Discussion Paper No. 28

An Integrated Model of Demand Allocation among
and within Product Categories

Wirawan Dony Dahana
Nobuhiko Terui
Nozomi Nakajima
November , 2014

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANAn Integrated Model of Demand Allocation among and within Product Categories

Wirawan Dony Dahana
Graduate School of Economics, Osaka University
dony@econ.osaka-u.ac.jp
Nobuhiko Terui*
Graduate School of Economics and Management, Tohoku University
terui@econ.tohoku.ac.jp
Nozomi Nakajima
Graduate School of Economics, Osaka University
nakajima@econ.osaka-u.ac.jp

Terui acknowledges the financial support of the Japanese Ministry of Education Scientific
Research Grants (A)25245054.

1

An Integrated Model of Demand Allocation among and within Product Categories

ABSTRACTThe authors propose an integrated model that links consumers’ budget allocation
decisions at two different stages. The first stage concerns consumers’ decisions toallocate budget among product categories, and the second stage concerns allocating theassigned budget among brands. In each stage, the authors assume that consumers
behave rationally by determining the optimal demand for the category as well as for thebrand to maximize utility. The authors use scanner data from purchases of instant coffee,
coffee cream, tea, and instant curry to investigate the role of consumer preference andprice in a hierarchical budget allocation problem. They also discuss some implicationsof anticipating a change in quantity demanded because of a change in expenditure andprice. Researchers can use the model to examine the marginal effects of expenditure not
only on category but also on brand purchase quantity.

Keywords: expenditure allocation, multi-stage decisions, reference price, heterogeneity

2

Consumers’ decision making is often considered a process of resource allocation
among two or more options. For example, a decision hierarchy often begins with aproblem of allocating time resources between leisure and labor, followed by a problemof allocating disposable income between consumption and savings. It continues with theallocation of the consumption budget among product categories. Ultimately, consumers
make a decision to allocate the budget assigned to each category among brands.
Undoubtedly, the outcome of a decision made at an earlier stage will influence decisions
at subsequent stages. We argue that a better understanding of how the decisions at
different stages relate with each other will provide useful insights for a marketer.
Therefore, we propose a framework for analyzing consumer decision making in
multiple stages. Considering data availability, we focus our study on two stages: one inwhich the budget is allocated among product categories and another in which the budget
assigned to each category is allocated among brands. In particular, we propose an
integrated model that links consumer decisions in both stages. In the present research,
we aim to determine how a change in consumer expenditure would affect demand for
product categories and brands. We also address how a change in the price of a brandcould influence demand allocation in both stages.
Modeling consumer demand allocation has been an area of interest for several
decades. Researchers have developed an ample number of models to accommodate
factors that are thought to be influential in consumer decision-making processes. A
so-called ideal model should be consistent with economic theory and facilitate easy
estimation and data fitting (Barten 1993). In general, it is difficult to generate a modelthat satisfies all of these conditions, as there is often a tradeoff among them.
Theoretically consistent models are usually associated with computational difficulty,

3and the resulting demand functions are often unappealing. On the other hand, simpler
models are usually easy to estimate, yet the derived demand function often falls short oftheoretical consistency.
One stream of research begins with the specification of direct utility function andderives Marshallian demand function from the first-order condition for utility
maximization (Stone 1954). Another approach utilizes a specified indirect utility
function and applies Roy’s identity to obtain estimable demand function (Christensen,
Jorgenson, and Law 1975). The third stream, which induces demand systems thatpossess useful properties, starts from a specified expenditure function and applies
Shephard’s lemma to arrive at a demand function. The almost ideal demand system(AIDS) of Deaton and Muellbauer (1980) is the best-known example of this type of
specification. Recently, Kim, Allenby, and Rossi (2002) employed a Bayesian inference
approach that, to some extent, solves the computational problem arising in theestimation of a demand function derived from a direct utility function.
Most of the models proposed thus far have focused on one of the stages in the
consumer decision-making process. One of the main objectives of these studies has
been to investigate the role of price and income in the distribution of budget shareamong product categories. While single-stage frameworks have contributed a great deal
to understanding consumer decision making, we expect to obtain richer findings byextending the scope of analysis to involve consumer decisions at several stages. For
example, a marketer might want to know how a change in the proportion of category
expenditures stemming from an increase in income would affect brand sales.
Marketing researchers have put considerable effort into analyzing consumer decisions
at multiple stages simultaneously. For example, Bucklin and Lattin (1991) proposed a4nested logit model of category purchase and brand choice to investigate planned andunplanned buying behavior. A model by Chib, Seetharaman, and Strijnev (2004)
integrated category purchase and brand choice in a framework that involves
no-purchase data. Models proposed by Mehta (2007) and Song and Chintagunta (2007)
deal with consumer decisions of category purchase and brand choice. The latter also
incorporated purchase quantity decisions into the model. However, all of these models
assume that consumers purchase only one brand within a category (i.e., the corner
solution) and, as such, are inapplicable in cases where consumers purchase multiple
brands simultaneously.
We summarize some related studies in Table 1. References 1 through 3 are
econometric models of demand systems dealing with budget allocation in one stage.
The demand model of Hauser and Urban (1986) is a one-stage model that
accommodates some findings in behavioral research. References 6 through 10 are
multistage models, yet their focus is on discrete consumer decisions, such as categorypurchase and brand choice decision. These types of models have some limitations in
explaining the effect of an expenditure change on category and brand spending. Lee etal. (2013) provided an inherently two-stage model that deals with the consumer problem
of demand allocation. However, the model carries out allocation according to a certain
sequence. That is, consumers first allocate expenditures between a category and theremainder, and then allocate the remaining amount between the next category and theremainder, and so on. While this modeling approach is useful for examining competition
among categories, we argue that it is unsuitable for analyzing the marginal effect ofexpenditures in multiple stages. Our modeling approach contributes to the literature in
the sense that it can provide a more comprehensive understanding of consumer decision

5

making, as it integrates expenditure allocation problems at both the category and brandlevels.
[Insert Table 1 about here.]
We have organized the remainder of this article as follows. The next section describes
the model for budget allocation in two stages. We then introduce the data used for
examining the proposed model. Subsequently, we provide the estimation results anddiscuss the implications of our model for the anticipation of demand changes. Finally,
we offer our concluding remarks.

MODEL
Expenditure Allocation among Categories
Suppose that a consumer is facing a problem of allocating budget among 𝑚 productcategories and the outside good at purchase occasion 𝑡. We assume that the overall
utility function of this consumer will increase with category expenditures at adecreasing rate, which is given by:𝑚

𝑈(𝐸1𝑡 , … , 𝐸𝑚𝑡 , 𝑧𝑡 ) = ∑ 𝜓𝑖𝑡 ln(𝐸𝑖𝑡 + 1) + ln(𝑧𝑡 ) ,

(1)

𝑖=1

where 𝐸𝑖𝑡 and 𝑧𝑡 denote the expenditure allocated to category 𝑖 and the outside good,
respectively. We can represent category expenditure as 𝐸𝑖𝑡 = 𝑝̃𝑖𝑡 𝑥𝑖𝑡 , where 𝑝̃𝑖𝑡 is theunit price and 𝑥𝑖𝑡 is the purchase quantity of category 𝑖. Here, we assume that productcategories are infinitely divisible, and, therefore, 𝑥𝑖𝑡 refers to the volume of category 𝑖purchased by a consumer at the time 𝑡1.
1

In our empirical analysis, we define 𝑥𝑖𝑡 as the purchased volume measured in grams. For most of
packaged goods, it would be more realistic to consider 𝑥𝑖𝑡 a discrete quantity, as in Lee and Allenby(2014). However, we treat it as a continuous variable in order to get a tractable link between thedecisions in different stages.

6

Parameter 𝜓𝑖𝑡 refers to category-specific attractiveness, whose value can vary acrosstime. For 𝑈 to be a valid utility function, it is necessary to impose a restriction suchthat 𝜓𝑖𝑡 ≥ 0. Without loss of generality, the attractiveness of the outside good is fixed toone. To accommodate market competition, we parameterize 𝜓𝑖𝑡 as a function of
baseline category preference and inventory effects.
𝑚𝜓𝑖𝑡 = exp (𝛼𝑖 + ∑ 𝛽𝑗𝑖 𝑠𝑗𝑡 )

(2)

𝑗=1

Here, 𝛼𝑖 represents baseline preference toward category 𝑖. 𝑠𝑗𝑡 denotes the inventory
level of category 𝑗. Parameter 𝛽𝑗𝑖 captures the inventory effect of category 𝑗 on
category 𝑖’s attractiveness. If 𝛽𝑗𝑖 takes a positive (negative) value, then we interpretthat category 𝑗 is a complement (substitute) for category 𝑖 (Lee, Kim, and Allenby2013). We let 𝛽𝑖𝑗 ≠ 𝛽𝑗𝑖 to allow for asymmetric substitution and complementary
effects. The specification of inventory level is given by:
𝑠𝑗𝑡 = 𝜆𝑗𝑑𝑡 (𝑠𝑗,𝑡−1 + 0.01𝑥𝑗,𝑡−1 )(3)where 𝑑𝑡 denotes the number of days elapsed since the last purchase (Ailawadi et al.
2007). The initial value of inventory level 𝑠𝑗0 is set to a hundredth of the averagepurchase quantity. 𝜆𝑗 is a parameter indicating the consumer consumption rate forcategory 𝑗. We fix 𝜆𝑗 = 0.8 to reduce the number of parameters. In this setting,consumers’ inventory depletes in about two weeks, which is approximately the same as
the average interpurchase time.
In the first stage, the consumer problem lies in determining how to allocate budget
among product categories to maximize utility. We denote the total expenditure spent by
a consumer at purchase occasion 𝑡 by 𝐸𝑡 . This utility maximization problem can be
written as follows:

7

𝑚

max

𝐸1𝑡 ,…,𝐸𝑚𝑡 ,𝑧𝑡

𝑈(𝐸1𝑡 , … , 𝐸𝑚𝑡 , 𝑧𝑡 ) = ∑ 𝜓𝑖𝑡 ln(𝐸𝑖𝑡 + 1) + ln(𝑧𝑡 )
𝑚

𝑖=1

(4)

s. t. ∑ 𝐸𝑖𝑡 + 𝑧𝑡 ≤ 𝐸𝑡
𝑖=1

Defining the Lagrange function by:𝑚

𝑉(𝐸1𝑡 , … , 𝐸𝑚𝑡 , 𝑧𝑡 ) = 𝑈(𝐸1𝑡 , … , 𝐸𝑚𝑡 , 𝑧𝑡 ) − 𝜆 (∑ 𝐸𝑖𝑡 + 𝑧𝑡 − 𝐸𝑡 ) ,

(5)

𝑖=1

we obtain the Kuhn–Tucker condition for this utility maximization problem as follows:𝜕𝑉𝜓𝑖𝑡=− 𝜆 = 0, if 𝐸𝑖𝑡 > 0
𝜕𝐸𝑖𝑡 𝐸𝑖𝑡 + 1𝜕𝑉
𝜓𝑖𝑡=− 𝜆 < 0, if 𝐸𝑖𝑡 = 0
𝜕𝐸𝑖𝑡 𝐸𝑖𝑡 + 1
𝜕𝑉1= −𝜆 =0
𝜕𝑧𝑡 𝑧𝑡(6)∗
We can write the solutions for this problem as 𝐸𝑖𝑡∗ = 𝑝̃𝑖𝑡 𝑥𝑖𝑡∗ , where 𝑥𝑖𝑡
is theoptimum purchase quantity. In addition, 𝑝̃𝑖𝑡 is not the actual price but the unit price∗level perceived by consumers. While we can observe 𝑥𝑖𝑡
from data, we cannot know

the actual value of 𝐸𝑖𝑡∗ because in general 𝑝̃𝑖𝑡 is unobservable. For this reason, wemust determine how to appropriately assign the value of 𝑝̃𝑖𝑡 . It is natural to posit that
consumers form their beliefs about category unit price based on the price of various
brands within the category. Accordingly, consumers use clues about the actual price ofvarious brands encountered during a past or present shopping trip as a reference pricerepresenting the category. In particular, we use some specifications of categoryreference price and choose the one that produces the best fit.

Expenditure Allocation within Categories
In the second stage, consumers move on to allocating budget among the brands

8within each category. Conditional on the optimum category expenditure 𝐸𝑖𝑡∗ , and brandprices {𝑝𝑖𝑘𝑡 }, this allocation problem reduces to a problem of determining which andhow much of the brands to purchase to maximize category level utility, which is definedas:𝐾𝑖

𝑈𝑖 (𝑦𝑖1𝑡 , … , 𝑦𝑖𝐾𝑖 𝑡 ) = ∑𝑘=1

𝜙𝑖𝑘ln(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡 + 1) .
𝛾𝑖𝑘

(7)

Here, 𝑦𝑖𝑘𝑡 denotes the purchase quantity2 of brand 𝑘 in category 𝑖. 𝜙𝑖𝑘 and 𝛾𝑖𝑘
stand for the brand attractiveness and satiation parameter, respectively. In contrast toEquation 1, we assume brand attractiveness is invariant with respect to purchaseoccasion. Instead, we include 𝛾𝑖𝑘 to accommodate consumers’ preference for variety,as in Hasegawa, Terui, and Allenby (2012).
Remember that the budget for each category that maximizes overall utility 𝑈 is
𝐸𝑖𝑡∗ = 𝑝̃𝑖𝑡 𝑥𝑖𝑡∗ . Consumers decide which and how much of the brands in category 𝑖 topurchase as long as the total amount spent does not exceed 𝑝̃𝑖𝑡 𝑥𝑖𝑡∗ . On the other hand,because 𝑥𝑖𝑡∗ is the optimum category purchase quantity, the quantity of brands
purchased must add up to 𝑥𝑖𝑡∗ . This implies that consumers have to maximize 𝑈𝑖
subject to budget and quantity constraints. This type of consumer problem is given as
follows:max

𝑦𝑖1𝑡 ,…,𝑦𝑖𝐾 𝑡
𝑖

s. t.

𝑈𝑖 (𝑦𝑖1𝑡 , … , 𝑦𝑖𝐾𝑖 𝑡 ) = ∑∑ 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡 ≤ 𝑝̃𝑖𝑡 𝑥𝑖𝑡∗
𝑘

𝑘

𝜙𝑖𝑘
ln(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡 + 1)
𝛾𝑖𝑘

and

∗
∑ 𝑦𝑖𝑘𝑡 = 𝑥𝑖𝑡𝑘(8)

In fact, utility maximization in Equation 8 can be subject to an awkward problem
whereby the solution could not exist. This occurs because in most cases, budget and2

In line with the definition of 𝑥𝑖𝑡 described previously, we define 𝑦𝑖𝑘𝑡 as the volume of brand 𝑘purchased by a consumer at the time 𝑡.

9quantity constraints cannot be simultaneously satisfied. Figure 1 illustrates this problem
when there are two brands in category 𝑖. Suppose that the price of brand 2 is higher thanthe price of brand 1 and that the reference price lies somewhere in between. The solidand dashed lines in the figure represent quantity and budget constraints, respectively. If
the indifference curve touches the quantity line at the point where the budget constraint
coincides with the quantity constraint (point i), a consumer can choose the productbundle at this point because it satisfies both constraints. However, if it touches thequantity line at any point in area ii, this bundle would be unaffordable, as the amountneeded to purchase it exceeds the budget constraint. On the other hand, if theindifference curve touches the quantity line at any point in area iii, the bundle isattainable; however, then the consumer would have some amount left over. The last twocases are the problems that must be handled. The problem in the second case is clear
because the solution does not exist. The third case must be excluded because we assume
that budget constraint is binding, provided that the marginal utility of the outside goodis greater than zero.
[Insert Figure 1 about here.]
We assume that consumers will make an adjustment if the optimum bundle does not
fall on the point where the budget line intercepts the quantity line. When the problem
depicted in area ii arises, a consumer will draw from the proportion allocated to the∗outside good as much as ∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡 − 𝑝̃𝑖𝑡 𝑥𝑖𝑡
to make any bundle in this areaaffordable. Consequently, the consumption of the outside good decreases, and the utility
obtained from consuming it becomes smaller. When consumers have more than enoughmoney to buy any bundle in area iii, the amount left over will be allocated to the outside
good. In this case, the budget for the outside good increases, and, therefore, consumers

10

gain additional utility.
Here, we provide a formal representation of this budget adjustment. Let 𝑐𝑖𝑡 =∗∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡 − 𝑝̃𝑖𝑡 𝑥𝑖𝑡
and 𝑧𝑖𝑡 = |𝑐𝑖𝑡 |. We define the two indicator variables 𝐼𝜌 and 𝐼𝜔

as follows:𝐼𝜌 = {

1 if 𝑐𝑖𝑡 > 0
1 if 𝑐𝑖𝑡 < 0, 𝐼𝜔 = {0 if 𝑐𝑖𝑡 ≤ 0
0 if 𝑐𝑖𝑡 ≥ 0(9)Then, after implementing budget adjustment, a consumer utility maximization problem
becomes:max

𝑦𝑖1𝑡 ,…,𝑦𝑖𝐾 𝑡𝑖𝑈𝑖∗ (𝑦𝑖1𝑡 , … , 𝑦𝑖𝐾𝑖 𝑡 ) = ∑𝑘𝜙𝑖𝑘ln(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡 + 1) + (𝐼𝜔 𝜔𝑖 − 𝐼𝜌 𝜌𝑖 ) ln(𝑧𝑖𝑡 + 1)
𝛾𝑖𝑘

s. t.

∑ 𝑦𝑖𝑘𝑡 = 𝑥𝑖𝑡∗
𝑘(10)

where 𝑈𝑖∗ is the adjusted category utility function, and 𝜔𝑖 and 𝜌𝑖 are parameters that
capture the sensitivity of consumer utility to an increase or a reduction in the outside
good expenditure. For this reason, we call them the gain and loss parameters.
The utility maximization in Equation 8 is similar to the problem of Satomura, Kim,
and Allenby (2011), which imposes both budget and quantity constraints. However, in∗contrast to their problem, the quantity 𝑥𝑖𝑡
is not the maximum quantity but the

optimum quantity. This implies that the equality in quantity constraint must hold.
Therefore, it is impossible to introduce a so-called outside good into the quantity
constraint and to define a Langrage function that includes this term. The idea behind thebudget adjustment assumption is to embed the budget constraint in the (adjusted) utility
function and construct a utility maximization problem subject to only quantity
constraint.
Defining the Lagrange function by:

11∗
𝑉𝑖 (𝑦𝑖1𝑡 , … , 𝑦𝑖𝐾𝑖 𝑡 ) = 𝑈𝑖∗ (𝑦𝑖1𝑡 , … , 𝑦𝑖𝐾𝑖 𝑡 ) − 𝜆 (∑ 𝑦𝑖𝑘𝑡 − 𝑥𝑖𝑡
),

(11)

𝑘

we determine the Kuhn–Tucker condition for Equation 10 as follows:(𝐼𝜔 𝜔𝑖 − 𝐼𝜌 𝜌𝑖 )𝑝𝑖𝑘𝑡
𝜕𝑉𝑖
𝜙𝑖𝑘
=
+
− 𝜆 = 0, if 𝑦𝑖𝑘𝑡 > 0𝜕𝑦𝑖𝑘𝑡 (𝛾𝑖𝑘 𝑦𝑖𝑘𝑡 + 1)
(𝑧𝑖𝑡 + 1)(𝐼𝜔 𝜔𝑖 − 𝐼𝜌 𝜌𝑖 )𝑝𝑖𝑘𝑡
𝜕𝑉𝑖
𝜙𝑖𝑘
=
+
− 𝜆 < 0, if 𝑦𝑖𝑘𝑡 = 0
𝜕𝑦𝑖𝑘𝑡 (𝛾𝑖𝑘 𝑦𝑖𝑘𝑡 + 1)
(𝑧𝑖𝑡 + 1)(12)

For the budget adjustment assumption to hold, it is necessary that 𝑈𝑖 (𝒚′𝑖𝑡 ) < 𝑈𝑖∗ (𝒚′′
𝑖𝑡 ),
′
′
′′
′′
) and 𝒚′′where 𝒚′𝑖𝑡 = (𝑦𝑖1𝑡, … , 𝑦𝑖𝐾𝑡
𝑖𝑡 = (𝑦𝑖1𝑡 , … , 𝑦𝑖𝐾𝑡 ) are the optimum bundles of therespective utility function. As already mentioned, consumers adjust their budget if either′′′′∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡
> 𝐸𝑖𝑡∗ or ∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡
< 𝐸𝑖𝑡∗ . Thus, we can formally express the condition forbudget adjustment by:3
∑
𝑘

𝜙𝑖𝑘
′′
′
′′
{ln(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡
+ 1) − ln(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡+ 1)} > 𝜌𝑖 ln(𝑧𝑖𝑡 + 1) , if ∑ 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡
> 𝐸𝑖𝑡∗
𝛾𝑖𝑘
𝑘

(13)
𝜙𝑖𝑘
′
′′
′′
{ln(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡∑
+ 1) − ln(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡+ 1)} < 𝜔𝑖 ln(𝑧𝑖𝑡 + 1) , if ∑ 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡
< 𝐸𝑖𝑡∗
𝛾𝑖𝑘𝑘{ 𝑘
The first inequality in Equation 13 implies that when the budget assigned to category 𝑖
is not enough to afford the bundle 𝒚′′𝑖𝑡 , consumers will draw some amount from thebudget allocated to the outside good so as to make 𝒚′′𝑖𝑡 affordable if the difference
between the utility of consuming 𝒚′𝑖𝑡 and 𝒚′′
𝑖𝑡 is greater than the loss of giving up
some amount of the outside good.4 The second inequality says that even when 𝒚′𝑖𝑡′′
(which is greater than 𝒚′′𝑖𝑡 ) is affordable, consumers would instead prefer 𝒚𝑖𝑡 if thedifference between the utility of consuming 𝒚′𝑖𝑡 and 𝒚′′
𝑖𝑡 is less than the gain from3

4

Although we assume that consumers implement budget adjustment, we do not restrict theparameters on the space bounded by Equation 13. However, we examine the plausibility of thisassumption by using the parameter estimates and describe the analysis later under Discussion.
′
Provided that the indifference curves are convex to the origin, we have 𝑈𝑖 (𝒚′′𝑖𝑡 ) > 𝑈𝑖 (𝒚𝑖𝑡 ) if
′′
∗
′′
′
∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡 > 𝑝̃𝑖𝑡 𝑥𝑖𝑡 because 𝒚𝑖𝑡 > 𝒚𝑖𝑡 .

12′′saving as much money as 𝐸𝑖𝑡∗ − ∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡
to get more of the outside good.

Statistical ModelIn this section, we aim to statistically represent the proposed model. We assume that
uncertainty stems from the misspecification of the utility function. That is, the utility
function specified in Equation 1 may be different from the actual one. To accommodate
this type of error, we employ the approach proposed by Kim, Allenby, and Rossi (2002).
They represented actual marginal utility as a function of model marginal utility and∗error term. We let 𝑢𝑖𝑡
be the actual and 𝑢𝑖𝑡 = 𝜕𝑈⁄𝜕𝑥𝑖𝑡 be the model marginal utility.

The relation between them is given by:∗) = ln(𝑢𝑖𝑡 ) + 𝜀𝑖𝑡 , 𝜀𝑖𝑡 ~𝑁(0,1)
ln(𝑢𝑖𝑡(14)

where 𝜀𝑖𝑡 represents the specification error which is independent across categories andpurchase occasions. Considering that budget constraint is binding and introducing theerror term into Equation 6, we obtain the link between category expenditures and errorterm:𝑚

𝐸𝑖𝑡 + 1𝜀𝑖𝑡 = ln () − ln (𝐸𝑡 − ∑ 𝐸𝑗𝑡 ) , if 𝐸𝑖𝑡 > 0
𝜓𝑖𝑡
𝑗=1

(15)

𝑚

𝜀𝑖𝑡 < ln (𝐸𝑖𝑡 + 1) − ln (𝐸𝑡 − ∑ 𝐸𝑗𝑡 ) ,𝜓𝑖𝑡

if 𝐸𝑖𝑡 = 0

𝑗=1∗
As for the second stage model, we represent the true marginal utility 𝑢𝑖𝑘𝑡
as afunction of the marginal utility of the model and error term, as in the first stage.
∗) = ln(𝑢𝑖𝑘𝑡 ) + 𝜖𝑖𝑘𝑡 , 𝜖𝑖𝑘𝑡 ~𝑁(0,1)
ln(𝑢𝑖𝑘𝑡(16)

where 𝑢𝑖𝑘𝑡 = 𝜕𝑈𝑖∗ ⁄𝜕𝑦𝑖𝑘𝑡 is the model marginal utility of brand 𝑘 in category 𝑖, and𝜖𝑖𝑘𝑡 is an independent and identically distributed error term following a standard

13

normal distribution. Thus, we substitute Equation 16 into Equation 12 to determine therelation between observed quantity and error term:(𝐼𝜌 𝜌𝑖 − 𝐼𝜔 𝜔𝑖 )𝑝𝑖𝑘𝑡
𝜙𝑖𝑘−)(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡 + 1)
(𝑧𝑖𝑡 + 1)(𝐼𝜌 𝜌𝑖 − 𝐼𝜔 𝜔𝑖 )𝑝𝑖𝑘𝑡
𝜙𝑖𝑘
𝜖𝑖𝑘𝑡 < ln(𝜆) − ln (−)(𝛾𝑖𝑘 𝑦𝑖𝑘𝑡 + 1)
(𝑧𝑖𝑡 + 1)
𝜖𝑖𝑘𝑡 = ln(𝜆) − ln (if 𝑦𝑖𝑘𝑡 > 0(17)
if 𝑦𝑖𝑘𝑡 = 0

We observe purchased quantity for each product category and brand. These quantities
are considered as the optimum values that maximize consumers’ utility at both levels∗(i.e., 𝑥𝑖𝑡∗ and 𝑦𝑖𝑘𝑡). In addition, we also observe the actual price of each brand 𝑝𝑖𝑘𝑡 andthe total expenditure 𝐸𝑡 . The likelihood function of the proposed model is a joint∗
probability of 𝐸𝑖𝑡∗ and 𝑦𝑖𝑘𝑡
. However, it can be considered as a probability of observing∗
𝑥𝑖𝑡∗ and 𝑦𝑖𝑘𝑡
given 𝑝̃𝑖𝑡 , 𝑝𝑖𝑘𝑡 , 𝐸𝑡 , and model parameters. Furthermore, this probability

can be decomposed into the probability of observing 𝑥𝑖𝑡∗ and the probability of∗observing 𝑦𝑖𝑘𝑡
conditional on 𝑥𝑖𝑡∗ .
∗
∗
∗
∗
∗
Pr({𝑥𝑖𝑡}, {𝑦𝑖𝑘𝑡}|𝛂, 𝛃, 𝛟, 𝛄, 𝛒, 𝛚) = Pr({𝑥𝑖𝑡}|𝛂, 𝛃)Pr({𝑦𝑖𝑘𝑡}|{𝑥𝑖𝑡}, 𝛟, 𝛄, 𝛒, 𝛚)∗∗}, {𝑥𝑖𝑡∗ })= 𝑙1𝑡 (𝛂, 𝛃|{𝑥𝑖𝑡 })𝑙2𝑡 (𝛟, 𝛄, 𝛒, 𝛚|{𝑦𝑖𝑘𝑡(18)

The first term in the last part of Equation 18 is the likelihood used to estimateparameters 𝛼 and 𝛽. Suppose that, without loss of generality, the first 𝑛 categories are∗purchased and the quantity of the remainder equals zero. Letting 𝑟𝑖𝑡 = ln(𝑝̃𝑖𝑡 𝑥𝑖𝑡+ 1) −ln(𝜓𝑖𝑡 ) − ln(𝐸𝑡 − ∑𝑚̃𝑗𝑡 𝑥𝑗𝑡∗ ), we can write the likelihood as follows:
𝑗=1 𝑝∗
𝑙1𝑡 (𝛂, 𝛃|{𝑥𝑖𝑡})= 𝚽(𝑟1𝑡 , ⋯ , 𝑟𝑛𝑡 )|𝐉1 | ∫𝑟(𝑛+1)𝑡

−∞

⋯∫

𝑟𝑚𝑡

−∞

𝚽(𝜀(𝑛+1)𝑡 , ⋯ , 𝜀𝑚𝑡 ) 𝜕𝜀(𝑛+1)𝑡 ⋯ 𝜕𝜀𝑚𝑡

(19)∗
where 𝐉1 is the Jacobian of change of variables from {𝜀𝑖𝑡 } to {𝑥𝑖𝑡
} and 𝚽 denotes

the multivariate normal density function.
The representation of the second likelihood is rather troublesome because we have 𝜆in Equation 17. To eliminate 𝜆, we define auxiliary variables as follows:14𝑠𝑖𝑘𝑡 = ln(𝑢𝑖𝑙𝑡 ) − ln(𝑢𝑖𝑘𝑡 )𝜐𝑖𝑘𝑡 = 𝜖𝑖𝑘𝑡 − 𝜖𝑖𝑙𝑡 ,(20)

where 𝑙 is the index for any brand having purchase quantity greater than zero. Again,
suppose that the first 𝑔 brands are purchased and the others are not. Therefore, thesecond likelihood is given by:∗}, {𝑥𝑖𝑡∗ })𝑙2𝑡 (𝛟, 𝛄, 𝛒, 𝛚|{𝑦𝑖𝑘𝑡= 𝚽(𝑠𝑖1𝑡 , ⋯ , 𝑠𝑖𝑔𝑡 )|𝐉2 | ∫𝑠𝑖(𝑔+1)𝑡

−∞

⋯∫

𝑠𝑖𝐾 𝑡

−∞

𝑖

𝚽(𝑣𝑖(𝑔+1)𝑡 , ⋯ , 𝑣𝑖𝐾𝑖 𝑡 ) 𝜕𝑣𝑖(𝑔+1)𝑡 ⋯ 𝜕𝑣𝑖𝐾𝑖 𝑡 (21)∗where 𝐉2 is the Jacobian of change of variables from {𝜖𝑖𝑘𝑡 } to {𝑦𝑖𝑘𝑡}.

Reference Price Specifications
Past studies have proposed several surrogates for reference price based on various
price clues that can be observed from data. Although most of the specifications
proposed have been applied to brand-level reference price (Kalyanaram and Winer
1995), we argue that the same approach can be applied to category level. Accordingly,
we utilize actual price data for approximating the true category reference prices. In
particular, we estimate the model with various reference prices specified as follows.
RP1: 𝑝𝑖𝑡 = 𝑝𝑖𝑏,(𝑡−1)RP2: 𝑝𝑖𝑡 = 𝜃𝑝𝑖,(𝑡−1) + (1 − 𝜃)𝑝𝑖𝑏,(𝑡−1)1

RP3: 𝑝𝑖𝑡 = 𝐾 ∑𝐾
𝑘=1 𝑝𝑖𝑘,(𝑡−1)1

RP4: 𝑝𝑖𝑡 = 𝐾 ∑𝐾
𝑘=1 𝑝𝑖𝑘𝑡
Here, 𝑏 denotes the brand purchased at the time 𝑡 − 1. RP1 is the price of brandpurchased in the previous purchase occasion. RP2 is the exponentially smoothedreference price. We determine the smoothing parameter 𝜃 by using a grid search. RP3
is the average price of focal brands in the previous purchase occasion. RP4 is the

15average price of focal brands on the day when the purchase is made. The first threereference price specifications are memory based, and the last is stimulus based.

Consumer Heterogeneity
To account for heterogeneity, we allow the parameters to vary across consumers.
Thus, the estimation is conducted for all individual-level parameters. Let ℎ be a suffix
representing the individual consumer. We define the random effects specification forheterogeneous parameters as follows.
̅ , 𝐃𝛼 ), 𝛂ℎ = (𝛼ℎ1 , 𝛼ℎ2 , … , 𝛼ℎ𝑚 )′− 𝛂ℎ ~𝑁(𝜶̅, 𝐃𝛽 ), 𝛃ℎ = vec{𝛽ℎ𝑖𝑗 }− 𝛃ℎ ~𝑁(𝛃∗∗∗∗̅ 𝑖 , 𝐃𝜙 ), 𝛟∗ℎ𝑖 = (𝜙ℎ𝑖1)′ , 𝜙ℎ𝑖𝑘 = exp(𝜙ℎ𝑖𝑘− 𝛟∗ℎ𝑖 ~ 𝑁(𝛟, 𝜙ℎ𝑖2, … , 𝜙ℎ𝑖𝐾
)
𝑖
∗
∗
∗
∗
)′ , 𝛾ℎ𝑖𝑘 = exp(𝛾ℎ𝑖𝑘− 𝛄∗ℎ𝑖 ~ 𝑁(𝛄̅𝑖 , 𝐃𝛾𝑖 ), 𝛄∗ℎ𝑖 = (𝛾ℎ𝑖1, 𝛾ℎ𝑖2, … , 𝛾ℎ𝑖𝐾)∗
̅, 𝐃𝜌 ), 𝛒∗ℎ = (𝜌ℎ1 , 𝜌ℎ2 , … , 𝜌ℎ𝑚 )′ , 𝜌ℎ𝑖 = exp(𝜌ℎ𝑖− 𝛒∗ℎ ~ 𝑁(𝛒)∗̅ , 𝐃𝜔 ), 𝛚ℎ∗ = (𝜔ℎ1 , 𝜔ℎ2 , … , 𝜔ℎ𝑚 )′ , 𝜔ℎ𝑖 = exp(𝜔ℎ𝑖− 𝛚∗ℎ ~ 𝑁(𝛚)Note that because parameters 𝜙, 𝛾, 𝜌, and 𝜔 are restricted to a non-negative value,
they are reparameterized so that the restriction can be satisfied. In the appendix, we
describe the procedure for estimating all of the hyper parameters.

DATA

We sourced the data from Customer Communications Co., Ltd, a marketing researchcompany in Tokyo, Japan. The data comprise purchases record of customers who madeshopping trips to a supermarket during one year from July 1, 2007, to June 30, 2008. We
chose four product categories for the analysis: instant coffee, coffee cream, tea, and16

instant curry. We selected 79 customers who made at least 3 purchases in each category
during the data period. This resulted in 3,821 observations in total. Table 4 shows asummary of category purchases. Among the four categories, coffee has the largest
budget share, followed by tea, curry, and cream. In each category, interior solutions
account for about one-third of purchase incidence.
[Insert Table 2 about here.]
Figure 2 shows monthly sales of the categories during the data period. The sales ofcoffee, cream, and tea appear to be correlated. In fact, the correlation coefficients aremoderately high among the three products: 0.53 for instant coffee vs. coffee cream, 0.62
for instant coffee vs. tea, and 0.66 for coffee cream vs. tea. The correlation between
each of these three products and instant curry is less than 0.29, revealing that instantcurry is independent from the remainder.
[Insert Figure 2 about here.]
We display basic information for the brands in each category in Table 5. We masked
actual brand names for confidentiality purposes. For the coffee, cream, and tea
categories, we chose five brands. For curry, we chose seven brands. We chose brands
based on the order of market share. In each category, the chosen brands account for
about 40 to 78 percent of the total sales. The last brands in the table are an aggregationof the remainder.
Most of the brands are available in various sizes. We observed that within a brand, theunit price varies across sizes. In most cases, a bigger size has a lower unit price than asmaller size does. We derived the unit price of each brand by averaging the unit pricesof different sizes. For brands in the curry category, most were priced in a similar range,
except for brand 5, which is a premium brand. This reflects a high competition intensity

17

in this category. In the coffee and tea category, the brands with the highest market share
have relatively higher prices. This indicates that consumers base their choice largely on
the quality preference rather than on price.
[Insert Table 3 about here.]

RESULT
Model Selection
The joint posterior density of the model was assessed by using Bayesian Markov
chain Monte Carlo (MCMC) methods. In Appendix A, we describe the algorithm used
for estimating model parameters in detail. For each parameter, we drew 20,000
iterations from a full conditional posterior distribution. We used the first 15,000 draws
during a burn-in period, and we kept the remainder for calculating posterior means. We
inspected chain convergence by splitting the samples kept for estimation into two
subsamples and testing the difference between the posterior means derived from them
(Geweke 1992). We confirmed that the difference was insignificant.
[Insert Table 4 about here.]
We estimated the proposed two-stage model with previously described reference
price specifications along with the one-stage model. The one-stage model corresponds
to the second stage in our proposed model, but has no any link to the first stage model.
We compared the predictive ability of each model based on three criteria:5 the log of
marginal likelihood (LML), deviance information criterion (DIC: Spiegelhalter et al.
2002), and root-mean-square error (RMSE). We computed LML and DIC using
5

We computed LML by using the overall likelihood from Equation 18 to compare the performance ofthe two-stage models. However, the computation of DIC is based only on the second term ofEquation 18 to compare the performance of all models. RMSE is computed as the root of meansquared differences between holdout sample of 𝑦𝑖𝑘𝑡 and their predictive values.

18

in-sample data to assess the goodness of fit. Furthermore, we computed RMSE using aholdout sample for validation purposes.
Table 4 presents the performances of the competing models. The values of DIC and
RMSE revealed that the two-stage models fit the data better than do the one-stage
models. The results indicate that involving consumer decisions in multiple stages leads
to a significant improvement in model fit. In addition, the model with the RP3
specification (i.e., the average price of focal brands in the previous purchase occasion)
outperforms the other two-stage models in terms of all criteria. However, the differences
are moderate, and one may find that a model with a different RP specification performs
better when applying it to different data. Next, we discuss the estimation results alongwith their implications for the best model.

First-Stage Model
We show the estimates of parameter 𝛼 in Table 5. As for the outside good, we fixed
product attractiveness at one for identification purposes, implying that the value of 𝛼
for the category equals zero. Compared to all of the categories except the outside good,
instant coffee has the greatest baseline attractiveness. With other things being equal, thisindicates that coffee is the most preferred category among the four under consideration.
The result is consistent with the fact that coffee accounted for a large portion of
consumers’ budget. However, the value of 𝛼 is not necessarily proportional to budget
share, as it also depends on category prices. A category with a relatively lower price cangain a high budget share despite having a small 𝛼. This is particularly true for the caseof curry, whose unit prices were lower than the unit prices of other categories. While its
𝛼 value is the smallest relative to the other categories, its budget share is slightly

19

greater than that of cream. In Figure 3, we display the subutility functions of thecategories when inventory levels are set to zero.
[Insert Table 5 about here.][Insert Figure 3 about here.]Now we consider the estimates of this parameter at a disaggregated level. Figure 4
shows the distributions of parameter 𝛼 over all customers. The figure shows that
customers differ in their preferences toward certain categories, indicating theimportance of accounting for consumer heterogeneity. For example, in the case of
instant coffee, the estimates range from −3.67 to −0.09. This means that although coffee
has the largest average value of 𝛼, some customers might have preferred tea to coffee.
In particular, the results reveal that the extent to which parameter 𝛼 varied overcustomers is relatively high for coffee, cream, and tea. On the other hand, the variationis smaller for curry than it is for the others.
[Insert Figure 4 about here.]
Table 6 shows the results for inventory effects 𝛽𝑗𝑖 . Self-inventory effects for instantcoffee and coffee cream are significant and have the expected signs. However, this is
not the case for the categories of tea and instant curry. The cross-inventory effect of
instant coffee on coffee cream demand is significant and has a positive sign. Thisimplies that consumers find coffee cream to be more attractive when they have enoughcoffee in storage. The effect, however, is asymmetric in the sense that the same effectdoes not apply to instant coffee when consumers have a higher inventory level of coffee
cream. The cross-inventory effect of tea to instant coffee is also significant but with anegative sign. This means that instant coffee is less attractive when consumers have ahigh inventory of tea, but not vice versa.

20Note that the cross-inventory effects of coffee cream on instant curry and of instantcurry on tea are also observed to be significant. However, we conjecture that this result
is a coincidence effect, as pointed out by Manchanda, Ansari, and Gupta (2007),
because curry is a frequently purchased category.
[Insert Table 6 about here.]

Second Stage Model
Next, we provide the parameter estimates for the second stage of the demand
allocation model. Table 7 presents the posterior means of brand attractiveness 𝜙 andsatiation parameter 𝛾. For the case of coffee, the magnitude of 𝜙 is roughly
proportional to market share. Therefore, in this category consumer preference plays asignificant role in the buying decision. In other words, consumers based theirpurchasing decision in the coffee category largely on their preference, not on brandprices. This might be because coffee is a hedonic good for which taste matters morethan price (Khan, Dhar, and Wertenbroch 2005). On the other hand, the relations
between brand attractiveness and market share for the other categories are not as strongas that observed for coffee. Consumers are more sensitive toward brand price for theseother, more utilitarian categories.
The satiation parameter varied across brands within a category. This parameter
governs the diminishing rate of marginal utility as quantity increases. The purchased
quantity of a brand with higher 𝛾 will be less than that of a brand with lower 𝛾, ceterisparibus. This is the case for both brand 1 and brand 2 in the tea category. Both brands
have similar brand attractiveness, and the former was priced higher than the latter.
However, the share of brand 1 is greater than that of brand 2 because the latter has a21

relatively higher satiation parameter.
[Insert Table 7 about here.]
Table 8 presents the estimates of parameter 𝜌 and 𝜔. The values of 𝜔 are greaterthan those of 𝜌 for the coffee, cream, and tea categories. This means that the gain fromsaving an additional dollar is greater than the loss of reducing one dollar from theoutside good expenditure. On the contrary, the result for curry is reversed. For thiscategory, consumers are more resistant to reallocate the outside good budget for curry.
We postulate that this is because curry was the least attractive category to consumers.
As a result, drawing an additional dollar from the expenditure allocated to outside goods
in order to satisfy the quantity constraint caused a significant loss in utility.
[Insert Table 8 about here.]DISCUSSION

Expenditure Effect
From the results of the first-stage model, we can analyze the marginal effect ofexpenditure on category demands. If consumers increase their expenditure by one unit,
how would this be distributed among product categories? The proportions can bẽ𝑡 ) becomputed using the Engel aggregation (Jehle and Reny 2011). Let 𝐱 𝑖𝑡 (𝐸𝑡 , 𝐩
̃𝑡 denotes a vector of theconsumers’ Marshallian demand system for the category and 𝐩̃𝑡 ∙ 𝐱 𝑖𝑡 (𝐸𝑡 , 𝐩̃𝑡 )reference prices. At optimum value, the budget constraint requires 𝐸𝑡 = 𝐩
̃𝑡 and 𝐸𝑡 . By differentiating both sides with respect to 𝐸𝑡 , we get:for all 𝐩
1 = ∑ 𝑝̃𝑖𝑡
𝑖

𝜕𝑥𝑖𝑡
𝜕𝐸𝑡

(22)

Each element in the summation refers to the proportion that will be allocated to category

22

𝑖 from the additional one-unit expenditure. Denoting each element by 𝜍𝑖 , we can
express it in terms of model parameters by using the category demand function given in
Appendix B as follows:̃𝑡 )𝜕𝑥𝑖𝑡 (𝐸𝑡 , 𝐩
𝜕𝐸𝑡
𝜓𝑖𝑡
= 𝑚
∑𝑗=1 𝜓𝑗𝑡 + 1

𝜍𝑖 = 𝑝̃𝑖𝑡(23)

Equation 23 indicates that the proportion of the additional one-unit expenditure to acategory is equal to the relative attractiveness of the category. We show the averagevalue 𝜍𝑖 of each category in Figure 5. The leftmost bar is the proportions averaged overall customers. As expected, a large portion of the expenditure is allocated to the outside
good. The proportions of other categories are relatively small and correspond withaverage budget shares.
We next split customers into three segments based on their average expenditure permonth in order to examine how the proportions relate to consumer spending. Segments
1, 2, and 3 are groups of customers with average expenditures of less than 4,000 yen,
between 4,000 and 8,000 yen, and above 8,000 yen per month, respectively. As we cansee, the segment with a greater average expenditure has a higher proportion of theoutside good and a lower proportion of the others. This means that while the amounts
spent on coffee, cream, tea, and curry increase as expenditure increases, the proportions
become smaller for customers spending more money. One explanation for this result isthat the consumption rates for these categories are relatively constant across customers,
which implies that those with higher expenditures have lower budget shares of thecategories. On the other hand, there is no significant correlation between expenditureelasticity and expenditure level. Another explanation is that “wealthier” customers do

23

not necessarily buy high-priced brands compared to their counterparts with lower
expenditure levels. Therefore, the amounts allocated to each category remain almost
unchanged in terms of total expenditure.
[Insert Figure 5 about here.]
Figure 6 shows the scatter plots between the proportions allocated to all categories
and the average expenditure per month of individual customers. The proportions
allocated to coffee, cream, and tea appear to be negatively correlated with the amounts
consumers spent per month. The correlation for curry is not as strong as that for theother categories, yet the proportion tends to decrease. On the other hand, the proportion
for the outside good approximates to one as expenditure increases. We can expect that
there is a point along the expenditure continuum beyond which the expenditure for the
focal categories will remain unchanged. We can take this for granted because there
would be a maximum quantity consumed by a consumer in a certain period.
[Insert Figure 6 about here.]
Next, we explore the effect of expenditure increase on brand purchases. We use theidentity 𝐸𝑡 = ∑𝑖 ∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡 (𝑥𝑖𝑡 , 𝑝̃𝑖𝑡 , 𝐩𝑖𝑡 ) + 𝑧𝑡 , where 𝑦𝑖𝑘𝑡 (𝑥𝑖𝑡 , 𝑝̃𝑖𝑡 , 𝐩𝑖𝑡 ) denotes thedemand function for brands given in Appendix B. We then compute the effect bydifferentiating the above identity with respect to 𝐸𝑡 . Denoting the proportion allocatedto brand 𝑘 in category 𝑖 from the additional one-unit expenditure by 𝜍𝑖𝑘 , we can
express it in terms of model parameters as follows:𝜕𝑦𝑖𝑘𝑡 (𝑥𝑖𝑡 , 𝑝̃𝑖𝑡 , 𝐩𝑖𝑡 )𝜕𝑝̃𝑖𝑡 𝑥𝑖𝑡
𝜓𝑖𝑡
1
= 𝑚
𝜙 𝛾
∑𝑗=1 𝜓𝑗𝑡 + 1{∑𝑙≠𝑘 (𝜙𝑖𝑙 𝛾𝑖𝑘 ) + 1}
𝑖𝑘 𝑖𝑙

𝜍𝑖𝑘 = 𝜍𝑖 𝑝𝑖𝑘𝑡

(24)

Now, suppose that all customers increased their expenditure by 1,000 yen on a certain

24

purchase occasion. Thus, the total increase in expenditure of 97 customers is 97,000 yen.
This induces an expenditure increase for each brand, which we can compute as 1000 ×
𝜍𝑖𝑘 . Figure 7 shows how this amount would be allocated among product categories andthen among brands. For example, out of the total increase in expenditure, 11,448 yen isallocated to the coffee category. This amount is further allocated to each brand within
the category: 2,687 yen to brand 1, 2,224 yen to brand 2, and so on. The Figure also
reveals that for coffee, the brand with the higher market share enjoys a larger portion of
the incremental expenditure. The underlying reason for this is that brands with a higher
market share have greater expenditure elasticity. However, this is not the case for the
categories of cream and tea. For curry, the incremental increase in expenditure is shared
about equally among all brands.
[Insert Figure 7 about here.]

Price Effect
While examining marginal effects of expenditure can provide insights into consumers’
decisions concerning demand allocation, expenditure is an exogenous variable. It isalmost impossible for a marketer to influence how much consumers will spend during aparticular shopping trip. Accordingly, it would be more useful if we could understandthe effect of a variable that is controllable to a marketer. Subsequently, we explore theimpact of price changes on brand demand.
Note that a change in the price of any brand can impact not only demand for thebrand but also the category reference price on the next purchase occasion. Consequently,
it can lead to a change in the budget allocated to that category. This will ultimately
induce a change in demand for the brand on the subsequent shopping trip. For this25

reason, we divide the price effect on demand into two periods; direct price effects occur
at the same purchase occasion (period 1), and indirect price effects occur through thereference price at the next purchase occasion (period 2).
To obtain direct and indirect price effects, we begin by computing the demand of eachbrand evaluated at its average price. Next, we reduce the prices by 10 percent and again
calculate demands under these new prices. We calculate the price effect in period 1 as adifference between the demand levels before and after the price reduction. We thencalculate the changes in reference price and category expenditure due to price reduction
to determine the quantity demanded on the next purchase occasion. We then reduced
this quantity by the quantity before the price reduction to obtain the price effect inperiod 2. Table 9 presents price effects in two periods for each brand in four productcategories. The numbers in the table represent an average price effect over all customers.
The results reveal that the effect in period 1 overwhelmed the effect in period 2. Thisimplies that a large portion of the effect occurs at the time the price changes, withalmost no carry over effect.
[Insert Table 9 about here.]

Budget Adjustment Assumption
We imposed the assumption of budget adjustment to ensure that the utility
maximization problems have some solutions. The condition for the assumption to holdis given in Equation 13. However, we do not restrict the parameters on the spacebounded by this condition. Instead, we estimate the parameters on unbounded space andexamine whether the results are consistent with the assumption. By doing so, we caninspect the plausibility of the assumption by assessing the probability that parameter

26

estimates lie on the subspace bounded by the assumption. To compute this probability,
we define 𝜑𝑖𝑡 as follows:′
𝜑𝑖𝑡 = 𝑈𝑖∗ (𝒚′′𝑖𝑡 )−𝑈𝑖 (𝒚𝑖𝑡 )′′𝜙𝑖𝑘
𝛾𝑖𝑘 𝑦𝑖𝑘𝑡+1=∑ln () + {𝐼𝜔 𝜔𝑖 − 𝐼𝜌 𝜌𝑖 } ln(𝑧𝑖𝑡 + 1)′𝛾𝑖𝑘
𝛾𝑖𝑘 𝑦𝑖𝑘𝑡 + 1

(25)

𝑘

′′
∗
′′
Equation 13 says that when either ∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡
> 𝑝̃𝑖𝑡 𝑥𝑖𝑡
or ∑𝑘 𝑝𝑖𝑘𝑡 𝑦𝑖𝑘𝑡
< 𝑝̃𝑖𝑡 𝑥𝑖𝑡∗ ,consumers will adjust their budget if 𝜑𝑖𝑡 > 0. Therefore, to determine whether theassumption holds, it is sufficient to assess the probability that the parameters satisfy the𝑐condition requirement (i.e., Pr(𝜑𝑖𝑡 > 0)). Letting 𝜑𝑖𝑡be the value of 𝜑𝑖𝑡 at the 𝑐-th

iteration, we compute the probability as follows:
𝐶

𝑇

1
𝑐
Pr(𝜑𝑖𝑡 > 0) ≅∑ ∑ 𝐼(𝜑𝑖𝑡
> 0)
𝐶𝑇

(28)

𝑐=1 𝑡=1

We observed that out of 97 customers, more than 90 have probabilities greater than
0.8. For the tea category, all customers have probabilities greater than 0.8. The results
reveal that even without restrictions imposed on the parameters, the estimates satisfy theassumption condition with high probability. Accordingly, we conclude that theassumption is highly plausible.

CONCLUDING REMARKSIn this paper, we proposed an integrated model of expenditure allocation amongproduct categories and brands. We applied the model to purchase history data andexamined its performance for different specifications of reference price. We also
compared its performance with the one-stage model and found that the proposed model
has a better predictive ability. The estimation results of category and brand27

attractiveness are consistent with our expectation about the roles of preference and pricein consumer purchase decisions. We observed that baseline preference explained a large
portion of variation in budget and market share. However, for some categories andbrands, their prices also substantially influenced marginal utility per dollar, which
ultimately affected their shares. In addition, we observed that the satiation parameter
had a significant effect on the sales of some brands, indicating the importance ofconsidering consumers’ preference for variety.
Our framework provides a tool for exploring the marginal effect of expenditure inseveral decision levels. The empirical results demonstrated how an incremental increasein expenditure would be distributed among categories and brands. The advantage of themulti-stage demand model is that it can aid marketers in anticipating demand changes
when expenditures increase due to a change in exogenous factors, such as economic
conditions. We also examined how a change in brand price could lead to a change in the
demand for both brand and category expenditures during the subsequent shopping trip.
Our analysis indicated that the price effect on category expenditure was small compared
to its effect on quantity demanded.
One of the major issues in modeling consumers’ decision making in multiple stages
is linking the decisions over different stages. An intuitive way to cope with this issue isto treat the solutions in the previous stage as additional constraints in the subsequent
stage. However, with this treatment, in most cases the solutions in the subsequent stage
that satisfy all constraints cannot be attained. We handled this problem by assuming that
consumers flexibly adjust the expenditure that has been allocated to the outside good toarrive at the solutions that satisfy all constraints. We argued that this is a reasonableassumption because the outside good usually contains some low-priority products

28

whose expenditure can be flexibly altered. Furthermore, there is a high probability thatthe parameters will take values that satisfy the condition required for the assumption to
hold.
The proposed model can be extended to handle consumer decisions in more than
two stages. However, data availability would be a major obstacle to doing so.
Furthermore, our model is built on the assumption that consumers first decide toallocate expenditure among categories and then among brands, which is a type ofplanned buying behavior. Additional research is necessary to examine how the results
would vary if we imposed the assumption of unplanned purchase behavior—that is, a
two-stage model in which consumers decide which and how much of the brand topurchase first and then subsequently decide the budget allocated to each category. We
leave these extensions to future research.

29

REFERENCES

Ailawadi, K. L., K. Gedenk, C. Lutzky, and S. A. Neslin (2007), “Decomposition of theSales Impact of Promotion-Induced Stockpiling,” Journal of Marketing Research, 44(3), 450–467.
Barten, A. P. (1993), “Consumer Allocation Models: Choice of Functional Form,”
Empirical Economics, 18, 129–158.
Bucklin, R., and Lattin, J. M. (1991), “A Two-State Model of Purchase Incidence andBrand Choice,” Marketing Science, 10 (Winter), 24–39.
Chib, S., P. B. Seetharaman, and A. Strijnev (2004), “Model of Brand Choice with a
No-Purchase Option Calibrated to Scanner-Panel Data,” Journal of Marketing
Research, 41 (2), 184–196.
Christensen, L. R., D. W. Jorgenson, and L. J. Law (1975), “Transcendental
Logarithmic Utility Functions,” American Economic Review, 5, 367–382.
Deaton, A., and J. Muellbauer (1980), “An Almost Ideal Demand System,” American
Economic Review, 70, 312–326.
Geweke, J. (1992), “Evaluating the Accuracy of Sampling-Based Approaches to the
Calculation of Posterior Moments,” In Bayesian Statistics 4, ed. J. M. Bernardo, J. O.
Berger, A. P. Dawid, and A. F. M. Smith, Oxford: Oxford University Press, 169–193.
Hasegawa, S., N. Terui, and G. M. Allenby (2012), “Dynamic Brand Satiation,” Journalof Marketing Research, 48, 842–853.
Hauser, J. R., and G. L. Urban (1986), “The Value Priority Hypotheses for Consumer
Budget Plans,” Journal of Consumer Research, 12, 446–462.
Jehle, G. A., and P. J. Reny (2011), Advanced Microeconomic Theory, 3rd edition. New

30

York: Prentice Hall.
Kalyanaram, G., and R. S. Winer (1995), “Empirical Generalization from Reference
Price Research,” Marketing Science, 14 (3), 161–169.
Khan, U., R. Dhar, and K. Wertenbroch (2005), in Inside Consumption: Frontiers of
Research on Consumer Motives, Goals, and Desires, ed. S. Ratneshwar and DavidGlen Mick, London: Routledge, 144-165.
Kim, J., G. M. Allenby, and P. E. Rossi (2002), “Modeling Consumer Demand for
Variety,” Marketing Science, 21 (3), 229–250.
Lee, S., and G. M. Allenby (2014), “Modeling Indivisible Demand,” Marketing Science,
Article in Advance, 1–18.
Lee, S., J. Kim, and G. M. Allenby (2013), “A Direct Utility Model for AsymmetricComplements,” Marketing Science, 32 (3), 454–470.
Manchanda, P., A. Ansari, and S. Gupta (1999), “The Shopping Basket: A Model for
Multicategory Purchase Incidence Decisions,” Marketing Science, 18, 95–114.
Mehta, N. (2007), “Investigating Consumers’ Purchase Incidence and Brand ChoiceDecisions Across Multiple Product Categories: A Theoretical and Empirical Analysis,”Marketing Science, 26 (2), 196–217.
Satomura, T., J. Kim, and G. M. Allenby (2011), “Multiple Constraints Choice Modelwith Corner and Interior Solutions,” Marketing Science, 30 (3), 481–490.
Song, I., and P. K. Chintagunta (2007), “A Discrete-Continuous Model for
Multicategory Purchase Behavior of Customers,” Journal of Marketing Research, 44,
595–612.
Spiegelhalter, D. J., N. G. Best, B. P. Carlin, and A. Van Der Linde (2002), “Bayesian
Measures of Model Complexity and Fit,” Journal of the Royal Statistical Society:31

Series B, 64 (4), 583–616.
Stone, R. (1954), “Linear Expenditure Systems and Demand Analysis: An Application
to the Pattern of British Demand,” The Economics Journal, 64, 511–527.

32

Table 1. Summary of the Selected Studies on Demand Model

No. ReferenceType ofdecision

NumberofRemarks
stages1

Stone (1954)

Demand allocation

1 stage

2

Christensen, Jorgenson,

Demand allocation

1 stage

Demand allocation

1 stageand Law (1975)
3

Deaton and Muellbauer(1980)

4

Hauser and Urban (1986)

Demand allocation

1 stage

5Kim, Allenby, and Rossi

Demand allocation

1 stage

Category purchase2 stages Corner solutions(2002)
6

Bucklin and Lattin (1991)

Brand choice7

Bucklin and Gupta (1992)

Category purchase2 stages Corner solutions

Brand choice89Chib, Seetharaman, andCategory purchase

Strijnev (2004)

Brand choice

Mehta (2007)

Category purchase
Brand choice

2 stages Corner solutions

2 stages Corner solutions
for brandpurchase

1011Song and Chintagunta

Category purchase(2007)

Brand choice

for brand

Purchase quantity

purchase

Demand allocation

2 stages Sequential

Lee, Kim, and Allenby(2013)

2 stages Corner solutions

allocation

33

Table 2. Purchase Summary
Product

Purchase

Total

Interior

category

incidence

expenditure

solution

Instant coffee

1,491

709,101

475

Coffee cream

1,172

250,242

377

Tea

916

325,813284

Instant curry

978251,816

253

34

Table 3. Basic Statistics of Brands

Brand

Instant coffee

Coffee cream

Tea

Share

Share

Share

Unit
price

Unit
price

Instant curry
Unit

Share

price

Unit
price

Brand 1

0.27

5.27

0.31

0.79

0.27

4.98

0.07

0.70

Brand 2

0.13

3.32

0.11

0.69

0.09

3.52

0.06

0.77

Brand 3

0.08

3.56

0.21

1.67

0.07

2.88

0.06

0.73

Brand 4

0.08

3.31

0.09

0.83

0.12

1.81

0.05

0.93

Brand 5

0.05

7.04

0.09

0.81

0.04

3.77

0.06

2.27

Brand 6

0.40

2.69

0.19

1.12

0.40

4.61

0.05

0.80

Brand 7

0.04

0.74

Brand 8

0.61

1.30

35

Table 4. Data Fit and Predictive Ability

Model

LML

DIC

RMSE

(In-sample)(In-sample)(Holdout
sample)Two-stage model:
with RP1−121,332

84,121

980.64

with RP2−122,125

84,063979.57

with RP3−119,319

83,913976.39with RP4−120,806

84,142981.2087,730992.08One-stage model:

36

Table 5. Parameter Estimates of Category Attractiveness𝑒𝑥𝑝(𝛼)

Product category

Post mean

Post STD

Instant coffee

−1.90

0.27

0.15

Coffee cream

−2.79

0.11

0.06

Tea

−2.43

0.10

0.09

Instant curry

−2.82

0.50

0.06

Outside good

0

fixed

1.00

37

Table 6. Parameter Estimates of Inventory EffectInstant

CoffeeTea

Instant currycoffeecream

Instant coffee−0.14 (0.05)

0.14 (0.06)−0.04 (0.07)

0.01 (0.06)

Coffee cream−0.09 (0.07)−0.21 (0.08)−0.11 (0.07)−0.27 (0.06)

Tea−0.21 (0.07)−0.08 (0.07)−0.07 (0.08)

0.13 (0.07)

Instant curry−0.01 (0.06)

0.00 (0.05)−0.20 (0.08)

0.07 (0.07)

Inventory

Posterior standard deviations are in parentheses. Bold indicates significant parameter.

38

Table 7. Brand Attractiveness and Satiation Parameters
Instant coffee

Coffee cream

Tea

Instant curry

Brand

Brand 1

Brand 2

Brand 3

Brand 4

Brand 5

Brand 6

𝜙

𝛾

𝜙

𝛾

𝜙

𝛾

𝜙

𝛾

3.47

1.45

1.09

1.41

1.04

0.68

0.90

1.43

(0.69)(0.15)(0.12)(0.24)(0.15)(0.08)(0.07)(0.17)

2.02

0.68

2.12

1.10

1.18

2.03

1.24

1.42

(0.41)(0.09)(0.22)(0.11)(0.10)(0.31)(0.10)(0.13)

2.31

0.90

0.85

2.24

3.12

0.85

1.66

1.92

(0.34)(0.14)(0.13)(0.40)(0.61)(0.05)(0.17)(0.25)

1.59

1.07

0.98

1.65

1.44

0.78

0.95

1.08

(0.27)(0.23)(0.11)(0.33)(0.16)(0.11)(0.07)(0.11)

1.23

1.08

2.47

1.27

1.43

1.25

1.43

1.45

(0.14)(0.18)(0.31)(0.26)(0.21)(0.08)(0.09)(0.21)

1.20

1.45

2.36

3.19

2.74

1.74

1.53

1.84

(0.01)(0.27)(0.27)(0.67)(0.39)(0.26)(0.15)(0.25)

1.67

2.07(0.09)(0.17)

2.78

1.04(0.27)(0.12)

Brand 7

Brand 8Posterior standard deviations are in parentheses.

39

Table 8. Estimates of Loss and Gain Parameters
Loss parameter 𝜌

Gain parameter 𝜔Posterior mean Posterior STD Posterior mean Posterior STD
Instant coffee

1.96

0.78

2.34

0.65

Coffee cream

0.04

0.03

0.89

0.23

Tea

0.70

0.14

0.94

0.30

Instant curry

1.58

0.36

0.43

0.17

40

Table 9. Price Effects
Instant coffee

Coffee cream

Tea

Instant curry

Brand
Period 1

Period 2

Total Period 1

Period 2

Total

Period 1

Period 2

Total Period 1

Period 2

Total

Brand 1

222

25

247

269

12

281

63

1

64

211

5

217

Brand 2

259

25

284

724

60

784

63

2

65

234

10

244

Brand 3

232

21

253

96

4

100

375

42

417

380

24

403

Brand 4

319

31

350

231

10

241

325

26

351

194

6

200

Brand 5

60

2

62

901

94

995

80

3

83

135

7

142

Brand 6

187

11

198

556

61

617

156

18

174

265

13

278

Brand 7

245

8

252

Brand 8

355

30

385

41

Figure 1. Budget and Quantity Constraint

42

Figure 2. Monthly Sales
90,00080,000Monthly Sales

70,00060,00050,000CoffeeCream

40,000Tea

30,000Curry

20,00010,000043Figure 3. Subutility Functions
0.50

Subutility

0.45

Coffee

0.40

Cream

0.35

Tea
Curry

0.30
0.25
0.20
0.15
0.10
0.05
0.00
0

5

10

15

Quantity

20

25

44

Figure 4. The Distribution of Baseline Preferences toward Category

45

Figure 5. The Allocation of Additional One-Unit Expenditure
Outside good
4.4%

Coffee

9.8%

5.2%

Curry

3.4%
4.5%
3.1%
7.3%

4.3%

7.5%

11.8%

Tea

4.1%
6.2%

5.3%

7.2%

Cream

10.1%

16.5%

81.7%

75.3%

71.4%
60.9%

All

Segment 1

Segment 2

Segment 3

46

Figure 6. Proportions Allocated by Individual Customers

47

Figure 7. Allocation of Additional Expenditure among Categories and Brands

48

APPENDIX A: MCMC ALGORITHM

We estimated the proposed model by using the MCMC method. We defined the priordistribution for each parameter in the model description. We assume that the
distributions are centered at the population mean. In particular, the hyper parameters are
updated at each step within the chain by using the current value of individual parameters.
Let 𝑐 = (1,2, … , C) be the number of MCMC iterations and 𝑑 and 𝐷 be prior values.
Thus, we can write the distributions of the hyper parameters at the 𝑐-th step as follows:

(𝑐)
∑𝐻
𝐃𝛼
(𝑐)
ℎ=1 𝛂ℎ̅|{𝛂ℎ }, 𝐃𝛼 ) = 𝑁 (
𝜋 (𝛂
, )

𝐻

(𝑐)

̅ ) = 𝐼𝑊 (𝑑𝛼 + 𝐻, 𝐷𝛼 + ∑𝜋 (𝐃𝛼 |{𝛂ℎ }, 𝛂(𝑐)

̅|{𝛃 }, 𝐃𝛼 ) = 𝑁 (𝜋 (𝛃ℎ(A. 1)

𝐻

𝐻

(𝑐)

(𝑐)

′

̅ )(𝛂ℎ − 𝜶̅) )(𝛂ℎ − 𝜶(A. 2)

ℎ=1

(𝑐)
∑𝐻
𝐃𝛽
ℎ=1 𝛃ℎ
, )
𝐻
𝐻

(𝑐) ̅
𝜋 (𝐃𝛽 |{𝛃ℎ }, 𝛃
) = 𝐼𝑊 (𝑑𝛽 + 𝐻, 𝐷𝛽 + ∑̅ 𝑖 |{𝛟∗(𝑐) }, 𝐃𝜙 ) = 𝑁 (
𝜋 (𝛟
ℎ𝑖
𝑖

𝐻

(𝑐)

′

̅)(𝛃 − 𝛃̅) )(𝛃ℎ − 𝛃ℎ(A. 4)

ℎ=1

∗(𝑐)
𝐃𝜙
∑𝐻
ℎ=1 𝛟ℎ𝑖, 𝑖)𝐻𝐻̅𝜋 (𝐃𝜙𝑖 |{𝛟∗(𝑐)ℎ𝑖 }, 𝛟𝑖 ) = 𝐼𝑊 (𝑑𝜙𝑖 + 𝐻, 𝐷𝜙𝑖 + ∑𝜋 (𝛄̅𝑖 |{𝛄∗(𝑐)ℎ𝑖 }, 𝐃𝛾𝑖 ) = 𝑁 ((𝑐)(A. 3)

𝐻(A. 5)′∗(𝑐)̅̅(𝛟∗(𝑐)ℎ𝑖 − 𝛟𝑖 )(𝛟ℎ𝑖 − 𝛟𝑖 ) ) (A. 6)

ℎ=1

∗(𝑐)
𝐃𝛾
∑𝐻
ℎ=1 𝛄ℎ𝑖, 𝑖)
𝐻𝐻̅ 𝑖 𝑖 ) = 𝐼𝑊 (𝑑𝛾𝑖 + 𝐻, 𝐷𝛾𝑖 + ∑𝜋 (𝐃𝛾𝑖 |{𝛄∗(𝑐)ℎ𝑖 }, 𝛄𝐻′̅ 𝑖 )(𝛄∗(𝑐)̅ 𝑖 ) ) (A. 8)(𝛄∗(𝑐)ℎ𝑖 − 𝛄ℎ𝑖 − 𝛄

ℎ=1

∗(𝑐)
∑𝐻
𝐃𝜌
ℎ=1 𝛒ℎ∗(𝑐)
̅|{𝛒ℎ }, 𝐃𝜌 ) = 𝑁 (
𝜋 (𝛒
, )

𝐻

(A. 7)

𝐻(A. 9)

49̅) = 𝐼𝑊 (𝑑𝜌 + 𝐻, 𝐷𝜌 + ∑𝜋 (𝐃𝜌 |{𝛒∗(𝑐)ℎ }, 𝛒

𝐻′

̅)(𝛒∗(𝑐)̅) )(𝛒∗(𝑐)−𝛒−𝛒ℎℎ(A. 10)

ℎ=1

∗(𝑐)
∑𝐻
𝐃𝜔
ℎ=1 𝛚ℎ∗(𝑐)̅ |{𝛚ℎ }, 𝐃𝜔 ) = 𝑁 (
𝜋 (𝛚
,
)

𝐻

̅ ) = 𝐼𝑊 (𝑑𝜔 + 𝐻, 𝐷𝜔 + ∑𝜋 (𝐃𝜔 |{𝛚∗(𝑐)ℎ }, 𝛚

(A. 11)𝐻

𝐻′̅ )(𝛚ℎ∗(𝑐) − 𝛚̅ ) ) (A. 12)(𝛚ℎ∗(𝑐) − 𝛚ℎ=1

We use the Metropolis–Hastings method to assess the posterior distributions. A new
candidate is augmented by using the random walk algorithm, where the innovation
variance was set as 0.9. The posterior means and standard deviations of the parameters
were obtained from their posterior distributions we listed below.
∗}, 𝛂̅ , 𝐃𝛼 , 𝛃 ℎ
1. 𝛂ℎ |{𝑥ℎ𝑖𝑡∗̅, 𝐃𝛽 , 𝛂ℎ}, 𝛃2. 𝛃ℎ |{𝑥ℎ𝑖𝑡∗∗̅ 𝑖 , 𝐃𝜙 , 𝛄∗ℎ𝑖 , 𝛒∗ℎ , 𝛚∗ℎ}, {𝑦ℎ𝑖𝑡}, 𝛟
3. 𝛟∗ℎ𝑖 |{𝑥ℎ𝑖𝑡𝑖∗∗}, {𝑦ℎ𝑖𝑡}, 𝛄̅𝑖 , 𝐃𝛾𝑖 , 𝛟∗ℎ𝑖 , 𝛒∗ℎ , 𝛚∗ℎ4. 𝛄∗ℎ𝑖 |{𝑥ℎ𝑖𝑡∗∗}, {𝑦ℎ𝑖𝑡}, 𝛒̅, 𝐃𝜌 , 𝛟∗ℎ𝑖 , 𝛄∗ℎ𝑖 , 𝛚ℎ∗5. 𝛒∗ℎ |{𝑥ℎ𝑖𝑡∗∗}, {𝑦ℎ𝑖𝑡}, 𝛚̅ , 𝐃𝜔 , 𝛟∗ℎ𝑖 , 𝛄∗ℎ𝑖 , 𝛒∗ℎ6. 𝛚∗ℎ |{𝑥ℎ𝑖𝑡50

APPENDIX B: DEMAND FUNCTION

We derived the Marshallian demand functions as the solutions of the direct utility
maximization problems in Equations 4 and 10. The demand for category 𝑖 is the
optimum quantity that maximizes overall utility whose function is given by:𝜓𝑗𝑡
𝐸𝑡 − ∑𝑗≠𝑖 (𝜓 − 1)𝜓𝑖𝑡𝑖𝑡𝑥𝑖𝑡 (𝐸𝑡 , 𝑝̃𝑖𝑡 ) = 𝑚,∑𝑗=1 𝜓𝑗𝑡 + 1
𝑝̃𝑖𝑡(B. 1)

if its marginal utility per dollar is equal to 𝜆, and 𝑥𝑖𝑡 (𝐸𝑡 , 𝑝̃𝑖𝑡 ) = 0 if otherwise. We
derive brand demand function when the difference between quantity and budget
constraint is set to zero (i.e., 𝑧𝑖𝑡 = 0). The demand function of brand 𝑘 is given by:𝑝𝑝̃𝑖𝑡 𝑥𝑖𝑡 + ∑𝑙≠𝑘 ( 𝛾𝑖𝑙𝑡 )𝜙∑𝑙≠𝑘 ( 𝑖𝑙 )𝛾𝑖𝑙𝑖𝑙𝑦𝑖𝑘𝑡 (𝑥𝑖𝑡 , 𝑝̃𝑖𝑡 , 𝐩𝑖𝑡 ) =−,𝜙𝑖𝑙 𝛾𝑖𝑘𝜙𝑖𝑙 𝛾𝑖𝑘1
𝑝𝑖𝑘𝑡 {∑𝑙≠𝑘 (𝜙 𝛾 ) + 1} {∑𝑙≠𝑘 ( 𝛾 ) + 𝜙 }
𝑖𝑘 𝑖𝑙
𝑖𝑙
𝑖𝑘
if its marginal utility per dollar is equal to 𝜆, and 𝑦𝑖𝑘𝑡 (𝑥𝑖𝑡 , 𝑝̃𝑖𝑡 , 𝐩𝑖𝑡 ) = 0 if
otherwise. 𝐩𝑖𝑡 = (𝑝𝑖1𝑡 , 𝑝𝑖2𝑡 , … , 𝑝𝑖𝐾𝑡 )′ is a vector of brand prices.

(B. 2)