Sales Tax: Specific or Ad Valorem Tax for a Non-renewable Resource?

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1 Sals Tax: Spcific or A Valorm Tax for a Non-rnwabl Rsourc? N. M. Hung 1 an N. V. Quyn 2 Absrac This papr shows ha for a im-inpnn spcific ax an a im-inpnn a valorm ax ha inuc h sam compiiv uilibrium in h Holling mol of rsourc xracion h a valorm ax yils a highr lvl of iscoun ax rvnus han h spcific ax. Morovr givn h sam lvl of iscoun ax rvnus h a valorm ax also yils a highr lvl of social wlfar. Finally for h im-pnn schuls of opimal a valorm ax an opimal spcific ax w show ha whn approprialy s hy ar uivaln in implmning h ynamic social opimum an proviing h sam iscoun ax rvnus. Kywors: Non-rnwabl Rsourcs A Valorm Tax Spcific Tax Wlfar JEL Classificaion: H21 Q3 In h liraur on axaion h form of h sals ax os no mar in saic analysis: h auhoriis can us ihr a spcific ax which cras a gap bwn proucr an consumr prics or an a valorm ax as a prcnag τ of h proucr pric o implmn h sam rsourc allocaion p whr p an no rspcivly h mark pric an uaniy wih p f f bing h invrs mark man funcion. Th spcific ax an h a valorm ax ar sai o b uivaln in ha hy cra h sam isorion an inuc h sam wlfar loss for an ual ax rcip τp. Dos his faur hol in a ynamic framwork? This is h main usion w ask in his papr. W firs focus on h im-inpnn ax schul bcaus of is wispra us in many ax lgislaions s Sarma an Narsh 21. In Scion 1 of h papr w show in h simpl Holling mol of compiiv rsourc xracion s Holling 1931 ha whn a im-inpnn spcific ax an a im-inpnn a valorm ax inuc h sam compiiv uilibrium h a valorm ax yils a highr lvl of ax rvnus han h spcific ax. In Scion 2 w riv h wlfar implicaions of hs 1 Déparmn économiu Univrsié Laval Qubc Canaa G1W 4R9 mail: nmhung@cn.ulaval.ca. H woul lik o hank Grar Gau for a iscussion on h subjc of his papr. 2 Déparmn scinc économiu Univrsié Oawa 55 Laurir E Oawa Onario Canaa K1N 6N5 mail: nvuyn@uoawa.ca. 1

2 wo im-inpnn forms of sals axs whn hy provi h sam iscoun ax rvnus an conclu ha h a valorm ax shoul b chosn as h appropria insrumn in ax sign. In Scion 3 w urn o h opimal im-pnn schul of h sals ax an argu ha h ynamic a valorm an spcific axs alhough hy xhibi ui iffrn characrisics ar uivaln in implmning h sam ynamic parn of rsourc allocaion an provi an inical lvl of iscoun ax rvnus. 1. Sals Tax in h Holling Mol Consir a compiiv mark for a non-rnwabl rsourc. Tim is coninuous an no by. Th oal rsourc sock of h inusry a im is S an h rsourc can b xrac a a consan uni xracion cos of γ. L p f b h invrs mark man curv for h rsourc a ach insan whr p is h pric pai by consumrs for a uni of h rsourc an is h mark man a pric p. W shall assum ha f > f ' < for all > an l im f. Th las assumpion implis ha h rsourc sock will no b pl in fini im bu asympoically. In wha follows w shall l obain whn u f x x no h insananous social wlfar unis of h rsourc ar consum. Th mark ra of inrs is r. Suppos ha h auhoriis lvy a im-inpnn spcific ax say on h firms in h inusry. To fin h compiiv uilibrium inuc by h spcific ax l us consir h following maximizaion problm: r 1 max u γ + ] subjc o 2 S. No ha h xprssion unr h ingral sign in 1 rprsns h iscoun valu of h sum of h consumr an proucr surplus a im. Invoking h maximum principl 2

3 w can assr ha h opimal xracion ra a im is h valu of ha saisfis h following firs-orr coniion: 3 r u' γ. In 3 is a posiiv consan which rprsns h iscoun scarciy rn shaow pric of h rsourc sock. W shall l no h valu of ha solvs 3; ha is 1 r 4 f γ + +. As fin is h opimal xracion ra a im. Furhrmor h opimal xracion program saisfis h following sock consrain: 1 r 5 f γ + + S. No ha h scon ingral in 5 is sricly crasing in. Furhrmor i ns o infiniy whn ns o an ns o whn ns o infiniy. Hnc by coninuiy hr xiss a uniu valu of say such ha 1 r 6 f γ + + S. Nx l p f no h rsourc pric along h opimal rajcory of h maximizaion problm consiu by 1 an 2. Euaion 3 now aks on h following form: r 7 p γ which can also b xprss unr h following form known as h Holling rul: 8 p r p γ ]. Now unr h compiiv uilibrium h n pric of on uni of h rsourc n of uni xracion cos an h spcific ax mus apprcia a las a h mark ra of inrs in orr for ha uni of rsourc o rmain in siu. Furhrmor if h n pric of h rsourc apprcias a a ra blow h mark ra of inrs hn h nir rsourc sock will b insananously xrac. Diffrnial uaion 8 hus scribs h voluion of h rsourc pric unr h compiiv uilibrium inuc by h 3

4 spcific ax. I follows ircly from 8 ha h rsourc pric incrass monoonically ovr im whil h corrsponing xracion ra crass an approachs zro asympoically. W now urn o h a valorm ax which is lvi as a consan prcnag τ of h rsourc pric. L no h rsourc pric a im unr h compiiv p uilibrium inuc by h im-inpnn a valorm ax τ. Th uilibrium coniion for h ass mark along h im pah of h compiiv uilibrium inuc by h consan a valorm ax τ is givn by r 9 1τ p γ ] 1 τ p γ. Diviing 9 by 1 τ w obain r τγ γ τγ 1 p γ. 1τ p 1τ 1τ Obsrv ha 1 also scribs h rsourc pric a ach insan unr h compiiv uilibrium inuc by h spcific ax ra τγ / 1 τ ; ha is h a valorm ax τ an h spcific ax wih τγ / 1 τ inuc h sam compiiv uilibrium. W now sablish: PROPOSITION 1: If a consan spcific ax an a consan a valorm ax inuc h sam compiiv uilibrium hn a ach insan h a valorm sal ax yils a highr lvl of ax rvnus han h spcific ax. PROOF: L b a im-inpnn spcific ax an τ b h im-inpnn a valorm ax whr τγ / 1τ. Thn τ / γ + an h wo sals axs inuc h sam compiiv uilibrium. Furhrmor h ax rvnu collc a im unr h spcific ax is whil h ax rvnus collc also a im unr h a valorm ax τ is τp p > γ + 4

5 whr h sric inualiy is u o h fac ha p > + γ. Q.E.D. 2. Tax Dsign an Wlfar Consiraion W now consir h isorion of a spcific ax. L r 11 W u γ. rprsn h iscoun social wlfar unr h compiiv uilibrium inuc by h spcific ax. W xpc ha a ris in by craing mor isorion in h compiiv xracion of h rsourc rucs W. Th proof of his rsul ruirs som mor chnical argumns. Firs w sablish som prliminary rsuls. CLAIM 1: A ris in h spcific ax i rucs h shaow pric of h rsourc i.. ' < an ii shifs proucion from h prsn o h fuur. Mor prcisly hr xiss a im Log ' ]/ r such ha / < for < / a an / > for >. PROOF: To sablish i no ha if riss or rmains h sam afr a ris in hn accoring o 4 h rsourc xracion a ach insan will b lowr afr h ris in han bfor h ris in an his mans ha h rsourc sock is no pl afr h ris in : a conraicion. To sablish ii firs no ha h xracion ra a im mus b lowr afr h ris in. In if h xracion ra a im is highr afr h ris in hn his rsul coupl wih a clin in h shaow pric of h rsourc imply ha h xracion a ach insan is highr afr h ris in : h cumulaiv rsourc xracion afr h ris in xcs h availabl sock an his is no possibl. Hnc / < for small valus of. Nx iffrnia 7 wih rspc o w obain r 12 p 1+ '. 5

6 W claim ha h righ si of 12 is posiiv for small valus of. In if his is no h cas hn 1 + r ' is ngaiv for all > an ns o mans ha h uilibrium rsourc pric a any im whn which > is lowr afr h ris in an his in urn implis ha h cumulaiv xracion afr h ris in xcs h availabl rsourc sock. Hnc hr xiss a uniu im Log ' ]/ r such ha 13 p > for < a < for >. Finally no ha ii follows from 13 f ' < an h fac ha p / f ' / ]. Q.E.D. Th claims w hav ma ar ui inuiivly appaling. A ris in h spcific ax is consir by proucrs as an incras in h cos of xracing h rsourc hus rnring i lss profiabl. Sinc h rsourc bcoms lss valuabl is shaow pric woul naurally fall. Furhrmor h incras in h cos in rm of ax paymns may b pospon: by rucing h xracion in h nar fuur small valus of an incrasing i in a isan fuur larg valus of rsourc proucrs will rais iscoun profis by frring ax paymns. W ar now ray o analyz h isorions gnra by h spcific ax. L us sa: CLAIM 2: Th highr h im-inpnn spcific ax h lowr will b h social wlfar associa wih h xracion of h rsourc. 6

7 PROOF: L us iffrnia 11 o obain afr som manipulaion: ] ' ' u W r r γ No ha h hir lin in 14 has bn obain wih h hlp of 7 an h las lin in 14 has bn obain by using h fac ha h rivaiv wih rspc o of h sock consrain 6 is ual o. Whhr ' W is posiiv or ngaiv pns on h sign of h ingral on h las lin of 14. Using Claim 1 w can assr ha his ingral saisfis h following inualiy:. + < + r r r r r r Q.E.D. ROPOSITION 2: If h auhoriis wan o rais a givn lvl of iscoun ax rvnus ROOF: L W ar now abl o monsra: P by imposing a sals ax hn i is mor fficin o us a consan a valorm ax han a consan spcific ax bcaus social wlfar is highr unr h formr sals ax han unr h lar sals ax alhough hy boh yil h sam lvl of iscoun ax rvnus. P no h spcific ax. From Proposiion 1 an a valorm ax / γ τ + ul yil a highr lvl of ax rvnus a ach insan han h spcific h iscoun ax rvnus collc unr h a valorm ax will xc ha collc unr h spcific ax i.. > p r r τ. L h ruir wo ax an hus 3. ] ] W' p r r r r r γ

8 iscoun ax rvnus b h righ-han ingral in h inualiy. I can b achiv wih a lowr a valorm ax ra say τ ' < / γ +. If w l ' b such ha τ ' ' / γ + ' hn ' <. Furhrmor h a valorm ax τ ' inucs h sam compiiv uilibrium as h spcific ax '. Hnc h ruir iscoun ax rvnus collc unr h spcific ax coul also b collc unr h a valorm ax τ ' which bcaus i inucs h sam compiiv uilibrium as h spcific ax ' yils a highr lvl of social wlfar han h spcific ax accoring o Claim 2. Q.E.D. 3. Tim-Dpnn Opimal Sals Tax W rsrv h las par of his no o iscuss h opimal sals ax in h Holling mol. A im-inpnn sals ax ncounr in pracic is isorionary an rucs iscoun social wlfar blow is opimal lvl s Nhr In hir insighful book on rsourcs conomics Dasgupa an Hal 1979 assr ha h im-pnn spcific ax schul r whr is a givn posiiv consan inucs a r r 15 p& r r p γ which can b simplifi o compiiv uilibriu m ha is socially opimal. In unr h compiiv uilibrium inuc by his im-pnn a x policy h voluion of h rsourc pric hrough im is govrn by h following iffrnial uaion: 16 p& r p γ. Th iffrnial uaion rprsn by 16 scribs h voluion hrough im of h rsourc pric unr h social opimum. I is clar ha h propos ax schul r which acs xacly lik a ax on rsourc rn or a ax on firms profi os no isor h social opimum. By varying w can vary h iscoun ax r r rvnus collc which is simply. Thus hr is no a uniu S 8

9 im-pnn schul for h opimal spcific ax. Th im-pnn opimal spcific ax pns on h iscoun ax rvnus h auhoriy wans o collc. Now w riv h opimal a valorm ax. Again rcall ha a consan a valorm ax inucs a compiiv uilibrium ha is no socially opimal. Hnc a socially opimal a valorm ax mus b im-pnn. Thus l τ no h a valorm ax ra a im. A any insan h n pric of h rsourc n of h uni xracion cos an h a valorm ax is givn by p γ τ p. Unr h compiiv uilibrium inuc by h a valorm ax τ h n rsourc pric mus apprcia a h mark ra of inrs i.. 17 p& & τ p τ p& r p γ τ p which can b rwrin as & τ p 18 rτ γ p& r p γ +. 1τ In orr for 18 o scrib h voluion of h rsourc pric unr h social opimum w shoul choos τ so ha & τ p rτ γ which in urn implis 19 & τ rγ. τ p Bcaus p riss wih im an ns o infiniy as h growh ra of h a valorm ax is crasing hrough im an ns o as. This rsul sans in conras wih h spcific ax h growh ra of which is consan an g ivn by & / r. Also as wih h opimal spcific ax h opimal im-pnn a valorm no isor h social opimum. W hav h following proposiion: ax os PROPOSITION 3: Whil h opimal spcific ax is of h form r for a givn > h opimal a valorm ax is givn by s γ + p γ τ τ whr τ is a consan r insi h inrval 1. Morovr if τ / p hn his im-pnn a valorm ax is opimal an yils h sam iscoun ax rvnus as h spcific ax.. γ r rs 9

10 PROOF: L p no h rsourc pric a im unr h compiiv uilibrium r wihou sals axs. Thn p saisfis 16 an can b rwrin as p γ p γ which can b us in 19 o yil rγ 2 Log τ ] τ r is givn. γ + p γ Ingraing 2 bwn im an im w obain rγ 21 Log τ ] Log τ ] s. rs γ + p γ I follows from 21 ha 22 τ τ r s rs p γ γ + γ. To sablish h scon samn of Proposiion 2 firs no ha for a givn valu of τ uaion 22 rprsns an opimal im-pnn a valorm ax. By varyingτ w obain iffrn opimal im-pnn a valorm axs ach of which allows us collc a givn iscoun ax rvnus. W now ruir ha h iscoun ax rvnus collc unr such an a valorm ax b ual o S h iscoun ax rvnus collc unr h spcific ax. L τ b such ha τ p i.. τ / p whr o an p no rspcivly h rsourc pric an h xracion boh a im unr h compiiv uilibrium. Clarly h ax rvnus collc a ach insan ar h sam unr h wo axs nsuring ha ou r ruirmn is saisfi. Diffrnia logarihmically τ wih rspc o im w g & τ / τ & / p& / p. No ha & / r an along h compiiv uilibrium pric pah w hav xacly h sam p& / p r1 γ / p. I now follows ha & τ / τ r γ / p which is 19 h uaion ha characrizs h opimal a valorm ax. W hav jus shown ha for any opimal im- pnn spcific ax hr is an opimal im-pnn a valorm ax ha yils h ax rvnu a ach insan an a foriori h sam iscoun ax rvnus ovr h nir im horizon. Th wo sals axs ar link by h rlaion r τ / p. Q.E.D. 1

11 4. Concluing Rmarks Th main rsuls w obain wih rspc o im-inpnn sal axs in his papr can b xn o h cas of imprfc compiiv rsourc marks. Tak for xampl h cas of rsourc monopoly. Again for h sam parn of rsourc xracion w may us h sam argumns o assr ha h a valorm ax is suprior o h spcific ax: ihr h a valorm ax yils a highr lvl of iscoun ax rvnus han h spcific ax whn boh axs inuc h sam compiiv uilibrium or for h sam lvl of iscoun ax rvnus h a valorm ax inucs lss wlfar loss han h spcific ax. Wih rspc o our iscussion on im-pnn opimal ax schuls h ask is mor cumbrsom. Hr monopoly is a sourc of sub-opimaliy so ha h opimal ax sign shoul oghr wih h obligaion o rais a crain lvl of ax rvnus also corrc a mark failur in orr o achiv a social opimum s Brsgrom Cross an Porr This problm alhough inrsing woul ak us oo far afil. L us finally no ha in many ynamic conomic analyss h ffcs of axs ar ofn sui as shifs of h say-sa uilibrium. No u anion is pai o h ransiory pahs owar a nw say sa an rsuls of h comparaiv saic analysis ar appli wihou any forhough. Rcall ha in comparaiv saic analysis givn a spcific ax on can always fin an a valorm ax ha implmns h sam uilibrium oucom an yils h sam ax rcip an vic vrsa. This uivalnc os no hol in h simpl Holling ynamic rsourc mol. W hav shown for h sam uilibrium rajcory an a valorm ax ha is consan ovr im yils a highr ax rvnu han a spcific ax. Furhrmor for a givn iscoun ax rvnu h social wlfar rsuling from compiiv uilibrium is highr wih h a valorm ax han wih h spcific ax. Also finally h uivalnc of hs wo forms of sal ax in a ynamic sing can b rsor only if hy ar im-pnn an w insis hy ar all opimal in h sns ha hy o no isor h compiiv uilibrium rajcory. 11

12 Th p rason for hs nw an surprising rsuls is h following. In h Holling mol of rsourc xracion w rfr o in his papr no non-gnra say sa xiss bcaus h rsourc sock is always pl. Thrfor h analysis shoul b conuc wih a sunc of ransiory sas an h appropria mho o look a h ffc of axs mus b h comparaiv ynamics no h simpl comparaiv saic analysis usually ncounr in h liraur of ax analyss. In his rgar w bliv w o call anion o a s of mor gnral problms no uly invsiga. Rfrncs Brgsrom T. C J. G. Cross an R. C. Porr 1981: Efficincy-Inucing Taxaion for a Monopolisically Suppli Dplabl Rsourc Journal of Public Economics 151 p Dasgupa P. an G. M. Hal 1979: Economic Thory an Exhausibl Rsourcs Cambrig Univrsiy Prss Cambrig. Holling H. 1931: Th Economics of Exhausibl Rsourcs Journal of Poliical Economy 39 2 p Nhr P. A. 1999: Economics of Mining Taxaion in Jron C. J. M. van r Brg. Hanbook on Environmnal an Rsourc Economics. Sarma J. V. M. an G. Narsh 21: Minral Taxaion aroun h Worl: Trns an Issus Asian-Pacific Tax Bullin Inrnaional Burau of Fiscal Documnaion Amsram. 12