One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Labor Supply James Heckman University of Chicago April 23, 2007 1 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models One period models: ( L < 1) � L ϕ − 1 � U ( C , L ) = C α − 1 + b α, ϕ < 1 α ϕ b ↑ = ⇒ taste for leisure increases 2 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models MRS at zero hours of work (Reservation Wage or Virtual Price): � ∂ U � ∂ L � | L = 1 , C = A R = � ∂ U ∂ C R = b L ϕ − 1 C α − 1 at L = 1 , C = A b R = A α − 1 ln R = ln b + (1 − α ) ln A 3 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Set: ln b = X β + ε b Assume: � � 0 , σ 2 ε b ∼ N b Assume: ln W ⊥ ⊥ ε b ( X , A , W ) ⊥ ⊥ ε b 4 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Assume wage is observed for everyone. Probability that a person with assets A , X , and Wage W works: Pr (ln R ≤ ln W | X , A ) = Pr( X β + (1 − α ) ln A + ε b ≤ ln W | X , A ) � ε b � ≤ ln W − X β − (1 − α ) ln A = Pr σ b σ b = Φ ( C ) where ln W − X β − (1 − α ) ln A C ≡ A > 0 σ b 5 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Let � D = 1 if person works ⇒ D = 1 [ln W � ln R ] D = 0 otherwise Pr(ln R ≤ ln W | X , A ) = Pr( D = 1 | X , A ) Take Grouped Data: Each cell has common values of W i , X i and A i . � P i = cell proportion working i � � Set � � P i = Φ C i C i = ln W i − X i β − (1 − α ) ln A i σ b inverse exists: C i = Φ − 1 � � � � P i (table lookup) 6 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Run Regression: C i on ln W i − X i β − (1 − α ) ln A i � σ b Coefficient on ln W i is 1 σ b Coefficient on X is β σ b Coefficient on ln A is 1 − α σ b Do for Logit � ε � e z Pr ≤ z = σ b 1 + e z 7 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Linear Probability Model � ε � z L ≤ ε z Pr ≤ z = ≤ z U σ b z U − z L σ b 8 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Micro Data Analogue: Sample size I, (Assumes we have symmetric ε around zero): I � L = Φ ( C i (2 D i − 1)) i =1 � � � β, � σ b , � α = arg max ln L consistent, asymptotically normal. (Likelihood is concave) Assumes we know wage for all persons, including those who work, but we don’t. Can be nonparametric about F ε b (Cosslett, Manski, Matzkin) 9 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Digression D ∗ = Z γ − V , D = 1( Z γ > V ), assume Var( V ) = 1. Can be nonparametric about V . Normality is not needed. Assume Z i , Z j are continuous: Pr ( D = 1 | Z ) = F V ( Z γ ) ∂ Pr ( D = 1 | Z γ ) f V ( Z γ ) γ i = γ i ∂ Z i = ∂ Pr ( D = 1 | Z γ ) f V ( Z γ ) γ j γ j ∂ Z j We can identify the coefficients up to scale. Back to text. 10 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Method II Don’t know wage, but ln W = Z γ + U ln R = X β + (1 − α ) ln A + ε � 0 � U � � 0 , σ UU σ ε U ∼ N ε σ ε U σ εε ln R − ln W ≥ 0 ⇐ ⇒ D = 0 Y 1 ≡ − X β − (1 − α ) ln A + Z γ − ( ε − U ) = ln W − ln R 11 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models ( X , ln A , Z ) ⊥ ⊥ ( ε − U ) ( ε − U ) ∼ N (0 , σ εε + σ UU − 2 σ ε U ) ( σ ∗ ) 2 Var ( ε − U ) = � σ ∗ ≡ σ εε + σ UU − 2 σ ε U Pr ( Y 1 ≥ 0 | X , A , Z ) = Pr ( D = 1 | X , ln A , Z ) 12 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Pr ( D = 1 | X , ln A , Z ) � − X β − (1 − α ) ln A + Z γ � ≥ ε − U = Pr σ ∗ σ ∗ � − X β − (1 − α ) ln A + Z γ � = Φ = Φ( C ) σ ∗ − X β − (1 − α ) ln A + Z γ C ≡ σ ∗ σ ∗ , β γ σ ∗ , 1 − α If Z and X distinct from each other and A , estimate σ ∗ , can’t estimate σ ∗ , ∴ get relative values. 13 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Suppose X and Z have some elements in common; X c = Z c elements in common X d , Z d are distinct elements in X , Z Y 1 σ ∗ = − X d β d − X c ( β c − γ c ) + Z d γ d + (1 − α ) ln A + ε − U σ ∗ σ ∗ σ ∗ σ ∗ σ ∗ ∴ identify β d σ ∗ , β c − γ c , γ d σ ∗ , 1 − α σ ∗ σ ∗ (The leading example of variables in common is education.) Allows U to be correlated with ε . (Method II may be required anyway.) 14 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Observe the wage only for working persons ln W = Z γ + U ln R = X β + (1 − α ) ln A + ε Assume ( X , Z , A ) ⊥ ⊥ ( ε, U ) Y 1 = ln W − ln R = Z γ − X β − (1 − α ) ln A + U − ε 15 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models λ ( q ) = φ ( q ) Letting ˜ Φ( q ), we have E (ln W | ln W − ln R ≥ 0 , X , Z , A ) � � Z γ − X β − (1 − α ) ln A � � σ ∗ = ln W E � ≥ ε − U � , X , Z , A � σ ∗ � Z γ − X β − (1 − α ) ln A � Z γ + σ UU − σ U ε ˜ = λ σ ∗ σ ∗ C ( X , A , Z ) = Z γ − X β − (1 − α ) ln A σ ∗ 16 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Remembering the Truncated Normal Random variable: Let: Z ∼ N (0 , 1) φ ( q ) φ ( q ) E ( Z | Z ≥ q ) = λ ( q ); λ ( q ) ≡ 1 − Φ( q ) = Φ( − q ) E ( Z | q ≥ Z ) = − E ( − Z | − Z ≥ − q ) 1 − Φ( − q ) = − φ ( q ) φ ( − q ) = − Φ( q ) λ ( q ) ≡ φ ( q ) � ⇒ Φ( q ) = − E ( Z | Z � q ) φ ( q ) Φ( − q ) = λ ( q ) = � and : E ( Z | Z ≥ q ) = λ ( − q ) 17 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Two Stage Procedures (1) Probit on Work participation Pr ( D = 1 | Z , X , A ) = Pr (ln W − ln R ≥ 0 | Z , X , A ) � Z γ − X β − (1 − α ) ln A � � � ≥ ε − U � = Pr � Z , X , A σ ∗ σ ∗ � Z γ − X β − (1 − α ) ln A � = Φ σ ∗ σ ∗ = [Var ( U − ε )] 1 2 ∴ we can estimate C ( X , A , Z ) Form ˜ (2) λ ( C ) 18 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Run Linear Regression Get Consistent Estimates of γ, σ UU − σ U ε σ ∗ With one exclusion restriction (one variable in Z not in X or ln A , say Z 1 ). 19 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Note that using Probit if X d , Z d are distinct elements in X , Z and β d σ ∗ , β c − γ c σ ∗ , γ d X c = Z c are elements in common we can identify σ ∗ , 1 − α σ ∗ . γ 1 Say we recover σ ∗ (by Probit) Note that we have γ (by Wage Regression on Z and � λ ) ∴ know σ ∗ λ is σ UU − σ U ε The estimated coefficient on � σ ∗ ∴ know σ UU − σ U ε 20 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Look at the residuals from equations � � Z γ + σ UU − σ U ε ˜ ≡ ln W − λ ( C ( X , A , Z )) V σ ∗ � � σ UU − σ U ε 1 2 ˜ = U − ( σ UU ) λ ( C ( X , A , Z )) 1 σ ∗ ( σ UU ) 2 ρ ≡ σ UU − σ U ε Let : 1 2 σ ∗ ( σ UU ) 1 2 ˜ = U − ρ ( σ UU ) λ ( C ( X , A , Z )) V = U − E ( U | ln W − ln R ≥ 0) ⇒ E ( V ) = 0 � V 2 � E = Var( V ) = Var ( U | ln W − ln R ≥ 0) 21 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models �� + ρ 2 � λ 2 �� � V 2 � 1 − ρ 2 � 1 + ˜ λ C − ˜ E = σ UU σ UU + σ UU ρ 2 � λ 2 � λ C − ˜ ˜ = Regress � λ C − C 2 � V 2 on � ˜ Get σ UU and σ UU ρ 2 ∴ know ρ 2 22 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Look at model: Wrong variables appear in wage equation 1 Errors heteroskedastic 2 Omitted variables 3 23 / 77
One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models Recovered Coefficients: � γ 1 σ ∗ (Probit) ⇒ σ ∗ γ (Wage Regression ) � σ UU − σ U ε (Wage Regression ) σ ∗ ⇒ σ UU − σ U ε σ ∗ � σ UU (Error 2 Regression ) ⇒ σ U ε σ UU − σ U ε 24 / 77
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