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1 Delivered Pricing, FOB Pricing, and Collusion in Spaial Markes Auhor(s): Maria Paz Espinosa Source: The RAND Journal of Economics, Vol. 23, No. 1, (Spring, 1992), pp Published by: Blackwell Publishing on behalf of The RAND Corporaion Sable URL: hp:// Accessed: 24/04/ :48 Your use of he JSTOR archive indicaes your accepance of JSTOR's Terms and Condiions of Use, available a hp:// JSTOR's Terms and Condiions of Use provides, in par, ha unless you have obained prior permission, you may no download an enire issue of a journal or muliple copies of aricles, and you may use conen in he JSTOR archive only for your personal, non-commercial use. Please conac he publisher regarding any furher use of his work. Publisher conac informaion may be obained a hp:// Each copy of any par of a JSTOR ransmission mus conain he same copyrigh noice ha appears on he screen or prined page of such ransmission. JSTOR is a no-for-profi organizaion founded in 1995 o build rused digial archives for scholarship. We enable he scholarly communiy o preserve heir work and he maerials hey rely upon, and o build a common research plaform ha promoes he discovery and use of hese resources. For more informaion abou JSTOR, please conac hp://

2 RAND Journal of Economics Vol. 23, No. 1, Spring 1992 Delivered pricing, FOB pricing, and collusion in spaial markes Maria Paz Espinosa* The aricle examines price discriminaion and collusion in spaial markes. The problem is analyzed in he conex of a repeaed duopoly game. I conclude ha he prevailing pricing sysems depend on he srucural elemens of he marke. Delivered pricing sysems emerge in equilibrium in highly monopolisic and highly compeiive indusries, while FOB is used in inermediae marke srucures. Thefac driving his resul is ha delivered pricing policies allow spaial price discriminaion ha faciliaes collusion, bu a he same ime hey have a very compeiive feaure: hey are he only pricing rules ha could be susained in a very compeiive marke srucure. 1. Inroducion * In some markes, sellers have he abiliy o discriminae among consumers by making price vary according o some characerisic of he buyer. Here I analyze a paricular ype of price discriminaion common in spaial markes where a consumer's locaion is observable. In his siuaion he choice variable for he firm is a price sysem ha specifies a price per uni of produc a each locaion: each firm mus choose a funcion p(x), where x is he disance beween he locaion of he consumer and he locaion of he firm. In pracice, differen indusries use differen ypes of pricing sysems, p(x), and his suggess ha he prevailing pricing sysem depends on he srucural elemens of he marke. When he pricing policy is FOB,' consumers can pick up he produc a he mill, paying he mill price p and incurring he ransporaion cos from he producer's o he consumer's locaion, i.e., p(x) = p + (x), or he seller may deliver he good o he buyer's locaion, as long as i charges mill price plus ransporaion coss. Delivered pricing policies are pricing rules p(x) ha are no based on consumers picking up he produc a he mill; he firm delivers he produc a he consumer's locaion. In a perfecly compeiive world wih a coninuum of firms, an FOB pricing sysem would be expeced: p(x) = c + (x), where c is he marginal cos of producion. In he case of a marke wih wo firms locaed a he * Universidad del Pais Vasco, Bilbao, Spain. This aricle is based on Chaper 1 of my Ph.D. hesis. I would like o hank Ramon Caminal, Richard Caves, Andreu Mas-Colell, Garh Saloner, and wo anonymous referees for heir commens. I am especially graeful o Michael Whinson for many helpful suggesions and advice. Needless o say, any remaining errors and shorcomings are mine. Financial suppor from Harvard Universiy Social Science Disseraion Fellowship is graefully acknowledged. ' Free on board. 64

3 ESPINOSA / 65 same poin, assuming here is no collusion, we would also expec p(x) = c + (x) (he Berrand soluion). A monopolis, however, could profi from price discriminaion and herefore would no use FOB. This associaion beween FOB and compeiion is par of he reason why oher pricing sysems have been labelled as "monopolisic" pracices and devices ha faciliae collusion.2 Several argumens have been given ha associae delivered pricing3 wih collusive pracices.4 Some of hem relae o incomplee informaion and he problem of deecion. Under an FOB pricing sysem firms could disguise price cus as lower ransporaion coss, while in a delivered pricing sysem his problem could no arise, since he implici agreemen is on final prices including freigh charges. This argumen does no seem o be a srong suppor for he idea of delivered pricing sysems as collusive devices, since i relies on he unlikely assumpion ha firms observe he price o he consumer in a delivered pricing sysem bu no when FOB is used.5 Sigler (1964) holds ha idenical delivered pricing sysems would make collusion easier in markes wih geographically unsable demand, by allowing firms in a erriory wih low demand o invade erriories wih high demand. This invasion would eliminae he need for frequen changes in he price schedule and make any collusive price more sable. I shall argue ha in an oligopolisic marke, when firms are spaially dispersed a delivered pricing sysem allows each seller o discriminae among consumers, charging a higher ne price (price minus cos) o consumers wih higher ne reservaion value (reservaion value minus cos) for he produc of ha paricular seller.6 This increases profis for he seller in he same way sandard price discriminaion increases he profis of a monopolis. On he conrary, an FOB pricing sysem enails he same ne price for all consumers, no maer heir locaion, and herefore i eliminaes he possibiliy of price discriminaion. Thus, here is a sense in which delivered pricing sysems can be considered "more collusive" han FOB. Since hey increase he profis o be made by colluding, hey ease collusion. The view ha delivered pricing policies are monopolisic pracices has been widely mainained (by Feer (1937), Machlup (1949), and Scherer (1980), among ohers), and i is also suppored by casual empirical evidence indicaing ha many indusries characerized by high concenraion and a spaially differeniaed produc (for example, cemen, seel, and corn in he Unied Saes) did use delivered pricing sysems.7 Sigler (1949) provides examples of indusries where carelizaion coincided wih he inroducion of delivered pricing sysems. The German cemen indusry was on an FOB price sysem unil a carel was formed, afer which i used delivered prices. The biuminous coal indusry in Grea Briain was using FOB prices before he compulsory carelizaion of 1930, and moved o a delivered price sysem afer ha dae. 2 "[TI]he delivered-price sysem as here used provides an effecive insrumen which, if lef free for use of he respondens, would resul in complee desrucion of compeiion and he esablishmen of monopoly in he cemen indusry" (Mr. Jusice Black, Federal Trade Commission v. Cemen Insiue e al., 333 U.S. 683 (1948)). 3There is a variey of delivered pricing policies. In an idenical delivered pricing sysem, all he firms quoe he same price o any one buyer. The mos common idenical delivered pricing rules are basing-poin pricing and uniform delivered pricing. In a uniform delivered pricing sysem, each firm quoes he same price o all he consumers. Under a basing-poin pricing rule, firms decide on he locaion of a base poin and a price a ha locaion; he price a any oher locaion is calculaed as he base price plus a ransporaion charge from he base poin. 4 Scherer ( 1980) argues ha delivered pricing sysems lessen compeiion because hey reduce a "complicaed price quoaion problem, if execued independenly, o a relaively simple maer of applying he righ formula." FOB mill pricing would make he "avoidance of independen pricing more difficul." See also Carlon ( 1983). 5 Carlon ( 1983) argues ha when firms do no observe prices and use shifs in business o deec deviaions, FOB would be a beer collusive device. 6 Cos includes ransporaion cos, so ha for a consumer, he ne reservaion value is differen depending on which seller he is considering buying from. 'See Greenhu, Greenhu, and Li ( 1980).

4 66 / THE RAND JOURNAL OF ECONOMICS The view of he Federal Trade Commission is ha delivered pricing sysems lessen compeiion. In Boise Cascade v. FTC ( 1980), he FTC challenged he pracice of a basingpoin pricing sysem in which ransporaion charges were always calculaed from he wes coas, regardless of he produc's origin. In anoher case, Ehyl e al. ( 1981), he FTC challenged he use of uniform delivered prices in which he price charged is independen of he buyer's locaion. However, idenical delivered pricing sysems have a very compeiive feaure. Given ha in an idenical delivered pricing sysem all firms are charging he same price a a given locaion, if a firm slighly decreased is price i would ge he whole marke, or a leas ha par of he marke ha i is profiable o sell o; whereas in an FOB pricing sysem, a small decrease on he price a he mill ges only he marginal consumers for he firm ha lowered is price, and o increase is marke share significanly a firm needs o decrease is price by a considerable amoun. In his sense, idenical delivered pricing sysems are a mehod of increasing he degree of compeiion hrough inerpeneraion of regional markes.8 This argumen seems o indicae ha a carel rying o enforce collusion will ry o avoid idenical delivered pricing sysems, and ha hese pricing sysems canno be considered as collusive pracices. And in fac here mus be oher reasons ha explain he use of delivered pricing, since in many highly compeiive indusries, such as reail drugsores, pizza deliveries, food reailing, furniure sores, and mail-order reailing, uniform delivered pricing (UDP), also called he posage samp sysem, is quie prevalen. The purpose of he aricle is o reconcile he wo argumens and explain why delivered pricing sysems (in paricular, UDP and basing-poin pricing) are used in highly monopolisic and in highly compeiive indusries. I find ha, under some resricions on he pricing rules, UDP appears as he equilibrium pricing policy in collusive and compeiive indusries, while FOB is he equilibrium policy for inermediae marke srucures. Wih no resricions on he pricing rules, I obain ha basing-poin pricing emerges in collusive equilibria bu is also he only pricing sysem ha could be susained in a very compeiive marke srucure; moreover, basing-poin pricing appears as he pricing rule in he opimal punishmen pah susaining he collusive equilibria. The aricle is organized as follows. In Secion 2, a repeaed duopoly game is used o compare UDP and FOB pricing sysems. Markes are characerized by wo parameer values: he discoun facor 6 and he ransporaion cos per uni value of he commodiies.9 For a given discoun facor, a higher ransporaion cos implies higher marke power for each seller in is local markes, and his makes possible more collusive oucomes. Noe ha a lower ransporaion cos means a more elasic demand funcion, and his we associae wih a more compeiive indusry. On he oher hand, for a given ransporaion cos, a higher discoun facor also implies ha more collusive oucomes are possible in equilibrium. I shall refer o a marke as "compeiive" if i has low ransporaion coss and a low discoun facor, and a marke will be said o be "monopolisic" when i has a high discoun facor and/or high ransporaion coss. For 6 = 0, (x) = 0, we ge he mos compeiive resul: price is equal o marginal cos and profis are zero (Berrand compeiion). For 6 = 1 we can ge he monopoly soluion, and for (x) high enough we also ge he monopoly soluion. The idea of compeiiveness is also usually relaed o he number of firms in he marke; however, our ineres is in indusries where, due o fixed coss for example, here is room for only a few firms. When he number of firms is no a variable, he degree of compeiiveness is bes measured by he values of 6 and. I sar by looking a he simples case in which only UDP and FOB are feasible (since here are coss associaed wih he implemenaion 8 Scherer ( 1980) expresses he belief ha his greaer inerfirm conac leads mainly o more inense nonprice rivalry, bu no o price cus as a compeiive weapon. 9 The ransporaion cos is assumed o be linear: (x) = x.

5 ESPINOSA / 67 of a complicaed pricing rule, firms may choose pricing sysems wih very few parameers). The resul is ha UDP will be used in very compeiive and also in very monopolisic markes. For marke srucures ha are neiher very monopolisic nor very compeiive, he model predics FOB. In Secion 3 I generalize he resul o linear pricing schedules. The main conclusion of Secion 2 sill holds: UDP will be used in very monopolisic and in very compeiive marke srucures. For marke srucures ha are neiher very compeiive nor very monopolisic, he model predics pricing sysems wih slope in he inerval [0, I, and he value of he slope is nonincreasing in (. This resul is consisen wih he empirical findings of Greenhu, Greenhu, and Li ( 1980), who find ha he more compeiive he marke is, he seeper is he delivered price schedule. I also obain he resul ha when firms are able o implemen marke share agreemens, he pricing sysem has slope in he inerval [-, 0 ], and he slope is nondecreasing in (. I find his resul o be consisen wih exising empirical evidence: in Japan and Wes Germany, counries where coordinaion on marke share is no as difficul as in he Unied Saes, delivered pricing policies end o have negaive slopes (see Greenhu (198 1 )), while for American firms he slope is posiive. Finally, Secion 4 considers he case of unresriced pricing policies. I is shown ha o susain any FOB price, he marke mus be monopolisic. If he possibiliies for collusion are limied, hen FOB will no emerge as he equilibrium of he game. This resul srongly conradics he idea of FOB as a compeiive pricing sysem. I shows ha, even hough FOB is he oucome of a perfecly compeiive marke wih a coninuum of firms, or a marke where he producs are no geographically differeniaed, his idea canno be carried over o markes where only a few firms are presen and where producs are spaially differeniaed. In his ype of marke, and for values of 6 such ha he monopoly soluion is no susainable, basing-poin pricing and, in general, idenical delivered pricing rules appear as he equilibrium policies. 2. FOB and UDP: compeiive versus collusive heories * The debae on he compeiive or collusive naure of delivered pricing has cenered on he comparison of very simple pricing rules and always has FOB as he reference poin. A reason why firms may be resriced o simple pricing rules (rules wih only a few parameers o opimize over) is ha here are coss associaed wih he implemenaion of a complicaed pricing rule. Since explici communicaion beween firms for price seing is illegal and he agreemens are implici, he greaer he number of parameers he firms have o agree on, he more difficul i will be o susain collusion; even in he absence of collusion, i may be cosly for a firm o implemen a complex pricing rule. A differen explanaion would be ha simple rules may be opimal. I sar by comparing UDP and FOB pricing rules. Laer on I shall analyze he case of more general pricing sysems. o A repeaed duopoly game. Since our main concern is o deermine he relaionship beween collusion among firms and he spaial pricing sysems ha hey use, we need a model ha allows for repeaed ineracion and in which producs are spaially differeniaed. In his secion I develop a simple model wih hese feaures. There are wo firms ha produce a homogeneous good. They are locaed a he exreme poins of he inerval [0, 1]. There are no coss of producion, and he ransporaion cos is equal o per uni of disance. I assume ha he firms inerac repeaedly wih an infinie horizon and maximize discouned profis. The discoun facor is denoed 6, 6 E [0, 1). Consumers are uniformly disribued wih a uni densiy along he inerval [0, 1]. Their preferences are as follows: each consumer has a reservaion value R for he good and consumes precisely one uni of he good per period of ime, buying from he firm ha has he lowes final (delivered) price, as long as he oal paymen does no exceed he reservaion

6 68 / THE RAND JOURNAL OF ECONOMICS value, and buys nohing oherwise. When he wo firms quoe he same delivered price a a given locaion, he consumer chooses he neares supplier. The good canno be sored, bu in a given period consumers may buy he produc and resell i for a profi o oher consumers. In each period, firms simulaneously announce a price funcion. The delivered price ha corresponds o FOB is p(x) = PC + x for all x E [0, 1]. Analogously, for UDP, p(x) = Pu for all x E [0, 1]. G( UDP, FOB) denoes he repeaed game in which firms are allowed o use eiher a UDP or an FOB pricing sysem. A UDP sraegy is a funcion ha selecs, for any hisory of play, a pair (Pu, a) E Go X [0, 1]. In a UDP pricing sysem a firm may no wan o sell o consumers locaed oo far away, since he firm has o pay he ransporaion coss and he price Pu may no be high enough o cover coss. Therefore, I assume ha firms may refuse o sell; ai denoes he fracion of he marke firm i is willing o sell o (whenever I do no specify he value of a i is supposed o be 1 ). An FOB sraegy is a funcion ha selecs, for any hisory of play, an elemen Pc E P. In an FOB pricing sysem firms are always willing o serve he enire marke, since he ransporaion coss are added on o he price.1 The payoff funcions of he firms in each period are given below. When boh firms use UDP sraegies, ll[(pui, ai), (puj, aj)] = (Pui - x)dx, where Z= 0 if pui > R min [1, ai] if Puj > Pui, Pui? R min [a, 1 - aj] if Puj <Pui, Pui? R min {ai, max [1/2, 1 - a]} if Puj = Pui, Pui? R. When boh firms use FOB sraegies, 1 (pc, Pc) Pci min [0 if Pci > PC1 + 1 ( c P if Pci < Pcj (1) Pci min Pcj pci+ (R pci)] if IPci -Pcj <?. When firm i uses a UDP sraegy and firm j uses an FOB sraegy, where ji ((Pui,?ai), Pcj) = (Pui - x)dx, (2) A max [min [ai 1 _ (PuiP)] 0] if p R 0 if pui > R I1j((Pui ai) Pc) j)= min{ I- max [min [i 1 - P ]?] Pcj This is rue as long as pc is greaer han or equal o marginal cos. Wih regard o equilibrium analysis, here is no loss of generaliy in resricing FOB prices o be greaer han marginal cos.

7 ESPINOSA / 69 Our focus will be on symmeric equilibria of G( UDP, FOB). I is worh noing ha when Pu? R is charged as he UDP price by boh firms, profis for each firm are rmin[ a,'/2 ] ll((pu, a), (Pu, a)) = J (Pu- x)dx, while when an FOB price of Pc is charged by boh firms, profis for each firm are { Pc. 2R- 2 2f Pc? 2 1c(PcPC)=l R-pC 2R- Pc if PC > 2 The explici consideraion of he repeaed naure of he game expands considerably he range of possible oucomes. In paricular, oucomes ha are more cooperaive han he saic soluion are aainable. Since our main concern is collusion, I focus on he opimal collusive equilibria of he game for given values of he marke parameers; i.e., among he symmeric equilibria, our ineres is in he maximal amoun of profis susainable and, mos imporanly, he form ha pricing sraegies ake in such equilibria. 0 Characerizaion of equilibria. In order o undersand he imporance of spaial differeniaion among firms, le us sar wih he case in which boh firms are locaed a he same poin in he inerval [0, Abreu ( 1988 ) has shown ha any given pah is susainable in a perfec equilibrium if and only if i can be susained by reversion, in case of a deviaion from ha pah, o a punishmen ha is he deviaor's wors possible perfec equilibrium. Therefore, knowing he wors perfec equilibrium for each player allows he characerizaion of all perfec equilibrium pahs. When here is no geographical differeniaion among he producs of he firms, seing an FOB mill price of zero repeaedly is a perfec equilibrium pah ha yields zero profis o boh players. Since in his model firms always have he opion o obain zero profis by no producing, he repeaed FOB soluion p(x) = (x) is he lowes-payoff perfec equilibrium pah. Any oher perfec equilibrium pah is susainable if and only if i can be susained by reversion o his FOB pah: if a player deviaes from he specified acions, he FOB policy p(x) = (x) sars he following period. Since he firms are locaed a he same poin, a deviaion consising of undercuing he opponen by some small amoun would yield wice as much profi as conforming o he acion ha gives he specified payoffs: li(p(x), p(x)). I follows ha o susain profis higher han he saic Nash equilibrium profis, 6 should be greaer han '/2: _ H(p(x), p(x)) > 2I1(p(x), p(x)) + 1 (30?6? V2. Moreover, for 6 2 1/2 he monopoly soluion is susainable. I is clear ha for a monopolis, he bes pricing policy would be a UDP p(x) = R.13 Since UDP exracs all he consumer surplus, no oher pricing sysem could give higher profis. Thus, when he producs are no spaially differeniaed, FOB appears as he equilibrium policy for compeiive markes, while UDP would require 6 2 1/2. Consider now he opposie case. Firms are locaed a he endpoins of he inerval [0, 1], and he ransporaion coss are very high. As a consequence, spaial differeniaion I Or he ransporaion cos is zero. 12 See Abreu ( 1988). 13 Noe ha his resul comes from our assumpions abou demand; when reservaion values are no idenical for all consumers, a monopolis who owns he wo plans may prefer a differen pricing sysem.

8 70 / THE RAND JOURNAL OF ECONOMICS is also high. In paricular, when > 2R, he ransporaion coss are so high ha boh firms are monopoliss in heir respecive local markes. Each firm has a fracion R of he marke and does no find i profiable o sell in he oher firm's local marke. There is no compeiive ineracion beween he firms. In his case, a UDP pricing sysem p(x) = R and any a E [1/2, 1] is he profi-maximizing sraegy for boh firms in G(UDP, FOB) for any value of a. Define w =. In he remainder of his secion he symmeric equilibria of G( UDP, FOB) are characerized for he case w? 1; ha is, given ha firms are locaed a he endpoins of he inerval [0, 1], any of he firms could find i profiable o sell in he oher firm's local marke. There is compeiive ineracion in any segmen of he marke. We sar by calculaing he values of 6 such ha he equilibria of G( UDP, FOB) coincide wih he monopolis soluion. For high enough values of he discoun facor, firms will be able o implemen he monopoly soluion, and herefore a UDP price p(x) = R will also be he soluion in his case. We now calculae he values of 6 ha allow he duopoliss o implemen a UDP pricing policy p(x) = R. For a UDP p(x) = R o be susainable in a perfec equilibrium of G( UDP, FOB), 6 should saisfy 1 1/2 e1 Zj a g (R-x)dx > (R-x)dx+1 a K(6,, R), (3) where K(a,, R) denoes he lowes payoff aainable in a perfec equilibrium of G(UDP, FOB): is value for he se of 6 ha susain p(x) = R is zero (see Appendix A). This inequaliy saes ha he profis from conforming o a UDP pricing policy p(x) = R are greaer han he profis from deviaing opimally and revering o an opimal punishmen pah forever afer. Noe ha when a firm is considering wheher o deviae from he pah [(R, 1 ), (R, 1 )], he bes deviaion is o undercu he oher firm slighly and sell o he enire marke (given our assumpion ha he ransporaion cos is no oo high, w? 1). From (3) we obain a 6 4? 8w - 4 This resul can be saed as he following: 4w - 3 Lemma 1. For every w? 1 and 6 2 8,4 a UDP policy Pu = R is susainable as a perfec equilibrium of G( UDP, FOB). This resul gives he values of 6 for which i is possible o susain full collusion. Now le us urn o he case in which only imperfec collusion is aainable. UDP policies. The opimal deviaion from a UDP policy is o undercu he opponen by some small amoun.14 Two cases mus be disinguished depending on wheher he opimal deviaion enails selling o he whole marke or o only a fracion of i. Case a: Pu?. The se of Pu susainable in a perfec equilibrium, condiioned on Pu 2, is given by he inequaliy 1 C1/2 a 1- ai (Pu-x)dx > J (Pu-x)dx + i K(6,, R). (4) 14 To deviae from a UDP pricing rule, firms will never use an FOB policy. The reason is ha FOB implies an unnecessarily low price in he local marke for he deviaing firm, while UDP is a more aggressive ype of

9 ESPINOSA / 71 In his case, given ha he price is higher han he cos of delivering he good o he consumers locaed a he oher end of he inerval, when a firm deviaes i finds i profiable o sell o he whole marke. From (4), and using he fac ha K( (,, R) 2 0, 3-46 PU < Since Pu is consrained o be greaer han, mus be a leas 1, which implies ha 6 2 1/4 is a necessary condiion for Pu?. We also mus check wha he value of K((6,, R) is; in Appendix A I show ha for 6 2 1/4, K((6,, R) = 0. Thus, 6 1/4 is sufficien as well as necessary for susaining Pu?. Case b: Pu <. Obviously, P5 = - is always susainable since i is a Nash equilibrium 2 of he one-sho game. We are ineresed in he se of Pu greaer han - ha are susainable 2 in a perfec equilibrium. The se of susainable Pu in he inerval [2 ) is given by he inequaliy 1 l1/2 Pu/l 6 1 6(J (Pux)dx 2 J (Pu-x)dx + 1 K(65, R). (5) In his case, when a firm deviaes i does no find i profiable o sell o he enire marke. The marginal consumer is a a disance Pu from he firm. From (5), 1+ 1/i PU<2( 1-6) Since we are condiioning Pu o be smaller han, his corresponds o he case 6 < 1/4. In Appendix A I show ha K((,, R) > 0 for 6 < 1/4, so ha he inequaliy is sric. The following able gives, as a funcion of he discoun facor, he maximum values of UDP prices and profis susainable in a symmeric perfec equilibrium.15 6 Pu llu E4w R- [8Jw 4 R 8 I4w-i3) l (6) l+1/ [0, 1/4) < V 2(1 -(5) 8(1 -(5) FOB policies. The above has assumed ha firms follow UDP rules. I now show he mos collusive FOB policies. deviaion given ha i allows a greaer peneraion of he rival's marke, wihou having o decrease prices in he home marke. 15 Noe ha for low values of he discoun facor, in paricular for 6 < 1/4, I give only he upper bounds for he susainable UDP price and profis (see Appendix A).

10 72 / THE RAND JOURNAL OF ECONOMICS Lemma 2. For w > 1, in G( UDP, FOB) a UDP deviaion is always an opimal deviaion from an FOB pricing sysem. The profis from using he opimal deviaion are (PC + )2 if PC ' 2, PC'R-- li[pu(pc) PC] = I 2 (7) Pc if PC>2, pc R--. PC2 2 Proof. Available on reques. Figure 1 illusraes he fac ha in G( UDP, FOB) he opimal deviaion is always a UDP deviaion. From Lemma 2 and assuming K(a,, R) = 0, he opimal collusive FOB prices are given by (2-PC (6p )2( ) for PC?2 p2?r-- (8) PC PC (PC - - (I - ) for PC>2, pc R- - (9) 2 2 ~~~~~~~~~~~~~2 Noe ha he consrain Pc < R - is always saisfied for w > 1. From (8), (6 1, and since we are consraining PC o be less han 2, we mus have less han 2, which implies 6 2( 1-6) - 6 > '/3. From (9), Pc = (1-2a) a 1/3. which corresponds o he case of Pc greaer han 2, i.e., Noe ha for 6 < 1/3, he expression for Pc has no meaning unless 6 1/4. As Lemma 4 in Appendix B shows, his is due o he fac ha for 6 < 1/4, FOB prices canno be susained in equilibrium. FIGURE 1 Pu (Pc) P (Pc) Pc 0 1

11 L ESPINOSA / 73 Therefore, he profis from using he opimal collusive FOB pricing sysem in G(UDP, FOB) are16 [min 1iC= f12a-3 2R-] if?4a?1/3 i 1/C R min [2(1 _ 26) R 4 ] if a> V3. (10) For 6 < 1/4 here is no FOB price susainable in G( UDP, FOB). The reason is ha for low values of he discoun facor, deviaions from he sysem are very profiable. The lack of effecive peneraion ino he opponen's marke (since each firm's price increases wih disance a a rae ) leaves he opponen wih he possibiliy of very profiable deviaions in is own local marke. Therefore, FOB is no likely o be observed in markes wih a spaially differeniaed produc where he possibiliies for cooperaion among he firms are limied. The profis from using he opimal collusive FOB pricing sysem in G( UDP, FOB) are given by (10), and he profis from using he opimal collusive UDP policy have been calculaed earlier. The main resul of his secion is summarized in Figure 2 (see also Proposiion 4 in Appendix B). This resul indicaes ha UDP is likely o be observed in very monopolisic indusries (defined as indusries where he ransporaion cos is high, w c 5/4, and/or he discoun facor is high, 6 2 6*( w)), bu also in very compeiive indusries (indusries wih a low discoun facor, 6 <5 k+ and a low ransporaion cos, w > 5h). For inermediae marke srucures he model predics FOB. This is consisen wih he observaion ha UDP is uilized in indusries wih he presumpion of collusive behavior bu also in very compeiive markes, and i may provide an explanaion for his apparen paradox. The resul derives from he fac ha, unless prices are low, he incenive o deviae is lower when he rival is using FOB raher han UDP. In a UDP sysem he wo firms are charging he same price a a given locaion, so ha if a firm slighly decreases is UDP price i ges he whole marke; his greaer inerpeneraion of markes makes deviaions very FIGURE 2 1/2 UDP 1/4 a+ UDP 0 5/4 16 See Appendix A for he value of K(B,, R).

12 74 / THE RAND JOURNAL OF ECONOMICS profiable. When prices are low, however, UDP is less profiable o deviae from, since for low prices a firm is no ineresed in serving he enire marke (i would make losses in he consumers locaed far away), while FOB leaves he opponen wih he possibiliy of very profiable deviaions in is own local marke. Thus, for high values of 6, a discriminaory pricing policy is preferable for a carel, since he enforcemen of collusion is no a pressing quesion, and price discriminaion allows he duopoliss o exrac a greaer proporion of he consumer surplus. However, as 6 goes below 0*( w), he enforcemen of collusion becomes harder and FOB looks more aracive as a way of sofening he incenive o chea and hence obaining higher prices. This is he case unil 6 reaches (+k. For lower values of he discoun facor, no FOB price can survive in equilibrium; UDP deviaions are oo profiable compared o he profis o be made along he FOB pah. Neverheless, below (+ UDP is sill susainable because for 6 < (+, UDP deviaions do no involve selling o he enire marke. Take for insance he case 6 = 0. For 6 = 0 he UDP equilibrium price is Pu =2 in a UDP pricing sysem he firm pays he ransporaion cos, so ha in ha equilibrium none of he firms has any incenive o gain addiional business (any sale furher han he midpoin of he inerval would generae a loss). Thus, for 6 = 0 he profis from deviaing from UDP are zero. However, for 6 = 0 he deviaion 1 ) profis for FOB are - (Pc + )2 (see (7)), which is sricly posiive for any PC 2 0; his is 6 due o he fac ha for any FOB price, here is always an incenive for a firm o deviae from he sysem by charging a higher price o he local cusomers, which makes i impossible o susain any FOB rule. The resul does no depend on he form of he punishmens used. If, for example, afer a deviaion he firms rever o he saic Nash equilibrium, Pu = 2 he same conclusion 2' obains, alhough he value of (*( w) and 6+ will be differen, and for each 6 equilibrium profis will be lower. The assumpion ha when consumers are indifferen hey go o he neares firm is crucial. Wihou i, we could obain nonexisence of equilibrium for some values of (; in paricular, for 6 = 0, if consumers randomize when indifferen, here is no equilibrium in pure sraegies. The assumpion ha firms can change he pricing rule (from FOB o UDP or vice versa) wihou incurring any cos is no crucial eiher. As long as he swiching cos is no oo high, UDP will sill be he bes deviaion from FOB, and herefore UDP will emerge as he equilibrium policy for low values of he discoun facor. Neverheless, if he cos of swiching pricing sysems is prohibiive, and if once in FOB firms canno use deviaions ou of he pricing sysem, hen we would observe FOB for low values of he discoun facor. Acually, if firms can credibly commi o a pricing policy a he beginning of he game, his amouns o resricing deviaions o follow he same pricing rule. In he case of UDP his is no a real consrain, since he bes deviaions from UDP are also UDP prices, bu in he case of FOB he consrain excludes he mos profiable deviaions. In Appendix B I show ha for w 2 w, wih an infinie cos of swiching pricing 2w -2 2w - 2 sysems, for 6 > 4 firms will choose UDP pricing policies, while for 6 < FOB 4w - 3 4w - 3 pricing rules are opimal. 3. Linear pricing sysems * The previous secion has ried o clarify he main argumens relevan in he discussion on he collusive naure of delivered pricing rules, which is usually presened as a comparison

13 ESPINOSA / 75 beween UDP (or anoher idenical delivered pricing policy) and FOB. However, firms may have oher policies available, even if hey are resriced o simple pricing rules; and he presence of hese policies could affec he opimaliy of UDP or FOB. In his secion I generalize he resuls o he class of linear pricing schedules. The main conclusion of Secion 2 sill holds: UDP pricing policies will be used in very compeiive and in very monopolisic marke srucures. For marke srucures ha are neiher very compeiive nor very monopolisic, he model predics pricing sysems wih slope ha is in he inerval [0, ] and is nonincreasing in 3. The range of 6 for which FOB is opimal is reduced. When marke share agreemens are allowed, he pricing sysem has slope in he inerval [-, 0), and he slope is nondecreasing in 3. Greenhu, Greenhu, and Li ( 1980) analyze empirically he influence of several facors on he slope of he delivered price schedules of he firms in heir sample and, in paricular, he effec of he degree of compeiiveness in he marke. Two measures of compeiiveness are used in heir sudy: he rank (or exen) of compeiion assigned by he seller as applying o his marke (hey call his a subjecive measure of he inensiy of compeiion) and he number of compeiors (an objecive measure). They obain he resul ha he more compeiive he marke, measured by he exen of compeiion assigned by he seller as applying o his marke, he seeper he delivered price schedule, and his relaionship has a high level of saisical significance. Their sudy provides some empirical suppor for our resul ha he slope of he pricing sysem is nonincreasing in 6 given ha marke share agreemens are difficul o implemen.17 Greenhu ( 1981 ) also observes ha in Japan and Wes Germany, delivered pricing sysems ended o have negaive slopes, while for American firms he slope was posiive. In our model his migh be explained by a sricer anirus legislaion in he Unied Saes making i harder for American firms o coordinae on marke sharing agreemens, or by he greaer difficuly in using pricing rules wih negaive slope (due o legislaion agains price discriminaion).18 o Descripion of he game. The descripion of he game is idenical wih ha in Secion 2, excep ha now firms' pricing policies mus be of he form p(x) = po + cx, po E R, c E OR, where x is he disance beween he locaion of he consumer and he locaion of he firm. The pricing sysems analyzed in he previous secion are paricular cases of hese pricing policies: when c = 0, we have UDP, and c = corresponds o FOB. Noe ha a linear pricing rule involves wo parameers, po and c. We do no allow in his secion piecewise-linear pricing; piecewise-linear would involve a leas hree parameers and be, in his sense, more complex. We are ineresed in he opimal collusive symmeric perfec equilibria of his game, which will be referred o as collusive equilibria. cl Collusive equilibria. Firs I characerize in Proposiion 1 he opimal collusive equilibria from he se of subgame perfec equilibria for which a = 1. In some collusive equilibria firms do no need o specify a < 1, since in equilibrium hey are selling in only half he marke. We are ineresed in hese equilibria because a = 1 in equilibrium implies here is no marke-sharing agreemen. I is worh noing ha firms can sill refuse o sell o ha 17 In our model he degree of compeiiveness is measured by he value of 6, which can be inerpreed as he exen of compeiion assigned by he seller as applying o his marke. 18 Greenhu ( 1981 ) aribues he differences in he sign of he slope of delivered price paerns o he Robinson- Paman Ac and poins ou ha "one spokesman inerviewed... also admied ha his company had followed his kind of pricing policy [charging a lower delivered price in a disan marke han ha charged in is home marke] bu upon he advice of lawyers eliminaed he lower (delivered) a disan marke poins in favour of a more moderae freigh absorpion pracice."

14 76 / THE RAND JOURNAL OF ECONOMICS porion of is demand ha is no profiable o supply; he resricion a = 1 is no a resricion on he sraegy ses of he firms, bu raher a crierion o selec equilibria where no markesharing agreemen is being used. Laer on I shall characerize he opimal collusive equilibria of he game (no only he equilibria wih a = 1) and show ha firms find i profiable o have marke sharing agreemens, i.e., in equilibrium hey will refuse o sell o cusomers locaed furher han he midpoin of he inerval even hough i is profiable (in he shor run) o serve hem a he announced pricing rule. We shall assume w 2 2 (ransporaion cos is no oo high). Le (pb(x), ab) be he bes response o (p0 + cx, a) and y he fracion of he marke capured by a firm when using he sraegy (pb(x), ab) agains (p0 + cx, a). Lemmas 5 and 6 in Appendix B give he minimum requiremens for a collusive symmeric equilibrium. The'slope of he price schedule canno be greaer han, given he possibiliy of arbirage among consumers, and i canno be negaive (if i were negaive, each firm would be selling in he oher firm's local marke). The mill price canno be negaive, and all he consumers have o be served. If any of hese requiremens is no me, hen i is possible o increase he equilibrium profis, conradicing ha firms were in a collusive equilibrium. Lemma 6 characerizes he opimal deviaion (pb(x), ab) from he symmeric policies [(p0 + cx, a), (p0 + cx, a)] when c 2 0. The opimal deviaion involves undercuing he opponen by a small amoun a every locaion: Pb(X) = p0 + c - cx, whenever his is compaible wih serving he enire marke, i.e., whenever c p < R - c (see Figure 3, op). If pb(x) = p0 + c - cx, his would imply prices higher han R in some areas, i.e., if p0? R - c, hen he slope is decreased unil he enire marke can be served: Pb(X) = R - (R - p0)x (see Figure 3, boom). Lemma 7 gives condiions ha he opimal pricing rule mus verify. Now we can esablish Proposiion 1. Assume w 2 2. In he opimal collusive equilibria in which a = 1, UDP pricing policies will be used for very high and also for low values of he discoun facor 3. FIGURE 3 R R po+ C Pb(x) = Po + c - cx - R-c Po 0 1/2 1 R R P + cx Pb (X) R-(RP )X Po - R-c 0 1/2 1

15 ESPINOSA / 77 For inermediae values of 6, he linear pricing sysem used in he opimal collusive equilibrium has slope in he inerval [0, ], and when c > 0, i is nonincreasing in 3. Proof. See Appendix B. For very high values of 6, a UDP price Pu = R prevails because i allows perfec discriminaion. As 6 decreases, Pu = R canno be susained, and hen i is opimal o have a pricing sysem wih a posiive slope. This pricing policy is equivalen (same profis and same profis from he opimal deviaion) o he opimal collusive UDP price Pu < R (see Figure 4). However, as 6 decreases furher, deviaions from a policy wih a posiive slope become oo profiable: a deviaing firm would undercu slighly he rival's price line, and his implies a high price in he deviaing firm's local marke; hen, UDP is sricly beer. We have characerized he opimal collusive equilibria from he se of subgame perfec equilibria for which a = 1. We also may consider he possibiliy of marke-sharing agreemens requiring a firm no o supply all of is demand ha is profiable o supply (in he shorrun sense). The case where firms are allowed o use marke-share agreemens is considered in he Appendix. The resul is Proposiion 2. For w> 2, when firms are allowed o use marke-share agreemens, UDP pricing policies will be used in he opimal collusive equilibrium for high values of he discoun facor 3. For lower values of 6, he linear pricing sysem has slope in he inerval [-, 0) and he slope is nondecreasing in 3. A marke-share agreemen a = 1/2 will be used whenever he slope is sricly negaive. Proof. Available on reques. When marke-share agreemens are possible, firms have pricing rules wih a negaive slope in an opimal collusive equilibrium. The reason is ha his ype of policy makes deviaions no as profiable as policies wih a posiive slope. Negaively sloped rules need a marke-share agreemen, however; oherwise, each firm would be selling in he rival's local marke, which is an inefficien arrangemen. 4. Unresriced pricing policies * Unil now we have been assuming ha he se of pricing policies available o a firm was somehow consrained o a family of simple pricing rules (UDP or FOB in Secion 2 and linear pricing policies in Secion 3). This may be a reasonable assumpion, since here are coss associaed wih he implemenaion of a complicaed pricing rule (we may hink ha pricing rules ha involve he choice of only one parameer, like FOB or UDP, are less complex han pricing policies ha involve more parameers). However, i is of ineres o FIGURE 4 R R 0p0 0 1~~~~~~~~~

16 78 / THE RAND JOURNAL OF ECONOMICS examine wha happens in he case where arbirary pricing sysems p(x) can be implemened wihou cos. In his secion we impose no paricular resricion on he pricing rule p(x). A pricing policy ha has been commonly used in pracice and will play an imporan role in his secion is he basing-poin pricing rule. When using his pricing policy, firms decide on he locaion of a base poin, Xb, and a price a ha locaion, Pb. The price a any oher locaion is calculaed as he base price plus he ransporaion cos from he base poin. Formally, a basing-poin pricing rule is a funcion p(x) such ha p(x) = Pb + Xb(X), where xb(x) expresses he disance from he consumer o he base poin as a funcion of he disance beween he consumer and he producer. Examples of his pricing policy are he Pisburgh Plus sysem used in he seel indusry and he Porland Plus sysem used for plywood. I was also implemened as a punishmen pah afer deviaions in he cemen indusry during he Grea Depression. The Supreme Cour described he punishmen as simple bu successful. Oher producers made he recalciran's plan an involunary base poin. The base price was driven down wih relaively insignifican losses o he producers who imposed he puniive basing-poin bu wih heavy losses o he recalciran who had o make all is sales on his basis. In one insance, where a producer had made a low public bid, a puniive base poin price was pu on is plan and cemen was reduced en cens per barrel; furher reducions quickly followed unil he base price a which his recalciran had o sell is cemen dropped o 75 cens per barrel, scarcely one half of is former base price of $1.45. Wihin six weeks afer he base price hi 75 cens, capiulaion occurred and he recalciran joined a Porland cemen associaion.19 Basing-poin pricing can be a very severe form of punishmen afer a deviaion, and herefore firms could susain higher profis hrough he hrea of making he deviaor an involunary base-poin. Acually, we can prove ha making he deviaor an involunary base-poin is he deviaor's wors possible perfec equilibrium, and hen any given pah is susainable in a perfec equilibrium if and only if i can be susained by reversion, in case of a deviaion from ha pah, o a basing-poin punishmen. Lemma 3. For any 6 E [0, 1], here is a subgame perfec equilibrium in basing-poin sraegies ha yields zero payoff o one of he players. Proof Consider he following pair of basing-poin sraegies: [(Pb, Xb)1, (Pb, Xb)210 = [(O, locaion of firm 2), (0, locaion of firm 2)]. Figure 5 illusraes his pair of idenical delivered pricing policies. Since consumers always buy from he neares supplier in an idenical delivered pricing sysem, he corresponding payoffs per period are - for firm 1 and zero for firm 2. 4 FIGURE 5 en Firm 1 1/2 Firm 2 19 As quoed in Machlup ( 1949).

17 ESPINOSA / 79 This is a subgame perfec equilibrium. Noe ha given firm 2's sraegy, firm 1 canno profiably deviae in any period (undercuing firm 2 only lowers oal revenue, by decreasing revenue from old cusomers and/or aracing cusomers for which he ransporaion cos is higher han he price, while raising price only makes firm 1 lose cusomers). The same is rue for firm 2. This is he subgame perfec equilibrium ha yields he lowes payoff o firm 2. Similarly, [(Pb, Xb)I, (Pb, Xb)2 180 = [(0, locaion of firm 1), (0, locaion of firm 1 )] is he subgame perfec equilibrium ha yields he lowes payoff o firm 1.20 Q.E.D. The following proposiion characerizes he collusive equilibria of he game. Le us define a basing-poinype policy as any p(x) = Pb + axb(x), i.e., firms can use freigh raes differen from he acual ones. Then, basing-poin appears as an opimal rule whenever he monopoly soluion is no susainable, i.e., for values of 6 less han 8w -3 Proposiion 3. In opimal collusive equilibria, when here is no resricion on he pricing policies, for 3 2 8w 4 X firms will use a UDP pricing rule. For values of 6 in he inerval (?' 8 ), a variey of idenical delivered pricing policies could be observed (including UDP, basing-poin pricing, ec.). Finally, for 6 = 0, he model predics basing-poin pricing. FOB sraegies are never used in an opimal collusive equilibrium. - 3 Proof. For he monopoly price p(x) = R exracs all he consumer surplus 8w-4' and i is susainable (see Secion 2). For 6 = 0, he bes susainable pricing sysem is basingpoin pricing: p(x) = p112 + X112(X) and P1/2 = 2 i.e., he base poin is a he middle poin of he inerval and he base price is /2, as indicaed in Figure 6. Profis are - for each firm. 4 To see ha he indicaed basing-poin rule is an equilibrium for 6 = 0, i is sufficien o prove ha none of he firms has any incenive o deviae. A profiable deviaion for firm 1 would involve eiher an increase in marke share from [0, 1/2] o [0, a], a e (1/2, 1 ] and/or an increase in he price o he exising cusomers in [0, 1/2]. An increase in marke share for firm 1 would require selling in [1/2, a ] a a price no higher han he ransporaion cos, given ha he prices of firm 2 in [1/2, 1 ] are p(x) = x, where x is he disance from he consumers o firm 1. Thus, an increase in marke share canno increase profis for firm 1. On he oher hand, an increase in he price o some of he consumers in [0, 1/2] is no profiable eiher: i would imply losing hose consumers o he compeiion, given ha in [0, 1/2] firm 2 is pricing a p(x) = - x, where x is he disance from he consumers o firm 1. Moreover, he indicaed basing-poin rule p(x) is he collusive equilibrium of he game for 6 = 0. Assume here is anoher pricing sysem p'(x) ha yields higher profis and is 20 This punishmen pah is saionary. If we inroduced producion coss, hen we could have a "sick and carro" basing-poin opimal punishmen pah, which would resemble more closely he punishmens implemened in he cemen indusry.

18 80 / THE RAND JOURNAL OF ECONOMICS FIGURE 6 0 1/2 1 susainable for 6 = 0. Then, in some inerval z, p'(x) > p(x) for x E z. The firm ha is no serving cusomers in z (firm j) would deviae selling in z since p'(x) > p(x) 2 x, where xis he disance from a consumer in z o firmj, hence conradicing ha p'(x) was susainable for 6 = 0. For inermediae values of he discoun facor, idenical delivered pricing sysems will be opimal. To see ha FOB (or any oher price rule differen from idenical delivered pricing) canno be observed in a collusive equilibrium, one need only noe ha when a a given poin he wo firms have differen prices, his only increases he deviaing profis for he firm ha has he lower price (compared o he siuaion in which boh firms have he lower price). This implies ha given a pricing sysem wih nonidenical prices a a given poin, i would be possible o increase profis, conradicing ha he pricing policy is used in a collusive equilibrium. Define B as he highes level of profis in excess of ( are he maximum profis when 6 = 0, wih a basing-poin pricing rule) ha can be made in equilibrium. Then, for each value of 6, he level of profis B is given by 1 4+ B = ~~~~~~ 4+ 2B5 ( 1 ) where - + B are he profis from conforming o he equilibrium price policies and - + 2B 4 4 are he profis from deviaing, given ha an idenical delivered price rule is opimal (see Figure 7) and he punishmen pah has a payoff of zero (by Lemma 3). B represens he profis due o he fac ha 6 > 0. From (11), B 4(1-2) B is increasing in 3 when 3 E [0, 1/2]; since we are focusing upon he case when 8w< -4 - and given ha 8-3 < - i follows ha B is increasing in 3 over he relevan range of discoun facors. The maximum level of profis ha can be susained is deermined by 6, and any idenical delivered pricing sysem ha yields B(b) + - is indif- 4 feren. Q.E.D.

19 ESPINOSA / 81 FIGURE 7. _ 0 1/2 1 Noe ha unless he discoun facor is very high (a leas /2 in his model), FOB canno be susained in equilibrium. The reason is ha when pricing policies are no resriced, deviaions can be very profiable, resuling in he impossibiliy of susaining any FOB price. This may explain he opinion ofen expressed ha delivered pricing sysems serve he purpose of sabilizing he indusry (if FOB canno be susained in equilibrium, any FOB price will be necessarily unsable). We have assumed ha whenever he firms quoe he same delivered price a a given locaion, he consumers choose he neares supplier. This assumpion is crucial o ge exisence of equilibrium. Consider 6 = 0 and assume, for example, ha consumers randomize among suppliers when he same delivered price is quoed; hen he game has no equilibrium in pure sraegies. The fac ha basing-poin pricing is used for low 6 and ha idenical delivered pricing policies are opimal for values of he discoun facor such ha he monopoly soluion canno be implemened does no depend on he assumed form of he demand funcion a each poin, he space considered (he Hoelling line), or he number of firms (as long as i is finie and firms are spaially differeniaed). However, he predicion of UDP for high 6 does depend on he demand funcion. Le D(p) be he demand funcion a each poin and assume ha pd(p) is concave. Then, for 6 high enough, he delivered pricing policy p(x), for x E [0,?/2], is implicily defined by D'(p)(p - x) + D(p) = 0, which corresponds o a price schedule p(x) increasing in x. When demand is linear, he slope of p(x) is Concluding remarks * This aricle has examined he quesion of wheher idenical delivered pricing policies should always be considered pracices ha impair compeiion. The resuls indicae ha he answer o his quesion is negaive. Alhough delivered pricing sysems allow spaial price discriminaion, which faciliaes collusion, hey are also a very aggressive form of compeiion, so ha for low values of he discoun facor FOB is no susainable in a subgame perfec equilibrium, and idenical delivered pricing policies will be observed. The opinion expressed by Scherer ( 1980) and ohers ha in order o increase he degree of compeiiveness in markes, firms should no be allowed o price differenly from FOB, excep perhaps o undercu anoher firm, is no suppored by he resuls obained here. To impose FOB allowing deviaions ou of he pricing sysem would be a soluion o he welfare loss problem only in monopolisic markes. If he marke srucure is compeiive (low 6 or low ), his policy would lead o nonexisence of equilibrium and presumably o grea insabiliy in he marke.

20 82 / THE RAND JOURNAL OF ECONOMICS Anoher opion is o enforce FOB sricly, i.e., only FOB is allowed, even for deviaions. The welfare effecs of his measure are ambiguous and can go in he wrong direcion, since banning he mos effecive deviaions from he pricing sysem means ha he level of delivered prices susainable in equilibrium would go up. This seems o be wha happened in he cemen and seel indusries when hey were ordered o disconinue he pracice of basingpoin pricing. In 1948 he Supreme Cour declared he basing-poin sysem unlawful, susaining he decision of he Federal Trade Commission. Afer he forced abandonmen of he sysem, cemen producers were able o raise delivered prices and revenues wih FOB prices. A he same ime, he seel indusry sared selling seel on an FOB basis, expecing a similar decision from he Federal Trade Commission, and again here was an increase in delivered prices and revenues for he indusry.21 This seems o be conradicory, for if FOB was more profiable han basing-poin, why was he indusry no using FOB before? The answer is ha when basing-poin is unlawful, FOB is more profiable han when firms are also allowed o use basing-poin (he opimal deviaion from an FOB price is o make he opponen a base poin, undercuing is mill price by some small amoun). When FOB is made compulsory, deviaions are less profiable, and hus higher FOB prices can be susained in equilibrium. The fac ha in a perfecly compeiive world wih an infinie number of firms FOB would prevail does no imply ha FOB is he mos compeiive pricing policy in spaial markes where producs are spaially differeniaed and, due o fixed coss for example, here is room for only a few firms. We have seen ha in hese markes, delivered pricing may be more compeiive han FOB. The conclusion is ha delivered pricing policies canno be labelled as faciliaing collusive pracices in all insances; he srucure of he marke should be aken ino consideraion. An ineresing quesion is how uncerainy of deecion of deviaions from a specified pricing policy can affec he incenives o use a given price rule. The presence of uncerainy would also bring argumens like he one given by Sigler ( 1964), who raionalizes he uilizaion of basing-poin pricing as a sabilizing device in a conex of geographically unsable demand. The inroducion of uncerainy would bring new elemens o he discussion and deserves furher research. Appendix A * Opimal punishmen pah. Le A (6,, R) be he opimal collusive oucome (per period) ha could be susained by reversion o a zero-payoff punishmen pah, and PA(6,, R) he price ha implemens ha oucome. Consider he following pair of UDP pricing policies: [(Pu, a)i, (pu, )2] = [(0, 1 ), (0, 1 )] (Al) The corresponding payoff is 8 per period for each firm. 8 The opimal punishmen pah has wo phases. For Tperiods, firms follow he pricing policy (A l ). Tis chosen such ha ( T-1I <x j j=tb1+a 2 b6=0. (A2) 8j=O j= T Afer ha firs phase of T periods, hey follow he policies (PA(6,, R), 1) ha yield A(6,, R) per period forever afer. We have o check wheher his "sick and carro" punishmen pah, sr,r I ), {as I\T, r[ (p, Il \A), (pa n I 21 "Irae consumers wroe leers o ediors, Congressmen, and Senaors, complaining abou he new sysem [FOB] which made hem pay prices higher han before," Machlup ( 1949).