Although this may seem simplistic, firms competing in prices can obtain three possible outcomes:

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1 Economics 101: handou par i) Hoelling s game Hoelling s game represens a form of compeiion. In is wo inerpreaions, wo enrepreneurs are choosing eiher he locaion of heir business fixing marke prices) or he price of heir goods fixing heir locaions). For he sake of argumen, we assume heir goods are perfec subsiues so ha his is compeiion in he mos sraighforward sense. Alhough Hoelling s iniial paper in 199 did no specify his, mos modern akes on he game represen i as wo merchans rying o arac cusomers from along a beach; we ll assume hey re selling ice cream, bu could jus as well be rashy -shirs or various chochkes. Firms face 0 coss of producion. The beach is represened as he real numbers from 0 o L ha is, he beach is of lengh L and consumers are spread evenly along i. Each consumer desires one serving of ice cream, and doesn care which sore he ges i from. Imporanly, cusomers dislike having o walk along he beach lazy Americans!) and if hey have o walk disance d o ge o he ice cream shop, hey resen i o he une of d where is he cos per uni walked here, we assume ha > 0 and ha he cos is subraced from uiliy; we could also assume ha < 0 and he cos is added o uiliy he mah is idenical bu I consider i simpler o hink abou subracing coss as posiive quaniies; i is ruly a maer of personal preference). To clarify wha is mean when we say, consumers are spread evenly along [he beach], le s use an example: suppose all consumers louging beween 0 and L decide o ge heir ice cream from firm A, while all consumers lounging beween L and L decide o ge heir ice cream from firm B. We hen say ha firm A sells L 0 = L unis of ice cream, while firm B sells L L = L unis. Mahemaically, each so-called consumer is infiniesimal, and we inegrae he mass of consumers going o a paricular ice cream shop o deermine ha firm s marke demand. However, since we assume ha cusomers are evenly disribued we can avoid any idea of inegraion and jus look a inervals conaining cusomers which go o one firm or anoher. Since cusomers don care one way or anoher which firm hey ge ice cream from, and hey really wan he one uni hey demand, heir choice of supplier will revolve around wo hings: he price he firms are charging, and he disance o walk o one firm or anoher. A his poin, i will be useful o assume ha firm A is locaed a poin a, and firm B is locaed a poin b along he boardwalk. To keep analysis simple, we ll assume ha a < b. Suppose ha some consumer is locaed a poin x along he beach. If his cusomer buys ice cream from firm A, she will have o pay price p A for he ice cream cone and will also have o walk x a o ge o he shop. Her ransporaion cos is hen x a, giving her a oal cos of p A + x a. Similarly, if she ges ice cream from firm B her oal cos will be p B + x b. Since she wans ice cream no maer wha, she will look only o minimize her cos and will buy from firm A if p A + x a < p B + x b, from firm B if p B + x b < p A + x a, and we will allow her behaviour o vary as fis our needs if she is indifferen beween he wo. Alhough his may seem simplisic, firms compeing in prices can obain hree possible oucomes: a) The firm can be oupriced by is rival, and obain 0 profis. b) The firm can ouprice is rival and receive he enire marke. Is profis are hen Lp i. c) The firm can share he marke wih is rival and receive m i as a fracion of he marke. Is profis are hen m i p i. Since hose facs are fairly obvious, a good quesion is when one siuaion or anoher migh obain. We ll assume ha we re alking abou firm A for now; hopefully i s obvious how his discussion will generalize o firm B s problem. a) Suppose firm A is oupriced by firm B, and π A = 0. Then by assumpion, no cusomer wans o purchase from firm A; formally, for all x, p A + x a > p B + x b. Suppose ha x > b; hen we February 1, 011 1

2 Economics 101: handou par i) have This ells us p A + x a) = p A + x a > p B + x b = p B + x b) p A > p B + a b) Similarly, suppose x < a; hen we have p A + a x) = p A + x a > p B + x b = p B + b x) This ells us p A > p B + b a) Lasly, suppose a < x < b; hen we have p A + x a) = p A + x a > p B + x b = p B + b x) This ells us p A > p B + a + b x) We can use algebra o see ha p B + b a) > p B + a + b x) > p B + a b) So for his zero-marke condiion o hold, we need p A > p B + b a) b) Suppose firm A ouprices firm B, and π A = Lp A. This is simply he converse of he firs siuaion above in which firm B ouprices firm A and so by analogy his will happen when p A < p B b a) c) Suppose he firms spli he marke. From he previous wo analyses, his will happen when p A p B b a), p B + b a) ) Firsly, noice ha since he marke is spli here mus be some cusomer x who is willing o buy from firm A, p A + x a < p B + x b. Suppose ha his cusomer is such ha x < b. Two cases arise if you like, skip pas his since i s jus a mess of algebra): x [a, b). Pick some x, a x < x. Then x a = x a < x a = x a, and x b = b x > b x = x b. So we have p A + x a < p A + x a < p B + x b < p B + x b and he cusomer a x prefers firm A o firm B on a price basis. x a. Pick some x < x. Then x a = a x = a x x + x = a x + x x and Then we see x b = b x = b x x + x = b x + x x p A + x a = p A + a x + x x < p B + b x + x x = p B + x b February 1, 011

3 Economics 101: handou par i) and so any consumer o he lef of x also prefers firm A o firm B. Pick some x, x < x a. Then x a = a x = a x x + x = a x x x and x b = b x = b x x + x = b x x x Then we see p A + x a = p A + a x x x < p B + b x x x = p B + x b and so any consumer beween x and a also prefers firm A o firm B. The moral of he sory is ha if he consumer a x prefers firm A o firm B, all consumers o he lef of x also prefer firm A o firm B. Similar logic will apply o hose consumers who prefer firm B coninuing o he righ of any one consumer who prefers firm B. This moral is derived from his: if he consumer a x prefers firm A o firm B, hen he consumer a a will prefer firm A o firm B. Then all cusomer o he lef of a prefer firm A o firm B. The analogous resul for firm B will hold, noing ha if here is a consumer wih x > b who prefers firm A o firm B hen no consumers wih x > b will prefer firm B as if here was such a cusomer x, all cusomers beween b and L would prefer B, conradicing he fac ha x > b prefers A). So in order o spli he marke, here mus be some consumer x and some consumer y, boh beween a and b, one of whom prefers firm A and one of whom prefers firm B. If his is he case, hen here is a consumer x who is perfecly indifferen beween purchasing ice cream from firm A and purchasing ice cream from firm B. This cusomer s coss mus be p A + x a) = p B + b x ) = x = 1 b + a + p ) B p A where we were able o do away wih absolue values since a < x < b. From he previous discussion, all consumers o he lef of x mus prefer firm A o firm B, and all consumers o he righ of x mus prefer firm B o firm A. Firm A s profis are hen π A = p A x = 1 p A b + a + p ) B p A So we now know profis in he hree cases, in which firm A obains none of he marke, some of he marke paricularly he lef porion of he marke), and all of he marke. Suppose for now ha he wo firms share he marke. We can compue bes responses in he usual way, by aking derivaives. Similarly, for firm B, 1 max π A p A, p B ) = max p A p A p A = 0 = b + a + p B p A = p A = 1 b + a) + p B) max p B π B p B, p A ) = max p B = 0 = L 1 p B b + a + p B p A p A ) b + a + p B p A L 1 b + a + p B p A = p B = L 1 b + a) p A) ) p B )) February 1, 011

4 Economics 101: handou par i) To find a candidae Nash equilibrium, we subsiue in o find he poin in prices) a which he wo besresponse funcions cross. p A = 1 b + a) + L 1 ) b + a) p A) 4 p A = 1 ) 1 b + a) + L p A = 1 L + b + a) To find candidae p B, we can eiher apply he same mehod o obain he crossing poin or jus subsiue in for wha we now know p A o be. p B = L 1 b + a) 1 ) L + b + a) p B = 4 L 1 b + a)) p B = 1 4L b + a)) So we now have he prices which will be used if firms A and B are forced/decide o share he marke. This allows us o explicily compue he locaion of he indifferen consumer, x = 1 b + a + p B ) p A x = 1 b + a + 1 ) 4L b + a) L + b + a)) x = 1 b + a + 1 ) L b + a)) From his, we obain x = 1 L + b + a) 6 π A p A, p B) = p Ax [ ] [ 1 1 = L + b + a) 6 πa = L + b + a) 18 ] L + b + a) π B p A, p B) = p B L x ) [ ] 1 = 4L b + a)) [L 16 ] L + b + a) = 4L b + a)) L 1 ) 6 b + a) πb = 4L b + a)) 18 Now we mus address he quesion of when he firms prefer o share he marke. Since 18 > 0, profis a he opimum are always posiive his is obvious for firm A; you can check algebraically ha a < b L implies 4L b + a) > 0). So we know ha neiher firm has incenive o deviae so ha i loses all of he marke o is rival and obains 0 profis. February 1, 011 4

5 Economics 101: handou par i) However, will a firm wan o undercu is rival o obain he enire marke? We can check his for firm A alone, as he logic will hold idenically for firm B. Recall ha firm A obains he enire marke if i ses p A < p B b a). When we seup he problem, we allowed for some fudging of consumer indifference condiions, so le s say his: if he consumer is indifferen beween firm A and firm B, he will buy ice cream from he firm wih he lower price for is good no considering ransporaion coss). Then firm A can obain he enire marke by seing p A p B b a); since any price under his boundary does no affec is demand curve, firm A will choose he larges price allowable and will se p A = p B b a). A his price, profis are π A = L p B b a) ) = L 4L b + a) b a)) = L L b + a) Wih hese profis in mind, will firm A ever wan o ouprice firm B? We check hese profis agains is profis from sharing he marke. L L b + a) L + b + a) 18 1L L b + a) L + b + a) 4L 4Lb + 1La 4L + 4Lb + 4La + b + ba + a 0L 8Lb + 8La b + ba + a If he lef-hand side is bigger, firm A prefers o seal he marke from B; if he righ-hand side is bigger, firm A prefers o share he marke. Since his mah is ugly, le s ry a few cases o see wha s going on. b = L. Then he relaion above is 0L 8L + 8La = 8L + 8La L + La + a Since a < b = L, he lef-hand side is always smaller han he righ-hand side and firm A prefers o share he marke regardless. Tha is, if firm B s posiion is disadvanageous enough firm A is willing o share raher han undercu, osensibly because firm A will obain very few cusomers from undercuing firm B. Noe ha his will hold if b is sufficienly close o L. a = 0. Then he relaion above is 0L 8Lb + 8La = 0L 8Lb b = b + ab + a Then he lef-hand side will be larger so long as b + 8Lb 0L < 0 According o he quadraic formula, he roos for his form are 8L ± 8L) + 80L = 14 ± ) 16 L Since his is an upward-facing quadraic in b, we see ha he lef-hand side is larger when b 14L 16L, 14L + 16L). Clearly, he lef-hand bound of his inerval is negaive and ou of he range of his quesion. So according o he righ-hand bound, if b < 0.697L he lef-hand side of he profi comparison inequaliy is larger, and firm A will prefer o seal he marke from firm B. February 1, 011 5

6 Economics 101: handou par i) Here s he kicker: suppose ha we have a siuaion in which firm A does prefer o seal he marke from firm B. Firm B has a well-defined bes response o his acion by firm A: since i chooses p B = L 1 b + a) + 1 p A is bes response will fall when p A drops. Noably, since b + a < L is price will be posiive so long as p A is posiive. So here is no poin a which A can seal he enire marke and earn posiive profis! This is a case in which Nash equilibrium does no necessarily exis; evenually, o coninue undercuing firm B firm A will need o se prices o be negaive, which is clearly a supid decision. Is opion hen is o share he marke, bu if he wo firms choose o share he marke firm A wans o undercu! There is no equilibrium his parameerizaion of his game. So in order o obain Nash equilibrium, we need b o be sufficienly large compared o a. If his condiion holds as represened in he profi comparison inequaliy above) hen he Nash equilibrium in prices will be p A = 1 L + b + a)) p B = 1 4L b + a)) Noe ha in class we did no have his issue: we made he seemingly innocuous assumpion ha every consumer o he lef of firm A will buy from firm A since she mus walk pas firm A o ge o firm B). While his assumpion isn unreasonable on is face, we can now see ha i changes he oucome of he game measurably noably, i makes Nash equilibrium in prices exis where i may no wihou he assumpion! We could, of course, inroduce mechanisms o suppor his decision in a fi of inervenion, he governmen mandaes ha you canno walk pas an ice cream shop wihou buying some ice cream) bu ha was no he spiri in which he assumpion was inroduced in secion. Imporanly, if he exisence of Nash equilibrium requires b o be significanly large relaive o a, all previous argumens from secion) abou making his a wo-sage game in which locaion is chosen firs fall apar. As you recall, we decided ha i was only reasonable for he wo firms o locae a he same poin; bu he small-deviaions argumen assumed price compeiion had a well-defined Nash equilibrium! Since we canno rely on his anymore, he locaion game argumen becomes somewha nonsensical. Wha causes Nash equilibrium o no exis here? In a word, disconinuiy. The individual firms besresponse funcions are disconinuous in he acions of heir opponens: if p B is high enough, firm A seals he enire marke. A he poin a which p B ransiions from firm A waning o share o firm A waning o seal, firm A s profis jump from he sharing level o he sealing level since i capures a nonnegligible marke segmen from an infiniesimal change in price. This jump causes issues wih he exisence of Nash equilibrium. Hoelling s locaion game To wrap up, le s discuss a simpler version of Hoelling s game. In his model, marke prices are fixed a p and he firms are choosing a and b, he locaion of heir ice cream sands. Since prices are idenical, consumers will walk o he firm which is closer. If firms are equidisan, we can hink of a consumer as having a 50% chance of going o eiher locaion his will make more sense when we apply i laer). Assume ha firms choose locaions a and b, a < b. Since prices are idenical, we see from above) ha x = 1 a + b + p ) B p A = 1 a + b) February 1, 011 6

7 Economics 101: handou par i) Tha is, he indifferen consumer is locaed exacly halfway beween firms A and B his holds regardless of wheher a < b or b < a, bu le s keep he convenion going). Then firm A s profis are π A = px = p a + b) Since a < b, here is some a such ha a < a < b. Then we have p a + b) < p a + b) so firm A prefers o deviae o locaion a raher han remain a a. This argumen holds for all a < b, so we canno have a < b in a Nash equilibrium. Noice ha we could say he same hing abou supporing a Nash equilibrium wih b < a. So our only candidae for Nash equilibrium is a = b, boh firms locaing a he same poin. If his is he case, each firm is equidisan o all consumers, so each consumer has a 50% chance of going o eiher firm. Think of i his way: you walk down he beach boardwalk o ge ice cream, and here are wo no-name ice cream shacks righ nex o each oher charging he same price. Absen any oher informaion Yelp is no allowed), you ll jus randomly choose which one you go o. Wih his consrucion, each firm receives 50% of he marke, or L Lp consumers. Profis o each firm are. Suppose ha a = b > L. Firm A can choose a such ha L < a < a, and p a + b) > Lp So by shifing a lile closer o he cener of he marke firm A is able o earn higher profis. Firm B faces he same decision, so a = b > L canno consiue a Nash equilibrium. Similarly, if a = b < L boh firms also face an incenive o shif closer o he cener of he marke, alhough in his case hey are deviaing righward. This ells us ha our only candidae equilibrium is a = b = L. Suppose firm A deviaes o some a < a from his sraegy. Then is profis are p a + b) = a p + Lp 4 < ap + Lp 4 = Lp So by deviaing i reduces is own payoff. I follows ha Nash equilibrium in he Hoelling locaion game where prices are exogenously fixed by he marke! is boh firms locaing a he same poin, each precisely halfway along he beach. Now, many people consider his o be a reason for why gas saions ofen crowd he same corner, or you find auo-repair shops all along he same drag in a own. There is some meri o his argumen, bu here are addiional complexiies regarding zoning and oher concerns which sar confounding he issue. Hoelling s analysis has found a much differen home, however, in he world of poliical heory. Consider he elecorae of a counry as exising along a one-dimensional coninuum of beliefs he poliical beach, if you will); a candidae who wans o win office mus selec a locaion along his coninuum find a poliical posiion o campaign for and hen arac voers. If voers are fixed in heir locaions along he coninuum, i urns ou o be opimal for poliical candidaes o locae precisely along he middle of he poliical specrum; since middle is a lile bi of a weasel word, i will help o menion ha his phenomenon is referred o as he Median Voer Theorem and is well-described on Wikipedia). Of course, i makes some silly assumpions poliical beliefs are no one-dimensional, alhough in a wo-pary sysem hey can appear o be bu we can acually see his inuiively in American elecions: in he primaries, candidaes end o sake ou more exreme posiions han in he general elecions. This is because in he primaries, he middle of he poliical specrum is exremely skewed Republican candidaes face only, roughly, he righ half of he specrum while Democraic candidaes face only he lef half) while in he general elecion a candidae will face he elecorae as a whole. Poliical commenaors ofen bring his up when discussing how a candidae s abiliy o appeal o her base may negaively reflec on her chances of winning he overall elecion. February 1, 011 7