Space Time Hotelling Model and Its Application to Retail Competition in a Duopoly

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1 Proeedings of the International MultiConferene of Engineers and Computer Sientists 5 Vol II, IMECS 5, Marh 8 -, 5, Hong Kong Spae Time Hotelling Model and Its Appliation Retail Competition in a Duopoly Hiroaki Sandoh, Member, IAEN, Takeshi Koide, Member, IAEN, and Jun Kiniwa Abstrat We propose a spae time Hotelling model that introdues a unit size of the vertial time axis in the lassial Hotelling unit interval model. The proposed model allows expliit onsideration of the probability that a onsumer arrives at a retail sre up time t purhase goods. The proposed model is useful in a variety of retailing problems. We briefly demonstrate an appliation of the proposed model retail ompetition in a duopoly. Index Terms spae time Hotelling model, retail ompetition, duopoly, business hours, departure time distribution, demand distribution up time t. I. INTRODUCTION SINCE Hotelling [] introdued the lassial [, ] unit interval represent the spatial onditions in his ompetition problem, the underlying idea of the [, ] unit interval has been used in the literature of a wide variety of researh pis e.g., [] []) due its simple struture and mathematial tratability. Among them, raitson [] presented a review of works up 98, while Bisaia and Mota [9] provided an extensive review of studies sine the 97s. The literature in these reviews is related ompetition problems. The Hotelling model has also been applied the sk of pollution [5], investigation of hannel performane [], faility loation problems [7], evaluation of influenes of sre-brand introdutions [6], and strategi outsouring for supply hain management [4]. The Hotelling model also influenes deriving demand funtions in deision siene [] and the theory of produt variety in eonomis [3]. When dealing with problems assoiated with retail sres under spatial onditions, it is important onsider the arrival times of individual onsumers beause onsumers not arriving during business hours annot purhase produts. Moreover, late-arriving onsumers might not be able purhase produts due sk outage. This study proposes a spae time Hotelling model that introdues a unit vertial time axis in the lassial Hotelling unit interval model. Some appliations retail ompetition in a duopoly are also disussed. II. NOTATIONS AND ASSUMPTIONS The notations and assumptions in this study are as follows: ) ) Homogeneous onsumers are uniformly distributed on the Hotelling unit interval [, ]. Manusript reeived Ober 9, 4; revised Ober 6, 4. This work was partially supported by JSPS KAKENHI rant number Hiroaki Sandoh is with raduate Shool of Eonomis, Osaka University, Toyonaka, OSAKA 5643 JAPAN sandoh@eon.osaka-u.a.jp. Takeshi Koide is with Department of Intelligene and Informatis, Konan University and Jun Kinwa is with Department of Eonomis, the University of Hyogo. ) Sre A is loated at, while B is at on the horizontal unit interval [, ]. 3) Sres A and B sell an idential produt at pries p A > ) and p B > ), respetively. 4) The time horizon of a single day is expressed by a vertial unit interval [, ]. 5) Individual onsumers on the horizontal [, ] interval depart only one per day travel either A or B, if the departure time permits arrival at the sre during its business hours. 6) Consumer departure times are independent and identially distributed i.i.d.) random variables having a general distribution with umulative distribution funtion t). 7) We assume ) and ), indiating that a onsumer annot depart with probability + ) ) for some arbitrary reason. 8) Consumer traveling veloity is > ) with travel ost per unit of time. 9) Both A and B are open during [t o, t ], where t o < t. To avoid analytial intriaies, t o. ) Consumer willingness pay for a produt is represented by u > ). ) Without loss of generality, we assume p A p B. A. Consumer best response III. MODEL When a onsumer at x [, ] visits sre A purhase a produt, her net utility is given by U A = u p A x, while purhasing a produt at sre B gives net utility U B = u p B x). Hene, by letting U A = U B, the boundary point, x, is given by, p A p B + x = p A p B ), p B < p A < p B +., p A p B ) Under the lassial Hotelling model, a onsumer on [, x] will visit sre A, while a onsumer on [ x, ] will travel sre B. This is not neessarily the ase in this study. Under the spae time Hotelling model, onsumer best responses are broadly lassified in the two ases shown in Figs. and. ISBN: ISSN: Print); ISSN: Online) IMECS 5

2 Proeedings of the International MultiConferene of Engineers and Computer Sientists 5 Vol II, IMECS 5, Marh 8 -, 5, Hong Kong Ω C Time Time Ω B Loation Loation Fig.. Consumer best response in the ase of p A > p B. Fig.. Consumer best response for p A = p B. Figure shows the best response of the market in the ase of p A > p B, where the regions of Ω i i = A, B, C) are defined by = Ω B = Ω C = x, t); x p A p B ), ) max t o x ), t t x, x, t); p A p B ) < x, 3) max t o x ), t t x, x, t); p A p B ) x, 4) max t x ), t t x. In Fig., onsumers with x, t) in travel sre A, beause sre B is not lose enough, while onsumers in Ω B visit sre B owing its lower prie p B. In Ω C, however, it is o late for onsumers travel B, so they visit A beause of its advantageous loation. It should be noted here that the lassial Hotelling model annot detet the existene of a region Ω C in Fig.. Asian ountries suh as Japan, Korea and Republi of China have many onveniene sres whih orrespond A. The onveniene sres are small format and ubiquius retail sres sell items at higher pries than normal supermarkets. The region Ω C explains the signifianes of onveniene sres. Figure depit the best response of the market for p A = p B, defining Ω i i = A, B) by = x, t) : x, 5) max t o x ), t t x, ISBN: ISSN: Print); ISSN: Online) Ω B = x, t) : < x, 6) max t o x ), t t x. Consumer behavior in Ω i i = A, B) is the same as that in Fig.. Note that in both Figs. and, onsumers with x, t) in the blank area below or Ω B ) an arrive at B or A ) during business hours, but we have exluded them beause ) onsumers in suh ases would like depart from their loations early in the morning, perhaps having plans in the daytime, and ) if they travel B, they might arrive there in the daytime. B. Probability of weloming a single onsumer Let ψ i t, x) i = A, B, C) be defined by ψ A t; x) = t x ) [ max t o x,, 7) ψ B t; x) = t x ) 8) [ max t o x,,, t o t < t x ψ C t; x) = ) t x ) x, 9) t x t t. Then ψ A t; x) signifies the probability that a onsumer at x [, x] will arrive at A up time t, while ψ B t; x) indiates the probability that a onsumer at x [ x, ] will visit B up time t. Finally, ψ C t; x) represents the probability that a onsumer at x [max p p B), ), ] reahes A up time t. We define ρ A t) by ρ A t) = x ψ C t; x)dx, ψ A t; x)dx + x p A p B +, ψ C t; x)dx, p B p A < p B +. ) IMECS 5

3 Proeedings of the International MultiConferene of Engineers and Computer Sientists 5 Vol II, IMECS 5, Marh 8 -, 5, Hong Kong Then, ρ A t) expresses the probability that an arbitrary onsumer on [, ] visits sre A up t. Likewise, let ρ Bt) be defined by ρ B t) ) ψ B t; x)dx, p A p B + =, ψ B t; x)dx, p B < p A < p B + x. These observations reveal that the share of sre A at time t and that of sre B are given by ξ A t) and ξ B t), respetively: ξ A t) = ρ A t) ρ A t) + ρ B t), ) ξ B t) = ρ B t) ρ A t) + ρ B t). 3) IV. APPLICATIONS A. Demand distribution One of the simplest appliations of the proposed model is obtaining demand distributions at A and B up time t. For this purpose, let us introdue i.i.d. random variables Y i t) i =,,, n) having mass probabilities as follows: Pr[Y i t) = ] = ρ A t) ρ B t), Pr[Y i t) = ] = ρ A t). Pr[Y i t) = ] = ρ B t), where n is the population size of onsumers on [, ]. Note that Y i t) = orresponds an event where the ith onsumer on [, ] visits neither A nor B up t. Likewise, Y i t) = signifies a visit A, while Y i t) = indiates reahing B up time t. Let indiar variables δ A) t) and δ B) t) be defined by δ A) i t) = δ B) i t) = i, Yi t) =,, Y i t),, Yi t) =,, Y i t). Then δ A) i t) and δ B) i t) are i.i.d. random variables that follow a binomial distribution with parameters ρ A t) and ρ B t), respetively, where the number of trials is given by n. Moreover, the number of onsumers visiting sres A and B up t an be respetively expressed by D A t) and D B t), where D A t) = D B t) = n i= n i= i δ A) i t), δ B) i t). When n is large, D i t) asymptially follows a normal distribution Nµ i t), σi t)), where for i = A and B. µ i t) = nρ i t), 4) σ i t) = nρ i t)[ ρ i t, 5) B. Newsvendor problem Another simple appliation of the proposed model is the newsvendor problem [] [6]). Let p and w respetively denote the selling prie and the raw prie per unit of produt, and let represent the opportunity loss per unit of produt. Then the expeted profit Π i Q i ) of sre i in the lassial newsvendor problem is given by Qi Π i Q i ) = wq i + + pfx)dx 6) Q i [pq i x Q i fx)dx, i = A, B, where Q i denotes the sking quantity at sre i, and fx) is the probability density funtion of Nµ i t ), σi t )). It is well known that the optimal sking quantity Q i = Q i is given by the solution ) Qi µ i t ) Φ σ i t ) = p w + p +, i = A, B, where Φ ) is the umulative distribution funtion of N, ). C. Business hours Business hours have traditionally been regulated, partiularly in many European ountries, although the liberalization of business hours has generated debates over the past three deades e.g., [7] []). We an, however, disuss business hours ompetition between two sres within regulations, where sre operation osts, suh as labor osts, play an important role. This subsetion briefly examines how derive the best response of eah sre against its ompetir in relation business hours, whih is the key obtaining a Nash equilibrium. This is beause the related mathematial analyses are very intriate. Figures 3 and 4 show the best onsumer response when sre B in Fig. hanges its losing time t t + t. In Figs. 3 and 4, Ω D shows the additional area of x, t) where onsumers visit B. We have seen through Eqs. 4) and 5) that the probability of aepting a single onsumer ρ B t) influenes the demand distribution at B through its mean and standard deviation. In the following, we onentrate upon the best response the losing time t of B against A. The additional probability ρb of weloming a single onsumer for B is given by ) When t p A p B, ρb is given by ρb = p A p B ) t x [ min t + t x, ). 7) ISBN: ISSN: Print); ISSN: Online) IMECS 5

4 Proeedings of the International MultiConferene of Engineers and Computer Sientists 5 Vol II, IMECS 5, Marh 8 -, 5, Hong Kong ) In the ase of t > p A p B, ρb beomes ρb = + p A p B ) t [ min t + t x t x, ) 8) [ min t + t x, t ) x. Let and w, respetively, denote the sre operation ost per unit of time and the raw prie of a produt. Then, if np B w) ρb > t, sre B has an inentive adopt t + t as its losing time; otherwise, t would not inrease. Hene, the best response t of B against A is obtained as the solution np B w) ρb = t. The best response of A with respet its losing time against B an be obtained in the same manner, and we may thereby obtain the Nash equilibrium based on the best responses of A and B. The opening times an also be obtained in an analogous manner. Under a monopoly, in ontrast, Hosseinipour and Sandoh [6] have disussed the optimal number of business hours and optimal sking quantity within the framework of the newsvendor problem. Fig. 3. ttime Ω D Loation t + t Area assoiated with marginal probability t p A p B ). V. CONCLUDIN REMARKS We proposed a spae time Hotelling model, where a unit vertial axis is used extend the lassial Hotelling unit horizontal interval [, ]. Then, the departure time distribution of eah individual onsumer was introdued express onsumer random behavior for arriving at a sre. The proposed model is useful in onsidering a wide variety of ompetitions between retail sres under a duopoly beause it an predit how many onsumers eah sre an expet during its business hours. We also applied the proposed model derive the demand distribution of eah individual sre, and then disuss the newsvendor problem for retail sres. Competition in relation business hours was also briefly onsidered. Fig. 4. ttime Ω D Loation t + t Area assoiated with marginal probability t > p A p B ). In the proposed model, the population size n of potential usmers is assumed be known. This may limit appliation of the proposed model in real irumstanes beause estimation of n is one of the most important problems. Bayesian approahes will be useful in oping with suh a problem at the expense of ompliating the model struture. A useful extension of our work would be introdue diret marketers with the intent omparing them and onventional retailers by regarding the distane diret marketers and the lead time at onventional retailers be negligible. REFERENCES [] H. Hotelling, Stability in ompetition, Eonomi Journal, vol. 39, pp. 4-57, 99. [] D. raitson, Spatial ompetition àla Hotelling: A seletive survey, The Journal of Industrial Eonomis, vol. 3, no. /, pp. -5, 98. [3] K. Lanaster, The eonomis of produt variety: A survey, Marketing Siene, vol. 9, no. 3, pp. 89-6, 99. [4] O. Shy and R. Stenbaka, Strategi outsouring, Journal of Eonomi Behavior & Organization, vol. 5, pp. 3-4, 3. [5] U. Chakravortya, B. Magne, and M. Moreau, A Hotelling model with a eiling on the sk of pollution, Journal of Eonomi Dynamis & Control, vol. 3, pp , 6. [6] A. roznik and H.S. Heese, Supply hain interations due srebrand introdutions: The impat of retail ompetition, European Journal of Operational Researh, vol. 3, pp ,. [7] M. T. astner, Saling and entropy in p-median faility loation along a line, Physial Review, vol. E84, pp. -7,. [8] F. C. Lai and T. Tabuhi, Hotelling meets Weber, Regional Siene and Urban Eonomis, vol. 4, pp. 7-,. [9] R. Bisaia and I. Mota, Models of spatial ompetition: A ritial review, Papers in Regional Siene, vo. 9, early view). [] Y. Fu, K. K. Lai and L. Lian, Briks or liks: The impat of manufaturer s enroahment on both manufaturer-owned and traditional retail hannels, Asia Paifi Journal of Marketing and Logistis, vol. 5, iss. 4, pp , 3. [] J. Huang, M. Leng and Ma. Parlar, Demand funtions in deision modeling: A omprehensive survey and researh diretions, Deision Sienes, vol. 44, no. 3, pp , 3. [] K. Arrow, T. Harris, and J. Marshak, Optimal invenry poliy, Eonometria, vol.9, pp. 5-7, 95. [3] T.M. Whittin, Invenry ontrol and priing theory, Management Siene vol., pp. 6-68, 955. [4] M. Khouja, The single-period news-vendor) problem: literature review and suggestions for future researh, Omega:International Journal of Management Siene, vol. 7, pp , 999. [5] N. C. Petruzzi and M. Dada, Priing and the newsboy problem: A review with extensions, Operations Researh, vol. 47, 83-94, 999. [6] A. Hosseinipour and H. Sandoh, Optimal business hours of the newsvendor problem for retailers, International Transations in Operational Researh, vol., iss. 6, pp , 3. ISBN: ISSN: Print); ISSN: Online) IMECS 5

5 Proeedings of the International MultiConferene of Engineers and Computer Sientists 5 Vol II, IMECS 5, Marh 8 -, 5, Hong Kong [7] D. De Meza, The fourth ommandment: is it Pare effiient? Eonomi Journal, vol. 94, pp , 984. [8] J. S. Ferris, Time, spae, and shopping: the regulation of shopping hours, Journal of Law, Eonomis, and Organization, vol. 6, pp. 55-7, 99. [9]. Clemenz, Competition via shopping hours: a ase for regulation? Journal of Institutional and Theoretial Eonomis, vol. 5, pp , 994. [] R. Inderst and A. Irmen, Shopping hours and prie ompetition, European Eonomi Review, vol. 49, pp. 5-4, 5. ISBN: ISSN: Print); ISSN: Online) IMECS 5