Discontinuous Extraction of a Nonrenewable Resource

Transcription

1 Dscontnuous Extracton of a Nonrenewable Resource Erc Iksoon Im 1 Professor of Economcs Department of Economcs, Unversty of Hawa at Hlo, Hlo, Hawa Uayant hakravorty Professor of Economcs Department of Economcs, Unversty of entral Florda, rlando, Florda James Roumasset Professor of Economcs Department of Economcs, Unversty of Hawa at Manoa, Honolulu, Hawa Abstract hs paper examnes the sequence of optmal extracton of nonrenewable resources n the presence of multple demands. We provde condtons under whch extracton of a nonrenewable resource may be dscontnuous over the course of ts depleton. JEL classfcaton: Q3; Q4 Keywords: Backstop technology; Dynamc optmzaton; Energy resources; Herfndahl prncple; Multple demands 1 Erc Iksoon Im, 200 W. Kawl Steet, Hlo, Hawa ; ph: (808) ; fax: (808) ; emal: em@hawa.edu. We would lke to thank an anonymous referee for valuable comments.

2 1 1. Introducton A fundamental theorem of resource economcs s that when there s a sngle end-use, the optmal sequence for extractng deposts of a nonrenewable resource should be n the order of ther effectve unt-costs of extracton (e.g., Herfndahl (1967), Solow and Wan (1976), Lews (1982). 2 hs prncple has been generalzed to the case of multple enduses. o wt, resources wll be extracted n order of ther end-use specfc prces, defned as the sum of extracton costs, converson costs, transportaton costs, and n stu shadow prces (hakravorty and Krulce, 1994, henceforth K; and Gaudet, Moreaux and Salant, 2001, GMS). 3 By addng set-up costs to the basc model, GMS prove a vacllaton result: one resource or landfll ste may be temporarly abandoned n favor of a hgher cost resource/ste and utlzed agan at a later date. In ths paper, we show that dscontnuous extracton of a nonrenewable resource s possble, even wthout setup costs, and provde condtons under whch the dscontnuty occurs. We modfy the framework of K who consder two nonrenewable resources, ol () and coal () and two end-uses, electrcty (E) and transportaton (), each characterzed by ther own demand. We add a thrd backstop resource (B) wth an nfnte supply (e.g., solar power). 4 Whle the assumpton of a constant unt extracton cost n K s retaned for each resource (c, =,, B ), we specfy converson costs as both resource and 2 For exceptons, see Kemp and Long (1980) and Amgues et al. (1998). 3 he Kemp/Long and Amgues exceptons to the Herfndahl prncple are consstent wth ths more general prncple. 4 GMS show that at least three resources are needed for dscontnuous extracton.

3 2 demand-specfc ( z, =,, B; = E, ) so that the net cost of resource n demand s w = c + z. he planner chooses nstantaneous extracton rates of resource n demand, denoted q (t), n order to maxmze the dscounted socal surplus, W: q rt 1 W e D = ( ) ( ) ( ) ( ) ( ) 0 0 x dx c + z q t λ tq t dt, (1) subect to q ( t) 0; Q ( t) 0; Q ( t) = q ( t), where r denotes the dscount rate, D the nverse demand functon for, Q (t) the stock 1 of resource avalable at tme t and λ (t) the co-state varable for resource. Defne the equlbrum prce for demand as 1 p ( t) = D ( q ( t)) and the prce of resource n demand as p ( t) = c + z + λ ( t) w + λ ( t). he necessary and suffcent condtons 5 are p ( t) p ( t) (f < then q ( t) = 0 ) ; (2) λ ( t) = rλ ( t) ; (3) lme t rt rt λ ( t) 0 ; lm e λ ( t) Q ( t) = 0. (4) t 5 he proof of suffcency s essentally the same as n K, hence suppressed.

4 3 ondtons (3) and (4) mply that lm ( t) = 0 Q t for nonrenewable resource, =,, and λ (0) = λ ( ) = 0 for the backstop resource whch s n nfnte supply. B B t 2. ptmal Extracton Sequence onsder the case n whch ol s the cheapest resource for both demands and the backstop s the most expensve. hat s, Assumpton: 0 < w < w < w <, = E,. (5) B As shown by hakravorty, Krulce, and Roumasset (2005), n general the orderng of the shadow prces s exactly the reverse of that of net costs, hence = < < < t [ 0, ) 0 λ ( t) λ ( t) λ ( t) B. Snce ol and coal are nonrenewable resources, they wll be eventually exhausted and the backstop s used for both demands. By vrtue of ther Proposton 7, the extracton sequence n each demand follows the order of the net costs,.e., ol followed by coal and then by the backstop resource (see Fg.1). Not every resource needs to be extracted for each demand. 3. ondtons for Dscontnuous Resource Extracton In ths secton, we demonstrate graphcally the possblty of dscontnuous extracton of a nonrenewable resource, and then provde necessary and suffcent condtons for the dscontnuty to occur. In Fg.1, the (energy) resource prce for each demand s depcted as an envelope curve: n bold sold for transportaton and n bold dash for electrcty. oal s extracted n phase II and agan n phase IV, but not n the ntermedate phase III.

5 4 p (t) : Energy prce for transportaton p E (t) : Energy prce for electrcty p B (t) p (t) (t) p backstop coal ol (t) p E (t) p E p B E (t) backstop coal ol t 2 t I II III IV V 0 t 1 E t 2 E t 1 Fg. 1: Dscontnuous oal Extracton: oal s extracted n phase II and IV, but not n III.

6 5 he swtch pont sequence n Fg. 1 s S1: 0 < t1e < t2e < t1 < t 2 < where t 1 and t 2 denote, respectvely, ol-to-coal and coal-to-backstop swtch ponts. If the p E (t) curve n the lower part of Fg. 1 shfts up, ol may not be used n electrcty, but coal extracton wll reman dscontnuous wth an altered swtch pont sequence S2: t 2 1E 0 < t2e < t1 < t <. Ether S1 or S2 s equvalent to the followng three nequaltes: ). 0 < t ; ( ). t < t ; ( ). t < t, E,. (6) ( 2E 2E = Under these two sequences, coal s extracted frst for electrcty and then for transportaton. We can now state PRPSIIN: Gven the Assumpton n (5), coal s extracted dscontnuously, frst for electrcty (E) and then for transportaton () after a tme delay, ff (I). λ (0) < w w ; BE E

7 6 w w λ (0) w w (II). 1 + max < < 1 + w w λ (0) w w B BE E, = E,. 6 Proof: At the swtch ponts for demand, p = p and pb p = whch yeld: 1 w w 1 w w B 1 = ln ; t2 = ln r λ (0) λ (0) r λ (0) t. (7) Substtutng (7) nto (6), notng 0 < λ (0) < λ (0) < from Secton 2, we can rewrte (), () and () n terms of λ (0) and w : ( ). 0 < t λ (0) < w w ; (8) 2E BE E λ (0) w ( ). t2e < t1 < 1+ λ (0) w w w BE E ; (9) w w wb w ( ). t1 < t2 < λ (0) λ (0) λ (0) λ (0) w w > 1+ max. λ (0) wb w (10) 6 here exsts a subset of w w, w, w w, w ) w = whch admts condtons (I) and (II), ( E, E BE, B e.g., w = (4,5,6,1,4,6 ). Gven w, λ (0) and (0) λ stll depend on other factors such as the ntal stocks of resources, the dscount rate and the magntude of demands, hence are not determned solely by w.

8 7 Notng that (9) and (10) ontly are equvalent to (II) n the Proposton completes the proof. he condtons n the Proposton can be re-expressed as three smple nequalty constrants: λ (0) < α, λ (0) < βλ (0) and λ (0) > γ λ (0) where α w BE w > 0 ; (11) E w w β 1 + > 1; (12) w w BE E w w γ 1+ max > 1, (13) wb w whch are graphcally depcted n Fg. 2. Let {( λ (0), λ (0)) 0 < λ (0) < λ (0) < } whch les above the 45 lne n the fgure. Set defnes the doman for ( λ (0), λ (0)) on whch the dscontnuous extracton of coal s feasble. he entre shaded area () represents an open subset that satsfes the condtons for ether S1 or S2 to occur. Note that t 1 E > 0 for S1 and t 1E 0 for S2. Usng 1 t n (7) for = E, we can restate these two nequaltes as λ (0) > µ + λ (0) for S1 and λ (0) µ + λ (0) for S2, respectvely, where µ w E w > 0. he lne λ (0) = µ + λ (0) splts the entre shaded area n the E fgure nto two parts. he dark shaded area ( 1) s an open subset of that admts only * sequence S1 and the lght shaded area ( 1) admts only S2. If µ µ α( β 1), there

9 8 exsts no( λ (0), λ (0)) whch admts S1, so that the entre shaded area admts only S2. λ (0) * µ + λ (0) β λ (0) µ + λ (0) 0 45 γ λ (0) 1 1 * µ µ λ (0) = α 0 α λ (0) Fg. 2: oal Extracton s Dscontnuous on the Shaded pen Set he Proposton s symmetrc wth respect to E and. Fg.1 shows that f E and were nterchanged, coal would stll be extracted dscontnuously wth S1 and S2 redefned as 0 t1 < t2 < t1 < t 2 < and t1 0 < t2 < t1 < t 2 <, respectvely. < E E E E

10 9 4. oncluson hs paper provdes condtons under whch optmal extracton of a nonrenewable resource s dscontnuous. At least three resources and two demands are necessary for dscontnuous extracton to occur. Wth many resources and demands, extracton patterns that appear chaotc may be consstent wth effcent resource use.

11 10 References Amgues, J. P., P. Favard, G. Gaudet and M. Moreaux, 1998, ptmal rder of Natural Resource Use When the apacty of the Inexhaustble Substtute s lmted, Journal of Economc heory 80, hakravorty, U. and D. L. Krulce, 1994, Heterogeneous Demand and rder of Resource Extracton, Econometrca 62, hakravorty, U., D.L. Krulce, and J. Roumasset, 2005, Specalzaton and Nonrenewable Resources: Rcardo meets Rcardo, Journal of Economc Dynamcs and ontrol (n press). Gaudet, G., Moreaux, M., Salant, S., 2001, Intertemporal Depleton of Resource Stes by Spatally Dstrbuted Users, Amercan Economc Revew 91 (4), Herfndahl,.., 1967, Depleton and Economc heory, n M. Gaffney, ed., Extractve Resources and axaton, (Unversty of Wsconsn Press), M.. Kemp and N. V. Long, n two folk theorems concernng the extracton of exhaustble resources, Econometrca 48 (1980), 663_673. Lews,. R., 1982, Suffcent ondtons for Extractng Least ost Resource Frst, Econometrca 50, Solow R., Wan, F.Y., 1976, Extracton osts n the heory of Exhaustble Resources, he Bell Journal of Economcs 7,