Джон М. Хартвик. 2009. Программы устойчивого потребления.

Ïðîãðàììû óñòîé÷èâîãî ïîòðåáëåíèÿ†
John M. Hartwick
18 îêòÿáðÿ 2007
5
Óñòîé÷èâîå ïîòðåáëåíèå íåâîçîáíîâëÿåìûõ
ðåñóðñîâ íà äóøó íàñåëåíèÿ1
Ìû ñåé÷àñ äâèíåìñÿ äàëüøå, èñïîëüçóÿ ïîñòîÿííîå ïîòðåáëåíèå íà äóøó íàñåëåíèÿ çà áëàãîïðèÿòíûé ðåçóëüòàò è èññëåäóåì ïîäðîáíîñòè ïðîãðàìì ïîñòîÿííîãî
ïîòðåáëåíèÿ (ìàêñèìèí). Ìîäåëü Ñîëîó [1974] áàçîâàÿ, íå ñîäåðæèò íè ðîñòà ÷èñëåííîñòè íàñåëåíèÿ, íè îáåñöåíèâàíèÿ ïðîèçâîäèòåëüíîãî êàïèòàëà, íè èçìåíåíèé â
òåõíîëîãèÿõ. Ïðèìåì ýòó ìîäåëü çà îòïðàâíóþ òî÷êó. Ýòà ìîäåëü ïîääåðæèâàåòñÿ
èíâåñòèðîâàíèåì ðåíòû, ïîëó÷åííîé îò íåâîçîáíîâëÿåìûõ ðåñóðñîâ, â íîâûé ïðîèçâîäñòâåííûé êàïèòàë (íóëåâûå ÷èñòûå èíâåñòèöèè [zero net investment] èëè ðåàëüíûå
ñáåðåæåíèÿ ðàâíû íóëþ [genuine savings at zero]). ×òîáû èñïîëüçîâàòü ïðîãðàììû
óñòîé÷èâîãî ïîòðåáëåíèÿ Ñîëîó íóæíî íà÷àòü ñ âûðàæåíèÿ
ðàçäåëèòü ïî÷ëåííî íà
K̇ = F (K, R, N )−C −δK ,
N:
k̇ = f (k, r) − c − δk − nk
ãäå ñòðî÷íûå áóêâû îçíà÷àþò ïðîïèñíûå áóêâû, äåëåííûå íà
:=
îçíà÷àåò ðàâíî ïî îïðåäåëåíèþ).
K
N
è
n := Ṅ /N
(Çíàê
- ýòî âåëè÷èíà êàïèòàëà, ïðîèçâîäèìîãî
ëþäüìè: ñòàíêè, çäàíèÿ, èíôðàñòðóêòóðà;
èñïîëüçóåìûõ â òåêóùèé ìîìåíò (R(t)
N,
R
- ýòî ïîòîê íåâîçîáíîâëÿåìûõ ðåñóðñîâ,
= −Ṡ(t), ãäå S(t) - ýòî îñòàâøååñÿ êîëè÷åñòâî),
- ýòî ÷èñëåííîñòü íàñåëåíèÿ (ðàâíàÿ ðàáî÷åé ñèëå),
C
- ýòî ñîâîêóïíîå ïîòðåáëåíèå
In: Handbook of Environmental Accounting. Eds. Thomas Aronsson, Karl-Gustaf L
ofgren, 2008, forthcoming.
1 Ïåðåâîä Áåëÿåâà Àëåêñàíäðà, 237 ãðóïïà, Èíñòèòóò Ìàòåìàòèêè è Êîìïüþòåðíûõ Íàóê, ÄÂÃÓ,
ôåâðàëü 2008
†
8
è
δ
nk ,
K(t).
- ýòî íåèçìåííûé óðîâåíü îáåñöåíèâàíèÿ
Âàæíî çàìåòèòü, ÷òî
ýòî ñëåäóåò èç äèôôåðåíöèðîâàíèÿ. Ïðåäïîëàãàëîñü, ÷òî
F (.)
K̇/N = k̇ +
èìååò ïîñòîÿííûé
F (.) := K α Rβ N 1−α−β ìû
α è β è α + β < 1. Ïîòîê äîáû÷è íåôòè R(t)
ýôôåêò ìàñøòàáà, à çíà÷èò, äëÿ íàøåé îñíîâíîé ôóíêöèè
ïîëó÷èì
f (.) := k α rβ
áåðåòñÿ èç çàïàñà
ñ ïîëîæèòåëüíûìè
S(t),
R(t) = −Ṡ(t).
ïðè ýòîì
Ïîëàãàåì, ÷òî äîáû÷à îïðåäåëÿåòñÿ
óñëîâèåì ðàâíîâåñèÿ àêòèâîâ (ïðàâèëî Õîòåëëèíãà)
F˙R /FR = FK ,
êîòîðîå íà äóøó
fr = βk α rβ−1
è
íàñåëåíèÿ ïðèìåò âèä:
f˙r /fr = fk .
×òîáû ïðîâåðèòü ýòî, ìû èìååì
Q
N
= kα rβ ,
çàòåì
fk = αk α−1 rβ .
Äðóãîé ÷àñòüþ ìîäåëè ÿâëÿåòñÿ ôóíêöèÿ ñáåðåæåíèé, òî åñòü èíâåñòèðîâàíèå ðåñóðñíîé ðåíòû èëè
K̇ = RFR + γQ,
ãäå ñëàãàåìîå
γQ
- ýòî äîáàâî÷íûå ñáåðåæåíèÿ â
äîïîëíåíèå ê òåêóùåé ðåñóðñíîé ðåíòå. Äîáàâî÷íûå ñáåðåæåíèÿ íóæíû äëÿ ñëó÷àåâ,
êîãäà ÷èñëåííîñòü íàñåëåíèÿ ðàñòåò è/èëè êàïèòàë óáûâàåò. Íà äóøó íàñåëåíèÿ èíâåñòèöèîííîå ïðàâèëî (savings rule) ïðèìåò âèä
k̇ + nk = rfr + γq.
Äëÿ ñëó÷àÿ ïðîèçâîäñòâåííîé ôóíêöèè Êîááà-Äóãëàñà
F (.) íàøè äâà êëþ÷åâûõ óðàâ-
íåíèÿ ÿâëÿþòñÿ ïðàâèëîì Õîòåëëèíãà è èíâåñòèöèîííûì ïðàâèëîì (savings rule) ñîîòâåòñòâåííî:
αq
q̇ ṙ
−
=
q r
k
è k̇ = βq
(7)
(8)
Îáðàòèòå âíèìàíèå, ÷òî ìû ïðèøëè ê èñïîëüçîâàíèþ òîëüêî ðåñóðñíîé ðåíòû äëÿ
k̇ .
(Ìû îòëîæèëè ñáåðåæåíèÿ
γq
äëÿ äðóãèõ íóæä)
2
Îñíîâíîé ðåçóëüòàò: êîãäà îáà óðàâíåíèÿ (7) è (8) ïîäñòàâëåíû â ïðîèçâîäíóþ
ïðîèçâîäñòâåííîé ôóíêöèè
èçìåíåíèÿ
c,
kα rβ
, òî ýòî äàåò
q̇ = 0.
×òîáû ïåðåéòè ê ðàññìîòðåíèþ
ìû äîëæíû âíèìàòåëüíî ñëåäèòü çà èçìåíåíèåì
N (t),
òî åñòü çà äè-
íàìèêîé ÷èñëåííîñòè íàñåëåíèÿ. Ñíà÷àëà çàìåòèì, ÷òî äëÿ íàøåé âûøåóïîìÿíóòîé
ñèñòåìû
c
îñòàåòñÿ êîíñòàíòîé äëÿ çíà÷åíèé
c
îïðåäåëåííûõ ôîðìóëîé
(1 − β − γ)q
Âîçíèêàþò òðè ñëó÷àÿ.
2 Ðîñò
÷èñëåííîñòè íàñåëåíèÿ, ìîäåëü ïîñòîÿííîãî ïîòðåáëåíèÿ íà äóøó íàñåëåíèÿ è ôóíêöèÿ
Êîááà-Äóãëàñà â ìîäåëè ABHMW èìååò òàêîå ñâîéñòâî, ÷òî ðåñóðñíàÿ ðåíòà ðàâíà íîâîìó êàïèòàëó
(βq = k̇ ), à îñòàâøèåñÿ ñáåðåæåíèÿ, òî åñòü γq , ðàâíû nk . Ñâÿçàí ñ ýòèì ðåçóëüòàò èíâåñòèöèîííîãî
ïðàâèëà βq = k̇ , à ïðàâèëî Õîòåëëèíãà ïîäðàçóìåâàåò q̇ = 0. Ýòè ñâîéñòâà ìîäåëè ABHMW
ìîòèâèðóþò ìàíåðó èçëîæåíèÿ íàøèõ ðåçóëüòàòîâ äëÿ ðîñòà ïðè ïîñòîÿííîì ïîòðåáëåíèè íà
÷åëîâåêà
9
(1) Ñîëîó [1974] (íóëåâîé ïðèðîñò íàñåëåíèÿ è íåò îáåñöåíèâàíèÿ
c
ïîñòîÿííî è ðàâíî
ìîäåëè
k̇
(1 − β)q .
ïîñòîÿííà èëè
K ).  ýòîì ñëó÷àå
Íåò äîáàâî÷íûõ ñáåðåæåíèé, ïîýòîìó
γ = 0.
 ýòîé
k(t) = k0 + β q̄t.
Òîãäà
1
ṙ
= −αq̄
r
k0 + β q̄t
è
ãäå
ζ
r(t) = ζ[k0 + β q̄t]−α/β
R
r := N
, ãäå N ïîñòîÿííàÿ â äàííîì
R∞
K(0) = K0 è 0 R(t)dt = S0 . Ýòî ïîçâîëÿåò íàì
ïîëîæèòåëüíàÿ êîíñòàíòà. Íàïîìíèì, ÷òî
ñëó÷àå. Äðóãèå íà÷àëüíûå óñëîâèÿ
Çàòåì çíà÷åíèå c âûðàæàåòñÿ â (1 − β)q̄ . Áåçóñëîâíî, äëÿ íåôòè
R∞
R(t)dt áûë êîíå÷íûì è ýòî ãàðàíòèðóåòñÿ óñëîâèåì
ìû òðåáóåì, ÷òîáû èíòåãðàë
0
α
> 1, îòìå÷åííûì Ñîëîó.
β
íàéòè
ζ
è
R(0).
(2) Mitra [1983] è Asheim, Buchholz, Hartwick, Mitra and Withagen [2007] (óñòîé÷èâûé
ðîñò íàñåëåíèÿ îïðåäåëÿåòñÿ äîáàâî÷íûìè ñáåðåæåíèÿìè). Çà îñíîâó áåðåì âñå òó æå
ñèñòåìó (7) è (8), ïîäðàçóìåâàþùóþ
íóëþ
(δ = 0).
Íàñ èíòåðåñóþò
ċ = 0
q̇ = 0.
èëè
Ïîëîæèì, ÷òî îáåñöåíèâàíèå
K
ðàâíî
c = (1 − β − γ)q .
Ṅ > 0 ìû äîëæíû ó÷åñòü äîáàâî÷íûå ñáåðåæåíèÿ â γq = nk äëÿ ïîëîæèòåëüíîé
êîíñòàíòû γ . Òî åñòü, äëÿ íåèçìåííûõ q è c ìû òðåáóåì, ÷òîáû ïðèðîñò íàñåëåíèÿ
Äëÿ
óäîâëåòâîðÿë óðàâíåíèþ
γ h αq i
Ṅ (t)
=
.
N (t)
α k
(9)
Îñíîâíîé ðåçóëüòàò ñòàòüè Asheim, Buchholz, Hartwick, Mitra, and Withagen [2007]
çàêëþ÷àåòñÿ â òîì, ÷òî ýòî óðàâíåíèå èíòåãðèðóåòñÿ äî
γ
N (t) = J · [A + Bt] αB .
ãäå
[ αq
]=
k
1
äëÿ ïîëîæèòåëüíûõ êîíñòàíò
A+Bt
A è B. J
òàêæå ïîëîæèòåëüíàÿ êîíñòàíòà.
Ýòî òàê íàçûâàåìûé êâàçè-àðèôìåòè÷åñêèé ðîñò íàñåëåíèÿ. Ñòàíåò ÿñíî, ÷òî äîëæíî
âûïîëíÿòüñÿ
ðåíòà
βq
γ < α. Âíèìàòåëüíîå ÷òåíèå âûøåíàïèñàííîãî ïîêàçûâàåò, ÷òî ðåñóðñíàÿ
â îäèíî÷êó ïîêðûâàåò ïðèðîñò êàïèòàëà
ïîêðûâàþò ñòîèìîñòü
nk .
k̇ ,
à äîáàâî÷íûå ñáåðåæåíèÿ
γq
Íàøå ðàñøèðåííîå ïðàâèëî èíâåñòèðîâàíèÿ ðåñóðñíîé
ðåíòû, èñïîëüçîâàííîå âûøå, òåïåðü ïðèíÿëî âèä
K̇ = (β + γ)Q.
K̇
(γ + β) h αq i
=
K
α
k
10
Ýòî äàåò íàì
Ñëåäîâàòåëüíî,
K(t) = L · [A + Bt](γ+β)/(αB) ,
L
ãäå
- ïîëîæèòåëüíàÿ êîíñòàíòà. Òàêæå
ìû ïîëó÷èëè
(α − γ) h αq i
Ṙ
=−
R
α
k
Çíà÷èò,
N
R(t) = M · [A + Bt](γ−α)/(αB) ,
ãäå
M
- ïîëîæèòåëüíàÿ êîíñòàíòà.
âîçðàñòàþò êâàçè-àðèôìåòè÷åñêè, â òî âðåìÿ êàê R óáûâàåò êâàçè-àðèôR∞
R(z)dz ñõîäèòñÿ, åñëè (α−γ)
ìåòè÷åñêè. Î÷åâèäíî, ÷òî
> 1. Òåïåðü íåïëîõî áû
αB
t
Q̇
Ṙ
îïðåäåëèòü çíà÷åíèÿ A è B . Ïðàâèëî Õîòåëëèíãà ìîæåò áûòü çàïèñàíî êàê
−R
=
Q
Q
Ṙ
α K . Ïîäñòàâëÿÿ çíà÷åíèÿ äëÿ R ïîëó÷åííûå âûøå, ìû ïîëó÷èì
è
K
Q̇ K̇
Q
−
= −β .
Q K
K
Ýòî ìîæåò áûòü âûðàæåíî êàê
ẏ/y = −βy
äëÿ
y ≡
Q
. Ýòî äèôôåðåíöèàëüíîå
K
óðàâíåíèå èìååò ðåøåíèå
y(t) =
ãäå
Ω = 1/y(0).
Ñëåäîâàòåëüíî,
íåôòè òðåáóåò
1
Ω + βt
Îòñþäà íåïîñðåäñòâåííî ñëåäóåò, ÷òî
A ðàâíî
α>β
K(0)
,à
αQ(0)
B ðàâíî
αQ
K
=
1
β
1
+α
t
p(0)
äëÿ
p(0) =
αQ(0)
.
K(0)
β
. Çíà÷èò, ñõîäèìîñòü èíòåãðàëà ïîòðåáëåíèÿ
α
è
α − γ > β.
Ýòî óñëîâèå ïðèâîäèòñÿ ê õîðîøî èçâåñòíîìó óñëîâèþ Ñîëîó [1974], êîãäà
γ = 0.
Çíà÷èò, ýòî îáîáùåíèå Ìèòðû ðàáîòû Ñîëîó [1974] òðåáóåò áîëüøåãî èíòåðâàëà
ìåæäó
è
α
è
β,
÷åì ïðè ïîñòîÿííîé ÷èñëåííîñòè íàñåëåíèÿ, êàê ó Ñîëîó.
Íàì íóæíî îïðåäåëèòü çíà÷åíèÿ
R(0), M , J
N0 .
è
Èç âûðàæåíèé äëÿ
N (t), R(t)
K(t)
è
L,
èñõîäÿ èç çàäàííûõ çíà÷åíèé
ïîëó÷èì
K0 γ/αB
]
αQ(0)
K0 γ−α
R(0) = M · [
] αB
αQ(0)
K0 γ+β
è K0 = L · [
] αB
αQ(0)
N0 = J · [
äëÿ
Q(0) = K0α R(0)β N01−α−β .
S0 =
×åòâåðòîå óðàâíåíèå
−M
K0 ∆+1
{
}
β αQ(0)
(∆ + 1) α
11
äëÿ
∆=
γ−α
αB
K0
Èìååì ÷åòûðå óðàâíåíèÿ äëÿ
çíà÷åíèÿ
L.
è
Ñëåäîâàòåëüíî, çàäàâàÿ íà÷àëüíûå
K0 , N0 è S0 , ìû ìîæåì íàéòè ðåøåíèå äëÿ ýòèõ êîíñòàíò. Çàòåì ìû ìîæåì
ïîëó÷èòü óñòîé÷èâûé óðîâåíü
(3) Îáåñöåíèâàíèå êàïèòàëà
îíè ïîêðûâàþò
q̇ = 0.
R(0), M , J
K̇ + δK .
C/N .
K
ñ ïîñòîÿííûì òåìïîì
δ . Ñáåðåæåíèÿ ðàâíû (β +γ)Q,
ßäðîì ñíîâà ÿâëÿåòñÿ ñèñòåìà èç (7) è (8), îáåñïå÷èâàþùàÿ
n(t)
Ìû ìîæåì çàíîâî ðåøèòü ýòó ìîäåëü äëÿ íåíóëåâûõ
è
δ.
òåïåðü äèíàìèêà ÷èñëåííîñòè íàñåëåíèÿ, êîòîðàÿ ïðèåìëåìà äëÿ
(1 − β − γ)Q.
Íàñ èíòåðåñóåò
ċ = 0,
ãäå
C =
Ìîäåëü ïðèìåò âèä
k̇ = βq
ṙ
−αq
=
r
k
Ṅ
γ h αq i
=
− δ.
N
α k
è
Íîâîå ñëàãàåìîå òåïåðü â âûðàæåíèè äëÿ ðîñòà íàñåëåíèÿ. Èíòåãðàë ýòîãî óðàâíåíèÿ
ðàâåí
N (t) = N0 e−δt (A + Bt)γ/(αB)
ãäå
A
è
B
ïîëîæèòåëüíûå êîíñòàíòû. Ýòà ôîðìà äëÿ
öèàëüíîå óìåíüøåíèå
N (t)
N (t)
ïîäðàçóìåâàåò ýêñïîíåí-
â ïðåäåëå. Òî åñòü, ÷èñëåííîñòü íàñåëåíèÿ äîëæíà ñòðå-
ìèòåëüíî óìåíüøàòüñÿ, ÷òîáû ñîõðàíèòü ïîñòîÿííîå ïîòðåáëåíèå, ïðè óñëîâèè, ÷òî
îáåñöåíèâàíèå ïðîèçâîäñòâåííîãî êàïèòàëà
δ
ïîä÷èíÿåòñÿ çàêîíó ðàäèîàêòèâíîãî
ðàñïàäà.
 ñëó÷àå ñ íåôòüþ ìû èìååì
Ṙ
R
= − αq
+
k
Ṅ
N
=
−[α−γ]
α
1
A+Bt
− δ,
÷òî èíòåãðèðóåòñÿ äî
R(t) = R0 e−δt (A + Bt)−(α−γ)/(αB) .
Òàêæå
K̇
K
=
βQ
K
+
Ṅ
N
=
[β+γ]
α
1
A+Bt
− δ,
ñëåäîâàòåëüíî
K(t) = K0 e−δt (A + Bt)−(β+γ)/(αB) .
K(t)
áîëüøå íå ÿâëÿåòñÿ ëèíåéíîé ôóíêöèåé îòíîñèòåëüíî âðåìåíè, èíòåãðèðîâàòü
òåïåðü ñëîæíåå, ÷åì ðàíüøå, ïðè
δ = 0 (Asheim, et. al. [2007]). Ýòè ðåøåíèÿ ïîõîæè íà
ïîëó÷åííûå âûøå äëÿ ìîäåëè Ìèòðû - ABHMW ñ êâàçè-àðèôìåòè÷åñêèì ðîñòîì ÷èñëåííîñòè íàñåëåíèÿ, çà èñêëþ÷åíèåì íîâîãî ìíîæèòåëÿ
óìåíüøåíèå ñ òåìïîì
δ
â ïðåäåëå äëÿ
K, N
è
R.
e−δt ,
êîòîðûé ïîäðàçóìåâàåò
Ýòî ïðåäïîëàãàåò, ÷òî óñòîé÷èâûé
óðîâåíü ïîòðåáëåíèÿ íà äóøó íàñåëåíèÿ â äàííîé ýêîíîìèêå áóäåò áåñêîíå÷íî ìàë, â
ëó÷øåì ñëó÷àå.
12
Çàìåòèì, ÷òî íàø ïîäõîä, ïðåäëîæåííûé âûøå, äëÿ ñëó÷àÿ ïðîèçâîäñòâåííîé ôóíêöèè Êîááà-Äóãëàñà, äîëæåí áûë áûòü íà÷àò ñ ôóíêöèè ñáåðåæåíèé è ôóíêöèè äèíàìè÷åñêîé ýôôåêòèâíîñòè (ïðàâèëî Õîòåëëèíãà), êîòîðûå îáåñïå÷èâàëè ïîñòîÿííûé
ÂÛÏÓÑÊ íà äóøó íàñåëåíèÿ, è çàòåì ðàáîòà ñ ñèñòåìîé áûëà ïðîäîëæåíà â òî
âðåìÿ, êàê ïîñòîÿííîå ïîòðåáëåíèå íà ÷åëîâåêà ïðåâðàòèëîñü â íîâîå òðåáîâàíèå.
Ìû èñïîëüçóåì ñåé÷àñ ýòîò ïîäõîä äëÿ áîëåå ñëîæíîãî ñëó÷àÿ, êîòîðûé âêëþ÷àåò
èçìåíåíèå ÷èñëåííîñòè íàñåëåíèÿ, îáåñöåíèâàíèå êàïèòàëà è ýêçîãåííûé ïàðàìåòð òåõíè÷åñêèé ïðîãðåññ.
(4) Ìîäåëü âûøå èìååò íåãàòèâíûé ðåçóëüòàò: äëÿ ëþáîãî ïîëîæèòåëüíîãî çíà÷åíèÿ îáåñöåíèâàíèÿ ïðîèçâîäñòâåííîãî êàïèòàëà
δ,
ïîñòîÿííûå ñáåðåæåíèÿ íå ñîâ-
ìåñòèìû ñ ÐÎÑÒÎÌ íàñåëåíèÿ, ïîêà ïîòðåáëåíèå íà äóøó íàñåëåíèÿ îñòàåòñÿ ïîñòîÿííûì. Ïîëîæèòåëüíîå çíà÷åíèå
δ
âëå÷åò ýêîíîìè÷åñêèé êðàõ îñîáîãî ðîäà. Ýòî
ðàñõîäèòñÿ ñ ðåçóëüòàòàìè ABHMW [2007], ãäå ëèíåéíîå èíâåñòèöèîííîå ïðàâèëî
áûëî ñîâìåñòèìî ñ ïîñòîÿííûì ïîòðåáëåíèåì íà ÷åëîâåêà, ðîñòîì ÷èñëåííîñòè íàñåëåíèÿ è èñ÷åðïàåìîñòüþ ïîòîêà íåôòè
R(t).
Âîçíèêàåò âîïðîñ, ìîæåò ëè ýêçîãåííûé
òåõíè÷åñêèé ïðîãðåññ îáðàòèòü íåãàòèâíûé ðåçóëüòàò ïðè ïîëîæèòåëüíîì
δ.
Ìû
îáðàòèìñÿ ê ýòîìó âîïðîñó â ýòîì ïàðàãðàôå (ýòè ðåçóëüòàòû âçÿòû èç ñòàòüè Cheviakov and Hartwick [2007]). Çàìåòèì, ÷òî â ýòîé ìîäåëè åñòü íîâûé, ýêçîãåííûé ïàðàìåòð - óðîâåíü òåõíè÷åñêîãî ïðîãðåññà
θ,
êîòîðûé ìîæåò îáðàòèòü ðåçóëüòàòû
ðàññìîòðåííîé âûøå ìîäåëè è, â íåêîòîðîì ðîäå, âîññòàíîâèòü ïîëîæèòåëüíûé
3
ðåçóëüòàò ìîäåëè ABHMW.
Ýêçîãåííûå òåõíè÷åñêèå èçìåíåíèÿ ïðîèñõîäÿò ñ ïîñòîÿííûì òåìïîì
òåëüíî,
θ.
Ñëåäîâà-
θt
K̇ = e F (K, R, N ) − C − δK , è äëÿ ñëó÷àÿ ïðîèçâîäñòâåííîé ôóíêöèè Êîááà-
Äóãëàñà, â ðàñ÷åòå íà äóøó íàñåëåíèÿ, ìû ïîëó÷èì
eθt F (K, R, N )
ïåðåõîäèò â
q = eθt k α rβ .
k̇ = eθt k α rβ − c − k[n + δ]. Q :=
Ôàêòè÷åñêè, áûëî áû ïîëåçíî ðàññìàòðèâàòü
òåõíè÷åñêèå èçìåíåíèÿ ñâÿçàííûìè ñ êàïèòàëîì, êàê â
ìû óâèäèì, ÷òî ýëåìåíò
θ
Q = [e α t K]α Rβ N 1−α−β :
äàëåå
θ
ÿâëÿåòñÿ êëþ÷åâûì. Ïðîäîëæàåì â òîì æå äóõå, ÷òî è
α
4
ðàíüøå. Èíâåñòèöèîííîå ïðàâèëî
(äîïîëíåííîå ïðàâèëî èíâåñòèðîâàíèÿ ðåñóðñíîé
3 Ñòèãëèö
[1974] ðàññìàòðèâàë ìîäåëü ñ ïîëîæèòåëüíûì ýêçîãåííûì òåõíè÷åñêèì ïðîãðåññîì
è ïîñòîÿííîé ñêîðîñòüþ ðîñòà ÷èñëåííîñòè íàñåëåíèÿ. Íàøå èññëåäîâàíèå ñòðåìèòñÿ óñòàíîâèòü,
êàêèì äîëæåí áûòü ðîñò íàñåëåíèÿ, êîòîðûé áûë áû ñîâìåñòèì ñ ïîñòîÿííûì ïîòðåáëåíèåì íà
÷åëîâåêà, ó÷èòûâàÿ ýêçîãåííûé ïîñòîÿííûé óðîâåíü òåõíè÷åñêîãî ïðîãðåññà è ñáåðåæåíèÿ, ëèíåéíûå
îòíîñèòåëüíî åãî âàëîâîãî ïðîäóêòà.
4 Ýòà ôîðìà èíâåñòèðîâàíèÿ ðåñóðñíîé ðåíòû âûáðàíà, ÷òîáû îáåñïå÷èòü q̇ = 0. Îáùèé óðîâåíü
s áóäåò áëèçîê ê β + γ , êàê â ñëó÷àå, êîãäà ê îñíîâíûì ñáåðåæåíèÿì ïðèáàâëÿþòñÿ äîáàâî÷íûå
ñáåðåæåíèÿ
13
ðåíòû
5
) ïðèìåò âèä
βq = k̇ +
è
αq
q̇ ṙ
−
=
q r
k
Îáà ýòè óñëîâèÿ ïîäðàçóìåâàþò, ÷òî
q̇
íîâêîé â
q
=θ
+ α kk̇
θk
α
ýòî ïðàâèëî Õîòåëëèíãà.
q̇ = 0 (Ýòî ìîæåò áûòü ïîëó÷åíî ïðîñòîé ïîäñòà-
+ β ṙr ). Ïðîäîëæèì ðàññìàòðèâàòü
c
ïðîïîðöèîíàëüíî
q,
åñëè íàñ
èíòåðåñóåò ïîñòîÿííîå ïîòðåáëåíèå íà äóøó íàñåëåíèÿ â ýòîé ýêîíîìèêå. Ýòî âåäåò
ê äîáàâî÷íûì ñáåðåæåíèÿì
γq = k[n + δ] −
ñ âåëè÷èíîé
s
â
sQ = K̇ + δK
è
s = β + γ.
θk
α
Ïîòðåáëåíèå íà äóøó íàñåëåíèÿ áóäåò
òàêæå ïðîïîðöèîíàëüíî òåêóùåìó âûïóñêó è, ñëåäîâàòåëüíî, ïîñòîÿííî. Çíà÷èò, ðîñò
÷èñëåííîñòè íàñåëåíèÿ äîëæåí óäîâëåòâîðÿòü óðàâíåíèþ
Ṅ
γ h αq i
δα − θ
n=
=
−
N
α k
α
êîòîðîå èíòåãðèðóåòñÿ äî
θ
s
γ
N (t) = N0 e[ α { β }−δ]t (k(t)) β
ãäå
k(t) = k0 e−(θ/α)/t +
αβq 6
. Âèäèì, ÷òî
θ
k(t)
â ïðåäåëå ñòðåìèòñÿ ê êîíñòàíòå. Ñëå-
äîâàòåëüíî, ÷èñëåííîñòü íàñåëåíèÿ óâåëè÷èâàåòñÿ â ïðåäåëå, åñëè
θ s
{ }
α β
− δ > 0.
Â
èçâåñòíîé ñòåïåíè, òåõíè÷åñêèé ïðîãðåññ çäåñü äîâîëüíî âûñîê, ÷òîáû ïîääåðæèâàòü
ýêîíîìèêó ñ ïîñòîÿííûì ïîòðåáëåíèåì íà ÷åëîâåêà è ðîñòîì ÷èñëåííîñòè íàñåëåíèÿ.
Äëÿ äèíàìèêè èñïîëüçîâàíèÿ íåôòè ó íàñ åñòü óðàâíåíèå
Ṙ
αq Ṅ
−[α − γ] h αq i
δα − θ
=−
+
=
−
R
k
N
α
k
α
êîòîðîå èíòåãðèðóåòñÿ
θ s−α
R(t) = R0 y 1/β e[ α { β }−δ]t (k(t))s/β .
Òàêæå
K̇
βQ Ṅ
[β + γ] h αy i
=
+
=
− δ.
K
K
N
α
k
5 θk
α - ýòî ñáåðåæåíèÿ, íàïðàâëåííûå íà ïîâûøåíèå ñòàðîãî k äî ýôôåêòèâíîñòè íàñòîÿùåãî
k . Ïî àíàëîãèè, íàïîìíèì, ÷òî nk - ýòî ñáåðåæåíèÿ äëÿ òîãî, ÷òîáû ó K ïîÿâèëàñü âîçìîæíîñòü
ñîõðàíèòü òåêóùåå k , êîãäà ðîñò ðàáî÷åé ñèëû îòðàæàåòñÿ íà óìåíüøåíèè êàïèòàëîâîîðóæåííîñòè
òðóäà.
6 Ìû òàêæå èìååì r(t) = r [k(t)]−(α/β) e−(θ/β)t , âûïîëíþùåå òðåáîâàíèå q̇ = 0
0
14
êîòîðîå èíòåãðèðóåòñÿ äî
θ s
K(t) = N0 e[ α { β }−δ]t (k(t))s/β .
Ñöåíàðèé, ñîâìåñòèìûé ñ êîíå÷íûì íà÷àëüíûì êîëè÷åñòâîì íåôòè òðåáóåò
h
θ s−α
α β
i
−δ <
0, ÷òîáû R(t) ñòðåìèëîñü ê 0 ïðè t → ∞. Ïîëó÷èëè âåðõíþþ è íèæíþþ ãðàíèöó òàêèõ
çíà÷åíèé
δ,
ïðè êîòîðûõ ýêîíîìèêà ðàçâèâàåòñÿ è ÿâëÿåòñÿ îñóùåñòâèìîé, à èìåííî
θ s−α
θ s
{
} < δ < { }.
α
β
α β
Èíòóèòèâíîå îáúÿñíåíèå çäåñü ñîñòîèò â òîì, ÷òî äîñòàòî÷íî âûñîêîå çíà÷åíèå îáåñöåíèâàíèÿ
δ
(ò.å.
h
θ s−α
α β
i
− δ < 0)
âûçûâàåò òîðìîæåíèå ýêîíîìèêè, ÷òîáû ñîõðàíèòü
äîëãîâðåìåííîå ïðîèçâîäñòâî â ïðåäåëàõ äàííîãî íà÷àëüíîãî çàïàñà ðåñóðñà, íî ñëèøêîì âûñîêèå çíà÷åíèÿ
δ
(ò.å.
δ > αθ { βs }) èñêëþ÷àþò âîçìîæíîñòü ñîâìåñòèìîñòè ðîñòà
âûïóñêà ïðîäóêöèè ñ óâåëè÷åíèåì ÷èñëåííîñòè íàñåëåíèÿ, êàê ïðîòèâîïîñòàâëåíèå
åãî ñîêðàùåíèþ.
Ñòèãëèö [1974] ðàññìîòðåë âàðèàíò ýòîé ìîäåëè, ïðè ïîñòîÿííîì
èíòåðåñîâàëî, êàêàÿ âåëè÷èíà
n
n
è
δ = 0.
Åãî
ñîâìåñòèìà ñ êîíå÷íûì íà÷àëüíûì çàïàñîì íåôòè,
ôèêñèðîâàííûì óðîâíåì ýêçîãåííîãî òåõíè÷åñêîãî ïðîãðåññà è àñèìïòîòè÷åñêè ïîñòîÿííûìè ïîòðåáëåíèåì íà äóøó íàñåëåíèÿ è èíâåñòèöèîííûì ïðàâèëîì. Íàøà öåëü
äðóãàÿ. Íàñ èíòåðåñóåò âèä
N (t),
ïðè ïîñòîÿííîì ïîòðåáëåíèè íà äóøó íàñåëåíèÿ,
ôèêñèðîâàííîì óðîâíå ëèíåéíûõ ñáåðåæåíèé è ïîñòîÿííûõ çíà÷åíèÿõ
ïîñëåäíåå îçíà÷àåò ýêçîãåííûé òåõíè÷åñêèé ïðîãðåññ. Ìû íàøëè
δ
è
θ,
ãäå
N (t) â ÿâíîì âèäå, â
îòëè÷èå îò ïîèñêà ðàçëè÷íûõ çíà÷åíèé èíâàðèàíòíîãî ïàðàìåòðà ðîñòà ÷èñëåííîñòè
íàñåëåíèÿ
6
n.
Çàòðàòû íà äîáû÷ó â áàçîâîé ìîäåëè Ñîëîó
Ñîëîó è Âîí [1975] èññëåäîâàëè çàòðàòû íà äîáû÷ó â ñëó÷àå íåâîçîáíîâëÿåìûõ ðåñóðñîâ. Ìîäåëü, êîòîðóþ îíè ïðåäëîæèëè, áûëà áàçîâîé ìîäåëüþ Ñîëîó [1974], â êîòîðîé
Q = F (K, R)−C −aR, ãäå a - ýòî ïîñòîÿííûå çàòðàòû íà äîáû÷ó åäèíèöû ðåñóðñà.Îíè
íå ðåøèëè ýòó ìîäåëü, ðàâíî êàê è Ñàòî è Êèì [2002], êîòîðûå òàêæå èññëåäîâàëè
âîïðîñ â äðóãîì êîíòåêñòå. Ìû ïðèâîäèì ðåøåíèå ýòîé ìîäåëè íèæå.
7
R - ïîòîê íåôòè, èñïîëüçóåìîé â òåêóùèé ìîìåíò, R(t) = −Ṡ(t), ãäå S(t) - îñòàâøååñÿ
êîëè÷åñòâî ðåñóðñà. ×èñëåííîñòü íàñåëåíèÿ ïîëàãàåòñÿ íåèçìåííîé â ýòîì èññëåäîâà-
7 Àëåêñåé
×åâÿêîâ ïðåäîñòàâèë ñâîþ ïîìîùü ïðè íàïèñàíèè ýòîé ãëàâû.
15
8
íèè.
Ìû èñïîëüçóåì ïðîèçâîäñòâåííóþ ôóíêöèþ Êîááà-Äóãëàñà,
F (K, R) := K α Rβ ,
0 < α, β < 1.
Ìîäåëü Ñîëîó-Âîíà ñîäåðæèò ñëåäóþùóþ ôóíêöèþ ñáåðåæåíèé è Ïðàâèëî Õîòåëëèíãà:
d
K = R(FR − a),
dt
(1)
d
(FR − a) = FK (FR − a)
dt
Çäåñü
a > 0
(2)
- ýòî ïàðàìåòð çàòðàò íà äîáû÷ó, êîòîðûé äåëàåò çàäà÷ó îòëè÷íîé îò
áàçîâîé ìîäåëè Ñîëîó.
Äëÿ ïðîèçâîäñòâåííîé ôóíêöèè Êîááà-Äóãëàñà ñîõðàíÿåìîå
êîëè÷åñòâî â ìîäåëè Ñîëîó-Âîíà ïðèíèìàåò ôîðìó
c = (1 − β)K α Rβ = const.
Ñëåäîâàòåëüíî, ìû ñðàçó ïîëó÷èëè çàâèñèìîñòü ôóíêöèé
(3)
K(t)
è
R(t):
α
R = R0 K − β ,
(4)
1
ãäå
R0 = (c/(1 − β)) β > 0
ïîñòîÿííî. ×òîáû ïîëó÷èòü ïîëíîå ðåøåíèå
(K(t), R(t)),
îñòàåòñÿ ðåøèòü òîëüêî îäíî óðàâíåíèå. Ïîäñòàâëÿåì âûðàæåíèå (4) â óðàâíåíèå äëÿ
ñáåðåæåíèé
d
K
dt
= (FR − a)R,
òîãäà
α
K̇ = βR0β − aR0 K(t)− β .
(5)
Óðàâíåíèå (5) ÿâëÿåòñÿ óðàâíåíèåì ñ ðàçäåëÿþùèìèñÿ ïåðåìåííûìè è åãî ìîæíî
ðåøèòü â êâàäðàòóðàõ. Ìû ïîëó÷èëè ñëåäóþùèé ðåçóëüòàò.
(K, R) = (K(t), R(t))
Òî÷íîå íåÿâíîå ðåøåíèå
òàìè íà äîáû÷ó
a 6= 0
ìîäåëè Ñîëîó-Âîíà (1), (2) ñ çàòðà-
è ïðîèçâîäñòâåííîé ôóíêöèåé Êîááà-Äóãëàñà èìååò âèä:
Z
K(t)
K0
dK1
βR0β
α
− aR0 K1 (t)− β
= t,
(6)
α
R(t) = R0 (K(t))− β .
 (6)
äëÿ
K1
- ýòî ïåðåìåííàÿ èíòåãðèðîâàíèÿ, à
K(t), R(t)
ïðè
t=0
K0 > 0
(7)
êîíñòàíòà. Íà÷àëüíûå óñëîâèÿ
ñîîòâåòñòâåííî ðàâíû
K(0) = K0 ,
−α
R(0) = R0 K0 β .
8 Äàñãóïòà
è Õèë [1979; ñòð. 305] ïîíèìàëè, ÷òî ìîäåëü Ñîëîó [1974] ìîæíî óñîâåðøåíñòâîâàòü,
÷òîáû äîïóñòèòü òàêèå äîïîëíèòåëüíûå ñáåðåæåíèÿ, ÷òî ñîâîêóïíîå ïîòðåáëåíèå ñìîæåò
óâåëè÷èâàòüñÿ âå÷íî, äàæå ïðè óñëîâèè ñóùåñòâåííîãî äëÿ ïðîèçâîäñòâà ðåñóðñà, çàïàñ êîòîðîãî
êîíå÷åí.
16
Î÷åâèäíî, ïðè
a=0
ðåøåíèå (6), (7) ñòàíîâèòñÿ õîðîøî çíàêîìûì ðåøåíèåì Ñîëîó.
Ìû ïðèøëè ê äðóãèì âîçìîæíûì ñöåíàðèÿì äëÿ ìîäåëè Ñîëîó-Âîíà. Ìû íà÷àëè ñ
àíàëèçà óðàâíåíèÿ (5). Ñ ýòîãî ìîìåíòà ïîëàãàåì
0 < β < α, êàê â ìîäåëè Ñîëîó.9 Ìû
óâèäèì, ÷òî ïîâåäåíèå ðåøåíèÿ ìîäåëè Ñîëîó-Âîíà ñóùåñòâåííî çàâèñèò îò îòíîøåíèÿ
ìåæäó ïàðàìåòðàìè çàäà÷è
a, α , β
è íà÷àëüíûõ óñëîâèé
R0 , K0 .
Âîçíèêàþò òðè
ðàçëè÷íûõ ñëó÷àÿ
1. K0 > K0∗ ;
2. K0 = K0∗ ;
3. K0 < K0∗ ;
ãäå
K0∗ =
a 1−β
R
β 0
Èç ïðàâîé ÷àñòè óðàâíåíèÿ (5), òàê êàê
dK(t)
dt
>0
t=0
â íà÷àëüíûé ìîìåíò
dK(t)
âî âòîðîì ñëó÷àå
dt
Ñëó÷àé 1.
=0
 ýòîì ñëó÷àå
äëÿ âñåõ
K0 >
α
β
αβ
.
> 1,
(8)
ëåãêî âèäåòü, ÷òî â ïåðâîì ñëó÷àå
è äàëåå â êàæäûé ìîìåíò âðåìåíè. Àíàëîãè÷íî,
t;
â òðåòüåì -
a 1−β
R
β 0
αβ
è
dK(t)
dt
dK(t)
dt
<0
> 0
äëÿ âñåõ
äëÿ âñåõ
t.
t.
Òîëüêî â ýòîì
a → 0 âîçìîæåí è ðåøåíèå (K, R) ñòðåìèòñÿ ê ðåøåíèþ ìîäåëè Ñîëîó.
Ðåøåíèÿ (K(t), R(t)), ñîîòâåòñòâóþùèå ïåðâîìó ñëó÷àþ, ñãåíåðèðîâàíû ÷èñëåííî äëÿ
ñëó÷àå ïðåäåë
α = 0.6, β = 0.3, R0 = 1
âñåõ ýòèõ çíà÷åíèé
÷àñòíîñòè, ïðè
K0 K0∗
è
K0 = 1.826
ìû èìååì
a = 0
ïðè çíà÷åíèÿõ
a = (0, 0.4, 0.9, 0.99, 1).
Äëÿ
K0∗ (óñëîâèå ïåðâîãî ñëó÷àÿ âûïîëíÿåòñÿ). Â
K0 >
ðåøåíèå Ñîëîó, ïðè
a = 1
ðåøåíèå áëèçêî ê êðèòè÷åñêîìó:
≈ 1.8257.
Ñëó÷àé 2. Âòîðîé ñëó÷àé, êàê è òðåòèé, ÿâëÿþòñÿ íîâûìè ïî ñðàâíåíèþ ñ ïðèâû÷íîé
ìîäåëüþ Ñîëîó. Âî âòîðîì ñëó÷àå ïîñòîÿííîå ðåøåíèå îáåñïå÷èâàåòñÿ òåì, ÷òî
K(0) = K0∗ = const,
α
R(t) = R0 (K0∗ )− β = const.
(9)
Ýòî ñîîòâåòñòâóåò çàñòîéíîé ýêîíîìèêå áåç èíâåñòèðîâàíèÿ, çàâèñÿùåé öåëèêîì îò
ðàçðàáîòêè äîñòóïíîãî íåâîçîáíîâëÿåìîãî ðåñóðñà, è ìîæåò îïèñûâàòü ðåàëüíîñòü
òîëüêî äëÿ êîíå÷íîãî âðåìåíè
t < T,
êîãäà öåíà äîáû÷è
a
íå ìåíÿåòñÿ.
Ñëó÷àé 3. Ýòîò ñëó÷àé õàðàêòåðèçóåòñÿ óìåíüøåíèåì ðàçìåðà êàïèòàëà èç-çà âûñîêîé ñòîèìîñòè äîáû÷è íåôòè
9 Íàøå
a,
÷òî ïðèâîäèò ê êðàõó ýêîíîìèêè çà êîíå÷íîå âðåìÿ.
èññëåäîâàíèå îñòàåòñÿ ñïðàâåäëèâûì è äëÿ ñëó÷àÿ α + β > 1.
17
 òðåòüåì ñëó÷àå ðåøåíèå ìîæåòü áûòü âû÷èñëåíî ñ ïîìîùüþ ñïåöèàëüíîé ôóíêöèè
Ëåð÷à
Φ(z, s, b). Óðàâíåíèå (6) òîãäà ïðèìåò ôîðìó, â êîòîðîé K(t) âñå åùå âûðàæåíî
íåÿâíî.
R0β αt
Ôóíêöèÿ Ëåð÷à
ñëó÷àå:
= K0 Φ
K0
K0∗
αβ
β
, 1, −
α
!
− K(t)Φ
K(t)
K0∗
αβ
β
, 1, −
α
!
.
(10)
Φ(z, s, b) - ýòî ñòåïåííîé ðÿä10 , ñõîäÿùèéñÿ ïðè |z| < 1 (êàê â äàííîì
K0 , K(t) < K0∗ )
è
b = − αβ 6= 0, −1, −2,. . . (âñåãäà
âåðíî äëÿ
0 < β < α).
K(t) è R(t) äëÿ ðàçëè÷íûõ íà÷àëüíûõ óñëîâèé áûëè ñãåíåðèðîâàíû
îãðàíè÷èâàþùèõ ïàðàìåòðîâ α = 0.6, β = 0.3, R0 = 1. Äëÿ ýòîãî âûáîðà
Ïðèìåðû êðèâûõ
äëÿ
a=1
K0∗
≈ 1.8257.
è
Ìû âçÿëè íåñêîëüêî ðàçëè÷íûõ çíà÷åíèé íà÷àëüíîãî êàïèòàëà, äëÿ
êàæäîãî èç ñëó÷àåâ 1, 2 è 3:
K0 = (3.1, 2, 1.71, K0∗ , 1.8, 1.6).
Ñëåäñòâèåì ÿâëÿåòñÿ
êðàõ ýêîíîìèêè.
Ñïèñîê ëèòåðàòóðû
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10 Φ(z, s, b)
=
zn
s
n=0 (n + b)
∞
P
18
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19
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