Should Pricing Policies be Regulated when Firms may Tacitly Collude?

Transcription

1 Should Prcng Polces be Regulated when Frms may Tactly Collude? George Norman Department of Economcs Tufts Unversty Medford, MA 0155 USA emal: Jacques-Franços Thsse CORE, Unversté Catholque de Louvan, 35, Voe du Roman Pays, B-1348 Louvan-la-Neuve, Belgum emal: Abstract: Actve anttrust polcy may result n perverse effects detrmental to consumer and socal welfare because such actve polcy affects market structure through ts mpact on the medum- and long-run decsons of frms. Ths paper nvestgates the effects of deregulaton of frms' prcng polces n a dynamc settng that allows for tact colluson. We show that, provded the consumer reservaton prce mposes no effectve constrants on frms, prce deregulaton wll lead to dscrmnatory prces that are almost always and almost everywhere lower than mll prces. By contrast, f the consumer reservaton prce s bndng, the welfare propertes of the two prcng polces wll be reversed wth consumers losng and frms ganng from dscrmnatory prcng. The requred degree of prce flexblty for ths to happen s lower n more concentrated market structures. Ths mples that f market concentraton s already hgh, deregulaton of prcng polces s less lkely to beneft consumers unless the regulatory authortes are wllng to mpose accompanyng restrctons ntended to reduce prce flexblty. Keywords: deregulaton, prce dscrmnaton, dynamc game JEL: L10, L13, L50.

2 1. Introducton Regulaton of the condtons under whch frms compete s common n the majorty of Western ndustralzed countres. Such regulaton s ntended to prevent or control, for example, explct or tact colluson among frms, mergers and acqustons, vertcal restrants and frms prcng polces, n partcular whether frms can employ dscrmnatory prcng. The common ratonale for these regulatory polces s that they are needed to protect consumers from the abuse of monopoly power by frms that supply them wth goods and servces. The purpose of ths paper s to suggest that care should be exercsed by anttrust authortes n ther desgn of polces ntended to promote competton n the marketplace. We do not deny the underlyng rason d'être of competton polcy but wsh to suggest that nave applcaton of the dea that competton s always and everywhere desrable may have unforeseen and harmful effects. Our analyss can be summarzed by the proposton that analyss of the effects of competton polcy should not take ndustry structure as gven. Polces that create too tough a compettve envronment may result n perverse effects detrmental to consumer and socal welfare because actve anttrust polcy affects market structure through ts mpact on the medum- and long-run decsons of frms. The stronger are the structural effects of regulatory polcy the more lkely s t that blnd adherence by the regulatory authortes to the benefts of competton wll be msguded. As a smple llustraton, consder the case for cartel laws. The receved wsdom has been that a hgh degree of coordnaton among frms benefts them through hgh profts but s detrmental to consumers through hgh prces, the losses of consumers generally outweghng the gans of producers. Ths lne of reasonng underpnned the tough ant-trust and merger polcy that characterzed regulaton n the Unted States and European untl the early 1980s. The desrablty and acceptablty of such tough polcy has, however, been questoned on several grounds. Wllamson (1968), for example, has argued that effcency consderatons may be grounds for defendng coordnaton among frms f such coordnaton allows them to perform cost-savng actvtes, the benefts of whch outwegh the prce-ncreasng or quantty-reducng effects of coordnaton. 1 1 German law allows "ratonalzaton" and "specalzaton" cartels on the grounds that they lead to effcency gans (see Kühn 1993).

3 Selten (1984) develops an argument that s more closely related to the deas we develop n ths paper. He shows that f the market wthn whch cartel laws are beng enacted s characterzed by free entry jont proft maxmzaton permts a greater number of compettors n a market than non-collusve behavor. (p. 183) As a consequence cartel laws are good for busness n the sense of greater average jont profts. (p. 14) Our focus n ths paper s on the regulaton of frms prcng polces, whch was wdely accepted up to the early 1980's to be n the publc nterest. Norman and Thsse (1996) provde several examples of such regulaton: of the prces charged by US and European arlne ndustres, the applcaton of resale prce mantenance, the polces artculated by the UK Prce Commsson and the Federal Trade Commsson n the US attackng dscrmnatory prcng. In a related context, the 1980s saw ncreasngly strct applcaton by the US and Europe of ant-dumpng legslaton (dumpng can be vewed as prce dscrmnaton n the nternatonal market-place). For example, from 1980 to 1987 the EC ruled aganst foregn frms n 49 cases affectng some three bllon ECUs of mports (Tharakan 1991, Table 1). Ths relatvely tough regulatory regme has been consderably relaxed over the past decade. The Robnson-Patman Act has not been appled n the US, the Prce Commsson was abolshed n the UK, and more generally frms have been left freer to choose ther prcng polces, as a result of whch (spatal) dscrmnatory prcng s a more lkely outcome (Thsse and Vves, 1988). Justfcaton for the relaxaton n polcy can be found n modern developments n spatal prcng whch argue that spatal prce dscrmnaton may be better for consumers than mll prcng: see, for example, Norman (1983), Thsse and Vves (1988). The way the argument runs s that dscrmnatory prcng s tougher for frms than mll prcng and so s pro- rather than antcompettve: "denyng a frm the rght to meet the prce of a compettor on a dscrmnatory bass provdes the latter wth some protecton aganst prce attacks. The effect s then to weaken competton, contrary to the belef of the proponents of nave applcaton of legslaton prohbtng prce dscrmnaton lke the Robnson-Patman Act, or smlar recommendatons of the Prce Commsson n the Unted Kngdom." (Thsse and Vves, 1988) 3

4 It s of nterest to note, for example, that dscrmnatory prces can be defended aganst acton under the Robnson-Patman Act or nternatonal ant-dumpng legslaton f they are ntended to "meet the competton". The benefts clamed for dscrmnatory prcng have been called nto queston n recent analyses by Armstrong and Vckers (1993) and Norman and Thsse (1996). The common theme of these works s that current polcy-based analyss s flawed because t takes no account of an mportant feedback loop between market conduct and market structure. If dscrmnatory prcng s, ndeed, more compettve for ncumbent frms t wll act as an entry deterrent. As a result, the essentally short-run benefts that the more compettve regme generates through lower prces may be more than offset by the longer-run effects t has on market structure: by reducng n the number of frms that wsh to enter the market or the scale at whch they enter. Our specfc contrbuton n ths paper s that we extend the Norman/Thsse analyss by relaxng an mportant assumpton: that post-entry prces are one-shot Bertrand equlbrum prces. We consder, nstead, the stuaton n whch the ncumbent frms recognze that the prce game s a repeated game. Before turnng to ths analyss, we develop some further prelmnary deas n the next secton. Our formal model s then presented n secton 3 and the prce equlbra for ths model are dentfed n secton 4. Secton 5 presents a welfare comparson of the two prcng polces and our man conclusons are summarzed n secton 6.. Some Prelmnary Analyss Let us try to make the deas presented above more concrete by examnng how we mght structure a general case capable of dentfyng the supposed benefts of tough regulatory polces. Assume that a market (as defned for polcy purposes) contans N actve frms, where N s to be determned endogenously through some entry and/or ext process. The pay-off to each frm s a functon of two sets of varables: () a set R descrbng the regulatory regme wthn whch the frms operate. R may be determned by Government, for example descrbng the class of acceptable prcng polces or the toughness of anttrust polcy, or by (a subset of) the ncumbent frms, for example, Armstrong and Vckers consder a stuaton n whch a monopolst serves two markets, one of whch s captve and the other subject to the threat of entry by a prce-takng entrant. They show that entry s more lkely and the scale of entry s larger when prce dscrmnaton s banned. Norman and Thsse consder a free-entry Salop-style market and show that prce dscrmnaton may nhbt entry suffcently as to be aganst the nterests of consumers and socety. See d'aspremont and Motta (1997) for a model n a related ven. 4

5 specfyng orderly prcng regmes such as basng pont prcng; consumer ncentves; prce-matchng guarantees. () a set P of strategc varables chosen by the frms more or less non-cooperatvely: prces, product specfcatons; locatons; quanttes. A subgame-perfect Nash equlbrum n N and P s dentfed from some m-stage game. For a gven regulatory regme R, ths equlbrum specfes a pay-off to each frm of the form: ( ) (1) ( ) Π P N, N : R for = 1,...N* whch wll have assocated welfare propertes. Now consder alternatve regulatory regmes, where we defne a tougher regme R R as beng a regme that s more compettve for the t frms n, or contemplatng entry to the market. For example, assume that the regulatory authorty tres to promote ferce competton n ths market by abolshng facltatng practces, or outlawng the trade assocaton to avod contacts among the frms, or warnng the frms that they are suspected of collusve behavor. For a gven N t s to be expected that: () Π ( ) ( ) < Π ( ) ( ) P N, N: R P N, N: R f R R wth the reducton n proft (partally) transferred to consumers. Equaton () would appear to justfy governments preference for tough rather than soft regulatory regmes. However, equaton () s essentally short-run, n that t takes market structure as gven. It s also to be expected that ncreased toughness of the regulatory regme wll reduce the number of actve frms n the market n the medum- and long-run. When frms have to decde whether to renew plants or to update ther products or to ncur other fxed costs, t may be that the decreased future stream of expected profts wll not cover these costs. The ext of some frms s to be expected, ncreasng proftablty for the remanng frms n the market: 3 (3) P ( N ) ( N : R) <Π ( P ( N ), N R) Π f N 1 > N 1, 1 : t The overall effect of an ncrease n the toughness of the regulatory regme s a combnaton of () and (3) and mght be expressed as follows. Let R > 0 be some measure of ncreased Π Π P Π toughness of R. Then = < 0 and < 0 but R P R N N R 5

6 Π Π P Π P Π N (4) = + + =? R P R P N N R (a) < 0 (b) > 0. In other words, ncreased toughness of R has: (a) a proft reducng effect n the short run through ts mpact on P for a gven N, and (b) a proft ncreasng effect n the longer run through nduced ext and stronger entry deterrence. For ncreased toughness to beneft consumers (a) must domnate (b). Otherwse ncreased market concentraton wll make consumers worse off, for example through reduced product varety or a longer travel dstance to shops, and more generally through hgher prces. Note the smlarty between ths analyss and Selten s argument that colluson may lead to lower concentraton and lower total profts n an ndustry. A qualtatvely smlar concluson would be reached f our measure of the desrablty of changes n R were some weghted average of producer and consumer surplus. 4 In short, polcy that establshes too tough a compettve envronment may have some unforeseen effects wth the (apparently) paradoxcal result of beng detrmental to the communty's welfare. An mportant lmtaton of almost all current analyss s that the equlbrum of the postentry subgame, on whatever strategc varables t s based, s assumed to be the outcome of a oneshot game. Ths, as we shall show, s an mportant assumpton. If prce, for example, s the strategc varable, dscrmnatory prcng s, ndeed, more compettve than non-dscrmnatory (referred to hereafter as mll) prcng n spatal markets, or ther non-spatal analogs, precsely because the prces that emerge are the result of a Bertrand-at-every pont process. But f the prce subgame s repeated t s not at all clear that the same concluson wll hold. We know that a monopolst always prefers dscrmnatory prcng to mll prcng and t mght be expected that the same wll be true when the prces charged by frms correspond to a tactly collusve outcome for an nfntely repeated prce subgame. 5 Assume, for example, as we do n the remander of our analyss, the famlar Hotellng/Salop spatal model n whch consumer demand s perfectly nelastc for prces below some reservaton prce. Wth demand of ths type t 3 A possble excepton would be f there were extensve network externaltes between frms. 4 There are further consderatons that we do not consder n ths paper. For example, prce dscrmnaton facltates market segmentaton by mult-product frms, ncreasng prces and profts but reducng consumer surplus. 6

7 trvally follows from the Folk Theorem that a set of dscrmnatory prces arbtrarly close to the consumers reservaton prce wll be subgame perfect equlbrum (SPE) prces for some dscount factor suffcently close to unty. Consequently, these prces wll be hgher than the SPE mll prces almost everywhere even f market structure s endogenously determned. It would appear, n other words, that the endogenous market structure argument s rrelevant to the choce of allowable prcng practces once we ntroduce the possblty that frms are aware that they are n competton over tme and ncorporate ths nto ther prcng decsons. But ths overlooks the mportance of a crucal part of the argument: that the dscount factor be suffcently close to unty. The queston we need to nvestgate s how close the dscount factor must be to unty for a partcular set of dscrmnatory or mll prces to be sustanable as an SPE. Ths queston arses from the nature of SPE prces. These are sustaned through some credble punshment strategy whch s desgned to ensure that the short-term gans from devatng from the (tactly) agreed prces are more than offset by the long-term losses that wll be ncurred once the punshment phase s mplemented. Lookng at ths another way, the ablty to rase prces above ther Bertrand-Nash equlbrum levels s constraned by the potental gans a frm wll make by devatng from these prces. Such gans are lkely to vary sgnfcantly wth dfferent prcng polces and so wll mpose dfferent lmts on the SPE prces. Ths suggests that we mght approach the problem from a somewhat dfferent perspectve by askng: For a gven dscount factor, what s the upper lmt on the dscrmnatory and mll prces that frms can charge? We can then nvestgate the welfare propertes of these prces. Two other remarks are n order at ths stage. Frst, the punshment strategy on whch we focus n ths paper s what s sometmes referred to as a "grm trgger strategy": any devaton from the tactly agreed prces nduces reverson to the one-shot Bertrand-Nash equlbrum prces forever. Other punshments are avalable to frms but t s reasonable to suggest that the qualtatve propertes of our analyss would reman vald for a larger class of such alternatve strateges. Secondly, t wll be shown below that the dscount factor s an amalgam of the dscount rate, whch s largely outsde the control of regulators (and frms), and frms' prce flexblty n response to devaton by ther rvals, whch s potentally controllable by regulators. Under what 5 We can assume the prce subgame to be nfntely repeated or to be repeated wth some known, constant 7

8 crcumstances (f any) mght the authortes wsh to mpose controls on the speed wth whch frms can change ther prces? Our analyss, n other words, ntroduces a further mportant element to regulatory control that has not been much dscussed, prmarly because the currently avalable analyss s statc. The dynamc approach followed here brngs wth t a new dmenson of competton n that the dscount factor can be made endogenous to the market. We show that, provded consumers reservaton prces mpose no effectve constrants on frms, the upper lmts on the tactly agreed prces are such that dscrmnatory prces are almost always and almost everywhere lower than mll prces. In other words, consumers gan and frms lose from dscrmnatory prcng. The ease wth whch frms can revse ther locaton decsons (the degree of spatal contestablty), whch was central to the welfare propertes of our 1996 analyss, now has a more lmted role. Frms gan less and consumers lose less from mll prcng when the market s spatally non-contestable. We also show, however, that explct account must be taken of the consumer reservaton prce snce ths mposes a more severe constrant on mll prces than on dscrmnatory prces. If ths constrant s strong enough and f prcng decsons are flexble enough, the welfare propertes of the two prcng polces wll be reversed wth consumers losng and frms ganng from dscrmnatory prcng. The requred degree of prce flexblty for ths to happen s lower n more concentrated market structures. Ths mples that f market concentraton s already hgh, deregulaton of prcng polces s less lkely to beneft consumers unless the regulatory authortes are wllng to mpose accompanyng restrctons ntended to reduce prce flexblty. 3. The Model We conduct our analyss n the context of the Hotellng/Salop locaton model, n whch the economy s assumed to be a crcle C over whch consumers are unformly dstrbuted at unt densty. The nterpretaton of ths model as a model of horzontal product dfferentaton s now so famlar that we need not repeat t here. We assume that each tme perod t s γ unts long, where γ measures the speed of response of ncumbent frms to any perceved prce devaton, and that the contnuous rate of tme dscount s ρ. Frms enterng ths market offer products that are dentcal n all characterstcs but ther locatons. Producton costs n any perod t are assumed to be dentcal for all frms, gven by: probablty n each perod t that the subgame wll contnue. 8

9 (5) C t (Q t ) = f + cq t where f can be thought of as the flow-equvalent present value of set-up costs -- f aggregate setup costs are F, then ( t+ 1) γ ρτ f = F e dτ = F ρ-- c are margnal costs and Q t s total output t = 0 tγ of the frm n perod t. Thus producton exhbts economes of scale, lmtng the number of frms that wll wsh to enter the market. We assume that f s suffcently small for us to be able to gnore nteger problems and we normalze c = 0 wthout loss of generalty. The set N = {1,,...n,...} denotes all potental entrants and the set ℵ denotes the frms that actually choose to enter. 6 Each frm chooses a locaton x C Φ, where x C f frm chooses to enter the market and x = Φ otherwse. Ths entry stage establshes a locaton confguraton denoted by the vector x = (x ) n whch we assume that the set ℵ corresponds to the frst #(ℵ) elements of x. It wll prove convenent below to refer to the tme-dependent locaton confguraton x t, where, unless otherwse stated, t s to be expected that x t = x for all t = Frms are assumed to make ther prcng decsons after they have made ther entry/locaton decsons. In each post-entry perod t, each frm ℵ establshes the prce t t schedule p ( r ) x gvng ts delvered prce to each consumer locaton r C. Transport costs are assumed to be lnear n dstance and quantty transported wth transport rate s, so that wth mll prcng the delvered prce schedule for frm n perod t s: t t (6) ( ) t p r x = m + s x r ℵ, r C. Wth dscrmnatory prcng we assume only that frms never prce below margnal costs: t t (7) ( ) p r x s x r ℵ, r C and we normalze s = 1 wthout loss of generalty. The frms prcng decsons made n each { } perod are denoted by the vector p t t t t ( x ) p ( r x ) t t =, ℵ, and P ( x ), ℵ, denotes the set of feasble delvered prce schedules for each entrant n perod t. Let s t = (x t,p t (x t )) and s = {s 0,s 1,...,s t,...}. 6 Ths mples that all frms make ther entry decsons smultaneously. In the analyss that follows we shall concentrate upon entry processes that lead to the loosest packng of frms consstent wth the relocaton costs that 9

10 Consumers purchase from the frm offerng the product at the lowest delvered prce. If there s a prce te, we assume that consumers act n a socally optmal manner by purchasng from the nearest frm. Consumer demand s perfectly nelastc at prces below a reservaton prce v, wth demand to frm ℵ from consumers at locaton r gven by: (8) [ ] t t t t t t t ( s ) = 1 f ( x ) and ( x ) = mn j ( x ) t q p r v p r p r j = 0 otherwse. The per-perod rate of proft to frm ℵ s: t+ γ t t (9) ( ) ( ) t [ ] ( ) ( 1 ) ρτ ( t+ 1 ) γ t t t ρτ π s = p r x x r q s e dτdr f e dτ C tγ [ [ p t ( r x t ) x r t ] q ( s t ) dr f C ] = 1 δ ρ δ t where δ = e -γρ s the dscount factor, wth 0 < δ < 1. The dscounted present value of frm s: π t (10) V ( s) = ( s t ) t= 0. Snce the dscount factor δ s determned by both γ and ρ t can be taken as a measure of prce flexblty n the relevant market whch, as we noted n secton, s potentally controllable by the regulatory authortes. In other words, the dscount factor can be made endogenous to the market by the regulator. In analyzng equlbrum n ths market we dstngush between the ntal entry stage and subsequent perods n whch ncumbent frms can take explct account of the repeated nature of prce competton between them. Snce n these later perods profts wll normally be shared on the bass of market areas... we may expect the process by whch market areas are determned to be a very aggressve one. (MacLeod et al., 1987, p. 19). Specfcally, we assume that market areas are determned by a two-stage entry/prce game n whch equlbrum for the prce-subgame t t s Bertrand-Nash. Ths s defned n the usual way as the set of prce functons pb ( x ) for all ℵ: t t t t t t t t ( x, p x ) π ( x, p ( x ), p b ( x )) t t (11) π b ( ) ( ) t t t t Denote sb x pb ( x ) =,. t t t t p ( x ) P ( x ) tγ. such that frms mght ncur. As a result, we could equvalently thnk of the entry process as beng sequental provded that 10

11 Equlbrum for the frst stage entry game s a locaton confguraton x such that: t t (1) π ( ) ( ) t ( ) π ( x t x, pb x, x ), pb ( x, x ) x C Φ and N. We follow a well-establshed tradton and confne our attenton to locaton confguratons n whch ncumbent frms are symmetrcally located on C. Once frms market areas have been determned by ths entry/prce process the denttes of the ncumbent frms are known and t becomes possble for them to coordnate ther prcng decsons to reflect the repeated nature of the prce game n whch they are now nvolved. We can relabel as t = 0 the begnnng of the post-entry repeated prce game. Gven the equlbrum t ( x pb ( x) ), dentfed by (11) and (1), t s always possble to fnd a set of prces ~ { ~ } ( b ) t t t N such that π ( x, ~ p) π x, p ( x) (13) P t* p = p for. MacLeod et al. (1987) show that the prcng strategy: τ τ ( x ) f ~ { x x} or { x = x and p p for some 0 τ t 1} p = ~ t p otherwse t t t b s a subgame perfect equlbrum prcng strategy provded that: t t t (14) ( ) ( ) ( b ) ( ) ( p ) π x, p ~ δπ x, p x 1 δ max π x,, p ~. p Equaton (14) ensures that devaton from the prces ~ p s not proftable for any ncumbent frm, whle the ntal entry process and the prcng strategy (13) guarantee that new entrants wll not be attracted to the market. MacLeod et al. do not dentfy what the prces ~ p mght be and t s to ths queston that we now turn. 4. Prce Equlbrum n the Repeated Prce Game In determnng the prces ~ p we need to dentfy frst, the shape of the frms delvered prce schedules and secondly, ther one-perod proft-maxmzng devaton from these prces. We smplfy the analyss by assumng that at the entry stage each frm s commtted to a partcular prcng polcy. In the non-dscrmnatory case, ths mples that all frms are commtted to mll prcng whether or not they abde by the agreed prces. 7 If the frms adopt dscrmnatory prcng, the post-entry prce game s played non-cooperatvely untl the entry process has been completed. 7 Ths mples that wth mll prcng the consumers control collecton of the goods. In the product dfferentaton analogy, t mples that the devatng frm s commtted at the entry stage to a specalzed technology that cannot produce a range of dfferentated products. 11

12 we assume that they are commtted to a unform delvered prcng polcy. 8 It should be clear that the optmal one-perod devaton from a tactly agreed unform delvered prce s a unform delvered prce. Assume for the moment that the market s of unt length 9 and that t contans n symmetrcally located frms, that no further entry s possble and that the consumer reservaton prce s suffcently hgh or the number of frms suffcently great that frms are n the compettve regons of ther demand curves (Salop 1979). For the moment we can suppose the no-entry condton to be determned ether exogenously by nsttutonal constrants or endogenously as the result of the free-entry process we have detaled n secton Mll Prcng The free-entry Bertrand-Nash equlbrum mll prces are: (15) m b = 1 n, = 1...n. After the entry stage has been completed we assume that the frms reach a tact agreement on a set of mll prces m ~ a = m ~ n, = 1...n, wth nstantaneous flow profts to each frm nclusve of fxed costs of: (16) π a = m ~ n = 1...n. We wsh to dentfy the hghest mll prces m a ( ) ~ δ whch can be an SPE for the dscount factor δ. As we have noted, these prces must exactly balance the one-perod rewards from cheatng on the agreed prces aganst the long-term losses when such devaton s punshed. The punshment phase s easly defned: we assume that frms revert to the one-shot Bertrand-Nash prces of equaton (15) forever. If frm decdes to devate from the tactly agreed prces more complex consderatons arse. In partcular, frm must decde how many of ts neghborng frms t wll undercut (stealng ther entre markets). It s convenent to use the followng defnton. 8 Ths s not unreasonable gven our assumpton that demand s perfectly nelastc for all prces less than the reservaton prce v. In the producton dfferentaton analog, the unform prcng assumpton mples that all frms are commtted to chargng the same prce for all of the product varants they offer. It s nterestng to note that Ford Motor Company, for example, s ncreasngly employng ths type of prcng polcy over parts of the product spectrum that they feel to be reasonably dstnct. 9 Ths normalzaton does not affect the analyss so long as the consumer reservaton prce s non-bndng. We shall relax ths assumpton n secton 4. when we consder the effects of a bndng reservaton prce. 1

13 (D) Let h(δ) = δ 1 + 3δ 3δ ( δ) and defne j [δ] to be the smallest nteger j greater than h(δ). Note that j [δ] s ncreasng n δ. We can show (see the Mathematcal Appendx) that f frm devates from the agreed prces t wll do so n such a way that t undercuts exactly j [δ] neghborng frms on each sde. Frm wll not, however, set prces such that t undercuts j [δ] neghbors and takes part of the markets of the (j [δ] + 1)th neghbors. We then have the followng: Proposton 1: 10 The hghest mll prce that can be an SPE for the repeated prce game when the market contans n symmetrcally located frms, no further entry s possble and the dscount factor s δ s: ~ (17) m a ( δ) ( 1 + j [ ] + j [ ] ) j [ ] ( 1 + j [ ] ) 1 δ δ δ δ δ 1 = mn v, n δ( 1+ j [ δ] ) j [ δ] n where j [δ] s defned by (D) and s such that 1 < j [δ] < n/. It s easy to show, as we would expect, 11 a that m ( ) ~ δ δ > 0. The upper lmt on the SPE mll prce s an ncreasng functon of the dscount factor, or equvalently, of the degree of prce flexblty. 4. Dscrmnatory Prcng We know 1 that provded no frm has monopoly power n any segment of ts market area the Bertrand-Nash equlbrum dscrmnatory prcng schedule for each entrant s: b (18) p ( r x ) = max x r, mn x r j j, = 1...n. Now assume that after the entry stage has been completed the frms tactly agree each to charge a unform delvered prce u ~. As wth dscrmnatory prcng, we wsh to dentfy the hghest unform Proofs of all propostons are gven n the Mathematcal Appendx. Ths s most easly done numercally snce, for any gven n, the mll prce s a functon only of δ. See, for example, Gee (1976), Lederer and Hurter (1986). 13

14 ~ delvered prces u a ( ) δ that can be an SPE for the dscount factor δ. Once agan, these have to balance reward and punshment, where now equaton (18) gves the prces that wll apply forever n the event of cheatng. Wth dscrmnatory prcng t s easy to dentfy the best possble devaton for frm : prce should be reduced margnally below the tactly agreed unform delvered prce ~u a ( δ ), gvng the cheatng frm a market area of ~ a ( ) ~ a u δ s u ( δ ) = 1. We then have: = wth our normalzaton s Proposton : The hghest unform delvered prce that can be an SPE when the market contans n symmetrcally located frms, no further entry s possble and the dscount factor s δ > 0.5 s: ~ (19) u a ( δ) δ( δ ) ( δ) = mn v, 1 1. n a As wth the mll prce, ~u ( δ) δ > 0 : the upper lmt on the unform delvered prce s an ncreasng functon of the dscount factor (or the degree of prce flexblty). 5. The Welfare Effects of Deregulatng Frms Prcng Polces The analyss of the prevous secton dentfes upper lmts on the prces that frms can charge but we know from the Folk Theorem that any prces between these upper lmts and the Bertrand-Nash prces can be sustaned as equlbrum prces for the relevant dscount factor δ. Ths makes welfare comparson of the alternatve prcng polces mpossble wthout the mposton of some further structure on the model. The approach we adopt s to assume that selfnterest among the ncumbent frms leads them to settle on the hghest possble prces that can be charged consstent wth there beng no ncentve to devate. In other words, for the remander of ths paper, we assume that equatons (17) and (19) descrbe the actual prces that wll be charged by the ncumbent frms after the entry stage has been completed. A very smple result characterzes the comparson of dscrmnatory and mll prces for the one-shot prce subgame, as llustrated by Fgure 1. If there are n actve frms wth each prcng polcy as n Fgure 1(a) no consumer loses from dscrmnatory prcng. By contrast, f there are n 14

15 actve frms wth mll prcng but only n/ wth dscrmnatory prcng as n Fgure 1(b) no consumer gans from dscrmnatory prcng. (Fgure 1 near here) It turns out that a smlar property holds for the SPE prces of equatons (17) and (19). Proposton 3: Assume that the reservaton prce s not bndng and that there are n actve frms wth mll prcng. u~ < m~ n : If there are also n actve frms wth dscrmnatory a a () ( δ) ( δ) n= n n= n prcng, all consumers gan from dscrmnatory prcng. () If no consumer s to gan from dscrmnatory prcng, there wll have to be no more than n / frms wth dscrmnatory prcng. These are actually stronger results than for the one-shot prce subgame. It would appear that the advantage to consumers of dscrmnatory prcng relatve to mll prcng s even greater wth the repeated prce game. Ths mght on frst sght seem surprsng snce, as Fgure 1(a) makes clear, the punshment phase wth dscrmnatory prcng s much more severe than wth mll prcng. The reason s to be found n the very dfferent rewards that accrue to prce-undercuttng wth the two prcng polces. When frms employ mll prcng, a frm that wshes to cut prce to one set of consumers must also offer reduced prces to all of ts exstng customers, dsspatng at least some of the benefts of the reduced prces. A prce-cuttng frm suffers no such loss of revenue wth dscrmnatory prcng snce prces to ts exstng customers are mantaned effectvely unchanged, strengthenng the temptaton to cut prces. The need to offset the oneperod gans to prce-cuttng mposes a suffcently strong constrant on the tactly agreed dscrmnatory prces as to lead to Proposton 3. In order to extend our comparsons of the two prcng polces we need to be more precse about the outcome of the entry stage detaled n secton 3. A key property that affects entry, market structure and so the effects of prce deregulaton s the degree of spatal contestablty. When ncumbents can relocate ther actvtes costlessly the ndustry s defned to be spatally contestable (SC). At the other extreme, f relocaton costs are prohbtve, makng locaton choce once-for-all, the ndustry s defned to be spatally non-contestable (SNC). 15

16 In a free-entry equlbrum wth SC each entrant frm just breaks even. We take as the freeentry SNC equlbrum the loosest packng of frms consstent wth there beng no further entry. At ths equlbrum the ncumbents fully explot the entry-deterrng advantage of beng commtted to ther locatons. As a result, the free entry number of entrants s: 13 (0) SC ( ) ; ( ) SC nd = f n f = f SNC 1 SNC nd = ( 8 f ) ; n f = / ( + 3) f. Snce n n > 1 we cannot conclude that the entry-deterrng effect of dscrmnatory prcng d f more than offsets ts pro-compettve benefts for consumers. We must, therefore, turn to more aggregate comparsons. In dong so, t turns out that we must also dstngush between cases n whch the reservaton prce s not bndng and those n whch t s. 5.1 Welfare Comparsons: Non-Bndng Consumer Reservaton Prce Throughout ths subsecton we shall assume that the consumer reservaton prce mposes no effectve constrant on frms prces. The complex nature of the prce equlbra makes analytcal results dffcult to derve. However, gven equatons (17), (19) and (0), set-up costs appear as a common multpler provded that the consumer reservaton prce s not bndng. As a result, all comparsons are determned by the dscount factor δ and numercal comparsons provde a complete descrpton of the relatve merts of the two prcng polces. Recall that δ = e -γρ, wth the result that, for a gven contnuous rate of tme dscount, δ wll be greater the faster the speed of response of ncumbents (the lower s γ). Fgure and Table 1 summarze the welfare effects of the two prcng polces. In Table 1 we also repeat our results for the one-shot prce subgame to facltate comparson. Consder, frst, the mpact of prcng polcy on delvered prces (Fgure (a)). If the market s SC every consumer s better off wth dscrmnatory prcng for any dscount factor greater than 0.5 and f the market s SNC every consumer s better off wth dscrmnatory prcng provded the dscount factor s greater than The benefts to consumers of the procompettve effects of dscrmnatory prces wth a gven market structure extend to endogenous 13 See Norman and Thsse (1996) for a detaled dscusson of these results. The subscrpt d denotes dscrmnatory prcng and the subscrpt f denotes mll (or fob) prcng. 16

17 market structures when prces are determned by equatons (17) and (19). Dscrmnatory prcng ncreases market concentraton and so ncreases prces (whch s why the dfference between the mll prce and unform delvered prce s lower when the market s SNC). However, ths s more than offset by the need to make prce-undercuttng unproftable whch, as we have already ndcated, mposes a stronger constrant on prcng when frms employ dscrmnatory prcng. The mpact of dscrmnatory prcng on aggregate consumer surplus s straghtforward. If frms total revenue s denoted TR(n) and aggregate consumer surplus s denoted CS(n) then t follows from our defnton of ndvdual consumer demand that CS(n) = v - TR(n). From Fgure (b) mll prcng generates hgher total revenues and so lower aggregate consumer surplus than dscrmnatory prcng no matter the degree of spatal contestablty. Ths s n contrast to the one-shot prce subgame, n whch mll prcng benefts consumers f the market s SNC. When the prce subgame s repeated, the ablty of frms to rase prces when they employ a mll prcng polcy s suffcently strengthened that, no matter the degree of spatal contestablty, consumers unambguously beneft from dscrmnatory prces. Now consder the mpact of prcng polcy on proftablty (Fgure (c)). If the market s SC (SNC) mll (dscrmnatory) prcng gves greater supernormal profts to the ndvdual frm. 14 Indvdual frm proftablty wth the two prcng polces s determned by the nterplay between the prce-reducng and entry-deterrng effects of allowng frms to employ a dscrmnatory prcng polcy. If the market s SC the prce-reducng effect s domnant. By contrast, f the market s SNC equatons (0) ndcate that the entry-deterrng effect of prce deregulaton s strengthened. Ths s suffcent, at the ndvdual frm level, to offset the proft-reducng effects of fercer prce competton. (Fgure and Table 1 near here) Such ambguty does not characterze aggregate profts: Fgure (d). Aggregate proft s greater wth mll prcng than dscrmnatory prcng no matter the degree of spatal contestablty. Ths s n sharp contrast to our analyss of SNC wth a one-shot prce subgame: see Table 1. The dfference derves once agan from our dscusson of Proposton 3 n whch we saw 14 There s a slght excepton to ths. Wth SC the dfference n profts between mll and dscrmnatory prcng s a seres of quadratcs, one for each value of j. Each such quadratc has a turnng pont at the dscount factor δ(j) = 4j /(4j + 4j -1) at whch ndvdual profts are equal for the two types of frm. 17

18 that the compettve dsadvantage of dscrmnatory prcng s strengthened when the prce subgame s repeated. The comparson of total surplus s just as n Norman and Thsse (1996). If total costs are TC(n) then total surplus s v - TC(n). Snce the entry process we have assumed s ndependent of the nature of the post-entry prce subgame, total costs are also ndependent of the nature of the post-entry prce subgame. If the market s SNC (SC) mll prcng wll generate hgher (lower) total surplus than dscrmnatory prcng. To summarze, when the prce subgame s repeated and the consumer reservaton prce s non-bndng, dscrmnatory prces beneft consumers almost always ndvdually and always n the aggregate whle mll prces beneft producers n the aggregate (and ndvdually f the market s SC). Although the mpact of dscrmnatory prces on aggregate surplus s determned by the degree of spatal contestablty when consumer surplus and producer surplus are weghted equally, t s clear that there wll be some addtonal weghtng of consumer surplus above whch regulators should unambguously favor dscrmnatory prces. It s mportant to note, however, that our results n ths subsecton are based upon the assumpton that the consumer reservaton prce mposes no effectve lmt on frms choce of prces as gven by equatons (17) and (19). We now examne the effect of relaxng ths assumpton. 5. Welfare Comparsons: Bndng Consumer Reservaton Prce It wll emerge below that f the consumer reservaton prce s bndng, welfare comparsons of the two prcng polces wll be affected by market length. As a result, assume that the market s of length Λ. Gven the entry process descrbed n secton 3, the free-entry number of entrants s now equal to: (0a) SC ( Λ ) ; ( Λ ) SC 1 1 nd = f n f = f SNC 1 SNC 3Λ / nd = ( Λ 8 f ) ; n f =. ( + ) f 3 We know that the hghest mll prce any frm wll wsh to charge s v - Λ/n and that the hghest unform delvered prce that can be charged s v. Also, an mportant mplcaton of subsecton 5.1 s that, no matter the degree of spatal contestablty, the dscount factor δ f above 18

19 whch the consumer reservaton prce s bndng s lower wth mll prcng than wth dscrmnatory prcng. We take advantage of these two propertes n ths secton. Our dscusson turns the analyss of the prevous secton on ts head to an extent by seekng to answer the followng questons: () Under what crcumstances wll all consumers be worse off wth dscrmnatory prcng as compared to mll prcng and how s ths affected by the degree of spatal contestablty? For ths to be the case t must be that the unform delvered prce s exactly equal to the consumer reservaton prce v, snce we know that the maxmum delvered prce wth mll prcng s v and ths apples to a zero measure of consumers. So what does the crtcal dscount factor δˆ d a have to be for a unform delvered prce u ( ) v ~ δˆ to be sustanable as an SPE? We can then d = show how ths crtcal dscount factor s affected by market sze, the consumer reservaton prce and the frms set-up costs. () Under what crcumstances wll consumers on average be worse off wth dscrmnatory prcng and how s ths affected by the degree of spatal contestablty? For ths to be the case the unform delvered prce must be greater than v - Λ/4n. So above what dscount factor wll a unform delvered prce of v - Λ/4n be sustanable as an SPE? Agan, we can nvestgate how s ths dscount factor s affected by market sze, the consumer reservaton prce and the frms set-up costs. () Under what crcumstances wll frms ether ndvdually or on aggregate beneft from prce deregulaton and how s ths affected by the degree of spatal contestablty? The approach we adopt s straghtforward. Equaton (19) dentfes the maxmum unform delvered prce that s sustanable as an SPE for a gven dscount factor. Invertng ths equaton, therefore, gves us the lowest dscount factor for whch a partcular unform delvered prce s sustanable as an SPE. Assume that the unform delvered prce s κ/n < v for some parameter κ > 1. From equaton (19), the dscount factor ( κ) ( δ d ( κ )) equals κ n s: ~ u a (1) ( κ 1) δ ( κ) κ. κ. d = 0 5 for 1 1 ( ) δ d for whch the unform delvered prce 19

20 It follows mmedately from (1) that ( ) δ d ( κ) s ncreasng n κ. δ κ κ 0. In other words, the crtcal dscount factor d Consder frst the crtcal dscount factor above whch all consumers wll be worse off wth ( d ) ~ a dscrmnatory prcng. We have already noted that for ths to be the case ( ) so that κ 1 = v.n, and from equaton (0a) we have: SC Λ SNC v Λ () κ 1 = v > κ 1 = u δ κ = κ n = v, 1 1 where v = v f. When the consumer reservaton prce mposes an effectve constrant on frms mll prces, the crtcal dscount factor above whch all consumers are worse off wth dscrmnatory prcng s lower when the market s SNC than when t s SC. For consumers n the aggregate to lose from dscrmnatory prcng t must be the case ( d ) ~ a that u δ ( κ) κ n = v Λ n, whch gves a lower lmt on κ of κ = v. n Λ 4. 4 Substtutng from equaton (0a) gves: SC Λ Λ SNC v Λ Λ (3) κ = v > κ = 4 4 and we obtan the same qualtatve concluson. Note further that κ v > 0 and κ Λ > 0 ( = 1,) 15 mplyng that when mll prces are constraned by the reservaton prce, consumers ndvdually and n the aggregate are more lkely to lose from dscrmnatory prcng f the market s small, the reservaton prce s low and/or frms set-up costs are hgh. For frms ndvdually to beneft from dscrmnatory prcng the lower lmt on κ s such that (see the Mathematcal Appendx): (4) κ κ SC 3 SNC 3 = 0. 5v Λ = v Λ SC κ > κ SNC 3 3 When mll prces are constraned by the reservaton prce, the crtcal dscount factor above whch frms ndvdually gan from dscrmnatory prcng s lower when the market s SNC than. 0

21 when t s SC. By the same argument as n footnote 14, we have κ v > 0 and κ Λ > 0, 3 3 mplyng that when mll prces are constraned by the reservaton prce, frms ndvdually are more lkely to gan from dscrmnatory prcng f the market s small, the reservaton prce s low and/or frms set-up costs are hgh. Fnally, we can show that for frms n the aggregate to gan from dscrmnatory prcng the lower lmt on κ s: (5) κ κ SC 4 SNC 4 = v Λ Λ = v Λ Λ SC κ > κ SNC 4 4 When mll prces are constraned by the reservaton prce, the crtcal dscount factor above whch frms wll gan from dscrmnatory prcng s lower when the market s SNC than when t s SC. Once agan, we have κ v > 0 and κ Λ > 0, mplyng that when mll prces are 4 4 constraned by the reservaton prce, frms n the aggregate are more lkely to gan from dscrmnatory prcng f the market s small, the reservaton prce s low and/or frms setup costs are hgh. These results take us back more nearly to those presented n Norman and Thsse for the one-shot prce subgame. When mll prces are constraned by the consumer reservaton prce, there wll be a degree of prce flexblty above whch consumers wll lose and producers wll beneft from dscrmnatory prcng. The degree of prce flexblty necessary for ths to be the case s lower f the market s SNC than f t s SC, reflectng the stronger entry-deterrng nature of SNC. We are also brought back to our queston n the ntroducton of whether there s a potental role for the regulatory authortes n nfluencng the degree of prce flexblty. Equatons (17) and (19) ndcate that, no matter the allowable prcng polcy, prces can be expected to be hgher the more flexble are frms' prcng decsons: snce a hgh degree of prce flexblty translates to a hgh dscount factor. We have shown that when the consumer reservaton prce does not mpose an effectve constrant on frms, consumers always beneft from prce deregulaton. We now see that these benefts wll be even greater f prce flexblty can be. 15 > Λ. It must be the case that v > Λ/n and we know from (0a) that n = (Λ/ηf) where η > 1. It follows that v/ f 1

22 reduced. When the consumer reservaton prce does mpose an effectve constrant, our analyss ndcates that consumers wll not gan from prce deregulaton, partcularly n more concentrated markets, unless the degree of prce flexblty s suffcently low, where the defnton of "suffcent" can be mpled from our formal analyss. It would appear, therefore, that there are, ndeed, consumer benefts to be had f prce deregulaton s accompaned by polces that reduce prce flexblty, perhaps by the regulator mposng some type of mnmum perod wthn whch prces cannot be changed. The ntuton behnd ths s famlar from the Folk Theorem. Low prce flexblty reduces the effectveness of the punshment phase that sustans the tactly collusve prces above the one-shot Bertrand-Nash level and so lowers the maxmum sustanable prces. 6. Conclusons In ths paper we have extended our earler analyss of the relatve merts of soft and tough prce regmes to the stuaton n whch post-entry prce equlbra reflect the repeated nature of the post-entry prce subgame. We derve welfare conclusons that contrast reasonably sharply wth our one-shot analyss. If the constrant mplct n the consumer reservaton prce s non-bndng then aggregate consumer surplus s greater and aggregate profts are lower wth dscrmnatory prcng than wth mll prcng no matter the degree of spatal contestablty of the market. In addton, provded that prce decsons are suffcently flexble (δ > 0.601) every consumer wll beneft from prce dscrmnaton. The entry-deterrng effect of dscrmnatory prcng s more than offset by ts pro-compettve effect. The reason for ths contrast les n the very dfferent constrants the two prcng polces mpose on frms abltes to rase ther prces above the Bertrand-Nash prces. Such constrants exst snce any set of tactly agreed prces must be senstve to the temptaton each frm has to cheat on these prces and so must be supported by some credble punshment strategy n the event that cheatng occurs. The punshment strategy on whch we have focused n ths paper s the grm trgger strategy n whch, f devaton occurs, frms move to the one-shot Bertrand-Nash prces forever. Gven the relatvely aggressve entry process we have proposed, we know from our prevous analyss that at these prces frms ndvdually and n the aggregate earn at least as great profts wth dscrmnatory prcng as wth mll prcng. Wth a mll prcng polcy any attempt by one frm to undercut ts rvals requres that the frm also offers the lower prces to all of ts

23 exstng customers. There s no such requrement wth dscrmnatory prcng as a result of whch the gans from cheatng are greater wth dscrmnatory prcng than wth mll prcng. Ths combnaton of weaker punshment of and greater gans to cheatng wth a dscrmnatory prcng polcy leads to the consumer gans and producer losses we have noted. If our analyss were to stop here t would appear that we could be much more sangune about the benefts of deregulaton of frms prcng polces even f ths were to affect market structure n the ways we have suggested. Our analyss has also shown, however, that the consumer reservaton prce has an mportant part to play n determnng the welfare propertes of alternatve prcng polces. If the reservaton prce effectvely constrans mll prces then there exsts a degree of prce flexblty above whch our welfare conclusons are reversed, wth frms ganng and consumers losng from dscrmnatory prcng. The degree of prce flexblty above whch ths wll apply s greater when the market s SNC; when market sze s small; when the consumer reservaton prce s low; and when frms set-up costs are hgh. Each of these condtons mples that there are less lkely to be benefts to consumers from prce deregulaton n markets already characterzed by hgh levels of market concentraton. They further mply a role for the regulatory authortes that has, so far as we are aware, not been consdered n the lterature, prmarly because most of the pre-exstng analyss s statc rather than dynamc. We have shown that the dscount factor has a crucal role to play n determnng the benefts that consumers are lkely to derve from prce deregulaton. Smply put, consumers are lkely to gan more from prce deregulaton f the dscount factor s low. The dscount factor s tself determned by the (contnuous) rate of tme dscount, whch s generally outsde the control of the regulatory authortes, and the speed wth whch frms can react to devaton from the tact agreed prces, whch s concevably wthn the control of the authortes. Our analyss ndcates that prce deregulaton s more lkely to be successful f t s accompaned by polces that mpose a perod over whch prces cannot be changed. Interestngly, ths s especally lkely to be the case n concentrated markets and these are the markets n whch response speeds mght be expected to be greatest. 3

24 Table 1: Welfare Comparsons Repeated Prce Game a Spatal Contestablty Spatal Non-Contestablty a a Prces m ~ ( δ) > u~ ( δ) f δ > 0.5 a ( ) a m δ > u~ ( δ ) ~ f δ > Consumer Surplus CS f > CS d CS f > CS d Supernormal Proft π f > π d π f < π d Π f > Π d Π f > Π d Total Surplus TS f < TS d TS f > TS d One-Shot Prce Game Consumer Surplus CS f < CS d CS f > CS d Supernormal Proft -- π f < π d Π f < Π d Total Surplus TS f < TS d TS f > TS d Notes: a The welfare comparsons relate to cases n whch the consumer reservaton prce s non-bndng. 4

25 (a) n Frms wth Each Prcng Polcy Prce 1 3 dscrmnatory prce mll prce (b) n Frms wth Mll Prcng; n/ Frms wth Dscrmnatory Prcng Prce 1 3 dscrmnatory prce mll prce Fgure 1: Comparson of Prce Equlbra 5

26 Spatal Contestablty Spatal Non-Contestablty (a) Dfference between Mll Prce and Unform Delvered Prce (b) Dfference n Total Revenue (c) Dfference n Indvdual Frm Profts 6