Labor Supply James Heckman University of Chicago April 23, 2007 1 - - PowerPoint PPT Presentation

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Labor Supply James Heckman University of Chicago April 23, 2007 1 - - PowerPoint PPT Presentation

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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

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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

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

One period models: (L < 1)

U (C, L) = C α − 1 α + b Lϕ − 1 ϕ

  • α, ϕ < 1

b ↑ = ⇒ taste for leisure increases

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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): R = ∂U ∂L

  • ∂U

∂C | L = 1, C = A R = b Lϕ−1 C α−1 at L = 1, C = A R = b Aα−1 ln R = ln b + (1 − α) ln A

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Set: ln b = Xβ + εb Assume: εb ∼ N

  • 0, σ2

b

  • Assume:

ln W ⊥ ⊥ εb (X, A, W ) ⊥ ⊥ εb

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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(Xβ + (1 − α) ln A + εb ≤ ln W | X, A) = Pr εb σb ≤ ln W − Xβ − (1 − α) ln A σb

  • =

Φ (C) where C ≡ ln W − Xβ − (1 − α) ln A σb A > 0

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Let D = 1 if person works D = 0

  • therwise
  • ⇒ D = 1 [ln W ln R]

Pr(ln R ≤ ln W | X, A) = Pr(D = 1 | X, A) Take Grouped Data: Each cell has common values of Wi, Xi and Ai.

  • Pi = cell proportion working i

Set Pi = Φ

  • Ci
  • Ci = ln Wi − Xiβ − (1 − α) ln Ai

σb inverse exists:

  • Ci = Φ−1
  • Pi
  • (table lookup)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Run Regression:

  • Ci on ln Wi − Xiβ − (1 − α) ln Ai

σb Coefficient on ln Wi is 1 σb Coefficient on X is β σb Coefficient on ln A is 1 − α σb Do for Logit Pr ε σb ≤ z

  • =

ez 1 + ez

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Linear Probability Model Pr ε σb ≤ z

  • =

z zU − zL zL ≤ ε σb ≤ zU

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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): L =

I

  • i=1

Φ (Ci (2Di − 1))

  • β,

σb, 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 Fεb (Cosslett, Manski, Matzkin)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Digression D∗ = Zγ − V , D = 1(Zγ > V ), assume Var(V ) = 1. Can be nonparametric about V . Normality is not needed. Assume Zi, Zj are continuous: Pr (D = 1 | Z) = FV (Zγ) ∂ Pr (D = 1 | Zγ) ∂Zi ∂ Pr (D = 1 | Zγ) ∂Zj = fV (Zγ) γi fV (Zγ) γj = γi γj We can identify the coefficients up to scale. Back to text.

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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 = Zγ + U ln R = Xβ + (1 − α) ln A + ε U ε

  • ∼ N

0 , σUU σεU σεU σεε

  • ln R − ln W ≥ 0 ⇐

⇒ D = 0 Y1 ≡ −Xβ − (1 − α) ln A + Zγ − (ε − U) = ln W − ln R

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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, σεε + σUU − 2σεU) Var (ε − U) = (σ∗)2 σ∗ ≡

  • σεε + σUU − 2σεU

Pr (Y1 ≥ 0 | X, A, Z) = Pr (D = 1 | X, ln A, Z)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Pr (D = 1 | X, ln A, Z) = Pr −Xβ − (1 − α) ln A + Zγ σ∗ ≥ ε − U σ∗

  • =

Φ −Xβ − (1 − α) ln A + Zγ σ∗

  • = Φ(C)

C ≡ −Xβ − (1 − α) ln A + Zγ σ∗ If Z and X distinct from each other and A, estimate

γ σ∗, β σ∗, 1−α σ∗ ,

can’t estimate σ∗, ∴ get relative values.

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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; Xc = Zc elements in common Xd, Zd are distinct elements in X, Z Y1 σ∗ = −Xdβd σ∗ − Xc (βc − γc) σ∗ + Zdγd σ∗ + (1 − α) σ∗ ln A + ε − U σ∗ ∴ identify βd σ∗ , βc − γc σ∗ , γd σ∗, 1 − α σ∗ (The leading example of variables in common is education.) Allows U to be correlated with ε. (Method II may be required anyway.)

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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 = Zγ + U ln R = Xβ + (1 − α) ln A + ε Assume (X, Z, A) ⊥ ⊥ (ε, U) Y1 = ln W − ln R = Zγ − Xβ − (1 − α) ln A + U − ε

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Letting ˜ λ(q) = φ(q) Φ(q), we have E (ln W | ln W − ln R ≥ 0, X, Z, A) = E   ln W

  • Zγ − Xβ − (1 − α) ln A

σ∗ ≥ ε − U σ∗ , X, Z, A    = Zγ + σUU − σUε σ∗ ˜ λ Zγ − Xβ − (1 − α) ln A σ∗

  • C (X, A, Z) = Zγ − Xβ − (1 − α) ln A

σ∗

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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) E (Z|Z ≥ q) = λ(q); λ(q) ≡ φ(q) 1 − Φ(q) = φ(q) Φ(−q) E (Z|q ≥ Z) = −E (−Z| − Z ≥ −q) = − φ(−q) 1 − Φ(−q) = − φ(q) Φ(q) ⇒

  • λ(q) ≡ φ(q)

Φ(q) = −E (Z|Z q) and : E (Z|Z ≥ q) = φ(q) Φ(−q) = λ(q) = λ(−q)

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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) = Pr Zγ − Xβ − (1 − α) ln A σ∗ ≥ ε − U σ∗

  • Z, X, A
  • =

Φ Zγ − Xβ − (1 − α) ln A σ∗

  • σ∗ = [Var (U − ε)]

1 2

∴ we can estimate C (X, A, Z) (2) Form ˜ λ (C)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Run Linear Regression Get Consistent Estimates of γ, σUU − σUε σ∗ With one exclusion restriction (one variable in Z not in X or ln A, say Z1).

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Note that using Probit if Xd, Zd are distinct elements in X, Z and Xc = Zc are elements in common we can identify

βd σ∗, βc−γc σ∗ , γd σ∗, 1−α σ∗ .

Say we recover

γ1 σ∗ (by Probit)

Note that we have γ (by Wage Regression on Z and λ) ∴know σ∗ The estimated coefficient on λ is σUU − σUε σ∗ ∴know σUU − σUε

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Look at the residuals from equations V ≡ ln W −

  • Zγ + σUU − σUε

σ∗ ˜ λ (C (X, A, Z))

  • =

U −

  • σUU − σUε

σ∗ (σUU)

1 2

  • (σUU)

1 2 ˜

λ (C (X, A, Z)) Let : ρ ≡ σUU − σUε (σUU)

1 2 σ∗

V = U − ρ (σUU)

1 2 ˜

λ (C (X, A, Z)) = U − E (U| ln W − ln R ≥ 0) ⇒ E (V ) = 0 E

  • V 2

= Var(V ) = Var (U| ln W − ln R ≥ 0)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

E

  • V 2

= σUU

  • 1 − ρ2

+ ρ2 1 + ˜ λC − ˜ λ2 = σUU + σUUρ2 ˜ λC − ˜ λ2 Regress

  • V 2 on
  • ˜

λC − C 2 Get σUU and σUUρ2 ∴ know ρ2

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Look at model:

1

Wrong variables appear in wage equation

2

Errors heteroskedastic

3

Omitted variables

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Recovered Coefficients:

γ1 σ∗ (Probit)

γ (Wage Regression )

  • ⇒ σ∗

σUU−σUε σ∗

(Wage Regression ) σ∗

  • ⇒ σUU − σUε

σUU (Error2 Regression ) σUU − σUε

  • ⇒ σUε

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

The term σUU − σUε σ∗ : σUU − σUε σ∗ is a Wage Regression coefficient ρ ≡ σUU−σUε

(σUU)

1 2 σ∗ (Error2 Regression )

σUU (Error2 Regression )

  • ⇒ σUU − σUε

σ∗ ⇒ 2 estimates of σUU − σUε σ∗

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

The term σ∗ :

γ1 σ∗ (Probit)

γ (Wage Regression )

  • ⇒ σ∗

σUU − σUε (Wage Regression & σ∗ above) ρ ≡ σUU−σUε

(σUU)

1 2 σ∗ (Error2 Regression )

σUU (Error2 Regression )      ⇒ σ∗ ⇒ 2 estimates of σ∗ To obtain σεε, we can solve (σ∗)2 = σUU + σεε − 2σUε ∴ (σ∗)2 + 2 σUε − σUU = σεε

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Suppose we have no exclusion restriction, just regressors. Then we can still estimate γ, σUU, σεε provided we substitute other information for exclusion restrictions. b = σUU − σUε σ∗ = σUU − σUε (σUU + σεε − 2σUε)

1 2

(coefficient on λ) E

  • V 2

= σUU + σUUρ2 ˜ λC − ˜ λ2 = σUU + σUU

  • σUU − σUε

(σUU)

1 2 σ∗

2 ˜ λC − ˜ λ2 = σUU + b2 ˜ λC − ˜ λ2 ⇒ σUU = E

  • V 2

− b2 ˜ λC − ˜ λ2

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Normalize variables: σεε = 1 or σUε = 0 Example: σUε = 0 Then know σUU (σUU + σεε)

1 2

∴can solve for σεε Alternatively, if σεε = 1 σUU − σUε (1 + σUU − 2σUε)

1 2 = known

solve for σUε, quadratic equation – sometimes get unique root. Note crucial role of regressor in getting full identification.

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Labor Supply – Hours of Work – Single Period Model

More Information: Direct Utility Function for non workers: V1 (A1, ϕ) A1 = unearned income if person works best attainable utility for a person who doesn’t work Indirect Utility Function: V2 (A2, W , ϕ) (W = wage) best available utility given that he “works”, (which may be V1). A2 is unearned income net of money costs of work

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

For person who works: If V2 > V1 person works Index Function: Y1 = V2 − V1 Y1 ≥ 0 person works Y2 = H = ∂V2 ∂W ∂V2 ∂A

  • = H (A2, W , ϕ)

Roy’s Identity: 3 types of labor supply functions: (a) participation (b) E(H|H > 0, W , A) (c) E(H|W , A) aggregate labor supply

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

None estimates a labor supply function (Hicks-Slutsky). Workers free to choose hours of work. Wage W is independent of hours of work. No fixed costs. Local comparison is global comparison. Consider a simple example based on Heckman (1974),

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

MRS Function: ln R = α0 + α1A + α2X + ηH + ε (1) ln W = Zγ + U ln R defines an equilibrium value of time locus. Labor supply H is the value that equates ln W = ln R: ln W = α0 + α1A + α2X + ηH + ε ∴ H = 1 η (ln W − α0 − α1A − α2X − ε) The “causal effect” of ln (wage) on labor supply is 1 η (holding A, X and ε constant). This is a Hicks-Slutsky effect.

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

E.g. ∂H ∂ ln W = S

  • substitution

effect

+ H ∂H ∂Y

income effect

, 1 η = ∂H ∂ ln W = WS + (WH) ∂H ∂Y . If η is constant, then as H ↑, for a fixed W , S ↑ (more substitution). As W ↑, S + H ∂H

∂Y ↓ ,

so the Hicks-Slutsky effect declines (net labor supply becomes more inelastic in this sense).

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Figure: Value of Time

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Define Y1 = ln W − ln R = Zγ + U − α0 − α1A − α2X − ε = (Zγ − α0 − α1A − α2X) + (U − ε). Hours of work then are: Y2 = H = 1 ηY1 if Y1 ≥ 0 Y3 = ln W = Zγ + U Var(U − ε) = (σ∗)2

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Population Labor Supply is generated from Y2 E (H|Y1 > 0, Z, A, X) = 1 ηE (ln W − ln R| ln W − ln R > 0, Z, A, X) = Zγ − α0−α1A − α2X η +1 ηE (U − ε | U − ε > −(Zγ − α0 − α1A − α2X)) = Zγ − α0−α1A − α2X η +1 ηE U − ε σ∗ | U − ε σ∗ > −(Zγ − α0 − α1A − α2X) σ∗

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Let C ≡ Zγ − α0 − α1A − α2X σ∗ E (H|Y1 > 0, Z, A, X) = 1 ηE (Y1|Y1 > 0, Z, A, X) = (Zγ − α0 − α1A − α2X) η +1 ηE (U − ε|U − ε > −(Zγ − α0 − α1A − α2X), Z, A, X)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

= C η σ∗ + σ∗ η E U − ε σ∗ |U − ε σ∗ > −C, Z, A, X

  • =

C η σ∗ + 1 η Cov(U − ε, U − ε) σ∗ ˜ λ(C) = C η σ∗ + 1 ησ∗˜ λ(C) = σ∗ η

  • C + ˜

λ(C)

  • E(ln W |Y1

> 0, Z) = E(ln W | ln W − ln R > 0, Z) = Zγ + σ∗E( U σ∗|U − ε σ∗ > −C, Z) = Zγ + Cov(U, U − ε) σ∗ ˜ λ(C)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Assume Regressors are available: ∴ We can estimate γ from linear regression ln W on Zγ and ˜ λ(C) using the known steps: (1) From participation equation, we can use probit to estimate Pr (D = 1 | Z, A, X) = Pr (Y1 > 0 | Z, A, X) = Φ Zγ − α0 − α1A − α2X σ∗

  • =

Φ(C) ∴ We know

γ σ∗, α0 σ∗, α1 σ∗, α2 σ∗ if X = A = Z or set of common and

distinct coefficients depending on X, A, Z elements. ∴ We know C. (2) Form λ(C). (3) From the Wage Regression of ln W on Z and ˜ λ(C).

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

∴we know γ, Cov(U,U−ε)

σ∗

. Thus we know Cov(U, U − ε) σ∗ = σUU − σUε (σUU + σεε − 2σUε)1/2. (4) From Error Regression :

  • V 2 on constant and
  • ˜

λC − C 2 , we estimate: E (V 2) = σUU + σUUρ2(˜ λC − ˜ λ2) ∴ know σUU, ρ2 Same position as before. Further identification of parameters is possible due to hours of work: (5) From hours of work data we have a proportionality restriction

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

E(H|Y1 > 0, Z, A, X) = σ∗ η

  • C + σ∗

η ˜ λ(C) but from employment (participation) equation we know C ∴ can estimate σ∗ η

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

(6) Using Cov(U,U−ε)

σ∗

from Wage Regression and covariance assumptions, we obtain: if σUε = 0 ⇒ Cov(U, U − ε) σ∗ = σUU (σUU + σεε)1/2 but σUU was obtained by Error Regression ∴ know σ∗ and σεε By The hours of Work Regression (5) we obtain σ∗ η ∴ know η Similarly if σεε = 1 ⇒ σUε known σUε known (sometimes; multiple roots) ∴ we have that all parameters are identified.

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

(7) If there is one variable in Z not in (X, A), say Z1, from coefficient on Z1 in E(H|Y1 > 0, Z, A, X), we obtain: E (H|Y1 > 0, Z, A, X) = σ∗ η

  • C + σ∗

η ˜ λ(C) = σ∗ η C + ˜ λ(C)

  • =

σ∗ η Zγ − α0 − α1A − α2X σ∗ + ˜ λ(C)

  • =

Z γ η + −α0 − α1A − α2X η + σ∗ η

  • ˜

λ(C)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

but from Wage Regression (3), we obtain γ ∴ can estimate η but the coefficient of ˜ λ(C) is σ∗ η

  • ∴

can estimate σ∗. Alternatively, we can determine η if Cov(U, ε) = 0

  • r

Var(ε) = 1.

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Selection Bias in Labor Supply Assume γj > 0 ∂E(H | Y1 > 0, Z, A, X) ∂Zj = ∂

  • σ∗

η

C + ˜ λ(C)

  • ∂Zj

= ∂

  • Z γ

η + −α0−α1A−α2X η

+

  • σ∗

η

  • ˜

λ(

Z γ

η + −α0−α1A−α2X η

σ∗

)

  • ∂Zj

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

But ∂˜ λ(q) ∂q = ∂ φ(q)

Φ(q)

∂q = −q φ(q) Φ (q) − φ2(q) Φ2 (q) = −˜ λ(q)

  • q + ˜

λ(q)

  • =

γj η − 1 η

  • λ(

λ + C)γj = γj η

  • (1 −

λ( λ + C)) < 1 − λ( λ + C) < 1 ∴ < γj η downward bias.

∂H ∂Zj ∂W ∂Zj

= ∂E(H | Y1 > 0, Z, A, X) ∂ ln W ≤ 1 η ∴ downward biased

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

E (H|Y1 > 0, Z, A, X) = σ∗ η

  • C + σ∗

η ˜ λ(C) = σ∗ η C + ˜ λ(C)

  • =

σ∗ η Zγ − α0 − α1A − α2X σ∗ + ˜ λ(C)

  • =

Z γ η + −α0 − α1A − α2X η + σ∗ η

  • ˜

λ(C)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

but from Wage Regression (3), we obtain γ ∴ can estimate η but the coefficient of ˜ λ(C) is σ∗ η

  • ∴

can estimate σ∗.

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Aggregate Labor Supply ALS ≡ Φ(C) · E (H|Y1 > 0) +Φ(−C) · E (H|Y1 < 0)

  • =

Φ(C) σ∗ η C + ˜ λ(C)

  • + Φ(−C) · [0]

= σ∗ η Φ(C)C + 1 √ 2π e−C 2/2

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

∂E(HALS|Z, A, X) ∂C = σ∗ η

  • Φ(C) + Ce− 1

2 C 2/2

√ 2π − Ce−C 2/2 √ 2π

  • =

σ∗ η [Φ(C)] ∂E(HALS|Z, A, X) ∂Zj = ∂E(H|Z, A, X) ∂C ∂C ∂Zj = γj η Φ(C)

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Obviously aggregate labor supply more elastic because of entry or exit: 1 E(H|Y1 > 0, W )P(Y1 > 0|W ) ·∂ {E(H|Y1 > 0, W )P(Y1 > 0|W )} ∂ ln W ≥ ∂E(H|Y1 > 0, W ) ∂ ln W

  • 1

E(H|Y1 > 0, W ), and ∂ ln E(H|Y1 > 0, W ) ∂ ln W + ∂ ln P(Y1 > 0|W ) ∂ ln W > ∂ ln E(H|Y1 > 0, W ) ∂ ln W Many ways to estimate model.

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Labor Supply with Optimal Wage-Hours Contracts (Lewis, 1969; Rosen, 1974; Tinbergen, 1951, 1956) Figure: Optimal Wage

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

If Y (h) is earnings, and Y ′ (h) is marginal wage, Virtual Income = Y (h) − Y ′ (h) h + A, where h = h (Y (h) , Y (h) − Y ′ (h) h + A, ν) Any equilibrium calculation use slope at zero hours of work. ln M(0, A) ≤ ln W (0) doesn’t work Equilibrium: ln M(h, A) = ln W (h) person works Can use estimated ln M(0, A) to price out goods that previously were not purchased.

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Fixed Cost Models: Fixed Money Cost (Cogan; 1981 Econometrica) Figure: Fixed Cost Models

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One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Introduce fixed money cost: given the wage, the worker selects a minimum number of hours. (1) Solve for Wd and Hd that causes the worker to be indifferent between work and no work V (A − F, Wd, ϕ) = U(A, 1, ϕ) Solve for Wd If no solution, person doesn’t work (2) Minimum number of hours Hd = VW /VA Hd = Hd(A − F, Wd, ϕ) Wd = Wd(A − F, ϕ) H = H(A − F, W , ϕ)

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SLIDE 56

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Index Function Model. Y1 = H − Hd Y2 = H Observe Y2 only when Y1 > 0 Example: (assume wage is known). H = Xβ + W η + ε1 functional form assumptions Hd = Xτ + ε2 Pr (consumer works) = Pr (H − Hd > 0|X, W ) = Pr(Xβ + W η + ε1 − Xτ − ε2 > 0) = Pr(X(β − τ) + W η > ε2 − ε1) σ∗ ≡

  • Var(ε2 − ε1)

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SLIDE 57

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Assume normality and we can identify β − τ σ∗ and η σ∗ From hours of work equation we know η ∴ know σ∗ E (H|H − Hd > 0, X, W ) = Xβ + W η + E (ε1|X(β − τ) + W η > ε2 − ε1, X, W ) = Xβ + W η + Cov(ε1,ε2) σ∗

  • λ(C)

C = X(β, τ) + W η σ∗

  • Know η, β ∴ know τ

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SLIDE 58

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Know Var(ε1), know σ∗ ∴ know Cov(ε1, ε2) ∴ know Var(ε2) = (σ∗)2 − Var(ε1) + 2 Cov(ε1, ε2)

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SLIDE 59

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Cogan doesn’t measure fixed costs

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SLIDE 60

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Figure: Fixed vs. Not Fixed

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SLIDE 61

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Null: Departure from simple proportionality model Cogan’s test conditional on function form.

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SLIDE 62

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Broken line Budget Constraint (2 part prices; negative income tax data) Two cases:

  • A. Know which interval person is in (no measurement error for

hours)

  • B. Don’t know which branch (income tax data)

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SLIDE 63

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Figure: Which Branch?

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SLIDE 64

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Take case 1: (1) M(A, 1, ε) ≥ W1 does a person work? (2) Person is interior in interval (0, h) if M(A + W1h, 1 − h, ε) ≥ W1 M(A, 1, ε) < W1 (3) In equilibrium at h if W2 ≤ M(A + W1h, 1 − h, ε) ≤ W1 (4) Works beyond h if M(A + W1h, 1 − h, ε) ≤ W2

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SLIDE 65

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Example: U = C α − 1 α + b Lϕ − 1 ϕ

  • α < 1, ϕ < 1

b Lϕ−1 C α−1 = MRS at zero hours work (L = 1)

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SLIDE 66

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

ln b = Xβ + ε ln b + (ϕ − 1) ln(1) − (α − 1) ln A ≥ ln W1 (doesn’t work) Xβ + ε − (α − 1) ln A ≥ ln W1 ε ≥ ln W1 + (α − 1) ln A − Xβ E(ε2) = σ2

ε

ε σε ≥ ln W1 + (α − 1) ln A − Xβ σε condition for not working estimate: σε, α, β

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SLIDE 67

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

(2) Interior in the first branch (0, h) Xβ + (ϕ − 1) ln(1 − h) − (α − 1) ln(A + W1h) − ln W1 σε ≥ −ε σε Xβ + (ϕ − 1) ln(1) − (α − 1) ln(A) − ln W1 σε ≤ −ε σε Use principle of index function, with variation in h, we identify ϕ and β.

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SLIDE 68

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

(3) Kink Equilibrium: ln W2 ≤ Xβ + ε + (ϕ − 1) ln(1 − h) − (α − 1) ln(A + W1h) ≤ ln W1 ln W2 − Xβ − (ϕ − 1) ln(1 − h) + (α − 1) ln(A + W1h) σε ≤ ε σε ≤ ln W1 − Xβ − (ϕ − 1) ln(1 − h) + (α − 1) ln(A + W1h) σε

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SLIDE 69

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Pr(h = h|W1, W2, X, A) = Φ ln W1 − Xβ − (ϕ − 1) ln(1 − h) − (α − 1) ln(A + W1h) σε

  • −

Φ ln W2 − Xβ − (ϕ − 1) ln(1 − h) + (α − 1) ln(A + W1h) σε

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SLIDE 70

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

(4) In second branch interior Pr Xβ + (η − 1) ln(1 − h) − (α − 1) ln(A + W1h) − ln W2 σε ≤ ε σε

  • = Φ

ln W2 − Xβ − (η − 1) ln(1 − h) + (α − 1) ln(A + W1h) σε

  • ∴ solve out hours of work.

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SLIDE 71

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Hours of work (standard case) U = C α − 1 α + b Lϕ − 1 ϕ

  • A = nonmarket income

W = wage C = W (1 − L) + A

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SLIDE 72

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Set ϕ = α, Set h = b

W

  • 1

α−1 − A

b

W

  • 1

α−1 + W

estimating equation: ln Wh + A 1 − h

  • = ln W

1 − α − Xβ 1 − α − ε 1 − α

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SLIDE 73

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Interior in Branch 1: E

  • ln

W1h + A 1 − h

  • |W1, A
  • = ln W1

1 − α − Xβ 1 − α − 1 1 − αE           ε

  • Xβ + (η − 1) ln(1 − h)

−(α − 1) ln(A + W1h) − ln W1

  • σε

≥ −ε σε ≥ Xβ + (α − 1) ln(1) −(α − 1) ln A − ln W1

  • σε

         

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SLIDE 74

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Last term is: −1 1 − α   

1 √ 2π

  • e−C 2

1 /σ2 ε − e−C 2 2 /σ2 ε

  • Φ
  • C1

σε

  • − Φ
  • C2

σε

  C1 = Xβ + (α − 1) ln(1 − h) −(α − 1) ln(A + (W1 − W2)h) − ln W1

  • σε

C2 = Xβ + (α − 1) ln(1) − (α − 1) ln A − ln W1 σε

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SLIDE 75

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

At h = h (with probability P = P3)Q E(h|h = h) = hP3 P3 = Pr      

1 σε

ln W2 − Xβ − (η − 1) ln(1 − h) +(α − 1) ln(A + W1h)

  • ≤

ε σε

≤

1 σε

ln W1 − Xβ − (η − 1) ln(1 − h) +(α − 1) ln(A + W1h)

     etc.

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SLIDE 76

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Branch 2 labor supply ln W2h + (W1 − W2)h + A 1 − h

  • = ln W2

1 − α − Xβ 1 − α − ε 1 − α E (W1 − W2)h + A 1 − h

  • h > h
  • = ln W2

1 − α − Xβ 1 − α − 1 1 − αE     ε

  • ln W2 + (α − 1) ln(A + W1h)

−Xβ − (α − 1) ln(1 − h)

  • σε

≥ ε σε     

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SLIDE 77

One period models Method II For working persons Labor Supply Optimal Wage-Hours Fixed Cost Models

Let Z (1) = ln W1h + A 1 − h

  • Z (2) = ln

W2h + (W1 − W2)h + A 1 − h

  • E(Z (1)|W1, A, 0 < h < h) = ln W1

1 − α − Xβ 1 − α − 1 1 − αE(ε|0 < h < h) E(Z (2)|W2, A, h > h) = ln W2 1 − α − Xβ 1 − α − 1 1 − αE(ε|h > h) Can estimate by 2 stage methods.

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