Strategic trade in pollution permits

Transcription

1 Strategc trade n polluton permts Alex Dckson and Ian A. MacKenze Department of Economcs, Unversty of Strathclyde, Glasgow, UK School of Economcs, Unversty of Queensland, Brsbane, Australa January 29, 2016 Abstract Markets for polluton have become a popular regulatory nstrument. Yet these markets are often hghly concentrated, whch may lead to strategc behavor by all partcpants. In ths artcle we nvestgate the mplcatons of strategc trade n polluton permts. The permt market s developed as a strategc market game, where all frms are allowed to behave strategcally and ther roles as buyers or sellers of permts are determned endogenously wth prce-medated trade. In a second stage, frms transact on a product market and we allow for a varety of market structures. Our framework establshes the endogenous determnaton of equlbrum prce, market structure, and levels of exchange n the permt market. Key words: Polluton market, Market power, Strategc market game. JEL classfcaton: C72, D43, D51, L13, Q53. 1 Introducton Markets for polluton permts have emerged as a manstream regulatory nstrument. Snce the early adopton of the US Acd Ran Program numerous schemes have been establshed to control polluton. 1 Behnd ths sprted regulatory response les the economc ratonale of least-cost polluton control: aggregate control costs are mnmzed when players trade polluton permts. Ths least-cost result reles on the exstence of low transactons costs as well as players actng compettvely. 2 Yet players strategc behavor n these markets and the resultng socal losses are a real concern (Montero, 2009; Hntermann, 2015). 3 The actons of large and nfluental players n the market have 1 Examples nclude the European Unon Emssons Tradng Scheme (EU ETS), Regonal Greenhouse Gas Intatve (RGGI), Western Clmate Intatve (WCI), and New Zealand Emssons Tradng Scheme (NZ ETS). Markets are also commencng n South Korea, Chna, and Inda. 2 In early schemes transacton costs appeared to be problematc, for example, n the Fox rver (O Nel et al., 1983) and RECLAIM (Foster and Hahn, 1995). Yet n most modern permt markets prohbtve transacton costs do not appear to be a sgnfcant problem. Asde from cost effectveness, a whole host of explanatons can be proposed for explanng neffcency wthn schemes, such as the poltcal economy aspects of regulaton, complance ssues, and uncertanty. 3 For example, Montero (2009) hghlghts these problems both wthn the U.S. sulfur permt market where 43% of permts allocatons were allocated to just four players as well as an nternatonal carbon market, where strategc behavor may exst between countres. More recently, Hntermann (2015) provdes evdence of prce manpulaton n the EU-ETS. 1

2 the potental to dstort the equlbrum permt prce, reduce the cost effectveness of polluton control, and nfluence the product market equlbrum. Although the exstence of market power and the assocated losses may be sgnfcant, the fundamental aspects of ths problem the nteractons between players n the permt market are not well understood. In partcular, very lttle s known about the formaton of equlbra n the permt market when all players behave strategcally. To address ths problem, we derve a strategc market game (Shapley and Shubk, 1977) that takes nto consderaton strategc behavor n the permt market. Our model comprses of two stages. In the frst stage, traders partcpate n a strategc permt market game. Traders n the permt market partcpate by submttng ether an offer (of permts) or bd (of money). A tradng post then aggregates the offers and bds and determnes the prce of permts n a way that clears the market. Trade s thus prce medated: whether a trader wshes to buy or sell permts depends on ther abatement technology and on ther conjecture of the prce n the market, whch s determned by ther belefs about the market actons of other traders. In the second stage, once frms receve ther fnal allocaton of permts from the permt market, they transact on the product market. We provde alternatve product market structures to assst n our nvestgaton; namely, we begn by allowng regulated frms to act as perfect compettors n the product market then advance our analyss so that the frms act as ndependent monopolsts, as well as consderng an olgopoly market structure. Industres regulated by cap-and-trade markets are often hghly concentrated (regonally segregated) markets, for example, ths has been evdent n the electrcty (Wolfram, 1999; Borensten et al., 2002; Bushnell et al., 2008) and cement (Ryan, 2012; Fowle et al., 2016) ndustres. Wth such a framework, we are thus nterested n the structure and formaton of equlbra n both the permt and product markets when all frms act strategcally. We focus our attenton on the ncentves for players to trade, the overall cost effectveness of regulaton, and the equlbrum structure of the markets. In our prce-medated model of permt exchange we consder the exstence of an equlbrum wth trade n permts, and demonstrate that the market equlbrum s always cost neffcent. Indeed, we fnd that even n the presence of gans from trade autarky may be the only outcome f the gans from trade are not suffcently large. Our framework also shows that strategc trade can alter the structure of the market, as the role of frms (buyers or sellers) and the equlbrum prce are now endogenously determned: buyers (sellers) n a compettve market can swtch ther role n a market wth strategc trade. When frms place bds or offers n the permt market, they also take nto account future strategc product market behavor. A frm holdng a permt not only has a drect effect on reducng ther abatement cost they wll also experence an ndrect effect n whch holdng the permt wll ncrease optmal producton as well as the ablty to rase rvals costs. The dea that frms attempt to manpulate the permt market prce has long been recognzed. A vast lterature has followed the contrbuton by Hahn (1984). 4 In hs 4 See Montero (2009) and Rechenbach and Requate (2013) for comprehensve lterature surveys on market power n polluton markets. Usng frameworks that model exhaustble resources, market power n polluton markets has also been consdered when polluton permts are storable (Lsk and Montero, 2006, 2011). 2

3 study, Hahn developed a permt market model wth a sngle large trader and a prcetakng compettve frnge of small traders. Ths framework, however, s restrctve. These tradtonal models exogenously mpose a behavoral restrcton on some agents: not all agents are permtted to behave strategcally and there s a requrement for an auctoneer. Thus, models that assume a monopolst and a compettve frnge framework have some pecular features. In such models, for example, the compettve frnge soaks up the excess demand from the monopolst and as the sze of the compettve frnge reduces, trade ceases to take place. Asde from these peculartes, the basc compettve frnge framework remans popular and has been extended n a number of drectons. Frst, models have accounted for addtonal players that act as olgopolsts competng aganst each other n the presence of a compettve frnge. 5 Yet precsely the same concluson holds (even though the olgopolsts could effectvely trade wth each other). Second, analyss has concentrated on establshng lnks between permt and product markets. The work by Msolek and Elder (1989) was the frst to present a model where a domnant frm can alter permt trades n order to manpulate rval frms costs n the product market. 6 Although the model provdes a lnk between both a permt and product market, the analyss suffers from the same weakness as other compettve frnge frameworks; namely, behavoral assumptons are stll requred to allow only one frm to manpulate the permt and product market. Thus, n general, these models are rather lmted n descrbng how trade mght take place n an economy populated by large frms n whch the assumpton of a substantal compettve frnge s not approprate. In order to provde nsght to ths problem, we must nvestgate strategc behavor from a dfferent perspectve. In ths artcle we provde a new framework to nvestgate strategc behavor n a permt market. We do ths by usng a strategc market game and provde a unfyng framework that ncorporates strategc behavor for all frms n the permt market as well as consderng the mpact of market power n the product market. To provde a full equlbrum characterzaton, we follow a three-step approach that explots the aggregatve propertes of the game played. In Step 1, we hypothesze a permt prce and consder whether frms would be (potental) buyers or sellers of permts. In Step 2, we consder the behavor of each sde of the market separately at the hypotheszed permt prce, deducng the aggregate supply of, and demand for, permts at that prce. Fnally, n Step 3, we check whether the hypotheszed permt prce s consstent wth aggregate demand and supply. If so, then we have dentfed a Nash equlbrum. Once a permt market equlbrum s determned, frms partcpate n the product market. We start by provdng a benchmark case of a perfectly compettve product market. We then develop our framework to allow each frm to be an ndependent monopolst wthn ther product market as well as provdng an olgopolstc product market structure. Allowng for such an approach provdes a comprehensve, realstc, and tractable structure to analyze strategc trade n permt markets (and the assocated product market). 5 See, for example, Westskog (1996) and, more recently, Hagem (2013) that dscusses the choce of strategc behavor. 6 See Salop and Scheffman (1983, 1987), and Rogerson (1984) for the underlyng framework. 3

4 1.1 Recent lterature To overcome the drawbacks of a compettve frnge framework, a small number of alternatve mechansms have been advocated. These alternatves use the supply functon approach of Klemperer and Meyer (1989) to model trade n polluton permts (Malueg and Yates, 2009; Wrl, 2009; Lange, 2012). 7 Usng ths supply functon approach dentfes the losses assocated wth strategc behavor and shows that, although a blateral olgopoly leads to the same equlbrum permt prce as the compettve soluton, tradng volume s lower. In supply functon frameworks, frms set up trade functons and specfy the number of permts that are to be bought or sold, condtonal on the equlbrum prce. The market maker then collects these schedules and determnes a market-clearng prce. Although ths approach does provde addtonal understandng of strategc permt trade, the man dsadvantage of usng a supply functon approach s that prce determnaton s a black box : the market maker determnes the equlbrum prce where aggregate net trades are zero wthout any attempt to focus on a prce-medated soluton. 8 Establshng a prce-medated soluton, therefore, may provde a rcher (and more plausble) approach to modelng strategc trade n permts somethng we consder n ths artcle. We provde a tradng mechansm n a blateral olgopoly framework that allows all traders to behave strategcally and n whch the sdes of the market (.e., the sets of buyers and sellers of permts) form endogenously, and s very much n the sprt of prce-medated trade va quantty competton à la Cournot. As such, the model does not take place n a black box wth a requrement for an auctoneer to clear the market, nstead, we outlne an explct prce-formaton mechansm. Our mechansm ncorporates a tradng post that aggregates the bds and offers of all players and the equlbrum prce s determned va the rato of total amount of money bd to the total number of permts offered. Any exchanges are therefore determned subject to the bds and offers made as well as the resultng permt market prce. We are able to drectly compare our framework wth the supply functon lterature. In drect contrast to the key fndngs of ths lterature, we show the equlbrum permt prce n our model of blateral olgopoly wll genercally be dfferent to the compettve equlbrum permt prce. Under certan condtons, therefore, some frms may swtch between sellng permts (n the compettve equlbrum) to buyng permts (n the blateral olgopoly), and vce versa. Moreover, we show that for trade to take place t s necessary that there are suffcent gans from trade, meanng autarky s the only equlbrum n some markets even though gans from trade may exst. By contrast, n supply functon models, autarky s only an equlbrum when the ntal permt allocaton s effcent somethng orgnally observed n Hahn (1984). 7 For a further dscusson see Godal (2011). For expermental fndngs of ths approach see Schner et al. (2014). 8 As noted by Malueg and Yates (2009), Wrl (2009), and Lange (2012), the method of obtanng unqueness n supply functon equlbra requres addtonal assumptons over-and-above the requrement of unspecfed prce determnaton. For example, Malueg and Yates (2009) requres that frms have dentcal margnal abatement cost slopes as well as sngle parameter lnear net-trade functons, whereas Lange (2012) requres that all strateges are consstent wth small (stochastc) changes n the demand functons. As our approach focuses on an explct tradng mechansm, whch produces a prce-medated soluton, we do not requre any of these addtonal assumptons. 4

5 In our general framework, we analyze frm behavor when the permt and product markets are lnked. In partcular, we combne the strategc permt market wth alternatve product market structures. 9 Earler lterature has also nvestgated the connectvty between the permt and product market, but ths has been framed through a tradtonal compettve frnge framework (along wth the subsequent weaknesses) (e.g., Sartzetaks, 1997; Hntermann, 2011). 10 Usng our framework, we show that the ntroducton of ndependent monopolsts n the product market unambguously lowers the equlbrum permt prce as the strategc supply (demand) of permts ncreases (decreases). When ths product market structure s replaced by an olgopoly product market, counterbalancng strategc effects occur such that there may be an ncrease n the demand for permts and upward pressure on the assocated permt prce. Our contrbuton s to provde a framework to model fully strategc trade n polluton permts, that s both realstc and tractable to allow for the full equlbrum characterzaton of the permt market. Ths, then, provdes a bass for the evaluaton of contemporary cap-and-trade markets when strategc behavor exsts for all market partcpants. By combnng our analyss wth alternatve product market structures, we also provde an encompassng model that ncorporates many current regulatory market structures. Our approach can be used to nest prevous attempts at strategc behavor n the product market (e.g., Msolek and Elder, 1989) as well as complementng the recent lterature on strategc permt markets that has yet to nvestgate the fundamental lnks between permt and product markets (Malueg and Yates, 2009; Wrl, 2009; Lange, 2012). Ths artcle s structured as follows. Secton 2 outlnes the economc envronment, determnes the equlbrum characterzaton of a strategc market game and product market equlbrum. Secton 3 provdes a dscusson of the permt market equlbrum. Secton 4 extends the framework to nclude strategc behavor n the product market. We then conclude n Secton 5. 2 The model 2.1 The economc envronment Consder an economc envronment that s populated by an ndex set of frms I = {1,..., N}, where frm I has an ntal stock of money m 0. Frms operate n a product market where the producton of goods generates polluton. Ths polluton s regulated by a cap-and-trade scheme. Frms have the opton to ether hold a permt to cover emsson labltes, or reduce emssons by utlzng (costly) abatement technologes. Before undertakng producton, frm s allocated an ntal endowment of permts ω > 0 wth the opportunty to engage n permt trade. 11 The regulator s polluton target s Ω = I ω. We consder a two-stage envronment: n the frst stage permt 9 Recently, Fowle et al. (2016) nvestgated the adopton of market-based nstruments (wthout market power) on a hghly concentrated product market (a regonally segregated cement ndustry). Fowle et al. (2016) fnds that the establshment of a market-based nstrument coupled wth the market-power dstortons n the product market generate losses over-and-above any benefts assocated wth emssons mtgaton. See also Ryan (2012). 10 For addtonal nsghts see De Feo et al. (2013). 11 The analyss can also allow for traders who have no ntal allocaton of permts but that mght want to transact n the permt market. 5

6 allocatons are determned n the permt market whch become common knowledge; n the second stage frms make producton decsons n the product market. Let x R be the number of permts that frm s allocated by the market after tradng: x > 0 for purchases of permts and x < 0 for sales. Let x denote the vector of fnal allocatons for all frms, and x the vector of all allocatons excludng that of frm. The fnal permt holdngs of frm are ω + x and we denote the prce of permts by p. In the product market frm s output s denoted by z, and the product prce φ s determned by an nverse demand relatonshp Φ(Z), where Z = I z s the aggregate supply of the good. Producton of the good generates polluton and the quantty of polluton emtted n producng z s gven by f (z ). Any polluton that s not covered by a permt must be abated; accordngly, polluton abatement requred by frm s a f (z ) (ω + x ). Frms undertakng producton ncur drect producton costs and polluton abatement costs, so frm s total cost of producton s gven by C (z, a ). Assumpton. For each frm I the functons f ( ) and C (, ) are twce contnuously dfferentable; f, f C zz C aa 0; C z, Ca 0 wth a strct nequalty f z > 0; C zz (C za ) 2 > 0; and fnally C z + f Ca = 0 when z = 0., C aa > 0 and C za 0; Frm s payoff s comprsed of any ntal wealth m, revenue or costs assocated wth permt market actvty x p, and, after accountng for all costs of producton, the proft from productve actvty: V = m x p + z φ C (z, f (z ) (ω + x )). Once ntal permt endowments have been set (whch are common knowledge), frms have the opportunty to trade permts and the market mechansm wll determne the fnal allocaton of permts. To capture frms strategc behavor n the market for polluton permts, we turn to a model of blateral olgopoly wth a market mechansm n whch market actons are quantty-based and trade s prce medated; no prce-takng assumptons are mposed ex ante and the role of frms as buyers or sellers of permts s determned endogenously n the market. Such strategc market games were ntroduced by Shapley and Shubk (1977) to model fully strategc behavor n general equlbrum settngs, whch we restrct to the case of two commodtes a good (permts) and money (see Dckson and Hartley, 2008). Trade takes place by way of an explct tradng mechansm: there s a tradng post to whch frms submt an offer of permts to be exchanged for money or a bd of money to be exchanged for permts, dependng on whether they want to sell or buy permts. 12 The tradng post aggregates the offers and bds of all frms and determnes the prce of permts as the rato of the total amount of money bd to the total number of permts offered. Exchanges are then determned accordng to the offers and bds made and the resultng market prce. Trade s therefore prce medated, and each ndvdual frm consders that ther actons nfluence ths prce. Whether a frm wshes to buy or sell permts wll depend on ther abatement technology and ther belef about the prce n the market. 12 Ths s n contrast to the exstng lterature on strategc trade n polluton permts (e.g., Hahn, 1984; Hntermann, 2011) that nvarably assumes the presence of a compettve frnge necesstatng a black box (auctoneer) approach to market clearng. 6

7 Formally, frm can make an offer of permts 0 q ω to be exchanged for money, or make a bd of money 0 b m to be exchanged for permts. 13 We assume frms only buy permts from ther ntal money holdngs and we rule out frms makng wash trades,.e., contemporaneously buyng and sellng permts. The set of strateges avalable to frm I s therefore S = {(b, q ) : 0 b m, 0 q ω, b q = 0}. The role of the tradng post s to aggregate the offers and bds and determne trades. Let the aggregate offer and the aggregate bd be Q = I q and B = I b, respectvely. If ether B or Q are zero then the tradng post s deemed closed and any offers or bds that are made are returned. So long as B, Q > 0, the prce of permts (denomnated n unts of money) s determned as p = B/Q, and the number of permts allocated to frm (n addton to ther ntal holdngs) s gven by { b /p f b > 0, q = 0 or x = (1) q f q > 0, b = 0. The change n frm s money holdngs s thus { b f b > 0, q = 0 or x p = q p f q > 0, b = 0. An ntutve nterpretaton of ths mechansm s as follows: the total supply of permts to the market from those that want to sell (Q) s shared among those traders that want to buy n proporton to ther bds (b /B), for whch a per-unt prce of p s transferred to the sellers. Once permt tradng has taken place, permt allocatons become common knowledge and frms engage n producton decsons n the product market. In our baselne model we assume that frms behave as prce-takers n the product market by modelng t as a perfectly compettve market. Later n the artcle, we explore the mplcatons of frms havng market power n the product market. 2.2 Product market decsons Let φ denote the product market prce that s set by the Walrasan auctoneer n a perfectly compettve product market. Then the proft of a typcal frm I from ther product market actvty s π (z ; x ) = z φ C (z, f (z ) (ω + x )). Once the permt market has cleared and frm has a permt allocaton x, the product market proft functon π (z ; x ) depends only on z. Frm I wll seek to choose z to 13 Throughout t s assumed that a suffcently large penalty can be leved on frms for offerng more permts than are n ther possesson, or makng bds that exceed ther money holdngs, that ths wll never consttute equlbrum behavor. For example, ths occurs wth the non-complance penalty n the EU-ETS, whch was set at e100 per tonne of CO 2 n 2013 and ncreases wth the Eurozone nflaton rate: sgnfcantly hgher than the equlbrum permt prce. 7

8 maxmze π (z ; x ), where the frst-order condton s d π (z ; x ) dz 0 C z (z, f (z ) (ω + x )) + f (z )C a (z, f (z ) (ω + x )) φ, (2) wth equalty f z > Snce we assume C z + f Ca = 0 when z = 0 the soluton wll always be nteror where the frst-order condton holds wth equalty, and we denote the soluton to (2) by z (φ; x ) > 0. The compettve equlbrum product prce determned by the Walrasan auctoneer must satsfy φ = Φ( n =1 z ( φ; x )). Ths, of course, depends on the dstrbuton of permt allocatons, so where approprate we wll wrte φ(x) as the soluton to ths equaton. 15 Wth a slght abuse of notaton we wrte z (x ) for the supply of frm to the product market n the compettve equlbrum, whch s derved by frms equatng ther overall margnal cost comprsed of the margnal cost of producton and abatement to the prce of the good. The relatonshp between a frm s behavor n the product market and ther actons n the permt market s gven by d z (x ) = d z (x ) dx, for s = {b, q}, (3) ds dx ds whch follows by vrtue of frm s product market strategy dependng only on ts own allocaton of permts. Implct dfferentaton of (2) yelds d z (x ) dx = C zz C za + C aa f + 2C za f + Caa ( f )2 + C a f > 0 (4) under our assumptons. Intutvely, f a frm acqures more permts n the permt market then less abatement s requred for a gven level of output. Ths has two effects relevant for product market decsons: snce C aa > 0 the margnal cost of abatement falls; and snce C za > 0 the margnal cost of producton falls. Both effects work to favor an ncrease n product market output when the frm s n possesson of more permts. To understand the effect of a change n the permt allocaton on a frm s proftablty n the product market let us, agan wth a slght abuse of notaton, wrte the optmzed proft functon n the product market as π (x ) = z (x ) φ C ( z (x ), f ( z (x )) (ω + x )). (5) Snce ths s only nfluenced by x, we can wrte d π (x ) = d π (x ) dx, (6) ds dx ds 14 Ths frst-order condton s both necessary and suffcent under the assumptons stated n Subsecton 2.1; the second-order condton s C zz C aa ( f )2 2C za f Ca f < Note that snce frms are assumed to be prce-takers n the product market no frm takes nto consderaton the effect of ther permt allocaton on the product prce. 8

9 n whch d π (x ) dx = d z (x ) dx [ φ C z C a f ] + Ca = C a ( z (x ), f ( z (x )) (ω + x )) (7) as the frst-order condton (2) mples φ C z C a f = 0. Equatons (6) and (7) show a drect lnk between the permt and product markets: ths wll be used to nvestgate the frm s actons wthn the permt market. 2.3 Permt market equlbrum Foreseeng the consequences of permt market actvty on actons n the product market, each frm I can be seen as solvng the problem max m x p + π (x ), (b,q ) S where x = b /p q, p = B/Q, and π s defned n (5). Ths problem s concave n both b and q so the frst-order condtons are both necessary and suffcent n dentfyng a best response. 16 When engagng n permt market actvty, a frm affects ts product market proftablty (accordng to (6)) and also ts expendture n the permt market. When choosng s = {b, q} S the frm wll balance the margnal change n product market proftablty wth the margnal change n permt market expendture, so that d π (x ) dx dx dx p, s = {b, q}, ds ds where the nequalty s replaced wth an equalty f s > 0. For a buyer of permts for whom s = b, x = b /p, and so t follows that dx (1 b /B)p 1 and dx p ds = 1. As such, the frst-order condton for a buyer of permts s ds = d π (b /p) dx (1 b /B) 1 p, (8) where the nequalty s replaced wth an equalty f b > 0. dx p ds = q and we have dx ds = (1 q /Q)p. Consequently, the frst-order condton s For a seller of permts for whom s = q, x = 1 and d π ( q ) dx (1 q /Q)p, (9) wth equalty f q > Ths follows by notng that for s = {b, q}, the frst dervatve of the payoff functon s dx p ds + d π (x ) dx ( ) and so the second dervatve s d2 x p + d2 π (x ) 2 dx ds 2 dx 2 ds + d π (x ) dx dx db = B b Q and d2 x B 2 ds 2 x p = (q /Q)B so dx p dq show n Lemma 1 that d2 π dx 2 = 2(B b ) Q; and x B 3 p = b so d2 x p db 2 dx ds d 2 x. ds 2 When s = b: x = (b /B)Q so = 0; and = 0. When s = q: x = q so d2 x dq 2 = Q q B and d2 x p = + 2(Q q ) B. As noted, d π (x ) Q 2 dq 2 Q 3 dx > 0 and we wll subsequently < 0, whch establshes the clam. 9

10 Snce frms are heterogeneous n ther cost structure, pursung a standard bestresponse analyss of ths game would be frutless as the dmensonalty of the problem makes t ntractable. Rather than mposng addtonal assumptons to nstl tractablty (e.g., restrctng frms to be one of two types), we follow an approach frst presented n Dckson and Hartley (2008) and later extended to the case of nteror endowments by Dckson and Hartley (2013) that explots the fact that frms payoffs depend only on ther own acton and the aggregaton of other frms actons n B and Q, whch themselves nfluence the prce p. Here we present the reasonng for permt exchange coupled wth subsequent product market decsons. The method allows the constructon of supply and demand functons n the permt market that account for strategc behavor and endogenous formaton of the sdes of the market, and can be used to dentfy a permt market equlbrum. The method proceeds as follows. Step 1: Hypothesze a permt prce p, and consder whch frms would act on each sde of the permt market f there was a Nash equlbrum wth ths prce. We defne p C a( z (0), f ( z (0)) ω ) (10) as frm s margnal abatement cost at ts ntal endowment and wll show (n Proposton 2) that frm wll be a buyer of permts only f p > p and a seller of permts only f p < p. When consderng behavor consstent wth a prce p, ths allows us to separate the set of frms nto those that wll potentally buy permts, and those that wll potentally sell. Step 2a: Hypothesze an aggregate supply of permts, Q, and consder the ndvdual supples of those frms that mght sell permts at prce p that are consstent wth a Nash equlbrum wth ths Q and p. Then ask whether frms ndvdual supples are consstent when aggregated,.e., that ndvdual supples aggregate to Q. Let q (p; Q) denote frm s supply consstent wth a Nash equlbrum n whch the aggregate supply s Q and the prce s p (whch s gven by the mnmum of ether the q that solves (9) or ω ). Then we seek the value of Q such that { I:p <p} q (p; Q) = Q, whch s the aggregate supply consstent wth a Nash equlbrum n whch the prce s p. Step 2b: Hypothesze an aggregate bd B from those frms that mght buy permts at prce p, and deduce ndvdual bds consstent wth ths aggregate bd, whch we denote b (p; B) (ths s gven by the mnmum of ether the b that solves (8) or m ). Seek consstency of the aggregate bd,.e., fnd the value of B such that { I:p >p} b (p; B) = B. Step 3: Seek a consstent prce,.e., a prce such that the consstent aggregate offer from Step 2a and bd from Step 2b satsfy B/Q = p, whch dentfes a Nash equlbrum. We begn by establshng Step 1. To do so, we frst requre the followng lemma. Lemma 1. For each frm I, d2 π (x ) dx 2 < 0. 10

11 Proof. Recall from (7) that d π (x ) dx = C a( z (x ), f ( z (x )) (ω + x )). As such, d 2 π (x ) dx 2 In (4) we deduced that d z (x ) dx = ( ) = C za d z (x ) + C aa f d z (x ) 1 dx dx = d z (x ) (C za + C aa f dx ) Caa. C zz C za +C aa f +2C za f +Caa ( f )2 +C a f, mplyng d 2 π (x ) dx 2 = C zz = (Cza C zz (C za + C aa f )2 + 2C za f + Caa ( f )2 + C a f ) 2 C zz + 2C za C aa f + Caa C acaa f )2 + C a f ( f, C aa whch s negatve as a result of our assumptons on cost and polluton generaton functons. We are now n a poston to complete Step 1. The followng proposton allows us to understand, once a permt prce has been hypotheszed, how frms are determned as ether buyers or sellers of permts. 17 Proposton 2. If there s a Nash equlbrum wth prce p then frm I wll be a buyer (seller) of permts only f p > (<)p. Proof of Proposton 2. Let p > p and assume, by contradcton, that sells permts. Then q > 0 and x = q < 0, so Lemma 1 mples d π (x ) dx > d π (x ) x dx p (see (10)). =0 d π But from the frst-order condton for sellers, (x ) dx = (1 q /Q)p < p. As such, p < d π (x ) dx < p, yeldng a contradcton. Thus, f p > p for frm then ths frm wll only buy permts n equlbrum. Demonstratng that f p < p then frm wll only sell permts s smlar and so omtted. Operatonally, the consstent behavor of frms s represented usng share functons. Take a typcal frm. If p > p then we know that the frm wll only be a seller of permts at such prces and we consder ther behavor consstent wth a Nash equlbrum n whch the permt prce s p and the aggregate supply of all potental sellers (those frms j = I for whom p > p j ) s Q > 0. Let σ = q /Q be frm s share of the total supply; then usng (9) we can deduce that frm s optmal share of the total supply s gven by ts sellng share functon s S (p; Q) = mn{σ, ω /Q} where σ s the soluton to wth equalty f σ > 0. l S (σ, Q, p) d π ( σ Q) dx (1 σ )p 0, (11) It s useful to ascertan the propertes of share functons. The share functon s S (p; Q) s mplctly defned, and mplct dfferentaton of (11) reveals that t s decreasng n 17 Ths s smlar to Dckson and Hartley (2013, Lemma 1), but s ncluded here for the case of permt exchange for a self-contaned treatment. 11

12 Q and non-decreasng n p; n addton, study of (11) reveals that lm Q 0 s S (p; Q) = 1 p p.18 Consder now the case where p < p : frm wll only be a buyer of permts. The behavor of frm consstent wth a Nash equlbrum n whch the prce s p and the aggregate bd s B > 0 s represented by ts buyng share functon s B (p; B) = mn{σ, m /B} where, usng (8), σ s the soluton to wth equalty f σ > 0. l B (σ, B, p) d π (σ B/p) dx (1 σ ) 1 p 0, (12) To deduce the propertes of a buyer s share functon, we note that f the aggregate bd s B and the prce s p, the mpled demand s B/p; thus, we wrte frm s share functon as s B (p, [B/p]p). Implct dfferentaton of (12) reveals that the share functon s strctly decreasng n [B/p] for fxed p, strctly decreasng n p for fxed [B/p], and has the property lm [B/p] 0 s B(p; B) = 1 p p These share functons represent each frm s consstent behavor at a partcular prce, wth partcular aggregate bds or offers. We now seek consstency of these aggregates to complete Steps 2a and 2b above. Consstency of the aggregate offer at prce p requres the sum of the ndvdual offers of all frms that wsh to sell at prce p to be equal to the aggregate offer, or, dvdng both sdes of ths equaton by Q, for the sum of the share functons to be equal to one. Defne S S (p; Q) { I: p <p} ss (p; Q). Then at prce p we dentfy the strategc supply, denoted by Q(p), as that level of Q where. 19 S S (p; Q) = 1. (13) For a gven p, all frms for whom p < p wll be ncluded n S S (p; Q) and snce each s S (p; Q) s contnuous and decreasng n Q, S S (p; Q) wll nhert ths property mplyng Q(p), where defned, s a functon. When p changes, the share functons of those frms who reman sellers change n a smooth way, and those frms who become sellers as the prce rses (or drop out of the set of sellers as the prce falls) agan do so n a smooth way, mplyng that S S (p; Q) s contnuous n p and consequently Q(p) vares contnuously n p. Moreover, consderaton of the equaton mplctly defnng Q(p) reveals t s nondecreasng n p. 20 The range of prces for whch Q(p) s defned s p > P S, where P S s 18 Recall from Lemma 1 that d2 π (x ) l < 0. As such, S dx 2 (σ,q,p) σ = Q d2 π (x ) + p > 0 so there s at most dx 2 one soluton to l (σ, Q, p) = 0: s S (p; Q) s a functon. Moreover, mplct dfferentaton of (11) gves s S (p;q) Q = l S (σ,q,p) Q l S (σ,q,p) σ = 19 The fact that d2 π (x ) dx 2 B/p d2 π (x ) dx 2 (1 σ ) 2 < 0, so we are ensured s B (p; B) s a functon. Undertakng mplct dfferentaton, s B (p;[b/p]p) [B/p] (1 σ ) 1 σ d 2 π (x ) dx 2 Q d2 π (x ) dx 2 +p < 0 and ss (p;q) p = l S (σ,q,p) p l S (σ,q,p) = (1 σ) Q d2 π (x ) σ dx 2 > 0. +p < 0 (Lemma 1) s agan mportant. Wth ths n mnd, note that l B (σ,b,p) σ = = l B (σ,[b/p]p,p) [B/p] l B (σ,[b/p]p,p) σ = σ d 2 π (x ) dx 2 B/p d2 π (x ) dx 2 (1 σ ) 2 < 0. In addton, s B (p;[b/p]p) p = l B (σ,[b/p]p,p) p l B (σ,[b/p]p,p) σ = < 0. The lmt s a consequence of takng lmts n (12) as [B/p] 0. B/p d2 π (x ) dx 2 (1 σ ) 2 20 Although { I: p <p} ss (p; Q) s contnuous n p, t s not dfferentable at values of p where new frms enter the set of sellers so mplct dfferentaton cannot be used. Rather, suppose by contradcton that for 12

13 unquely defned by the equaton { I: p < P S } 1 p = 1. (14) P S For p P S, the aggregate share functon S S (p; Q) takes a value less than one when Q s close to zero and, snce t s decreasng n Q, ths s also true for hgher values of Q; accordngly, t s never equal to one. Conversely, for p > P S t exceeds one when Q s small enough and snce t s contnuous and decreasng n Q t s equal to one at exactly one value of Q: the strategc supply. On the buyers sde, we seek to fnd the consstent level of [B/p], whch s the aggregate demand for permts. Ths requres that ndvdual bds when aggregated exactly equal the aggregate bd B, or that the sum of share functons equals one. Defnng S B (p; [B/p]p) { I: p >p} sb (p; [B/p]p), the strategc demand for permts, denoted by D(p), s that level of [B/p] whch satsfes S B (p; [B/p]p) = 1. (15) Contnuty of the strategc demand functon follows by smlar deductons to those made for strategc supply, and study of the condton mplctly defnng strategc demand allows us to deduce that strategc demand s decreasng (strctly) n p. 21 range of prces for whch D(p) s defned s p < P B, where P B s unquely defned by the equaton 1 P B { I: p > p P B } The = 1. (16) For reasons that are smlar to those elucdated for strategc supply, f p P B then the aggregate share functon s less than one for all values of [B/p] so for these prces strategc demand s undefned whereas t takes postve values for p < P B. Turnng fnally to Step 3, a permt prce p s consstent wth a Nash equlbrum n whch trade n permts takes place f and only f strategc supply and demand are equal at that prce, for only then wll the aggregate offer of permts and bd of money be consstent wth the prce. Snce strategc demand s strctly decreasng n p and strategc supply s non-decreasng n p, f strategc supply and demand cross they do so only once, mplyng that there s at most one Nash equlbrum n whch trade n permts takes place. Ths wll be the case so long as P S < P B. Under such crcumstances p > p we have Q(p ) < Q(p). Then the fact that share functons are decreasng n Q and non-decreasng n p mples 1 = { I: p <p} s S (p; Q(p)) s S(p; Q(p)) < { I: p <p } s S(p ; Q(p )) = 1, { I: p <p } a contradcton. 21 Suppose by contradcton that p > p and D(p ) D(p). Then the facts prevously deduced that the share functon s strctly decreasng n p (and [B/p]) mples a contradcton. 1 = { I: p >p} s B (p; D(p)p) s B(p; D(p)p) < { I: p >p } s B(p ; D(p )p ) = 1, { I: p >p } 13

14 let ˆp be the equlbrum prce at whch Q( ˆp) = D( ˆp) then the equlbrum aggregate supply of permts to the market s ˆQ = Q( ˆp); the equlbrum aggregate bd of money s ˆB = ˆp ˆQ; the equlbrum supply of each frm for whom p < ˆp s ˆq = ˆQ s S( ˆp; ˆQ) and the equlbrum bd of each frm for whom p > ˆp s ˆb = ˆB s B( ˆp; ˆB). Equlbrum permt allocatons are ˆx = ˆb / ˆp ˆq. If P S P B then there s no Nash equlbrum n whch trade n permts takes place; n such crcumstances the only Nash equlbrum s autarky (whch s always an equlbrum n blateral olgopoly) and each frm s fnal allocaton of permts s ther ntal endowment Features of the permt market equlbrum Wth our framework establshed n the prevous secton, t s pertnent to consder features of the permt market equlbrum and the consequences of strategc behavor. In partcular wthn ths secton we wll focus on the exstence, structure, and cost effectveness of the permt market equlbrum as well as the comparatve statcs of the model. 3.1 Exstence of equlbrum In blateral olgopoly, as just noted, there s always an autarkc Nash equlbrum n whch no trade takes place. An mportant queston s whether t s the only equlbrum. The exstence of a non-autarkc Nash equlbrum and therefore whether any trade takes place n the market for permts hnges on whether P S defned n (14) s less than P B defned n (16). To better understand the relatonshp between these two objects we next elucdate the detals of ther constructon. Recall that p C a( z (0), f ( z (0)) ω ) s frm s margnal abatement cost wth ts ntal endowment of permts. Gven an ntal dstrbuton of permt endowments we can, wthout loss of generalty, re-order frms accordng to the magntude of ther margnal abatement cost: p 1 p 2 p N. Now we construct two functons that each depend on p. The frst functon, that dentfes P S, s { I: p <p} 1 p p, (17) whch s ncreasng n p. For p p 1 the functon s undefned; for p 1 < p p 2 t takes the value 1 p 1 p ; for p 2 < p p 3 t takes the value 2 p 1 + p 2 p ; for p n < p p n+1 t takes the value n n =1 p p. The second functon, whch wll dentfy P B, s 1 p p, (18) { I: p >p} whch s decreasng n p and pecewse lnear. Workng from large values of p to smaller values, for p p N the functon s undefned; for p N 1 p < p p N t takes the value 1 p ; N for p N 2 p < p p N 1 t takes the value 2 ; and for p N n p < p N n+1 t takes the value n N =N n+1 p p. 22 It s readly verfed by nspecton of payoffs that f the bds and offers of all other frms are zero then any postve bd or offer gves a lower payoff than beng nactve, makng autarky a Nash equlbrum. p N p p N 1 14

15 R 1 p 1 P S P B p N p R 1 p 1 P B P S p N p Fgure 1: The constructon of P S and P B. The upward-slopng functons are { I: p <p} 1 p p, whch dentfes P S, and the downward-slopng functons, whch dentfy P B, are { I: p >p} 1 p p. P S s dentfed by the value of p where (17) s equal to one; P B s gven by the value of p where (18) s equal to one. Fgure 1 plots these functons for two dfferent economes. In the upper panel the p s are wdely dspersed and t s clear that n ths case P S < P B and therefore a non-autarkc Nash equlbrum n whch trade n permts takes place exsts n ths economy. In the lower panel, however, the p s are less dspersed and n ths case P S > P B, so the only equlbrum here nvolves no trade n permts. The dsperson of the p s measures the gans from tradng permts: f they are all equal there are no gans from trade and as they become more dspersed the gans from trade ncrease. As our llustraton makes clear, the exstence of gans from trade s not suffcent to ensure trade wll take place: p 1 < p N does not mply P S < P B. Rather, for a non-autarkc permt market equlbrum to exst there must be suffcent gans from tradng permts. 3.2 Cost effcency of equlbrum If a non-autarkc equlbrum does exst (.e., the economy s such that P S < P B ) wll ths equlbrum reduce polluton levels to Ω n a cost-effectve way? If we were wllng to assume that frms act as prce-takers then the standard Walrasan equlbrum of the permt market would be used to descrbe equlbrum. Well-known results tell us that at the Walrasan equlbrum margnal abatement costs wll be equalzed; thus, whenever gans from trade n permts exst trade wll take place, and emsson reductons wll be 15

16 acheved n a cost-effectve manner (Montgomery, 1972). In our model, consder two frms and j that are actve n a non-autarkc equlbrum wth permt prce ˆp, where s a seller of permts ( p < ˆp) and j s a buyer of permts ( p j > ˆp). Then t follows from (8) and (9) that (1 ˆσ ) 1 C a( z ( ˆx ), f ( z ( ˆx )) (ω + ˆx )) = ˆp = (1 ˆσ j )Cj a( z j( ˆx j ), f j ( z j ( ˆx j )) (ω j + ˆx j )). (19) From (19), the followng proposton s mmedate. Proposton 3. In any permt market equlbrum n whch trade takes place there exst, j I for whom C a( z ( ˆx ), f ( z ( ˆx )) (ω + ˆx )) < Cj a( z j( ˆx j ), f j ( z j ( ˆx j )) (ω j + ˆx j )), so emssons reductons are not acheved n a cost-effectve manner, unless all frms are neglgble (so ˆσ 0 for all I). Ths mples that between any buyer and seller (wth non-neglgble market share), further cost reductons are possble by transferrng more permts from the seller to the buyer. All frms n blateral olgopoly behave strategcally; those that sell permts wll restrct supply to try to ncrease the prce, those that buy wll restrct ther bds to put downward pressure on the prce. These strategc tensons combne to result n generc neffcences n the fnal allocaton of permts. 3.3 Structure of the market In the permt tradng model developed n ths artcle the sdes of the market form endogenously: whether a frm becomes a seller or buyer of permts n equlbrum depends on ther margnal abatement cost at ther endowment n relaton to the permt prce, whch depends on the actons of all frms. Snce there s nothng n our model to suggest that the permt prce wll be the same wth strategc behavor as wth prcetakng frms n a Walrasan model of permt exchange, prma face t s unclear whether frms wll take the same role as seller or buyer n these two market structures. Proposton 4. Suppose P S < P B so there s a permt market equlbrum wth trade. Let p W be the prce of permts n a compettve market, and suppose that n the permt market equlbrum ˆp < [>]p W and there s a frm for whom ˆp < p < p W [ ˆp > p > p W ]. Then n a compettve market frm would be a seller [buyer], but when frms are modeled as behavng strategcally the same frm, f actve, s on the opposte sde of the market. Proof. Let ˆp < p < p W. If frm was a buyer n a compettve market then x > 0 and C a( z (x ), f ( z (x )) (ω x )) = p W. But the fact that d2 π (x ) < 0 (Lemma 1) dx 2 mples that p C a( z (0), f ( z (0)) ω ) > C a( z (x ), f ( z (x )) (ω x )) = p W, a contradcton. Thus, n a compettve market, frm s a seller. In a strategc market, f frm s also a seller then x < 0 and C a ( z (x ), f ( z (x )) (ω x )) = (1 ˆσ ) ˆp. But then Lemma 1 agan mples C a ( z (0), f ( z (0)) ω ) < C a ( z (x ), f ( z (x )) (ω x )) so we have the nequalty p < (1 ˆσ ) ˆp < ˆp, a contradcton. The proof of the case ˆp > p > p W s smlar and so omtted. 16

17 Malueg and Yates (2009) present a competng model of fully strategc trade n permts that reles on the supply functon approach of Klemperer and Meyer (1989). Although ther focus s on the role of prvate nformaton n permt markets, to ensure tractablty of the model they must restrct supply functons to be lnear. Ths has the consequence that, regardless of the dstrbuton of market power, the equlbrum prce wll be equvalent to the compettve permt prce (ther Proposton 1). The equvalence of the equlbrum prce between a strategc framework and a perfectly compettve framework, regardless of the dstrbuton of market power, s a rather unrealstc feature of the supply functon approach. In our blateral olgopoly framework, the equlbrum prce under strategc behavor s only equal to the compettve prce f there s a perfect balance n strategc manpulaton between both sdes of the market, whch, genercally, wll not be the case. 3.4 Comparatve statcs As observed throughout ths artcle, a number of fundamentals determne how frms trade permts: frms endowments; ther producton (and abatement) technologes; as well as the demand n the goods market. We now consder the nfluence of these features on the permt market equlbrum. Recall that the equlbrum n the permt market s dentfed by the ntersecton of the strategc supply and demand functons, the constructon of whch reles on aggregatng frms share functons defned n (11) and (12). A mert of the approach s that the propertes of these share functons are relatvely straghtforward to deduce, allowng a comparatve statc analyss of equlbrum. A frm s sellng share functon s determned by the frst-order condton d π ( σ Q) dx (1 σ )p = 0, the left-hand sde of whch s ncreasng n σ (by Lemma 1). As such, anythng that ncreases [decreases] d π ( σ Q) dx wll decrease [ncrease] the share functon. Also note that strategc supply s determned by { I: p <p} s S (p; Q) = 1, the left-hand sde of whch s decreasng n Q. Consequently, f a frm s sellng share functon decreases [ncreases] then, other thngs equal, strategc supply wll decrease [ncrease], for the range of prces where ths frm would be a seller. A smlar ratonale can be made for buyers share functons. A frm s buyng share functon s determned by d π (σ B/p) dx (1 σ) 1 p = 0, the left-hand sde of whch s decreasng n σ. Thus anythng that ncreases [decreases] d π (σ B/p) dx wll ncrease [decrease] the share functon. Agan recall that strategc demand s determned by { I: p >p} s B (p, [B/p]p) = 1, the left-hand sde of whch s decreasng n [B/p]. It follows that f a frm s buyng share functon ncreases [decreases] then strategc demand wll ncrease [decrease], over the range of prces where ths frm would be a buyer. Now, from (7) we know that d π (x ) dx = C a( z (x ), f ( z (x )) (ω + x )). Our assumptons on frms cost functons then mples that d π (x ) dx wll ncrease [decrease] f (a) there s an ncrease [decrease] n demand n the product market that results n z (x ) ncreasng [decreasng] for all x ; (b) the polluton generated from a gven level of producton ncreases [decreases], where a reducton may be due to, for example, mprovements n abatement technology; and (c) the frm s permt allocaton decreases [ncreases]. Consder a stuaton, then, where demand ncreases n the product market, whch 17

18 nfluences all frms. Each frm s sellng share functon wll decrease, whch decreases the strategc supply of permts, and each frm s buyng share functon wll ncrease, whch ncreases the strategc demand for permts (recall that strategc supply s an ncreasng functon of p, and strategc demand s strctly decreasng n p). Consequently, an ncrease n demand n the product market ncreases the equlbrum prce of permts. The effect on the equlbrum volume of permts traded s unclear snce, whle supply has contracted, the permt prce has ncreased. Consder next a stuaton where abatement technologes become more effcent so less polluton s generated from the producton of goods and suppose ths nfluences all frms equally. Then sellng share functons wll decrease, whch wll result n an ncrease n the strategc supply of permts, and buyng share functons wll decrease resultng n a reducton n strategc demand for permts. The effect of more effcent abatement technologes s to reduce the equlbrum prce of permts, but the effect on the quantty of permts traded s unclear. If the regulator wshes to reduce total emssons Ω, whch t mplements by reducng the endowment of all frms, then the effect s to decrease all frms sellng share functons whch reduces strategc supply, and ncrease ther buyng share functons whch ncreases strategc demand. The consequence wll be upward pressure on the equlbrum prce of permts. Note, however, that changes n permt endowments are often not undertaken n a unform way. For example, we may consder a stuaton where a regulator changes polcy from an equtable dstrbuton of permts to a dstrbuton where more hghly pollutng frms receve more permts. Suppose that wth an equtable dstrbuton of permts the equlbrum prce s ˆp and suppose further that the regulator ncreases the endowment of permts to those who are buyers (.e., for whom p > ˆp) and reduces the endowment of permts to sellers (.e., those frms for whom p < ˆp). For those frms that receved a greater [smaller] endowment, ther buyng share functon reduces [ncreases] and ther sellng share functon ncreases [reduces], wth the necessary mplcaton that for all p ˆp strategc demand s lowered and, lkewse, for all p ˆp strategc supply s also lowered. Consequently, the equlbrum quantty of permts traded wll declne under the new regulaton. In fact t s even possble that under an equtable dstrbuton of permts where P S < P B, a change to the ntal endowment towards a grandfathered dstrbuton of permts contracts both the strategc supply and demand enough to make P S P B, so no trade n permts takes place: referrng back to Fgure 1, grandfatherng may shft the economy from a stuaton depcted n the upper panel, to that depcted n the lower panel. 23 The effect on the equlbrum permt prce when there remans a non-autarkc equlbrum s unclear, and even f the aggregate endowment of permts declnes t does not necessarly follow that the permt prce wll ncrease. 23 Note that f a frm s p under an egaltaran dstrbuton of permts s low then t wll ncrease under grandfatherng, whereas f p s hgh t wll decrease under grandfatherng, thus reducng the gans from trade. 18

19 4 Market power n the product market We now turn to consder non-compettve product market structures and the effect on the permt market equlbrum. In the prevous framework t was assumed that frms were prce-takers n the product market; yet t s possble that some element of market power may exst. Ths s, n fact, qute lkely as many ndustres regulated by a cap-andtrade scheme are hghly concentrated, such as the electrcty market (Wolfram, 1999; Borensten et al., 2002; Bushnell et al., 2008) and the cement ndustry (Ryan, 2012; Fowle et al., 2016). The manfestaton of market power n a product market s the restrcton of supply to ncrease the prce. We deduced n our comparatve statcs exercse that there s a postve relatonshp between frms supply n the product market and ther net demand for permts, and therefore wth the equlbrum permt prce. As such, f frms have market power n the product market and the supply of goods to the market reduces, the market-power effect wll put downward pressure on permt prces relatve to the stuaton where frms are assumed to be prce takers. If frms are ndependent monopolsts n the product market whch would be the case f the output of ther producton process was suffcently dfferentated, or frms served regonal markets then the market-power effect s the only addtonal consderaton, the detals of whch we elucdate n the next subsecton. After ths we consder the ssues assocated wth mperfect competton n a product market, where the strategc mportance of a frm s cost functon (n relaton ts compettors) provdes a rcher lnk between the product and permt markets. 4.1 Independent monopolsts Consder a stuaton n whch frms serve ndependent monopoles followng the concluson of the permt market: frms have market power n the product market, but there s no strategc nteracton. Ths structure may occur, for example, when electrcty companes partcpate n a permt market and are subsequent natural monopolsts for ther electrcty supply (e.g., Ellerman et al., 2000). To begn, let Φ (z ) be the nverse demand functon n the market of frm, then each frm s payoff functon takes the form V = m x p + π (z, x ) where π (z, x ) = z Φ (z ) C (z, f (z ) (ω + x )). To ensure that we can compare behavor n an ndependent monopoly market structure wth that n a compettve product market we requre some equvalence between the markets. Thus, we assume that f a frm supples an dentcal quantty ether as a monopolst or n a compettve market, then the prce t wll receve wll be the same. Wth the functons we have defned, ths requres that for any vector of permt allocatons x, Φ s such that Φ ( z (x )) φ(x, x ). We also assume Φ < 0 and 2 Φ + z Φ < 0, whch are the standard monotoncty and decreasng margnal revenue assumptons that ensure concavty of frms product 19