Theory Appendix. Model Setup. V - the value function of an unemployed worker at the beginning of the UI spell.

Transcription

1 Theory Appendix A Model Setup V - the alue function of an unemployed worker at the beginning of the UI pell. t - earch effort / normalized to the hazard of exiting unemployment at each point in time. earching at a gien intenity come at a cot of ψ t t where ψ t. may be arying oer time. S t - the urial function for remaining in unemployment The indiidual can chooe earch effort in a way that will determine the urial function. S t exp t tdt We can model thi a the indiidual deciding oer a urial function S where S repreent the whole function while S t repreent the alue of the urial function at t directly. A gien urial function come at a per period cot of earch effort ψ t S. b t - Flow of UI benefit at point in time t b t b until P Potential enefit uration b t after P T - Time horizon u. - the flow utility function when unemployed. - the flow utility function when employed depend on net wage w τ The alue function i gien a: V S t ub t + [ S t ]w τ S t ψ t t dt S t ub dt + S t u dt + P [ S t ]w τ dt S t ψ t t dt 2 udget contraint of the goernment: ˆ S t b dt [ S t ]τ dt 3 b T τ therefore: τ T b 4

2 dτ T + T d b + T + ε,b + ε,b T T 2 b 5 dτ d T + T T d b + τ b 6 Note that P S t dt and therefore d S P + P dt and d P dt Approximation with Contant Hazard If the hazard of exiting from unemployment i contant t, then unemployment duration follow an exponential ditribution where all releant tatitic can be calculated from the earch intenity : Surial function: S t e t,. Furthermore: S t dt e t dt ep S p [ S p ] S P e P Furthermore we can define: ξ + P e P For the cae of a contant hazard P ξ and P Proof: dt ξ, dt i proportional to the nonemployment effect: P ξ + P e P dt 2

3 ˆ T dt d d de t dt t d te dt te t dt + T et 2 For T going to infinity: Note that: and therefore: dt d d de t dt t d te dt te t dt + P ep 2 2 e T T + + P e P + T e T dt ep P P e P ξ d S P + ξ ξ And for d we get imilarly: ˆ d P Social Planner problem: d S P dt ξ max b,p,τ V b, P, τ.t. udget Contraint 7 3

4 Howeer the budget contraint implicitely define τ a a function of b, o we can intead maximize the uncontrained problem: max b,p V b, P, τb 8 FOC for b Uing the enelope condition, the marginal effect of increaing UI benefit by $ i gien a: S t dt u b [ S t ] dt dτ u b T dτ 9 Plugging in dτ [ ] u b [T ] + ε,b + ε,b T T [ u b w τ ] [ ] w τ ε,b + ε,b T Recaling: Or uing marginal effect: w τ u c u,t P c e c e u c u,t P c e d c e + ε,b + ε,b T b T τ T b u c u,t P c e d c e b + τ dw c e u c u,t P c e c e of $ add. tranfer u c u,t P c e c e of $ add. tranfer d b + τ per $ add. Tranfer τ η,b + η,b b per $ add. Tranfer 4

5 Uing the contant hazard approximation: FOC for P u c u,t P c e ξb + τ c e S P ub u [ S t ] dt dτ S P ub u T dτ ˆ P S P ub u S P b + S P ub u b τ τ ˆ ub u b P S P τ Uing the contant hazard approximation we get: ub u b w τ S P w τ S P ub u b w τ w τ ξb + τ ξb + τ 2 If we let define ũ c u,t>p b b u kdk ubu b a the aerage marginal utility between conumption leel of b and for a UI exhautee, we can write thi a: bũ b w τ S P ξb + τ 3 w τ 5

6 dw SP b c e ũ c u,t>p c e c e of $ add. tranfer ũ c u,t>p c e c e of $ add. tranfer P τ SP b per $ add. Tranfer ˆ P τ dt + S P b per $ add. Tranfer 4 5 Two equation to highlight: FOC for b aily-chetty Formula: FOC for P SW Formula: u b w τ w τ Mechanical increae in pending of $ add. tranfer } {{ } Mechanical Tranfer to Unemployed ξb + τ } {{} 6 ũ w τ S P b ξb + τ w τ } {{} Mechanical increae in pending of $ add. tranfer } {{ } Mechanical Tranfer to Unemployed 7 Calculating comparable dicincentie effect from literature FOC for b aily-chetty Formula: u c u,t P c e c e of $ add. tranfer u c u,t P c e c e of $ add. tranfer ξb + τ } {{ } per $ add. Tranfer η,b ξ + q ρ per $ add. Tranfer 8 9 6

7 u c u,t P c e c e of $ add. tranfer η,b ξ + q S P ρ per $ add. Tranfer 2 where q i the payroll tax rate τ qw and ρ i the replacement rate: b ρw and we aume that pre-unemployment wage equal pot unemployment wage. FOC for SW Formula: S P b ũ c u,t>p c e c e of $ add. tranfer ũ c u,t>p c e c e of $ add. tranfer ξb + τ S P b per $ add. Tranfer ξ + q S P ρ per $ add. Tranfer