Equations need to be viewed in Intenet Explorer. Symbol fonts are not available in Firefox.

Chapter 3
Describing Markets and Competition

3.1  Market and Competitive Structure

In the preceding chapter we viewed competition among brands in an industry in the simplest possible way, that is, with the assumption that every brand is directly competing against all other brands in the industry. But competition in the actual market place may take more complex patterns. It is not unusual to find some grouping of brands such that within a group competition among its members are intense, but competition is not intense or even nonexistent between the groups.

Consider the toothpaste industry. Brands belonging to it may be subdivided into at least three groups: family use, breath care, and tar removers (for smokers). Each group emphasizes a different product attribute: the first group emphasizing decay-preventive ingredients, the second, breath fresheners, and the third, tar control, etc. The three groups serve different segments of buyers, and therefore do not directly compete with each other, although brands in a group tend to be highly competitive in the sense that the action of one brand affects the market shares of others.

Then why can't we treat the three brand groups of the toothpaste industry as three separate industries? It is because, at the retail level at least, the shares of three groups are often observed to be interdependent. A price deal for a brand in the family-use group, say, may affect the demand of brands in other brand groups. The reason is not difficult to find. Though three groups of toothpaste are used by different user groups, they are often bought by the same person in a family. A mild degree of interdependencies among three groups makes it necessary to treat them as a single industry.

Our purpose in this chapter is to create a framework that may be used to describe the market and competitive structures existing in an industry. By a market and competitive structure we mean the structure of interdependencies among competitors in an industry, as expressed by grouping patterns of brands. We could simply call such a pattern a competitive structure, but a competitive structure may be a reflection of the underlying pattern of buyer demand in the market. If it were found that brand A directly competed with brand B but not with brand C, it might be because the buyers perceive brand A and B as alternatives, but not brand A and C. Since we mostly utilize aggregate market-share data in determining grouping patterns of brands, we will not know if such a conjecture is true or false. The choice of the cumbersome expression market and competitive structures reflects our lack of information about the underlying perception of buyers.

In this chapter we will examine various approaches for looking at and modeling market and competitive structures among brands. Of particular interest are the phenomena: differential effectiveness of marketing actions, asymmetry of competitive interactions, and variations between market segments. We will also deal with two topics of considerable importance - the distinctiveness of brands and time-series issues - related to the description of market and competitive structures.

3.2  Asymmetries in Market and Competition

An even cursory observation of competitive interactions in the market place reveals that some firms (brands) are capable of exerting inordinately strong influence over the shaping of demand and competition, while other firms (brands) are not. One often sees some price leaders who can reduce their prices and obtain a large gain in market share, seemingly oblivious to their competitors' reactions. Other firms can create strong buyer loyalties, albeit for only a short term, with splashy promotional campaigns. It appears that such notable marketing clout is not possessed by all firms in an industry and, interestingly, are not the sole property of larger firms. There are fairly large firms in many industries which have difficulties in increasing their shares, even if they reduce prices below those of their price-aggressive competitors. Furthermore, the impact of one competitor's action may affect one rival or one group of rivals more than another. For example, a price cut by Hewlett-Packard in the personal computer market may disproportionately draw more market share from Apple than from another major brand, say, IBM.

The imparity in competitive interdependencies are perhaps more pronounced in the retail trade. Retailers with experience know that which brands they should use for loss-leaders (i.e., brands whose price is cut to increase store traffic) and which brands to feature in newspapers for the maximum effect. And they do not cut prices or feature arbitrarily, either. Retailers seem to share pretty much the same opinion as to which brands are candidates for loss-leaders or newspaper features.

Those observations illustrate both differential effectiveness of brands and asymmetries in market and competitive structures. Differential effectiveness among brands reflects that firms (brands) have different degrees of effectiveness in carrying out their marketing activities. That such differences exist in real markets is obvious, but differential effectiveness alone does not create or reflect asymmetries. Asymmetries are reflected in differential crosseffects among brands. Firms are, it appears, differentially effective not only with respect to their own shares and sales, but also with respect to their ability (that is, clout) to influence the shares and sales of other brands. Furthermore, firms seem to differ in the degree to which they are influenced by other brands' actions (that is, vulnerability). We will deal with those two aspects of competition in the following sections.

3.3  Differential Effectiveness

We have already discussed this issue of differential effectiveness in Chapter 2, and a solution at that time was to include some parameters in market-share models to take account of the overall marketing effectiveness of each brand. The reader will recall, in the now-familiar specification of attraction models,

where parameters a_i (i = 1, 2, ¼, m) represent the marketing effectiveness of respective brands.

The inclusion of the a 's in attraction models, however, does not fully account for differential effectiveness among brands. The differential effectiveness may be specific to each marketing instrument, such as a brand which has a particularly effective pricing policy or an effective advertising campaign. The a_i 's do not appear in the elasticity formulas for a particular marketing instrument, X_k (namely, e_si = b_k (1 -s_i) for MCI models and e_si = b_k X_ki (1 - s_i) for MNL models). The marketing-effectiveness parameters may reflect differences in the brand franchise or brand loyalty . Literally, they are the constant component of each brand's attraction, but they have nothing to do with elasticities. As a result, elasticity formulas for simple attraction models do not reflect differential effectiveness. If we are to insist that share elasticities must also reflect differential effectiveness, those elasticity formulas will have to be modified.

If it is decided to modify market-share elasticities to account for differential effectiveness, the reader will find that this may be achieved in only one way, that is, by specifying parameters b_k 's in such a manner that each brand has a special parameter, b_ki, for variable X_k . The attraction models will have to be respecified as follows.

This is the differential-effects version of attraction models. This modification does not change the basic structure of direct and cross elasticities for attraction models. For example,

MCI Model:

MNL Model:

As variable X_ki increases, the elasticity decreases for MCI models, but it increases and then decreases for MNL models. By expanding the parameterization of the model we are now able to capture brand-by-brand differences in market responsiveness to each element of the marketing mix. If all brands are equally effective then b_ki = b_kj = b_k    "  i,j , and the elasticity expressions reduce to those for simple attraction models.

3.4  Differential Cross Elasticities

In the preceding chapters we presented elasticities as a key concept in market-share analysis, but what do they tell us of the effects that a firm may exert with its actions on the sales and shares of other firms in the same industry? The share, sales, and industry-volume elasticities described in Chapter 2 - known as direct elasticities - are not sufficient, if the analyst is interested in knowing what effects other brands' actions will have on his/her brand's share, or what effects his/her actions will have on other brands' shares. For the purpose of analyzing differential crosseffects among brands we will need a new concept - cross elasticities . Let us give a more precise definition to this new concept.

Suppose that brand j changed variable X_kj by a small amount DX_kj . The cross elasticity of brand i 's (i ¹ j) share with respect to variable X_kj may be verbally expressed as ``the ratio of the proportion of change in market share for brand i corresponding to the proportion of change in variable X_k for brand j ,'' and is defined as follows.

Note that e_si.j has two subscripts: the first indicates the brand which is influenced and the second, the brand which influences. This is an arc cross-elasticity formula and the point cross elasticity is defined as:As in the case of direct elasticities, the above formula is for variable X_kj , but for the sake of simplicity no superscript or subscript k will be attached to e_si.j . It will be clear from the context which variable is being referenced.

We now turn to the forms of point elasticities for specific market-share models. Point cross elasticities for differential-effects attraction models take the following forms.

MCI Model:

MNL Model:

It may be added that, for simple-effects attraction models, b_kj in the above formulas are replaced by a common parameter b_k .

Let us consider what the above formulas imply. For the raw-score versions of both MCI and MNL models cross elasticities with respect to variable X_kj are constant for any brand i ( i ¹ j). This means that the relative changes of other brands' shares (i.e.,¶s_i / s_i) caused by brand j 's actions are the same for any brand, though actual changes in shares (i.e.,   ¶s_i) are different from one brand to another, depending on the current share level for each brand (i.e.,   s_i). A numerical example may help to illustrate this point. Suppose that there are four brands and their respective shares are 0.3, 0.1, 0.2 and 0.4. Assuming that an MCI model is applicable and b_k1 is 0.5, the value of cross elasticity for brand i (i ¹ 1) with respect to the change in X_k1 is given by

If no other variables have been changed, a 10% increase in variable X_k1 will bring about a 1.5% reduction in any other brand's share, i.e.,

If brand 2 has a 20% share of market, the actual loss of its share will be by 0.3 percentage points, i.e.,

These calculations are summarized in Table 3.1.

Table 3.1: Numerical Example of Cross Elasticities for MCI Model

Though share cross elasticities are equal for brands 2, 3, and 4, the actual change in a brand's share varies, reflecting its current level. (The sum of actual changes in shares is zero for all brands in the industry, as it should be.) Note that the competitive positions of brands 2, 3, and 4 relative to each other have not changed by the reduction in their shares. In fact, the new share of any brand other than 1 may be simply calculated by

New Share of Brand i = (1 - New Share of Brand 1) × Old Share of Brand i .

We may add that, for an MNL model, relative and actual changes in s_i are a function of the current value of X_kj , but the value of e_si.j for this model is identical for any other brand i .

From those calculations one can see that simple attraction models specify a rather peculiar pattern of competition in that the relative effects of a brand's actions on another brand's share are identical for any brand in the industry. This equality of cross elasticities implied by such simple attraction models does not fit what we observe in the marketplace. There are brands which seem to be nearly immune from other brands' price changes; some firms seem to be able to ignore promotional activities of other brands with little loss of their shares, while others seem to be greatly affected by such activities, and so forth. Examples of this kind may be found in many industries. It is therefore desirable to introduce in market-share models the inequality of cross elasticities, if we are to analyze differential cross effects among brands. There are two ways to attack this problem. On one hand, we could reflect the asymmetries which might arise from the temporal distinctiveness of marketing efforts. This is pursued in section 3.8. On the other hand, we could extend the parameters of the attraction model to reflect asymmetries due to systematic and stable cross-competitive effects . Fortunately, this can be accomplished with relative ease within the framework of attraction models as shown below.

where b_kij is the parameter for the cross-competitive effect of variable X_kj on brand i .

Equation (3.4) is called an attraction model with differential cross-competitive effects or a fully extended attraction model to distinguish it from a differential-effects attraction model (3.1). The most important feature of the fully extended model is that the attraction for brand i is now a function not only of the firm's own actions (variables X_ki 's, k = 1, 2, ¼, m) but also of all other brands' actions (variables X_kj 's, k = 1, 2, ¼, K ; j = 1, 2, ¼, m). The b_kij 's for which i is different from j are the cross-competitive effects parameters, which partly determine cross elasticities. The b_kij 's for which j equals i (i.e., b_kii) are direct-effects parameters and are equivalent to the b_ki 's in the differential-effects model (3.1). This notation is cumbersome, but it is necessary to keep track of who is influencing whom. Note that the fully extended model has many more parameters (with m×2×K b_kij 's and m a_i 's) than the original attraction model (with K + m parameters) and the differential-effects model (with mK + m parameters). We will take up the issues related to estimating b_kij 's in Chapter 5.

3.5  Properties of Fully Extended Models

In order to see what market and competitive structures implied by the fully extended model (3.4) , let us look at the direct and cross elasticities for this model.

MCI Model:

MNL Model:

These formulas are common for both direct and cross elasticities; if i is equal to j , the above formulas give direct elasticities for brand i , otherwise they give cross elasticities.The elasticity formulas may be more succinctly written in matrix notation.

MCI Model:

MNL Model:

where:

E = m ×m matrix with elements { e_si.j }

I = m ×m identity matrix

J = m ×m matrix of all 1's

D_s = m ×m diagonal matrix of market shares {s₁, s₂,¼, s_m }

B = m ×m matrix with elements { b_kij }

D_X = m ×m diagonal matrix of variables {X_k1, X_k2,¼, X_km } .

e_si.j is the elasticity of market share for brand i with respect to changes in marketing variable X_kj for brand j , and it is given by b_kij minus the weighted average of b_khj's over h , where the weights are the market shares of respective brands (s_h). Figure 3.1 illustrates the competitive pattern implied by the elasticity formulas.

Figure 3.1: Cross Elasticities in the Fully Extended Model

Let's assume that variable X_kj in this case is the price for brand j . Then parameter b_kij for which i is not equal to j tends to take a positive value. In other words, when brand j reduces its price the share of brand i tends to decrease. This effect of brand j 's price change on brand i 's share is depicted as the direct effect in Figure 3.1. Note that the direct effect is modified by the size of brand i 's share. When brand i 's share is nearly one, brand i is little affected directly by the moves by brand j .This statement is true for the relative changes (¶s_i / s_i) in brand i 's share. In terms of absolute sales volume, the impact of brand j 's price change may be substantial.The influence of brand j 's price change is not limited to the direct effect to brand i , however. When brand j reduces its price, its own share should increase. Furthermore, the market shares of brand 1 through m (other than brand i and j ) will also receive a negative effect, which in turn should have a positive effect on brand i 's share. Indirect effects in Figure 3.1 depict influences of those kinds.

In order to examine formally the points raised above, rewrite the cross-elasticity formula for MCI models as follows.

The first term, of course, represents the direct effects. The second term shows the indirect effects through brand j . The last term consists of indirect effects through all other brands. If X_kj is brand j 's price, one expects that b_kjj < 0 and b_kij > 0 (for i ¹ j). Since the first and last terms are expected to be positive and the second term negative, the sign of e_si.j is indeterminate, and dependent on the relative size of (1 - s_i)b_kij - s_j b_kjj and å_{h ¹ i,j}^m s_h b_khj .

Consider the following special cases.

Case 1: All cross-elasticity parameters (b_kij,  i ¹ j) are zero. In this case, e_si.j = - s_j b_kjj , This is the same as the cross-elasticity formula for the differential-effects MCI models.

Case 2: All cross-elasticity parameters (b_kij,  i ¹ j) are approximately equal. In this case,

Then

This suggests that e_si.j has the same sign as b_kij .

Case 3: b_kij is nearly zero, but

In this case e_si.j may have a sign different from b_kij .

Case 3 is an interesting situation because, in this case, it is possible that brand i even gain a share when brand j reduces its price. For case 3 to occur brand j 's share should be relatively small, but the impact of its actions on brands other than i must be large. (This brings to our mind an image of an aggressive small brand j which is frequently engaged in guerilla price-wars.) In addition, brand i must be reasonably isolated from the rest of the market, implying that it is a niche -er. This case illustrates the richness of the description of market and competitive structures offered by the fully extended attraction models.

It may be added that if i = j , we may write

The first term represents the direct effect of X_ki on brand i 's share; the second term gives the sum of all indirect effects on brand i 's share through influences on all other brands. This formula suggests a possibility that, even if the direct effect is negligible (e.g., b_kii is small), direct elasticity, e_si.i , may be sizeable due to the combination of indirect effects. In other words, a brand may be able to increase its share merely by reducing other brands' shares. The reader should note that simple-effects or differential-effects attraction models do not allow such a possibility. This is another indication of descriptive richness of the fully extended attraction models.

To summarize, the fully extended (i.e., differential cross elasticity) attraction models offer an enormous advantage over many market-share models in that it is capable of describing the complexity of market and competitive structures with relative ease. Since the simple-effects and differential-effects models may be considered as the special cases of the fully extended models,In technical jargon, we say that the simple-effects and differential-effects models are nested within the fully extended models.we will adopt the latter models as the basic models of market shares in this book.

3.6  Determining Competitive Structures

Once cross-elasticity parameters are introduced in market-share models, it becomes possible to specify market and competitive structures on the basis of cross elasticities among brands. An example will serve to illustrate this concept. Suppose that the marketing variable in question is price. One may estimate share elasticities with respect to price using a differential cross-elasticities market-share model. Table 3.2 shows the matrix of direct and cross elasticities among seven brands in a hypothetical industry.

Table 3.2: Direct and Cross Elasticities for Seven Brands

At the first glance, the existence of a market and competitive structure may not be apparent from Table 3.2(a). However, if we rearrange the table both row- and column-wise, we obtain Table 3.2(b), in which the existence of submarkets or brand groups is more apparent. Because of mutually large cross elasticities, brands 1, 2, 6 and 7 form a group. Brands 4 and 5 form another. Brand 3 is more or less isolated. In this example the groups are rather distinct in that the cross elasticities between the brands in the first group and those in the second and third groups are small. Though the brands in each group are highly interdependent, price competition between the first group and the second and third groups is expected to be moderate or virtually nonexistent.

It would be too hasty for one to say that this industry consists of three brand groups on the basis of Table 3.2 alone, since we have no knowledge of what market structure(s) may be suggested with respect to other marketing variables. It may turn out that another structure is suggested by share elasticities with respect to product quality or promotional outlays. It is necessary to look at the whole complex pattern of interdependencies between brands before one is able to say how submarkets are formed. But the principle which we will follow in determining market structures in the remaining part of this book will be the same as the illustrative example: we will look at the tables of direct and cross elasticities for relevant marketing variables and reorganize them in such a manner that brands which have mutually large cross elasticities are collected in a group. In this task multivariate techniques based on factor analysis will be employed. We will turn to the procedures actually used in determining market structures in Chapter 6.

A word of caution is in order. Analyzing tables of share elasticities may not yield a clear-cut pattern of brand grouping in some situations. Brand groups may be partially interlocked. Or groups may be nested (or contained) within larger groups. Table 3.3 gives an example of interlocking brand groups.

Table 3.3: Interlocking and Nested Brand Groups

For the sake of clarity the entries which have insignificant cross elasticities are not shown. In this example, group 2 (brands 3, 4, and 5) is interlocked with group 1 (brands 1 through 4) and groups 3 (brands 5, 6, and 7). In this situation one may interpret that elasticities in the table are produced by three distinct buyer segments, each of which perceives a different set of brands as relevant alternatives. But it is also plausible to think that there are only two segments in the market, and that price changes by brand 5 for some reasons affect only brands 3 and 4 in the first group. Those two interpretations are not separable from the table.

3.7  Hierarchies of Market Segments

Ambiguities in interpreting the nature of competition from the tables of elasticities are often caused by the aggregation, that is, by not explicitly recognizing segments of buyers in the market. As was already pointed out, the overt pattern of brand grouping does not necessarily give hints about the underlying patterns of buyer demand.

There are two possible interpretations on the nature of brand groups in Table 3.2, for example. One interpretation is that there exist three distinct market segments and the cross elasticities reflect the difference in product perception among segments. The buyers who belong to the first segment may consider brands 1, 2, 6, and 7 as relevant alternatives either because of their product attributes or their collective availability at the retail level; those who belong to the second segment may consider only brands 4 and 5 as relevant alternatives; and so forth. In this interpretation brand groups correspond one to one with market segments of buyers.

The second interpretation of brand groups is that brands tend to be grouped in accordance with different types of buyer needs they serve. Suppose that the consumer uses regular and instant coffee for different occasions (e.g., regular coffee with meals, but instant coffee for other occasions). This will cause the coffee market to be divided into the regular and instant brand groups, and minor price differences between the two groups will not affect demands of either. This type of segmentation on the basis of needs, or a benefit segmentation, does not produce distinct buyer segments in the market. Of course if two brand groups serve two entirely isolated buyer needs, they should be treated as two distinct industries rather than one. But if the price of regular coffee is drastically reduced, the demand for instant coffee may be affected. A regular brand with an aggressive price policy may have some cross elasticities with instant brands, or vice versa. Moderate cross elasticities between groups would force one to treat them as a single industry.

As in the above example, if the elasticities are measured only for the entire market, it will be impossible to establish the propriety of the above two interpretations solely on the basis of tables such as Table 3.2. In order to evaluate the correctness of these two interpretations, one will need data sets such as consumer panels (either diary or scanner panels). Moreover, it is desirable to have accompanying data on the buyer perception of alternative brands. Lacking such detailed data sets, however, one should at least understand well the aggregate implications of variabilities in elasticities among buyer segments. We will first look at the nature of elasticities in a multisegmented market.

Suppose that there are two segments in the market, containing N₁ and N₂ buyers, respectively. We will use the notation q_i(l) and s_i(l) to indicate, respectively, the sales volume and market share of brand i in the l^th segment. Since

where:

q_(l) = sales volume in segment l (l = 1, 2)

q = total sales volume ( q₍₁₎ + q₍₂₎ ).

The point share elasticity of brand i with respect to X_kj is given by

This shows that an overall elasticity is the weighted average of corresponding segment elasticities, weights being the relative sales volumes for respective segments. If we write the segment elasticity as e_si.j(l) , then the general expression for e_si.j is given by

where L is the number of segments and q_i is the sales volume for brand i for the entire market. This expression gives one the means to compute the overall elasticity matrix from matrices for segments.

3.8  Distinctiveness of Marketing Activities

Fully extended attraction models have advanced our ability to deal with the complexity of market and competitive structures, but there are other aspects of competition which have not been properly dealt with even in the fully extended models. We will turn to the some of the more critical issues in this section and the next one. Here we will take up the issue of distinctiveness of marketing activities by competing brands.

The main thesis of this section is that a brand's marketing actions may or may not influence the behavior of buyers depending on the degree to which its actions are distinguishable from the actions of its competitors. This issue is very much related to the distinction between importance and salience of product attributes in buyers' choice. For example, any consumer will say that being nutritious is one of the important attributes in his/her choice of bread. But, if all brands of bread available in the market have the same nutritional value (or at least are perceived so by consumers), being nutritious will not affect the consumer's choice of brands of bread. Instead consumers may decide on the basis of the color of package, position of store shelf, or likes and dislikes of the persons who appeared in television commercials. In a fiercely competitive industry the pressure of competition usually works to equalize the products offered by firms with respect to those attributes which buyers perceive important. Thus, as an economist put it, consumers tend to make their choice on the basis of the least important (yet salient) attributes of the product.

This phenomenon is not limited to product attributes. The authors posit that the effectiveness of any marketing activity would be dependent on the degree that it is distinct from those of competitors. Even casual observations bear out this proposition. Price reduction by a firm would have more effects on market shares when other brands' prices are kept high than it would when all competitors also reduce their prices. The market-share impact of one firm's promotion would be significantly greater when the firm is alone in promotion than it would when all firms engage in promotional activities. Advertising activities by firms competing in an oligopolistic industry tend to cancel out each other's effect, so much so that they have little influence on market shares. (When the Surgeon General of the United States prohibited cigarette commercials on television, it was rumored that the parties who were most pleased by the decree were the competing cigarette manufacturers.)

If we take the position that it is the differences between brands, rather than the absolute levels of marketing activities that materially affect buyers' preference, then we will have to devise a scheme to bring the distinctiveness of marketing activities among brands into market-share analysis. Luckily, attraction models handle the distinctiveness issue quite naturally. Consider the general form of attraction models.

It is obvious that the value of market share for brand i , s_i , will not change if we divide the numerator and denominator of the second equation above by a constant. Specifically, if we divide each  A_i by the geometric mean of  A_i over i , [ A\tilde] , namely

the operation will not affect the value of s_i , since

Let us look at the specific forms of [ A\tilde] for MCI and MNL models.

MCI Model:

MNL Model:

where:

[`(a)] = the arithmetic mean of a_i over i ( i = 1, 2, ¼, m)

[`(e)] = the arithmetic mean of e_i over i

[X\tilde]_k = the geometric mean of X_ki over i

[`X]_k = the arithmetic mean of X_ki over i .

Using the above results we may write MCI and MNL models as follows.

MCI Model:

MNL Model:

and for both modelsIf one divides s_i by the geometric mean of s_i over i , the result would be equal to  A_i^* . In other words, if we let [s\tilde] be the geometric mean of s_i over i , that is,

then

This fact will be extensively utilized in the estimation procedure in Chapter 5.

Consider the implication of the foregoing analysis. One may express the variables in MCI and MNL models in a deviation form without changing the properties of the models. In other words, we may express a variable either as

or

and substitute X_ki^* for X_ki in MCI or MNL models, respectively. This property of attraction models does not change if we move from the simple-effects form to differential-effects models and fully extended models.If monotone transformations (f_k) other than identity or exponential are used, substituting

where [f\tilde]_k(X_k) is the geometric mean of f_k(X_ki) over i , for f_k(X_ki) in an attraction model, will not change the nature of the model.This shows that the variables in attraction models may be replaced by some equivalent form of deviations from the industry mean and that those models in essence operate on the principle of distinctiveness. Take an MCI model, for example. If X_k is price, each brand's price may be expressed as deviations from the average price for the industry. If all brands charge the same price, X_ki^* will be equal to one, and price will not affect the shares of brands. Only when the prices for some brands deviate from the industry mean do they influence market shares of themselves and others.

The handling of distinctiveness by attraction models becomes a technically difficult issue when the variable in question is a qualitative one. Product attributes are the example of variables of this type. A make of refrigerator may or may not have an ice-maker; an automobile model may or may not have an automatic transmission; a brand of toothpaste may or may not have tar-control ingredients, etc. Such a variable may take only two values, namely, one if the product (or brand) has an attribute and zero if it does not. Of course, one may compute the industry average for a binary (two-valued) variable (which is the same as the proportion of products or brands which have that attribute) and subtract it from the value for each product/brand. But by this operation the transformed variable may take either positive or negative values, and hence it may be used only with an MNL model (or some monotone transformation f which allows negative values). In order to incorporate binary variables in an MCI model a simple but effective transformation - the index of distinctiveness - was developed.Nakanishi, Masao, Lee G. Cooper & Harold H. Kassarjian [1974], ``Voting for a Political Candidate Under Conditions of Minimal Information,'' Journal of Consumer Research , 1 (September), 36-43.

Suppose that X_k is a variable associated with the possession or non-possession of an attribute. Let the proportion of products (or brands) in this industry which have the attribute be r . If there are 10 brands and two brands have the attribute, r will be 0.2. The value of the index of distinctiveness for each brand is determined by the following simple operation.

If brand i has the attribute, X_ki = 1/r .

If brand i does not have the attribute, X_ki = 1 - r .

Thus if r equals 0.2, those brands with the attribute are given the value of 5 and those without the attribute will be given the value of 0.8. Note that the smaller r , the greater the value of X_k for those brands that have the attribute. This represents in essence the effect of the distinctiveness of a brand. If a brand is only one which has the attribute the index value (1/r) becomes maximal.

It is interesting to note that this index has a rather convenient property that it is ratio-wise symmetrical to the reversal of coding a particular attribute. If we reversed the coding of possession and nonpossession of an attribute in the previous numerical example, r would be 0.8, and the value of X_k for those brands with the attribute would be 1.25 (= 1/0.8) and that for the brands without the attribute would be 0.2 (= 1/5). In other words, those brands without the attribute become distinctive in the reverse direction.

The index of distinctiveness shown above transforms a binary variable such that it is usable in an MCI model. Cooper and NakanishiCooper, Lee G. & Masao Nakanishi [1983], ``Standardizing Variables in Multiplicative Choice Models,'' Journal of Consumer Research , 10 (June), 96-108. ound that this index is a special case of a more general transformation applicable not only for qualitative variables but also for any quantitative variable. First, convert any variable X_ki to a standardized score by the usual formula.

where:

[`X]_k = the arithmetic mean of X_ki over i

s_k = the standard deviation of X_ki over i .

Since standardized z-scores (z_ki 's) may take both positive and negative values, they may be used in an MNL model in the form of exp(z_ki) , but cannot be used in an MCI model. To create a variable usable in the latter model transform z-scores in turn in the following manner.

This new transform, z_k , (to be called the zeta-score for X_k ) takes only positive values and has a property that it is ratio-wise symmetrical when the positive and the negative directions of variable X_k are reversed. For example, let the value of z_ki be 2.5. If X_ki is multiplied by -1 , z_ki will take a value of 0.4 (= 1/2.5). It may be easily shown that the zeta-score includes the index of distinctiveness as a special case for binary variables.For a binary variable X_k , [`X]_k = r and s_k = r(1 - r) . Hence

Substitution of the z_ki² 's in the zeta-score formula yields squared roots of distinctiveness indices.

The zeta-score is based on the ratio of the noncentral moment of inertia about brand i to the central moment of inertia on measure X_k (namely, the variance of X_k ) - thus reflecting how an object stands out from a group relative to the variability of the group. This ratio is not affected by a general linear transformation of X_k , making it an appropriate transformation of interval-scale ratings - thus allowing interval-scale rating to be used in MCI as well as MNL models. The ratio has a minimum value of one for brands at the center (i.e., the mean of X_k ), and increases as a particular brand gets farther away from the center. To translate this ratio into a usable index we invert it at the mean of the underlying variable. This allows us to tell if a brand is distinctively high or distinctively low in an attribute compared to the other brands in the competitive offering. Figure 3.2 gives the comparison of the zeta-score with the exp(z_ki) transform.

Figure 3.2: Comparison of Zeta-Score and Exp(z-Score) Transforms

Although the shape of two transforms are quite similar, the choice between the two may be made by the form of the elasticities. The direct and cross elasticities for the exp(z_ki) transforms are given by

and those for the zeta-transforms are given by

where:

S = the m ×m matrix with elements {¶z_kj/ ¶X_ki} , i.e.,

S = 1 sk [ I - 1 m J - 1 m ZZ¢]

D_s = an m ×m diagonal matrix with the i^th diagonal element s_i

D_X = an m ×m diagonal matrix with the i^th diagonal element X_ki

s_k = the standard deviation of X_k over i

J = an m ×m matrix of 1's

Z = an m ×1 vector of standardized scores (i.e., z_ki = (X_ki - [`X]_k)/ s_k)

D_z = an m ×m diagonal matrix with the i^th diagonal element

| zki |/(1 + zki2)     .

Figure 3.3 compares the elasticities of the zeta-score with the exp(z_ki) transform.

Figure 3.3: Comparison of Zeta-Score and Exp(z-Score) Elasticities

The dip in the middle of the elasticity plot for zeta-scores corresponds to the flat portion of the zeta-score function depicted in Figure 3.2. With zeta-scores, change is always depicted as slower in the undifferentiated middle portion of the distribution. Consider what this might imply for a frequently purchased branded good (FPBG ). If it establishes an initial sale price about one-half a standard deviation below the average price in the category, the price is distinctively low and market-share change is relatively rapid. If the price drops further from this point, market share increases, but at a slower and slower rate. Bargain-hunting brand switchers have already been attracted to the brand, and little more is to be gained from further price cuts. If the price increases from this initial sale price, market share drops rapidly at first, as the value of being distinctively low priced is dissipated. At the undifferentiated position at the middle of the price distribution, market share is changing least rapidly as minor changes on either side of the average price go largely unnoticed. This indistinct region is similar to what DeSarbo, et al.DeSarbo, Wayne S., Vithala Rao, Joel H. Steckel, Yoram Wind & Richard Columbo [1987], ``A Friction Model for Describing and Forecasting Price Changes,'' Marketing Science , 6, 4 (Fall), 299-319.represent in their friction-pricing model and similar to what Gurumurthy and LittleGurumurthy, K. & John D. C. Little [1986], ``A Pricing Model Based on Perception Theories and Its Testing on Scanner Panel Data,'' Massachusetts Institute of Technology Working Paper Draft, May.discuss in their pricing model based on Helson's adaptation-level theory. On the high-priced side and analogous series of events happen. Small price increases around the average price are not noticed, but once the brand price is high enough to be distinguished from the mass, the loss of market share becomes more rapid. At some point, however, the change in market share must decline, as the brand looses all but its most loyal following.

In many categories of FPBG 's the brands pulse between a relatively high shelf price and a relatively low sale price. In such cases the middle of the elasticity curve is vacant and the values of the elasticities for zeta-scores and exp(z-scores) might be quite similar. The exp(z-score) elasticities might be most descriptive of the path of market-share change from aggregate advertising expenditures, with increasing market-share growth as the expenditures move from zero up to the industry average, and diminishing growth rate for additional expenditures. If the analyst has reasons for preferring one form of elasticities to the other, he/she should choose the one which fits his/her needs best.

An advantage zeta-scores or exp(z-scores) have relative to raw scores is due to their role in separating the underlying importance of a feature from the particular pattern of shared features in any given choice context. If two brands are both on feature in a store they do not each get the same boost in market share as if they were featured alone. By specifically modeling such contextual effects we overcome the limitations imposed by the IIA (context-free) assumption of Luce choice models discussed in Chapter 2. The IIA assumption does not recognize that the value of a major promotion is somehow shared by all the brands on sale in that time period. The parameters of a raw-score (Luce-type) model will always reflect the underlying value of the feature commingled with the particular pattern of shared features in the contexts used for calibration. By explicitly modeling the pattern of feature sharing with a distinctiveness index, the parameters are free to reflect the underlying value of a feature. In forecasting, one again uses either zeta-scores or exp(z-scores) to help translate the underlying value of a feature to the particular pattern of shared features in new time periods. Whatever the value of a feature, we know that the per-brand worth is diluted as the number of brands on feature increases. When all brands are on sale, there will be no differential market-share benefit - just an increase in the cost of doing business. There are many situations in which marketing actions must be distinct to be effective.

Transformations such as exp(z-scores) and zeta-scores not only highlight the differences among brands but serve to standardize variables. This reduces the multicollinearity inherent in the differential-effects forms of market-response models as will be shown in Chapter 5.

3.9  Time-Series Issues

Up to this point we have discussed competition only in the static sense, in that brands are competing within a single period of time. Gains and losses of market shares for competing brands are viewed as the joint market responses to marketing activities among competitors in the same time period. But in the real world we shall have to be more concerned with the dynamic aspects of competition. A firm's marketing activities affect the performance of its product, as well as that of its competitors, not only in a single period but also over many periods. A new or improved product may not achieve immediate market acceptance; promotional activities may have delayed (or lagged) effects; efforts to secure channel cooperation may only bear fruit a long time afterwards, etc. Furthermore, there are seasonal and cyclical fluctuations in market demand which may have important competitive implications and affect the performance of all brands in an industry. If market-share analysis is to be meaningful, it is necessary to introduce elements in the analysis to account for those dynamic competitive phenomena.

The developments in data-gathering techniques (to be described in Chapter 4) in the recent years have given the analyst a new impetus to perform time-series analysis of market-share data. Store audits have been the traditional source of bi-monthly market-share data, but it would have taken too many periods for the analyst to obtain from this source a sufficient number of observations for standard time-series analysis, such as the Box-Jenkins procedure. But the introduction of optical scanners at the retail level has completely changed the picture. With proper care, the analyst will be able to obtain weekly, or even daily, market-share figures at some selected stores. This drastic improvement in data collection has opened a new avenue of time-series analysis for market-share data.

Major considerations about time-series analysis may be summarized in the following manner.

1. Seasonal and other regular fluctuations in industry sales.
2. Seasonal and other regular fluctuations in market shares.
3. Delayed effects of marketing variables.

Each will be discussed in turn.

First, the industry sales (i.e., market demand) for many products are clearly subject to seasonal and other types of regular or cyclical fluctuations over time. Ice cream in summer and toys in December are prime examples of highly seasonal concentrations in sales. For another example, some products might show a high rate of sales in the first week of the month, following the payday. This tends to produce a regular pattern of within-month cyclical fluctuations for weekly data. We will need to take account of regular fluctuations in the industry sales in order to be able to predict brand sales accurately. The theory in time-series analysis tells us to model regular fluctuations of this sort by the following form.

where:

Q_(t) = the industry sales in period t

w_t = the weight attached to Q_(t-t) ( t = 1, 2, ¼, T).

Seasonal variations may be handled by a heavy weighting of w_t for suitable months or weeks. The beginning-of-the-month surge in some brand shares due to paydays may be handled by heavy weighting of w_t in weekly data.

Second, the market shares of a brand may also exhibit clear seasonal or cyclical fluctuations. Unlike the industry sales, market shares are expected to be less subject to seasonal variations. However, there still may be some situations in which relatively regular fluctuations in market shares appear. Suppose that the sales of a brand in a month were especially high because of a special promotional offer and that the average purchase cycle for the product class is three months. This brand's share may show a peak in every third month after that, creating a pattern resembling a seasonal fluctuation, provided that the initial offer created a loyal group of customers. For whatever the real reasons, relatively regular fluctuations in market shares may be expressed as a function of weighted averages of past market shares, that is,

where s_i(t) is the market share of brand i in period t and

f_t 's (t = 1, 2, ¼, p) are weights attached to past shares.

Third, we need to recognize the delayed effects of marketing variables on market shares of competing brands. Much evidence has been accumulated regarding the fact that advertising and other promotional effects tend to be felt not only in the period of execution but also in the subsequent periods. Such delayed effects may be expressed as the functional relationships between past marketing activities and the current market shares. Mathematically stated, those relationships may be written as:

where X_(t) is the vector of marketing variables in period t for all brands in the industry, that is,

where m is the number of brands in the industry and K is the number of relevant marketing variables.

If we combine the last two formulations, we obtain the following dynamic market-share model.

The main question is how to specify the function f in the above equation. One may, of course, think of using a linear formulation such as

where a, b_i0, ¼, b_ir are parameters and e_(t) is the error term. Note that b_iv (

v = 0, 1,2, ¼, r) is a row vector of (K ×m) parameters { b_kijv } , each of which shows the effect of X_kj(t-v) on s_i(t) and that the symbol ``¢ '' indicates the transpose of a row vector to a column vector.

Unfortunately, this formulation does not do because it does not satisfy the logical-consistency conditions discussed in Chapter 2. In other words, there is no guarantee that the values of s_i(t) estimated by this model will be contained in the range between zero and one and the sum of market shares will be equal to one. There may be other formulations which satisfy the logical-consistency conditions, but in this book we will propose a special form that is based on the log-centering transform discussed in Chapter 2.

Let us redefine variables as follows.

s_i(t)^* = the log-centered market-share for brand i in period t i.e.,

log( si(t)/ ~ s   (t)  )

[s\tilde]_(t) = the geometric mean of s_i(t) over i in period t

X_(t)^* = a vector with elements {log(X_ki(t)/[X\tilde]_k(t)) } (k = 1, 2, ¼, K; i = 1, 2, ¼, m)

[X\tilde]_k(t) = the geometric mean of X_ki(t) over i for period t .

The proposed time-series model is expressed as

The reader will note that equations (3.6) and (3.7) are remarkably similar, except that in (3.7) log-centered variables are used. Yet equation (3.7) is based on the attraction models of market share, namely, MCI and MNL models, and yields logically consistent estimates of market shares. Furthermore (3.7) is linear in its (log-centered) variables, and therefore its time-series characteristics are well known. For those reasons time-series analysis may be performed using log-centered variables. The future values of the s_i(t)^* 's may be computed from (3.7) and then transformed back to the s_i(t) 's. For a further justification of equation (3.7), see Appendix 3.10.1.

There are some more issues which must be discussed before we close this section. In proposing equation (3.7) we did not touch on the property of the error term, e_(t) . One of the major issues in time-series analysis is the handling of the error term which may be correlated with its own past values. In particular, it has been suggested to define e_(t) as a weighted average of present and past values of another variable, u_(t) , a white-noise error term which has zero mean and variance s² .

where q₁, ¼, q_q are parameters. In this specification it is clear that e_(t) is correlated with e_(t-1), e_(t-2), ¼,e_(t-q) because they share common terms in variable u . If we combine this specification of the error term with equation (3.7), we will have a time-series model known as an ARMAX(p,q) model.If we assume that e_(t) are independently distributed with zero mean and a constant variance, (3.7) is called an ARX(p) model.

Another issue that must be taken into account is the obvious fact that market shares of different brands are not independent of each other. The joint determination of market shares may be modeled by the following scheme. Let

be the column vector of log-centered values of market shares for period t , and

be the m ×(m ×K) matrix with each row identical to X_(t)^* . Using this notation the ARMAX model is written as

where:

F_l = the m ×m matrix with elements { f_lij }

f_lij = the parameter for the effect of s_j(t-l)^* on s_i(t)^*

B_v = the m ×(m ×K) matrix with elements { b_kijv }

E_(t) = ( e_1(t) e_2(t) ¼e_m(t))¢

e_i(t) = the error term for s_i(t)^* .

The model specified by equation (3.8) is known as a vector-valued ARMAX model in the literature of time-series analysis. In the theoretical sense it is a most comprehensive formulation of time-series properties associated with market-share analysis. In the practical sense, however, it is too unwieldy for the analyst to utilize. For one thing, it is extremely difficult to specify the model correctly in terms of the lag structure in the model. Unless there are some a priori grounds that give the number of lags, such as p, q, r , those numbers will have to be specified by a tedious trial-and-error process. For another thing, even if the model is correctly specified in terms of the lag structure, the model has been known to pose serious estimation problems, especially when many parameters are involved. With relatively moderate numbers of brands, variables and/or the lags, the total number of parameters which must be estimated will become large. For example, if m = 5 , K = 3 , p = 2 , r = 2 , and q = 1 , the total number of parameters (q 's, b 's and f 's) to be estimated is

excluding the number of variance components for E_(t) . It can be seen that even a reasonable attempt for realistic modeling based on the vector-valued ARMAX model is bound to be frustrated for those reasons.

In the remaining part of this book we will take a pragmatic stance in that the auto-correlations of the error term e_i(t) are negligible except for perhaps the first-order correlations (i.e., corr[ e_i(t),e_i(t-1)]). Furthermore, by adopting a simplifying assumption that u_i and u_j are uncorrelated within a period as well as over periods, we introduce a model called a seemingly uncorrelated ARMAX model whose parameters may be more easily estimated. This model seems to capture the basic properties of over-time competition between the brands without unduly complicating the analysis.

3.10  Appendix for Chapter 3

3.10.1  *Log-Linear Time-Series Model

In the last section for time-series issues, we proposed the following log-linear model of time-series analysis in lieu of a linear model.

To convince the reader of the compatibility of this time-series model with the market-share models in this book, the MCI and MNL models in particular, we first begin with the following specification of attraction for brand i in period t .

This is a straightforward extension of the cross-effects model in (3.4). One may apply the log-centering transformation to the above model to obtain

where:

A_i(t)^* = the log-centered attraction for brand i in period t
i.e., log(  A_i(t) / [ A\tilde]_(t))

[ A\tilde]_(t) = the geometric mean of  A_i(t) over i in period t

X_(t)^* = a vector with elements { log(X_ki(t) /[X\tilde]_k(t)} (k = 1, 2, ¼, K ; i = 1, 2, ¼, m)

[X\tilde]_k(t) = the geometric mean of X_ki(t) over i for period t

e_i(t)^* = the log-centered value of e_i(t)

b₀, b₁, ¼,b_r are parameters vectors.

The above equation looks much the same as equation (3.7) except that the latter is defined for the log-centered values of s_i(t) rather than those of  A_i(t) .

Fortunately, it may be easily proved that s_i(t)^* is equal to A_i(t)^* . Since s_i(t) is proportional to   A_i(t) , one may write s_i(t) = c  A_i(t) , the constant of proportionality, c , being the sum of  A_j(t) over j . Applying the log-centering transformation to s_i(t) we have

But since s_i(t) = c  A_i(t) and hence [s\tilde]_(t) = c[ A\tilde]_(t) , implying s_i(t)^* =  A_i(t)^* . Substituting s_i(t)^* for  A_i(t)^* in equation (3.10), we obtain equation (3.7). Note that model (3.8) is a multivariate extension of model (3.7) and no additional justification is necessary. Thus we have shown that the log-linear time-series models (3.7) and (3.8) are logical extensions of the attraction models of this book.