Consumption and Labor Supply with Partial Insurance: An Analytical Framework

Transcription

1 Consumption and Labor Supply with Partial Insurance: An Analytical Framework Jonathan Heathcote Federal Reserve Bank of Minneapolis, CEPR Kjetil Storesletten Federal Reserve Bank of Minneapolis, CEPR Gianluca Violante New York University, CEPR, and NBER Conference in Honor of Thomas Sargent and Christopher Sims Federal Reserve Bank of Minneapolis, May Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 1 /20

2 Measurement of risk sharing Three broad questions: 1. Fraction of individual shocks that transmits to consumption 2. Insurability nature of the recent increase in US inequality 3. Life-cycle shocks vs. initial conditions in determining inequality Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 2 /20

3 Measurement of risk sharing Two complementary approaches: 1. Structural model risk sharing as equilibrium outcome Sensitive to assumed market structure and insurance channels Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 3 /20

4 Measurement of risk sharing Two complementary approaches: 1. Structural model risk sharing as equilibrium outcome Sensitive to assumed market structure and insurance channels 2. Quantify overall risk sharing from data agnostic about sources Requires long, high-quality panel data on (c, y) Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 3 /20

5 Our approach: Bewley meets Deaton 1. Structural equilibrium model with non-contingent bond, labor supply, and redistributive taxation 2. Flexible financial market structure that does not hardwire agents access to insurance Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 4 /20

6 Our approach: Bewley meets Deaton 1. Structural equilibrium model with non-contingent bond, labor supply, and redistributive taxation 2. Flexible financial market structure that does not hardwire agents access to insurance Analytical tractability Closed-form equilibrium cross-sectional (co-)variances of (w, h, c) Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 4 /20

7 Our approach: Bewley meets Deaton 1. Structural equilibrium model with non-contingent bond, labor supply, and redistributive taxation 2. Flexible financial market structure that does not hardwire agents access to insurance Analytical tractability Closed-form equilibrium cross-sectional (co-)variances of (w, h, c) Labor supply data informative about risk-sharing Like c, h react differently to insurable vs. uninsurable shocks to w Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 4 /20

8 ECONOMIC ENVIRONMENT Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 5 /20

9 Demographics and preferences Perpetual youth demographics with constant survival probability δ Preferences over sequences of consumption and hours worked: E b (βδ) t b u(c t,h t ;ϕ) t=b u(c t,h t ;ϕ) = c1 γ t 1 1 γ exp(ϕ) h1+σ t 1+σ where ϕ F ϕ,b is distaste for work relative to consumption Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 6 /20

10 Technology and individual endowments Technology: linear in aggregate effective labor Competitive labor market: wage = individual productivity Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 7 /20

11 Technology and individual endowments Technology: linear in aggregate effective labor Competitive labor market: wage = individual productivity Individual wage: sum of two orthogonal components (in logs): logw t = α t +ε t Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 7 /20

12 Technology and individual endowments Technology: linear in aggregate effective labor Competitive labor market: wage = individual productivity Individual wage: sum of two orthogonal components (in logs): logw t = α t +ε t α t = α t 1 +ω t with ω t F ω,t Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 7 /20

13 Technology and individual endowments Technology: linear in aggregate effective labor Competitive labor market: wage = individual productivity Individual wage: sum of two orthogonal components (in logs): logw t = α t +ε t α t = α t 1 +ω t with ω t F ω,t ε t = κ t +θ t with θ t F θ,t κ t = κ t 1 +η t with η t F η,t At labor market entry, agents draw α 0 F α0,b and κ 0 F κ0,b Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 7 /20

14 Private risk-sharing 1. Non-state-contingent bond traded in zero net supply 2. Insurance claims tarded against shocks to ε only Captures other (residual) insurance arrangements: financial markets, spousal labor supply, family transfers, etc. Partial insurance: between bond economy and complete markets Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 8 /20

15 Government Government: runs a progressive tax/transfer scheme Redistribution and financing of (non-valued) expenditures G t Two-parameter function maps pre-government earnings (y = wh) to disposable earnings (ỹ) ỹ = λy 1 τ τ measure the degree of progressivity Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 9 /20

16 EQUILIBRIUM Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 10 /20

17 Equilibrium In equilibrium, there is no bond trade among households Sharp dichotomy between shocks: (α t,ϕ) uninsured privately, while ε t perfectly insured Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 11 /20

18 Equilibrium In equilibrium, there is no bond trade among households Sharp dichotomy between shocks: (α t,ϕ) uninsured privately, while ε t perfectly insured We can solve for equilibrium allocations and prices in closed-form Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 11 /20

19 Link to Constantinides and Duffie (1996) (i) CRRA prefs, (ii) unit root shocks to log disposable income, (iii) zero initial wealth, (iv) wealth in ZNS no bond-trade equilibrium Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 12 /20

20 Link to Constantinides and Duffie (1996) (i) CRRA prefs, (ii) unit root shocks to log disposable income, (iii) zero initial wealth, (iv) wealth in ZNS no bond-trade equilibrium Our environment micro-founds unit root disposable income: 1. Primitive exogenous process: wages 2. Labor supply: exogenous wages endogenous earnings 3. Non-linear taxation: pre-tax earnings after-tax earnings 4. Private risk-sharing: earnings post-trade disposable income 5. No bond-trade: disposable income = consumption Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 12 /20

21 Hours worked logh a t (ϕ,α,ε) = ˆϕ+ ( ) 1 γ σ +γ α+ 1 σ ε+ha t where ˆϕ ϕ σ+γ and 1 σ 1 τ σ+τ Hours worked decrease in effort cost ϕ Response to ε proportional to tax-modified Frisch elasticity Response to α depends on γ which controls wealth effect Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 13 /20

22 Consumption logc a t (ϕ,α,ε) = (1 τ) ˆϕ+(1 τ) ( ) 1+ σ σ +γ α+c a t Independent of the insurable shock ε Effect of ˆϕ mediated by tax progressivity Response to α mediated by labor supply and tax progressivity Random walk, displays excess smoothness relative to PIH Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 14 /20

23 ANSWERS TO THE THREE QUESTIONS Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 15 /20

24 Pass-through coefficient Pass-through from permanent wage shocks to consumption: φ w,c t cov( c t,ω t +η t ) var(ω t +η t ) Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 16 /20

25 Pass-through coefficient Pass-through from permanent wage shocks to consumption: φ w,c t cov( c t,ω t +η t ) var(ω t +η t ) = (1 τ) 1+ σ σ +γ v ωt v ωt +v ηt progressive taxation 0.73 labor supply 0.87 private insurance 0.63 Overall, we estimate: φ w,c t = 0.40 Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 16 /20

26 Risk-sharing over time 0.55 Variance of Log Wages 0.3 Variance of Log Consumption 0.6 Insurable Fraction of Cross sectional Wage Dispersion Data Model 0.1 Data Model Year Year Year var t (w) = var t (α)+var t (ε)+v µy +v µh ( ) 2 var t (c) = (1 τ) 2 var t (ˆϕ)+(1 τ) 2 1+ σ var t (α)+v µc σ +γ Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 17 /20

27 Risk-sharing over time 0.55 Variance of Log Wages 0.1 Correlation between Log Wages & Log Hours 0.6 Insurable Fraction of Cross sectional Wage Dispersion Data Model Data Model Year Year Year var t (w) = var t (α)+var t (ε)+v µy +v µh cov t (w,h) = ( ) 1 γ var t (α)+ 1 σ σ +γ var t(ε) v µh Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 18 /20

28 Lifecycle inequality decomposition Total Variance Percent Contribution to Total Variance Initial Heterogeneity Life-Cycle Measurement Preferences Productivity Shocks Error var(log w) var(log h) var(log c) All components are orthogonal decomposition is unique Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 19 /20

29 Why preference heter. is a source of inequality cov t (w,h) = ( ) 1 γ var t (α)+ 1 σ σ +γ var t(ε) v µh < 0 cov t (h,c) = (1 τ)var t (ˆϕ)+ (1 τ)(1+ σ)(1 γ) ( σ +γ) 2 var t (α) > 0 γ = 1.5 var t (ˆϕ) > 0 Heathcote-Storesletten-Violante, Consumption and Labor Supply with Partial Insurance p. 20 /20