Appendix 6.1 The least-cost theorem and pollution control
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1 Appendx 6.1 The least-cost theorem and polluton control nstruments Ths appendx s structured as follows. In Part 1, we defne the notaton used and set the scene for what follows. Then n Part 2 we derve a necessary condton for polluton control to be cost-effectve: that s, to attan any gven target at least cost. An EPA has several nstruments avalable for attanng a polluton (or polluton abatement) target. Here we consder three classes of nstrument: quanttatve regulatons (a varant of command and control) n Part 3; an emssons tax (Parts 4 and 5); an emssons abatement subsdy (Part 6); and transferable emssons permts (Part 7). Collectvely, Parts 3 to 7 take the reader through what an EPA would need to know, and how t could operate each of those nstruments, n order to acheve a target at least cost. Fnally n Part 8, we generalse prevous results to the case of a non-unformly-mxng pollutant. Part 1 Introducton There are pollutng frms, ndexed = 1,...,. Each frm faces a fxed output prce and fxed nput prces, and maxmses profts by an approprate choce of output level (Q ) and emsson level ( ). Emssons consst of a unformly mxng pollutant, so that the source of the emsson s rrelevant as far as the polluton damage s concerned. Let ^ be the maxmsed proft of the th frm n the absence of any control over ts emsson level and n the absence of any charge for ts emssons. Ths s ts unconstraned maxmum proft level. At ths unconstraned proft maxmum the frm s emsson level s ˆ. 1
2 Let be the maxmsed proft of the th frm when t s requred to attan a level of emssons < ˆ. Ths s ts constraned maxmum level of profts. To reduce emssons, some addtonal costs wll have to be ncurred or the frm s output level must change (or both). The constraned proft level wll, therefore, be less than the unconstraned proft level. That s, < ^. We next defne the frm s abatement costs, C, as unconstraned mnus constraned profts: C = ^ Abatement costs wll be a functon of the severty of the emssons lmt the frm faces; the lower s ths lmt, the greater wll be the frm s abatement costs. Let us suppose that ths abatement cost functon s quadratc. That s C = + 2 (6.4) We llustrate ths abatement cost functon n Fgure ote that that the abatement cost functon s defned only over part of the range of the quadratc functon. Abatement costs are zero when the emsson lmt s set at ˆ, the level the frm would have tself chosen to emt n the absence of control. Abatement costs are maxmsed when = 0, and so the frm s prohbted from producng any emssons. 2
3 C 2 C α β δ 0 ˆ Fgure 6.14 The frm s abatement cost functon. Two thngs should be sad about equaton 6.4. Frst, as each parameter s ndexed by, abatement costs are allowed to vary over frms. Second, the arguments that follow do not depend on the abatement cost functon beng quadratc. We have chosen that functonal form for expostonal smplcty only. Part 2 The least-cost theorem We now consder the problem of an envronmental protecton agency (EPA) meetng some standard for total emssons (from all frms) at the least cost. Let denote the predetermned total emsson target. In the expressons that follow, the varables are to be nterpreted as endogenous, the values for whch are not predetermned but emerge from the optmsng exercse beng undertaken. The problem can be stated as 1 n C subect to (6.5) 1 3
4 The Lagrangan for ths problem s L C 1 1 (6.6) The necessary condtons for a least-cost soluton are L β 2 0, 1,2,..., (6.7) and L 0 (6.8) ] Equatons 6.7 and 6.8 gve + 1 equatons n + 1 unknowns. Solvng these smultaneously gves each frm s emsson lmt, (whch now should be regarded as the optmsed emssons lmt for the frm), and the optmsed shadow prce of the polluton constrant (the Lagrange multpler). Snce s constant over all frms, t can be seen from equaton 6.7 that a least-cost polluton abatement programme requres that the margnal cost of abatement be equal over all frms. There s a trcky ssue relatng to sgns n equaton 6.7. otce that an ncrease n corresponds to a relaxaton of a polluton target (a decrease n requred abatement) so the term ( 2 ) s the margnal cost of a reducton n polluton abatement beng requred of frm. It wll therefore be a negatve quantty. Ths can be verfed by lookng at the slope of the C functon n Fgure By multplyng equaton 6.7 through by mnus one, we obtan 2 = (6.7 ) 4
5 Here the term on the left-hand sde ( 2 ) s the frm s margnal cost of an ncrease n polluton abatement, a postve quantty. It follows from 6.7 that s also a postve quantty. Ths s consstent wth the text of ths chapter and the prevous one. Part 3 Least-cost polluton control usng quanttatve regulaton If the EPA knew each frm s abatement cost functon (that s, t knew C for = 1,..., ), then for any total emsson standard t seeks,, the system of equatons 6.7 and 6.8 could be solved for for each frm. The EPA could then tell each frm how much t could emt. The total quantty of emssons would, from equaton 6.8, be reached exactly, and the target would, as the above theorem shows, be attaned at least cost. Part 4 Least-cost polluton control usng an emssons tax As an alternatve to settng quanttatve emssons controls on each frm, an emsson tax could be used. If the EPA knew each frm s abatement cost functon, then for any total emsson standard t seeks,, the system of equatons 6.7 and 6.8 could be solved for the value of the shadow prce of the polluton constrant,. ote that, unlke, ths shadow prce s constant for each frm. The EPA could then set a tax at a rate of t per unt of emssons and charge each frm ths tax on each unt of polluton t emtted. Proft-maxmsng behavour would then lead each frm to produce emssons, the least-cost soluton. To see why ths should be so, note that n the absence of any quantty constrant on emssons, proftmaxmsng behavour n the face of an emssons tax mples that the frm wll mnmse the sum of ts abatement costs and polluton tax costs. That s, the frm chooses to mnmse CT, the total of ts abatement and tax costs: 5
6 CT = C + t = t The necessary condton s CT t 2 0, 2,..., (6.9) Clearly, f t n equaton 6.9 s set equal to n equaton 6.7, the necessary condtons 6.7 and 6.9 are dentcal, and so the tax nstrument acheves the total emssons target at least cost. Part 5 What role s there for a tax nstrument where each frm s abatement cost functons are not known? In general, the EPA wll not know abatement costs. However, f an arbtrarly chosen tax rate, say t, s selected, and each frm s charged that rate on each unt of emsson, then some total quantty of emssons, say, wll be realsed at least cost. Of course, that amount wll n general be dfferent from. Only f t = t wll be dentcal to. An teratve, tral-and-error process of tax rate change may enable the EPA to fnd the necessary tax rate to acheve a specfc target. Part 6 Least-cost polluton control usng an emssons-abatement subsdy Another method of obtanng a least-cost soluton to an emssons target s by use of abatement subsdes. Suppose a subsdy of s s pad to each frm on each unt of emssons reducton below ts unconstraned proft-maxmsng level, ˆ. Then proft-maxmsng behavour mples that the frm wll maxmse total subsdy recepts less abatement costs. That s, the frm maxmses CS = s( ˆ ) C = s( ˆ ) ( + 2 ) 6
7 The necessary condton s CS 2 s 0, 1,2,..., (6.10) whch, after multplyng through by 1, s dentcal to equaton 6.9 f s = t. So, once agan, f s n equaton 6.10 s set equal to n equaton 6.7, the necessary condtons 6.7 and 6.10 are dentcal, and so the subsdy nstrument acheves the total emssons target at least cost. oreover, ths result demonstrates that n terms of ther effects on emssons, a tax rate of t per unt of emssons s dentcal to a subsdy rate of s per unt of emssons abatement, provded s = t. Part 7 Least-cost polluton control usng transferable emssons permts Suppose that the EPA ssues to each frm lcences permttng L 0 unts of emssons. Frms are allowed to trade wth one another n permts. The th frm wll trade n permts so as to mnmse the sum of abatement costs and trade-acqured permts: CL C P L L 0 2 P L L 0 (6.11) where P s the market prce of one emsson permt. Gven that L s the quantty of emssons the frm wll produce after trade we can wrte ths as 0 CL C P( L L ) L L P( L L ) 2 0 (6.12) The necessary condton for mnmsaton s 7
8 CL L L P β 2 0, 1,2,..., (6.13) whch can be nterpreted as the frm s demand functon for permts. If the EPA sets a total emssons target of then s the total supply of permts and 1 L 0 L P 1 L (6.14) ow compare equatons 6.13 and 6.14 wth equatons 6.7 and 6.8. These are dentcal f P = (rememberng that L = ). oreover, comparson of equaton 6.13 wth equatons 6.9 and 6.10 shows that P = t = s. So by an ntal ssue of permts (dstrbuted n any way) equal to the emssons target, the EPA can realse the target at least cost. oreover, t can do so wthout knowledge of ndvdual frms abatement cost functons. Part 8 Least-cost abatement for a non-unformly-mxng pollutant The target of the EPA s now n terms of ambent polluton levels rather than emsson flows. Specfcally the EPA requres that A A for 1,... 1 d (6.15) The problem for the EPA s to attan ths target at least cost. We deal wth the case where the same ambent target s set for each receptor area. Ths problem can be stated as n C subect to A d A 1 1 (6.16) for 1,..., The Lagrangan for ths problem s 8
9 L C μ 1 A d μ J A d1 1 (6.17) where C = + 2 The necessary condtons for a least-cost soluton are L 1,2,..., and (6.18) J β 2 1 (μ d) 0, l L (6.19) μ A d 0 for 1,..., J 1 The system of equatons 6.18 and 6.19 conssts of + J equatons whch can be solved for the + J unknowns (, = 1,..., and, = 1,..., J). Equaton 6.18 can be wrtten as J β 2 1 (μ d ), 1,2,..., (6.20) Then after multplyng through by 1, usng C to denote the th frm s margnal cost of abatement, and expandng the sum on the rght-hand sde, we obtan C μ d μ d...μ d, J J 1, 2,..., (6.21) The par of equatons 6.20 and 6.21 can be compared wth the soluton for the unformly mxng polluton case, equaton 6.7 multpled by 1. 9
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