Monetary Policy with Heterogeneous Agents: … Policy with Heterogeneous Agents: Insights from Tank Models Davide Debortoli Jordi Galí October 2017 Davide Debortoli, Jordi Galí Insights

Monetary Policy with Heterogeneous Agents:Insights from Tank Models

Davide Debortoli Jordi Galí

October 2017

Davide Debortoli, Jordi Galí () Insights from TANK October 2017 1 / 23

Motivation

Heterogeneity in Monetary Models

- idiosyncratic shocks- incomplete markets- borrowing constraints- supply side as in conventional NK models ! HANK

Main lessons:

- heterogeneity in MPCs ! role of redistribution- direct vs. indirect e¤ects

Drawbacks

- computational complexity- limited use in classroom or as a policy input

Wanted: a simple tractable framework that is a good approximation


Present Paper

An Organizing Framework

- heterogeneity between constrained and unconstrained households- heterogeneity within unconstrained households

Baseline Two-Agent New Keynesian model (TANK)

- constant share of hand-to-mouth households- redistribution through �scal rule

Question: How well does a TANK model capture the mainpredictions of HANK models regarding the aggregate economy�sresponse to aggregate shocks?


Main Findings

A baseline TANK model with a suitably calibrated transfer ruleprovides a good approximation to a prototypical HANK model

- e¤ects of monetary policy and other shocks- role of �scal policy

Isomorphic to a modi�ed RANK model

Application: Optimal monetary policy in TANK

- heterogeneity as a source of a policy tradeo¤- calibrated model: price stability nearly optimal (as in RANK!)


Some References

HANK models: Gornemann, Kuester and Nakajima (2016), Kaplan,Moll, Violante (2016), McKay and Reis (2016), McKay Nakamuraand Steinsson (2016), Werning (2015), Auclert (2016), Ravn andSterk (2013, 2017), Sterk and Tenreiro (2013), etc.

TANK models: Campbell and Mankiw (1989), Galí, Lopez-Salido,Vallés (2007), Bilbiie (2008), Cúrdia and Woodford (2010), Broer,Hansen, Krussel and Oberg (2016), Nisticó (2016), Bilbiie (2017),etc.



Continuum of heterogeneous households, indexed by s 2 [0, 1].Preferences: E0 ∑∞

t=0 βtU(C st ,Nst ;Zt) where

U(C ,N;Z) ��C1�σ � 11� σ

� N1+ϕ

1+ ϕ

�Z

Θt � [0, 1] : set of unconstrained ("Ricardian") households inperiod t

- measure 1� λt- access to one-period bonds with riskless return Rt .

Goal: derive a (generalized) Euler equation for aggregateconsumption

- in the spirit of Werning (2015), but di¤erent emphasis



Individual Euler equation: For all s 2 Θt

Zt(C st )�σ = βRtEt

�Zt+1(C st+1)

�σ



Averaging over all s 2 Θt and letting CRt � 11�λt

Rs2Θt

C st ds yields:

Zt(CRt )�σ = βRtEt

nZt+1(CRt+1)

�σVt+1o

where

Vt+1 � CRt+1jtCRt+1

!�σ "2+ σ(1+ σ)vars jtfcst+1g2+ σ(1+ σ)vars jtfcst g

#with

CRt+1jt � 11� λt

Zs2Θt

C st+1ds

vars jtfcst+kg � 11� λt

Zs2Θt

(cst+k � cRt+k )2ds

Vt : index of heterogeneity within unconstrained householdsDavide Debortoli, Jordi Galí () Insights from TANK October 2017 8 / 23


Recall:Zt(CRt )

�σ = βRtEtnZt+1(CRt+1)

�σVt+1o

Letting Ct �R 10 C

st ds :

ZtH�σt (Ct)�σ = βRtEt

�Zt+1H�σ

t+1(Ct+1)�σVt+1

where

Ht �CRtCt

Ht : index of heterogeneity between constrained and unconstrainedhouseholds



Imposing market clearing and log-linearizing:

yt = Etfyt+1g �1σbrt � 1

σEtf∆zt+1g+Etf∆ht+1g �

1σ

Etfbvt+1gRepresentation in levels:

byt = � 1σ

�brLt � zt�� bht � bft ,brLt = EtfbrLt+1g+brt

bft = Etfbft+1g+ 1σ

Etfbvt+1gComparison to RANKHow important are �uctuations in bht and bvt (and bft)? Open questionNext: A baseline TANK model (bft = bvt = 0)Davide Debortoli, Jordi Galí () Insights from TANK October 2017 10 / 23

A Baseline TANK Model: Households

Two types of households: s 2 fK ,RgRicardian: measure 1� λ

CRt +BRtPt+QtSRt =

BRt�1(1+ it�1)Pt

+WtNRt +(Dt +Qt)SRt�1+T

Rt

Keynesian: measure λ

CKt = WtNKt + TKt


A Baseline TANK Model: Policy

Fiscal Policy

- transfer rule:TKt = τ0D + τd (Dt �D)

- balanced budget:

λTKt + (1� λ)TRt = 0.

Special cases:(i) laissez-faire (τ0 = τd = 0)(ii) full steady state redistribution (τ0 = 1, τd = 0)(iii) full redistribution period by period (τ0 = 1, τd = 1) ) RANK

Monetary Policy: Taylor-type rule


A Baseline TANK Model: Supply Side

Wage equationWt =MwCσ

t Nϕt

with NRt = NKt = Nt , andMw > 1.

TechnologyYt (i) = AtNt(i)1�α

In�ation equation (Rotemberg pricing):

Πt (Πt � 1) = Et

�Λt,t+1

�Yt+1Yt

�Πt+1 (Πt+1 � 1)

+ε

ξ

�1Mp

t� 1Mp

��whereMp

t � (1� α)AtN�αt /Wt andMp � εp

εp�1


A Baseline TANK Model

Goods market clearing

Ct = Yt∆p(Πt)

where ∆pt (Πt) � 1� (ξ/2) (Πt � 1)2 and Ct � (1� λ)CRt + λCKt



Aggregate Euler equation

1 = β(1+ it)Etn(Ct+1/Ct)

�σ (Ht+1/Ht)�σ(Zt+1/Zt)Π�1t+1

owhere

Ht � CRtCt

=1

∆p(Πt)

�WtNtYt

+1

1� λ

DtYt+TRtYt

�= 1+

1∆p(Πt)

�λ(1� τd )

1� λ

�∆p(Πt)�

1� α

Mpt

��λ(τ0 � τd )

1� λ

�1� 1� α

Mp

�YYt

�Davide Debortoli, Jordi Galí () Insights from TANK October 2017 15 / 23


Log-linearized equilibrium conditions

πt = βEtfπt+1g+ κeytbyt = Etfbyt+1g � 1

σ(bit � Etfπt+1g) +Etf∆bht+1g+ 1

σ(1� ρz )ztbht = χyeyt + χnbyntbit = φππt + φybyt + νt

where eyt � byt � bynt , and bynt � 1+ϕσ(1�α)+α+ϕ

at

χn �λ(τ0 � τd )

(1� λ)H

�1� 1� α

Mp

�χy � χn �

λ(1� τd )(1� α)

(1� λ)HMp

�σ+

α+ ϕ

1� α

�Davide Debortoli, Jordi Galí () Insights from TANK October 2017 16 / 23


Three-equation representation

πt = βEtfπt+1g+ κeyteyt = Etfeyt+1g � 1

σ(1+ χy )(bit � Etfπt+1g �brnt )

bit = φππt + φybyt + νt

where brnt � (1� ρz )zt + σ (1+ χn)Etf∆bynt+1g) isomorphic to a modi�ed RANK model

χy ∂y/∂r

positive dampenedzero unchanged

negative ampli�ed


A Baseline TANK Model: Calibration andSimulations

Preferences: β = 0.9925, σ = ϕ = 1, ε = 9

Technology: α = 0.25, ξ = 372

Financial frictions: λ = 0.21

Policy rule: φπ = 1.5, φy = 0.125

Monetary policy shocks

- the role of �scal policy- the role of �nancial frictions


Monetary Policy Shocks and Transfer Rules

Monetary Policy Shocks and the Share of Constrained Households

A Baseline HANK Model

Continuum of heterogeneous households, indexed by s 2 [0, 1]Labor income: WtNt expfest g, where est = 0.966est + 0.017εstAssets: one-period nominal bond, in zero net supply.

Borrowing limit: Bst � �ψY

Alternative calibrations: ψ = f0.5, 1, 2g, implyingλ = f0.36, 0.21, 0.11gFiscal policy: government takes all pro�ts and redistributeslump-sum, same transfer rule as in TANK


HANK vs. TANK Comparison

After solving HANK model (Reiter (2010)), TANK calibrated tomatch

λ : fraction of constrained householdsβ : steady state real rate

Simulations


Interest Rate Elasticity of Output Type I Fiscal Policy

The Effects of a Monetary Policy Shock The Case of Rigid Prices

The Effects of a Monetary Policy Shock

The Effects of a Preference Shock

The Effects of a Technology Shock

Optimal Monetary Policy

Welfare losses (second order approximation)

Lt '12

E0

∞

∑t=0

βtn

π2t + αy y2t + αhbh2towhere αy =

�σ+ φ+α

1�α

�1ξ and αh �σ

ξ

�1�λ

λ

�Optimal monetary policy problem

min12

E0

∞

∑t=0

βtn

π2t + αy y2t + αhbh2tosubject to

πt = βEtfπt+1g+ κeytbht = χyeyt + χnbyntλ(τ0 � τd ) 6= 0) χn 6= 0 ) policy tradeo¤


Optimal Monetary Policy

Representation of the optimal policy:

pt � pt � p�1 = �1κ

�αy yt + χyαhht

�Impulse responses


Optimal Monetary Policy in TANK

Conclusions

A baseline TANK model with a suitably calibrated transfer ruleprovides a good approximation to a prototypical HANK model

- e¤ects of monetary policy and other shocks- role of �scal policy

Isomorphic to a modi�ed RANK model

Application: Optimal monetary policy in TANK

- heterogeneity as a source of a policy tradeo¤- price stability nearly optimal (as in RANK!)


Documents

Monetary Policy with Heterogeneous Agents: … Policy with Heterogeneous Agents: Insights from Tank Models Davide Debortoli Jordi Galí October 2017 Davide Debortoli, Jordi Galí Insights