Task 1: Connection to previous framework (20min) ¶ Write down the marginal treatment response functions m 0 , m 1 m_0, m_1 m 0 , m 1
Hints:
You will need to use the decomposition of ϕ i \phi_i ϕ i
Feel free to use the glossary below to help you with the notation / copy & paste
elements.
No need to take “same notation” too strictly: Feel free to replace the i i i
Task 2: Examples for different individuals (40min) ¶ Consider the two paragaphs of page 8 (starting with the second line) and, in particular,
footnote 5.
Provide examples of individuals (“personas”) who differ in their utility from working,
their cost of participation, and their response to treatment. Locate them in Figure 3.
Hint:
A persona is a fictional character that represents a certain type of individual. It
can be helpful to give them a name and a brief description.
An example could start with: “Paula is a single mother of two children who worked as a
waitress before becoming a mother. She thoroughly enjoyed her job. Paula does not have
relatives in the area and her children’s fathers are not available, either. This puts
considerable constraints ...”
Task 3: Estimation Strategy (40min) ¶ 3.1 Role of identifying assumptions ¶ Explain how A1 and A2 help in enabling estimation of (13) and (14).
Hint:
3.2 Role of programme parameters G , r , t G, r, t G , r , t ¶ How do these programme parameters enter the estimation? Discuss the merits and pitfalls
of this approach.
Task 4: Results (30min) ¶ Explain
whether the spline estimation strategy is important as opposed to a linear one
why there is evidence for heterogeneity
whether you find the results convincing
Glossary ¶ Utility function:
U ( H i , Y i ; θ i ) − ϕ i P i U(H_i, Y_i; \theta_i) - \phi_i P_i U ( H i , Y i ; θ i ) − ϕ i P i Welfare benefit formula:
B i = G − t W i H i − r N i B_i = G - t W_i H_i - r N_i B i = G − t W i H i − r N i Budget constraint:
Y i = { W i ( 1 − t ) H i + G + ( 1 − r ) N i if P i = 1 W i H i + N i if P i = 0 Y_i =
\begin{cases}
W_i (1 - t) H_i + G + (1 - r) N_i & \text{if } P_i = 1 \\
W_i H_i + N_i & \text{if } P_i = 0
\end{cases} Y i = { W i ( 1 − t ) H i + G + ( 1 − r ) N i W i H i + N i if P i = 1 if P i = 0 Labor supply function:
H i = H [ W i ( 1 − t P i ) , N i + P i ( G − r N i ) ; θ i ] H_i = H[W_i (1 - t P_i), N_i + P_i (G - r N_i); \theta_i] H i = H [ W i ( 1 − t P i ) , N i + P i ( G − r N i ) ; θ i ] Participation function:
P i ∗ = V [ W i ( 1 − t ) , G + N i ( 1 − r ) ; θ i ] − V [ W i , N i ; θ i ] − ϕ i P_i^* = V[W_i (1 - t), G + N_i (1 - r); \theta_i] - V[W_i, N_i; \theta_i] - \phi_i P i ∗ = V [ W i ( 1 − t ) , G + N i ( 1 − r ) ; θ i ] − V [ W i , N i ; θ i ] − ϕ i Parametrization of the costs:
ϕ i = m ( Z i ) + ν i \phi_i = m(Z_i) + \nu_i ϕ i = m ( Z i ) + ν i Participation indicator:
P i = 1 ( P i ∗ ≥ 0 ) P_i = 1(P_i^* \geq 0) P i = 1 ( P i ∗ ≥ 0 ) Set of budget constraint variables:
C i = [ W i , N i , G , t , r ] C_i = [W_i, N_i, G, t, r] C i = [ W i , N i , G , t , r ] Labor supply response:
Δ i ( θ i ∣ C i ) = H [ W i ( 1 − t ) , G + N i ( 1 − r ) ; θ i ] − H [ W i , N i ; θ i ] \Delta_i(\theta_i | C_i) = H[W_i (1 - t), G + N_i (1 - r); \theta_i] - H[W_i, N_i; \theta_i] Δ i ( θ i ∣ C i ) = H [ W i ( 1 − t ) , G + N i ( 1 − r ) ; θ i ] − H [ W i , N i ; θ i ] Definition of marginal individuals:
0 = V [ W i ( 1 − t ) , G + N i ( 1 − r ) ; θ D ] − V [ W i , N i ; θ D ] − ϕ D = d V ( θ D ∣ C i ) − ϕ D 0 = V[W_i (1 - t), G + N_i (1 - r); \theta_D] - V[W_i, N_i; \theta_D] - \phi_D = dV(\theta_D | C_i) - \phi_D 0 = V [ W i ( 1 − t ) , G + N i ( 1 − r ) ; θ D ] − V [ W i , N i ; θ D ] − ϕ D = d V ( θ D ∣ C i ) − ϕ D Mean labor supply responses:
Δ M T E ( C i ) = E ϕ D [ Δ [ θ D ( ϕ D , C i ) ∣ C i ] ] \Delta_{MTE}(C_i) = E_{\phi_D} \left[ \Delta[\theta_D(\phi_D, C_i) | C_i] \right] Δ MTE ( C i ) = E ϕ D [ Δ [ θ D ( ϕ D , C i ) ∣ C i ] ] Mean effect of the transfer program:
Δ ˉ P i = 1 = E ( Δ i ∣ C i , P i = 1 ) = 1 P ∫ S θ ϕ Δ i ( θ i ∣ C i ) d J ( θ i , ϕ i ) \bar{\Delta}_{P_i = 1} = E(\Delta_i | C_i, P_i = 1) = \frac{1}{P} \int_{S_{\theta \phi}} \Delta_i(\theta_i | C_i) dJ(\theta_i, \phi_i) Δ ˉ P i = 1 = E ( Δ i ∣ C i , P i = 1 ) = P 1 ∫ S θϕ Δ i ( θ i ∣ C i ) dJ ( θ i , ϕ i ) Participation rate:
P = E ( P i ∣ C i ) = ∫ S ϕ ∫ S θ 1 { V [ W i ( 1 − t ) , G + N i ( 1 − r ) ; θ i ] − V [ W i , N i ; θ i ] − ϕ i } d J ( θ i , ϕ i ) P = E(P_i | C_i) = \int_{S_{\phi}} \int_{S_{\theta}} 1\{V[W_i (1 - t), G + N_i (1 - r); \theta_i] - V[W_i, N_i; \theta_i] - \phi_i\} dJ(\theta_i, \phi_i) P = E ( P i ∣ C i ) = ∫ S ϕ ∫ S θ 1 { V [ W i ( 1 − t ) , G + N i ( 1 − r ) ; θ i ] − V [ W i , N i ; θ i ] − ϕ i } dJ ( θ i , ϕ i ) Mean effect over the entire population:
Δ ˉ = E ( Δ i P i ∣ C i ) = ∫ S θ ϕ Δ i ( θ i ∣ C i ) d J ( θ i , ϕ i ) \bar{\Delta} = E(\Delta_i P_i | C_i) = \int_{S_{\theta \phi}} \Delta_i(\theta_i | C_i) dJ(\theta_i, \phi_i) Δ ˉ = E ( Δ i P i ∣ C i ) = ∫ S θϕ Δ i ( θ i ∣ C i ) dJ ( θ i , ϕ i ) Marginal treatment effect (MTE):
∂ Δ ˉ ∂ P \frac{\partial \bar{\Delta}}{\partial P} ∂ P ∂ Δ ˉ Labor supply equation:
H i = β i + α i P i H_i = \beta_i + \alpha_i P_i H i = β i + α i P i Fixed costs equation:
P i ∗ = m ( Z i ) + δ i P_i^* = m(Z_i) + \delta_i P i ∗ = m ( Z i ) + δ i Participation indicator:
P i = 1 ( P i ∗ ≥ 0 ) P_i = 1(P_i^* \geq 0) P i = 1 ( P i ∗ ≥ 0 ) Population mean of H i H_i H i
E ( H ∣ W , N , G , t , r , Z ) = E θ [ H ( W , N ) ∣ W , N ] + E θ , ν [ Δ ∣ P = 1 , W , N , G , t , r , Z ] E θ , ν ( P ∣ W , N , G , t , r , Z ) E(H | W, N, G, t, r, Z) = E_\theta[H(W, N) | W, N] + E_{\theta, \nu}[\Delta | P = 1, W, N, G, t, r, Z] E_{\theta, \nu}(P | W, N, G, t, r, Z) E ( H ∣ W , N , G , t , r , Z ) = E θ [ H ( W , N ) ∣ W , N ] + E θ , ν [ Δ∣ P = 1 , W , N , G , t , r , Z ] E θ , ν ( P ∣ W , N , G , t , r , Z ) Reduced form equation for H H H
E ( H ∣ Z i = z ) = E ( β i ∣ Z i = z ) + E ( α i ∣ P i = 1 , Z i = z ) E ( P i ∣ Z i = z ) E(H | Z_i = z) = E(\beta_i | Z_i = z) + E(\alpha_i | P_i = 1, Z_i = z) E(P_i | Z_i = z) E ( H ∣ Z i = z ) = E ( β i ∣ Z i = z ) + E ( α i ∣ P i = 1 , Z i = z ) E ( P i ∣ Z i = z ) Participation probability:
E ( P i ∣ Z i = z ) = Pr [ δ i ≥ − m ( z ) ] E(P_i | Z_i = z) = \text{Pr}[\delta_i \geq -m(z)] E ( P i ∣ Z i = z ) = Pr [ δ i ≥ − m ( z )] Identifying restrictions:
E ( β i ∣ Z i = z ) = β E(\beta_i | Z_i = z) = \beta E ( β i ∣ Z i = z ) = β E ( α i ∣ P i = 1 , Z i = z ) = g [ E ( P i ∣ Z i = z ) ] E(\alpha_i | P_i = 1, Z_i = z) = g[E(P_i | Z_i = z)] E ( α i ∣ P i = 1 , Z i = z ) = g [ E ( P i ∣ Z i = z )] Final estimating equations:
H i = β + g [ F ( Z i ) ] F ( Z i ) + ϵ i H_i = \beta + g[F(Z_i)] F(Z_i) + \epsilon_i H i = β + g [ F ( Z i )] F ( Z i ) + ϵ i P i = F ( Z i ) + ξ i P_i = F(Z_i) + \xi_i P i = F ( Z i ) + ξ i Five-knot natural cubic spline:
g ( F ^ ) = g 1 + g 2 F ^ + g 3 S 3 + g 4 S 4 + g 5 S 5 S 3 = d 1 − d 4 S 4 = d 2 − d 4 S 5 = d 3 − d 4 d 1 = max ( 0 , F ^ − F 1 ) − max ( 0 , F ^ − F 5 ) / ( F 5 − F 1 ) d 2 = max ( 0 , F ^ − F 2 ) − max ( 0 , F ^ − F 5 ) / ( F 5 − F 2 ) d 3 = max ( 0 , F ^ − F 3 ) − max ( 0 , F ^ − F 5 ) / ( F 5 − F 3 ) d 4 = max ( 0 , F ^ − F 4 ) − max ( 0 , F ^ − F 5 ) / ( F 5 − F 4 ) \begin{align*}
g(\hat{F}) & = g_1 + g_2 \hat{F} + g_3 S_3 + g_4 S_4 + g_5 S_5 \\
S_3 & = d_1 - d_4 \\
S_4 & = d_2 - d_4 \\
S_5 & = d_3 - d_4 \\
d_1 & = \max(0, \hat{F} - F_1) - \max(0, \hat{F} - F_5) / (F_5 - F_1) \\
d_2 & = \max(0, \hat{F} - F_2) - \max(0, \hat{F} - F_5) / (F_5 - F_2) \\
d_3 & = \max(0, \hat{F} - F_3) - \max(0, \hat{F} - F_5) / (F_5 - F_3) \\
d_4 & = \max(0, \hat{F} - F_4) - \max(0, \hat{F} - F_5) / (F_5 - F_4)
\end{align*} g ( F ^ ) S 3 S 4 S 5 d 1 d 2 d 3 d 4 = g 1 + g 2 F ^ + g 3 S 3 + g 4 S 4 + g 5 S 5 = d 1 − d 4 = d 2 − d 4 = d 3 − d 4 = max ( 0 , F ^ − F 1 ) − max ( 0 , F ^ − F 5 ) / ( F 5 − F 1 ) = max ( 0 , F ^ − F 2 ) − max ( 0 , F ^ − F 5 ) / ( F 5 − F 2 ) = max ( 0 , F ^ − F 3 ) − max ( 0 , F ^ − F 5 ) / ( F 5 − F 3 ) = max ( 0 , F ^ − F 4 ) − max ( 0 , F ^ − F 5 ) / ( F 5 − F 4 )