1. INTRODUCTION

In marketing management it helps to know what consumers buy and to understand how they do so. Managers therefore usually track their brand’s sales or market share. They may also use scanner-panel data to tell how many customers they have in a year, say, and how often they buy the brand, how many other brands they buy as well, and which other brands. But how and why do these and other such measures vary from brand to brand?

Many academics picture competitive markets as being highly complex. They expect no simple general answers. Practitioners often concur, believing their own brand to be different, if not unique. Consumers and markets are often seen as relentlessly dynamic and continually buffeted by what marketers do: when they buy a different brand, they must have re-evaluated their previous brand and decided that the other one is better.

Our observation is different and simpler. How often people buy a product and what brands they choose appear to be largely habitual at least for the time being. Consumers seem to have small personal portfolios of brands from which they consistently choose, typically buying one brand more often than another. Within such a framework of steady but divided loyalties, individual purchases then occur in an apparently irregular or even “as-if random” manner.

The make-up of these personal portfolios of brands differs from one consumer or household to the next. Yet this heterogeneous behavior aggregates to measures of brand performance that have been found to follow much the same simple and lawlike patterns in over 50 varied product categories from soap to soup and beyond (including some durables and services). Examples are (a) the similar buying rates of different brands, (b) the high incidence of light buyers, and (c) the low incidence of 100%-loyal ones.

These apparently lawlike patterns are in turn closely predictable from a single and parsimonious model, the Dirichlet. This is defined for steady-state markets with no market partitioning, in terms of four statistical distributions for individual consumers’ buying propensities (Poisson, Gamma, Multinomial, and Beta). No explanatory (e.g. marketing mix) variables are needed because in the model nothing is systematically changing and brands are assumed to be equally substitutable. The Dirichlet model in no way predicts that all markets should be near-stationary and non-partitioned in this way (although many often are). It only describes what markets are like when they are steady and non-partitioned. The model also highlights any deviations from the normal patterns.

These patterns and the associated model can still cause surprise. Can there be “lawlike” regularities (or even just a “nearsteady- state”) in markets which are constantly subjected to dynamic marketing inputs? And if near-steady-state markets do exist, can they be of interest to marketing practitioners who are invariably trying to change that steady-state? Our answers to these questions are a clear “yes”: the Dirichlet framework can be used to address many practical questions in marketing management.

Because of our emphasis on the use of the Dirichlet framework, the next section presents a variety of such applications before any detailed description of the general patterns or of the model. Performance audits and cases of market partitioning are described. So too are cases in evaluating new brands and the analysis of dynamic markets. These cases illustrate how the framework can be applied well beyond the model’s seemingly very restrictive conditions of steady-state markets with no partitioning.

The underlying patterns are described next. A formal outline of the Dirichlet model follows, noting also its limitations and biases. This sequence, patterns —> model reflects that it is their existence that enables the practical use of the model and of the whole Dirichlet framework. The distinctiveness of this approach is touched upon at the end of the modelling section.

Knowledge of such regular and predictable findings seems key to an understanding of brand loyalty, differentiation, and positioning theories, and also of advertising, promotions, and competitiveness more generally. These broader implications are developed in the final section.

2. PRACTICAL APPLICATIONS

In this section we illustrate the large number of applications of the Dirichlet framework – in particular, how it has been applied to new brands, brand audits, market partitioning, and dynamic markets.

2.1 New Brand Norms

Dirichlet-type patterns impose constraints on the common “Anything Goes” kinds of marketing planning. To illustrate we describe a case of pre-launch new brand evaluation, as an amalgam of practical experience. We then briefly cover two cases of post-launch new brand evaluation.

A Pre-Launch Evaluation

In considering a new instant coffee, “X”, a US company had decided on a market share of 5%, or 8 purchases a year per 100 households. This target had been iterated on the chosen marketing-mix expenditures and estimated revenues. Management was then given the choice of two marketing policies, A and B.

POLICY A was to position Brand X as a niche brand, since 16% of consumers had rated it very high in placement tests. X was therefore to be targeted at a small heavy-buying segment who really liked it, with some 1% of the population buying X about 8 times a year (thus satisfying the sales target of 8 purchases per 100 households). Given its special product-positioning, X was to be premium-priced, with loyalty-building promotions and advertising in up-market media. Distribution was also to be up-market, without expensive trade deals, relying on product and advertising pull rather than trade push.

POLICY B was the extreme alternative to such niching. X would be an add-on or variety brand, based on its special product formulation. It would reach the pre-set sales target of 8 by being bought by some 8% of the population but only about once a year. It would be competitively priced, with awareness-building promotions, “try something-different” advertising in mass media, and trade-support bought by slotting allowances and eye-catching merchandising.

Management then asked for a check against consumer panel data. This showed that both policies A and B were totally out of step with the market. Table 1 sets out the observed and predicted results for eight of the leading brands. The penetrations differed some 50-fold (from 11% down to 0.2%). But the brands’ purchase rates were very similar – all roughly 3 (with an exceptional 4.5 for Maxim), and a small downward “Double Jeopardy” trend from 3.2 for the leader down to 2.1.

Interpolating, the analysts concluded that if Brand X did reach its ultimate target market-share of 5%, or 8 purchases per 100 households, it would be bought about 2.7 times a year by its buyers. And hence by 3% of all consumers, i.e. 8/2.7 = 3. The “Sales Equations” (buyers x rate = sales) for the three scenarios were therefore

Policy A

A Niche Brand: 1% buying 8 times: 1×8=8

Policy B

An Add-on-Brand: 8% buying once: 8×1=8

Market

A Normal Brand 3% buying 2.7 times: 3×2.7=8

The NPD team now queried whether the similar average purchase frequencies of about 3 in Table 1 were not just an unexpected coincidence, with at least one big exception anyway. The analysts countered that the close fit of the well-established theoretical norms (T) showed that the pattern was neither accidental nor unexpected. They added that to break out of this pattern, X would have to differ far more from the existing brands than they differed from each other. Yet the marketing plan for X had claimed nothing of the kind.

Management concluded that the new brand would not reach its 5% sales target via either POLICY A or B, and also that the allocated budget was insufficient for a me-too 3 x 2.7 target. They dropped plans for X, forestalling that “4 out of 5 new brands fail”.

Post-Launch Evaluations

Knowledge of Dirichlet-type patterns has also been telling in post-launch cases of both near-failure and economic success:

• A year or two after the launch of Campbell’s new premium-priced “Tastes of the World” soup in Britain, the brand showed low loyalty-related measures compared with all its bigger competitors. “TotW” seemed doomed, until Campbells were made aware of the Double Jeopardy phenomenon. This is that average purchase frequencies and repeat-buying levels differ little between brands other than for a small but regular and predictable downward trend with market-share as already illustrated here in Table 1. Consequently, TotW’s low loyalty proved to be normal (and broadly unalterable) for any brand with TotW’s low share. TotW’s problem was just too few customers. This could in principle be remedied since the leading brands (even premium-priced ones) were already attracting far more customers and their policies could be emulated.
• At the other extreme, Unilever’s new toilet soap Shield achieved a remarkable 20% share within the first few weeks of its launch. Moreover, repeat-buying and switching were immediately as for an established 20% brand in that market (Wellan and Ehrenberg 1988). This implied that Shield’s high share would continue at least for a time, as it did. It is now useful to know that this can happen. (There was much inconclusive argument at the time as to why and how it had happened).

2.2 Performance Audits in Established Markets

A fundamental use of the Dirichlet framework is to assess the performance of existing brands in established markets. To illustrate, Table 2 gives some observed and theoretical performance metrics for Folgers, P&G’s 24% US brand of instant coffee. This market was typically near-steady at the time, but had been highly dynamic.

 

In Table 2 we can see how the market penetration of Folgers increased much less than pro rata, from 1% in a week to 11% in the year; how infrequently many of its customers bought the brand (46% once in the year and only about 3 times on average); how they bought other brands in total just about as often (6.4 – 3.2 = 3.2) as they bought Folgers itself (3.2), thus giving Folgers a Share of Category Requirements (SCR) of 50% (3.2/6.4); and which specific other brands they also bought. Such figures may seem threatening (e.g. “half your annual customers buying you only once”). But the theoretical benchmarks show the measures to be near to normal, and hence typical for brands that survive in competitive markets (such as those listed in Table 7, on page 12).

Deviations from the general patterns and theoretical predictions might signal a possible threat or opportunity, or simply be due to sampling effects. For instance, in Table 1 Maxim’s very high annual purchase rate (4.5) was found to be due to two “outliers” (two households making over 30 purchases each).

A related use is to provide a range of interpretative norms for new but similar data (another year, country, or category). Instead of blindly data-mining with the unfocused question “What is this market like?”, one can simply check whether the already familiar Dirichlet patterns (as in Table 1 and 2) recur or not:

• In Table 1, Brim’s somewhat low average purchase frequency of 2.1 had not occurred the year before (the first thing to check). It was therefore not a brand characteristic but a one-off, due perhaps to a fire in a warehouse (“bad”) or a promotion with an extra-large bonus of once-only buyers (“good” presumably, as an analysis leaving out the promoted month would have shown more explicitly).
• In the UK instant coffee market, the market-leader Nescafé has a high buying rate. But this is “normal”, given Nescafé’s high 34% share there (in contrast to its US share).
• Doctors’ medical prescriptions generally follow the Dirichlet patterns (despite doctors neither consuming nor paying for the drugs, and being professionally trained in making their choices – Stern 1995). However, in the early ’90s, an unprecedentedly high prescription rate occurred for Bristol-Myers’ cardio-vascular drug Capoten: 10 a year per prescribing doctor, instead of the norm of 5. This was traced to frequent Capoten prescribers being offered a free PC. The prescribing rate fell back when the inducement was withdrawn (Stern and Ehrenberg 1995).
• For private labels, is their loyalty high or low? Do they appeal to a special segment? In making comparisons with manufacturers’ brands, the Dirichlet can allow for market-share differences and for the very restricted availability of private labels. In the outcome, PLs show normal brand loyalty and no special proneness across different stores (e.g. Uncles and Ellis 1989, Bound and Ehrenberg 1997).

2.3 Partitioning In Established Markets

Markets for substitutable brands are usually non-partitioned; that is, there are no special groupings of the brands. Thus the observed purchase duplications of Folgers in the instant coffee market with other brands in Table 3 showed no clustering. An algebraic approximation of the Dirichlet, the “Duplication of Purchase Law”, summarises this. It says that the percentage of buyers of Folgers who in the chosen time-period also buy Maxwell House, Tasters Choice, Nescafe, etc. is simply proportional to these other brands’ penetrations (with some wobbles), as in Table 3 where the correlation is 0.99.

For partitioned markets (e.g. ground vs instant coffee, or caffeinated vs decaffeinated), the Dirichlet-type non-partitioned “Duplication of Purchase Law” often continues to apply for the separate partitions: Table 4 illustrates for the 1990 UK gasoline market between Unleaded and Leaded (using just 4 brands here by way of example, see Scriven and Ehrenberg 1994).

The purchase duplications between pairs of brands were:

• Much higher – at about 30% to 10% – for Unleaded brands (in the North-West quadrant), and also for the Leaded ones (the SE quadrant).
• Much lower – at well under 10% –  between two different brands of either type (in the SW and NE quadrants), but not zero mainly because of 2+ car families where one car still requires leaded.
• Middling – at about 10% – for the same brand Unleaded and Leaded (in the SW and NE diagonals), mainly because the 2+ car families may buy unleaded and leaded at the same local gas station (plus brand-loyalty elsewhere).
• Always in line with the brands’ overall penetrations (shown in the bottom row, i.e. the standard Dirichlet-like Duplication of Purchase Law – the correlation is .8).*
• Without the accumulated prior knowledge of Dirichlet-type patterns and theory, the brand duplication data are likely to have been left in a more opaque form like Table 5.

2.4 The Analysis Of Dynamic Markets

Substantial changes in markets are also of much interest. The steady-state Dirichlet norms can then identify relevant sameness or deviations. Our impression so far is that the “steady-state” loyalty related measures mostly persist. Only brand penetrations and brand-switching levels change. Some specific cases are:

Two Dramatic Long-Term Trends

The 1992 US Instant Coffee market as in Tables 1 and 2 had changed radically in two ways from 10 years before: (i) category sales had slumped by two-thirds, and (ii) Folger’s brand share had doubled (and other shares had been literally decimated e.g. Brim’s from 4% to 0.3%). Nonetheless, the loyalty-levels were Dirichlet both in 1992 and 1981 (Ehrenberg 1997). Only the market shares and penetrations were very different.

In another long-run study, the UK toothpaste market was found to be Dirichlet in 1967/68 and 1990/91, despite a major decline in sales of Gibbs and the introduction of Crest, Aquafresh and Sensodyne (Ehrenberg, Uncles and Carrie 1994).

A Short-Term Trend

When the Quarter I to Quarter II sales of US laundry detergents dropped by 15% in the early ’90s, quarter-by-quarter repeat-buying was tracked in a “Conditional Trend Analysis” (CTA), i.e. repeat-buying conditional on each consumer’s previous purchasing status (an extension of the original Negative Binomial Distribution or NBD model, Ehrenberg 1972/88, Goodhardt and Ehrenberg 1967, Morrison 1969). The sales drop was not due to existing QI buyers buying less in QII, but to too few “new” buyers coming in (only 25%), as summarized in Table 6.

The Seasonal Soup Market

The CTA norms here showed that the winter peak was half due to all-year-round buyers buying more, and half to peak-season-only buyers. Such an insight can affect the timing of advertising and any attempts to spread peak-demand (Wellan & Ehrenberg 1990).

Stock-outs

The analysis of temporary out of stocks need not lead to a longer-term loss of sales: CTA showed that before-to-after repeat-buying was normal, as if there had been no out-of-stocks, with no “learning” from the forced switching (Charlton, Ehrenberg and Pymont 1972).

Price Promotions

Analyses of 150 price promotions in Britain, Germany, Japan and the US has led to three generalisable results: (i) before-to-after repeat-buying is unaffected (i.e. again no “learning”); (ii) no before-to-after sales increase; (iii) virtually no new buyers (Ehrenberg, Hammond and Goodhardt 1994; also asides by Abraham and Lodish 1987; McQueen, Foley and Deighton 1993; and Jedidi, Mela and Gupta 1999). Price promotions therefore mainly affect the purchase timing of brands from each consumer’s purchase repertoire.

A more specific case is when the giant pack of Unilever’s leading UK laundry detergent was promoted with extra product, management expected this to appeal to the brand’s heavy buyers. But the CTA norms showed the opposite – the offer had attracted recent non-buyers and light buyers (Ehrenberg 1972).

2.5 Further Applications

Dirichlet norms have also been used as benchmarks in analyzing other diverse issues in marketing management, such as:

• cannibalization in the German and UK detergent markets (Lomax et al. 1996);
• price-sensitivities (Ehrenberg, Scriven and Barnard 1997; Scriven and Ehrenberg 1995);
• consumer loyalty programs in Australasia (Dowling and Uncles 1997; Sharp and Sharp 1997, 1999); and
• subscription markets (Sharp and Wright 1999).

These cases, and those discussed above at greater length, illustrate the scope of benchmarking applications to date but are by no means exhaustive of what can be analysed (e.g. much more work at the level of specific stock-keeping units, Fader and Hardie 1996).

3. GENERALIZABLE PATTERNS

Use of the Dirichlet findings rests on the accumulated knowledge of patterns derived over many years from undigested  tabulations of scanner or diary-panel data for individual households or people. This involved data mining, pattern recognition, empirical generalization, and theoretical modeling (e.g. Ehrenberg 1982; Lindsay and Ehrenberg 1993).

3.1 The Range Of Conditions

Generalisation across packaged goods product categories is now well established (e.g. a study of 34 such categories is reported in Uncles et al.1994). But the Dirichlet patterns extend far beyond these to all the conditions in Table 7.

The more striking extensions include consumers’ choice of products at different prices, retail outlets, TV programs, industrial products and even cars (which may cost 10,000 times as much as some groceries) (e.g. Barwise and Ehrenberg 1988; Brewis- Levie and Harris 1999; Ehrenberg 1959, 1972/88; Ehrenberg and Bound 1999; Kau and Ehrenberg 1984; Scriven and Ehrenberg 1994, 1995; Uncles and Ehrenberg 1990a, 1990b; Uncles and Hamond 1995; Wrigley and Dunn 1984; and company-specific reports). There is scope for more, such as recent geographical extensions to Australasia (Wright, Sharp and Sharp 1998) and Japan (Kau et al. 1998) and to markets in developing countries. On-line electronic shopping too. The factor that seems to enable the Dirichlet patterns to emerge is that in all cases an individual’s choices are mainly between items which to him or her appear at that time directly substitutable.

3.2 Specific Regularities

Table 8 illustrates the main patterns numerically for 13 categories that were in near-steady-state at the time of analysis. It also gives the average of the theoretical Dirichlet predictions as discussed in the next section. The ten main patterns are as follows:

1. Annual penetrations (% buying) are much lower for smaller brands (dropping from an average of 46% for the brand leader down to 8% for the eighth brand).
2. In contrast, average purchase frequencies are very similar from brand to brand (all about 3 or 4 on average in a year say) unless a brand’s penetration were to be very high. (The amounts bought per purchase – not shown here – vary even less by brand and are often about 1.1.)
3. Such average purchase frequencies however decrease as a small “Double Jeopardy” (DJ) trend with decreasing market share. McPhee’s DJ phenomenon says that a smaller brand is “punished twice” in that fewer people buy it and they buy it somewhat less often. DJ is not an attribute of specific brands – possibly reflecting “Brand Equity” – but a statistical selection effect (e.g. Ehrenberg, Goodhardt and Barwise 1990; McPhee 1963). McPhee’s mathematical mechanism is, at its simplest, that customers of a small brand have more chance of buying the bigger brands, all other things being equal, and will therefore buy the smaller brand itself slightly less often.
4. Individuals’ buying rates differ greatly. Few of a brand’s customers buy it often (broadly in line with Pareto’s classic 80:20 rule). Here again there is a DJ effect, with smaller brands having even more once-only buyers.
5. 100%-loyalty (or near 100%-loyal) over a series of purchases is relatively rare. In Table 8, only some 11% of the annual customers of a brand bought it exclusively, with a DJ trend again. They bought on average about 3 times. This purchase rate is much the same as that for all buyers of the brand (3.7 here), but much less than their category buying rate (on average 15). Contrary to general belief, 100%-loyal buyers are not especially worth having.
6. Most customers of a brand are multibrand buyers with steady split-loyalties. Over a number of category purchases (say 10+), they buy other brands in total more often than the brand itself. The typical brand in Table 8 accounted for only a 15% “Share of Category Requirements” (SCR) for its annual customers (e.g. 3.7/15). Lowish SCRs occur even among a brand’s heavy buyers and for people’s favorite brands (Hammond 1997). There is also a small systematic subpattern, McPhee’s “Natural Monopoly” increase in category buying rates with decreasing share shown in Table 8.  This is again a consistently predictable statistical selection effect.
7. Which specific other brands these multibrand buyers also purchase is again  similar from brand to brand in a non-partitioned market, as illustrated in Table 8 (and also within the partitions in Table 4). Thus the first brand in Table 8 is bought by about 55% of each of the other brands’ customers, the fifth by about 21%.
8. These duplications of purchase between brand A with another brand B vary with how many people buy brand B at all (i.e. B’s penetration). This is the so-called Duplication of Purchase Law, discussed earlier (Table 3). 
9. A numerically very strong pattern is with the length of the chosen analysis period, e.g. typically from a week (the shortest period in which purchasing is usually stationary) to a year or more. Penetration and repeat-buying is much lower in a week, as briefly shown in Table 2 and documented in much detail in the earlier Repeat-Buying text (Ehrenberg 1972/88). 
10. Loyalty-related measures (e.g. the incidence of 100%-loyal buyers, or a brand’s SCR) are however much higher in a shorter period, e.g. near-100% in a single week when most consumers make at most only a single purchase in an undifferentiated category and so do not have the chance to be disloyal. (Labeling consumers as “loyal” based on their few purchases in quite a short period like a quarter can therefore, we think, be questionable, as for example in Mela, Gupta and Lehmann 1997.)

Other regular patterns (e.g. for period-by-period repeat-buying, or duplicated buyers’ buying rates) are described in the earlier literature (e.g. Ehrenberg 1972/88). Anyone with access to consumer-panel or similar data can check on these various patterns and/or expand on them.

4. THE DIRICHLET MODEL

Historically, the Dirichlet model was developed by our colleagues Chris Chatfield and Gerald Goodhardt (1975) to account for these already known empirical near-steady state patterns, and also almost in parallel from a theoretical “utility” basis (Bass, Jeuland and Wright 1976, see also Rungie 1999).

4.1 The Basic Proposition

The Dirichlet proposition is that experienced consumers behave as if, for the time being, they have steady and personal split-loyalty purchase propensities (probabilities in the model) for when they buy and what brand to choose. That is what the data “looked like” and what more formal analysis had long supported. The model consists of five distributional assumptions about these propensities:

A Gamma Distribution

Individual consumers’ as-if-long-run category buying-rates are assumed to follow a smooth “Gamma” type distribution. There are strong theoretical reasons for this relating to the near independence of brands (Goodhardt and Chatfield 1973). Like Pareto’s 80:20 rule, the Gamma has many light (or non-) buyers and few heavy ones, unless the mean is very high.

Poisson Distributions

Specific purchases are spread as-if randomly over time about each consumer’s long-run probability, and independently of when the previous purchase was made (a “zero-order” process). This specifies “Poisson” distributions, as has been widely tested (Bass et al. 1984; Chatfield and Goodhardt 1973; Dunn, Reader and Wrigley 1983; Ehrenberg 1959; and Schmittlein, Bemmaor and Morrison 1985).

The Poisson and Gamma assumptions combine to give the Negative Binomial Distribution (NBD) model. This has been widely used in repeat-buying studies and conditional trend analyses, where each brand has been considered on its own (Ehrenberg 1959, 1972/88). In the Dirichlet, the NBD is used for the category distribution of purchases, not individual brand distributions.

A Multivariate Beta Distribution

Consumers’ differing brand-choice probabilities are assumed to follow smooth Beta distributions of a multivariate “Dirichlet” type (named after a French-named German mathematician). Strong theoretical backing again derives from the near independence of brands or “Independence from Irrelevant Alternatives” (IIA) assumption (Goodhardt, Ehrenberg and Chatfield 1984; Luce 1959; Morrison and Schmittlein 1988).

Multinomial Distributions

On any one purchase-occasion, consumers are assumed to choose each brand as if randomly with their own fixed brand choice probabilities. This is the widely-used zero-order “Multinomial” distribution of brand-choice.

Independence of Brand-Choice and Purchase-Incidence

This is in line with observed market-shares being similar for light, medium, and heavier category-buyers.

4.2 Numerical Inputs and Outputs

Five input variables have to be calibrated to model any given near-steady-state market, as shown in Table 9: category penetration (e.g. 31%) and average purchase frequency (e.g. 5.0) in a base-period; the average number of brands bought by each category buyer (e.g. 1.6 brands, as 5.6 x 9/31); the size of one or more brands as the only brand-specific input (e.g. Folger’s 24% share); and the chosen length of analysis period (e.g. a year). The effects of varying marketing-mix inputs would be subsumed in changes in the market-shares (and hence in all other measures), and/or show up as discrepancies (e.g. Bhattacharya et al. 1996; Bhattacharya 1997; Kahn, Kalwani and Morrison 1988).

Each possible purchase made by any consumer is therefore specified probabilistically – a “stochastic” representation. These probabilities can be aggregated to estimate chosen performance measures, using either custom-built software (e.g. Hewitt 1990; Kearns 1999; Uncles 1989) or simulated individual Dirichlet-type consumers (Goodhardt 1995).

There are no “closed-form” algebraic formulae to show how the output measures interrelate, since the Dirichlet mathematics involves infinite series. In principle, much data-mining on the undigested theoretical predictions would therefore be needed. But in practice we can turn to the empirical patterns which were discerned earlier, as outlined in the previous section. Not having usable “if this then that” is no different from how we describe the Normal distribution, say – we rarely refer to the formula but routinely describe its patterns (i.e. as being symmetrical, hump-backed, with 95% lying between ±2 standard deviations).

There are fortunately also some more explicit algebraic approximations that predate the derivation of the Dirichlet model. One, for the Double Jeopardy effect on average purchase rates w, is w = w_o / (1-b). This shows how w varies non-linearly with the brand’s penetration b, expressed as a proportion. Here w_o is a constant which can be estimated as the average observed w(l-b) for any chosen brands; w_o can also be thought of as the limiting value of w as b tends to zero.

Another very helpful approximate formula is the “Duplication of Purchase Law” b_Y/X x Db_Y . This says that the number of buyers of brand X who in the chosen time period also buy Y (denoted by b_Y/X) is simply proportional to brand Y’s penetration b_Y, as in Table 3. The simplest estimate of D is average duplication/average penetration.

 

4.3 The Fit of The Model

Judging the model is simplified by having varied “point estimates” to check against observed values, as in the tables here (rather than just assessing the overall fit using a measure such as an R² or a log-likelihood). Furthermore, the data tend to relate to many brands, products, time periods, and countries, often with very large individual samples (e.g. 10,000 or more nowadays). With only five input measures, many output ones, and no “explanatory variables” there is no question of “over fitting”. Almost any reported discrepancy is “significant” (i.e. real), but not necessarily large, important, or systematic.

In the event, discrepancies occur and some show up as systematic sub-patterns. But there are few consistent biases overall, as is illustrated by the bottom line of Table 8. The irregular scatter of the O-T differences is generally small, on average often at about 5% or less (e.g. a mean absolute deviation of .2 for average purchase rates of about 3.6 in Table 8). A 5% difference from a predicted buying rate of 3.6 would of course matter to sales. However, the model’s main mission is not to predict already-known sales volumes, but to reflect how similar the average purchase rates in Table 8 are, compared with the 10-fold variation in market-share.

One early systematic deviation from the model was the “Variance Discrepancy”, a shortfall of very frequent buyers which has since been “explained away” as an artefact of our calendar: most people buy a typical grocery product at most once a week (Chatfield 1967; Ehrenberg 1959). Other systematic deviations are:

• Quarter-by-quarter repeat-buying can be over-predicted (by 5 to 10 percentage points – Ehrenberg 1972). Repeat-rates also tend to erode somewhat over non-adjacent periods (e.g. by 8 percentage points over a year, East and Hammond 1996). This can be regarded as a “slightly leaky bucket”.
• Annual purchase frequencies are a unit or so higher than predicted for some market-leaders, perhaps because they are more likely to be in-stock and more widely available (Fader and Schmittlein 1993; Reibstein and Farris 1995).
• The annual purchase rates of 100%-loyal buyers are consistently under-predicted by a purchase or so (e.g. for each of the 100 brands in Table 8). This is unexplained. But since the discrepancy hardly varies it has not been of diagnostic marketing value so far.
• The distribution of light, medium, and heavy category buyers is sometimes a little “flatter” than predicted by the Poisson-Gamma NBD (see above) for reasons that are not understood. The more limited “Empirical” Dirichlet model can then be fitted instead (Ehrenberg 1988; Uncles 1989).

4.4 Other Modelling Approaches

The Dirichlet model predicts stable brand performance measures from stable market-shares. This provides norms and insights against which marketing issues can be evaluated, as we have seen. In contrast, econometric response models (e.g. logits) aim to predict changes in market shares from changing marketing-mix inputs like advertising and price (e.g. Bucklin and Gupta 1998; Hanssens, Parsons and Schultz 1990; Lilien, Kotler and Moorthy 1992; Lilien and Rangaswamy 1998). Such models are often re-specified on each application. Some assume turbulent (continually changing) choice probabilities for each consumer (e.g. Erdem 1996; Erdem and Keane 1996; and Guadagni and Little 1996), while others attempt to model generally non-stationary conditions (Lenk, Rao and Tibrewala 1993; Vilcassim and Jain 1991; Wagner and Taudes 1986). They all require many parameters to be estimated and interpreted – involving both conceptual and computational complexity. In any practical application involving many brands and stock keeping units (SKUs) this may run to hundreds of parameters (as discussed by Fader and Hardie 1996).

Economists and psychologists have also carried out much work in the last few decades into broader issues of consumer decision-making, often along experimental and/or mathematical lines and frequently with roots in Game Theory (e.g. Kagel and Roth 1995; Thaler 1994).

A direct structural alternative to the Dirichlet is first-order Markov processes. This was popular in the 1960’s and 70’s (e.g. Kuehn 1962; Massy, Montgomery and Morrison 1970), and occasionally since then (e.g. Bronnenberg 1998). Markov assumes homogeneous consumers and fixed switching and repeat-purchase probabilities for each brand. That is directly contrary to the Dirichlet-type findings and much of the response modelling literature. Another purely structural model is the Hendry system (Butler 1966). This uses the Duplication of Purchase Law formulation for pairs of purchases but with different assumptions from the Dirichlet and very different outcomes (Ehrenberg and Goodhardt 1974).

The Dirichlet approach has also been elaborated in two main ways. Either using “mass points” to model consumer heterogeneity, rather than smooth Gamma and Beta distributions (Colombo and Morrison 1989; Reader and Uncles 1988; Fader and Hardie 1996). Or relating heterogeneity to its possible sources such as demographics (Allenby and Lenk 1994; Bass 1993; Bhattacharya et al. 1996; Chintagunta, Jain and Vilcassim 1991; Ehrenberg 1959; Fader 1993; Fader and Lattin 1993; Jones and Zufryden 1980; Russell and Kamakura 1994; Vilcassim and Jain 1991; Wrigley and Dunn 1985). So far, however, no major gains in predictive power or parsimony have been claimed.

5. IMPLICATIONS

Earlier we showed how the Dirichlet framework can be used in specific applications. There are broader implications too, for our understanding of consumers, brands and marketing management more generally. We conclude with a discussion of these.

5.1 Understanding Consumers 

Polygamous Consumers

A brand’s customers are mostly polygamous, rather than either monogamous or promiscuous. They typically have several steady partners, at least for the time being, one or two being favorites (e.g. Hammond 1997). To be “loyal” to a brand does not require a consumer being a heavy or an exclusive buyer of it, or having a unique “commitment” or “involvement” (e.g. McWilliam 1994).

Nor do users of substitutable brands generally differ in their attitudes to them – there is much accumulated evidence of that (Barnard and Ehrenberg 1997; Barwise and Ehrenberg 1985; Dall’Olmo Riley et al. 1997; Ehrenberg, Barnard and Scriven 1997, Franzen 1994). This is consistent with the small role brands play when consumers consume the product: for coffee say, the common questions are “Milk or Sugar?” or “Decaffeinated or Regular?”, not “Would you like Maxwell House or a Nescafe?”. Nevertheless, consumers may come to identify with their habitual brands (“I use it, therefore I like it”).

A Dirichlet consumer’s propensity to buy a brand would not be affected by previous purchases of the brand (the model’s “zero-order” assumption). Being experienced, any such “learning” would already have occurred in the past. Little observed purchase feedback has in fact been claimed in the literature, and may even then have been due to some overlooked non-stationarity (e.g. Shoemaker et al. 1977).

Precisely when a consumer buys – whether this week or next – and which repertoire brand is then chosen are both assumed in the model to be as-if random. This is not to say consumers literally toss mental pennies. Instead, specific choices are governed by a variety of reasons, motives, and feelings – such as habits, out-of-stocks, promotions, special displays, moods, the mother-in-law coming, etc. This is broadly as discussed in consumer behavior studies (Engle, Blackwell, and Miniard 1995; McAlister and Pessemier 1982; Rossiter and Percy 1997; Solomon, Bamossy, and Askegaard 1998), in choice modeling (Horowitz and Louviere 1995), in qualitative studies (Fournier and Yao 1997, Gordon 1994), and in the context of limited problem solving (East 1997, Foxall and Goldsmith 1994, Olshavsky and Granbois 1979; Weilbacher 1993). In practice, consumers’ varying reasons appear mostly to be sufficiently idiosyncratic and irregular to be successfully treated as being as-if random in the Dirichlet model.

The findings for price promotions fit in. As noted, their large short-term up-and-down sales blips stem almost entirely from past customers, who only need to change their quasi-random purchase-timing of the promoted brand rather than switch to an unfamiliar brand.

More generally, consumers could switch far more often to other similar brands just insofar as they are similar. But by the same token there is no great motivation for them to do so. Having small habitual brand repertoires requires less mental effort than continually switching brands. Yet it enables consumers to exercise some choice without having to re-evaluate all the available criteria, or somehow weigh up the expected utilities at each purchase (as is supposed in “rational” economics – but see Thaler 1994).

Segmentation

Brand segmentation into distinct and relatively homogeneous sub groupings of consumers for each brand is not directly allowed for in the Dirichlet framework. In practice, the customers of closely substitutable brands are in fact very similar in their make-up (Collins 1971; Hammond, Ehrenberg and Goodhardt 1996), and also in their attribute beliefs (as noted earlier).

There can nonetheless be segmentation at the category level – larger households being heavier buyers, or some consumers being more profitable (e.g. Day, Shocker and Srivastava 1979; Ehrenberg 1959, 1988). Again, users of leaded gasoline had older car engines; petfood buyers have pets; and pre-sweetened cereals are eaten more by children. But such sub-patterns then tend to hold equally for all the substitutable brands competing in that category or sub-category.

5.2 Understanding Brands

Brand Differentiation

Brands are identical in the Dirichlet theory except for their names and market shares – brands are brands. This is also largely so in practice, with sales-effective product advantages and innovations usually soon being copied. Even early-mover advantages may not last (Chiang and Robinson 1997; Roquebert, Phillips and Westfall 1996; Szymanski, Troy and Bharadwaj 1995). Sustainable product differentiation therefore seldom occurs between brands.

But differentiation does exist within brands: products have different pack-sizes or flavors; shampoos are for oily, dry, and even normal hair; interest rates vary by withdrawal notice; cars have 2, 3, 4 or 5 doors; and so on (see also Fader and Hardie 1996).

Consumers may well spend more time choosing between these different functional specifications than between brands as such (Heath 1999; Moran 1990). But competitive brands in a category tend all to have much the same range of line-extensions. In that sense, nearly all brands are umbrella brands and similar to each other as such – which is where the non-partitioned Dirichlet model then applies directly.

Minor differences between competing brands do occur (e.g. the bottle top or car door handles – Carpenter, Glazer and Nakamoto 1994). But even when they are occasionally featured “on-pack” or in the advertising, they tend mostly to be noticed only after consumers have started to use the brand, or perhaps as a result of word-of-mouth discussion. Such minor differences may then lead to longer-term brand preferences for otherwise similar brands. For brands therefore to gain very different penetrations or shares of the market is consistent with the basic “Brands are Brands” formulation of the Dirichlet.

Brand Loyalty

Brand loyalty has been measured in many different ways. But in the Dirichlet framework each aggregate loyalty measure tends to be similar for competing brands. Insofar as the measure does vary, it correlates with market share (in accord with Double Jeopardy): a big brand has more customers, who are on average fractionally more “loyal” – but not by much. Changing sales do not arise from a changing loyalty level among the brand’s customers but from a changing number of customers.

This Dirichlet portrayal of brand loyalty is out of kilter with the widely-held view that “brand equity” is an idiosyncratic property of an individual brand (e.g. Aaker 1996).

Instead, there are big brands and smaller brands, rather than some strong ones and other weak ones (Ehrenberg 1993; Feldwick 1996; Goodhardt 1999). Differing market shares are then due to the very different numbers of people to whom each brand is “salient”, i.e. who feel positive about each brand (Ehrenberg, Barnard and Scriven 1997).

The similarity of different brands’ loyalty levels also explains the apparent failure of loyalty programs to raise the loyalty greatly if at all (Dowling and Uncles 1997; Sharp and Sharp 1997, 1999; Uncles and Laurent 1997). In principle, increased loyalty could occur even in a stationary market – Folgers instant coffee drinkers in Table 1-3 could buy it twice as often without having to drink more coffee (Folger’s share of category requirements was 50%.) But both in theory and in practice this does not seem to happen.  Instead, the normal split-loyalty and related Dirichlet patterns tend to persist.

Brand Advertising

Brand advertising has scope mainly because of the lack of much sustainable differentiation at the overall brand level (i.e. the umbrella over subvariants with the same name). Consumers want the product or service (some coffee, a hotel room, or whatever). Hence advertising does not have to try to persuade consumers that brand A really differs from the similar B. They just have to choose one of the available options of the right type. Consumers then seem to find it convenient and reassuring to have developed habitual choice propensities.

The implication is that advertising an established brand needs mainly to publicize it, mostly by reminding experienced consumers: “Here I am – Remember me”. Publicizing the brand is what much advertising appears in fact to do, including announcing new brands or a special price – often in highly creative ways for impact and memorability (Ehrenberg et al. 1999). Insofar as advertising seldom increases sales in the longer-term, it is not because it cannot do so but because competition stops it. Advertising on this view is therefore mainly defensive.

5.3 Marketing Management

The ubiquity of predictable patterns of buying behavior and of more or less stationary markets could add to the existing doubts about the role of marketing and of inputs such as advertising. But there is scope within the Dirichlet-type constraints: brands do differ in their market shares and penetrations, and they can gain sales. Such scope however exists also for one’s competitors. With intense competition, the outcome of marketing activity is therefore usually a more or less competitive equilibrium rather than gains for any or all.

Hence a successful outcome is maintenance of one’s sales and the reinforcement of one’s loyal customers (both consumer and trade). Competition means running hard to stand still, with profitable survival being greatly preferable to the most likely alternative.