The socially optimal and equilibrium locations of two stores or libraries with consumer search

Abstract

This paper examines the socially optimal (and also equilibrium) locations of two stores or libraries on a linear market of unit length. If each consumer has probability $w$ of finding a desired product at each store, then the socially optimal locations are never completely centralized for full information, but are completely centralized for $w \le 0.5$ when costly visit search is necessary. The Nash equilibrium locations of two stores, and various alternative models for the socially optimal locations of two stores, are also examined.

Appendix

This Appendix derives Eqs. (13) and (14), which are used in Propositions 7 and 8 of Sect. 7. In the full-information case of Sect. 3, if the two stores (libraries) are at the locations described in Proposition 1, it follows by substituting these locations into (3), multiplying by tD and adding 2$f$, that aggregate social costs are

$$\begin{aligned} \hbox {ASC} = 2f + (\mathrm{wtD}/2)[2 - w - (1/2)w^2]. \end{aligned}$$

(15)

If, on the other hand, there is a single combined store (library) at location 0.5, the aggregate expected travel distance is the probability that a consumer makes a trip, which is $2w - w^{2}$, multiplied by the average round-trip travel distance of consumers who do make the trip, which is 0.5. Thus, aggregate social costs are simply

$$\begin{aligned} \hbox {ASC}= F + (\mathrm{tD}/2)(2w - w^2) = F + (\mathrm{wtD}/2)(2 - w). \end{aligned}$$

(16)

Comparing (15) and (16), it follows that one single combined store in the middle is better than two separate stores if and only if

$$\begin{aligned} (2f - F)/(\mathrm{tD}) > (1/4)w^3, \end{aligned}$$

(17)

which is the same as (13) of the text.

In the costly visit-search case of Sect. 4, if $0 < w \le 0.5$ and the two stores (libraries) are at the locations described by (12) in Proposition 2, it follows by substituting these locations into (8), multiplying by tD and adding 2$f$, that aggregate social costs are

$$\begin{aligned} \hbox {ASC} = 2 f + \mathrm{tD}(1/2) \quad \hbox {for}\; 0 < w \le 0.5. \end{aligned}$$

(18)

If there is a single combined store (library) at location 0.5, the aggregate expected travel distance is the probability that the consumer makes a trip, which is 1 (since consumers do not know ahead of time whether they will find what they want), multiplied by the average round-trip travel distance of consumers who do make the trip, which is 0.5. Aggregate social costs are thus

$$\begin{aligned} \hbox {ASC }= F + \mathrm{tD}(1/2). \end{aligned}$$

(19)

Comparing (18) and (19), the single combined store at location 0.5 is always better than two separate stores at the same location 0.5 as long as $F < 2f$, which is an obvious result. Consider now the other case in Proposition 2. If $0.5 \le w \le 1$, and if the two separate stores are at the locations described by (11) in Proposition 2, then substituting these locations into (8), multiplying by tD, and then adding 2$f$, shows that aggregate social costs are

$$\begin{aligned} \hbox {ASC} = 2 f + (\mathrm{tD}/2)[(1 + w)^{-2} (5w + w^2 - 4w^3)]\quad \hbox {for}\; 0.5 \le w \le 1. \end{aligned}$$

(20)

If there is a single combined store at location 0.5, the aggregate social costs are given by (19). Comparing (19) and (20), it is seen that one single store in the middle is better than two separate stores if and only if

$$\begin{aligned} (2f - F)/(\mathrm{tD}) > (1/2) (1 + w)^{-2} (1 - 3w + 4w^{3})\quad \hbox {for}\; 0.5 \le w \le 1. \end{aligned}$$

(21)

which is the same as (14) of the text.