Some Extensions * Juan-Pablo Montero Pontificia Universidad Católica de Chile. 1. Introduction

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1 Cuadernos de Economía, Vol. 44 (Novembre), pp , 2007 An Aucton Mechansm for the Commons: Some Extensons * Juan-Pablo Montero Pontfca Unversdad Católca de Chle Effcent regulaton of the commons requres nformaton about the regulated frms that s rarely avalable to regulators (e.g., cost of polluton abatement). Montero (2008) proposes a smple mechansm for nducng frms to truthfully reveal ther prvate nformaton: a unform prce sealed-bd aucton of an endogenous number of (transferable) lcenses wth a fracton of the aucton revenues gven back to frms. Ths paper dscuses further propertes of the mechansm ncludng ts extenson to the possblty of prvate externaltes and non-transferablty of lcences. JEL: D44, D62, D82 Keywords: Externaltes, Asymmetrc Informaton, Unform-Prce Aucton. 1. Introducton Regulatory authortes generally fnd that part of the nformaton they need for mplementng an effcent regulaton s n the hands of those who are to be regulated. Regulatng externaltes such as access to common resources (e.g., clean ar, water streams, fsheres, etc.) s not the excepton. Envronmental regulators, for example, know lttle about frms polluton abatement costs, so wthout communcatng wth frms they would be unable to establsh the effcent level of polluton. Dfferent mechansms have been proposed for nducng frms to reveal ther prvate nformaton but for dfferent reasons, these mechansms has been of lmted use. In a recent paper, Montero (2008) proposes a smpler and more * Assocate Professor of Economcs at the Pontfca Unversdad Católca de Chle and Research Assocate at the MIT Center for Energy and Envronmental Polcy Research. I would lke to thank semnar partcpants and anonymous referees for several comments and Fondecyt (Grant # ) for fnancal support. Emal: jmontero@faceapuc.cl Some of these mechansms nclude Roberts and Spence (1976), Kwerel (1977), Dasgupta et al. (1980), Spulber (1988), Varan (1994) and Duggan and Roberts (2002).

2 142 Cuadernos de Economía Vol. 44 (Novembre) 2007 effectve mechansm: a unform prce sealed-bd aucton of an endogenous number of (transferable) lcenses wth a fracton of the aucton revenues gven back to frms 2. The mechansm s developed under the addtonal assumpton that frms know nothng about the other frms characterstcs (they may be even unaware of the number of frms beng regulated). The aucton s man ngredents endogenous supply of lcenses and paybacks enter nto the unform-prce format n a way that the resultng aucton mechansm s both ex-post effcent and strategy-proof (.e., tellng the truth s a domnant strategy). The supply curve of lcenses reflects the cost to socety (other than frms) from allocatng these lcenses to frms. Paybacks, on the other hand, are such that the total payment for lcenses of each frm s exactly equal to the damage t exerts upon all the other agents (.e., other regulated frms and the rest of socety). Hence, the aucton mechansm follows a Vckrey-Clarke-Groves (VCG) payoff rule n that t makes each frm to pay exactly for the externalty t mposes on the other agents. The purpose of ths paper s to present some addtonal, yet mportant, propertes of the mechansm not ncluded n Montero (2008) and to show how the mechansm can be extended to other externalty problems such as those nvolvng non-unformly mxed pollutants (.e., frms pollutants are not perfect substtutes n the damage functon) and prvate externaltes. In so dong, the rest of paper s organzed as follows. The next secton provdes a bref descrpton of the aucton mechansm of Montero (2008) and the followng secton presents the propertes and extensons. 2. The Aucton Mechansm Ths secton, whch closely follows Montero (2008), ntroduces the aucton mechansm for the case of a classcal polluton externalty (.e., homogeneous pollutant). 2.1 Notaton and frst-best allocaton Consder n frms ( =,..., n ) to be regulated. All frms are assumed to have nverse demand functons for polluton of the form P ( x ) wth P ( x ) < 0, where x s frm s polluton level that s accurately montored by the regulator (In some cases I wll work wth the demand functon, whch s denoted by X ( p ) wth X ( p ) < 0, where p s the prce of polluton). Functon P ( ) s only known by frm, nether by the regulator nor by the other frms. The aggregate demand curve for polluton s denoted by P(x), where x = = x s total polluton. The socal n 2 Lcenses are generally refered to as permts or allowances n water and ar polluton control, as rghts n water supply management and as quotas n fsheres management. In ths paper, I wll use the term lcense throughout.

3 An Aucton Mechansm for the Commons: Some Extensons 143 damage caused by polluton x s D(x) wth D(0) = 0, D ( x) > 0 and D ( x) 0 D ( x) can be nterpreted more generally as the regulator s supply functon for lcenses S(p), where D ( S( p)) = p. We may want to assume that D(x) s publcly known but t s actually not necessary. In the absence of regulaton frm would emt x 0 0, where P ( x ) = 0. Hence, frm s cost of reducng emssons from x 0 to some level x < x 0 s x C ( x ) = 0 x P ( z) dz note that C x ( ) P ( x ) and the mnmum total cost of achevng polluton level x < x 0 x s C( x) = 0 x P( z) dz. The regulator s objectve s to mnmze the sum of clean-up costs and damages from polluton,.e., C(x) + D(x). Therefore, the socally optmal or frstbest polluton level x < x 0 satsfes (1) P( x ) = D ( x ) = P ( x ) for all =,..., n But the regulator cannot drectly mplement the frst-best allocaton because he does not know the demand functons P ( ). He must then look for mechansms n whch t s n the frms best nterest to communcate ther prvate nformaton to hm. Montero s (2008) aucton scheme s one of such mechansms. 2.2 The aucton scheme Consder n frms. The aucton scheme operates as follows. Frms are nformed n advance about the aucton rules (ncludng the way the aucton clears and the paybacks are computed). Frm (= 1, 2,, n) s asked to bd a non-ncreasng nverse demand schedule P ˆ ( x ) (or, equvalently, a non-ncreasng demand schedule X ˆ ( p)). Based on ths nformaton, the regulator computes the resdual supply functon (.e., resdual margnal damage functon) for each frm usng the other frms reported demand schedules, that s (2) S ( p) = S( p) Xˆ ( p) where X ˆ ( ) ˆ p = j X j ( p) and D x = S ( ) ( p). As shown n Fgure 1, the resdual margnal damage functon D x = S ( ) ( p) s only defned at and above the pont at whch D ( x) = Pˆ ( x ) = pˆ. The regulator clears the aucton by determnng a prce p and number lcenses l for each bdder accordng to (3) p = Pˆ ( l ) = D ( l ) or, equvalently, l = S ( p ) = Xˆ ( p ). Thus, frm receves l lcenses and pay p for each lcense. Soon after the frm gets a fracton a (l ) of the aucton revenues back (.e., payback s a ( l ) p l ). Snce the effcent equlbrum prce, gven Xˆ ( p) and X ˆ ( p), solves Xˆ ( ˆ) ( ˆ) ˆ p = S p X ( pˆ), by makng frm face the margnal damage curve (2), we are bascally nformng the frm that for whatever demand report t chooses to submt to the regulator/auctoneer, ts report, together wth those of the other

4 144 Cuadernos de Economía Vol. 44 (Novembre) 2007 p Fgure 1 Aucton equlbrum prces, lcenses and payments D' ( x ) D' ( x ) pˆ pˆ Pˆ ( x ) l Pˆ ( x ) x ˆ xˆ Pˆ ( x ) x, x, x frms, wll be used effcently. In addton, Montero (2008) shows that f a ( l ) s equal to D l (4) a ( l ) = ( ) D ( l ) l l where D ( l ) = D 0 ( z) dz s s resdual damage functon, then t s optmal for each frm to bd ts true demand curve P ( x ) regardless of what other frms bd. Ths effcent and strategy-proof result s not surprsng n that the aucton mechansm follows a VCG payoff rule: t makes each frm pay for ts (resdual) damage D ( l ) to all other agents. Ths resdual damage, whch s the shaded area n Fgure 1, ncludes both the pecunary externalty mposed upon other regulated frms and the polluton externalty mposed upon socety. Fgure 1 also helps to see that the aucton scheme mplements the frst-best wth each frm facng the same prce at the margn (.e., p = p for all ) and gettng exactly the frst-best allocaton of lcenses (.e., l = x ): f P ˆ ( x ) = P ( x ) and Pˆ ( x ) = P ( x ), then l = x, l = x and ˆp = p. Although n prncple the regulator goes bdder after bdder determnng ndvdual prces p, these prces are all the same regardless of how truthful frms are (n terms of Fgure 1: p =... = pn = p ˆ ). But unless frms have dentcal demand curves, fnal prces,.e., ( a ) p, wll dffer across frms (n and off equlbrum).

5 An Aucton Mechansm for the Commons: Some Extensons Extensons Ths secton presents some addtonal propertes of the mechansm, not dscused n Montero (2008), and then shows how the mechansm easly accommodates to other externalty problems such as those nvolvng non-unformly mxed pollutants (.e., frms pollutants are not perfect substtutes n the damage functon) and prvate externaltes. 3.1 Evoluton of paybacks As we ncrease the number of frms, frm has vrtually no effect on the equlbrum prce, so D ( x ) D ( 0 ) and a ( l ) 0 ; hence, the aucton scheme has converged to the Pgouvan prncple for taxng externaltes. To llustrate how rapdly the aucton s payment rule approaches Pgou, let us consder a numercal example. Suppose there are n symmetrc frms wth lnear demand curves. The aggregate demand curve s P( x) = p( x / x 0 ), where p s the choke prce (.e., the prce at whch demand goes to zero) and x 0 s the unregulated level of polluton. The margnal damage functon s D ( x) = hx. Solvng as a functon of the number of frms, we obtan (5) p a( n) = 0 2 ( n ) hx + np If we further let the slopes of the aggregate demand and margnal damage curves be the same (.e., X ˆ ( p) = l ), then equaton (5) reduces to a( n) = / ( 4n 2 ), where n. The rebate for three frms s 10 percent, for ten frms s 2.6 percent, and for 100 frms s less than 0.3 percent. 3.2 Off-equlbrum behavor If we have a sngle frm ( = 1) to be regulated and ths frm knows D(x), t should be notced that the frm does not need to truthfully bd ts entre demand schedule but only the porton relevant to the aucton clearng. It could for nstance submt the perfectly nelastc demand schedule around ts frst-best allocaton,.e., X ˆ ( p) = l. In the context of multple frms ( = 1,, n), however, t s n each frm s best nterest to bd truthfully not only that porton of the demand curve around ts frst-best allocaton x but rather a large porton of ts demand curve. Even f a frm knows D(x), t can no longer antcpate l = x wth precson because t does not know other frms demand curves P ( x ) (t may be even unaware of the number of frms beng regulated). To be more precse, a frm wll only fnd t strctly optmal to bd truthfully the porton of ts demand curve that s relevant for the aucton clearng. Thus, f frms assgn zero probablty to the event that the clearng prce wll fall below some value, say p, frms can just bd an almost perfectly elastc (or nelastc for that matter) demand curve for p p. Whle ths off-equlbrum behavor has no consequences on the clearng

6 146 Cuadernos de Economía Vol. 44 (Novembre) 2007 prce, and hence, on mplementng the frst-best allocaton, t does have an effect on frms total payments. But because demand schedules are non-ncreasng n p, a frm s total payment wll never be greater (and generally smaller) than the Pgouvan payment. 3.3 Budget balancng The aucton mechansm s, lke any other VCG mechansm, a non-budgetbalanced mechansm both n and off equlbrum (unless X ˆ ( p) = 0 for all ). Although there s no effcency reasons for balancng the budget there may be poltcal economy reasons for dong so (Tetenberg, 2003) 3. As frst ponted out by Groves and Ledyard (1977), f there are at least three agents t should be possble to balance the budget for a varety of mechansms. The basc dea s to dstrbute the surplus or defct generated by each agent (D (l ) n our case) among the other agents n some lump-sum manner as to avod any ncentve effects. Behnd ths dea les an mplct separablty condton that n our case would allow us to ether make the payment (D (l ) ndependent of some frm j s report (.e., P ˆ j ( x j ) ), as n Duggan and Roberts (2002), or to perfectly dsentangle the contrbuton of each frm j s report to frm s payment, as n Varan (1994). By constructon, the aucton mechansm lacks of such separablty; hence, there s no way n whch the mechansm can be modfed to acheve perfect budget-balancng whle retanng ts frst-best propertes 4. There exsts, however, an approxmate soluton. Buldng upon the dea of Groves and Ledyard (1977), let denote by D j ( l j ) the total payment that frm j would have hypothetcally faced under the same aucton mechansm but n the absence of frm s demand schedule, where l j s the correspondng number of lcenses allocated to j. The regulator can thus fashon a lump-sum compensaton refundng R for frm usng these nfluence-free hypothetcal payments. For example, (6) R Dj = l j n n ( ) where 2 j Ths soluton assures a perfectly balanced budget (.e., n = R = n = D ( l )) only n the lmtng case of a large number of frms; otherwse, Σ R could be smaller, greater or equal than Σ D ( l ). The rato r Σ R / Σ D ( l ) wll ultmately depend on the number of frms and shape of the demands and margnal damage curves. For lnear curves, for example, t can be shown that for three (symmetrc) 3 We may also want to take nto consderaton the general equlbrum reasons of Bovenberg and Goulder (1996) for not balancng the budget. 4 The reason why the mechansms of Duggan and Roberts (2002) and Varan (1994) can balance the budget s because they are based on dscrete announcements by frms. In the former frms announce quanttes whle n the latter they announce prces. In the aucton mechansm frms announce a contnuum of quantty-prce pars.

7 An Aucton Mechansm for the Commons: Some Extensons 147 frms ρ can be anywhere between 0.60 and 1.50, for ten frms anywhere between 0.90 and 1.11 and for 100 frms anywhere between 0.99 and Thus, a regulator that cannot run a defct,.e., constraned to return at most D ( l ) to frms, can nform n advance that t wll return only some fxed fracton of the total D ( l ) (n the case of 10 frms ths fracton could be 90 percent). 3.4 Imperfect substtutablty of lcenses Consder the case n whch socal damage s no longer a functon of total polluton but, as n Dasgupta et al. (1980) and Duggan and Roberts (2002), of the frms polluton vector. There are n 2 frms wth (prvately known) demand and cost functons P ( x ) and C ( x ), respectvely, where = 1,, n. Polluton damages are denoted by the dfferentable and convex functon D(x), where x = (x 1,, x n ) s the polluton vector. Wthout perfect substtutablty of pollutants, and hence of lcenses, we do not want to nsst on a unform-prce aucton because t may be socally optmal that each frm faces a dfferent prce for lcenses at the margn. For the same reason the regulator wants to make lcenses to be frm-specfc as to prevent any tradng of lcenses after the aucton. Let x = ( x,..., x n ) be the frst-best allocaton vector (whch s nteror and unque); then x* satsfes the frst-order condtons (7) = D( x ) C ( x ) P ( x ) for all =,..., n. x For the aucton mechansm to delver the frst-best allocaton, the payment rule dentfed n Secton 2 mples that frm s resdual damage curve as a functon of x must be x (8) D( x ( y),..., x ( y), y, x ( y),. D ( x ) +.., x n( y)) dy y 0 where x j ( y ) s the frst-best allocaton to frm j when y lcenses are allocated to frm. It s easy to see that f frm s total payment s gven by (8), the soluton to frm s problem,.e., fnd the number of lcenses l that mnmzes C ( l ) + D ( l ), satsfes the frst-order condton (7). To compute frm s resdual damage curve the auctoneer/regulator wll use the bds from the remanng n 1 frms to solve a system of n 1 rst-order condtons (9) ˆ D( x,..., x, x, x,..., x ) P ( x ) n j j = + x j for j = 1,, n and j. Solvng the system of equatons (9) leads to n 1 functons of the form x j ( x ) for all j. These functons are then entered nto D( x, x ( x )) to fnally obtan frm s resdual damage functon (8).

8 148 Cuadernos de Economía Vol. 44 (Novembre) 2007 Gven defnton (8), the aucton works exactly as before. The regulator clears the aucton by determnng a prce p and number lcenses l for each bdder accordng to D l l (10) p = Pˆ l = D l (, x ( ) ( ) ( ) l and soon after gves a rebate of a ( l ) p l, where a ( l ) = D ( l ) / l D ( l ) wth 0 ( l ). a 3.5 Prvate externaltes Consder now the case n whch frms not only mpose costs on socety but also mpose costs (or benefts) on other frms. Fshng n open sea and grazng goats n publc land are two commons examples but the analyss here apples more generally to any prvate externalty problem. There are n 2 frms. Frm s producton s denoted by x and ts (dfferentable) proft functon by where 2 2 Π ( x,..., x,..., xn) where Π ( ) / x > 0 and Π ( ) / x < 0. For concreteness, let us focus on the case of pure prvate negatve externaltes,.e., Π ( ) / x j < 0 for all j (t s relatvely straghtforward to generalze the scheme to the presence of both socal and prvate externaltes). Let x = ( x,..., x n ) be the frst-best or jont-proft-maxmzng allocaton vector (whch s nteror and unque); then x* satsfes the frst-order condtons Π Π ( x ) j ( x ) (11) + = 0 for all =,...,n x j x Had the regulator known the sze of the externalty exerted by each frm at the frst-best level,.e., j Π j ( x ) / x, he would have just charged a Pgouvan tax equal to τ j τ j j Π j ( x ) / x to frm s output, where τ j measures the (frst-best) margnal damage that mposes on j. But snce regulators generally do not have such nformaton, Varan (1994) has provded them wth the followng smple multstage mechansm. Frst, all frms smultaneously announce the magntude of the vector of Pgouvan taxes to be faced by each frm (ncludng tself). Then the regulator uses frms announcements to compute transfers from/to frms as a functon of the producton vector x. Fnally, output x s decded. Varan shows that transfers can be structured n a way that the (unque) subgame-perfect equlbrum of ths game s that each frm reports the frst-best Pgouvan tax vector and that x = x*.as explaned by Varan (1994) n the concludng paragraph of hs paper, however, the man problem wth ths multstage mechansm s that t requres complete nformaton by the frms. The aucton mechansm proposed n ths paper does not requre frms to possess any such nformaton. It assumes that Π ( x ) s frm s prvate nformaton. In the specfc context of prvate externaltes, the aucton mechansm operates as follows. Frms are asked to submt (non-ncreasng) demand

9 An Aucton Mechansm for the Commons: Some Extensons 149 schedules P ˆ ( x,..., x,..., xn) for = 1,, n 5. The regulator/auctoneer uses that nformaton to recover reported proft functons x Π (12) ˆ ( x, x ) ˆ = P ( x,..., x, y, x +,..., 0 xn) dy whch then he uses to compute the resdual damage functons as dctated by Proposton 4 (13) D ( x ) ˆ ( x,..., x, 0, x,..., x ) Πˆ ( x ( x ),..., x,..., x ( x )) Π j + j n j j n for all j = 1,, n and j. The frst sum n (13) s the reported frst-best profts of all frms but n the absence of frm and the second sum s the frst-best profts of all frms but when frm s allowed to produced x > 0. As n the basc model, expresson (13) tracks down the (frst-best) proft losses that the presence of frm, as measured by x, causes on all other agents. Agan, t s not dffcult to see that f frm s total payment s gven by (13), the soluton to frm s problem,.e., fnd the number of lcenses l = x that maxmzes Π ( l, x ) D ( l ), satsfes the frst-order condton (11). The computaton of functons x j ( x ) for all j s as n the prevous secton: the auctoneer wll use the bds from the j frms and solve the n 1 frst-order condtons as a functon of x. A smple example may help here (to make t more nterestng I wll allow for corner solutons). Consder two frms 1 and 2 (or j and ) wth proft functons Π ( x, x j ) = ( θ x x j ) x 0, where the value of θ s frm s prvate nformaton. For θ > θ j the socally optmal soluton s θ (14) x = and x j = 0 2 (and for θ = θ j = θ the effcent soluton s x + x j = θ / 2). In the absence of regulaton, frms wll produce beyond ths jont-proft maxmzng level (we may have a total collapse of the resource n that θ < x + x j for = 1, 2) 6. The aucton mechansm corrects the externaltes as follows. Frms are asked to report ther demand curves or types to the auctoneer, say ˆθ and ˆθ 2, knowng beforehand that the regulator/auctoneer wll use ths nformaton to determne allocatons (15) θˆ / 2 f θˆ > θˆ j l = ˆ 0 f θ < θˆ j 5 Note that n many commons problems these demand schedules wll reduce to P ˆ ( x, x ), where x j x j. 6 Suppose that s common nformaton that θ s are..d. over the support [ θ, θ ], the Bayesan Nash equlbrum s θ 2E[ θ E ] [ θ j ] x = 2 6 for = 1, 2 and where E[ ] s the expected value operator.

10 150 Cuadernos de Economía Vol. 44 (Novembre) 2007 and total payments ˆ 2 θ j / 4 f θˆ > θˆ j (16) D = ˆ 0 f θ < θˆ j for = 1, 2. If θ ˆ θ ˆ θ ˆ = j = the regulator flps a con for decdng who gets the θ ˆ / 2 lcenses for a total payment of θ ˆ 2 / 4 (we assume that the wnnng frm opts to produce despte makng zero profts). By lettng frm face a payment equal to frm j s (frst-best) profts had frm not exsted (.e., Π j = θ 2 j / 4 ), t s n frm s best nterest to submt a truthful bd (.e., ˆθ = θ ) regardless of what frm j bds. Ths s not surprsng snce the aucton mechansm has collapsed to a sngle-object second-prce aucton 7. ReferencEs Bovenberg, L. and L. Goulder (1996), Optmal envronmental taxaton n the presence of other taxes: general equlbrum analyss, Amercan Economc Revew 86, Dasgupta, P., P. Hammond and E. Maskn (1980), On mperfect nformaton and optmal polluton control, Revew of Economc Studes 47, Duggan, J. and J. Roberts (2002), Implementng the effcent allocaton of polluton, Amercan Economc Revew 92, Groves, T. and J. Ledyard (1977), Optmal allocatons of publc goods: A soluton to the free rder problem, Econometrca 45, Kwerel, E. (1977), To tell the truth: Imperfect nformaton and optmal polluton control, Revew of Economc Studes 44, Montero, J.-P. (2008), A smple aucton mechansm for the optmal allocaton of the commons, Amercan Economc Revew, forthcomng. Roberts, M. and M. Spence (1976), Effluent charges and lcenses under uncertanty, Journal of Publc Economcs 5, Spulber, D. (1988), Optmal envronmental regulaton under asymmetrc nformaton, Journal of Publc Economcs 35, Tetenberg, T. (2003), The tradable-permts approach to protectng the commons: Lessons for clmate change, Oxford Revew of Economcs Polcy 19, Varan, H. (1994), A soluton to the problem of externaltes when agents are well-nformed, Amercan Economc Revew 84, Snce there are multple socally optmal solutons for the case n whch θ = θ j, one may be nclned to replace the con-flppng allocaton by a more equtable allocaton such as the followng: f θˆ θˆ θˆ = j =, then l = l j = θ ˆ / 4 and D = Dj = θ ˆ 2 / 8. Ths allocaton rule stll yelds a truth-tellng Nash equlbrum but no longer n domnant strateges. If, for example, frm beleves ˆθ j > θ > θ j, t may be optmal for to bd θ ˆ = θ ˆ > θ. j