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