Stock-Dependent Extraction Costs and the Technological Efliciency of Resource Depletion

Sinn, Hans-We ne A icle S ock-Dependen Ex ac ion Cos s and he Technological E liciency o Resou ce Deple ion Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik P o ided in Coope a ion wi h: Duncke & Humblo , Be lin Sugges ed Ci a ion: Sinn, Hans-We ne (1981) : S ock-Dependen Ex ac ion Cos s and he Technological E liciency o Resou ce Deple ion, Zei sch i ü Wi scha s- und Sozialwissenscha en (ZWS) - Vie eljah essch i de Gesellscha ü Wi scha s- und Sozialwissenscha en, Ve ein ü Socialpoli ik, ISSN 0342-1783, Duncke & Humblo , Be lin, Vol. 101, Iss. 5, pp. 507-517, h ps://doi.o g/10.3790/schm.101.5.507 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/291503 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ S ock-Dependen Ex ac ion Cos s and he Technological E iiciency o Resou ce Deple ion By Hans-We ne Sinn* A g ea numbe o s udies on op imal esou ce ex ac ion in he p esence o ex ac ion cos s ha e been ca ied ou , and he e also exis some s udies whe e he ealis ic assump ion o s ock-dependen ex ac ion cos s is made. Howe e , he li e a u e is no e y explici abou pu e e iciency condi ions in he Pa e o sense, i.e., condi ions ha a e independen o special assump- ions abou in e empo al p e e ences and ma ke s uc u es. The p esen pape add esses he e iciency p oblem explici ly and, in pa icula , ies o emo e some con usion emaining in a ecen pape by Heal. 1. In oduc ion In his lec u e gi en o he 1979 con e ence o he Ve ein ü Social- poli ik1, Geo ey Heal p esen ed an e iciency condi ion o in e - empo al esou ce ex ac ion in he p esence o ex ac ion cos s. This pape illus a es ha HeaVs condi ion is allacious and co ec s he mis ake. In addi ion, i demons a es he compa ibili y be ween he co ec ed condi ion and he op imali y condi ions de i ed in Rawlsian and u ili a ian amewo ks by Solow/Wan (1976) and Heal (1976), espec i ely. 2. HeaFs E iciency Condi ion Conside an economy p oducing a single composi e commodi y. A each poin in ime ou pu Y is gi en by (1) Y = G(K,R, ) ; Gk> > 0 , GKK> GRR < 0 » and esou ce ex ac ion cos F in e ms o he composi e commodi y is2 * This pape was w i en in associa ion wi h he Sonde o schungs- be eich 5, P ojec II/B. I g a e ully acknowledge commen s by John McMil- lan, Ho s Siebe and Wol gang Vog . Remaining sho comings a e en i ely mine. 1 Heal (1980). 2 Unlike (2), Heal assumes a ma ginal ex ac ion cos unc ion C (S, R). I e e o a o al cos unc ion because his seems mo e sys ema ic and a oid he 32 Zei sch i ü Wi scha e- und Sozialwissenscha en 1981/5 OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.101.5.507 | Gene a ed on 2023-04-04 11:58:54 508 Hans-We ne Sinn (2) F = X (S, R) , Xs<0 , XR > 0 , XBB>0 , X îS<0 , whe e K is he capi al s ock, R he a e o esou ce ex ac ion and S he s ock o he deple able esou ce. Ou pu is comple ely used o consump ion C, in es men in he capi al s ock I and ex ac ion cos F: (3) y = C + I + F . The a e o change o e ime3 in he capi al s ock is (4) K = I and he a e o change in he s ock o he deple able esou ce is (5) S = -R. Toge he wi h he ini ial s ocks K0 and So (Ko, So > 0) he ime pa hs o I and R comple ely de e mine he in e empo al alloca ion pa e n in he economy. The ques ion is unde which condi ions hese ime pa hs cons i u e an in e empo ally e icien alloca ion o esou ces. The alloca ion is said o be e icien i he e is no ime in e al whe e i is possible o inc ease consump ion wi hou dec easing i a he same ime in ano he in e al. O cou se one could bypass he e iciency ques ion by adding o he abo e o mulas a well speci ied wel a e unc ional and calcula ing he op imal alloca ion explici y. Bu in iew o he di icul ies o in e gene a ional wel a e compa isons i seems use ul o sepa a e e iciency and dis ibu ion p oblems analogously o he p ocedu e in s a ic alloca ion heo y. symbol C because i is also used o consump ion. Bu o cou se he e does no R emain a subs an ial di e ence i we de ine X (S, R) == / C (S, u) du. In ligh o o he ime dependence o he p oduc ion unc ion i would seem mo e sys- ema ic o use an ex ac ion cos unc ion X (S, R, ). Bu his e sion would no be compa ible wi h Heal. Ou esul s would, howe e , s ill go h ough. Pe haps he unc ion X (S, R) should be called a ac o -inpu unc ion a he han a cos unc ion since he e a e no p ices in ol ed. On he o he hand, wi h ou pu Y as he numé ai e, he p ice by which we had o mul iply in o de o ha e a cos unc ion is uni y. Xs < 0 e lec s he assump ion ha he o de o ex ac ion om di e en deposi s is such ha he lowes cos deposi is ex ac ed i s , he o he s ollowing in s ic sequence. This assump ion by i sel can be shown o ollow om he equi emen o e iciency i he a e o e u n on capi al is s ic ly posi i e and i he ex ac ion indus y uses a pa o ou pu y as an inpu . C . Kemp/Long (1980 a, b, d) and Sinn (1981). 3 We de ine Z = dZ/d whe e is a ime index. OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.101.5.507 | Gene a ed on 2023-04-04 11:58:54 S ock-Dependen Ex ac ion Cos s 509 Among he condi ions necessa y o e iciency we should pa icula ly be in e es ed in a ma ginal condi ion ela ing o each o he he e ec s o capi al in es men and esou ce ex ac ion. Unde he absence o ex ac ion cos s Heal (p. 42 - 44) shows ha he g ow h a e o he ma ginal p oduc o he esou ce should be equal o he a e o e u n on capi al: (6) GK=.A-inGR . This condi ion has also been de i ed by Solow (1974) and S igli z (1974) in Rawlsian and u ili a ian amewo ks, and in a compe i i e economy i would au oma ically be sa is ied since i would hen be he same as he Ho elling ule wi h Gk as he ma ke a e o in e es and G as he ma ke p ice o he esou ce. I is no su p ising ha (6) does no hold any mo e i he e a e ex ac ion cos s. Wi hou p oo Heal claims (pp. 46, 48) o his case ha he ne ma ginal p oduc o he esou ce, i.e., i s ma ginal p oduc minus i s ma ginal ex ac ion cos , should change a a a e gi en by he a e o e u n on capi al: (7) GK=-^ln(GR-XR) . Fu he mo e he s esses (pp. 46, 80) ha (7) implies ha he g ow h a e o he ma ginal p oduc i i y o he esou ce will be equal o a weigh ed a e age o he e u n on capi al, Gk, and he a e o change o ma ginal ex ac ion cos s, Gjj / X^j X^ Xjj GR GR J XR GR (8) whe e he change in ma ginal ex ac ion cos s can i sel be explained by a change in he a e o ex ac ion and he s ock o he esou ce4: /q XR_ RR SR Al hough hese e iciency condi ions migh ha e some in ui i e appeal a i s glance, hey look suspicious i con as ed wi h a well- known ex ension o he Ho elling ule o he case o a compe i i e ma ke wi h posi i e ex ac ion cos s5: 4 In Heals pape his equa ion shows a ious yping e o s. 6 See Le ha i/Li ia an (1977) equ. (15), Dasgup a/Heal (1979) p. 169 and Kemp/Long (1980 c) equ. (15 b). C . also Pindyck (1978 a and b). In (1978 b) Pindyck allows o explo a ion cos s in addi ion o ex ac ion cos s, bu in 32* OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.101.5.507 | Gene a ed on 2023-04-04 11:58:54 510 Hans-We ne Sinn (10) T = P P He e deno es he ma ke a e o in e es and p he ma ke p ice o he ha es ed esou ce ne o ma ginal ex ac ion cos s which unde compe i i e condi ions equals he ne ma ginal p oduc i i y o he esou ce, GR — XR. Ob iously condi ion (10) is only compa ible wi h (7) i X8 = 0, i.e., i , in con as o HeaVs assump ion, ex ac ion cos s do no depend on he emaining s ock o he esou ce6. Thus, i (7) we e eally ue, he compe i i e economy would no ensu e he in e empo al e iciency o esou ce alloca ion. In he ligh o he undamen al heo em o s a ic wel a e heo y his appea s o be a a he s ange implica ion. And in ac condi ions (7) and consequen ly (8) a e w ong. I can easily be shown ha pu e echnological e iciency conside a ions equi e an ex ension o hese condi ions in a way o make hem compa ible wi h (10). The p oo is simila o HeaVs p oo o (6). S a ing wi h a gi en ime pa h o he economy we conduc ma ginal a ia ions in he con ol a iables wi hin a limi ed ime span wi hou howe e changing he in e empo al alloca ion elsewhe e. We can hen easily see which condi ions ha e o be sa is ied such ha no Pa e o imp o emen is possible. We assume o a while ha he economy ope a es in disc e e ime wi h pe iods o leng h 6>, <9 > 0, while he e e ence pe iod o de ining he low a iables is uni y. A poin in ime and he subsequen ime in e al belong oge he , such ha hey can be named by he same ime index. As he ime span o conduc ing he a ia ions we choose he pe iods and + 0 and assume ha The eason o dC = 0 is ha we wan o examine whe he an inc ease in second-pe iod consump ion is possible wi hou changing consump ion bo h pape s he excludes he possibili y ha ma ginal ex ac ion cos s depend on he speed o ex ac ion, R. The connec ion be ween (10) and his esul can easily be unde s ood a e ha ing ead his pape and compa ing ou equa ion (21) wi h equa ion (A.9) in (1978 a) and (9) in (1978 b) o he case o ze o ex- plo a ion cos s. ® C . Weins ein and Zeckhause (1975) esp. pp. 379-381. These au ho s de i e equa ion (7) unde he assump ion ha ex ac ion cos s depend only on he a e o ex ac ion al e na ely o he case o a ma ke equilib ium and a u ili a ian planning p oblem. 3. De i a ion o he Co ec Condi ion (11) dLC = dK = dK + 2Q=dS = dS + 2e = 0 . OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.101.5.507 | Gene a ed on 2023-04-04 11:58:54 S ock-Dependen Ex ac ion Cos s 511 in he i s pe iod. I i is possible o a gi en in e empo al alloca ion, hen his alloca ion is ine icien . Cons an s ocks a he beginning and he end o he wo-pe iod in e al a e equi ed by ou assump ion ha he alloca ion is o be unchanged ou side his in e al. Since o mulas (1) - (3) imply (12) [(G (K , R , T) - X (S , R ) — Ix — CT] = 0 , = , +6, he ollowing wo equa ions ha e o be sa is ied o he a ia ions ca ied ou : (13) cLI = (GR -XR )dR (14) dC + Q = GK + QdK + Q - dI + 9 - XS +QdS + e + (GR + e ~ xR + e) dR + e • Now, K + e = K +I 0 and S + e = S - R 0; hence dK + & = dl 0 and dS +e = —cLR 6. Fu he mo e (11) implies ha dl = —dl + e and dR + g = —dR . Thus, (13) and (14) can be combined o (15) dC + 9= [(1 + G GK + Q) (GR - XR ) + exS +Q-(GR +Q-xR +e)]dR . This o mula shows by how much consump ion in he second pe iod can ise, i capi al in es men and esou ce ex ac ion a e inc eased in a way ha keeps i s -pe iod consump ion unchanged. Ob iously, i he a ia ion is conduc ed a ound an e icien ime pa h, we ha e dC = 0 by de ini ion. Hence i is eadily appa en om pa h, we ha e dC +e= 0 by de ini ion. Hence i is eadily appa en om (15) ha ^ (GR +e ~ xR + e^ ~ ~ xs +e (GR -xRje GR -xR is a necessa y condi ion o an e icien in e empo al alloca ion. So a , he a gumen has been ca ied ou o 0 > 0. Bu by choosing 0 su icien ly small we can app oach he con inuous case as closely as we wish. Acco dingly he condi ion can hen also be w i en as (17) In (G - XR) - d GR — XR OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.101.5.507 | Gene a ed on 2023-04-04 11:58:54 512 Hans-We ne Sinn o , equi alen y, as (18) + + GR GR I XR R UR whe e all a iables e e o he same poin in ime. Equa ion (17) is, as we expec ed, indeed analogous o he compe i i e condi ion (10) p o ing he in e empo al e iciency o he compe i i e ma ke allo- ca ion. Since ex ac ion cos s ise wi h a all in he s ock o he esou ce, Xs < 0, (17) implies ha , unlike Heals con en ion [equ. (7)], he ne ma ginal p oduc o he esou ce should change by a a e less han he a e o e u n on capi al. In addi ion, a compa ison be ween (8) and (18) shows ha he g ow h a e o he g oss ma ginal p oduc i i y o he esou ce is no jus a weigh ed a e age o he a e o e u n on capi al and he g ow h a e o ma ginal ex ac ion cos s, bu smalle han his by XS/GR. An in ui i e explana ion o ou esul can be gi en as ollows: The e a e wo ools o shi ing consump ion om he i s pe iod o he second. The i s is an inc ease in in es men . I one uni o consump ion is subs i u ed by capi al in es men , hen second-pe iod consump ion can be inc eased by one uni plus he a e o e u n o e u n on capi al. The second ool is a educ ion in he a e o esou ce ex ac ion. Suppose esou ce ex ac ion in he i s pe iod alls by an amoun gi en by he ecip ocal alue o he ne ma ginal p oduc i i y o he esou ce, such ha consump ion in his pe iod is educed by a uni . Then, second- pe iod consump ion can be inc eased by a uni plus he pe cen age inc ease in he ne ma ginal p oduc i i y plus, and his is he new elemen , he dec ease in second-pe iod ex ac ion cos s e ec ed by he a ailabili y o a highe esou ce s ock7. I he in e empo al alloca ion is o be Pa e o op imal hen he possible inc ease in second-pe iod consump ion mus be he same o each ool, o only hen i is impossible o al e bo h esou ce ex ac ion and in es men in a way ha keeps i s -pe iod consump ion cons an , bu inc eases consump ion in he second pe iod. 7 To p o ide u he in ui ion o he esul , suppose, be o e he a ia ion is conduc ed, he e is a cons an uni ex ac ion cos wi hin each o he wo pe iods conside ed ha depends only he esou ce s ock a ailable a he beginning o he co esponding pe iod. Then a esou ce uni he ex ac ion o which is shi ed om he i s pe iod o he second can be ex ac ed a a cos ha is below he p e ious uni (and ma ginal) ex ac ion cos in he second pe iod by he amoun — XS +e. Hence, i he ex ac ion o 1 /(GR — XR ) o he esou ce is shi ed om he i s pe iod o he second, he inc ease in ex ac ion cos s in he second pe iod would be o e es ima ed by — XS + Q I {GR — X lP i i would be aken o be 1 /(GR — XR ) imes he uni ex ac ion cos in he second pe iod be o e he a ia ion. OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.101.5.507 | Gene a ed on 2023-04-04 11:58:54 S ock-Dependen Ex ac ion Cos s 513 Equa ions (17) and (18) ha e been de i ed o a e y gene al ex ac- ion cos unc ion. I is howe e wo h o conside he special, al hough s ill plausible, case8 (19) X(S,R)=R i (S), g'< 0, whe e uni ex ac ion cos s g depend only on he emaining s ock o he esou ce. Since he simpli ied ex ac ion cos unc ion implies ha XR = g (S),XR = gS = -g'R andXs = Rg', (17) and (18) can be educed o and in his case. Hence, he absolu e a e o change o he g oss ma ginal ;p oduc i i y o he esou ce ela i e o he ne ma ginal p oduc i i y equals he a e o e u n o capi al, and he ela i e a e o change o he g oss ma ginal p oduc i i y is a sha e o he a e o e u n on capi al whe e he sha e is gi en by he a io o ne o g oss ma ginal p oduc i i y. The eade should con as (20) and (21) wi h HeaVs equa ions (7) and (8). 4. Compa ison wi h he U ili a ian Op imum As suppo o he allacious equa ion (8) Heal ci es his 1976 pape in he Bell Jou nal9. The pape does no di ec ly add ess he e iciency p oblem since he analysis is ca ied ou in a u ili a ian amewo k. Bu e iciency is a necessa y condi ion o a u ili a ian op imum. Thus we should elabo a e b ie ly upon he ela ionship o ou esul s. The p oblem s udied in he Bell pape is o ind op imali y condi ions unde he u ili a ian aim 00 (22) max J u{C^e-^d o s A unc ion o his ype has equen ly been used. Pe haps he i s p omo e was Go don (1954). Recen examples a e Heal (1976), Pindyck (1978 a and b) and Solow/Wan (1976). • Heal (1980), 80: "I ... can only men ion b ie ly ha he cha ac e iza ion I ga e o e icien p ice pa hs — p ice changes equal o a weigh ed a e age o in e es a es and ma ginal cos changes — can also be shown o hold o a esou ce a ailable in a ange o deposi s o di e en quali ies. This is shown in an a icle o mysel in he Bell Jou nal 1976." C . Heal (1976). OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.101.5.507 | Gene a ed on 2023-04-04 11:58:54 514 Hans-We ne Sinn whe e u is a s ic ly conca e u ili y unc ion and d he a e o ime p e e ence. O he wise he model is (in he ele an aspec s) he same as he e. The ex ac ion cos unc ion is o ype (19). Heal shows ha he solu ion o his p oblem indeed p o ides an op imali y condi ion some- wha simila o (8): He e he g ow h a e o he ma ginal alue p oduc is shown o be a weigh ed a e age o he discoun a e and he ela i e change o he ou pu p ice in u ili y e ms, whe e he weigh s a e he same as hose in (8). The o mal simila i y is, howe e , meaningless. No e ha o a Hamil onian o he kind H = e~8 {u(C) + p [G (K, R) ! — g (S) R — C] + q(— R)} he equa ion + GK = d is a necessa y condi ion o an in e iou op imum. Thus (23) can be w i en as Ano he s udy in esou ce ex ac ion wi h s ock-dependen ex ac ion cos s is ha o Solow and Wan (1976). Inspi ed by Rawls' minimax ule hese au ho s examine he condi ions o maximizing he le el o a s eady, ime-in a ian low o consump ion. Since echnological e ici- ency is a necessa y condi ion o a Rawlsian op imum, we again should be able o demons a e he compa ibili y wi h ou esul s. Al hough o mally somewha di e en , he echnological assump ions o Solow and Wan a e hose o his pape wi h X (S, R) = R g (S)10. Thei app oach can be s a ed as ollows. The dual p oblem o maximizing consump ion o a gi en s ock o he esou ce is o minimize accu- mula ed esou ce ex ac ion o a gi en le el o consump ion C. Hence we (23) p = U'G , c = u'g . (24) which is he same as (21). 5. Compa ison wi h he Rawlsian Op imum oo (25) S. . io C . Solow/Wan (1976) n. 3. OPEN ACCESS | Licensed unde CC BY 4.0 | h ps://c ea i ecommons.o g/abou /cclicenses/ DOI h ps://doi.o g/10.3790/schm.101.5.507 | Gene a ed on 2023-04-04 11:58:54