Kirk Hamilton and Giovanni Ruta Accounting price of an exhaustible resource: response and extensions

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1 Kirk Hamilon and Giovanni Rua Accouning price of an exhausible resource: response and exensions Aricle (Acceped version) (Refereed) Original ciaion: Hamilon, Kirk and Rua, Giovanni (2016) Accouning price of an exhausible resource: response and exensions. Environmenal and Resource Economics. ISSN DOI: /s Springer Science+Business Media This version available a: hp://eprins.lse.ac.uk/66941/ Available in LSE Research Online: June 2016 LSE has developed LSE Research Online so ha users may access research oupu of he School. Copyrigh and Moral Righs for he papers on his sie are reained by he individual auhors and/or oher copyrigh owners. Users may download and/or prin one copy of any aricle(s) in LSE Research Online o faciliae heir privae sudy or for non-commercial research. You may no engage in furher disribuion of he maerial or use i for any profi-making aciviies or any commercial gain. You may freely disribue he URL (hp://eprins.lse.ac.uk) of he LSE Research Online websie. This documen is he auhor s final acceped version of he journal aricle. There may be differences beween his version and he published version. You are advised o consul he publisher s version if you wish o cie from i.

2 Accouning Price of an Exhausible Resource Response and Exensions 1 Kirk Hamilon and Giovanni Rua 2 Absrac. Wei (2015) presens a novel derivaion of he accouning price for an exhausible resource in a non-opimal economy subjec o an allocaion mechanism. We show ha Wei (2015) and Hamilon and Rua (2009) are in fac employing differen and muually exclusive allocaion mechanisms for he economy, and his explains he differences beween he respecive accouning prices. Because accouning prices mus be defined subjec o he allocaion mechanism for he economy, he prices derived in he wo papers are equally valid wihin heir respecive allocaion domains. Furher analysis shows ha if here is declining marginal produc of facors, a Harwick invesmen rule for he model economy (se invesmen jus equal o depleion, valued a he accouning price) will lead o declining consumpion for he Wei (2015) accouning price, and increasing consumpion for he Hamilon and Rua (2009) accouning price. This resul is exended o consider he accouning sandards recommended in he UN SEEA(2012), as well as accouning for environmenal exernaliies from resource use. Keywords: exhausible resource, accouning price, resource depleion, Harwick Rule, environmenal exernaliy JEL codes: Q01 Q32 Q53 Q56 1 Inroducion Hamilon and Rua (2009) se ou a model of a simple exracive economy wih he aim of esablishing he accouning price of an exhausible resource (his appears in Secion 5 of heir paper). If S is a finie sock of an exhausible resource and N is is economic value (he presen value of he rens generaed over he finie lifeime of he resource), hen he accouning price of he resource is equal o is marginal social value as measured by N S. As Dasgupa and Mäler (2000) show, accouning prices are he key o measuring susainabiliy in a non-opimal economy subjec o an allocaion mechanism. 3 1 The commens of peer reviewers are acknowledged wih hanks. The usual caveas apply. 2 Granham Research Insiue, LSE, and World Bank respecively. Conac k.e.hamilon@verizon.ne 3 Roughly speaking, an allocaion mechanism is an algorihm or se of rules ha maps iniial endowmens of asses ino a unique fuure pah for he economy. The allocaion need no be opimal. 1

3 Wei (2015) offers an alernaive way o define he accouning price for his exracive economy and suggess ha he price derived by Hamilon and Rua is incorrec. The purpose of his noe is o show ha Hamilon and Rua (2009) and Wei (2015) are employing differen and muually exclusive allocaion mechanisms, which explains why he accouning prices differ beween he wo papers. More imporanly, his noe exends he wo papers o esablish wheher he alernaive accouning prices can suppor a version of he Harwick Rule (Harwick 1977) in he non-opimal exracive economy. Comparisons wih he accouning sandard esablished in he UN Sysem of Environmenal-Economic Accoun (SEEA 2012) are derived, as well as an exension of SEEA (2012) o deal wih polluion exernaliies. We firs clarify he allocaion mechanisms in Wei (2015) and Hamilon and Rua (2009). Secion 3 explores he Harwick Rule under he alernaive assumpions made in he wo papers. Secion 4 relaes Hamilon s (2015) analysis of he SEEA (2012) o he measuremen of susainabiliy in Hamilon and Rua (2009), and exends he analysis o include a polluion exernaliy. The final secion concludes. 2 Alernaive allocaion mechanisms For an exracive economy wih iniial socks of produced asses K(0) and exhausible resource S(0), an allocaion mechanism α defines a mapping α: {K(0), S(0)} {K(), S(), R(), C()} =0 Over his possibly infinie ime horizon he pahs of produced capial K, resource sock S, resource exracion R and consumpion C are uniquely defined by α. 4 The allocaion mechanism is feasible if K() 0, S() 0. 4 In wha follows, all variables are assumed o be funcions of ime, unless oherwise saed. 2

4 To make his more concree, i is useful o ake a subse of α ha concerns only he evoluion of he resource sock and is economic value N. The basic accouning rule for he resource is, S = R (1) In Wei (2015) he allocaion mechanism α W given resource sock S() consiss of: (W1) Choose an exracion pah such ha R() = R W is consan (W2) Assume ha he uni resource ren is also consan, n() = n W Under hese assumpions i follows, as shown in Wei (2015), ha he dae of exhausion T of he resource sock is a funcion of S() given by, T(S()) = S() + (2) R W and he value of he resource sock, given consan discoun rae r is, N W = T(S()) n WR W e r(z ) dz (3) As Wei esablishes, under his allocaion mechanism he accouning price is measured as, q NW S = n W e r(t(s()) ) (4) In Hamilon and Rua (2009) he allocaion mechanism α HR given resource sock S() is defined as, (HR1) Choose a fixed exhausion ime T (HR2) Choose consan R HR o saisfy S() = R HR dz T 3

5 (HR3) Assume a consan uni resource ren n() = n HR From assumpion (HR2) i follows ha R HR = S() T (5) Under hese assumpions he value of he resource sock is N HR T = n HRR HR e r(z ) dz = n HR S() T e r(z ) dz T (6) As Hamilon and Rua show, he accouning price under his allocaion mechanism is p NHR S = n HR T 1 r (1 e r(t ) ) (7) As expressions (3) and (6) show, he effec of he alernaive allocaion mehods is o make he resource value N an explici funcion of he resource sock S. I is worh exploring he inuiion behind hese resuls for alernaive accouning prices. Under α W an incremen o he resource sock S mus resul in an exension of he exhausion dae o T(S()) + T, owing o he fixed quaniy of resource exracion a each poin in ime (W1). The resul is ha he change in resource value N is effecively he presen value of he las uni of he resource exraced his is he inerpreaion of accouning price q as seen in expression (4). Conversely, under α HR he exhausion dae is fixed (HR1). As a resul a small incremen S in he resource sock does no affec he accouning price p (expression 7) and he change in he value of he sock is given by N = p S. As should be clear, hese allocaion mechanisms are muually exclusive. You eiher choose a fixed quaniy of exracion a he ouse, which makes he exhausion ime a funcion of he sock of resource (Wei), or you choose a fixed exhausion ime a he 4

6 ouse, which makes he fixed quaniy exraced a funcion of he sock of resource (Hamilon and Rua). As should also be clear, if boh allocaion mechanisms use he same consan uni ren, say n, hen i is possible o choose exracion R = R W = R HR such ha he exhausion ime is T = S() +. In his case he value of he resource sock is he same under eiher R W approach, N W = N HR, bu he wo accouning prices q and p are sill disinc because hey are defined subjec o heir corresponding allocaion mechanisms. Assuming equal values of he resource sock under each approach permis us o analyze he relaionship beween he wo accouning prices. For Wei (2015) we have, qs = qr = N (8) qs = N (T(S()) ) (9) For Hamilon and Rua (2009) we have, ps = N (10) ps + p S = N (11) Combining expressions (8) and (11) we derive, p = q + p S R (12) Expression (8) is derived in Wei (2015), while expression (11) is derived in Hamilon and Rua (2009). As expressions (8) and (9) sugges, accouning price q only gives economically meaningful values when muliplied by a flow raher han a sock. Expression (11) embodies boh he real change in resource wealh defined by Hamilon 5

7 and Rua, ps = pr, and he corresponding capial gain linked o resource exracion, p S. Expression (12) is paricularly helpful, because i says ha he Hamilon and Rua accouning price p is equal o he Wei accouning price q plus he capial gain per uni of exracion. I is worh noing ha in valuing exhausible resources, naional accounans generally calculae running averages for recen annual quaniies of resource exraced and forecas ha he mos recen average is he consan quaniy ha will be exraced up o he poin of exhausion of he resource. The ime o exhausion T is hen calculaed as he raio of economic reserves o he forecas annual quaniy exraced. This approach nowihsanding, he implicaion of he foregoing analysis is ha he accounan mus assume eiher ha he quaniy exraced is fixed or ha he erminal dae is fixed in deermining he accouning price for he resource. The nex secion shows ha his choice has imporan consequences when applying policies for susainabiliy in an exracive economy. 3 Alernaive implemenaions of he Harwick Rule Harwick (1977) esablishes ha a closed exracive economy wih fixed echnology, consan populaion and a finie resource ha is a necessary inpu o producion can enjoy consan consumpion over an infinie horizon if invesmen in produced capial jus equals he value of resource depleion a each poin in ime. 5 The economy is susainable under his rule. In he Harwick model, resource depleion equals he marginal renal value of he resource, and he marginal renal rae is assumed o increase a he rae of ineres (he Hoelling Rule). Here we wish o explore he obvious generalizaion of he Harwick Rule o he models of Wei (2015) and Hamilon and Rua (2009). The basic idea is o se invesmen in 5 In addiion, he elasiciy of subsiuion beween produced capial and he exhausible resource mus be equal o 1 in he Harwick (1977) model, and he elasiciy of oupu wih respec o produced capial mus be greaer han he elasiciy for he resource. 6

8 produced capial equal o he value of resource depleion derived from he respecive allocaion mechanisms. 6 We generalize he models o an economy wih a neoclassical producion funcion such ha, F(K, R) = C + K + f(r) (13) F K > 0, F KK < 0, F R > 0, F RR < 0 (14) The producion funcion exhibis declining marginal produc wih respec o facors, and f(r) is he exracion cos funcion for he resource. The ineres rae F K is no assumed o be consan over ime, which has implicaions for he resuls which follow. Firs, he general expression for he value of he resource sock becomes, T N = n R e z F K (τ)dτ dx (15) Under Wei s allocaion mechanism he accouning price is herefore, q = n e T(S()) F K (τ)dτ Under Hamilon and Rua s allocaion mechanism he accouning price is, p = n T 1 F K () (1 e F T K (τ)dτ ) Finally, from expression (15) we derive he insananeous change in he value of he resource sock as a resul of resource exracion, T F K (τ)dτ N = n R e (16) 6 The analysis in his secion is a special case (assuming consan exracion and consan uni ren) of he more general resuls in Hamilon (2015). We make comparisons wih Hamilon (2015) in secion 4 below. 7

9 We can now exend he analysis in he preceding secion by inroducing a Harwick invesmen rule ino he allocaion mechanisms of Wei (2015) and Hamilon and Rua (2009). Expression (13) indicaes ha oupu can be consumed, invesed or spen on resource exracion. Since allocaion rules W1 and HR2 specify he pah for resource exracion, hey deermine exracion coss f(r). The inroducion of an invesmen rule ino he respecive allocaion mechanisms herefore deermines he fuure pah of consumpion, yielding a unique fuure pah for he economy as a whole. For he model of Wei he invesmen rule is (W3) K = qr = N Since R is assumed o be consan (W1), i follows from expression (13) ha, K = F K K C = d d (N ) = d d (F KN n R ) = F KN F K N And herefore, subsiuing (W3), C = F KN (17) For he model of Hamilon and Rua he invesmen rule is (HR4) K = pr = N S R From (HR4) we have K = N S+NR S s R = F KN n R S+NR S 2 R Given R consan (HR2), i follows ha, 8

10 N C = F K K K = F K R F KN n R S+NR R S S 2 Collecing erms his reduces o, C = (n S N) R 2 S 2 (18) The resul for he Wei allocaion mechanism, expression (17), is negaive under he assumpion of declining marginal produc of facors. 7 Invesing qr a each poin in ime resuls in declining consumpion. Conversely, since n S is he undiscouned value of resource flows over he life of he resource, and N is he discouned presen value of hese resource flows, expression (18) esablishes ha he Hamilon and Rua allocaion mechanism yields growing consumpion a each poin in ime. 4 Exensions: he UN SEEA (2012) and exernaliies from resource use Having esablished how he Wei (2015) and Hamilon and Rua (2009) models perform under a Harwick invesmen rule, hese resuls can be made more general by considering how he Hamilon and Rua model relaes o he Sysem of Environmenal-Economic Accouning (SEEA 2012). A furher exension analyzes how o accoun for an environmenal exernaliy from resource use, combining he assumpions of fixed exhausion ime T and consan resource exracion R wih SEEA (2012) accouning convenions. An imporan sep in sandardizing environmenal and resource accouning pracice was he adopion of he SEEA (2012) as a UN saisical sandard. Hamilon (2015) analyzes 7 To be precise, (W3) implies ha K > 0 since N < 0 (expression 16), and so F K = F KK K < 0. 9

11 he applicaion of SEEA (2012) o he problem of measuring he susainabiliy of developmen, using he generalized Harwick rule of Hamilon and Harwick (2005). 8 Expression (10) above implies ha he uni value of depleion of an exhausible resource in Hamilon and Rua (2009) is given by p N. This corresponds exacly o he sandard S se ou in SEEA (2012), and so i is useful o compare he measuremen of susainabiliy presened in secion 3 o ha derived in Hamilon (2015) for he SEEA. There are wo differences beween he models of Hamilon and Rua (2009) and Hamilon (2015). Firs, Hamilon (2015) assumes ha resource exracion declines a a consan rae, R R = φ; his compares wih he consan level of exracion R associaed wih a fixed exhausion ime T in Hamilon and Rua. Second, Hamilon (2015) assumes ha marginal resource exracion coss are consan, which implies ha uni rens will vary wih he resource price F R ; Hamilon and Rua assume ha uni rens n are consan. Hamilon (2015) shows ha, given he assumpions abou declining exracion and consan marginal exracion coss, seing genuine saving G K pr equal o 0 (he sandard Harwick Rule) implies ha consumpion is insananeously consan. If, insead, here are increasing marginal resource exracion coss hen he sandard Harwick Rule implies ha C is proporional o he (posiive) inframarginal rens on exracion; he sandard Harwick Rule implies increasing consumpion in his insance. Expression (18) above shows ha he sandard Harwick Rule in he Hamilon and Rua (2009) model also implies increasing consumpion. An imporan advanage eviden in expression (18), however, is ha i is simple o measure he amoun by which consumpion increases under he sandard Harwick Rule. In conras, o measure his increase in he Hamilon (2015) model wih increasing marginal exracion coss requires 8 Hamilon and Harwick (2005) derive a generalized Harwick Rule, showing ha C = F K G G for genuine saving G. The sandard Harwick Rule is a special case for G = G = 0; a pah where genuine saving is idenically 0 a each poin in ime will exhibi consan consumpion. 10

12 specificaion of he exracion cos funcion, knowledge ha may no be forhcoming in many circumsances. An environmenal exernaliy from resource use If we assume ha healhfulness H is a sock ha conribues o wellbeing, hen uiliy can be measured as U = U(C, H). In wha follows we examine he quesion of measuring susainabiliy using SEEA (2012) convenions when exracion of he resource leads o healh damage measured as d(r ); formally, d(r ) is represened as a deducion from he sock of healh H, while exracion R is held consan for an assumed exhausion dae T. The opimal growh problem is o maximize V = T U(C, H) e ρ(s ) ds (19) subjec o accouning ideniy (13) and he following: H = d(r ) (20) S = R (21) If we hink of local air polluion as an example of an exernaliy associaed wih use of he resource in producion, hen d(r ) encompasses he whole sequence from resource use o polluan emission, dispersion, and human exposure, finally resuling in damage o healh. In wha follows we make he simplifying assumpion ha he marginal exracion cos for he resource f is consan, as is he marginal healh damage from resource use d. 9 We herefore relax he assumpion of consan uni resource rens in Wei (2015) and Hamilon and Rua (2009). 9 This simplificaion is no necessary, bu i sreamlines he derivaion of he main resuls. 11

13 The basic growh heory leading o he expression for genuine saving in an opimal exracive economy wih a polluion exernaliy is derived in he Annex. Taking he resuls for he opimal economy as a model, we derive he generalized Harwick Rule for he non-opimal economy under he assumpions of consan resource exracion and SEEA accouning convenions for measuring resource depleion. A key parameer derived in he Annex is z, he shadow price of a uni of healh damage, which equals he presen value of he insananeous willingness o pay for a uni of healhfulness U H /U C, as seen in Annex expression (A4). Because of he exernaliy, he shadow price of he resource has o include he value of marginal damage o healh from resource use, as seen in he Annex expression (A3). Because marginal exracion coss and marginal healh damages are consan, he value of he resource sock N is equal o he presen value of ne resource rens, T N = (F R f zd )R e z F K (τ)dτ dz This corresponds o expression (15) in secion 3. Genuine saving in his economy is given by, G K pr zd (22) Tha is, ne saving equals invesmen in produced capial, minus depleion of he resource sock, minus he value of damage o he sock of healh from he polluion exernaliy. Here p N/S, per SEEA (2012) convenions. The generalized Harwick Rule is derived as follows: G = K z d p R = F K K C (F K z U H U C ) d (F K p (F R f zd )) R S + N R S 2 F K G = F K K F K pr F K zd 12

14 Collecing erms as in secion 3 yields he generalized Harwick rule: C U H U C d = F K G G + ((F R f zd )S N) R2 S 2 (23) In his expression C U H U C d is he dollar-valued insananeous change in wellbeing (aking accoun of he healh damages from resource use). The erm ((F R f zd )S N) is he difference beween he value of he resource sock in he opimal economy 10 and N, which is he value of he resource sock in he non-opimal economy being modeled. The final erm in expression (23) is herefore posiive. Under he sandard Harwick Rule (G = G = 0) wellbeing is herefore increasing, closely paralleling secion 3 wih he excepion ha he healh exernaliy has o be included. If genuine saving is non-negaive and growing a a rae less han he ineres rae F K a each poin in ime, hen wellbeing is everywhere increasing, which implies ha social welfare V is also everywhere increasing. 5 Conclusions Wei s (2015) conribuion o he lieraure is o show ha here is an alernaive allocaion mechanism ha can be applied o he model economy of Hamilon and Rua (2009), and ha his mechanism leads o a novel accouning price for he resource. The foregoing analysis shows ha he allocaion mechanisms employed in each paper are disinc and he accouning prices derived are equally valid wihin he domain defined by heir respecive allocaion mechanisms. In his response we have derived he relaionship beween he accouning prices in he wo papers. Neiher Wei (2015) nor Hamilon and Rua (2009) go on o esablish he behavior of he economy under a Harwick invesmen rule derived from heir respecive accouning 10 Recall ha he Hoelling rule, expression (A3), applies in he opimal economy. As a resul he growh in uni rens is compleely offse by he discoun rae. 13

15 prices for he resource. This response o Wei fills he gap by showing ha, under sandard assumpions regarding declining marginal produc of facors, he Wei accouning price resuls in declining consumpion a each poin in ime, while he Hamilon and Rua accouning price leads o increasing consumpion. If he policy goal for he economy is o achieve consan consumpion, hen he Wei approach is clearly under-invesing while he Hamilon and Rua approach is over-invesing. If he policy goal is non-declining consumpion a each poin in ime, hen he Hamilon and Rua approach is a sufficien condiion o reach he goal. Looking forward, i is clear ha much more rigor is required in he applicaion of Dasgupa and Mäler s (2000) concep of accouning prices for asses. In paricular, i needs o be made clear for models such as he above ha N S S N(S; α). Tha is, accouning prices can only be measured wih respec o he assumed allocaion mechanism α. And α needs o be fully specified. We also show ha he Hamilon and Rua (2009) model of exracion implicily employs he valuaion of resource depleion adoped in SEEA (2012). As a resul, here are srong parallels beween he sandard Harwick Rule, derived using he Hamilon and Rua (2009) assumpions, and he Hamilon (2015) analysis of he sandard Harwick Rule using SEEA (2012) accouning convenions. If resource use resuls in healh damages (for example from polluion emissions), hen we show ha under SEEA (2012) accouning convenions he generalized Harwick Rule has o accoun boh for insananeous healh damages and he reducion in resource rens resuling from he persisen loss of healhfulness associaed wih resource use. In his model he sandard Harwick Rule leads o increasing wellbeing. References 14

16 Dasgupa, P. and K.-G. Mäler (2000). Ne Naional Produc, Wealh, and Social Wellbeing. Environmen and Developmen Economics 5, pars 1&2, Hamilon, K. (2015). Measuring Susainabiliy in he UN Sysem of Environmenal- Economic Accouning. Environmenal and Resource Economics, 2015, DOI: /s y. Hamilon, K. and J.M. Harwick, (2005) Invesing Exhausible Resource Rens and he Pah of Consumpion, Canadian Journal of Economics, 38(2), Hamilon, K. and G. Rua (2009). Wealh Accouning, Exhausible Resources and Social Welfare. Environmenal and Resource Economics 42: Harwick, J.M. (1977). Inergeneraional Equiy and he Invesing of Rens from Exhausible Resources. American Economic Review 66, SEEA (2012). Sysem of Environmenal-Economic Accouning 2012: Cenral Framework. Unied Naions, European Commission, Food and Agriculure Organizaion of he Unied Naions, Inernaional Moneary Fund, Organisaion for Economic Cooperaion and Developmen, World Bank. Wei, T. (2015). Accouning Price of an Exhausible Resource: A Commen. Environmenal and Resource Economics 60: Annex. Opimal resource exracion wih a healh exernaliy For healh sock H, healh damage funcion d(r), uiliy U(C, H) and accumulaion equaions (13), (20) and (21), he objecive is o maximize, V = T U(C, H) e ρ(s ) ds (A1) 15

17 for consan pure rae of ime preference ρ. The Hamilonian funcion is given by, H = U + γ 1 K + γ 2 H + γ 3 S where he γ i are he corresponding shadow prices. From he firs order condiion on consumpion ( U C = 0) i follows ha γ 1 = U C, while he dynamic firs order condiion (γ 1 = ργ 1 H K ) on U C yields he sandard Ramsey equaion, F K = ρ ( U C U C ) (A2) Defining γ 2 U C z, where z is he value of a uni of he healh sock H, he firs order condiion on exracion yields, γ 3 = U C (F R f zd ) The dynamic firs order condiion for γ 3 herefore gives he Hoelling rule for his economy, d d (F R f zd ) F R f zd = F K (A3) Marginal rens on exracion herefore deduc he marginal damage o healh zd. Nex, he dyamic firs order condiion on γ 2 gives, d d (U Cz) = ρu C z U H and subsiuing (A2) yields, 16

18 z = F K z U H U C This differenial equaion has soluion, z = U H (s) e F K (τ)dτ ds U C s (A4) Genuine saving G is herefore derived from he Hamilonian funcion as, G = K zd(r) (F R f zd )R (A5) 17