how much insurance in bewley models? - princeton...

How Much Insurance in Bewley Models?

Greg Kaplan

New York University

Gianluca Violante

New York University, CEPR, IFS and NBER

Boston University – Macroeconomics Seminar Lunch

Kaplan-Violante, ”Insurance in Bewley Models” – p. 1/43

Consumption Insurance

• To what degree are households insulated from idiosyncraticincome fluctuations?

• Most empirical studies “reject” full insurance

1. Consumption responds to individual income shocks:Cochrane (1991), Attanasio-Davis (1996)

2. Consumption mobility exists:Fisher-Johnson (2006), Jappelli-Pistaferri (2006)

• Two key difficulties in identifying the degree of transmission ofshocks into consumption:

1. No panel data combining info on income and comprehensiveconsumption measure

2. Shocks are not observable

Recent Progress

• Blundell-Pistaferri-Preston (2007, 2008)

• Developed data by merging PSID and CEX

• Developed empirical methodology to distinguish consumptioninsurance against shocks with different “durability”

• Their findings:

1. The insurance coefficient with respect permanent shocks toafter-tax household earnings shocks is estimated at 0.36

2. The insurance coefficient with respect to transitory shocks toafter-tax household earnings is estimated at 0.95

Recent Progress

• Blundell-Pistaferri-Preston (2007, 2008)

• Developed data by merging PSID and CEX

• Developed empirical methodology to distinguish consumptioninsurance against shocks with different “durability”

• Their findings:

1. The insurance coefficient with respect permanent shocks toafter-tax household earnings shocks is estimated at 0.36

2. The insurance coefficient with respect to transitory shocks toafter-tax household earnings is estimated at 0.95

Importance of “BPP Facts” for Macroeconomics

• Can our incomplete-markets models replicate the BPP facts?

1. Degree of consumption insurance is a direct summary statisticin incomplete-markets model

2. Policy evaluation depends on insurance channels available tohouseholds (“crowding-out”)

• We begin exploring this question within the standardincomplete-markets model (“Bewley model”)

Importance of “BPP Facts” for Macroeconomics

• Can our incomplete-markets models replicate the BPP facts?

1. Degree of consumption insurance is a direct summary statisticin incomplete-markets model

2. Policy evaluation depends on insurance channels available tohouseholds (“crowding-out”)

• We begin exploring this question within the standardincomplete-markets model (“Bewley model”)

“Bewley Models”

• Key ingredients:

I Continuum of households with concave preferences overstreams of consumption

I Households face idiosyncratic exogenous earningsfluctuations

I They can borrow/save through a one-periodnon-state-contingent asset, subject to a borrowing limit

I Equilibrium in the asset market determines interest rate

• A workhorse of quantitative macroeconomics: precautionarysaving, wealth inequality, labor supply, asset pricing, fiscal policy,welfare costs of business cycles, inflation

Three Questions

1. How much consumption insurance is there in Bewley models withrespect to transitory and permanent shocks ?

2. Is this amount high or low relative to the data?

• Given the discrepancy, how do we reconcile model and data?

3. Is the BPP methodology reliable?

Three Questions

1. How much consumption insurance is there in Bewley models withrespect to transitory and permanent shocks ?

2. Is this amount high or low relative to the data?

• Given the discrepancy, how do we reconcile model and data?

3. Is the BPP methodology reliable?

Outline

1. General framework for the identification and measurement ofconsumption insurance

• BPP methodology as a special case

2. Outline and calibration of life-cycle Bewley model

• Calculation of insurance coefficients in the model

• Assessment of bias in BPP methodology

• Argument that model-data discrepancy is large

3. Reconciliation of model vs. data discrepancy

• Advance information

• Lower durability of shocks

• Other possibilities left for future work

Outline

A Framework for Measuring Insurance

• Detrended log-earnings yit for individual i of age t:

yit =

t∑

j=0

a′jxi,t−j

where xi,t−j is an (m× 1) vector of orthogonal i.i.d. shocks, andaj is an (m× 1) vector of coefficients

• Definition: insurance coefficient φx with respect to shock x

φx = 1−cov (∆cit, xit)

var (xit)

• Identification problem: realized shocks xit not directly observable

yit =

t∑

j=0

a′jxi,t−j

var (xit)

yit =

t∑

j=0

a′jxi,t−j

var (xit)

Identification Strategy

• Let yi be the entire lifetime history of income realizations forindividual i, from t = 1, ..., T

• Suppose there exist functions of observable histories of individualincome gxt (yi) such that:

cov (∆cit, xit) = cov (∆cit, gxt (yi))

var (xit) = cov (∆yit,gxt (yi))

• Then, we can identify φx as:

φx = 1−cov (∆cit, g

xt (yi))

cov (∆yit, gxt (yi))

• BPP is a special case of this strategy

Identification Strategy

• Let yi be the entire lifetime history of income realizations forindividual i, from t = 1, ..., T

• Suppose there exist functions of observable histories of individualincome gxt (yi) such that:

cov (∆cit, xit) = cov (∆cit, gxt (yi))

var (xit) = cov (∆yit,gxt (yi))

• Then, we can identify φx as:

φx = 1−cov (∆cit, g

xt (yi))

cov (∆yit, gxt (yi))

• BPP is a special case of this strategy

The BPP Methodology

1. Permanent + Transitory earnings process

• Recall our log-earnings representation:

yit =

t∑

j=0

a′jxi,t−j

• Set m = 2, xit = (ηit, εit)′, a0 = (1, 1)

′ and aj = (1, 0)′, j ≥ 1

∆yit = ηit +∆εit

• MaCurdy (1982), Abowd-Card (1989), Carroll (1997)

The BPP Methodology (transitory shocks)

2. Identify φε through the function:

gεt (yi) = ∆yi,t+1

= ηi,t+1 + εi,t+1 − εit

and note that:

cov (∆yit,∆yi,t+1) = −var (εit)

cov (∆cit,∆yi,t+1) = −cov (∆cit, εit)

where the second equality requires:

A1 [no advanced info]: cov (∆cit, ηi,t+1) = cov (∆cit, εi,t+1) = 0

The BPP Methodology (transitory shocks)

2. Identify φε through the function:

gεt (yi) = ∆yi,t+1

= ηi,t+1 + εi,t+1 − εit

and note that:

cov (∆yit,∆yi,t+1) = −var (εit)

cov (∆cit,∆yi,t+1) = −cov (∆cit, εit)

The BPP Methodology (permanent shocks)

3. Identify φη through the function:

gηt (yi) = ∆yi,t−1 +∆yit +∆yi,t+1

= ηi,t−1 + ηit + ηi,t+1 + εi,t−2 + εi,t+1

and note that:

cov (∆yit,∆yi,t−1 +∆yit +∆yi,t+1) = var (ηit)

cov (∆cit,∆yi,t−1 +∆yit +∆yi,t+1) = cov (∆cit, ηit)

A2 [short memory]: cov (∆cit, ηi,t−1) = cov (∆cit, εi,t−2) = 0

The BPP Methodology (permanent shocks)

3. Identify φη through the function:

gηt (yi) = ∆yi,t−1 +∆yit +∆yi,t+1

= ηi,t−1 + ηit + ηi,t+1 + εi,t−2 + εi,t+1

and note that:

cov (∆yit,∆yi,t−1 +∆yit +∆yi,t+1) = var (ηit)

cov (∆cit,∆yi,t−1 +∆yit +∆yi,t+1) = cov (∆cit, ηit)

A2 [short memory]: cov (∆cit, ηi,t−1) = cov (∆cit, εi,t−2) = 0

BPP Estimation: Main Results

1. The insurance coefficient with respect permanent shocks toafter-tax household earnings shocks is estimated to be φη = 0.36

2. The insurance coefficient with respect to transitory shocks toafter-tax household earnings is estimated to be φε = 0.95

3. The estimated age profile of φηt is roughly flat

A Life-cycle Bewley Economy

• Demographics: Overlapping generations of households who liveup to T periods: work until age T ret, and retire thereafter.Unconditional survival rate ξt < 1 after retirement

• Preferences: E0∑T

t=1 βt−1ξt

C1−γ

it−1

1−γ

• Idiosyncratic households (after-tax) earnings process:

log Yit = κt + yit = κt + zit + εit

zit = zi,t−1 + ηit

I κt common deterministic experience profile

I zit permanent component, εit transitory component

I zi0 is drawn from a given initial distribution

t=1 βt−1ξt

C1−γ

it−1

1−γ

t=1 βt−1ξt

C1−γ

it−1

1−γ

t=1 βt−1ξt

C1−γ

it−1

1−γ

• Markets: Households can borrow (up to A ≤ 0) and save throughrisk-free bond. Perfect annuity markets.

• World interest rate: r

• Government: Social security benefits P (Yi) paid to retirees

• Budget constraints:

Cit +Ai,t+1 = (1 + r)Ait + Yit, if t < T ret

Cit +ξt

ξt+1Ai,t+1 = (1 + r)Ait + P (Yi), if t ≥ T ret

Cit +ξt

Calibration

• Preferences:

I Relative risk aversion coefficient: γ = 2

I Discount factor β to replicate aggregate net-worth-incomeratio of 2.5 for bottom 95% of US households

• Interest rate: r = 3%

• Earnings process:

I Rise in earnings dispersion over lifecycle: ση = 0.01

I Initial earnings dispersion: σz0 = 0.15

I BPP estimate: σε = 0.05

Calibration

• Debt limit: Natural or no-borrowing constraints

• Initial wealth: Zero or calibrated to net-worth distribution of 20-30years-old

• Social security:

1. Net earnings ⇒ gross earnings by inverting Gouveia-Strausstax function

2. Benefits modelled as concave function of gross averagelifetime earnings, as in US two-bendpoint system

3. Benefits partially taxed

Lifecycle Implications

30 40 50 60 70 80 90

0

0.5

1

1.5

2

x 105

Age

$ (0

0,00

0)

Lifecycle Means

Natural BCZero BC

30 40 50 60 70 80 900.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Age

Var

Log

s

Lifecycle Inequality

Natural BCZero BCWealth

ConsumptionNet earnings

Net benefits

Consumption

Net earnings

Net benefits

Baseline Economy

Permanent Shock Transitory ShockData Model Model Data Model ModelBPP BPP TRUE BPP BPP TRUE

Natural Borrowing Limit0.36

(0.09)0.22 0.23

0.95

(0.04)0.94 0.94

Zero Borrowing Limit0.36

(0.09)0.07 0.23

0.95

(0.04)0.82 0.82

• Model has right amount of insurance wrt transitory shock(if borrowing limit is loose)

• Model has less insurance than data wrt permanent shock

Baseline Economy

(0.09)0.22 0.23

0.95

(0.04)0.94 0.94

(0.09)0.07 0.23

0.95

(0.04)0.82 0.82

Baseline Economy

(0.09)0.22 0.23

0.95

(0.04)0.94 0.94

(0.09)0.07 0.23

0.95

(0.04)0.82 0.82

Baseline Economy

(0.09)0.22 0.23

0.95

(0.04)0.94 0.94

(0.09)0.07 0.23

0.95

(0.04)0.82 0.82

• BPP coefficient for transitory shocks are unbiased

• BPP coefficient for permanent shocks are downward biased

I Bias massive for no-borrowing economy

Baseline Economy

(0.09)0.22 0.23

0.95

(0.04)0.94 0.94

(0.09)0.07 0.23

0.95

(0.04)0.82 0.82

Baseline Economy

(0.09)0.22 0.23

0.95

(0.04)0.94 0.94

(0.09)0.07 0.23

0.95

(0.04)0.82 0.82

Age profile of φε

30 35 40 45 50 55

0.4

0.5

0.6

0.7

0.8

0.9

1

Age

φε

Natural BC

TRUEBPP

30 35 40 45 50 55

0.4

0.5

0.6

0.7

0.8

0.9

1

Age

φε

Zero BC

TRUEBPP

• Ability to borrow crucial to smooth transitory shocks at young ages

Age profile of φη

30 35 40 45 50 55

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Age

φη

Natural BC

30 35 40 45 50 55

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Age

φη

Zero BC

TRUEBPP

•• Age profile of insurance coefficients against permanent shocks(φηt ) in the model is increasing, whereas in the data it is flat

Age profile of φη

30 35 40 45 50 55

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Age

φη

Natural BC

30 35 40 45 50 55

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Age

φη

Zero BC

TRUEBPP

• Age profile of insurance coefficients against permanent shocks(φηt ) in the model is increasing, whereas in the data it is flat

Age profile of φη

30 35 40 45 50 55

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Age

φη

Natural BC

30 35 40 45 50 55

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Age

φη

Zero BC

TRUEBPP

• Bias in BPP estimator large when agents are close to theconstraint

Why the Downward Bias in BPP Estimator?

• From the definition of φηBPP :

φηBPP = 1−cov (∆cit,∆yi,t−1 +∆yit +∆yi,t+1)

cov (∆yit,∆yi,t−1 +∆yit +∆yi,t+1)

= 1−cov (∆cit, ηi,t−1 + εi,t−2 + ηit + ηi,t+1 + εi,t+1)

var (ηit)

= φη +cov (∆cit, ηi,t−1 + εi,t−2)

var (ηit)︸︷︷︸

A2: short memory

+cov (∆cit, ηi,t+1 + εi,t+1)

A1: no adv. info

= φη +cov (∆cit, εi,t−2)

¿0

• Last term large when agent close to borr. constr. at t− 2

Sensitivity Analysis (Natural BC)

Permanent Shock Transitory ShockTRUE (0.23) BPP (0.22) TRUE (0.94) BPP (0.94)

Initial Wealth Dist. 0.23 0.23 0.94 0.94γ = 5 0.27 0.24 0.93 0.93

γ = 10 0.32 0.29 0.92 0.92

Rep. ratio = 0.25 0.19 0.17 0.93 0.93Rep. ratio = 0.65 0.27 0.26 0.94 0.94ση = 0.02 0.25 0.24 0.93 0.93

ση = 0.005 0.22 0.20 0.94 0.94

σz0 = 0.2 0.23 0.22 0.94 0.94

σz0 = 0.1 0.24 0.22 0.94 0.94

σε = 0.075 0.24 0.22 0.94 0.94

σε = 0.025 0.23 0.22 0.94 0.94

Sensitivity Analysis (K/Y and r)

1 2 3 40.05

0.1

0.15

0.2

0.25

0.3

0.35

Wealth−Income Ratio

φη

− In

s. c

oeff.

for p

erm

. sho

ck

Natural BC

1 2 3 40.05

0.1

0.15

0.2

0.25

0.3

0.35

Wealth−Income Ratio

φη

− In

s. c

oeff.

for p

erm

. sho

ck

Zero BC

r=2%r=3%r=4%r=5%

A Welfare Calculation

• Economy where agents survive with probability ζ = 1/ (1 + π),discount future at rate β = 1/ (1 + ρ)

• Consumption allocation:

cit = (1− φη) zit

• Log-Normal shocks

• The welfare cost of going from φη = 0.36 to φ̃η = 0.23 is:

ω 'γ

2

[

(1− φη)2−(

1− φ̃η)2]

σηρ+ π

• With γ = 2, ρ = 0.03, π = 0.0286, and ση = 0.01: ω = −3.1%

ω 'γ

2

[

(1− φη)2−(

1− φ̃η)2]

σηρ+ π

• With γ = 2, ρ = 0.03, π = 0.0286, and ση = 0.01: ω = −3.1%

ω 'γ

2

[

(1− φη)2−(

1− φ̃η)2]

σηρ+ π

• With γ = 2, ρ = 0.03, π = 0.0286, and ση = 0.01: ω = −3.1%

ω 'γ

2

[

(1− φη)2−(

1− φ̃η)2]

σηρ+ π

• With γ = 2, ρ = 0.03, π = 0.0286, and ση = 0.01: ω = −3.1%

Advance Information

• Model I: households observe, one period in advance, a fraction ofthe permanent shock

• Model II: households know their own deterministic income profileat age t = 0 (e.g., Lillard-Weiss, 1979)

• Given BPP identification method, neither form of advanceinformation can reconcile model and data

Advance Information

Preempting the permanent shock

• Permanent income growth in period t comprises of two orthogonaladditive components, ηsit and η

ait

• The component ηait is already in the information set of the agent attime t− 1

• From the definition of insurance coefficient:

φη = 1−cov (∆cit, ηit)

var (ηit)= 1−

cov (∆cit, ηsit + η

ait)

var (ηsit + ηait)

=var (ηsit)

var (ηit)φη

s

+var (ηait)

var (ηit)

[

1−cov (∆cit, η

ait)

var (ηait)

]

≈ (1− α)φηs

+ α

increasing in α, since with loose borrowing limits cov (∆cit, ηait) ≈ 0

• Permanent income growth in period t comprises of two orthogonaladditive components, ηsit and η

ait

• The component ηait is already in the information set of the agent attime t− 1

• From the definition of insurance coefficient:

φη = 1−cov (∆cit, ηit)

var (ηit)= 1−

cov (∆cit, ηsit + η

ait)

var (ηsit + ηait)

=var (ηsit)

var (ηit)φη

s

+var (ηait)

var (ηit)

[

1−cov (∆cit, η

ait)

var (ηait)

]

≈ (1− α)φηs

+ α

increasing in α, since with loose borrowing limits cov (∆cit, ηait) ≈ 0

• Ignoring the usual downward bias, the BPP methodology yields:

= 1−cov

(∆cit, η

sit + η

ai,t + η

ai,t+1

)

var (ηsit + ηait)

≈ (1− α)φηs

+ α

[

1−cov

(∆cit, η

ai,t+1

)

var (ηait)

]

≈ φηs

• BPP estimator is independent of the amount of advanceinformation

• Simulations confirm this finding

• Ignoring the usual downward bias, the BPP methodology yields:

= 1−cov

(∆cit, η

sit + η

ai,t + η

ai,t+1

)

var (ηsit + ηait)

≈ (1− α)φηs

+ α

[

1−cov

(∆cit, η

ai,t+1

)

var (ηait)

]

≈ φηs

• BPP estimator is independent of the amount of advanceinformation

• Simulations confirm this finding

Predictable individual income profile

• Generalize log-earnings (deviations from common age-profile) to:

yit = βit+ zit + εit

zit = zi,t−1 + ηit,

with E [βi] = 0 in the cross-section, and SD [βi] = σβ

• The individual-specific slope βi is learned at time zero

• Lillard-Weiss (1979), Baker (1997), Haider (2001),Guvenen (2007)

• When we increase σβ , we decrease ση accordingly to keep thetotal rise in lifetime earnings inequality constant

Permanent Shock Transitory ShockData 0.36 (0.09) 0.95 (0.04)

Model Model Model ModelTRUE BPP TRUE BPP

Natural BC40% 0.23 0.25 0.94 0.94

60% 0.23 0.28 0.94 0.94

80% 0.22 0.37 0.94 0.94

Zero BC40% 0.23 −0.01 0.82 0.82

60% 0.23 −0.10 0.82 0.82

80% 0.23 −0.31 0.82 0.82

• Upward bias in BPP coefficient with natural BC

Permanent Shock Transitory ShockData 0.36 (0.09) 0.95 (0.04)

Model Model Model ModelTRUE BPP TRUE BPP

Natural BC40% 0.23 0.25 0.94 0.94

60% 0.23 0.28 0.94 0.94

80% 0.22 0.37 0.94 0.94

Zero BC40% 0.23 −0.01 0.82 0.82

60% 0.23 −0.10 0.82 0.82

80% 0.23 −0.31 0.82 0.82

• Additional downward bias in BPP coefficient with zero BC

Why the Upward Bias in the BPP Estimator?

= 1−cov (∆cit, ηi,t−1 + εi,t−2 + ηit + ηi,t+1 + εi,t+1 + 3βi)

var (ηit) + 3var (βi)

• Ignoring usual downward bias due to binding constraint:

φηBPP ≈

[var (ηit)

]

φη+

[3var (βi)

] [

1−cov (∆cit, βi)

var (βi)

]

= (1− α)φη + αφβ

φβ ≈ 1 with loose borrowing constraints (upward bias)

φβ ≈ 0 with tight borrowing constraints (downward bias)

Why the Upward Bias in the BPP Estimator?

= 1−cov (∆cit, ηi,t−1 + εi,t−2 + ηit + ηi,t+1 + εi,t+1 + 3βi)

• Ignoring usual downward bias due to binding constraint:

φηBPP ≈

[var (ηit)

]

φη+

[3var (βi)

] [

1−cov (∆cit, βi)

var (βi)

]

= (1− α)φη + αφβ

φβ ≈ 1 with loose borrowing constraints (upward bias)

φβ ≈ 0 with tight borrowing constraints (downward bias)

Persistent (rather than permanent...) shocks

• Generalize log-earnings process to AR(1) + transitory:

yit = zit + εit

zit = ρzit−1 + ηit, with ρ < 1

• BPP instruments no longer valid [misspecification]

• Define quasi-difference: ∆̃yt ≡ yt − ρyt−1

• Identification of (φη, φε) can still be achieved by setting

gεt (yi) = ∆̃yt+1

gηt (yi) = ρ2∆̃yt−1 + ρ∆̃yt + ∆̃yt+1

under same assumptions A1 & A2

yit = zit + εit

gηt (yi) = ρ2∆̃yt−1 + ρ∆̃yt + ∆̃yt+1

yit = zit + εit

gηt (yi) = ρ2∆̃yt−1 + ρ∆̃yt + ∆̃yt+1

Persistent shocks

Persistent Shock Transitory ShockData 0.36 (0.09) 0.95 (0.04)

TRUE BPP BPP (missp.) TRUE BPP BPP (missp.)Natural BCρ = 0.99 0.30 0.28 0.28 0.93 0.93 0.93

ρ = 0.97 0.39 0.39 0.39 0.93 0.93 0.92

ρ = 0.95 0.47 0.46 0.46 0.93 0.93 0.90

Zero BCρ = 0.99 0.27 0.17 0.16 0.82 0.82 0.82

ρ = 0.97 0.33 0.28 0.27 0.81 0.81 0.81

ρ = 0.95 0.38 0.35 0.33 0.81 0.81 0.80

ρ = 0.93 0.42 0.42 0.38 0.81 0.82 0.78

• Reconciliation of model and data for ρ ∈ (0.93, 0.97)

Persistent shocks

ρ = 0.97 0.39 0.39 0.39 0.93 0.93 0.92

ρ = 0.95 0.47 0.46 0.46 0.93 0.93 0.90

Zero BCρ = 0.99 0.27 0.17 0.16 0.82 0.82 0.82

ρ = 0.97 0.33 0.28 0.27 0.81 0.81 0.81

ρ = 0.95 0.38 0.35 0.33 0.81 0.81 0.80

ρ = 0.93 0.42 0.42 0.38 0.81 0.82 0.78

• Misspecification bias in BPP estimator is small

Persistent shocks

ρ = 0.97 0.39 0.39 0.39 0.93 0.93 0.92

ρ = 0.95 0.47 0.46 0.46 0.93 0.93 0.90

Zero BCρ = 0.99 0.27 0.17 0.16 0.82 0.82 0.82

ρ = 0.97 0.33 0.28 0.27 0.81 0.81 0.81

ρ = 0.95 0.38 0.35 0.33 0.81 0.81 0.80

ρ = 0.93 0.42 0.42 0.38 0.81 0.82 0.78

• Usual downward bias in BPP estimator

Persistent shocks

ρ = 0.97 0.39 0.39 0.39 0.93 0.93 0.92

ρ = 0.95 0.47 0.46 0.46 0.93 0.93 0.90

Zero BCρ = 0.99 0.27 0.17 0.16 0.82 0.82 0.82

ρ = 0.97 0.33 0.28 0.27 0.81 0.82 0.81

ρ = 0.95 0.38 0.35 0.33 0.81 0.81 0.80

ρ = 0.93 0.42 0.42 0.38 0.81 0.82 0.78

• Insurance coefficients for transitory shocks unaffected

Age profile of φη

30 35 40 45 50 55

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Age

φη

Natural BC

TRUEBPP missp.

30 35 40 45 50 55

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Age

φη

Zero BC

TRUEBPP missp.

ρ=0.97

ρ=1

ρ=0.93

ρ=1

• In the model, age profile of insurance coefficients wrt to persistentshocks is flatter, hence closer to the data

Relationship with STY

0.92 0.94 0.96 0.98 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Autocorrelation coefficient (ρ)

1−φη

One minus the insurance coeff.

NBCZBC

0.92 0.94 0.96 0.98 1

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Incr

ease

from

age

25−

60

Increase in variance of log cons.

NBCZBC

• These two norms of consumption insurance can disagree

Relationship with STY

0.92 0.94 0.96 0.98 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

1−φη

One minus the insurance coeff.

NBCZBC

0.92 0.94 0.96 0.98 1

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Incr

ease

from

age

25−

60

Increase in variance of log cons.

NBCZBC

• These two norms of consumption insurance can disagree

Age profile of wealth: model vs. data

30 40 50 60 70 80 90

0

0.5

1

1.5

2

x 105

Age

$ (0

0,00

0)

Lifecycle wealth profiles

Natural BC (ρ=0.97)

Zero BC (ρ=0.93)

SCF Net worth (89−92)

• A version of the model with more realistic age profile of wealthwould be also more successful in replicating the BPP facts

Age profile of wealth: model vs. data

30 40 50 60 70 80 90

0

0.5

1

1.5

2

x 105

Age

$ (0

0,00

0)

Lifecycle wealth profiles

Natural BC (ρ=0.97)

Zero BC (ρ=0.93)

SCF Net worth (89−92)

• A version of the model with more realistic age profile of wealthwould be also more successful in replicating the BPP facts

Conclusions

1. We generalized BPP methodology, and argued that insurancecoefficients should become a key summary statistic of IM models

2. BPP estimator downward biased when BC tight

3. Plausibly calibrated Bewley model has too little insurance

4. Ins. coeff. 6= rise in consumption inequality over life cycle

5. Advance information does not reconcile model and data

6. A (very) persistent income shock goes a long way

7. Modifications of model that get age-wealth profile right promising

Consumption InsuranceConsumption InsuranceConsumption InsuranceConsumption Insurance

Recent ProgressRecent Progress

Importance of ``BPP Facts'' for MacroeconomicsImportance of ``BPP Facts'' for Macroeconomics

``Bewley Models"Three QuestionsThree Questions

OutlineOutlineOutline

A Framework for Measuring InsuranceA Framework for Measuring InsuranceA Framework for Measuring Insurance

Identification StrategyIdentification Strategy

The BPP MethodologyThe BPP Methodology (transitory shocks)The BPP Methodology (transitory shocks)

The BPP Methodology (permanent shocks)The BPP Methodology (permanent shocks)

BPP Estimation: Main ResultsA Life-cycle Bewley EconomyA Life-cycle Bewley EconomyA Life-cycle Bewley EconomyA Life-cycle Bewley Economy

A Life-cycle Bewley EconomyA Life-cycle Bewley EconomyA Life-cycle Bewley EconomyA Life-cycle Bewley Economy

CalibrationCalibrationLifecycle ImplicationsBaseline EconomyBaseline EconomyBaseline Economy

Baseline EconomyBaseline EconomyBaseline Economy

Age profile of $phi ^{varepsilon }$Age profile of $phi ^{eta }$Age profile of $phi ^{eta }$

Age profile of $phi ^{eta }$Why the Downward Bias in BPP Estimator?Sensitivity Analysis (Natural BC)Sensitivity Analysis (K/Y and r)A Welfare CalculationA Welfare CalculationA Welfare CalculationA Welfare Calculation

Advance InformationAdvance InformationAdvance InformationAdvance Information

Preempting the permanent shockPreempting the permanent shock

Predictable individual income profilePredictable individual income profilePredictable individual income profile

Predictable individual income profilePredictable individual income profileWhy the Upward Bias in the BPP Estimator?Why the Upward Bias in the BPP Estimator?

Persistent (rather than permanent...) shocksPersistent (rather than permanent...)shocksPersistent (rather than permanent...)shocks

Persistent shocksPersistent shocksPersistent shocksPersistent shocksAge profile of $phi ^{eta }$Relationship with STYRelationship with STY

Age profile of wealth: model vs. dataAge profile of wealth: model vs. data

Conclusions

how much insurance in bewley models? - princeton...

Documents