Chapter 2
Understanding Market Shares

2.1  Market Shares: Definitions

What do we mean by market share ? An obvious definition of a firm's market share might be ``that share of the market commanded by a firm's product (or brand).'' But this is merely a tautology and not a definition, and therefore does not help us understand market shares. Its basic difficulty lies in the ambiguity of the term market . One normally thinks of a market being a collection of persons (or institutions) who are likely to purchase a certain class of product. For consumer products and services, the market is a group of consumers who are potential buyers of a product or service, such as detergent, air travel, or coffee. (For simplicity, the term product is to be understood to encompass services hereafter, unless otherwise specified.) Those consumers who never buy a product are out of the market . If we define market in this manner, market shares should mean shares of potential consumers .

However, clearly this is not the common usage of the term. In most cases, market shares mean shares of the actual sales (either in quantity sold or dollar volume) for a product in a given period and in a given geographical area. Market in those situations should be taken as the sales performance of a product class in the market, rather than a collection of buyers for the product. In this book the term market shares will be used mostly in this latter sense. This concept of market shares may be more explicitly stated in the following manner.

where:

s_i = the market share of firm i

Q_i = the sales (quantity sold or dollar volume) of firm i 's product

Q = the total sales (quantity sold or dollar volume) for the market

Q = å_{j = 1}^m Q_j

m = the number of competing firms.

The quantity Q in the above equation is commonly called the industry sales , and we will follow this convention. Note that in this formulation market shares are a temporally and spatially specific concept, that is, defined and measured only for a specific period and a specific geographical area. It does not make much sense for one to talk about a firm's share in general; one must say instead a firm's share of the U. S. market in 1986, that of the European market in 1987, etc. This is because both the numerator and denominator of equation (2.1) are time- and area-specific, and will have to be matched for the same period and geographical area.

We shall have occasion also to refer to shares of potential buyers, but they will be denoted as consumer shares or buyer shares , so that there will be no confusion. Note that television audience ratings are consumer shares in this sense. In addition, a retail store's share of consumers who come to the shopping district in which it is located will be called a shopper share to distinguish it from its share of consumers from a certain geographical area or segment (e.g., consumer share). Most analytical techniques for market-share analysis will be directly applicable to the analysis of consumer shares with slight changes in the interpretation of equation (2.1), where Q_i and Q are replaced by the number of consumers, N_i and N, respectively. These points will be further clarified in a later section.

A further issue which should be addressed at this point is concerned with the choice of the level of distribution channel from which sales figures are obtained for computing market shares. When a manufacturer uses both wholesalers and retailers in its channel of distribution, it may possible to obtain three sets of sales figures, one each at the factory, wholesale, and retail levels. Complications arise since the sales figures at those three levels do not generally coincide. It is, of course, easy for the manufacturer to get sales (or shipment) data at the factory level, but are they meaningful for the purpose of market-share analysis? Some channels of distribution are notorious for their insensitivity to the changes in consumer demand. After all, aren't retail-sales figures more directly indicative of the firm's performance in the market? Isn't one of the most basic questions in market-share analysis how consumers are influenced by prices and promotions at the retail level? Or lacking retail-sales figures (which is not an uncommon condition for many firms), are wholesale (withdrawal) figures more appropriate than factory-shipment figures? Those are the questions which must be answered by the analyst before market shares may be computed. Much is dependent on the nature of products and services and the firm's ability to collect appropriate data. But, since assessing the effectiveness of marketing instruments in influencing consumer demand is one of the most importance uses of market-share analysis, the general principle should be to measure market shares as close as possible to the consumer-demand level.

2.2  Defining Industry Sales

We have defined market shares on the basis of sales performance of products and services, but this definition is not concrete enough. In order for one to be able actually to compute market shares for one's firm or brand, it is necessary to measure the denominator of equation (2.1), which is commonly called the industry sales. There are several problems associated with this measurement. First and foremost, what is a relevant industry ? As the first approximation, let's say that an industry is composed of a set of firms which are competing for the same group of (potential) buyers.Note that the number of competitors in an industry is given by m in equation (2.1).But how does one know which firms are competing for the same group of buyers? Here we will have to leave the matter pretty much to the experience and judgment of the analyst.

In many business contexts the market boundaries are known quite well. But even in the absence of such knowledge, or in times when it is proper to question prior assumptions, there is a principle to guide the analyst in delineating the boundaries for an industry, that is, choosing a set of firms which compete against each other. We know that the industry boundaries change, depending on what level(s) of buyer needs and wants for which firms compete. The more basic the needs for which firms compete, the greater will be the number of firms in an industry. Conversely, the more specific the attributes (of products and services) with which firms choose to compete, the smaller will be the number of firms in an industry. To illustrate, if we assume that firms compete for the basic human need of personal transportation, all firms which produce automobiles, motorcycles, and bicycles (even skateboards!) compete against each other and therefore form an industry. On the other hand, if we narrowly select those firms which choose to compete solely on the basis of the luxury of interior fixture of a vehicle, only a small number of firms will be in an industry.

But the principle is not easily translated into practice. There are techniques (such as multidimensional scaling) which may be used for delineating industry boundaries, but in the authors' opinion they are more exploratory than definitive. Instead, we will propose in this book a more developmental approach. Since the models in this book are able to give the analyst a basis for evaluating industry boundaries, why not use them empirically to form the relevant boundaries for an industry? In this approach, the analyst will first proceed with tentative boundaries for an industry, and, with the increase in experience and additional data, keep modifying the boundaries to converge eventually on a workable definition of an industry. If the focus is on understanding the effectiveness of one's marketing efforts, one wants an industry definition broad enough to include all threats to one's marketing program, but narrow enough that the same set of measures of marketing effort and performance can be applied to all competitors. We will illustrate this with the Coffee-Market Example presented in Chapters 5 - 7.

Second, even if one is successful in defining an industry, it may not be possible to know the sales (quantity sold or dollar volume) of firms other than one's own. Fortunate are those few industries for which trade associations or governmental agencies regularly gather data on industry sales. (The automobile industry is a prime example.) There are also those consumer products for which some agencies maintain continuous measurement of sales at the retail level for nearly all competing firms in the market (see Chapter 4). Lacking such associations/agencies, the estimation of industry sales may pose to the analyst a difficult research problem. One may survey buyers for their purchases (in a period) and estimate industry sales by multiplying average purchase size by the number of potential buyers. Or one may deduce industry sales from an indicator which is known to have a high correlation with them. With whatever technique one chooses, it may be only possible to estimate one's own share, but impossible to estimate the shares of competitors. As is shown in later chapters, this inability to estimate competitors' shares seriously compromises the efficacy of market-share analysis.

To summarize, the definition of relevant industry boundaries is not always a simple and precise operation. It requires subjective judgment of the analyst based on his or her experience and thorough knowledge of product, market, and competitors. But if industry definition is not a straightforward operation, one might be better off to select models which do not heavily depend on the correct definition of a relevant industry. As will be shown later, the MCI model and its relatives are less sensitive to how an industry is defined than the linear or multiplicative models are, since the former is adaptable to the so-called hierarchical market segments (see Chapter 3).

2.3  Kotler's Fundamental Theorem

With the preliminaries in the last two sections, we are now in a position to explore further the relationship of a firm's market shares with its marketing activities. For the time being we shall assume that a relevant industry is defined and industry sales are measured. KotlerKotler, Philip [1984], Marketing Management: Analysis, Planning, and Control , Fifth Edition, Englewood Cliffs, NJ: Prentice-Hall, Inc.posits that a firm's market share is proportional to the marketing effort of its product. In mathematical notation, this supposition may be written as:

where:

M_i = the marketing effort of the product of firm i

k = a constant of proportionality.

This is not a bad assumption. If a firm's marketing effort were measurable, one would think (hope?) that the greater the marketing effort of one's firm the greater should be its market share.

For equation (2.2) to be useful, setting aside for the time being the question of how one might measure marketing effort, one must know the value of proportionality constant, k . But market shares for an industry must sum to one, i.e.,

This implies that

or

Hence

By substituting this value of k in equation (2.2), we have

This last equation says that the market share of firm i is equal to the firm's marketing effort divided by the sum of marketing effort for all competitors in the industry. In other words, it says that a firm's market share is equal to its share of marketing effort, a statement which certainly seems plausible. Equation (2.3) is what Kotler calls the fundamental theorem of market share (Kotler [1984], p. 231).

It is interesting to note that this formulation is so basic that a whole series of variations may be devised from it. For example, if firms tended to differ in terms of the effectiveness of their marketing effort, one may write

where a_i is the effectiveness coefficient for firm i 's marketing effort. This implies that, even if two firms expend the same amount of marketing effort, they may not have the same market share. If one firm's marketing effort is twice as effective as that of the other, the former will achieve a market share twice as large as the other's share.For other variations, see Kotler [1984], pp. 224-237.

So far we stepped aside the question of how one goes about measuring the marketing effort for a firm's product. Here Kotler additionally assumes that a firm's marketing effort is a function of its marketing mix, both past and current. Mathematically, we may write,

where:

P_i = the price of firm i 's product

A_i = the advertising expenditures of firm i

D_i = the distribution efforts (e.g., trade allowances given by firm i).

There are wide choices in the specification of the functional form for equation (2.5). For example, if we choose it to be a multiplicative function

where p_i , a_i , d_i are parameters to be estimated, and substitute this expression in (2.3) or (2.4), the resultant market-share model will be an MCI model (see Chapter 1). Or if we choose an exponential function

the market-share model is called the multinomial logit (MNL for short) model. We will have more to say on the choice of functional forms.

2.3.1  A Numerical Example

At this point a numerical example may help the reader understand the nature of the models proposed by Kotler. Table 2.1 gives a hypothetical industry with three firms, with the values of the marketing mix for each and computed market shares.

Table 1: Numerical Example of Kotler's Fundamental Theorem

The model used here is a basic MCI model (equation (2.4) and the multiplicative function for M_i). The share of firm 1 is computed in the following manner.

In this example price is the most dominant factor. Firm 2 has a higher marketing-effort effectiveness, but its share is less than firm 3 because of a higher price. Compare Table 2.1 with Table 2.2 in which firm 2 reduced its price to $10.00. Now its share is the largest.

Table 2: Numerical Example - The Effect of Reducing Price

2.4  *Market-Share Theorem

Kotler's fundamental theorem gives us one justification for accepting equation (2.3) as a valid representation of the relationship between a firm's marketing mix and its market share. This market-share-as-share-of-marketing-effort representation makes a lot of intuitive sense, but there are other ways than Kotler's to derive such a representation. We will review some of them in a later section, and only look at one important theorem derived by Bell, Keeney and LittleBell, David E., Ralph L. Keeney & John D. C. Little [1975], ``A Market Share Theorem,'' Journal of Marketing Research , XII (May), 136-41.here.

Bell, Keeney, and Little (BKL hereafter) consider a situation where, in making a purchase of a product, consumers must choose one brand from a set of alternative brands available in the market. They posit that the only determinant of market shares is the attraction which consumers feel toward each alternative brand, and make the following assumptions about attractions. Letting  A_i be the attraction of brand i ( i = 1, 2,¼,m) and s_i be its market share,

A_i ³ 0 for all i and å_{i = 1}^m  A_i > 0 (i.e., attractions are nonnegative and their sum is positive).

A_i = 0 Þ s_i = 0 . (The symbol Þ should read ``implies,'' i.e., zero attraction implies zero market share.)

A_i =  A_j Þ s_i = s_j     (i ¹ j) (i.e., equal attraction implies equal market share).

When  A_j changes by D , the corresponding change in s_i     (i ¹ j) is independent of j (e.g., a change in attraction has a symmetrically distributed effect on competitive market share).

From those four axioms they show that the following relationship between attractions and market shares may be derived.

Perhaps no one would argue the fact that equation (2.6) and equation (2.3) are extremely similar. True, equation (2.3) and (2.6) represent two rather distinct schools of thought regarding the determinants of market shares (a firm's marketing effort for the former and consumer attraction for the latter). But an additional assumption that the attraction of a brand is proportional to its marketing effort (which is not unreasonable) is all that is required to reconcile two equations. It is rather comforting when the same expression is derivable from different logical bases.Axiom A 2.4 is has been the subject of critical discussion, as will be developed later.

BKL also show that a slightly different set of assumptions also yield equation (2.6). Let C be the set of all alternative brands from which consumers make their choice.

A_i ³ 0

The attraction of a subset S ( Í C) is equal to the sum of the attractions of elements in S .

A_i is finite for all i and non-zero for at least one element in C .

If the attractions of subsets S⁽¹⁾ and S⁽²⁾ are equal, their market shares are equal.

The last axiom establishes the relationship between attractions and market shares. BKL observe that, if we add an assumption that

in lieu of B 2.4,  A_i in this set of axioms satisfies the assumptions for probabilities in a finite (discrete) sample space. Because of this BKL suggest that attractions may be interpreted as unnormalized probabilities . However, this in turn suggests that if attractions were to follow axioms B 2.1 through B 2.4, by normalizing the  A_i 's through (2.6), market shares (s_i) may be interpreted as probabilities. This latter interpretation seems to confuse an aggregate concept (that is, market shares) with an individual (or disaggregated) concept (that is, probabilities). Only when the market is homogeneous (i.e., not composed of systematically different consumer segments), can market shares and choice probabilities be used interchangeably. We will return to this point in section 2.8.

2.5  Alternative Models of Market Share

The previous two sections gave the rationales behind the MCI model and its close cousin, the MNL model. We now give explicit specifications to those models.

MCI Model:

MNL Model:

where:

a_i = a parameter for the constant influence of brand i

e_i = an error term

and the other terms are as previously defined in Chapter 1. In the following parts of this book, we will use attraction , rather than marketing effort , to describe  A_i , because it is a more accepted terminology, keeping in mind that this implies the assumption that attraction is proportional to marketing effort. Note that the MCI model above is a version of the general MCI (attraction) model in Chapter 1 in that the monotone transformation, f_k , is an identity transformation. The MNL model is another version of the general model where f_k is an exponential transformation.Those models will be referred to raw-score MCI or MNL models in later chapters of this book.

But the MCI and MNL models are not the only models of market shares. A common formulation is that of the linear model which assumes simply that a brand's market share is a linear function in marketing-mix variables and other relevant variables. Another common form is the multiplicative model , where market shares are given as a product of a number of variables (raised to a suitable power). Although there are other more complicated market-share models, for our purposes at present it is sufficient to define explicitly the following three alternative models.

Linear Model:

Multiplicative Model:

Exponential Model:

The reader should note that the five models - MCI, MNL, linear, multiplicative, and exponential - are closely related to each other. For example, if we take the logarithm of both sides of either the multiplicative or exponential model, we will have a linear model (linear in the parameters of the respective models, and not in variables). In other words, the difference between the linear model and the multiplicative and exponential models is merely one of the choice of transformations for variables, that is, whether or not the logarithmic transformation is applied to variables. (The specification for the error term may be different in those three models, but this is a rather fine technical point which will be addressed in Chapter 5.)

The most interesting relationship is, however, the one between the MCI and multiplicative models, which is also duplicated between the MNL and exponential models. The multiplicative model, of course, assumes that market shares are a multiplicative function in explanatory variables, while in the MCI model attractions are multiplicative in variables and market shares are computed by normalizing attractions (i.e., by making the sum of market shares to be equal to one). Obviously, the key difference between the two is normalization. In this connection, Naert and Bultez [1973] proposed the following important condition for a market-share model.

1. Estimated market shares from the model are nonnegative.
2. The sum of estimated market shares is greater than zero and less than or equal to one.

These conditions, commonly known as the logical-consistency requirements , are clearly not met by either the multiplicative or exponential model, but are met by their respective normalized forms (i.e., MCI and MNL), which must be a clear advantage for MCI and MNL models. Note that the linear model does not satisfy the logical-consistency requirements.

Why, then, are the MCI and MNL models not used more extensively? The answer is that for a time both of those models were considered to be intrinsically nonlinear models, requiring estimation schemes which were expensive in analysts' time and computer resources. This, however, turned out to be a hasty judgment because those models may be changed into a linear model (in the model parameters) by a simple transformation. Take the MCI model, for example. First, take the logarithm of both sides.

If we sum the above equation over i (i = 1, 2, ¼,m) and divide by m , we have

where [s\tilde] , [X\tilde]_k and [(e)\tilde] are the geometric means of s_i , X_ki and e_i , respectively. Subtracting the above from the previous equation, we obtain

where:

a_i^* = (a_i - [`(a)])

e_i^* = log(e_i / [(e)\tilde]) .

The last equation is linear in model parameters a_i^* ( i = 1, 2,¼, m) and b_k (k = 1, 2, ¼, K). (In addition, there is another parameter s_e² , the variance of e_i , to be estimated, but this parameter does not concern us until Chapter 5.) This transformation will be called the log-centering transformation hereafter.The importance of this transformation is that we can estimate the parameters of the original nonlinear model using linear-regression techniques.Note that, if we apply the log-centering transformation to the MNL model, we obtain the following linear form.

where [`(a)] , [`X]_k and [`(e)] are the arithmetic means of a<sub>i</sub> , X<sub>ki</sub> and e<sub>i</sub> , respectively. If we let a<sub>i</sub><sup>*</sup> = (a<sub>i</sub> - [`(a)]) and e_i^* = (e_i -[`(e)]) ,

Both equations (2.12) and (2.13) are linear functions in model parameters, and hence called log-linear models. Recall that the multiplicative and exponential models are also log-linear models. In other words, both the MCI and MNL models are really special cases of log-linear models. This point may be dramatically illustrated by the following comparisons among the reduced forms of these models.The reduced forms of these models contain variables transformed so that they made be directly submitted to a multiple-regression routine.:

Multiplicative Model:

Exponential Model:

MCI Model:

MNL Model:

In the above, the multiplicative and exponential models are shown in the log-linear form. In all five equations the right-hand side is linear in both a_i (i = 1, 2,¼, m) and b_k (k = 1, 2,¼,K). The left-hand side is either a market share, the logarithm of a market share, or a log-centered form of a market share. We will defer the discussion on the specification of the error term ( e_i , loge_i and e_i^* ) till Chapter 5, but, since the number of parameters in the five formulations are the same, one would expect that those models would be just as accurate in predicting the dependent variable , namely, the left-hand side of each equation. Many readers would then ask, ``Which one makes the most accurate prediction of market shares?'' We report that many studies on predictive accuracy of market-share modelsSee Brodie, Roderick & Cornelius A. de Kluyver [1984], ``Attraction Versus Linear and Multiplicative Market Share Models: An Empirical Evaluation,'' Journal of Marketing Research , 21 (May), 194-201. Ghosh, Avijit, Scott Neslin & Robert Shoemaker [1984], ``A Comparison of Market Share Models and Estimation Procedures,'' Journal of Marketing Research , 21 (May), 202-10. Leeflang, Peter S. H. & Jan C. Reuyl [1984], ``On the Predictive Power of Market Share Attraction Models,'' Journal of Marketing Research , 21 (May) 211-15.found the logical-consistency property of the MCI and MNL models to produce only marginally better predictions than the linear and multiplicative models. Why then all this fuss about the MCI and MNL models? First, these test did not include the more sophisticated versions of the models (see Chapter 3). And second, as was stated in Chapter 1, we do not believe that predictive accuracy is the only important criterion for judging the value of a model. We would rather find the answer in the construct validity (i.e., intrinsic meaningfulness) of those models, which is discussed in the next section.

2.6  Market-Share Elasticities

There is a common yet perplexing question which often arises on the part of a product/brand manager, that is, ``How much will our brand share change if we change a marketing-mix variable by a certain amount?'' The answer to this question is obviously vital to those who develop short-term marketing plans. After all, a brand manager must decide what price to charge and how much the firm should spend in advertising, sales promotion, trade allowances, etc. If the market responses to the changes in marketing-mix variables were known, his/her job would become in many ways immensely simpler.The planning process would still require a substantial effort as is discussed in Chapter 7.But this is one of the most difficult pieces of information to obtain. It might be said that in a sense this entire book is devoted to answering this difficult question.

Before we begin the discussion of how one predicts actual changes in market shares, let us first look into a theoretical concept, market-share elasticity , which will help us in measuring responses toward marketing-mix variables. Simply stated, market-share elasticity is the ratio of the relative change in a market share corresponding to a relative change in a marketing-mix variable . Expressed mathematically,

where s_i is the market share, and X_ki is the value of the k^th marketing-mix variable, for brand i . The symbol D indicates a change in respective variables. There is nothing conceptually difficult in market-share elasticity. For example, if a brand's share increased 10% corresponding to a price reduction of 5%, the above equation would give a (price) elasticity of -2 ; or if advertising expenditures were increased by 3% and as a result the share increased by 1%, the (advertising) elasticity would be 0.33; and so forth. We say that a brand's market share is elastic with respect to X_ki if the (absolute) value of e_si is greater than one; inelastic if it is less than one. It is also obvious that one may predict the changes in market share from the knowledge of market-share elasticity. If e_si is 0.5, then we know that a 10% increase in X_ki will produce a 5% increase in market share. In absolute terms, if the current share is 30% and the current advertising expenditure (per period) is $1 million, a $100,000 increase in advertising expenditure will result in a 1.5% increase in market share.

No one would deny that share elasticities would give one a clear perspective on the effect of marketing-mix variables on market shares. And this is not just a theoretical concept, either. It is not far-fetched if we said that, even if product/brand managers might not know the exact magnitude of market response to the change in a marketing-mix variable, they might have a reasonably good idea of market-share elasticities for their product/brand, at least to the extent that market shares are either elastic or inelastic to changes in marketing-mix variables. As will be shown later, it is sometimes difficult for one to separate market-share elasticities from the elasticities regarding the industry sales, but, even so, experienced managers should have a fair grasp of market-share elasticities. If a manager had no idea of whether a 5% change in price or advertising expenditures would bring about a less than, equal to, or more than 5% change in the market share, how could he/she even approach to making short-term marketing plans?

No matter how experienced a manager is, however, he/she seldom knows the exact magnitude of elasticities. One of the reasons for such general states of uncertainty is that share elasticities change over time depending on the share levels and the intensity of competitive activities at that time. A manager might be able to get a reasonable idea of share elasticities for his/her brand through experience if they were relatively stable over time, but the fact is that elasticities are not constant over time. In general, the greater the share of a brand, the smaller one expects the elasticities for its share to be. (The reader will find that this is the basic reason for rejecting the Multiplicative model because it implies constant elasticities.) It is obvious that, if a brand has a 95% share of market, say, it cannot gain more than 5 percentage points even if the magnitude of X_ki is increased by more than 5%. Also one would generally expect that a brand's share changes will be affected by what other brands do. If the competitors are relatively inactive, a brand may gain a large share by lowering its price. But, if competitors retaliate quickly, a brand may not gain any share at all for the same amount of price reduction. To summarize, the necessity to take share levels and competitive reactions into account puts the manager at a severe disadvantage in knowing market-share elasticities accurately.

Market-share models discussed in the preceding section will help managers by providing them with the estimates of elasticities. This, we believe, is one of the most important contributions of those models. Stated differently, there is no way to estimate elasticities directly from empirical data without adopting a model. This may not be intuitively clear because the formula for computing elasticities (2.14) appears to contain only those terms which may be empirically measurable. But note that the Ds_i term in equation (2.14) must correspond to the change in a specific marketing-mix variable, DX_ki . Suppose that one observed that a brand's share increased 3% in a period. How does one know how much of that increased share is due to price reduction? Or due to increased advertising? To assess those so-called partial effects one needs a market-share model.

The reader may be cautioned at this point that the estimated values of elasticities vary from one model to the other, and hence one must choose that model which fits the situation best. To illustrate, we will derive the share elasticity with respect to X_ki for the simplest version of each model. For that purpose, however, one needs another concept of share elasticity which is slightly different from the one defined by (2.14). Technically, (2.14) is called the arc elasticity . This is because both Ds_i and DX_ki span a range over the market-response curve which gives the relationship between market shares and marketing-mix variables. The other elasticity formula is called the point elasticity and takes the following form.

Note that the only difference between the two formulas is that ( Ds_i /DX_ki) in equation (2.14) is replaced by ( ¶s_i / ¶X_ki) in (2.15). Formula (2.15) utilizes the slope of the market-response curve at a specific value of X_ki . The reason for using the point-elasticity formula than the arc formula is that the former gives much simpler expressions of share elasticity. We may add that (2.15) is a close approximation of (2.14) for a small value of DX_ki , that is, when the change in X_ki is very small. The point elasticity for each model is given below. For convenient formulas for computing point elasticities see Appendix 2.9.1.

Linear Model:

Multiplicative Model:

Exponential Model:

MCI Model:

MNL Model:

Though the five market-share models are similar in the sense that they are either linear or log-linear models, the share elasticities implied from the models are quite different. One may wish to disqualify some models on the basis of those expressions on some a priori grounds. In the following we will only present verbal discussions, but the interested reader is referred to a more mathematical treatment of the properties of market-share elasticities given in Appendix 2.9.2.

First, one would expect that a brand's share elasticity approaches zero as the share for that brand approaches one. The Multiplicative model implies that share elasticity is constant regardless of the share level, and therefore seems rather inappropriate as a market-share model.

Second, it is generally accepted that it becomes harder to gain market shares as a firm increases its marketing effort. In other words, one would expect market-share elasticity to approach zero as X_ki goes to infinity (or minus infinity depending on the variable in question). But the Exponential model implies an opposite: share elasticity may be increased indefinitely as the value of X_ki increases. This is an uncomfortable situation, especially if variable X_ki is a promotional variable (such as advertising expenditures, number of salesmen, etc.). In addition, the Exponential model has the same problem as the Multiplicative model: for a fixed value of X_ki , e_si is constant for all levels of s_i .

Note that the elasticity expression for the Linear model reflects that share elasticity declines as the share increases, but, when the share approaches one, the elasticity does not approach zero. In fact, share elasticity approaches 1 as X_ki increases to infinity (or minus infinity, as the case may be). Thus the Linear model produces a highly unreasonable share-elasticity expression.

Considering what we expect from share elasticities, one may conclude that the Linear, Multiplicative, and Exponential models are not proper market-share models for use in marketing decision making. This leaves us the MCI and MNL models as feasible alternatives. Figure 2.1 shows the change in share elasticity over the positive range of X_ki values.

Figure 2.1: Share Elasticities for MCI and MNL Models

The share is assumed to increase as X_ki increases over the range. Accounting for this share increase, the share elasticity for the MCI model monotonically declines as X_ki increases (Figure 2.1 (a)), while that for the MNL model increases to a point and then declines.

The reader will ask which expression is a better one for share elasticity. The answer is, ``It depends on variable X_ki .'' The relevant issue here is how share elasticity should behave for low values of the variable. If X_ki is product price, for example, it is more likely that share elasticity is fairly large even when price is near zero. Hence, one would be inclined to use the MCI model for price. On the other hand, if the variable is advertising expenditure, it is not unreasonable to assume that, at an extremely low level of expenditure, advertising is not very effective. (This is to say that there is a threshold effect for advertising.) This assumption, of course, leads to the adoption of the MNL model for advertising expenditure. Indeed, it is the authors' position that the choice between the MCI and MNL models is not one or the other; both models may be mixed in a single formulation for market shares. The reader may recall that a general MCI (attraction) model of the following type was presented in Chapter 1.

General MCI (Attraction) Model:

where f_k is a monotone transformation of X_ki . It is obvious that, if one chooses an identity transformation for k (that is, f_k(X_ki) = X_ki), (2.16) becomes the MCI model; if f_k is an exponential function (that is, f_k(X_ki) = exp(X_ki)), then (2.16) becomes the MNL model. But there is no reason for one to have to choose either the identity or exponential transformation for all the f_k 's in (2.16). Depending on the nature of variable X_k , one should be free to choose either the identity or exponential transformation (or any other appropriate monotone transformation, for that matter). This is why in (2.16) f_k has subscript k . For the remainder of this book, we will use a mixture of MCI and MNL models.

2.7  Sales-Volume Elasticities

The reader is perhaps convinced by now that market-share elasticities play a significant role in market-share analysis, but we must add that they are not the only important concept in marketing planning. It may be superfluous to say that the product/brand manager is interested not in forecasting market-share changes themselves, but in forecasting the changes in sales volume of the firm's product corresponding to the changes in marketing-mix variables. For this latter purpose, knowing share elasticities is not enough. Since a brand's sales volume is a product of its market share and the relevant industry sales, one would also need information on how much industry sales change due to the firm's marketing activities.

There is a rather interesting relationship between the share, industry-sales, and sales-volume elasticities for a firm. One may define point sales elasticity in the manner analogous to (2.15).

where Q_i is the sales volume (in units sold) for firm i . Similarly, point industry-sales elasticity may be defined as follows.

where Q is the sales volume for the entire industry. It is well known that the following simple relationship exists among e_si , e_Qi , and e_Q .Lambin, Naert & Bultez [1975], ``Optimal Marketing Behavior in Oligopoly,'' European Economic Review , 6, 105-28.

This equation shows that sales elasticity for firm i is merely the sum of the firm's share elasticity and industry-sales elasticity.This assumes no systematic competitive reactions. This assumption is relaxed in section 6.3.For example, if share elasticity is 0.6 and industry-sales elasticity is 0.1, the sales elasticity is 0.7. In other words, if a firm increased a marketing-mix variable by 10%, say, and as a result increased its share by 6% and at the same time succeeded in expanding the industry sales by 1%, then its sales volume should increase by approximately 7%.The numerical result is approximate because equation (2.19) holds for point elasticities only. If the change in X_ki is relatively large as in this example, the relationship should be expressed by arc elasticities as follows:

When [(DQ)/ Q] is relatively small, e_Qi » e_si + e_Q as in the case of point elasticities.Equation (2.19) combines the essential elements of interbrand competition. In order to expand a firm's brand sales, one will have to increase either industry sales or the market share for the brand, or both. But one would expect that in many cases the industry sales will be rather inelastic to a single firm's marketing activities. If this were indeed the case, brand sales could be expanded only if the firm's share could be increased. Suppose that e_si = 0.6 and e_Q = 0.1 . In order to increase the brand sales by 10%, the market share will have to be increased by more than 8.5% (10 ×0.6 / 0.7). On the other hand, if industry sales were reasonably responsive to a firm's marketing mix, the increase in brand sales could be achieved with relatively small increase in market share. If e_si = 0.6 and e_Q = 0.3 , say, a 6.7% ( 10 ×0.6 / 0.9) share increase results in a 10% increase in brand sales.

To summarize, when industry-sales elasticities are near zero, it is clear that firms in an industry will be fighting for a share of a fixed pie . The reliance on market-share increases in improving a firm's sales will undoubtedly leads to a more intense competition. If industry-sales elasticities are reasonably large, firms will not have to be too sensitive about taking shares from others (or others taking one's share), resulting in more complacent competitive relationships among firms. Thus the knowledge of industry-sales elasticities is helpful to marketing managers in assessing competitive pressures existing in an industry. We will turn to the estimation of industry-sales elasticities in Chapter 5 and the effects on brand planning are discussed in Chapter 7.

2.8  *Market Shares and Choice Probabilities

So far we have chosen to treat market shares as an aggregate quantity, namely, the ratio of a firm's (brand) sales to the relevant industry sales. But, since aggregate sales are composites of many purchases made by individual buyers (consumers and industrial buyers), market-share figures must be related to individual buyers' choices of various brands. In fact, one frequently encounters measures of market shares which are based on data obtained from individual buyers. The market-share figures computed from the so-called consumer-diary panels or optical-scanner panelsThese are consumer panels whose purchase records are maintained by utilizing optical scanners located at selected retail stores.are prime examples of such individual-based market-share measures. But such individual-based market-share measures are estimates of actual market shares, and, if one had actual share figures, why should one bother with those measures? In this section we will look at the relationships between market shares and choices made by individual buyers and examine the importance of individual-based market-share data.

In analyzing the relationships between market shares and individual choice probabilities, we will have to consider the variability of two factors - choice probabilities and purchase frequency for individual buyers - over the population of buyers in the market. Let us first define those two concepts.

Suppose that each buyer purchases a number of units of the product per period. We will assume that the purchase frequency (i.e., the number of purchases per period) by an individual buyer is a random variable which has a statistical distribution. We shall call this distribution an individual purchase-frequency distribution, since it is defined for each buyer in the market. The specific form of this distribution does not concern us here except that it has its mean (mean purchase frequency).

Let's assume, not unreasonably, that a buyer does not always purchase the same brand from a set of alternative brands (this is the relevant industry in our previous definition). In addition, it is assumed that a buyer's choice of a brand at one purchase occasion is made independently from his/her previous purchase and the buyer's selection is governed by probabilities specific to each brand.This is to say, each buyer's brand selection in a period follows a Bernoulli process. The set of probabilities for alternative brands in the industry is called individual choice probabilities .

In our view, whether or not the buyer's behavior is truly probabilistic or deterministic is not an issue here. A buyer's choice behavior may be totally deterministic, but the environmental conditions surrounding purchase occasions may be such that they involve probabilistic elements which, from the viewpoint of an outside observer, make the buyers' choices appear probabilistic.

We also posit that the attractio

ns of alternative brands affect choice probabilities. This is to be consistent with our position that brand attractions are the determinants of market shares. We refer the reader to Appendix 2.9.3 for how individual choice probabilities are determined from attractions of alternative brands.

We distinguish four cases regarding the homogeneity (or heterogeneity) of the buyer population with respect to individual choice probabilities and purchase frequencies .

Case 1: Homogeneous Purchase Frequencies and Choice Probabilities

If mean individual purchase frequencies are equal for all buyers and the brand selection of every buyer in the market is governed by the same set of choice probabilities, it is rather obvious that the market share of a brand will be approximately equal to the choice probability for the brand. (Actually, it is the expected value of the market share of a brand which will be equal to the choice probability for the brand.) In this case market shares may be interpreted as individual choice probabilities without much difficulty. For example, if the market share for a brand is 0.3, one may say that each buyer chooses this brand with a 0.3 probability.

Case 2: Homogeneous Purchase Frequencies and Heterogeneous Choice Probabilities

The interpretation of market shares will have to be changed a little if each buyer has a different set of choice probability values for alternative brands. We still assume that mean purchase frequencies are equal for all buyers in the market. Under those assumptions it is easy to show that the expected value of a brand's market share is equal to the (population) average of choice probabilities for that brand. In other words, a market share of 0.3 may be interpreted that the average of choice probabilities across the buyer population is 0.3.

Case 3: Heterogeneous Purchase Frequencies and Homogeneous Choice Probabilities

This is the case where, while a common set of choice probabilities is shared by all buyers, mean purchase frequencies vary over buyers and have a statistical distribution over the buyer population. In this case the expected value of a brand's market share is still equal to its choice probability.

Case 4: Heterogeneous Purchase Frequencies and Choice Probabilities

In this case both choice probabilities and purchase frequencies are assumed to be variable over the buyer population. We need to distinguish further two cases within this.

(a) Uncorrelated Case: choice probabilities and purchase frequencies are uncorrelated (i.e., independently distributed) over the buyer population.

(b) Correlated Case: choice probabilities and purchase frequencies are correlated over the buyer population.

Let's first look at the uncorrelated case. If purchase frequencies and choice probabilities are uncorrelated, the expected value of market shares are, as is shown later, still equal to population averages of choice probabilities (as in the case of homogeneous purchase frequencies). Turning to the correlated case, one finds that market shares are no longer directly related to choice probabilities. This is perhaps more realistic for most products, since one often hears that so-called heavy users and light users exhibit remarkably different purchase behavior. Heavy users are said to be more discriminating in taste, more price conscious, and tend to purchase family-size or economy packages, etc. It is not surprising, then, to find heavy users preferring some brands or brand/size combinations to those preferred by light users. If there were differences in the value of choice probability for a brand between heavy and light users, individual purchase frequencies and choice probabilities would be correlated and the market share for the brand will be biased toward the choice probability values for heavy users simply because they purchase more units of the brand. Thus market shares and choice probabilities generally do not coincide in this case. Table 2.3 illustrates those points.

Table 2.3: Effect of Correlation Between Purchase Frequencies and Choice Probabilities

Table 2.3 (a) shows the case where purchase frequencies and choice probabilities are uncorrelated. In this case the expected market shares are equal to the averages of choice probabilities. In Table 2.3 (b) there is a moderate degree of correlation between purchase frequencies and choice probabilities, as heavy buyer 3 prefers brand 1 while light buyers 1 and 2 prefer brand 2. In this case market share for brand 1 is greater than the average of choice probability for the brand. The results above may be stated more formally as follows.

The expected value of unit sales for brand i is obtained by averaging (over the buyer population) individual purchase frequencies multiplied by the individual's choice probability for the brand. Hence the expected value of market share for brand i is given by:

or

where:

E(s_i) = the expected value of market share for brand i

m = the mean purchase frequency per period (per individual)

[`(m)] = the population mean of m

p_i = the individual choice probability for brand i

g(m, p_i) = the joint density function for m and p_i .

Equation (2.20) shows that the expected value of market share for brand i is a weighted average of choice probabilities (weights are individual mean purchase frequencies) divided by average mean purchase frequency [`(m)] . From (2.20) we directly obtain the following result.

where:

[`(p)]_i = the population mean of p_i

cov(m, p_i) = the covariance of m and p_i .

This is because, by definition, cov(m, p_i) = [`(m)] E(s<sub>i</sub>) -[`(m)] [`(p)]<sub>i</sub> . This equation shows that in general E(s<sub>i</sub>) is not equal to [`(p)]_i . Since cov(m, p_i) may be positive or negative, one cannot say if market shares are greater or smaller than population mean of choice probabilities. But if m and p_i are positively correlated, E(s_i) is greater than [`(p)]<sub>i</sub> . If the correlation is negative, E(s<sub>i</sub>) is less than [`(p)]_i . Note also that, if cov(m, p_i) = 0 (that is, if there is no correlation between the market share and choice probability), then the expected market share and the average choice probability are equal. In other words, in the uncorrelated case the expected value of a brand's market share is equal to its average choice probability ([`(p)]_i). The foregoing results are summarized in Table 2.4.

Table 2.4: Relations Between Market Shares and Choice Probabilities

It is apparent from the table that the only case where there is no correspondence between market shares and choice probabilities is Case 4 (b). This fact might tempt one to look at this case as an exception or anomaly, but this is probably the most prevalent condition in the market. A practical implication of the preponderance of Case 4 (b) is that, for the purpose of market-share forecasts, it is not sufficient for one to be able to predict the choice behavior of individuals accurately; rather it becomes necessary for one to be able to predict choice probabilities for each different level of purchase frequencies.

Of course, the situation cannot be changed by merely assuming that m and p_i are uncorrelated over the buyer population (Case 4 (a)). Since m and p_i are arithmetically related, that is, p_i = m_i/ m where m_i is the expected number of units of brand i purchased by an individual, and å_{i = 1}^m m_i = m where m is the number of alternative brands in the industry, the assumption that cov(m, p_i) = 0 (for all i) implies a very restrictive form of joint distribution for m and p_i . Indeed, it may be shown that m is distributed as a Gamma function and the p_i 's are jointly distributed as a Dirichlet distribution. No other distributional assumption will give cov(m, p_i) = 0 . See Appendix 2.9.4 for the proof of this result. The reader will find that the assumption that the processes which generate individual purchase frequencies are independent of the processes which determine individual choice probabilities is very common in modeling consumer choice behavior. But note that this practice is equivalent to assuming no correlation between m and p_i and hence should be made only in limited circumstances where those restrictive distributional assumptions are justified.

What does all this argument about the relationship between purchase frequencies and choice probabilities suggest to analysts and marketing managers? If the correlation between purchase frequencies and choice probabilities are suspect, it is clearly advisable to segment the market in terms of purchase frequencies and analyze each segment separately. One may discover that marketing instruments have different effects on different segments and may be able to allocate marketing resources more efficiently. Also forecasting of brand sales and market shares will become more accurate if market shares are forecast for each segment and weighted by the mean purchase frequencies for the segments to obtain the estimate of overall market shares. Segmentation analysis of this type, however, requires more refined data than usual aggregate market share data, such as consumer-diary or scanner-panel data. Individual-based market-share estimates obtained from such data sets may become increasingly important if the data are accompanied by measures of the individual buyers' characteristics and allow the analyst to look into how buyer profiles are related to purchase frequencies. We will deal with the issues related to market-share data more extensively in Chapter 4.

2.9  Appendices for Chapter 2

2.9.1  *Calculus of Market-Share Elasticities

The point elasticities of market share s_i with respect to a marketing variable X_kj are given by

The calculation of elasticities may be rather cumbersome for specific market-share models. In this appendix several formulas which are useful in calculating market-share elasticities are derived. In the following e_y.x indicates the elasticity of variable y with respect to variable x .

Chain Rule for a Compound Function:

If Y = f(X) and Z = g(Y) , then

Elasticity for a Sum of Variables:

If W = Y + Z , then

Elasticity for a Product of Variables:

If W = Y·Z , then

Elasticity for an Inverse of a Variable:

If Y = 1/Z , then

2.9.2  *Properties of Market-Share Elasticities

Market-share elasticities have the following properties.

1. Since Q_i = Q ·s_i , we readily see that e_Qi.x = e_Q.x + e_si.x , from the elasticity formula for a product of variables in Appendix 2.9.1.
2. If s_i is an increasing function in X , e_si.x® 0 as s_i ® 1 . This is because, as s_i ® 1 , Q_i ® Q and therefore e_Qi.x ® e_Q.x in (1).
3. If s_i is a strictly increasing function in X , e_si.x® 0 as X ® ¥ . This is derived from (2), since as X ® ¥ , s_i ® 1 . If s_i is an increasing function in X but approaches a constant ( < 1) as X ®¥ , then e_si.x ® 0 (since ds_i/dX ® 0 ).

Not all market-share models satisfy the above three properties of market-share elasticities. Compare the following five models and the corresponding elasticities with respect to variable X .

Linear Model:

Multiplicative Model:

Exponential Model:

MCI Model:

MNL Model:

If b is positive, it is clear that neither the linear, the multiplicative, nor the exponential model satisfies property (2) and (3). Only the MCI and MNL models satisfy all properties. This is one of the basic justifications for our choice of the latter models for market-share analysis.

2.9.3  *Individual Choice Probabilities

We have not discussed how individual choice probabilities are determined. The focus of this appendix is to relate individual choice probabilities to attractions of alternative brands. We can of course assume that the choice probability for a brand is proportional to its attraction, and obtain a result similar to Kotler's fundamental theorem discussed in section 2.3. But there are other more axiomatic approaches to deriving choice probabilities, and here we will be dealing with two basic models which are closely related to each other. It may be added that the terms attraction and utility will be used interchangeably in this appendix.

2.9.3.1  *Constant-Utility Models

The simplest model for choice probabilities is the constant-utility model which is also called the Luce model or Bradley-Terry-Luce model. Its basic assumption (or axiom) may be stated as follows. Axiom 1 Let an object, x , be an element of the choice set (i.e., set of choice alternatives), C , and also of a subset of C , S (i.e., S Í C). The probability that x is chosen from C is equal to the product of the probability that x is chosen from S and the probability that (an element of) S is chosen from C.

Luce calls this assumption the individual choice axiom , which may be expressed mathematically as:

where Prob (x \mid C) is read ``the probability that x is chosen from C .''

This axiom for choice probabilities leads to results similar to that of the Market-Share Theorem for market shares. If we let

for an arbitrary object in C , then for two objects x and y in C

and this ratio does not change with the choice of z . Also, since

we have

The quantity u_x is called the constant utility of object x , and presumably determined for each individual as a function of marketing activities for x .

This model formed a basis of various models of individual choice behavior,Huff, David L. [1962], Determination of Intraurban Retail Trade Areas , Los Angeles: Real Estate Research Program, University of California, Los Angeles. Haines, George H., Jr. Leonard S. Simon & Marcus Alexis [1972], ``Maximum Likelihood Estimation of Central-City Food Trading Areas,'' Journal of Marketing Research , 9 (May), 154-59. Nakanishi, Masao & Lee G. Cooper [1974], ``Parameter Estimation for a Multiplicative Competitive Interaction Model - Least Squares Approach,'' Journal of Marketing Research , 11 (August), 303-11.and was also implicitly adopted for many market-share models. But the model exhibits the so-called Independence from Irrelevant Alternatives (IIA) property which produces some quite counterintuitive results. From the axiom we have

for any subset of S Í C which contain both x and y . Since this relationship must hold for set {x , y},

This ratio is independent of the choice of z . Since z is supposedly irrelevant to the odds of choosing x over y , this has been called the Independence of Irrelevant Alternatives (IIA) property. The classic counter examples are from Debreu.Debreu, Gerard [1960], ``Review of R. D. Luce's Individual Choice Behavior: A Theoretical Analysis , American Economic Review , 50 (1), 186-8.Although Debreu proposed a record-buying situation, the issues are more clearly illustrated using a transportation-choice example. Suppose a person is indifferent between riding on a red bus (RB) versus a blue bus (BB) if offered just these two alternatives, but prefers riding a taxi (T) four-to-one over a red bus , if offered this pair or four-to-one over a blue bus , if offered that pair of alternatives. The choice axiom would summarize this case by noting that Prob(RB \mid {RB, BB}) = .5 , Prob(T\mid {T, RB}) = .8 , and Prob(T \mid {T, BB}) = .8 . While it seems clear that the probability of choosing a taxi shouldn't decrease if it is offered in a choice set along with both a red bus and a blue bus (i.e., Prob(T \mid {RB, BB, T}) should still be .8 ), the choice axiom insists that Prob(T \mid {RB, BB, T}) = .67 and Prob(RB \mid {RB, BB, T}) = .16 . The choice axiom forces this so that the ratio of the utility of RB to T is constant regardless of the choice set in which they are offered.

The concept of constant utility and the IIA property are really two sides of the same coin. If we think of utility is a inherent property of an object which doesn't change regardless of the context in which a choice is made, we will be trapped by the IIA property into counterintuitive positions.

There are two ways out of this problem. First, we can explicitly consider how the context in which choices are made affect the attraction of the alternatives. This is the path we follow in Chapter 3 when discussing the distinctiveness of marketing efforts (see section 3.8). Second, we can consider utility to be a random variable, rather than a constant. This is the topic of the next section.

2.9.3.2  *Random-Utility Models

A broad group of choice models is based on the assumption that utilities which an individual feels toward various objects (in our application, brands in an industry) at each purchase occasion are random variables, and the individual selects that brand which happens to have the largest utility value among the alternatives at that occasion. This model, known as a random-utility model , is defined as follows. Let U₁, U₂, ¼, U_m be the utilities for alternative brands where m is the number of brands in the choice set, C ,C is the set of all competing brands in the industry.and g(U₁, U₂, ¼, U_m) be the joint density function for them. The probability that brand i is chosen at a purchase occasion is given by

In order to evaluate this probability, however, one must evaluate an integral function. For three brands, the probability that brand 1 is chosen is given by the following integral.

Similarly, Prob (2 \mid C) and Prob (3 \mid C) are given by suitably changing the upper limits of integration. Integral (2.22) may be defined for any number of choice objects (e.g., brands).

A large number of variants of random utility models may be created from this definition by selecting different specifications for g . However, the usefulness of random utility models is limited because, unless the density function g is so very special as to give an analytical solution, the evaluation of this integral will in general require numerical integration. For example, if g is a joint normal density (known as a probit or multivariate-probit model), there is no analytical (or closed-form) solution to this integral. The probit model is a reasonable model for many applications, but its use has been hampered by the fact that the evaluation of (22) for a large number of objects involves tedious numerical integration.

There is one exception, however, to the need for cumbersome numerical integration of (2.22). McFaddenMcFadden, Daniel [1974], ``Conditional Logit Analysis of Qualitative Choice Behavior.'' In Paul Zarembka (editor), Frontiers in Econometrics , New York: Academic Press, 105-42. showed that, if the joint distribution for random utilities {U₁, U₂, ¼, U_m} is a so-called multivariate extreme-value distribution of type I, then integral (2.22) has a closed-form solution. A multivariate extreme-value distribution takes the following form.

where a_i (i = 1, 2, ¼, m) are parameters. This distribution is convenient because the maximum value among a sample of random utilities {u₁, u₂, ¼, u_m} from this distribution is also distributed as an extreme-value distribution of the following form.

where u_max is a realization of a new random variable,

Using this property, the distribution function for random variable

is given by

Then the probability that brand i is chosen at a purchase occasion is given by

The reader will recall that, if we let the attraction of brand i ,   A_i , be equal to exp(a_i) , this expression is similar to an MNL model. Indeed the foregoing argument has been used to derive MNL models for individual choice behavior. However, one may derive an expression similar to more straightforward attraction models (2.6), if, instead of an extreme-value distribution of type I, one chooses an extreme-value distribution of type II, namely,

where  A_i (i = 1, 2, ¼,m) are parameters. To show this, first note that the distribution function for random variable

is given by

Using this,

This demonstrates the fact that MCI models as well as MNL models are derivable if extreme-value distributions are assumed for the joint distribution of random utilities.

Although both equations (2.24) and (2.25) are derived for individual choice probabilities, one may derive an attraction model for aggregate market shares, if the definition of distribution functions is slightly changed. Suppose that random utilities for alternative brands, U₁ , U₂ , ¼ , U_m , are jointly distributed over the population of individual buyers, rather than within an individual. Each individual has a set of realized values for utilities, u₁, u₂, ¼,u_m , and will select that brand which has the maximum utility value among m brands. Cast in this manner, the problem is to find the proportion of buyers who will purchase brand i , but equations (2.24) and (2.25) give precisely this proportion (that is, market share) for two extreme-value functions. Thus McFadden's argument may be used to give another justification for using an attraction model.

Although random utility models in general do not have the IIA property, it should be noted that some random utility do have it. YellottYellott, John I. [1977], ``The Relationship between Luce's Choice Axiom, Thurstone's Theory of Comparative Judgment, and the Double Exponential Distribution,'' Journal of Mathematical Psychology , 15, 109-44.proved that a random utility model is equivalent to a constant utility model (and hence possesses the IIA property) if and only if the joint distribution of random utilities follows a multivariate extreme-value distribution. The basic forms of MNL and MCI models happen to belong to this special case. But we wish to emphasize that it is possible to construct attraction-type models of probabilistic choice which do not have the IIA property. We will discuss two such models - the fully extended and distinctiveness models - in Chapter 3.

2.9.4  *Multivariate Independent Gamma Function

It was stated in section 2.8 that the condition that cov(m, p_i) = 0 occurs for a very restricted case where m is distributed as a Gamma function and the p_i 's are jointly distributed as a Dirichlet function. In this appendix we will give a proof for this statement. Let m_i be the expected number of purchases of brand i by an individual. By definition m_i = mp_i (or p_i = m_i / m). We seek a joint distribution of m_i (i = 1, 2, ¼, m), where m is the number of alternative brands, for which m and each m_i / m( = p_i) are mutually independent. But, since m and its inverse are usually negatively correlated, one would expects that m and m_i / m will be correlated for most random variables. This suggests that we are indeed looking for a rather special distribution for the m_i 's.

LukacsLukacs, E. [1965], ``A Characterization of the Gamma Distribution,''Annals of Mathematical Statistics , Vol. 26, 319-24.showed that, for mutually independent nonnegative random variables X₁ and X₂ , if (X₁ + X₂) and X₁/(X₁ + X₂) are mutually independent, both X₁ and X₂ must be Gamma variates with a common shape parameter (but not necessarily a common location parameter). We will use this result to show the following propositions. Proposition 1 If the m_i 's ( i = 1, 2,¼, m) are mutually independent Gamma variates with density function

then m and the p_i 's are independently distributed.

Proposition 2 If the m_i 's are mutually independent nonnegative random variables and m and m_i / m( = p_i) are independent, then the m_i 's are jointly distributed as multivariate (independent) Gamma function.

Proof 1 By the change-of-variable formula, the joint density of m and p₁, p₂, ¼, p_m-1 is given by

where J is the Jacobian determinant associated with the transformation of the m_i 's to m and the p_i 's. Hence

where

Thus g(m, p₁, p₂, ¼, p_m-1) can be factored into two parts: the first is a Gamma density function for m with parameters a_. and b , and the second is a Dirichlet density function for p₁, p₂, ¼, p_m-1 (and p_m = 1 - p₁ - p₂ - ¼-p_m-1) with parameters a_i (i = 1, 2, ¼, m). Therefore m and p₁, p₂, ¼, p_m are independent.

Proof 2 Let, for an arbitrary index j^* (j^* Î C) , M_j* be the sum of the m_i 's except m_j , i.e.,

By the assumption, m_j and M_j* are independently distributed nonnegative random variables and also m_j and m_j / m = m_j/(m_j + M_j*) are independent. By Lukacs' result m_j and M_j* are both Gamma variates with a common shape parameter. Since index j was chosen arbitrary, this result applies for all j , that is, all m_j 's are Gamma variates sharing a common shape parameter. Q.E.D.