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Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

The workhorse brand choice models in marketing are the multinomial logit (MNL) and nested multinomial logit (NMNL). These models place strong restrictions on how brand share and purchase incidence price elasticities are related. In this paper, we propose a new model of brand choice, the “price consideration” (PC) model, that allows more flexibility in this relationship. In the PC model, consumers do not observe prices in each period. Every week, a consumer decides whether to consider a category. Only then does he/she look at prices and decide whether and what to buy. Using scanner data, we show the PC model fits much better than MNL or NMNL. Simulations reveal the reason: the PC model provides a vastly superior fit to inter-purchase spells. Copyright © 2009 John Wiley & Sons, Ltd.


1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

The workhorse brand choice model in quantitative marketing is unquestionably the multinomial logit (MNL). It is often augmented to include a no-purchase option. Occasionally, nested multinomial logit (NMNL) or multinomial probit (MNP) models are used to allow for correlations among the unobserved attributes of the choice alternatives, or the models are extended to allow for consumer taste heterogeneity. There have been rather strong arguments among proponents of different variants of these models.

But in two fundamental ways, all these workhorse brand choice models are more alike than different: (1) they assume essentially static behavior on the part of consumers, in the sense that choices are based only on current (and perhaps past) but not expected future prices, and (2) they all make strong (albeit different) assumptions about when consumers see prices, and when they consider purchasing in a category.

For instance, any brand choice model that does not contain a no-purchase option is a purchase timing/incidence model—of a very strong form. This incidence model says that a random and exogenous process1 determines when a consumer decides to buy in a category, and that the consumer only sees prices after he/she has already decided to buy. On the other hand, standard brand choice models that contain a no-purchase option make an opposite and equally strong assumption: that consumers see prices in every week,2 and decide whether to purchase in the category based on these weekly price vectors.3 And, in either case, whether one assumes consumers see prices always or only if they have already decided to buy, standard models assume that consumers then make decisions solely based on current (and perhaps past) prices.

In this paper we propose a fundamentally different model of brand choice that we call the “price consideration” or PC model. The difference between this and earlier brand choice models is that, in the PC model, consumers make a weekly decision about whether to consider a category, and this decision is made prior to seeing any price information. Of course, the decision will depend on inventory, whether the brand is promoted in the media, and so on. Only after the consumer has decided to consider a category does he/she see prices. In this second stage, the consumer decides whether and what brand to buy. Thus, the PC model provides a middle ground between the extreme price awareness assumptions that underlie conventional choice models (i.e., always vs. only when you buy) because in the PC model consumers see prices probabilistically.

We estimate the PC model on Nielsen scanner data for the ketchup and peanut butter categories. We compare the fit of the PC model to both MNL with a no-purchase option, and NMNL with the category purchase decision at the upper level of the nest. All three models incorporate (i) state dependence in brand preferences a la Guadagni and Little (1983), (ii) dependence of the value of no-purchase on duration since last purchase (to capture inventory effects), and (iii) unobserved heterogeneity in brand intercepts.4

We find that the PC model produces substantially superior likelihood values to both the MNL and NMNL, and dominates on the AIC and BIC criteria. Simulation of data from the models reveals that the PC model produces a dramatically better fit to observed inter-purchase spell lengths than do the MNL and NMNL models. In particular, the conventional models greatly exaggerate the probability of short spells. For the PC model, this problem is much less severe.

To our knowledge, this severe failure of conventional MNL and NMNL choice models to fit spell distributions has not been previously noted—or, even if it has, it is certainly not widely known. We suspect this is because it is common in the marketing literature to evaluate models based on in-sample and holdout likelihoods, and fit to choice frequencies, while fit to choice dynamics is rarely examined. Since the PC model is as easy to estimate as the conventional models,5 we conclude it should be viewed as a serious alternative to MNL and NMNL.

2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

Keane (1997a) argued that conventional MNL and NMNL choice models (with or without no-purchase) could produce severely biased estimates of own and cross-price elasticities of demand, if inventory-planning behavior by consumers is important. To understand his argument, consider this simple example. Say there are two brands, A and B, and that consumers are totally loyal to either one or the other. The following table lists the number of consumers who buy A, B or make no-purchase over a 5 week span under two scenarios. First, a no promotion environment:

Week12345
A1010101010
B1010101010
No-purchase8080808080

And second a scenario where Brand A is on promotion (say a 20% price cut) in week 2:

Week12345
A10205510
B1010101010
No-purchase8070858580

Now, notice that brand A's market share goes from 50% to 67% in week 2. Brand B's market share drops to 33%, even though no consumer switches away from B. Thus, a brand choice model without a no-purchase option would conclude that the cross-price elasticity of demand is substantial (i.e., 34/.20 = 1.7), even though it is, in reality, zero.6

Interestingly, including a no-purchase option does not solve the basic problem. Notice there is a post-promotion dip, presumably arising from inventory behavior, that causes the number of people who buy A to drop by 50% in weeks 3 and 4, before returning to normal in week 5. If we took data from the whole 5-week period, we would see that Brand A's average sales when at its “regular” price are 7.5, increasing to 20.0 when it is on promotion. Thus, a conventional static choice model with a no purchase option would conclude the price elasticity of demand is (20.0/7.5 − 1)/(.20) = 8.3. The correct answer is that the short run elasticity is 5 and that the long run elasticity is zero, since all the extra sales come at the expense of future sales.

As Keane (1997a) notes, the exaggeration of the elasticity could be even greater if consumers are able to anticipate future sales. For example, suppose a retailer always puts brand A on sale in week 2 of every 5 week period. As consumers become aware of the pattern, they could concentrate their purchases in week 2, even if their price elasticity of demand were modest. Conventional choice models, however, would infer an enormous elasticity, since a modest 20% price cut causes sales to jump greatly in week two.

An important paper by Sun et al. (2003) shows that the problems with conventional choice models noted by Keane (1997a) are not merely academic. They conduct two experiments. First, they simulate data from a calibrated model where the hypothetical consumers engage in inventory-planning behavior. Thus, Sun et al. know (up to simulation noise) the true demand elasticities. They estimate a set of conventional choice models on this data; i.e., MNL with and without no-purchase, NMNL with no-purchase. Each model includes consumer taste heterogeneity and state dependence. The MNL without no-purchase exaggerates cross-price elasticities by about 100%, while MNL and NMNL with no-purchase do so by about 50%. A dynamic structural model of inventory-planning behavior fit to the same data produces accurate elasticity estimates—as expected since it is the “true” model that was used to simulate the data.

Second, Sun et al. estimate the same set of models on Nielsen scanner data for ketchup. Strikingly, they obtain the same pattern. Estimated cross-price elasticities from the MNL and NMNL models that include no-purchase are about 50% greater than those implied by the dynamic structural model that accounts for inventory-planning behavior. And those from the MNL without no-purchase are about 100% greater.

Thus, marketing research needs to confront the fact that own and cross-price elasticities of demand are much greater when estimated from pure brand choice models than from models that include a no purchase option. What accounts for this phenomenon? As we've seen, one explanation is the inventory-planning behavior that Keane (1997a) predicted would cause such a problem.7 In response, several authors, such as Erdem et al. (2003), Sun et al. (2003) and Hendel and Nevo (2004), have proposed abandoning conventional static choice models (i.e., MNL and NMNL) in favor of dynamic structural inventory-planning models. Of course, the difficultly with these models is that they are extremely difficult to estimate.

In this paper, we adopt a different course of action. We seek to develop a simple model—no more complex than MNL or NMNL—that does not suffer from the problems noted by Keane (1997) and Sun et al. (2003). The motivation for our approach is two fold: First, we conjecture that a brand choice model with a more flexible representation for category consideration/purchase incidence may provide a more accurate reduced form approximation to consumer's purchase decision rules than the conventional models like MNL and NMNL. Second, we note there is a simpler way (than estimation of complex dynamic inventory models) to reconcile the pattern of elasticities across models noted by Sun et al. (2003). The idea is simply that consumers do not choose to observe brand prices in each period.

More specifically, we propose a two-stage “price consideration” (PC) model in which a consumer decides, in each period, whether to consider buying in a category. This decision may be influenced by inventories, advertising, feature and display conditions. If a consumer decides to consider a category, he/she looks at prices and decides whether and what brand to buy.8 Such a model can generate a pattern where consumers are more sensitive to prices when choosing among brands than when deciding whether to buy in a category. The positive probability that consumers do not even consider the category (i.e., they do not even see prices) creates a wedge between the brand choice price elasticity and the purchase incidence price elasticity.

While this model is extremely simple, it has not, as far as we know, been used before in the marketing literature. It is important to note that the PC model is quite different from a nested multinomial logit (NMNL) model, where consumers first decide whether to buy in a category and then, in a second stage, decide which brand to buy. In the first stage of the NMNL, the decision whether to purchase in the category is a function of the inclusive value from the second stage, which is, in fact, a price index for the category. Thus, the NMNL model assumes that consumers see all prices in all periods. The second stage is different as well, because in our model the second stage includes a no purchase option. That is, if a consumer decides to consider a category, he/she may still decide, upon seeing prices, to choose no-purchase. A NMNL model can certainly generate a pattern that brand choice price elasticities substantially exceed purchase incidence price elasticities, but it would do so by assuming that brands are very similar. As noted above, such an assumption is inconsistent with data on brand switching behavior.

There are a number of ways we can test our model against alternative models in the literature. First, we can simply ask whether it fits better than simple MNL and NMNL models that include a no purchase option. Second, our model can be distinguished from the alternative dynamic inventory story for the same phenomenon by looking at categories where inventories are not important. If brand choice price elasticities exceed purchase incidence price elasticities even for non-storable goods, it favors the simple story.

3. THE PRICE CONSIDERATION (PC) MODEL

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

3.1. Basic Structure and Properties of the PC model, and Comparison to MNL and NMNL

The simplest version of the PC model takes the following form: Consider a category with J brands. In each time period t, prior to seeing prices, a consumer decides whether to consider the category. Let PCt denote the probability the consumer considers the category in week t. If the consumer decides to consider the category, then he/she looks at prices, and a MNL model with a no-purchase option governs choice behavior. Let Ujt = αj − βpjt + ejt denote utility of purchasing brand j at time t, where ejt is an extreme value error. Then, letting Pt(j | C) denote the probability, conditional on considering the category, that the consumer chooses brand j at time t, we have:

  • equation image(1)

Let option J + 1 be no-purchase. Normalizing the deterministic part of its utility to zero, we have:

  • equation image(2)

Then, the unconditional probability that the consumer buys brand j at time t is:

  • equation image(3)

and the unconditional probability of no purchase is:

  • equation image(4)

The derivative of the log odds ratio between brands j and k with respect to price of j is simply:

  • equation image(5)

which is identical to what one obtains in the MNL model, because the PCt cancel out. However, the derivative of the log odds ratio between brand j and the no purchase option J + 1 is:

  • equation image(6)

This expression lies in the interval (−β, 0), approaching − β as PCt[UPWARDS ARROW]1. Thus, if PCt < 1, our model implies a greater price elasticity of the brand choice log odds than that for purchase incidence.

Comparison of (5) and (6) reveals the intuition for identification of PCt in our model. (5) implies that the independence of irrelevant alternatives (IIA) property holds among brands, but (6) shows that it does not hold between brands and the no purchase option. Thus, this type of departure from IIA can be explained by our model with PCt < 1, but not by a standard MNL model with a no purchase option. Thus, the PC model is more flexible than the MNL in terms of how price can affect purchase incidence vs. choice amongst brands.

Next, we examine how price elasticities for purchase incidence vs. brand choice differ between our model and a nested logit model (see Maddala, 1983, p. 70). The difference here is more subtle, because a NMNL with a first stage category purchase decision, followed by a second stage brand choice decision, can also generate brand choice price elasticities that exceed purchase incidence price elasticities. What is required is that the coefficient on the inclusive value be less than one in the first stage.

In the nested logit we have:

  • equation image(7)

and, defining the inclusive value as:

  • equation image(8)

we have:

  • equation image(9)

where 1 − ρ is (approximately) the correlation among the extreme value error terms in the second stage (see McFadden, 1978). Thus, the unconditional probability of purchase for brand j is:

  • equation image(10)

This gives that:

  • equation image(11)

and therefore:

  • equation image(12)

Thus, by setting ρ< 1, the nested logit also allows the elasticity of the purchase incidence log odds with respect to price to be less than the elasticity of the brand choice log odds.

However, there is a crucial difference between how the NMNL and PC models achieve this divergence of purchase vs. brand choice elasticities. By setting ρ well below 1, the NMNL model forces brands to be close substitutes, implying frequent brand switching by individual consumers. Intuitively, as ρ[DOWNWARDS ARROW]0, the scale of β must increase to maintain any given sensitivity of total category sales to price. But as the scale of β increases, Pt(j|Buy) approaches a step function, equal to 1 if j is the lowest priced brand and zero otherwise. In contrast, the PC model can generate a large divergence between the price sensitivities of purchase incidence and brand choice, without requiring that brands be close substitutes—making it more flexible.9

The simple price consideration model described in (1)-(6) can be elaborated in obvious ways. We can allow for consumer heterogeneity in the brand intercepts {αj}, and we can also let the category consideration probability PCt be a function of feature and display indicators, ad exposures, and time since last purchase. If there is a great deal of taste heterogeneity, in the form of very different α vectors across consumers, this model can generate simultaneously the patterns that (i) brand switching is infrequent and that (ii) price elasticities of demand appear to be much greater when estimated from pure brand choice models than from either purchase incidence models or choice models that include a no purchase option.

It is also interesting to examine the relation between the NMNL and PC models when there is consumer taste heterogeneity. This is difficult to do in general, but to get some intuition for this case, consider a consumer who so strongly prefers a single brand, let's say brand 1, that his/her probability of buying any other brand is negligible. In most frequently purchased categories, loyal consumers like this appear to comprise a substantial share of the population. For such a consumer, our model implies that:

  • equation image(13)

while the nested logit gives:

  • equation image(14)
  • equation image(15)

It is obvious that, in the nested logit model, it is not possible to separately identify ρ from α and β if all consumers are of this type. The price consideration model does not have the same problem, since there is a loss of generality in setting PCt = 1 in equation (3). To see this, suppose that we try to find parameter values α1′ and β′ such that:

  • equation image(16)

for all values of p1t. If there are at least 3 observed price levels in the data, then this gives three equations in only two unknowns and a solution will not generally be possible.

Intuitively, a model with a fairly small value of PCt can generate sales approaching zero for moderately high prices, without simultaneously requiring that purchase probabilities become very large at low prices. In other words, when setting PCt = 1, we have that the log odds is a linear function of price:

  • equation image(17)

while the PC model generalizes this to:

  • equation image(18)

giving:

  • equation image(19)

Finally, it is important to note that NMNL also restricts the log odds to be a linear function of price. In the nested logit we have:

  • equation image(20)

Thus, in the PC model the log odds ratio is a more flexible function of price than in the NMNL.

This result does not only arise for loyal consumers. Returning to the case where all brands are considered in the second stage, we can derive analogous expressions. We consider the derivative of the log odds of purchase vs. no purchase with respect to a constant shift of the whole price vector, which we denote by ∂[·]/∂. For the price consideration model we have:

  • equation image(21)

while for the nested logit we have:

  • equation image(22)

Again, the first expression is a nonlinear function of prices if PCt < 1, while the later does not depend on prices. Another interesting distinction between the two models is that the former expression approaches − β[1 − Pt(Buy|C)]< 0 as PCt[DOWNWARDS ARROW]0, while the latter approaches 0 as ρ[DOWNWARDS ARROW]0.

In summary, we have described several ways that the PC model is more flexible than either MNL or NMNL. In the empirical section, we'll see that this added flexibility is empirically relevant - the PC model provides a significantly better fit to choice data from two product categories (i.e., ketchup and peanut butter) than do the conventional models.

3.2. Econometric Model Specification for the PC Model

Now we turn to our detailed specification of the PC model. In week t, consumer i's probability of considering a category depends on a vector of category promotional activity variables (Xct), household size (memi) and time since last purchase (purch_gapit). Specifically, let

  • equation image(23)

We let Xct include indicators for whether any brand in the category is on feature or display, the idea being that these promotional activities may draw consumers' attention to the category. We include household size (memi) and time since last purchase (purch_gapit) to capture inventory effects: a longer time since last purchase means the consumer is more likely to have run out of inventory, and hence more likely to consider the category, especially if household size is large.10 γc, γmem and γpg are the associated coefficients of Xct, memi and purch_gapit, respectively.

Finally, γi0 is a random coefficient that captures unobserved consumer heterogeneity in the likelihood of considering a category. This may arise because some consumers/households have higher usage rates than others. We assume that γi0 is normally distributed.

Now we turn to the second stage, where a consumer has decided to consider (but not necessarily buy) the category. Let Uijt denote utility to consumer i of purchasing brand j at time t. We allow this utility to depend on observed and unobserved characteristics of the consumer, and interactions among consumer and brand characteristics. Specifically, for j=1, …, J, let:

  • equation image(24)

For j=J+1 (i.e., the no purchase option), let:

  • equation image(25)

The αij for j = 1, …, J are a vector of brand intercepts that capture consumer i's tastes for the unobserved attributes of brand j. As utility is measured relative to the no purchase option, an intercept in (25) is not identified. Xjt is a vector of observed attributes of brand j at time t, and β is a corresponding vector of utility weights. pjt is price of brand j at time t. We allow for observed heterogeneity in the marginal utility of consumption of the outside good. Thus, Zit is a vector of observed characteristics of consumer i at time t, and the price coefficient is given by φp + Zitφ1.

The term GL(Hijt, δ) in (24) is the “brand loyalty” or state dependence variable defined by Guadagni and Little (1983) to capture the idea that a consumer who bought a brand frequently in the past is likely to buy it again. Here, Hijt is consumer i's purchase history for brand j prior to time t, and δ is the exponential smoothing parameter; λ is the coefficient mapping GL into the evaluation of utility. Thus, we have GLijt = δGLij, t−1 + (1 − δ)dij, t−1, where dij, t−1 is the indicator function which equals one if consumer i bought brand j at t − 1, and zero otherwise.11

To capture possible correlation of consumer tastes among brands, the distribution of the vector αi is assumed to be multivariate normal. Finally, eijt is an i.i.d. extreme value error term that captures the idiosyncratic taste of consumer i for brand j at time t. Thus, (24)-(25) is what is known as a “heterogeneous” MNL model (see, e.g., Harris and Keane, 1999) that allows for both heterogeneity and state dependence in choice behavior.

3.3. Specification of the Alternative MNL and NMNL Models

We also estimate MNL and NMNL models in order to compare their fit to that of the PC model. The specification of MNL is essentially the same as the price consideration model except that (i) the probability of considering a category is assumed to be 1; (ii) the utility of choosing the no-purchase option is assumed to be,

  • equation image(26)

Here, we let the value of the no-purchase option depend on the same category feature, display, household size and purchase gap variables that influence the decision to consider the category in the PC model. This is critical in order to allow a fair comparison between the two models. As in (25), the intercept in (26) is normalized to zero for identification.

In NMNL, the utility of choosing the no purchase option is also given by equation (26). Thus, the conditional choice probability in the second stage is given by:

  • equation image(27)

The inclusive value conditioning on (αi1, …, αiJ) now becomes:

  • equation image(28)

The probability of buying in a category is therefore given by,

  • equation image(29)

where 0 < ρ< 1. Notice that if ρ = 1, the NMNL model is equivalent to MNL.

3.4. Estimation Issues

We face an initial conditions problem as we do not observe consumers' choice histories prior to t = 1 (see Heckman, 1981). This creates two problems: (1) we cannot construct the initial values of GL(Hij1, δ) and purch_gapi1, and (2), even if we observed these variables they would be correlated with the brand intercepts αij and the usage rate parameter γi0. We deal with this problem in two ways: First, we hold out n weeks of individual choice histories from the estimation sample and use them to impute the initial value of GL(Hij1, δ) and purch_gapi1 for all consumers.12 Second, we integrate over the joint distribution of GL(Hij1, δ), purch_gapi1, αi and γi0 using a computationally convenient procedure proposed by Wooldridge (2003a, b).

In this approach, the distributions of the unobserved heterogeneity terms are allowed to be functions of the initial values of GL and purch_gap, as follows; in all three models we have:

  • equation image(30)

where εij is multivariate normal with mean zero, and, in the PC model only, for γi0, we also have:

  • equation image(31)

where υi is normal with mean zero and variance equation image.

Next, consider construction of the likelihood function. There is no closed form for the choice probabilities, since αi is a J·1 multivariate normal vector, and γi0 is normally distributed. Thus, the probability of a consumer's choice history is a J + 1 dimensional integral. For the categories we consider, J = 4. We have data on many consumers, so many such integrals must be evaluated, rendering traditional numerical integration infeasible. Instead, we use simulated maximum likelihood, in which Monte Carlo methods are used to simulate the integrals (see, e.g., McFadden, 1989; Pakes, 1986; Keane, 1993, 1994). We use 200 draws for (αi, γi0).

4. CONSTRUCTION OF THE DATA SETS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

We use the Nielsen scanner panel data on ketchup and peanut butter for Sioux Falls, SD and Springfield, MO. The sample period begins in week 25 of 1986 for both categories. It ends in week 34 of 1988 for ketchup, and in week 23 of 1987 for peanut butter. The ketchup category has 3189 households, 114 weeks, 324,795 store visits, and 24,544 purchases, while peanut butter has 7924 households, 51 weeks, 258,136 store visits, and 31,165 purchases.13

During this period, there were four major brands in ketchup: Heinz, Hunt's, Del Monte, and the Store Brand; four major brands in peanut butter: Skippy, JIF, Peter Pan and the Store Brand. There are also some minor brands with very small market shares, and we dropped households that bought these brands from the sample.14 As a result, we lose 558 (out of 7924) households in peanut butter, and 101 (out of 3189) households in ketchup. The number of store visits is reduced to 236,351 in peanut butter, and 314,417 in ketchup. To impute the initial value of GL and purch_gap, we use the first 10 weeks of individuals' choice history for peanut butter, and 20 weeks for ketchup. This reduced the number of store visits in the data to 175,675 and 259,310 for peanut butter and ketchup, respectively.

Household characteristics included in Zit are household income (inci) and household size (memi). Attributes of alternatives included in Xjt are an indicator for whether the brand is on display (displayjt), whether it is a featured item (featurejt), and a measure of coupon availability (coupon_avjt). Attributes of the category included in Xct are a dummy for whether at least one of the brands is on display (Idt), and a dummy for whether one of the brands is a featured item (Ift).

A difficulty arises in forming the price variable because we model only purchase timing and brand choice, but not quantity choice. Yet each brand offers more than one package size, and price per ounce varies across package sizes. We need to have the price variable be on an equal footing across brands and weeks.15 Thus, we decided to always use the price of the most common size package when estimating our model - 32 oz for ketchup, 18oz for peanut butter. Admittedly, this introduces some measurement error into prices, but the problem cannot be resolved by a different construction of the price variable—only by introducing quantity choice.

Another problem in constructing the price and promotion variables is that in scanner data we only observe the price paid by the consumer for the brand he/she actually bought. Similarly, we only observe whether a brand is on display or feature when the consumer chooses the brand. Therefore, prices, displayj and featurej for other brands and weeks must be inferred. Erdem et al. (1999) discuss this “missing prices” problem in detail.

To deal with this problem, we use the algorithm described in Keane (1997b). It works as follows: (1) Sort through all the data for a particular store on a particular day. If a consumer is found who bought a particular brand, then use the marked price he/she faced as the marked price for that brand in that store on that day. (2) If no one bought a particular brand in a particular store on a particular day, use the average marked price in that store in that week to fill in the price. (3) If no one bought a particular brand in a particular store in a particular week, then use the average marked price of the brand in that store over the whole sample period to fill in the price.

The missing displayj and featurej variables are inferred in a similar fashion. Of course, the observed displayj's and featurej's are dummies equal to 0 or 1. However, since we may use weekly average values or store average values to fill in the missing displayj's and featurej's, some of them end up falling between 0 and 1 in our data set.

Next we turn to our coupon variable. Keane (1997b) and Erdem et al. (1999) discuss the extremely severe endogeneity problem—leading to extreme upward bias in price elasticities—created by use of price net of redeemed coupons as the price variable. Instead, we use a brand/week specific measure of coupon availability, coupon_avjt. Basically, this variable measures the average level of coupon redemption for a particular brand in a particular week. The algorithm for constructing it, which is rather involved, is also described in Keane (1997b).

A final problem is that a small percentage of observations have unreasonably high prices, presumably due to measurement/coding errors. So we created a maximum “plausible” price in each category—$ 3 for peanut butter and $ 2 ketchup (in 1985-88 nominal dollars)—and replaced prices above this maximum with the mean observed price of the brand. This procedure affected 2% of the observations for peanut butter, and 1.4% of the observations for ketchup.

Summary statistics for households are given in Table I, while those for brands are in Table II. Notice that the two categories are rather different. In peanut butter, the four brands have similar market shares (ranging from 29.9% for Skippy to 19.8% for Peter Pan), while in ketchup Heinz is dominant (with a market share of 63.1%). Households buy peanut butter and ketchup on 11.66% and 7.35% of shopping occasions, respectively. These figures give face validity to the idea of the PC model: when purchase frequency in a category is this low, it seems unlikely that households would check on peanut butter and ketchup prices in every week.16

Table I. Summary statistics of the data and household characteristics
 Peanut ButterKetchup
  • *

    Household income ranges from 1 to 14: 1 = less than $ 5,000; 2 = $5, 000–9, 999; 3 = 10, 000–14, 999; …; 10 = $45, 000–49, 999; 11 = $50, 000–59, 999; 12 = $60, 000–74, 999; 13 = $75, 000–99, 000; 14 = $100, 000 or more.

#Households7,3663,088
Max #weeks observed per household4194
#Store visits175,675259,310
#Purchases20,47819,044
Average household income*5.945.99
Average household size2.842.73
Table II. Summary statistics of product characteristics
Peanut Butter
AlternativeNo purchaseSkippyJIFPeter PanStore Brand
#observations155,1976,1374,8674,0615,413
share (%)88.343.492.772.313.08
mean(pjt)n.a.1.8421.9171.8871.366
mean(featurejt)n.a.0.02110.00330.03800.0298
mean(displayjt)n.a.0.00360.00380.03170.0085
mean(coupon_avjt)n.a.0.10300.0630.2150.0025
Ketchup
AlternativeNo purchaseHeinzHunt'sDel MonteStore Brand
#observations240,26612,0423,2121,5282,262
share (%)92.654.641.240.590.87
mean(pjt)n.a.1.1511.1461.0860.874
mean(featurejt)n.a.0.04010.04330.05830.0236
mean(displayjt)n.a.0.02820.03290.02660.0183
mean(coupon_avjt)n.a.0.12460.08350.02400.0045
 Peanut ButterKetchup
mean(Ift)0.09040.1602
mean(Idt)0.04630.1027

5. ESTIMATION RESULTS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

Table III presents results obtained by estimating the multinomial logit model (MNL), nested multinomial logit model (NMNL) and price consideration model (PC), using data for the peanut butter category. We estimate two versions of the PC model. In PC-I, decisions to consider the category depend on category feature and display indicators, the purchase-gap, and household size, while in PC-II these factors influence the utility of the no- purchase option as well.17

Table III. Estimates for the Peanut Butter Category
 MNLNMNLPC IPC II
 Estimates.eEstimates.e.Estimates.e.Estimates.e.
α1 (Store Brand)− 5.4650.148− 6.6730.212− 3.7910.078− 4.9720.158
α2 (JIF)− 5.3090.159− 6.4740.220− 3.5910.097− 4.8040.173
α3 (Peter Pan)− 6.0030.161− 7.2170.224− 4.3170.102− 5.4990.175
α4 (Skippy)− 5.1150.157− 6.2510.216− 3.3870.094− 4.5960.169
αinit, GL106.1706.061108.4715.55823.5481.70225.0022.223
αinit, pg− 0.0450.004− 0.0820.007− 0.0410.004− 0.0370.005
βd(displayjt)1.3770.0541.4710.0561.7280.0481.4140.055
βf(featurejt)2.0610.0392.2010.0422.2590.0322.0840.040
βc(coupon_avjt)1.5900.1571.9860.1721.6050.1641.5990.166
φp(pjt)− 0.2390.085− 0.3460.092− 0.8460.050− 0.2260.090
φinc(pjt·inci)0.0170.0030.0240.0030.0180.0030.0180.003
φmem(pjt·memi)− 0.0470.022− 0.0710.0240.1120.006− 0.0710.023
State dependence: GLijt = δ*GLijt−1 + (1 − δ)*dijt−1
λ(GLijt)− 1.8690.584− 0.4510.7233.1790.3102.6920.320
δ0.9890.0010.9890.0010.9490.0040.9530.005
Utility of no purchase:
βfc(Ift)− 0.3910.030− 0.5790.034  − 0.2890.036
βdc(Idt)− 0.6080.038− 0.7120.038  − 0.4930.046
βmem(memi)− 0.3060.039− 0.3090.031  − 0.3300.041
βpg(purch_gapit)− 0.0230.002− 0.0190.001  0.0010.002
η  − 0.8550.122    
ρ( = 1/(1 + exp(η)))  0.702     
Probability of considering a category: Pit(C) = exp(Lit)/(1 + exp(Lit)) where: Lit = γi0 + γf·Ift + γd·Idt + γmem·memi
+ γpg·purch_gapit, and γi0 = γ0 + γinitial·purch_gapi1 + υi
γ0    − 1.5940.140− 1.5920.162
γinitial    − 0.0290.012− 0.0420.013
γmem    0.1180.0300.1210.033
γf    1.3760.1021.0150.107
γd    1.7480.1831.0450.157
γpg    0.7170.0350.8680.055
συ    1.1630.0551.3120.088
log-likelihood74596.5074546.88273983.1173869.12
2(log-likelihood)149193.00149093.764147966.23147738.23
AIC149249.00149151.764148028.23147808.23
BIC149539.45149452.583148349.79148171.29

One striking aspect of the results is that coefficient αinit, GL from equation (30), that captures how the pre-sample initialized values of the GL variables are related to the brand intercepts, is very large and significant in all the models.18 Thus, the pre-sample purchase behavior conveys a great deal of information about brand preferences.

Strikingly, there is no evidence for state dependence in the MNL and NMNL models. The coefficient on GL is the wrong sign (and insignificant in NMNL), and the parameter δ is close to one, implying a brand purchase hardly moves GL. The PC-I and PC-II models, on the other hand, imply significant state dependence. The coefficients on GL are 2.7 to 3.2, and the estimates imply a value of (1 − δ), the coefficient on the purchase dummy, of around. 05. Thus, e.g., in PC-II a lagged purchase raises the next period utility for buying a brand by 0.127.

All four models imply similar average price coefficients. The PC-I and PC-II models imply that, for the “average” household the price coefficients are −.421 and −.321, respectively. Thus, e.g., in PC-II, the effect of a lagged brand purchase on current utility from the brand is equivalent to the effect of a .127/.321 = 39 cent price reduction. Mean prices are in the $ 1.36 to $ 1.92 range, so this is a substantial state dependence effect.

Part of why the PC models imply substantial state dependence while MNL and NMNL do not is that the coefficient αinit, GL, while still substantial, is smaller in the PC models by a factor of 4. Thus, the MNL and NMNL models ascribe all the pre-sample differences in purchase behavior to heterogeneity, while the PC models do not.

As we would expect, the display, feature and coupon availability variables all have large and highly significant positive coefficients in the brand specific utility functions in all four models. In the MNL and NMNL models, the utility of the no-purchase option depends negatively on the category feature and display indicators, duration since last purchase in the category (purch_gap), and on household size. The latter two findings are consistent with inventory behavior. Analogously, in the PC model, the probability of considering the category depends positively on the category feature and display indicators, duration since last purchase in the category, and household size. All these effects are highly significant.

It is useful to examine what the PC models imply about the probability of considering the category under different circumstances. Consider a baseline situation with no brand on display or feature, and a household of size 3 that just bought last period (i.e., purch_gapit = 1).19 Then, the PC-II model estimates imply the probability of considering the peanut butter category is 39.7%, on average.20 The estimates of γf and γd are 1.015 and 1.045, respectively. This implies that the consideration probability increases to 75.6% if one or more brands is on display and feature.21 [Note that, in peanut butter, the category display and feature indicators equal 1 in 4.63% and 9.04% of weeks, respectively]. Finally, the estimate of γpg is 0.868. This implies that, starting from the baseline, if we increase the purchase gap to 5 weeks, the probability of considering the category increases to 90.7%.22

The PC-II model implies that, even conditional on having decided to consider the category, the category feature and display indicators, and household size, have significant negative effects on utility of no-purchase. The log-likelihood improvement from PC-I to PC-II is 114 points, while AIC and BIC improve 220 and 179 points, respectively. This suggests feature and display have some influence on brand choice beyond just drawing attention to the category.23

Finally, we turn to our main goal, which is to investigate whether the PC model fits the data better than MNL and NMNL. The bottom panel of Table III presents log-likelihood, AIC and BIC values for the four models. The likelihood value for NMNL is modestly better than that for MNL (i.e., 50 points). The ρ on the inclusive value is 0.70, which implies that brands are closer substitutes than suggested by the MNL model (see equation (9), and recall that if ρ = 1.0 the models are equivalent). As we discussed in Section 2 and 3.1, NMNL can generate brand choice price elasticities exceeding those for purchase incidence by assuming brands are similar.

However, the log-likelihoods for the PC-I and PC-II models are superior to NMNL by 563.8 points and 677.8 points, respectively. The AIC and BIC produce very similar comparisons. This is to be expected as the models have similar numbers of parameters (i.e., the MNL, NMNL, PCI-I and PC-II models have 28, 29, 31 and 35 parameters, respectively). Thus, the PC models clearly produce better fits to the peanut butter data than the MNL and NMNL models.

Table IV presents estimates for the ketchup category. Here the estimate of ρ in the NMNL model is essentially 1, so MNL and NMNL are essentially equivalent. Hence, the parameters estimates and likelihood values for MNL and NMNL are essentially the same. The qualitative results for ketchup are quite similar to those for peanut butter. One small difference is that, in ketchup, MNL and NMNL do generate a positive coefficient λ on the GL variable (about 1.3), but it is not statistically significant. Furthermore, the state dependence implied by the point estimates is very weak. The estimates imply a value of (1 − δ), the coefficient on the purchase dummy, of around 0.005. Thus, a lagged purchase raises the next period utility for buying a brand by only about 0.007. As the price coefficient at the mean of the data is −.797, the effect of a lagged purchase on the current period utility evaluation for a brand is equivalent to a price cut of less than 1 cent.

Table IV. Estimates for the Ketchup Category
 MNLNMNLPC IPC II
 Estimates.eEstimates.eEstimates.eEstimates.e.
α1 (Heinz)− 3.9020.170− 3.9020.176− 2.8600.084− 3.6930.181
α2 (Hunt's)− 5.2510.171− 5.2520.176− 4.0840.087− 4.9150.182
α3 (Del Monte)− 6.1300.173− 6.1290.181− 4.9590.095− 5.7860.187
α4 (Store Brand)− 5.9660.166− 5.9690.174− 4.7990.082− 5.5930.182
αinit, GL86.1057.69586.2454.26916.7201.92116.3271.013
αinit, pg− 0.0370.002− 0.0370.002− 0.0300.002− 0.0310.003
βd(displayjt)1.0970.0411.0970.0421.1280.0321.1370.042
βf(featurejt)2.2380.0322.2380.0322.1360.0242.2960.034
βc(coupon_avjt)1.5830.1051.5830.1071.8350.1111.8360.111
φp(pjt)− 1.0010.148− 1.0000.154− 1.6800.080− 0.9630.154
φinc(pjt·inci)0.0090.0050.0090.0050.0060.0050.0040.005
φmem(pjt·memi)0.0550.0410.0550.0420.2490.0120.0370.042
State dependence: GLijt = δ*GLijt−1 + (1 − δ)*dijt−1
λ(GLijt)1.3380.8001.3460.9704.2350.4534.3230.438
δ0.9950.00050.9950.00030.9730.0040.9720.002
Utility of no purchase:
βfc(Ift)− 0.0370.029− 0.0370.034  0.2440.035
βdc(Idt)− 0.1050.034− 0.1050.038  0.0320.040
βmem(memi)− 0.2170.045− 0.2170.031  − 0.2590.048
βpg(purch_gapit)− 0.0110.001− 0.0110.001  − 0.0020.001
η  − 10.2120.632    
ρ( = 1/(1 + exp(η)))  0.999     
Probability of considering a category: Pit(C) = exp(Lit)/(1 + exp(Lit)), where: Lit = γi0 + γf·Ift + γd·Idt + γmem·memi
+ γpg·purch_gapit, and γi0 = γ0 + γinitial·purch_gapi1 + υi
γ0    − 1.0430.163− 0.9370.157
γinitial    − 0.0150.008− 0.0090.008
γmem    − 0.0360.035− 0.0700.033
γf    1.5270.1071.9250.161
γd    1.0350.1411.2740.182
γpg    0.4750.0280.4210.025
συ    0.8660.1140.7160.107
log-likelihood71945.20471945.19671357.80471313.949
− 2(log-likelihood)143890.408143890.393142715.608142627.899
AIC143946.408143948.393142777.608142697.899
BIC144244.845144257.489143108.021143070.945

As with peanut butter, the PC models again imply much stronger state dependence. The price coefficient for an “average” household is −.964 for PC-I and −.838 for PC-II, which is similar to the values produced by the logit models. But the PC models generate values of λ·(1 − δ) of about (4.3)(.028) = .12, so a lagged purchase is comparable to about a .12/.84 = 14 cent price cut. Mean prices are $ 0.87 to $ 1.15, so this is a substantial effect.24

Another difference is that category feature and display variables have positive effects on the value of no-purchase in the PC-II model, whereas in peanut butter they had negative effects. Thus, in ketchup, feature and display make a consumer more likely to consider the category, but, conditional on consideration, they make the consumer less likely to actually buy. In peanut butter, they made both consideration and conditional purchase probabilities higher.

To get a sense of how likely a household is to consider ketchup during a store visit, take a situation with no brand on display or feature, and a household of size 3 that bought last period.25 The estimates of PC-II imply the probability the household considers buying ketchup is 33.4%, on average. This increases to 90.2% if one or more brands is on display and feature, a larger effect than in peanut butter. [Note that category display and feature indicators equal 1 in 10.3% and 16.0% of weeks, respectively]. Finally, if we increase the purchase gap to 5 weeks, the consideration probability increases from 33.4% to 69.4%, a smaller effect than in peanut butter.

Turning to the issue of model fit, the MNL and NMNL produce essentially identical log-likelihoods, for the reason noted earlier. The PC-I model is superior by 587.4 points, and the PC-II model is superior by 631.3 points. Again, the PC-I and PC-II models have only 2 and 6 more parameters that NMNL, so the AIC and BIC tell a very similar story. Thus, the PC model also produces a clearly better fit to the ketchup data than do the MNL and NMNL models.

6. MODEL SIMULATIONS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

6A. Model Simulations: Inter-Purchase Spells

We found in section 5 that the PC models fit better than MNL and NMNL based on likelihood and information criteria (AIC, BIC). This fact alone provides no insight into what aspect(s) of the data are better captured by the PC models. Simulation of the models can shed light on this issue. Thus, we simulated several million hypothetical households from each model. To do this, we simulated 2000 artificial households for each household in the data, using the observed characteristics and forcing variables (price, display and feature) for that household.

Table V reports the fit of each model to unconditional choice frequencies. Here, there is little to choose from amongst the models, as all seem to fit choice frequencies well. Interestingly, in the marketing literature it is unusual to report any aspect of the fit of brand choice models other than the likelihood, information criteria and unconditional choice frequencies.26 Rarely are other aspects of fit, like conditional choice frequencies or inter-purchase times, reported. We now turn to such measures.

Table V. Model Fit: Simulated vs. Actual Brand Choice Frequencies
Peanut Butter
 No purchaseSkippyJIFPeter PanStore Brand
  1. Note: The baseline simulation takes the price and promotion variables in the data as given. We simulate 2,000 hypothetical consumer choice histories based on the characteristics of each sample household, as well as the price and promotion history the household faced. Thus, we simulate 7, 366 × 2, 000 hypothetical choice histories for peanut butter, and 3, 088 × 2, 000 for ketchup.

Data0.88340.03490.02770.02310.0308
MNL0.87610.03710.02900.02600.0318
Nested MNL0.87680.03660.02890.02600.0317
PC I0.87770.03630.02840.02630.0313
PC II0.87800.03620.02810.02600.0316
Ketchup
 No purchaseHeinzHuntsDel MonteStore Brand
Data0.92660.04640.01240.00590.0087
MNL/NMNL0.92350.04940.01270.00620.0082
PC I0.92340.04910.01280.00620.0085
PC II0.92370.04900.01280.00620.0084

Table VI reports choice frequencies conditional on choice in the previous period. Here, the superior fit of the PC models becomes quite apparent. For example, in the data, the probability a household buys Skippy at t given that it bought it at t − 1 is 12.5%. The MNL and NMNL put this probability at about 24% to 25% - roughly double the true value. The PC models also predict too high a frequency, 16.6% to 16.9%, but this is only one-third too high, a much smaller error.27 The story is similar for ketchup. Thus, we see that a key problem with the MNL and NMNL models is that they exaggerate probabilities of very short (i.e., one period) inter-purchase spells.

Table VI. Choice Probabilities Conditional on Lagged Choice: Model vs. Data
Peanut Butter
 Choice at time t − 1
SkippyJIFPeter PanStore BrandAny brand
  1. Note: See Table V.

Data0.12520.14560.11060.14530.1325
MNL0.25310.26600.23390.25530.2527
Nested MNL0.24010.25170.21960.24350.2394
PC I0.16870.17820.16720.16640.1700
PC II0.16620.17540.16250.17130.1689
Ketchup
 Choice at time t − 1
 HeinzHunt'sDel MonteStore BrandAny brand
Data0.08540.08050.10850.08010.0858
MNL/Nested MNL0.17680.14610.13860.14430.1651
PC I0.11660.09390.08960.09580.1083
PC II0.11610.09570.09250.09720.1087

Figure 1 shows that PC model is far superior to MNL and NMNL in fitting the overall inter-purchase spell distribution. In peanut butter, the logit models roughly double the true frequency of one period spells, as already noted. They get the frequency of two period spells about right, and then underestimate the frequency of 3 through 9 period spells. In contrast, while the PC models slightly exaggerate the probability of one period spells (as already noted), for periods 2 onward their predicted spell frequencies are essentially indistinguishable from those in the data. As we see in the bottom panel of Figure 1, the story is very similar for ketchup.

thumbnail image

Figure 1. Inter-Purchase Time Distribution. This figure is available in color online at www.interscience.wiley.com/journal/jae

Download figure to PowerPoint

Figure 2 reports hazard rates, which make clear some additional interesting patterns. Essentially, we see that the MNL and NMNL models completely miss the fact that the hazard rate is low immediately after purchase, rises for a few weeks to a peak at about week 4, and then gradually declines. Instead, they predict that the hazard rate is falling throughout. In contrast, the PC models do capture the basic pattern, although, as noted earlier, they predict too high a purchase hazard in the first period after purchase.

thumbnail image

Figure 2. Purchase Hazard. This figure is available in color online at www.interscience.wiley.com/journal/jae

Download figure to PowerPoint

These results are consistent with our intuition for why the PC model should fit better than the MNL and NMNL logit models, discussed in Sections 1 and 2. The latter models are constrained to have a close link between the sensitivity of brand choice to price/promotion and the sensitivity of category purchase incidence to price/promotion. The PC models break this link, allowing it to provide a much better fit in inter-purchase spells while still fitting brand choices.28

Specifically, we see that the PC model does not exaggerate the frequency of short inter-purchase spells nearly so badly as the MNL and NMNL models. Often, deals/promotions last for more than one period. The logit models have difficultly explaining why a consumer would not then be likely to buy in consecutive periods, unless they are rather insensitive to price/promotion in general.29 The PC model can avoid this problem by saying a consumer is likely to not even consider a category a week after having just bought.

Finally, an interesting question is whether the PC model outperforms the MNL/NMNL models only in the aggregate or also in sub-samples. Looking at sub-samples is also a useful way to uncover evidence of misspecification. Thus, we looked at subgroups based on city (Sioux Falls vs. Springfield), income (above or below $ 40,000), household size (above or below 3), and education of the household head (high-school, some college or college). It was clear that the PC model beat the MNL/NMNL models in every subgroup, both in terms of the likelihood/AIC/BIC and the fit to the purchase hazard, particularly for very short spells. This was true for both the peanut butter and ketchup categories. We do not report these results in detail in the interest of space, but they are available upon request.

6B. Model Simulations: Promotion Effects

In this section we look at the models' predictions for the impact of promotions. Table VII reports the impact of a temporary 10% price cut for Skippy, and compares the predictions of the NMNL and PC models.30 As price cuts are often accompanied by feature/display activity, we also turned on 0.5% of the display dummies and 2.9% of the feature dummies, as described in detail in the footnote to the table.

Table VII. Peanut Butter: Effects of Temporary 10% Price Decrease for Skippy
Nested MNL
WeekSkippyJIFPeter PanStore BrandTotal
  1. Note: The table reports the percentage change in purchase frequencies for each brand by week, following a temporary 10% price cut for Skippy in week 34. Thus “Week 1” of the simulation is actually week 34 in the data.

  2. We choose week 34 as the base period for simulation because there was no promotion at all (i.e., Skippy's display and feature dummies equal zero for all observations). But price cuts are often accompanied by display and/or feature. To simulate this, we use the following procedure: First, we calculated the frequency with which display and feature are 1 when Skippy is on “deal.” These figures were 0.0052 and 0.02895, respectively. Then, in the simulation, we set Skippy's display dummy to 1 with probability 0.0052, and its feature dummy to 1 with probability 0.02895, in week 34.

  3. When we set the brand specific display (feature) dummy to 1, we also adjust the categorical level display (feature) dummy accordingly (setting it to 1 if it had been 0 in the data). In week 34, the average frequency of the categorical display and feature dummies in the baseline are 0.0631 and 0.0265, respectively; their average frequencies in the simulation are 0.0681 and 0.0548, respectively. Their average frequencies for the whole sample are 0.0469 and 0.0905, respectively.

121.2841− 0.2710− 0.31320.07925.7300
2− 0.1458− 0.0969− 0.31330.1314− 0.1353
3− 0.33900.0184− 0.3173− 0.1583− 0.1862
40.10210.1595− 0.23670.09590.0564
50.3194− 0.1401− 0.73850.1823− 0.0183
6− 0.1932− 0.07660.33840.20070.0237
70.1253− 0.10740.3088− 0.18530.0582
8− 0.21280.1744− 0.1635− 0.1546− 0.0967
90.2779− 0.2627− 0.4009− 0.2207− 0.1168
100.20640.37190.10100.06860.1865
Price Consideration II
 SkippyJIFPeter PanStore BrandTotal
120.42850.21980.17980.34395.8103
2− 0.2137− 0.5287− 0.3159− 0.4538− 0.3646
3− 0.5506− 0.3232− 0.3659− 0.4585− 0.4308
40.12840.6119− 0.14960.13040.1808
50.3795− 0.4063− 0.11840.27440.0850
60.01970.10100.2580− 0.07610.0607
70.3277− 0.25660.3436− 0.26610.0765
8− 0.03320.1211− 0.3092− 0.1890− 0.0925
90.0997− 0.04530.0915− 0.02530.0280
100.36640.3144− 0.36290.07220.1128
Table VIII. Ketchup: Effects of Temporary 10% Heinz Price Decrease
Nested MNL
WeekHeinzHunt'sDel MonteStore BrandTotal
  1. Note: The table reports the percentage change in purchase frequencies for each brand by week, following a temporary 10% price cut for Heinz in week 44. Thus “Week 1” of the simulation is actually week 44 in the data.

  2. We choose week 44 as the base period for simulation because there was (almost) no promotion at all (i.e., Heinz' display dummies equal zero for all observations, and feature was 1 for only 0.7% of observations). But price cuts are often accompanied by display and/or feature. To simulate this, we use the following procedure: First, we calculated the frequency with which display and feature are 1 when Heinz is on “deal.” These figures were 0.0357 and 0.0489, respectively. Then, in the simulation, we set Skippy's display dummy to 1 with probability 0.0357, and its feature dummy to 1 with probability 0.0489, in week 44.

  3. When we set the brand specific display (feature) dummy to 1, we also adjust the categorical level display (feature) dummy accordingly (setting it to 1 if it had been 0 in the data). In week 44, the average frequency of the categorical display and feature dummies in the baseline are 0.0858 and 0.0551, respectively; their average frequencies in the simulation are 0.1184 and 0.1014, respectively. Their average frequencies for the whole sample are 0.1018 and 0.1606, respectively.

135.6246− 1.6961− 2.0031− 0.537923.0642
2− 0.2775− 1.28250.06991.3003− 0.2368
3− 0.4627− 0.0037− 0.80160.8059− 0.2022
40.06450.2452− 0.20970.92600.1177
50.06790.67370.2956− 0.03730.1877
6− 0.23610.4360− 0.86970.0605− 0.1738
70.3613− 1.29930.92010.01270.1404
8− 0.10330.10210.16550.39580.0233
9− 0.1407− 0.38510.33430.6866− 0.0271
100.20570.8462− 0.65480.38820.2549
Price Consideration II
 HeinzHunt'sDel MonteStore BrandTotal
136.4323− 1.2495− 1.8488− 0.499323.5150
2− 0.5822− 2.0916− 1.08331.1292− 0.6591
3− 0.9858− 0.3131− 0.74860.3057− 0.6102
4− 0.15130.7361− 1.07770.6298− 0.1177
50.05220.1579− 0.19110.07130.0593
6− 0.13310.1108− 0.9187− 0.2845− 0.1586
70.6772− 1.13870.7906− 0.40790.3240
80.1858− 0.49940.4755− 0.00570.1356
90.0434− 0.07390.20760.72330.1110
100.24870.8157− 0.33680.17540.2843

The similarity of the models' predictions is remarkable. The PC-II model predicts a 20.4% increase in Skippy sales in the week of the promotion, while the NMNL model predicts a 21.3% increase. Thus, both models imply short run “price” elasticities of demand of roughly − 2, although it should be stressed this is not a conventional price elasticity because we include the additional promotion activity that typically accompanies a price cut.

Each model implies little brand switching. The NMNL and PC-II models imply that total category sales increase by 5.7% and 5.8%, respectively. The NMNL says 98% of the increase in Skippy sales is from category expansion, while only 2% is switching from other brands. The PC-II model actually says that sales of the other brands increase slightly (about 0.2 to 0.3%), so all the increased Skippy sales is from category expansion. This phenomenon is a spillover effect of promotion that arises in the PC models. If one brand uses features and/or displays, it can attract a consumer's attention to the category, but, when he/she checks out prices, he/she may decide to buy some other brand besides the one that was promoted.

We can also examine the long run effects of the price cut. Their sign is theoretically ambiguous. A price cut may lead to purchase acceleration, where a consumer buys today rather than waiting to buy in a future period (an inventory planning phenomenon). On the other hand, increased sales today can lead to greater future sales through habit persistence/enhanced brand equity, captured by the GL variables.

The NMNL and PC-II make opposite predictions about the direction of these effects, but both agree they are small. According to NMNL, the drop in Skippy sales in weeks 2 through 10 cancels out 3.4% of the increased sales created by the price cut. In contrast, PC-II says Skippy sales increase by roughly 0.52% in weeks 2 through 10. A notable difference between the models is that PC-II implies a post-promotion dip for all brands in weeks 2 to 3, with total category sales falling about 0.4% in each week. The comparable figure for NMNL is only bout 0.16%.

The results for ketchup are a bit different. When we simulate a 10% price cut for Heinz, accompanied by turning on display (feature) with a probability of 3.6% (4.9%), we get much larger increases in own sales, 35.6% for NMNL and 36.4% for PC-II. Thus, each model implies the own “price” elasticity of demand of roughly − 3.6.31 The PC-II model again implies a larger post-promotion dip, with sales in the category falling about 0.63% in weeks 2 and 3, while NMNL has them falling only about 0.22%. In ketchup, both models say little of the short run increase in Heinz sales is due to brand switching, 1.9% in NMNL and 1.6% in PC-II. Also, the category spillover effect of promotion that we saw for the PC-II model in the peanut butter category is not so strong in the ketchup category. Finally, the NMNL says 2.7% of the increase in Heinz sales is due to purchase acceleration, while the PC-II says it is essentially zero.

7. CONCLUSION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

In this paper we have proposed a new model of brand choice we call the “price consideration” (PC) model, and shown that is it a viable alternative to the workhorse MNL and NMNL models of the marketing literature. The distinguishing feature of the PC model is that it introduces a weekly decision whether or not to consider a product category, and models this decision as depending only on non-price factors. Only if a household decides to consider a category does it look at prices and decide whether and what brand to buy.

The PC model can accommodate a more flexible relationship between purchase incidence and brand share price elasticities than conventional MNL and NMNL models. Using data from the peanut butter and ketchup categories, we show that the PC model produces a much better fit, particularly to inter-purchase spells. The standard models greatly overstate the frequency of short spells, because they have difficulty reconciling the observed high sensitivity of brand shares to price with low sensitivity of purchase incidence to price in the period shortly after a purchase (when inventory is high). Moreover, the PC model is as simple to estimate as standard models.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

Ching's work on this project was supported by Connaught Start-up Grant at the University of Toronto. Keane's work on this project was supported by NSF grants SBR-9511280 and SBR-9812067, and ARC grant FF0561843. The authors thank (without intending to implicate) participants in the marketing and economics seminars at Duke University, the 2006 Marketing Science Conference, the 2006 Marketing Dynamics Conference and the 7th Invitational Choice Symposium held at Wharton in June 2007 (especially Michael Hanemann and Bob Meyer), for helpful comments.

Appendix

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information
Appendix: Variances and Correlations of the Brand Intercepts
Peanut Butter
 MNLNMNLPC IPC II
  1. Σjj is the variance of αij. The ψij's are the corresponding correlation matrix elements.

Σ112.3563.2751.9882.029
Σ222.0913.0131.8081.799
Σ332.2102.9342.0031.975
Σ441.8572.5951.5231.523
ψ21− 0.0540.114− 0.023− 0.053
ψ310.0930.2190.1350.131
ψ320.0860.2330.1830.183
ψ410.1920.3690.1860.159
ψ420.2340.4130.2720.247
ψ430.1280.3000.2070.193
Ketchup
 MNLNMNLPC IPC II
Σ110.8110.8110.6360.664
Σ221.6961.6961.5451.571
Σ332.1792.1772.0452.057
Σ442.4202.4192.3192.243
ψ210.2000.2000.1850.204
ψ310.0700.0700.0670.077
ψ320.6920.6940.7730.766
ψ410.0320.0310.0380.074
ψ420.3900.3880.5610.536
ψ430.4430.4400.5250.509

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information
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  • 1

    By “exogenous” we mean unrelated to prices, promotion, tastes and inventories.

  • 2

    For expositional convenience, throughout the paper we will treat the week as the decision period.

  • 3

    This is also true in a nested logit model, where at the top level consumers decide whether to buy in the category, and in the lower level choose amongst brands. In such a model, the category purchase decision is based on the inclusive value from the lower level of the nest, which is in turn a function of that week's price vector.

  • 4

    We assume a multivariate normal distribution of brand intercepts, so in fact we are estimating what are known as “heterogeneous logit” models (see, e.g., Harris and Keane, 1999).

  • 5

    In fact, the PC model may be easier to estimate than the NMNL, as the latter often has a poorly behaved likelihood.

  • 6

    Erdem et al. (2003) describe the problem more formally as “dynamic selection bias.” Suppose there are several types of consumers, each with a strong preference for one brand. Suppose further that consumers engage in optimal planning behavior. Each period, a consumer considers his/her level of inventories and the vector of brand prices, and decides if it's a good time to buy. In this framework, if brand A has a price cut in week t, consumers who buy in week t will contain an overrepresentation of the type that prefers A. Thus, in the self-selected subset of consumers who buy in the category in any given week, there is a negative correlation between brand prices and brand preference (i.e., in weeks when price of a brand is low, the sample contains an over-representation of the type of consumers who prefer that brand). This causes the price elasticity of demand to be exaggerated.

    Interestingly, this negative correlation between brand prices and brand preference induced by inventory behavior is opposite in sign to the positive correlation that has been the concern of the literature on “endogenous prices.” That literature, stemming from Berry et al. (1995), deals with a fundamentally different type of data, where prices and sales are aggregated over long periods of time - in contrast to the high frequency (i.e., daily) price variation observed in scanner data. Then, brands with high unobserved quality will tend to be high priced, inducing a positive correlation between brand prices and preferences. This biases price elasticities of demand towards zero. In scanner data, this problem of unobserved brand quality can be dealt with simply by including brand intercepts.

  • 7

    Another potential explanation is that brands are viewed as very close substitutes. That would explain why, conditional on purchase in a category, consumers' choice among competing brands appears to be much more sensitive to price than their decision to purchase in the first place. But a well-known fact about consumer behavior is that in many frequently purchased product categories brand loyalty is high (i.e., rates of brand switching are low), implying that brands are not viewed as close substitutes.

  • 8

    This two-stage decision process may be (an approximation to) optimal behavior in a version of an inventory-planning model where there is some fixed cost of examining prices, and it is only optimal to examine prices if inventories fall below some critical level, or if consumers observe some advertising/display/feature signal that reduces the cost of observing prices or conveys a signal that prices are likely to be low.

    Alternatively, the PC model may also be viewed as departing from the optimal backward induction process (from the whole vector of prices back to the expected utility of buying in the category) that consumers are assumed to follow in “rational” choice models. In PC, consumers use a forward looking heuristic (that does not depend on prices) to decide whether to consider a category, but then, once they do decide to consider, they engage in fully rational calculations to make the brand choice/category purchase decision.

  • 9

    Nevertheless, we were concerned that the PC and NMNL models would be difficult to distinguish empirically. The reason is that we only observe purchases, not whether or not a consumer sees prices/considers a category. And each model has a mechanism to generate brand choice price elasticities that exceed purchase incidence price elasticities. In our empirical application, we found that distinguishing between the two models is not a problem—the PC model clearly fits the data better.

  • 10

    A longer gap may also indicate a consumer is losing interest in the category, as suggested by the empirical death models of Schmittlein, Morrison, Ehrenberg, etc. (see Helsen and Schmittlein, 1993, for a review).

  • 11

    Thus, GL is an exponentially smoothed weighted average of lagged purchase indicators for brand j by consumer i. Of course, other forms of state dependence are possible, but this is the most common form in marketing.

  • 12

    At t = 1 − n we assume that GL(Hij1−n, δ) = 0 and purch_gapi1−n = 1. We set n = 10 in peanut butter and 20 in ketchup.

  • 13

    We assume store visit decisions are exogenous (e.g., a no-visit week arises if the family leaves Sioux Falls on a vacation, which we do not model), and drop weeks when a consumer did not visit a store. If we instead treat these as no-purchase weeks it has little effect on the results, since households visit a store in the large majority of weeks.

  • 14

    If we combine these small brands into “other” their combined market share is only 0.55% and 0.06% in peanut butter and ketchup, respectively. Since these brands are rarely purchased, their price and promotional activity information is generally missing, given how these variables are constructed in the Nielsen data (see Erdem et al., 1999, for a discussion). To avoid these problems, we decided to drop these small brands from the model, and to drop households who have ever chosen “other” in their choice histories.

  • 15

    The obvious alternative is, on purchase occasions, to use the vector of prices for the size the consumer actually bought, converted to price per ounce. But this begs the question of what price vector to use on occasions when the consumer chose not to buy. Suppose that in non-purchase weeks we use the vector of prices for the most common size (e.g., 32oz for ketchup), also converted to price per ounce. To see the problem this approach creates, assume there are just two sizes, 32oz and 64oz. Furthermore, lets say that on some fraction of purchase occasions consumers buy the 64oz, and that its price per ounce tends to be lower. This will introduce a systematic bias whereby the price per ounce vector tends to be lower on purchase occasions that on non-purchase occasions (since the average price on purchase occasions is a weighted average of 32 and 64 oz prices, while that on non-purchase occasions is just the average 32oz price). Hence, the price elasticity of demand is exaggerated. It is to avoid this problem that we chose to base the price vector on a common size for both purchase and non-purchase occasions.

  • 16

    We would argue that the PC model is still appropriate for certain categories like milk, which certain households (e.g., those with young children) buy very frequently. This can be captured by heterogeneity in γi0.

  • 17

    The PC-II model nests MNL. By setting Pit(C) = 1 for all i and t, PC-II becomes MNL. Compared with MNL, PC-II has five more parameters, which generate Pit(C). One can send Pit(C) to 1 by sending the mean of γi0 to infinity.

  • 18

    The coefficient αinit, pg, which captures how the pre-sample purchase gap is related to the brand intercepts, is negative and significant, as expected. That is, households with a longer initial purchase gap tend to like the entire category less. Hence they have generally lower intercepts, making them more likely to choose the no-purchase option.

  • 19

    We also specify that the initial value of purch_gapi1 was 5. This influences the mean of γ0 as in (31).

  • 20

    We integrate out the unobserved heterogeneity, γi0, when obtaining the average probability of considering.

  • 21

    That is, when the category display and feature dummies are both set to 1. The predicted effect of feature and display may seem high, but it should be emphasized that this is only the probability that a consumer will check the prices of peanut butter. After checking prices, the consumer may still choose not to buy in the category.

  • 22

    The estimate of γmem is 0.121, which implies the effect of household size is fairly small. If we reduce household size from 3 to 1, the consideration probability drops from 39.7% to 35.4%.

  • 23

    The Appendix contains variances and correlations of the brand intercepts. We estimate the Cholesky parameters, but report the implied variances and correlations, as they are more informative. In peanut butter, the NMNL model implies larger variances of and larger correlations among the brand specific intercepts than the other models.

  • 24

    The correlations of the brand specific intercepts in the ketchup category show large positive correlations amongst brands 2, 3 and 4 but not with brand 1 (Heinz). This suggests there is basically a Heinz type and a type that regularly buys the other, lower priced, brands.

  • 25

    As in the peanut butter simulation, we assume the initial value of purch_gapi1 = 5, and integrate over γi0.

  • 26

    For example, in the last eight years of Marketing Science, the only paper that reports the fit to inter-purchase times for a brand choice model is Sun (2005). Of course, papers that specifically study purchase incidence do report predicted inter-purchase times (see, e.g., Fok et al., 2002), but they abstract from brand choice.

  • 27

    Similarly, the probability a household buys any brand in the peanut butter category at t given that it bought it at t − 1 is 13.3%. The MNL and NMNL put this probability at about 24% to 25%, while the PC models put it at about 17%.

  • 28

    As we noted in Section 3.1, while NMNL can generate a price elasticity of demand for brand choice much larger than for purchase incidence, it must do it within the inclusive value framework. That means NMNL would require that brands be very similar on unobservables. This has implications for other of aspects of behavior, such as brand switching and sensitivity of purchase incidence to prices of different brands. But the PC model can generate a brand choice elasticity greater than the purchase incidence elasticity without constraining substitutability among brands.

  • 29

    In other words, the standard models have difficulty reconciling the observed sensitivity of brand choice to price with the low frequency with which consumers buy again after a short spell.

  • 30

    The MNL and PC-I models make similar predictions, so we do not report them. They are available on request.

  • 31

    This figure is remarkably close to estimates obtained for the same category by Erdem et al. (2003), using a structural inventory model, and Erdem et al. (2006), using a learning model.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEMS WITH CONVENTIONAL CHOICE MODELS, AND THE PC ALTERNATIVE
  5. 3. THE PRICE CONSIDERATION (PC) MODEL
  6. 4. CONSTRUCTION OF THE DATA SETS
  7. 5. ESTIMATION RESULTS
  8. 6. MODEL SIMULATIONS
  9. 7. CONCLUSION
  10. Acknowledgements
  11. Appendix
  12. REFERENCES
  13. Supporting Information

The JAE Data Archive directory is available at http://qed.econ.queensu.ca/jae/datasets/ching001/ .

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.