How Much Insurance in Bewley Models?

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 idiosyncratic income fluctuations? Kaplan-Violante, Insurance in Bewley Models p. 2/43

Consumption Insurance To what degree are households insulated from idiosyncratic income 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 of shocks into consumption: 1. No panel data combining info on income and comprehensive consumption measure Kaplan-Violante, Insurance in Bewley Models p. 2/43

Consumption Insurance To what degree are households insulated from idiosyncratic income 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 of shocks into consumption: 1. No panel data combining info on income and comprehensive consumption measure 2. Shocks are not observable Kaplan-Violante, Insurance in Bewley Models p. 2/43

Recent Progress Blundell-Pistaferri-Preston (2007, 2008) Developed data by merging PSID and CEX Developed empirical methodology to distinguish consumption insurance against shocks with different durability Their findings: 1. The insurance coefficient with respect permanent shocks to after-tax household earnings shocks is estimated at 0.36 2. The insurance coefficient with respect to transitory shocks to after-tax household earnings is estimated at 0.95 Kaplan-Violante, Insurance in Bewley Models p. 3/43

Importance of BPP Facts for Macroeconomics Can our incomplete-markets models replicate the BPP facts? Kaplan-Violante, Insurance in Bewley Models p. 4/43

Importance of BPP Facts for Macroeconomics Can our incomplete-markets models replicate the BPP facts? 1. Degree of consumption insurance is a direct summary statistic in incomplete-markets model 2. Policy evaluation depends on insurance channels available to households ( crowding-out ) We begin exploring this question within the standard incomplete-markets model ( Bewley model ) Kaplan-Violante, Insurance in Bewley Models p. 4/43

Bewley Models Key ingredients: Continuum of households with concave preferences over streams of consumption Households face idiosyncratic exogenous earnings fluctuations They can borrow/save through a one-period non-state-contingent asset, subject to a borrowing limit Equilibrium in the asset market determines interest rate A workhorse of quantitative macroeconomics: precautionary saving, wealth inequality, labor supply, asset pricing, fiscal policy, welfare costs of business cycles, inflation Kaplan-Violante, Insurance in Bewley Models p. 5/43

Three Questions 1. How much consumption insurance is there in Bewley models with respect 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? Kaplan-Violante, Insurance in Bewley Models p. 6/43

Three Questions 1. How much consumption insurance is there in Bewley models with respect 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? Kaplan-Violante, Insurance in Bewley Models p. 6/43

Outline 1. General framework for the identification and measurement of consumption insurance BPP methodology as a special case Kaplan-Violante, Insurance in Bewley Models p. 7/43

Outline 1. General framework for the identification and measurement of consumption 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 Kaplan-Violante, Insurance in Bewley Models p. 7/43

Outline 1. General framework for the identification and measurement of consumption 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 Kaplan-Violante, Insurance in Bewley Models p. 7/43

A Framework for Measuring Insurance Detrended log-earnings y it for individual i of age t: y it = t a jx i,t j j=0 where x i,t j is an (m 1) vector of orthogonal i.i.d. shocks, and a j is an (m 1) vector of coefficients Definition: insurance coefficient φ x with respect to shock x φ x = 1 cov ( c it, x it ) var (x it ) Kaplan-Violante, Insurance in Bewley Models p. 8/43

A Framework for Measuring Insurance Detrended log-earnings y it for individual i of age t: y it = t a jx i,t j j=0 where x i,t j is an (m 1) vector of orthogonal i.i.d. shocks, and a j is an (m 1) vector of coefficients Definition: insurance coefficient φ x with respect to shock x φ x = 1 cov ( c it, x it ) var (x it ) Identification problem: realized shocks x it not directly observable Kaplan-Violante, Insurance in Bewley Models p. 8/43

Identification Strategy Let y i be the entire lifetime history of income realizations for individual i, from t = 1,..., T Suppose there exist functions of observable histories of individual income g x t (y i ) such that: Then, we can identify φ x as: cov ( c it, x it ) = cov ( c it, g x t (y i )) var (x it ) = cov ( y it,g x t (y i )) φ x = 1 cov ( c it, g x t (y i )) cov ( y it, g x t (y i )) Kaplan-Violante, Insurance in Bewley Models p. 9/43

Identification Strategy Let y i be the entire lifetime history of income realizations for individual i, from t = 1,..., T Suppose there exist functions of observable histories of individual income g x t (y i ) such that: Then, we can identify φ x as: cov ( c it, x it ) = cov ( c it, g x t (y i )) var (x it ) = cov ( y it,g x t (y i )) BPP is a special case of this strategy φ x = 1 cov ( c it, g x t (y i )) cov ( y it, g x t (y i )) Kaplan-Violante, Insurance in Bewley Models p. 9/43

The BPP Methodology 1. Permanent + Transitory earnings process Recall our log-earnings representation: y it = t a jx i,t j j=0 Set m = 2, x it = (η it, ε it ), a 0 = (1, 1) and a j = (1, 0), j 1 y it = η it + ε it MaCurdy (1982), Abowd-Card (1989), Carroll (1997) Kaplan-Violante, Insurance in Bewley Models p. 10/43

The BPP Methodology (transitory shocks) 2. Identify φ ε through the function: g ε t (y i ) = y i,t+1 = η i,t+1 + ε i,t+1 ε it Kaplan-Violante, Insurance in Bewley Models p. 11/43

The BPP Methodology (transitory shocks) 2. Identify φ ε through the function: gt ε (y i ) = y i,t+1 = η i,t+1 + ε i,t+1 ε it and note that: cov ( y it, y i,t+1 ) = var (ε it ) cov ( c it, y i,t+1 ) = cov ( c it, ε it ) where the second equality requires: A1 [no advanced info]: cov ( c it, η i,t+1 ) = cov ( c it, ε i,t+1 ) = 0 Kaplan-Violante, Insurance in Bewley Models p. 11/43

The BPP Methodology (permanent shocks) 3. Identify φ η through the function: g η t (y i ) = y i,t 1 + y it + y i,t+1 = η i,t 1 + η it + η i,t+1 + ε i,t 2 + ε i,t+1 Kaplan-Violante, Insurance in Bewley Models p. 12/43

The BPP Methodology (permanent shocks) 3. Identify φ η through the function: g η t (y i ) = y i,t 1 + y it + y i,t+1 = η i,t 1 + η it + η i,t+1 + ε i,t 2 + ε i,t+1 and note that: cov ( y it, y i,t 1 + y it + y i,t+1 ) = var (η it ) cov ( c it, y i,t 1 + y it + y i,t+1 ) = cov ( c it, η it ) where the second equality requires: A1 [no advanced info]: cov ( c it, η i,t+1 ) = cov ( c it, ε i,t+1 ) = 0 A2 [short memory]: cov ( c it, η i,t 1 ) = cov ( c it, ε i,t 2 ) = 0 Kaplan-Violante, Insurance in Bewley Models p. 12/43

BPP Estimation: Main Results 1. The insurance coefficient with respect permanent shocks to after-tax household earnings shocks is estimated to be φ η = 0.36 2. The insurance coefficient with respect to transitory shocks to after-tax household earnings is estimated to be φ ε = 0.95 3. The estimated age profile of φ η t is roughly flat Kaplan-Violante, Insurance in Bewley Models p. 13/43

A Life-cycle Bewley Economy Kaplan-Violante, Insurance in Bewley Models p. 14/43

A Life-cycle Bewley Economy Demographics: Overlapping generations of households who live up to T periods: work until age T ret, and retire thereafter. Unconditional survival rate ξ t < 1 after retirement Preferences: 1 1 γ T E 0 t=1 βt 1 C ξ 1 γ it t Idiosyncratic households (after-tax) earnings process: log Y it = κ t + y it = κ t + z it + ε it z it = z i,t 1 + η it κ t common deterministic experience profile z it permanent component, ε it transitory component z i0 is drawn from a given initial distribution Kaplan-Violante, Insurance in Bewley Models p. 14/43

A Life-cycle Bewley Economy Markets: Households can borrow (up to A 0) and save through risk-free bond. Perfect annuity markets. Kaplan-Violante, Insurance in Bewley Models p. 15/43

A Life-cycle Bewley Economy Markets: Households can borrow (up to A 0) and save through risk-free bond. Perfect annuity markets. World interest rate: r Kaplan-Violante, Insurance in Bewley Models p. 15/43

A Life-cycle Bewley Economy Markets: Households can borrow (up to A 0) and save through risk-free bond. Perfect annuity markets. World interest rate: r Government: Social security benefits P (Y i ) paid to retirees Budget constraints: C it + A i,t+1 = (1 + r) A it + Y it, if t < T ret C it + ξ t ξ t+1 A i,t+1 = (1 + r) A it + P (Y i ), if t T ret Kaplan-Violante, Insurance in Bewley Models p. 15/43

Calibration Preferences: Relative risk aversion coefficient: γ = 2 Discount factor β to replicate aggregate net-worth-income ratio of 2.5 for bottom 95% of US households Interest rate: r = 3% Earnings process: Rise in earnings dispersion over lifecycle: σ η = 0.01 Initial earnings dispersion: σ z0 = 0.15 BPP estimate: σ ε = 0.05 Kaplan-Violante, Insurance in Bewley Models p. 16/43

Calibration Debt limit: Natural or no-borrowing constraints Initial wealth: Zero or calibrated to net-worth distribution of 20-30 years-old Social security: 1. Net earnings gross earnings by inverting Gouveia-Strauss tax function 2. Benefits modelled as concave function of gross average lifetime earnings, as in US two-bendpoint system 3. Benefits partially taxed Kaplan-Violante, Insurance in Bewley Models p. 17/43

Lifecycle Implications x 10 5 Lifecycle Means Lifecycle Inequality 2 1.5 Natural BC Zero BC Wealth 0.55 0.5 0.45 Net earnings Natural BC Zero BC $ (00,000) 1 Var Logs 0.4 0.35 0.3 Consumption 0.25 0.5 Net earnings Consumption 0.2 0.15 0 Net benefits 0.1 Net benefits 30 40 50 60 70 80 90 Age 0.05 30 40 50 60 70 80 90 Age Kaplan-Violante, Insurance in Bewley Models p. 18/43

Baseline Economy Natural Borrowing Limit Zero Borrowing Limit Permanent Shock Transitory Shock Data Model Model Data Model Model BPP BPP TRUE BPP BPP TRUE 0.36 (0.09) 0.36 (0.09) 0.22 0.23 0.07 0.23 0.95 (0.04) 0.95 (0.04) 0.94 0.94 0.82 0.82 Model has right amount of insurance wrt transitory shock (if borrowing limit is loose) Kaplan-Violante, Insurance in Bewley Models p. 19/43

Baseline Economy Natural Borrowing Limit Zero Borrowing Limit Permanent Shock Transitory Shock Data Model Model Data Model Model BPP BPP TRUE BPP BPP TRUE 0.36 (0.09) 0.36 (0.09) 0.22 0.23 0.07 0.23 0.95 (0.04) 0.95 (0.04) 0.94 0.94 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 Kaplan-Violante, Insurance in Bewley Models p. 19/43

Baseline Economy Natural Borrowing Limit Zero Borrowing Limit Permanent Shock Transitory Shock Data Model Model Data Model Model BPP BPP TRUE BPP BPP TRUE 0.36 (0.09) 0.36 (0.09) 0.22 0.23 0.07 0.23 0.95 (0.04) 0.95 (0.04) 0.94 0.94 0.82 0.82 BPP coefficient for transitory shocks are unbiased Kaplan-Violante, Insurance in Bewley Models p. 20/43

Baseline Economy Natural Borrowing Limit Zero Borrowing Limit Permanent Shock Transitory Shock Data Model Model Data Model Model BPP BPP TRUE BPP BPP TRUE 0.36 (0.09) 0.36 (0.09) 0.22 0.23 0.07 0.23 0.95 (0.04) 0.95 (0.04) 0.94 0.94 0.82 0.82 BPP coefficient for transitory shocks are unbiased BPP coefficient for permanent shocks are downward biased Bias massive for no-borrowing economy Kaplan-Violante, Insurance in Bewley Models p. 20/43

Age profile of φ ε 1 Natural BC 1 Zero BC 0.9 0.9 0.8 0.8 φ ε 0.7 TRUE BPP φ ε 0.7 TRUE BPP 0.6 0.6 0.5 0.5 0.4 0.4 30 35 40 45 50 55 Age 30 35 40 45 50 55 Age Ability to borrow crucial to smooth transitory shocks at young ages Kaplan-Violante, Insurance in Bewley Models p. 21/43

Age profile of φ η 1 Natural BC 1 Zero BC 0.8 0.8 0.6 0.6 0.4 0.4 φ η 0.2 φ η 0.2 0 0 0.2 0.2 0.4 TRUE BPP 0.4 TRUE BPP 0.6 0.6 30 35 40 45 50 55 Age 30 35 40 45 50 55 Age Kaplan-Violante, Insurance in Bewley Models p. 22/43

Age profile of φ η 1 Natural BC 1 Zero BC 0.8 0.8 0.6 0.6 0.4 0.4 φ η 0.2 φ η 0.2 0 0 0.2 0.2 0.4 TRUE BPP 0.4 TRUE BPP 0.6 0.6 30 35 40 45 50 55 Age 30 35 40 45 50 55 Age Age profile of insurance coefficients against permanent shocks (φ η t ) in the model is increasing, whereas in the data it is flat Kaplan-Violante, Insurance in Bewley Models p. 22/43

Age profile of φ η 1 Natural BC 1 Zero BC 0.8 0.8 0.6 0.6 0.4 0.4 φ η 0.2 φ η 0.2 0 0 0.2 0.2 0.4 TRUE BPP 0.4 TRUE BPP 0.6 0.6 30 35 40 45 50 55 Age 30 35 40 45 50 55 Age Bias in BPP estimator large when agents are close to the constraint Kaplan-Violante, Insurance in Bewley Models p. 23/43

Why the Downward Bias in BPP Estimator? From the definition of φ η BP P : φ η BP P = 1 cov ( c it, y i,t 1 + y it + y i,t+1 ) cov ( y it, y i,t 1 + y it + y i,t+1 ) = 1 cov ( c it, η i,t 1 + ε i,t 2 + η it + η i,t+1 + ε i,t+1 ) var (η it ) = φ η + cov ( c it, η i,t 1 + ε i,t 2 ) var (η it ) }{{} A2: short memory = φ η + cov ( c it, ε i,t 2 ) var (η it ) }{{} 0 + cov ( c it, η i,t+1 + ε i,t+1 ) var (η it ) }{{} A1: no adv. info Last term large when agent close to borr. constr. at t 2 Kaplan-Violante, Insurance in Bewley Models p. 24/43

Sensitivity Analysis (Natural BC) Permanent Shock Transitory Shock TRUE (0.23) BPP (0.22) TRUE (0.94) BPP (0.94) Initial Wealth Dist. 0.23 0.23 0.94 0.94 γ = 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 0.93 Rep. ratio = 0.65 0.27 0.26 0.94 0.94 σ η = 0.02 0.25 0.24 0.93 0.93 σ η = 0.005 0.22 0.20 0.94 0.94 σ z0 = 0.2 0.23 0.22 0.94 0.94 σ z0 = 0.1 0.24 0.22 0.94 0.94 σ ε = 0.075 0.24 0.22 0.94 0.94 σ ε = 0.025 0.23 0.22 0.94 0.94 Kaplan-Violante, Insurance in Bewley Models p. 25/43

Sensitivity Analysis (K/Y and r) 0.35 Natural BC 0.35 Zero BC 0.3 0.3 φ η Ins. coeff. for perm. shock 0.25 0.2 0.15 r=2% r=3% r=4% r=5% φ η Ins. coeff. for perm. shock 0.25 0.2 0.15 r=2% r=3% r=4% r=5% 0.1 0.1 0.05 1 2 3 4 Wealth Income Ratio 0.05 1 2 3 4 Wealth Income Ratio Kaplan-Violante, Insurance in Bewley Models p. 26/43

A Welfare Calculation Kaplan-Violante, Insurance in Bewley Models p. 27/43

A Welfare Calculation Economy where agents survive with probability ζ = 1/ (1 + π), discount future at rate β = 1/ (1 + ρ) Consumption allocation: c it = (1 φ η ) z it Log-Normal shocks Kaplan-Violante, Insurance in Bewley Models p. 27/43

A Welfare Calculation Economy where agents survive with probability ζ = 1/ (1 + π), discount future at rate β = 1/ (1 + ρ) Consumption allocation: c it = (1 φ η ) z it Log-Normal shocks The welfare cost of going from φ η = 0.36 to φ η = 0.23 is: ω γ 2 [ (1 φ η ) 2 ( 1 φ η) 2 ] σ η ρ + π Kaplan-Violante, Insurance in Bewley Models p. 27/43

A Welfare Calculation Economy where agents survive with probability ζ = 1/ (1 + π), discount future at rate β = 1/ (1 + ρ) Consumption allocation: c it = (1 φ η ) z it Log-Normal shocks The welfare cost of going from φ η = 0.36 to φ η = 0.23 is: ω γ 2 [ (1 φ η ) 2 ( 1 φ η) 2 ] σ η ρ + π With γ = 2, ρ = 0.03, π = 0.0286, and σ η = 0.01: ω = 3.1% Kaplan-Violante, Insurance in Bewley Models p. 27/43

Advance Information Kaplan-Violante, Insurance in Bewley Models p. 28/43

Advance Information Model I: households observe, one period in advance, a fraction of the permanent shock Kaplan-Violante, Insurance in Bewley Models p. 28/43

Advance Information Model I: households observe, one period in advance, a fraction of the permanent shock Model II: households know their own deterministic income profile at age t = 0 (e.g., Lillard-Weiss, 1979) Kaplan-Violante, Insurance in Bewley Models p. 28/43

Advance Information Model I: households observe, one period in advance, a fraction of the permanent shock Model II: households know their own deterministic income profile at age t = 0 (e.g., Lillard-Weiss, 1979) Given BPP identification method, neither form of advance information can reconcile model and data Kaplan-Violante, Insurance in Bewley Models p. 28/43

Preempting the permanent shock Permanent income growth in period t comprises of two orthogonal additive components, η s it and ηa it The component ηit a time t 1 is already in the information set of the agent at From the definition of insurance coefficient: φ η = 1 cov ( c it, η it ) var (η it ) = var (ηs it ) s var (η it ) φη + var (ηa it ) var (η it ) (1 α) φ ηs + α = 1 cov ( c it, ηit s + ηa it ) var (ηit s + ηa it [ ) 1 cov ( c it, ηit a ) ] var (ηit a ) increasing in α, since with loose borrowing limits cov ( c it, η a it ) 0 Kaplan-Violante, Insurance in Bewley Models p. 29/43

Preempting the permanent shock Ignoring the usual downward bias, the BPP methodology yields: φ η BP P = 1 cov ( c it, y i,t 1 + y it + y i,t+1 ) cov ( y it, y i,t 1 + y it + y i,t+1 ) = 1 cov ( c it, ηit s + ηa i,t + ) ηa i,t+1 var (ηit s + ηa it [ ) (1 α) φ ηs + α 1 cov ( )] c it, ηi,t+1 a var (ηit a ) φ ηs Kaplan-Violante, Insurance in Bewley Models p. 30/43

Preempting the permanent shock Ignoring the usual downward bias, the BPP methodology yields: φ η BP P = 1 cov ( c it, y i,t 1 + y it + y i,t+1 ) cov ( y it, y i,t 1 + y it + y i,t+1 ) = 1 cov ( c it, ηit s + ηa i,t + ) ηa i,t+1 var (ηit s + ηa it [ ) (1 α) φ ηs + α 1 cov ( )] c it, ηi,t+1 a var (ηit a ) φ ηs BPP estimator is independent of the amount of advance information Simulations confirm this finding Kaplan-Violante, Insurance in Bewley Models p. 30/43

Predictable individual income profile Kaplan-Violante, Insurance in Bewley Models p. 31/43

Predictable individual income profile Generalize log-earnings (deviations from common age-profile) to: y it = β i t + z it + ε it z it = z i,t 1 + η it, with E [β i ] = 0 in the cross-section, and SD [β i ] = σ β The individual-specific slope β i is learned at time zero Lillard-Weiss (1979), Baker (1997), Haider (2001), Guvenen (2007) Kaplan-Violante, Insurance in Bewley Models p. 31/43

Predictable individual income profile Generalize log-earnings (deviations from common age-profile) to: y it = β i t + z it + ε it z it = z i,t 1 + η it, with E [β i ] = 0 in the cross-section, and SD [β i ] = σ β The individual-specific slope β i is learned at time zero Lillard-Weiss (1979), Baker (1997), Haider (2001), Guvenen (2007) When we increase σ β, we decrease σ η accordingly to keep the total rise in lifetime earnings inequality constant Kaplan-Violante, Insurance in Bewley Models p. 31/43

Predictable individual income profile Permanent Shock Transitory Shock Data 0.36 (0.09) 0.95 (0.04) Model Model Model Model TRUE BPP TRUE BPP Natural BC 40% 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 BC 40% 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 Kaplan-Violante, Insurance in Bewley Models p. 32/43

Predictable individual income profile Permanent Shock Transitory Shock Data 0.36 (0.09) 0.95 (0.04) Model Model Model Model TRUE BPP TRUE BPP Natural BC 40% 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 BC 40% 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 Kaplan-Violante, Insurance in Bewley Models p. 33/43

Why the Upward Bias in the BPP Estimator? From the definition of φ η BP P : φ η BP P = 1 cov ( c it, y i,t 1 + y it + y i,t+1 ) cov ( y it, y i,t 1 + y it + y i,t+1 ) = 1 cov ( c it, η i,t 1 + ε i,t 2 + η it + η i,t+1 + ε i,t+1 + 3β i ) var (η it ) + 3var (β i ) Ignoring usual downward bias due to binding constraint: [ ] φ η BP P var (η it ) var (η it ) + 3var (β i ) = (1 α) φ η + αφ β [ φ η + 3var (β i ) var (η it ) + 3var (β i ) ] [ 1 cov ( c ] it, β i ) var (β i ) φ β 1 with loose borrowing constraints (upward bias) φ β 0 with tight borrowing constraints (downward bias) Kaplan-Violante, Insurance in Bewley Models p. 34/43

Persistent (rather than permanent...) shocks Kaplan-Violante, Insurance in Bewley Models p. 35/43

Persistent (rather than permanent...) shocks Generalize log-earnings process to AR(1) + transitory: y it = z it + ε it z it = ρz it 1 + η it, with ρ < 1 BPP instruments no longer valid [misspecification] Define quasi-difference: y t y t ρy t 1 Identification of (φ η, φ ε ) can still be achieved by setting g ε t (y i ) = y t+1 g η t (y i ) = ρ 2 yt 1 + ρ y t + y t+1 under same assumptions A1 & A2 Kaplan-Violante, Insurance in Bewley Models p. 35/43

Persistent shocks Persistent Shock Transitory Shock Data 0.36 (0.09) 0.95 (0.04) TRUE BPP BPP (missp.) TRUE BPP BPP (missp.) Natural BC ρ = 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 BC ρ = 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) Kaplan-Violante, Insurance in Bewley Models p. 36/43

Persistent shocks Persistent Shock Transitory Shock Data 0.36 (0.09) 0.95 (0.04) TRUE BPP BPP (missp.) TRUE BPP BPP (missp.) Natural BC ρ = 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 BC ρ = 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 Kaplan-Violante, Insurance in Bewley Models p. 37/43

Persistent shocks Persistent Shock Transitory Shock Data 0.36 (0.09) 0.95 (0.04) TRUE BPP BPP (missp.) TRUE BPP BPP (missp.) Natural BC ρ = 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 BC ρ = 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 Kaplan-Violante, Insurance in Bewley Models p. 38/43

Persistent shocks Persistent Shock Transitory Shock Data 0.36 (0.09) 0.95 (0.04) TRUE BPP BPP (missp.) TRUE BPP BPP (missp.) Natural BC ρ = 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 BC ρ = 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 Kaplan-Violante, Insurance in Bewley Models p. 39/43

Age profile of φ η 1 Natural BC 1 Zero BC 0.8 0.8 0.6 0.6 0.4 ρ=0.97 0.4 ρ=0.93 φ η 0.2 φ η 0.2 0 ρ=1 0 ρ=1 0.2 0.2 0.4 TRUE BPP missp. 0.4 TRUE BPP missp. 0.6 0.6 30 35 40 45 50 55 Age 30 35 40 45 50 55 Age In the model, age profile of insurance coefficients wrt to persistent shocks is flatter, hence closer to the data Kaplan-Violante, Insurance in Bewley Models p. 40/43

Relationship with STY 0.8 One minus the insurance coeff. 0.5 Increase in variance of log cons. 0.75 NBC ZBC 0.45 1 φ η 0.7 0.65 0.6 0.55 0.5 Increase from age 25 60 0.4 0.35 0.3 0.25 NBC ZBC 0.45 0.2 0.4 0.92 0.94 0.96 0.98 1 Autocorrelation coefficient (ρ) 0.92 0.94 0.96 0.98 1 Autocorrelation coefficient (ρ) Kaplan-Violante, Insurance in Bewley Models p. 41/43

Relationship with STY 0.8 One minus the insurance coeff. 0.5 Increase in variance of log cons. 0.75 NBC ZBC 0.45 1 φ η 0.7 0.65 0.6 0.55 0.5 Increase from age 25 60 0.4 0.35 0.3 0.25 NBC ZBC 0.45 0.2 0.4 0.92 0.94 0.96 0.98 1 Autocorrelation coefficient (ρ) 0.92 0.94 0.96 0.98 1 Autocorrelation coefficient (ρ) These two norms of consumption insurance can disagree Kaplan-Violante, Insurance in Bewley Models p. 41/43

Age profile of wealth: model vs. data x 10 5 Lifecycle wealth profiles 2 1.5 $ (00,000) 1 0.5 0 Natural BC (ρ=0.97) Zero BC (ρ=0.93) SCF Net worth (89 92) 30 40 50 60 70 80 90 Age Kaplan-Violante, Insurance in Bewley Models p. 42/43

Age profile of wealth: model vs. data x 10 5 Lifecycle wealth profiles 2 1.5 $ (00,000) 1 0.5 0 Natural BC (ρ=0.97) Zero BC (ρ=0.93) SCF Net worth (89 92) 30 40 50 60 70 80 90 Age A version of the model with more realistic age profile of wealth would be also more successful in replicating the BPP facts Kaplan-Violante, Insurance in Bewley Models p. 42/43

Conclusions 1. We generalized BPP methodology, and argued that insurance coefficients 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. 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 Kaplan-Violante, Insurance in Bewley Models p. 43/43