Abstract and Applied Analysis

Volume 2016, Article ID 3254240, 8 pages

http://dx.doi.org/10.1155/2016/3254240

## Pricing Strategy versus Heterogeneous Shopping Behavior under Market Price Dispersion

^{1}Department of Economics, Universidad Complutense de Madrid, Madrid, Spain^{2}Department of Economic Analysis, Universidad Complutense de Madrid, Madrid, Spain^{3}Department of Psychology, Harvard University, Cambridge, MA, USA^{4}Department of Economics, ICADE, Universidad Pontificia Comillas, Madrid, Spain^{5}Instituto Complutense de Estudios Internacionales (ICEI), Universidad Complutense de Madrid, Madrid, Spain

Received 8 August 2016; Accepted 10 November 2016

Academic Editor: R. Company

Copyright © 2016 Francisco Álvarez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the ubiquitous problem of a seller competing in a market of a product with dispersed prices and having limited information about both his competitors’ prices and the shopping behavior of his potential customers. Given the distribution of market prices, the distribution of consumers’ shopping behavior, and the seller’s cost as inputs, we find the computational solution for the pricing strategy that maximizes his expected profits. We analyze the seller’s solution with respect to different exogenous perturbations of parametric and functional inputs. For that purpose, we produce synthetic price data using the family of Generalized Error Distributions that includes normal and quasiuniform distributions as particular cases, and we also generate consumers’ shopping data from different behavioral assumptions. Our analysis shows that, beyond price mean and dispersion, the shape of the price distribution plays a significant role in the seller’s pricing solution. We focus on the seller’s response to an increasing diversity in consumers’ shopping behavior. We show that increasing heterogeneity in the shopping distribution typically lowers seller’s prices and expected profits.

#### 1. Introduction

More often than not, economic agents must operate in the market under lack of relevant information. Pricing is the primary key strategic variable for any profit maximizing seller with some market power. Each seller sets his own price—typically different across sellers—and each buyer searches for the lowest price while having a limited capacity to visit shops and learn different prices.

In this paper we adopt the point of view of a single seller and analyze his optimal pricing policy under limited knowledge both about its competitors’ prices and the shopping behavior of the typical buyer. The seller has overall information about rival prices represented by a continuous probability distribution , which is nondegenerate. This fact is referred to in the literature as market* price dispersion*. Additionally, the seller has an idiosyncratic lower bound : he is not willing to sell at any price below . The parameter can be interpreted as the seller’s unitary production cost, assuming linear production technology. More generally, represents the seller’s valuation of not selling. Also, we will consider a single representative buyer, who eventually visits a (random) number of shops from a total sample of size that has been previously drawn from . Notice that considering a representative consumer is not a restrictive condition for our analysis, since the relevant decision variable for the seller is the price he should offer to a buyer visiting his shop. The rival prices observed by the buyer are private information; the seller only knows that prices are drawn from .

The basic setting above can be framed within the literature of price dispersion in economics, business, or marketing. A first issue is why price dispersion should exist in a market for a homogeneous good. Classical economic theory postulates the “law of one price” since the seminal competition model by Bertrand [1]. If buyers look at all prices, technologies are linear, and the good on sale is homogeneous, only the shop offering the lowest price operates in the market. However, evidence from decades of empirical studies denies the law of one price, see, for example, the exhaustive survey by Baye et al. [2] or, more recently, the systematic study by Kaplan and Menzio [3]. Stigler [4] introduced search costs on the buyer’s side: if each additional price observation is costly, buyers will generally check only a subset of all existing prices. Later research by Burdett and Judd [5] completed the picture by showing that, under the existence of search costs, buyers will only check a subcollection of all existing prices, which in turn induces some price dispersion among sellers.

While theoretical economics may explain* why* price dispersion occurs—justifying the underlying assumptions in this paper—there is still the issue of* how* to behave under price dispersion. The optimal behavior of buyers under search cost has been analyzed in the literature with the focus on characterizing a (nondegenerate) price distribution that conforms an equilibrium. Here we opt for a more practical view: a seller in the market must define a pricing policy* even* off equilibrium. This angle turns out to be more relevant in real markets with large and persistent disequilibria.

A key feature in our pricing model is the fact that shopping behavior is diverse (or heterogeneous), that is, the number of sellers visited to make their purchase varies across consumers. Alternatively, a representative consumer selects the number of sellers to visit according to some (nondegenerate) probability distribution . The seller’s problem thus consists of selecting the price that maximizes his expected profit in an environment of price dispersion and, also, of diverse shopping behavior.

In the proposed model, a basic pricing policy maps an input quadruple to the price that yields maximal expected profits, namely, , where is the seller’s unit cost, is the market* price distribution*, is the* shopping distribution*, that is, the distribution of the number of shops which are visited by a representative consumer, and is the size of the sample of prices.

We solve numerically the seller’s problem for different costs, price distributions, and shopping distributions. We consider to be a Generalized Error Distribution (GED hereafter), which allows us to consider a wide variety of price distributions by changing the first, second, and higher order moments. The GED family includes normal and quasuniform distributions as particular cases. We are particularly interested in the role of the shape of in the seller’s solution. Apparently, the question whether the shape of the price distribution does or does not condition the optimal pricing behavior has not been considered in the literature. Furthermore, it was shown in Alvarez et al. [6] that changes in the first two moments of , but not in its shape, affect the consumer’s shopping behavior (determined via efficient time allocation). We will show below that this is not the case for the seller’s pricing problem.

We assume that is a discrete distribution defined on the set , so that a representative consumer selects* a priori* a sample of shops and then visits a number of them with probability (to keep notation simple we will use interchangeably for the shopping distribution and its probability mass function). This amounts to consider that the consumer’s behavior is perceived as probabilistic by the seller. In turn, this probabilistic perception might be exclusively due to the seller’s lack of knowledge or, alternatively, it might be that the buyer’s behavior is intrinsically probabilistic. Our main finding here is that, once a buyer goes shopping, more diversity in his shopping behavior, or, alternatively, more uncertainty in the knowledge of the seller about the consumer’s behavior, entails lowering prices and expected profits.

The pricing model above can be implemented in real markets once accurate estimates of market prices and consumer behavior are available. Estimates of the two basic inputs, and , of the pricing model above can in principle be obtained for specific markets. Price histograms for a large number of goods and markets can be obtained from available consumer data sets and they can be typically fitted by some within the GED family, as shown by Kaplan and Menzio [3]. Shopping data required to estimate are not commonly available in the literature or public databases. Yet, some studies on choice overload can provide partial information about (see, e.g., the references in the survey by Scheibehenne et al. [7]).

The manuscript is organized as follows. In Section 2 we describe the problem of a seller maximizing his expected profits in an environment of dispersed prices, which we solve numerically as explained in Section 3. In Section 4 we discuss the results of the numerical analysis. We conclude in Section 5 with final remarks.

#### 2. The Seller’s Problem

We adopt the point of view of a seller (or a firm) with linear production technology that must decide the price of his product in a market with an indeterminate number of competitors. The main elements of the scenario in which the seller makes his pricing decision are the existence of market price dispersion and imperfect knowledge about rivals’ prices and about the shopping behavior of his potential customers.

Specifically, the seller sets his price in a market in which a representative consumer with a fixed demand of, say, one unit, goes shopping. The seller has information about rivals’ prices represented by a probability distribution . The consumer selects* a priori * sellers or shops to visit, obtained as a random draw from . The products offered by the different sellers are indistinguishable for the consumer; that is, the market product is homogeneous. Consequently, he just searches for the lowest price. The consumer eventually observes price quotes, obtained as a subcollection of the preliminary sample of size .

In general, shopping behavior is diverse, so that a population of consumers is expected to be nonhomogeneous in terms of the number of shops to be visited. In turn, the shopping behavior of a representative consumer can be understood as probabilistic and heterogeneous. This heterogeneity is represented by a distribution supported on the set , so that a consumer will visit shops with probability . We assume that in the shopping behavior: if , the seller acts as a monopolist regardless of its rivals prices (regardless of ) since the consumer only checks one price. From the seller’s perspective, this probability distribution also reflects his uncertainty about the type of consumer who will visit his shop: he just knows that a consumer that visits shops will show up with probability .

The seller’s problem thus consists of finding the mapping , where is the optimal price; that is, maximizes the seller’s expected profits. Expected profits are defined bywhere is the probability of success in selling at price . Specifically, is the probability of making the sale at price , given that a consumer will buy at the lowest price after having visited a number of shops, , (with probability ), from a sample of size obtained from . Notice that can be written aswhere is the cumulative distribution function of the distribution . Now, given a price , the first term in equation (1) is simply the* mark-up*, the difference between price and unitary cost. The chances that the seller earns his mark-up—making the sale—competing in price with other sellers require that (i) the consumer visits the seller’s shop given that he will visit shops out of , and (ii) the seller’s price is the lowest among those prices checked by the consumer. The second factor in each term of the sum in (2) corresponds to (i), whereas the third factor corresponds to (ii). Given that the consumer will visit shops with probability , (2) gives the probability of making the sale at price . In the case that any of the events (i) or (ii) does not occur, , and consequently the seller makes zero profits, so that (1) can be interpreted as expected profit.

Notice that the seller faces a trade-off when choosing his price: increasing the price (the mark-up) implies increasing effective profits in the case he sells the unit, but also lowering the probability of selling the unit.

Shopping behavior is understood here as the choice of the number of shops to visit (alternatively, prices to check). Even in the case that the consumer has selected a total number of shops to visit* a priori*, he may actually end up visiting a smaller number . The classical literature of search cost (e.g., Stigler [4]) gives a rational explanation for this shopping behavior: assuming that the consumer minimizes her total expected cost—search cost plus expected price to be paid—the optimal may be typically lower than . Alternatively, the consideration of time as a fundamental constraint affecting consumer’s behavior can also explain the rational choice of a number of visits below . This applies in particular when the total number of shops is large and the consumer optimally allocates his time among several alternative uses. Indeed, it was shown in Sanchis et al. [8] and Alvarez et al. [6] that assuming that is determined from the time allocation that maximizes well-being, a consumer might choose not to visit the total number of available shops.

No particular optimal behavior on the consumer’s part will be assumed here. In fact, behavioral scientists claim that consumers’ behavior may be far from rational in this kind of shopping environment. The so-called choice overload phenomenon is a focal example in which consumers are worse off when they visit all available shops (see, e.g., Schwartz [9] and Iyengar and Lepper [10]). A consumer who visits* all* available shops is called a* maximizer* by social psychologists; otherwise he may be generically called a* satisficer* [11]. A consumers’ population is not expected to be composed by maximizers only, but rather by a mixture of maximizers and different types of satisficers. This mixture is represented in this paper by the shopping distribution : given a sample of shops in the market, is the probability that the representative consumer is a maximizer, whereas , for , are the probabilities that he is some type of satisficer.

#### 3. Numerical Analysis: Input Data and Problems under Study

We follow a numerical approach for the analysis of the seller’s problem described above. Given the unitary cost , a distribution of market prices , the number of sampled sellers, and a shopping distribution , a core computational routine must produce a numerical solution to problem (1). Since an explicit solution can only be obtained in very special cases, the numerical analysis becomes essential. We have used R as a programming environment (R Core Team [12]).

We consider the market price distribution to be a Generalized Error Distribution GED , whose parameters are location (), scale (), and shape () [13]. GED is a family of symmetrical unimodal distributions with domain in whose probability density function is given bywhere denotes the Gamma function. The parameter locates the mean, the median, and the mode of the distribution. The variance of the distribution depends on both scale and shape , as follows:Our choice is due to the fact that GED is a flexible family of distributions that includes Laplace (double exponential) distributions if , normal distributions if , or quasiuniform distributions when is large enough. In fact, the GED converges pointwise to a uniform distribution on as . Several GED probability densities for different shapes are displayed in Figure 2(a).

The skewness of a GED is zero, while the excess of kurtosis depends exclusively on the shape . In fact, GEDs with shape are leptokurtic. This is a significant case, since distribution of prices for individual goods appears to be leptokurtic in many cases (see Kaplan and Menzio [3]). In the model analysis with respect to shape below both leptokurtic (e.g., Laplace) and platykurtic distributions (e.g., quasiuniform) are considered. We set the mesokurtic case (normal distributions) as the benchmark case below for other analyses where shape is constant.

The solution to our seller’s problem can be written in parametric form as a mapping where is the maximum of (1) and is the seller’s optimal (expected) profit. We are mainly interested in analyzing the response of the output variables as the diversity of satisficers in the market increases. This amounts to study the seller’s solution with respect to suitable variations in while keeping everything else constant. Notice that a homogeneous population of maximizers can be defined by . When the probability mass is spread over the set , the heterogeneity of the distribution increases with respect to the maximizer case . It will be useful to characterize the degree of heterogeneity using some sensible parameter. We analyze the effects on price and profits of an increase in shopping heterogeneity which is due to two different sources, in turn controlled by two different parameters. First we consider in Section 4.2.1 uniform spreads over an increasing range of ’s, , with , which can be parameterized using the Shannon’s entropy of the distribution. Second, in Section 4.2.2, we analyze distributional spreads generated by shopping fatigue that can be parameterized by the probability of not visiting a new shop after having visited a number of them.

As a preliminary study, in Section 4.1, we consider a homogeneous population of maximizers, that is, , and analyze the effect of a variation in some basic parameters of the problem on the seller’s solution. We are particularly interested in learning whether the seller’s solution depends only on the mean and dispersion—typical deviation—of prices or whether the shape of the price distribution also matters.

#### 4. Results and Discussion

Initially, we assume a homogeneous population of maximizers, so that , and we analyze the seller’s response with respect to changes in the cost or in a distributional parameter of prices, while keeping everything else constant except for the market size —the number of available sellers* a priori*. The price distribution is assumed to be GED with shape , which corresponds with a normal distribution. The results do not differ if a different GED is considered. We are particularly interested in the effect of the shape of the price distribution on price and profits. Typically, for a set of values of , we compute and compare the seller’s solution within a range of equispaced values of each parameter, while keeping the rest of parameters at their benchmark values. The benchmark case and the range of variation considered for each parameter are shown in Table 1. The benchmark value of in Table 1 corresponds to the price level which is higher than 15% of all prices.