Essays on competition and returns of inside money

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(5)

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(9)

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(10)

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(12)

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u(0) = 0* # . y >˜ 0 # # u(˜y) = ˜y*

,# $ $ # %# $# # - *

' '& + D _{* *} _& _' _)' _/

uj(yc) = 1−e−A y

$# Aj=eγ j∈ {l $", h # %#"}*

& + # + '& j y_j∗ # . uj(yc)−yp

log(A)

A *

! _{!(((" # $} _# _# _& _%

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(13)

γ_l = 3.00 γ_h = 2.503_* _# _# _# _# _%

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nh & & $ # %# *

3 _# _& _y∗

low< y∗high ≤γhigh< γlow*

(14)

, & + + P # .

P

(n+n ).(n+n )=

   

Π1 Π2

Ψ2 Ψ1

    $# Π1 (n.n)

=                    

0 π 0 0 0

0 0 π 0 0 **

* *** *** * ** *** ***

0 0 0 *** ***

0 0 0 0 π

                    Π2 (n.n )

=                    

1−π 0 0 0 0

1−π 0 0 0 0 **

* *** *** * ** *** ***

1−π 0 0 *** ***

1−π 0 0 0 0

                    Ψ1 (n .n )

=                    

0 ψ 0 0 0

0 0 ψ 0 0 **

* *** *** * ** *** ***

0 0 0 *** ***

0 0 0 0 ψ

                    Ψ2 (n .n)

=                    

1−ψ 0 0 0 0

1−ψ 0 0 0 0 **

* *** *** * ** *** ***

1−ψ 0 0 *** ***

1−ψ 0 0 0 0

                    *

# # l # & & # % % + & h (1−π) #

& & % # # % % . & l π* # $ $ ψ #

h*

< & $# # - ) % * $ + #

* ,# & & # *

; $ * < # # & & )

*

* < # & % # #

(15)

# * ,# & ) # % & # {0,1}* G

& ) # ) $ # $ # & ) $ & 0_*

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% & ) $ &

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# % ' % * $ $# # #

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% ' % & ykm

ij<p>∈ ℜ+$# #

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l # # # ' # & & ) '

i, j ∈ {0,1}$# i # + & # # j # # # '

< p > p∈ {l1, ..., ln, h1, ..., hn }*

& # & & ) # & & ) *

xk

i $ #k∈ {b, n} i∈ {0,1} # # '

$# # k & ) $# i* G # /

xb=B xn₀+xn₁ = (1−B) !"

< (xk_i, y_ij<p>km ) i∈ {0,1} k, m∈ {b, n} #

+ $ * ,# + $ & ) % # # # #

$ # - * ,# % & $ & ) & ) $ #

-0_< _# _& _%# _{H #} _& ₎ _$ _& _&

# & ) $ % & # & ) & # $ # $

$# # & & & *H

F_E _# _% _& _& ₎ _& _& ₎ _& ₎ _$ _$

(16)

# # *

E # # # $# & ) * ; # %

# % *

$ $ $ %

& $ / % ' % & $ & ) #

. # % # $ # # % #

% # & # % * < & ) & )

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< # % $ & & + *

9 % $ $ # # # # & # )

%*

. . $ + +

vk

i<p> # . $ # k$# & % p i*

,# + & & )

−ybm_j<l>+β πvb_<l′_>+ (1−π)vb_{<h >} ≥ 0 "

−ybm_j<h>+β ψvb_<h′_>+ (1−ψ)vb_{<l >} ≥ 0, D"

l∈ {l1, ..., ln} h∈ {h1, ..., hn} l′ % # $ & l h′ % # # %# &

h (m, j)∈ {(b, ),(n,0),(n,1)}*

,# %# '# " D" & # # %# & # & )

% * ,# & # & ) & & *

$ & # & $ # * # % $ # & )

- & # # #

) $ % # + & # # # # # * #

(17)

& - & ) #

# # # *

# & ) # % % $# # * ,# .

# 2

ul(yikn1<l>) +β πv0n<l′_>+ (1−π)v₀n_{<h >} ≥ β πvn₁_<l′_>+ (1−π)vn₁_{<h >} 3"

uh(ykni1<h>) +β ψv0n<h′_>+ (1−ψ)vn₀_{<l >} ≥ β ψv₁n_<h′_>+ (1−ψ)v₁n_{<l >} 4"

ul(ybn0<l>) +β πv0n<l′_>+ (1−π)v₀n_{<h >} ≥ β πvn₀_<l′_>+ (1−π)vn₀_{<h >} 0"

uh(ybn0<h>) +β ψv0n<h′_>+ (1−ψ)vn₀_{<l >} ≥ β ψv₀n_<h′_>+ (1−ψ)v₀n_{<l >} F"

l∈ {l1, ..., ln} h∈ {h1, ..., hn} l′ % # $ & l h′ % # # %# &

h (k, i)∈ {(b, ),(n,0)}*

,# . # & ) # % % $# #

−y₁nm_j<l>+β πv₁n_<l′_>+ (1−π)v₁n_{<h >} ≥ β πv₁n_<l′_>+ (1−π)v₁n_{<h >} 2"

−y₁nmj<h>+β ψv1n<h′_>+ (1−ψ)v₁n_{<l >} ≥ β ψv₁n_<h′_>+ (1−ψ)vn₁_{<l >} ("

−y₀nm_j<l>+β πv₁n_<l′_>+ (1−π)v₁n_{<h >} ≥ β πv₀n_<l′_>+ (1−π)v₀n_{<h >} ! "

−y₀nm_j<h>+β ψv₁n_<h′_>+ (1−ψ)v₁n_{<l >} ≥ β ψv₀n_<h′_>+ (1−ψ)vn₀_{<l >} , !!"

l∈ {l1, ..., ln} h∈ {h1, ..., hn} l′ % # $ & l h′ % # # %# &

h (m, j)∈ {(b, ),(n,1)}*

$ # & ) * < &

2" (" & ) $# $ # %

*

,# # # = $ 1 # = $ 1

2 _3" _4" _& _% _$# _0" _F" _# _ybn

<l> ybn<h>≥

(18)

# * * & $ # (_/

xn₀ xb+xn₁ =xn₁ xb+xn₀ ! "

9 (1),

xn₀ =xn₁ =(1−B)

2 !D"

,# # $ $ $ /

vn₀_<l> = x

b .

Sul y

bn

0<l> +

xb .

S −y

nb

0<l> +

xn

1 S (−y

nn

01<l>) +

+β x

b₊_xn

1

S πv

n

1<l′_>+ (1−π)vn₁_{<h >} + 1−

xb₊_xn

1

S πv

n

0<l′_>+ (1−π)vn₀_{<h >} !3"

vn₁_<l> = x

b

Sul y

bn

1<l> +

xn₀ S ul(y

nn

01<l>) +

+β x

b₊_xn

0

S πv

n

0<l′_>+ (1−π)vn₀_{<h >} + 1−

xb₊_xn

0

S πv

n

1<l′_>+ (1−π)vn₁_{<h >} !4"

vb_<l> = x

b

Sul y

bb ..<l> +

xn

0 S ul y

nb

0<l> +

xb .

S −y

bb <l> +

xn

0

S −y

bn

0<l> +

+x

n

1

S −y

bn

1<l> +β πvb<l′_>+ (1−π)vb_{<h >} , !0"

l_{∈ {}l₁, ..., ln} l′ % # $ & l*

# $ # %# /

vn₀_<h> = x

b

Suh y

bn .0<h> +

xb

S −y

nb

0<h> +

xn₁ S (−y

nn

01<h>) +

+β x

b₊_xn

1

S ψv

n

1<h′_>+ (1−ψ)v₁n_{<l >} + 1−

xb₊_xn

1

S ψv

n

0<h′_>+ (1−ψ)v₀n_{<l >} !F"

( _# _& _{! " #} _& ₎ _$# _# _%

$# # *

(19)

vn₁_<h> = x

b

Suh y

bn

1<h> +

xn

0 S uh(y

nn

01<h>) +

+β x

b₊_xn

0

S ψv

n

0<h′_>+ (1−ψ)v₀n_{<l >} + 1−

xb₊_xn

0

S ψv

n

1<h′_>+ (1−ψ)v₁n_{<l >} !2"

vb_<h> = x

b

Suh y

bb <h> +

xn

0 S uh y

nb

0<h> +

xb

S −y

bb <h> +

xn

0

S −y

bn

0<h> +

+x

n

1

S −y

bn

1<h> +β ψvb<h′_>+ (1−ψ)vb_{<l >} !("

h_{∈ {}h1, ..., hn } h′ % # $ & h*

E # % !3" !(" 3(nl+nh) .

* ,# # * < % # $ & # # &

% % *

E $ $ + # (5(nl+nh).1) y /

y=     yl yh   

 "

yl

(5n.1)

= ybb

′

yh

(5n .1)

= ybb

′

*

$ # v # (3(nl+nh).1) $# # # .

. . & & /

v(y) =

   

vl(y)

vh(y)

  

=M−1Ar(y) $# !"

vl(y)

(3n.1)

= vn

0<l >. . . v0n<l > v1n<l >. . . vn1<l > vb<l >. . . vb<l >

(20)

vh(y)

(3n .1)

= vn

0<h >. . . vn0<h > vn1<h >. . . v1n<h > vb<h >. . . vb<h >

′

*

# . M

(3(n+n )x3(n+n ))=I3(n+n )−T1−T2

$#

I₃₍n+n ) # (3(nl+nh).3(nl+nh)) .

T1

(3(n+n )x3(n+n ))

= β_S

                   

−(xb₊_xn

1)Π1 (xb+xn1)Π1 0 −(xb+xn1)Π2 (xb+xn1)Π2 0

(xb₊_xn

0)Π1 −(xb+xn0)Π1 0 (xb+xn0)Π2 −(xb+xn0)Π2 0

0 0 0 0 0 0

−(xb₊_xn

1)Ψ1 (xb+xn1)Ψ1 0 −(xb+xn1)Ψ2 (xb+xn1)Ψ2 0

(xb₊_xn

0)Ψ1 −(xb+xn0)Ψ1 0 (xb+xn0)Ψ2 −(xb+xn0)Ψ2 0

0 0 0 0 0 0

                    T₂

(3(n+n )x3(n+n ))

=β

   

I3⊗Π1 I3⊗Π2

I3⊗Ψ2 I3⊗Ψ1

    # . A

(3(n+n )x10(n+n ))= 1 S    

Al 0

0 Ah

   , $# Al

(3n.10n)

=        

0 xb_I

n 0 0 0 0 0 0 xbIn xn₁In

0 0 xb_I

n 0 xn0In 0 0 0 0 0

xb_I

n 0 0 xn0In 0 xbIn xn0In xn1In 0 0

        Ah

(3n .10n )

=        

0 xb_I

n 0 0 0 0 0 0 xbIn xn1In

0 0 xb_I

n 0 xn0In 0 0 0 0 0

xb_I

n 0 0 xn0In 0 xbIn xn0In xn1In 0 0

       

# (10(nl+nh).1) '

-r(y) =

   

rl(y)

rh(y)

   ,

$#

rl(y)

(10n.1)

=

   

ul(yl)

−yl

  

 rh(y)

(10n .1)

=

   

uh(yh)

−yh

   

$# ul uh yl yh '$ *

(21)

!

,# & . $ # &

* ,# ' %

w=

k,i,p

ϕ_<p>xk_ivk_i<p> "

$# ϕ # & P (k, i)∈ {(b, ),(n,0),(n,1)} p_{∈ {}l1, ..., ln, h1, ..., hn}*

$ "

w=ϑ′v(y) D"

$# # (1.3(nl+nh)) ϑ′ /

ϑ′ = _x_⊗_ϕ

l x⊗ϕh

# # & & ) & ) x

(1I3)= x

n

0 xn1 xb

# ϕl

(1In)

= _ϕ

# ϕ_h

(1.n )

= _ϕ

<h > ϕ<h > ϕ<h > $# # $ ) # &

ϕ P*

< # $ $ # $ & !D" !3"' !("

w=

p

    

k,m,i,j

xk ixmj

S z y

km ij<p>

 χ_p

 

 3"

$# (k, m, i, j) ∈ {(b, b, , ),(b, n, ,0),(b, n, ,1),(n, b,0, ),(n, n,0,1)} p _{∈ {}l1, ..., ln, h1, ..., hn}

z ykm

ij<p> =u ymkji<p> −ykmij<p> χp % # β, π ψ #

= $ $# p $#

# # % + ! _*

,# 3" + # & & % ' % # '

-! _E _#

p

(22)

# % # * w & # . #

*

, $# $ $ nl=nh= 1 # $ $ #

χ_l = (1−ψ) [(1−βψ) +β(1−π)]

(2−π−ψ) (1−βπ) (1−βψ)−β2(1−π) (1−ψ) 4"

χ_h = (1−π) [(1−βπ) +β(1−ψ)]

(2−π₋ψ) (1−βπ) (1−βψ)−β2(1−π) (1−ψ) . 0"

# $ # %# $ * * π $ ψ= 1 # χ_l = 0 χ_h = ₁₋1_β* ,#

# # # %# *

; # # # # $ $ $ * * ψ $ π= 1 # χ_l = ₁₋1_β

χ_h = 0* ,# # # $ *

"

θ= (B, π, ψ, β)∈Θ = [0,1]3×[0,1)

" # < . <*

3" # y∗ _{# +} _'& _y∗

l =

log(A)

A $# $

y∗_h=log(_AA ) $# # %#*

(23)

$

%

# # $ # ' @ # )

$ # π=ψ= 0 nl=nh= 1*

$# # # ) * * γ_l=γ_h=γ*

$ β % % 0.4 0.9 # $

# & # # % %# # & & )

* # % . # $# # $ & #

*

&

# $ %# # %

& ) % * $ & # $ & # %

# . *

& % # $ # nl=nh= 1 γl=γh=γ $

* ,# + $ & $ # $

*

# 3"' 0" # $ w # # /

S(1−β)w=xbxb u ybb −ybb +xbxn₀ u y₀nb +u ybn₀ −ybn₀ −y₀nb +

+xb_xn

1 u ybn1 −ybn1 +xn0x1n[u(y01nn)−ynn01]*

,# "' !!"/

g1=−ybb+βvb≥0

g2=−ybn0 +βvb≥0

g3=−ybn1 +βvb≥0

g4=−y0nb+β(v1n−v0n)≥0

g5=−y01nn+β(v1n−v0n)≥0

g₆=ybn

(24)

!3" ' !(" /

S(1−β)vb₌_xb _{u y}bb ₋_ybb ₊_xn

0 u ynb0 −ybn0 +xn1 −ybn1

S(1−β) +β 1 +xb ₍_vn

1−vn0) =xb u ybn1 −u ybn0 +y0nb +xn0[u(y01nn)] +xn1[y01nn]

,# $

∂v ∂y =

x _[u′₍_y ₎₋₁_]

S(1−β)

∂v ∂y =−

x S(1−β)

∂v ∂y =−

x S(1−β)

∂v ∂y =

x u′₍_y ₎

S(1−β)

∂v ∂y = 0 ∂(v −v )

∂y = 0 ∂(v −v )

∂y =−

x u′₍_y ₎

S(1−β)+β(1+x)

∂(v −v )

∂y =

x u′₍_y ₎

S(1−β)+β(1+x )

∂(v −v )

∂y =

x S(1−β)+β(1+x )

∂(v −v )

∂y =

x u′₍_y ₎₊_x

S(1−β)+β(1+x )

% !" % γ = 3.0 S = 3 % % 5,782 β _∈[0.40,0.98]

B _∈ [0.01,0.98] +% " % γ = 2.5 S = 3 % % 4,998

β_∈[0.48,0.98] B_∈[0.01,0.98]*

(25)

% ! ' % γ= 3.0 S = 3 % % 5,782

β∈[0.40,0.98] B ∈[0.01,0.98]*

% ' % γ= 2.5 S = 3 % % 4,998

β_∈[0.48,0.98] B_∈[0.01,0.98]*

(26)

B

β B

I

A

R

C

D E

1

nonbank product ion

bank product ion f or nonbanks wit h money

bank product ion f or banks bank

product ion

bank product ion f or nonbanks wit hout money

nonbank product ion f or banks

nonbank product ion f or nonbanks

consumpt ion of t he nonbank wit hout money

0 1

% D ' % *

, & !" " # y∗ % #

*

%

< 9

B 0.50 0.02 0.02 0.50 0.50 0.50 0.02 0.10

β 0.79 0.57 0.54 0.60 0.48 0.45 0.50 0.50

ybb ₁₀₀_.₀₀₀₀ ₉₉_.₄₅₁₆ ₉₁_.₃₉₄₃ ₁₀₀_.₀₀₀₀ ₁₀₀_.₀₀₀₀ ₉₉_.₂₀₉₂ ₇₈_.₉₂₅₁ ₈₃_.₀₂₆₃

ybn

0 100.0000 99.4516 91.3943 65.2353 43.9983 37.8337 78.9251 83.0263

ybn

1 100.0000 99.4516 91.3943 116.6436 111.4107 99.2092 78.9251 83.0263

ynb₀ 100.0000 100.1701 101.4593 65.9738 52.6100 49.9180 86.4856 76.2535

ynn

01 100.0000 100.0000 100.0139 65.9738 52.6100 49.9180 86.4856 76.2535

, & ! ' 5 B β$ #γ= 3.0 S= 3 *

(27)

%

< 9 !

B 0.50 0.02 0.02 0.50 0.50 0.50 0.02 0.02

β 0.92 0.69 0.65 0.79 0.60 0.50 0.60 0.48

ybb ₁₀₀_.₀₀₀₀ ₉₇_.₃₇₆₇ ₈₇_.₅₇₅₆ ₁₀₀_.₀₀₀₀ ₁₀₀_.₀₀₀₀ ₇₇_.₅₂₈₀ ₇₂_.₃₈₁₇ ₄₄_.₂₁₀₂

ybn

0 100.0000 97.3767 87.5756 79.2592 45.2889 22.5670 72.3817 0.0000

ybn

1 100.0000 97.3767 87.5756 113.5901 113.2598 77.5280 72.3817 44.2102

ynb

0 100.0000 101.1487 102.5543 80.7831 56.9754 51.0179 83.2648 44.2102

ynn

01 100.0000 100.0000 100.0429 80.7831 56.9754 51.0179 83.2648 44.2102

, & ' 5 B β$ #γ= 2.5 S= 3 *

+ r≡y_y −1 !"* % 3" #

r γ= 3.0 *

% 3 ' # r(%) γ= 3.0 S= 3 % %

5,782 β_∈[0.40,0.98] B_∈[0.01,0.98]*

% ) $ % . $ # + '& y∗

r= 0* $ $ # β $ $ # % < $ B %

(28)

% & ) g₄ g₅)& + $ #

y∗_* _# _& ₎ _& _{% #}

& ) $ # * * % % # & ) # % !!_{* <}

r >0 ynb

0 =ynn01* ,# & ) & & #

% *

,# & % $ ) %β* ,# - #

% & ) g3 & & ) #

% & ) g1 & & ) #

y∗ _# _& _)*

% < * * $ B & ) g1g2 g3)& + & ) $

# y∗_* _# _$ _{$ &} ₎ _& ₎ ₎

. & ) & % # # ! _{* <} _{r <} ₀

ynb

0 =ynn01*

,# & % 9 $ ) %β* ,# - # % 9

& ) g4& & ) # #

& ) $ # & # & ) # & ) # % !D_*

% & * % ! & ) % % & & )

$ # * * g6 & & %* #

!" J '% H % *

' (

# $ # ' @ # )

$ # π=ψ= 0 nl=nh = 1* ,# # )

* * γ_l> γ_h*

!!_E _# ∂ v −v ∂y

y y∗

∂ v −v ∂y

y y∗

< ∂ v −v

∂y

y y∗

> & )$ ) % & ) *

! _E _# ∂v

∂y y y∗

∂v ∂y

y y∗

> *

!D_E _# ∂ v −v

∂y

y y

> *

(29)

# 3"' 0" # $ w # # /

S(1−β)w= 1₂

p∈{l,h}

xb_xb _u

p ybb<p> −ybb<p> +

+xb_xn

0 up y0nb<p> +up ybn0<p> −ybn0<p>−ynb0<p> +

+xb_xn

1 up ybn₁<p> −ybn1<p> +x0nxn1 up ynn₀₁<p> −ynn01<p> *

,# "' !!"/

g1=−ybb<l>+βvb<h>≥0

g2=−ybn0<l>+βvb<h>≥0

g3=−ybn1<l>+βvb<h>≥0

g4=−y0nb<l>+β vn1<h>−vn0<h> ≥0

g5=−ynn01<l>+β v1n<h>−v0n<h> ≥0

g6=ybn0<l>≥0

g7=−ybb<h>+βvb<l>≥0

g8=−ybn0<h>+βvb<l>≥0

g9=−ybn1<h>+βvb<l>≥0

g10=−ynb0<h>+β vn1<l>−v0n<l> ≥0

g11=−y01nn<h>+β vn1<l>−vn0<l> ≥0

g12=ybn0<h>≥0

!3" ' !(" /

S(1−β2)vb

<p>=xb up ybb<p> −ybb<p> +xn0 up y0nb<p> −ybn0<p> +xn1 −ybn1<p> +

+β xb _u

p′ ybb_<p′_> −ybb_<p′_> +xn₀ up′ y₀nb_<p′_> −ybn₀_<p′_> +xn₁ −ybn₁_<p′_>

S 1−λ2β2 vn

1<p>−v0n<p> =xb up ybn1<p> −up ybn0<p> +y0nb<p> +xn0 up y01nn<p> +xn1 ynn01<p> +

λβ xb _u

p′ ybn₁_<p′_> −up′ ybn₀_<p′_> +y₀nb_<p′_> +xn₀ up′ y₀₁nn_<p′_> +xn₁ ynn₀₁_<p′_>

$# λ= 1−1+Sx p, p′∈ {l, h} p=p′*

(30)

∂v ∂y =

x [u′₍_y ₎₋₁_]

S(1−β )

∂v

∂y =− x S(1−β )

∂v

∂y =− x S(1−β )

∂v ∂y =

x u′₍_y ₎

S(1−β )

∂v

∂y = 0 ∂v ′

∂y =β

x_[u′₍_y ₎₋₁_]

S(1−β )

∂v ′

∂y =−β x S(1−β )

∂v ′

∂y =−β x S(1−β )

∂v ′

∂y =β

x u′₍_y ₎

S(1−β )

∂v ′

∂y = 0

,# & )

∂₍v −v ₎

∂y = 0

∂(v −v )

∂y =−

x u′₍_y ₎

S(1−λ β )

∂(v −v )

∂y =

x u′₍_y ₎

S(1−λ β )

∂(v −v )

∂y =

x S(1−λ β )

∂(v −v )

∂y =

x u′₍_y ₎₊_x

S(1−λ β )

∂₍v ′ −v ′ )

∂y = 0

∂(v ′ −v ′ )

∂y =−λβ

x u′₍_y ₎

S(1−λ β )

∂(v ′ −v ′ )

∂y =λβ

x u′₍_y ₎

S(1−λ β )

∂(v ′ −v ′ )

∂y =λβ

x S(1−λ β )

∂(v ′ −v ′ )

∂y =λβ

x u′₍_y ₎₊_x

S(1−λ β )

% 4" % γ_l= 3.0 γ_h= 2.5 S= 3 % % 4,661 β_∈[0.40,0.98]

B ∈[0.20,0.98]*

(31)

% 4 ' % γ_l= 3.0 γ_h= 2.5 S= 3 % %

4,661 β∈[0.40,0.98] B∈[0.20,0.98]*

+% 0" $ # $ # $# * ,# # # +%

# % # +% *

high low

B β

I

Ch

0 1

Cl1 Cl2

Dh

Eh

Dl

El

% 0 ' % *

, & D" # y_l∗ y∗_h %

(32)

%

B 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

β 0.95 0.87 0.82 0.70 0.58 0.53 0.50 0.45

ybb

<l> 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 88.9072

ybb

<h> 100.0000 100.0000 100.0000 100.0000 100.0000 94.9716 84.6871 69.8548

ybn

0<l> 100.0000 97.0189 93.9973 71.4410 50.6976 42.0257 36.7862 27.5739

ybn

0<h> 100.0000 98.4830 97.1475 75.1305 53.4247 42.1545 35.0419 23.2037

ybn

1<l> 100.0000 102.7363 105.0849 115.1518 118.6979 117.3722 109.4564 88.9072

ybn

1<h> 100.0000 101.4616 102.6626 115.2193 115.5372 94.9716 84.6871 69.8548

ynb₀_<l> 100.0000 102.9811 106.0027 84.1184 66.7842 60.9185 58.2567 54.9445

ynb

0<h> 100.0000 90.5658 82.1723 63.1603 50.4756 46.6139 44.6148 41.6981

ynn

01<l> 100.0000 105.7174 110.3580 84.1184 66.7842 60.9185 58.2567 54.9445

ynn

01<h> 100.0000 90.5658 82.1723 63.1603 50.4756 46.6139 44.6148 41.6981

, & D ' 5 B β$ #γl= 3.0 γh= 2.5 S = 3 *

$ + $ # %# r<l> ≡ _yy −1

r<h>≡ y_y −1* % F" 2" # r<l> r<h> #

% % 5,782 β∈[0.40,0.98] B∈[0.01,0.98]*

(33)

% F ' # rl (%) γl= 3.0 γh= 2.5 S= 3 % %

5,782 β∈[0.40,0.98] B∈[0.01,0.98] $ # *

% 2 ' # rh (%) γl= 3.0 γh= 2.5 S= 3 % %

5,782 β_∈[0.40,0.98] B_∈[0.01,0.98] $ # *

$ $ $ $# # # % # $# & )

& + *

% ) $ % . $ # + '& y∗

l y∗h

$ # %# * r<l>= y

∗

y∗ −1 r<h>= y

∗

y∗−1*

(34)

# %# (g₁₀ g₁₁) & + * # $ & )

& # !3 _& ₎ _# _y∗

l $ !4* <

r<l>$ & r<h> *

,# $ & % $ ) %β* g5

& g4 g5& * # & )

& & # % *

% & ) & # $ % / g9 g7 g3

g1*

,# . $ & # % $# # $# $

& *

!3

E # ∂ v_∂y −v

y y∗

∂ v −v

∂y

y y∗

∂ v −v

∂y

∗

∂ v −v

∂y

∗

<

∂ v −v

∂y

y y∗

∂ v −v

∂y

y y∗

> & )$ ) % & ) *

!4_E _# ∂ v −v ∂y

∂ v −v

∂y > $ ) % & ) *

(35)

"

)

"

# # $ # ' @ # )

$ # *

# . $ $ # % % # % # # .

& * # # $ $ % π ψ∈ {0.01,0.25,0.50,0.75,0.99}

# *

*

) . # % % + & .

L(y, )) =ϑ′v(y)−1 c)

′_J₍_g₍_y₎₎ _F"

$# # (5(nl+nh).1) y % & " # (3(nl+nh).1) .

& . . & & v(y) =M−1_Ar₍_y₎

$# # (1.6(nl+nh)) % % )′ j(g(y)) # .

/

j(g(y))

(6(n+n ).1)

=

           

e−cg (y)₋₁

** *

e−cg (y)₋1

           

# (6(nl+nh).1) g(y) =e(y) +βF v(y) $# # # J )H #

" D" 0" F" ! " !!" $#

e(y) =

   

el(y)

eh(y)

  

(36)

el(y)

(6n.1)

=                

−yl

− − −

ybn

0<l >

** *

ybn

0<l >

               

eh(y)

(6n .1)

=                

−yh

− − −

ybn

0<h >

** *

ybn

0<h >

               

# (6(nl+nh).3(nl+nh)) .F

F =     Fl Fh     / Fl

(6n.3(n+n ))

=                    

0 0 Π1 0 0 Π2

−Π1 Π1 0 −Π2 Π2 0

0 0 0 0 0 0

                    Fh

(6n .3(n+n ))

=                    

0 0 Ψ2 0 0 Ψ1

−Ψ2 Ψ2 0 −Ψ1 Ψ1 0

0 0 0 0 0 0

                    *

< c >0 & *

9 ) !0 _# _%

# %)' $ # . % & /

)(_jm+1)=)(_jm)e−˜cg 1≤j _≤6(nl+nh)

$# (m) m' # *

!0₉ ₎

(37)

5 # % % ) y & $ # # c ˜c

# & % !F _{E $} _'& _# _# _{$ )} _{$ *}

G & % 21" 27" $ % /

L(y, )) =ϑ′M−1Ar(y)−1 c)

′_j₍_g₍_y₎₎

- % $ # y′ /

[L(y, ))]

(1I5(n+n ))

=ϑ′M−1A [r(y)]−1 c)

′ _[_j₍_g₍_y_))] _2"

$# & # # !2_/

[j(g(y))]

(6(n+n )I5(n+n ))

= [j(g)] [g(y)] /

[j(g)]

(6(n+n )I6(n+n ))

=−c

       

e−cg ₀

**

* * ** ***

0 e−cg

       

[g(y)]

(6(n+n )I5(n+n ))

= [e(y)] +βF M−1_A _[_r₍_y_)]

8 % 2" - % % $ # y′ # L/

[L(y, ))]

(5(n+n )I5(n+n ))

= I5(n+n )⊗ϑ′M−1A [r(y)]−

1

c I5(n+n )⊗)

′ _[_j₍_g₍_y_))]

$# & # # !(_/

[j(g(y))]

(30(n+n )I5(n+n ))

= I₆₍n+n )⊗ [g(y)] [j(g)] [g(y)] +

+ [j(g)]⊗I5(n+n ) [g(y)]

[j(g)]

(36(n+n ) I6(n+n ))

=c2

       

e−cg _.ε

1 0

**

* * ** ***

0 e−cg .ε₆₍n+n )

       

$# εj #

!F_,# _& _# _# _# _{π, ψ, n}

l, nh, S, β B* 9

@ %c c c ym _% _{* * $#} _{E $} _@ _% _*

!2 _% _E ₎ _{!(22 (!*}

(38)

6(nl+nh) $# # # 1 # j # $#

[g(y)]

(30(n+n ) I5(n+n ))

=β(I₅₍n+n )⊗F M−1A) [r(y)]*

,# m # # # yopt & % # /

H L(y(m), )(m)) y(m+1)−y(m) =D L(y(m), )(m)) ′. ("

< $ % ( ! _G _{& #} _%

& ' = %' & D_*

# $ # B % % # 3

(29) + & 4 _{D L}₍_y(m)_{, )}(m)₎ _{H L}₍_y(m)_{, )}(m)₎

% ;; "*

, # + % +% (" ! " # [j(g(y))]

[j(g(y))] *

% ( ' [j(g(y))] nl=nh= 2*

E # # + e y # e nl nh I nl nh .*

! ₅ _{3 !*F 6} _{F* *}

G 02 G5< F4 # G (

G 5 K & 9K<G G <K<5< >"

D _# _% _' ₌ _%' ₊ _# _. _# _*

3_,# _& _$ ₎ _E _{( *}

4_,# _& _$ ₎ ₎

. *

(39)

% ! ' [j(g(y))] nl=nh= 2*

"

# & # $ ) $ #

nl = nh = 1 # * * S = 3

B = 0.50 β= 0.79 γl= 3.0 γh= 2.5 % π ψ∈ {0.01,0.25,0.50,0.75,0.99}*

$ + $ $# $ # %# r<l,h>≡

y

y −1

$ $# $ $ r<l,l>≡ y

y −1* # $ $ +

# %# $# $ $ r<h,l> ≡ y

y −1

# %# $# $ # %# r<h,h>≡ y

y −1*

, & 3" # y_l∗ y∗_h % #

$# # ) 0 _{* *} _π₊_ψ_{= 1}_*

(40)

π 0.01 0.25 0.50 0.75 0.99

ψ 0.99 0.75 0.50 0.25 0.01

ybb

<l> 100.0000 100.0000 100.0000 100.0000 100.0000

ybb_<h> 100.0000 100.0000 100.0000 100.0000 100.0000

ybn

0<l> 82.8749 86.9911 91.7414 96.6209 99.9006

ybn₀_<h> 79.4499 84.3894 90.0896 95.9451 99.8807

ybn

1<l> 111.2575 109.3332 106.6134 103.0678 100.0991

ybn

1<h> 113.5089 111.1998 107.9361 103.6814 100.1189

ynb

0<l> 110.8154 107.0639 104.2583 102.7023 100.0994

ynb

0<h> 80.6555 77.9250 75.8830 74.7505 74.1719

ynn

01<l> 110.8154 107.0639 104.2583 102.7023 100.1985

ynn

01<h> 80.6555 77.9250 75.8830 74.7505 74.1719

, & 3 ' 5 B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S= 3

$ #nl=nh= 1.

, & 4" & 3"*

< $# # ) & ) & %

$ * * # *

π 0.01 0.25 0.50 0.75 0.99

ψ 0.99 0.75 0.50 0.25 0.01

r<l,h> 0.40 2.12 2.26 0.36 −1.77

r<l,l> 0.40 2.12 2.26 0.36 0.00

r<h,l> 40.73 42.70 42.24 38.70 37.42

r<h,h> 40.73 42.70 42.24 38.70 34.98

, & 4 ' L" B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S= 3

$ #nl=nh= 1.

, & 0" # y_l∗ y∗_h % #

$# & P # +. ϕ= [ 0.5 0.5 ]* * * π=ψ*

(41)

π 0.01 0.25 0.50 0.75 0.99

ψ 0.01 0.25 0.50 0.75 0.99

ybb

<l> 100.0000 100.0000 100.0000 100.0000 100.0000

ybb_<h> 100.0000 100.0000 100.0000 100.0000 100.0000

ybn

0<l> 89.3783 90.1835 91.7414 95.0001 99.7770

ybn₀_<h> 92.7732 91.8701 90.0896 86.0945 79.4354

ybn

1<l> 108.0425 107.5736 106.6134 104.3468 100.2215

ybn

1<h> 106.1187 106.7534 107.9361 110.2986 113.5151

ynb

0<l> 102.4802 103.1965 104.2583 104.9999 100.2230

ynb

0<h> 76.6543 76.2870 75.8830 76.0764 80.4265

ynn

01<l> 102.4802 103.1965 104.2583 105.4163 100.4445

ynn

01<h> 76.6543 76.2870 75.8830 76.0764 80.4265

, & 0 ' 5 B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S = 3

$ #nl=nh= 1.

, & F" & 0"*

< $# # ) $ $

# * * # *

,# # % $ # K !(( " + % * K F

$# # $ $ % * # #' '

% ' ) - # - & # * G

# # % # & % " % & )

PL=

   

κ 1−κ

1−κ κ

  

 # $ #

/

# . * $ #

. % % # ' $ $ & % #

(42)

& ' 2 _{* * *} _G ₃ _% _# _#

# ) * ,# # & $# #

# $# $ # # - $ & % *

5 # # ) # ) $# # $ & # & $ & *H

π 0.01 0.25 0.50 0.75 0.99

ψ 0.01 0.25 0.50 0.75 0.99

r<l,h> 2.59 2.63 2.26 −0.17 −9.30

r<l,l> 5.43 4.24 2.26 −0.62 0.00

r<h,l> 42.27 42.13 42.24 44.33 55.62

r<h,h> 38.44 39.94 42.24 44.98 41.14

, & F ' L" B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S= 3

$ #nl=nh= 1.

)

# $ ) $ # #

# # * * S= 3 B= 0.50 β= 0.79 γ_l= 3.0 γ_h= 2.5 %

π ψ_{∈ {}0.01,0.25,0.50,0.75,0.99}*

$ + + & $(l1)$# $ & # %#(hn)

r<l ,h > ≡ y

y −1 n∈ {1,2, ..., nh} & $(ln)$#

$ # & $(ln−1) r<l ,l − >≡

y y

−

−1 n∈ {2, ..., nl}*

# $ $ + + & # %#(h1)$# $ &

$(ln) r<h ,l >≡ y

y −1 n∈ {1,2, ..., nl} & # %#(hn)$#

$ # & # %#(hn−1) r<h ,h − >≡

y y

−

−1 n∈ {2, ..., nh}*

# & ) % nl = 1 nh = 5 # - % &

# %# *

, & 2" # y∗_l y∗_h % #

2_,# _& _. _# _# _{% %} _%*

(43)

$# π # +. 0.25 $ # - ψ*

π 0.25 0.25 0.25 0.25 0.25

ψ 0.01 0.25 0.50 0.75 0.99

ybn

0<l > 91.4708 90.2941 88.8965 87.3057 85.5310

ybn

0<h > 94.2444 93.5642 92.7975 91.9647 91.0638

ybn

0<h > 88.1996 87.7610 87.3263 86.9003 86.4430

ybn

0<h > 83.2700 83.2417 83.3153 83.4521 83.5582

ybn

0<h > 79.3573 79.8537 80.5160 81.2475 81.9055

ybn

0<h > 76.3030 76.8538 77.6150 78.4635 79.2305

ybn

1<l > 106.7855 107.5078 108.3146 109.1711 110.0561

ybn₁_{<h >} 105.0306 105.5424 106.1014 106.6880 107.2998

ybn

1<h > 109.1006 109.3584 109.6096 109.8516 110.1070

ybn

1<h > 111.7605 111.7744 111.7383 111.6709 111.6184

ybn

1<h > 113.5484 113.3354 113.0452 112.7165 112.4132

ybn₁_{<h >} 114.7801 114.5676 114.2672 113.9226 113.6022

ynb

0<l >=y01nn<l > 102.3762 102.8740 103.5312 104.3139 105.1536

ynb

0<h >=y01nn<h > 76.2533 76.4963 76.8750 77.3598 77.8711

ynb

0<h >=y01nn<h > 76.2653 76.8155 77.5358 78.3601 79.1794

ynb

0<h >=y01nn<h > 76.2753 77.0651 78.0226 79.0596 80.0583

ynb

0<h >=y01nn<h > 76.2835 77.2634 78.4022 79.6000 80.7371

ynb

0<h >=y01nn<h > 76.2835 77.2634 78.4022 79.6000 80.7371

, & 2 ' 5 B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S = 3

$ #nl= 1 nh= 5 π= 0.25.< & $ & ) 100.0000*

% !!" # $ & ) # %# & 2"*

% # π ψ*

< +% !!" $ # # ( _{# %#}

(_{< $ $} _# _$# _$ ₎ _n

(44)

ψ* # % # & & # % + $ * * #

ψ # % *

# $# ψ_−→1 & # %# r<h , > 40.61%.

,# # $# # %#* ,# & . & " #

& *

Interest rates in period state high

40 41 42 43 44 45 46 47 48 49 50 51

(%

)

0.01 0.25 0.50 0.75 0.99

r

h1,l1 r h2,h1

r

h3,h2

r

h4,h3

r

h5,h4

r

h5,h5

% !! ' 6 # L" B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S= 3

$ #nl= 1 nh= 5 π= 0.25 ψ∈ {0.01,0.25,0.50,0.75,0.99}*

E $ ) nl= 5 nh= 1 # - % & $*

% ! " # $ & ) $ #

$# ψ # +. ψ = 0.25 $ # - π* $ %

$# # # $# $* ,# & . & !"

# & *

(45)

Interest rates in period state low

-2 -1 0 1 2 3 4

(%

)

0.01 0.25 0.50 0.75 0.99

r

l1,h1

r

l2,l1

r

l3,l2

r

l4,l3

r

l5,l4

r

l5,l5

% ! ' 6 # L" B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S= 3

$ #nl= 5 nh= 1 ψ= 0.25 π∈ {0.01,0.25,0.50,0.75,0.99}*

,# & $ $ # %# = # # $# # %# & )@

& # & # % % * * & ) $ ' ) %*

; # # # $# $ & )@ & )$ ' ) %*

$ ) nl= 5 nh= 10 $# & $

& $# # % D _*

$ % > > !" $ + . #

* K y<p> P<p> # % % p*

# # % /

S y<p>=xbxbybb<p>+xbxn0ybn0<p>+xbx1nybn1<p>+xn0xby0nb<p>+xn0xn1y01nn<p>

p_{∈ {}l1, ..., ln, h1, ..., hn }*

) P<p> & # = 1 Y<p> &

y<p>* , $ %# # & % $# * ,#

S Y<p>=xbxb+xbxn0+xbx1n+xn0xb+xn0xn1 =B+( 1−B )

4 .

D _$ _n

(46)

< $ # %# %# # - +% !D" !3" $# +% !4"

!0" # $ # . & # $ # %# $# # ) *

Total output in period state low

0.0342 0.0343 0.0344 0.0345 0.0346

y<l1> y<l2> y<l3> y<l4> y<l5>

π = 0.25 ; ψ = 0.75 π = 0.50 ; ψ = 0.50 π = 0.75 ; ψ = 0.25

% !D ' 6 # B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S= 3

$ #nl= 5 nh= 10$# # ) *

Total output in period state high

0.0431 0.0432 0.0433 0.0434 0.0435 0.0436 0.0437 0.0438 0.0439 0.0440 0.0441

y<h1> y<h2> y<h3> y<h4> y<h5> y<h6> y<h7> y<h8> y<h9> y<h10>

π = 0.25 ; ψ = 0.75 π = 0.50 ; ψ = 0.50 π = 0.75 ; ψ = 0.25

% !3 ' 6 # B= 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S= 3

$ #nl= 5 nh= 10$# # ) *

(47)

Price index in period state low

0.0663 0.0664 0.0665 0.0666 0.0667 0.0668 0.0669 0.0670

P<l1> P<l2> P<l3> P<l4> P<l5>

π = 0.25 ; ψ = 0.75 π = 0.50 ; ψ = 0.50 π = 0.75 ; ψ = 0.25

% !4 ' 6 # . B = 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S = 3

$ #nl= 5 nh= 10$# # ) *

Price index in period state high

0.0520 0.0521 0.0522 0.0523 0.0524 0.0525 0.0526 0.0527 0.0528 0.0529 0.0530 0.0531

P<h1> P<h2> P<h3> P<h4> P<h5> P<h6> P<h7> P<h8> P<h9> P<h10>

π = 0.25 ; ψ = 0.75 π = 0.50 ; ψ = 0.50 π = 0.75 ; ψ = 0.25

% !0 ' 6 # . B = 0.5 β= 0.79 γ_l= 3.0 γ_h= 2.5 S = 3

$ #nl= 5 nh= 10$# # ) *

< . % * ,# $

> > !" $# % & # ) *

< # % # & $ % # #

(48)

+

#

%

# # $ % # ' @ " - # * ,# %

$ & ) # # & ) # & $ $ - & ) %

* # & ) ) & ) $ # * * % $ *

,# # +

% $ # # % *

# & ) % # # & ) #

. # % $ # # & ) # & ) % & *

,# # & ) % # % % # . # % %

& $ # % $# * 6 & )

$ # & ) # *

G & & ) % & & % #

# & # # *

,# & ) # & $ .' * * *

,# $ % ' % & ykm

ij ∈ ℜ+ $#

# 'k m_{∈ {}R & ) "G % & ) " n & ) "}$# k # +

# # l # # # ' # &

% & ) 'i, j_{∈ {}, R_} & ) 'i, j_{∈ {}0, G, R_}$# i # + &

# # j # # # *

xk

i $ #k∈ {R, G, n} i∈ {, R}$# k=G i∈ {0, G, R}$# k=n #

# ' $# # k % & )

& ) $# i* G # /

xR=BR xG+xGR=B−BR xn0+xnG+xnR= (1−B) D "

0≤BR≤B≤1*

G % #

(49)

% & & $ / % ' % & $ & ) #

. # % # $ # # % #

% # & # % * < & ) # #

& ) & # & )* < & ) # # & )

# - * < & ) & ) # & ) # $# #

# & ) * < % & ) & ) # % % $# # #

& ) & ) # % $# # # & )

# # & ) # * < % & ) $ # % & )* #

# # # $ # # * % & )

& ) # # * & ) $

# & ) # & ) % * # % # % # *

,# & + * .

. $ + + vk

i # .

$ # k$# & % i*

,# + & % & ) $ # /

−yGG₊_βvG _≥ ₀ _D!"

−yGG_R +βv_RG ≥ 0 D "

−yGR+βv_RG _≥ 0 DD"

−yGn_G +βvG _≥ 0 D3"

−yGn_R +βv_RG ≥ 0 D4"

,# . & % & ) $ # /

−yGG_R +βv_RG _≥ 0 D0"

−yGG_RR+βv_RG ≥ 0 DF"

(50)

,# . & & ) /

−yRG_R +βvR _≥ 0 D("

−yRR+βvR ≥ 0 3 "

−yRn_R +βvR ≥ 0 3!"

,# %# '# D!" ' 3!" & # # %# & # & )

% *

# & ) # % % $# # * ,# .

#

u(ynn₀_G) +βv₀n _≥ βvn_G 3 "

u yGn

G +βv0n ≥ βvnG 3D"

u y_RGGn +βv₀n ≥ βvn_G 33"

u(ynn₀_R) +βv₀n ≥ βvn_R 34"

u(yGRnn) +βvnG ≥ βvnR 30"

u yGn_R +βvn_G ≥ βvn_R 3F"

u yRn_R +βv₀n _≥ βvn_R 32"

(51)

,# . # & ) # % % $# #

−ynn₀_G+βvn_G _≥ βvn₀ 3("

−y₀nn_R+βvn_R ≥ βvn₀ 4 "

−ynG₀ +βvn_G ≥ βvn₀ 4!"

−ynG₀_R +βvn_G _≥ βvn₀ 4 "

−y₀nR+βvn_R ≥ βvn₀ 4D"

−y_GRnn +βvn_R ≥ βvn_G 43"

−y_GRnG+βvn_R _≥ βvn_G 44"

−y_GnR+βvn_R ≥ βvn_G 40"

$ # & ) * < & ) $#

$ # % *

,# # # = $ 0 G # = $

0 G & ) # = $ R # = $ R

% & ) * */

xn_G[B−BR] +xnRBR = xn0B 4F"

xn₀[B−BR] +xGxnR = xnG B+xGR 42"

xG[BR+xnR] = xGR[xnG+BR] 4("

D " 4F"' 4(" $# # %

. 9*

,# # $ $ /

v₀n = x

n G

S (−y

nn

0G) +

xn R

S (−y

nn

0R) +

xG

S −y

nG

0 +

xG R

S −y

nG

0R +

xR

S −y

nR

0 +

+β x

n

G+xG+xGR

S v

n G+

xn R+xR

S v

n

R+ 1−

xn

G+xG+xGR+xnR+xR

S v

n

(52)

v_Gn = x

n

0 S u(y

nn

0G) +

xG

S u y

Gn G +

xG R

S u y

Gn RG +

xn R

S (−y

nn GR) +

xG R

S −y

nG GR +

xR

S −y

nR G +

+β x

n

0+xG+xGR

S v

n

0 + xn

R+xGR+xR

S v

n

R+ 1−

xn

0 +xG+ 2xGR+xnR+xR

S v

n G 0!"

v_Rn = x

n

0 S u(y

nn

0R) +

xn_G S u(y

nn GR) +

xG

S u y

Gn R +

xR

S u y

Rn R +

+β x

n

0 +xR

S v

n

0 + xn

G+xG

S v

n

G+ 1−

xn

0+xR+xnG+xG

S v

n

R 0 "

vG = x

n

0

S u y

nG

0 +

xG

S u y

GG ₊xGR

S u y

GG R +

+x

n G

S −y

Gn G +

xn R

S −y

Gn R +

xG

S −y

GG ₊xGR

S −y

GG R +

xR

S −y

GR ₊

+β x

n

R+xGR+xR

S v

G

R+ 1−

xn

R+xGR+xR

S v

G _0D"

v_RG = x

n

0

S u y

nG

0R +

xn G

S u y

nG GR +

xG

S u y

GG R +

xG R

S u y

GG RR +

xR

S u y

RG R +

+x

n G

S −y

Gn RG +

xG

S −y

GG R +

xG R

S −y

GG RR +

+β x

n

G+xG+xR

S v

G_{+ 1}₋xnG+xG+xR

S v

G

R 03"

vR = x

n

0

S u y

nR

0 +

xn G

S u y

nR G +

xG

S u y

GR ₊xR

S u y

RR ₊

+x

n R

S −y

Rn R +

xR

S −y

RR ₊xGR

S −y

RG

R +βvR 04"

E # % 0 " 04" 6 .

* ,# # * < % # $ & # # &

% % *

,# & . $ # &

(53)

& # % * ,# ' %

w=

k,i

xk_ivk_i 00"

$# (k, i)∈ {(n,0),(n, G),(n, R),(G, ),(G, R),(R, )}

< # $ $ # $ & !D" !3"' !("

w=

k,m,i,j

xk ixmj

S(1−β)z y

km

ij 0F"

$# (k, m, i, j)∈ {(n, n,0, G), . . . ,(R, G, , R)} z ykm

ij =u ymkji −ykmij *

w & # . # *

"

θ= (B, BR, β)∈Θ =

[0,1]2×[0,1)

" # < . *

0F" # y∗ _{# +} _'& _{# #} _z′₍_y∗_{) = 0}_*

$ % # $ # $# & + '& *

,# & # $# # % % % $ & C #

# & $ # $*

G & ) # .' & $ % & ) & .' # % #

& - # B # & ) % * ; # $ & ) B #

% & ) % *

,# $ # & ) % & $ / & ) B #

% & ) % # % - # % & - * * RA(B, BR)<0 $#

RA(B, BR)≡vR−

xG_vG₊_xG RvRG

xG₊_xG R

(54)

# # B # & ) % # % - # & - * *

RA(B, BR)>0*

< & & $# RA(B, BR) = 0*

" ! " #

$ % B ∈ (0,1) & 0 < B < B˜ ≃ 0.12405 u(_yy∗∗) <

B+1 2B

2

lim

B →0 RA(B, BR)>0 Blim→B−

RA(B, BR)<0

$ % B ∈ (0,1) & 0.12405 ≃ B < B <˜ 1 u(_yy∗∗) >

B+1 2B

2

lim

B →0 RA(B, BR)<0 Blim→B−

RA(B, BR)>0

" # < . *

% !F"' " # % # %

-& B = 0.1 B = 0.6* +% !F" !2" $ %

& & & ) B = 0.1 β = 0.98 u(_yy∗∗) = 6.36

S = 3 & & [ xn₀ = 0.3049, xn_G= 0.1547, x_Rn= 0.4403, xG= 0.0140, xG_R= 0.0334, xR= 0.0526

y_jimk=y∗,∀(k, m, i, j)∈ {(n, n,0, G), . . . ,(R, G, , R)} ]*

% !F ' 6 # % -(RA)

B= 0.1 β= 0.98 u(_yy∗∗) = 6.36 S = 3 BR∈[0,0.1]*

(55)

% !2 ' 6 # &

B= 0.1 β= 0.98 u(_yy∗∗) = 6.36 S = 3 BR∈[0,0.1]*

# +% !(" " $ % & & & )

B = 0.6 β = 0.98 u(_yy∗∗) = 6.36 S = 3 &

& [ xn

0 = 0.1342, xnG= 0.0826, xnR= 0.1833, xG= 0.1296, xGR= 0.1631, xR= 0.3074

ymk

ji =y∗,∀(k, m, i, j)∈ {(n, n,0, G), . . . ,(R, G, , R)} ]*

% !( ' 6 # % -(RA)

(56)

% ' 6 # &

B= 0.1 β= 0.98 u(_yy∗∗) = 6.36 S = 3 BR∈[0,0.6]*

+% !" # # M=xn

G+xnR+xGR - #

& ) B # % % # & ) Br∈[0, B]*

% ! '6 # # BR∈[0, B]

β= 0.98 u(_yy∗∗) = 6.36 S= 3*

ρG₌ x

x +x ρ G R=

x

x +x * - % 02" $ # BR /

(57)

∂RA(B, BR)

∂BR

= ∂v

R

∂BR −

ρG∂v

G

∂BR

+ρG_R∂v

G R

∂BR

+ ∂ρ

G

∂BR

vG+ ∂ρ

G R

∂BR

vG_R 0("

G _∂B∂ρ =−∂B∂ρ

∂RA(B, BR)

∂BR

= ∂v

R

∂BR −

ρG∂v

G

∂BR

+ρG_R∂v

G R

∂BR

+ ∂ρ

G R

∂BR

v_RG−vG F "

& ) $# % & ) ) * # B %# & )

# $ * ,# & ) % & # % & ) %

# & & ) * * & ) $ # * 5 # & +

# % * * # $ & $ % & ) % %

& ) * ,# . # D * #

& ) # %# '# F " ∂RA_∂B(B,B ) <0 # &

& * # . *

# B % %# # & + # % % % & ) $ - $#

# & ) % * ,# . # D * < #

& ) # %# '# F " % ∂RA_∂B(B,B ) >0 # &

& * ; # $ $ # . *

,# . # & & * # # .

- # % $ ' # % & ) % *

E # +% !(" $ & & * # % & )

$ % $ # * ; # # # #

& ) $ % . % & ) %

# *

$# # $ & & & & ) %

$# # & & # . # $ # *

(58)

% *

E $ $ # . & & & & *

" % B _∈(0,1) & β_∈ (0,1) u(_yy∗∗) &

" # vn

R> vGn> v0n>0 vR>0 vRG> vG>0

" # < . *

" + % B _∈(0,1) & β_∈(0,1) & 3< u(_yy∗∗)

'

" # B → 0 β → 1 3 < u(_yy∗∗) # . & & & ) &

3 vn

R > vGn > vn0 >0 vR >0* vGR > vG >0 & + '& * ;

& + '& B+1 2B

2

& % & D RA(B,0+) > 0

RA(B, B−)<0*

" , % B_∈(0,1) β_∈(0,1) u(_yy∗∗) & "

# vn

R> vGn> v0n>0 vR>0 vRG> vG >0

" # < . *

" - % B _∈ (0,1) β _∈(0,1) & 3< u(_yy∗∗)

'

" # B _→ 1 β _→ 1 3 < u(_yy∗∗) # . & & & ) &

0 vn

R > vGn > vn0 > 0 vR > 0* vGR > vG > 0 & + '& *

; & + '& B+1

2B

2

% 1 & D RA(B,0+) < 0

RA(B, B−)>0*

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(59)

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(60)

% D ' 6 # w Bg∈[0, B−BR]$ #BR +.

β= 0.98 u(_yy∗∗) = 6.36 S= 3*

% 3 ' 6 # w B $ β= 0.98 u(_yy∗∗) = 6.36 S= 3*

(61)

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