On Environmental Taxation under Uncertainty *

Transcription

1 Augus 1999 On Environmenal axaion under Uncerainy * by homas Aronsson a and Sören Blomuis b Absrac his paper addresses opimal axaion, when he relaionship beween consumpion and environmenal damage is uncerain and reaed as a random variable by policy makers. he main purpose is o analye how addiional uncerainy abou his relaionship affecs he opimal uni ax on he consumpion good ha is causing environmenal damage. We find ha he opimal response o his ax depends on (i) he aiudes owards risk and (ii) how oher policy insrumens affec he demand for he good ha is causing damage o he environmen. JEL-classificaion: H21, D62, D8, Q2 Keywords: Green axes, Uncerainy, Opimal axaion * he auhors would like o hank Magnus Wiksröm for helpful commens and suggesions. α Deparmen of Economics, Universiy of Umeå, S Umeå, Sweden. homas.aronsson@econ.umu.se β Deparmen of Economics, Uppsala Universiy, Box 513, Uppsala, Sweden. Soren.Blomuis@nek.uu.se

2 1. Inroducion Much research effor has been pu ino sudying green axes as a means of improving he resource allocaion. I is now recognied ha he proper design of environmenal axaion does no only depend on he environmenal damage caused by e.g. producion or consumpion; i also depends on oher disorions in he economy 1. However, as far as we kno here are very few aemps o incorporae uncerainy ino he analysis of environmenal axaion 2. his is somewha surprising, since he environmenal effecs of producion and consumpion are almos always uncerain. In he conex of he relaionship beween consumpion/producion and polluion, here are several ypes of uncerainy ha may be of ineres o sudy. Firs, here can be uncerainy abou he magniude (or severiy) of he environmenal damage caused by cerain polluans. We migh be able o perfecly observe he uaniy of a poenial polluan while, a he same ime, no knowing is exac impac on he environmen. A freuen example is he effecs of carbon dioxide emissions, where he major issue is wheher or no global warming may have severe impacs on he climae and, herefore, on he living condiions of mankind. A second ype of uncerainy occurs when he emissions hemselves are no perfecly observed. For example, i would be very hard o observe he exac amoun of nirogen and oher emissions from individual cars. his paper analyes opimal environmenal axaion under he firs ype of uncerainy menioned above, where he environmenal conseuences of economic behavior are no known wih perfec cerainy. o operaionalie his idea, we will assume ha he relaionship beween he aggregae consumpion of a cerain good - o be called diry good - and he resuling environmenal damage is uncerain, meaning ha his relaionship will have he characer of a random variable. By assuming ha he probabiliy disribuion for his random variable is known, we can formulae he opimal ax problem in erms of maximiaion of expeced uiliy. he main purpose is o analye how a mean preserving increase in he spread of he random environmenal damage affecs he opimal choice of environmenal axaion. 1 See e.g. Bovenberg and de Mooij (1994, 1996) and Bovenberg and van der Ploeg (1994). 2 Excepions are Aronsson (1998), dealing wih he role of environmenal axaion under uncerain iming of he developmen of new abaemen echnologies, and Aronsson e al. (1998), who address exernal effecs relaed o he likelihood of nuclear accidens. 2

3 When he environmenal conseuences of consumpion are no known wih perfec cerainy, he aiudes owards risk become imporan in he conex of environmenal policy. If agens do no become less risk averse when he environmenal damage increases, i is no difficul o imagine a precauionary moive for environmenal axaion. his means ha uncerainy abou he relaionship beween consumpion and environmenal damage may work o increase axes, which are moivaed by concern for he environmen. A he same ime, i is imporan o recognie ha he governmen does no only impose axes in order o inernalie exernaliies; i also uses axaion from various sources o finance public goods. As a conseuence, he opimal level of environmenal axaion is likely o depend on how he oher ax insrumens available affec he demand for goods ha are causing environmenal damage. herefore, he aiudes owards risk are, in general, no sufficien for deermining he policy implicaions of he ype of uncerainy o be analyed here. o simplify he analysis as much as possible, we shall disregard disribuional policy objecives as well as he inheren dynamic naure of many polluans by using a saic represenaive agen model 3. Environmenal axaion will be inroduced in erms of a uni ax on he diry good, he consumpion of which is causing he environmenal damage. In he benchmark version of he model examined in Secion 2, he governmen is assumed o be able o raise lump-sum axes, meaning ha he ax insrumens available are he ax on he diry consumpion good and he lump-sum ax. In Secion 3, he lump-sum ax will be replaced by a labor income ax. Finally, Secion 4 concludes he paper. 3 he more recen research has exended he sudy of environmenal axaion o address ineremporal choice and disribuional objecives. For example, Bovenberg and de Mooij (1997) and Aronsson (1999) analye welfare implicaions of green ax reforms in a dynamic general euilibrium framework, whereas Pirillä and uomala (1997) sudy opimal enviromenal axaion in an economy where he ax sysem is designed o fulfill boh efficiency and disribuional objecives. 3

4 2. A Benchmark Model he benchmark seing means ha he governmen can finance is consumpion by lump-sum axes. his assumpion will be relaxed in he nex secion, where he lumpsum ax is replaced by a labor income ax. Since we disregard disribuional issues, he populaion is normalied o one. he budge consrain of he represenaive agen is wrien c + x + wl = wh (1) where c and x are consumpion of clean and diry privae goods, respecively, and L is leisure. he price of he diry good is defined as = p +, where p is he fixed producer price and a uni ax. We shall refer o as an environmenal ax. he price of he clean good has been normalied o one. he righ hand side of euaion (1) measures full income, where w is he wage rae, H a ime endowmen and a lump-sum ax. o be able o derive clear-cu resuls relaing o he conseuences of uncerainy, i has no been uncommon o assume an addiive uiliy funcion 4. We shall follow his approach in par and wrie he direc uiliy funcion of he represenaive individual as follows: U = u( c, x, L) + ϕ( g) + v( βd) (2) where g is public consumpion and D he aggregae consumpion of he diry privae good or exernaliy base. Since he populaion has been normalied o eual one, we have D = x. Noe ha he represenaive individual akes he exernaliy base as exogenous when solving his/her opimiaion problem. he erm β D is inerpreed o measure he environmenal damage caused by he consumpion of diry goods. he parameer β will be explained below. We assume ha u ( ) is increasing in each argumen, wice coninuously differeniable and sricly uasiconcave. Regarding he 4 See e.g. Johansson and Löfgren (1985) and Koskela and Ollikainen (1997, 1998). 4

5 oher pars of he uiliy funcion, we assume ϕ g ( g) >, ϕ gg ( g) <, v ( ) < and v ( ) <, where = βd. From euaion (2) i follows ha he represenaive individual behaves as if he/she is maximiing u ( c, x, L) subjec o euaion (1), which means ha he privae opimiaion gives c = c(, ) (3a) x = x(, ) (3b) L = L(, ) (3c) and we assume ha c, x and L are all normal goods. Even if he exernaliy base, D, is perfecly observed, he relaionship beween his exernaliy base and he environmenal damage is uncerain o policy makers. According o euaion (2), he environmenal damage is proporional o he aggregae consumpion of diry goods, and β is he facor of proporionaliy. We shall define β = β + θε (4) 2 wih E( ε ) =, σ ε = 1 and Pr( ε > β / θ) = 1. We can inerpre β( ε) as a posiive random variable wih mean β and sandard deviaion θ. A mean preserving increase in he spread of his random variable is, herefore, measured by an increase in he sandard deviaion, θ. By subsiuing euaions (3a), (3b) and (3c) ino euaion (2), and hen aking expecaions, we obain he expeced indirec uiliy funcion V ] = Ω(, ) + ϕ( g) + v( β x(, ))] (5) 5

6 he governmen chooses, and g such as o maximie he expeced indirec uiliy subjec o he budge consrain x(, ) + = g (6) Since our main purpose is o analye he opimal ax par of he governmen s problem, we subsiue euaion (6) ino euaion (5) Max V ] = MaxΩ(, ) +, ϕ( x(, ) + ) + v( β x(, ))] and he firs order condiions can be wrien as 5 V ] = Ω + ϕ ( x + x ) + x β v ( βx)] = (7a) g V ] = Ω + ϕ ( x + 1) + x β v ( βx)] = (7b) g where = βx and subindices denoe parial derivaives. We assume ha he ax revenue is an increasing funcion of and, i.e. x x > and 1 x >, a he euilibrium. + Given euaions (7a) and (7b), which implicily define he opimal and, one can use euaion (6) o solve for he opimal public expendiure. We shall also reuire ha he second order sufficien condiions for maximiaion are fulfilled. + Le us begin by inerpreing he firs order condiions. By solving euaion (7b) for ϕ g, subsiuing ino euaion (7a) and using Roy s ideniy, we can rewrie euaion (7a) in erms of an analogue o a well known opimal ax rule; = βv ( βx)] Ω (8) 5 he firs order condiion for public consumpion is implici in euaions (7a) and (7b). his condiion means ha he marginal uiliy of he public good is eual o he marginal cos of public funds. 6

7 Euaion (8) means ha he opimal environmenal ax euals he expeced marginal exernaliy in real erms, where Ω < is he negaive of he marginal uiliy of income. his resul is a conseuence of he assumpion ha he governmen can use lump-sum axes o finance public consumpion. I means ha i is opimal o fully inernalie he expeced exernal effec. Noe finally ha euaion (8) only gives he opimal environmenal ax on an implici form; boh x and Ω depend on. Le us now urn o he main issue of his secion; how a mean preserving increase in he spread of he (random) environmenal damage affecs he opimal environmenal ax. Differeniaing euaions (7a) and (7b) wih respec o, and θ, we find V ] E [ V ] V V ] / ] = / x { βv x { βv } } (9) and we can solve for / and / as follows; [ x V ] x V ]]{ βv } = (1a) A [ x V ] x V ]]{ βv } = (1b) A where A is he deerminan corresponding o he Hessian marix of euaion sysem (9). Noe ha E [ ] <, E [ ] < and A V ] V ] { V ]} 2 > from he V V second order sufficien condiions for maximiaion. = o be able o analye he implicaions of uncerainy for he opimal environmenal ax, one has o specify he aiudes owards risk in greaer deail. Since he wors case scenarios for some polluans are doomsday-like, i appears o be reasonable o assume ha he consumer will a leas no become less risk averse when he environmenal damage (or exernaliy base) increases. Formally, he preferences are characeried by nondecreasing absolue risk aversion if 7

8 v v 1 ( ) 1 ( ) v ( ) for > v ( ) 1 where v / v >, since v < and v <. Consider he following resul; Lemma 1: Wih nondecreasing absolue risk aversion, β v ( βx)] / <. Proof: see he Appendix. Lemma 1 is a conseuence of he assumpion ha he exernaliy par of he uiliy funcion is addiive, implying ha he impac of he variance parameer θ on he marginal uiliy of he exernaliy base is aribuable o risk version. his is so because he consumpion of diry goods does no depend explicily on θ. We can now inerpre euaion (1a) as follows; Proposiion 1: Wih nondecreasing absolue risk aversion, an increase in θ will increase (decrease) he opimal environmenal ax if, and only if, x E V ] x V ] is negaive (posiive). [ o fully undersand his resul, noe ha since x <, hen x ] >. his erm V reflecs he precauionary moive for environmenal axaion menioned in he inroducion. Given he assumpion abou nondecreasing absolue risk aversion, his effec will work o increase he environmenal ax as a response o addiional uncerainy. he inuiion is ha one can counerac he disuiliy from addiional uncerainy by reducing he consumpion of diry goods. However, he consumpion of diry goods will also be reduced by higher lump-sum axaion, and even if x < he erm x E ] can, in general, go in eiher direcion. [ V herefore, wihou furher assumpions, we canno rule ou he siuaion where he governmen responds o addiional uncerainy by increasing he lump-sum ax and reducing he environmenal ax. o rule ou his possibiliy, one would have o impose resricions on he magniude of he income effecs ; if x is sufficienly small, he 8

9 opimal response o addiional uncerainy will be o increase he environmenal ax. By aking his argumen o is exreme, we can derive; Proposiion 2: Wih nondecreasing absolue risk aversion, and if he uiliy funcion is uasi-linear in he sense u ( c, x, L) = c + f ( x, L), hen / >, / < and g / =. Proof: Wih he assumpion of uasi-lineariy, we can rewrie euaions (1a) and (1b) o read = = x { v x [ x + } ϕ A x ]{ v A gg > } ϕ gg < herefore, by using he budge consrain i follows ha g / =.g he inuiion behind Proposiion 2 is as follows. Since uasi-lineariy implies ha he marginal uiliy of public consumpion is eual o one a he opimum, and by he assumpion ha public consumpion eners he uiliy funcion addiively, i follows ha public consumpion will no be affeced by addiional uncerainy. In addiion, he only policy insrumen ha can be used o reduce he consumpion of diry goods is he environmenal ax. herefore, he opimal response will be o increase he environmenal ax and reduce he lump-sum ax so as o mainain he public secor euilibrium a he iniial level of public expendiures. 3. Disorionary Labor Income axaion he budge consrain of he represenaive consumer is now given by c + x + ω L = ωh (11) 9

10 where ω = w ( 1 τ) is he ne wage rae and τ he labor income ax rae. he oucome of privae opimiaion changes o read (if we suppress he ime endowmen for noaional convenience) c = c(, ω ) (12a) x = x(, ω ) (12b) L = L(, ω ) (12c) By subsiuing euaions (12a), (12b) and (12c) ino euaion (2), and hen aking expecaions, we obain he expeced indirec uiliy funcion E [ V ] = Ω(, ω ) + ϕ( g) + v( β x(, ω ))] (13) he governmen chooses, τ and g such as o maximie he expeced indirec uiliy funcion subjec o he budge consrain. By using he shor noaion R(, τ) = x(, ω ) + τw[ H L(, ω )], he public secor budge consrain can be wrien as R (, τ ) = g (14) By subsiuing he budge consrain ino he expeced indirec uiliy funcion, we can wrie he opimiaion problem in a way similar o ha of he previous secion Max V ] = MaxΩ(, ω ) +, τ ϕ( R(, τ)) + v( β x(, ω ))] and he firs order condiions can be wrien as V ] = Ω + ϕ R + x β v ( βx)] = (15a) g Vτ ] = Ωω w + ϕ g Rτ xω w βv ( βx)] = (15b) 1

11 We assume ha he second order sufficien condiions for maximiaion are fulfilled. As in he previous secion, he main concern is o analye how a mean preserving increase in he spread of environmenal damage affecs he opimal environmenal ax. Differeniaing euaions (15a) and (15b) wih respec o, τ and θ, we obain V ] E [ Vτ ] V V τ τ τ ] / = x{ βv ] τ / x ω w{ βv } } (16) We can hen solve for he wo parial derivaives as follows; [ xω w V τ ] + x Vτ τ ]]{ βv } = (17a) B τ [ x Vτ ] + xω w V ]]{ βv } = (17b) B where E [ ] <, E [ ] < and B V ] V ] { V ]} 2 > according o he V V τ τ = ττ τ second order sufficien condiions for maximiaion. We have derived he following resul wih respec o he environmenal ax; Proposiion 3: Wih nondecreasing absolue risk aversion, an increase in θ will increase (decrease) he opimal environmenal ax if, and only if, x E V ] + x w V ] is posiive (negaive). [ τ τ ω τ he erm x V τ τ ] is unambiguously posiive and conribues o increase he opimal environmenal ax, as long as he assumpion abou nondecreasing absolue risk aversion applies. By analogy o he previous secion, his effec can be hough of as reflecing he precauionary moive for environmenal axaion. A he same ime, an increase in θ will also affec he opimal labor income ax which, in urn, influences he consumpion of diry goods. he sign of x ω w Vτ ] is ambiguous, since boh x ω and E [ V τ ] can go in eiher direcion. his makes he impac of θ on ambiguous as well. herefore, even if nondecreasing absolue risk aversion provides an incenive o increase axes on goods 11

12 whose conseuences for he environmen are uncerain, i is no necessarily opimal o do so. 4. Discussion his paper analyes opimal ax policy in a model, where he relaionship beween consumpion and environmenal damage is uncerain. he main purpose is o examine how addiional uncerainy abou his relaionship affecs he opimal environmenal ax, where he laer is defined as a uni ax on he consumpion good ha is causing damage o he environmen. As i urns ou, he resuls depend primarily on (i) he aiude owards risk and (ii) how he oher ax insrumens affec he consumpion good ha is causing environmenal damage. If he privae agens do no become less risk averse when he environmenal damage increases - meaning ha he preferences are characeried by nondecreasing absolue risk aversion - here is a precauionary moive for environmenal axaion. his effec will work o increase he opimal environmenal ax as a response o addiional uncerainy. However, he influence of oher ax insrumens may offse he precauionary moive o increase he environmenal ax. herefore, even if nondecreasing absolue risk aversion provides an incenive o increase axes on such goods, whose conseuences o he environmen are uncerain, i may no be opimal o do so in an economy wih several ax insrumens and oher policy objecives han inernaliing exernaliies. Appendix Proof of Lemma 1: Since β = β + θε, we can define E βv ( βx)] = εv [ 2 ( βx)] + βx εv ( βx)] + θx ε v ( βx)] (A1) he firs erm on he righ hand side of euaion (A1), E ε v ( βx)] = cov( ε, v ), is [ negaive, because dv ( ) / dε = θ x v ( ) <. Similarly, he hird erm on he righ hand side is nonposiive, since θ >, x >, ε 2 and v ( ) <. 12

13 o sign he second erm on he righ hand side of euaion (A1), le us define = βx. Consider firs he case when hen implies >, so ε >. Nondecreasing absolue risk aversion v ( ) v ( ) v ( ) (A2) v ( ) or, if we muliply boh sides by v ( ) <, v ( ) v ( ) v ( ) (A3) v ( ) Muliplying euaion (A3) by ε > and hen aking expecaions on boh sides, we find v ( ) ε v ( )] εv ( )] < (A4) v ( ) If, on he oher hand, < and ε <, he euivalen of euaion (A3) akes he form v ( ) v ( ) v ( ) (A5) v ( ) Finally, by muliplying euaion (A5) by ε < and hen aking expecaions, we will again obain euaion (A4). herefore, ε v ( βx)] and he second erm on he righ hand side of euaion (A1) is nonposiive.g 13

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15 Koskela, E. and Ollikainen, M. (1998) imber Supply, Ameniy Values and Biological Uncerainy. Discussion papers no 439, Deparmen of Economics, Universiy of Helsinki. Pirillä, J. And uomala, M. (1997), Income ax, Commodiy ax and Environmenal Policy, Inernaional ax and Public Finance 4,