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PRICE COMPETITION AND SOCIAL WELFARE COMPARISONS BETWEEN LARGE-SCALE AND SMALL-SCALE RETAILERS

In some localities, a large-scale chain retailer competes against a small-scale local independent retailer that specializes in, for instance, vegetables, fruits, and flowers produced locally for local consumption. The former usually attracts consumers by emphasizing its width and depth of products variety, whereas the latter seeks to overcome its limited products assortment by offering lower prices for them than the chain store. This is possible for the local store partly because of lower labor costs and for various other reasons. This study employs the Hotelling unit interval to examine price competition in a duopoly featuring one large-scale chain retailer and one local retailer. To express differences in their product assortments, we assume that the large-scale retailer denoted by A sells two types of product, G1 and G2, whereas the local retailer denoted by B sells only G1. Moreover, we assume that all the consumers purchase G1 at A or B after comparing prices and buy G2 at A on an as-needed basis. We examine both Nash and Stackelberg equilibrium to indicate that the local retailer can survive competition with the large-scale chain retailer even if all the consumers purchase both G1 and G2. We also reveal that a monopolistic market structure, not duopoly, can optimize the social welfare if consumers always purchase both G1 and G2.


Introduction
A large-scale chain retailer store customarily offers a wide and deep product variety to attract more consumers over a wider area. Product width is a term signifying the different types of products a retailer offers; product depth refers to the variety of a product offered. It is a tenet in business economics that social welfare gains and revenue gains from product width and depth must be balanced against the prospect of lower consumer prices and lessened product choice.
Lancaster [13] has surveyed the issue of product variety from an economist's point of view. The term product variety in his study corresponds broadly to the number of "brands" as that term appears in the marketing literature. Kök, Fischer, and Vaidyanathan [12] have extensively reviewed literature on assortment planning. Kök and Fischer [11] have proposed a method by which retailers could optimize product assortment and estimate consumer demand. Cachon and Kök [3] have examined the determination of product assortment among multiple merchandise categories and basket shopping consumers. Toporowski and Lademann [18] have reviewed the literature that examines assortment, price, and location in food retailing.
Some chain retailers set prices according to local markets (Dobson and Waterson [4]) to compete against local independent retailers. Focusing on grocery retailers, Leszczyc, Shinha, and Timmermans [15] have estimated a dynamic hazard model to understand factors influencing timing of consumer purchases, store choices, and the competitive dynamics of retail competition. Swoboda, Berg, Schramm-Klein, and Foscht [17] have discussed the relative importance of retail brand equity and store accessibility for determining store loyalty in different local competitive contexts. In contrast, local independent stores are usually small-scale, for example, discount supermarkets offering everyday-low-pricing and minimal service. They offer fewer product groups and variants within groups, but they acquire neighborhood consumers who want to reduce their dependence on automobile (Handy and Cliftono [8]) by offering lower prices than competitors on targeted groups of products (Kahn and McAlister [10]; Guptill and Wilkins [7]). Lal and Rao [16] have investigated the factors underlying the success of everyday-low-pricing. Bell and Lattin [1] have linked consumer preference for every-day-low-price retailers versus high as well as low-price retailers to the expected dollar cost of a household's shopping basket.
In some areas of Japan, local independent retailers sell vegetables, fruits, flowers, and fish produced locally for local consumption. They sell their limited assortments of product groups at lower prices than those offered by large-scale chain retailer and cater to customers for whom freshness and prices are essential. They can do so because they buy directly at lower prices from local growers, farms, and cooperatives and their labor cost are often below those of chain retailers.
This study uses the Hotelling unit interval model to address price competition in a duopoly featuring a large-scale chain retailer and a small-scale local independent retailer. We first discuss Nash equilibrium and then investigate Stackelberg equilibrium to indicate that the local independent retailer can survive competition with the chain retailer. We examine social welfare and its optimality at Nash and Stackelberg equilibrium. All the propositions in this study are straightforwardly proven and therefore omitted.

Model 2.1. Notation and assumptions
Our notations and assumptions are as follows: (1) Consumers, who are homogeneous except locations, are uniformly distributed along the Hotelling unit interval [0, 1] (Hotelling [9], Biscaia and Mota [2]). (2) Large-scale retailer A is located at 0 and a small-scale local independent retailer B is at 1 on the horizontal unit interval [0, 1]. (3) To express the differences in the product breadth, we assume that A sells two types of products (G 1 and G 2 ), whereas B sells only G 1 . (4) Consumers purchase only G 1 with probability α and purchase G 1 and G 2 simultaneously with probability 1 − α. Since we concentrate upon the price competition between A and B, p 2 is assumed to be determined by some suitable criterion and thus p 2 is given. (7) Consumer willingness to pay (WTP in short) for G 1 and G 2 are u 1 and u 2 , respectively. (8) Traveling cost for each consumer is c per unit of distance. (9) u 2 − p 2 > 0.
2 . In assumption (6), we assume that the raw price of G 1 at A is identical to that at B. In the real circumstances, however, the latter could be smaller than the former. If B can survive the competition under assumption (6), B can survive more easily in the actual environment. Besides, assumption (9) ensures consumers an incentive to purchase G 2 , while assumption (10) is introduced to assure that all the consumers over [0, 1] have positive utility in purchasing G 1 occasionally with G 2 at equilibrium prices, which will be discussed more precisely later.

Indexes 2.2.1. Boundaries
When a consumer at location x ∈ [0, 1] purchases G 1 only at A , her utility U 1A is given by while if she visits B to buy G 1 , her utility U 1B is given by Hence, when a consumer purchases G 1 only, the boundary by which consumers are categorized delineatesx where consumers with x ∈ [0,x 1 ] travel to retailer A , and consumers having x ∈ (x 1 , 1] visit B. Note above that we take into account the consumers' round-trip travel costs from their home to the store unlike the usual Hotelling models (see, e.g., [2,5,6,14]). This is to shed light upon the difference in the traveling costs between the two types of consumers specifically; consumers visiting A only and those visiting both A and B.
If a consumer at x ∈ [0, 1] purchases G 1 and G 2 at A , her utility U 2A is given by while if she visits B to buy G 1 and then A to obtain G 2 , her utility U 2B becomes Then, the boundaryx 2 in the case of purchasing both G 1 and G 2 is given bỹ where consumers with x ∈ [0,x 2 ] travel to retailer A to buy both G 1 and G 2 , and consumers with x ∈ (x 2 , 1] visit B to obtain G 1 and then A to purchase G 2 .
In the following, we concentrate upon the case of 1 − 2c ∈ [0, 1] for simplicity, and hence we additionally introduce assumption (11).

Expected profits
When consumers behave as observed above, expected profit to A is given by and B earns his expected profit given by

Best responses
The derivative of Π A ( p 4c .

By solving
1 , we obtain the best response of A against B which is written as The best response of B against A , which is a solution to Copyright c ⃝ by ORSJ. Unauthorized reproduction of this article is prohibited.

Consumer surplus
Consumer surplus is an essential index for evaluating the equilibrium. Its intricate structure is given by 3. Equilibrium 3.1. Nash equilibrium Solving Eqs. (11) and (12)  simultaneously, we obtain the Nash equilibrium with regard to price as follows: By substituting Eqs. (14) and (15) into (7) and (8), the boundariesx 1 andx 2 becomẽ At present, however, these solutions are, actually, candidates for the Nash equilibrium because we have not yet verified that all the consumers over [0, 1] can have positive utility through the behaviors described by these solutions.
Let us examine the case where consumers purchase G 1 only. A consumer at location x ∈ [0,x * 1 ] has the minimum utility U * 1A when x =x * 1 , whereas the utility of a consumer at x ∈ (x * 1 , 1] shows its infimum U * 1B when x →x * 1 + 0. They are given by Assumption (10) assures U * 1A = U * 1B > 0 due to 0 ≤ α ≤ 1. In case consumers purchase G 1 and G 2 , a consumer at x ∈ [0,x * 2 ] encounters her minimum utility when x =x 2 , and a consumer at x ∈ (x * 2 , 1] has a constant utility as long as x ∈ (x 2 , 1]. They are given by Copyright c ⃝ by ORSJ. Unauthorized reproduction of this article is prohibited. Assumptions (9) and (10) Expected profits to A and B at the Nash equilibrium are , and the consumer surplus is given by ] .
The indexes derived above initiate Proposition 3.1. ) decreases with increasing α. ) to A gained from G 1 increases with α, whereas its share S * A decreases with α. In addition, the profit Π * B to B and its corresponding share S * B increases with α. (iv) Boundaryx * 1 increases from 1 3 to 1 2 as α increases, whereas boundaryx * 2 is increasing in α from 4 6 to 1. Proposition 3.1-(ii) implies that B should refrain its price low to compete with A when more consumers purchase both G 1 and G 2 , and consequently A also makes its price lower in accordance with B. It also suggests that if more consumers purchase G 1 only, B can raise its price along with A to a certain degree because A becomes less advantageous. Further, their prices become closer to each other as α increases. Proposition 3.1-(iii) reveals that when more consumers purchase G 1 only, A can increase his profit gained from G 1 by raising its price against its decreasing share. Proposition 3.1-(iii) also indicates that B can earn more profit and share by raising its price when more consumers buy G 1 only. Proposition 3.1-(iv) shows the range ofx * 1 andx * 2 .

Stackelberg equilibrium
In the real circumstances, the large-scale chain retailer can be considered a price leader and the small-scale local retailer a price follower within a Stackelberg game framework. Substituting Eq. (12) into Eqs. (7), (8), and (9) gives 4c .
Hence, the solution to which is the optimal price of G 1 for A against the best response of B.
When A adopts the selling price in Eq. (17), the optimal price of G 1 for B becomes The boundariesx 1 andx 2 in Eqs. (7) and (8) becomẽ In the same manner as we observed in 3.1., let us examine here whether or not all the consumers can accept the behaviors suggested by the above solution.
Firstly, if consumers purchase G 1 only, a consumer's minimum utility U * * 1A at location x =x * * 1 and her infimum utility U * * 1B when x →x * * 1 + 0 are expressed by From assumption (10), we have U * * 1A = U * * 1B > 0. Secondly, if consumers buy G 1 and G 2 , a consumer's minimum utility U * * 2A at location x =x * * 2 and her constant utility U * * 2B at x →x 2 + 0 are given by Assumptions (9) and (10)  These observations reveal the shares S * * A and S * * B of G 1 held by A and B, respectively, are given by Expected profits to A and B are , Finally, the consumer surplus at the Stackelberg equilibrium is ] .
The indexes in this subsection suggest Proposition 3.2.  b) to A from G 1 increases with α, its corresponding share S * * A decreases with α though. The profit Π * * B to B and his share S * * B increases with α. (iv) When α increases from 0 to 1,x * * 1 increases from 1 4 to 3 8 , andx * * 2 increases from 1 2 to 3 4 .

Comparison
This subsection compares the Nash and Stackelberg equilibrium. Table 1 summarizes the indexes derived above. Comparison between the Nash equilibrium with the Stackelberg equilibrium presents Proposition 3.3.
Copyright c ⃝ by ORSJ. Unauthorized reproduction of this article is prohibited.

Consumer surplus
Copyright c ⃝ by ORSJ. Unauthorized reproduction of this article is prohibited. Proposition 3.3-(iv) reveals that the local independent retailer B can survive competition against the large-scale chain retailer A because it earns a profit both at the Nash and Stackelberg equilibrium. In addition, A can earn more at the Stackelberg equilibrium than at the Nash equilibrium because A offers a higher price with a larger share of G 1 at the Stackelberg equilibrium. On the other hand, B can also earn more at the Stackelberg equilibrium than at the Nash equilibrium although its share at the Stackelberg equilibrium is smaller than that at the Nash equilibrium. This is because B offers a higher price which can absorb the influence by its decreasing share. Proposition 3.3-(v) is directly led by (i). Moreover, the difference in consumer surplus (CS * − CS * * ) between the Nash and Stackelberg equilibrium is decreasing in α.

Social Welfare
This section confines itself to social welfare to explore socially optimal prices of the product G 1 . Social welfare is given by At the Nash equilibrium, it becomes and at the Stackelberg equilibrium, it is given by .
Hence, we have W * > W * * , which is a common understanding among economists. Now, let us maximize the social welfare with respect to p Consequently, we reach Proposition 4.1. to maximize social welfare, consumers buying both G 1 and G 2 simultaneously visit only A . Accordingly, social welfare is maximized not in a duopoly but in a monopoly dominated by A when α = 0. Proposition 4.1 indicates that when maximizing the social welfare, it becomes more difficult for B to enter the market as α decreases. However, monopoly is generally considered undesirable for other economic reasons, which suggests the maximization of social welfare would be meaningless. Table 2 compares the social welfare at Nash and Stackelberg equilibrium along with socially optimum.

Conclusion
This study employed the Hotelling unit interval model to examine price competition between a large-scale chain retailer A and a small-scale local independent retailer B. To represent the difference in product assortment between A and B, we assumed that A sells product G 1 and G 2 and that B deals in only product G 1 . We also assumed homogeneous consumers purchase G 1 from A or B and buy G 2 at A . Then we focused on price competition over G 1 . Moreover, we assumed that each individual consumer purchased G 1 only with probability α and purchased both G 1 and G 2 with probability 1 − α. Nash and Stackelberg equilibrium were examined, and the main results in this study are as follows: (1) The local retailer can earn profits both at the Nash and Stackelberg equilibrium and survive competition with a large-scale chain retailer even when all the consumers purchase both G 1 and G 2 . (2) Both A and B earn more at the Stackelberg equilibrium than at the Nash equilibrium.
However, consumer surplus diminishes more at the Stackelberg equilibrium than at the Nash equilibrium. (3) Maximization of social welfare suggests A and B adopt a same price for G 1 , where consumers buying both G 1 and G 2 visit only A . Particularly when all the consumers purchase both G 1 and G 2 , social welfare is maximized not in a duopoly but in a monopoly dominated by A .
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In this study, we concentrated on the case where the second boundaryx 2 when consumers purchase both G 1 and G 2 is given by an interior solution under assumption (11). One of useful extensions of our work is to relax the conditions in assumption (11). Furthermore, all the consumers over [0, 1] are assumed to have positive utility through behaviors according to the Nash and Stackelberg equilibrium in terms of assumption (10). Another useful extension is to relax this assumption so that we can encounter the case where some consumers in a specific area of [0, 1] have negative utility at the Stackelberg equilibrium, they have positive utility at the Nash equilibrium though. These extensions are currently under investigation.