DISCUSSION PAPERS IN ECONOMICS

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1 STRATHCLYDE DISCUSSION PAPERS IN ECONOMICS THE WELARE IMPACTS O DISCRIMINATORY PRICE TARIS BY NIKOLAOS DANIAS AND J KIM SWALES NO 6-09 DEPARTMENT O ECONOMICS UNIVERSITY O STRATHCLYDE GLASGOW

2 The welfare mpacts of dscrmnatory prce tarffs Nkolaos Danas a J. Km Swales a Department of Economcs, Unversty of Strathclyde, Glasgow, UK, G4 0QU. E- mal: nkolaos.danas@strath.ac.uk Department of Economcs, Unversty of Strathclyde, Glasgow, UK, G4 0QU. E-mal: j.k.swales@strath.ac.uk June 8, 06 Astract Ths paper looks at the use of asymmetrc tarffs as a regulatory nstrument. We use a monopolstc market setup wth two markets and we ntroduce prce controls n one of the two. The purpose of the regulator s to maxmse consumer welfare through ths prce dscrmnatory practce. We consder cases where the welfare of the consumers n the two markets s weghted equally and cases where t s not. In some cases we allow for the two markets to e lnked through a monopsonstc nput market. The paper focuses on the welfare mplcatons of ths regulatory approach, wth the frm operatng under a proft restrcton. Results suggest that havng only one prcecontrolled market s n certan cases a good opton from a welfare perspectve. Keywords: monopoly, asymmetrc regulaton, tarffs, welfare JEL classfcaton: D4, D6, I3, I38, L, L5 --

3 . Motvaton The motvaton ehnd ths paper s to shed some lght on, and to help understand, the mechansms of asymmetrc regulaton. Polcy makers are faced wth questons of how and to what extent to regulate markets. The dlemma s to alance the ncreased levels of control offered y stronger and tghter regulaton aganst the outcomes that result when markets are allowed to operate ased on ther own devces. Relyng on market operaton allows proft-maxmsng frms to address the challenges n the usness envronment ether usng a long or short-term perspectve, whch can have advantages, snce they have the opportunty to ase ther decson-makng on ther knowledge of the market and cost structures. Addtonally, f the frm s properly ncentvsed to act n a way that serves ts proft-maxmzng nterests, then enforcement of regulaton may e less expensve and less complcated. Ths paper addresses the ssue of asymmetrc regulaton. In a smple model where a monopolstc/monopsonstc frm purchases n one wholesale market and sells n two retal markets, we examne the outcomes when one of the retal markets operates under prce controls, whereas the other faces no regulaton. An mportant restrcton assumed for oth markets s that the frm ears an olgaton to meet the market demand n the controlled market, meanng that quantty ratonng s not allowed n that market. The asymmetrc regulatory approach (where consumers face prcedscrmnatory tarffs) presented n ths paper could e a realstc and sensle opton wth real-world applcatons f we consder the posslty of t s eng adopted : where there s lack of knowledge aout costs; where there s not enough poltcal wll to ntroduce full tarffs; and where socal polces targeted on specfc populatons are ntroduced. In ths paper we assume that the am of the regulator s to maxmze consumer welfare suject to constrants. We focus specfcally on the case where there s dscrmnatory tarff settng. By separatng markets n prce-controlled and non-prce-controlled ones, the regulator can engage n actvtes that result n ndrect ncome redstruton, as an --

4 mpled cross-susdzaton proaly takes place etween the markets. In our paper these markets are modelled usng two markets. Market s a prce-controlled market whch can represent a market servng vulnerale consumers needng protecton from hgh prces, and Market s a non-prce-controlled market whch can represent a market servng the remander of the market. An mportant element of our model s that t allows us to consder the posslty that the two markets are not only served y the one frm ut they are also nter-connected through ther nput prces. Ths s ecause the output sold to each market s suppled usng nput purchased n one common wholesale market. Wth ascendng wholesale nput prces, the two retal markets are nterrelated as larger actvty n one of them ncreases costs for the other one. The model presented n ths paper and the understandngs that t provdes are applcale n cases such as tarff-settng for transportaton servces (regulated as aganst unregulated fares), prce capped fnance costs for mortgages for certan groups, such as frst tme uyers) and control of energy prces for certan users. In these cases, asymmetrc or partal tarffs can e set to serve ncome redstrutng purposes through prce dscrmnaton drven y consumer welfare concerns for protected customer groups. Regulaton s a very popular topc n the economcs lterature (Bradley and Prce, 988; Vogelsang, 00; Dos, 004; Stgler and redland, 96; Laffont and Trole, 986; Averch and Johnson, 96; Baldwn and Cave, 999; Newery, 00). Prcecap regulaton s a well-known and wdely used nstrument of regulaton whch has een examned prevously n the lterature (Cowan, 00; Parker, 997; Braeutgam and Panzar, 993).Ths paper takes the approach of the regulator nto account and therefore the latter sectons examne consumer welfare, amng to apprecate mpacts of regulaton for socety as a whole. The effect of regulaton on consumer welfare has een examned n papers n the pulshed lterature (Clemenz, 99; Kang et al., 000; Sappngton and Sley, 99; Eaton and Grossman, 986). -3-

5 Secton outlnes the model wth no regulaton. Secton 3 ntroduces a tarff n a unfed retal market and dscusses nterventons that optmze the susequent welfare mpact. Secton 4 dscusses the ntroducton of dfferental tarffs n the two separate retal markets, frst where unt wholesale prces are fxed and then where they rse as total output rses. Secton 5 s the concluson.. The asc model wth no regulaton In order to ease the analyss, a numer of smplfyng assumptons are made. The monopoly electrcty suppler faces two dentcal markets, each characterzed y the lnear nverse demand functon: () p a q, where p and q are the prces and quanttes n market, and a and are parameters whch take postve values. or heurstc reasons, that the two markets are ntally assumed to e dentcal, so that the values of a and do not vary across the two markets. The value taken y the parameter a s the maxmum prce that the monopolst can charge n ether market and have non-negatve sales. Therefore n all the analyss the tarff s never set aove a. The parameter s the (negatve) slope of the nverse demand curve. The frm s total cost, C T, s made up of a fxed cost, Γ, and the cost of purchasng electrcty n the wholesale market, n whch t acts as a monopsonst. The wholesale prce of electrcty, p W, s agan assumed to e a lnear functon of total electrcty supply, q T. Ths mples: () CT Γ+ pwqt where (3) pw c + dqt and -4-

6 (4) qt q q+ q Agan c and d are parameters whch take postve values; c s the mnmum prce n the wholesale market that would generate a non-negatve supply and d s the slope of the wholesale electrcty supply curve. We assume that oth c and d are non-negatve. Where d s zero, there s a constant wholesale prce for electrcty equal to c. Before the ntroducton of any prce caps, the frst order condtons for proft maxmzaton for the monopolst suppler s to set margnal revenue equal to margnal cost n each market. Usng equaton (), margnal revenue n market (MR ) s gven as: (5) d( pq ) MR a q dq rom the standard monopolstc argument, p > MR. The lnear margnal revenue curve has the famlar characterstc that t meets the prce axs at the same pont as the demand curve, ut has a negatve slope that s twce as steep. Usng equatons (), (3) and (4), the margnal cost n market (MC ) s derved as: dc qt (6) MC MC c + dq dq q T T T Equaton (6) s mportant for three reasons. rst, the margnal cost s aove the average varale cost, reflectng the monopsonstc poston of the sngle uyer. As a result, agan the margnal cost curve cuts the prce axs at the same pont as the wholesale electrcty supply curve ut s twce as steep. Second, the margnal cost s the same n each market. Thrd, the level of the margnal cost s dependent on the total supply of electrcty. Ths mples that the margnal cost n one market s dependent on the output levels n oth markets. or susequent analyss ths means that the margnal cost n market s a functon of the output n market and vce versa. Through ths lnk nterventons n one market affect actvty n the second. -5-

7 In the case wth no regulaton, each market faces the same margnal cost and has an dentcal margnal revenue curve. Usng equatons (5) and (6), ths mples that the unconstraned proft-maxmzng outputs n markets one and two are gven as: (7) q U a c, ( + d) These quanttes are postve as long as producton s fnancally vale at all (that s as long as a > c). The correspondng prces n the ndvdual markets can e otaned from susttutng equaton (7) nto equaton (). Ths gves: (8) p U 4ad + a + c ( + d) The general expresson for total profts for the company takes the form (9) Π pq p q Γ T W T where the T suscrpts represents the frm total summng actvty n oth retal markets. In the specfc unconstraned case, susttutng equatons (3), (7) and (8) nto equaton (9) produces, wth some manpulaton: (0) U ( a c) Π T Γ ( + d) In the short run producton s proftale as long as a > c. In the long-run revenue earned needs to e enough to cover the fxed costs Γ n order that non-negatve profts can e made. Each market makes half of the revenue, so that the short-run profts made n each market, U Π, can e dentfed as: -6-

8 () U ( a c) Π 4( + d) Ths gves the current proft n each market ut does not allocate the fxed costs across markets. 3. Introducton of a tarff n a sngle, unfed market Prmarly for pedagogc reasons, t s useful to egn y analyzng the mpact of the ntroducton of a prce tarff p n a sngle unfed market and to trace the welfare mplcatons. To e clear, n ths case there s only one retal market whch, for convenence, s dentfed as market. rom equatons (), () and (3), we can determne the output and cost as lnear functons of the prce tarff. Applyng equaton (9) and settng the fxed cost, Γ, equal to zero gves profts as: () (( + d) p ( ad + c))( a p ) Π ( p pw ) q dπ ( a+ c) + ad ( + d) p where and dp d Π + d ( ) d p ( ) < 0 Where p a, quantty demanded s zero, so that profts are therefore also zero for tarffs n ths range. Smlarly, where p p W, so that the market prce equals average varale cost, agan profts fall to zero. rom equaton (), ths tarff s ad + c. + d Gven that equaton () s quadratc, the tarff level whch maxmzes the profts (and therefore also the prce level whch maxmzes profts n a sngle unfed market) s ad + ( a + c) the md-pont of these two tarff values and s calculated as p, wth ( + d) a c a correspondng output of q. ( + d) -7-

9 The full welfare mplcatons of mposng the tarff can e analysed as three elements: the consumer surplus; the producer surplus to the compettve wholesale supplers; and the profts to the retal monopolst/monopsonst. The consumer surplus ( S ) can e derved, usng equaton (), as: where: ( a p ) (3) S ds ( a p ) d S (4) < 0, > 0 dp d( p ) Smlarly, from equatons (), (3) and (4) the producer surplus n the wholesale market equals: so that da ( p ) (5) P dp d( a p ) (6) < 0 dp Equaton () gves the profts to the retaler, wth dπ ad + ( a+ c) ( + d) p (7) dp If weghts ω, ω and ω Π are attached to the consumer surplus, producer surplus and S P profts, the change n welfare as the tarff changes s gven as: (8) dw ds dp dp ω ω ω P dp dp dp dp S + P + Susttutng expressons (4), (6) and (7) nto equaton (8) gves: -8-

10 dw (9) dp ( ωpd + ωs)( a p ) + ω P ( ad + ( a+ c) ( + d) p ) where ( ω d + ω ) ω P ( + d). dw P S ( ) d p Applyng the frst and second order condtons mples that welfare s maxmzed where the tarff s set at the optmal level, ndcated y a star superscrpt (0) p ω ( ad + ( a + c) a( ω d + ω ) * P P S ωp ( + d) ωpd ωs as long as the second order condtons hold, that s: ωpd + ωs () ω P >. ( + d) Expresson () llustrates an mportant prolem for the analyss. A reducton n the tarff wll always ncrease the producer and consumer surplus. But where p s less than the proft maxmsng level, a reducton n the tarff wll reduce profts. If the welfare weght on profts s so low that expresson () does not hold, the frst order condton s a welfare mnmum and welfare wll e ncreased wthout lmt as the tarff falls. Smlarly, f () a( ωpd + ωs) ωpd + ωs > ω P >. ad + ( a+ c) ( + d) then although there s a determnate welfare maxmzng prce tarff, t s negatve. The senstvty of the welfare optmal tarff to the weghts gves: * p ωpd( a c) (3) < 0 ω ( ω ( + d) ω d ω ) P P P S * p ωp ( a c) (4) < 0 ω ( ω ( + d) ω d ω ) S P P S -9-

11 Equatons (3) and (4) mply that the greater the weght on consumer surplus and the producer surplus n the wholesale market, the lower the welfare maxmsng tarff. Some specal cases produce famlar results. Maxmsng welfare where ω, ω 0, so that no concern s pad to the welfare of consumers or wholesale producers, generates a tarff set at the proft maxmzng prce dscussed n Secton 3: * ad + ( a + c) (5) p ( + d) When ω P 0 and ωs ω Π, the welfare maxmzng pont s set where the margnal cost curve for the monopsonstc retaler cuts the demand curve, so that S P * ad + c (6) p + d. Alternatvely, f ω S 0, the welfare maxmzng pont s where the margnal revenue curve cuts the wholesale supply curve, so that: * a + ad + c (7) p + d. nally, f the welfare of all actors s weghted equally, so that ω, ω, ω P, the welfare maxmzng tarff s the compettve prce, where the consumer demand curve cuts the wholesale compettve supply curve, so that: S P * ad + c (8) p + d. At ths pont retal profts are zero. Ths mples that n a unfed market, f the welfare of consumers and wholesale (compettve) producers are weghted hgher than the welfare of the monopolst, then the welfare optmzng prce tarff leaves the monopolst makng a loss. Ths rases the queston of how ths s sustaned. How does the government force the proft maxmzng frm to mantan producton? Ths s another aspect of the ssues rased n -0-

12 the dscusson around expressons () and (). In the analyss n susequent sectons, we therefore take the non-negatve profts as a constrant n order to mantan the contnung exstence of the frm. Also for pedagogc reasons we also typcally gve zero weght to the producer surplus n the wholesale market. Although the am of ths secton s manly ntroductory, where tarffs are ntroduced n one market ut the frm sells n two, there are condtons under whch output n the second market falls to zero. The second market collapses. Ths s a topc we consder n greater detal later n the paper. When ths occurs and only the controlled market remans operatve, under susequent reductons n the tarff rate, equatons outlned n the secton aove wll apply to the remanng sngle controlled market. 4. Impact of separate tarffs n each of the two markets Where there are two retal markets, markets and, the regulator can dscrmnate and dfferentate the tarffs set n each. In general t wll prove useful to defne all the relevant varales as functons of the two prce tarffs and the demand and cost parameters. We egn wth the consumer surplus. Usng equaton (3) and defnng the total consumer welfare, S T markets and, for values of p, p a:, as the weghted sum of the consumer surplus n (9) S ω S ω ( a p ) T S S,, where: S p T ωs( a p ) 0 and ωs s the weght put on the consumer surplus n market. --

13 The consumer s welfare s mnmzed at the value zero where p, p a. Where the consumer s welfare s fxed at some postve level, S T, equaton (9) can e nterpreted as the assocated so-consumer welfare functon, whch for convenence, we refer to susequently as the so-welfare functon. Dagrammatcally, these functons take the form of segments of an ellpse whch s centred on pont A n p, p space. Pont A has co-ordnates (a,a) and the segments are values where p, p a. One of the so-welfare curves (where ω ω ) s shown as W n gure. In general the slope of the so-consumer s welfare functon s gven as: S S dp ST / p ωs ( a p ) dp (30) ST / p ωs( a p ) Broadly, the lower are the values of the tarffs elow a, the hgher the level of consumer welfare. or all so-welfare functons, where p vertcal and where p a t s horzontal. a, the functon s Where the weghts for each market are equal, the so-welfare functons are segments of crcles, as shown n gure. Ths s llustrated y W and, from equaton (30), all so-welfare functons where the consumer surplus n each market s weghted equally have a slope equal to - where p p. If the consumer s surplus n market s weghted more heavly than n market, the so-consumer s welfare functons have a shape smlar to W and W n gure. Whlst the curves wll stll e vertcal and horzontal where the relevant prces take ther maxmum value, a, the slope where p p now has a lower asolute value equal to ω / ω. S S --

14 Adaptng equaton (5) to the case where there are two retal markets, the producer surplus, PT s gven as PT (3) d( a p p ) so that PT d( a p p ) (3) < 0 p Any ncrease n prce n ether retal market reduces total output and therefore reduces producer surplus and at pont A, where p p a, the producer surplus s zero, gven that output s zero. The slope of the so-producer surplus lnes s gven as: p PT / p (33) p P / p T -3-

15 The susequent so-producer surplus functons are therefore negatve 45 degree straght lnes, where further to the SW of pont A, the hgher the producer surplus. nally, to calculate the monopolst/monopsonst s proft, output n each market s determned y equaton (), average wholesale prce y equaton () and the monopolst s profts y equaton (9), where n each case a tarff s mposed as the prce. (34) ( ( ))( ) ( ( ))( p c d a + p + p a p p c d a + p + p a p ) Π T + whch gves (35) ( ( ( ))) ( ( ( )))( ) (( ) ( a c + d a p + p c + a + d a p + p p + p p + p ) ) Π T + Ths mples that (36) ΠT ad + c + a d p + p p Π d +, < 0 p p 4 ( ) T ( ) ( ) Usng expresson (36), the frst and second order condtons mply that profts are maxmzed where: 4 (37) ad + c + p a p + 4d Ths set of tarffs equal the unconstraned monopoly prces where there are two retal markets. Iso-proft curves for the frm are the locus of values for p and p whch produce a constant proft for the frm. These are mpled y equaton (35). In general, ths takes the form of an elpse. Intally the no-fxed-cost zero-proft so-proft case s taken as a -4-

16 enchmark. Where the markets are treated dentcally, so that the prce tarff s the same n oth, zero profts occur where: p, p aor c + ad. Where oth prces + d equal a, there s zero demand, therefore zero profts. Agan note that a s an effectve upper ound for the prce tarff, as hgher values mply negatve demand. Where the c + ad two tarffs take the lower value,, whch s susequently denoted as z, the + d tarff s set at the wholesale prce. Note that wth dentcal markets and tarffs set n oth markets, ther dentcal proft maxmzng value wll e denoted as k, the mean value etween the two zero proft values. 4. The ntroducton of tarffs where the wholesale prce s constant (d 0) It s useful to ntally take the specal case where the wholesale prce s fxed at the level c and s nvarant to the output level so that d 0. rom equaton (35) the total profts are now gven as: ( p c)( a p ) ( p c)( a p ) (38) Π T + Equaton (38) can e reformulated as: (39) a c a c ( a c) + ΠT + + p p Ths s the equaton for the so-proft curve where profts equal case t represents a crcle whose center s ( a c) Π. T a+ c Π T. In ths specal and whose radus, r, s -5-

17 Imposng equal tarffs, the lowest value on the zero so-proft curve, whch we call more generally z, s equal to c and from equaton (37) the proft maxmzng prces are: (40) a + p c p gure shows the zero so-proft curve n ths case. Gven the constrant that output cannot take negatve values. The zero so-proft curve therefore comprses an solated pont, A, and the arc BZC. At ponts A and Z, the tarffs n oth markets take the value a and z respectvely. In ths case the segment BZC comprses one half of a crcle whose center s the proft maxmzng tarffs (k,k). Ths zero so-proft curve goes through the ponts (c,c), (c,a) and (a,c) and has a radus equal to f, where a c f so that the radus can also e expressed as a c. One key element of the analyss s the locus of tarff values where the so-proft curves are horzontal or vertcal. These ponts dentfy the proft-maxmsng prce n one market, gven a specfc tarff n the second market. However, a second nterpretaton s that these ponts show the mnmum values of each tarff that wll support a gven proft level. The mnmum level of p s shown where the so-proft curve s horzontal, the mnmum level of p where t s vertcal. The general slope of the so-proft curve, not restrcted to the case where d 0, s gven as: dp Π / p 4ad + c + a d( p + p ) p T (4) dp ΠT / p 4ad + c + a d( p + p ) p Where the tarffs are equal n oth markets, the so-proft curve wll have a -45% slope. Ths reflects the symmetry n the model. It mples that the mpact on profts of a change n the ether prce would e just offset y an equal and opposte change n -6-

18 P P the other prce. The so-proft curve s horzontal where dp / dp 0. rom equaton (4) ths requres that ΠT / p 0whch occurs where (4) 4ad + c + a d( p + p ) p 0. P Rearrangng equaton (4) produces the proft-maxmsng value of p, gven a tarff P p. 4ad + c + a d (43) p p ( + d) + d Equaton (43) can e seen as a reacton functon: t s the prce set n the unregulated P market, p, that wll e the est response to the regulator settng a specfc tarff, P p, n market. In the specal case where d 0, ths reacton functon reduces to a+ c p. Ths smply means that f a tarff s set n market, the proft maxmzng tarff n market s the ntal monopoly prce and that as the tarff n market falls, the maxmum proft prce for market remans fxed at the vertcal lne KE n gure. a+ c. Ths s represented y Wth no dscretonary weghts (that s, settng all weghts to unty) the frst order condtons for maxmzng welfare are that: W S PT PT (44) p p p p In ths case, usng equatons (9) and (36) and notng that settng d 0 mples that the producer surplus s zero, equaton (44) can e expressed as: (45) W ( ) ( ) ( ) a p a p p c + 0 p -7-

19 whch mples p, p c. In ths case the welfare maxmzng poston s clear. The regulator should mpose a tarff equal to c n each market and the welfare maxmzng pont s at Z. The monopolstc frm makes zero profts. gure shows that the zero so-proft curve as the half crcle, center K, whch goes through Z. If the welfare of consumers n each market s gven the same weght, the hghest attanale consumer so-welfare functon, consstent wth non-negatve profts s the quarter crcle, center A, whch agan passes through Z. Ths s the optmal poston n ths case. If the consumer welfare weghts dffer across the two retal markets, the optmal outcome s changed. Increasng the weght on market shfts the consumer sowelfare functon so that t s tangent to the zero so-proft curve along the segment ZE. Ths s llustrated n gure. The so-welfare curve W that passes through pont Z s no longer tangent to the zero so-proft curve. The hghest (weghted) consumer surplus s now found at pont H on the so-welfare curve W. The most extreme case s where the welfare of consumers n market two s gven zero weght y the regulator. In ths case, the so-welfare curves ecome straght horzontal lnes, wth lower lnes producng hgher consumer welfare. Ths would lead to an optmal outcome gven y pont E. The monopoly prce s stll charged n market and the profts generated n that market are wholly used to susdse the consumers n market. The segment ZE n oth gures and mght nclude a range where the tarff n market s negatve. Take, for example, the stuaton where there are no wholesale cost so that c 0. In ths case, all the ponts on the zero so-proft lne, apart from ponts A and Z, have one negatve prce tarff. In prncple the exstence of negatve prces s not prolematc (although negatve output clearly s). Negatve prces could -8-

20 represent the frm payng susdes to consumers n one market, rather than chargng a postve prce. 4.. A tarff n only one of the two markets, zero proft constrant Ths paper focusses on stuatons where there are restrctons on the extent to whch the separate markets can e regulated and, n partcular, the extreme case where tarffs can only e set n one market. A specfc concern s the sze of the loss n effcency that such a constrant would mply, compared to the outcome where tarffs are optmally set n oth markets. rom equaton (9), where consumers n each market are gven a weght of, the soconsumer surplus curve s gven y the formula: -9-

21 (46) ( ) ( S T a p a p ) + Expresson (46) s the equaton of a crcle whose center s pont A (rememerng that ths only operates for values of can e replaced y: p a). If R s the radus of that crcle, equaton (46) (47) S T R Usng Pythagoras s theorem, the so-welfare functon that passes through the pont that maxmzes the monopoly proft has a radus f (recallng that we defned f, n the dscusson followng equaton (40), as consumer surplus s therefore f a c ). rom equaton (47) the assocated. Wth optmal regulaton, gven the zero proft constrant, tarffs are set at Z. The radus of the so-welfare functon passng through that pont s f, so that the assocated consumer surplus s 4 f. Movng to the optmal regulaton mproves consumer welfare, as measured y the consumer surplus, y 400%. However, f the regulator s restrcted to settng a tarff n only one market, how does ths affect the resultng welfare? Wth only one tarff n operaton, consumer welfare s optmzed at pont E, and wth equal consumer weghts n each market the radus of the so-welfare functon that goes through ths pont s gven as: (48) R ( f + f) + f ( + ) f Susttutng ths result nto equaton (47) gves a consumers surplus of ( + ) f. + Ths value, expressed as a rato of the optmal welfare s Alternatvely, f the changes n welfare, rather than the asolute values, are compared -0-

22 the rato s Therefore although there s a loss n effectveness n only 3 eng ale to target one market, the ncrease n welfare s only reduced y 0% and the consumer surplus n the market not recevng the tarff s unchanged. Moreover, n the case where d 0, f the regulator can decde whch market to target, 0% represents the maxmum lost potental welfare. Where there s greater weght placed on the enefts to one set of consumers, the resultng loss n effectveness, as aganst a stuaton where the same tarff s appled to oth sets of consumers, wll e lower. In fact we can use the analyss to fnd the relatve weghts that have to e placed on the consumer surplus n the two markets that would lead the regulator to actually prefer mposng a tarff n only one market, rather than the same tarff n oth. Where the two markets are weghted unequally, the maxmum consumer surplus where oth tarffs are constraned to take the same value s f ω + ω. Where the tarff can only apply n market, the maxmum consumer surplus s f ω (3 + ) + ω. Therefore f the regulator s faced wth ths choce, the optmal decson would e to apply the tarff only n market f: f ω (3 + ) + ω ω + ω ω 3 > f.64 >. ω Therefore as long as consumer surplus n market s weghted more than 64% greater than n market, t s etter to place an optmal tarff only n market than have to mpose equal tarffs n oth markets. --

23 4.. A tarff n only one of the two markets, postve proft constrant Up to now we have only consdered the case where the constrant operates wth zero profts. However, f fxed costs are postve, so that Γ > 0, or f the company has some knd of power wth whch t can push ack aganst the regulator, the proft constrant wll e postve. The maxmum, unregulated, profts, Π MAX, are determned y susttutng the value, a + p p c nto equaton (38). Ths gves the result: (49) a c f Π MAX Usng equaton (49), t s convenent to express the actual proft constrant, Π T, as (50) ρ f Π T ρπ MAX --

24 where ρ 0. Susttutng ths nto the expresson assocated wth equaton (38) for the radus of the so-proft curve, r, gves the result that (5) r ( r) f r ( r) f. Therefore f ρ 0 we have the zero proft case where r f, whereas ρ produces the proft maxmzng case where r 0. or values of π etween and 0, the radus of the so-proft lne les etween 0 and f. As the proft constrant ncreases the crcular so-proft curve moves closer to the proft-maxmsng pont (k,k). Note that now the proft constrant no longer goes through pont A (representng prces a,a). Also the pont Z whch mnmzes prces n oth markets, consstent wth the profts constrant and prce equalty, now has prces greater than c. We can use equatons (46) and (50) to calculate the new maxmum welfare where there s a postve proft dentfed y ρ. The maxmum value of R s now ( ρ) + f. The maxmum consumer surplus, where consumers n oth markets are gven a weght of unty equals: (5) S T + ( ρ) f ρ + ( ρ) f where ( ρ) f + ST < 0. ρ We can calculate the mplcaton of ntroducng the tarff n only one market and mposng the proft constrant. Agan f the tarff s mposed n market, the output n market remans at the monopoly level and the excess profts n that market susdse output n market. The maxmum consumer surplus (oth markets wth a untary weght) s now: Note that we use the upper-case, R, for the radus of the so-welfare functon and the lower case, r, for the so-proft functon. -3-

25 (53) S T + + ( ρ) f ρ + ( ρ) f (( ρ)) f + ST Wth ρ. Smlarly, the rato, Ω, of the consumer welfare where one tarff s mposed, as aganst a common tarff n oth markets, s: (54) ρ + ( ρ) Ω <. ρ + ( ρ) Ω However, > 0, so that ths rato gets hgher, and approaches as the value of ρ ρ approaches. That s to say, the proportonate loss n welfare n an optmal sngle market tarff, as aganst a common tarff n oth markets falls as the proft constrant tghtens. Where the proft constrant s postve, there s an ncrease n welfare where the weghts for market and are such that: (55) 3 ( ) f f + > ω ρ ( ρ) f ω ρ ( ρ) + + f + ω ρ + ρ ω ω ρ + ( ρ) whch mples >. ω ρ + ( ) ( ρ) ω Ths means that, for example, f ρ equals 0.5, then f.38 ω >, then t s etter for the regulator to mpose a tarff solely n market, rather than mpose a common tarff n oth markets. -4-

26 Ths secton essentally analyses the stuaton where the postve profts made n the unregulated retal market (market ) are used to susdse consumers n the regulated market. There s no detrmental mpact on consumers n market. However, on the other hand, market consumers receve no eneft from the regulaton. In the next secton, where we ntroduce a postvely slopng wholesale supply curve, the analyss ecomes more complex. 4. The ntroducton of tarffs where the wholesale prce ncreases wth output (d > 0) In the general case, where the wholesale cost rses as total output ncreases the soproft curves are gven y the expresson (35). The adopton of a postve value for the wholesale supply parameter d has a numer of mportant mplcatons. Consder the zero so-proft curve, and compare t to the crcular curve, where d 0, dscussed n detal n Secton 4.. The new so-proft curve, where d > 0, s llustrated n gure

27 The zero so-proft curve now has the form of an oval. The pont A (a,a) s stll on the curve ut the pont Z (z,z), where wholesale cost equals retal prce, has moved further up the 45 degree lne through the orgn: that s to say, z > c. The unconstraned proft maxmzng pont, K (k,k), s stll at the md-pont etween A and Z. The zero so-proft curve s symmetrc around the two perpendcular axes that have negatve and postve 45 degree slopes and pass through the pont K. The wdth on the 45 degree lne through the orgn, AZ, s now shorter than the dstance along the other axs, G. Ths s also verfed y the fact that the ponts (z,a) and (a,z) are no longer on the zero so-proft curve, ut generate postve profts. rom equaton (43), f the tarff s mposed only n market, and the frm sets the proft maxmzng prce n market, the market prces le on a lne wth a slope equal to + d. Ths s shown as KE n gure 4 for values of p k. That s to say, f the d tarff n market falls y unt, the proft maxmzng prce n market wll ncrease d y + d. There s now a clearer trade-off etween the consumers n the separate markets. A tarff mposed solely n market n a prevously unregulated system means that the consumer surplus n market now falls as the consumer surplus n market rses, One potental ssue s whether the restrcton that the prce n market cannot rse aove the value a acts as a constrant n ths case. Essentally ths s askng the queston: can the regulator set a tarff n market such that the proft-maxmsng response would e to produce zero output n market, yet the frm would stll e make postve profts? or ths to occur mples the followng dual restrctons. rst, the tarff n market must e greater than the wholesale prce for supplyng the output of market. Smultaneously, the margnal wholesale cost at that output must e greater than a, and therefore greater than the hghest margnal revenue n market. We proceed y -6-

28 fndng the hghest tarff, p * wholesale cost just equals a, so that usng equaton (6):, where ths apples. Ths requres that the margnal (56) a c + dq * Ths rearranges to (57) * a c q d The tarff at whch ths occurs s gven as: (58) * * ad ( a c) p a q d Ths s therefore the hghest tarff n market at whch the frm wll voluntarly leave market. The key ssue s whether the company s proftale at that tarff. Ths requres the tarff n market to e greater than the wholesale prce, wholesale prce s gven as: * p W. The (59) * * a c a+ c pw c + dq c + Therefore, for proftalty: (60) * * ad ( a c) a + c p pw d d Ths s smply the condton that the asolute slope of the supply functon s greater than the slope of the demand functon. If ths expresson holds, the mnmum prce tarff p consstent wth zero profts and zero proft-maxmsng output n market ** would e where the tarff n market just equals the wholesale prce. Ths mples that: (6) * ** ** * a p ad + c p pw c+ dq c+ d + d Therefore, where d, we can defne a tarff range for market where the proft maxmsng output for market s zero. Ths range s gven y: -7-

29 ad ( ) (6) a c ad c p + d + d If the tarff s set n ths range, the analyss reverts to that outlned n Secton 3. The frm wll only operate n the regulated market. It s nterestng to consder the specal case where d. In ths case the zero so-proft s horzontal at E where t cuts the vertcal lne through A. Ths mples that the mnmum prce tarff n market consstent wth non-negatve profts just leads to output n market ecomng unproftale. Ths s llustrated n gure 5. In ths case the lne KE s the lne gven n equaton (43) where n ths case the slope equals. Usng the expressons around equaton (37), n ths case the values of z and k are (a+c)/3 and (5a +c)/6. It s useful to adopt the followng notaton: (63) a c f g 6 3 Ths means that the dstances AK, AZ and AE take the values g, g and 3g respectvely. If the only concern s the consumer surplus wth dfferental weghtng -8-

30 etween markets, suject to a proft constrant, then t s straghtforward to analyse ths as n su-secton 4. where the wholesale prce was fxed. Where there s no nterventon, the outcome s at K and, usng equaton (47), the consumer surplus s g /. Where the tarff s ntroduced wth the same value n oth markets, the outcome s at z and the consumer surplus equals 4g /d. As n the case where d 0, ntroducng the unform mnmum tarff ncreases consumer surplus to 4 tmes ts orgnal value. However, f the mnmum tarff s only mposed n one market, n ths case the consumer surplus s gven as 9g /. Ths gves a hgher value than ether of the other optons. Of course, ths mples that f the consumer surplus n market were gven a greater weght than those n market, ths would furnsh an even stronger argument for favourng the ntroducton of a sngle tarff n only that market. We know that for d, then the zero proft outcome means settng prce equal to the wholesale prce n market. In that case the consumer surplus, S, s gven as: (64) a ( c) S ( + d) However, the consumer surplus where the tarff s ntroduced n oth markets smultaneously, S, equals: (65) a ( c) S ( + d) or S S, then d >. But where market s reduced to an output of zero, then d >, so that wherever ths occurs, the choce etween equal mnmum tarffs n oth markets and concentratng n just one market means that the consumer surplus would e maxmsed y just applyng the tarff to one market. It also means that there s a range of values for d gven y d where a greater consumer surplus would e generated, wth a zero proft constrant, y settng a tarff n market equal to the -9-

31 5. Conclusons It appears often to e the case that n markets wth a degree of monopoly power, regulators wsh to mpose prce controls n only part of. Ths s typcally for redstrutve reasons, wth the welfare of one set of consumers weghted more heavly than others. In the case of the electrcty market ths could e motvated y concerns over fuel poverty. In ths paper we formally analyse ths ehavour usng a very stylsed model n whch a monopolst servng two separate retal markets s a monopsonst n an otherwse compettve wholesale market. The paper focuses on the consumer welfare mplcatons of mposng a prce tarff n only one market, aganst havng to mpose a unform tarff n oth markets. The analytcal results suggest that the welfare costs of mposng prce constrants n only one market are relatvely low, and ths s especally the case where the enefts to the favoured market are weghted more heavly. Where the mnmum proft constrant s ncreased, the relatve welfare loss from prce control only operatng n one market s further reduced. Moreover, wth a degree of scarcty n the wholesale market we get the potental for a counterntutve result. Ths s that there are condtons where t s etter to only control one tarff, rather than mpose a unform tarff n oth markets, even wth neutral consumer welfare weghts. average wholesale prce and a tarff hgher than the margnal wholesale cost n the second market, ather than a common tarff n oth markets. -30-

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