A Model of Intertemporal Emission Trading, Banking, and Borrowing*

Transcription

1 Ž. JOURAL OF EVIROMEAL ECOOMICS AD MAAGEME 31, ARICLE O A Model of Intertemporal Emsson radng, Bankng, and Borrowng* JOAA D. RUBI Department of Economcs and Energy, Enronment and Resources Center, Unersty of ennessee, Knoxlle, ennessee Receved October 30, 1995; revsed ovember 13, 1995 hs paper provdes a general treatment of emsson tradng, bankng, and borrowng n an ntertemporal, contnuous-tme model. Usng optmal-control theory, the decentralzed behavor of frms s shown to lead to the least-cost soluton attanable under jont-cost mnmzaton. Explct solutons for the tme paths of emssons and permt prces are derved when frms are allowed to both bank and borrow and when frms are only allowed to bank emsson permts. he polcy mplcatons of emsson bankng and borrowng are dscussed Academc Press, Inc. 1. IRODUCIO Marketable emsson permts are an economc nstrument used to attan a predetermned level of envronmental qualty. he basc concept behnd emsson tradng, orgnated by Dales 5, s a smple one. Intally, a regulatory agency lmts the overall level of emssons, ether by settng a standard or allocatng emssons, and then t allows frms to trade ther emsson allocatons or surplus permts. It s now wdely agreed among economsts that marketable emsson permts can be a cost-effectve strategy for controllng envronmental pollutants, and an extensve lterature on ther propertes has developed Žsee etenberg 12 and Cropper and Oates 4 for thorough revews.. hs work has dentfed three sources of potental cost savngs: emsson tradng between frms, emsson averagng between sources wthn a frm, and emsson tradng through tme. Despte common reference to these three components, prevous theoretcal and emprcal research has focused almost exclusvely on the frst two tems, tradng and averagng. he small amount of research nto emsson bankng and borrowng s regrettable snce publc polcymakers have already begun to ncorporate bankng and borrowng rules nto law. For example, the bankng Ž and tradng. of sulfur doxde s now authorzed by the Clean Ar Act Amendments of 1990, and Calforna allows manufacturers of passenger cars to bank Ž and trade. hydrocarbon emssons. In the context of fuel effcency, rather than emssons, the borrowng and bankng of *he author apprecates helpful comments on earler versons of ths paper from Mchael Caputo, Amy Farmer, Glora elfand, Catherne Klng and two anonymous referees $18.00 Copyrght 1996 by Academc Press, Inc. All rghts of reproducton n any form reserved.

2 270 JOAA D. RUBI CAFE Ž corporate average fuel economy. credts for up to three years has been allowed snce 1980 ŽCongressonal Quarterly 1980 Almanac 2, pp. 487, owever, the concept of marketable permts, and ntertemporal tradng n partcular, goes far beyond the realm of envronmental cleanup or fuel economy standards. It apples to the trade of goods whose exstence s statutorly generated but prvately transferable. For example, the concept apples to the allocaton of resdental or commercal property development rghts or to the buyng and sellng of federal funds between banks who must meet reserve requrements. In the latter case, conservatve banks who choose to carry surplus reserve requrements act as sellers of overnght loans Ž e.g., tradng. and, n addton, may choose to hold addtonal excess reserves speculatvely over several tme perods Ž e.g., bankng. n the hope that a lucratve loan opportunty arses. hroughout ths paper, however, the dscusson of bankng wll be couched n terms of standard usage wthn the envronmental feld, whch refers to emsson of pollutants by frms. Wthn the envronmental lterature ths paper flls the theoretcal gap by provdng a comprehensve treatment of ntertemporal emsson tradng. 2 In partcular, t explores the problem of mnmzng the cost of ntertemporal emsson control by heterogeneous frms n the presence of polluton standards and emsson permts that are tradable across frms and through tme. In such a settng, frms may drectly reduce emssons, and they may purchase, sell, bank and borrow emsson permts n order to meet applcable standards or to take advantage of any speculatve opportuntes that may arse. o generate permts, a frm may choose to pollute less than the current standard. hese generated permts may be sold to a dfferent frm or deposted n an emsson bank to be used later by the frm or sold at a later tme to another frm. he borrowng of permts occurs when a frm pollutes more than ts current standard, but the cumulatve defct must be repad by the end of the plannng horzon. Bankng and borrowng lower the cost of complance wth emsson standards by allowng frms to adjust ther emsson stream more flexbly through tme. he model used here bulds on the semnal statc paper by Montgomery 7 on marketable permts and extends the work of etenberg 12 and Cronshaw and Kruse 3, who examne some aspects of permt markets wth bankng n dscrete tme. etenberg characterzes the jont least-cost allocaton of polluton for a cumulatve pollutant gven a constrant on the total amount of polluton over tme. In ths cumulatve emsson-permt system t appears that all permts are ssued at the begnnng of the tme horzon. Some permts are therefore always banked untl used up. Before the permts are used up ther prce must rse at the rate of nterest. Cronshaw and Kruse examne a permt system wth bankng where permts are allocated to frms n each of perods. hey show that n a market for transfer- 1 Wth the borrowng of emsson credts there s the obvous concern about frms borrowng and then declarng bankruptcy. In the CAFE program, enforcement of the CAFE standards takes the form of fnes and a revew of company product plannng to determne the lkelhood of future complance ŽShaffer 10.. he enforcement problem s nversely related to the cost of entry and ext. For captal-ntensve ndustres such as electrcal power producers or automoble manufacturers, enforcement should not be a problem. For ndustres characterzed by a rapd turnover of frms, enforcement of all envronmental regulatons s a more serous ssue. 2 See Rubn and Klng 8 for an emprcal examnaton of emsson bankng and borrowng for lght-duty vehcle manufacturers.

3 IEREMPORAL EMISSIO RADIG 271 able and bankable permts, a compettve equlbrum exsts and acheves aggregate emsson targets at least system cost. Cronshaw and Kruse also show that f frms do not desre to bank, permt prces wll rse by less than the rate of nterest. hs paper extends the work of etenberg and Cronshaw and Kruse by provdng a more general treatment of permt tradng n contnuous tme through the use of optmal-control theory. Instead of lmtng ntertemporal tradng to bankng, ths paper starts from the context of allowng both borrowng and bankng wth restrctons on borrowng as a specal case. he formulaton allows the dervaton and extenson of the results of etenberg and Cronshaw and Kruse, but nterpretatons are necessarly somewhat dfferent. Importantly, the contnuous-tme framework allows for the explct soluton for the tme path of emssons, whch, cost savngs notwthstandng, may be ntertemporal emsson tradng s most sgnfcant aspect. ether etenberg s nor Cronshaw and Kruse s paper examnes ths ssue n detal. 3 In addton, ths analyss extends the model of Cronshaw and Kruse by consderng the qualtatve mpacts of emsson borrowng on permt prces, the emsson stream, and total socal damages. 2. E FORMAL MODEL Consder a frm,, that has two control varables, e Ž. t, the quantty of emssons each perod, and yž. t, the quantty of emsson permts bought Ž y 0. or sold Ž y 0. at prce Pt, Ž. whch wll be determned by the equlbrum condtons of all frms n the permt market over the plannng horzon of length. he level of emssons that are n the bank, BŽ. t, s a state varable. Followng Montgomery 7, the abatement-cost functon for frm, C Ž e., s equal to the dfference between the unconstraned profts and profts n whch the frm adopts an emsson level S Žwhere S s less than the unconstraned emsson level, e. and adjusts ts output n order to obtan maxmum profts for the constraned level of emssons. It s assumed that C s twce contnuously dfferen- table and convex n e and that margnal abatement costs, C Ž e., are postve and strctly ncreasng, that s, C Ž e. 0 and C Ž e. 0. Frms choose abatement levels and permt purchases and sales gven that permt prces, Pt, Ž. mght change over tme. hs makes the formal model nonautonomous. In addton, frms are assumed to be makng decsons over a fnte tme horzon of length. he length of can, of course, be relatvely short or very long. 3. E JOI-COS PROBLEM In the jont-cost problem, the regulator s goal s to mnmze the total costs of polluton abatement of heterogeneous frms subject to reachng a regonal emsson standard, St. Ž. he regulator assgns each frm a standard or endowment Ž. of emssons, S t, so that Ý S Ž. t St. Ž. ote that emsson bankng and 1 3 etenberg 12, p. 29 notes that hs cumulatve-emsson permt system does not regulate emsson rates but only total emssons. Cronshaw and Kruse 3, p. 6 note that bankng could lessen envronmental damages by delayng emssons or could allow frms to possbly emt large amounts of emsson usng prevously banked permts.

4 272 JOAA D. RUBI borrowng, as opposed to tradng, change the nature of standards snce frms can emt above the standard at some ponts n tme. What s requred s that frms have to fulfll the cumulatve standard over the entre plannng horzon. In the jont-cost problem, the regulator s able to acheve cost savngs by optmally adjustng the emsson stream. Defnng the optmal-value functon as J**, B ÝB as the aggregate stock of banked emssons, B as the rate of change of aggregate banked emssons, and as the termnal tme perod, the problem can be wrtten as 4 rt e Ý 0 Ý 1 Ž. J** Mn e C e Ž t. dt Ž 1. s.t. B Ž S Ž t. e Ž t.. Ž 2. BŽ 0. 0, BŽ t. 0 Ž 3. ež t. 0,. Ž 4. he objectve functonal Ž. 1 mnmzes the sum of the frms present dscounted emsson-abatement costs. Equaton Ž. 2 s the state equaton that shows that the changes n the aggregate stock of banked emssons equal the aggregate dfferences between all frms standards and emssons. he standards, S, can be nterpreted ether as frm-specfc emsson standards or as ntal endowments of permts where the emsson standards are understood to be equal to zero. Equaton Ž. 3 says that the ntal aggregate stock of banked emssons s zero, the aggregate stock of permts may never have negatve balances Žths constrant s later relaxed to allow borrowng., and at the termnal perod the aggregate stock of emsson permts s non-negatve. Equaton Ž. 4 requres that frms cannot have negatve emsson levels. he jont-cost problem can be solved through use of optmal-control theory. Usng ths framework, defnng Ž. t as the costate varable on the state equaton and as the multpler functon for the bank non-negatvty constrant, the present value amltonan s and the Lagrangan s rt e Ý CŽ e t. Ý Ž S t e t., Ž. Ž. Ž. Ž. rt Le Ý CŽ e t. Ý Ž S t e t. B Ž. Ž. Ž. Ž. hs problem yelds the followng necessary condtons for an optmal soluton Ž. Kamen and Schwartz 6, Part II, Secton 17 : L Ḃ Ž S Ž t. e Ž t.. Ž 7. Ý 1 L ; B0, 0, B 0 Ž 8. B 4 If both bankng and borrowng are always allowed, then Ž. 3 and Ž 13. should be replaced wth the smpler constrant BŽ. 0 0 and B Ž. 0, rather than BŽ. 0 0 and Bt Ž. 00,.

5 IEREMPORAL EMISSIO RADIG 273 L L rt e CŽ e. 0; e 0, e 0, Ž 9. e e BŽ. 0, Ž. 0, BŽ. Ž. 0. Ž 10. Economc Interpretaton of ecessary Condtons for Optmzaton of the Jont-Cost Problem Equaton Ž. 7 smply restates the state equaton. ext, Ž. 9 says that all frms that dscharge some emssons should have present dscounted margnal abatement Ž rt costs e C Ž e.. equal to the margnal cost of an addtonal unt of banked emssons Ž., and that all frms that dscharge emssons should have equal margnal abatement costs. hs equaton also shows that Ž. t, the present value costate varable, whch s the margnal or shadow value of a unt of emssons n the bank, s negatve. hs reflects the fact that f there were an addtonal unt of emssons n the systemwde bank, systemwde total abatement cost would be lower because frms could then dscharge more emssons, whch would lower ther costs. For later use denote the vector of optmal systemwde emssons from ths problem as E Ž. t Že Ž. t,...,e Ž.. 1 t. Equaton Ž. 8 says that f, n total, frms bank a postve quantty of emssons over some tme nterval, then the margnal value of an addtonal unt of emssons n the systemwde bank s constant for that tme nterval. hs mples that the margnal value of a unt of emsson n the bank would always be a negatve constant f frms were allowed to borrow emssons Ž.e., f Bt Ž. were unrestrcted n sgn, mplyng that 0.. If frms, n total, would lke to borrow emssons, but are not allowed to do so, the margnal value of an addtonal unt of emssons n the bank would be decreasng Ž.e., gong to zero from the left snce the multpler, t, Ž. s postve. he above condtons tell us what necessary condtons must hold for the mnmzaton of the present value of total costs. he suffcent condtons needed to guarantee an optmum to ths problem are that be convex n e and B and that the above necessary condtons hold; addtonal contnuty and regularty condtons that are met for ths problem are gven n full n heorem 1 of Seerstad and Sydsæter 9, pp. 317, In partcular, the suffcent condtons requre that each frm s cost functon be convex n emssons. Moreover, exstence follows from heorems 2.1 and 3.1 of Stenberg and Stalford 11, snce the state equaton Ž. 7 s lnear n the e, and the ntegrand s strctly convex n the e. ext, t s necessary to look at how ndvdual frms wll make ther decsons to optmally control abatement costs and permt purchases and sales, gven that frms take permt prces as exogenous. 5 Ž Accordng to Seerstad and Sydsæter, for optmal-control problems wth pure state constrants e.g., fžbž t.,t. 0, or n ths case Bt Ž. 0., the costate varable may be dscontnuous at the termnal perod. Gven approprate regularty condtons an optmal soluton to the problem stll exsts, but ths possblty wll not be consdered n the analyss.

6 274 JOAA D. RUBI 4. E FIRM S PROBLEM Formally, defnng the optmal value functon as J, and lettng Pt Ž. be the nstantaneous prce of permts yž. t purchased or sold by frm, and A Ž. t and DŽ. t be the bounds on the nstantaneous purchase and sale of permts, frm s problem s rt Ž. y, e 0 J Mn e C e Ž t. PŽ t. y Ž t. dt Ž 11. s.t. B S Ž t. e Ž t. y Ž t. Ž 12. B Ž 0. 0, B Ž t. 0 Ž 13. ež t. 0 Ž 14. AŽ t. yž t. DŽ t., AŽ t. 0, DŽ t. 0. Ž 15. he objectve functonal Ž 11. says that frm chooses the level of emssons and purchases or sales of permts to mnmze ts present dscounted costs. Permts may be nstantaneously purchased or sold wth no transacton costs. Equaton Ž 12. s the state equaton, whch says that changes n the bank of emssons equal the dfference between the standard and the level of emsson plus any permts bought or sold. Equaton Ž 13. says that the ntal stock of the bank s zero, frms may never have a negatve bank balance Ž ths constrant s later relaxed., and at the termnal perod the frm s free to hold any non-negatve stocks of emsson permts. Equaton Ž 14. requres that frms cannot have negatve emsson levels. Fnally, Eq. Ž 15. bounds the nstantaneous purchase or sales of permts. hs last constrant, Eq. Ž 15., s a necessary techncal requrement that arses f one wants to consder arbtrary prce paths, snce the objectve functon s lnear n y. he soluton to ths type of control problem s straghtforward but unnecessarly complcated to get at the economc ssues at hand. 6 An alternatve approach s to consder prce paths for whch a soluton to the frms problem, wthout bounds, exsts. hs s equvalent to assumng that the frm has an nternal soluton. he economc ntuton of ths assumpton s dscussed below. owever, there are two cases n whch a frm would not have an nternal soluton. One stuaton s when the least-cost strategy for the frm s to sell a bounded maxmum number of permts at any pont n tme. he other stuaton s when a frm s margnal abatement costs are so hgh that t would rather take no abatement actvtes, but smply buy as many permts as possble. he maxmum number of permts a frm could buy at any pont n tme s lkely to be controlled by the regulatory agency to prevent localzed hot spots of concentrated emssons. hus, n both these cases, a frm could potentally desre to sell, borrow, or buy some maxmum number of permts at any pont n tme. By assumng that frms do not buy or sell permts at a bounded rate, ths problem can be solved through the use of an optmal-control-theory framework by defnng Ž. t as the present-value costate varable on the state equaton and Ž. t as the multpler functon for the bank non-negatvty constrant for frm, 6 See Kamen and Schwartz 6, Part II, Secton 12 for the soluton technque sutable to ths problem.

7 IEREMPORAL EMISSIO RADIG 275 1,...,. 7 hen, the present-value amltonan s Ž. Ž. Ž. rt e C e Py S e y. 16 he amltonan yelds the followng necessary condtons: 8 Ḃ Ž S e. y Ž 17. ; B0, 0, B 0 Ž 18. B e rt e CŽ e. 0; e 0, e 0 Ž 19. e rt e P0 Ž 20. y BŽ. 0,Ž. 0, BŽ. Ž. 0. Ž 21. Economc Interpretaton of the Frm s ecessary Condtons Equaton Ž 17. smply restates the state equaton. ext, lookng at Eq. Ž 19., f rt e 0, then e 0 and f e 0, then e CŽ e.. hus, f the frm emts some polluton, then the present dscounted margnal abatement costs equal the margnal value of an addtonal unt of emssons n the frm s emsson bank account. If present dscounted margnal abatement costs Ževaluated at the complete abatement level of emssons. are less than the present value of an addtonal unt of emssons n the bank, then the frm dscharges no emssons. Furthermore,, the margnal value of a unt of banked emssons for frm, s negatve. hs reflects the fact that f the frm had an addtonal unt of emssons n the bank, abatement costs would be lower, as would the present value of total costs. Equaton Ž 18. says that the margnal value of a unt of emssons n the bank s ncreasng Ž becomng less negatve, or smaller n absolute value. f the nonnegatvty constrant on the bank s bndng. 9 hat s, the value to the frm of an addtonal unt of emssons n the bank s decreasng f the frm desres to borrow a unt of emssons but t cannot. If the constrant aganst borrowng were not present, then the margnal value of a unt of banked emssons would be constant whenever the frm was not sellng permts at ts maxmum level. Lookng now at Ž 20., we see that the frm wll purchase or sell y permts so that the dscounted margnal cost of a unt n the bank s equal to ts dscounted prce, 7 he more general case requres the ntroducton of addtonal multpler varables for the permt bounds gven n Ž See Seerstad and Sydsæter 9, pp for a complete lst of the necessary condtons. Almost necessary condtons bascally requre that the Lagrangan multpler,, only has jumps at a fnte number of ponts and has pecewse contnuous dervatves elsewhere. 9 It s also possble for the bank constrant to be bndng just enough that the value of a unt of banked emssons s constant and the stock n the bank s equal to zero. hs case s not fundamentally dfferent and wll not be explctly addressed.

8 276 JOAA D. RUBI e rt P. Permt prces are determned by equlbrum condtons of all frms n the market for permts. For frm Ž as well as all other frms. to have a soluton wthout bounds on permt purchases or sales over the entre tme horzon, permt prces must follow along a sngular path defned by e rt P 0. hen, rt along the sngular path, 0, so the followng must hold: re P ert P 0. Substtutng n for from Eq. Ž 18., the followng equaton results: rt 0re P ert P. After some manpulaton, the followng restrctons on the percentage rate of growth n the prce of emsson permts for the frm to reman on the sngular path are derved: 5 r Ṗ rt B 0 e whenever ½. Ž 22. P r 0 P If frm ether wants to bank emssons or does not face a restrcton on the borrowng of emsson credts, then a nonbounded soluton requres that the prce path of permts follow otellng s rule and grow at the rate of return earned from rt Ž. holdng any other asset. Equvalently, the present value prce of permts, e P t, s constant. If a frm faces a bndng constrant on the borrowng of emsson credts, then 0, and the rate of growth n prces must be less than the comparable nterest rate. In ths case the present-value prce of permts s decreas- rt ng through tme snce e PŽ. t Ž. t, Ž. t Ž. t 0, and Ž. t 0. Sum- marzng, for a partcular frm to have a non-bounded soluton, the present-value prce of permts must be constant when each frm can bank and borrow emssons. If the bank s requred to be non-negatve, and the frm desres to borrow, then a non-bounded soluton requres that permt prces be decreasng. he transversalty condton, Ž 21., mples that f a frm holds a permt at the termnal perod, ts value must be zero. Combnng condtons Ž 19. and Ž 20., under the condton that a frm emts a postve quantty of polluton, then C Ž e. P. hat s, under normal condtons a frm wll equate the margnal cost of polluton abatement wth the prce of a permt. herefore, present-value margnal abatement costs are constant when each frm can freely bank and borrow emssons, and may declne f borrowng s restrcted and frms desre to borrow. 5. EQUILIBRIUM CODIIOS As dscussed above, the regulator s goal s to mnmze the jont total costs of polluton abatement of heterogeneous frms subject to reachng a regonal emsson standard, S, over the -perod horzon so that B Ž. 0. It s necessary to show that an equlbrum n a market for permts exsts and s effcent n the sense that t s equal to the system-cost mnmzaton by regulators who know the cost functons of all the frms. Montgomery s 7 defnton of a market equlbrum for polluton permts s used, but modfed to account for the dynamc problem at hand. Especally, but not exclusvely, n problems wth lnear-control varables, the optmal path of the lnear control y Ž. t can be dscontnuous n t. herefore n Lemma 1, whch appears n the Appendx, t s necessary to assume that the

9 IEREMPORAL EMISSIO RADIG 277 optmal control y Ž. t s contnuous n the prce of permts, P, even at partcular nstances, t, where y Ž. t s dscontnuous n t. hs assumpton s not too restrctve; ts economc content s smply that at a certan pont n tme f a frm suddenly Ž e.g., dscontnuously. decdes to buy or sell a dfferent quantty of permts, the decson s based on a contnuous functon of the permt prce. An ntertemporal equlbrum n the permt market also requres that frms have an nternal soluton. DEFIIIO. An ntertemporal market equlbrum n emsson permts over a perod horzon conssts of the vectors Y* Ž. t Žy Ž. t,..., y Ž.. t and E* Ž. 1 t Že Ž. t,...,e Ž.. t and the scalar P* Ž. t 0 such that Y* Ž. t and E* Ž. 1 t mnmze each frm s costs gven P* Ž. t, rt Ž. 0 e C e Ž t. P* Ž t. y Ž t. dt Ž 23. Ž. Ž. subject to each frm s constrants for each t 0,, and the market clearng condton on permts Ý 1 y Ž t. 0. Ž 24. he market-clearng condton s a flow condton that smply requres that at any pont n tme a permt purchased by one frm be sold by some other frm. In addton, cost mnmzaton wll also mpose the followng termnal stock condton: Ý 1 P* Ž. B Ž. 0. Ž 25. Cost mnmzaton requres that the prce of a unt of polluton n the bank be zero at the termnal perod or that the stock of permts from all frms be zero. EOREM 1. An ntertemporal market equlbrum n emsson permts oer a -perod horzon exsts when frms are not buyng and sellng permts at ther maxmum nstantaneous rate. Proof. See the Appendx. he Market Equlbrum s the Least-Cost Soluton Before provng that the market soluton s at least as nexpensve as that from the socal planner, t s nterestng to thnk about what bankng and borrowng mean n the context of a market. As shown above, the prce path of permts must be ether constant or decreasng. hus, for all frms that have not shut down or are rt not emttng at ther maxmum rate, e PŽ. t Ž. t. hus must be equal for all frms and be of ether constant or declnng value. hs says that the value of a unt of banked emsson s the same for all frms that have an nternal soluton for permt purchases, even though t s possble for some frms to borrow whle others bank and yet others do nether. Furthermore, f e rt P s fallng over some perod, then the result of all frms buyng and sellng permts s an overall net desre to borrow.

10 278 JOAA D. RUBI ow t s necessary to show that the market soluton s as least as good as the soluton from the socal planner. In general, permt prces are some functon of all frms cost functons of emssons abatement, P gc Ž Ž e.,...,c Ž e Gven the assumpton of nteror solutons, Ž 19. and Ž 20. can be equated to yeld C Ž e. Pt Ž. 0. Snce C 0, the mplct functon theorem mples that e e ŽPŽ t.. s the soluton to ths margnal condton, where EPŽ. t 1C0. Over the -perod horzon, the total quantty of emssons must be Ý Ž. Ý e P* Ž z. dz S Ž z. dz Ž snce the stock of emssons n the bank s defned to be zero at t 0 and wll be chosen to be zero at t. An mplcaton of the equlbrum condtons s a relatonshp between the prce path of permts and equlbrum bankng. At any tme t the aggregate change n the bank can be expressed as Ý 1 Ž. Ž. Ḃ S Ž t. e PŽ t. y PŽ t., Ž 27. and the stock n the bank at any tme t equals t 0 1 Ý Ž Ž. Ž.. BŽ t. S Ž z. e PŽ z. y PŽ z. dz. Ž 28. Gven that the permt market clears n each perod, ths reduces to t 0 1 Ý Ž Ž.. BŽ t. S Ž z. e PŽ z. dz. Ž 29. hs says that the stock of all frms permts equals the cumulatve number of permts that have been banked. If borrowng s allowed, then Bt Ž. can be negatve, ndcatng that, cumulatvely, frms are borrowng emssons. When borrowng s not allowed, then t must be non-negatve. In ether case, over the whole tme horzon the sum of all frms stocks must be non-negatve 0 1 Ý Ž Ž.. BŽ. S Ž z. e PŽ z. dz0. Ž 30. he last result wll be used n the proof that the market-equlbrum soluton from frms ndvdually buyng and sellng permts s effcent n the sense that t s at least as good as the systemwde cost mnmzaton by regulators who know the cost functons of all frms. hs proof s based on a proof by Cronshaw and Kruse 3, who show that a dscrete-tme permt system wth bankng attans the mnmum jont-cost soluton. EOREM 2. he market-equlbrum soluton Ž e Ž. t,...,e Ž.. 1 t, and Ž y Ž. t,..., y Ž.. 1 t s at least as nexpense as the system-cost mnmzaton, so that Ý J J**, where J* and J** are the optmal-alue functons defned n Ž and Ž. 1,respectely. Proof. See the Appendx.

11 IEREMPORAL EMISSIO RADIG CARACERIZIG E PA OF EMISSIOS From an envronmental perspectve, perhaps the most mportant result of allowng frms to ntertemporally move permts, as opposed to smply tradng them nstantaneously among themselves to attan the least-cost allocaton, s the effect that bankng and borrowng can have on the stream of emssons through tme and the resultng health and envronmental mpacts. If frms use bankng to smooth decreasng emsson standards over tme and f margnal damages from polluton are ncreasng, then bankng generates lower total polluton damages when consderng the ntegral over the whole tme horzon of each perod s damage. From the formal model we can derve some nsght nto how frms adjust ther emsson streams when they can take advantage of the ablty to bank and borrow. For all frms not buyng or sellng permts at ther maxmum rate, but able to equate margnal abatement costs wth the margnal value of banked emssons, t rt was shown earler that e C Že. 0. If ths can be sustaned for some nonzero tme nterval, then along ths nterval the followng condton on the tme path of e Ž. t s derved by dfferentatng Ž 25. wth respect to t and collectng terms: Ž. e, Ž 31. Ž. Ž. rṫ rc e e C e C e Ž. whch after substtutng from Eq. 18 gves rc Ž e. e rt e. Ž 32. C Ž e. C Ž e. Gven the sgn condtons descrbed earler, the frst term on the rght n Ž 32. s negatve and the second term s postve or zero. he second term s zero whenever all the frms desre only to bank emssons or are allowed to borrow emssons. hs means that f frms never desre to borrow emssons or desre to borrow and are allowed to do so, so that Ž. t 0, then the frms emsson streams wll declne through tme. When borrowng s not allowed and frms desre to borrow, the second term on the rght s postve and could be larger n magntude than the frst term, whch s negatve. hus, n prncple, aggregate emssons could ncrease through tme when borrowng s not allowed. hs would occur, however, only when emsson standards are becomng less strngent through tme. he dscountng of abatement costs causes frms to delay abatement and abatement expendtures beyond the tme frms would have engaged n them had costs not been dscounted. When standards are constant through tme, Ý S 1 0, then the least-cost behavor of frms s to borrow emssons, Ý1 B 0, n early years and pay them back n later years. hs s true snce, as was shown above, the present-value margnal cost of abatement must be constant when bankng and borrowng are allowed. When frms choose to bank emssons, Ý1 B 0, t must be because they expect future Ž undscounted. abatement costs to rse by more than the rate of dscount; they would not otherwse bear the addtonal earler abatement expendtures. he only way for undscounted abatement costs to rse through

12 280 JOAA D. RUBI tme s for the level of abatement to rse through tme, snce C Ž e. 0, C Ž e. 0. Snce constant standards engender borrowng, t must be that bank- 10 ng arses from standards that tghten, Ý1S 0, over some nterval of tme. If damages from emssons are a convex functon and standards are tghtenng through tme, then allowng frms to bank lowers socal damages. Bankng allows frms to smooth ther emsson stream through tme. For a convex damage functon, where cumulatve damage s the ntegral of damages n all tme perods, total damages are reduced f the level of emssons s more constant through tme. hs result s one of the qualtatve dfferences between a statc and a dynamc marketable permt system. By gvng frms an ncentve Ž cost savngs. to more than meet current standards, frms wll ntertemporally move emssons to lower socal damages. hs occurs when frms are allowed to bank emssons and when standards are becomng strcter through tme. Emsson standards becomng strcter through tme s, however, often the norm. If polluton damages are convex, then allowng frms to shft emssons nto the present wll ncrease envronmental damages. Whether or not frms should be allowed to borrow, at the expense of ncreasng envronmental damages, depends on balancng the cost savngs to frms wth the addtonal harm caused by borrowng. Snce frms dscount future costs, they wll want to borrow when permt prces grow at less than the rate of nterest, are constant, or actually decrease over tme. In equlbrum, permt prces wll equal margnal abatement costs. hus frms wll want to borrow emssons when standards are constant or not becomng more strngent at a suffcently hgh rate. ence, f standards are constant through tme so that frms desre to borrow, but are not allowed to do so, then the rate of emssons wll reman constant through tme. In ths case we can gve an economcally nterestng nterpretaton to, the multpler on the non-negatvty con- strant on the bank for frm. Settng e 0 n Eq. Ž 32. and rearrangng yelds Ž. rt C e e r. Substtutng n PŽ. t for C Ž e. and rearrangng, t s seen that Ž. rt t re PŽ. t. herefore, Ž. t, the multpler on the bank s non-negatvty constrant, can be nterpreted as the perodc payment that frm would be wllng to make on a perpetual annuty whose purchase prce s equal to the present dscounted cost of an emsson permt. A specal case arses when r 0. In ths case frms have no ncentve to put off costs untl the future, and gven the assumpton of strct convexty of the emssoncontrol functon, frms want to emt polluton at the same rate over the entre tme horzon, regardless of whether emsson standards are becomng strcter. If emsson rates are becomng strcter through tme and the dscount rate s zero, frms wll want to save or buy permts n the begnnng tme perods for later use. gher dscount rates lower the value of future cost savngs and decrease the ncentve for frms to bank emssons. hs analyss has been framed n a world of certanty. In actualty, margnal abatement costs and margnal damages are uncertan. In partcular, a frm mght use banked permts to cover a perod of partcularly hgh emsson levels because of 10 hs concluson needs to be tempered when consderng a more general settng than s consdered n ths model. For example, a frm may wsh to bank emssons f nvestments n abatement equpment are coordnated wth other nonabatement nvestments. Addtonally, frms may want to bank emssons n antcpaton of future growth, especally when there exsts consderable uncertanty about the future prce of permts.

13 IEREMPORAL EMISSIO RADIG 281 a sudden ncrease n the demand for output. If margnal damages are ncreasng, ths excess burst of emssons could cause unacceptably large damages. An envronmental regulator, therefore, may wsh to put a celng on the total allowable dscharge of emssons n any one gven perod. 7. FIAL REMARKS hs paper characterzes the optmal behavor of frms that face emsson standards set by regulators, but are allowed to purchase, sell, bank, and borrow emsson permts gven a fnte plannng horzon of length. It was shown that an equlbrum soluton to the frms problems exsts and s effcent n the sense of achevng the least-cost soluton Žthe soluton attaned by a socal planner who knows the cost functons of all frms.. he equlbrum permt-prce path was shown to be constant or decreasng dependng on whether frms desred to bank or were allowed to borrow permts and whether frms faced a bndng constrant on borrowng. Exstence of an equlbrum permt-prce path also requres that frms not buy and sell permts at ther maxmum nstantaneous rate. Strategc behavor and prce and cost uncertanty were not taken nto consderaton. As argued n the body of the paper, perhaps the most mportant consequence of emsson bankng and borrowng s the ablty of frms to shft ther emsson stream through tme. In partcular, when socal damages are an ncreasng functon of the level of polluton emtted at any one tme, then t s good publc polcy to allow frms to bank emssons when standards are becomng strcter through tme. In contrast, when standards are constant or easng Ž S 0., then allowng frms to borrow wll rase socal damages whle lowerng frms costs. At least for now our socety s becomng ncreasngly concerned wth envronmental qualty, and emsson standards are becomng more strngent. hs paper has shown that n these crcumstances, bankng of emsson permts can lower socal damages. Bankng also provdes cost savngs to frms by allowng them to adjust ther own nternal rates of emsson reductons to an externally set standard. Along wth averagng and tradng, then, bankng can also lower the monetary costs of complance. Bankng, therefore, along wth averagng and tradng, should be consdered by publc polcymakers n charge of ensurng the safety of our envronment. Proof of heorem 1 APPEDIX he defntons, lemmas, and proofs that follow are structured after those of Montgomery 7, who proves the exstence of an equlbrum n a permt market for a statc problem. he strategy s to derve the suffcent condtons that guarantee a market equlbrum gven the market-clearng prce P* Ž. t. ext, the suffcent condtons for the exstence of a jont-cost mnmum are derved and shown to exst. It s then demonstrated that the condtons that satsfy the jont-cost mnmum also satsfy the condtons for the market equlbrum where P* Ž. t s not dentfed as the negatve of the present-value costate varable from the jont-cost rt problem,.e., e ** Ž. t P* Ž. t.

14 282 JOAA D. RUBI DEFIIIO. An ntertemporal market equlbrum n emsson permts over a -perod horzon conssts of the vectors Y* Ž. t Žy Ž. t,..., y Ž.. t and E* Ž. 1 t Že Ž. t,...,e Ž.. t and the scalar P* Ž. t 0 so that Y* Ž. t and E* Ž. 1 t mnmze each frm s costs gven P* Ž. t rt Ž. 0 e C e Ž t. P* Ž t. y Ž t. dt Ž A1. Ž. Ž. subject to each frm s constrants for each t 0,, and the marketclearng condton on permts Ý 1 y Ž t. 0. Ž A2. he market clearng condton s a flow condton that smply requres that at any pont n tme a permt purchased by one frm must be sold by some other frm. In addton, cost mnmzaton wll also mpose the followng termnal stock condton: Ý 1 P* Ž. B Ž. 0. Ž A3. Cost mnmzaton requres that the prce of a permt be zero at the termnal perod or that the stock of permts from all frms be zero. LEMMA 1. An ntertemporal market equlbrum n emsson permts oer a -perod horzon exsts f there exst optmal shadow alues for the emssons for each frm, *Ž Ž. t,..., Ž.. 1 t 0; non-negatty multplers for the permt bank, * Ž,...,.; an optmal permt prce P* Ž t. 0 for all t 0, 1, and f, n addton, frms are not buyng and sellng permts at ther maxmum nstantaneous rate so that L Ḃ Ž Se. y Ž A4. L ; B 0, 0, B 0 Ž A5. B L e L rt e CŽ e. 0; e 0, e 0 Ž A6. e and L rt e P* 0 Ž A7. y B 0, 0, B Ž. Ž. Ž. Ž. 0 Ž A8. Ý y Ž t. 0, P* Ž. Ý B Ž.. Ž A Proof. hs strategy of ths proof s to show that frms that mnmze costs gven P*t Ž. 0 do ndeed generate the condtons lsted above and that the condtons lsted above are suffcent for each frm to mnmze costs.

15 IEREMPORAL EMISSIO RADIG 283 Form the Lagrangan Ž. Ž. Ž. rt Le C e P*y Se y B. A10 For all, the soluton Ž,, e, and y. to Ž A1. subject to Ž 12. Ž 15. exsts by heorem 2.1 and 3.1 of Stenberg and Stalford 11. Moreover, the suffcent condtons of heorem 3.1 of Seerstad and Sydsæter 9, Chapter 5 are met snce the amltonan s convex n Ž e, y, B.. Dfferentaton yelds the condtons lsted above. DEFIIIO. An ntertemporal emsson-constraned jont-cost mnmum over a -perod horzon s the vector of emssons E** Ž. t Že Ž. t,...,e Ž.. t, whch mnmzes the jont costs of all frms rtý e 0 Ž. J** Ž. Mn e C e Ž t. dt Ž A11. subjet to the constrants for the jont cost problem Ž.Ž hs defnton requres the same aggregate level of emsson abatement and the same restrctons on the bankng of emssons as requred by the sum of the abatement actvtes by each frm n the ntertemporal market equlbrum defned above. he followng lemma gves suffcent condtons for the exstence of an ntertemporal jont-cost mnmum to exst. hs lemma s then used to prove the exstence of an equlbrum n the polluton-permt market. LEMMA 2. An emsson ector E** Ž. t s an ntertemporal emsson-constraned jont-cost mnmum oer a -perod horzon f there exst optmal shadow alues for the systemwde emssons, ** 0, and a non-negatty multpler for the emsson bank, ** 0, so that L Ḃ** Ž S e. Ž A12. ** Ý 1 L ** **; B** 0, ** 0, **B** 0 Ž A13. B** L L rt e CŽ e. ** 0; e 0, e 0, Ž A14. e e B** Ž. 0, ** Ž. 0, B** Ž. ** Ž. 0. Ž A15. Proof. Defnng as the costate varable on the state equaton and as the multpler functon for the emsson bank non-negatvty constrant, the Lagrangan for the jont cost mnmum s rt Le Ý CŽ e t. Ý Ž S t e t. B. A Ž. Ž. Ž. Ž. Usng the same arguments as n the proof for Lemma 1, the soluton Ž**, **, e. to Ž A11. subject to Ž.Ž. 1 4 exsts.

16 284 JOAA D. RUBI EOREM 1. An ntertemporal market equlbrum n emsson permts oer a -perod horzon exsts when frms are not buyng and sellng permts at ther maxmum nstantaneous rate. Proof. hs proof proceeds by usng the suffcent condtons for the jont-cost Ž. Ž. Ž. rt problem, A12 A15, to show that e e, y B S e, e P* **, and ** wll also satsfy the suffcent condtons for decentralzed problem Ž A4. Ž A9. for all 1,..., and t 0,. Equaton Ž A5.. Ž A5. s met by settng ** and ** gven n Ž A13.. Equaton Ž A6.. From Ž A14. L L rt e CŽ e. ** 0; e 0, e 0,. e e It s true, therefore, that e e and ** also satsfy Ž A6. for all and t. Ž. rt Equaton A7. Let ** e P* for all and t; then Ž A7. s satsfed. Equaton Ž A4.. From Ž A12. we have 1 1 Ý Ý Ž. B** B S e. Let B B, and snce at ** e P*, e e, then n addton we know, usng Ž A4., that rt B S e y. Summng over all, ths gves Ý 1 B B** y. herefore, Ý y 1 0, whch s necessary and feasble. It s also the market- clearng condton. Equaton Ž A8.. Let B Ž. B Ž. and Ž. ** Ž.. Let no frm hold any stock of pollutants n the jont-cost problem at t ; then Ý B Ž. 1 B Ž. 0. By settng B Ž. B Ž. 0 we have a feasble soluton that satsfes the equalty requrement of Ž A8.. ow let only one frm, frm, hold a stock of permts at t. hen Ý B Ž. 0. Lettng B Ž. B Ž. 1,we have a feasble soluton that satsfes the nequalty part of Ž A8.. In addton, let P* Ž. ** Ž., and we get the cost-mnmzng condton on the termnal-perod Ž. stock, whch requres that P* Ý B Ž herefore, we have found a feasble soluton that exsts and satsfes all the condtons of Lemma Proof of heorem 2. Let e Ž. t and y Ž. t be the cost-mnmzng emsson and permt paths for frm. Suppose, however, that e Ž. t and y Ž. t do not mnmze the system costs. hen there exst some other e Ž. t and yž. t that are feasble and 11 hs proof s a contnuous-tme adaptaton of a proof by Cronshaw and Kruse. Ý 1

17 IEREMPORAL EMISSIO RADIG 285 produce a lower system cost. hat s, Ý Ý Ž e z. dz S Ž z. dz Ž feasblty., Ž A rt rt Ý Ý Ž. Ž. e C e Ž z. dz e C e Ž z. dz Ž lower cost.. Ž A18. ow, defnng alternatve permt purchases and sales so that the frst 1 frms have enough permts for ther emssons and so that the permt market clears e S 1 Ý j j1 for 1,...,1 y. Ž A19 y for. Ž. In addton, from A19, frm also has a suffcent ntal endowment and permts to cover ts emssons Ž. Ý ž / y Ž z. S e Ž z. dz S e Ž z. dz0. Ž A20. Snce Ý y 0, then Ý Py 0 because Pt s the same for all frms. herefore, upon usng the last observatons n Ž A18. and Ž A19., the cost of the alternatve program s less than that of the equlbrum, Ž. 1 1 rt rt Ý Ž. Ý Ž Ž. Ž. e C e Ž z. Py Ž z. dz e C e Ž z. Py Ž z. dz. So for some frm, t must be true that Ž. Ž. rt rt Ž. Ž. 0 0 Ž A21. e C e Ž z. Py Ž z. e C e Ž z. Py Ž z. dz, Ž A22. but ths contradcts the optmalty of e Ž. t and y Ž. t for the frm. hs proves that the market equlbrum s effcent. REFERECES 1. Calforna Ar Resources Board, Proposed Regulatons for Low-Emsson Vehcles and Clean Fuels, staff report, August Congressonal Quarterly 1980 Almanac, Congressonal Quarterly Inc., Washngton, D.C. Ž Mark B. Cronshaw and Jame B. Kruse, Permt Markets wth Bankng, workng paper, Dept. of Economcs, Unversty of Colorado, Boulder Ž Maureen Cropper and Wallace Oates Ž 1992., Envronmental economcs: A survey, J. Econ. Lt. 30, Ž J.. Dales, Polluton, Property, and Prces, Unversty of oronto Press, oronto, Canada Ž Morton I. Kamen, and ancy L. Schwartz, Dynamc Optmzaton: he Calculus of Varatons and Optmal Control n Economcs and Management, Second Edton, Elsever, ew York Ž Davd W. Montgomery, Markets n lcenses and effcent polluton control programs, J. Econ. heory 5, Ž

18 286 JOAA D. RUBI 8. Jonathan Rubn and Catherne Klng, An emsson saved s an emsson earned: An emprcal study of emsson bankng for lght-duty vehcle manufacturers, J. Enron. Econom. Manag. 25Ž. 3, Ž Atle Seerstad and Knut Sydsæter, Optmal Control heory wth Economc Applcatons, ŽAd- vanced extbooks n Economcs, 24. Elsever, ew York Ž Bran Shaffer, Regulaton, competton, and strategy: he case of automoble fuel economy standards, , Markets, Poltcs and Socal Perform., 13, Ž A. M. Stenberg and. L. Stalford, On exstence of optmal controls, J. Optm. h. Appl. 11Ž Ž homas etenberg, Emsson radng, An Exercse n Reformng Polluton Polcy, Resources for the Future, Washngton, D.C. Ž