Insurance and Opportunities: The Welfare Implications of Rising Wage - - PowerPoint PPT Presentation

Insurance and Opportunities: The Welfare Implications of Rising Wage Dispersion

Jonathan Heathcote (Georgetown University) Kjetil Storesletten (University of Oslo) Gianluca Violante (New York University)

Money and Banking Workshop, University of Chicago, March 14 2006

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 1/30

Motivation

• Welfare analysis in heterogeneous agents models with incomplete

insurance against idiosyncratic risk is central to macroeconomics

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 2/30

Motivation

• Welfare analysis in heterogeneous agents models with incomplete

insurance against idiosyncratic risk is central to macroeconomics

• Examples: welfare effects of a...
• 1. change in the amount of risk (technology)

◮ Attanasio-Davis (1996), Blundell-Preston (1998), Krueger-Perri (2003),

Heathcote-Storesletten-Violante (2005)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 2/30

Motivation

• Welfare analysis in heterogeneous agents models with incomplete

insurance against idiosyncratic risk is central to macroeconomics

• Examples: welfare effects of a...
• 1. change in the amount of risk (technology)

◮ Attanasio-Davis (1996), Blundell-Preston (1998), Krueger-Perri (2003),

Heathcote-Storesletten-Violante (2005)

• 2. change in the insurability of risk (markets)

◮ Levine-Zame (2002), Kubler-Schmedders (2001)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 2/30

Motivation

• Welfare analysis in heterogeneous agents models with incomplete

insurance against idiosyncratic risk is central to macroeconomics

• Examples: welfare effects of a...
• 1. change in the amount of risk (technology)

◮ Attanasio-Davis (1996), Blundell-Preston (1998), Krueger-Perri (2003),

Heathcote-Storesletten-Violante (2005)

• 2. change in the insurability of risk (markets)

◮ Levine-Zame (2002), Kubler-Schmedders (2001)

• 3. change in redistributive policies (government)

◮ Long list... related to welfare costs of business cycles (Lucas, 2003)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 2/30

Contributions

• 1. Tractable framework delivering transparent mapping between

primitives of economy (preferences, risk, insurance market structure) and welfare expressions

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 3/30

Contributions

• 1. Tractable framework delivering transparent mapping between

primitives of economy (preferences, risk, insurance market structure) and welfare expressions

• 2. Role of flexible labor supply: insurance vs. opportunities

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 3/30

Contributions

• 1. Tractable framework delivering transparent mapping between

primitives of economy (preferences, risk, insurance market structure) and welfare expressions

• 2. Role of flexible labor supply: insurance vs. opportunities
• 3. Alternative representation of welfare effects based on changes in
• bservable cross-sectional moments

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 3/30

Outline of the Talk

• 1. Baseline economy with Cobb-Douglas preferences and simple

statistical representation of individual risk

• Equilibrium allocations
• Welfare expressions for 3 thought experiments
• Alternative representation for welfare effects
• 2. Some illustrative calculations
• 3. Extension to richer process for individual risk
• Tractability preserved by no-bond-trade equilibrium

(Constantinides-Duffie, 1996)

• 4. Separable preferences

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 4/30

The Economy

• Demographics and preferences: Continuum of agents with

time-separable preferences E0

∞

• t=0

βt (cη

t (1 − ht)1−η)1−θ − 1

1 − θ

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 5/30

The Economy

• Demographics and preferences: Continuum of agents with

time-separable preferences E0

∞

• t=0

βt (cη

t (1 − ht)1−η)1−θ − 1

1 − θ

• Endowments: initial wealth is zero for all agents and assets are

in zero net supply

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 5/30

The Economy

• Demographics and preferences: Continuum of agents with

time-separable preferences E0

∞

• t=0

βt (cη

t (1 − ht)1−η)1−θ − 1

1 − θ

• Endowments: initial wealth is zero for all agents and assets are

in zero net supply

• Technology: linear in aggregate hours weighted by

efficiency-units of labor

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 5/30

The Economy

• Demographics and preferences: Continuum of agents with

time-separable preferences E0

∞

• t=0

βt (cη

t (1 − ht)1−η)1−θ − 1

1 − θ

• Endowments: initial wealth is zero for all agents and assets are

in zero net supply

• Technology: linear in aggregate hours weighted by

efficiency-units of labor

• Labor market: competitive, hourly wages equal individual labor

productivities

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 5/30

Individual Productivity Shocks

• Two orthogonal log-Normally distributed components

log w = α + εt α ∼ N

• −vα

2 , vα

• ,

εt ∼ N

• −vε

2 , vε

• i.i.d.
• Hence:

log w(α, εt) = (α + εt) ∼ N

• −v

2, v

• ,

with E[w] = 1

• We model α as a permanent individual effect and εt as i.i.d. shock

(Gottschalk-Moffitt, 1994)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 6/30

Asset Market Structure

• Three distinct structures:
• 1. Autarky: no financial instruments

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 7/30

Asset Market Structure

• Three distinct structures:
• 1. Autarky: no financial instruments
• 2. Complete markets: complete insurance against either

component of the wage shock ◮ Trade opens before the realization of α

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 7/30

Asset Market Structure

• Three distinct structures:
• 1. Autarky: no financial instruments
• 2. Complete markets: complete insurance against either

component of the wage shock ◮ Trade opens before the realization of α

• 3. Incomplete markets: no insurance against the permanent

component of wages, complete insurance against transitory shocks ◮ Trade opens after the realization of α

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 7/30

Incomplete-Markets Economy: Interpretations

• Households literally have access to insurance against some

shocks, but not others ◮ Cochrane (1991), Altonji-Hayashi-Kotlikoff (1991),

Guiso-Pistaferri-Schivardi (2005)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 8/30

Incomplete-Markets Economy: Interpretations

• Households literally have access to insurance against some

shocks, but not others ◮ Cochrane (1991), Altonji-Hayashi-Kotlikoff (1991),

Guiso-Pistaferri-Schivardi (2005)

• Ex-post complete markets with ex-ante heterogeneous agents

◮ Cunha-Heckman-Navarro (2005)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 8/30

Incomplete-Markets Economy: Interpretations

• Households literally have access to insurance against some

shocks, but not others ◮ Cochrane (1991), Altonji-Hayashi-Kotlikoff (1991),

Guiso-Pistaferri-Schivardi (2005)

• Ex-post complete markets with ex-ante heterogeneous agents

◮ Cunha-Heckman-Navarro (2005)

• Economy with a non-contingent bond (i.e., “Bewley economy")

where precautionary saving and borrowing allow smoothing shocks that aren’t too persistent ◮ Numerical comparison of two economies → good approximation

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 8/30

Properties of Cobb-Douglas Preferences

u (c, h) = (cη(1 − h)1−η)1−θ − 1 1 − θ

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 9/30

Properties of Cobb-Douglas Preferences

u (c, h) = (cη(1 − h)1−η)1−θ − 1 1 − θ

• Coefficient of relative risk aversion: ¯

γ ≡ 1 − η + ηθ

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 9/30

Properties of Cobb-Douglas Preferences

u (c, h) = (cη(1 − h)1−η)1−θ − 1 1 − θ

• Coefficient of relative risk aversion: ¯

γ ≡ 1 − η + ηθ

• Frisch labor supply elasticity: φ ≡ λ 1−h

h

◮ where λ ≡ 1−η+ηθ

θ

is the Frisch elasticity of leisure ◮ Non-stochastic Frisch labor supply elasticity: ¯ φ = λ · 1 − η η > 1 − η

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 9/30

Properties of Cobb-Douglas Preferences

u (c, h) = (cη(1 − h)1−η)1−θ − 1 1 − θ

• Coefficient of relative risk aversion: ¯

γ ≡ 1 − η + ηθ

• Frisch labor supply elasticity: φ ≡ λ 1−h

h

◮ where λ ≡ 1−η+ηθ

θ

is the Frisch elasticity of leisure ◮ Non-stochastic Frisch labor supply elasticity: ¯ φ = λ · 1 − η η > 1 − η

• (c, 1 − h) substitutes when θ > 1(λ < 1)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 9/30

Equilibrium Allocations

• Autarky

log cAUT (α, ε) = log(η) + α + ε log lAUT (α, ε) = log(1 − η)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 10/30

Equilibrium Allocations

• Autarky

log cAUT (α, ε) = log(η) + α + ε log lAUT (α, ε) = log(1 − η)

• Complete markets

log cCM (α, ε) = log(η) + λ(1 − λ)v 2 + (1 − λ) (α + ε) log lCM (α, ε) = log(1 − η) + λ(1 − λ)v 2 − λ(α + ε)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 10/30

Equilibrium Allocations

• Autarky

log cAUT (α, ε) = log(η) + α + ε log lAUT (α, ε) = log(1 − η)

• Complete markets

log cCM (α, ε) = log(η) + λ(1 − λ)v 2 + (1 − λ) (α + ε) log lCM (α, ε) = log(1 − η) + λ(1 − λ)v 2 − λ(α + ε)

• Incomplete markets

log cIM (α, ε) = log(η) + λ(1 − λ)vε 2 + α + (1 − λ) ε log lIM (α, ε) = log(1 − η) + λ(1 − λ)vε 2 − λε ◮ Individual wealth is always zero

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 10/30

Welfare Analysis

• ωm: welfare change of a change in labor market risk (∆vα, ∆vε)
• A×E

u ((1 + ωm) cm, hm) d f(α, ε) =

• A×E

u

• ˆ

cm, ˆ hm

• d ˆ

f(α, ε)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 11/30

Welfare Analysis

• ωm: welfare change of a change in labor market risk (∆vα, ∆vε)
• A×E

u ((1 + ωm) cm, hm) d f(α, ε) =

• A×E

u

• ˆ

cm, ˆ hm

• d ˆ

f(α, ε)

• χm: welfare change from completing markets

(∆vα = −vα, ∆vε = vα)

• A×E

u ((1 + χm) cm, hm) d f(α, ε) =

• A×E

u (cCM, hCM) d f(α, ε)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 11/30

Welfare Analysis

• ωm: welfare change of a change in labor market risk (∆vα, ∆vε)
• A×E

u ((1 + ωm) cm, hm) d f(α, ε) =

• A×E

u

• ˆ

cm, ˆ hm

• d ˆ

f(α, ε)

• χm: welfare change from completing markets

(∆vα = −vα, ∆vε = vα)

• A×E

u ((1 + χm) cm, hm) d f(α, ε) =

• A×E

u (cCM, hCM) d f(α, ε)

• κm: welfare change from eliminating risk (∆vα = −vα, ∆vε = −vε)
• A×E

u ((1 + κm) cm, hm) d f(α, ε) = u

• ¯

c, ¯ h

• Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 11/30

Welfare Expressions

• Welfare effect of a change in labor market risk

ωAUT ≃ −¯ γ ∆v 2 ωCM ≃ ¯ φ∆v 2 ωIM ≃ ¯ φ∆vε 2 − ¯ γ ∆vα 2

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 12/30

Welfare Expressions

• Welfare effect of a change in labor market risk

ωAUT ≃ −¯ γ ∆v 2 ωCM ≃ ¯ φ∆v 2 ωIM ≃ ¯ φ∆vε 2 − ¯ γ ∆vα 2

• Welfare gain from completing markets

χAUT ≃ (¯ φ + ¯ γ)v 2 χIM ≃ (¯ φ + ¯ γ)vα 2

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 12/30

Welfare Expressions

• Welfare effect of a change in labor market risk

ωAUT ≃ −¯ γ ∆v 2 ωCM ≃ ¯ φ∆v 2 ωIM ≃ ¯ φ∆vε 2 − ¯ γ ∆vα 2

• Welfare gain from completing markets

χAUT ≃ (¯ φ + ¯ γ)v 2 χIM ≃ (¯ φ + ¯ γ)vα 2

• Welfare change from eliminating risk, e.g. through progressive

taxation system τ(w) = 1 − 1/w κAUT ≃ ¯ γ v 2 κCM ≃ −¯ φv 2 κIM ≃ ¯ γ vα 2 − ¯ φvε 2

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 12/30

A Caveat

• Fix η = 1/3, vary θ
• ¯

γ = 1 − η + ηθ

• ¯

φ = ¯

γ θ

0.5 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Fraction of lifetime consumption Frisch elasticity (φ)

Welfare gain of completing markets (variance of shocks normalized to 1)

0.5 1 1.5 2 2.5 3 2 4 6 8 10

Coefficient of risk aversion (γ) Frisch elasticity (φ)

Coefficient of Risk Aversion as a function of φ

lower bound for φ is 1−η = 0.666 Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 13/30

Equilibrium cross-sectional moments

• Closed-form cross-sectional variances and covariances of the

joint distribution of (w, h, c) in IM economy cov (log w, log h) = ¯ φvε var (log h) = ¯ φ2vε var (log c) = vα + (1 − λ)2 vε cov (log c, log h) = (1 − λ) ¯ φvε

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 14/30

Alternative representation of welfare change

• Use cross-sectional moments to map our expression for ωm into

an alternative representation based on “observables” ωm ≃ ∆cov (log w, log h) − ¯ γ 2 ∆var (log c) − 1 2¯ φ∆var (log h) + ¯ γ − 1 2 ∆cov (log c, log h)

• Alternative approach to welfare calculations

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 15/30

Alternative representation of welfare change

• Use cross-sectional moments to map our expression for ωm into

an alternative representation based on “observables” ωm ≃ ∆cov (log w, log h) − ¯ γ 2 ∆var (log c) − 1 2¯ φ∆var (log h) + ¯ γ − 1 2 ∆cov (log c, log h)

• Alternative approach to welfare calculations
• Percentage change in aggregate labor productivity

∆ log(Y/H) = ∆cov (log w, log h)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 15/30

Welfare calculations I

• Preferences: η = 1/3, θ = 4

◮ Risk aversion coefficient ¯ γ = 2 ◮ Frisch labor supply elasticity ¯ φ = 1

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 16/30

Welfare calculations I

• Preferences: η = 1/3, θ = 4

◮ Risk aversion coefficient ¯ γ = 2 ◮ Frisch labor supply elasticity ¯ φ = 1

• Individual Risk

◮ PSID (1968-1997) ◮ Residual wage dispersion vw : 0.25 → 0.35 ◮ Transitory component vε : 0.08 → 0.13 ◮ Permanent component: vα : 0.17 → 0.22

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 16/30

Welfare calculations I

Welfare change of Welfare gain from Welfare gain from rise in wage dispersion completing markets eliminating risk ωIM χIM→CM κIM

• 2.47% (-2.50%)

+39.1% (+33.0%) +16.9% (+15.5%) Volat. Level Volat. Level Volat. Level

• 7.50%

+5.00% +11.0% +22.0% +28.5%

• 13.0%
• “Level” component: increase in aggregate productivity that

mitigates the loss

• Bounds: ωAUT = −10% and ωCM = +5%

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 17/30

Welfare calculations I

Welfare change of Welfare gain from Welfare gain from rise in wage dispersion completing markets eliminating risk ωIM χIM→CM κIM

• 2.47% (-2.50%)

+39.1% (+33.0%) +16.9% (+15.5%) Volat. Level Volat. Level Volat. Level

• 7.50%

+5.00% +11.0% +22.0% +28.5%

• 13.0%
• Productivity gain twice as big as insurance gain

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 18/30

Welfare calculations I

Welfare change of Welfare gain from Welfare gain from rise in wage dispersion completing markets eliminating risk ωIM χIM→CM κIM

• 2.47% (-2.50%)

+39.1% (+33.0%) +16.9% (+15.5%) Volat. Level Volat. Level Volat. Level

• 7.50%

+5.00% +11.0% +22.0% +28.5%

• 13.0%
• Policies eliminate also the “good risk”

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 19/30

Welfare calculations II

• Same preference parameters:

¯ γ = 2, ¯ φ = 1

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 20/30

Welfare calculations II

• Same preference parameters:

¯ γ = 2, ¯ φ = 1

• From PSID data:

∆cov (log w, log h) ≃ +0.012 ∆var (log h) ≃ +0.010

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 20/30

Welfare calculations II

• Same preference parameters:

¯ γ = 2, ¯ φ = 1

• From PSID data:

∆cov (log w, log h) ≃ +0.012 ∆var (log h) ≃ +0.010

• From CEX data:

– Slesnick (2001), Krueger-Perri (2005), Attanasio-Battistin-Ichimura (2005): ∆var (log c) ∈ (0.01, 0.05) – Krueger-Perri (2005): ∆cov (log c, log h) ≃ −0.007

• Result: ω = −2.65%

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 20/30

Economy with permanent shocks

• Individual risk process: log wt = αt + εt
• Uninsurable component:

αt = πt + ψ, ψ ∼ N

• −vψ

2 , vψ

• πt

= πt−1 + ωt, ωt ∼ N

• −vω

2 , vω

• and π0 = 0

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 21/30

Economy with permanent shocks

• Individual risk process: log wt = αt + εt
• Uninsurable component:

αt = πt + ψ, ψ ∼ N

• −vψ

2 , vψ

• πt

= πt−1 + ωt, ωt ∼ N

• −vω

2 , vω

• and π0 = 0
• Insurable component:

εt = π∗

t + ζt,

ζt ∼ N

• −vζ

2 , vζ

• i.i.d.

π∗

t

= π∗

t−1 + ω∗ t ,

ω∗

t ∼ N

• −vω∗

2 , vω∗

• and π∗

0 = 0

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 21/30

Market Structure: An Island-Economy Interpretation

• At t, agents born onto islands indexed by (ψ, {ωs}), s = t, ..., ∞
• Within island: agents can trade full set of Arrow securities paying
• ne unit of consumption at t + 1, for each (ωt+1, ω∗

t+1, ζt+1)

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 22/30

Market Structure: An Island-Economy Interpretation

• At t, agents born onto islands indexed by (ψ, {ωs}), s = t, ..., ∞
• Within island: agents can trade full set of Arrow securities paying
• ne unit of consumption at t + 1, for each (ωt+1, ω∗

t+1, ζt+1)

• Between islands: agents can trade non-contingent bond
• Model nests complete markets (vψ = vω = 0) and Bewley

economy (vω∗ = vζ = 0)

• Perfect annuity markets

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 22/30

No-bond-trading Equilibrium

• Expected growth in marginal utility of consumption is the same

across all islands

• Let βδ = 1/(1 + ρ), then the equilibrium interest rate satisfies:

ρ − r∗ ≃ ¯ γ(1 + ¯ γ)vω 2 + ¯ γ(λ − 1)vω∗ 2

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 23/30

No-bond-trading Equilibrium

• Expected growth in marginal utility of consumption is the same

across all islands

• Let βδ = 1/(1 + ρ), then the equilibrium interest rate satisfies:

ρ − r∗ ≃ ¯ γ(1 + ¯ γ)vω 2 + ¯ γ(λ − 1)vω∗ 2

• Constantinides and Duffie (1996) extended to:

◮ flexible labor supply ◮ groups of population allowed to perfectly pool risk

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 23/30

No-bond-trading Equilibrium

• Expected growth in marginal utility of consumption is the same

across all islands

• Let βδ = 1/(1 + ρ), then the equilibrium interest rate satisfies:

ρ − r∗ ≃ ¯ γ(1 + ¯ γ)vω 2 + ¯ γ(λ − 1)vω∗ 2

• Constantinides and Duffie (1996) extended to:

◮ flexible labor supply ◮ groups of population allowed to perfectly pool risk

• Same allocations: c (αt, εt), l (αt, εt) and zero wealth

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 23/30

Welfare Effect of Change in Labor Market Risk

ωIM = ¯ φ ∆vζ 2 + µ∆vω∗ 2

• − ¯

γ ∆ψ 2 + µ∆vω 2

• where the lifetime multiplier µ is given by:

µ = 1 1 − βδ exp

• (¯

γ − 1) ¯ φ vω∗

2

− ¯ γ vω

2

• Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 24/30

Welfare Effect of Change in Labor Market Risk

ωIM = ¯ φ ∆vζ 2 + µ∆vω∗ 2

• − ¯

γ ∆ψ 2 + µ∆vω 2

• where the lifetime multiplier µ is given by:

µ = 1 1 − βδ exp

• (¯

γ − 1) ¯ φ vω∗

2

− ¯ γ vω

2

• As households age:

◮ Expected labor productivity grows thanks to vω∗ > 0 ◮ Size of uninsurable uncertainty grows due to vω > 0

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 24/30

Welfare Effect of Change in Labor Market Risk

ωIM = ¯ φ ∆vζ 2 + µ∆vω∗ 2

• − ¯

γ ∆ψ 2 + µ∆vω 2

• where the lifetime multiplier µ is given by:

µ = 1 1 − βδ exp

• (¯

γ − 1) ¯ φ vω∗

2

− ¯ γ vω

2

• As households age:

◮ Expected labor productivity grows thanks to vω∗ > 0 ◮ Size of uninsurable uncertainty grows due to vω > 0

• With β = 1 and separability (¯

γ = λ = 1): µ =

1 1−δ

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 24/30

Separable Preferences

u (c, h) = c1−γ 1 − γ − ϕ h1+σ 1 + σ

• 1/σ is the Frisch (compensated) elasticity of labor supply
• 1−γ

γ+σ is the Marshallian (uncompensated) elasticity of labor supply

• ϕ measures the relative taste for leisure: WLOG, ϕ = 1

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 25/30

Incomplete Markets Allocations

log c (α) = 1 + σ σ + γ vε 2σ + 1 + σ σ + γ

• α

log h (α, ε) = − γ 2σ2 1 + σ σ + γ

• vε +

1 − γ σ + γ

• Marshallian

α + 1 σ

• F risch

ε

• Individual consumption is increasing in vε and α
• Insurable shock ε and uninsurable shock α have different effects
• n labor supply decision

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 26/30

Welfare Effects of Change in Labor Market Risk

ωCM ≃ 1 σ ∆v 2 ωAUT ≃ −(γ − 1) + γ (1 + σ) σ + γ ∆v 2 ωIM ≃ 1 σ ∆vε 2 − (γ − 1) + γ (1 + σ) σ + γ ∆vα 2

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 27/30

Separable vs. CD Preferences

• 1. Productivity gain: with CD taken only as higher average leisure,

with separability taken also as higher average consumption

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 28/30

Separable vs. CD Preferences

• 1. Productivity gain: with CD taken only as higher average leisure,

with separability taken also as higher average consumption

• 2. Autarky: with CD there is always a welfare loss, with separability

there could be a welfare gain ωAUT ≃ −(γ − 1) + γ (1 + σ) σ + γ ∆v 2

• γ ∈ [0, 1/(2 + σ)] → ωAUT > 0
• by continuity, since as γ → 0, ωAUT → ωCM > 0

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 28/30

Welfare Calculations with Separability

1 2 3 4 5 6 7 8 9 10 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01

Fraction of lifetime consumption Inverse of Frisch elasticity (σ)

Welfare cost of change in wage dispersion ( γ = 2 )

Based on estimates of individual risk Based on changes in cross−sectional moments

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 29/30

Next?

• Tractable equilibrium framework to study consumption and labor

supply with partial insurance → analytical solution

• Extensions include:

◮ preference heterogeneity/shocks ◮ time-varying risk and aggregate uncertainty ◮ measurement error in (c, h, w)

• Given panel data on (w, h) (from PSID) and cross-sectional data
• n consumption (from CEX), we can identify and estimate all the

structural parameters of model

Heathcote-Storesletten-Violante, ”Insurance and Opportunities” – p. 30/30