ECON Lecture 5 (OB), Sept. 21, 2010

Transcription

1 1 ECON Lecure 5 (OB), Sep. 21, 2010 axaion of exhausible resources Perman e al. (2003), Ch INODUCION he axaion of nonrenewable resources in general and of oil in paricular has generaed a lieraure of is own. Oil is a paricularly profiable nonrenewable resource, heavily axed in mos counries. he purpose of axaion can be said o be direced o collec a large share of he resource ren ( oil ren ) for he public. he comparison of differen kinds of axaion from wo perspecives, fiscal efficacy, ha is how successful or useful axaion is for collecing he resource ren, and creaing/avoiding disorions in he marke, measured as deviaions from a non-ax siuaion. Anoher fiscal aspec of nonrenewable resources is he use of he resource ren, hereunder Duch Disease, ren seeking e al. For many oil counries he fiscal managemen of he oil ren has major imporance for naional economic developmen as well as for he disribuion of he benefis from oil exracion.

2 he analysis of axaion issues and efficiency aspecs can be very complicaed, we will discuss axaion of a few sylized kinds wihin our simple models. he focus in he lieraure will be on how he axaion rules affec he price and exracion profiles, relaive o he compeiive soluion. In more recen years he climae effecs have of major ineres. 2 Perman e al. (2003) has a shor and somewha unsaisfacory secion 15.7 on axaion, wih erminology differing from ours. [Noe ha he book uses royaly ax for wha we call resource ren ax and revenue ax for wha we call royaly ax, while Sinn 2008 used cash flow ax for resource ren ax.] he mehod we apply is o derive he impac of a ax hrough comparing wih he compeiive soluion, suppored by inuiive reasoning. We do his primarily in he simple sandard Hoelling model. We shall also allow cos o be a funcion of he exracion rae. In boh seings he shadow price of he remaining resource will be posiive and hus he enire sock will be exraced. For he analysis below we inroduce he following ypes of axes wrien as funcion σ(.) of one or more of he following variables: exracion ( ), price (p ), resource sock (S ), ime (): σ(, ) = s() royaly / (severance ax σ(,p,) = s()p ad valorem royaly / (severance) ax σ() = f() franchise ax/license fee σ(s,) = g()s propery (resource sock) ax σ(,p,) = u()[p -b] resource ren / cash flow ax

3 3 AXAION UNDE CONSAN UNI COSS We sar wih he simple Hoelling model wih consan uni coss and a ime invarian demand schedule wih a choke poin. We mark he compeiive nonax soluion wih as he reference case: (1) p = α β (2) q = p b (3) q = rq (4) S 0 = 0 d (5) p = α Le us consider a resource ren ax. In his case only (3) ges a differen specificaion. he arbirage principle in his case will be (6) dq [ (1 u ( ))]/ d= rq(1 u ( )) We see ha u() consan implies ha q grows a rae r, hence no disorion. I also follows easily ha u > 0 implies ha q grows a a rae higher han r, and vice versa when u < 0 (why?). wih exracion shifed owards he presen/fuure, respecively. By varying he iniial rae u(0) any par of he resource ren can be colleced, as long as u(0) p b. Inroducing royaly ax implies ha he arbirage principle insead of (3) is (7) dq [ s ( )]/ d= rq [ s ( )] If s() increases wih rae r hen (1)-(5) are sill fulfilled and he soluion he same as he reference case, and in fac he same as a consan rae resource ren ax. Hence, no disorion under royaly ax when he ax rae increases by r. 0

4 4 If on he oher hand he royaly ax rae s() is consan over ime, i follows ha q mus increase a a rae less han r (why?), causing a similar effec as a lower ineres rae, namely flaening of he exracion profile hrough a higher iniial price and a correspondingly longer exracion period. he imposiion of such a ax is easily seen o have he same effec as increased uni cos. Le us go back o he resource ren ax and follow Sinn 2008 in denoing (1-ax rae) as he ax facor, i.e. θ = (1 u ( )) and (6) becomes (6 ) [ θ ]/ d q d = rq θ ˆ Le θ change exponenially, θ = θ0e θ wih ˆ θ consan. ˆ θ > 0 means reduced ax over imes and vice versa, so le us hink of ˆ θ as negaive, increasing axaion wheher for fiscal or climae reasons. Wih he ax rule (6 ) pu ino he original problem, we find ha we ge he reference case soluion wih r changed o r θˆ. Increased resource ren ax over ime by exponenially reduced ax facor has he same effec on he price and depleion profiles as increased ineres rae. Le us check ha he effecs we have found are suppored by inuiive reasoning! Consan royaly ax is reduced in real value by posponing producion, while an increasing resource ren ax rae can be couneraced by moving producion owards he presen. easonable?

5 5 AXAION UNDE VAYING UNI COSS Le us have ago a a more ambiious analysis by allowing uni coss o depend upon he exracion rae, replacing b wih b ( ). I will sill be he case ha all of he resource is depleed. he equaions (1)-(5) for he reference sill holds, excep ha (2) will have o be replaced by (2 ) q = p b ( ) he maximizaion problem wih he ax funcion in all-encompassing form as σ(,p,s,) now resuls in he Hamilonian in curren value as c (8) H = p b( ) σ (, p, S,) µ By he maximum principle we have (9) p = µ + b ( ) + σ and he shadow price equaion (10) µ = rµ + σs he ransversaliy condiion, H c =0 a, becomes (11) p b ( ) σ = µ + + which combined wih he firs order condiion a gives ha (12) b ( ) σ b ( ) + σ = + (9)-(12) can be considered as he equaions holding for a single resource producing firm. Assuming all firms idenical he equaions can be inerpreed as hose holding for he resource producing indusry. We hen have o endogenize he price via he demand equaion and hus replace p in (9) wih p = p( ). We hen differeniae (9) oally wr., using (10) and (9) o eliminae respecively. Afer rearranging we arrive a µ and µ,

6 6 (13) ( p b ( ) σ σ p) = rp [ b ( ) σ ] + σ + σ σ, p S S he differenial equaion for, useful for sudying he effecs of axes. In he non-ax case he equaion simplifies o ( ( )) [ ( )], (14) p b = rp b (noe ha decreases over ime). Le us consider he axes menioned above: he franchise ax, σ = f(), a fixed amoun, possibly varying over ime, for he enire period exracion akes place. We may expec his ax o give an incenive o shoren he exracion period? he royaly ax, σ(, ) = s(). Here each uni of oil depleed will be axed. If he ax rae is consan over ime, wha can he agen do o reduce he ax burden? he depleion profile can be shifed o reduce he burden. Less producion early and more laer will reduce he presen value of he axaion and hus is wha we would expec. If he ax rae varies he agen will naurally ry o deplee more when (he presen value of) he rae is low and less when (he presen value of) he rae is high. he ad valorem royaly ax has similar effecs and we shall no discuss i separaely. he propery ax, σ(s,) = g()s. Here he remaining resource sock is axed, hence he ax burden can be reduced by shifing he producion forward. Finally, he resource ren ax, i.e. σ(,p,)= u()[p -c]. As i is neural, i is an aracive ax o apply. Bu also emping for he auhoriies o apply i progressively, ha is a higher ax rae when he resource ren is higher. he ask is hus o find he impac upon he producion profile using (13), aking noe of he opimal no-ax soluion (14). he ax issues in he real world of resource depleion are, however, more complicaed han deal wih here.

7 7 HE FANCHISE AX Noe ha (13) hen simplifies o: (15) ( p b ( )) = rp [ b ( )], idenical o ha of he non-ax case (14)! his means ha if he wo exracion pahs for some had he same depleion level hey would coincide. From ransversaliy (12) follows ha he marginal cos a is b ( ) f( ) b ( ) = + and hus higher han ha of he non-ax case a. hus he erminal exracion level mus be higher han As he oal exracion is he same wih and wihou axaion he only possible profile for exracion under franchise axaion is wih a higher iniial exracion, hence lower iniial price, and a shorer exracion period, see Figure 1. Figure 1. he effec of a franchise ax S 0 () () S 0

8 8 HE OYALY AX We have σ = s(). From he ransversaliy condiion (12) we noe ha as σ in his case is equal o σ, we mus have =, (bu no necessarily = ). he differenial equaion (13) is simplified o (16) ( p b ( )) = r[ p b ( ) s( )] + ds / d If he royaly ax rae is consan, he las erm in (16) vanishes. We can hen make a comparison wih he non-ax case in he following way. If a some =, hen mus be flaer han, in oher words i mus cu from below. Hence, he only possible soluion is ha sars ou lower (price higher) han, cus from below and coninues beyond o ending wih exracion rae equal o, see Figure 2. Figure 2 he effec of a consan royaly ax () () hus a any iniial sock level (=0) a consan royaly ax causes o be reduced relaive o he non-ax soluion. Furhermore, he exracion period is lenghened. We can explain or inerpre his by noing ha he ax increases marginal cos, hus i seems reasonable ha depleion is lower. Bu he ax canno affec he cumulaive exracion, hence he reduced depleion rae implies a longer horizon. (Noe ha figure 2 incorrecly depics he (idenical) resource socks in he wo cases.)

9 9 We can also noe ha he case we discussed above under he assumpion of consan uni cos, namely s() increasing by rae r, is neural. Insead of (16), we ge (16 ) ( p b ( )) = r[ p b ( ) s( )] + ds / d = rp [ b ( )] We can also see ha such an increasing royaly ax will yield a consan proporion of he resource ren by comparing he ransversaliy condiions for he ax and non-ax case, cf. (12). b ( ) p ( ) µ s ( ) b p ( ) = + = + + vs. µ ( ) As =, µ + s is equal o µ a = and as µ, µ and s grow a rae r, he ax will be a consan proporion of he resource ren a all imes. I is in fac a consan rae resource ren ax.

10 10 HE POPEY AX (ESOUCE SOCK AX) Here σ(s,) = g()s and (13) becomes (17) ( p b ( )) = rp [ b ( )] + g ( ) I follows ha if he depleion rae for any is equal o ha of he non-ax case, he ax case mus have a seeper fall in he depleion rae ha he non-ax case, i.e. <. From he ransversaliy condiion (12) follows furhermore, as 0 S =, ha =. hus he only possible profile is somehing like Figure 3. As we see he depleion is faser wih ax han wihou. Figure 3. he effec of a propery ax (based on remaining resource sock). () ()

11 11 WHA ABOU A POGESSIVE ESOUCE EN AX? Le us finally elaborae furher on he resource ren case. We have already found ha a consan rae of resource ren ax is somewha of an ideal as i is perfecly neural and expropriaes a corresponding share of he cumulaed resource ren. I may be emping o apply i as a progressive ax such ha wih ax rae higher he higher he resource ren, ha ax will cach a higher share of windfall income (for whaever reason) in erms of high resource prices. Define π = p b( ) and respecify he resource ren ax as σ(, p,) = s( π) π. We have sudied s consan above, wha if s is no consan? Inroducing his specificaion in (9) we can work ou he expression for correspnding o (13) as he following: (18) [ p b ( ) s ( p b ( ))( p p b ( )) s ( p b ( )) s ( p b ( ))( p p b ( )) s ( p b ( ))( p b( )) s ( p b ( ))( p p b ( )) = r( p b ( ) s ( p b ( )) s ( p b ( ))( p b( ))) which can be simplified o (18 ) [( p b ( ))(1 ( s+ πs )) (2 s + s π)( p b ( ))( p+ p b ( ))] = r( p b ( ))(1 ( s+ s' π )) Sudying (18 ) (and he ransversaliy condiion) we can easily conclude as seen above ha when s s 0, his ax is non-disoring. Bu also if 2s + s π = 0, (18 ) is non-disoring! Wha hen are he implicaions of 2s + s π = 0? From his follows ha he ax rae can be wrien as s( π ) = A B 1 or σ= s( ππ ) = Aπ B, a linear resource π ren ax. hus, progressiviy in he resource ren ax is only neural when he ax is linear in he oal resource ren, which again implies ha he ax is negaive for small enough resource rens. If he resource ren ax is always posiive, increasing and convex, i.e. s+ π s > 0 and 2s + π s 0, hen i can be shown from elaboraing (18 ) ha he ax slows down he depleion of resources wih high resource ren and high demand elasiciy and speeds up he depleion of resources wih low resource ren and low demand elasiciy.