Monopolistic competition with a mail order business

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1 Economcs Letters 66 (000) locate/ econbase onopolstc competton wth a mal order busness Jan Bouckaert* Unversty of Ghent, Dept. of Economcs, Hovenersberg 4, B-9000 Ghent, Belgum Receved 15 January 1999; accepted 13 July 1999 Abstract I analyse free-entry competton between stores and mal order busnesses. Consumers purchasng at stores ental dstance related transportaton costs. Purchase from a mal order busness (OB) nvolves a fxed cost. Compared to alop s model [alop,.c., onopolstc competton wth outsde goods. Bell Journal of Economcs 10, ], fewer frms are actve wth free-entry. At most one OB enters. 000 Elsever cence.a. All rghts reserved. Keywords: (Non-)localzed monopolstc competton JEL classfcaton: D11 1. Introducton Ths paper studes free-entry competton when frms can choose between two alternatves to sell a homogeneous good at mll prces. The frst alternatve conssts of openng a retal store, whch consumers can vst by payng a lnear transportaton cost. The second alternatve nvolves settng up a mal order busness (OB), where consumers receve the good by payng an exogenous fxed cost, rrespectve of ther locaton. The OB serves ts consumers usng ths fxed cost technology e.g. a (electronc) postal servce. Its locaton, therefore, becomes completely rrelevant. Total expendtures from buyng at the retal store equal the prce at retal plus the transportaton cost to the retal store. In contrast, all consumers buyng from the OB have the same total expendture. The analyss adds a OB to alop s (1979) crcle model. If the fxed cost technology s too expensve and the set-up cost s large relatve to the margnal transportaton cost, no OB appears. Otherwse, at most one OB pops up n equlbrum. The OB competes n a non-localzed fashon wth all stores. The retal stores, however, compete n a localzed way wth the OB. The *Tel.: ; fax: E-mal address: jan.bouckaert@rug.ac.be (J. Bouckaert) / 00/ $ see front matter 000 Elsever cence.a. All rghts reserved. PII: (99)

2 304 J. Bouckaert / Economcs Letters 66 (000) ntroducton of a OB mples that a smaller number of frms are actve n equlbrum compared to 1 alop s orgnal model.. The model Consder a market for a homogeneous product. argnal cost of producton s constant and normalzed at zero. Each frm chooses one of two possble strateges to market the product. The frst s the tradtonal way of openng a store where consumers are charged a unform mll prce p $ 0. A purchase at ths store for a consumer located at dstance z mples a lnear transportaton cost tz $ 0. I use alop s crcle model, where frms are located equdstantly from each other. The second strategy conssts of a OB where consumers can order the product (by mal) at mll prce q $ 0 plus a fxed cost w $ 0 (e.g. the prce of the stamp or the electronc orderng costs) for sendng the product to the consumer s locaton. Ths exogenous fxed cost w s ndependent of one s locaton. One nterpretaton s that the OB s located at the center of the crcle. The radus of the crcle then represents the fxed cost w. There s a unt mass of consumers located unformly on the crcle. Consumers have the same reservaton prce r and unt demand. They buy from the frm offerng the lowest full prce,.e. mll prce plus lnear/fxed (transportaton) cost. Let the number of frms n the market be N $, ndexed 3 by 5,...,N. Consder the followng three-stage game. In stage one, each frm decdes to enter the market or not (ecton 4). In stage two, havng observed the number of frms that entered the market, they choose to become a tradtonal store or a OB. In stage three, havng observed each other s decson n the second stage, frms compete n prces (ecton 3). I solve the game for ts ubgame Perfect Nash Equlbra n pure strateges by the method of backward nducton. 3. Prcng Consder frst the case n whch more than one frm operates a OB. A standard Bertrand result appears for these frms, snce they are not dfferentated at all wth respect to each other. Prce competton results n chargng a zero prce. nce set-up costs are strctly postve, at most one frm wll open a OB. Ths results n two possble cases: () no frm operates a OB, or () only one frm sells through the mal. The frst case s dentcal to alop s crcle model. If frm sets prce p, and ] p s the prce charged by the other frms, a consumer located at dstance x from frm, wth x [ [0,1/N], s ndfferent between buyng from frm or ts neghbour f p 1 tx 5p ] 1 t(1/n x). Defne profts as demand at both sdes tmes prce, and frm s proft equals p ( p, ] p) 5 xp 5 [(p ] p 1 t/n)/t] p.in 1 In Thsse and Vves (1988) frms make strategc choces n terms of spatal prce polcy; unform FOB prcng or dscrmnatory prcng. They remark: [L]et us emphasze (... ) that (... ) unform prcng s dfferent from unform delvered prcng as defned n postage stamp systems. I take those two varants of unform prcng as frms avalable strateges. Other related papers are Heal (1980), pegel (198), Furlong and lotsve (1983), and Henret and Rochet (1991). The model assumes that prce dscrmnaton based on the consumer s address s llegal. Ths seems reasonable f the analyss concentrates on competton wthn one country. 3 The analyss rules out that only one frm enters the market n equlbrum. ee also footnote 6.

3 J. Bouckaert / Economcs Letters 66 (000) the symmetrc soluton every frm s market share s 1/N and p* 5 p* 5 t/n s the equlbrum prce. Every frm s gross proft, expressed as a functon of the number of frms N, equals t p * (N) 5 ]. (1) N whch wll be referred to as the -equlbrum proft. In the second case, exactly one frm operates a mal order busness. The other frms are equally spaced around the crcle at dstance 1/(N 1) from each other. Each of them faces three compettors: ts two negbours on the crcle and the OB. In between every two neghborng frms on the crcle, there are two ndfferent consumers. One s ndfferent between frm and ts neghborng frm on the crcle. Gven p] charged by ths compettor on the crcle, ths ndfferent consumer s located at y, where p 1 ty 5p] 1 t[(1/(n 1)) y) f y # 1/(N 1). The other s ndfferent between frm and the OB. Gven a prce ] q charged by the OB, ths ndfferent consumer s located at z such that p 1 tz 5q] 1 w. Ify # z, the OB has no market share and zero profts. If y. z, the mal order busness serves part of the market. Defne total demand for frm on the crcle by y f 0 # p # (q] 1 w) (p] 1 t/(n 1)) D ( p, ]] p, q) ; z f (q] 1 w) (p] 1 t/(n 1)) # p #q] 1 w () 50 f ] q 1 w # p. Profts for frm on the crcle are then p ( p, p, ]] q) 5 D ( p, p, ]] q)p. (3) nce the OB s locaton s n the center of the crcle, t faces (N 1) neghbours. For a gven prce ]p, the OB competes for the more dstant consumers n between every two frms on the crcle. The most dstant consumer s at dstance 1/((N 1)) from every frm on the crcle. The consumer who s ndfferent between buyng from frm or from the OB, s located at z such that ] p 1 tz 5 q 1 w. Ths expresson apples for each sde of all (N 1) frms on the crcle. Therefore, the OB s total demand D s defned as 0 f q $p] 1 t/(n 1) w 5 ] (N 1) ] t D (p, q) ; ]]]D p w q 1]]] f p] w # q #p] D 1 t/(n 1) w (4) t (N 1) 1 f q #p] w. The proft for the OB equals p (p ], q) 5 D (p ], q)q. (5) Eq. (5) s contnuous and quas-concave n q. Optmzng (3) wth respect to p, and (5) wth respect to q, the frst-order condtons f all frms have a postve market share, are p 5 0.5[q] 1 w] and q 5 0.5[p] w 1 t/(n 1)]. Usng the assumpton of symmetry ( p 5p ] 5 p) and ] q 5 q for the 4 Ths analyss also assumes that the market equlbrum les n the compettve regon of frm s demand curve. That s, the reservaton prce r $ 3t/ (see alop (1979) for the exposton).

4 306 J. Bouckaert / Economcs Letters 66 (000) OB, the Nash-equlbrum of the prcng-game equals ( p *, q *) where p* 5 [w 1 t/(n 1)]/6 and 5 q* 5 [t/(n 1) w]/3 at every store on the crcle and the OB, respectvely. ubsttuton of these prces nto (3) and (5) leads to profts expressed as a functon of the number of frms N: 1 t p * C(N) 5] ]]] 1w 18t (N 1) for every frm on the crcle, and D D (N 1) t p * (N) 5]]] ]]] w (7) 9t (N 1) for the OB. Eqs. (6) and (7) wll be referred to as the -equlbrum profts. Defne D 1 3 h(n) ; t ]] ]]]]. (8) N 1 Nœ(N 1) h(n) s non-negatve for all N $ 3; furthermore, h(3) 5 0, h(`) 5 0 and h(), 0. Proposton 1. (a) If w # h(n), we have an -equlbrum. (b) Otherwse, the -equlbrum s the unque equlbrum n pure strateges. Proof. Frm s proft equals t/n f all frms locate on the crcle. If only frm operates a OB, ts proft s ((N 1)/9t)(t/(N 1) w) by (7). Therefore, frm fnds t optmal to start up a OB f t/n # ((N 1)/9t)(t/(N 1) w). Ths condton s equvalent to w # h(n). From (6) all other frms on the crcle have a proft of p * C(N) 5 (1/18t)(t/(N 1) 1 w). The standard Bertrand argument says that swtchng to the center yelds zero profts. nce p *(N). 0, these frms reman on C the crcle. Ths proves part (a). If, however, t/n. ((N 1)/9t)(t/(N 1) w), the opposte nequalty holds,.e. w. h(n). Frm locates on the crcle and no other frm swtches to the center. Ths proves part (b). h Proposton 1 tells that n any -equlbrum the cost of sendng the good through the mal should be small enough. nce 0 # w # h(n), the lower bound on the number of frms n an -equlbrum s N $ 3. The ntuton s that a frm has an ncentve to open a OB only f ts proft as a frm on the crcle s relatvely small. In an -equlbrum, the OB foregoes some market power by a decrease n the equlbrum prces. Therefore, a sngle frm on the crcle has suffcent ncentves to become a OB f the gan n market share s large enough. Calculatons show that the OB has a larger market compared wth the -equlbrum. Fnally, each frm competes only wth ts two neghbours n the -equlbrum. The cross-prce elastctes are postve for neghbourng frms, but zero for all other frms. That s, there s localzed competton. In the -equlbrum, the frm n the center competes wth every frm on the crcle. Clearly, ths generates some form of nonlocalzed competton, as the cross-prce elastcty ( D / q)(q/d ) s postve (and dentcal) for all. The OB shoulders tself n (6) 5 If t/(4(n 1)). w, the OB charges a hgher prce than the frms on the crcle. For hgher values of w, lower prces result. The OB and the frms on the crcle charge always lower prces compared to the stuaton n whch frms operate only on the crcle.

5 J. Bouckaert / Economcs Letters 66 (000) Fg. 1. arket shares n an -equlbrum wth N 5 7. between every frm on the crcle. The frms on the crcle have only one drect compettor; the OB. A small change n ther own prce, affects only the OB s market share. The cross-prce elastcty ( D / p )( p /D ) s postve (and dentcal) for all. The cross-prce elastcty ( D / p j)( p j/d ) equals zero for all j ±. They are engaged n some form of localzed competton. Fg. 1 shows an example wth N 5 7. The bold lnes represent the retal stores market share. Proposton. Frms on the crcle earn hgher profts n the -equlbrum compared to the -equlbrum: p *(N). p *(N). C Proof. (1) strctly exceeds (6) f and only f w, t(3/ Œ N ] 1/(N 1)). Compare the rght-hand sde Œ] Œ] of ths nequalty wth h(n) to see that t(3/ N 1/(N 1)). h(n) f and only f. N/(N ]]] 1) œ/(n 1). For all N $, the rght-hand sde of the latter nequalty s an ncreasng functon. By applyng l Hoptal s ˆ rule, t reaches ts maxmum of 1 for N `. nce w # h(n) n the -equlbrum, the result follows. h 4. Free-entry Ths secton adds an entry stage to the above analyss. Thus, each frm decdes frst whether or not to enter the market. Havng observed the number of frms entered the market, the entrants play the two-stage game of the prevous secton. Those who do not enter receve zero profts. Consder a fxed cost of entry F. 0. ecton 3 establshed that the - and -equlbrum are possble canddates satsfyng the subgame perfectness condton. Our concept of free-entry equlbrum requres that enterng frms earn non-negatve profts, and all other frms antcpate non-postve profts when enterng (see Anderson et al. (199)). Ths motvates the followng two defntons: Defnton 1. N* s the number of frms n a free-entry -equlbrum f () p (N * ) 5 F; and () p (N *) $ p (N *). Condton () ensures that all frms make zero profts. It mples that N* 5Œ ] t/f, by (1). Condton

6 308 J. Bouckaert / Economcs Letters 66 (000) () guarantees that wth the equlbrum number of frms n the market, no frm wants to swtch to a OB. The condton s equvalent to w $ h(n * ) (.e., the condton n Proposton ). In the free-entry -equlbrum, therefore, w $ h( Œ ] t/f ). Defnton. N* s the number of frms n a free-entry -equlbrum f () p ) 5 F; and () p (N *) $ p (N *). The frst condton ensures that all frms on the crcle make zero profts. Proposton establshed that p C(N), p (N). It follows that only the frms on the crcle must satsfy the zero-proft condtons for free-entry. The second condton guarantees that wth the equlbrum number of frms n the market, exactly one frm wants to swtch to the OB. Defne the followng functon: D 1 Œ ]] t g(n * ) ;] 18tF ]]] *. (9) N 1 The functon g(.) s ncreasng and, by (9), the equalty g(n * ) 5 w represents the zero-proft condton for the frms on the crcle. Condton () n Defnton s equvalent to w # h(n * ). Therefore, N* 6 satsfes the requrements () and () of Defnton f and only f g(n *) 5 w # h(n *). Proposton 3. Let p (N * ) 5 p ) 5 F; then N *. N *. That s, f N* and N* are determned by the zero-proft condton, the number of frms n the -equlbrum s hgher than t would be n the -equlbrum. Proof. uppose N * # N *. nce (6) s decreasng n N, p * ) # p * ). It follows from Proposton that p * ), p * (N * ). The free-entry -equlbrum requres that p * (N * ) 5 F. But then p * ), F, and N* cannot be the number of frms under a free-entry equlbrum. A contradcton. h Proposton 3 states that the number of frms n the free-entry -equlbrum s larger than n the free-entry -equlbrum. Therefore, the market wth a OB s more compettve. Ths accords wth the result that nonlocalzed competton yelds fewer frms n a free-entry equlbrum than wth 7 localzed competton (Deneckere and Rothschld, 199). The condtons for an - and -equlbrum are now analyzed. Lemma. () h(3) 5 0 and h(n). 0 for all N. 3; () g(3) $ 0 f and only f t/f # 7; () g9(n). h9(n) for all N. 3; (v) g(n). h(n) for all N large enough. Œ]] Œ] Proof of Lemma. () h(3) 5 0, obvous. h(n). 0 for all N. 3 f and only f N/( N 1). 3/. Œ]] Œ] nce N/( N 1) s strctly ncreasng n N and equals 3/ at N 5 3, the result follows. () g(3) $ 0 f and only f t/f # 7. From evaluaton of Eq. (9) at N 5 3, we fnd that t/f 5 7. nce g(n) s strctly ncreasng, the result follows. () g9(n) h9(n). 0 for all N. 3 f and only f Œ] 3/ 3t/((N 1) ). 3 t(3n )/(4N(N 1) N). It can easly be checked that ths holds for all 6 Assume that F, t/18, such that N * ee, however, Anderson et al. (199) p. 194 for a crtcal assessment of ths nterpretaton.

7 J. Bouckaert / Economcs Letters 66 (000) N. 3. (v) From () and (), h(3) 5 0 and g(3) # 0 f and only f t/f $ 7. nce g9(n). 0 for all fnte N and g9(n) h9(n). 0 for all N. 3 from (), g(n). h(n) for some N. 3. If t/f, 7, then g(3). h(3) 5 0. h Proposton 4. () N* exsts, then N * $ 3 s ncreasng n w, and decreasng n F; () If a free-entry -equlbrum Proof. () Inspecton of (9) yelds the comparatve statc results; () From the Lemma, g9(n). 0 and g(n * ) 5 w # h(n * ) cannot be satsfed f g(3). 0. nce w $ 0, N * $ 3 f an -equlbrum exsts. h An ncrease n w mples more frcton n the market and prevents the OB from decreasng the prces drastcally. Therefore, more frms can enter the market. Proposton 5. () Let F # t/7; then there exsts a ] w. 0, such that an -equlbrum wth free-entry exsts f and only f 0 # w #w. ] () If F. t/7, free-entry does not result n an -equlbrum. ] ] ] Proof. () By the Lemma, there exsts a N such that h(n) 5 g(n) ;w. ] nce g(3) # 0 and g9(n). h9(n) for all N. 3 wth g9(n). 0, for w #w] there s a unque N such that g(n) 5 w, h(n); () nce h(3) 5 0 and g9(n). h9(n) for all N $ 3, the condton for a free-entry -equlbrum 0 # g(n) 5 w # h(n) (as stated n Defnton ) can never be satsfed. h If the fxed set-up cost s too large compared to the margnal cost of transportaton, the zero-proft condton for frms on the crcle cannot be satsfed. Proposton 6. Let N* and N* satsfy the zero-proft condtons of the free-entry equlbrum. (a) Let h(n * ), h(n * ). Then, () the -equlbrum s unque f h(n * ) # w; () f w # h(n * ), the - equlbrum s unque; () f h(n * ), w, h(n * ), no pure strategy equlbrum exsts. (b) If h(n * ) # h(n * ), then for all () w, h(n * ), the -equlbrum s unque; () w. h(n * ), the -equlbrum s unque; () h(n * ) # w # h(n * ) both the -equlbrum and the -equlbrum coexst. Proof. (a) () from Defnton 1, a free-entry -equlbrum exsts, snce w $ h(n * ) holds, whle condton () of Defnton s volated; () mlarly, no free-entry -equlbrum exsts, snce condton () of Defnton 1 s volated, whle Defnton holds; () In the same fashon, both condtons for the free-entry - and -equlbrum are volated f h(n * ), w, h(n * ). (b) can be proven n a smlar fashon. h 5. Concluson Ths paper uses alop s crcle model where frms ether operate a store or a OB. Consumers ental dstance related transportaton costs when buyng at a store. Buyng from a OB mples a fxed cost, rrespectve of the consumer s locaton. Wth free-entry, at most one OB pops up. Compared to alop s model, the number of frms enterng the market s lower. The OB competes

8 310 J. Bouckaert / Economcs Letters 66 (000) wth every frm on the crcle, and therefore engages n non-localzed competton. The stores on the crcle face only one local compettor the OB and are engaged n localzed competton. Acknowledgements I would lke to thank Helmut Bester and Erc van Damme for ther advce. I also benefted from helpful comments made by Hans Degryse, Dave Furth, Andreas Ortmann, Johan tennek, and Frank Verboven. References Anderson,.P., de Palma, A., Thsse, J.-F., 199. Dscrete Choce Theory of Product Dfferentaton, The IT Press, Cambrdge, assachusetts; London, England. Deneckere, R., Rothschld,., 199. onopolstc competton and preference dversty. Revew of Economc tudes 59, Furlong, W.J., lotsve, G.A., Wll that be pckup or delvery?: an alternatve spatal prcng strategy. Bell Journal of Economcs 13, Heal, G., patal structure n the retal trade: a study n product dfferentaton wth ncreasng returns. Bell Journal of Economcs 11, Henret, D., Rochet, J.C., croeconome de l assurance (Collecton Econome et statstques avancees, Edtons Economca). Thsse, J.-F., Vves, X., On the strategc choce of spatal prce polcy. Amercan Economc Revew 78, alop,.c., onopolstc competton wth outsde goods. Bell Journal of Economcs 10, pegel,., 198. Prcng polces under condtons of spatal competton. The Journal of Industral Economcs 31,