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Monetary Policy and Bubblesin a New Keynesian Modelwith Overlapping Generations
Jordi Galí
CREI, UPF and Barcelona GSE
August 2017
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Motivation
Asset price bubbles: ubiquitous in the policy debate...
- key source of macro instability- monetary policy: cause and cure
...but absent in workhorse monetary models- no room for bubbles in the New Keynesian model- no discussion of possible role of monetary policy
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Motivation
Asset price bubbles: ubiquitous in the policy debate...
- key source of macro instability- monetary policy: cause and cure
...but absent in workhorse monetary models- no room for bubbles in the New Keynesian model- no discussion of possible role of monetary policy
Present paper: modification of the basic NK model to allow for bubbles
Key ingredients:
(i) overlapping generations of finitely-lived agents(ii) transitions to inactivity ("retirement")
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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) ⇒ flexible prices
New Keynesian models with overlapping-generations: Piergallini (2006),Nisticò (2012), Del Negro et al. (2015) ⇒ no discussion of bubbles
Monetary policy and bubbles in sticky price models:
- Bernanke and Gertler (1999, 2001): ad-hoc bubble- Galí (2014): 2-period OLG, constant output- Present paper:
- many-period lifetimes- variable employment and output- nests standard NK model as a limiting case
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A New Keynesian Model with Overlapping Generations
Individual survival rate: γ (Blanchard (1985), Yaari (1965))
Size of cohort "at birth": 1− γ
Total population size: 1
Two types of individuals:
- "Active": manage own firm, work for others.- "Retired": consume financial wealth
Probability of remaining active: υ (Gertler (1999))
Labor force (and measure of firms): α ≡ 1−γ1−υγ ∈ (0, 1]
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Consumers
Complete markets (including annuity contracts)
Consumer’s problem:
maxE0∞
∑t=0(βγ)t logCt |s
1Pt
∫ α
0Pt (i)Ct |s (i)di + Et{Λt ,t+1Zt+1|s} = At |s [+WtNt |s ]
At |s = Zt |s/γ
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Firms
Technology:Yt (i) = ΓtNt (i)
where Γ ≡ 1+ g ≥ 1.
Price-setting à la Calvo
- incumbent firms: a fraction θI keep prices unchanged- newly created firms: a fraction θN set price equal to Pt−1- average price rigidity index: θ ≡ υγθI + (1− υγ)θN
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Labor Markets and Inflation
Wage equation:
Wt =
(Ntα
)ϕ
where Wt ≡ Wt/Γt and Nt ≡∫ α0 Nt (i)di .
Natural level of output: setting 1/Wt =M
Y nt = ΓtY
with Y ≡ αM− 1ϕ . Remark : invariant to bubble size.
New Keynesian Phillips curve
πt = ΦEt{πt+1}+ κyt
where Φ ≡ βγθI/θ, κ ≡ λϕ, and yt ≡ log(Yt/Y nt ).
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Asset Markets
Aggregate stock market
QFt =∞
∑k=0
(υγ)kEt{Λt ,t+kDt+k}
Remark : same discount rate as labor income.
Bubbly assetQBt |s = Et{Λt ,t+1QBt+1|s}
with QBt |s ≥ 0 for all t and s ≤ t.Aggregate bubble:
QBt = Bt + Ut
where Bt ≡ ∑t−1s=−∞ Q
Bt |s ≥ 0 and Ut ≡ QBt |t ≥ 0
Equilibrium condition:
QBt = Et{Λt ,t+1Bt+1}
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Characterization of Equilibria
Balanced Growth Paths
Equilibrium Dynamics around a Balanced Growth Path
Remark: key role for consumption function (individual and aggregate) inthe determination of equilibria
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Balanced Growth Paths
Consumption function (consumer of age j ; normalized by productivity)
(i) active individuals (letting Λ ≡ 11+r ):
Cj = (1− βγ)
[Aaj +
11−ΛΓυγ
(WNα
)](ii) retired individuals
Cj = (1− βγ)Arj
Aggregate consumption function
C = (1− βγ)
[QF +QB + WN
1−ΛΓυγ
]= (1− βγ)
[QB + Y
1−ΛΓυγ
]using QF = D/(1−ΛΓυγ) and Y =WN +D.
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Balanced Growth Paths
In equilibrium (C = Y):
1 = (1− βγ)
[qB +
11−ΛΓυγ
]where qB ≡ QB/Y .Bubbleless BGP (qB = 0)
ΛΓυ = β
or, equivalently,r = (1+ ρ)(1+ g)υ− 1 ≡ r
Remark #1 : r increasing in υ
Remark #2 : υ < β ⇔ r < g
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Balanced Growth Paths
Bubbly BGP:
qB =γ(β−ΛΓυ)
(1− βγ)(1−ΛΓυγ)> 0
u =(1− 1
ΛΓ
)qB ≥ 0
whereΛΓ ≥ 1⇔ r ≤ gΛΓυ < β⇔ r > r
Existence condition:υ < β
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g+ρ
υβ 1
Figure 1. Balanced Growth Paths
Bubbly BGP without bubble creation Bubbly BGP with bubble creation Bubbleless BGP
-1
g
0
r
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Some Numbers
Life expectancy (at 16): 63× 4 = 252 quarters ⇒ γ ' 0.996Average employment rate: 0.6 ⇒ υ = 0.9973
Condition for existence of bubbles: β > 0.9973
Average real interest rate (1960-2015): r = 1.4%÷ 4 = 0.35%Average growth rate (1960-2015): g = 1.6%÷ 4 = 0.4%
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Equilibrium Dynamics (I)
Goods market clearing:yt = ct
Aggregate consumption function:
ct = (1− βγ)(qBt + xt )
where
xt = ΛΓυγEt{xt+1}+ yt −ΛΓυγ
1−ΛΓυγ(it − Et{πt+1})
Aggregate bubble dynamics:
qBt = ΛΓEt{qBt+1} − qB (it − Et{πt+1})
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Equilibrium Dynamics (II)
New Keynesian Phillips curve
πt = ΦEt{πt+1}+ κyt
Monetary Policyit = φππt + φq q
Bt
Assumption: no fundamental shocks, focus on bubble-driven fluctuations
Stationary solutions?
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Figure 4. Determinacy and Indeterminacy Regions
around Bubbly BGPs
r = 0.00335 r = 0.0035
r = 0.0037 r = 0.004
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Bubble-driven Fluctuations around the Bubbleless BGP
Assumed bubble process:
qBt ={ υ
βδqBt−1 + ut with probability δ
ut with probability 1− δ
where {ut} > 0 is white noise with mean u & 0.Remark : it satisfies the relevant bubble condition
qBt = (β/υ)Et{qBt+1}
Equilibrium dynamics:
yt = Et{yt+1} − (it − Et{πt+1}) +ΘqBt
πt = ΦEt{πt+1}+ κyt
it = φππt + φq qBt
where Θ ≡ (1− βγ)(1− υγ)/βγ > 0.
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Bubble-driven Fluctuations around the Bubbleless BGP
Remark: alternative representation of the dynamic IS equation
yt = Et{yt+1} − (it − Et{πt+1} − rnt )
wherernt = ΘqBt
is the natural rate of interest.
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Bubble-driven Fluctuations around the Bubbleless BGP
Determinacy condition (for any given bubble):
φπ > max[1,1κ(Φ− 1)
]Remark : independent of φq
Equilibrium output and inflation (assuming determinacy)
yt = (1− υγθI/θ)Ψ(Θ− φq)qBt
πt = κΨ(Θ− φq)qBt
where Ψ ≡ 1(1−υγθI /θ)(1−υ/β)+κ(φπ−υ/β)
> 0.
Simulated bubble driven fluctuations (φπ = 1.5, φq = 0) (*)
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Figure 2. Simulated Bubble-Driven Fluctuations around the Bubbleless BGP
0 50 100 150 200 250 300 350 400
bubble size
0
0.1
0.2
0.3
0 50 100 150 200 250 300 350 400
output
0
2
410 -4
0 50 100 150 200 250 300 350 400
inflation
0
0.005
0.01
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Bubble-driven Fluctuations around the Bubbleless BGP
An optimal "leaning against the bubble" monetary policy:
φq = Θ > 0
⇒ yt = πt = 0
Remark : in the present environment, bubble fluctuations are not affectedby "leaning against the bubble" policies
Remark : same outcome can be attained by directly targeting inflation(φq = 0, φπ → +∞), with no need to observe the bubble or knowing Θaccurately
Monetary policy and macro volatility (*)
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Figure 3. Bubble-driven Fluctuations: Monetary Policy and Macro Volatility in a Neighborhood of the Bubbleless BGP
2
4
1.5
10 -3
6
12
8
0.8
10 -40.62.5
0.40.2
3 0
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Bubble-Driven Fluctuations around a Bubbly BGP
Equilibrium dynamics
yt =ΛΓυ
βEt{yt+1} −
Υυ
β(it − Et{πt+1}) +ΘqBt
qBt = ΛΓEt{qBt+1} − qB (it − Et{πt+1})πt = ΦEt{πt+1}+ κyt
it = φππt + φqqBt
where Θ ≡ (1−βγ)(1−υγ)βγ > 0 and Υ ≡ 1+ (1−βγ)(ΛΓ−1)
1−ΛΓυγ ≥ 1
Remark : {qBt } no longer independent of monetary policy.
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Bubble-driven fluctuations around a Bubbly BGP
An optimal "leaning against the bubble" policy
φq =Θβ
Υυ> 0
φπ > 1−(1−Φ)(1−ΛΓυ/β)
κΥυ/β' 1
then:⇒ yt = πt = 0
but no elimination bubble fluctuations:
⇒ qBt = (1/χ)qBt−1 + ξt
where ξt ≡ bt − Et−1{bt}+ ut and χ ≡(
ΛΓυβ
)ΛΓ(1−βγ)+γ(β−ΛΓυ)
1−ΛΓυγ > 1if r < r .Remark: same outcome with strict inflation targeting policy.Simulated bubble-driven fluctuations
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Figure 5. Simulated Bubble-driven Fluctuations Around a Bubbly BGP
20 40 60 80 100 120 140 160 180 200
time
0
0.05
0.1
bubble size
20 40 60 80 100 120 140 160 180 200
time
0
5
10
15
10 -7 output
20 40 60 80 100 120 140 160 180 200
time
-2
0
2
4
6
8
10
10 -6 inflation
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Figure 6.b Bubble-driven Fluctuations: Monetary Policy and Bubble Volatility in a Neighborhood of a Bubbly BGP
0.3864
3
0.38645
12.5 0.8
0.3865
0.6
0.38655
10 -420.4
1.5 0.20
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Conclusions, Caveats and Possible Extensions
Bubbly equilibria may exist in the NK model once we depart from theinfinitely-lived representative consumer assumption. Room forbubble-driven fluctuations.
More likely in an environment of low natural interest rates.
No obvious advantages of "leaning against the bubble" policies (relative toinflation targeting), plus some risks (e.g. may amplify bubble fluctuations)
Caveats/potential extensions
(i) Rational bubbles. But non-rational bubbles can be readilyaccommodated.(ii) ZLB has been ignored. Potential interesting interaction with bubbles(e.g. by raising underlying natural rate, bubbles may lower the risk ofhitting the ZLB).(iii) No role for credit supply factors; may be needed to boost "bubblemultiplier".
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Bubbleless Equilibria
Equilibrium dynamics
yt = Et{yt+1} − (it − Et{πt+1})
πt = ΦEt{πt+1}+ κyt
it = φππt
Monetary policy and equilibrium determinacy:
φπ > max[1,1κ(Φ− 1)
]
Thus, if θ/θI <βγ1+κ ⇒ φπ > (βγθI/θ − 1)/κ > 1
⇒"reinforced Taylor principle"
Remark: forward guidance puzzle still present in equilibrium despitefinitely-lived agents.
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Figure 6.a Bubble-driven Fluctuations: Monetary Policy and Output Volatility in a Neighborhood of a Bubbly BGP
0.5
3
1
1
1.5
2
10 -5
2.5
2.5 0.8
3
3.5
0.610 -4
2 0.41.5 0.2
0