NBER WORKING PAPER SERIES PRICES VS. QUANTITIES VS. TRADABLE QUANTITIES. Roberton C. Williams III. Working Paper

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1 NBER WORKING PAPER SERIES PRICES VS. QUANTITIES VS. TRADABLE QUANTITIES Roberton C. Wllams III Workng Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambrdge, MA 0138 October 00 The author would lke to thank Don Fullerton, Larry Goulder, Preston McAfee, Robert Mohr, Jonathan Rubn, Tom Tetenberg, Marty Wetzman, and semnar partcpants at the Unversty of Texas, the Lonestars Conference, and the NBER Summer Insttute for helpful comments. The vews expressed heren are those of the author and not necessarly those of the Natonal Bureau of Economc Research 00 by Roberton C. Wllams III. All rghts reserved. Short sectons of text, not to exceed two paragraphs, may be quoted wthout explct permsson provded that full credt, ncludng notce, s gven to the source.

2 Prces vs. Quanttes vs. Tradable Quanttes Roberton C. Wllams III NBER Workng Paper No. 983 October 00 JEL No. L51, D81, Q ABSTRACT Ths paper extends Wetzman s (1974) semnal paper comparng prce and quantty nstruments for regulaton to consder a thrd opton: tradable quantty regulatons, such as tradable permts. Contrary to what pror work has suggested, fxed quanttes may be more effcent than tradable quanttes f the regulated goods are not perfect substtutes, even when tradng ratos are based on the rato of expected margnal benefts between goods, not smply one-for-one. Indeed, when benefts are ndependent across goods, or when the goods are complements, tradable quanttes are never the most effcent nstrument. Ths theory s appled to dynamc polluton problems, and suggests that permt bankng should be allowed for stock pollutants, but not for flow pollutants. These results ndcate that many regulatons, ncludng the current sulfur doxde tradng program and proposed greenhouse gas regulatons, are neffcent. Roberton Wllams Department of Economcs Unversty of Texas Austn, TX 7871 and NBER rwllam@eco.utexas.edu

3 I. Introducton Ths paper examnes the effcency of tradable quantty regulatons: regulatons that fx the aggregate quantty of a set of goods, but provde flexblty n how to dvde that aggregate quantty among the goods n the set. Such polces are typcally mplemented va a system of tradable permts; n the case of polluton regulaton, for example, the goods n queston are the polluton emssons from each of a group of frms, and a system of tradable emssons permts would cap total emssons. Tradable quanttes are already wdely used n envronmental regulaton, and are ncreasngly consdered as an opton n other regulatory settngs. 1 Ths ncreasng popularty s due n part to effcency consderatons. Economsts typcally vew tradable quanttes as beng more effcent than fxng the quantty of each good. However, ths paper shows that s not generally true; unless the goods are perfect substtutes, fxed quanttes may be more effcent than tradable quanttes, even f tradng occurs at ratos that reflect the expected ratos of margnal benefts between goods, not at a one-to-one rato. These results ndcate that many exstng and proposed regulatons are neffcent, ncludng the current sulfur doxde tradng program and proposed regulatons for ntrous oxdes and greenhouse gases. The key dfference between ths paper and pror research on the effcency of tradable permts s that ths paper explctly models asymmetrc nformaton; frms know ther costs, but the regulator does not. To do so, t uses a framework based on Wetzman s (1974) semnal paper, whch compared two archetypal regulatons: settng a prce for each good versus fxng the quantty of each good. Ths paper extends that comparson to nclude tradable quanttes. Asymmetrc nformaton s an essental part of ths comparson; wthout t, prce, quantty, and tradable quantty nstruments yeld dentcal outcomes. 1 Uses of tradable permts nclude lmts on foregn mports to New Zealand (see McAfee et al., 1999), sulfur doxde under the 1990 Clean Ar Act Amendments, and smog under the Los-Angeles-area RECLAIM program. Such permts have also been proposed for a wde range of other pollutants, ncludng greenhouse gases and ntrous oxdes, as well as for non-envronmental problems, ncludng prce ncreases leadng to nflaton (Vckrey, 199 and 1993) and natonal budget defcts wthn the European Unon (Casella, 1999). Furthermore, some other polces are functonally equvalent, n that they fx an aggregate quantty but allow the market to determne the allocaton across partcular goods. One example would be a purchase contract that would be satsfed by provdng a certan combned total of any of several dfferent goods; ths could be done explctly, or smply by wrtng a contract wth very loose specfcatons. 1

4 Wetzman showed that the relatve effcency of prce regulaton versus quantty regulaton depends on the relatve slopes of the margnal beneft and margnal cost curves. If the margnal cost curve s steeper than the margnal beneft curve, prce regulaton wll be more effcent, whereas f the margnal beneft curve s steeper, quantty regulaton wll be more effcent. The ntuton for ths result s that prce regulaton provdes frms wth more flexblty. Ths flexblty reduces expected costs, but also reduces expected benefts. The steeper the margnal cost curve, the greater the expected cost savngs, but the steeper the margnal beneft curve, the greater the reducton n expected benefts. Wetzman mentons tradable quanttes n a footnote (p. 490), statng that they would be better than fxed quanttes n the case n whch all goods are dentcal, and that n ths case the queston reduces to one of whether the aggregate quantty should be controlled va prce or quantty regulaton. However, the paper does not explctly show ths, and does not dscuss the effcency of tradable quanttes when the goods are not dentcal. Subsequent work ether gnores tradable quanttes altogether, or t assumes that tradable quanttes wll be superor to fxed quanttes, and then apples a Wetzman-style analyss at the aggregate level to compare prces and tradable quanttes. 3 Nether approach s satsfactory; the former gnores a potentally attractve opton, whle the latter as ths paper wll show s not vald except n the specal case n whch all the goods are perfect substtutes. Ths paper explctly compares tradable quanttes to prces and to fxed quanttes n a settng wth asymmetrc nformaton. It shows that the choce between tradable quanttes and prces or fxed quanttes depends on the relatve slopes of the margnal cost and margnal beneft curves as was true for the choce between prces and fxed quanttes and also on the degree of substtutablty or complementarty between the goods. If the goods are perfect substtutes, tradable quanttes are always Adar and Grffn (1976) provde a clear graphcal analyss of ths result. 3 For example, Newell and Pzer (00) and Hoel and Karp (1999) take the former approach, whle Oates, Portney, and McGartland (1989) take the latter. An excepton s the smultaneous and ndependent paper by Yates (000), whch explctly compares tradable and non-tradable polluton permts n a framework smlar to the one used n ths paper. However, ths paper consders a broader set of nstruments and a broader range of cases, whch allows t to develop an ntuton that s absent n Yates.

5 more effcent than fxed quanttes. In ths case, the decson between tradable quanttes and prces depends on the relatve slopes of the aggregate margnal cost and margnal beneft curves. When the goods are substtutes, but not perfect substtutes, then each of the three nstruments could be the most effcent. If the margnal cost curve s substantally steeper than the margnal beneft curve, prces wll be most effcent. If the margnal beneft curve s substantally steeper, then fxed quanttes wll be most effcent. In an ntermedate range, where the slopes are smlar, tradable quanttes wll be most effcent. The less substtutable the goods are, the smaller that range wll be. Fnally, f margnal benefts are ndependent across goods, or f the goods are complements, then tradable quanttes are never the most effcent of the three nstruments. These results represent a sgnfcant theoretcal contrbuton, and have substantal practcal mportance. To provde one example of the latter, ths paper brefly consders the queston of whether polluton permts should be bankable (whether frms should be allowed to emt less polluton n one year n return for beng allowed to emt more n future years). It shows that several current and proposed polces (ncludng the sulfur doxde permt program and the Kyoto global warmng agreement) are neffcent, n that they allow bankng n cases when t shouldn t be allowed, or don t allow t n cases when t should be allowed. The next secton of the paper develops a model of regulaton of producton of a set of goods, derves expressons for the relatve effcency of the three nstruments, and then consders the mplcatons of these expressons. The thrd secton demonstrates an applcaton of these results to the problem of whether polluton permts should be bankable. The fnal secton offers conclusons and suggestons for future research. II. The Model Ths secton develops a smple model of regulaton of producton of a set of goods, and uses that model to nvestgate the relatve effcency of three types of regulatory nstruments: prces, fxed quanttes, and tradable quanttes. The structure of ths model s smlar to the model used n 3

6 Wetzman (1974), though t consders tradable quanttes n addton to prces and fxed quanttes. A. Assumptons A set of N goods s assumed. The cost of producng good s gven by C ( q,θ ), where q s the vector of quanttes of each good and θ s a vector of random varables. The total beneft s gven by Bq (). 4 The functons C () and B() are assumed to be contnuous and twce-dfferentable. To assure a unque nternal soluton, I adopt the standard assumptons on C and B; specfcally, C s ncreasng and strctly convex n q, B s ncreasng and concave, and for any θ, margnal costs exceed margnal benefts for q suffcently large, and are less than margnal benefts for q = 0. 5 Frms are assumed to mnmze costs, subject to any regulatory constrants. In the absence of regulaton, then, producton wll be set such that the margnal cost s zero. The regulator can choose one of three regulatory nstruments a set of prces, a set of quanttes, or a system of tradable quanttes. 6 Frms wll set producton to mnmze costs, subject to meetng the regulatory constrant. Whle frms have perfect nformaton about ther costs, the regulator does not know the realzaton of θ when settng the regulaton. For the prce nstrument, ths mples the frst-order condton 4 For smplcty, I gnore uncertanty n benefts. Wetzman (1974) showed that f cost uncertanty and beneft uncertanty are not correlated, then such uncertanty has no effect on the choce between prce and quantty regulaton. Stavns (1996) further analyzed the case n whch the uncertanty s correlated. Analogous results hold here, so ths paper s results would be unchanged f benefts were also uncertan, as long as the beneft uncertanty s not correlated wth the cost uncertanty. 5 For consstency wth pror work, especally Wetzman (1974), ths model assumes that each good has a postve margnal cost and beneft; thus, for polluton regulaton, each good s not polluton from a partcular frm, but the reducton n polluton by that frm. 6 Several papers (for example, Roberts and Spence, 1976, Wetzman (1978), and Kaplow and Shavell, 1997), have suggested more complex regulatory nstruments that wll generally be more effcent than the smple nstruments consdered here. Greenwood and McAfee (1991) derve the optmum over all possble regulatory mechansms, though they consder only the specal case n whch the beneft functon s addtvely separable. In practce, those more complex nstruments are rarely used, and so ths paper focuses only on smple regulatory nstruments. 4

7 (1) C q = p where p s the vector of prces set by the regulator. Under the quantty nstrument, frms set producton equal to the requred quantty. () q = q where q s the vector of quanttes set by the regulator. Under the system of tradable quanttes, the frst-order condton s (3) C q = r λ and the overall quantty constrant s (4) r q = Q where r s a vector of tradng ratos 7 (wth r unts of good beng tradable for r unts of good j), Q s the total quantty requred, and λ s the shadow prce for that total quantty constrant. 8 Note that r and Q are set by the regulator, whle λ wll be determned by r, Q and the cost functons. The goal of the regulator s to maxmze the expected value of benefts mnus costs EBq () ( ) C q,θ. 9 In the absence of uncertanty on the part of the regulator, all three nstruments would be set to 7 Many studes have shown that when the regulated goods are not dentcal for example, when polluton damages dffer by locaton or tme perod tradable quantty regulatons should not allow one-to-one trades, but nstead should allow trades at a rato that reflects the rato of margnal benefts between goods. See, for example, McGartland and Oates (1985) n the context of emssons permt tradng, Klng and Rubn (1997) or Leby and Rubn (000) n the context of emssons permt bankng, or Casella (1999) n the context of tradable lmts on EU budget defcts. All of these studes assume that the regulator has perfect nformaton. 8 In the case of a tradable emssons permt program, Q s the total number of permts ssued, λ s the market prce of a permt, r s the number of permts requred for one unt of emssons, and equaton (4) s the market-clearng condton for the permt market. In other contexts, such as a contract wth loose specfcatons, λ would be a shadow prce wthn the frm. 9 Note that dstrbutonal or other consderatons could be ncorporated nto the beneft and cost functons, so the assumpton that the regulator maxmzes benefts mnus costs does not necessarly mply that the regulator s concerned only wth economc effcency. 5

8 acheve the same optmal vector of quanttes. For the prce nstrument, ths would requre the regulator to set the prce for each good such that the prce equals the margnal beneft from that good. (5) p = B q q=q * = C q q=q * Under the system of tradable quanttes, the tradng ratos would be set proportonal to the margnal beneft from each good; thus, wthout loss of generalty, they can be set equal to the margnal beneft (6) r = B q q= q The optmal total quantty would be gven by (7) Q = r q * Wthout uncertanty, then, each nstrument would acheve a frst-best outcome. 10 Wth uncertanty, however, the three nstruments wll have dfferent effects. Let q denote the ex ante optmal vector of quanttes the vector of quanttes that maxmzes the expected value of benefts mnus costs defned by (8) E C q q =q = B q q =q In order to proceed further, assume that the uncertanty s suffcently small to justfy a secondorder approxmaton for the cost and beneft functons n the neghborhood of q. Wthout loss of generalty, quanttes wll be normalzed such that at q, the margnal beneft and the expected margnal cost each equal 1. Ths mples that (9) Cq,θ ( ) C( q,θ)+ ( 1 + α () θ )ˆ q + 1 and γ ˆ q 10 The prce nstrument could also acheve a frst-best outcome f prces were not fxed, but could vary based on the quantty of producton of each good. Note, however, that ths would be more complex than just allowng a nonlnear prce schedule, because the prce for a partcular good would have to depend not just on the quantty of that good, but also on the quanttes of all other goods as well. 6

9 q + 1 (10) Bq () B()+ q ˆ j β j q ˆ q ˆ j where ˆ q s the devaton n quantty from q, gven by (11) ˆ q = q q β and γ are the matrces of the second dervatve of the beneft functon and of the expected value of the second dervatve of the cost functon, respectvely, each evaluated at q. (1) γ = E C q q =q (13) β j = B q q j q =q and α () θ s a functon that translates the vector of random varables θ nto a vertcal shft n the margnal cost curve for good. The unt normalzaton mples that α () θ has an expected value of zero. (14) E( α () θ )= 0 Fnally, the model assumes that that the dstrbuton of α () θ s ndependent across goods. B. Quanttes Under Dfferent Instruments Under fxed quanttes, producton s smply equal to q. But producton wll generally devate from q under each of the other two nstruments. Takng a dervatve of the approxmaton to the cost functon (9) gves an approxmaton to margnal cost for good. (15) C q 1 + α ()+γ θ q ˆ Combnng the approxmaton to margnal cost (15) wth the frst-order condton for the prce nstrument and rearrangng gve an approxmaton for the devaton from q under the prce nstrument (16) q ˆ p 1 α θ γ () The optmal prce s equal to the margnal beneft at quantty q, whch, as noted earler, s 7

10 normalzed to 1. Thus, (17) p 1 Substtutng (17) nto (16) yelds () (18) q ˆ p α θ where ˆ q p γ s the devaton from q that wll result under the ex ante optmal prce. A smlar process yelds an expresson for quantty under a system of tradable quanttes. Substtutng the expresson for margnal cost (15) nto the frst-order condton for tradable quanttes (3) and rearrangng yeld (19) q ˆ r λ 1 α θ γ () The tradable quantty nstrument wll set r (the vector of tradng ratos) to be proportonal to the margnal beneft at quantty q, whch was normalzed to 1. Wthout loss of generalty, then, the tradng ratos are also normalzed to 1. (0) r 1 The optmal total quantty wll then equal the sum over all goods of q. (1) Q = q Substtutng (19), (0), and (1) nto (4) and rearrangng (usng (11)) yeld an expresson for the shadow prce of the quantty constrant under the tradable quantty system (whch wll be the equlbrum permt prce n the case of tradable permts) () λ 1 +ψ α () θ γ where ψ s the slope of the aggregate margnal cost curve, whch s equal to the horzontal sum of the margnal cost curves over all goods (n the case of tradable permts, ths wll be the same as the market demand curve for emssons permts). 8

11 (3) ψ = 1/ 1 γ Substtutng (0) and () nto (19) yelds t (4) q ˆ t where q ˆ ψ α j () θ γ j γ jj α () θ γ s the devaton from q that wll result under the system of tradable quanttes. Note that f γ s equal for all (an assumpton that wll be mantaned for much of the analyss that follows), ths t expresson reduces to qˆ ( α ( θ ) s) γ, where s s the sample mean of the α s, gven by s = j α j ( θ ) N. Ths s smlar to the quantty devaton under the prce nstrument (18). However, under the prce nstrument, the quantty devaton for good depends on the margnal cost devaton for good, whereas under the tradable quantty nstrument, the quantty devaton for good depends on the dfference between the margnal cost devaton for good and the average of the margnal cost devatons over all goods. C. Comparatve Advantages of Dfferent Instruments Followng Wetzman (1974), defne the comparatve advantage of one nstrument over another as the dfference between the two nstruments of the expected value of benefts mnus costs. 11 The comparatve advantage of one of the other two nstruments relatve to fxed quanttes s thus gven by [ ( () C( q,θ))] (5) =E( B() q Cq,θ ( )) B q Substtutng the approxmatons to the cost and beneft functons (1) and (13) nto (5) and smplfyng, usng (8) and (11) gve 11 In calculatng the comparatve advantage of one nstrument over another, the model assumes that the regulatory parameters quota levels, tax rates, tradng ratos, and the number of permts allocated are set optmally ex ante. If the parameters for one or more of the nstruments are not set optmally, ths may change the relatve effcency of the varous nstruments. 9

12 (6) E α ()ˆ θ q + 1 β j q ˆ q ˆ j 1 j q γ ˆ For now, assume that β and γ are symmetrc across goods. Ths assumpton wll be relaxed later, but t s useful now n yeldng clear, smple results. Specfcally, assume that γ = γ, that β = β, and that β j = φβ j. 1 The parameter φ determnes whether each good s a complement or a substtute for each of the other goods. When φ s postve, the goods are substtutes. For φ = 1, they are perfect substtutes. When φ = 0, the margnal beneft of a partcular good s ndependent of the quanttes of the other goods; thus, the goods are nether complements nor substtutes. Fnally, f φ s negatve, the goods are complements. Usng these assumptons, substtutng the expresson for the quanttes under the ex ante optmal prces (18) nto (6) and smplfyng gve an expresson for the comparatve advantage of prces relatve to quanttes. (7) PQ β + γ σ γ where σ s the varance of the margnal cost of good at quantty q, whch s approxmated by the varance of α () θ (8) σ = E C q q=q E C q q=q E[ α () θ ] Equaton (7) corresponds to the expresson for the comparatve advantage of prces over quanttes from Wetzman (1974). Ths expresson mples prces wll be more effcent than quanttes when the slope of the margnal cost curve s greater than the slope of the margnal beneft curve. Quanttes wll be more effcent when the margnal beneft curve s steeper. Prces allow frms more flexblty, whch reduces expected costs. Ths s especally mportant when margnal cost s very senstve to quantty. However, to the extent that margnal beneft s senstve to quantty, ths flexblty 1 Note that, because unts are normalzed such that the frst dervatves of the beneft and cost functons are equal across goods, ths assumpton that the second dervatves are equal s less restrctve than t mght at frst appear. 10

13 reduces expected benefts. Substtutng (4) nto (6) and smplfyng, usng the assumpton that costs and benefts are symmetrc across goods, gve an expresson for the comparatve advantage of tradable quanttes relatve to fxed quanttes. (9) TQ β( 1 φ)+γ N 1 σ γ N Ths expresson s somewhat smlar to the expresson for the comparatve advantage of prces relatve to quanttes (7); when the margnal beneft curve s relatvely steep, fxed quanttes are more effcent, whereas when the margnal cost curve s relatvely steep, tradable quanttes are more effcent. Ths reflects the fact that, lke prces, tradable quanttes offer frms more flexblty than do fxed quanttes. Unlke n expresson (7), however, the degree of complementarty or substtutablty between dfferent goods matters. The more substtutable the goods are for each other (the closer φ s to 1), the more effcent tradable quanttes wll be relatve to fxed quanttes. Ths occurs because under tradable quanttes, producng more of one good allows less producton of other goods. No such lnkage occurs under prces or fxed quanttes. If the goods are close substtutes for each other, ths s effcent; the more of one good s produced, the lower the effcent quantty wll be for each of the other goods. If the margnal beneft of a partcular good s ndependent of the quanttes of the other goods (φ s 0) ths lnkage provdes no advantage. And f the goods are complements, then ths lnkage s a dsadvantage; n that case, the more of a partcular good s produced, the hgher wll be the effcent quanttes of the other goods. Fnally, the comparatve advantage of tradable quanttes relatve to prces s equal to the dfference between the comparatve advantage of tradable quanttes relatve to fxed quanttes and the comparatve advantage of prces relatve to fxed quanttes. (30) TP = E[ ( B() q t Cq ( t,θ )) ( Bq ( p ) Cq ( p,θ )]= TQ PQ 11

14 Substtutng the expressons for PQ (7) and TQ (9) nto (30) and smplfyng gve (31) TP 1 β[ 1 +φ( N 1) ]+ γ N γ σ Agan, ths expresson depends on the relatve slopes of the margnal beneft and margnal cost curves. Tradable quanttes provde less flexblty than do prces (because whle the amount of each good can vary, the total amount of all goods cannot), and thus tradable quanttes are relatvely effcent when the margnal beneft curve s steep relatve to the margnal cost curve. And agan, the more substtutable the goods are for each other, the more effcent tradable quanttes wll be. These three expressons for comparatve advantage (expressons (7), (9), and (31)) represent the key fndngs of ths paper. Comparng these three expressons shows whch nstrument wll be the most effcent n any partcular stuaton. Frst, consder the case n whch the goods are substtutes for each other, so φ > 0. In ths case, each of the three nstruments could be the most effcent, dependng on the relatve slopes of the margnal beneft and margnal cost curves. When the margnal cost curve s steep relatve to the margnal beneft curve when γ > β[ 1 +φ( N 1) ] prces wll be the most effcent nstrument. When the margnal beneft curve s relatvely steep when γ < β( 1 φ) fxed quanttes wll be the most effcent nstrument. And for some mddle range, when the slopes of the two curves are smlar when β[ 1 +φ( N 1) ]> γ > β( 1 φ) tradable quanttes wll be most effcent. Note that the sze of ths range depends on how substtutable the goods are for each other. When they are very close substtutes, there wll be a wde range of slopes for whch tradable quanttes would be most effcent. In the extreme case n whch the goods are perfect substtutes, φ = 1. In ths case, tradable quanttes wll always be more effcent than fxed quanttes, regardless of the slopes of the margnal beneft and margnal cost curves. The ntuton behnd ths result s smple. When benefts take ths form, only the expected cost dffers between the two nstruments; the benefts are the same. Tradable quanttes have a lower expected cost, because they equalze margnal cost across goods, whereas fxed quanttes do 1

15 not allow such flexblty. 13 In ths perfect substtutes case, tradable quanttes wll be more effcent than prces f γ N < β. The left-hand term, γ, s the slope of the market-wde margnal cost curve, whch s the horzontal sum N of the margnal cost curves over all goods. Gven the symmetry assumpton, that slope s equal to the slope of the margnal cost curve for each good, dvded by the number of goods. Thus, n ths case, tradable quanttes are more effcent than prces f the margnal beneft curve s steeper than the marketwde margnal cost curve. If the margnal cost curve s steeper, then prces are more effcent. Ths s an analogue of Wetzman's (1974) result comparng prce and quantty nstruments; quantty regulaton s more effcent f the margnal beneft curve s steeper than the margnal cost curve, whle prce regulaton s more effcent f the margnal cost curve s steeper. The dfference s that for multple goods regulated through a system of tradable quanttes, the approprate margnal cost curve s the market-wde margnal cost curve, whereas when usng a fxed quantty nstrument to regulate a sngle good, t s the margnal cost curve for that good. In contrast, when goods are only very weakly substtutable for each other, there wll be only a very small range of slopes for whch tradable quanttes would be the most effcent nstrument. Indeed, when margnal benefts are ndependent of the quanttes of other goods (φ = 0 ) or the goods are complements (φ < 0 ) tradable quanttes wll never be the most effcent nstrument, regardless of the relatve slopes of the margnal beneft and margnal cost curves. In ths case, when the margnal beneft curve s steeper than the margnal cost curve, fxed quanttes wll be the most effcent nstrument, and when the margnal cost curve s steeper, prces wll be the most effcent nstrument. In such cases, the way that tradable quanttes lnk the producton of the dfferent goods provdes 13 An alternatve ntuton s that under tradable quanttes, the goods are perfect substtutes to the frms n meetng the regulaton. Thus, when the goods are perfect substtutes n the beneft functon, tradable quanttes match frms ncentves wth margnal benefts. 13

16 no advantage (and s actually a dsadvantage when the goods are complements). 14 Thus, when flexblty s benefcal (when the margnal cost curve s relatvely steep), the most effcent nstrument wll be the one that provdes the most flexblty prces. When flexblty s harmful (when the margnal beneft curve s relatvely steep), the most effcent nstrument wll be the one that provdes the least flexblty fxed quanttes. Of course, ths assumes that all three types of regulaton are feasble. If, for example, prce nstruments are nfeasble (due to poltcal consderatons, perhaps), then tradable quanttes may be the best remanng opton even when the goods are complements. These results dffer substantally from pror work n the context of tradable permts. 15 That lterature has suggested that as long as tradng ratos are set based on the rato of margnal benefts between the goods not smply allowng one-for-one trades tradable quanttes wll always domnate fxed quanttes, even when the goods are not perfect substtutes. The problem s that unless the goods are perfect substtutes, the rato of margnal benefts between any two goods wll depend on the quanttes of those two (and perhaps other) goods. Thus, even f the tradng ratos are set optmally ex ante, they wll n general not be optmal ex post. Because none of that pror work explctly ncorporated asymmetrc nformaton, t dd not recognze ths ssue. These results have far-reachng mplcatons. They suggest, for example, that tradable polluton permts wll be very effcent for a pollutant wth global effects, such as greenhouse gas emssons, where emssons from dfferent locatons are perfect substtutes, but generally should not be used to regulate 14 When the goods are complements, t mght be effcent to lnk the producton of dfferent goods, but not n the way that tradable quanttes lnk producton. Ths could be accomplshed by settng a prce for producton of a bundle consstng of a certan amount of each good, rather than a separate prce for each good. Thus, to the regulated frms, the goods would be perfect complements, just as tradable quanttes cause the goods to be perfect substtutes to the regulated frms. Such a bundle prce polcy logcally completes the set of archetypal nstruments. Fxed quanttes fx both the total amount of producton and the composton of that producton, prces allow both to vary, tradable quanttes fx the total amount, but allow the composton to vary, and ths bundle prce nstrument fxes the composton but allows the total amount to vary. Such a polcy would be more effcent the more complementary the goods are to each other. When the goods are perfect complements, t would domnate prces, n the same way that tradable quanttes domnate fxed quanttes when the goods are perfect substtutes. Ths nstrument s rare n envronmental polcy, but s common n some other contexts, such as purchase contracts for complementary goods. 15 See, for example, McGartland and Oates (1985), Klng and Rubn (1997), Leby and Rubn (000), or Casella (1999). 14

17 pollutants wth very localzed effects. Smlarly, frms should be allowed to trade reduced emssons of one pollutant for ncreased emssons of a dfferent pollutant only f the two are close substtutes. 16 In the context of purchasng contracts, these results suggest that frms should wrte very flexble contracts when purchasng a set of closely substtutable goods, but that such contracts would be hghly neffcent when purchasng complementary goods. D. Comparatve Advantages of Dfferent Instruments n the General Case Removng the assumpton that benefts and costs are symmetrc across goods causes the expressons for the comparatve advantages of the dfferent nstruments to become somewhat more complex. Followng the same steps used to derve equaton (7) yelds an expresson for the comparatve advantage of prces over fxed quanttes n ths general case (3) PQ σ γ ( β +γ ) Ths expresson s very smlar to (7), but now, because the slopes of the margnal cost and beneft functons dffer across goods, the comparatve advantage of prces relatve to fxed quanttes depends on the weghted averages of those slopes. The basc ntuton behnd the result s unchanged; prces provde more flexblty than fxed quanttes, and when the average margnal cost curve s steeper than the average margnal beneft curve, that flexblty s an advantage. Followng the same steps used to derve equaton (9) yelds an expresson for the comparatve advantage of tradable quanttes over fxed quanttes n ths general case (33) TQ σ γ ( β +γ ) 1 ψ γ + ψ β γ j β j γ jj ψ j k β j β jk γ jj γ kk The frst term n ths expresson s the value of the extra flexblty afforded by tradable quanttes relatve to fxed quanttes, and s generally smlar to expresson (3), dependng on the weghted 16 Montero (001) analyzes regulaton of multple pollutants through tradable permts. Its model and results correspond to those n ths paper, for the specal case n whch φ=0, and t does not consder polluton taxes. 15

18 averages across goods of the slopes of the margnal cost and margnal beneft curves. When the average margnal cost curve s steeper than the average margnal beneft curve, that flexblty s an advantage, and so tradable quanttes are relatvely effcent. The second term reflects whether the goods are complements or substtutes for each other. When the goods are substtutes, ths term wll be postve, reflectng the fact that tradable quanttes are relatvely effcent n ths case. When the margnal beneft from a partcular good s ndependent of the quanttes of other goods, ths term wll equal zero. And when the goods are complements, ths term wll be negatve, because n ths case, the way that tradable quanttes lnk producton across goods s a dsadvantage. The thrd term reflects devatons from symmetry across goods n benefts, costs, and the degree of uncertanty. Ths term s dffcult to nterpret except n specal cases, but wll go to zero f the rato of benefts to costs equal for all goods, f the degree of uncertanty s equal for all goods, or f those two parameters are uncorrelated across goods. For the specal case n whch benefts are ndependent across goods, ths term s proportonal to the weghted covarance across goods between σ γ and β γ, weghted by 1 γ. Fnally, substtutng (3) and (33) nto (30) yelds an expresson for the comparatve advantage of tradable quanttes relatve to prces. (34) TP σ γ β + γ ( ) ψ γ +ψ β γ j β j γ jj ψ j k β j β jk γ jj γ kk Ths expresson and ts nterpretaton are very smlar to expresson (33), except that prces provde more flexblty than tradable quanttes, and thus the frst term wll be negatve when flexblty s an advantage when the average margnal cost curve s steeper than the average margnal beneft curve. The second and thrd terms are the same as n expresson (33). When the goods are perfect substtutes, the thrd term n (33) and (34) equals zero, and the second 16

19 ψ term n each expresson reduces to β 1. Thus, expresson (33) wll reduce to TQ γ σ γ 1 ψ γ γ, whch s always postve. Thus, just as n the smpler case consdered earler, when the goods are perfect substtutes, tradable quanttes are more effcent than fxed quanttes, regardless of the slopes of the margnal cost and margnal beneft curves. And expresson (34) wll reduce to TP σ [ ] β +ψ. Agan, the sgn of ths expresson depends on whether the margnal γ beneft curve s steeper than the market-wde margnal cost curve. E. Caveats A few caveats are n order, due to the smplfyng assumptons that have been made. Frst, ths analyss assumes that the margnal cost of each good does not depend on quantty of any other good. If ths assumpton s volated, then ths wll affect the effcency of tradable quanttes. If ncreasng the quantty of one good were to lower the margnal cost of other goods, then the optmal quantty of those goods would rse. However, under tradable quanttes, producng more of one good wll decrease the requred quantty of the other goods, and so tradable quanttes would be less effcent than ths model ndcates. In the opposte case f producng more of one good ncreases the margnal cost of the other goods then tradable quanttes wll be more effcent than ths model ndcates. Second, the analyss assumes that the cost shocks are uncorrelated across goods. Postvely correlated shocks would tend to make tradable quanttes behave more lke fxed quanttes. If cost shocks are perfectly correlated, then they wll cause the shadow prce on the overall quantty constrant (the permt prce under tradable permts) to change, but wll have no effect on the quantty of each good; thus, the effcency of tradable quanttes wll be the same as that of fxed quanttes. Negatvely correlated shocks would have the opposte effect, magnfyng the dfference between fxed and tradable quanttes. 17

20 III. An Applcaton to the Problem of Bankng and Borrowng of Polluton Permts Ths secton apples the earler results to the problem of whether polluton permt programs should allow permt bankng and borrowng (emttng less polluton now n exchange for beng allowed to emt more n the future, or vce-versa). These are, n effect, permt trades between dfferent tme perods. Thus, the dfferent goods n ths case are reductons n polluton n dfferent tme perods. Ths secton presents a smple model of polluton regulaton n a dynamc settng, frst for a flow pollutant and then for a stock pollutant, and uses the tools developed n the prevous secton to examne the effcency of bankable/borrowable permts (tradable quanttes) relatve to polluton taxes and fxed quotas. A. Regulaton of a Flow Pollutant Frst, consder regulaton of a flow pollutant that s, a pollutant that causes damage only n the perod n whch t s emtted. The dscounted benefts and costs of polluton abatement are gven by ( ) (35) B = δ Bq = 0 ( ) (36) C = δ Cq,θ =0 where δ s the dscount factor, whch s assumed to be the same for frms and for the regulator. The cost and beneft functons are assumed to be tme-statonary. Substtutng the second dervatves of (35) and (36) nto the defntons of γ (1) and β (13) yelds (37) β () () = B q γ C q Because the statonarty of the cost and beneft functons n each perod mples that q wll be constant over tme, the rato of γ to β wll also reman constant, so β γ = β jj γ jj, j. Together wth the fact that n ths case β j = 0 j, that mples that the second and thrd terms n (33) and (34) wll equal zero. 18

21 Then a comparson of the frst terms of (33) and (34) shows that ether taxes or quotas wll be more effcent than tradable permts. If β γ < 1 f the margnal cost curve for abatement n each perod s steeper than the margnal beneft curve then taxes wll be more effcent than bankable permts. If the margnal beneft curve s steeper, then quotas wll be more effcent. Thus, as a general rule, permt bankng should not be allowed for flow pollutants; f t s better to use a quantty nstrument than a prce nstrument, then separate quotas for each perod wll be more effcent than bankable permts. In practce, nearly all emssons permt programs allow frms to bank permts. For example, the 1990 Clear Ar Act Amendments, Calforna's Low-Emsson Vehcle Program, the CAFE vehcle fuel-economy standards, and the leaded gasolne phaseout program have all allowed permt bankng n some form. There are some exceptons. If the beneft and cost functons are suffcently non-statonary that the rato β γ vares sgnfcantly across perods, and that rato s hgher n perods for whch the regulator has more uncertanty about costs, then the thrd term n (33) and (34) wll be postve and thus t s possble for bankable permts to be the most effcent of the three nstruments. A more promsng argument for the use of bankable permts to regulate a flow pollutant would be f β γ < 1, and thus emssons taxes are the most effcent nstrument, but taxes are not poltcally feasble. In ths case, bankable permts would be the next best choce. 17 B. Regulaton of a Stock Pollutant Consder nstead the case of regulaton of a stock pollutant one for whch polluton damages n a 17 Adjustment costs are sometmes rased as another argument for allowng bankng. A full analyss of the mpact of adjustment costs s beyond the scope of ths model, but the ntuton for ths case s relatvely smple. If adjustment costs are sgnfcant, the margnal cost of abatement n one perod wll depend on abatement n other perods; abatement n a gven perod lowers the margnal cost of abatement n later perods. As dscussed n secton II.E, however, ths non-separablty of the cost functon wll actually work aganst tradable permts. It mples that abatement n any gven perod wll rase the optmal amount of abatement n later perods. However, wth permt bankng, more abatement n one perod mples just the opposte; less abatement s requred n later perods. 19

22 gven perod depend on the amount of emssons n all prevous perods. In ths case, the beneft functon wll take the form. (38) B = δ B s j q j =0 j=0 where s s the stock decay factor. The term n parentheses s the stock of polluton at tme. Unlke n the case of the flow pollutant, benefts are not ndependent across tme perods. Nor s abatement n any gven perod a perfect substtute for abatement n any other perod, though t s close. To see ths, consder the rato of the margnal benefts of abatement between two perods. (39) B = δ k s k B q k = B q B q = s j + max(,j ) k=mn(,j ) k = δ k s k B δ k s k B The rato of the margnal beneft from abatement n perod to that n perod j s a constant plus the dscounted value of avoded damages between the two perods dvded by the margnal beneft n perod. Ths rato s the margnal rate of substtuton of abatement between the two perods. If t s constant, then abatement s perfectly substtutable between the two perods. Thus, f the second term n (39) were zero, then abatement would be perfectly substtutable between the two perods. Ths s the case n the lmt as both the stock decay factor and the dscount factor go to one. In that case the damages that occur after perod j for whch abatement s perfectly substtutable between the two perods swamp the effects that occur n the nterval between perod and perod j. Thus, n the lmt as the stock decay factor and dscount factor go to one, the results from secton II.C for the case of perfect substtutes wll apply. Whle nether factor wll actually equal one n practce, as long as both factors are close to one, abatement n one perod wll be a close substtute for abatement n other perods. In that case, bankable permts wll typcally be more effcent than emssons quotas. Ths suggests that bankng and tradng should be allowed n mplementng the Kyoto protocol on greenhouse gas emssons. Smlarly, the choce between bankable permts and taxes would depend on the relatve slopes of 0

23 the margnal beneft and margnal cost curves, where the approprate margnal cost curve s the margnal cost curve for emssons reductons n all perods. Ths result s also mportant for polcy decsons. Newell and Pzer (00) and Hoel and Karp (1999) each found that taxes are generally superor to emssons quotas for the regulaton of long-lved stock externaltes, because the margnal beneft curve s relatvely flat compared to the margnal cost curve for a sngle perod. However, nether study consdered bankable permts. The margnal cost curve for abatement over all perods wll be much flatter than the curve for any sngle perod, snce t s the horzontal sum (weghted by dscounted margnal damages) of the margnal cost curves across all perods. As a result, whle taxes wll generally be superor to emssons quotas, there s a sgnfcant range of cases n whch bankable emssons permts wll be more effcent than taxes. Thus, the argument for usng taxes to regulate long-lved stock pollutants s not so clear-cut once bankable permts are consdered. IV. Conclusons Ths paper develops a framework to evaluate the relatve effcency of three dfferent types of nstruments prces, fxed quanttes, and tradable quanttes n controllng producton of a set of goods. It then llustrates these results by consderng the problem of whether polluton permts should be bankable. The paper shows that f the goods are substtutes, each of the three nstruments could be the most effcent, dependng on the relatve slopes of the margnal beneft and margnal cost curves. When the margnal cost curve s steep relatve to the margnal beneft curve, prces wll be the most effcent nstrument. When the margnal beneft curve s relatvely steep, fxed quanttes wll be the most effcent nstrument. And for some mddle range, when the slopes of the two curves are smlar, tradable quanttes wll be the most effcent nstrument. The more substtutable the goods are for each other, the larger s the range n whch permts are the most effcent nstrument. When the goods are perfect substtutes, tradable quanttes wll always be more effcent than fxed quanttes. The choce between tradable quanttes and prces n ths case depends on the relatve 1

24 slopes of the aggregate margnal cost and margnal beneft curves. If the aggregate margnal cost curve s steeper, then prces are more effcent, whle f the aggregate margnal beneft curve s steeper, tradable quanttes are more effcent. Ths s the aggregate-level analogue of Wetzman s (1974) result. In contrast, when the margnal beneft from a partcular good s ndependent of the quanttes of the other goods, or the goods are complements, ether fxed quanttes or prces are typcally more effcent than tradable quanttes. Only f the goods are suffcently asymmetrc, both n the amount of uncertanty about costs and n the slopes of the margnal beneft and margnal cost curves, and these asymmetres are strongly correlated, can tradable quanttes be the most effcent nstrument. These results hold even when tradng ratos are set based on the rato of expected margnal benefts between the goods, rather than merely allowng one-for-one trades. Thus, ths result dffers sharply from past work, whch has suggested that tradable quanttes wll domnate fxed quanttes even when goods are not perfect substtutes, as long as tradng ratos are proportonal to margnal benefts. The queston of whether frms should be allowed to bank and borrow emssons permts s used to llustrate the mplcatons of these results. The case of a flow pollutant where polluton damage depends only on the current flow of polluton emtted, not on past or future emssons corresponds to the case n whch the margnal beneft from a partcular good s ndependent of the quanttes of other goods produced. Thus, n ths case, ether non-bankable permts (fxed quanttes) or polluton taxes (prces) wll be more effcent than bankable permts (tradable quanttes). For a stock pollutant where polluton damage comes from the stock of accumulated past emssons polluton reducton n one tme perod s a close substtute for reductons n other perods. Thus, bankable permts wll generally be more effcent than non-bankable permts, though ths wll not always be true; non-bankable permts wll be more effcent f the stock decay rate and dscount rate are suffcently low, and the margnal beneft curve s substantally steeper than the margnal cost curve. The choce between bankable permts and polluton taxes wll depend on the relatve slopes of the ntertemporal margnal cost curve (whch wll be substantally flatter than the margnal cost curve n a sngle perod) and the margnal beneft curve.

25 These results could have tremendous mportance for envronmental polcy. They show that many current and proposed envronmental regulatons are not optmal. Emssons permt programs for flow pollutants typcally allow bankng (for example, the lead phase-out and sulfur doxde permt programs), whle the Kyoto agreement to reduce carbon doxde emssons a stock pollutant does not yet allow for such ntertemporal flexblty. These results also show that emssons permt programs should allow tradng for pollutants wth global effects, such as greenhouse gases, where polluton n one locaton s a perfect substtute for polluton elsewhere, but not for pollutants wth very localzed effects. But most current proposals for mplementng the Kyoto agreement lmt emssons tradng between countres, whle some exstng and proposed US regulatons allow tradng even for relatvely localzed pollutants (such as the proposed use of tradable permts to regulate emssons of ntrogen oxdes). 18 There are a number of promsng drectons for future research on ths topc. The tradable quantty regulatons analyzed n ths paper allow unlmted tradng, but the framework developed here could also be used to consder polces that lmt tradng n some way. Such polces are common n practce. For example, some emssons tradng programs allow tradng only wthn a local area, rather than across the entre unverse of polluton sources. The queston of how large these tradng areas should be s qute nterestng. Smlarly, Roberts and Spence (1976) and Wetzman (1978) showed that hybrd nstruments nstruments that combne elements of both prces and quanttes, such as emssons permts wth a prce cap wll always be more effcent than a pure prce or pure quantty nstrument. A smlar result should hold for hybrds between fxed and tradable quanttes. And fxed/tradable hybrd nstruments are relatvely common compared to prce/quantty hybrds; most emssons permt programs allow permt bankng, but not permt borrowng, and Los Angeles s RECLAIM smog-tradng program allows permts to be traded from coastal locatons to nland locatons, but not vce-versa. These programs behave lke 18 Note, however, that f polluton taxes and other prce nstruments are poltcally nfeasble, tradable permts may be the best avalable opton even for a localzed pollutant. 3

26 tradable quanttes when frms want to trade one drecton (bankng, or tradng nland), but behave lke fxed quanttes when frms would lke to trade the opposte drecton (borrowng, or tradng to the coast). Further research could determne whether such hybrd nstruments are more effcent, as s the case for prce/quantty hybrds. Fnally, emprcal applcatons of ths framework could be very useful for polcy. Gven the ncreasngly rch data on envronmental regulaton that s becomng avalable, t should not be too dffcult to estmate the parameters of nterest for a partcular case. 4

27 References Adar, Zv and James M. Grffn, Uncertanty and the Choce of Polluton Control Instruments, Journal of Envronmental Economcs and Management, 3: Casella, Alessandra, Tradable Defct Permts: Effcent Implementaton of the Stablty Pact n the European Monetary Unon Economc Polcy, 9:33-61 Greenwood, Jeremy and R. Preston McAfee, Externalttes and Asymmetrc Informaton, Quarterly Journal of Economcs, 106:103-1 Hoel, Mchael and Larry Karp, Taxes Versus Quotas for a Stock Pollutant, workng paper, Unverstes of Olso and Calforna-Berkeley Kaplow, Lous and Steven Shavell, On the Superorty of Correctve Taxes to Quantty Regulaton, NBER Workng Paper #651 Klng, Catherne and Jonathan Rubn, 1997, Bankable Permts for the Control of Envronmental Polluton, Journal of Publc Economcs, 64: Leby, Paul and Jonathan Rubn, 000, Bankable Permts for the Control of Stock and Flow Pollutants: Optonal Intertemporal Greenhouse Gas Emsson Tradng, Envronmental and Resource Economcs, forthcomng McAfee, R. Preston, Wendy Takacs and Danel Vncent, 1999, Tarffyng Auctons, RAND Journal of Economcs 30: McGartland, Albert and Wallace Oates, 1985, "Marketable Permts for the Preventon of Envronmental Deteroraton," Journal of Envronmental Economcs and Management 1:07-8 Montero, Juan-Pablo, 001, Multpollutant Markets, RAND Journal of Economcs, 3: Newell, Rchard and Wllam Pzer, 00, Regulatng Stock Externaltes Under Uncertanty, Journal of Envronmental Economcs and Management, forthcomng Roberts, Marc and Mchael Spence, 1976, Effluent Charges and Lcenses Under Uncertanty, Journal of Publc Economcs5: Rubn, Jonathan, "A Model of Intertemporal Emsson Tradng, Bankng, and Borrowng," Journal of Envronmental Economcs and Management, 31:69-86 Stavns, Robert, 1996, Correlated Uncertanty and Polcy Instrument Choce, Journal of Envronmental Economcs and Management, January 1996 Vckrey, Wllam, 199. Chock-Full Employment wthout Increased Inflaton: A Proposal for Marketable Markup Warrants Amercan Economc Revew 8: Vckrey, Wllam, Today s Task for Economsts Amercan Economc Revew: 83:1-10 Wetzman, Martn, 1974, Prces vs. Quanttes, Revew of Economc Studes 41: