Non-price spatial competition on retail market

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NON-PRICE SPATIAL COMPETITION ON RETAIL MARKET



Introduction

The retail market, commonly understood as the intermediary between producers and final consumers, has been intensively developing and complicating especially for the last hundred years. It can be explained, at least, by a following simple, but probably not complete, logic. Scientific and technological progress always gives new opportunities to producers who would like to gain innovations either for development of new varieties that could raise the demand, or for improvement their current production process in terms of costs' reduction. No doubt, these incentives lead to increased competition in most of industries, especially in large FMCG (Fast Moving Consumer Goods) sector. This, in turn, induces plenty of varieties and brands, so that it evidently leads to what we can call a negative network effect: the more varieties an industry provides, the weaker information and signals consumers get about each of producing goods. Eventually, it seems relevant to all firms to realize their production in an intermediate platform. The latter creates the retail market itself, which today concentrates practically all producers' competition in ordinary supermarkets.

The complexity of vertical relations between wholesalers, retailers and final consumers has been one of the most challenging parts of economic reality. It induced generations of deep theoretical researches, so that since times of Cournot publication (1838), which is known by the first formal model of imperfect competition, market theory has developed and spread incredibly. This particularly occurred in large fields of imperfect markets analysis and urban economics, both straightforwardly related to retail market formation - the object of the current paper. So, below I provide a short guide line along these scientific destinations.

As a rising competition between retailers consistently led to geographical spread of this market, understanding its economic nature requires a spatial competition analysis, pioneered by Hotelling, albeit not focused on the retail market exactly [Hotelling, 1929]. The main contribution of his (today classical) work, “Stability in Competition”, is thought to be in modeling one side of reality by which a perfectly competitive market is impossible. Now we call it the spatial differentiation of a product, when the same good supplied by various firms from different locations cannot be considered, by any agent's point of view, a unique, non-differentiated good. A simple reason lies on some extra costs consumers bear to purchase it. This is captured by the Hotelling model by introduction of linear transportation costs that decrease a benefit of an atomic consumer.

Moreover, except well-known Principle of Minimum Differentiation, thought to be the main conclusion of Hotelling's work, there is also one structural pattern that explains recognition of his model. An object of one-dimensional “linear” town, taken as a foundation for the model, appeared to be highly convenient to further analysis and some variations, at least in consumers' distribution. Evidently, it sufficiently illustrates an urban structure in a way that simplifies technical part of analysis, but does not break significant conclusions.

Half a century later, the Hotelling model was thoroughly investigated and reconsidered in a number of papers. As a sequence, a lot of critics for the original work followed. The main ground for criticism appeared to be the Principle of Minimum Differentiation itself, mentioned before, which induces lots of doubts in the model validity. d'Aspremont et al. argue that there is no price equilibrium when the sellers are located too close to each other [d'Aspremont et al., 1979]. Moreover, the research provides conditions to existence of price solution for any firms' location, and, to the contrary to Hotelling's analysis, a tendency to the maximum differentiation is observed there.

Another way to think, why the Hotelling model might be actually misspecified, is to recall on sequential nature of the most of market games rather than simultaneous one. This view is given by Anderson provided Stackelberg leadership model on one-dimensional spatial structure [Anderson, 1987]. The author also suggests general discussions, concerning possible sources of the Hotelling model's invalidity, and emphasizes some ways that other researchers provide to restore the existence of equilibrium:

a) restrictions on the mill-price undercutting;

b) introducing alternative specification of transportation costs;

c) mixed strategies approach;

d) introducing heterogeneity in consumer demand.

One can meet the second innovation more often than the others that is justified by empirical examination, in particular. Tabuchi and Thisse, using quadratic transportation costs, came to amazing conclusions [Tabuchi, Thisse, 1995]. Introducing triangular density for consumers, they went through original two-stage location-price game and, surprisingly, proved non-existence of symmetric equilibrium under described structure. Moreover, being in identical conditions, firms are asymmetric in their profits.

That above is thought to be a sufficiently complete review over the branch of Hotelling-like models. Now the other kind of logic, predominantly considering some peculiarities of consumers' preferences, should follow. In the context of attending retail trading places, most of us prefer a wide range of goods, delivered by a shop, instead of a couple of substitutes. Particularly for this reason, today we can observe different sizes of supermarkets, some of which may occupy an area comparable with one of a whole shopping center. This tendency also leads to similarity of goods' variety consumers can find in competing outlets that, actually, does not allow retailers set prices significantly different from ones provided by others. As a sequence, we might say pricing is not a primary strategy of retailers. The significance of consumers' love to variety in terms of a retailer's size and the consideration of some extra costs people bear to reach a shopping center or a supermarket were elegantly captured and modeled by Huff [Huff, 1963]. He noticed, referring to empirical evidence, that only two parameters of retailing organizations are constantly considered by consumers - the size and the distance. Criticizing former gravity models, originated by Reilly in spatial economics field, Huff proposed more elegant probabilistic approach by a simple formulation of consumers' preferences based on mentioned variables only. This eventually allows determining trading areas for retailers that completes the analysis.

By this point, it should be fairly mentioned that the Huff model has got some successful empirical applications, for example, in Drezner and Drezner [Drezner, Drezner, 2004]. However, the practical implementation of the model became representative and consistent only after various sophisticated generalizations. One, and the first, was suggested by Griffith in 1982, being worked out predominantly due to the author's critics for the absence of traditional economic patterns, such as prices, demand and elasticities [Griffith, 1982]. All these are properly included in the original Huff structure, keeping attractiveness and distance variables.

Above all, the retail market, in any case functioning in urban environment, is seriously affected by its inseparable patterns, such as transport infrastructure, population distribution, presence of free area in various districts, and others.

These discussions have encouraged an idea to analyze spatial competition in retail market in the simplest way, nevertheless designed to combine Hotelling and Huff approaches. The main and common objective of the current research is to understand a specificity of retail market formation in an arbitrary agglomeration in terms of optimal sizes and location retailers would choose. According to argumentation above, pricing, unlike in lots of remarkable papers, is not of a current interest. Therefore, a final comparative analysis in terms of agents' profits is not valid, so I pay all attention to the market's spatial structure itself and to the population welfare. I provide a Hotelling-like duopoly model, in which non-cooperating retailers simultaneously enter and choose their size and location, where a size could be also interpreted as a production variety parameter. Population density is not a deterministic function, but it relates to triangular density family describing different degree of town centricity. Consumers are supposed to be homogenous, each having a utility increasing in the products' variety parameter (“love-to-variety” parameter) and decreasing in the distance to a shop chosen. What extremely important in the suggested model is the rental cost, proportional to people's density, divided to a fixed rate and a variable part. Traditionally using the indifferent consumer problem, the static equilibrium is derived directly.

The structure of the work's body is as follows. The first, and the largest, section provides the core of the current research that is a basic model of duopolistic retail market with mentioned endogenous variables and exogenous structure. Being a formalized strategic game, it accomplishes by the static equilibrium. The next is dedicated to comparative statics to follow, how any changes in exogenous urban environment affect the equilibrium market state. And the third one works out, whether the derived market equilibrium could be socially optimal under some sets of parameters, so a social welfare analysis is provided there. These chapters complete the model itself. One may think that the theoretical research might not be exhaustive without modeling a monopolistic retail market and comparing its results to the optimal oligopolistic outcome. However, at the beginning of the third section I provide a necessary explanation that justifies a disregard of monopoly analysis. In the final part I present conclusions, based on results from all previous sections, and provide some either theoretically strong or just intuitive comments. The paper ends up with common discussions and a perspective view to the modeling of retail market formation under more general oligopolistic structure.



1. Basic model: duopoly in the retail market

Before mathematical formalization of the basic model, one issue concerning the urgent terminology should be clarified. There might be some works, also devoted to imperfect market analysis, that provide a sensible separation of various names referring to any trading points in the retail market. Indeed, there is a difference between a shop, a supermarket, a shopping center and, probably, some other related terms not only in scientific fields, but just in the common reality. Despite this notification, I use these terms equally meaningful in the current paper due to the point that its underlying idea might concern the retail market, in general.

A common line of the basic model has been, in fact, presented in the introductory part. As the goal is to analyze the results of some agents' strategic behaviour in a market, I formalize it as a non-cooperative one-step static game in normal form:

The set of agents, I, includes two competitive retailers, so

Since this point, any retailer is noted by index , while index is related to any representative consumer.

Then, strategies, also called as endogenous variables of the system, are modeled as

where - the size of i-th retailer's shop; - i-th retailer's location on the linear town figure.

As for the first variable, it is necessary to consider its other interpretations that seem to be more convenient. Particularly, for a further modeling of consumers' preferences, we should think of the term as a “love-to-variety” parameter. An underlying logic is simple here: the larger a supermarket is, the more differentiated products it provides, being more attractive to consumers. Although the latter may seem not always convincing, we take the attractiveness of goods' variety as a crucial assumption lied in the current framework, just to avoid consumers' heterogeneity problematic. Therefore, it is correct to treat as a share of wholesalers, whose production is present in i-th retailer's outlet. This treatment straightforwardly explains a normalization of the variable, but also requires an important restriction: The evidence is simple: we let all producing varieties be supplied in the intermediary market, otherwise an issue of wholesalers' activity and their relations with retailers, currently ignored, becomes inevitable.

The second variable, is based on a classical abstract figure, introduced by Hotelling, that describes an urban space in the simplest way. Following by the first part of the strategies' set, I normalize the length of this linear town construct to one, without loss of generality.

At this point, before presenting the last term of the game, two crucial parts of the model, referring exactly to urban patterns, must be discussed: distribution of consumers and retailers' cost function. There is a branch of works, particularly in urban economics field, that are aimed to study the dynamics of population distribution, modeling this endogenously with a link to changes in a labour market due to various people's incentives. From this side, one can find a joint process of the retail market formation and the development of urban infrastructure (including construction and disappearance of living areas) too complicated for its modeling in a way that would eventually provide clearly interpreted results. Moreover, the current purpose is a building just the simplest basic model of duopolistic market, so we can hardly think of any significant changes in citizens' distribution due to the activity of two, even huge, supermarkets. At least, we are interested in simultaneous decision making, thus the endogeneity of any urban environment patterns is not relevant in the static game.

Unlike it is done in the original Hotelling model, I would like to enrich the structure by a variable distribution function. I see an elegant approach in setting a triangular density function, defined symmetrically in the linear town, so that the function is able to capture the degree of the town centricity. At the same time, values the function takes in points 0 and 1, which we call the edges, should also proportionally vary, keeping their non-negativity. I provide related calculations in Appendix A, so a figure, satisfied these requirements, is as follows:

where is a parameter of centricity of an abstract town, and the range always keeps the density non-negative and not higher in the edge values than at the very centre, i.e. in the point Particularly, the expression describes a uniformly distributed population in the only case when the function is differentiable in every point of definition.

By the mentioned argumentation, related to exogeneity of consumers' distribution, I keep the structure of costs, that any retailer bears to enter a market, also as given. Moreover, just out of a permanently fair evidence, concerning pricing in land and property markets, these costs are considered to be proportional to citizens' density, derived above. So, the rental costs of i-th agent is taken as:

This way of modeling might also implicitly compensate a disregard of the point that, in reality, retailers are always limited in their location choice. Inaccessibility of a desired area is most likely to occur in downtown and districts close to the very center. But at the same time, despite a high demand, expected in this area, its occupation is probably enough to cost too much to be optimal in terms of pure profits. Therefore, even being accidentally free, a prestigious area at the center of town would not be rented by a competitive retailer, especially implying the optimization problem just in static. This logic seems to be illustrated by the proposed function, namely, by multiplication of density, connected with a location variable, and a supermarket's size.

The introduction of a fixed rate of rental cost, r, just captures a real structure of this type of costs and technically makes always positive, actually, being a term of the entrance barrier to i-th retailer.

Overall, mathematical description of the specificity of consumers' distribution and the rental cost structure completes the model's body. As previously mentioned and justified, it does not assume any terms related to pricing in the retail market, as well as any ordinary expenses tied to the scale of production or to its factors. The latter is ignored due to the fact that the current research concerns a traditional intermediary market, so wholesalers get all the production load and costs connected with it. By this, albeit acceptable, simplification, we raise a following issue.

Strictly, ignoring the pricing problematic in the market of matter, we have to imply, at least, a perfect substitutability of producers' goods, in order to avoid a concept of monopolistic competition, in particular. On the other hand, Huff logic, which claims the sustainability of two factors only, i.e. the distance to a retailer and the size of the latter, is still actual. Furthermore, a piece of idea, currently modeling, implies that a representative consumer is not responsive to a small variation in prices of similar goods: only present variety affects a supermarket's attractiveness. Therefore, under this reasoning, and also following a previous manner of normalization, I set: without a loss of economic sense. In addition, it might be said that pricing factors of market power in the modeled retail market are not under our consideration.

Now I follow particularly the Hotelling model in building agents' revenue. Each atomistic consumer provides a unit demand that, in the context of the retail market supply, means a consumer's choice of a unit of any substitutes provided in a shop. Thus, keeping in mind a specific consumers' distribution in its general parametrized form, the last part of the static game formalization, U, should be written in a following way:

where F(•) indicates a cumulative distribution function, and denotes the indifferent consumer that separates two trading areas due to the spatial competition.

Somebody can notice that F(•) always does not exceed one, but a value of the density function, included in the rental cost expression, is larger than one in an interval Moreover, a constraint, concerning the cover of all producing goods, do not allow for too small values of these variables. These observations may imply rather strict limitations to both optimal strategies and exogenous parameters. In particular, it should require very insignificant value of the fixed rate of rent tariff, r, and make retailers to occupy areas in periphery, just in order to survive in the market.

Finally, to complete the system, we have to model the preferences of homogenous consumers over the factors that are thought to determine attractiveness of i-th retailer. As widely confessed, I am using a convenient object of the utility function. At this point, I have to provide probably the most significant underlying assumption, straightforwardly determining both the structure of a representative utility and, thus, retailers' strategies.

Assumption: consumers sacrifice their love to goods' variety in order to reduce time and transportation costs, connected with the geographical access to a chosen retailer.

Literally, an average consumer would better prefer purchasing in a small supermarket, located close to her, than bear essential extra costs in order to reach a large one. Despite the assumption has appeared just out of intuition and some regular observations, this point is likely to be realistic and even might be a reason for the actual tendency of development and spread of tiny sales outlets.

Therefore, a structure of the utility function is as follows:

where is a function of transportation cost, generally depending on an atomic consumer's and i-th retailer's location, and reflects the specificity of a consumer's love to variety, which is provided by the same i-th retailer. Taking all terms normalized as described before, I include these subfunctions additively, without any harm to further calculations.

Evidently, the assumption itself is captured by different signs of the second derivatives of subfunctions. Following this condition, I propose quadratic metrics for consumers' dissatisfaction from overcoming the necessary distance - as the most natural and simple one, also used by Tabuchi and Thisse, in particular. Here the parameter should be interpreted as a quality of transport infrastructure in an abstract town. The second subfunction is taken concave in precise form: it gets its global extreme value in for the illustration of possibly maximum enjoyment from the choice within a complete set of producing varieties. So, for the convenience, we can express the utility function as follows:

Now, and finally, it is necessary to derive an expression of the indifferent consumer treated as a unique atomic citizen equiprobably choosing either of supermarkets. Below I present just an equation, satisfying this consumer's position, and the result, keeping some lines of simple calculations in Appendix B.

This completes a theoretical framework and its mathematics.

As mentioned in the introductory chapter, I let our duopolists strategically define their location and size at the same time. It leads to simultaneous decision making over all endogenous variables included in the model: So, the corresponding optimization problem in terms of retailers' profits should be expressed as:

As the described model does not imply any other incentives and strategies of agents, perhaps existing in a real intermediary market, the system satisfying any static equilibrium has to consist of four expressions of retailers' reaction functions, also called best responses. However, the duopolists' best replies do not close the system; some comments and details should be mentioned beforehand. Firstly, and just technically, we bear in mind that a derivative of a continuous cumulative distribution function, F(•), is just a density, f(•), which was, precisely for the model, derived before. Secondly, discussing different interpretations of the strategies' terms, I have set a constraint: , that must be considered. And thirdly, the system should be added by a couple of trivial constraints, mentioned along formalization, that relate to our exogenous parameters. So, the complete system of matter in its general form is as follows:

Using an expression for the location of the indifferent consumer and the proposed rental cost function, we can easily derive its partial derivatives, necessary in the explicit form for a further solution:

Therefore, the system may be slightly improved as follows:

One can notice that I still keep implicit a couple of terms. Precisely, the density, is left just for the reason of visual simplicity, as it is duplicated in the profits' derivatives. However, terms according to the specificity of supposed distribution, can take two possible values with different signs that, in fact, turns out to be a source of calculating difficulty. Further I show that this point also becomes critical for the model results, i.e. for the spatial structure of the retail market.

Before proceeding the solution, I suggest considering the first retailer located always to the left from the second one, without loss of generality. Thus, in particular, for any acceptable values of it entails: , the term, present in the first and the third equations of the system above.

Also, though the indifferent consumer problem gives a correct expression for her location, we missed a crucial detail here. Indeed, there is an unpleasant term, included in the expression of namely, a denominator of the first component, Thus, the figure makes the equation indefinite for every case of the locations' matching. However, it is evident that, as soon as competitive retailers set at the same point, they just divide a market equally. In the current framework, this evidence may be easily destroyed by a difference in the second variable. Further in this section and more detailed in Appendix C I prove that the matching in the agents' locations is always followed by choosing equal size.

Now, keeping in mind a duality of derivatives we have to take into consideration three possible spatial structures of the retail market:

1) (the left side of the linear town)

2) (the right side of the linear town)

3) (different sides)

Apparently, a symmetric equilibrium may arise only in the last case, whereas the others can potentially reveal a spatial asymmetry of the market. Furthermore, the solution process may be significantly reduced, if one might remember that the left and the right sides of an abstract town are modeled analogically. So, since it is sufficient to go through the first case above, which is related to the left part, missing the other one.

The following content presents the existence of both symmetric and asymmetric equilibria, holding all necessary computations for Appendix C. Surprisingly, the latter accident turns out to be the set (but not a continuum) of optimal outcomes. However, some of them are left disregarded in the current research, so that I provide only the simplest related condition. Overall, the main interest lies exactly on the symmetric equilibrium, which is unique in proposed model, and thus the most of informative conclusions are devoted to the symmetric type of results.

Let's start from an asymmetric case.

Location I:

Some precise manipulations, provided in Appendix C, lead to the following important equation:

This is an expression, promised to be derived, which does not allow for various supermarkets' sizes in equilibrium until retailers strategically choose the same point for their outlets. Trivially, the reverse is also fair. Since this point, I specifically call it as “matching strategies condition”.

Further mathematical manipulations eventually lead to a couple of possible alternatives for the optimal goods' variety variables:



Actually, I take these conditions as the sufficient intermediate results for asymmetric location case. However, may be easily worked out further up to the corresponding equilibrium. As a matching strategies condition is valid, necessarily leads to and, therefore, the competitors equally divide the market: It incredibly simplifies profits' expressions and practically makes agents to minimize their rental costs:

Keeping in mind a constraint for covering all wholesalers' production and remembering that just a left part of town is currently considered, we can immediately find a solution for the system of linear optimization problems:

So, the first way for an asymmetric equilibrium existence leads to spatial concentration of the retail market exactly in the very edge of agglomeration. And there is a clear argument for explanation of this strategic behaviour. Precisely, agents, tied by the matching of both strategies of the choice, appear to be less competitive, particularly in the sense of spatial differentiation. As the case is also restricted by just a half of town's area, the only incentive the retailers get is still a reduction of their costs. The latter straightforwardly leads to the derived equilibrium, evidently implied the possibly largest profits in a competitive market:

Remarkably, the second alternative condition, reveals the same, namely the poorest, aggregate goods' variety, so that retailers must supply only different wholesalers' substitutes. This low, related to any asymmetric equilibrium in the created model, calls into question a possibility of the appearance of such a market structure in reality. Indeed, to find out an asymmetric equilibrium, I have technically limited an available space by considering just a half of an abstract city. In the current framework it seems to be the only opportunity for this purpose, just because the density and the rental cost functions are not differentiable in Unfortunately, this inconvenience and the mentioned mechanism, used to avoid it, might turn into interpretation, not quite realistic: one may treat it as a practical unavailability of the dropped part of a city for the retail market formation.

Comments above complete analysis of asymmetric equilibria. In addition, due to the symmetry, we could derive analogical outcome for the disregarded asymmetric case, in particular:

The condition for another equilibrium, is also supposed to be valid in this case.

Finally, we are going on consideration of spatially symmetric market.

Location II:

This time we can claim that the whole city retail market is proportionally distributed between two agents. So, within calculations, presented in Appendix C, as previously, we would prefer using a natural condition, corresponding to the symmetric case: Remarkably, related manipulations with expressions in the system above, reveal the matching in the second variable: In fact, these two conditions, covering the main patterns of a competitive symmetric market, allow us for solving just for one retailer's optimization problem:

As the market is symmetrically separated and the agents strategically choose the same size for their outlets, we can simplify the system by setting and rewrite, as follows:

As shown in Appendix, this object explicitly provides a solution for “the left” retailer:

So, the rest, and critical, detail we need to capture is satisfying the total mass of wholesalers:

This completes a view to formation of a spatially symmetric retail market, resulting by the following unique equilibrium:

Thinking strictly, one might point out that the analysis does not consider a specific case, which is likely to hide other equilibria, and she would be correct. Exactly, the case of uniformly distributed consumers is disregarded along calculations. Obviously, confusingly simplifies both density function and rental cost structure, thus changing the whole mathematical form of the competitors' optimization problem. However, I must confess that this case has been dropped in principal. By simple reasoning, a zero value of either of exogenous parameters is thought to impoverish the model by losing a part of its structure, describing a pattern of an urban environment. Setting we follow maths, lied in the original Hotelling model, and automatically reject minimally necessary specificities of an arbitrary agglomeration that are captured by the proposed model. Though a uniform distribution is theoretically justified, I can not feel any interest in its consideration under the created model's framework.

At this point, the current chapter, dedicated to the basic model of competition in the retail market, is finished. Then, I enjoy some technical investigation of the derived results in the following section, planning all conclusions, which provide the economic sense, after a social welfare analysis.

2. Technical analysis: comparative statics

This short section is planned as a logical continuation of the research, just succeeded in finding equilibria in the created static model. Precisely, a current need is to deepen our understanding of the achieved results in terms of how the latter are affected by exogeneous parameters. A particular interest lies in investigation of such changes for the unique symmetric equilibrium, as one can remember that an asymmetric case has revealed an independent edge solution. Evidently, it is sufficient to explore the parameters' affect just on the first retailer's optimal strategies:

Calculating derivatives is known to be a direct way to accomplish the stated goal. According to the supposed content of the research, I provide just these derivatives below and some necessary comments, in order to significantly base on them in my conclusions.

Notice that the square of the first term of nominator is included in the discriminant. But, as the latter also contains another positive figure, we confirm: Therefore,

for any acceptable values of the parameters.

For a convenient illustration of the next two derivatives, I suggest a following notation:



And then:

Unfortunately, sophistication of the expression for a partial derivative does not allow for defining its sign: it depends nontrivially on a complete set of the parameters. But the other two derivatives always take values below zero.

As for the products' variety optimal value, its partials are easier to derive:

Despite an analysis of competitive profits has been stated to be not our primary goal, we might have, at least, a brief look at them, as their expressions are missed in the previous section:

Using expressions, recently gained for the partials, one may easily get the following important results:



The effect of population distribution parameter, on retailers' profits remains obscure.

3. Social welfare analysis: a centralized planning of the retail market

The main objective of this chapter is to find out a socially optimal retail market in terms of both its spatial structure and products' variety supplied by two retail trading outlets. However, as promised in the introduction, I provide now brief discussions concerning a comparison of so-called social planner problem and monopolistic behavior in the context of an arbitrary retail market.

There are two mutually connected reasons for ignoring a consideration of monopolistic market in the current setting. The first lies on practical impossibility of a spontaneous appearance of a monopoly in an intermediary market. In other words, we might have lots of doubts concerning the entrance of the only independent agent, who possesses such marketing and technological power that may allow herself both to hold the whole market continuously enough and to concentrate mostly on her profit's maximization problem. By this argument, we can hardly justify the relevance of modeling exactly the static game, as it would evidently miss the perspective of a market development that is likely to imply an occurrence of competition. The second remark, thought to be a sequence from the first one, means that a constantly powerful monopolist may arise only in a case of the state monopoly. Indeed, the state may have an incentive to monopolize a retail market, also possessing an opportunity to change an urban infrastructure in a way that would make her monopolistic profits better off. But even this scenario does not allow the state monopoly to think exclusively about her utility. More likely, this agent would primarily care of consumers' preferences in order to cover all the population demand and, therefore, raise her own revenue.

According to this logic, I consider a type of “caring” state monopoly that fulfils exactly the same optimization task as a social planner, namely, maximization of consumers' welfare. So, the problem should be modeled as follows:

The expression above concerns a Pareto-optimal supermarket in the left side of a town. Due to the symmetry of consumers' distribution, it is sufficient to define the optimal outlet also to the right from the center. Overall, we will get a geographically symmetric solution, in a manner of the outcome derived in the basic model.

Appendix D provides more detailed calculations, and below I present only the final explicit optimization expression and the results.

Remember that a parameter of distribution centricity, is limited by the interval [0;4]. It easily implies definite districts, possibly occupied by socially optimal supermarkets:



The ranges might be intuitively clear: the more populated is the downtown, the closer to the very center a social planner should set the retail agents in order to satisfy the central mass of consumers, in particular.

Going on a discussion of the location outcome, we are interested whether these values may match to the derived in the basic model. Precisely, this is captured by a following equation:

As illustrated in the same Appendix, it leads to a determined relation between all three parameters, conveniently expressed as a polynomial by argument

Unluckily, this sophisticated equation can be hardly simplified or interpreted deeply in any economic sense. However, it seems to be consistent and have real roots for some sets of parameters, that is why we accept it just as the unique condition for matching Pareto-optimal and equilibrium location in the retail market.

Also, one can notice that the expression for a socially efficient location do not include the other two parameters, though the latter are present in the derived Nash-Hotelling equilibrium. Literally speaking, this evidence reflects a power of a central planner who is, along its optimization problem, caring by only the market availability to every atomistic consumer. Facing neither competition, nor any exogenous restrictions, the “caring” monopolist just ignores all the patterns of urban environment, except the specificity of citizens' distribution.

As for the second term, it has been technically expected to turn into the maximum products' variety, provided by each shop. A comparison with the optimal values, derived in the basic model, leads to the following result:

To sum up, the results above give a remarkable and also the most general conclusion of the current research.

Lemma: A competitive retail market is possible to be socially optimal under realization of two definite conditions to exogeneous parameters:

According to the conditions, the matching of Pareto-optimal and equilibrium outcomes require a certain pattern of consumers' distribution, but allow for a variance in fixed rate of rental cost and transport infrastructure parameter, binding them by the first equation.



4. Conclusions

retail market duopoly location

This final chapter gathers different economic implications, provided by the basic model of duopoly in the retail market. Some comments and informative findings have been partially mentioned in the previous sections, so below I provide conclusions, mostly corresponding to the main purpose of the research, namely, to the analysis of outcomes, possible in an arbitrary competitive intermediary market.

It may be generally confirmed that mathematical formalization, proposed to the economic sense of interest, appears to be valid. Precisely, basing on a convenient object of linear town, created structure both follows a classical manner of modeling the spatial competition, introduced particularly by Hotelling, and holds specific patterns, necessary for the economic concept of the research. In short, potential risks of mathematical invalidity, which, anyway, can not be excluded in such type of works, are not, fortunately, realized.

Overall, we get some competitive equilibria as a realization of the agents' simultaneous strategic behaviour. However, as I commented in section 2, the existence and the derived form of asymmetric equilibria leaves doubts, as its derivation initially required the restriction to the space availability. One might regard this crucial detail, as setting an artificial (and severe) limitation to the market space, and, actually, is likely to be fair. Therefore, it is not anymore surprising a lot to get the edge solution in asymmetric case: retailers, deprived of a half of town's area, practically lose a competitive spirit, just minimizing their costs for functioning in the market. This equilibrium is suspicious also out of another sense: consumers, settled close to the opposite edge of town, would bear huge transportation costs, which, according to the suggested utility function, could not be covered enough by offered variety of goods. It is quite clear, retailers would not stay in an area (at least, in static game), that does not allow them to get a possibly maximum demand. That is why, if we replace the density function by completely differentiable one, which does not require such technical restrictions, a possibility of any asymmetric equilibrium still remains doubtful.

For this reasoning, the spatial symmetry, as a result of the created model, is more expected and explainable, though its necessary condition and the competitors' separation to different sides of the linear town, as well, have been also implemented in purpose. Remarkably, the model provides a unique symmetric equilibrium, coexisted with a couple of trivial, but essential, constraints, as the covering wholesalers' market, for instance. It would not be a waste of time to duplicate it again:

Briefly looking at these results, one may conclude that a strategic choice of location is much more complex for retailers than defining an optimal variety. Indeed, according to equilibrium expressions for the outcomes of spatial competition itself lie on the whole set of parameters, i.e. depends on each considered pattern of urban environment. Moreover, this dependence appears to be highly sophisticated, in spite of simplicity of the basic model. This common observation is likely to reflect the complexity of finding an optimal location, occurred in real intermediary markets. It is not a secret that today so called geomarketing stands at the head of the corner for every ambitious retailer, as a company's development in this specific market is, obviously, unimaginable without a carefully planned geographical expansion. Thus, plenty of highly complicated empirical methods and instruments are used for this goal, and no significant pattern of the reality is allowed to be dropped out from the spatial analysis. Literally, the created model confirms it by the fact that a complete exogeneous structure is included in the derived equilibrium location values.

The more detailed analysis, presented in section 2, reveals another type of complexity, related to the effect of town's centricity to the competitors' optimal location. This effect remains uncertain, that might be explained by two contradicting points. On the one hand, the higher concentration of consumers at the center of an abstract town would push retailers toward the main mass of people in order to satisfy them. But on the other hand, a better satisfaction of the population's preferences is not a primary incentive of the duopolists, who are initially concerned by arisen market competition. Furthermore, setting close to the very center becomes as expensive, as crowded the latter is. However, the expression for a competitive location shows retailers' urge to stay in the downtown, i.e. nearly in the interval for the first duopolist and in for the second one (as the last term in the expression for seems to exceed the second by their absolute values). In fact, it makes the values of competitive profits to be in danger of negativity for the majority of parameters' sets.

Nevertheless, the effect of two other introduced parameters has appeared to be clear. A relatively large value of fixed rate of rental cost reduces any competitor's incentive to locate closer to the center. This result has been intuitively expected. As for the rest parameter, there might be a controversial logic, similar to discussion of the distribution parameter's effect on the location. Remember that retailers do not explicitly care of consumers' satisfaction, therefore, they are not encouraged to move toward the center in a case of high transportation costs.

A simpler result is derived for the second retailers' strategy, and this one provides the certain remarkable treatment. It has appeared that the optimal size competitors choose to function in the market is affected just by a degree of the population centricity. Precisely, the more crowded is the city center, the less goods' variety a retailer offers to consumers. If we bind it to the results, derived to the other variable, an elegant conclusion might be suggested here: as supermarkets, according to the symmetric equilibrium, are set close to the most populated part of town, retailers have to care of their costs' reduction by optimizing their size. Literally, as parameter is involved in the rental cost function, the agents can not allow for establishing large outlets in the downtown; otherwise, it is not beneficial to enter the market. Anyway, the discovered relationship between the supplied variety and the centric urbanization might remarkably describe a real tendency of setting smaller shops instead of hypermarkets.

As a logical sequence, the initial range of the town centricity parameter, is severely restricted by the equilibrium expression to The interval emerges due to the necessity of presenting all wholesalers' products to the final consumers. So, there is an evidence of impossibility of a spatially symmetric retail market in a city with too populated center. Actually, this conclusion seems to be a bit principal for an urban reality. But in fact, the source of this limitation lies in a load retailers bear to cover a complete set of producing substitutes. Precisely, the basic model proposes a duopolistic retail market, implying that two competitive supermarkets would satisfy the entire population. It seems quite challenging, as even small urban settlements are likely to have more than two outlets. Explicitly, an increase in the retailers' number would allow for their size reduction, and, probably, for a raise of their profits till a certain threshold; furthermore, it would accept a larger range of the population specificity.

Overall, it may be concluded that the model successfully provides a view to an arbitrary retail market formation. Some theoretical findings, worked out above, confirm that the analysis in terms of chosen endogenous variables, is able to be informative and, probably, perspective, whereas the parametrized patterns of an urban environment are selected in appropriate way to get various conclusions. An exclusion of either of them would make the model unrepresentative, whereas its complication by other terms might lead to lots of uninterpreted results. However, one might get an intuition that the achieved results would have been more visually attractive and conceptually deeper under slightly more elegant mathematical structure. Truly, there is a place for the technical improvement here.



Discussions

The results above allow thinking that the simplest model, created in the current work, provides some theoretical implications to understanding both a competitive retail market formation and relations between urban economics' patterns. However, though keeping the whole idea and the proposed manner of modeling, there are some ways for improvement. Once mentioned, an inconvenience of a chosen density function might be overcome by an appropriate smooth function. As a sequence, there would be no need in a separate consideration of symmetric and asymmetric cases, as both retailers would get the whole populated space while going through their optimization problem. And secondly, the model might be enriched by introducing specific areas, for some reasons (there could be a lot) unavailable to building retailing outlets. For instance, there is a logic in linking such areas to density values in order to capture a realistic pattern of the downtown's spatial occupation.

Moreover, the model might be progressively and prospectively generalized. Precisely, it is supposed to make a static model with an arbitrary number of retailers in order to follow how equilibria values change according to this number of agents. This time, the idea might require more sophisticated type of modeling, as an object of the indifferent consumer seems to be invalid here. Intuitively, a probabilistic approach, suggested by Huff, could be implemented into formalization of a normalized continuous two-dimensional space of consumers and further - into a system of agents' optimization problems. Presumably, such a mathematical structure would keep all necessary patterns and even might be able to expand some of them.



References

1. Anderson, S. Spatial competition and price leadership. International Journal of Industrial Organization, 1987, vol. 5, no. 4, pp. 369-398.

2. d'Aspremont, C., Gabszewicz, J. J., & Thisse, J. F. On Hotelling's “Stability in competition”. Econometrica: Journal of the Econometric Society, 1979, pp. 1145-1150.

3. Drezner, T., &Drezner, Z. Finding the optimal solution to the Huff based competitive location model. Computational Management Science, 2004, vol. 1, no. 2, pp. 193-208.

4. Griffith, D. A. A generalized Huff model. Geographical Analysis, 1982, vol. 14, no. 2, pp. 135-144.

5. Hotelling, H. Stability in competition. The Collected Economics Articles of Harold Hotelling, Springer, New York, NY, 1990, pp. 50-63.

6. Huff, D. L. A probabilistic analysis of shopping center trade areas. Land Economics, 1963, vol. 39, no. 1, pp. 81-90.

7. Tabuchi, T., & Thisse, J. F. Asymmetric equilibria in spatial competition. International Journal of Industrial Organization, 1995, vol. 13, no. 2, pp. 213-227.



Appendix A

This part of computation work is devoted to derivation of the density function that must satisfy the requirements claimed along the model formalization. At first, I have supposed its general form, keeping the maximum value in and allowing for an arbitrary slope, and arbitrary minimum values, d, in the edge points,

As a cumulative distribution function returns its unity value on its complete definition space, we use the following equation to derive a relation between introduced parameters:

Replacing parameter d by the derived linear expression in the initial general function, we gain:

that is, due to non-negativity of term d, is restricted by a trivial condition:



Appendix B

Below I present a solution for the location of the indifferent consumer as a function of endogenous variables:



Appendix C

The current mathematical application covers the solution for the basic model. According to the content, it is divided to parts C.1 and C.2, respectively related to asymmetric and symmetric location cases.

C.1:

Expressions (I.1) and (I.3) lead to a new one: