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 4-5 2012 Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 1 /20
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
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
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
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
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
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
E CONOMIC E NVIRONMENT Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 5 /20
Demographics and preferences • Perpetual youth demographics with constant survival probability δ • Preferences over sequences of consumption and hours worked: � ∞ ( βδ ) t − b u ( c t , h t ; ϕ ) E b t = b u ( c t , h t ; ϕ ) = c 1 − γ − exp( ϕ ) h 1+ σ − 1 t t 1 − γ 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
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
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): log w t = α t + ε t Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 7 /20
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): log w 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
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): log w t = α t + ε t α t = α t − 1 + ω t with ω t ∼ F ω,t = κ t + θ t with ε t 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 α 0 ,b and κ 0 ∼ F κ 0 ,b Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 7 /20
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
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 ( ˜ y ) y = λy 1 − τ ˜ τ measure the degree of progressivity Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 9 /20
E QUILIBRIUM Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 10 /20
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
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
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
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
Hours worked � 1 − γ � α + 1 log h a σ ε + H a t ( ϕ, α, ε ) = − ˆ ϕ + t σ + γ � � ϕ σ ≡ 1 − τ 1 where ˆ and ϕ ≡ σ + γ 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
Consumption � 1 + � � σ log c a α + C a t ( ϕ, α, ε ) = − (1 − τ ) · ˆ ϕ + (1 − τ ) · t σ + γ � • Independent of the insurable shock ε • Effect of ˆ ϕ 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
A NSWERS TO THE T HREE Q UESTIONS Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 15 /20
Pass-through coefficient • Pass-through from permanent wage shocks to consumption: ≡ cov (∆ c t , ω t + η t ) φ w,c t var ( ω t + η t ) Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 16 /20
Pass-through coefficient • Pass-through from permanent wage shocks to consumption: ≡ cov (∆ c t , ω t + η t ) = (1 − τ ) · 1 + � σ v ωt φ w,c σ + γ · t var ( ω t + η t ) � v ωt + v ηt ◮ progressive taxation → 0 . 73 ◮ labor supply → 0 . 87 ◮ private insurance → 0 . 63 • Overall, we estimate: φ w,c = 0 . 40 t Heathcote-Storesletten-Violante, ”Consumption and Labor Supply with Partial Insurance” p. 16 /20
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