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the relative distance between retailers. Function b(θ) summarisestheimpact of the characteristics of the consumer on her perception of relative distance, and influences the probability to buy in each retailer. Note that, for this expression of the probability distribution µ c ,thesales of retailer r under location pattern (c 1 ,c 2 )isgivenby Q r (c 1 ,c 2 )= d(c, c r ) −b(θ) n c π c (θ) (2) d(c, c c∈C 1 ) −b(θ) + d(c, c 2 ) −b(θ) θ∈Θ 3.2.1 The functional form of b(θ) Expression (2) shows the total sales of each retailer given a location pattern, a population distribution and a distance function. All the elements in this expression are measured and known, but the specific form of the function relating the attitude towards relative distance of a consumer of type θ, δc −b(θ) needs to be established. To this end, some assumptions are made on the functional form of b(θ). First, consider that type θ refers to a one-dimensional characteristic of the consumer. Being sex, age and educational level the available characteristics at the constituency level, since b(θ) isrelatedto the consumers’ mobility, we choose θ as the characteristic potentially more related with attitude towards mobility, i.e. age. Hence, we may think of b(θ) as a function relating the age of the consumer to her perception of (relative) distance between the location of the two retailers. We will assume that there are two different attitudes towards distance depending on age: the attitude of young people, less reluctant to walking or driving a longer distance to go shopping, and older people, more reluctant to distance. It is highly probable that most of the young people buy in their working-area rather than on the living-area and may choose a retailer farther away if it suits them. Older people use to buy several times along the week, but do not commute out of their living-area. Then, to simplify the model we consider the existence of two type of consumers Θ = {θ y , θ o } where θ y refers to young consumers and type θ o to the older ones. Let us denote by πc y = π c (θ y )and(1− πc y )=π c (θ o ). Finally, assume the following functional form for b(θ): −2+β b(θ) = −2 − β if θ = θ y if θ = θ o 14
to reproduce a model close to that of Huff (1963) where the attitude towards mobility varies with the age of the consumer. Under this assumption, the probability of a representative consumer living in c to buy in retailer c 1 is given by: p c (c 1 |θ y ,c 2 ) = p c (c 1 |θ y , δ c )= p c (c 1 |θ o ,c 2 ) = p c (c 1 |θ o , δ c )= 1 1+δ c −2+β 1 1+δ c −2−β for young and older consumers respectively, where δ c = d(c,c 2) d(c,c 1 ) . Figure 2 shows the shape of the probability function when β = .5 asafunctionofthe distance ratio δ c , which corresponds to a S-shaped function. It can be checked that d(p c (c 1 |θ y −3+β , δ c )) δ c = (2− β) dδ c (1 + δ −2+β c ) 2 d(p c (c 1 |θ o −3−β , δ c )) δ c = (2+β) dδ c (1 + δ −2−β c ) 2 are positive whenever β < 2. Moreover p c (c 1 |θ y , δ c )=p c (c 1 |θ o , δ c )ifandonly if δ c =0orδ c = 1. Finally, note that d(p c (c 1 |θ y , δ c )) = dδ c δc=1 1 2 − 1 4 β and d(p c (c 1 |θ o , δ c )) = dδ c δc=1 1 2 + 1 4 β In conclusion, independently of β, p c (c 1 |θ y , δ c ) > p c (c 1 |θ o , δ c )ifδ c > 1and p c (c 1 |θ y , δ c ) < p c (c 1 |θ o , δ c )if0< δ c < 1 Hence, when both retailers are at the same distance of a consumer, δ c =1, the slope of the probability function is greater for older consumers than for younger ones showing an bigger aversion to the distance for the formers, i.e. for a given increase in the ratio between the distances d(c, c 2 )andd(c, c 1 ), the increase in the probability to buy to the closer retailer in c 1 is bigger for consumer of type θ o than for consumer of type θ y . 15
Page 1 and 2: LOCATION STRATEGIES BASED ON DISCRE
Page 3 and 4: 1 Introduction Much attention has b
Page 5 and 6: is available are constituencies 2 a
Page 7 and 8: individuals making choices (the con
Page 9 and 10: supermarket placed in the same area
Page 11 and 12: Finally, let Q r c denote the amoun
Page 13: uy in a retailer located in c r is
Page 17 and 18: We present the case of three middle
Page 19 and 20: Requena Segorbe Utiel Location Shar
Page 21 and 22: Table 1: Payoff Matrix (normalized)
Page 23 and 24: shares are presented in the table,
Page 25 and 26: 5 Concluding remarks In this paper
Page 27 and 28: 7 Appendix: empirical data In Spani
Page 29 and 30: 25 26 27 31 32 33 41 42 25 3.8 26 1
Page 31: Segorbe Segorbe is divided in nine

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