PROOF FOR A CASE WHERE DISCOUNTING ADVANCES THE DOOMSDAY. T. C. Koopmans

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1 PROOF FOR A CASE WHERE DISCOUNTING ADVANCES THE DOOMSDAY T. C. Koopmans January 1974 WP-74-6 Working Papers are no inended for disribuion ouside of IIASA, and are solely for discussion and informaion purposes. The views expressed are hose of he auhor, and do no necessarily reflec hose of IIASA.

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3 PROOF FOR A CASE WHERE ADVANCES THE DOOMSDAY DISCOUNTING by Tjalling C. Koopmans* In a previous paper (Koopmans [1973J), I considered some problems of "opimal" consumpion ~ over ime of an exhausible resource of known finie oal availabiliy R. In one of he cases sudied, consumpion of a minimum amoun of he resource is assumed o be essenial o human life, in such a way ha all life ceases upon is exhausion a ime T. Assuming a consan populaion unil ha ime, and denoing by r he posiive minimum consumpion level needed for survival of ha populaion, he survival period T is consrained by (1) o < T < R/r _ T Here equaliy (T=T) can be aained only by consuming a he minimum level (r=~) a all imes, 0 ~ < T. However, opimaliy is defined in erms of maximizaion of he inegral over ime of discouned fuure uiliy levels, (2) *This research was sared a he Cowles Foundaion for Research in Economics a Yale Universiy, New Haven, Conn., USA, wih he suppor of he Naional Science Foundaion and he Ford Foundaion, and compleed a he Inernaional Insiue for Applied Sysems Analysis in Laxenburg, Ausria. I am indebed o John Casi for valuable commens.

4 2. where p is a discoun rae, p ~ 0, applied in coninuous ime o he uiliy flow v(r ) arising a any ime from a consumpion flow r of he resource. The uiliy flow funcion v(r) is defined for r ~!, is wice coninuously differeniable and saisfies (3a,b,c,d) v'(r) > 0, v"(r) < for r >!' v(!) = 0, lim v'(r) = 00 r~r Tha is, v(r) is (a) sricly increasing and (b) sricly concave. The sipulaion (c) anchors he uiliy scale. Some such anchoring, hough no neces~arilyhe given one, is needed whenever populaion size is a decision variable. The las requiremen (d) simplifies a sep in he proof, and can be secured if needed by a disorion of v(r) in a neighborhood of r ha does no affec he soluion. The paper referred o gives an inuiive argumen for he following Theorem: For each p ~ here exiss a unique opimal pah r = r, < ~ ~ T p ' maximizin~ (2) subjec o I(4a) (4) (4b) r is a coninuous funcion on [O,TJ,

5 3. For p = 0, he opimal pah (;' io '$ < TO) is defined by,..,..,.. (Sa) r = r, a consan, for 0 < < TO, (S) (Sb) vcr) = rv' (r), (Sc),..", rt O = R For p > 0 i is defined by (6) (6a),.. -p,.. -pt,.. e v' (r ) = e Pv'(r), o < < T p,,.. r as in (Sb), (6b) J~p,.. rd = R The diagram illusraes he soluion. For p = 0, (6) implies (5), and consumpion of he resource is consan during,.. s urvival. Is opimal level r is obained in (5b,c) by balancing he number of years of survival agains he consan level of uiliy flow ha he oal resource sock makes possible during survival. Since ;. ~ ' he opimum survival period TO is shorer han he maximum T defined by (1). For p >0, he opimal pah r follows a declining curve,.. given by (6a), which sars from a level r o such ha, when,.. resource exhausion brings life o a sop a ime = T p, he level r T = r is jus reached. p Since he decline is seeper when P is larger, he survival period is shorer, he larger is p - which explains he ile of his noe.

6 4. The inuiive argumen already referred o gives insigh ino ne heorem; he following proof esablishes is validiy. Proof: We firs consider pahs opimal under he added consrain of some arbirarily fixed value T = T* of T saisfying 0 < T* < T. Assume ha such a " T* - opimal" pah r exiss and ha (7) r > r + 0 for 0 ~ ~ T* and some 0 > 0 Then, if S is a coninuous funcion defined for 0 < < T* such ha (8 ), he pah (9) o < < T* is T*-feasible for 1 1 < 1 and saisfies V(p,T*,(r )) - V(p,T*,(r ) = I(lOa) T* (10) = Joe- p (v(r ) - v(r*))d = (lab) T* \ = EJ e-pv' (r)sd + R( ), a

7 5. where he remainder R( ) is of second order in. I is herefore a necessary condiion for he T*-opimaliy of r ha (11), say, because, if we had p ' # p '" 0 ~ '," ~ T*, we could by choosing S of one sign in a neighborhood in [O,T*] of ', S of he opposie sign in one of il and zero elsewhere while preserving (8) make he las member of (10) posiive for some wih 1 1 ~ 1. In he ligh of (3a,b), (11) jusifies our assumpion ha r is a coninuous funcion of. We now find ha r is consan for p = 0, sricly decreasin~ for p > O. Given r T *, say, he soluion r of (11) is uniquely deermined, and, for each, r is a sricly increasing differeniable funcion o f he glven. r * T *. Also, by (3d), lim IoT rd = ITo rd = T*r < ~r = R r T *-+.!: Whereas, for sufficienly large r T *, Therefore here is a unique number a* > r such ha he unique soluion r of (11) wih r * T = a* saisfies

8 6. T* (12) rd = R f o From here on r will denoe ha pah for he chosen T*. Noe ha his pah also saisfies (7). To prove he unique T*-opimaliy of r, le r be any T*-feasible pah such ha'r o r o for some OE[O,TJ. Then, by he coninuiy of r, r, r # r for all in some neighborhood or 0 f o in [0,T*]. By (3b), for all E[0, T*], v(r ) - v(r*) [<] (r - r*) v' (r*) ~ for c[:.] where T* = [O,T*] - T. Therefore, we have from (loa), (11), (4b) wih T = T*, and (12) ha V(p,T*,(r» - V(p,T*,(r» = < T* (r - r*)e-pv' (r*)d = o f = e-pt* v'(r ft* T **) (r - r*)d < 0 o Hence r is uniquely T*-opimal.

9 7 We now make T* a variable, wriing T insead of T* and r~ insead of r. Noe ha, for each, 0 ~ < T, r~ is a differeniable funcion of T for < T < T. Therefore =ro -p e T v(~' )d is a differeniable funcion of T for 0 < T < T, and by (1). Bu, by (12), o = dt dr = r + lot dr T dt d T Therefore, Bu hen, from (Sb), since d~ (v(r) - rv' (r)) = -rv"(r) > 0 for r > 0, by (3b),

10 8. Finally, since 0 T' T' < T < T' < T implies r T, ~ r T for Thus, which V T reaches is unique maximum for ha value T p T " r T = r. of T for This esablishes he second par of he heorem. par follows by specializaion when p = o. The firs REFERENCE Koopmans, T.C., "Some observaions on 'opimal' economic growh and exhausible resources", in Bos, Linnemann and de Wolff, Ed~, Economic Srucure and Developmen, essays in honour of Jan Tinbergen, Holland PUblishing Co., 1973, pp

11 9. o I' ()(l rc:v" <ro II (Yl