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† The authors are Assistant Professor of Marketing, Philip L. Siteman Professor of Marketing at Washington University

in St. Louis, and Assistant Professor of Marketing at University of Texas at Dallas. Authors are listed in an alphabetical

order. Corresponding author: Qin Zhang, ZhangQ@utdallas.edu, Tel: 972-883-6525.

‡The authors thank the late Dick Wittink for his encouragement and wisdom. The authors also would like to thank Joel

Huber and the two anonymous JMR reviewers, participants at the Second Quantitative Marketing and Economics

conference, and Scott Neslin for their excellent comments and suggestions.

DECOMPOSING PROMOTIONAL EFFECTS WITH A DYNAMIC

STRUCTURAL MODEL OF FLEXIBLE CONSUMPTION

Tat Chan

Chakravarthi Narasimhan

Qin Zhang†, ‡

November, 2007

Forthcoming, JMR

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ABSTRACT

In this article, the authors offer a methodology to decompose the effects of price

promotions into brand switching, stockpiling, and change in consumption by explicitly

allowing for consumer heterogeneity in brand preferences and consumption needs. They

develop a dynamic structural model of a household that decides when, what, and how

much to buy, as well as how much to consume, to maximize its expected utility over an

infinite horizon. By making certain simplifying assumptions, the authors reduce the

dimensionality of the problem. They estimate the proposed model using household

purchase data in the canned tuna and paper towels categories. The results from the model

offer insights into the decomposition of promotional effects into its components. This

could help managers make inferences about which brand‟s sales are more responsive to

stockpiling or increase in consumption expansion and how temporary price cuts affect

future sales. Contrary to previous literature, the authors find that brand switching is not

the dominant force for the increase in sales. They show that brand-loyal consumers

respond to a price promotion mainly by stockpiling for future consumption, whereas

brand switchers do not stockpile at all. The authors also find that heavy users stockpile

more, whereas light users mainly increase consumption when there is a price promotion.

Key words: decomposition of promotional effects, dynamic structural model, flexible

consumption, consumer heterogeneity.

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It is well known that store-level sales respond positively to short-term price

promotions. Because different households respond to promotions differently, researchers

have developed household-level models to understand the heterogeneity in purchase

behavior. Following the seminal work of Guadagni and Little (1983), researchers (e.g.,

Bell, Chiang, and Padmanabhan 1999; Bucklin, Gupta, and Siddarth 1998; Chiang 1991;

Chintagunta 1993; Gupta 1988) have explored the effects of price promotions using

household scanner data and have decomposed the short-term effects into brand switching,

purchase acceleration, and an increase in purchase quantity, with an increase in purchase

quantity typically arising out of stockpiling behavior by households. All this research has

assumed that the consumption rate of households is invariant to changes in prices.

However, recent research has shown that households‟ consumption rates are also affected

positively by price promotions (Ailawadi and Neslin 1998; Bell, Iyer, and Padmanabhan

2002; Sun 2005). The increase in consumption rate could be due to cross-category

substitution as well as to a desire to consume more in the category. The latter effect may

be caused by an income effect due to a lower price or may simply be induced by having a

larger inventory. The drivers of the increase in sales due to a temporary price promotion

need to be identified to understand the impact on manufacturers‟ and retailers‟ profits.

For example, if the increase in sales is primarily due to an increase in consumption, both

manufacturers and retailers might be better off from a temporary price promotion. If the

effect is mostly brand switching, the manufacturer of the promoted brand is better off, but

the retailer may or may not be. If the increase in sales comes mostly from stockpiling, the

impact on profitability is ambiguous for both the manufacturer and the retailers, and

further investigation is needed (Van Heerde, Leeflang, and Wittink 2004). Furthermore,

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if households that have a high preference for a particular brand stockpile the product for

future use, it represents a loss in profits to the manufacturer. Similarly, if households that

already have a high consumption level stockpile rather than expand their consumption,

this again could represent a loss in profits. Thus, modeling flexible consumption in

household behavior is important because, otherwise, an increase in sales may erroneously

be attributed to other factors, resulting in misleading implications for manufacturers and

retailers.

The foregoing discussion identifies two important research issues that need to be

addressed. First, a model of household behavior in which consumption can be flexible

based on external environment (e.g., prices, display, feature advertisements) and internal

resources (e.g., income, inventory holding costs) should be developed to quantify the

sources of increase in sales of a product due to temporary price promotions. Second, the

differences in household behavior (e.g., increasing consumption, brand switching,

stockpiling) based on brand preferences and overall usage rates need to be incorporated

to identify correctly which households respond in what way to temporary promotions.

Such a model has potential implications for managers deciding whether to offer any

promotions, in what form they should be offered, and to whom they should be potentially

targeted. Our goal in this article is to develop a household-level model that enables us to

draw inferences to address these important questions.

We develop a dynamic structural model of a household that maximizes

discounted utility from its consumption in a category. Consumption in each period is

endogenous and is based on current marketing mix, inventory levels, preferences, and the

household‟s expectations about future prices. In deciding whether to buy today, a

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household trades off inventory cost against the opportunity cost of buying the product in

the future at a possibly higher price. Our model incorporates heterogeneity across

households in inventory cost, price sensitivity, and underlying preferences. We propose a

solution to overcome the dimensionality problem in our dynamic optimization model by

imposing some simplifying assumptions. We apply our model to household-level scanner

panel data in the canned tuna and paper towels categories.

We run simulations using the estimates from our model. We are able to

decompose the total effect of a price promotion into its components of increase in

consumption, brand switching in current and future periods, and stockpiling. Our analysis

yields several notable insights. First, contrary to what is shown in most previous literature

(e.g., Chiang 1991; Gupta 1988; Sun 2005), brand switching is not the dominant effect.1

Second, for larger-share brands, the majority of a promotion-induced sales increase is

attributed to stockpiling. Third, for smaller-share brands, the consumption effect is

greater. Furthermore, we show that a household‟s brand preference has a significant

impact on its stockpiling and flexible consumption behavior. Brand loyals mainly

respond to a price promotion by stockpiling, whereas brand switchers do not stockpile.

We also find that heavy users stockpile more for future consumption, whereas light users

have a larger consumption increase under the price promotion. These findings have

important implications for pricing and promotion strategies for manufacturers and

retailers.

We organize the rest of the article as follows: In the next section, we describe

related research and position our contribution with respect to this literature. Following

this, we describe our dynamic structural model. Then, we discuss the data and details of

our model. We also discuss the results of policy simulations and then provide some

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suggestions to managers based on our findings. Finally, we conclude with some

extensions and directions for further research. Technical details of the estimation are

provided in the appendices.

LITERATURE REVIEW AND CONTRIBUTIONS

Gupta (1988) was the earliest to decompose the promotional responses into brand

switching, purchase (incidence) acceleration and stockpiling effects using household-

level data in the coffee category. He found that the dominant force was brand switching

accounting for 84% of the change in response, while purchase acceleration accounted for

14% and stockpiling accounted for 2%. Similar results were reported by Chiang (1991)

and in the cross-category study by Bell, Chiang and Padmanabhan (1999). Van Heerde,

Gupta and Wittink (2003) propose a different decomposition measure based on unit sales.

Using the same dataset as Gupta (1988) did but at the store-level, they found that only

33% of unit sales increase is due to brand switching. Sun, Neslin and Srinivasan (2003)

find that the brand-switching elasticities are overestimated in reduced-form models and to

correct for this bias they develop a dynamic structural model that accommodates

consumers‟ forward-looking behavior under promotion uncertainty. These papers

assume that consumption rate of a household is a constant and does not change as a result

of price reduction. Van Heerde, Leeflang and Wittink (2004) propose a regression model

to decompose store-level sales increase due to price promotions into cross-period

(stockpiling), cross-brand (brand switching), and category expansion (consumption)

effects. They find that each effect accounts for one third on average in the four

categories (tuna, tissue, shampoo and peanut butter) they examined. Such store-level

models help us to understand aggregate effects of price promotions but lack the

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advantages of household-level models that can explicitly account for observed and

unobserved household heterogeneity in inventory cost, price sensitivity and underlying

preferences. These models may suffer from the “over-parameterization” problem if one

wants to estimate cross-substitution patterns among many products. The proposed

household-level model enables us to explore the differences in promotional responses

across households even when there are a large number of products in the choice set.

Thus, we are able to add to this literature by documenting the link between promotional

responses, especially changes in consumption, and household behavior.

There is a stream of literature that models flexible consumption under price

uncertainty using dynamic structural models based on household-level scanner data.

Erdem, Imai and Keane (2003) assume that households have an exogenous usage

requirement in each period that is revealed to households after their purchases. The focus

of their model is to study how inventory and future price expectations affect a

household‟s purchase decisions. Sun (2005) models consumption as an endogenous

decision and explores how price promotions affect this. While we also model

consumption as an endogenous decision our focus on promotional effects and the

decomposition leads us to interesting insights about how heterogeneity in brand

preferences and consumption needs affects promotional responses. Moreover, to

overcome the problem of “curse of dimensionality” in the dynamic programming due to a

large choice set and a large number of panel members, we adopt a hedonic approach in

modeling households‟ utility and invoke a few assumptions in the decision process of

households to simplify the optimization problem.

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MODEL

The Household’s Problem

Let J be the number of products and H the number of households in the market. Let

yht be the J×1 vector of household h‟s consumption quantity, xht a J×1 vector of quantity

purchased, both assumed to be continuous,2 and uht(·) be the utility function of

consumption, in period t .

At time t household h decides on whether to purchase, which product to purchase,

how much to purchase, and how much to consume. Since products can be bought today

for consumption in future periods, purchasing and consumption decisions in current

period will affect the inventory that a household holds and change the implicit cost of

future consumption. This creates a dynamic linkage among decisions across periods.

Formally, the dynamic planning problem at time t for the household h can be stated as

follows:

sup{xs, ys}Et{

ts

tsγ

[uh(yhs) − λh ps′ xhs − ch (

J

1j

hsj

I

)]|ht ]} (1)

s.t. Ihs = xhs + Ih,s-1 – yhs

xhs, yhs, Ihs 0

where ps is a J×1 vector of prices, and Ihs a J×1 vector of the inventory levels of products

in period s. We use λh to denote the marginal utility of income and ch the inventory cost

of one standardized unit for household h, and γ is the discount factor that is common for

all households. Et{|ht} is the expectation operator conditional on the information set at t,

ht. The information set includes the inventory inherited from previous period, Ih,t-1,

2 This is an approximation of the fact that households only make discrete unit purchases. Similar assumption has been

made by Kim, Allenby and Rossi (2002) and Chan (2006).

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current and past marketing mix variables such as prices, features and displays, and

household demographic variables. The endogenous decision variables in Equation 1

include {xht, xh,t+1,…; yht, yh,t+1,…}, where each component is a J×1 vector, which is subject

to the non-negativity constraints.

The Indirect Utility Function

We use the hedonic approach to model a household‟s expected utility of

consumption at some future period s evaluated at time t, s≥t, as follows:

Etuh(yhs) = ( Ψht′ Α′ yhs + φh)

h

(1)

where, A is a J×(C+J) characteristic matrix. The first C columns represent observable

product attributes such as brand names and flavors. The last J×J sub-matrix in A is an

identity matrix each diagonal element of which indicates the existence of the

corresponding unobserved product attribute. Ψht is a vector of household-specific and

time-varying coefficients consisting of a C×1 vector of household h‟s preferences for the

observed attributes, ψht, and a J×1 vector of preferences for the unobserved attributes, ξht.

Hence, Ψht = (ψht| ξht ). We do not model state-dependence that might arise, for example,

due to variety seeking behavior (see Seetharaman (2004)) or the increase of repeat future

purchases of promoted brands (see Ailawadi et al. (2007)), to keep the model tractable

and not let the dimensionality explode.

Given Ψht and A, h and φh determine the curvature and intercept of the marginal

utility − the marginal utility with respect to consumption of product j is

h (Ψht′ Αj′) (Ψht′ Α′ yhs + φh )

1

h

, where Aj is the jth row of the characteristic matrix,

A. When the consumption is zero, i.e., yhs = 0, it equals h (Ψht′ Αj′) φh

1

h

. To

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guarantee diminishing marginal returns (i.e., concavity), h is restricted to be between 0

and 1. A household with a larger is likely to consume more of the product category

than those with a smaller . To allow for corner solutions (i.e., zero purchases), φh is

restricted to be positive. Since it is difficult to separately identify both h and φh, we

follow Kim, Allenby and Rossi (2002) and fix φh to be 1.

Let the household-specific coefficients in Equation 1 and 2 be θh ≡ (h, λh, ch), and

let Zh be a vector of demographic variables for household h, we assume that θh = g(Zh, θ0),

where g() is a vector of functions, and θ0 is a vector of parameters to be estimated. We

will discuss the model details later.

The Proposed Solution to the Dynamic Optimization Problem

It is difficult to solve the dynamic optimization problem in Equation 1 when there

are a large number of products in the choice set because of the “curse of dimensionality”.

Previous empirical research typically relies on either product aggregation or some other

simplifying assumptions.3 In our application to the tuna category we have 12 products

with different combinations of product attributes. Product aggregation at brand level will

mask some interesting cross-substitution patterns that exist at a more disaggregate level.

We propose a solution to overcome this dimensionality problem. Assume ch > 0 for all h,

ptj > 0 for all t and j. Let

j p be the highest price that product j could possibly charge. As

j p – ptj is finite for all t and j, we show in Appendix B that there exists a finite time period,

T, such that, households do not expect to purchase at t and stockpile for the consumption

3 An example of the former is Sun (2005) who models purchases of only two products (aggregated at brand level). An

example of the latter is Hendel and Nevo (2006) who assume that the utility from a brand is derived entirely at the

moment of purchase; hence, brand and quantity choices can be separated. Their assumption does not apply to our case

since the utility from product attributes in our model is derived at the moment of consumption.

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in periods beyond t+T no matter how much inventory they are holding. Thus, we can

rewrite the problem in Equation 1 to a finite horizon problem as below (for simplicity we

omit the subscript h hereafter):

sup{xs, ys}Et{

Tt

ts

tsγ

[u(ys) − λ ps′ xs − c (

J

1j

sj I )]|t ]} (3)

s.t. Is = xs + Is-1 – ys

xs, ys, Is 0

The optimal purchase and consumption levels in period t, {xt*, yt*}, in Equation 3 are

equivalent to the optimal solutions we would obtain from solving the infinite horizon

problem in Equation 1. This implies that empirical researchers could start from a

reasonably large number for T and solve this finite horizon problem.

To solve the finite horizon dynamic optimization problem above one could use

algorithms such as the backward induction method. But, with a large number of state

variables, the problem is still too complicated to be solved. We therefore impose the

following two assumptions to further simplify the problem:

(A.1) A household consumes each product in its inventory proportionately. That

is, given its inventory after the purchase at time t, It-1 + xt, the household plans to

consume a proportion s of its inventory in a future period s, where s≥t.

(A.2) In period t, after observing current prices, a household updates its

expectation about future prices. Specifically, we assume that after observing the

current price of product j in period t, ptj, a household updates its expected prices

for product j for future periods as follows:

0

tj

p =

0

j 1, - tp

+ ω (ptj –

0

j 1,- tp

) (4)

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where

0

j 1,- tp

is the expected price before the household observes ptj, and

0

tj

p is the

updated expected price after ptj is observed,4 and ω is a parameter to be estimated.

We expect 0 < ω <1.

With these two assumptions, we can simplify and rewrite Equation 3. In period t,

under Assumption A.1 given previous inventory It-1 and current purchases xt, the

household‟s consumption at t, yt, can be written as t (It-1 + xt), and the total inventory,

j

J

1

tjI after the consumption is (1–t)

J

1j

j 1, - t

(I

+ xtj). For a future planning period s, since

under Assumptions A.2 the expected future price is assumed to be constant over all

planning periods, the expected purchase at s, xs, will then only be used for consumption

at s but not for stockpiling beyond s.5 Thus, the consumption at s, ys, can be written as s

(It-1 + xt) + xs and the inventory at s can be written as (1–t–…–s) (It-1 + xt). Hence,

Equation 3 now can be rewritten as follows:

sup{xt,…,xt+T;δt,…,δt+T }{Etu(t(It-1 + xt)) – λ pt′ xt − c (1–t)

J

1j

j 1,- t

(I

+ xtj) (5)

+

s

Tt

1t

tsγ

[Etu(s(It-1 + xt) + xs) – λpt0′ xs − c (1–t–…–s)

J

1j

j 1, - t

(I

+ xtj)]}

s.t. xt,…,xt+T, δt,…,δt+T 0,

Tt

ts

s δ = 1

Comparing Equation 5 with Equation 3, first, we note that the space of

consumption decisions is reduced from J(T+1) to 1(T+1); second, the purchase in

period s, xs, only affects the expected consumption in that period and does not affect the

4 We assume that a household‟s expected price for product j before it observes the price in the first period in our data,

0

0j

p , is the regular price at the store most frequently visited by this household. The regular price is defined as the most

frequently charged price in the store that is no less than its average price.

5 This implicitly assumes that all products are perfectly dividable, which could pose a potential problem when applying

to bulky products.

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inventory. Finally, the future expected price at s, ps is the same for all planning periods

and equal to pt0, which is the expected price formed in period t.

We note that Assumption A.1 and A.2, while vastly simplifying the dynamic

programming problem, impose some restrictions on the households‟ consumption and

price expectations. Below we examine the implications of these assumptions and

compare them with those in literature.

Assumption A.1 describes how different products in the inventory are depleted.

This assumption will have impact on our inference about the quantity and identities of

household inventory over time, and on the expected utility of consumption. In

Assumption A.1 we assume homogeneity in the consumption of inventory across

products. Similar assumption is used in Erdem, Imai and Keane (2003) and Hendel and

Nevo (2006). Alternatively, we could assume that a household consumes its most

favorite product first (Sun (2005)), or that it consumes the inventory in the order they

were bought, i.e., first-in-first-out (FIFO) rule. If the product category is highly

perishable, FIFO would be the most reasonable assumption. However, this does not

apply to our empirical applications in either canned tuna or paper towel category. The

extent of the impact of this assumption would depend on the existence of multiple

products in household inventories. We further discuss this issue in the section of

Estimation Results.

Assumption A.2 is about a household‟s formation of price expectations. Since

expected price

0

tj

p is assumed to be constant over all future planning periods, this

assumption implies that at time t it is the household‟s expectation that if it were to

purchase at some time s>t then it would be only for consumption in period s. To buy at

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time t and hold the product until s, a household trades off the cost of holding inventory

until s to the savings that is realized in period t. In prior research (e.g., Erdem, Imai and

Keane (2003) and Sun (2005)) consumer expectations are assumed to be fully rational.

In such a case one would first estimate the price generating process from data and assume

that households‟ expectations conform to this process. In our model we do not assume

this but allow households to update their price expectations every time they observe a

new price. This vastly simplifies the dynamic optimization problem. However, if

households buy from promotion to promotion and stockpile on each promotion just

enough to last until the next promotion, our model will interpret such behavior as an

outcome of high inventory cost or households‟ lowering their expectations, i.e., a lower

value for

0

tj

p . Thus, we would expect to see an unreasonably high estimate for the

inventory cost from our estimation. In the section of Estimation Results we conduct a

simulation to examine the impact on our conclusions when households in fact expect

periodic price promotions.

We provide more details on how to derive households‟ optimal decisions on

whether to buy, which brand to buy, how much to buy and consume in Appendix C. We

use the Simulated Method of Moments (SMM) in our estimation. The estimation

procedure involves a nested algorithm for estimating θ: an “inner” algorithm that

computes a simulated quantity purchased to solve the problem in Equation 5 for a trial

value of θ, and an “outer” algorithm that searches for the value of θ that minimizes a

distance function between the simulated and observed quantity. We repeat the inner

algorithm until the outer algorithm converges. Details of the method are discussed in

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Appendix C. Identification issues related to the model estimation are discussed in

Appendix D.

EMPIRICAL ANALYSIS

Data Description

We estimate the proposed model using the A. C. Nielsen scanner panel data on

canned tuna from January 1985 to May 1987 in Sioux Falls, SD. The reason we choose

this category is that canned tuna is easily storable and potentially a good candidate for

stockpiling and flexible consumption. The sample consists of 74,795 observations from

1,000 households drawn randomly from a panel of 3,250 households. The selected

households made 13,394 purchases during the sample period of 123 weeks and bought

exclusively the size 6.5 oz of canned tuna. We focus on purchases of 6.5 oz since 94.7%

of the total quantity sold is of this package. There are three main product attributes:

brand, water/oil based, and light/regular in fat content. The grouping of the total 33

SKUs by product attributes generates 12 product alternatives. The first 11 products are

based on SKUs that share the same three attributes: brand name, water or oil, and light or

regular, and the last product consists of SKUs that belong to other brands. Henceforth,

we will use the term “product” to refer to one of these 12 alternatives. For each purchase

occasion, we construct the price, feature and display of the product bought as the

weighted average over the SKUs that belong to this product alternative. The weight used

is the quantity of sold. For a product that a household does not purchase in a week, the

price, feature and display are constructed as the numerical average over all the SKUs that

belong to the product alternative in the household's most frequently visited store.