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 Davide Debortoli, Jordi Galí () Insights from TANK October 2017 2 / 23
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 main predictions of HANK models regarding the aggregate economy s response to aggregate shocks? Davide Debortoli, Jordi Galí () Insights from TANK October 2017 3 / 23
Main Findings A baseline TANK model with a suitably calibrated transfer rule provides 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!) Davide Debortoli, Jordi Galí () Insights from TANK October 2017 4 / 23
Some References HANK models: Gornemann, Kuester and Nakajima (2016), Kaplan, Moll, Violante (2016), McKay and Reis (2016), McKay Nakamura and Steinsson (2016), Werning (2015), Auclert (2016), Ravn and Sterk (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. Davide Debortoli, Jordi Galí () Insights from TANK October 2017 5 / 23
An Organizing Framework Continuum of heterogeneous households, indexed by s 2 [0, 1]. Preferences: E 0 t=0 β t U(C s t, N s t ; Z t ) where C 1 σ 1 U(C, N; Z) 1 σ N 1+ϕ Z 1 + ϕ Θ t [0, 1] : set of unconstrained ("Ricardian") households in period t - measure 1 λ t - access to one-period bonds with riskless return R t. Goal: derive a (generalized) Euler equation for aggregate consumption - in the spirit of Werning (2015), but di erent emphasis Davide Debortoli, Jordi Galí () Insights from TANK October 2017 6 / 23
An Organizing Framework Individual Euler equation: For all s 2 Θ t Z t (C s t ) σ = βr t E t Zt+1 (C s t+1) σ Davide Debortoli, Jordi Galí () Insights from TANK October 2017 7 / 23
An Organizing Framework Averaging over all s 2 Θ t and letting Ct R 1 1 λ t Rs2Θ t Ct s ds yields: n o Z t (Ct R ) σ = βr t E t Z t+1 (Ct+1) R σ V t+1 where V t+1 C t+1jt R Ct+1 R! σ " 2 + σ(1 + σ)varsjt fct+1 s g # 2 + σ(1 + σ)var sjt fc s t g with Z Ct+1jt R 1 C 1 λ t+1ds s t s2θ Z t var sjt fct+k s g 1 (ct+k s ct+k R 1 λ )2 ds t s2θ t V t : index of heterogeneity within unconstrained households Davide Debortoli, Jordi Galí () Insights from TANK October 2017 8 / 23
An Organizing Framework Recall: Letting C t R 1 0 C s t ds : Z t (C R t ) σ = βr t E t n Z t+1 (C R t+1) σ V t+1 o where Z t Ht σ (C t ) σ = βr t E t Zt+1 Ht+1 σ (C t+1) σ V t+1 H t C R t C t H t : index of heterogeneity between constrained and unconstrained households Davide Debortoli, Jordi Galí () Insights from TANK October 2017 9 / 23
An Organizing Framework Imposing market clearing and log-linearizing: 1 y t = E t fy t+1 g σ br t Representation in levels: by t = 1 σ E tf z t+1 g + E t f h t+1 g 1 σ br L t z t bh t bf t, 1 σ E tfbv t+1 g br L t = E t fbr L t+1g + br t bf t = E t fbf t+1 g + 1 σ E tfbv t+1 g Comparison to RANK How important are uctuations in bh t and bv t (and bf t )? Open question Next: A baseline TANK model (bf t = bv t = 0) Davide Debortoli, Jordi Galí () Insights from TANK October 2017 10 / 23
A Baseline TANK Model: Households Two types of households: s 2 fk, Rg Ricardian: measure 1 λ C R t + BR t P t + Q t S R t = BR t 1 (1 + i t 1) P t + W t N R t + (D t + Q t )S R t 1 + T R t Keynesian: measure λ C K t = W t N K t + T K t Davide Debortoli, Jordi Galí () Insights from TANK October 2017 11 / 23
A Baseline TANK Model: Policy Fiscal Policy - transfer rule: - balanced budget: T K t = τ 0 D + τ d (D t D) λt K t + (1 λ) T R t = 0. Special cases: (i) laissez-faire (τ 0 = τ d = 0) (ii) full steady state redistribution (τ 0 = 1, τ d = 0) (iii) full redistribution period by period (τ 0 = 1, τ d = 1) ) RANK Monetary Policy: Taylor-type rule Davide Debortoli, Jordi Galí () Insights from TANK October 2017 12 / 23
A Baseline TANK Model: Supply Side Wage equation W t = M w C σ t N ϕ t with N R t = N K t = N t, and M w > 1. Technology Y t (i) = A t N t (i) 1 In ation equation (Rotemberg pricing): Yt+1 Π t (Π t 1) = E t Λ t,t+1 + ɛ ξ 1 M p t Y t 1 M p α Π t+1 (Π t+1 1) where M p t (1 α)a t N α t /W t and M p ɛ p ɛ p 1 Davide Debortoli, Jordi Galí () Insights from TANK October 2017 13 / 23
A Baseline TANK Model Goods market clearing C t = Y t p (Π t ) where p t (Π t ) 1 (ξ/2) (Π t 1) 2 and C t (1 λ)c R t + λc K t Davide Debortoli, Jordi Galí () Insights from TANK October 2017 14 / 23
A Baseline TANK Model Aggregate Euler equation o 1 = β(1 + i t )E t n(c t+1 /C t ) σ (H t+1 /H t ) σ (Z t+1 /Z t )Πt+1 1 where H t C t R C t 1 Wt N t = p + 1 D t + T R t (Π t ) Y t 1 λ Y t Y t 1 λ(1 τd ) = 1 + p p 1 α (Π t ) (Π t ) 1 λ M p t λ(τ 0 τ d ) 1 α Y 1 1 λ M p Y t Davide Debortoli, Jordi Galí () Insights from TANK October 2017 15 / 23
A Baseline TANK Model Log-linearized equilibrium conditions by t = E t fby t+1 g π t = βe t fπ t+1 g + κey t 1 σ ( bi t E t fπ t+1 g) + E t f bh t+1 g + 1 σ (1 ρ z )z t bh t = χ y ey t + χ n by t n where ey t by t by n t, and by n t bi t = φ π π t + φ y by t + ν t χ n λ(τ 0 τ d ) (1 λ)h 1+ϕ σ(1 α)+α+ϕ a t 1 χ y χ n λ(1 τ d )(1 α) (1 λ)hm p 1 α M p σ + α + ϕ 1 α Davide Debortoli, Jordi Galí () Insights from TANK October 2017 16 / 23
A Baseline TANK Model Three-equation representation ey t = E t fey t+1 g π t = βe t fπ t+1 g + κey t 1 σ(1 + χ y ) ( bi t E t fπ t+1 g br n t ) bi t = φ π π t + φ y by t + ν t where br n t (1 ρ z )z t + σ (1 + χ n ) E t f by n t+1 g ) isomorphic to a modi ed RANK model χ y positive zero negative y/ r dampened unchanged ampli ed Davide Debortoli, Jordi Galí () Insights from TANK October 2017 17 / 23
A Baseline TANK Model: Calibration and Simulations Preferences: β = 0.9925, σ = ϕ = 1, ɛ = 9 Technology: α = 0.25, ξ = 372 Financial frictions: λ = 0.21 Policy rule: φ π = 1.5, φ y = 0.125 Monetary policy shocks - the role of scal policy - the role of nancial frictions Davide Debortoli, Jordi Galí () Insights from TANK October 2017 18 / 23
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: W t N t expfet s g, where et s = 0.966et s + 0.017ε s t Assets: one-period nominal bond, in zero net supply. Borrowing limit: Bt s ψy Alternative calibrations: ψ = f0.5, 1, 2g, implying λ = f0.36, 0.21, 0.11g Fiscal policy: government takes all pro ts and redistributes lump-sum, same transfer rule as in TANK Davide Debortoli, Jordi Galí () Insights from TANK October 2017 19 / 23
HANK vs. TANK Comparison After solving HANK model (Reiter (2010)), TANK calibrated to match λ : fraction of constrained households β : steady state real rate Simulations Davide Debortoli, Jordi Galí () Insights from TANK October 2017 20 / 23
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) L t ' 1 2 E 0 where α y = σ + φ+α 1ξ 1 α and α h σ ξ Optimal monetary policy problem subject to min 1 2 E 0 n β t π 2 t + α y ỹt 2 + α h bh t 2 t=0 1 λ λ n β t π 2 t + α y ỹt 2 + α h bh t 2 t=0 π t = βe t fπ t+1 g + κey t bh t = χ y ey t + χ n by n t o o λ(τ 0 τ d ) 6= 0 ) χ n 6= 0 ) policy tradeo Davide Debortoli, Jordi Galí () Insights from TANK October 2017 21 / 23
Optimal Monetary Policy Representation of the optimal policy: Impulse responses ˆp t p t p 1 = 1 κ α y ỹ t + χ y α h ĥ t Davide Debortoli, Jordi Galí () Insights from TANK October 2017 22 / 23
Optimal Monetary Policy in TANK
Conclusions A baseline TANK model with a suitably calibrated transfer rule provides 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!) Davide Debortoli, Jordi Galí () Insights from TANK October 2017 23 / 23