Controlling Urban Air Pollution Caused by Households: Uncertainty, Prices, and Income

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1 March, 2011 Controllng Urban Ar Polluton Caused by Households: Uncertanty, Prces, and Income Carlos A. Chávez a,*, John K. Stranlund b, Walter Gómez c a Unversdad de Concepcón, Departamento de Economía, Vctora 471, Barro Unverstaro, Concepcón, Chle. b Unversty of Massachusetts-Amherst, Department of Resource Economcs, 214 Stockbrdge Hall, Amherst, MA 01003, USA. c Unversdad de la Frontera, Departamento de Ingenería Matemátca, Avenda Francsco Salazar 01145, Temuco, Chle. Abstract: We examne the control of ar polluton caused by households burnng wood for heatng and cookng n the developng world. Snce the problem s one of controllng emssons from nonpont sources, regulatons are lkely to be drected at household choces of wood consumpton and combuston technologes. Moreover, these choces are subtractons from, or contrbutons to, the pure publc good of ar qualty. Consequently, the effcent polcy desgn s not ndependent of the dstrbuton of household ncome. Snce t s unrealstc to assume that envronmental authortes can make lump sum ncome transfers part of control polces, effcent control of ar polluton caused by wood consumpton entals a hgher tax on wood consumpton and a hgher subsdy for more effcent combuston technologes for hgher ncome households. Among other dffcultes, mplementng a polcy to promote the adopton of cleaner combuston technologes must overcome the seemngly paradoxcal result that effcent control calls for hgher technology subsdes for hgher ncome households. Keywords: Effcency, urban ar polluton, nonpont polluton, envronmental polcy, uncertanty * Correspondence to: Carlos Chávez, Departamento de Economía, Unversdad de Concepcón, Vctora 471, Barro Unverstaro, Concepcón, Chle. Phone: (56-41) , Fax: (56-41) , E-mal: cchavez@udec.cl. 1

2 Controllng Urban Ar Polluton Caused by Households: Uncertanty, Prces, and Income 1. Introducton Ar polluton caused by households burnng wood for heatng and cookng s a serous concern n many urban areas of the developng world. To llustrate the problem consder the cty of Temuco, the captal cty of the Araucanía regon n southern Chle. Ths cty contans about 350,000 people n about 86,000 households. About 80% of the households n the cty report usng wood as an energy source. The consumpton of wood s not partcularly concentrated at any ncome level. The 30% of the cty s households n the mddle of the ncome dstrbuton also account for 30% of the cty s annual wood demand. Smlarly, the 20% of households wth the hghest ncomes consume about 22% of the total wood fuel annually (Gómez et. al 2009). It has been estmated that 90% of total emssons of suspended partculate matter n Temuco s caused by households burnng 500,000 cubc meters of wood annually. There are about 100,000 smoke stacks connected to cookng and heatng stoves n the cty (Comsón Naconal del Medo Ambente, CONAMA 2007, Chávez et. al 2009). The number of days that the concentraton of total suspended partculate matter (PM 10 ) exceeded the 24-hour average Chlean legal lmt of 150 µg/m 3 was 11 n 2005, 15 n 2006, 21 n 2007, 36 n 2008, and 37 n Furthermore, durng the 2009 season, the maxmum daly average concentraton on a 24 hour bass for the cty was n the range of µg/m 3. Durng the worst day of the 2009 season, the concentraton of PM 10 was measured at about 6090 µg/m 3 at 5 p.m., ncreasng to 6240 µg/m 3 by 11 p.m. that same day. To put these fgures n context, the Ar Qualty Gudelnes of the World Health Organzaton (WHO) call for lmtng the mean 24-hour concentraton of PM 10 n urban areas to 50 µg/m 3 (WHO 2005). 2

3 Moreover, there are at least two reasons to be pessmstc about ar qualty n Temuco. Frst, generatng household energy wth kerosene or lquefed gas the two closest substtutes for wood n central-southern Chle s about 5 to 8 tmes more expensve than usng fuel wood (Gómez-Lobo 2005). Second, the supply of wood from natve forests surroundng the ctes s also ncreasng, as many small-scale farmers harvest wood to sell n urban areas. Even though an offcal fgure of the number of supplers of wood to Temuco s not avalable, the Natonal Forest Servce had regstered about 470 producers. Most of these producers are owners of small plots of land less than 100 hectares (Lobos 2001 and Von Baer et. al. 2002). 1 Stuatons lke ths pose major challenges for envronmental regulatory authortes at local and natonal levels. The great number of ndvdual sources of polluton makes drect emssons montorng mpractcal; thus, ar polluton from households s best characterzed as a nonpont polluton problem. The nablty to montor emssons mples that regulaton s lkely to be drected at emssons nputs, n partcular wood consumpton and household combuston technologes. In addton, regulators face a great deal of uncertanty because of stochastc weather effects on the concentraton of ar polluton and human health, and because of lmted nformaton about how households use combuston technologes and the wood nput. Fnally, choces of wood consumpton or more effcent combuston technologes are subtractons from or contrbutons to a pure publc good (.e., ar qualty). It s well known by publc economsts that one cannot separate effcent provson of a publc good from the dstrbuton of ncome. (For 1 Ar polluton problems caused by households burnng wood for heatng have also occurred n some regons of developed countres. Examples nclude the cty of Chrstchurch n New Zealand (Barna and Gmson 2002, Envronment Canterbury 2009, and Wlton et. al. 2006), the cty of Launceston n Australa (Kesby et. al. 2002, Luhar et. al. 2006), Sacramento Calforna (Sacramento Metropoltan Ar Qualty Management Dstrct 2006 and 2008), the town of Lbby Montana (HPBA 2008), and Brtsh Columba n Canada (Mnstry of Envronment-Brtsh Columba 2005). 3

4 example, see Laffont 1988, chapter 2). The man objectve of ths paper s to examne the role that ncome dstrbuton plays n the determnaton of polces to control urban ar polluton from households. 2 Our approach s to consder optmal taxes for wood consumpton for household energy and subsdes for more energy effcent (less pollutng) combuston technologes. We recognze that wood taxes or technology subsdes may not be mplementable n partcular nstances, because of montorng, other nformaton problems, and poltcal realtes. Nevertheless, dervng the optmal taxes and subsdes can yeld mportant nsghts nto the problem of controllng ar 86 polluton from households. 3 In partcular, we show that f authortes are able and wllng to make unrestrcted lump sum transfers of ncome among households, then these prces should be roughly equal across households. However, t s unrealstc to assume that lump sum ncome redstrbutons can be made part of a polcy to control household ar polluton. In the absence of lump sum transfers and assumng dmnshng margnal utlty of consumpton of a prvate good, an optmal polcy wll force more of the burden of emssons control onto wealther households. The reason s that dong so serves to reduce the expected costs of the polcy by equalzng the weghted expected margnal dsutlty of control across households. Ths s analogous to the standard prescrpton to mnmze aggregate abatement costs of statonary ndustral polluton sources by equalzng ther margnal abatement costs. Snce t s effcent for hgher ncome households to take on relatvely more of the abatement burden n our context, an optmal polcy 2 Our approach mght apply, wth sutable modfcatons, to other household polluton ssues, ncludng automoble emssons, and households garbage dsposal and recyclng behavor. Addressng how ncome dstrbuton affects optmal polcy n these other contexts may be frutful areas for future research. 3 There s a proposed plan for the cty of Temuco that offers subsdes to nduce voluntary adopton of more effcent combuston technologes. The man feature s a subsdy-based stove exchange program to nduce the renovaton of 12,000 stoves over a ten year perod (CONAMA 2007, Chapter II, Artcle 10). 4

5 wll charge a hgher wood tax on wealther households, as well as offer them a hgher subsdy for the purchase of more effcent combuston technologes. Ths last result may seem paradoxcal, but t s mportant to realze that the technology subsdy s not meant to correct ncome dspartes ts purpose s to ad n the effcent control of household polluton. Whle the role that ncome dstrbuton plays n the settng of envronmental taxes has been well-studed by envronmental economsts, t s usually n the context of unform taxes. Therefore, the queston of how ncome dstrbuton affects the dstrbuton of control responsbltes s usually not addressed. 4 One excepton s Chchlnsky and Heal (1994) who examne how the global dstrbuton of ncome affects the effcent dstrbuton of greenhouse gas abatement to confront clmate change. They show that the famlar prescrpton that margnal abatement costs should be equal across countres only holds f countres commt themselves to large-scale transfers of ncome from rcher to poorer natons. In the absence of these transfers the effcent dstrbuton of abatement requres that rcher countres undertake more abatement than would be mpled by equalzng margnal abatement costs. 5 Our contrbuton s that we examne the effect of ncome dstrbuton on the control of a nonpont polluton problem generated by households. Chchlnsky and Heal (1994) assume perfect nformaton about all benefts and costs of greenhouse gas control, whle regulators have only lmted nformaton about these elements n the control of ar polluton caused by 4 For example, ths s true of Sandmo s (1975) classc artcle on optmal commodty taxaton n the presence of an externalty. Other examples nclude Bovenberg and Goulder (2002, secton 5) and Fullerton and Wolverton (2005). Income dstrbuton s not an ssue n these latter artcles because they assume dentcal households. 5 Also see Heal, Chchlnsky and Starret (1993). Sheeran (2006) seeks to clarfy certan aspects of Chchlnsky s and Heal s analyss. There are a number of polces that charge dfferental tax rates, but we are not aware of any that vary by ncome levels. Dfferental carbon emssons taxes that vary by ndustry have been ntroduced n parts of Europe (Baranzn et. al and Bye and Nyborg 2003). Snce these taxes are not drected at households they are very dfferent from the taxes we model n ths paper. 5

6 households. More mportantly, Chchlnsky and Heal (1994) assume that greenhouse gas emssons are perfectly observable. For our problem t s not practcal to montor household emssons; thus, control polces are lkely to focus on controllng the nputs of the producton of polluton, namely the fuel nput and the combuston technology. There s an extensve lterature on nonpont polluton control, of whch Shortle and Horan (2001) have provded a valuable revew. However, we are not aware of any study that consders the mpact of ncome dstrbuton on the optmal control of a nonpont pollutant. Our work makes ths contrbuton to the nonpont control lterature n general, as well as to the study of the control of ar polluton caused by households n the developng world n partcular. The rest of the paper proceeds as follows. In the next secton we lay out a model of the control of ar polluton caused by households burnng wood for energy, and derve the optmal taxes on wood consumpton and subsdes for more effcent combuston technologes. The man results of the paper are contaned n secton 3 where we examne the nterdependence between optmal polces and ncome dstrbuton. We conclude n secton 4 wth an extended dscusson of several mplementaton ssues that our results generate A model of regulatng ar polluton from households Consder an urban area consstng of a large number of households that produce energy by burnng wood. Each household makes a small contrbuton to polluton levels, but the resultng aggregate level s dangerous for the communty. Due to the large number of pollutng households, an envronmental authorty s unable to measure emssons from each household. 6

7 Consequently, we explore the desgn of taxes to nduce lower wood consumpton and subsdes to promote the adopton of cleaner-burnng wood combuston technologes Bascs of the model Let there be n households ndexed by. Each household produces energy by combnng a combuston technology, whch we denote as x 1, and the amount of wood used, x 2. Thus, the producton of energy n a household s gven by [1] c = c( x 1, x2). Assume that c s ncreasng n fuel use x 2. We nterpret x 1 as an ndex of avalable wood combuston technologes and order the technologes accordng to ther effectveness n producng energy gven an amount of fuel. Assume that more effectve combuston technologes are ndcated by hgher levels of x 1 so that c s ncreasng n x 1. For analytc convenence we assume that x 1 s a contnuous varable. 7 There may very well be uncertanty from a regulator s perspectve about c, perhaps because of unobservable skll levels or wood qualty, but we gnore ths possblty because households energy producton s not our prmary concern. 6 Another opton would be to pursue an ambent polluton tax and subsdy as frst proposed by Segerson (1988). Ths polcy would nvolve household-specfc penaltes f the ambent concentraton of ar polluton surpasses some lmt and subsdes f the concentraton s lower than that lmt. Despte the nterest n these mechansms for controllng nonpont polluton, we are not aware of an nstance n whch they have been appled. Shortle and Horan (2001) dscuss several practcal lmtatons of these mechansms. 7 Assumng that the combuston technology s a contnuous varable may not be too far from the truth. Combuston technologes can vary along several dmensons ncludng type, sze, vntage, and so on. Treatng each combnaton of characterstcs as a dstnct technology can produce a large number of technologes that, when ordered accordng to energy-producng effcency, can be modeled as beng on a contnuum. 7

8 Our man concern s that energy producton creates emssons of polluton, r, as a byproduct. Thus, household emssons depend on the combuston technology, the wood nput, 153 and a random parameter σ (from a regulator s pont of vew) that captures unobserved varaton n how the combuston equpment s actually used: [2] r = r( x 1, x2, σ ). Suppose that r s ncreasng n the amount of wood used, but s decreasng n the combuston technology (under the assumpton that a more productve combuston technology burns more cleanly and uses less wood for the amount of energy generated). The random parameter σ represents households preferences and sklls that affect how the combuston equpment s used, and consequently the producton of emssons. For example, emssons are an ncreasng functon of the mosture content of the wood used, and households choose wood wth 162 varyng mosture content. 8 Furthermore, households can adjust the amount of wood burned per perod of tme by varyng the ar flow n and out of the combuston equpment. Reducng arflow ncreases burn tme but also ncreases emssons. Overfllng the combuston chamber wth wood to avod frequent refllng can also produce hgher emssons (Klppel and Nussbaumer 2007; Nussbaumer 2003 and 2006). Envronmental qualty n a cty depends on the ambent concentraton of polluton. The man pollutants produced from burnng wood are ntrogen oxde, carbon monoxde, and fne partculate matter. To smplfy matters, we assume that household emssons produce a sngle pollutant. In addton we assume that ths pollutant s unformly mxed; that s, the ambent concentraton of the pollutant depends only on the sum of household emssons. Ths assumpton 8 It has been reported n the cty of Temuco that some households actually prefer to use wetter wood even though t has lower calorc content, because most wood burns slower and lasts longer (Chávez et. al. 2009, CONAMA-DICTUC 2008, CONAMA 2007, Nussbaumer 2006). 8

9 makes the locaton of emssons rrelevant, whch s a reasonable approxmaton for polluton problems n the ctes that motvate ths work. 9 Let the ambent concentraton of the pollutant be n [3] a= a rk( xk1, xk2, σk), θ, k= 1 whch s ncreasng n aggregate household emssons so t s ncreasng n ndvdual household emssons as well. (We use k to ndex households n specfc summatons, n ths case the sum of emssons s over households, to avod potentally confusng notaton at certan ponts n the paper. The populaton of households consumng wood s the same as the populaton affected by exposure to the pollutant). The parameter θ s a random factor that captures the effect of weather condtons on ar qualty. For example, wndy days result n lower ambent polluton for a gven level of emssons because polluton s blown away and dspersed. However, cold can produce a thermal nverson that traps pollutants at ground level, resultng n hgher ambent concentraton of polluton. The utlty functon for a household s denoted by [4] u = u ( c, y, a, µη, ). Suppose that utlty s ncreasng n energy use c and the consumpton of a prvate commodty y, but s decreasng n ambent polluton a. Note that snce the ambent concentraton of polluton affects each household s utlty, ther choces of combuston technology and wood consumpton can be vewed as contrbutons to and subtractons from the local publc good of ar qualty. The varable µ s a random parameter that captures the noton 9 As noted n the ntroducton our work s motvated by household polluton problems n urban areas of the central-southern regon of Chle. Most of these areas are located n the central valley, on relatvely flat land surrounded by small hlls, away from the Los Andes mountan range. The many emsson ponts are qute unformly dstrbuted wthn each cty. We should note that the model can be easly modfed to consder non-unformly mxed pollutants. 9

10 that weather affects household energy choces. Thnk of households generatng more energy to heat ther homes when t s colder. We assume for smplcty that the dstrbutons of the random weather parameters, µ and θ, are known to all households as well as to the regulator. The last term n a household s utlty functon, η, s a random term from the regulator s perspectve that represents unobservable household characterstcs that affect ts producton of energy, lke the nsulaton of the house, preferences for the type of combuston equpment, and preferences for warmth. Each household faces a set of prces for the combuston technology and wood nput, whch we denote as p1 and p 2, respectvely. To keep the model tractable and focused we assume that these prces are fxed throughout. Relaxng ths assumpton would not change the fundamental nsghts of ths paper about the role of ncome dstrbuton n the desgn of polces to control household ar polluton. The prce of the prvate consumpton good s equal to one. We further smplfy the analyss by assumng that each household has exogenous ncome w, whch s taxed at an exogenous rate z. Income taxes are fxed because real envronmental agences do not have the authorty to make ncome transfers a part of envronmental regulatons. We wll see later that ths assumpton plays a very mportant role n polcy formaton. We can obtan prncples for controllng urban ar polluton from households by dervng the optmal subsdes on combuston technologes and taxes on wood. Antcpatng that optmal taxes/subsdes could vary across households, denote the subsdy on household s combuston technology as t 1 0, and ts tax on each unt of wood used as t 2 0. Household specfc after subsdy/tax prces on combuston technologes and wood are p 1 + t 1 and p2 + t 2, = 1,..., n, respectvely. 10

11 Gven the after subsdy/tax prces on combuston and wood, a household s budget constrant s [5] w(1 z) y ( p1 t 1) x 1 ( p2 t2) x2 = We do not examne how ncome taxes affect the optmal prces on combuston technologes and wood nput, but we do assume that the government can fund the household polluton control program. To that end assume n n n wz t x + t x = 1 = 1 = 1 ; that s, the government s ncome tax recepts are suffcent to meet the revenue requrements of the household polluton control program. If aggregate subsdy payments exceed the aggregate taxes on the wood nput, then the dfference s fnanced out of ncome tax recepts. If wood tax recepts exceed subsdes for more effcent combuston technologes, then the excess s smply added to the government s budget Household energy nput choces A household s decson problem s to choose a combuston technology, wood nput, and consumpton of the prvate good to maxmze ts expected utlty subject to [1], [3], and [5]. That s, a household chooses x 1, x 2, and y to solve: [6] max E ( u ( c, y, a, µη, )) s.t. c = cx ( 1, x2) w = y + ( p + t ) x + ( p + t ) x n a = a rk( xk1, xk2, σk), θ, k= 1 11

12 where w = w(1 z ) s the household s after tax ncome, E denotes the expectaton operator for household. Ths expectaton s wth respect to the jont dstrbuton of ( µθη,, 1,..., η 1, η+ 1,..., ηn, σ1,..., σ 1, σ+ 1,..., σn), condtonal on η and σ. Substtutng the constrants of [6] nto ts objectve allows us to wrte the household s problem as choosng x 1 and x 2 to maxmze E u c x x w p t x p t x a r x x σ θ µη. n [7] ( 1, 2), ( 1+ 1) 1 ( 2 + 2) 2, k( k1, k2, k),,, k= 1 Assume that [7] s contnuously dfferentable, that t s strctly concave n x 1 and x 2, and that optmal choces of these nputs are not zero. Then, the followng frst-order condtons determne each household s optmal choces of combuston technology and wood consumpton, gven these choces by all the other households n the cty: u c u u a r [8] E ( pj + tj ) + = 0, = 1,..., n, j = 1, 2. c xj y a r xj These frst order condtons mplctly defne Bayes-Nash best response functons, and the soluton to these 2n equatons, assumng that one exsts, gves us a Bayes-Nash equlbrum dstrbuton of wood use and combuston technologes n the urban area. The frst-order condtons reveal that each household wll optmally choose the level of combuston technology and nput use consderng three elements; the margnal utlty of the combuston technology or wood use n the generaton of energy, the margnal reducton n utlty from the reducton n spendng on other prvate goods, and the margnal mpact that the choce of technology or wood use has on the polluton damage the household experences. To the extent that the household can detect a change n ambent polluton from ts own emssons (.e., a r > 0 ), the choce of a more effcent combuston technology reduces the polluton damage t suffers (because 12

13 r < ) whle an ncrease n ts use of wood ncreases the damage t suffers (because x1 0 r / > 0 x2 ), holdng the choces of all other households constant. Note that a r > 0 s the same for all, because of our assumpton that ambent polluton depends only on the sum of households emssons Effcent household choces Optmal wood taxes and combuston technology subsdes nternalze the external costs and benefts of the households choces. To determne the taxes and subsdes that wll nduce an effcent allocaton of energy choces, we frst derve an effcent allocaton of wood use and combuston choces by maxmzng an expected Bergson-Samuelson socal welfare functon. 10 That s we choose ( x 1, x 2), = 1,,n, to solve 267 [9] max E n λ u ( c, y, a, µη, ), g = s.t. c = cx ( 1, x2), = 1,..., n, w = y + px 1 1+ px 2 2, = 1,..., n, n a = a rk( xk1, xk2, σk), θ, k= 1 In the objectve of [9], λ > 0, = 1,..., n, are exogenous household utlty weghts. If every household s utlty has equal weght n the socal welfare functon, then the utlty weghts are λ = 1/ n for = 1,,n, but other dstrbutons of the weghts are possble. 10 The alternatve method for fndng effcent allocatons s to maxmze the expected budgetconstraned utlty of one household whle holdng the expected budget-constraned utltes of the other households constant. Our results do not depend on maxmzng a socal welfare functon, because the alternatve methods are functonally equvalent. 13

14 Eg refers to the expectaton operator for the envronmental authorty, whch s wth respect to the jont dstrbuton of µ, θ, σ,..., σ, η,..., η ). Our ncluson of household budget constrants n ( 1 n 1 n ths problem (nstead of, for example, posng producton functons for combuston technologes and fuel wood) reflects the noton that an envronmental agency wth lmted authorty must desgn a control polcy, gven the exstng dstrbuton of ncome and ncome taxes, and the supples of combuston technologes and fuel wood. In partcular, we do not aggregate the communty s ncome because the authorty cannot make lump sum transfers of ncome among households. Gven the utlty weghts, f a soluton to the program exsts t wll dentfy one of the many possble effcent allocatons. Utlty weghts are mportant, because partcular effcent outcomes are assocated wth partcular weghts. That s, all of the effcent allocatons obtanable gven the exstng dstrbuton of ncome can be dentfed by varyng the utlty weghts. For the exstng dstrbuton of ncome and utlty weghts, assume that the soluton to [9] s characterzed by the followng frst order condtons: u c u u a r uk a r [10] λeg pj + + λkeg = 0, c xj y a r x j k a r x = 1,..., n and j = 1,2. j To nterpret the frst order condton t may be more nformatve to rewrte them n the followng way: 292 [11] λ u c u u a r, = 1,..., n and j = 1,2. n k Eg pj = Eg λk c xj y k= 1 a r xj Ths s a modfcaton of the usual Lndahl-Bowen-Samuelson condtons for the effcent provson of a publc good. The modfcatons come from two sources: (1) the context of 14

15 household heatng and cookng decsons that affect the publc good of urban ar qualty n the developng world, and (2) the uncertanty n the model the stochastc weather effects on ar polluton and household utlty as well as uncertanty about the use of combuston technologes and wood nput. The left sde of [11] s the government s weghted expectaton of a household s margnal non-envronmental net beneft of employng nput j. On the rght sde of [11] s government s expectaton of the mpact of that decson on the weghted sum of margnal dsutltes from urban ar polluton. Note that the sgn of the rght sde of [11] depends on whether the energy nput s the combuston technology or the wood nput. For the combuston k k < 0, a r > 0, n technology (j = 1), the rght sde of [11] s negatve because λ ( u a) and r x1 < 0. The negatve sgn ndcates that the envronmental authorty s expectaton of aggregate polluton damage s decreasng when a household employs a more effcent combuston technology. On the other hand, the rght sde of [11] s postve for the wood nput (j = 2), because r x 2 > 0. The postve sgn ndcates that expected aggregate polluton damage s ncreasng n a household s use of wood. k= Effcent wood taxes and combuston technology subsdes Havng characterzed effcent allocatons of combuston technologes and wood consumpton, we now determne optmal taxes on wood consumpton and subsdes for more effcent combuston technologes that wll nduce these choces. Clearly, these wll be second-best optmal polces because of the authorty s nablty to make lump sum ncome transfers. A smple modfcaton of [8] gves us the government s expectaton of how households wll respond to taxes and subsdes, t j = 1,..., n and j = 1,2. Smply replace the E 's n [8] wth 15

16 Eg to reflect the fact that government uses ts own expectaton of households decson crtera to determne optmal taxes and subsdes. 11 After dong ths substtute the result nto [10] and rearrange terms to obtan 321 [12] t j = k Eg λk k a r xj E g u u λ y a r, = 1,..., n and j = 1, The denomnator of [12] s the regulator s expectaton of household s margnal utlty of consumpton of the prvate good tmes the weght assgned to that household. Ths term s postve. The numerator s the envronmental authorty s expectaton of the margnal mpact of household s choce of nput j on weghted aggregate damage experenced by all the other households. Ths s the expected external cost (n the case of wood consumpton) or beneft (n the case of combuston technology) from household s decson. Ths term s negatve f j s the combuston technology, confrmng that t 1 < 0 s a subsdy for the purchase of more effcent combuston technologes. The numerator s postve f j s the wood nput, confrmng that t 2 > 0 s a tax on wood use The control of urban ar polluton and the dstrbuton of household ncome. The presence of welfare weghts and the margnal utlty of prvate good consumpton n the taxes/subsdes n [12] means that the dstrbuton of ncome wll play an mportant role n optmal polces to control urban ar polluton. To understand how ncome dsparty affects polcy desgn, use [12] to subtract t hj from 11 Obvously, we requre that the government and households hold symmetrc belefs about the stochastc relatonshps between polluton damage, household emssons, and household choces. 16

17 t mj for an arbtrary par of households h and m and for both energy nputs j = 1, 2. Carryng out ths subtracton and rearrangng terms yelds um uh u h a r m um a r h [13] tmjλmeg thjλheg = λheg λmeg. ym yh a rm x mj a rh x hj On the rght sde of [13], λheg (( uh a)( a rm )( rm xmj )) s the authorty s expectaton of the weghted mpact of household m s choce of wood consumpton or more effcent combuston technology on household h s dsutlty from polluton. Therefore, one potental source of varaton of wood taxes and combuston technology subsdes across households stem from dfferences between the mpacts of each household s choces on the polluton damage suffered by every other household. Snce we are motvated by md scale urban areas lke Temuco, Chle wth more than 80,000 households, the margnal mpact of one household s choces on some other s utlty s probably very small. 12 Hence, we thnk t s reasonable to assume that rght hand sde of [13] s approxmately zero so that [14] t λ E ( u y ) t λ E ( u y ), for all household pars h and m, and j = 1, 2. hj h g h h mj m g m m For some household and wth j beng the wood nput, t E ( u y ) λ s the j g government s weghted expectaton of the household s margnal cost of the wood tax n terms of utlty of consumng the prvate good. For the combuston technology, the term s the authorty s weghted expectaton of the household s margnal beneft of the technology subsdy. The result n [14] ndcates that the weghted expectaton of the margnal cost of the wood tax n utlty terms should be approxmately equal across households. The same s true of the combuston 12 Ths, of course, does not mply that the expected margnal damage from a sngle household s emssons (.e., the numerator of [12]) s also small. 17

18 technology subsdy. Ths s remnscent of the requrement to equate margnal abatement costs of commercal pont polluton sources to mnmze the aggregate abatement costs of polluton control. Our result n [14] also ndcates that taxes and subsdes vary across households as ( ) λ E u y vares over households. Of course, the margnal utlty of consumpton of the g prvate good, u y, vares wth household ncome. Dmnshng margnal utlty of consumpton of the prvate good mples that u y decreases as household s ncome ncreases. Therefore, [14] ndcates that the dstrbuton of ncome plays an mportant role n formulatng polces to control urban ar polluton from households. In fact, E ( u y ) λ only vares across households f the authorty s unable or g unwllng to make unrestrcted lump sum ncome transfers. When an authorty makes these 367 transfers, t hj t for all household pars h and m, and j = 1, 2. To see ths, modfy the socal mj decson problem [9] by elmnatng the ndvdual household budget constrants and replacng them wth the sngle aggregate ncome constrant, n n n n [15] w 1 = y 1 + p1 x 1 1+ p2 x = = = = 1 2 Ths modfcaton allows an authorty to dstrbute the aggregate ncome of the communty n any way t wants. Let φ > 0 be the multpler attached to the aggregate wealth constrant for the Lagrange equaton for the problem. Then, the frst order condtons for determnng the allocaton of wood use and combuston technologes are: u c u a r uk a r [16] λeg + + λkeg φpj = 0, c xj a r x j k a r x = 1,..., n and j = 1, 2, j and the frst order condtons for determnng consumpton of the prvate good are. 18

19 [17] λe ( u y ) φ = 0, = 1,..., n. g Combne [16] and [17] to obtan u c u a r uk a r u [18] λeg + + λkeg λeg pj = 0, c xj a r x j k a r x j y = 1,..., n and j = 1,2. Note that [18] s the same as [10]. Therefore, combnng ths wth [8] wth E replaced by E g yelds [12] and ultmately [14]. However, the major dfference n assumng the government has the unrestrcted ablty to make lump sum transfers comes from [17], whch mples [19] λ E ( u y ) = λ E ( u y ), for all household pars, h and m. h g h h m g m m Ths mples that the effcent polluton control polcy would nclude ncome transfers so that households weghted expected margnal utltes of consumpton of the prvate good are equal. For gven utlty weghts and dmnshng margnal utlty of prvate good consumpton, these ncome transfers would tend to be from rcher households to poorer ones. If these transfers are made, [19] ndcates that E ( u y ) λ plays no role n how an optmal control polcy treats g dfferent households. That s, effcent lump sum ncome transfers would mply that household wood taxes and combuston technology subsdes satsfy t hj m, and j = 1, 2. t mj, for all household pars h and However, as we noted earler t s unlkely that these ncome transfers would be made a part of polces to control household ar polluton. In ths case, the effect of ncome dsparty on the effcent polluton control polcy cannot be dealt wth drectly, but nstead must be dealt wth through the specfcaton of household utlty weghts or through the varaton n wood taxes and technology subsdes across households. The frst opton would nvolve choosng a wood tax and 19

20 technology subsdy that s unform across households and adjustng the utlty weghts so that ( ) λeg u y s the same for all households. Snce u y wll be hgher for lower ncome households, ths strategy would assgn lower weghts n the socal welfare functon to these households. It s hard to magne a strategy that s more arbtrary and unfar. However, our analyss suggests that we should recognze that the choce of unform taxes nvolves an mplct assgnment of utlty weghts that are based aganst low ncome households. Varyng the wood taxes and combuston technology subsdes seems to us to be more defensble. From [14] t s easy to see that both the wood tax and the subsdy for more effcent combuston technologes wll tend to be hgher for hgher ncome households. That s, f h s a wealther household than m, then the combuston technology subsdes satsfy th1 > tm1, and the wood consumpton taxes satsfy th2 > tm2. At frst glance t may seem paradoxcal that ncome dfferences call for a hgher subsdy for wealther households. However, the reason that effcency calls for wealther households to take on more of the burden of reducng ar polluton than less wealthy households s to dstrbute the expected margnal utlty costs of wood taxes and benefts of technology subsdes so they are equal. Wth lump sum transfers ths equlbraton s accomplshed by ncome redstrbuton. In the absence of these transfers t s accomplshed by pushng more of the control burden onto 415 wealther households. 13 To see why wealther households take on more of the burden let us 13 Our result that the subsdy s larger for hgh ncome households does not depend on any assumpton about the margnal mpact of a household s combuston technology on ts emssons; n partcular, we do not assume that dfferences n ths value are ndependent of ncome. The same s true of the wood nput. These dfferences are elmnated because they are part of how emssons of one household affect the damage experenced by one other household, whch s lkely to be small n large communtes. In the case of a small number of households, we acknowledge that optmalty mght call for makng the low ncome households bear most of the burden of polluton control. Ths could occur f low ncome households own the least effcent 20

21 assume that we can tax household emssons drectly. We wll show that the emssons tax s hgher for wealther households n the absence of lump sum ncome transfers, thereby demonstratng that effcency calls for wealther households to take on more of the burden of ar polluton control. Assume that the th household faces an emssons tax t on ts emssons r, nstead of tax on ts wood consumpton and a subsdy for more effcent combuston technologes. Then the household s decson problem s to choose x 1, x 2, and y to solve [20] max E ( u ( c, y, a, µη, )) s.t. c = cx ( 1, x2), w = y + px + p x + tr( x, x, σ ), n a = a rk( xk1, xk2, σk), θ. k= 1 The frst-order condtons are: u c u r u a r [21] E pj + t + = 0, = 1,..., n, j = 1, 2. c xj y x j a r xj As before, replace the E 's n [21] wth Eg to reflect the fact that government uses ts own expectaton of the households decson crtera to determne optmal taxes and subsdes. After dong ths, substtute the result nto [10] and rearrange terms to obtan combuston technologes and that a subsdy for an upgrade can produce a larger reducton n emssons than f the subsdy was gven to hgher ncome households wth more effcent combuston devces. We also acknowledge that optmal taxes and subsdes mght vary wth ncome f polluton s not unformly mxed and the spatal dstrbuton of damages s correlated wth ncome. Our model does not account for ths possblty, but t can be modfed n a straghtforward way to do so. 21

22 432 [22] t = λ k keg k a r x j λ E g u u y a r r x j, = 1,..., n To smplfy matters, assume that the uncertanty n r x, j = 1,2 s uncorrelated wth the uncertanty n u y and n ( u a)( a r ), k for all = 1,..., n. Ths allows us to elmnate r xj from [22]. 14 Once ths has been done, subtract t hj j from t mj to obtan um uh uh a um a [23] tmλmeg thλheg = λheg λmeg. ym yh a rm a rh Agan, the rght sde expresson s lkely to be very small, so [23] mples t E ( u y ) ( ) m m g m m λ h h g h h t λ E u y, for all household pars h and m. If the authorty can tax household emssons but cannot make lump sum transfers of ncome across households, [23] suggests that the effcent tax on emssons s hgher for hgher ncome households. Thus, optmalty calls for makng hgher ncome households bear more of the burden of controllng household ar polluton. Of course, we have mantaned that the man dffculty n ths polcy problem s that household emssons cannot be observed. When an authorty can control wood consumpton and combuston technologes t pursues polces that place more of the ar polluton control burden on hgher ncome households by placng a hgher tax on ther wood consumpton and offerng a hgher subsdy for ther purchase of more effcent combuston technologes. It s mportant to 14 Ths lack of correlaton could come about f, for example, there was no regulatory uncertanty about how household choces of wood consumpton and combuston technology produce emssons. If we are not able to elmnate r xj from the rght sde of [22], a household s emsson tax would depend on the nput j. In ths case, no emssons tax could smultaneously satsfy [21] for both the wood nput and the combuston technology. Ths problem s dscussed n another context by Shortle and Horan (2001). 22

23 realze that a hgher technology subsdy for wealther households s not meant to correct ncome nequalty. The purpose of the subsdy s to motvate the purchase of more effcent combuston technologes. A hgher technology subsdy for wealther households s a part of how more control burden s optmally placed on wealther households. It bears repeatng, however, that the technology subsdy s only needed because household emssons cannot be controlled drectly. Thus, t s the nonpont nature of the problem combned wth the nablty of an authorty to make unrestrcted lump sum ncome transfers that lead to hgher technology subsdes for wealther households. 4. Concludng remarks about mplementaton We have derved a set of effcent household-specfc taxes on wood consumpton and subsdes for more effcent combuston technologes. Our most mportant result s that these nterventons are dependent of the dstrbuton of ncome; n fact, effcency requres that these taxes and subsdes be structured so that wealther households take on more of the control burden. In ths secton we dscuss some practcal mplementaton ssues assocated wth our results. Although we have assumed that taxes on wood consumpton and subsdes for combuston technologes are avalable, n many settngs n the developng world wood consumpton s not observable. Ths may be due to the absence of formal markets for wood for heatng, whch s the case of wood used by urban households n central southern Chle. The market for wood s manly nformal and no regulatory authorty has actual control or transacton records. In a recent survey of a sample of urban households n the cty of Temuco, about 90% of the respondents acknowledged buyng wood wthout payng taxes (CONAMA-DICTUC 2008). In the absence of the ablty to montor wood consumpton, ar polluton control polcy would then focus on promotng cleaner combuston technologes. However, our result that 23

24 ncome dspartes mply that hgher technology subsdes should be provded to hgher ncome households becomes problematc. It s hard to magne that there would be much poltcal support for our recommendaton. In fact, t may be the case that subsdes are only feasble f they are targeted at lower ncome groups. Ths s lkely to be true n Chle where subsdes that are part of socal welfare programs are targeted at the poor. We are not aware of any envronmental polcy nterventon n Chle that uses subsdes for household choces; however, t seems lkely that such a polcy would be mplemented n concert wth socal polces that defne how subsdes are allocated. Provdng a hgher technology subsdy to hgher ncome households mght appear to be at odds wth other socal welfare objectves. Whle hgher technology subsdes for hgher ncome households may be part of an effcent control program, they may not be part of a control polcy that pursues other reasonable objectves. For example, an authorty may be motvated to get the largest mprovement n ar qualty wth a lmted budget to pay subsdes for more effcent combuston technologes. Then t may be the case that these subsdes should be drected manly at poorer households f ths s where the margnal reducton n emssons from a dollar of technology subsdy s hghest. Pursung the bggest envronmental mprovement for a fxed mplementaton budget s a reasonable polcy objectve, even though t wll not lead to the theoretcally effcent soluton. However, t may be easer to understand than the dea of effcency, and hence, easer to sell to lawmakers and the publc. There s another reason to target combuston technology subsdes to low ncome households, whch s perhaps more specfc to the developng country context of the problem we explore. Whle n our model households are able to choose or may be nduced to choose the 24

25 optmal combuston technology, lqudty constrants can make that choce dffcult for some households. Chávez, et al. (2010) conducted a survey of a sample of Temuco s households to determned ther wllngness to change combuston equpment wth a government subsdy. They found that responses depended on whether or not there was credt avalable for the part of the cost of the combuston equpment not covered by the subsdy. Even though we have assumed that combuston technologes are observable, and therefore can be subsdzed, there s stll a costly enforcement problem to manage. A montorng and penalty program needs to be desgned along wth the ncentve polcy to make sure that those who take advantage of the subsdy actually purchase approved equpment and use t properly. Our results suggest that optmal technology subsdes and wood taxes vary contnuously accordng to household ncomes. Whle contnuty s possble because authortes are lkely to have ncome nformaton for tax purposes, t s more lkely that authortes wll group households nto a relatvely small number of ncome classes and apply dfferent subsdes and taxes to each class. Dfferentated after-tax or after-subsdy prces produce the rsk of developng so-called black markets n wood or combuston equpment. Black markets could also develop across communtes that move effcent combuston equpment ntended for one communty to another. If ths problem proves dffcult to deal wth, then authortes could be forced to consder mplementng a unform technology subsdy or wood tax despte the effcency consequences. Our results also suggest that the effcent subsdes (and wood taxes f they are avalable) should vary across ctes. Although we have llustrated the problem of concern wth the case study of Temuco n southern Chle, the same type of ar polluton s a serous problem n several medum and small sze ctes n the valley south of the Chlean captal of Santago. Because of the heterogenety across these ctes n terms of populaton, ncome dstrbuton, dstrbuton of 25

26 combuston technologes, the use of wood, and general envronmental condtons, t s hghly unlkely that the same control polcy wll be approprate for dfferent ctes. A coordnated plan may also be requred to manage the flow of retred equpment across communtes. An effectve equpment subsdy program wll generate a stock of dscarded stoves. If these are not destroyed, they may be avalable to other communtes at sgnfcantly reduced prces. Because of ths a stove replacement program n one communty can have envronmental mpacts n other communtes; hence, the potental need for coordnated ar qualty programs. Acknowledgments We gratefully acknowledge fnancal support from Concyt-Chle under Project Fondecyt No and Fondecyt Internatonal Cooperaton

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