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.  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).  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 [1] Alvarez-Cuadrado, Francisco and Ngo Van Long [2007] A Mixed Bentham-Rawls Criterion for Intergenerational Equity: Theory and Implication typescript. [2] Asheim, Geir B., Wolfgang Buchholz, John M. Hartwick, Tapan Mitra and CeesWithagen [2007] Constant Saving Rates and Quasi-arithmetic Population Growth under Exhaustible Resource Constraints, Journal of Environmental Economics and Man- agement, 53, 2, pp. 213-239. [3] Asheim, Geir B., Tapan Mitra and Bertil Tungodden [2006] Sustainable Recursive Social Welfare Functions, typescript. [4] Baranzini, Andrea and Francois Bourguignon [1995] Is Sustainable Growth Optimal? International Tax and Public Finance, 2, pp. 341-56. [5] Cairns, Robert and Ngo Van Long [2006] Maximin: A Direct Approach to Sustainability Environment and Development Economics, vol. 11, no. 3, June, pp. 275-300. [6] Cheviakov, Alexei F. and John M. Hartwick [2007] Constant Consumption with Exhaustible Resources: New Scenarios typescript, presented at the Canadian Economics Association meetings, Halifax, Nova Scotia, May, 2007. 10 Φ(z, s, b) = zn s n=0 (n + b) ∞ P 18 [7] Chichilnisky, Graciella [1996] An Axiomatic Approach to Sustainable Development, Social Choice and Welfare, 13, 3, pp. 231-257. [8] Dasgupta, Partha and Georey M. Heal [1979] Economic Theory and Exhaustible Resources, New York: Cambridge University Press. [9] Dixit, Avinash K., Peter Hammond, and Michael Hoel [1980] On Hartwick's Rule for Regular Maximin Paths of Capital Accumulation and Resource Depletion, Review of Economic Studies, 47, 3, pp. 551- 56. [10] Figuiere, Charles and Mabel Tidball [2006] Sustainable Exploitation of Natural Resource: a Satisfying Use of Chichilnisky's Criterion UMR LAMETA, Research Paper, Montpellier, France. [11] Hamilton, Kirk and John Hartwick [2005] Investing Exhaustible Resource Rents and the Path of Consumption, Canadian Journal of Economics, 38,2, pp.615-21. [12] Hamilton, Kirk and Cees Withagen [2006] Savings Growth and the Path of Utility Canadian Journal of Economics, forthcoming. [13] Hamilton, Kirk and David Ulph [1995] The Hartwick Rule in a Greenhouse World, unpublished manuscript, University College, London. [14] Hartwick, John M. [1977] Intergenerational Equity and the Investing of Rents from Exhaustible Resources American Economic Review, 66, pp. 253-56. [15] Hartwick, John M. [2004] Sustaining Periodic Motion and Maintaining Capital in Classical Mechanics Japan and the World Economy, vol. 16, no. 3, Special Issue August, pp. 337-58. [16] Leonard, D. and N. V. Long [1992] Optimal Control Theory and Static Optimization in Economics, Cambridge: Cambridge University Press. [17] Li, Chuan-Zhong and Karl-Gustaf Lofgren [2000] Renewable Resources and Economic Sustainability: A Dynamic Analysis with Heterogeneous Time Preferences Journal of Environmental Economics and Management, 40, 3, November, pp. 236-50. [18] Ludwig, Donald [1995] A Theory of Sustainable Harvesting SIAM Journal of Applied Mathematics, 2, April, pp. 564-75. [19] Martinet, Vincent and Gilles Rotillon [2007] Invariance in Growth Theory and Sustainable Development, Journal of Economic Dynamics and Control, in press. 19 [20] Nordhaus W.D. and . Boyer J. [2000] Warming the World: Economic models of global warming. Cambridge Massachusetts: MIT Press. [21] Sato, Ryuzo and Youngduk Kim [2002] Hartwick's Rule and Economic Conservation Laws Journal of Economic Dynamics and Control, vol. 26, no. 3, March, pp. 437-49. [22] Solow, Robert M. and F.Y. Wan [1975] Extraction Costs in the Theory of Exhaustible Resources Bell Journal of Economics, 7, 2, pp. 359-370. [23] Stollery, Kenneth R. [1998] Constant Utility Paths and Irreversible Global Warming, Canadian Journal of Economics, 31, 3, August, pp. 730-42. [24] Withagen, Cees and Geir B. Asheim [1998] Characterizing Sustainability: The Converse of Hartwick's Rule, Journal of Economic Dynamics and Control, 23, 1, September, pp. 159-65. 20