UPF and Barcelona GSE) Monetary Policy and Bubbles August 2016 15 / 22

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Balanced Growth Paths

Bubbly BGP with bubble creation (Q^B >0, ut =u >0 all t):

q^B = ^γ(β ΛΓυ) (1 βγ)(1 ΛΓυγ) u= 1 1

ΛΓ q^B where

ΛΓ>1,^r <g ΛΓ< ^β

υ ,^r >(1+ρ)(1+g)υ 1

Remark #1: necessary and su¢ cient condition for existence: υ< β Remark #2: continuum of bubbly BGPs f^q^B^,^ug^{indexed by} r 2 ((1+ρ)(1+g)υ 1,g)

Remark #3: q^B increasing inr, with limr!^gq^B =q^B

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+ g ρ

β 1 υ

Figure 1. Balanced Growth Paths

Bubbly without creation Bubbly with creation Bubbleless

-1 g

0

r

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Figure 2. Balanced Growth Paths: Bubble Size

Bubbly without creation

Bubbly with creation

Bubbleless

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Some Numbers

Life expectancy (at 20): (80 20) 4=240 quarters) ^γ=0.9958 Average retirement age: (63 20) 4=172 quarters ) ^υ=0.9983 (conditional on survival)

Condition for existence of bubbles: β>0.9983

Average real interest rate (1960-2015): r =1.4% 4=0.35%

Average growth rate (1960-2015): g =1.6% 4=0.4%

Consumers’discount factor on future income: ΛΓυγ'^0.995<1

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Equilibrium Dynamics (I)

Aggregate consumption function:

b

c_t = (1 βγ)(_bq_t^B +_bx_t) where

b x_t =

∑

∞ k=0

(_ΛΓυγ)^kE_tfby_t₊_kg ₁ ^ΛΓυγ_ΛΓ υγ

∑

∞ k=0

(_ΛΓυγ)^kE_tf^bⁱ^t⁺^k ^π^t⁺^k⁺¹g

= ΛΓυγEtfbxt+1g+_byt ΛΓυγ

1 ΛΓυγ(bit Etf^π^t⁺¹g)

)solution to the forward guidance puzzle? (Del Negro et al. (2016)) Aggregate bubble dynamics:

b

q_t^B =ΛΓE_tfbq^B_t₊₁g ^q^B(bi_t E_tf^π^t⁺¹g) ) role of monetary policy (Galí (2014)):

E_tf∆bq_t^B₊₁g= 1 1

ΛΓ bq_t^B + ^q

B

ΛΓ(bi_t E_tf^π^t⁺¹g)

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Equilibrium Dynamics (II)

New Keynesian Phillips curve

πt =ΛΓE_tf^π^t⁺¹g+κby_t Monetary Policy

bi_t =φ_ππt+φ_qbq_t^B Goods market clearing

b ct =y_bt

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Equilibrium Fluctuations: The Bubbleless Case

Equilibrium dynamics b

y_t =E_tfby_t₊₁g (bi_t E_tf^π^t⁺¹g) πt = ^β

υE_tf^π^t⁺¹g+κby_t bi_t =φ_ππt

Local uniqueness

φ_π >max 1,1 κ

β υ

1

υ< ₁₊^β_κ ) "reinforced Taylor principle"

Forward guidance puzzle remains

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Figure 3a

Monetary Policy and Equilibrium Uniqueness

around the Bubbleless BGP

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Figure 3b

Monetary Policy and Equilibrium Uniqueness

around the Bubbleless BGP

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Bubbly Equilibrium Fluctuations

Equilibrium dynamics b

yt = ^ΛΓυ

β Etfbyt+1g+Φbq_t^B Υυ

β (bit Etf^π^t⁺¹g) πt =ΛΓE_tf^π^t⁺¹g+κby_t

b

q_t^B =_ΛΓEtfbq^B_t₊₁g ^q^B(bit Etf^π^t⁺¹g) bit =φ_ππt+φ_qbq_t^B

where Φ ⁽¹ ^βγ_βγ⁾⁽¹ ^υγ⁾,Υ 1+ ⁽¹ ₁^βγ_ΛΓυγ⁾⁽^ΛΓ ¹⁾ ,q^B = ₍₁ ^γ⁽^β ^ΛΓυ⁾

βγ)(1 ΛΓυγ)

Particular case #1 (no bubble creation): ΛΓ=Υ=1 ; q^B =q^B Particular case #2 (about bubbleless BGP):ΛΓ= Υ= ^β_υ ;q^B =0 Intermediate cases: ΛΓ 2 ^1,^β_υ ^,^q^B 2 ^0,^q^B

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Figure 4

Monetary Policy and Equilibrium Uniqueness:

The Case of No Bubble Creation (r=g=0.004)

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Figure 6a

Monetary Policy and Equilibrium Uniqueness

around a Bubbly BGP with Bubble Creation (r=0.003935 )

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Figure 6b

Monetary Policy and Equilibrium Uniqueness

around a Bubbly BGP with Bubble Creation (r=0.003931 )

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Figure 5

Monetary Policy and Equilibrium Uniqueness

around the Bubbleless BGP

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Figure 8

Macro Volatility and Leaning against the Bubble Policies

(type II bubbles, r=0.39% )

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Main Messages and Next Steps

Reminder of the possibility of bubbly equilibria once we depart from the in…nite-lived representative consumer framework

- more likely in an environment of low natural interest rates Perils of using interest rate policy to tame asset price bubbles

- indeterminacy more likely - risk of larger ‡uctuations Caveats

- rational bubbles

- no role for credit supply factors Next steps:

- Welfare and role of monetary policy

- Global equilibrium dynamics (nonlinearities, switching equilibria)