The Distribution of Wealth andthe Marginal Propensity to Consume

Christopher Carroll1 Jiri Slacalek2 Kiichi Tokuoka3 Matthew N. White4

1Johns Hopkins University and NBER

2European Central Bank

3Ministry of Finance, Japan

4University of Delaware

Household Wealth Data and Public Policy

IFS/Public Economics UK Conference

Marginal Propensity to Consume

The Question: How Large Is the MPC (≡ κ)?

If households receive a surprise one-off EUR 1 in income,how much more will be in aggregate spent over the next year?

Why Do We Care?

I MPC reflects size of economy’s response to fiscal stimulusI Crude Keynesianism: Transitory tax cut multiplier = 1

1−κ − 1I If κ = 0.75, then multiplier is 4− 1 = 3

I Some micro estimates of κ are this large

I If κ = 0.05, then multiplier is only ≈ 0.05I This is about the size of κ in RBC models

Our Claim: Heterogeneity Is Key To Modeling the MPC

Need to Consider:

I Households are heterogeneous

I Wealth is unevenly distributed

I C function is highly concave

I ⇒ Distributional issues matter for aggregate CGiving EUR 1 to the poor 6= giving EUR 1 to the rich

Concavity of Consumption Function and Wealth Heterogeneity

Consumption�HquarterlyL permanentincome ratio Hleft scaleL¯

mt�HptWtL

Histogram: empirical HSCF1998L density ofmt�HptWtL Hright scaleL

¯

0 5 10 15 200.0

0.5

1.0

1.5

0.

0.05

0.1

0.15

0.2

To-Do List

1. Calibrate realistic income process

2. Match empirical wealth distribution

3. Back out optimal C and MPC out of transitory income

4. Is MPC in line with empirical estimates?

Our (Micro) Income Process

I Motivated by Friedman’s (1957) Permanent Income Hypothesis

I Idiosyncratic (household) income process:

yyy t+1 = pt+1ξt+1W

pt+1 = ptψt+1

pt permanent income ψt+1 permanent shockξt transitory income W aggregate wage rate

I ξt incorporates unemployment insurance (Carroll (1992)):

ξt = µ with probability u

= (1− τ)¯̀θt with probability 1− u

µ UI when unemployedτ rate of tax collected for the unemployment benefits

Decision Problem

v(mt) = max{ct}

u(ct) + β�DEt

[ψ1−ρt+1v(mt+1)

]s.t.

at = mt − ct

at ≥ 0

kt+1 = at/(�Dψt+1)

mt+1 = (k + r)kt+1 + ξt+1

r = αZ (KKK/¯̀LLL)α−1

Variables normalized by ptW

Parameter Values

I β, ρ, α, δ, ¯̀, µ , and u taken from JEDC special volume

I Key new parameter values:

Description Param Value Source

Prob of Death per Quarter D 0.00625 Life span of 40 yearsVariance of Log ψt σ2

ψ 0.016/4 Carroll (1992); SCFDeBacker et al. (2013)

Variance of Log θt σ2θ 0.010× 4 Carroll (1992)

Annual Income, Earnings, or Wage Variances

σ2ψ σ2

ξ

Our parameters 0.016 0.010

Carroll (1992) 0.016 0.010Storesletten, Telmer, and Yaron (2004) 0.008–0.026 0.316Meghir and Pistaferri (2004)? 0.031 0.032Low, Meghir, and Pistaferri (2010) 0.011 −Blundell, Pistaferri, and Preston (2008)? 0.010–0.030 0.029–0.055DeBacker, Heim, Panousi, Ramnath, and Vidangos (2013) 0.007–0.010 0.15–0.20

Implied by KS-JEDC 0.000 0.038Implied by Castaneda et al. (2003) 0.03 0.005

?Meghir and Pistaferri (2004) and Blundell, Pistaferri, and Preston (2008) assume that the transitory component is serially correlated (an MA

process), and report the variance of a subelement of the transitory component. σ2ξ for these articles are calculated using their MA estimates.

Typology of Our Models—Three Dimensions

1. Discount Factor βI ‘β-Point’ model: Single discount factorI ‘β-Dist’ model: Uniformly distributed discount factor

2. Empirical Wealth Variable to MatchI Net WorthI Liquid Financial Assets

3. Life CycleI Perpetual Youth (a la Blanchard)I Overlapping Generations

Dimension 1: Estimation of β-Point and β-Dist

‘β-Point’ model

I ‘Estimate’ single β̀ by matching the capital–output ratio

‘β-Dist’ model—Heterogenous Impatience

I Assume uniformly distributed β across households

I Estimate the band [β̀ −∇, β̀ +∇] byminimizing distance between model (w) and data (ω) net worthheld by the top 20, 40, 60, 80%

min{β̀,∇}

∑i=20,40,60,80

(wi − ωi )2

s.t. aggregate net worth–output ratio matches steady-state value from the perfect foresight model

Results: Wealth Distribution

Percentile

US data HSCFLKS-JEDC

Β-Point

Β-Dist

0 25 50 75 1000

0.25

0.5

0.75

1F

Results: Wealth Distribution

Micro Income Process

Friedman/Buffer Stock KS-JEDC KS-Orig�

Point Uniformly Our solution HeteroDiscount DistributedFactor‡ Discount

Factors? U.S.β-Point β-Dist Data∗

Top 1% 8.6 28.4 3. 3.0 24.0 29.6Top 20% 54.3 83.4 39.5 35.0 88.0 79.5Top 40% 76.6 93.8 65.4 92.9Top 60% 90. 97.4 83.6 98.7Top 80% 97.5 99.3 95.1 100.4

Notes: ‡ : β̀ = 0.9894. ? : (β̀,∇) = (0.9867, 0.0067). Bold points are targeted. KKK t/YYY t = 10.3.

Marginal Propensity to Consume and Net Worth

Most Impatient Hleft scaleL ¯

Identical PatienceHleft scaleL ¯

­ Most Patient Hleft scaleL

mt

ctf

Histogram: empirical density ofnet worth Hright scaleL

¯

Representative agent's net worth ®

0 5 10 15 200.0

0.5

1.0

1.5

0.

0.1

0.2

0.3

0.4

0.5

0.6

Results: MPC (in Annual Terms)

Micro Income Process

Friedman/Buffer Stock KS-JEDC

β-Point β-Dist Our solution

Overall average 0.1 0.23 0.05By wealth/permanent income ratioTop 1% 0.06 0.05 0.04Top 20% 0.06 0.06 0.04Top 40% 0.06 0.08 0.04Top 60% 0.07 0.12 0.04Bottom 1/2 0.13 0.35 0.05

By employment statusEmployed 0.09 0.2 0.05Unemployed 0.23 0.54 0.06

Notes: Annual MPC is calculated by 1− (1−quarterly MPC)4.

Estimates of MPC in the Data: ∼0.2–0.6

Consumption Measure

Authors Nondurables Durables Total PCE Horizon Event/Sample

Blundell et al. (2008b)‡ 0.05 Estimation Sample: 1980–92Coronado et al. (2005) 0.36 1 Year 2003 Tax CutHausman (2012) 0.6–0.75 1 Year 1936 Veterans’ BonusJohnson et al. (2009) ∼ 0.25 3 Months 2003 Child Tax CreditLusardi (1996)‡ 0.2–0.5 Estimation Sample: 1980–87Parker (1999) 0.2 3 Months Estimation Sample: 1980–93Parker et al. (2011) 0.12–0.30 0.50–0.90 3 Months 2008 Economic StimulusSahm et al. (2009) ∼ 1/3 1 Year 2008 Economic StimulusShapiro and Slemrod (2009) ∼ 1/3 1 Year 2008 Economic StimulusSouleles (1999) 0.045–0.09 0.29–0.54 0.34–0.64 3 Months Estimation Sample: 1980–91Souleles (2002) 0.6–0.9 1 Year The Reagan Tax Cuts

of the Early 1980s

Notes: ‡: elasticity.

Typology of Our Models—Three Dimensions

1. Discount Factor βI ‘β-Point’ model: Single discount factorI ‘β-Dist’ model: Uniformly distributed discount factor

2. Empirical Wealth Variable to MatchI Net WorthI Liquid Financial Assets

3. Life CycleI Perpetual Youth (a la Blanchard)I Overlapping Generations

Dimension 2: Matching Net Worth vs. Liquid Financial Assets

­ Most impatient Hleft scaleLMost patient Hleft scaleL

¯

ctfct

mt

Histogram: empirical density ofnet worth Hright scaleL

¯

¬ Histogram: empirical density ofliquid financial asset + retirement assetsHright scaleL

0 5 10 15 200.0

0.5

1.0

1.5

0.

0.1

0.2

0.3

0.4

0.5

0.6

Liquid Assets ≡ transaction accounts, CDs, bonds, stocks, mutual funds

Matching Net Worth vs. Liquid Financial Assets

I Buffer stock saving driven by accumulation of liquidity

I May make more sense to match liquid (and retirement) assets(Hall (2011), Kaplan and Violante (2011))

I Aggregate MPC Increases Substantially: 0.23 ↑ 0.43

β-DistNet Worth Liq Fin and Ret Assets

Overall average 0.23 0.43

By wealth/permanent income ratioTop 1% 0.05 0.12Top 20% 0.06 0.13Top 40% 0.08 0.2Top 60% 0.12 0.28Bottom 1/2 0.35 0.59

Notes: Annual MPC is calculated by 1− (1−quarterly MPC)4.

Distribution of MPCsWealth Heterogeneity Translates into Heterogeneity in MPCs

Annual MPC

KS-JEDC

Matching net worth

Matching liquid financial + retirement assets

0 0.25 0.5 0.75 10

25

50

75

100Percentile

Typology of Our Models—Three Dimensions

1. Discount Factor βI ‘β-Point’ model: Single discount factorI ‘β-Dist’ model: Uniformly distributed discount factor

2. Empirical Wealth Variable to MatchI Net WorthI Liquid Financial Assets

3. Life CycleI Perpetual Youth (a la Blanchard)I Overlapping Generations

Dimension 3: Overlapping Generations

Realistic Life-Cycle Model

I Three education levels: e ∈ {D,HS ,C}I Age/education-specific income profiles

yyy t = ξtpppt = (1− τ)θtpppt ,

pppt = ψtψespppt−1

I Age-specific variances of income shocksI Transitory unemployment shock with prob u

I Household-specific mortality Des

Household Decision Problem

ves(mt) = maxct

u(ct) + β�DesEt

[ψ1−ρt+1ves+1(mt+1)

]s.t.

at = mt − ct ,

kt+1 = at/ψt+1,

mt+1 = (k + r)kt+1 + ξt+1,

at ≥ 0

Calibration

Description Parameter Value

Coefficient of relative risk aversion ρ 1Effective interest rate (r − δ) 0.01Population growth rate N 0.0025Technological growth rate Γ 0.0037Rate of high school dropouts θD 0.11Rate of high school graduates θHS 0.55Rate of college graduates θC 0.34Average initial permanent income, dropout pppD0 5000Average initial permanent income, high school pppHS0 7500Average initial permanent income, college pppC0 12000Unemployment insurance payment µ 0.15Unemployment rate u 0.07Labor income tax rate τ 0.0942

Results: Wealth Distribution in Life Cycle Model

Percentile

US data HSCFLKS-JEDC

Β-Point

Β-Dist

0 25 50 75 1000

0.25

0.5

0.75

1F

Results: MPC (in Annual Terms)

Micro Income Process Life-Cycle Model

KS-JEDC FBSOur solution β-Dist β-Point β-Dist β-Dist

Wealth Measure NW NW NW NW Liquid

Overall average 0.05 0.23 0.11 0.29 0.42By wealth/permanent income ratioTop 1% 0.04 0.05 0.08 0.07 0.07Top 20% 0.04 0.06 0.09 0.07 0.07Top 40% 0.04 0.08 0.08 0.07 0.11Top 60% 0.04 0.12 0.08 0.10 0.20Bottom 1/2 0.05 0.35 0.13 0.49 0.70

By employment statusEmployed 0.05 0.2 0.10 0.28 0.42Unemployed 0.06 0.54 0.13 0.39 0.56

Notes: Annual MPC is calculated by 1− (1−quarterly MPC)4.

Results: MPC by Age

Most patient

Most impatient

Population average

30 40 50 60 70 80 90 100Age0

0.25

0.5

0.75

1MPC

I Initial drop in MPC: Build-up of buffer stock

I Rise while rapid income growth, fall before retirement, then increasing mortality risk

Empirical Wealth Distribution Across Countries

Eurosystem Household Finance and Consumption Survey (HFCS)

I Detailed wealth data from 15 euro area countries

I Ex ante harmonized, country-representative

I 62,000 households

I First wave released in April 2013, second planned for 2016

I http://www.ecb.europa.eu/home/html/researcher_hfcn.en.html

European Evidence—Wealth Distribution

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100

150

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Austri

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Belgium

Cypru

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Spain

Finlan

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mbo

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Malt

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Portu

gal

Sloven

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Slovak

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excludes outside values

Net Wealth Liquid Assets

MPC Across European Countries—A bit Lower than US

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10.

20.

30.

40.

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All Cou

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Notes: The figure shows the range of aggregate MPCs spanned by the estimates based on the distribution of net wealth (lower bound) and of liquidassets (upper bound).

Wealth Inequality and MPCI W inequality implies higher MPC, especially for liquid assets

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0.4 0.5 0.6 0.7 0.8Gini Coefficient

Net Wealth Liquid Assets

Conclusions

What We Do

I We solve a model that matchesI Income dynamicsI Wealth distribution

Results

I The model produces more plausible implications aboutI Aggregate MPCI Distribution of MPC across households

References I

Blundell, Richard, Luigi Pistaferri, and Ian Preston (2008): “Consumption Inequality and Partial Insurance,” American Economic Review,98(5), 1887–1921.

Carroll, Christopher D. (1992): “The Buffer-Stock Theory of Saving: Some Macroeconomic Evidence,” Brookings Papers on EconomicActivity, 1992(2), 61–156, http://econ.jhu.edu/people/ccarroll/BufferStockBPEA.pdf.

Castaneda, Ana, Javier Diaz-Gimenez, and Jose-Victor Rios-Rull (2003): “Accounting for the U.S. Earnings and Wealth Inequality,”Journal of Political Economy, 111(4), 818–857.

DeBacker, Jason, Bradley Heim, Vasia Panousi, Shanthi Ramnath, and Ivan Vidangos (2013): “Rising Inequality: Transitory orPermanent? New Evidence from a Panel of US Tax Returns,” mimeo.

Friedman, Milton A. (1957): A Theory of the Consumption Function. Princeton University Press.

Hall, Robert E. (2011): “The Long Slump,” AEA Presidential Address, ASSA Meetings, Denver.

Kaplan, Greg, and Giovanni L. Violante (2011): “A Model of the Consumption Response to Fiscal Stimulus Payments,” NBER WorkingPaper Number W17338.

Krusell, Per, and Anthony A. Smith (1998): “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, 106(5),867–896.

Low, Hamish, Costas Meghir, and Luigi Pistaferri (2010): “Wage Risk and Employment Over the Life Cycle,” American Economic Review,100(4), 1432–1467.

Meghir, Costas, and Luigi Pistaferri (2004): “Income Variance Dynamics and Heterogeneity,” Journal of Business and Economic Statistics,72(1), 1–32.

Storesletten, Kjetil, Chris I. Telmer, and Amir Yaron (2004): “Cyclical Dynamics in Idiosyncratic Labor-Market Risk,” Journal of PoliticalEconomy, 112(3), 695–717.

Friedman (1957): Permanent Income Hypothesis

Yt = Pt + Tt

Ct = Pt

Progress since then

I Micro data: Friedman description of income shocks works well

I Math: Friedman’s words well describe optimal solution to dynamic stochasticoptimization problem of impatient consumers with geometric discounting underCRRA utility with uninsurable idiosyncratic risk calibrated using these microincome dynamics (!)

Our (Micro) Income Process

Idiosyncratic (household) income process is logarithmic Friedman:

yyy t+1 = pt+1ξt+1W

pt+1 = ptψt+1

pt = permanent incomeξt = transitory incomeψt+1 = permanent shockW = aggregate wage rate

Further Details of Income Process

Modifications from Carroll (1992)

Transitory income ξt incorporates unemployment insurance:

ξt = µ with probability u

= (1− τ)¯̀θt with probability 1− u

µ is UI when unemployedτ is the rate of tax collected for the unemployment benefits

What Happens After Death?

I You are replaced by a new agent whose permanent income is equal to thepopulation mean

I Prevents the population distribution of permanent income from spreading out

What Happens After Death?

I You are replaced by a new agent whose permanent income is equal to thepopulation mean

I Prevents the population distribution of permanent income from spreading out

Ergodic Distribution of Permanent Income

Exists, if death eliminates permanent shocks:

�DE[ψ2] < 1.

Holds.

Population mean of p2:

M[p2] =D

1−�DE[ψ2]

Dimension 2.a: Adding KS Aggregate Shocks

Model with KS Aggregate Shocks: Assumptions

I Only two aggregate states (good or bad)

I Aggregate productivity Zt = 1±4Z

I Unemployment rate u depends on the state (ug or ub )

Parameter values for aggregate shocks from Krusell and Smith (1998)

Parameter Value

4Z 0.01ug 0.04ub 0.10

Agg transition probability 0.125

Dimension 2.b: Adding FBS Aggregate Shocks

Friedman/Buffer Stock Shocks

I Motivation:More plausible and tractable aggregate process, also simpler

I Eliminates ‘good’ and ‘bad’ aggregate state

I Aggregate production function: KKKαt (LLLt)1−α

I LLLt = PtΞtI Pt is aggregate permanent productivityI Pt+1 = PtΨt+1

I Ξt is the aggregate transitory shock.

I Parameter values estimated from U.S. data:

Description Parameter Value

Variance of Log Ψt σ2Ψ 0.00004

Variance of Log Ξt σ2Ξ 0.00001

Dimension 2.b: Adding FBS Aggregate Shocks

Friedman/Buffer Stock Shocks

I Motivation:More plausible and tractable aggregate process, also simpler

I Eliminates ‘good’ and ‘bad’ aggregate state

I Aggregate production function: KKKαt (LLLt)1−α

I LLLt = PtΞtI Pt is aggregate permanent productivityI Pt+1 = PtΨt+1

I Ξt is the aggregate transitory shock.

I Parameter values estimated from U.S. data:

Description Parameter Value

Variance of Log Ψt σ2Ψ 0.00004

Variance of Log Ξt σ2Ξ 0.00001

Results

Our/FBS model

I A few times faster than solving KS model

I The results are similar to those under KS aggregate shocks

Results: MPC Over the Business Cycle

Model: β-Dist Krusell–Smith (KS) Friedman/Buffer Stock (FBS)

Scenario Large Bad Large BadBase Recssn Expnsn Base Perm Shock Trans Shock

Overall average 0.23 0.25 0.21 0.20 0.20 0.21By wealth/permanent income ratioTop 1% 0.05 0.05 0.05 0.05 0.05 0.05Top 10% 0.06 0.06 0.06 0.06 0.06 0.06Top 20% 0.06 0.06 0.06 0.06 0.06 0.06Top 40% 0.08 0.08 0.08 0.06 0.06 0.06Top 50% 0.09 0.10 0.09 0.06 0.06 0.09Top 60% 0.12 0.12 0.11 0.09 0.09 0.09Bottom 50% 0.35 0.38 0.32 0.32 0.32 0.32

By employment statusEmployed 0.20 0.20 0.20 0.19 0.19 0.19Unemployed 0.54 0.56 0.51 0.41 0.41 0.41

Results: MPC Over the Business Cycle

Krusell–Smith

I Aggregate and idiosyncratic shocks positively correlated

I Higher MPC during recessions, especially for the unemployed

Friedman/Buffer Stock

I Shocks uncorrelated

I MPC essentially doesn’t vary over BC

Macro Dynamics in Life-Cycle Model

I Population growth N, technological progress Γ

I Tax rate to finance social security and unemployment benefits: τ = τSS + τU

I τSS =

∑e∈{D,HS,C}

[θepppe0

∑384t=164

(((1+Γ)(1+N))−t

∏ts=0(ψes�Des)

)]∑

e∈{D,HS,C}

[θepppe0

∑163t=0

(((1+Γ)(1+N))−t

∏ts=0(ψes�Des)

)]I τU = uµ