Classical Labor Supply: Partial Insurance ECON 34430: Topics in - - PowerPoint PPT Presentation

Classical Labor Supply: Partial Insurance

ECON 34430: Topics in Labor Markets

• T. Lamadon (U of Chicago)

Winter 2016

Heathcote Storesletten and Violante (2013) Consumption and Labor Supply with Partial Insurance

Intro

HSV 2013

• The paper wants to answer 3 broad questions:

1 Fraction of individual shocks that transmits to consumption 2 Insurability nature of the recent increase in US inequality 3 Life-cylce shocks vs initial conditions in determining inequality

Intro

HSV 2013

• To do so they develop an island model
• two sources of permanent shocks: island level and individual
• allow for certain insurance claimed to be traded within and

between

• solve for the equilibrium, show that close form solution exist
• estimate on data

The Model: preferences

HSV 2013

• perpetual youth model, constant survival probability δ
• the economy is composed of an infinite number of individuals

in an infinite number of islands

• preferences over consumption and hours

Eb

∞

• t=b

(βδ)t−bu(ct, ht; φ) u(ct, ht; φ) = c1−γ

t

− 1 1 − γ − exp(φ) h1+σ

t

1 + σ

• cohort born at time t draws φt ∼ Fφt

The Model: productivity

HSV 2013

• productivity is composed of an idiosyncratic and island specific

components: log wt = αt

• island

+ ǫt

• ind.
• the island level follows a random walk

αt = αt−1 + ωt with ωt ∼ Fωt

• the individual component is formed by a random walk and an

iid transitory ǫt = κt + θt with θt ∼ Fθt κt = κt−1 + ηt with ηt ∼ Fηt

• agents entering at time t draw α0, κ0, φ from cohort specific

distributions

The Model: production and taxes

HSV 2013

• large number of individual and island means no aggregate

uncertainty

• production of the final good takes place through a constant

return to scale technology → individuals are paid their productivity

• given gross income yt = wtht net earnings are given by:

˜ yt = λ(yt)1−τ

• this achieves some redistribution and approximates the US tax

system

The Model: market structure

HSV 2013

• all assets are in 0 net supply
• at birth agents have 0 financial asset
• α0 and φ are drawn before trading starts
• agents are attached to an island with unknown ωt realized

sequence

• within island, agents can trade a complete set of insurance

contracts

• at t ≥ b they can trade st+1 = (ωt+1, ηt+1, θt+1)
• contracts pay δ−1 unit of consumption in state st+1
• between island, more limited
• at t ≥ b they can trade st+1 = (ηt+1, θt+1)
• can’t condition on ωt+1
• note that agents can trade a risk free bond by buying δ of

each (ηt+1, θt+1) contracts

The Model: agent’s problem

HSV 2013

• call st = sb, sb+1, ...st the individual history of shocks

sj = (b, φ, α0, κ0, θb) for j=b (ωj , ηj , θj ) for j > b

• Qt(S; st) is price of insurance bought at time t by agent with

st for event set S ⊆ S.

• Bt(st+1; st) is the quantity purchased
• Q∗

t (Z; st) and B∗ t (zt+1; st) are equivalent for price and

quantity for agents from other islands.

• Z and z do not include ω.

The Model: agent’s budget constraints

HSV 2013

λ

• wt(st)ht(st))

1−τ + dt(st) = ct(st) +

• S

Qt(st+1; st)Bt(st+1; st)dst+1 +

• Z

Q∗

t (zt+1; st)B∗ t (st+1; st)dzt+1

where the realized wealth is given by dt(st) = δ−1 Bt−1(st; st−1) + B∗

t−1(zt; st−1)

The Model: equilibrium

HSV 2013

• Given sequences {Fφt, Fα0t, Fκ0t, Fωt, Fηt, Fθt} a competitive

equilibrium is a set of allocations {ct(st), ht(st), dt(st), Bt(·, st), B∗

t (·, st)} and prices

{Bt(·, st), B∗

t (·, st)} such that 1 allocation maximize expected utility 2 insurance markets clear 3 final good and labor market clear

• oveview of the results:

1 no bond traded in equilibrium between islands 2 close form solution for hours and consumption 3 close form solution for insurance claims prices 4 sharp dichotomy between individual shocks and island shocks

Main result

HSV 2013

• No insurance trade between islands B∗

T(Z; st) = 0 for all Z

• Consumption and hours are given by:

log ct(st) = −(1 − τ)ˆ φ + (1 − τ)1 + ˆ σ ˆ σ + γ αt + Ca

t

log ht(st) = −ˆ φ + 1 − γ ˆ σ + γ αt + 1 ˆ σǫt + Ha

t

where Ca

t and Ha t are age-date specific constant, ˆ

σ ˆ φ are tax-modified constants.

• bond prices are given by

where ∆Ca

t is independent of a and Fst integrates over

(ω, η, θ)

Intuition behind no trade

HSV 2013

• first, agents can trade a risk free bond across island if they

want to

• however they all value this bond in the same way, why?
• multiplicative and i.i.d. structure makes claim price history

independent

• the idiosyncratic part (ηt, θt) is perfectly insured within island,
• the common part ωt is shocking everyone within island
• the island shock ωt can’t be traded across Island
• the island level shocks are the same in all Islands
• nevertheless, we get an environment with labor supply,

insurable and no insurable shocks, and every thing in close form.

Intuition behind asset no entering the state space

HSV 2013

• within island, allocation can be derived using the planner

solution (no reference to individual asset position)

• yet asset holding is non-degenerate
• between island, we have the no-trade outcome

Consumption and Labor Supply Decisions

HSV 2013

log ct(st) = −(1 − τ)ˆ φ + (1 − τ)1 + ˆ σ ˆ σ + γ αt + Ca

t

log ht(st) = −ˆ φ + 1 − γ ˆ σ + γ αt + 1 ˆ σǫt + Ha

t

• hours and ǫt = κt + θt
• hours work is increasing in the insurable component ǫt
• the response is scaled by the Frisch elasticity
• insurability of ǫ rules out income effect
• hours and αt
• γ > 1 means income effect dominates substitution: hours ց
• consumption is independent of ǫt because it is fully insured
• consumption follows a random walk
• pass-through depends on σ, γ and tax-schedule

Data and estimation

HSV 2013

• Use a combination of CEX and PSID
• Use PSID with fine age groups for
• moments in level involving hours and wages
• same moments in difference
• same moments in second difference
• Use CEX with fine age groups for
• moments in level involving consumption
• estimate parameters using minimum distance (11, 532

moments and 164 parameters)

Some Moments

HSV 2013

More Moments

HSV 2013

Parameter estimates

HSV 2013

• δ = 0.996 and τ = 0.185 ( R2 = 0.92)
• 1/ˆ

σ = 0.35 broadly consistent with the litterature

• 45% of permanent innovation appears to be insurable

Passthrough coefficient

HSV 2013

• pass-through from permanent wage shocks to consumption
• progressive taxation 0.815
• labor supply 0.845
• private insurance 0.63
• overall φw,c

t

= 0.386

• the pass-through from pre-tax earnings is 0.272, very similar

to Blundell, Pistaferri and Preston which found 0.225

Wages and consumption growth variance

HSV 2013

• the ratio of consumption growth to wage growth:
• we see even more smoothing due to taxes and labor supply

here

• at baseline, increase in variance of consumption is 25% of

increase in log-wages even though 40% of wage shocks transmit to consumption.

Life cycle variance decomposition

HSV 2013

• initial heterogeneity accounts for 40% to 50% for all variables
• insurable versus uninsurable differ for different variables
• no simple answer, depends on var of interest
• when simulating life-time earnings, they find that initial

conditions account for 63% of the dispersion

Conclusion

HSV 2013

• developed a micro-founded very tractable model of partial

insurance

• includes labor-supply decision
• close form consumption and hours
• do we learn more than BPP?
• how realistic/useful is the market structure?

Arellano, Blundell, Bonhomme (WP) Earnings and Consumption Dynamics: A Nonlinear Panel Data Framework

Model for earnings

ABB WP

• Earnings follow:

yit = ηit + ǫit ηit = Qt(ηi,t−1, uit)

• note that unit root is a particular case:

ηit = ηi,t−1 + F −1(uit)

• Estimation and identification:
• identification is akin Hidden Markov Chains
• estimation is akin the EM but
• latent variable is continuous and dim ≥ 1
• uses Gibbs sampling in the E-step
• uses quantile regression in the M-step (not strictly likelihood)

Non-linear dynamics

ABB WP

• the paper allows for the persistence of the sock to differ at

different quantiles ρt(ηi,t−1, τ) = ∂Qt(ηi,t−1, τ) ∂η

• the persistence of η can differ depending on the values of η

and the shock u.

• the model allows for conditional skewness and conditional

kurtosis.

• in the unit root model ρ = 1 everywhere.

Results - Earning process

ABB WP

Results - Error terms

ABB WP

Results - Earning process - conditional skewness

ABB WP

Results - Earning process - conditional skewness

mobility

Consumption rule

ABB WP

• estimating the consumption response:

cit = gt(ait, ηit, ǫit, νit)

• where ait are assets, νit is some unobserved shock (possibly

correlated).

Results - Consumption rule

ABB WP

Results - Consumption rule

ABB WP

Results - Consumption rule

ABB WP

References