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 ∞ � ( 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 • cohort born at time t draws φ t ∼ F φ t
The Model: productivity HSV 2013 • productivity is composed of an idiosyncratic and island specific components: log w t = α t + ǫ t ���� ���� island ind. • the island level follows a random walk α t = α t − 1 + ω t with ω t ∼ F ω t • the individual component is formed by a random walk and an iid transitory ǫ t = κ t + θ t with θ t ∼ F θ t κ t = κ t − 1 + η t with η t ∼ F η t • agents entering at time t draw α 0 , κ 0 , 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 y t = w t h t net earnings are given by: y t = λ ( y t ) 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 • α 0 and φ are drawn before trading starts • agents are attached to an island with unknown ω t realized sequence • within island, agents can trade a complete set of insurance contracts - at t ≥ b they can trade s t +1 = ( ω t +1 , η t +1 , θ t +1 ) - contracts pay δ − 1 unit of consumption in state s t +1 • between island, more limited - at t ≥ b they can trade s t +1 = ( η t +1 , θ t +1 ) - can’t condition on ω t +1 • note that agents can trade a risk free bond by buying δ of each ( η t +1 , θ t +1 ) contracts
The Model: agent’s problem HSV 2013 • call s t = s b , s b +1 , ... s t the individual history of shocks � ( b , φ, α 0 , κ 0 , θ b ) for j = b s j = ( ω j , η j , θ j ) for j > b • Q t ( S ; s t ) is price of insurance bought at time t by agent with s t for event set S ⊆ S . • B t ( s t +1 ; s t ) is the quantity purchased t ( Z ; s t ) and B ∗ t ( z t +1 ; s t ) are equivalent for price and • Q ∗ quantity for agents from other islands. • Z and z do not include ω .
The Model: agent’s budget constraints HSV 2013 � 1 − τ + d t ( s t ) = c t ( s t ) � w t ( s t ) h t ( s t )) λ � Q t ( s t +1 ; s t ) B t ( s t +1 ; s t ) ds t +1 + S � t ( z t +1 ; s t ) B ∗ t ( s t +1 ; s t ) dz t +1 + Q ∗ Z where the realized wealth is given by d t ( s t ) = δ − 1 � � B t − 1 ( s t ; s t − 1 ) + B ∗ t − 1 ( z t ; s t − 1 )
The Model: equilibrium HSV 2013 • Given sequences { F φ t , F α 0 t , F κ 0 t , F ω t , F η t , F θ t } a competitive equilibrium is a set of allocations { c t ( s t ) , h t ( s t ) , d t ( s t ) , B t ( · , s t ) , B ∗ t ( · , s t ) } and prices { B t ( · , s t ) , B ∗ t ( · , s t ) } 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 ; s t ) = 0 for all Z • Consumption and hours are given by: φ + (1 − τ )1 + ˆ σ log c t ( s t ) = − (1 − τ )ˆ σ + γ α t + C a t ˆ φ + 1 − γ σ + γ α t + 1 log h t ( s t ) = − ˆ σǫ t + H a t ˆ ˆ σ ˆ where C a t and H a t are age-date specific constant, ˆ φ are tax-modified constants. • bond prices are given by where ∆ C a t is independent of a and F st 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 ( η t , θ t ) is perfectly insured within island, - the common part ω t is shocking everyone within island - the island shock ω t 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 φ + (1 − τ )1 + ˆ σ log c t ( s t ) = − (1 − τ )ˆ σ + γ α t + C a t ˆ φ + 1 − γ σ + γ α t + 1 log h t ( s t ) = − ˆ σǫ t + H a t ˆ ˆ • hours and ǫ t = κ t + θ t - hours work is increasing in the insurable component ǫ t - the response is scaled by the Frisch elasticity - insurability of ǫ rules out income effect • hours and α t - γ > 1 means income effect dominates substitution: hours ց • consumption is independent of ǫ t 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 ( R 2 = 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 φ w , c = 0 . 386 t • 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: y it = η it + ǫ it η it = Q t ( η i , t − 1 , u it ) • note that unit root is a particular case: η it = η i , t − 1 + F − 1 ( u it ) • 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 ρ t ( η i , t − 1 , τ ) = ∂ Q t ( η i , 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
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