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1 " ! H L(

' ! ! ! θA L/" %

! H/" % (θL< θH).

8_{? 4 - . ! ' 1 / 0}

2 $ " f θil =θil,

" l ( $ ! $ % "

" % & ! ! θ( C "

" % & ! ( ? C - ===. " "

! ! ! !

! (

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) $ /! / ' (E _{2 $}

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* ! $ "

- ( ( /! '

" * ! .(

) % & ! ! ! /

! " ! ! 1A

U(c )₋ϕ(l ) +U(c )₋ϕ(l ), - .

" cj lj ! ! j(

) U ϕ A

2 " ! ! $

" ( ) ! $

" ! $ " ! * ( !

" " ! ! ! (

<_{) ! " % & ! "}

! ( 2 ! θ

" ! /θ " % ! "

$ " '/! /# ! $

- " ! $ " .(

E_{2 ! C 6 - . @ - ==E.(}

U lim

x→ U

′_{(x) = +∞}

ϕ lim

x→ ϕ

′_{(x) = 0}

lim

x→ −

ϕ′(x) =_∞.

! /i " % /

" ! A

max

{s,l ,l }u(c

i₎

−ϕ(li) +u(ci)₋ϕ(li), - .

( ( ci _≤liw ₋si -8.

ci _≤liwi +si - .

" # 1 si ₍

{lij, cij, w , wi} i L,H

j ,

{li

j, cij}j , i

! "

# $ w = θ l_l θ l_l ,

wi _=θi_.

4 " % & ! A

u′(ci₎

u′_(ci₎ = 1, -<.

u′(ci)w =ϕ′(li), -E.

u′(ci)θi =ϕ′(li). -;.

:1 -<. 1 " /

-1.( :1 -E. -;. 1 "

$ - .

-. ! 1 2 ! (

B θL < w < θH ! "/! " %

$ ! " /! /

" % ( 2 * "

! /θL" % $ "

! (

!

? ! /$ " %

' # " & ( /

" ! % "

" % & ! -$ / .(

Y_ij ! /j " % ! iA

Y_ij _≡ w l

j

i, i= 1

θjl_ij, i= 2 .

? ! "

! ' A

YL+YH +YH+YL_≥cL+cL+cH+cH.

" ! "

" " !

% ' ! % (;

2 $ / !

max

{c ,Y }

u(cL)₋ϕ Y

L

θL +u(c

L₎

−ϕ Y

L

θL ]+[u(c

H₎

−ϕ Y

H

θH +u(c

H₎

−ϕ Y

H

θH

( ( YL+YH +YH+YL_≥cL+cL+cH+cH

2 - 0 . $ / A

u′(cL) =u′(cL) =u′(cH) = u′(cH) = u′(c∗), ->.

ϕ′_(li j)

θi =u

′_(ci_{), i=L, H, j = 1,2. -=.}

:1 ->. 1 # " % & ! (

:1 -=. " %

(ϕ′_(li_{)) 1 $ (θ}

iu′(c∗)). 2 !

! " %

! ( 7 / !

" " % & ! ! /θH " % "

% ! /θL(

;_{2 ! ! $ 5 $}

' ( ?

" " ! " % ! /

! ( C " "

H 0 & " -" ! 1 .( 2 !

! !! / "

* " ( C " !! "

! " " # (

!

" ! !

' " ! (

2 * ! /

' ( ) $ "

" % & ! !

" $ ! ! ( C "/

" !

" % & ! ! ( 2

" % ( 2

" " ! '

! / ! !

ci j, Yji

i L,H

j , (

? " $ ! " /

! '! ! (

w = θ

L_lL_+θH_lH

lL_+lH =

θL Y_w +θH Y_w

Y w + Y w - . = θ

L_YL_+θH_YH

YL_+YH .

4 $ % " " % & ! ! "

wi =θi.

? " !! ' !

! " % ! (

C / ! "

! ! /θH " % ! /θL " % & ( 2

/ ! ' # " 5

/ ! 1

- . " " $ ! A

max

{c ,w ,Y }

u(cL)₋ϕ Y

L

w +u(c

L₎

−ϕ Y

L

θL +u(c

H₎

−ϕ Y

H

w +u(c

H₎

−ϕ Y

H

θH

( ( YL+YH+YH+YL_≥cL+cL+cH+cH, - .

u(cH)₋ϕ Y

H

w +u(c

H₎

−ϕ Y

H

θH ≥u(c

L₎

−ϕ Y

L

w +u(c

L₎

−ϕ Y

L

θH , - .

w = θ

L_YL_+θH_YH

YL_+YH . - 8.

λ, η, ! " 1 - .

- . - 8. ! ( 2 " $ /

* " ! cL _cH

u′(cL) =u′(cL) = λ

1₋η , - .

u′(cH) =u′(cH) = λ

1 +η . - <.

:1 - . ! ! /

L " % ! λ

! ! /H " %

- " ! / ! η.( :1 /

- <. 0 ! 1 ! /H " % !

λ !

(

9 1 "

! !

-cL=cL< cH =cH.(

* " ! YL _A

ϕ′ Y

L

θL 1 θL −ηϕ

′ YL

θH 1

θH =λ. - E.

:1 - E. " %

! /L ! A ! ! θL /

λ " % '

! ( B 1 !!

" ! !

ϕ′ Y_θ

ϕ′ Y

θ

θH θL > η.

2 " ! " %

! ' * ! "

" -! .(

* " ! YL _A

ϕ′ Y

L

w =w

λ

1₋η −w 1₋η w −θ

L

. - ;.

9 1 - ;. " " % ! /L

$ ! !

A

! 1 " w - 1 /

* .( 2 * ! ! * "

! w ! /L& ! θL * !

" ! ( ? w > θL " 1

* " " ! " % ! /L

$ ! (

C " * !!

* " ! "

" -" " %

.(

ϕ′ lL < w u′ cL > θLu′ cL .

* " ! YH _YH_{, "}

" $ / A

ϕ′ Y

H

θH 1

θH =

λ

1 +η, - >.

ϕ′ Y

H

w =w

λ

1 +η +w 1 +η θ

H

−w . - =.

2 ! 1 - >. 1 - E.(

2 " % ! /H ! /

!

" % (

1 - <. " " % 1 $

/! ! (

:1 - =. 1 - ;.( 2 *

* !! " " ! /H " % & ! /

(

"

#

" "

" $ " % & ! $ !

-" ! % .( 2 $ * "

" 1 " % ! ! ( 2

" * ( ! * "

- " % " 1 ! /

! ! .( C "

! " " ( 2 *

" A !

" % (

2 " ! ! # (

% # $

& # $

& &

# . Vx_(lj

, lj) ! /j " %

" " % (lj, lj) " x = 0 x = 1

! ( l_ij lj_i ! " % "

" ! ( 4 $ /! !!

lj =lj′, j, j′.

) " " 1 A

1 2V (l

H_{, l}H_{) +}1

2V (l

L_{, l}L₎

≥ 1₂V (lH, lH) +1 2V (l

L_{, l}L_).

? /! " " % & !

" li ˜_li _{( C !}

" A

V (li, li)_≥V (li, li). - .

B ' " " " " 1 A

1 2V (l

H_{, l}H_{) +}1

2V (l

L_{, l}L₎

≥ 1

2V (l

H_{, l}H_{) +}1

2V (l

L_{, l}L_{). - .}

2 - . " " (

$ U U " !!

" ! " " (lj_i)A

U = 1

2V (l

H_{, l}H_{) +}1

2V (l

L_{, l}L_{) =} 1

2[2u(c

H₎

−v(lH)₋v(lH)]

+1 2[2u(c

L₎

−v(lH)₋v(lH)],

U = 1

2V (l

H_{, l}H_{) +}1

2V (l

L_{, l}L_{) =} 1

2[2u(c

H₎

−v(lH)₋v(lH)]

+1 2[2u(c

L

" " ! " !

" ! (

cL _{< c}H _cH ₋ l θ −w _{= c}H _{> c}L ₌

cL+l θ −w - lH =lL $ /! !!

.( C " A

U ₋U =

θ −w

[u′(cH₋ l

H_θ

2 ) +u

′_(cL

+ l

H_θ

2 )]l

H

dθ >0,

" ! u.

- . ) %/ # "

u(x) =Kx,

K >0.

) " -w =θi. 1 - <. - E.

ϕ′(li j)

θi =K,

i, j.

1 ->. -=. " # $ /

"

ϕ′_(li j)

θi =K.

2 ! 1 " $ / (

1 - <.

K = ϕ

′_(li₎

w .

2 w = θi $ / 1

" . " "

( . . .

? 4 ( $ / " ! /

! $ !/ ' 1 # ! /

" % ( C " ! / ! (

2 / " % /

( 2

" A ' /

* 0 /

, ' ! ! ! # " /

" " ! ! (

2 ! " !

" " ! " ! /

( / !

! " (

"

# _{c˜j_i,Y˜_ij_}j L,H_{i ,} ! / '/

! " " !

{c˜j_i,˜l_ij_}j L,H_{i ,} ( 9 " " " " (

) " $ ! 1

! ( C "

$ !

Yj = ˜Yj w θj .

? "

! _{˜cj_i, Y_ij_}j L,H_{i ,} $ (

B ' " " " _{c˜j_i, Y_ij_}j L,H_{i ,} / ! "

/ !

(

4 !! / !

" 1 " _{c˜j_i,˜lj_i}j L,H_{i ,}

! A

u(˜cH) +u(˜cH)₋ϕ Y

H

w −ϕ

˜

YH

θH ≤u(˜c

L

) +u(˜cL)₋ϕ Y

L

w −ϕ

˜ YL

θL .

2 ! /θH " !! "

$ ! A

YL

w =

˜ YL

θL > ˜ YL

θH. θH > θL(

2 "

˜ YL

θH =

YL

w θL θH <

YL

w .

C ! /θH " ! ! /

θL " ! _{˜cj_i,Y˜_ij_}j L,H_{i ,} !

/ ' ! " " ( . . .

2 ! "

/ % "/ % " % ! ( 2

" " (

9 1 # $ /! " / % " % &

" "/ %

" % & (

$

4 " " * " $ &

( 2 /

! " % " # "

D " % & ! * (

7 ! 1 " ! " %/

1 2 $ " " 1

2( 2 ! (

? ! /1 " % ! θ > 0" " %

1 ! /2 " % ! θ+ε 2( C

$ " % & ! "

! " % " ! 2(

$ i _{∈ {}1,2_} ! /j " %

j =i, c > 0. 2 ! /i " %

! " $ ! ! /i (

* * !

" " % ! ( C "

ε >0 0 (

? ! $ " % & !

( ! ! #

* - ' ! .(

) ! $ * "

! " % ( 2 $ " ! j

$ ! ! ! /j " %

$ ! " "

1 l > l , - .

2 l < l ,

" li _{!! ! /i ! 1(}>

) $ !

w =θ₋ c

l +l .

) " $

1 " % ! 1,

2 " % ! 2.

2 " wi = θ > w i = 1,2. ? !!

" % " "

" (

" 6 / ( ) #

" ! ! A

" " '

# ) " " % "

" A θ( 2 $ / ( 9 "

" % " " 1 w < θ

$ ! - 1 θ ! .( C "

$ / ( . . .

"

%&

"

" 4 ' !

! " % &

>_{2 " ε 0}

! - ! ! . " " ( 2 % 1 {ε _}," ε >

n ε . T ! n.2

{T } 1 - .(

! ( 2 ! "

& ! /

! ( , % ! /

! / ! "

" % & ! (

) ! ! $'

! β( 2 ! ! /θj " %

$' (1 +β)θj( )

! " % & ! (

" ! 0

" ( C " ! /

" " 1 " ( 2

/ * " 0 1 - *

' .(

0 ! 0 " (

C " ! " ( C " 0 /

! " ! - "

4 (8.( 2 # " ! ! A

' % β _∈(0,θ _θ−θ )

β > β,

β < β.

# 6 ! ! β = 0,

! " ( 9 ' 2 "

I J ! !

! " β # (

7 " β " (

" β " "

β > β _{c˜j_i,˜lj_i}j L,H_{i ,} ! ! "

β =β . 2 ! " β =β !

(β ₋βo)[θL(˜lL_{+ ˜l}L_{) +θ}H

(˜lH_{+ ˜l}H_{)]>0( C}

! " (

9 " β = θ −θ

θ ,! ! ! /θ

L

" %

" ! θH. C " "

" (

2 " " " /

β " "

β > β. . . .

C " /

! ( 4 " ! /

! " " " ! /

! " (

" ( C " ! /

( !

" ( 2 *

β " (

"

&

) # /

" ' ( 2

! ! " " ! /

D (

! !

! ( 9 ' ! " /% "

" % & ! /

! (

? 4 " " "

1 2( 2 1 ! " % 1 2

3( ? ! /i " % ! θH " " %

i _{∈ {}1,2_}. $ i _{∈ {}1,2_} " % "

! j _{∈ {}1,2_} j =i, c >0.C

" ! i_{∈ {}1,2_} " "

! i(

? ! /3" % ! θL < θH ( 2

! /3" % " ! ! /i" % !! /

! i_{∈ {}1,2_}.

2 ! " % # 4 ( ?

4 ! " %

" ! 1 2( ? 4 ! " %

(

2 " " ( ! $

* " (

2 " % " " % " (

" " " % !

! ( )

" $ /! " 1 '! !

$ ! '! A

w = (l +l )θ

H_{+l θ}L

−c

l +l +l ,

/! " 1 " % & ! (

! /

! "

( 2 0 "

! " ( 2 "

! ! A

( % c_∈(0,6√2σ(θ))

c > c

c < c,

σ(θ) θ.

# ) -c= 0. /

! 4 " ! (

C 6 ! " ( 2

"

c= 0(

# 7 " c"

( 9 $ "

" " " c (

2 " "

c >0 ! ( 2 % c= 4(θH ₋θH). 2 1 -)).

" ! ! "

θL( C "

( 4 ! " " 4(θH ₋θH) 6√2σ(θ). . . .

. . .

! ! "

! " % 0 " ! " ( ? /

" 0 " ! /

( C / * " 1 0 A

(

6 ! < " 0

- =; .( 2 !

/ ' ( " !

$ " ! - !

. ! (

2 0 % " !

1 * FB - ===.G(

$ & ! (

" ! ' /

! ( 2 " 0

! -c < c. ! " % !

(

" "

2 ! ! " ( B /

( )

$ - .(

2 ! ! (

C " 0 !

! (

2 !

! ( ' " $ % "

" % $ ! % "

! ( 9 " % $

! / '

-" ! " % .( 2 " C - =>;.

" % " I 0 J % " "

(

2 ! ' "

" A ' ( 2

" ! !

! " % - $ &

! ! . - ( ( " % &

/ ' .(

$ /! " ' $ "

" % / " ( 2 " *

" !

! ( "/!

-% . ! " "

( ! !

" " 5 4 ( C " /!

! " " "

" ( 2 ! '

" ' ( !

' !

" % " ! ! A $ (

'

$

: ! /

" ( :' ! ! "/

$ A ! ' !! +

1 ! ! " % & ( 2

" $ " % & " /

(

! ! " " ! " /

$ ( 2 $ !

! " % " ! -! /

.( 2 5 / %

' + ( 2 ! /

! " % "

! - ! .(

) " " ! ! "

' / ( 2 "

" !

! 0 ( 2 0

5 % ! '

( 2 /

! - =<E.A

2 !

1 /

" ! 6 /

6

(

) ! ! " %

-! / ! / ' ! . $

! !

) " ! ! !

" ! 0

-.(

" ! A

! !

( ! $

" - . ! !

0 - ".(

, " ' ! /

" $ ! $ " % &

! " ( 2 ' +

/ : ' ! ( B !

$ " % & ! ! # /

" !

(

&

2 ! / !

! /θH " ! ( 2 /

! ! /θL

u(cL)₋ϕ Y

L

w +u(c

L₎

−ϕ Y

L

θL ≥u(c

H₎

−ϕ Y

H

w +u(c

H₎

−ϕ Y

H

θL . - 8.

λ, η, , ξ ! " 1 - .

- . - 8. - 8. ! ( 2 $ " /

" 1 " % !

! ! (

* ( YH ₌

YL_.

# 4 !! ! ( 2

"

2u(cH)₋v(lH)₋v Y

H

θH = 2u(c

L₎

−v(lL)₋v Y

L

θH - .

2u(cL)₋v(lL)₋v Y

L

θL = 2u(c

H₎

−v(lL)₋v Y

H

4 - <. - .

v Y

L

θH −v

YH

θH =v

YL

θL −v

YH

θL .

?!! 9 2 7 "

Y

Y

v′ s

θH 1 θHds=

Y

Y

v′ s θL

1

θLds. - E.

4 !! YL_=YH

∂ ∂θ[v

′ s

θ 1

θ] =−v

′′ s

θ s

θ −v

′ s

θ 1

θ . - ;.

2 "

Y

Y θ

θ

[₋v′′ s θ

s

θ −v

′ s

θ 1

θ ]dθds= 0 - >.

" ! YH _=YL_{. . . .}

2 " / !

(

* )

# 4 !! / ! ( 2

" YH _=YL_{=Y (}

9 $ / "

ϕ′ Y

L

θL 1

θL =

λ 1 +ξ₋η,

ϕ′ Y

H

θH 1

θH =

λ 1₋ξ+η.

C "

ϕ′ Y

θL > ϕ

′ YH

θH ⇒ϕ

′ Y

θL 1 θL > ϕ

′ YH

θH 1 θH

⇒ λ

1 +ξ₋η > λ

7 cH _YH _{! ! ( 2}

* / ! ! /θH

∆cH u′ cH ₋ϕ′ Y

H

θH 1

θH = 0.

/ ! ! /θL *

∆cH ₋u′ cH +ϕ′ Y

H

θL 1

θL = ∆c

H

−u′ cH +ϕ′ Y θL

1 θL

= ∆cH ₋ λ 1₋ξ+η +

λ

1 +ξ₋η >0.

2 / !

' - "

.( C / !

( . . .

2 ' / ! ! /

θL / ! (

* " % θL

# 4 !! / !

! /θL ( 9 $ / "

u′(cL) = u′(cL) =ϕ′ Y

L

θL 1 θL =

λ 1 +ξ,

u′(cH) = u′(cH) =ϕ′ Y

H

θH 1

θH =

λ 1₋ξ.

2 "

ϕ′ Y

L

θL < ϕ

′ YH

θH θL θH < ϕ

′ YH

θH ⇒l

L_{< l}H_,

cL> cH,

ϕ′ Y

L

θL < ϕ

′ YH

θH θL θH < ϕ

′ YH

θH ⇒l

H _{> l}L_.

2 / ! ! /θH (

. . .

2 " " ! " 6 ! A

+ %

θH

# 8 ! / ! /

/ ! /θH′s (

2 - 1 . $ / / !

! ( . . .

, #

F G ?K: ,9 ( ?( - =; .( I2 % J JA + 3 /

% J * + ,

-' >>/< (

F G ? 2,B ( ( 6 : :2 7( ( - .( I: !

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