Informality in labor market and welfare

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max

{ct,kt ,Lt }∞t ∞

t

βt_u(c t)

s.t  



ct+kt =wt+rtkt+ (1−δ)kt+qtLt+κt

ct, kt 0 0 Lt 1 t

k >0 " %

# ct $ G $ $ tH kt

$ # $ tH Lt $

-# ( $ qtH κt

$-$ " % H wt rt

# " % % $ %

$ $ %

! 4 $ {(qt, wt, rt)}∞_t

$ {κ_t}∞

t " % # " $ 4

-$ $ # #

$

. % % G $ # $

# " $ " C 4

v(k) =

 



maxh,c[u(c) +βv(h)]

s.t c+h=w+rk+ (1−δ)k+q+κ

c, h 0

 

 < =

! ( " %

−u′(c(k)) +βv′(h(k)) = 0 < =

# h(k) c(k) $ $ $ $

$ $ %

C * % $ ! # ( &$ ( % %

%

v′(k) = (1 +r−δ)u′(c(k)) <:=

" <:= < = # 4

-G $ * 4 ! 4 "

# G " ( $

-$ $

u′(c(k)) =β(1 +r′−δ)u′(c(h(k))) <@=

c(k) +h(k) =w+rk+ (1−δ)k+q+κ < =

# " 7

u(c) = 1

1−γc

−γ _<D=

B " # # 4 <@= < = # 7

c(k)−γ =β(1 +r′−δ)c(h(k))−γ <F=

c(k) +h(k) =w+rk+ (1−δ)k+q+κ <I=

! # 7 # &

" # & " % !

" $ &

$ < $ $ =

# " & "

$ % $

&

$-$ * $ %

(

! " %

# "

F (Kf, Nf) =AK

αf

f N

−αf

f

G(Ki, Ni, L) =AB KiψL −

ψ αi

N −αi

i

# 0 < αf < 1 0 < αi < 1 0 < ψ < 1 Kf Ki

$ $ % H Nf Ni

$ % H L

$ ! A $ $ %

B $ %

$ % ! $

" % αf " " %

αiψ " %

1−αi " " % 1−αf #

%

wf rf # " $ $ %

( " % ! $ ( $ %

( " %

πf = (1−τva)AK

αf

f N

−αf

f −(1 +τl)wfNf −(1 +τk)rfKf <J=

# τva & % < 0< τva <1= τk

& % $ % <τk >0= τl &

% % <τl >0=

( T−

va ≡ (1−τva) Tl ≡ (1 +τl) Tk ≡ (1 +τk)

4 <J= #

πf =Nf Tva− Ak

αf

f −Tlwf −Tkrfkf < =

# kf ≡

Kf

Nf

! ( ( G $ ( & ; $

; $ ( $ # " $ 7

rf =

T−

va

Tk

Aαfk

αf−

f < =

wf =

T−

va

Tl

A(1−αf)k

αf

f < =

! ; $ $

(

! $ ( $ % ( " %

πi =AB K

ψ i L

−ψ αi_N −αi

i −wiNi−riKi−qL < :=

# q % $ $

( " % # " wi

ri % $ q

! ( ( G $ ( & ; $

; $ ( $ # " 4 7

ri =

ABαiψ kiψl

−ψ i

αi

ki

< @=

q= ABαi(1−ψ) k

ψ i l

−ψ i

αi

li

< =

wi =AB(1−αi) kiψl

−ψ i

αi

< D=

# ki ≡ K_Ni

i li ≡

L Ni

! % ( $ $

$ (

$ G !

-$ $ # " 7

w=wf =wi

r =rf =ri

B " 4 < = < = < @= < D= # ( # " 7

w

r =

Tk

Tl

1−αf

αf

kf

= 1−αi

αiψ

ki < F=

! $ - - $ $

ki =

Tk

Tl

αi

αf

1−αf

1−αi

ψkf < I=

% " % & (

$

$-$ ! " % " " %

κ =τ_vaAKαf

f N

−αf

f +τlwNf +τkrKf < J=

# τva τl τk & "

!

"

#

$%

4 G $ 7

c(k) +h(k)−(1−δ)k =AKαf

f N

−αf

f +AB K ψ i L

−ψ αi_N −αi

i < =

# Kf Nf $ $ $ $

$ $ $ % (

! " G

4 7

Nf +Ni = 1 < =

Kf +Ki =k < =

L= 1 < :=

# ; k $ $

# " % " " $

6 $ % 4 (

{(ct, kt )}∞t $ % {(Kf t, Nf t)}∞_t

( {(Kit, Nit, Lt)}∞_t (

4 $ {(rt, wt, qt)}∞t 7

< = {(ct, kt )}∞t % <F= <I= " % $ {(rt, wt, qt)}∞t

" % k H

< ={(Kf t, Nf t)}∞_t % < = < = " % $ {(rt, wt, qt)}∞t H

< ={(Kit, Nit, Lt)}∞t % < @= < = < D= " % $ {(rt, wt, qt)}∞t H

< %= 7 < = < = < = < := t

! & ' ( $%

" # (

r = 1

β −(1−δ) < @=

! $

-" % $ %

kf =

Tva−

Tk

Aαf

β

1−(1−δ)β

−_αf

< =

ki =ψ

Tk

Tl

αi

αf

1−αf

1−αi

Aαf

T−

va

Tk

β

1−(1−δ)β

−_αf

1 # " 4 ; # ( " %

Ni = B

Tl

T−

va

1−αi

1−αf

kψαi

i

kαf

f

−_{ψ αi}

< F=

8 % " %

Nf = 1−Ni

! ( $%

( 4 # $

(& - 4

$ - - kf $ (

$ k "

Niki+ (1−Ni)kf =k < I=

# ki " % < D= Ni " % < F=

"

)

$

! # % " $ $

# & " ! $ δ

% - - $ 0.2 $

0.35 0.95 #

# $ %

$ $ γ + 0 ! % < @=

* "

αf 0.35

αi 0.05

δ 0.04

γ 2.5

! $ ψ ( "" " # $

-$ ! % $

( % $ . % # $

-# &-#34; " " " % $ ψ

% % ψ # $ $ % "

#

% " # % ψ

! $ <A B β= # (

% $ " # $ B

# 4 < I= < = < D= < F= < =

" # % τk= 0.25<# $ & 0.2

$$ $ =: _τ

l = 2/3 <# $ & 0.4

$$ = τva = 0.03@ "

# "7 $ - - $ 2.5

0.1 $ " 4 < =

< D= < F= # ( # " 7

Ni =

          

B Tl

T−

va −αi −αf×

× Tk

Tl

αi

αf −αf

−αiψ Aαf

T−

va

Tk r

−_αf ψαi

×

× AαfT

−

va

Tk r

− −αf_αf

          

−_{ψ αi}

< J=

# &$ " # < = < D= < I=

( 4 # ( %

$ A B β ! 4 "

< = < D= < J= < = ! #

% $ ψ

* " + ψ

ψ = 0.1 ψ = 0.3 ψ = 0.5 ψ = 0.7 ψ = 0.9

A F@ F@ F F F

B : F J D

β J@ J@ J@ J@ J@

$ A β $ % #

" % ψ ! : #

:_{6$$ & C # 4 % # &}

& $$

@ _{+ < @= * < =}

$ % - - $ % $ ψ

* ( * *, # (

ψ = 0.1 ψ = 0.3 ψ = 0.5 ψ = 0.7 ψ = 0.9

yi/y @ @ @ @ @

χ/y :: :: :: :: ::

# % yi/y χ/y % ψ " 0.1

0.9 ! % # $ % H *

< =

-

.

$

! C ; $ # "

# $ $ β " $

-αi αf 4 B !

# $ 7 A B #

4 < J= < = ! " # "

% C ; 0.2 $

-0.04 ! ) '

0.3 C ; $ 0.4

B & 1 <#

$ & 0.5 $$ = ! @ #

% $ % $ A B C ;

% $ ψ

* " + ψ

ψ = 0.1 ψ = 0.3 ψ = 0.5 ψ = 0.7 ψ = 0.9

A @D @D @D @D @D

B : :F :I @ @

! # % 4 yi/y χ/y k/y

% $ ψ

! * / ( * *, # ( / " # * *, # (

ψ = 0.1 ψ = 0.3 ψ = 0.5 ψ = 0.7 ψ = 0.9

yi/y

χ/y :D :D :D :D :D

k/y J : : :

$$ &

? 4 yi/y χ/y $ % ψ "

0.1 0.9 8 $ - - $ % %

$ ψ

* &

' (

)

$

8 # $

# " &$ E &

τl & % τva %

" % % κ _{. $ # %}

% $

ψ 1 % # % τl

τl τl % "

# Ni # # τl

# * &

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

l

0.08 0.1 0.12 0.14 0.16

I

N

ψ=0.1

ψ=0.3

ψ=0.5ψ=0.7

ψ=0.9

. $ # $ " % τl

$ # τl τva )

" % % ! % $

# # &$ 1

-% # % τl $

# $

-% τl % " $

A # %

$ $

$ $ D

# * , #

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

l 0.95

0.975 1 1.025

1.05 1.075 1.1 1.125

y

ψ=0.1 ψ=0.3

ψ=0.5 ψ=0.7

ψ=0.9

! % $ &

% &

$ " " % % "- $

B

% τl % "

-

.

$

$ &$ C ; !

% 4 B ! %

-" $ ψ " 0.1 0.9 #

# # ψ = 0.1 ! % Ni #

$ % B 1 % # % τl

D _{$$ & E # "- 4}

τl τva )

" % % τl % "

$$ . $ : #

# * &

0.5 1 1.5 2 2.5

l 0.26

0.28 0.3 0.32 0.34

i

N

ψ=0.1

! $ " # . $ @

" % τl E

B $

#

# * , #

0.5 1 1.5 2 2.5

l

0.4 0.42 0.44 0.46 0.48

y

ψ=0.1

! # B C ;

-" ! $ "

# "

!

*

(

!

)

$

$ % # % ' &$ #

& τl # &

% τva # ) " % % κ

# $ &$

$ # # $ #

$ " "

# ; 4 &$

$ ! ' # # $

" C " #

% ' $ &$

E τl τva % " %

% κ _{$ k . $ # #}

% $

$ % ψ = 0.9 # %

&$ % % $ τl = 2/3

# τl τva )

" % % κ _{$ k}

# ! * &

0 . 4 0 . 4 5 0 . 5 0 . 5 5 0 . 6 0 . 6 5 0 . 7 l

0 . 0 2 5 0 . 0 5 0 . 0 7 5 0 . 1 0 . 1 2 5 0 . 1 5 0 . 1 7 5 0 . 2

I

N

ψ =0 . 1 ψ =0 . 3

ψ =0 . 5

ψ =0 . 7

. $ D # # $$ $ #

$ (& # % $ #

# 0 * , #

0 . 4 0 . 4 5 0 . 5 0 . 5 5 0 . 6 0 . 6 5 0 . 7

l

0 . 9 8 0 . 9 9 1 1 . 0 1 1 . 0 2 1 . 0 3 1 . 0 4

y

ψ =0 . 1

ψ =0 . 3

ψ =0 . 5

ψ =0 . 7

# ( ' B !

$ # # $$

"

!

-

.

$

# C ;

$ # ψ = 0.5 1 $

% &$ % . $ F # %

τl τva )

" % % κ _{$ k (}

$ % B

# 1 * &

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

l

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

i

N

ψ=0.5

. $ I # # $$ $

% $ # # %

# 2 * , #

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

l

0.4 0.42 0.44 0.46 0.48

y

ψ=0.5

# ( $$ % "

! B C ;

0

3

0

# # % $ $ &$

$ '

"-- $ τl

τva ) " % % κ $ k

" # % ) $$ #

$ " ! #

! F

! # " %

W =

∞

t

βtu(ct)

! $ τl W " %

dW dτl _∗

=

∞

t

βt_u´(c∗₎ dct

dτl _∗

=u´(c∗₎ ∞

t

βt dct

dτl _∗

! 4 4 " %

ct+kt −(1−δ)kt=yt

u´(ct) =β(1 +rt −δ)u´(ct )

#

yt=Ak

αf

f,t(1−Ni,t) +ABk αiψ

i,t N

−αi −ψ

i,t

ki,t =

1 +τk

1 +τl

αi

αf

1−αf

1−αi

ψkf,t

Ni,t= B

1 +τl

1−τva

1−αi

1−αf

kψαi

i,t k

−αf

f,t

−_{ψ αi}

rt =

1−τva

1 +τk

Aαfk

αf−

f,t

Ni,tki,t+ (1−Ni,t)kf,t =k

% # ki Ni τva τl kf

! 4 4 Ni,tki,t+ (1−Ni,t)kf,t = k (

$ kf k ! # % # y r

k % #

ct+kt −(1−δ)kt=yt(kt, τl)

u´(ct) =β[1 +rt (kt , τl)−δ]u´(ct ) <: =

% # $ τl

 



dct

dτl _∗+

dkt

dτl _∗ =

∂yt

∂kt _∗+ 1−δ

dkt

dτl _∗+

∂yt

∂τl _∗

dct

dτl _∗−

dct

dτl _∗ = β

u c∗ u c∗

∂rt

∂kt _∗

dkt

dτl _∗+β

u|c∗ u c∗

∂rt

∂τl _∗

<: =

( # "

Cτl(s)≡ ∞

t

st dct

dτl _∗

Kτl(s)≡ ∞

t

st dkt dτl _∗

<::=

# 0< s <1 ?

∞

t

st dct

dτl _∗

= 1

s Cτl(s)−

dc dτl _∗

<:@=

∞

t

st dkt

dτl _∗

= 1

sKτl(s) <: =

dk

dτl _∗ = 0

# $ 4 <: = st _{# 0 < s <}

1 ; ( # # "

4 7





∂yt

∂kt _∗+ 1−δ− s −1

−βu c_{u c}∗∗

∂rt

∂kt _∗ s 1− s 

 Kτl(s)

Cτl(s)

= − −s ∂yt

∂τl _∗

lhs <:D=

#

lhs= 1

1−sβ u´(c∗₎

u´´(c∗₎

∂rt

∂τl _∗

− 1

s dc dτl _∗

% Kτl(s) Cτl(s)

Kτl(s) =

1 ∆

1 1−sβ

u´(c∗₎

u´´(c∗₎

∂rt

∂τl _∗

− 1

s dc dτl _∗

− ∂yt

∂τl _∗

<:F=

Cτl(s) =

1 ∆          ∂yt

∂kt _∗+ 1−δ− s ×

× _−sβu c_{u c}∗∗

∂rt

∂τl _∗− s

dc dτl _∗ +

− _−s ∂yt

∂τl _∗β

u c∗ u c∗

∂rt

∂kt _∗ s          <:I= #

∆≡ ∂yt

∂kt _∗

+ 1−δ− 1

s 1−

1

s − β

u´(c∗₎

u´´(c∗₎

∂rt

∂kt _∗

1

s

& " % % & & > 1

1 4 # %

&dc dτl

|∗ = β

u´(c∗₎

u´´(c∗₎

∂rt

∂kt _∗

&Kτl &

− _−(1−&)C

τl &

− ₊ &

&−1β

u´(c∗₎

u´´(c∗₎

∂rt

∂τl _∗

0< &− _{<1 ! % & # " 4}

∂yt

∂kt _∗

+ 1−δ−& (1−&) = βu´(c

∗₎

u´´(c∗₎

∂rt

∂kt _∗

&

$ " % 4 Kτl(&

− _{) "}

( 4 # (

(1−&) Cτl &

− ₋ &

&−1

∂yt

∂τl _∗

= βu´(c

∗₎

u´´(c∗₎

∂rt

∂kt _∗

&Kτl & −

# &$ ) $ $

$ (

dc dτl

|∗ =

1

&−1 β

u´(c∗₎

u´´(c∗₎

∂rt

∂τl _∗

+ ∂yt

∂τl _∗

# $ τl #

dW dτl _∗

=u´(c∗)Cτl(β) <:J=

( dW

dτl _∗ # $ # " % % 7

∂rt

∂τl _∗

∂rt

∂kt _∗

∂yt

∂τl _∗

∂yt

∂kt _∗ $ $

$ ! 4 4 #

ki(τl, τva, kf) =

1 +τk

1 +τl

αi

αf

1−αf

1−αi

ψkf

Ni(τl, τva, kf) = B

1 +τl

1−τva

1−αi

1−αf

[ki(τl, τva, kf)]ψαik

−αf

f

−_{ψ αi}

Ni(τl, τva, kf)ki(τl, τva, kf) + [1−Ni(τl, τva, kf)]kf =k

! 4 ( kf $ τl, τva k

kf =kf(τl, τva, k) "

C $ 1 ! # (

∂kf

∂τl _∗

=−

∂Ni

∂τl (ki−kf) +Ni

∂ki

∂τl

∂Ni

∂kf (ki−kf) + 1−Ni 1−

∂ki

∂kf _∗

<@ =

∂kf

∂τva _∗

=−

∂Ni

∂τva(ki−kf) +Ni

∂ki

∂τva

∂Ni

∂kf (ki−kf) + 1−Ni 1−

∂ki

∂kf _∗

<@ =

∂kf

∂k _∗ =

1

∂Ni

∂kf (ki−kf) + 1−Ni 1−

∂ki

∂kf _∗

<@ =

# kf " % < =

! " % " #

κ _{= [1−Ni(τl, τva, kf)]Ak}αf

f τva+τl

1−τva

1 +τl

(1−αf) +τk

1−τva

1 +τk

αf

# kf = kf(τl, τva, k) $ #

-τl τva ) " % % κ

! % 4 ( τva $

τl k τva = τva(τl, k)

" (

H(τl, τva, k) ≡ [1−Ni(τl, τva, kf)]k αf

f A×

× τva+τl

1−τva

1 +τl

(1−αf) +τk

1−τva

1 +τk

αf −κ

# kf =kf(τl, τva, k) $ $ % % 7

∂H ∂τl

= −Akαf

f τva+τl

1−τva

1 +τl

(1−αf) +τk

1−τva

1 +τk

αf ×

× ∂Ni

∂τl

+ ∂Ni

∂kf

∂kf

∂τl

+

+ (1−Ni)Aαfk αf−

f τva+τl

1−τva

1 +τl

(1−αf) +τk

1−τva

1 +τk

αf ×

×∂kf

∂τl

+ (1−Ni)Ak αf

f

(1−τva)

(1 +τl)

∂H

∂τva

= −Akαf

f τva +τl

1−τva

1 +τl

(1−αf) +τk

1−τva

1 +τk

αf ×

× ∂Ni

∂τva

+ ∂Ni

∂kf

∂kf

∂τva

+

+ (1−Ni)Aαfk αf−

f τva+τl

1−τva

1 +τl

(1−αf) +τk

1−τva

1 +τk

αf ×

×∂kf

∂τva

+Akαf

f (1−Ni)×

× 1−τl

1−αf

1 +τl

−τk

αf

1 +τk

∂H

∂k = τva+τl

1−τva

1 +τl

(1−αf) +τk

1−τva

1 +τk

αf Ak

αf

f

∂kf

∂k (1−Ni)αfk

−

f −

∂Ni

∂kf

C $ 1 ! # (

∂τva

∂τl _∗

=−

∂H ∂τl _∗

∂H ∂τva _∗

<@:=

∂τva

∂k _∗ =−

∂H ∂k ∗

∂H ∂τva _∗

<@@=

# kf " % < =

# $ % % ∂r

∂k ∗

∂r ∂τl _∗

∂yt

∂τl _∗

∂y ∂k ∗

1 4 4 < = # (

∂r

∂k _∗ = −

1 1 +τk

∂τva

∂k Aαfk

αf−

f

∗

+

− 1−τva

1 +τk

Aαf(1−αf)k

αf−

f ∂kf ∂k + ∂kf ∂τva ∂τva

∂k _∗ <@ =

∂r ∂τl _∗

= − 1

1 +τk

∂τva

∂τl

Aαfk

αf−

f

∗

+

− 1−τva

1 +τk

Aαf (1−αf)k

αf−

f

∂kf

∂τl

+ ∂kf

∂τva

∂τva

∂τl _∗

<@D=

# kf " % < =

! $ dy_{dk ∗ ∂τ}∂y

l _∗ $ " %

y(τl, τva, k) = Ak αf

f [1−Ni(τl, τva, kf)] +

+AB[ki(τl, τva, kf)] αiψ_[N

i(τl, τva, kf)] − αi −ψ

# kf = kf (τl, τva, k) $ ( % !

% % y # $ k τl % " %

∂y

∂k _∗ = Aαfk

αf−

f (1−Ni)

∂kf

∂k +

∂kf

∂τva

∂τva

∂k _∗+

−Akαf

f ∂Ni ∂τva ∂τva ∂k + ∂Ni ∂kf ∂kf ∂k + ∂kf ∂τva ∂τva

∂k _∗+

+ABαiψk αiψ−

i N

−αi −ψ

i ×

× ∂ki

∂τva ∂τva ∂k + ∂ki ∂kf ∂kf ∂k + ∂kf ∂τva ∂τva

∂k _∗+

+AB[1−αi(1−ψ)]kiαiψN

−αi −ψ

i ×

× ∂Ni

∂τva ∂τva ∂k + ∂Ni ∂kf ∂kf ∂k + ∂kf ∂τva ∂τva

∂k _∗ <@F=

∂y ∂τl _∗

= Aαfk αf−

f (1−Ni)

∂kf

∂τl

+ ∂kf

∂τva

∂τva

∂τl _∗

+

−Akαf

f

∂Ni

∂τl

+ ∂Ni

∂τva

∂τva

∂τl

+∂Ni

∂kf

∂kf

∂τl

+ ∂kf

∂τva

∂τva

∂τl _∗

+ +ABαiψkαiiψ− N

−αi −ψ

i ×

× ∂ki

∂τl

+ ∂ki

∂τva

∂τva

∂τl

+ ∂ki

∂kf

∂kf

∂τl

+ ∂kf

∂τva

∂τva

∂τl _∗

+ +AB[1−αi(1−ψ)]kiαiψN

−αi −ψ

i ×

× ∂Ni

∂τl

+ ∂Ni

∂τva

∂τva

∂τl

+∂Ni

∂kf

∂kf

∂τl

+ ∂kf

∂τva

∂τva

∂τl _∗

<@I=

# kf " % < =

0

4

(

# $ % dW

dτl _∗ ' $ (τl, τva)

! $ # 7 # τl% )

τva " % % ? $

(τl, τva) ( ' ! "

" % %

0 ) $

. $ J # % dW

dτl _∗ ' $ (τl, τva)# ψ = 0.1

I

! # # τl !

( $ (τl, τva)

. $ J - 6 7 B * <ψ .=

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 l

-160 -140 -120 -100 -80 -60

W

d

d

l

6$$ & * # # ψ = 0.9 % % dW

dτl _∗

$ % " τl # ! % ψ = 0.9

$ % " $ "

$ 1 %

ψ # ( " % % dW

dτl _∗ $ (τl, τva)

$ & &

I _{6$$ & * # # % ψ}

% $ " " % % #

0 - . $

. $ # % dW_dτ

l _∗ ' $ (τl, τva)# ψ = 0.1

! # # τl !

( $ (τl, τva)

. $ - 6 7 C ; * <ψ . =

0.8 0.9 1 1.1 1.2 1.3 1.4 l

-370 -360 -350 -340

W

d

d

l

! # % B

C ; 7 &

% & %

$ " % %

1

"

" # % $ &$ " τl

) "τva " % %

$ " % $ ψ

% " % τl C

# % τl # %

# $ (& # % #

$ # ! B

C ;

6 " # % $$ " 4

-# # # # τl ! $ $

& &

% " % %

#

2

-

'

#

C + + 9 K - " E $ 6

-L *+.*-1.2 @

* 1 K # * 7 ;

E E 4 L 2 MMM2

: 1 * N O

-+ K " " . " A 7 B > 6

%-DJ E L FD $ " @ J-@J:

@ A . + * # K #L

% J ? @ $ " - F

N K6 % - E $ +

-1 " L JI

D N K! E 1 ! & +

-1 " L J JIF

F + * # K 6 !

* $ PL %

I ? @

I + 0 E ! % $ K -C 8$

! & E $ % $ " * L

B % " @

J " KC E $ 6 6"" "

! 1 + % L ! % @ ?

1 @

6$$

& 6 - !

6 #

8$

# " # $ % " % τva = τl = τk = 0% " &Yi >0'(

! ( (

% # " ; $ 7

min

Ki,Ni

{rKi+wNi+q}

s.t Yi =ABK

ψαi

i N

−αi

i

! " " $ " %

L(Ki, Ni, λ) =rKi+wNi+q−λ ABKiψαiN

−αi

i −Yi

( " %

∂L

∂Ki

= r−λψαiABK ψαi−

i N

−αi

i = 0

∂L ∂Ni

= w−λ(1−αi)ABKiψαiN

−αi

i = 0

∂L

∂λ = − ABK

ψαi

i N

−αi

i −Yi = 0

#

Ki

Ni

= w

r ψαi

(1−αi)

" %

Ni(w, r, Yi) =

w r

ψαi

(1−αi)

− −_αiψαi−ψ

(AB)− −_αi −ψ _Y −αi −ψ

i

Ki(w, r, Yi) =

w r

ψαi

(1−αi)

−_αi −_αi −ψ

(AB)− −_αi −ψ Y −αi −ψ

i

! # " 7

ci(w, r, Yi) = w

−_αi −_αi −ψ r

ψαi

−_αi −ψ ×

× ψαi

1−αi

− −_αiψαi−ψ

+ ψαi 1−αi

−_αi −_αi −ψ

×

×(AB)− −_αi −ψ _Y −αi −ψ

i +q

(

ai ≡w

−_αi −_αi −ψ r

ψαi

−_αi −ψ 

 

ψαi −αi

− −_αiψαi−ψ

+ + ψαi

−αi

−_αi −_αi −ψ





(AB)− −_αi −ψ

! #

ci(w, r, Yi) =aiY

−_αi −ψ

i +q <@J=

# " " %

cmgi(Yi)≡

∂ci(w, r, Yi)

∂Yi

= 1

1−αi(1−ψ)

aiY

αi −ψ

−_αi −ψ

i < =

( (

min

Ki,Ni

{rKf +wNf}

s.a Yf =AK

αf

f N

−αf

f

! " " ; $ " %

L(Ki, Ni, λ) =rKf +wNf −λ AK

αf

f N

−αf

f −Yf

! ( " %

∂L

∂Kf

= r−λαfAK αf−

f N

−αf

f = 0

∂L

∂Nf

= w−λ(1−αf)AK αf

f N

−αf

f = 0

∂L

∂λ = − AK

αf

f N

−αf

f −Yf = 0

#

Kf

Nf

= w

r αf

(1−αf)

" %

Nf(w, r, Yf) =A−

w r

−αf α_f

1−αf

−αf

Yf

Kf(w, r, Yf) = A−

w r

−αf α_f

1−αf

−αf

Yf

! # " 7

cf(w, r, Yf) =A− w −αfrαf

αf

1−αf

−αf

+ αf 1−αf

−αf

Yf

( af ≡A− w −αfrαf αf −αf

−αf

+ αf −αf

−αf

-( #

cf(w, r, Yf) =afYf < =

A ( <$ % = " 7 cmgf(Yf)≡

∂cf(w,r,Yf)

∂Yf =af >0 # % Yi cmgi(Yi)<

af Yi ∈(0, Yi) cmgi(Yi) "

(0,∞) limYi→ cmgi(Yi) = 0 ! $ ( $

6$$

& C - *4 %

C #

! &

1

! &

$$

1

# 4 % # & $

& $$ $ <

$ % $$ = E

# # $ # &

$ % $$

! G $ # # " # 7

max

{(ct,ki,t ,kf,t ,ni,t,nf,t,Lt)}

∞

t ∞

t

βtu(ct)

s.t               

ct+ki,t +kf,t = (1−τls)wf,tnf,t+wi,tni,t+

(1−τs

k)rf,tkf,t+ri,tki,t+ (1−δ) (ki,t+kf,t) +qtLt+κt

ki,t+kf,t=kt

ni,t+nf,t = 1

ct, ki,t , kf,t , ni,t, nf,t 0 0 Lt 1 t

k >0 " %

! C 4 $

v(k) =

      

maxc,h,nf,kf logc+βv(h)

s.t  



c+h= (1−τs

l)wfnf +wi(1−nf) +

(1−τs

k)rfkf +ri(k−kf) + (1−δ)k+q+χ

c 0, h 0,0 kf k, 0 nf 1

      

! ( * % $ " %

1

c(k) =βv

′_(g(k))

1

c(k)[(1−τ

s

k)rf −ri] = 0

1

c(k)[(1−τ

s

l)wf −wi] = 0

v′(k) = 1

c(k)[(1−δ) +ri]

! * 4

1

c(k) =β

1

c(g(k))[(1−δ) +r

′

i]

(1−τs

k)rf = ri (1−τsl)wf = wi (

" $ $

1 $ ( & ; $ $ % (

#

wf =A(1−αf)K

αf

f N

−αf

f

rf =AαfK

αf−

f N

−αf

f

1 $ ( & ; $ $ % (

#

wi =AB(1−αi) KiψL −

ψ αi

N−αi

i

ri =ABαiψ KiψL

−ψ αi_N −αi

i K

−

i

q =ABαi(1−ψ) KiψL −

ψ αi

N −αi

ii L−

4 # % # " 7

AB(1−αi) K

ψ i L −ψ

αi

N−αi

i = (1−τsl)A(1−αf)K αf

f N

−αf

f

ABαiψ KiψL −

ψ αi_N −αi

i Ki− = (1−τ s

k)AαfK αf−

f N

−αf

f

! $ % # " #

$ - - 7

1−αi

αiψ

Ki

Ni

= 1−τ

s l

1−τs k

1−αf

αf

Kf

Nf

< =

# 1− τs

k = τk 1 −τ

s

l = τl # 4

< I= ! 4 % # &

& $$

6$$

& E -

"

6

% # $ - - "

τva τl A # ( kf : (0,1)×(0,∞)→R

kf (τva, τl) = A

1−τva

1 +τk

αf

β

1−(1−δ)β

−_αf

ki : (0,1)×(0,∞)→R

ki(τva, τl) =

1 +τk

1 +τl

αi

αf

1−αf

1−αi

ψ 1−τva

1 +τk

Aαf

β

1−(1−δ)β

−_αf

r= _β −(1−δ) ! # % # "

∂kf

∂τl

= 0

∂kf

∂τva

=− αf

1−αf

1 1 +τk

1

rAk

αf

f <0

∂ki

∂τl

=− 1

1 +τl

ki <0

∂ki

∂τva

=−ki

kf

αf

1−αf

1 1 +τk

1

rAk

αf

f <0

(τva, τl)∈(0,1)×(0,∞) ( Ni

τva τl Ni : (0,1)×(0,∞)→R " %

Ni(τva, τl) =

1 +τl

1−τva

B1−αi

1−αf

kψαi

i k

−αf

f

−_{ψ αi}

# kf = kf(τva, τl) ki = ki(τva, τl) ( $

% % Ni

∂Ni

∂τva = −ψ αi B

τl −τva

−αi −αfk

ψαi

i k

−αf

f

−_{ψ αi} ×

× −τva +ψαik

−

i ∂ki

∂τva+

−αfk−_f

∂kf

∂τva

−ψ αi

τl −τvaB

−αi −αfk

ψαi

i k

−αf

f

−_{ψ αi}

$ %

# - "

% % ∂Ni

∂τva ? # # # " # 7

−τva +ψαik −

i ∂ki

∂τva −αfk −

f ∂kf

∂τva = −τva +

αf −αf τk

A rk

αf−

f (αf −ψαi)

αf −ψαi > 0 % % _∂τ∂Ni

va $ % ?

$ $

$ %

∂Ni

∂τl = −ψ αi B

τl −τva

−αi −αfk

ψαi

i k

−αf

f

−_αi −ψ

−_{ψ αi} ×

×B _−τ

va −αi −αf k

ψαi

i k

−αf

f 1 +ψαi(1 +τl)k

−

i ∂k∂τil

#

1 +ψαi(1 +τl)ki−

∂ki

∂τl

= 1−ψαi

0 < ψαi < 1 # ∂N_∂τi

l > 0 ! Ni

" τva τl 8 % Nf : (0,1)×(0,∞) → R

(

Nf(τva, τl) = 1−Ni(τva, τl)

" τva τl

6 # % # $

τva τl ( Kf : (0,1)×(0,∞) → R Ki :

(0,1)×(0,∞)→R

Kf(τva, τl) =Nf(τva, τl)kf (τva, τl)

Ki(τva, τl) =Ni(τva, τl)ki(τva, τl)

$ % ! $ % % " %

∂Kf

∂τl

=kf

∂Nf

∂τl

<0

∂Kf

∂τva

=Nf

∂kf

∂τva

+kf

∂Nf

∂τva

<0

∂Ki

∂τl

=Ni

∂ki

∂τl

+ki

∂Ni

∂τl

∂Ki

∂τva

=Ni

∂ki

∂τva

+ki

∂Ni

∂τva

? " % % ∂Ki

∂τl

∂Ki

∂τva "

6$$

&

- !

8 $

C ;

*

B *

Kus "" " $ B E

" $ $

hus % " " B ! $ " %

Yus = [Aexp (φhus)Lus] − α

Kusα

# A ' % Lus # (

yus ≡

Yus

Lus

%

ybrexp (φhbr)

yusexp (φhus)

= 0.2

exp [φ(hus−hbr)] 2 # ( ybr/yus

0.4

6$$

& * -

ψ = 0.9 ψ = 0.95

# % dW

dτl _∗ % $ ψ

. $ : @ B . $ D F I

J C ;

. $ - 6 7 B * <ψ . =

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

l -80

-70 -60 -50 -40

W

d

d

l

. $ - 6 7 B * <ψ . =

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

l -60

-55 -50 -45 -40

W

d

d

l

. $ : - 6 7 B * <ψ . =

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

l -54

-52 -50 -48 -46 -44 -42 -40

W

d

d

l

. $ @ - 6 7 B * <ψ . =

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

l 0

25 50 75 100 125 150

W

d

d

l

. $ - 6 7 B * <ψ . =

0.6 0.7 0.8 0.9 1 1.1

l -10

-7.5 -5 -2.5 0 2.5 5

W

d

d

l

. $ D - 6 7 C ; * <ψ . =

0.8 0.9 1 1.1 1.2 1.3 1.4

l -390

-385 -380 -375

W

d

d

l

. $ F - 6 7 C ; * <ψ . =

0.8 0.9 1 1.1 1.2 1.3 1.4

l -4000

-3000 -2000 -1000 0

W

d

d

l

. $ I - 6 7 C ; * <ψ . =

0.8 0.9 1 1.1 1.2 1.3 1.4

l -120

-110 -100 -90 -80 -70

W

d

d

l

. $ J - 6 7 C ; * <ψ . =

0.8 0.9 1 1.1 1.2 1.3 1.4

l -200

-190 -180 -170 -160 -150 -140

W

d

d

l

. $ - 6 7 C ; * <ψ . =

0.6 0.7 0.8 0.9 1 1.1

l -230

-220 -210 -200 -190 -180

W

d

d

l