Costly location in Hotelling duopoly

Transcription

1 Cosly locaion in Hoelling duopoly Jeroen Hinloopen and Sephen Marin November 01 Absrac We inroduce a cos of locaion ino Hoelling s (199) spaial duopoly. We derive he general condiions on he cos-of-locaion funcion under which a pure sraegy price-locaion Nash equilibrium exiss. Wih linear ransporaion cos and a suiably specified cos of locaion ha rises oward he cener of he Hoelling line, symmeric equilibrium locaions are in he ouer quariles of he line, ensuring he exisence of pure sraegy equilibrium prices. Wih quadraic ransporaion cos and a suiably specified cos of locaion ha falls oward he cener of he line, symmeric equilibrium locaions range from he cener o he end of he line. Key words: Horizonal produc differeniaion, spaial compeiion, cos of locaion JEL Classificaion: D1, D4, L1. Thanks are due o Chrisian Ruzzier for useful commens and consrucive suggesions received a he 11h Annual Inernaional Indusrial Organizaion Conference, Boson, May 01, o Bill Novshek and oher seminar aendees a Purdue Universiy, and o seminar paricipans a he European Universiy Insiue. The usual disclaimer applies. Universiy of Amserdam, FEB/ASE, Roeerssraa 11, 1018 WB Amserdam, The Neherlands; J.Hinloopen@uva.nl. Deparmen of Economics, Kranner School of Managemen, Purdue Universiy, Wes Lafayee, Indiana , USA; smarin@purdue.edu. 1

2 1 Inroducion To paraphrase Sigler (1964), no one has he righ o invie aenion o anoher exension of Hoelling (199) wihou advance indicaion of he jusificaion for doing so. Our jusificaion is he observaion ha in a lieraure where wha maers is locaion, locaion, locaion, 1 locaion iself has been reaed as a free good. Since economics is someimes referred o as he science of scarciy, his seems an odd specificaion for economiss o make, and we inroduce a renal cos of locaion ha varies wih locaion on he Hoelling line. One inerpreaion of he received approach is ha Hoelling implicily assumed he cos of locaion o be independen of locaion, normalized i o 0 for simpliciy, and ha he lieraure has followed his approach. Bu renal coss ypically differ by locaion. In Europe, cener-ciy locaions are ypically more expensive han hose on he periphery. The same was rue of he Unied Saes hrough he mid-1950s, and he opposie may be he case for some U.S. ciies oday. Hoelling (199) rebelled agains he assumpion of homogeneous producs because of is implicaion, in he Berrand duopoly model, ha all demand would swich from one supplier o anoher in response o an infiniesimal difference in price. Inroducing produc differeniaion and making, as he hough, demands coninuous funcions of prices, he reached he Principle of Minimum Differeniaion, ha Buyers are confroned everywhere wih an excessive sameness. In a celebraed commen, d Aspremon e al. (1979) showed ha if firms locae oo close o he cener of he Hoelling line, here is no pure-sraegy equilibrium in prices, because of precisely he kinds of disconinuiies in demand ha Hoelling had hough o avoid. 4 Numerical analysis (Osborne and Pichik, 1987) indicaes ha equilibrium locaions in he wo-sage Hoelling game are wihin he region where here are no pure-sraegy equilibria in prices. d Aspremon e al. (1979) furher show ha if, keeping all oher aspecs of Hoelling s specificaion, ransporaion cos is made quadraic raher han linear in disance, duopoliss will choose maximum raher han minimum differeniaion. We consider a wo-sage model, wih cosly locaion choice in he firs sage, followed by a price-seing sage. Throughou he paper, we emphasize he inerpreaion of locaion cos c(y) as a renal cos ha varies wih disance y from he end of he line. This inerpreaion naurally suggess he polar opposie cases ha cos of locaion rises moving from he ends o he cener and alernaively ha cos of locaion rises moving from he cener o he end of he line. One migh insead inerpre c(y) as an R&D or produc-developmen cos ha mus be incurred o bring a produc of aribue y o marke, in which case he Hoelling line represens produc characerisic space raher han geographic space. Produc developmen cos may vary wih horizonal produc characerisics, in a way ha is conex-specific; he ruly black ulip, despie much effor, does no exis. We emphasize he geographic inerpreaion of he Hoelling line because i suggess naural specificaions for c(y). For he firs sage, we derive he general condiions on he cos of locaion

3 funcion ha mus be me for an equilibrium o exis. Tha is, we find condiions under which locaion coss ensure he exisence of a pure-sraegy locaion-price equilibrium. Wih linear ransporaion cos, hese condiions require ha cos of locaion rise moving oward he cener of he line a an increasing rae. Wih quadraic ransporaion cos, he condiions for exisence of an equilibrium require ha ransporaion cos do no fall moving oward he ends of he line. We illusrae our general resuls for specific cos-of-locaion funcions. In Secion we give references o he spaial oligopoly lieraure. In Secion we presen a Hoelling Main Sree duopoly model wih linear ransporaion cos and locaion cos rising oward he cener of he line. In Secion 4 we presen he model wih quadraic ransporaion cos and locaion cos rising oward he ends of he line. Secion 5 concludes. Lieraure The lieraure ha flows from Hoelling (199) is vas. 5 Exensions include 6 a finie reservaion price (Lerner and Singer (197), Salop (1979), Economides (1984), Hinloopen and Van Marrewijk (1999)), a circular road (Chamberlin (195), Vickrey (1964/1999), Samuelson (1967), Salop (1979), Economides (1989)), graphs (Soeeven (010)), nonlinear ransporaion coss (d Aspremon e al. (1979), Capozza and Van Order (198), Economides (1986)), more han wo firms (Chamberlin (19), Lerner and Singer (197), Shaked (198)), quaniy compeiion (Hamilon e al., (1994), Gupa e al. (1997)), sequenial enry wih no relocaion (Presco and Visscher (1977), Eaon and Ware (1987)), price-aking firms (Anderson and Engers (1994), Hinloopen (00)), compeiion in n > 1 dimensions (Irmen and Thisse (1998)), and non-uniform disribuions of consumers along he Hoelling line (Anderson, Goeree and Ramer (1997)). If nohing else, his lieraure esablishes ha he equilibrium predicions of a spaial oligopoly model are highly sensiive o he deails of he specificaion. Inroducing a cos of locaion is no an excepion o his characerisic of he lieraure. Linear ransporaion cos We firs consider he case of linear ransporaion cos. Assuming ha he locaion cos funcion c (y) is wice coninuously differeniable, we show ha exisence of a subgame perfec price-locaion Nash equilibrium requires ha he firs and second derivaive of c (y) be posiive. We illusrae his general resul for a specific funcional form of he cos-of-locaion funcion..1 Sage : price seing The marke consiss of a line of lengh l = 1, along which consumers are uniformly disribued. There are wo firms, A and B, locaed a disances a and b respecively from he lef and righ ends of he line wih a + b l. The firms

4 supply a homogeneous produc ha yields gross surplus v. Consumers have uni demand, and a consumer locaed a x (measured from he lef side of he marke) has ne surplus U(x; a) = v x a p A, (1) if buying from firm A, where > 0 is he ransporaion rae, and p i is he price charged by firm i, i = A, B, and ne surplus U(x; b) = v 1 b x p B. () if he produc is bough from firm B. v is large enough ha all consumers always buy. 7 The boundary consumer has idenical ne surplus from eiher firm, U(x; a) = U(x; b), and is a disance x = 1 [p B p A + (1 + a b)] () from he lef end of he line. Fixed and marginal cos of producion are consan, and (wihou loss of generaliy) normalized o be zero. Le c(y) > 0 be he locaion cos funcion, where y is he disance from he firm s locaion o he neares end of he line (y = a for firm A, y = b for firm B). We assume ha c is wice coninuously differeniable. In a familiar way (see e.g. Marin (00)), now allowing for he cos of locaion, he objecive funcions of firm A and B (condiional on prices and locaions) are π A (p A, p B, a, b) = p A c (a) p A < p B (1 a b) and 1 p A [(1 + a b) + p B p A ] c (a) p A p B (1 a b) c (a) p A > p B (1 a b) π B (p A, p B, a, b) = p B c (b) p B < p A (1 a b) 1 p B [(1 a + b) + p A p B ] c (b) p A p B (1 a b) c (b) p B > p A (1 a b) respecively. The exisence of locaion cos rules ou back-o-back, zero-price equilibria (which would imply negaive payoffs). Condiions for he exisence of puresraegy price equilibria wih firms a differen locaions are due o d Aspremon e al. (1979); we sae hem in he form given by Marin (00). (4) (5) 4

5 Proposiion 1 (d Aspremon e al. (1979)) For a+b < 1, a pure-sraegy price equilibrium exiss if, and only if a + b 6 b (6) b + a 6 a. (7) If (6) and (7) are saisfied, equilibrium prices and payoff s, given locaions, are ( p A(a, b) = 1 + a b ), (8) ( p B(a, b) = 1 a b ), (9) π A = 1 (p A) c (a), (10) π B = 1 (p B) c (b), (11) where aserisks denoe second-sage equilibrium values, aking locaions as given. Proof. d Aspremon e al. (1979). As d Aspremon e al. remark, for symmeric locaions, (6) and (7) simplify o a = b 1 4. (1). Sage 1: choice of locaion Le π A (a, b) denoe firm A s sage 1 payoff funcion in he Hoelling model wihou locaion cos, so ha π A (a, b) = π A (a, b) c (a). Proposiion Necessary and suffi cien condiions for he exisence of a subgame perfec pure-sraegy locaion-price equilibrium are (a) ha (6) and (7) be saisfied for locaions saisfying he locaion firs-order condiions and (1 + a b (1 + b a (provided he implied a 0, b 1), (b) he locaion second-order condiions and ) c (a ) 0 (1) ) c (b ) 0, (14) 9 c (a ) < 0 (15) 9 c (b ) < 0; (16) 5

6 (c) he paricipaion consrains π A (a, b ) = 1 c (a ) 0 (17) π B (a, b ) = 1 c (b ) 0; (18) and (d) c (a) rises more rapidly han π A (a, b ) over he range 1 4 a ã (b ), where ã (b ) is A s bes-response locaion o b in he game wihou locaion cos. Proof. The presence of cos of locaion ha is sunk in he pricing sage does no aler he region where here is a pure-sraegy price equilibrium or he region where here is a mixed-sraegy price equilibrium, save ha i rules ou back-o-back, zero-price equilibria. Subsiue (8) and (9) ino (10) and (11), respecively, o obain expressions for he firs-sage objecive funcions. The firs- and second-order condiions are immediae. The firs-order condiions imply ha equilibrium is symmeric. Then p A(a, b ) =. (19) Wih (10) and he requiremen ha equilibrium payoffs be nonnegaive, one obains he paricipaion consrains (17) and (18). This esablishes ha (a, a ) is a local maximum pair of locaions on 0 a 1 4. Turning o condiion (d), he locaion firs-order condiions (1), (14) imply ha if c (a) is suffi cienly small on 1 4 a ã (b ), here is a local maximum pair of locaions in he range 1 4 < a < 0.7 (Osborne and Pichik (1986)), and his is he global maximum; if c (a) is suffi cienly large, here is no local maximum pair of locaions in he cenral quariles. For inermediae values of c (a), payoff funcions have wo local maxima, and he global maximum is in he ouer quarile; condiion (d) is suffi cien for he laer case o hold. 8 The firs-order condiions imply c > 0; he second-order condiions imply c > 0. Tha is, wih linear ransporaion coss for a pure-sraegy pricelocaion equilibrium o exis i is necessary ha he cos of locaion rises oward he cener of he Hoelling line a an increasing rae. Corollary (a) Locaion bes-response lines have negaive slope, da db = brf π A (a,b) a b = π A (a,b) a 1 9 π A (a,b) < 0 (0) a (he denominaor on he righ is posiive by he second-order condiion), and (b) Increases in move he symmeric equilibrium locaion oward he cener of he line, da d = 1 c (a > 0. (1) ) 6

7 . Example I Le he locaion cos funcion ake he form 9 c (y) = y β, () for y = a, b (now omiing aserisks where possible wihou confusion, for noaional compacness). We show below ha β > 1 is one of he condiions for he exisence of a pure-sraegy price-locaion equilibrium. For β > 1, he locaion cos funcion () is a proper fracion raised o a power greaer han 1. Larger values of β hen imply smaller locaion cos (see Figure 1). 0.0 a β β = β = 1.8. β = a Figure 1: Renal cos funcion, example I, β = 1., 1.8,....1 Equilibrium locaions For firm A, (1) gives he firs-order condiion (similarly for firm B) π A (a, b) = 1 ( a 1 + a b ) βa β 1 0, () from which he symmeric equilibrium locaions are ( ) 1 a β 1 =. (4) β For (4) o be valid, he equilibrium i idenifies mus saisfy four condiions (i) he second-order condiion of he locaion sage, (ii) he d Aspremon e al. symmeric-equilibrium ouer-quarile condiion (1), (iii) he firm paricipaion (nonnegaive profi) condiion, and (iv) he sabiliy condiion in locaion space. 7

8 .. Second-order condiions Evaluaing he second derivaive for he equilibrium locaion (4), he secondorder condiion is π A a = 1 ( ) β β 1 β(β 1) < 0. (5) 9 β (5) defines he region in (, β)-space where he second-order condiion is saisfied. β > 1 is necessary for he second-order condiion o be saisfied. For = 1, he second-order condiion is saisfied for all β > 1... Ouer-quarile condiion The ouer-quarile condiion, which we assume is me, is a = ( ) 1 β 1 1 β 4. (6) (6) defines he region in (, β)-space where he ouer-quarile condiion is saisfied. If in (6) we normalize = 1, hen, numerically, 10 he ouer-quarile condiion is saisfied for 1 < β.44. (7)..4 Firm paricipaion consrain From (8) and (9), symmeric equilibrium prices are p A = p B =. (8) In equilibrium, each firm supplies half he marke, ha is, 11 Then he paricipaion consrain is q A = q B = 1. (9) π A = 1 ( ) β β 1 0, (0) β which defines he region in (, β)-space where he firm paricipaion condiion is saisfied. For, he firm paricipaion consrain is me for all β > Sage 1 sabiliy Necessary condiions for sabiliy are ha he race of he marix of second-order parial derivaives of payoffs funcions be negaive, and he deerminan posiive, when evaluaed a equilibrium values. The assumpion ha he second-order condiions are me means ha he race condiion is me. 8

9 The sabiliy marix is wih deerminan ( ) β 1 9 β(β 1) β 1 β 1 9 ( β(β 1) ( ) β β 1 β(β 1) β [ β(β 1) 9 ( β The deerminan of he sabiliy marix is posiive if β ) β β 1, (1) ) β ] β 1. () ( ) β β 1 β(β 1) < 0, () 9 β a sronger condiion han (5). If we normalize = 1, condiion () is me for all β > Locaion bes-responses 0.5 b A s brf: β = A s brf: β = (a, b ), β =. B s brf:β = (a, b ), β = B s brf: β = a Figure : Locaion bes response curves wih linear ransporaion cos and cos of locaion c(y) = y β. Firm A s firs-order condiion () can be solved for b, 9

10 b = a + 9 β aβ 1, (4) o obain an equaion for firm A s locaion bes-response equaion, wrien in inverse form. Figure shows bess response curves and equilibrium locaions for wo values of β, β = and β =. 1 Bes-response lines slope downward: locaion choices are sraegic subsiues. A larger value of β means smaller locaion cos, all else equal, and (as one expecs), he equilibrium locaion shifs oward he cener of he line as β increases. In sum, for ransporaion cos = 1 and for β saisfying ) condiion (d) of proposiion, he price-locaion pair (p, a ) = (1, (β) 1 1 β is a pure sraegy subgame perfec Nash equilibrium for he Hoelling model wih linear ransporaion cos and cos of locaion (4) for β (1,.44]. Figure shows he equilibrium locaions over he admissible range of β..4 β. 1 1/4 a, b Figure : Equilibrium locaions in he Hoelling model as a funcion of β, cos of locaion c(x) = x β ( = 1). 4 Quadraic ransporaion cos 4.1 Sage : price seing Following d Aspremon e al. (1979), assume now ha a consumer locaed a x has ne uiliy U(x; a) = v x a p A (5) if buying from firm A, U(x; b) = v 1 b x p B (6) 10

11 if buying from firm B. The locaion of he marginal consumer is x = a + p B p A (1 a b) + 1 a b, (7) from which he profis of boh firms condiional on price and locaion follow 1 and π A (p A, p B, a, b) = p A c(a) a + p B p A (1 a b) + 1 a b > 1; ] p A [a + p B p A (1 a b) + 1 a b c(a) 0 a + p B p A (1 a b) + 1 a b 1; c(a) a + p B p A (1 a b) + 1 a b < 0, π B (p A, p B, a, b) = p B c(b) b + p A p B (1 a b) + 1 a b > 1; ] p B [b + p A p B (1 a b) + 1 a b c(b) 0 b + p A p B (1 a b) + 1 a b 1; (8) c(b) b + p A p B (1 a b) + 1 a b < 0. (9) Absen cos of locaion, d Aspremon e al. (1979) show ha for his siuaion, a unique price equilibrium exiss for any locaions a and b, and ha i is given by ( p A(a, b) = (1 a b) 1 + a b ), (40) ( p B(a, b) = (1 a b) 1 a b ). (41) 4. Sage 1: choice of locaion Equilibrium prices (40), (41) fall as a firm approaches he rival s locaion. d Aspremon e al. (1979) show ha in he wo-sage game wih quadraic ransporaion cos and wihou cos of locaion, firms locae a he ends of he line, each supplying half he marke a he maximum noncooperaive equilibrium price. Resuls wih locaion cos are given below, and for a simple parameerizaion, equilibrium locaions can be arbirarily close o he cener of he line. Proposiion 4 Necessary and suffi cien condiions for he exisence of a subgame pure-sraegy locaion-price equilibrium are he locaion firs-order condiions 18 ( + a b )(1 + a + b ) c (a ) 0, (4) 11

12 and 18 ( + b a )(1 + b + a ) c (b ) 0, (4) (provided he implied a 0, b 1), he locaion second-order condiions 9 (5 + a b ) c (a ) < 0, (44) and 9 (5 + b a ) c (b ) < 0, (45) and he firm paricipaion consrains π A (a, b ) = (1 a b ) c (a ) 0, (46) and π B (a, b ) = (1 a b ) c (b ) 0. (47) Proof. Subsiue (40) and (41) ino (8) and (9), respecively, o obain expressions for he firs-sage objecive funcions. The firs- and second-order condiions are immediae. The firs-order condiions imply ha equilibrium is symmeric. Then boundary consumers are locaed a he cener of he line, x = 1/, and equilibrium prices are p A(a, b ) = p B(a, b ) = (1 a b ), (48) from which he paricipaion consrains (46) and (47) follow. The firs-order condiions imply ha c < 0. The second-order condiion may be saisfied for c posiive or negaive, provided i is no below a lower bound. Tha is, wih quadraic ransporaion coss for a pure-sraegy pricelocaion equilibrium o exis i is necessary ha he cos of locaion does no rise oward he cener of he Hoelling line. Proposiion 4 implies Corollary 5 (a) Locaion bes-response lines have negaive slope, da db = brf π A (a,b) a b π A (a,b) a = 1 9 (1 a b) π A (a,b) < 0 (49) a (he denominaor on he righ is posiive by he second-order condiion), and (b) Increases in move he symmeric equilibrium locaion oward he end of he line, a = a 6 + c < 0. (50) (a) (c (a) /9(5 + a b by he second-order condiion, so he denominaor on he righ is posiive). 1

13 4. Example II Le he locaion cos funcion ake he form 14 c (a) = ρ ( ) 1 a, (51) where ρ > 0 is a locaion cos scale parameer. For his specificaion, locaion cos akes is maximum value a he ends of he line, and falls o 0 a he cener of line (Figure 4). 0.5 c(a) 0.0 c (a) = ( 1 a) a Figure 4: Renal cos funcion, example II, ρ = 1. I is immediae ha he locaion second-order condiions (44) and (45) are saisfied for (51) Equilibrium locaions For he locaion cos funcion (51), firm A s sage 1 objecive funcion is π A (a, b) = 1 ( (1 a b) 1 + a b ) ( ) 1 ρ a. (5) Firm A s locaion firs-order condiion is π A (a, b) = ( 1 + a b a 6 In symmeric equilibrium, ) (1 + a + b) + ρ ρ ( ) 1 a 0. (5) a = 1 ρ 6 ρ + 1 = ρ + 1. (54) 1

14 a 0 requires ρ ρ min 6, (55) and we assume his condiion is me. From (54), a increases as ρ increases. Tha is, a increases as ρ increases and falls as increases, ceeris paribus. (54) implies ha a is never a he cener of he line, alhough i approaches he cener of he line asympoically as ρ, or 0 for a given ρ (Figure 5). a a = 1 ρ 1 6 ρ ρ/ Figure 5: Equilibrium locaion as a funcion of ρ/. 4.. Firm paricipaion consrain The symmeric equilibrium payoff is π A = ( ) 16 (9ρ + 4) > 0. (56) ρ + The firm paricipaion consrain is always me. 4.. Sage 1 sabiliy The marix of equilibrium second derivaives of payoff funcions is ( ) 1 (6ρ+)(ρ+) ρ+ 1 6 ρ+ 1 6 ρ+ 1 (6ρ+)(ρ+) ρ+. (57) 14

15 The race is negaive. The deerminan of he sabiliy marix is he sabiliy condiion is saisfied. ( 18ρ + 1ρ + ) > 0; (58) Locaion bes-responses The firs-order condiion (5) implicily defines firm A s locaion bes-response funcion. Bes response curves slope downward and, as ρ/ rises, equilibrium locaions move oward he cener of he line (Figure 6). b 0.5 A s brf,. ρ = 1 A s. brf, ρ = (a, b ), ρ =. B s brf, ρ = 0.1 (a, b ), ρ = 1. B s brf, ρ = a Figure 6: Locaion bes-response curves wih quadraic ransporaion cos and cos of locaion c(y) = ρ(1/ y), ρ = 1 and ρ =. In sum, adding a cos of locaion ha declines moving oward he cener of he line o he Hoelling model wih quadraic ransporaion coss can reverse he conclusion of d Aspremon e al (1979). In paricular, i can yield a siuaion approaching endogenous minimum spaial differeniaion ha does no suffer from he problems in Hoelling (199). 5 Conclusion We have moivaed our specificaion wih he observaion ha ren varies wih locaion. In he Hoelling Main Sree model wih linear ransporaion cos, 15

16 if locaions oward he cener of he line command suffi cienly higher ren, ren acs as a cenrifugal force ha induces firms o locae ouside he region where equilibrium prices are in mixed sraegies. In he Hoelling Main Sree model wih quadraic ransporaion cos, if locaions oward he ends of he line command suffi cienly higher ren, ren acs as a cenripeal force ha induces firms o locae oward he cener of he line. We envisage exending he presen work o derive he equilibrium renlocaion relaionship from he disribuion of he populaion. This will permi us o examine condiions leading o he hollowing-ou of cener ciies ha is observed in he Unied Saes from he laer par of he wenieh cenury. Noes 1 Safire (009) aribues he phrase o he lae Lord Harold Samuel, a Briish real esae ycoon. Cos of locaion is somehing oher han relocaion coss. The laer sars from a paricular siuaion, and hen ells a dynamic sory (alhough ofen wihin a saic framework). Our design corresponds o he case of a firm ha mus incur a locaion-dependen renal cos o se-up before i can se price. See Karmon (01) for a comparison of cener-ciy and suburban renal coss for seleced U.S. ciies, and Bernard (010) for a ciy-suburbs comparison of he overall cos-of-living. 4 See heir foonoe 1. See also Vickrey (1964, 1999). 5 For surveys, see Archibald e al. (1986), Morris (1997). 6 These references do no include he closely-relaed lieraure ha models basing-poin and oher spaial pricing policies. 7 Tha is, he marke is covered. The analysis produces condiions on v for he marke o be covered. 8 If he payoff funcion is differeniable, he condiion is c (a) π A(a,ã) a for 0.5 a 0.7. The condiion can be expressed in erms of discree changes. 9 Qualiaively similar resuls are obained for he quadraic locaion cos funcion c (y) = ry, where r > The upper limi solves (β) 1 β 1 = 1/4. 11 Since firms locae in he ouer quariles, a firm s mos disan cusomers are locaed a he cener of he line. The ne uiliy of such a consumer is 16

17 ] v [/ (/β) 1 β 1, and he marke coverage condiion is ha his be nonnegaive. The marke coverage condiion on v, which we assume is me, can be wrien v ( / (/β) 1/(β 1)). 1 The corresponding equilinrium locaions are 1/1 and 1/6, respecively. 1 The marke coverage condiion on v is v 5/4. 14 Qualiaively similar resuls are obained for he linear locaion cos funcion c(a) = γ (γ a), wih γ > 0. References [1] Anderson, Simon P. and Engers, Maxime (1994). Spaial compeiion wih price-aking firms. Economica 61: [] Anderson, Simon P., Goeree, Jacob K. and Ramer, Roald (1997). Locaion, Locaion, Locaion. Journal of Economic Theory 77: [] Anderson, Simon P. and Neven, Damien J. (1991). Courno compeiion yields spaial agglomeraion. Inernaional Economic Review : [4] Archibald, G. C., B. Curis Eaon and Richard G. Lipsey (1986). Address models of value heory, in New Developmens in he Analysis of Marke Srucure, Joseph E. Sigliz and G. Frank Mahewson (ediors). Cambridge, Massachuses: MIT Press, pp. 47. [5] d Aspremon, Claude, Gabszewicz, Jean Jaskold and Thisse, Jacques François (1979). On Hoelling s Sabiliy in compeiion. Economerica 47: [6] Bernard, Tara Siegel ( July 010). High-rise, or house wih yard? New York Times. Downloaded July 01 from URL hp:// [7] Capozza, Dennis R. and Van Order, Rober (198). Produc differeniaion and he consisency of monopolisic compeiion: a spaial perspecive. Journal of Indusrial Economics 1: 7 9. [8] Chamberlin, Edward H. (19). The Theory of Monopolisic Compeiion. Cambridge, Massachuses: Harvard Universiy Press. [9] (195). The produc as an economic variable. Quarerly Journal of Economics 67: 1 9. [10] Eaon, B. Curis and Ware, Roger (1987). A heory of marke srucure wih sequenial enry. Rand Journal of Economics 18: [11] Economides, Nicholas (1984). The Principle of Minimum Differeniaion revisied. European Economic Review 4:

18 [1] (1986). Minimal and maximal produc differeniaion in Hoelling s duopoly. Economics Leers 1: [1] (1989). Symmeric equilibrium exisence and opimaliy in differeniaed produc markes. Journal of Economic Theory 47: [14] Gupa, B., Pal, Debashis, and Sarkar, J. (1997). Courno compeiion and agglomeraion in a model of locaion choice. Regional Science and Urban Economics 7: [15] Hamilon, Jonahan H., Klein, James F., Sheshinski, Eyan, and Slusky, Seven M. (1994). Quaniy compeiion in a spaial model. Canadian Journal of Economics 7: [16] Hinloopen, Jeroen (00). Price regulaion in a spaial duopoly wih possible non-buyers. Annals of Regional Science 6:19 9. [17] Hinloopen, Jeroen and van Marrewijk, Charles (1999). On he limis and possibiliies of he principle of minimum differeniaion. Inernaional Journal of Indusrial Organizaion 17: [18] Hoelling, Harold H. (199). Sabiliy in compeiion. Economic Journal 9: [19] Irmen, Andreas and Thisse, Jacques François (1998). Compeiion in mulicharacerisics spaces: Hoelling was almos righ. Journal of Economic Theory 78: [0] Karmon, Jennifer (15 November 01). Renal coss, ciy vs. suburbs: a handy infographic. Downloaded July 01 from URL hp://homes.yahoo.com/blogs/spaces/renal-coss-ciy-vs-suburbshandy-infographic hml. [1] Lerner, Abba P. and Singer, H. W. (197). Some noes on duopoly and spaial compeiion. Journal of Poliical Economy 45: [] Marin, Sephen (00). Advanced Indusrial Economics. Blackwell Publishers. [] Morris, Claire (1997). Address models of produc differeniaion: a survey, Sudies in Economics 971, Deparmen of Economics, Universiy of Ken, December. [4] Osborne, Marin J. and Pichik, Carolyn (1987). Equilibrium in Hoelling s model of spaial compeiion. Economerica 55: pp [5] Presco, Edward C. and Michael Visscher (1977). Sequenial locaion among firms wih foresigh. Bell Journal of Economics 8:

19 [6] Safire, William On language: locaion, locaion, locaion. (June 6, 009). New York Times (hp:// [7] Salop, Seven C. (1979). Monopolisic compeiion wih ouside goods. Bell Journal of Economics 10: [8] Samuelson, Paul A. (1967). The monopolisic compeiion revoluion. Pp in Kuenne, Rober E., edior. Monopolisic Compeiion Theory: Sudies in Impac. New York: John Wiley & Sons. [9] Shaked, A. (198). Exisence and compuaion of mixed sraegy Nash equilibrium for firms locaion problem. Journal of Indusrial Economics 1: [0] Soeeven, Adriaan R. (010). Price compeiion on graphs. Tinbergen Insiue Discussion Paper /1. [1] Sigler, George J. (1964) A heory of oligopoly. Journal of Poliical Economy 7: 44 61; reprined in Sigler, George J. (1968) The Organizaion of Indusry. Homewood, Illinois: Richard D. Irwin, Inc [] Vickrey, William S. (1964). Microsaics. Harcour, Brace and World, New York. [] (1999). Spaial compeiion, monopolisic compeiion, and opimum produc diversiy. Inernaional Journal of Indusrial Organizaion 17: