Regulating monopoly price discrimination

Transcription

1 Journal of Regulatory Economcs (2018) 54: ORIGINAL ARTICLE Regulatng monopoly prce dscrmnaton Smon Cowan 1 Publshed onlne: 2 July 2018 The Author(s) 2018 Abstract A monopolst sells ts product n separated markets. The effects of requrng a unform proft margn nstead of monopoly prcng are assessed. A margn equal to the output-weghted arthmetc mean of the monopoly margns rases consumer surplus but reduces total output. When the margn equals the (lower) harmonc mean total output exceeds the monopoly level f the demand functons are convex, and socal welfare rses. Extensons cover a unform prce-margnal cost rato and a unform margn when the ntal prce s unform and costs dffer. The analyss uses convexty relatons and the mplcatons of proft-maxmzaton. Keywords Prce dscrmnaton Monopoly Margn regulaton JEL Classfcaton D42 L12 L13 1 Introducton Frms wth market power that sell ther products n several markets often set dfferent proft margns for the same product. What would happen f such a frm was requred to set the same proft margn n every market, wth the level of ths margn beng determned by a regulator as an average of the ntal margns? Ths paper presents a general theoretcal analyss to address ths queston. Such a polcy would effectvely ban prce dscrmnaton and would determne the level of proftablty of the frm. In the UK the level and the structure of proft margns n domestc energy supply have became an ssue for the ant-trust authorty and poltcans. In 2016 the Competton and Markets Authorty found that electrcty and natural gas supplers were over-chargng customers on default tarffs compared to those who have sgned up for B Smon Cowan smon.cowan@economcs.ox.ac.uk 1 Department of Economcs, Unversty of Oxford, Manor Road Buldng, Oxford OX1 3UQ, UK

2 2 S. Cowan fxed prce tarffs and who thus have shown a hgher senstvty to prce. 1 Margns for the supply of the same product dffer markedly wthn each frm. See Waddams Prce (2018) for a recent assessment. Smlarly a perceved rse n market power has become a matter of general concern to polcy-makers. De Loecker and Eeckhout (2017)show that the the average rato of prce to margnal cost n the USA rose from 1.18 n 1980 to 1.67 n 2014, havng had no secular trend between 1950 and Weche and Wambach (2018) perform a smlar analyss for European countres, where the recovery of market power after the Great Recesson of the late 2000s has been slower than n the USA. The model has a monopolst sellng a sngle product n separated markets. Demand functons and margnal costs can dffer across markets. Intally the frm sets ts monopoly prce n each market. The margn s prce mnus margnal cost, and the monopoly margns dffer. As a result output s neffcently dstrbuted as far as socety s concerned. A gven output s allocated effcently when the proft margns are equal, because the margnal socal value of output equals prce mnus margnal cost. The man am n ths paper s to analyse the mplcatons for consumer surplus and socal welfare of requrng a unform margn. Two levels of the margn are consdered: the arthmetc mean of the monopoly margns and the harmonc mean. In each case the weghts are the shares of total output. A unform margn equal to the arthmetc mean benefts consumers n aggregate. The argument for ths s general, as t depends only on demand curves slopng downwards, and t does not depend on the ntal margns beng the monopoly ones. An mplcaton of ntal proft maxmzaton, however, s that total output s lower wth ths unform margn and the effect on socal welfare can be negatve or postve. The harmonc mean s below the arthmetc mean, so all consumers are better off wth the former. The lower margn has two addtonal desrable features: t guarantees that total output s above the monopoly level when the demand functons are convex, and t s the proft-maxmzng unform margn f the demand functons are lnear (so the frm prefers t to the arthmetc mean). An alternatve form of regulaton, whch equalzes the rato of prce to margnal cost, s also explored. Ths s partcularly relevant when the products dffer across markets and thus cannot be drectly aggregated. Mravete et al. (2017) analyse the effects of a regulaton that requres state-run alcohol retalers n Pennsylvana to set a unform rato of prce to wholesale cost of 1.3. A regulator wth complete nformaton and a full set of nstruments would set each prce equal to margnal cost and would fnance any resultng losses by a lump-sum transfer. If the regulator cannot make transfers then the second-best soluton s Ramsey prcng. Ths equalzes the product of the mark-up (the margn dvded by the prce) and the prce elastcty and yelds zero proft to the frm. The focus nstead here s on pecemeal reforms to monopoly prcng that have robust welfare propertes and can be mplemented wthout a large nformaton requrement. In the analyss of classc prce dscrmnaton margnal cost s common across markets. The frm chooses dscrmnatory prces that vary wth demand dfferences. Alternatvely t chooses a unform prce, whch mples a unform margn. In contrast 1 See the Energy Market Investgaton, Fnal Report, June 2016, avalable at government/organsatons/competton-and-markets-authorty.

3 Regulatng monopoly prce dscrmnaton 3 n ths paper the level of the unform margn s determned by a regulator. Classc prce dscrmnaton can be negatve or postve for socal welfare, and knowledge of the shapes of the demand functons s needed to determne the sgn of the welfare effect (Agurre et al. 2010). An mportant feature of the analyss here s that t makes mnmal assumptons about the shapes of the demand functons. When margnal costs dffer prce dscrmnaton may be sad to hold ether when margns dffer or when prce-cost ratos vary. Clerdes (2004) dscusses and compares these defntons. The analyss here covers both defntons. DellaVgna and Gentzkow (2017) show that unform prcng s very common amongst US retalers, and Cavallo et al. (2014) show that four large global retalers set unform prces for tens of thousands of products wthn currency unons. Chen and Schwartz (2015) have a monopolst ntally choosng a unform prce whle margnal costs dffer and demand functons have the same shape. The frm s then allowed to use monopoly prcng n each market. Aggregate consumer surplus s hgher wth monopoly prcng under mld condtons on the demand functon, and socal welfare s hgher under condtons that are weaker (because profts rse). The technques used n the man part of ths paper are appled to assess the welfare effects of a unform margn rather than a unform prce. A strkng result s that f demand functons are convex a unform margn equal to an average of the ntal margns guarantees that both aggregate consumer surplus and profts are hgher than wth the unform prce. Ths works for any ntal unform prce provded the mpled margn s postve. Unform margn regulaton s examned n Sect. 2. Secton 3 assesses a unform prce-cost rato. In Sect. 4 the unform margn and unform prce-cost rato are compared. Secton 5 analyses a unform margn when costs dffer and ntally the frm s constraned to set a unform prce. Secton 6 concludes. 2 Margn regulaton A monopolst supples more than one market. The markets are separated, so consumers n one market cannot buy from another and thus prces can dffer. Demand n market s q (p ), whch s decreasng n the prce, p, and s twce-dfferentable. The functonal form of demand can dffer across markets. In ths secton the goods are dentcal and can be aggregated. The regulator observes prces, margnal costs and outputs. Demand functons are not necessarly known. Margnal cost, c 0, s constant. The cost levels can dffer across markets. Intally for each market the frm chooses p to maxmze ts proft, (p c )q (p ), whch s assumed to be a sngle-peaked functon. The frstorder condton that determnes the monopoly prce s (p c )q (p ) + q (p ) = 0. The monopoly prce s p, the assocated margn s p c > 0, and output s q = q (p )>0. There s assumed to be some varaton n monopoly margns across markets. The arthmetc mean of the monopoly margns, denoted by the superscrpt a, wth weghts equal to the shares n total output, s m a = (p c )q. q

4 4 S. Cowan Ths s total proft per unt of output wth monopoly prcng. The frst form of regulaton requres the frm to prce so that the margn equals m a n each market: p a = c + m a. Total output s effcently allocated across markets because the margnal socal value of output, whch equals the margn, s equalzed. The level of the margn determnes how much output n total there s. Movng to a unform margn of m a reduces prces n hgh-margn markets and ncreases them n low-margn markets. Perhaps surprsngly, the effect on consumers n aggregate s certan to be postve. Standard consumer theory can be used to show ths. Consumer surplus n market, v (p ), s a decreasng and convex functon of the prce. Roy s Identty for quas-lnear utlty yelds v (p ) = q (p ).Atthemargn the loss of surplus from a prce rse equals the ncreased cost of purchasng the orgnal quantty. The second dervatve s v (p ) = q (p )>0. Convexty mples that the change n aggregate consumer surplus has a lower bound: v v (p0 )(p1 p 0 ) = (p 0 p 1 )q (p 0 ) (1) where superscrpts 0 and 1 denote the values n perods 0 and 1 respectvely and v s the change n consumer surplus. Ths lower bound s the reducton n expendture f consumers contnued to purchase the old quanttes. Proposton 1 Unform margn prcng, wth the margn equal to the weghted arthmetc mean of the monopoly margns, generates hgher total consumer surplus than monopoly prcng. Proof The lower bound n (1) n the move to unform margn prces s (p p a )q (p ) = (p c m a )q (p ) = 0 usng the defnton of m a. The cost to consumers, n aggregate, of the monopoly quanttes when facng the unform margn of m a equals the cost at the monopoly prces. 2 Aggregate consumer surplus can be thought of as the surplus of a representatve consumer. Ths consumer cannot be worse off, because the orgnal bundle s affordable, and n fact s better off as quanttes are adjusted n response to the relatve prce changes. The assumpton that the frm maxmzes profts ntally s not used n Proposton 1: t holds for any startng set of prces yeldng dfferent margns. It also holds whether or not all markets are served when the unform margn apples. The effect of the unform margn on total output can be determned. Proft maxmzaton and the defnton of m a mply that (p c )q > m a q (p a ) = 2 In prce ndex terms the Laspeyres prce ndex equals 1. (p c )q q (p a ), q

5 Regulatng monopoly prce dscrmnaton 5 and so q (p a )< q. Remark 1 Total output s strctly lower wth a unform margn of m a than wth monopoly prcng. If total output were to be hgher wth the unform margn then profts would be larger, whch contradcts the assumpton of ntal proft maxmzaton. In the case of classc prce dscrmnaton, wth margnal cost that s common to all markets, the mposton of the unform margn causes prces to fall n low-elastcty markets and to rse n hgh-elastcty markets, so t s ntutve that the effect on total output s negatve. 3 The effect on welfare of a unform margn of m a can be negatve or postve: there s less output, but t s effcently dstrbuted across the markets. Consumer surplus s hgher but profts are lower. Formally, welfare n market s the sum of consumer surplus and profts: w (p ) = v (p ) + (p c )q (p ). The effect of a hgher prce on socal welfare n one market equals the socal value of addtonal output, p c, tmes the effect of the prce ncrease on output (the slope of demand) and the second dervatve s w (p ) = (p c )q (p ), w (p ) = q (p ) (1 + (p c ) q (p ) q (p ) Welfare s concave n the prce f the expresson n large brackets s postve. Concavty of the aggregate welfare functon holds f the demand functons are all concave (for example wth lnear demands), as long as all markets are served wth unform margn prcng. If the demand functons have constant elastctes, wth q (p ) = s p ɛ for s > 0 and ɛ > 1, welfare s convex when prces are above c (1 + ɛ )/ɛ.the monopoly prce p = ɛ c /(ɛ 1) satsfes ths. Wth cost and elastcty dfferences that are not too large the aggregate welfare functon s convex. Proposton 2 If socal welfare s concave (convex) n prces then welfare s lower (hgher) wth a unform margn equal to the weghted arthmetc mean of the monopoly margns than wth monopoly prcng. Proof When socal welfare s concave n prces the upper bound to the change n total welfare s (p c )q (p )(pa p ) = (p p a )q (p ) = 0, usng the frst-order condton for proft-maxmzaton for the frst equalty and the proof of Proposton 1 for the second. Wth convex welfare ths s the lower bound. ). 3 I am grateful to a referee for suggestng ths ntuton.

6 6 S. Cowan The welfare bound s the same as the lower bound to the change n consumer surplus once account s taken of proft maxmzaton. The purpose of Proposton 2 s to establsh that welfare can go down or up rather than to present a complete welfare analyss. Whle a unform margn of m a s good for consumers t can reduce socal welfare, for example when all demand functons are lnear. The second form of regulaton sets the unform margn equal to the weghted harmonc mean of the monopoly margns: m h = q q p c. (2) The harmonc mean, denoted by the superscrpt h, s lower than the arthmetc mean strctly so wth monopoly margns that dffer so each prce s below the level wth the arthmetc mean. 4 All consumers are thus better off, and socal welfare s hgher, than wth m a. When all demand functons are convex the lower margn ensures that socal welfare s above the level wth monopoly prcng, because total output rses. Proposton 3 Suppose all the demand functons are convex. A unform margn equal to the weghted harmonc mean of the monopoly margns yelds total output above the level wth monopoly prcng. Proof Wth convex demand functons there s a lower bound to the change n total output n the move from monopoly prces to a unform margn of m h : q q (p )(mh + c p ) = m h q (p ) p + q (p c ) = 0 where the frst-order condton, (p c )q (p ) + q (p ) = 0, has been used twce to obtan the frst equalty, and the defnton of m h n (2) s appled to obtan the second. Convexty of demand mples that n a market where the prce falls there s a relatvely large ncrease n demand, whle n a market wth a prce ncrease the reducton n demand s relatvely small. Proposton 3 apples whether or not all markets are served wth postve outputs at the unform margn prce. Wth convex demands socal welfare s hgher wth the unform margn than wth monopoly prcng because total output s hgher strctly so f any demand functon s strctly convex and the output s effcently dstrbuted. The formal analyss of the welfare effect uses the general framework of Varan (1985). Addng the change 4 The arthmetc mean can be wrtten as the expected value of the margns, E[p c], and the harmonc mean s 1/E[1/(p c)]. The harmonc mean s smaller by an applcaton of Jensen s Inequalty.

7 Regulatng monopoly prce dscrmnaton 7 n profts to the lower bound to the change n consumer surplus n (1) gves a lower bound to the change n aggregate welfare: w (p 1 c ) q. (3) When the new margn, p 1 c, s unform and postve the lower bound n (3) has the same sgn as the change n total output. An addtonal reason to requre a unform margn of m h rather than m a when the demand functons are lnear s that the lower margn generates larger profts, provded all markets are served wth the unform margn. Let q (p ) = a b p for a > 0, b > 0, and a b c > 0. The monopoly margn s p c = a b c 2b and the monopoly output s q = a b c 2. The harmonc mean of the margns s m h = (a b c ) 2 b. Suppose the frm could choose ts unform margn, m, to maxmze proft m (a b c b m), whch s a strctly concave functon of m gven the assumpton that all markets are served. The frst-order condton s (a b c ) 2m b = 0 and the soluton s m h. Profts are hgher because the lower margn equals the proftmaxmzng unform margn. Remark 2 When the demand functons are lnear, and all markets are served when the unform margn m h s set, a unform margn of m h yelds a Pareto mprovement compared to a unform margn of m a. When demands are lnear, and all markets are served, total output wth the unform margn of m h equals that wth monopoly prcng. The same total output s now effcently allocated, so socal welfare s hgher wth m h than wth monopoly prcng. Ths, and the fact that the frm would choose m h, lnks the analyss to that of Pgou (2013) for classc prce dscrmnaton. Pgou shows that when margnal cost s common, demand functons are lnear, and all markets are served, the chosen unform prce yelds the same total output as dscrmnaton. Remark 2 depends on all markets beng served. Suppose there are two markets, the margn s hgher n market 1, so p 1 c 1 > m a > m h > p 2 c 2, and wth the unform margn m h demand n market 2 s zero. The frm would earn hgher profts wth m a as ths s closer to p 1 c 1, the optmal margn n the market that contnues to be served, and the proft functon n ths market s strctly concave n the margn. Any requred postve unform margn could nstead be be regarded as a maxmum. If the frm chooses a margn that s lower then all consumers wll be better off and, by revealed preference, the frm earns more proft. Treatng m h as a cap makes a welfare ncrease (compared to monopoly prcng) more lkely when the condton n Proposton 3 does not hold, wthout guaranteeng one. Requrng a unform margn that s capped by the harmonc mean has attractve features. Consumers, n aggregate, gan, and socal welfare rses under plausble condtons.

8 8 S. Cowan 3 Regulatng the prce-cost rato An alternatve s to regulate the rato of prce to margnal cost. Ths may be used when the goods cannot be aggregated, for example f they have dfferent qualtes. The analyss, though, s general and apples equally to the case where the goods can be aggregated. Assume that () margnal cost s strctly postve n each market and () the ratos of prce to cost wth monopoly prcng dffer across markets. The analyss s smlar to that for the unform margn. Suppose frst that the frm must have a unform prce-cost rato equal to monopoly revenue dvded by monopoly cost, r a = p q c q, (4) wth p r margn. = r a c. The next proposton mrrors Propostons 1 and 2 for the unform Proposton 4 Suppose that the frm sets a unform rato of prce to margnal cost equal to monopoly revenue dvded by monopoly cost. () Consumer surplus s hgher than wth monopoly prcng. () If aggregate welfare s concave (convex) n prces then welfare s lower (hgher) than wth monopoly prcng. Proof () The lower bound to the change n consumer surplus s (p r a c )q = p q p q c q c q = 0. () When welfare s concave n prces the upper bound to the change n welfare s (p c )q (p )(r a c p ) = (p r a c )q (p ) = 0, usng the frst-order condton for proft-maxmzaton for the frst equalty and the result n () for the second. Wth convex welfare ths s the lower bound. To understand the role of equal prce-cost ratos, suppose there s a fxed budget that can be spent on the costs of producng the outputs. Wrte socal welfare as a functon of output as w (q ), a strctly concave functon whose dervatve equals the margn p c. Consder the problem of choosng the outputs to maxmze aggregate welfare w (q ) subject to the constrant that total cost, c q, equals the budgeted amount. The soluton to ths classcal constraned maxmzaton problem s that the ratos of the margn to the cost, {(p c )/c }, are equal, and that the level of the common rato s the value of the Lagrange multpler on the budget constrant. 5 It follows that the prce-cost ratos are equal. Suppose, n contrast, the prce-cost ratos are 2 n market and 4/3 nmarket j, wth c = 1, p = 2, and c j = 3 and p j = 4. 5 The Lagrangan s w (q ) + λ(c c q ) where C > 0 s the budget. The frst-order condton for market s p c λc = 0.

9 Regulatng monopoly prce dscrmnaton 9 Reducng output n j by one unt enables producton of three extra unts of output n market for the same total cost and leads to a net ncrease n welfare of 2 (because the margns each equal 1). The rato defned n (4) can be wrtten as r a = 1 + (p c ) c c q c q, whch s 1 plus the arthmetc mean of the margn-cost ratos usng the cost shares as weghts. Ths leads to the second form of regulaton, whch requres a unform prce-cost rato equal to 1 plus the harmonc mean of the margn-cost ratos: r h = 1 + c q c 2 q p c. (5) Ths s strctly below r a, so all consumers are better off than wth the hgher rato, and socal welfare s hgher. There s a parallel result to Proposton 3. Demand convexty and the lower rato mply that socal welfare s hgher than wth monopoly prcng. Proposton 5 Suppose the demand functons are convex. If the frm sets a unform rato of prce to margnal cost equal to 1 plus the harmonc mean of the monopoly margn-cost ratos then welfare s above the monopoly level. Proof The lower bound to the welfare change, gven generally n (3), s (r h 1) c q, whch has the same sgn as the sum. Convexty of the demand functons mples c q c q (p )(r h c p ) = c q (p )((r h 1)c + c p ) = (r h 1) = 0. c 2 q p c + c q In the frst equalty c s subtracted and added wthn the prce dfference term. The second takes r h 1 outsde the summaton and uses the frst-order condton twce. The thrd uses the defnton n (5). When the demand functons are lnear profts are hgher wth the lower requred rato (provded all markets are served). Wth q (p ) = a b p as the generc demand functon r h =. Consder the frm choosng ts unform rato, r, to maxmze c (a +b c ) 2 b c 2 (r 1) c (a b rc ), a strctly concave functon of r gven the assumpton that all markets are served. The frst-order condton s c (a b rc ) (r 1) b c 2 = 0,

10 10 S. Cowan whose soluton s r h. The frm prefers the lower prce-cost rato when the demand functons are lnear. Requrng a common prce-cost rato (based on the harmonc mean) s good for consumers and often good for welfare. As wth the unform margn nothng s lost, and there are potental gans, from treatng ths as a maxmum rather than as a requred prce-cost rato. 4 Comparng the unform margn and prce-cost rato The unform margn and unform prce-cost rato, based on harmonc means, can be compared. If costs are dentcal then the unform margn and unform prce-cost rato yeld the same outcomes: the proft margn wth the common rato gven n (5) s (r h 1)c = m h when each cost equals c. Costs must vary across the markets for there to be a dfference. Assume, then, that costs dffer but the demand functons are dentcal n shape, though they may dffer n multplcatve market sze parameters. For two demand functons the comparson s mmedate because one of the forms of regulaton does not bte. Demand wth constant elastcty s q (p ) = s p ɛ where s > 0 s the market sze parameter. Proposton 3 apples because ths s a convex functon. The unform margn yelds hgher socal welfare and consumer surplus than monopoly prcng. For ths functon prce-cost rato regulaton yelds the same outcomes as monopoly prcng, wth p /c = ɛ/(ɛ 1). The opposte happens for exponental demand, q (p ) = s e p /λ for λ > 0. The demand functon s convex and thus Proposton 5 apples: welfare and consumer surplus are strctly hgher wth the unform prce-cost rato than wth monopoly prcng. The monopoly prce s p = c + λ, so the monopoly margns are equal and unform margn regulaton does not constran the frm. Whle the unform prce-cost rato mples neffcency n the allocaton of output across markets, t nduces a large enough ncrease n total output that socal welfare rses. When the demand functons are lnear the same result as for exponental demand apples: both socal welfare and consumer surplus are hgher wth the unform prcecost rato. Let demand be q (p ) = s (a bp ). Wthout loss of generalty let s = 1 and defne c s c as the expected value of the margnal costs. The margn and rato are m h = a b c 2b and r h = 1 + a c b s c 2 2b s c 2 respectvely. Varan s lower bound to the change n welfare, gven n (3), n the move from m h to r h s (r h 1) c q. Ths has the same sgn as c q = c s b(m h (r h 1)c ) = b s (c c) 2 > 0. 2 The second equaton uses the values of m h and r h. The expresson s (c c) 2 s the varance of margnal costs. Drect calculaton shows that the change n profts s

11 Regulatng monopoly prce dscrmnaton 11 negatve. 6 Because socal welfare ncreases the change n consumer surplus must be postve. The next remark summarzes the three results n ths secton. Remark 3 Suppose that margnal costs dffer across markets and that demand functons dffer only n multplcatve sze parameters. () For constant elastcty demand, margn regulaton wth the harmonc mean rases consumer surplus and welfare above the monopoly levels, whle prce-cost rato regulaton has no effect. () For exponental demand prce-cost rato regulaton based on the harmonc mean rases consumer surplus and welfare above the monopoly levels, whle margn regulaton has no effect. () For lnear demand both forms of regulaton rase consumer surplus and welfare, wth the unform prce-cost rato yeldng hgher consumer surplus and welfare. The comparson of the two forms of regulaton requres knowledge of the shape of the demand functons. 5 Startng wth a unform prce Let the frm ntally set a unform prce, p, whch need not be the proft-maxmzng prce. Costs dffer so the ntal margns are not equal. The quantty sold n market at ths prce s q (p). The weghted average of the margnal costs s c c q (p)/ q (p). Now suppose the frm must set a unform margn equal to p c, assumed to be strctly postve, so the prce n market s p = c + p c. The unform margn yelds hgher aggregate consumer surplus than the unform prce. The lower bound to the change n consumer surplus, n lne wth Proposton 1, s (p c p + c)q (p) = ( c c )q (p) = 0. To obtan addtonal results assume, as do Chen and Schwartz (2015), that demand functons dffer only n ther sze parameters, so q (p ) = s q(p ) and consumer surplus s s v(p ). Set s = 1 as a normalzaton. The average of the margnal costs becomes c = s c. Wth ths form of demand addtonal nsght for the consumer surplus result s avalable. Aggregate consumer surplus s s v(p ), whch s equvalent to expected consumer surplus. Convexty of v(p ) mples that f the new prces have an expected value whch s the same or lower, and a varance whch s larger, then expected consumer surplus wll be hgher. Prce dsperson s good for consumer surplus because the gan n surplus from a prce reducton s larger than the loss n surplus from an equal prce ncrease when demand slopes down. The unform margn, by constructon, keeps the expected prce constant and mples that the varance of prces equals the varance of costs, whle the varance wth the unform prce s zero. 6 The change n profts has the same sgn as ( c q / q )2 s c 2 < c 2 s c 2 = s (c c) 2 < 0. Both nequaltes are by the fact that the varance of margnal costs s postve.

12 12 S. Cowan If demand s convex then total output rses, q s q (p)(c c) = q (p) s (c c) = 0, (6) as the prce change s c c and by the defnton of c. There are two mplcatons. Frst, welfare rses when demand s convex. There s at least as much output and t s now effcently dstrbuted. Second, and more notably, profts are hgher than wth the unform prce because output s hgher whle the average proftablty of each unt of output s unchanged. Wth the unform prce profts are (p c )s q(p) = (p c)q(p), whle wth the unform margn profts are (p c) s q(c + p c). The frm, and consumers n aggregate, are better off wth the unform margn than wth the unform prce. Ths does not depend on the level of the unform prce provded t exceeds the average of the margnal costs. Collectng these results gves the proposton for ths secton. Proposton 6 Suppose that ntally a unform prce s set and margnal costs dffer. () A unform margn, equal to the weghted arthmetc average of the ntal margns, rases aggregate consumer surplus. () If demands dffer only n sze and are convex then profts and socal welfare are also hgher wth ths unform margn (provded t s postve). Statement () n Proposton 6 depends only on the convexty of consumer surplus, whle statement () holds wth demands that dffer only n sze and that are convex. Nether depends on proft maxmzaton, so the proposton could apply, for example, to a state-owned utlty that sets a unform prce for socal or poltcal reasons. The unform margn does not yeld a Pareto mprovement consumers n hgh-cost markets face hgher prces but t s noteworthy that aggregate consumer surplus and profts can both be ncreased by a smple polcy. There are two extensons of the analyss. Frst, suppose the demand functons are concave and the unform margn s negatve. Contnue to assume that the demand functons dffer only n sze. A subsdy from the government mght allow the frm to operate whle makng a loss. Consumers, on average, stll beneft from the unform margn: part () of Proposton 6 holds and does not depend on the level of the ntal prce. Concavty of demand means that total output falls. The lower bound to the change n total output n (6) becomes the upper bound when demand s concave. Wth lower output, and the same negatve average proft margn, the loss the frm ncurs s reduced so socal welfare rses. Second, f p = c profts are zero wth both the unform prce and the unform margn. The latter entals that prces equal margnal costs so socal welfare, whch equals consumer surplus, s maxmzed. Ths does not depend on the shapes of the demand functons. By contnuty f the ntal unform prce s suffcently close to c welfare wll be hgher wth the unform margn ndependently of the shape of demand.

13 Regulatng monopoly prce dscrmnaton 13 6 Concluson Smple forms of regulaton of the level and structure of a monopolst s prces have been shown to yeld clear results for socal welfare and consumer surplus n qute general crcumstances. Unform margns and unform prce-cost ratos work n smlar ways. When ntally the frm sets a unform prce and costs dffer the unform margn ensures that consumer surplus rses and, f demands are convex, profts ncrease n addton, provdng a powerful argument aganst unform prcng n such crcumstances. Acknowledgements I am grateful to two anonymous referees, Iñak Agurre, Marus Schwartz and partcpants at the 2017 UIBE Workshop on IO and Competton Polcy for helpful comments. Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made. References Agurre, I., Cowan, S., & Vckers, J. (2010). Monopoly prce dscrmnaton and demand curvature. The Amercan Economc Revew, 100(4), Cavallo, A., Neman, B., & Rgobon, R. (2014). Currency unons, product ntroductons, and the real exchange rate. The Quarterly Journal of Economcs, 129(2), Chen, Y., & Schwartz, M. (2015). Dfferental prcng when costs dffer: A welfare analyss. The RAND Journal of Economcs, 46(2), Clerdes, S. K. (2004). Prce dscrmnaton wth dfferentated products: Defnton and dentfcaton. Economc Inqury, 42(3), De Loecker, J., & Eeckhout, J. (2017). The rse of market power and the macroeconomc mplcatons. Techncal report, Natonal Bureau of Economc Research. DellaVgna, S., & Gentzkow, M. (2017). Unform prcng n US retal chans. Workng Paper 23996, Natonal Bureau of Economc Research. Mravete, E. J., Sem, K., & Thurk, J. (2017). One markup to rule them all: Taxaton by lquor prcng regulaton. Workng Paper 24124, Natonal Bureau of Economc Research. Pgou, A. C. (2013). The economcs of welfare. Basngstoke: Palgrave Macmllan. Varan, H. R. (1985). Prce dscrmnaton and socal welfare. The Amercan Economc Revew, 75(4), Waddams Prce, C. (2018). Back to the future? Regulatng resdental energy markets. Internatonal Journal of the Economcs of Busness, 25(1), Weche, J. P., & Wambach, A. (2018). The fall and rse of market power n Europe. Workng Paper Seres n Economcs 379, Unversty of Lüneburg, Insttute of Economcs.