LOCATION CHOICE OF FIRMS UNDER STACKELBERG INFORMATION ASYMMETRY. Serhij Melnikov 1,2

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1 TRANPORT & OGITI: he Inernaonal Journal Arcle hsory: Receved 8 March 8 Acceed Arl 8 Avalable onlne 5 Arl 8 IN 46-6 Arcle caon nfo: Melnov,., ocaon choce of frms under acelberg nformaon asymmery. Transor & ogscs: The Inernaonal Journal, 8; Volume 8, Issue 44, Arl 8, IN 46-6 OATION HOIE O IRM UNDER TAKEBERG INORMATION AYMMETRY erh Melnov, Insue of Marme Busness, Odessa Naonal Marme Unversy, Urane, el: , e-mal: nfn333@ur.ne Dearmen of Transor ervces Mare, Insue for Mare Problems and Economcand-Ecologcal Research, Urane, el: , e-mal: nfn333@ur.ne Absrac: Ths aer develos he saal duooly model [ang, W.J., Hwang, H. and Ma,.., 6, aal dscrmnaon: Berrand vs. ourno wh asymmerc demands, Regonal cence and Urban Economcs, 36,. 7 8] o analyze he locaon choce of frms under acelberg nformaon asymmery. The comeve game consss of wo sages. In he frs sage, he frms smulaneously selec her locaons. In he second sage, a he gven locaon decsons, he frms smulaneously choose her suled quanes. The equlbrum of he model s solved by bacward nducon. I s obaned ha under ceran condons he nformaon asymmery effec domnaes he mare sze effec and boh frms agglomerae n he small mare. In hs case he omal locaon choce of frms deends on who maes he frs move n he game. Key words: saal duooly, asymmery, agglomeraon INTRODUTION Afer he emergence of he famous Hoellng's wor [], he roblems of agglomeraon and dserson of frms n sace have become a consan subec of economss sudy. In case of rce comeon frms wll dserse, as under agglomeraon her rofs wll decrease unl hey become zero due o he Berrand arado []. In case of quanave comeon, frms wll end o agglomerae [3], [4]. The aer [5] sudes he effecs of saal rce dscrmnaon on ouu, welfare and locaon of a monools n he cone of saal economy. I s shown ha a monools wll be locaed n dfferen mares under dfferen rcng schemes. In arcular, f he sloe of he demand funcon n one mare s hgher han n he oher one, he monools wll be Volume 8, Issue 44, Arl 8 35

2 . Melnov ocaon choce of frms under acelberg nformaon asymmery T& locaed n he frs mare under smle rcng, or n he second mare under dscrmnaory rcng. Invesgaon of agglomeraon and dserson of frms deendng on ransor coss and mare szes s carred ou n he aer [6]. Ths aer [6] develos a barbell model [5] wh homogeneous roduc and asymmerc demands o comare rces, aggregae rofs and socal welfare beween ourno and Berrand comeon, and o analyze he frms' equlbrum locaons. I focuses on he macs of he saal barrer generaed from ransor coss, and he mare sze effec resulng from asymmerc demands. I shows ha he mare-sze effec s crucal n deermnng frms' locaons under ourno comeon, bu nsgnfcan under Berrand comeon. The aer [7] consders a saal dscrmnaon ourno model wh asymmerc demand. The model uses a geograhc nerreaon of he lnear mare and deals wh dfferenaed roducs. The aer analyzes he quanave and saal soluons of frms and shows ha agglomeraon or dsersed locaon may occur deendng on he combnaon of arameers. As s nown, he effecs of asymmery are romsng areas for sudyng saal models [8]-[]. The am of hs arcle s o develo he saal duooly model [6] and o analyze he locaon choce of frms under acelberg nformaon asymmery. THE MODE uose here are wo mares, whch are locaed a he endons of he lne wh a un lengh. The mares are conneced by road or hghway. There s a sze asymmery beween mares. Assume ha he sze of he lef mare (-mare) eceeds he sze of he rgh mare (-mare). There are wo comeng frms, whch can be locaed a any on along a lne. In boh mares, frms sell homogeneous goods and arbrage among consumers s ecluded. Each frm faces lnear ransoraon coss of o move one good un er one un of dsance. A dsance of he h frm o he -mare s, =, (g.). The locaon of frms relave o each oher s no secfed, can be equal o, greaer or less han. Each frm chooses an omal locaon whch can be n one of he wo mares or a a on on he lne. The barbell model fs he realy well and can be used o eamne he rade beween wo counres as well. arge mare s frm nd frm - - mall mare g. The saal duooly model (barbell model) ource: auhor's develomen The lnear demand curves a each mare (): q,, q,, () Volume 8, Issue 44, Arl 8 36

3 . Melnov ocaon choce of frms under acelberg nformaon asymmery T& where, he mare rces, q, q he quanes suled of h frm, a mnmum rce, a whch here s no demand (mare oenal), s equal o, he coeffcen of rce sensvy, γ he mare sze asymmery coeffcen (g.). b q, q, q, g. The mare sze asymmery ource: auhor's develomen The comeve game consss of wo sages. In he frs sage, he frms smulaneously selec her locaons. In he second sage, a he gven locaon decsons, he frms smulaneously choose her suled quanes. The equlbrum of he model s solved by bacward nducon. The rof of h frm s defned as he sum of s rofs from boh mares: q q q q q q ma, THE OURNOT EQUIIBRIUM,, 3.,q, q e us fnd he ourno equlbrum. The analyss sars wh he second sage. rs, we fnd he omal quanes of sules. olvng he frs-order condons yelds he reacon curves: () q q q, q q, q, q q, (3) he second-order condons (4): q, q,,, 3. (4) Volume 8, Issue 44, Arl 8 37

4 . Melnov ocaon choce of frms under acelberg nformaon asymmery T& Volume 8, Issue 44, Arl 8 38 olvng he sysem of equaons (3) yelds he ourno equlbrum quanes of sules:, 3 q. 3 q (5) The equaons (5) show ha he omal volumes of he h frm ncrease when hey are aroachng he mare and he comeors are dsancng from he mare (6): q, q, q, q. (6) The omal rces and rofs are:, 3,,, 3,, (7). In he frs sage each frm selecs a rof-mamzng locaon a a gven locaon of he comeor. ubsuon of (7) no () and dfferenaon wh resec o locaon gves:, 4 4 (8) he second-order condon:. 8 () rom he second-order condon () follows ha he rof funcon of h frm () s srcly conve wh resec o locaon. Thus, a equlbrum sae frms wll be locaed only n he mares,.e. e = or e =. e us noe ha hs resul was frs obaned n [5]. There are four ossble locaons for frms: n he same mares: (, ) = (, ), (, ), and n he dfferen mares: (, ) = (, ), (, ). e us assume ha here s no mare sze asymmery,.e. γ =. Then he rofs of frms (7) n all cases are:,,,, 5,, (). 4,,,, I follows from () ha n he absence of any asymmeres, frms wll choose dfferen mares: (, ) or (, ). Ths s he comeon effec. The locaon of a comeor

5 . Melnov ocaon choce of frms under acelberg nformaon asymmery T& n he oher mare enhances he frm's mare ower. e us noe ha wh he growh of he ransor arff, he mac of he comeon effec s nensfed, >. e us elore how some asymmeres affec he locaon choce of frms. e us consder hree asymmeres: locaon asymmery, mare sze asymmery and acelberg nformaon asymmery. The locaon asymmery s gven n he form of condon: (g.). The locaon asymmery leads o a sngle saal equlbrum: (, ). The mare sze asymmery s gven n he form of condon: γ > (g. ). Under mare sze asymmery, he -mare begns o "arac" boh frms o self. Ths s he mare sze effec. Thus, under he acon of wo asymmeres, wo saal equlbrum saes are ossble: (, ) and (, ). e us analyze a locaon decson of he nd frm:,, 4 f., () Eresson () shows he mnmal ransor arff a whch he nd frm wll reman n he -mare. Wh he reducon of he ransoraon arff, < (γ ) / γ, he -mare wll "arac" he nd frm o self and he agglomeraon of frms wll ae lace. The resul () was frs obaned n [6]. 3 THE TAKEBERG EQUIIBRIA The acelberg nformaon asymmery arses when one of he frms (leader) becomes aware of he comeor's sraegy (follower). Under acelberg nformaon asymmery here are four ossble equlbrum saes (Table ). Tab. The acelberg equlbra Equlbrum I II IV Mares arge mall ource: auhor's develomen leader s follower nd leader s follower nd leader nd follower s leader nd follower s leader s follower nd leader nd follower s nce he frm can be a leader n one mare and a follower n he oher one, le us consder he funcons of frms' rofs searaely n each mare. Assume ha he h frm s a leader, and he h frm s a follower. ubsung he reacon curves of he h frm (3) no he rof funcons of he h frm (), we oban: q ma,,,,,5 q 3 q ma,,, 3.,5 q q q () Volume 8, Issue 44, Arl 8 3

6 . Melnov ocaon choce of frms under acelberg nformaon asymmery T& The frms rofs n he acelberg equlbrum are derved by usng sandard rocedure (Table ). Tab. The frms rofs n he acelberg equlbrum leader -mare 8, -mare 8, follower 3 6, 3 6,,, 3. ource: auhor's develomen,, 3. We wll hen analyze he frms omal locaon under he mac of hree asymmeres. 3. Equlbrum I: s frm leader, nd frm follower n boh mares The frms equlbrum rofs are (4): 8 8, I I (4) If he nd frm chooses he -mare, hen he s frm wll be locaed n he -mare because of he locaon asymmery. e us fnd a locaon decson of s frm f he nd frm s locaed n he -mare (5): I,, I. (5) Thus, n he Equlbrum I he s frm wll always be locaed n he -mare. e us analyze a locaon decson of he nd frm:,, 3 3 6, f I 3. (6) I follows from (6) ha he level of he mnmal arff has decreased (7): 3 3. I (7) Thus, he follower's oson n boh mares srenghens he comeon effec for he nd frm and weaens he mare sze effec. Ths s due o he mac of he nformaon asymmery effec. Volume 8, Issue 44, Arl 8 4

7 . Melnov ocaon choce of frms under acelberg nformaon asymmery T& 3. Equlbrum II: s frm leader, nd frm follower n he -mare; nd frm leader, s frm follower n he -mare The frms equlbrum rofs are (8): 8 3 6, II II (8) If he nd frm chooses he -mare, hen he s frm wll be locaed n he -mare because of he locaon asymmery. e us fnd a locaon decson of he s frm f he nd frm s locaed n he -mare (): II,, II. () Thus, n he Equlbrum II he s frm wll always be locaed n he -mare. e us analyze a locaon decson of he nd frm: II,, II, f II 6 8. () rom () follows ha he level of he mnmal arff has furher decreased (): II I () Thus, he leader's oson n he -mare has furher srenghened he comeon effec for he nd frm. rom () also follows ha he mare sze effec s ossble only a γ > 4/3. A γ 4/3, nd frm wll always be locaed n he -mare. 3.3 Equlbrum : nd frm leader, s frm follower n he -mare; s frm leader, nd frm follower n he -mare The frms equlbrum rofs are (): 3 6 8, () If he nd frm chooses he -mare, hen he s frm wll be locaed n he -mare because of he locaon asymmery. e us fnd a locaon decson of he s frm f he nd frm s locaed n he -mare: Volume 8, Issue 44, Arl 8 4

8 . Melnov ocaon choce of frms under acelberg nformaon asymmery T&,, f ,. (3) I follows from (3) ha for γ < 4/3 here s ransor arff a whch he s frm wll choose he -mare. A γ 4/3, s frm wll always be locaed n he -mare. If he s frm chooses he -mare, hen he nd frm wll be locaed n he -mare because of he locaon asymmery. e us fnd a locaon decson of he nd frm f he s frm s locaed n he -mare (4):,, , f (4) I s found ha n he equlbrum he locaon of frms sgnfcanly deends on he level of he mare sze asymmery. A γ 4/3, agglomeraon s ossble n he -mare, and a γ < 4/3, agglomeraon s ossble n he -mare. The fnal locaon of frms wll deend on who maes he frs move n he game. Thus, n he equlbrum, hree saal equlbrum saes are ossble: (, ), (, ) and (, ). rom (4) follows ha he level of he mnmal arff has ncreased (5): I.(5) Thus, he leader's oson n he -mare has decreased he mac of he comeon effec for he nd frm. 3.4 Equlbrum IV: s frm follower, nd frm leader n boh mares The frms equlbrum rofs are (6): , IV 8 8. IV (6) If he nd frm chooses he -mare, hen he s frm wll be locaed n he -mare because of he locaon asymmery. e us fnd a locaon decson of s frm f he nd frm s locaed n he -mare (7): IV,, IV. (7) Thus, n he Equlbrum IV he s frm wll always be locaed n he -mare. e us analyze a locaon decson of he nd frm: IV,, 8 IV f IV,. (8) Volume 8, Issue 44, Arl 8 4

9 . Melnov ocaon choce of frms under acelberg nformaon asymmery T& e us noe ha soluon (8) concdes wh he locaon soluon of he nd frm n he ourno equlbrum (), IV. Thus, when he nd frm s a leader n boh mares, he nformaon asymmery effec does no affec he omal locaon soluons of he frms. Thus, he omal locaons are deermned by hree effecs he mare sze effec, he comeon effec and he nformaon asymmery effec. The mare sze effec aracs frms o he large mare, he comeon effec ushes he frms away from each oher, and he nformaon asymmery effec aracs frms o he mare where hey are leaders. Possbly of he frms agglomeraon n he small mare eends he resuls derved by ang e al. (6), who consdered wo effecs: he mare sze effec and he comeon effec. 4 ONUION Ths aer has develoed a barbell model under acelberg nformaon asymmery o analyze he frms' equlbrum locaons. A comarave analyss of he locaon choce of he frms s carred ou. In he ourno equlbrum he omal frms locaons are deermned by wo facors: he mare sze effec and he comeon effec. The mare sze effec moves frms o he large mare whle he comeon effec ushes he frms away from each oher. When one mare s suffcenly larger han he oher one or ransor arff s suffcenly low, he mare sze effec domnaes he comeon effec and boh frms are locaed n he large mare. In he acelberg equlbrum he omal frms locaons are deermned by an addonal facor he nformaon asymmery. Under he nformaon asymmery one of he frms has nformaon abou a comeor s sraegy. I s found ha an ncrease of he ransoraon arff conrbues o he dserson of frms, and he growh of he sze mare asymmery conrbues o he agglomeraon of frms n he large mare. I s obaned ha under ceran condons he nformaon asymmery effec domnaes he mare sze effec and boh frms agglomerae n he small mare. In hs case he omal locaon choce of frms deends on who maes he frs move n he game. In furher research, he barbell model can be generalzed o he case of a se of mares and frms. I would be neresng o nvesgae how an omal locaon of frms wll deend on oher asymmery yes, e.g. of qualy, coss, ec. Acnowledgemen I would le o han he wo anonymous referees for very helful commens and suggesons. I would le o use hs ooruny o eress my arecaon o he Insue of Marme Busness of Odessa Naonal Marme Unversy; Dearmen of Transor ervces Mare of he Insue for Mare Problems and Economc-and-Ecologcal Research for he encouragemen and suor hroughou he sudy. Volume 8, Issue 44, Arl 8 43

10 . Melnov ocaon choce of frms under acelberg nformaon asymmery T& References [] Hoellng, H.,, ably n omeon, The Economc Journal, 3 (53), [] D'Asremon,., Jasold-Gabszewcz, J. and Thsse, J.-., 83, Produc Dfferences and Prces, Economcs eers,,. -3. [3] Anderson,. and Neven, D.,, ourno omeon Yelds aal Agglomeraon, Inernaonal Economc Revew, 3(4), [4] Hamlon, J., Thsse, J.-. and Wesam, A., 8, aal dscrmnaon, Berrand vs. ourno n a model of locaon choce, Regonal cence and Urban Economcs,, [5] Hwang, H. and Ma,..,, Effecs of saal rce dscrmnaon on ouu, welfare, and locaon, Amercan Economc Revew, 8, [6] ang, W.J., Hwang, H. and Ma,.., 6, aal dscrmnaon: Berrand vs. ourno wh asymmerc demands, Regonal cence and Urban Economcs, 36, [7] Andree K. and ala J.,, Produc dfferenaon n a saal ourno model wh asymmerc demand, Economcs Bullen, 3(), [8] crmore, M.,, ymmerc and Asymmerc Equlbra n a aal Duooly, Avalable a RN Elecronc Journal, Rereved from hs://do.org/.3/ssrn.4484 [] Esel, H. A.,, Equlbra n omeve ocaon Models, n boo: oundaons of ocaon Analyss, [] Esel, H.A. and Maranov V., 7, Asymmeres n omeve ocaon Models on he ne, n boo: aal Ineracon Models. rnger Omzaon and Is Alcaons, vol 8, rnger,.5-8. Volume 8, Issue 44, Arl 8 44