Online appendix for Household heterogeneity, aggregation, and the distributional impacts of environmental taxes

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1 Onine appendix for Houseod eterogeneity, aggregation, and te distributiona impacts of environmenta taxes Tis onine appendix contains suppementary anaysis () for Section 4.3 exporing te distributiona impacts among ouseods under aternative revenue recycing scemes and (2) for Section 5 providing additiona propositions tat are anaogous to ropositions 6 for mutipe extensions of te core mode (pre-existing taxes on capita and abor, non-separabe utiity in poution, abor-eisure coice, an arbitrary number of commodities, and aternative revenue recycing scemes).

2 . Incidence for aternative carbon tax revenue recycing scemes Tis section provides suppementary resuts for Section 4.3 in te main text focusing on incidence for a U.S. carbon tax for two aternative revenue recycing scemes. A first case assumes tat ouseods receive te revenue in proportion to teir consumption of te dirty good refecting concerns about o setting adverse impacts for poor ouseods (see Figure A.2). A second case considers equay distributing te carbon revenue on a per capita basis (see Figure A.3). Figure A.2: Wefare impacts ( ) of increased poution tax across annua expenditure decies; revenues aocated in proportion to dirty good consumption (a) Aternative assumptions about ouseod caracteristics (b) Aternative assumptions about production caracteristics

3 Figure A.3: Wefare impacts ( ) of increased poution tax across annua expenditure decies; revenues aocated on per-capita basis (a) Aternative assumptions about ouseod caracteristics (b) Aternative assumptions about production caracteristics

4 2. Extensions 2.. Aternative revenue recycing scemes Tabe B.6 reports price canges for aternative revenue recycing scemes. As is evident te price canges for bot ˆr and ˆp Y are very simiar among aternative revenue recycing cases indicating tat te impact of ouseod eterogeneity on te equiibrium outcome is argey independent of te way te environmenta tax revenue is redistributed. Tabe B.6: rice canges for aternative revenue recycing scemes cov Base cov Low cov Hig Base Low Hig Low Hig ˆp Y ˆr ˆp Y ˆr ˆp Y ˆr ˆp Y ˆr ˆp Y ˆr Redistribution proportiona to income Redistribution proportiona to dirty good consumption Redistribution on per capita basis Notes: ˆr and ˆp Y are expressed as te percentage cange reative to te price eve before te poution tax increase. Te resuts in te tabe are based on te centra case assumptions for production side caracteristics re-existing, non-environmenta taxes Accounting for pre-existing taxes on capita and abor in te bencmark, anaogous to Fuerton & Heute (2007), modifies te cost sares now incuding tax payments ( YK r(+ K )K Y p Y, and simiary for Y YL, K and L ) as we as te ouseods income constraint now incuding tax revenues as new sources of income: = w L + r K + Z Z + K KK + L LL, were K and L denote te ad vaorem tax rate on capita and abor, respectivey, and, K, and L are te sares of tota revenue from poution, capita and abor taxes redistributed to ouseod, respectivey. We find tat as ong as te revenue from capita and abor taxes is aso distributed in proportion to income, tere is no additiona e ect of ouseod eterogeneity on price canges as eterogeneity in terms of bot uses and sources side is uncanged. In tis case, a ropositions 6 remain vaid. Distributing capita and abor tax revenue in a non-neutra way wi introduce additiona eterogeneity on te sources side. In tis case, ropositions and 6 sti od true and price canges for ˆr and ˆp Y are quantitativey simiar (anaogousy to Section 5.). Tis can be seen as foows. Te budget cange foowing a cange in te poution tax is now given by: Mˆ = ŵ w L M + ˆr r K M + Z Z (ˆ Z + Ẑ) + K r KK (ˆr + ˆK) + L w LL (ŵ + ˆL). Since te tota amounts of capita and abor in te economy are assumed to be exogenousy given and fixed, it foows tat ˆK = 0 and ˆL = 0. Hence: ˆ = ŵ w( L + L LL) + ˆr r( K + K KK) + Z Z (ˆ Z + Ẑ). Tis expression is formay identica to te budget cange witout capita and abor taxes, wit L repaced by L + L LL and K repaced by r( K + K KK). It terefore foows tat te mode resuts are identica to (5a)-(5c), wit te foowing canges: L! OT,L ( )E,M w( L + L LL) p Y Y + Y Y (E Y,M E,M ) w( L + L LL) (.OT) K! OT,K ( r( K + )E,M K KK) + Y p Y Y Y (E Y,M E,M ) r( K + K KK). (2.OT)

5 From te above considerations, it is straigtforward to derive te foowing propositions wic are anaogous to ropositions 6 in te paper. We use te abe OT to enabe comparison between origina propositions and te propositions based on te mode wit pre-existing, oter taxes. roposition.ot. Assume non-zero ad vaorem taxes on capita and abor inputs in production. Ten roposition ods. roof. Anaogous to te proof of roposition. roposition 2-5.OT. Assume non-zero ad-vaorem taxes on capita and abor inputs in production. Assume tat a tax revenue is redistributed in proportion to bencmark income: = K = L. Ten ropositions 2 5 od. p +p Y Y roof. Wit OT,L as in (.OT) and OT,K as in (2.OT) it foows tat OT,L = ( ) w L and p Y Y OT,K = ( ) r K. p Y Y Tese expressions are identica to te case wit omotetic preferences and zero taxes on capita and abor. Hence te price canges are aso identica. roposition 6.OT. Assume non-zero ad vaorem taxes on capita and abor inputs in production. Ten roposition 6 ods. roof. Since for identica ouseods te te resuts. OT expressions are zero, it foows tat te taxes on capita and abor ave no impact on 2.3. Non-separabe utiity in poution Wit non-separabe utiity, consumption of cean and dirty goods in genera depends on te eve of poution: = (p, p Y,, Z) and Y = Y (p, p Y,, Z). Equations (A.) and (A.2) ten become ˆ = ( E,M + ( ) )ˆp (( )E,M ( ) )ˆp Y + E,M ˆ + E,ZẐ Ŷ = ( E Y,M )ˆp (( )E Y,M + )ˆp Y + E Y,M ˆ + E Y,ZẐ, were E,Z Z and E Y,Z Z Z Y can, respectivey, be interpreted as te poution easticity of cean and dirty consumption. Equations (0) and () can ten be written as: ˆ Ŷ = (ˆp Y ˆp ) + (EY, E, )( ˆp + ( )ˆp Y ˆ ) + (E,Z EY,Z )Ẑ (.NS) ˆ = ( E,M + ( ) )ˆp (( )E,M ( ) )ˆp Y + E,M ˆ + E,ZẐ. (2.NS) In order anayze our propositions, we first need to derive te price canges for te case wit non-separabe utiity in poution. Tey turn out to be identica to tose for separabe preferences, up to te coe cients A, B and C, wic become A! A NS A K NS B! B NS B L C! C NS C NS, (3.NS) NS wit NS = M p Y Y ( )E,Z + Y Y (E Y,Z E,Z ). Te next subsection derives tis resut Derivation of price canges for non-separabe utiity in poution Sove (.NS) for Ŷ and insert te resut into (B.4). Rearrange to obtain: Y Y (ˆp Y ˆp ) + (EY, E, )( ˆp + ( )ˆp Y ˆ ) + (E,Z EY,Z )Ẑ = Y Y ˆ YK ˆK Y YL ˆL Y YZ Ẑ. From (B.3) insert te foowing on te rigt-and side of te equaity: +0 = K ˆK + L ˆL te numéraire, tus yieding: Y ( (ˆp Y ) + (EY, E, Y )(( )ˆp Y ˆ ) + (E,Z EY,Z )Ẑ) = ˆ and use te fact tat is p Y Y ( ) ˆ + K ˆK + L ˆL YK ˆK Y YL ˆL Y YZ Ẑ. (4.NS)

6 Eiminate ˆ from equation (4.NS) by using equation (2.NS), ten insert te expicit expression for te budget cange ˆ : ˆp Y = L ŵ + K ˆr + Z ˆ Z, + K ˆK + L ˆL YK ˆK Y YL ˆL Y + ( NS + Z ) YZẐ. (5.NS) wit NS = M ( p Y )E Y,Z + Y Y (E Y,Z E,Z ). Next, sove equations () and (2) for ˆK and ˆL, and insert tem into (5.NS). Furtermore, insert equation (B.2) to eiminate ˆp Y, tus obtaining: ( ( NS + Z ) YZ)Ẑ = ( YK K )ˆr + ( YL L )ŵ + ( YZ Z )ˆ Z + ˆK Y ( K K + YK ) + ˆL Y ( L L + YL ). (6.NS) Sove equations (4) and (5) for ˆK Y and ˆL Y, and insert tem into equation (6.NS). Tis yieds: C NS Ẑ = ( K + YK ( + K (e KK e ZK ) + L (e LK e ZK )))ˆr + ( + ( L + YL ( + K (e KL e ZL ) + L (e LL e ZL )))ŵ Z + YZ( + K (e KZ e ZZ ) + L (e LZ e ZZ )))ˆ Z, (7.NS) wit C NS = K + L + YZ ( Z + NS ). Next eiminate Ẑ. To acieve tis, substitute equations () and (2) into (3), obtaining: Substituting equations (4) and (5) into (8.NS) yieds: K ˆK Y + L ˆL Y = (ŵ ˆr). (8.NS) (ŵ ˆr) = ( L K )Ẑ + YK ( L (e LK e ZK )ˆr K(e KK e ZK ))ˆr YL ( L (e LL e ZL )ŵ K(e KL e ZL ))ŵ+ YZ ( L (e LZ e ZZ )ˆ Z K (e KZ e ZZ ))ˆ Z. (9.NS) Now sove equation (9.NS) for Ẑ and equate to equation (7.NS): ( K L )( K + YK ) + C NS + YK [ A NS (e KK e ZK ) + B NS (e LK e ZK )] ˆr + ( K L )( = ( L K )( L + YL ) C NS + YL [ A NS (e KL e ZL ) + B NS (e LL e ZL )] ŵ Z + YZ ) + YZ [ A NS (e ZZ e KZ ) + B NS (e ZZ e LZ )] ˆ Z, (0.NS) wit A NS ( K L ) K + C NS K = A K NS and B NS ( K L ) L + C NS L = B L NS.(0.NS) is formay identica to (B.), wit te coe cients A, B and C repaced by A NS, B NS and C NS. It terefore foows tat price canges wi aso be identica up to te vaue of tese coe cients Resuts As can be seen from above, te cange in te poution eve foowing a poution tax increase can a ect price canges and ence te equiibrium beavior of ouseods. Tis introduces an additiona dimension of eterogeneity to te extent tat ouseods ave di erent preferences about poution. From te above considerations, it is straigtforward to derive te foowing propositions wic are anaogous to ropositions 6 in te paper. We use te abe NS to enabe comparison between te origina propositions and te propositions based on te mode wit non-separabe poution. Equa factor intensities in production roposition.ns. Assume non-separabe utiity from poution. Ten roposition ods.

7 roof. If K = L, ten from te proof of roposition, we know tat it ten foows tat A = B = K C. Tis impies tat A K NS = B L NS = K (C NS ) wic in turn is equivaent to A NS = B NS = K C NS. It ten foows tat a te terms containing ouseod caracteristics in te expressions for te price canges drop out. Heterogeneous ouseods wit omotetic preferences In tis paragrap, assume tat te poution tax revenue is returned to ouseods in proportion to income. Now define te e ective poution easticity of cean consumption E,Z. It ten foows tat: NS = using te fact tat, from te budget constraint, te foowing ods: EY,Z = 2 5 od., E,Z. Using tis, te anaogues to ropositions roposition 2.NS. Assume non-separabe utiity from poution. It ten foows tat, in addition to cov(, L ) and, roposition 2 is extended to incude te e ective poution easticity of cean consumption,. roof. Te proof is anaogous to te one for roposition 2 accounting for te new term. roposition 3.NS. Assume non-separabe utiity from poution. Ten, in addition to and, te singe ouseod wit omotetic preferences in roposition 3 is aso caracterized by a poution easticity of cean consumption equa to te e ective easticity. roof. For te assumptions in roposition 3 it is straigtforward to see tat price canges are identica to tose derived for an economy wit a singe consumer wit omotetic preferences, cean good expenditure sare, easticity of substitution in utiity, and poution easticity of cean consumption. roposition 4.NS. Assume non-separabe utiity from poution. Ten te anaogue of roposition (4) ods, wit te singe consumer being caracterized by a poution easticity of cean consumption given by te e ective poution easticity. roof. Te proof of roposition 4.NS carries troug anaogousy to te one for roposition (4). However, since now A NS,H YL + B NS,H YK coud in principe be equa to zero, it is necessary to additionay sow tat te oter two coe cients mutipying te e s in D NS,H,2 cannot aso be zero. It ten foows tat, if A NS,H YL + B NS,H YK = 0, ten one can construct te exampe anaogousy, based on e LZ or e KZ. To sow tis, assume tat A NS,H YL +B NS,H YK = 0. It foows tat B NS,H K ( YZ + YL ) A NS,H K YL = K B NS,H and A NS,H L ( YZ + YK ) B NS,H L YK = L A NS,H, terefore in order to be bot zero, te foowing must od: A NS,H = 0 and B NS,H = 0. Tis in turn impies A H = K NS and B H = L NS, wic in turn impies A H = B H. Inserting te expicit expressions for A H K L and B H deivers: L L K + L K L = K K L + K L K, ( L K ) K = ( K L) L, ( L K)( K L K + YK ) = ( K L)( K L + YL ). L Since we are assuming tat K, L, it foows tat te ast equaity is equivaent to ( K + YK ) = ( L+ YL ), wic is a contraction. K L roposition 5.NS. Assume non-separabe utiity from poution. Ten roposition 5 ods. roof. Since ony A, B and C are a ected by tis mode extension, and since tey a mutipy wit easticities tat are zero, it foows tat price canges in tis specia case are identica to tose in te origina mode. Identica ouseods wit non-omotetic preferences roposition 6.NS. Assume non-separabe utiity from poution. Ten roposition (6) ods, wit te coe cients in Condition are generaised as foows: A ID! A ID + K E,Z,B ID! B ID + L E,Z and C ID! C ID + E,Z. roof. Tis foows from equation (3.NS), using te fact tat E Y,Z = E,Z. roposition 6.NS iustrates tat wie extending te mode to aow for non-separabiity of utiity in poution can a ect te quantitative parameter vaues at wic te mode beavior switces, it does not cange te quaitative beavior of te mode.

8 2.4. Labor-eisure coice An important dimension aong wic ouseods can di er is teir vauation of eisure time resuting in di erences wit respect to te easticity of abor suppy. Incorporating endogenous abor suppy significanty enances te compexity of studying te impact of ouseod eterogeneity of equiibrium outcomes as it a ects bot ow income is earned and spent. To keep te teoretica anaysis tractabe, we restrict our attention to Cobb-Dougas utiity: U (, Y, ) = Y Y Y, were income is given by = w(t )+r K + Z Z. T represents ouseod s endowment of productive time. 24 We furter assume tat in te bencmark, ouseods dedicate an equa fraction of teir productive time to eisure: L, 8. T Using te first-order conditions, te demand functions are: = p (wt + r K + Z Z) Y = Y (wt + r K + Z Z) = Y (wt + r K + Z Z). p Y w It ten foows tat ˆ = ˆp + ŵ wt + w + ˆr r K + w + Z Z + w (ˆ Z + Ẑ) Ŷ = ˆp Y + ŵ wt + w + ˆr r K + w + Z Z + w (ˆ Z + Ẑ) ˆ = ŵ + ŵ wt + w + ˆr r K + w + Z Z + w (ˆ Z + Ẑ). We now need to modify our mode equations as foows: () is repaced by (.LL), and (2) is repaced by: ˆL L L + ˆL Y L Y L = ˆ. L In order anayze our propositions, we first need to derive te price canges for te case wit abor-eisure coice. Tey turn out to be identica to tose for a mode wit exogenous abor suppy, up to te vaue of te parameters, wic are extended as foows: K! LL,K w w r K wt + r K + Z Z K wt + r K + Z Z p L! LL,L w wt wt + r K + Z Z w(t ) L + w r K + Z Z p wt + r K + Z Z Z! LL,Z Te next subsection derives tese resuts Derivations for abor-eisure coice w wt + r K + Z Z Z w Z Z p wt + r K + Z Z. rice canges Up unti equation (B.5), te derivation is anaogous, yieding te foowing expression: (.LL) (2.LL) (3.LL) ˆp Y = p Y Y ( ) ˆ + K ˆK + L ˆL YK ˆK Y YL ˆL Y YZ Ẑ. (4.LL) Eiminate ˆ from equation (4.LL) by using equation (.LL) : wt ˆp Y = ŵ + w wl L + ˆr + w K + ˆ Z + w M + K ˆK + L ˆL YK ˆK Y YL ˆL Y + Ẑ + w Z YZ, (5.LL) Z 24 Di erences in T coud be viewed as refecting di erences in abor productivity across ouseods.

9 were L = T and te s are evauated for omotetic preferences. Next, sove equations () and (3.LL) for ˆK and ˆL, and insert tem into (5.LL). Furtermore, insert equation (B.2) to eiminate ˆp Y, tus obtaining: + w Z YZ Ẑ = YK + w Now eiminate ˆ by substituting (2.LL) into (6.LL) : YZ Ẑ = YK LL,Z M K ˆr + wt YL + w wl + ˆK Y ( K K + YK ) + ˆL Y ( L L + YL ) + L L LL,K ˆr + YL ˆ. LL,L ŵ + YZ M L ŵ + YZ + w LL,Z ˆ Z + Z ˆ Z (6.LL) + ˆK Y ( K K + YK ) + ˆL Y ( L L + YL ), (7.LL) were LL,K M p, +w LL,L M and wl +w LL,Z M p. Equation (7.LL) +w is formay identica to (B.7), wit te exception of te s. It terefore foows tat te resuting price canges are aso formay identica to 5a (5c) up to te vaue of te parameters. +w K w r K +w wt L + w r K + Z Z p +w Z w Houseod parameters Using te fact tat cov(, ) = M and = cov(,, ) + cov(, ) + cov(, ) + cov(, ) + M. ence Anaogousy: and L LL,L = p Y Y ( ) wt w(t ) ( ) w L p Y Y + cov(, K ) p L cov( L, ) L p Y Y L LL,L = cov( L,, ) + L cov(, ) + ( )cov( L, ) ( L) p Y Y Using te above expressions consider te LL,K = cov( L,, ) + K cov(, ) ( )cov( L, ) p Y Y + K + ZZ p = cov( L, ) + cov(, K ) L p Y Y p (p + p Y Y) L + ( L ), p Y Y cov(, L ) p + cov(, L ) p Z Z ( L ). K. parameters as tey appear in te expressions for te price canges: LL,Z = L Z Z p p Y Y cov(, ) LL,K K LL,L = ( L L )cov( L p Y Y ( L),, ) + ( K (L L ) + K ( Z ))cov(, ) + ( )( L L )cov( L, ) + cov(, L ) p Z Z p (8.LL) ( K + Z K ), (9.LL) were w, wt +r K + Z Z = p +p Y, and te covariance of tree variabes is defined anaogousy to Y te definition for two variabes in our paper. Note tat, in te foowing, we wi refer to as ouseod s expenditure sare on eisure. 0 M0

10 Resuts We find tat resuts are mainy simiar wit new parameters summarizing te additiona cannes of ouseod eterogeneity as we as te aggregate impact of abor-eisure coice on te genera equiibrium. roposition is identica, roposition 3 is anaogous wit presence of a term tat refects te impact of average expenditure sare of eisure on aggregate outcomes, and roposition 2 is anaogous accounting in addition for interactions between eisure coice and expenditure and income patterns. ropositions 4 and 5 are anaogous, too. For te specia case of Cobb-Dougas utiity, we tus find tat e ect of ouseod eterogeneity is simiar to te case witout abor-eisure coice; were it di ers it can be understood in terms of additiona terms refecting interactions between te various types of eterogeneity (abor-eisure coice, expenditures and income patterns). Weter or not te aggregation bias is quantitativey smaer or arger depends on specific parametrization. Te foowing subsection provide detaied anaysis supporting te above statements. We use te abe LL to enabe comparison between te origina propositions and te propositions based on te mode wit eisure. Equa factor intensities in production roposition.ll. Assume te mode wit abor-eisure coice and Cobb-Dougas utiity. Ten roposition ods. roof. If K = L, ten from te proof of roposition, we know tat it ten foows tat A = B = K C. Tis impies tat A LL = B LL = K C LL. It ten foows tat a te terms containing ouseod caracteristics in te expressions for te price canges drop out. Heterogeneous ouseods wit omotetic preferences roposition 2.LL. Assume te mode wit abor-eisure coice, equa bencmark sare of eisure time across ouseods ( L = L, 8 ), and Cobb-Dougas utiity. Ten, in addition to cov(, L ) and, roposition 2 is extended to incude cov(, ), cov( L, ) and cov( L,, ). roof. Equations (8.LL) and (9.LL). roposition 3.LL. Assume te mode wit abor-eisure coice, equa bencmark sare of eisure time across ouseods ( L = L, 8 ), and Cobb-Dougas utiity. If income sares are identica across ouseods ( L = L, 8 ), ten output and factor price canges are identica to tose for a singe ouseod caracterized by Cobb-Dougas preferences, cean good expenditure sare, an easticity of substitution between cean and dirty goods in utiity equa to te e ective easticity, and expenditure sare on eisure given by te income-weigted average of te sares across ouseods,. roof. Equations (8.LL) and (9.LL). Consider furtermore te foowing: = w L ( L L + L +w = w = w ( ) = L wl ( ) +w L using = L L L. Rewrite te above equaity, terefore obtaining: =. It terefore foows tat, in te case were abor ) income sares are identica across ouseods ( L = L, 8), te same ods for te s, tus impying cov(, ) = 0. roposition 4.LL. Assume te mode wit abor-eisure coice, equa bencmark eisure time across ouseods ( = L, 8 ), L and Cobb-Dougas utiity. Assume di erent factor intensities (i.e., K, L ), constant expenditure sares across ouseods (i.e., =, 8) and non-zero covariance between abor income sares and expenditure sares on eisure (i.e., cov(, L ), 0). Ten, for any observed consumption and production decisions before te tax cange, tere exist production easticities (i.e., and e i ) suc tat te reative burden on factors of production is opposite compared to te mode wit a singe consumer, couped to te same production side data and caracterized by an expenditure sare on eisure given by te income-weigted average of sares across ouseods,. roof. For te above assumptions, te cange in te return on capita is given by: ˆr = L YZ D LL, " A LL, (e ZZ e KZ ) B LL, (e ZZ e LZ ) + ( K L )( + # ) were A LL, = L K + K ( L + YZ + ZZ ), B p LL, = K L + L ( K + YZ + Z Z ), C p LL, = K + L + YZ + Z Z, D p LL, = C LL, + e KL [A LL, YL +B LL, YK ]+e LZ [B LL, K ( YL + YZ ) A LL, K YL ]+e KZ [A LL, L ( YK + YZ ) B LL, L YK ]+( K L )( L YK K YL + ( K + Z K )) ( K L ) cov(, L ) p. Anaogousy to te proof of roposition 4, one parameter coice tat eads to te reversa of factor price canges between te eterogeneous ouseod mode and te singe ouseod mode is te foowing: = e KZ = e LZ = 0 and [A LL, YL + B LL, YK ]e KL 2 min[( K L )( L YK K YL + ( K + Z K )) ( K L ) cov(, L ) p, ( K L )( L YK K YL + ˆ Z,

11 ( K + Z K ))], max[( K L )( L YK K YL + ( K + Z K )) ( K L ) cov(, L ) p, ( K L )( L YK K YL + ( K + Z K ))]. roposition 5.LL. Assume te mode wit abor-eisure coice, equa bencmark eisure time across ouseods ( = L, 8 ), L and Cobb-Dougas utiity. Assume Leontief tecnoogies in cean and dirty good production (i.e., = e i = 0), and tat te dirty sector is reativey more capita intensive (i.e., K > L ), suc tat te foowing ods: ( YL K YK L ) p Y Y + p ( K + Z K ) = 0. Ten: (i) if consumers are identica on te sources and uses side of income: ˆp Y = 0, ŵ > 0 and ˆr < 0. (ii) if consumers are identica on te uses side of income, and te L and s ave ow covariance (i.e., D LL > 0), ten ˆp Y < 0, ŵ > 0 and ˆr < 0. (iii) if consumers are identica on te uses side of income, and te L and s ave ig covariance (i.e., D LL < 0), ten ˆp Y > 0, ŵ < 0 and ˆr > 0. roof. rice canges assume te foowing form: ˆp Y = YZ D LL ( YL K YK L ) + ( K + Z K ) ˆr = cov(, L ) ˆ Z p L YZ ( + p YY D LL p )ˆ Z, were D LL = ( L YK K YL ) + ( K + Z K ) cov(, L ) p More tan two sectors Our anaysis so far assumed a igy aggregated sectora representation. Incuding more sectors can obviousy a ect te aggregation bias as it enabes representing ouseod eterogeneity aong more dimensions. Wit a finer sectora resoution, it is, for exampe, conceivabe tat poorer ouseods ave iger expenditure sares on some dirty goods and ower expenditure sares on some oters wen compared to ricer ouseods. Te probem is furter compounded by te possibiity tat di erent pouting goods are ikey to be produced wit di erent capita and abor intensities. As te aggregation bias is determined by te interaction between ouseod and production side caracteristics, te impact of going from two to mutipe sectors on te aggregation bias is tus in genera not cear-cut. We sow for a specia case wit Leontief tecnoogies in production tat te aggregation bias can sti be important for assessing te incidence of environmenta taxes in a setting wic incudes an arbitrary number of of dirty sectors J, denoted by te index. Anaogous to roposition 5 wit Leontief tecnoogies, we find tat te covariance between te ownersip of abor and consumption of eac dirty good across ouseods can reverse te sign of te factor price canges. We use te abe MC to enabe comparison between te origina propositions and te propositions based on te mode wit mutipe pouting commodities. roposition 5.MC. Assume Cobb-Dougas preferences and Leontief tecnoogies in cean and dirty production sectors. Assume furtermore tat eac dirty sector is more capita-intensive tan te economy-wide average (i.e., K Y L Y >, 8 ), and tat every K L dirty sector is more capita intensive tan te cean sector (i.e., K Y L Y > K L, 8 ). Ten, te foowing ods: (i) If consumers are identica on te sources or uses side of income, or bot: ŵ > 0 and ˆr < 0. (ii) If abor ownersip and dirty good consumption (for eac dirty good ) ave a positive covariance, ten ŵ > 0 and ˆr < 0. (iii) If abor ownersip and dirty good consumption (for eac dirty good ) as a negative covariance, ten ŵ > 0 and ˆr < 0 if covariance is ow (i.e. D J > 0), and ŵ < 0, ˆr > 0 if covariance is ig (i.e. D J < 0). roof. For J, we derive in te subsection Derivations beow te foowing expression for te renta rate of capita: ˆr = J L n= KYn L Yn Y nz n K L D J ˆ Z, were D J J KYn L Yn n= K L L YnKn K YnLn cov( p n p Yn Y n, L. ) For J =, tis is identica to te case considered in roposition 5. Consider te above equation for ˆr, bearing in mind tat if abor ownersip and dirty good consumption ave a positive covariance for eac good, ten te abor ownersip and cean good consumption ave negative covariance, since =.

12 Furtermore L Y K K Y L = L K ( Y K K is more capita intensive tan te cean sector. Y L L ) = L K p p Y Y ( K Y K L Y L ). Tis expression is positive if every dirty sector Derivations Te equiibrium conditions () (3) for te mode wit J dirty sectors and one cear sector are given by: ˆK K K + ˆL L L + ˆK Y K Y K = 0 (.MC) ˆL Y L Y L = 0 (2.MC) ˆK ˆL = 0 (3.MC) ˆK Y Ẑ = 0 8 (4.MC) ˆL Y Ẑ = 0 8 (5.MC) ˆp + ˆ = K (ˆr + ˆK ) + L (ŵ + ˆL ) (6.MC) ˆp Y + Ŷ = Y K (ˆr + ˆK Y ) + Y L (ŵ + ˆL Y ) + Y Z (ˆ Z + Ẑ ) 8 (7.MC) ˆ = K ˆK + L ˆL (8.MC) Ŷ = Y K ˆK Y + Y L ˆL Y + Y Z Ẑ 8 (9.MC) ˆ Ŷ = ˆp Y 8, (0.MC) ˆ = ˆ 8 (.MC) ˆ = ˆ (2.MC) Ŷ = Y Yˆ Y wit ˆ = ŵ w L + ˆr r K + ZZ (ˆ p +p Y Y Z + unknowns ( ˆK, J ˆK Y, ˆL, J ˆL Y, ŵ, ˆr, ˆ, p x, J ˆp Y, J Ŷ, J Ẑ, H ˆ, J H Ŷ 8, (3.MC) Z Ẑ ). Equations (.MC) (3.MC) are 6 + 5J + H + JH equations in 6 + 5J + H + JH Z ). Foowing Waras Law, one of te equiibrium conditions is redundant, tus te e ective number of equations is 5 + 5J + H + JH. We coose as te numéraire good, tus deivering a square system of equations. Te equiibrium soutions are terefore fuy determined as functions of te exogenous tax increase ˆ Z > 0. In order to derive te factor price canges, start by subtracting (8.MC) from (6.MC) an (9.MC) from (7.MC): Substitute (2.MC) into (8.MC) and (3.MC) into (9.MC): Sove (0.MC) for Y and insert into (7.MC): Y Y 0 = Kˆr + L ŵ (4.MC) ˆp Y = Y K ˆr + Y L ŵ + Y Z ˆ Z 8. (5.MC) Y Y ˆp Y = Y ˆ = K ˆK + L ˆL (6.MC) Yˆ = Y K ˆK Y + Y L ˆL Y + Y Z Ẑ 8. (7.MC) Y ˆ Y K ˆK Y Y L ˆL Y Y Z Ẑ 8. (8.MC) From (6.MC) insert te foowing on te rigt-and side of te equaity: 0 = K ˆK + L ˆL Y Y ˆp Y = ˆp Y, tus yieding: ˆp Y = ( Y Y ˆ and use te fact tat ) ˆ + K ˆK + L ˆL Y K ˆK Y Y L ˆL Y Y Z Ẑ 8. (9.MC)

13 Eiminate ˆ from equation (9.MC) by using equation (.MC), ten insert te expicit form of te budget cange ˆ : ˆp Y = ŵ L + ˆr K + K ˆK + L ˆL Y K ˆK Y Y L ˆL Y Y Z Ẑ 8, (20.MC) were L = ( Y Y ) w L, K = ( Y Y ) r K and using te fact tat ( Y Y ) Z Z p +p Y Y = 0. Now sove equations (.MC) and (2.MC) for ˆK and ˆL and insert tem into equation (20.MC). Furtermore, insert equation (5.MC) to eiminate ˆp Y, tus obtaining: Y Z Ẑ = ( Y K K )ˆr + ( Y L L )ŵ + Y Z ˆ Z + ˆK Y Y K + K Y L Y ˆK Y K + ˆL Y K Y L + ˆL Y L L Sove equations (4.MC) and (5.MC) for ˆK Y and ˆL Y, and insert tem into equation (2.MC). Tis yieds: Ẑ ( K K Y K + L L Y L )Ẑ = ( K + Y K )ˆr + ( Next eiminate te Ẑs. To acieve tis, substitute (.MC) and (2.MC) into (3.MC), obtaining: Substituting equations (4.MC) and (5.MC) into (23.MC) yieds: Now insert (24.MC) into (22.MC): ˆr( Y K ( K ) + ŵ( Y L K Y L Y ˆK Y + ˆL Y = 0. K L K Y K + L Y L )Ẑ = 0. L ) + ˆ Z Y Z = 8. (2.MC) L + Y L )ŵ + Y Z ˆ Z 8. (22.MC) (23.MC) (24.MC) K Y K Ẑ Ẑ 8. (25.MC) Now combine te above J equations in (25.MC) in order to be abe to sove for te factor price canges. To do so, mutipy eac equation by an unknown parameter A and sum over : ˆr (A Y K A K ) + ŵ (A Y L A L ) + ˆ Z A Y Z = (( A ) K Y + A )Ẑ. (26.MC) K It terefore foows tat if te rigt-and side is zero, ten te Ẑs drop out of (26.MC). As an ansatz, require te foowing, wic wi ten make te rigt-and side of (26.MC) zero due to (24.MC): ( A ) K Y K + A = K Y K L Y L 8. (27.MC) In order to sove for te set of As tat satisfies (27.MC), sum (27.MC) over, and reabe indices to obtain te foowing (using te notation K Y K Y and anaogous notation for te oter aggregate variabes): Insert (28.MC) back into (27.MC), and sove for A : For te A coe A = K Y K L Y L ( A ) = K K ( K Y K K Y K ( K Y K cients as in (29.MC), (26.MC) ten becomes: A ( Y K K ) ˆr + A ( Y L L Y L ). L Y ) = L KY L L K L ) L Y L ŵ = ˆ Z A Y Z. (28.MC) 8. (29.MC) (30.MC)

14 Sove (4.MC) for ŵ and substitute into (30.MC), tus obtaining: L A Y Z ˆr = A L Y K K Y L L K + K L ˆ Z. (3.MC) Tis ten deivers te expression for ˆr, using te fact tat L K + K L = ( Y Y )( L L ) = cov( p Y Y, L ) p cov(, L ). (32.MC)