РАВНОВЕСИЕ ПО БЕРЖУ В МОДЕЛИ ОЛИГОПОЛИИ КУРНО ½

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+ % ! '4( !"0%3 !!,% %'+ ++% % 7%- + .
& 4(% % +"4, ++% % :#> "+% (& + ! '%-+!, "
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½
! "#" $!%
9
=+ 3& : '"+?%+& G J!+
!" #$%"
@ ! % ?/ N ?& !/ '% !-'% !+"!, !%-%
Γ = N, {Xi }i∈N , {fi (x)}i∈N .
(1.1)
K'% , -% !+ "'+ %+ + N = {1, 2, . . . , N }& (!% N > 1L -'3 4 N
+&
% M%'""
, ' + ?/& +%! +/ !!%/ $'%3 !+%* xi ∈ Xi ⊆
n
k
⊆ R $ + R , k 1& 4'% , '%% 4(%! k.%% %+'+ '%3 !+!%,% ).
%!(% % ! !+& #%%! ! "+"/! " "'(%% 4 k '%3 !+.
!%, ( %& 4 +% + +'% !?+& !'! " 4+%'%% %+'+3 3*L + %4,!!% ! + 4%! " i
x = (x1 , . . . , xN ) ∈ X =
Xi ⊆ Rn (n =
i∈N
ni );
i∈N
-% !+% X %'%% )?" +> fi(x)& ( % ?%+/0" (% !+ ).
?+" i. (i ∈ N)
L e'%% (xz
i ) = (x1 , . . . , xi−1 , zi , xi+1 , . . . , xN ) f = (f1 , . . . , fN )
e
e
#$% 1 (x , f ) = ((x1 , . . . , xeN ), (f1 (xe ), . . . , fN (xe ))) ∈ X × RN 4+%! "
+ % (1.1)& % max fi (xe xi ) = fi (xe ) (i ∈ N);
xi ∈Xi
(1.2)
-% xe 4+%! " + % (1.1)
B
B
B
N
#$% & 1 (xB , f B ) = (xB
1 , . . . , xN ), (f1 (x ), . . . , fN (x )) ∈ X × R 4+%!.
" + % (1.1)& % B
max fi (xxB
i ) = fi (x ) (i ∈ N);
x∈X
(1.3)
'%% xB 4+%! " + % (1.1)
& %'#%" ($ $ )$ )'
+ !% 56 !% & ! '%! % , .
+& 4+'"0 ' !! -% !+ 56 !%! "& (! ( ! /! 4 (%! (% !+ !+"% !+ N% -% % !+.
+%! " + %4,!!% ++%>+" %'-%" " '%, + %' !+ 4+ !& !,/ '%-+" , #!3 '% ?%4+"& ! /
%& '' '!& ! +4'%3 !+/! N + 2
4+'!%%3 7'% (!!,& (! + % +% "'+% % ! 1 ' N -%.
!+ + {1, 2, . . . , N } 4( (%%4 N BM% +0%3 i. 4+'!%% (i ∈ N)
4 %!3 $4'3 * %-! +%% '? 4( (%%4 qi 1
#! -'3 4 + % -%! !+!, !+ + (% !+% %,>%& (% α > 0&
,>%& (% β& ! % !, +% %+% !+
α qi β (i = 1, . . . , N ).
(2.1)
1+% %+% !+ + $* ! 4(%! !! )!& (! 4+' !+%% 0 ! -'
4 4+'!%%3 )4(% (% K'% , '" !! (!%& (! + % #! 0 !
'+ G!, %+ %+% !+ 4 $* + !& (! + '% ! !+%! %3 !
$%& ' !+ % #%!#%* C!! ! ' %! !,
' !!( + %& !%& ! !%!& %4+ ! -+>%3 " % ?%& !+ !+ + 4%% % %%% 4%% %'%%3 +%( α
@++% % 7%- + '% 9
4+' !+ i. (i ∈ N) %'/! " %3 4+ ! .
(% !+ +0%3 '? qi ! !, %' !+% + +'% cqi + d& 4'% , c d !.
+%! !+% %'% %%%% !"% 4'%- $ %%% 4'%- ! "! "&
%& 4!! 4! (& 4 ,"& !4?/ '+"&
!" 2 %' %0%3& 4%& !+& ?%43 !*
: % + 4+ ! ! !++%! " ?% '?& !/ !-% (!%
%3 4+ "0%3 ! 0% (% !+ q̄ = q1 +q2 +. . .+qN !+>% '- !+
G(!%& (! ?% p !+ %3 4+ ! ! %'-%"& %& %' !+"% %% + +'%
p(q̄) = a − bq̄,
(2.2)
'% a = const > 0 2 (," ?% !+& !"3 -!%,3 #))?%!
# !( ! b > 0 4+%!& , ;'%!< ?% !% + '- %'?
'?
1%'-& (! ?% %'%"%! " !& (! ++% !, %'-%% C!
4(%!& (! -'3 4 4+'!%%3 '%! + %& (! 4+'! ( i. (i ∈ N) #! !+"%!
p(q̄)qi = (a − bq̄)qi = a − b
qk qi ,
k∈N
% $+( 4 +(%! 4'%-%* '%!
πi (qi , . . . , qN ) = a − b
qk qi − (cqi + d).
(2.3)
k∈N
1%'%! " !-%& (!& %'%"" +3 M% 4+' !+& +' !+ -'3
).4+'!%" %!%! " ;?,%< +%'%% + %!+
!%!(% " '%, !% +4'%3 !+" %' !+"%! 3 % ?.
/ N ?A
N, {Qi = [α; β]}i∈N , {πi (q1 , . . . , qN )÷(2.3)}i∈N .
(2.4)
K'% ,& + $*& N = {1, 2, . . . , N } 2 -% !+ "'+ %+ +& Qi = [α; β] 2
-% !+ !!%3 i (i ∈ N) G!? q = (q1, . . . , qN ) ∈ Q = Q1 × Q2 × . . . × QN &
)?" +> i. πi(q) = πi(q1, . . . , qN ) %'%% + $*
)$* !
a > c" (2.4) q B =
B )" q B = α (i ∈ N)" #
= (q1B , q2B , . . . , qN
i
πiB = πi (q B ) = [a − N bα]α − (cα + d) = [a − c]α − bN α2 − d.
4 ! % , ! + G!?" ++% " 7%- + % $9* %'%"%! " !%3
4 N %+% !+A
B
B
πi (qqi ) πi (q )
∀q∈Q
'%& &
=
$* %+% !+ $
* /! +'
(qqiB )
(i ∈ N),
(2.5)
(q1 , q2 , . . . , qi−1 , qiB , qi+1 , . . . , qN )
B
B
⎧ +
q
+
.
.
.
+
q
a
−
b
q
q1 − (cq1B
⎪
2
N
1
⎪
⎨ a − b q1 + q2B + . . . + qN q2B − (cq2B
. . . . . . . . .
⎪
⎪
⎩ B
B − (cq B
qN
a − b q1 + q2 + . . . + qN
N
+ d) + d) . .
+ d) B
B
B + . . . + qB
a
−
b
q
+
q
q1 − (cq1B + d),
1
2
N
B
B
a − b q1 + q2B + . . . + qN
q2B − (cq2B + d),
. . . .
. . . B . B . . . B B − (cq B + d)
a − b q1 + q2 + . . . + qN qN
N
!/! " +%'+ + % qi ∈ Qi (i ∈ N)
O% 4%!!,& (! !?%3 ++%B " 7%- + $9* '%! qB = (α, α, . . . , α)& #! ++% % 7%- +> πi (i ∈ N) !+"!
πiB = πi (q B ) = [a − N bα]α − (cα + d) = [a − c]α − bN α2 − d.
=+ 3& : '"+?%+& G J!+
%3 !+!%,& , 0" !+ !+%3 '?& -'3
i.3 (i ∈ N) !% +%(+%! , + % !, ( !+ $9* 1%%3'% !%%, ++% / :#> + % $9*
)$* & !
a > c" (2.4) e
q e = (q1e , q2e , . . . , qN
),
i ∈ N ⎧
a−c
⎪
α,
⎪
α,
⎪
⎪
b(N
+ 1)
⎪
⎪
⎨ a−c
a−c
, α <
< β,
qie =
b(N
+
1)
b(N
+ 1)
⎪
⎪
⎪
⎪
a−c
⎪
⎪
⎩ β,
β.
b(N + 1)
$
(i ∈ N)
(2.6)
+
⎧
⎪
(a − c)α − bN α2 − d,
⎪
⎪
⎪
⎪
⎪
⎨
(a − c)2
e
e
πi = πi (q ) =
− d,
⎪
(N + 1)2 b
⎪
⎪
⎪
⎪
⎪
⎩ (a − c)β − bN β 2 − d,
% % % a−c
α,
b(N + 1)
a−c
< β,
α<
b(N + 1)
a−c
β.
b(N + 1)
4 ! % , ! + G!?" ++% " :#> + $9* %'%"%! " !%3 .
+% !+ $ %'%%% *
e )]q − (cq + d)},
π1 (q e ) = max π1 (q e q1 ) = max {[a − b(q1 + q2e + . . . + qN
1
1
q1 ∈[α;β]
q1 ∈[α;β]
q2 ∈[α;β]
q2 ∈[α;β]
e )]q − (cq + d)},
π2 (q e ) = max π2 (q e q2 ) = max {[a − b(q1e + q2 + . . . + qN
2
2
. . . . . . . . . . . . . . . . . . . .
πN (q e ) = max πN (q e qN ) = max {[a − b(q1e + q2e + . . . + qN )]qN − (cqN + d)}.
qN ∈[α;β]
(2.7)
qN ∈[α;β]
$D*& ! -% +>%&
'" / i ∈ N (%%4 (qeqi) 4(% !?" qe& + !3
e
!!%" i. qi 4%% qi
" -' i ∈ N )? πi(qeqi) %%%3 qi ' !%! " +.
% '+ !%+3A
∂πi (q e qi ) e
e
e )−c = a − 2bqi − b(q1e + . . . + qi−1
+ qi+1
+ . . . + qN
= 0,
qi =qie
∂qi
qi =qie
∂ 2 πi (q e qi ) = −2b < 0.
∂qi2
qi =q e
i
(2.8)
!% +% 4 $* %%! % !& , #))?%! # !( ! b > 0& 4 %+
+% !+ -' i ∈ N (% !% N %3 +%3A
⎧
⎪
e = a − c,
⎪
2q1e + q2e + q3e + . . . + qN
⎪
⎪
b
⎪
⎪
⎨ e
e = a − c,
q1 + 2q2e + q3e + . . . + qN
b
⎪
⎪
. . . . . . . . .
⎪
⎪
⎪
⎪ q e + q e + q e + . . . + 2q e = a − c ,
⎩
1
2
3
N
b
@++% % 7%- + '% %>%% !3 '%! !?"
e
q =
e
(q1e , q2e , . . . , qN
)
=
a−c
a−c
a−c
,
,...,
.
(N + 1)b (N + 1)b
(N + 1)b
1 +% +" α < b(Na −+c1) < β 3'%% qie (i ∈ N) ' !+"/! ,%
4(%% )?3 πi(qeqi) !%4% [α; β]& %'+!%,& "+"/! " ++% :#>
!!%" + % $9*
-% b(Na −+c1) α& !& + +! %+% !+ 4 $*& !%4% [α; β] -'" 4
)?3
πi (q e qi ) (i ∈ N) ! +%! G%'+!%,& + $D* ' !/! "
e
qi = α (i ∈ N)
(% ' b(Na −+c1) β& )? πi(qeqi) + % i ∈ N '! -% !
+4 !/0 [α; β] G!+%! !+%& +% !+ $D* +"/! " qie = β (i ∈ N)
%%& M%'"" + % ! +%'% ("& +'& (! ++% % :#> !!%
+ $9* %'%"/! " )3 $F*
1%%3'% !%/ +>%3 πi(qe) + 3'%3 !? $F* (i ∈ N) 1' !++
$F* + )? +> $*& '" -' i. (i ∈ N) ( ++% % :#> +> + % $9* %& b(Na −+c1) α ' πie = πi (α, α, . . . , α) = [a − bN α] α − (cα + d) = (a − c)α − bN α2 − d.
b(Na −+c1) β& ! ++% 3 :#> +> i. πie '%!
πie = πi (β, β, . . . , β) = [a − bN β] β − (cβ + d) = (a − c)β − bN β 2 − d.
:%?& + (% α < b(Na −+c1) < β !+!
πie
a−c
a−c
a−c
a−c
a−c
a−c
,
,...,
= a − bN
− c
+d =
(N + 1)b (N + 1)b
(N + 1)b
(N + 1)b (N + 1)b
(N + 1)b
= πi
=
(a − c)2
N
(a − c)2
(a − c)2
−
·
−d=
− d.
(N + 1)b (N + 1)b N + 1
(N + 1)2 b
4& +> i. + !? ++% " :#> %%! +'
⎧
a−c
⎪
(a − c)α − bN α2 − d, % α,
⎪
⎪
b(N + 1)
⎪
πie
e
= πi (q ) =
⎪
⎪
⎨
(a − c)2
− d,
⎪
(N + 1)2 b
⎪
⎪
⎪
⎪
⎪
⎩ (a − c)β − bN β 2 − d,
% % a−c
< β,
b(N + 1)
a−c
β.
b(N + 1)
α<
+ $ $ , # $*( '$-. $ / # 0-(
#! )% +%'% +%% +>%3& !% (! & + ,4++.
> , !?%3 ++% " 7%-& !% +>& !% -'/! + !?
++% " :#> " #! "!,.! ! ! ("
=+ 3& : '"+?%+& G J!+
a−c
α ++% % :#> !!% xei & %'%%% + $F*&
%(! . 1 1
b(N + 1)
+'/! ++% 7%- !!%" xBi = α (i ∈ N) G%'+!%,& .
& '%-+" , !? ++% " :#>& (! !% -% +>& % ( & '%-+" , !? ++% " 7%-& %A
a−c
α.
(3.1)
πie = πiB b(N + 1)
+% %+% !+ α < b(Na −+c1) < β& ! +> i. (i ∈ N)
+ !? ++% " :#> '%!
%(! . 11
πie = πi (q e ) =
++% 3 7%- +> !+!
(a − c)2
− d,
(N + 1)2 b
πiB = πi (q B ) = α[a − c − N bα] − d.
4 !,
(a − c)2
− d − (a − c)α − bN α2 − d =
−
=
2
(N + 1) b
a−c
a−c
(a − c)2
=
bN
·
α
−
·
α
−
.
= bN α2 − (a − c)α +
(N + 1)2 b
(N + 1)b
N (N + 1)b
πie
πiB
1 , ( + N > 1 a − c > 0& !
a−c
a−c
<
.
N (N + 1)b
(N + 1)b
+ !& (! α > 0 #))?%! # !( ! b > 0& ! 4 !, πie − πiB -!%,
a−c
0<α<
!?!%, N (N + 1)b
,
a−c
a−c
<α<
N (N + 1)b
(N + 1)b
+ /& % α=
4& '" + % i ∈ N
⎧
⎪
⎪
πie > πiB ,
⎪
⎪
⎪
⎨
πie = πiB ,
⎪
⎪
⎪
⎪
⎪
⎩ πie < πiB ,
% % % a−c
.
N (N + 1)b
a−c
a−c
< β,
N (N + 1)b
(N + 1)b
a−c
a−c
< β,
α=
N (N + 1)b
(N + 1)b
a−c
a−c
<α<
< β.
N (N + 1)b
(N + 1)b
α<
(3.2)
:%?& ! %'3
(3
a−c
%(! . 111 1 α < β ++% 3 :#> !!%%3 i. 4+'!%" '%!
(N + 1)b
!+ , +4- (% !+ !+& ! % !, xei = β (i ∈ N) %
, + !? ++% " :#> !+!
πie = (a − c)β − bN β 2 − d.
@++% % 7%- + '% K
III
II
L
I
M
G+%% +>%3 + !?" ++% " 7%- :#>
!? ++% " B 7%- '-% 4!, !+ '! ' ,
4%>%& % xi = α (i ∈ N) ++% 3 7%- +> #! '%!
πiB = (a − c)α − bN α2 − d.
@ ! 4 !, + +>& !/ (! ( ! $9* + !.
?" ++% " :#> 7%- & '" i (i ∈ N) !+!
πie − πiB = (a − c)β − bN β 2 − d − (a − c)α − bN α2 − d =
= (a − c)(β − α) − bN (β 2 − α2 ) = (β − α) · [a − c − bN (β + α)].
β > α& ! 4 +%'%3 4 ! +'%! 4 +/0%3 %33
)?
a − c − bN (α + β).
a−c
α+β =
bN
a−c
α+β <
bN
C! )?" %"%! 4 K(!& 4 !, πie − πiB + / α + β =
a−c
$!%4 LM * 1
4 !, πie − πiB -!%,& + (%
=
bN
a−c
α+β >
'%! !?!%,3
bN
c
G%'+!%,& α < β (Na +− 1)b
'" +>%3 i. + !? ++% "
7%- πiB + !? ++% " :#> πie %%! % ! !>%%
⎧
⎪
% α + β < aN−bc α < β,
π e > πiB ,
⎪
⎪
⎨ i
(3.3)
% α + β = aN−bc α < β,
πie = πiB ,
⎪
⎪
⎪
a
−
c
⎩ πe < πB ,
% α + β > N b α < β.
i
i
9
=+ 3& : '"+?%+& G J!+
%%& M%'"" + % ! ("& 4 ) $*& $* $* ( % +%%
++% 7%- :#> +>%3 '" i.e (i ∈ N) + % $9* $ *
! P& 4-%3
& % +> πi + !? ++% " :#> '%!
,>%& (% +> πiB & ++% 3 7%-
! PP& !& ++% % 7%- ' !+"%! i (i ∈ N) +> ,>3&
(% +> + !? ++% " :#>
! PPP 3 KLM +> + + !?" ++% " 7%-
:#> '! '+
2 /! !+%-'%" & Q ++/! %'/03 !(% 3 % +.
%>%" $ !!%3 +* + '% 1 0) # ,4" !"% a& b& c N & 3! 9 ( A
a−c
,
N (N + 1)b
a−c
,
2N b
a−c
,
(N + 1)b
a−c
.
Nb
G 0,/ #! ( %∗ !!,
& '% +'%!, ! P& PP& PPP
" !, 4(%" α β∗& 4/0% ;' !+< qi
:3! !( (α∗, β∗∗) !% % K!% +" !,& + 3 4 !%
!%3 -% !( (α , β∗)
:%?& 0,/ !+%-'%3 & 4 Q + !, "+3 +' ++% %>%"&
! % !, ++% / !?/ +> + + %3
11 0) #
111 0) #
13 0) #
2 4%5!
4/(%% !%! %0% 4A '" ! ;%!< ;+"43< ! I.
!%%! !& (! ;,! !(% "< ?%?" ++% " 7%- % %
+ 4'( # '%% "& (! #! !& +%+% 4+>"& (! ++% % 7%- + #(% '%" -%! !, %% #))%!+& (% ++% % :#>& %
' '%! !+!, " %' !+% -%% ?%? ++% " 7%- #.
% ++% % 7%- 43%! + #% ' !3% % !& !% -%& % -% 4%! ? $ & %& 4 56*
.
G
F
S
V
Q
R
E
T
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..
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