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Capital, Income Inequality, and Consumption:the Missing Link
Florin O. Bilbiie Diego Känzig Paolo Surico
Univ. of Lausanne London Business School London Business School
Banque de France-CEPR Workshop on Heterogeneity, 5-6 December 2019
A topic with some tradition ...
Motivation
I Investment in Physical Capital (K) central to macro:growth, inequality, optimal taxation, business cycles
I Real Business Cycles (RBC): General Equilibrium model
I K investment ∼ 4 times more volatile than GDP, etc.
I New Keynesian (NK) ... No K
I side show, does not matter much (Appendix G, robustness)
I ingredient in DSGE versions, but back seat ... for CCaveat: "financial accelerator" notwithstanding
I Missing Link RBC—NK: "to K or not to K"
Monetary Policy Transmission
I What happens in response to monetary policy?
I Recent extension of Representative-Agent (RA)NK:
I Heterogeneous-agent (TA&HA)NK: Consumption CI common theme: general equilibrium GE of the essence
I "GE" = core of RBC ... through K
Our Result(s)
I Heterogeneity puts K back at the center in NK!
I Novel complementarity K – income (Y) inequality
Amplification of Mon. Pol. Effects on Cno Y Inequality Y Inequality
no K =repres.-agent smallK small LARGE
I Interaction w/ fiscal redistribution: essential what K income
I opposite effects of redistributing physical vs monopoly K
I T(ractable)-HANK w/ K→ analytics (novel even for RANK)
I Role of wage rigidity
*ANK: Heterogeneity/Inequality MattersI Empirical: Campbell Mankiw; Hall; ...; Kaplan Violante; Cloyne Ferreira Surico
I RANK+K: Dupor, Carlstrom Fuerst, Sveen Weinke, Woodford, Rupert Sustek...
I 2000s TANK: Galí Lopez-Salido Vallés; Bilbiie...
I borrower-saver: Iacoviello; Nistico; Eggertsson Krugman; Bilbiie Monacelli Perotti
I 2010s HANK focus on C: Auclert; Kaplan, Moll, Violante; Gornemann Kuester
Nakajima; McKay Nakamura Steinsson; Guerrieri Lorenzoni; Bayer Luetticke Pham-Dao Tjaden; Auclert
Rognlie; Debortoli Galí; Ravn Sterk; Den Haan Rendahl Riegler; Luetticke; McKay Reis; Challe Matheron
Ragot Rubio; Oh Reis; Hagedorn Manovskii Mitman; Auclert Rognlie Straub; Acharya Dogra; Werning;
Cantore Freund; Bilbiie Ragot; Cui Sterk; Bhandari Evans Golosov Sargent; Bilbiie
I now: HANK with focus on KI Auclert Rognlie Straub; Alves Kaplan Moll Violante; this paper ...
I Sticky Wages: TANK Colciago ... HANK Broer, Hansen, Krusell, Oberg ...
A Tale of Two Inequalities
I This paper:
Cj + Sj = (WN)j + (rK+D)j + Tj ≡ Yj
I no K, Income (Y) inequality only: RHS
Cj = (WN)j +Dj + Tj ≡ Yj
I Bilbiie 2008 JET ("TANK"), many HA models
I K, no Y inequality (but still MPC heterogeneity): LHS
Cj + Sj = Y
Saver-spender: Gali Lopez-Salido Valles 2007, Mankiw 2000, any HA model with
assets in positive net supply
A Tale of Two Inequalities
I This paper:
Cj + Sj = (WN)j + (rK+D)j + Tj ≡ Yj
I no K, Income (Y) inequality only: RHS
Cj = (WN)j +Dj + Tj ≡ Yj
I Bilbiie 2008 JET ("TANK"), many HA models
I K, no Y inequality (but still MPC heterogeneity): LHS
Cj + Sj = Y
Saver-spender: Gali Lopez-Salido Valles 2007, Mankiw 2000, any HA model with
assets in positive net supply
A Tale of Two Inequalities
I This paper:
Cj + Sj = (WN)j + (rK+D)j + Tj ≡ Yj
I no K, Income (Y) inequality only: RHS
Cj = (WN)j +Dj + Tj ≡ Yj
I Bilbiie 2008 JET ("TANK"), many HA models
I K, no Y inequality (but still MPC heterogeneity): LHS
Cj + Sj = Y
Saver-spender: Gali Lopez-Salido Valles 2007, Mankiw 2000, any HA model with
assets in positive net supply
A Tale of Two Inequalities
I This paper:
Cj + Sj = (WN)j + (rK+D)j + Tj ≡ Yj
I no K, Income (Y) inequality only: RHS
Cj = (WN)j +Dj + Tj ≡ Yj
I Bilbiie 2008 JET ("TANK"), many HA models
I K, no Y inequality (but still MPC heterogeneity): LHS
Cj + Sj = Y
Saver-spender: Gali Lopez-Salido Valles 2007, Mankiw 2000, any HA model with
assets in positive net supply
A(n Outrageous) ModelI λ hand-to-mouth H, 1− λ savers S w/ bonds Euler (loglin)
cSt = EtcS
t+1 − rt
+ Solow (1956) investment It =αβYt
αβ < 1 is the savings rateit = yt
+ (Loglin) individual:
(1− αβ)cSt +
αβ1− λ
it = ySt
(1− αβ)cHt = yH
t = χyt,
the key χ = a model of the income distribution.I Aggregate Euler (use ct = it = yt)
ct = Etct+1 −(1− αβ)(1− λ)
1− (αβ+ λχ)rt.
Isolating Income (Y) InequalityI no K (αβ= 0), all RHS: Bilbiie 2008→2019
cHt = χyt; cS
t =1− λχ
1− λyt.
I Aggregate Euler ∼ Aggregate Demand
ct = Etct+1 −1− λ
1− λχ︸ ︷︷ ︸multiplier
rt
amplification iff χ>1 (counter-cyclical Y ineq.)full-HANK general. Auclert JMP; evidence: Patterson 19; Slacalek Tristani Violante 19; Heathcote Perri Violante 10
I New Keynesian Cross:
Aggregate MPC(ish): λ× 1×χ+ ...
"indirect effect" Kaplan Moll Violante
The New Kenesian Cross ct= ωyt− (1−ω)Ωrt+...
ctERC: ct = ŷt
PE: ct = c(ŷt,rt,gt)
ΩD
ΩI
Ω
ω
ŷt
Isolating K Inequality (LHS): This PaperI assume that income is perfectly redistributed χ = 1:
(1− αβ)cSt +
αβ1− λ
it = yt;
(1− αβ)cHt = yt.
I Aggregate Euler - Demand
ct = Etct+1 −(1− αβ)(1− λ)
1− (αβ+ λ)︸ ︷︷ ︸multiplier
rt
I Another Keynesian-cross multiplier (λ < 1−αβ):
the savings rate (of S) acts as an MPC (of H)
S’s saving-investment→ K income, redistribution→ H, not saving
I Novel analytical isolation, translates to any HA w/ some K (net saving)I ex.: Auclert Rognlie Straub: turn off χ but other channels (inattention, liquidity)
The Multiplier ... of the Multiplier
I both K and Y inequality∣∣∣∣∂ct
∂rt
∣∣∣∣ = (1− αβ)(1− λ)
1− (αβ+ λχ)=
1− λ
1− λχ 1
1−αβ
I Complementarity if Y ineq. counter-cyclical χ> 1:∣∣∣∣∂ct
∂rt
∣∣K, Y ineq
∣∣∣∣ > ∣∣∣∣∂ct
∂rt
∣∣no K, Y ineq
∣∣∣∣× ∣∣∣∣∂ct
∂rt
∣∣K, no Y ineq.
∣∣∣∣I Intuition 1: aggreg MPC∼ λχ+αβ so multiplier 1
1−(αβ+λχ)
I Intuition 2: K multiplier 1
1−αβ"inside" Y ineq. multiplier
1−λ1−λχ at each round
A picture worth 1/(1-x) words
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
1
2
3
4
5
6
7
Figure: C multipliers as a function of λ (α = 0.33, β = 0.99, χ = 1.7).
Simple Testable Predictions1. Y and C inequality:
ySt − yH
t =1−χ1− λ
yt
cSt − cH
t =1
1− αβ(yS
t − yHt )−
αβ(1− αβ)(1− λ)
it
Solow=
1−χ− αβ(1− λ)(1− αβ)
yt
both counter-cyclical iff χ > 1−αβ. General condition:
χ+IY
∂it∂yt
> 1
2. C ineq. more counter-cyclical than Y ineq. if it procyclical (dah)
I compare to evidence (Coibion Gorodnichenko Kueng Silvia 2017)
A Tractable HANK (THANK) with K: This Paper
I extend Bilbiie (2019) with illiquid K
I idiosyncratic uncertainty: two states S1−h1−s
H
λ =1− s
2− s− h
I assets: liquid bonds, illiquid K & shares (claims to profits)
I labor union (sticky wages later)
I the government redistributes K and/or profits
I Reminder: THANK matches micro moments: iMPCs Auclert
Rognlie Straub, income risk, leptokurtosis, left-skewness, cyclical skewness, etc.
I Novel analytical solution in special case (incl. RANK!) analytics
A Tractable HANK (THANK) with K: This PaperI Illiquid K (Qt ≡ (Φ′ (.))−1 Tobin’s marginal Q)
Qt=βEt(CS
t+1
CSt)−
1σ [(1− τK)RK
t+1+Qt+1(1− δ+Φt+1−It+1
Kt+1Φ′t+1)]
Kt+1 = (1− δ)Kt +Φ(
It
Kt
)Kt
I Liquid bonds
(CSt )− 1
σ = βEt
1+ rn
t1+ πt+1
[s(CS
t+1)− 1
σ + (1− s)(CHt+1)
− 1σ
]I Redistribution CH
t = wtNt + THt
λTH,t = τDDt + τKrKt Kt
I No Y ineq. χ = 1: perfect redistribution
τD = τK = λ
Parameterization: Standard
Quantifying the Complementarity
Amplification of Mon. Pol. Effects on CNo Y ineq. Y ineq.
no-K 1(=RANK) 1.51 (Bilbiie 2008)
K 1.15 2.25
Quantifying the Complementarity
Amplification of Mon. Pol. Effects on CRANK No Y ineq. Y ineq. + Risk
no-K 1 1(=RANK) 1.51 1.60K 0.66 1.15 2.25 2.62
I RANK w/ K dampening: real (long) rate increases Iresponse, less C
I uniform across models: iid shocks: .89→1.55→3.95(w/o K : 2.02)
I idiosyncratic risk: adds prec. saving in liquid assetsI K→ long-run amplification (across models) Long-Run
Fiscal Redistribution (of Financial Income)
Income Redistribution DYes No
K Yes 1.15 4.34No 0.50 2.62
I D countercyclical→ YH less cyclical→dampening
I rKK highly procyclical→ YH more cyclical→ ++amplification
I But D procyclical (even wrt MP). One avenue:
Sticky Wages: Some Nuisance
I extend our novel framework of isolating K and Yinequality channels
I derive testable predictions
I role of redistributing capital income and profits
Sticky Wages
Amplification of Mon. Pol. Effects on CRANK No Y ineq. Y ineq. + Risk
no-K 1.37 (37%) 1.37 (37%) 1.38 (-9%) 1.40 (-13%)K 1.29 (95%) 2.10 (89%) 2.43 (8%) 2.67 (2%)
I RA: Further break of neutrality→ higher MP effect
I No Y ineq.: also large amplification; conditional-on-Kamplification similar to flex-W (2.1/1.29)
I Conditional on Y ineq. less amplification: dampening w/o K!
I →Less difference in fiscal redistributions (different Kincomes more similar over cycle)
Empirical Relevance
1. Peak responses to MP
dc < dy < di
match both absolute and relative.
2. Both C and Y ineq. are counteryclical, with C ineq. more so
Testable Prediction 1: Peak Responses to MP
0 1 2 3 4 5 6 7 8
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
Figure: IRFs to expansionary interest rate shock of 25 basis points.
Testable Prediction(s) 2: C & Y Inequality
0 1 2 3 4 5 6 7 8
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 1 2 3 4 5 6 7 8
-0.5
-0.4
-0.3
-0.2
-0.1
0
Figure: IRFs to expansionary interest rate shock of 25 basis points.
I Only HA with both K and Y inequality fits evidence by i.a.Coibion et al (2017)
Conclusions
I Further step toward Macro convergenceI K back in policy-relevant, monetary models
I Through a multiplier of the multiplier:I complementarity K - Y inequality
I Key for MP what K income is fiscally redistributed,machines vs monopolyI optimal policy?
I Missing link: 1. RBC–NK & 2. quant.–tractable HANK
Analytics: Closed-form multipliers
I Novel analytical solution:δ = 1; ϕ = 0; σ = 1; κ → ∞; φπ = 1
I RANK Multiplier
∂ct
∂ (−rt)= 1−
(1− α2β
)+ (1− α)ψ−1 α2β
1+ψ−1
(1− α)ψ−1 + 1− α2β≤ 1
I TANK fixed-price Multiplier
∂ct
∂ (−rt)=
1− λ
1− λχ+
αβ
1− αβ
(1− α) λ
1− (1− α) λχ
I complementarity proof in this special caseI THANK
Long-run Amplification
0 20 40 60
0
0.2
0.4
0.6
0.8
1
PDV Consumption: Flexible-Wage
0 20 40 60
0
0.2
0.4
0.6
0.8
1
PDV Consumption: Sticky-Wage
Table