Lecture notes on the Theory of Non-renewable Resources

Transcription

1 1 Updaed EON4925 Resource economics, Spring 24 Olav Bjerkhol: Lecure noes on he heory of Non-renewable Resources 2. he Hoelling rule for prices of exhausible resources In LN-1 we found ha he resource ren increases by rae r, cf. relaion (1.4). his equilibrium law is in he lieraure commonly referred o as he Hoelling rule. In his frequenly cied work from 1931, discussing he developmens of prices in resource markes, Harold Hoelling arrived a a condiion for he marke price ha is formally idenical o (1.4). hus, a main conclusion in Hoelling's sudy was ha in equilibrium he resource ren (he ne price), defined as he difference beween he marke price of he resource and marginal exracion coss, mus increase a a rae equal o he rae of ineres. he underlying assumpion is again ha he producers in he marke possess exclusive righs o non-renewable naural resources. he only way of having a reurn on preserving he resource sock, is ha he ne price of he resource increases over ime. In order for he asse marke o be in equilibrium, he growh rae for he resource ren mus equal he opporuniy cos, i.e. he ineres rae or he reurn on invesmens. Hoelling (1931) also showed ha he compeiive equilibrium pah for he ne price coincide wih he condiions for opimal allocaion of he oal sock of he resource, o be shown below. he Hoelling rule derived from social opimizaion when uni coss are consan Formally, he Hoelling rule in is simples version can be derived in a way ha is quie similar o he soluion of Gray's problem. Le S denoe he oal resource sock in he economy, R is he oal exracion in he marke a ime, U(R ) is he uiliy of he consumpion of he resource a ime (in money erms). Exracion is assumed o be carried ou wih consan uni coss, b. he opimal exracion pah for he sociey as a whole is found by maximizing (2.1) r [ U( R ) br ] e d, R wih respec o R, over a possibly infinie ime horizon consrained by he condiion (2.2) R d = S [,]. Again, he maximizaion is Like in Gray's problem, he soluion o (2.1) and (2.2) can be found by using opimal conrol heory. he Hamilonian is r (2.3) HS (,, R, λ ) = [ UR ( ) bre ] λ R

2 2 According o he necessary condiions, he opimal exracion pah mus fulfil he following relaion: r (2.4) [ U ( R) b] e λ (= λ for R > ) he shadow price of he resource sock a ime, λ, is in he presen case, wih exracion coss no depending on he accumulaed producion, consan. We have / S =. For posiive exracion, (2.4) can hen be wrien as (2.5) U ( R ) b = λe Using ha in a marke economy, U ( R ) = p, i is seen ha (2.5) expresses he Hoelling rule: along he opimal pah he marginal ne price, which is idenical o he resource ren, should increase a he rae of discoun. he inerpreaion of he Hoelling condiion as a marke equilibrium condiion is similar o he case of he resource exracing firm: along he socially opimal exracion pah he owners of he resource socks are indifferen beween exracing and leaving he resource in he ground. If his arbirage condiion a some poin is no fulfilled, some agens will be able o increase heir profis by changing he speed of exracion or by aucioning he complee resource sock on he marke. In order o deermine he complee equilibrium pahs for he ne price and he level of exracion, i remains o solve for he iniial price p. We assume here ha he demand is such ha here exiss a choke price, p max, for which demand equals zero.he erminal condiion in his case yields (2.6) [ ] [ r max p be = p b] where p max is he inercep of he demand schedule. his equaion, ogeher wih he resource resricion (2.2) in he opimizaion problem and he demand equaion, deermine he iniial values for he exracion level and he resource ren. onsisency beween socially opimal depleion and a compeiive soluion Hoelling (1931) showed no only ha here is a socially opimal depleion profile, bu also ha here is a compeiive soluion consisen wih he socially opimal depleion profile. We approach his by solving he social planning problem for he opimal depleion profile for n idenical naural resource firms and hen showing ha he opimaliy problem for one of hem (essenially Gray s problem) gives opimaliy condiions consisen wih hose of he social planning problem. We hus assume ha here are n idenical naural resource firms. he socially opimal rae of depleion is conceived as he rae ha maximizes he gross surplus (consumers surplus plus producers surplus) derived from he demand funcion given as p(.). he amoun depleed (per uni of ime) from each firm a ime is R while he amoun of unexraced resource in each firm a ime is S. he cos of exracion is given for each firm by he funcion b(r, S ) wih b R >, b R > and b S <. he rae of discoun is, r, he same for he social planning problem and in he compeiive soluion. r

3 3 (2.7) nr r max [ p( xdx ) nbr (, S)] e d R ns& = nr, S = S, S = S, R he sae variable in his problem is he amoun of remaining resource ns, while he conrol variable is he rae of depleion nr. he Hamilonian of his problem is wih adjoined price (shadow price) λ - as follows: (2.8) nr r λ = H (, ns, nr, ) [ p( x) dx nb( R, S )] e λnr Assume ha S * and R * solves he problem. hen i follows from he maximum principle ha R * maximizes he Hamilonian, which implies ha when coninuiy, differeniabiliy and concaviy of he Hamilonian hold, we have r (2.9) = [ pnr ( ) b R( R, S)] e λ = ( nr ) Furhermore, he rae of change of he shadow price is given by (2.1) & r λ = = b S( R, S) e ( ns ) Alernaively, for he curren value problem he Hamilonian is nr (2.11) H (, ns, nr, µ ) = p( x) dx nb( R, S ) µ nr he firs order condiion is now (2.12) = pnr ( ) b ( R, S) µ = ( nr ) While he rae of change of he shadow price in curren values is given by (2.13) & µ rµ = = b S( R, S ) ( ns ) hen, he remainder of Hoelling s proof is o consider he profi maximizing problem of one of he n firms facing a given price pah (in fac he price pah resuling from he given demand curve and he depleion of he n firms). We hus have he problem: (2.14) r max [ p R b( R, S )] e d R S& = R, S = S, S = S, R he curren value Hamilonian of his problem is (2.15) H (, S, R, κ ) = p R b( R, S ) κ R

4 4 Assume ha S * and R * solves he problem. hen i follows from he maximum principle ha R * maximizes he Hamilonian, i.e. (2.16) = p b R( R, S) κ = R Furhermore, he rae of change of he adjoined price in curren values is given by (2.17) & κ rκ = = b S( R, S ) S As he price p is assumed o be p(nr * ), i.e. n idenical firms producing he same amoun, hese condiions are exacly he same as in he social planning problem and in equilibrium κ is equal o µ. Will he correc equilibrium be reached in he marke? Hoelling hus derived condiions for he exisence of equilibrium in markes for depleable resources. In general, equilibrium heory does no ell how equilibrium is brough abou. Neiher does he heory give guidelines for he marke developmen in cases where prices and quaniies for some reason have reached values ouside he equilibrium pah. More specifically, i is worh noing ha due o he dynamic naure of he problem, he soluion demands significanly more from he marke agens han in he radiional saic marke equilibrium. Sricly, marke equilibrium in he Hoelling model requires he exisence of fuure markes. Individual resource owners mus have perfec foresigh as o he evoluion of ne price of he resource, ( p b ). I is no sufficien ha he producers have assessed correcly he rae of growh in reurns on differen asses (i.e. a Hoelling rule prevails); oal exracion mus also saisfy he end poin condiions (2.2) and (2.6). If he iniial price is oo high, here is oo much conservaion in early years and a par of he resource sock will be lef in he ground a he ime when he demand schedule reaches he choke price, p max. If, on he oher hand, p is oo low, here is over-exploiaion iniially, and he resource sock will be depleed oo early. here is also he possibiliy ha in he laer case, he pressure agains he resource sock, a a higher level of demand han along he opimal pah, will induce he price of he resource o increase a a rae higher han wha follows from he r % rule. onsan, bu differen exracion coss Implici in he derivaion of he simples Hoelling model above was an assumpion of marke consising of price aking firms, wih homogeneous and idenical resource socks and cos srucure. In pracice, exracion coss may vary considerably beween differen areas and producers. In he more realisic case of heerogeneous producers and socks, an ineresing quesion relaes o he succession of depleion of he various resource deposis. Assume ha he supply side of he marke consiss of wo resource owners, wih uni coss b 1 and b 2, respecively. From he derivaion of he Hoelling rule above i should be clear ha in he presen case marke equilibrium mus fulfill he following relaions:

5 5 (2.18) p b e r λ ( i = 1,2) i i where λ i is he shadow price relaed o he resource sock of producer i. As before, he sric inequaliy in (2.18) prevails for R i =. Wih consan uni cos funcions, marke equilibrium requires sequenial exracion of each resource sock. Assume oherwise ha for some ime inerval we have simulaneously R, R. his implies ha he following equaion holds: 1 2 r r (2.19) b1+ e λ1 = b2 + e λ2 b1 b2 = ( λ2 λ1) e r While he lef hand side in he lower par of (2.19) is a consan, he righ hand side grows a a rae equal o r. hus, by assuming simulaneous exracion for some, we have arrived a a conradicion, he premise mus be false, and we conclude ha given consan, bu differen uni coss, marke equilibrium requires sequenial exracion of each resource sock. In a firs ime inerval, [, 1 ], he marke is supplied by he cheapes resource (say producer group 1). he corresponding resource ren mus increase a he rae of discoun. In he same ime period, producer group 2 wih he higher coss reaps a reurn on leaving his resource sock in he ground ha exceeds he ineres rae, since we have (2.2) p& p& p b p b > = 2 1 I should be noed ha price pah is coninuous, bu wih a "kink" in 1, when producer 2 "akes over" he marke. A he same ime, he resource ren jumps o a lower level, before coninuing o increase a a rae equal o r. Wih (consan) higher uni coss of producion, here is no way producer 2 can escape his fall in he resource ren; he bes he can achieve is o pospone exracion unil 1. Wih a more general cos srucure, he resul ha socks of differen qualiy should be exraced sequenially, does no hold, however. If for insance he exracion coss vary wih curren producion, i.e. bi= bi(r i ), he complee marke equilibrium pah will generally involve periods of overlapping exracion. o wha exen exracion of differen socks will be simulaneous, depend on he flexibiliy in he cos srucure of he producers and on he size of he differen deposis. Marke equilibrium requires ha resource rens, i.e. he difference beween he marke price and marginal coss, grows a rae r for all exracing firms. Wih flexibiliy in he cos srucure, producers may equalize heir reurns by operaing a differen levels of exracion. Formally, he condiion for simulaneous exracion is ha he sum of marginal cos and he resource ren should be equalized beween producers, which in he wo-deposi case is equivalen o he following relaion: (2.21) b ( R1 ) ( 2 ) ( 2 1) r b R = λ λ e I may be noed ha in general exracion of individual resource deposis may ake place a differen marginal producion coss. r

6 6 Wih variable exracion coss, he inerpreaion of "high" and "low" cos deposis may no be unique. From (2.21) i is clear ha he level of exracion, in addiion o he cos srucure, also depends on he size of he iniial resource sock hrough is shadow price, λ. he larger is he deposi, he lower is he value of he shadow price, and he higher is he speed of exracion. here will be a endency ha "high cos producers" ener he marke laer (a higher prices), or resric heir exracion o a larger exen, han producers wih large socks and/or low producion coss. Summing up he equilibrium soluion and he price pah in he Hoelling model presened above is remarkably simple. I is perhaps his simpliciy ha has led many auhors and commenaors o compleely rejec he heory by referring o lack of empirical evidence 1. In paricular, i is hard o poin a acual resource markes wih observed exponenial growh for he ne price. learly, he relevance of he Hoelling heory should be quesioned. However, i is imporan o be aware of ha since he pioneering works of Gray and Hoelling he heory of exhausible resources has been furher developed and a number of complicaing elemens have been added. Wih more realisic assumpions regarding echnology and marke srucure, he classical and clearcu r%-rule for he developmen of he resource ren does no survive. One imporan case was sudied in he previous secion, wih exracion coss depending on remaining reserves. Exending he Hoelling model similarly, yields exacly he same conclusion, namely ha he resource ren in equilibrium increases a a rae less han he rae of ineres, cf. equaion (1.17). Moreover, we may exend he marke model furher, e.g. by assuming ha boh he cos funcion and he demand funcion shif over ime, due o changes in oher variables, such as echnological improvemens and income growh. Such changes severely complicae he soluion of he model. However, i has been shown ha wih he demand and cos funcions shifing over ime, he equilibrium pah for he resource ren may even fall over some ime inervals (see e.g. Farzin (1992)). Finally, in a world involving imperfec compeiion and uncerainy, as will be discussed below, he "simple and smooh" Hoelling resuls may be significanly affeced. learly, as a consequence of increased complexiy, i also becomes more difficul o underake empirical ess of he heory. Hoelling versus Ricardian ren Harold Hoelling and David Ricardo are probably he mos widely quoed economiss wihin he field of resource economics. Boh were sudying markes for naural resources, and each ended up by concluding ha in marke equilibrium here will be excess profis - or rens - in producion. However, he concep of Ricardian ren is principally very differen from he resource or Hoelling ren concep ha is involved in he heory of exhausible resources. Ricardo did no focus explicily on resource scarciy. Raher, he Ricardian view is ha naural resources such as land consis of heerogeneous unis. Given ha he bes qualiy pieces of land are culivaed firs, as demand increases lower 1 An early empirical sudy of resource markes was Barne and Morse (1963). Among more recen analysis are Smih (1979) and Slade (1982).

7 7 qualiy land is brough ino producion. Assuming ha he marke price has o equal uni coss of he marginal land, he price will exceed coss of all more producive unis of he resource. hus, he laer will earn Ricardian ren. he marke equilibrium for a good involving Ricardian ren can be illusraed by a supply curve, which is equal o he marginal coss for he marke as a whole. I increases wih increasing volume since increased producion implies use of resource unis of descending qualiy. Marke equilibrium is found in he inersecion beween he supply and demand curve. On he marginal uni of producion, here is no pure profi, while Ricardian ren is earned on all inramarginal resource unis. urning o a marke for an exhausible resource, we have already seen ha he marke equilibrium may involve some elemen of Ricardian ren. More specifically, if he exracion cos funcion of a resource owner is of he form described by relaion (1.15), Ricardian ren will be presen. Hoelling ren or resource ren may come on op of Ricardian ren. As discussed above, his is due o resource scarciy in a sric sense, so ha here is an absolue limi o accumulaed resource exracion. As a resul, in equilibrium he marke price equals marginal coss plus he Hoelling ren, q, i.e. (2.22) p = b ( R, S ) R + q In accordance wih he resuls derived above, a he chosen level of exracion he marke price exceeds marginal coss. hus, in addiion o he Ricardian ren indicaed producion now requires an addiional reurn, since curren exracion ineviably is a he expense of fuure producion. he resource (Hoelling) ren is he difference beween he marke price and marginal coss. In a compeiive marke for an ordinary good all producers ha supply he marke operae a idenical marginal coss. In a marke for an exhausible resource, on he oher hand, marginal exracion coss a he chosen levels of exracion may in general differ, since individual resource rens vary. From he discussions above we know ha when producion coss depend solely on curren exracion, he resource ren in equilibrium will rise over ime a a rae equal o he rae of ineres. However, in he more general case where he remaining reserve sock is included in he cos funcion, his resul is alered; he rae of increase in he resource 2 ren will in general be lower han he ineres rae (see (1.17)). Since b/ R S <, he cos funcion shifs upwards as he sock is depleed, and he resource ren may herefore even diminish over ime. he fac ha fuure marginal coss increase as a direc consequence of curren producion aciviy implies ha he disincion beween Ricardian and Hoelling rens may no be quie clear cu. By inegraing (2.13) we find a decomposiion of he resource ren, cf. Lasserre (1991, p.1): (2.23) giving r( s ) r( s ) = S s s ( & µ rµ ) e ds b ( R, S ) e ds

8 8 (2.24) r ( ) r ( s) br ( s, Ss) µ = µ e e ds S his relaion expresses he resource ren a ime as he sum of he discouned ren a ime plus he cumulaed presen value of all fuure effecs on exracion cos of exracing he marginal uni of he resource a ime. Because he presence of S in he cos funcion should reflec differen qualiies of he sock, he second componen may be inerpreed as a Ricardian componen of he (resource) ren, while he (pure) scarciy ren is conained in he firs par of equaion (2.24). he implicaion of adoping his "dynamic" inerpreaion of Ricardian ren is ha a relaively smaller share of he excess profi a ime is classified as being of he Hoelling ype. Sill, common erminology is o denoe he omplee value of µ in (2.24) as he Hoelling ren. c