A Dynamic Structural Model of Addiction, Promotions, and Permanent Price Cuts pot

A Dynamic Structural Model of
Addiction, Promotions, and Permanent Price Cuts



Brett R. Gordon
*

Graduate School of Business
Columbia University
3022 Broadway, Uris 511
New York, NY 10027
Email:
Tel: 212-854-7864
Fax: 212-854-7647
Baohong Sun
Tepper School of Business
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA15213
Email:
Tel: 412-268-6903
Fax: 412-268-7357

First Draft: December 11, 2008
Current Draft: August 17, 2009

Abstract

Addictive goods fundamentally differ from non-addictive goods: consuming more of an
addictive good today reinforces the addiction and increases the likelihood of future consumption.
Thus, addiction creates an intertemporal link between a consumer’s past and present decisions,
altering their incentives to purchase and to hold inventory. Despite the influence of addiction, its
impact on consumer purchase strategies and its implications for firms remain unclear.

We construct a dynamic structural model with rational addiction and endogenous consumption to
investigate how consumers respond differently to temporary versus permanent price promotions
for addictive and non-addictive goods. We apply our model to unique consumer panel data on
purchases of cigarettes, crackers, and butter. We find that addiction accumulated through past
consumption affects decisions for cigarettes but not the two non-addictive categories. Ignoring
addiction for cigarettes leads to biased estimates of price sensitivity, inventory holding costs, and
stock-out costs. For cigarettes, we find an interesting asymmetry: the temporary consumption
elasticity is smaller than the permanent consumption elasticity, but the converse is true for the
purchase elasticities. No such asymmetry exists for crackers or butter. We discuss additional
implications for retailer and manufacturer pricing strategies.

Keywords: rational addiction; dynamic structural model; endogenous consumption; price cut;
permanent price cut

*
Brett Gordon is Assistant Professor of Marketing at the Graduate School of Business at Columbia University.
Baohong Sun is Professor of Marketing at the Tepper School of Business at Carnegie Mellon University. We
appreciate comments from Ron Goettler, Avi Goldfarb, Wes Hartmann, Ran Kivetz, Oded Netzer, and participants
at the 2007 Marketing Science Conference. All remaining errors are our own.
2


1. Introduction
Addictive products are fundamentally different from non-addictive products. Consuming more of

an addictive good today reinforces addiction and increases the likelihood of future consumption.
Thus addiction influences consumers’ decisions by creating a link between past and present
consumption utility, which alters their incentives to purchase more and to hold inventory.
Consumers must manage their purchase and consumption decisions in order to avoid the
negative consequences of excess addiction. Despite the influence of addiction on consumers’
decisions, its impact on retailer and manufacturer pricing strategies remains unclear.
To address these issues, we construct a dynamic structural model of addiction with
endogenous consumption and future price uncertainty. We use the model to investigate how
consumers respond differently to temporary versus permanent price promotions for addictive and
non-addictive goods, to understand the empirical implications of ignoring addiction, and to
examine the consequences for firm’s promotional policies. Unlike most past work with non-
addictive goods that assumes consumption is exogenous or coincides with purchase (Erdem and
Keane 1996, Gonul and Srinivasan 1996, Erdem, Imai, and Keane 2003), we explicitly model
purchase and consumption as separate decisions.
1
Distinguishing between them is necessary
because stockpiling causes short-term consumption and purchases to differ, and addiction is a
function of consumption and not purchases. Although we do not observe inventories, we can
distinguish between them through joint variation in inter-purchase times and quantities.
Endogenizing consumption also allows us to incorporate the key features of addictive
products that separate them from non-addictive products. Consumers possess a stock of addiction
that depends on their past consumption and that affects their present marginal utility of
consumption. Addiction decays over time, and current consumption replenishes it. We use a
flexible form for utility and unobserved heterogeneity to permit varying levels of addiction (if
any) to exist among consumers. Our specific formulation for addiction is consistent with

1
Two recent exceptions are Hendel and Nevo (2006) and Hartmann and Nair (2008).
3

theoretical (Becker and Murphy 1988), empirical (Tauras and Chaloupka 1999), and
experimental work (Donegan et al 1983, Peele 1985) on the relationship between addiction and
consumer behavior.
Although the economics literature on addiction often uses the terms “addiction” and
“habit persistence” interchangeably (Pollack 1970, Iannaccone 1986), the marketing literature
usually takes habit persistence to mean the effect of past propensities to choose a specific brand
on current choice probabilities (Heckman 1981, Roy, Chintagunta, and Haldar 1996,
Seetharaman 2004, Dube, Hitsch, and Rossi, forthcoming). For example, in Roy, Chintagunta,
and Haldar (1996), habit persistence makes the last brand-size combination purchased more
likely to be purchased again.
2

Addiction, however, differs from this notion of habit persistence in three critical ways.
First, the reinforcing effect of addiction implies that past purchase quantities can increase current
purchases (Ryder and Heal 1973, Boyer 1978, Becker and Murphy 1988, Orphanides and Zervos
1995), whereas existing marketing models of habit persistence do not explicitly model purchase
quantity and make the last brand or size purchased more likely to be purchased again. Second,
addiction operates at the category level, whereas past work formulates habit persistence at the
brand level. Category-level consumption is the most relevant input to determine addiction as
opposed to any brand-level factors (Mulholland 1991). Third, our modeling approach uses
specific behavioral processes, such as the reinforcement effect of consuming an addictive good,
to motivate the source and nature of choice dynamics.
We apply our model to unique consumer panel data on cigarette purchases. Cigarettes
are an ideal category for our purposes because there is strong evidence that smoking is addictive
(Chaloupka and Warner 2000). For comparison, we apply the model to two non-addictive
categories, crackers and butter, using the same consumer sample. We estimate the model with

2
Similarly, the model in Guadagni and Little (1983) implies that the last brand-size purchased is more likely to be

purchased in the future. However, this outcome is due to positive state dependence in the form of brand and size
loyalty terms. In contrast, Roy, Chintagunta, and Haldar (1996) use serial correlation in the errors terms of the
utility-maximizing alternatives across periods to induce the persistence in choices.
4


unobserved heterogeneity and with/without addiction on all three categories, and calculate the
elasticity of consumption and purchase with respect to temporary and permanent price changes.
Our key results are the following. First, consumers make different purchase and
consumption decisions for addictive products and non-addictive products. Addiction
accumulated through past consumption not only generates direct utility but it also enhances the
marginal benefit of consumption. Consumers with higher addiction are less price sensitive and
have higher stock-out costs. Second, we find the dynamic addiction model fits best for cigarettes,
whereas the dynamic model without addictions is a better fit for crackers and butter owing to
being more parsimonious, consistent with the intuition that such a model should be preferred for
categories that are truly not addictive. Third, ignoring addiction for cigarettes leads to biased
estimates of inventory holding costs, stock-out costs, and price sensitivity. For a temporary price
cut, the model without addiction overestimates consumption and underestimates short-term
inventories, producing a downward bias in the price coefficient since consumers anticipate the
cost of sustaining the higher consumption after the price reverts.
We also find an asymmetry in cigarette elasticities: temporary consumption elasticities
are smaller than permanent consumption elasticities due to the smoothing of consumption via
addiction, but temporary purchase elasticities are larger than permanent purchase elasticities
because addiction creates strong stockpiling incentives to avoid stock-outs. In contrast, for non-
addictive goods both consumption and stockpiling inventories are higher for temporary changes
than for permanent changes. We decompose the impact of temporary and permanent price
changes on purchase patterns, consumption, and displacement. We find that temporary price
changes are less effective at inducing switching between product tiers compared to permanent
price cuts due to the interaction between addiction and inventory. These results demonstrate the
importance of recognizing the dynamics introduced by addiction and stockpiling in the context

of addictive products.
5


We contribute to the existing literature in the following ways. From a theoretical
perspective, we adapt the rational addiction model of Becker and Murphy (1988) to a dynamic
structural model that accounts for inventory dynamics and embeds a continuous consumption
choice within a dynamic discrete-choice framework.
3
Two assumptions of the Becker-Murphy
model are that consumers are forward-looking and have time-consistent preferences. Numerous
papers find strong evidence in support of the forward-looking behavior in the context of
cigarettes (Chaloupka 1991, Becker, Grossman, and Murphy 1994, Arcidiacono, Sieg, and Sloan
2005, Wan 2005). Despite some evidence in support of time-inconsistency (O’Donoghue and
Rabin 1999), time-consistent preferences are used to empirically model addiction in a variety of
settings.
4
The formal identification of time-consistent versus time-inconsistent preferences from
data remains unclear. Fang and Wang (2008) show that the discount parameters in time-
inconsistent models are only partially identified under certain exclusion restrictions and in the
absence of consumer heterogeneity.
5
Relaxing the time-consistency assumption would be a
valuable extension especially from a public policy perspective (Gruber and Koszegi 2001).
6

On the empirical side, despite the long tradition in economics of using rational addiction
models to study cigarette consumption, most of the work uses large-scale surveys and reduced-
form models (Chaloupka 1991, Becker, Grossman, and Murphy 1994, Coppejans et al 2007).
This approach restricts the range of possible policy experiments and often relies on aggregate

(e.g., state level) price data to conduct inference. Our structural model enables us to perform a
number of counterfactual simulations and uses rich, individual-level panel data. We perform a
cross-category analysis and demonstrate that consumers respond differently to price cuts for

3
See Dockner and Feichtinger (1993) and Orphanides and Zervos (1995) for two extensions.
4
See Waters and Sloan (1995) for an application to alcohol, Olekalns and Bardsley (1996) for caffeine, and Choo
(2000) and Arcidiacono, Sieg, and Sloan (2005) for cigarettes.
5
Rust (1994a, 1994b) show that the discount factor is generically not identified for standard dynamic discrete choice
models, and Magnac and Thesmar (2002) generalize these results to dynamic single-agent models.
6
Machado and Sinha (2007) develop an analytical model of time-inconsistent smokers’ participation and cessation
decisions, and using a hazard-rate model estimated on survey data, find support for their model.
6


addictive and non-addictive goods. We compare the effects of temporary and permanent price
cuts on addictive and non-addictive goods.
Our model makes several methodological contributions relative to existing research. In
marketing, our work draws on the broad class of dynamic consumer models applied to frequently
purchased products (Erdem and Keane (1996), Gonul and Srinivasan (1996), Sun, Neslin, and
Srinivasan 2003, Sun 2005). In particular, our model most closely relates to the dynamic
stockpiling models of Erdem, Imai, and Keane (2003) and Hendel and Nevo (2006), who
examine ketchup and laundry detergent, respectively. Chen, Sun, and Singh (2007), who
examine how consumers adjusted their cigarette brand choices following Philip Morris’s
permanent price cut in response to the growth of generic brands, do not model the purchase
quantity and consumption decisions. To our knowledge, there is no research that examines
consumer purchase and consumption decisions in the presence of addiction and inventory

dynamics. In addition, we explicitly compute the optimal consumption path as a function of
inventory and addiction.
Finally, as consumers continue to embrace healthier lifestyles and consider more products
containing unhealthy ingredients (e.g. nicotine, caffeine, sugar, and salt) as “products of vice,”
we make a first attempt to understand how the unique features of addictive goods affect purchase
and consumption decisions and the implications on price and promotion effects.
The rest of the paper proceeds as follows. Section 2 presents the model and estimation
approach. Section 3 discusses the data, identification, parameter estimates, and model fit. Section
4 compares the resulting consumer policy functions for purchase and consumption and presents
the results of the pricing simulations. Section 5 concludes with a discussion of limitations of the
present work and avenues for future research.



7


2. Model
This section develops a dynamic model of rational addiction where consumers face uncertainty
about future prices and store visits. Consumers must optimally balance the impact of current
consumption on future addiction and inventory levels. The model does not impose addiction by
assumption, and relies on the intertemporal relationship between past consumption and present
decisions to identify the addiction process. We explicitly model consumption and purchase as
separate decisions, and later show that distinguishing between them is necessary to understand
the consumer decision process for addictive goods and has important policy implications.

2.1. Period Utility
There are I consumers who make periodic (e.g., weekly) decisions about which product to
purchase, how much to purchase, and how much to consume at T equally spaced time periods.
There are

0, ,jJ= 
product alternatives where choice
0j =
represents a no-purchase decision.
Let
ijt
c
be consumer i’s consumption of product j in week t and define the dummy variable
1
ijqt
d =
as a choice of product j and quantity q. Then
1
J
it ijt
j
cc
=
=
∑
is the category consumption
choice,
{ }
,
it ijqt
jq
dd=
is the vector of purchase quantity indicators, and
,
1

ijqt
jq
d =
∑
.
A consumer’s period (indirect) utility in state
{ }
,,
it it it t
s aIP=
is the sum of consumption
utility, purchase utility, and inventory costs:
(1)
(, ,;) (, ; ) (,;,) (;)
it it it it i c it it i p it t i i it i
Ucds uca udP CIh
θ α βξ
=+−

where the stock of addiction
0
it
a
≥
summarizes the cumulative effect of past consumption,
0
it
I ≥
is the consumer’s inventory,
{ }

1
, ,
t t Jt
PP P=
is a vector of prices, and
{ }
,,,
i i iii
h
θ αβξ
=

is the parameter vector. We discuss each component of the utility function in turn.
For consumption utility, we require a form that can capture the distinct features of
addictive goods: reinforcement, tolerance, and withdrawal (Peele 1985, Chaloupka 1991).
8


Reinforcement implies that greater past consumption raises the marginal utility of present
consumption. Tolerance suggests that a given level of consumption yields less satisfaction as
cumulative past consumption rises. Finally, withdrawal refers to the negative reaction from a
decrease or interruption in consumption due to a stock-out or intentional cessation.
7

A convenient form that can encompass these effects—without imposing them by
assumption—is a quadratic utility function. It allows for the necessary complementarity between
consumption and addiction and satisfies standard regularity assumptions found in the habit
formation literature (Stigler and Becker 1977). Thus the period utility from consumption is
(2)
22

0 1234 5
( , ; ) 1{ 0}
c it it i i it i it i it i it i it i it it
u c a c c c a a ac
αα α α α α α
= =++++ +
.
If consumption is zero, the first coefficient,
0i
α
, is the cost of a stock-out, or withdrawal.
Consumption may be zero when the inventory is exhausted and the consumer is unable to make a
purchase (no store visit). The next two coefficients,
1i
α
and
2i
α
, represent the instantaneous
utility of consumption independent of addiction. For the following two coefficients,
3i
α
captures
the direct utility from addiction and
4i
α
allows for the tolerance effect. The last term represents
the reinforcement effect: if
5
0

i
α
>
, then addiction increases the marginal utility of consumption.
Addiction plays an important role because it creates an intertemporal link between past
consumption and current decisions. We use this simple law of motion to govern a consumer’s
stock of addiction:
(3)
,1
(1 )
i t i it it
a ac
δ
+
=−+
,
where
01
i
δ
≤≤
is the constant rate of depreciation of addiction over time. Overall cigarette
consumption strengthens addiction regardless of a cigarette’s brand, and addiction directly

7
Although we do not explicitly model the cessation decision, our model partially captures it because implied
consumption for a consumer could be zero in a period. Such periods may or may not indicate a decision to quit
smoking depending on whether the consumer obtains cigarettes from a source outside our data set or fails to
properly record their purchases. Choo (2000) who uses a dynamic model of rational addiction and annual survey
data to examine smoking and quitting decisions in response to changes in the smoker’s health.

9


influences consumers’ preferences by changing the marginal utility of consumption.
8
Our
formulation of addiction is theoretically (Iannaccone 1986, Becker and Murphy 1988),
empirically (Becker, Grossman, and Murphy 1994), and experimentally (Peele 1985, Rose 2004)
consistent with prior work on the relationship between addictive goods and consumer behavior.
Although the literature contains numerous formulations for habit persistence (Heckman 1981,
Erdem 1996, Roy, Chintagunta, and Haldar 1996, Seetharaman 2004), none would produce a
pattern consistent with addiction because they do not explicitly model purchase quantity
decisions. These approaches make a consumer more likely to repeatedly purchase the same
brand-size combination, but not more likely for the consumer to increase their purchase quantity.
In addition to consumption, consumers simultaneously choose to purchase from among a
discrete set of product-quantity combinations for each product j. Purchase utility is given by:
(4)
2
1 23
,
( ,;,) ( )
p it t i i ijqt i jqt ijt i ijt i ijt ij ijqt
jq
udP d pq q q
βξ β β β ξ ε
= + + ++
∑

where
ijt

q
denotes the purchase quantity,
jqt
p
denotes the per-unit price for quantity q, and
jqt ijt
pq
is the total expenditure. The parameter
1i
β
measures consumer’s price sensitivity. We
account for product-level differentiation through the fixed-effects
ij
ξ
and quantity-related
differences through the linear and quadratic quantity terms. The squared term on quantity allows
for a non-linear relationship between purchase size and utility. The variable
ijqt
ε
is a random,
unobserved shock to utility that affects consumer i's decision, distributed i.i.d. extreme value
distribution to obtain the usual multinomial logit choice probabilities.
Quantities purchased in the current period are available for immediate consumption.
Products not consumed are stored at a holding cost of
i
h
, such that
(;)
it i i it
CI h h I= ⋅

. Inventory is
not product specific, and evolves according to

8
We could extend the model to allow the evolution of addiction to depend on brand-specific characteristics such as
tar and nicotine levels, but we would not expect this to have a significant impact on our results.

10


(5)
1
,
it it ijqt ijt it
jq
I I dq c
+
=+−
∑
.
The inventory state variable is important because it creates another intertemporal link between
purchase and consumption: the cost of holding additional inventory must be balanced against the
desire to avoid a costly stock-out.
In summary, collecting the formulations for the individual components of utility, the
indirect utility function is:
(6)
22
0 12345
2
1 23

,
( , , ; ) 1{ 0}
()
it it it it i it i it i it i it i it i it it
ijqt i jqt ijt i ijt i ijt ij ijqt i it
jq
U c d s c c c a a ac
d p q q q hI
θα α α α α α
β β β ξε
= =+++ + +
+ + + ++ −
∑

Next we discuss how consumers form expectations about future prices and store visits, and then
formulate the consumer’s dynamic decision problem.

2.2. Price Expectations
Stockpiling is common for many consumer-packaged goods, including cigarettes, and consumers
make decisions based on their expectations of the future distribution of prices (Erdem and Keane
1996, Gonul and Srinivasan 1996, Sun, Neslin, and Srinivasan 2003). A key component of the
model is the random process governing future prices. In order to generate robust predictions, the
price process should be realistic and capture several features of real world prices, such as the
dependence of current prices on competitors’ prices and own lagged prices.
Let
jt
P
be the aggregate price of product j, which we distinguish from
jqt
p

, the price for
a particular product-quantity combination. Similar to Erdem, Imai, and Keane (2003), we assume
logged aggregate prices follow a first-order Markov process,
(7)
1 2 1 3 ,1
1
ln ln ln
1
jt j j jt j j t jt
lj
PP P
J
γγ γ η
−−
≠
=++ +
−
∑
,
where
1jt
P
−
is the past price of product j at time t – 1. Price competition enters through the
inclusion of the mean log price of competing tiers. The variable
jt
η
is the random shock of
11

product j at time t, which follow a multivariate normal distribution,
(0, )
jt
N
η
η
Σ
. The diagonal
elements denote the corresponding variance of
jt
η
, and the off-diagonal elements denote the
covariance between the prices of different products. The correlation across products helps further
capture the co-movement of prices of competing tiers.
The system above describes the process governing product-level prices. We model the
price process for product-quantity specific prices as contemporaneous functions of the aggregate
product price
jt
P
. The price process for a given quantity q in product j is:
(8)
12
ln ln ,
jqt jq jq jt jqt
pP
λλ ν
=++

where we also assume

2
(0, ).
jqt v
N
νσ

This formulation reduces the state space of the dynamic
consumer problem from containing
JQ
product-quantity prices to
J
product prices, while still
allowing the per-unit prices to vary by product.

2.3. Store visits
In the data we observe that consumers sometimes do not visit a store. Modeling store visits is
important because random store visits create an extra precautionary incentive to hold inventories
and consumption varies with duration between visits.
We allow the probability of a store visit to depend on whether the consumer visited a
store in the previous period. We use a binomial distribution to model store visit behavior. Let
it
π
be a binary variable that indicates whether the consumer visits the store in the current period.
Then
11
Pr( 1| 1)
i it it
ρπ π
+
= = =

denotes the probability of visiting a store next period conditional
on visiting a store this period. Similarly,
01
Pr( 0| 0)
i it it
ρπ π
+
= = =
denotes the probability of not
visiting a store next period conditional on not visiting a store this period. We estimate these
probabilities at the consumer level directly from the observed store visit frequencies and treat
their values as known in the dynamic estimation.
12


2.4. Dynamic Decision Problem
Given their current state, period utility function, and expectations about future prices and store
visits, consumers simultaneously make their optimal product-quantity,
*
ijqt
d
, and consumption,
*
it
c
, decisions. The value function when a consumer visits a store is
()
it
Vs
and the value function

without a store visit is
()
it
Ws
. We assume the discount factor is fixed and known at
0.995
β
=
.
9

Given the period utility function, the Bellman equation during a period with a store visit is:
(9)
{ }
11 1 1
,
()max (, ,;) [ ( )(1 )( )|]
s.t. 0 , for all , , and 1.
it it
it it it it it i it i it it
cd
it it ijqt it ijqt
jq
Vs U c d s E Vs Ws s
c I d q jq d
θ βρ ρ
++
= + +−
≤≤+ =
∑

The operator
t
E
denotes the conditional expectation over future prices given the consumer's
state
it
s
. During a period without a store visit, the consumer’s value function is:
(10)
{ }
0 10 1
( ) max ( , ; ) ( ; ) [(1 ) ( ) ( ) | ]
s.t. 0
it
it c it it i it i i it i it it
c
it it
Ws u c a CI h E Vs Ws s
cI
α βρ ρ
++
= − +− +
≤≤

We solve the value functions for the optimal consumption conditional on a product choice:
(11)
{ }
**
11 1 1

*
argmax ( , , ; ) [ ( ) (1 ) ( )| ]
s.t. 0
it
it it it it it i it i it it
c
it it it it
c U c d s E Vs Ws s
c I dq
θ βρ ρ
++
= + +−
≤≤+

where
*
it
d
has a one in the position where
*
ijqt ijqt
dd=
and zero elsewhere. This requires a one-
dimensional optimization at each choice-specific value function, which greatly increases the
computational burden of estimation.

2.5. Heterogeneity, Initial Conditions, and Estimation
We account for unobserved heterogeneity by assuming that each consumer belongs to one of M
unobserved preference segments (Kamakura and Russell 1989). We denote
m

i
φ
as the probability

9
The discount factor in dynamic models is generally not identified (Rust 1994a, Rust 1994b), hence we fix its value
to yield a sensible annual discount rate. A model with time-inconsistent preferences requires two discount factors to
be specified, significantly increasing the difficulty of proper parameter identification.
13


that consumer
i
belongs to preference segment
m
. More formally, the probability that a
consumer in the sample is of type
m
is
(12)
'
'1
exp( )
exp( )
m
m
i
M
m
m

δ
φ
δ
=
=
∑
.
where
, for 1, , ,
m
mM
δ
= 
are a set of parameters to be estimated.
We estimate the model using maximum likelihood. Let
it
D
denote the observed product-
quantity decision at time t, let
*
i
TT⊂
denote the subset of all time periods in which consumer
i

made a store visit, and
{ }
1
, ,
M

θθ θ
=
denotes the dynamic parameters of interest. Given the
extreme value distribution of the error term, the probability of observing consumer
im∈
making
decision
ijqt
d
at time
*
i
tT∈
is
(13)
*
1
,
exp( )
Pr( | , ; )
exp( )
m
M
ijqt
m
ijqt ijqt it it i
m
m
ijqt
jq

V
d d aI
V
θφ
=
= =
∑
∑
,
where
m
ijqt
V
is the value function for choice j,q for consumer i in segment m at time t:
(14)
{ }
*
11 1 1
()max (, ,; ) [ ( )(1 ) ( )| ]
it
m mm
ijqt it it it it it m i it i it it
c
V s Ucds E V s W s s
θ βρ ρ
++
= + +−
.
For
*

i
tT∉
, we do not observe a store visit, but we must compute the laws of motion for
addiction and inventory levels given the implied consumption in these periods.
To calculate the likelihood, let
t
D
be the vector of choices over households in period
t
.
We must calculate the likelihood
( | ;)
tt
LD s
θ
, which contains the unobservable state variables
it
a

and
it
I
. Since both of these evolve deterministically, we can calculate their law of motions given
some initial distribution for
1i
a
and
1i
I
. Then for a given value of the parameter vector, the log-

likelihood function of the sequence of choices over all the households is
14


(15)
( ) ( )
( )
( )
( )
*
1 1 111 111 1
( , , | , , ; ) log Pr | , , ; , , , ; dF ,
i
T T it it it i i it it i i it i
iI
tT
LD Ds s DaD aI ID aI aI
θ θθ
−−
∈
∈
=
∑∑
∫


where
11
(,)
ii

Fa I
is the initial joint density of addiction and inventory levels. We must integrate
over
11
(,)
ii
Fa I
to resolve the dependence of addiction and inventory on their initial values.
We estimate the system of simultaneous equations in (7) using SUR and the individual
product-quantity processes in equation (8) using OLS regression. To estimate the price process
for a particular product-quantity, we first construct the history of prices for that product-quantity
each consumer observed over the weeks of the sample period. We then pool together these
individual price sequences in the estimation. We estimate the price process parameters using the
price data prior to the estimation of the dynamic model.
Several issues are worth note concerning the solution and estimation of the model. First,
we face an initial conditions problem since we do not observe the inventory and addiction levels
at the start of the data. We start with initial inventory and addiction levels of zero and solve the
dynamic programming problem, and then we simulate the model 50 times for 2T periods. This
process generates an initial bivariate density over inventory and addiction that we use to simulate
the initial states for each consumer. Second, all the state variables are continuous, making it is
impossible to solve exactly for
ijqt
V
and
ijqt
W
at every point in the state space. We discretize the
state space using a sufficiently fine grid, use local polynomial interpolation to calculate the
expected continuation values on points off the grid, and use Monte Carlo simulation with Halton
draws to approximate the integral. In evaluating the likelihood, we recalculate the optimal

purchase and consumption decisions for each consumer and period to avoid approximation error
in using the estimated policy functions. More computational details are in Appendix A.



15


3. Empirical Application
This section describes the data set, provides an informal discussion of the model’s identification,
and discusses the model fit and parameter estimates. Due to space considerations, additional
details are in Appendices B and C.

3.1. Data
The data come from an ACNielsen Wand panel collected from two separate submarkets in a
large Midwestern city during the 118 weeks from January 1993 to August 1995. Our data consist
of detailed purchase histories for 2,100 households across multiple categories. The purchase
history is fairly complete: purchases are recorded from all outlets, including convenience stores
and gas stations. This broad inclusion is particularly important because small retail outlets
account for 26% of cigarette sales in our data.
The cigarette category contains several hundred distinct products with variants in terms
of strength (regular/light), size (e.g., 100s), and flavor (e.g., menthol). To keep our study
manageable, we classify products into three quality tiers based on common industry
classifications of premium, generic, and discount value products. We aggregate to the tier level
instead of the brand level for two reasons. First, the taste of cigarettes is more likely to differ
across tiers than within a tier due to varying levels of tar and nicotine (Mulholland 1991 and
Viscusi 2003). Second, our focus is not on inter-brand competition. Although modeling brand-
level dynamics, such as loyalty, would no doubt improve the model’s predictive ability,
addiction exists independently of brand choice.
Figure 1 plots the distribution of purchase quantities of cigarettes in our sample. The

large spikes at 10, 20, and 30 correspond to purchases of cartons each containing ten packs.
Based on this distribution, we discretize purchase quantity into five bins of {1-2, 3-4, 5-9, 10-19,
20+} and use each bin’s midpoint in our model. Thus, we have 15 tier-size combinations. On
average, nearly 80 percent of average weekly purchases were for nine packs or less.
16


[Figure 1 about here]
In addition to cigarettes, we consider purchases from the crackers and butter categories.
Crackers are a particularly good comparison category because, like cigarettes, they are storable
and purchased with some frequency (compared to, say, detergent) but are not addictive. We
include butter for comparison to a less frequently purchased category. We avoided perishable
products, such as yogurt or milk, because this would introduce a product characteristic not
observed in cigarettes and could confound the comparison.
Choice models applied to panel data typically define and estimate the utility function at
the household level. However, members of a household may have different preferences and
consumption patterns. Defining addiction at the household-level would inevitably understate or
overstate the importance of addiction for some household members, introducing a potential bias
into our estimation. To avoid this issue, we split the household-level observations into individual
observations based on the gender and age of the purchaser, recorded with each purchase.
To facilitate these cross-category comparisons, we use the same sample of individuals
across the three categories. We select those individuals who made at least ten cigarette
purchases, ten crackers purchases, and four butter purchases. Of the 1,351 individuals defined at
the household-gender-age level who purchased cigarettes at least once, 584 satisfy all these
criteria. We have no reason to believe that smokers who purchased sufficient quantities of
crackers and butter differ on some unobserved dimension from smokers who did not meet our
sample criteria. We randomly select 300 individuals for estimation and use 50 for a hold-out
sample. The individuals in our estimation sample made an average of 42 cigarette purchases
across a total of 12,689 purchase observations. We construct similar product tiers for crackers
and butter based on product prices and brands.

[Table 1 about here]
Table 1 provides sample descriptive statistics about the categories and product
aggregates. For cigarettes, the average purchase quantity per incidence was 13.39, and
17


consumers' average consumption per week is 4.93 packs. The premium tier is the market leader,
capturing approximately 51 percent of the total market. The average price of cigarettes ranges
from $1.63 for the premium tier to $1.12 for the discount tier. Crackers and butter both contain
three large brands that occupy different price tiers and account for more than 80% of sales. For
crackers, Nabisco is the dominant brand with a market share of nearly 50% and a sales-weighted
price of $1.95 for a 16-ounce box, compared to a 22% share for the private label brand with an
average price of $1.10 per 16-ounce box.

3.2 Identification
We provide an informal discussion of identification and offer some descriptive statistical
evidence. The panel aspect of our data greatly facilitates the identification of preferences and
heterogeneity. The identification of the purchase utility coefficients is straightforward. The price
parameter,
1i
β
, is identified through variation in prices over time. The quantity parameters,
2i
β

and
3i
β
, and the fixed effects,
ij

ξ
, are jointly identified through differences in the purchase
probabilities across product tiers and purchase sizes.
The arguments behind the identification of the consumption, storage cost, and addiction
parameters are more subtle since the corresponding quantities are unobservable. However, it is
important to note that given a set of initial conditions, the structural model provides a direct
mapping from the observed purchase quantity to the current consumption, which in turn
determines next period addiction and inventory. Thus we identify the consumption parameters,
0i
α
,
1i
α
, and
2i
α
, through the joint distribution of inter-purchase times and quantity choices; that
is, a large number of cigarettes purchased over a short period of time implies a high rate of
consumption. For the inventory parameter,
i
h
, consider the case of two consumers who face the
same (stationary) price process over time, purchase the same quantity during this period, but one
consumer purchases more frequently than the other. This variation leads us to conclude that the
18


one who purchases more frequently has higher storage costs than the other. A similar argument
exists to help identify the addiction parameters,
3i

α
,
4i
α
, and
5i
α
. Suppose two consumers
purchase the same quantity of cigarettes at the same frequency, but one consumer makes
additional visits to the store without purchasing anything during those visits. This variation lets
us conclude this consumer gained less utility due to addiction because she was more able to resist
the temptation to purchase cigarettes during the additional store visits.
As discussed earlier, the fundamental distinction between addiction and other forms of
state dependence is the emphasis on quantities: more consumption in the past implies more
consumption in the present. We present reduced-form evidence that the cigarette data contain
patterns consistent with this notion of addiction, and that these patterns differ from those found
in the non-addictive categories. More specifically, we estimate a joint model of purchase
incidence and purchase quantity along the lines of Gupta (1988) and Bucklin and Gupta (1992).
A logit model determines purchase incidence, and conditional on a purchase occasion, a
truncated-at-zero Poisson model determines the number of units bought. We use the flexible
consumption rate function found in Ailawadi and Neslin (1998), and calculate inventory
recursively based on the implied weekly consumption
it
c
. In addition to explanatory variables
such as inventory and mean consumption, we include a consumption stock variable, defined as
, ,1 ,1
(1 )
it it it
s sc

δ
−−
=−+
. This variable helps capture persistence in consumption over time and
loosely proxies for the addiction state variable in our structural model. The model specification,
parameter estimates, and further details are in Appendix C.
The estimates in Table C1 suggest a difference between cigarettes on the one hand and
crackers and butter on the other in terms of the importance of the consumption stock. In
particular, the consumption stock has a positive and significant effect on purchases of cigarettes
but it has no effect for crackers and butter. This suggests that past consumption quantities have a
positive influence on current purchase decisions for cigarettes, which is consistent with addictive
behavior. Further, the estimates in Table C1 show most of the other variables are of the expected
19


sign and magnitude: for example, inventory exerts a negative influence on purchases for all
categories, but its effect is greatest for cigarettes and smallest for butter.
To be clear, we view these results as purely descriptive evidence for purchase behavior
consistent with addiction. We must use the structural model to specify a causal link between
observed purchases and unobserved consumption and addiction. Although this result only
provides descriptive evidence for some form of consumption dynamics, it helps identify the
fundamental correlations in the data that we require to estimate our structural model.

3.3. Model Evaluation and Comparison
In order to demonstrate the importance of the key components of our model—forward-looking
consumers, endogenous consumption, and addiction—we estimate three models for comparison.
Model 1 removes forward-looking behavior, endogenous consumption, and addiction from the
full model. This myopic model is commonly adopted to study consumer brand choice behavior
for non-addictive products. Model 2 adds forward-looking behavior and endogenous
consumption, but still excludes addiction. Model 3 is the full model with forward-looking

behavior, endogenous consumption, and addiction.

[Tables 2A, 2B, and 2C about here]
Table 2A reports the model fit statistics for the estimation and holdout samples for
cigarettes. The fit statistics show that the full model with two segments fits the data the best.
Estimating the model with additional segments did not sufficiently improve the performance.
Table 2B compares the tier transition matrix in the data to the simulated switching probabilities
the model generates. In general, the simulated transition probabilities do a good job of
replicating those found in the data. Table 2C compares simulated values of choice probabilities,
average purchase quantity, purchase incidence, and the average inter-visit duration with those
from the sample. The full model fits remarkably well on all these dimensions, indicating the
20


model is able to replicate several key features of the data.
[Table 2D about here]
For consistency, we use two segments to estimate Models 1 – 3 on crackers and butter.
The results in Table 2D show that the dynamic model with addiction fits best for the addictive
category and that the dynamic model without addiction fits best for the two non-addictive
categories. The addiction process fails to provide additional explanatory power for crackers and
butter. In contrast, the addictive process for cigarettes alters consumers’ decisions and, as we
later show, their responses to price changes.

3.4. Parameter Estimates
Table 3 reports the parameter estimates for the utility function for Model 2 (without addiction)
and Model 3 (with addiction) for each category.
10
First, we discuss the results for Model 3 with
cigarettes, and then compare them with the estimates for Model 2 to demonstrate the bias from
ignoring addiction. Finally, we contrast the cigarette estimates to those from crackers and butter.

All the price process estimates are in Appendix D.
[Table 3 about here]
For cigarettes, the addiction depreciation coefficient,
i
δ
, is significant for both segments,
indicating that past consumption quantities affect current decisions. The linear and quadratic
coefficients on consumption and addiction are positive and negative, respectively, in both cases
and the coefficient on the interaction between consumption and addiction is positive. These
estimates suggest that addiction plays an important role in explaining consumer purchase and
consumption behavior of addictive goods. The estimates imply that the consumption utility
function is concave, and that past consumption reinforces the marginal utility of present

10
We omit the estimates from Model 1 because the primary comparison is between the dynamic model with and
without addiction.
21


consumption. Thus, the parameter estimates in the utility function satisfy the conditions required
for addiction (Becker and Murphy 1988).
The parameters differ between consumer segments. Consumers in segment one receive
less instantaneous utility from consumption, have a higher marginal utility for addictive
consumption, are less price sensitive, and have higher stock-out costs. Addiction plays a larger
role for segment one: the mean (standard deviation) of the addiction level for a consumer in
segment one is 4.63 (1.65) and in segment two is 2.27 (1.38). For example, if we consider
representative consumers from each segment, the interaction term between consumption and
addiction accounts for 73% and 21% of consumption utility for segments one and two,
respectively. In economic terms, a one-unit increase in consumption for both consumers would
produce a period surplus increase of $5.90 for segment one versus $1.41 for segment two.

The stock-out coefficient implies a cost of roughly $8.90 for segment one and $3.88 for
segment two, with the more addicted segment suffering the higher stock-out cost. The inventory
holding costs of $0.31 and $0.27, roughly the same for each segment, imply that higher
inventory reduces the probability of purchasing and increases the rate of consumption.
Comparing the results for Model 2 and Model 3, there are several points to note. First,
including addiction increases the price coefficients for both segments by roughly 30%. Ignoring
addiction leads the model to underestimate price sensitivity because addiction helps account for
some lack of responsiveness in demand to price changes (Keane 1997). Second, the linear
consumption parameters decrease in magnitude by about 25%, mostly due to the new interaction
term between consumption and addiction. Ignoring the effect of past consumption and present
consumption, Model 2 places more weight on the instantaneous benefits of consumption and
overestimates many of the consumption utility parameters. Third, including addiction in the
model raises the stock-out cost estimates and lowers the holding cost estimates. Without
addiction, the model partially rationalizes a given level of consumption with higher inventory
holding costs and lower stock-out costs. The presence of addiction provides an alternative
22


mechanism for the model to explain why a consumer might, say, decrease consumption in the
presence of inventory. A low rate of consumption need not imply low holding costs because the
negative effect of consumption on addiction, through
4i
α
, may outweigh the costs of holding
additional inventory,
i
h
.
Comparing the results above for cigarettes to those obtained using the crackers and butter
categories, consumers appear to shop differently for addictive and non-addictive products in a

few ways. First, and most importantly, consistent with the model fit statistics in Table 2D, the
estimates for both categories do not change significantly after adding addiction into the model.
All the addiction terms are insignificant in Model 3 with the sole exception of the linear
addiction term for crackers, which is modestly significant but small in magnitude. Although this
suggests the addiction process may be picking up some other type of state dependence in
crackers in addition to stockpiling, the underlying process does not appear consistent with a
model of addiction because the coefficient on the interaction between addiction and consumption
is insignificant. These results demonstrate that the model is suitable for both addictive and non-
addictive products because it finds evidence of addiction when appropriate and produces
reasonable parameter estimates on categories that lack addiction.
Second, the inventory holding cost parameters for all three categories are significant,
demonstrating the importance of accounting for state dependence in the form of stockpiling for
these categories. Third, the average estimates for the monetary cost of stock-outs are $0.91 and
$0.32 for crackers and butter, respectively, which are considerably lower than the stock-out
estimate for cigarettes. Relative to cigarettes, crackers or butter are less likely to induce a store
trip, and thus the cost of a trip for cigarettes is less likely to be distributed over multiple products.
In addition, the stock-out cost estimates for cigarettes might include a psychological cost
component associated more with addictive goods.


23


4. Implications on the Effectiveness of Price and Promotion
In this section we discuss how the consumers’ policy functions for purchase and consumption
vary depending on their addiction and inventory levels. We also present purchase and
consumption elasticity estimates for temporary and permanent price changes, and decompose the
results the effects of the price changes into tier switching, consumption, and displacement.

4.1. Optimal Purchase and Consumption Decisions

We study the optimal decisions for purchase and consumption as a function of inventory and
addiction to understand how consumers’ decisions are driven by the endogenous state variables.
For cigarettes, both inventory and addiction affect the shape of the policy functions. To highlight
the impact of the addiction state variable, we compare the decision functions from the cracker
category estimated with addiction to those from cigarettes, and show that the addiction state
variable plays almost little in influencing cracker purchases and consumption.
11

[Figures 2A, 2B, 3A, and 3B about here]
Figure 2A plots the optimal consumption averaged over both segments as a function of
inventory and addiction.
12
Figures 3A and 3B depict the policy functions for crackers from
Model 3. First, we focus on how consumption varies with inventory for a given level of
addiction. As the utility function (Equation 6) shows, consumers must balance the instantaneous
benefits of consumption—including the reinforcement effect—with the consequences of excess
addiction. At low levels of addiction, the results are similar to those in Ailawaide and Neslin
(1998) and Sun (2005) for non-addictive products: optimal consumption increases at a declining
rate with inventory due to inventory pressure and consumers strategically smoothing
consumption to preserve inventory for the future in the event of no store visit. However, when

11
A comparison with the decision functions from butter yields similar conclusions.
12
To construct this decision function, we simulate the model over the data set. This generates a sequence
*
{, ,}
it it it
caI

of consumption, addiction, and inventory levels for each consumer in each period, representing the empirical
decision function. We use local polynomial regression to smooth the decision function to create the 3-D plots.
24


addiction becomes sufficiently high, the reinforcement effect and inventory holding costs
increase the level of consumption. Optimal consumption is not only significantly higher than
when addiction is low, but consumption also increases monotonically with inventory. However,
the consumption policy function for crackers in Figure 3A does not exhibit any significant
variation in the addiction dimension.
Holding inventory constant at a low level, optimal consumption as a function of addiction
resembles an inverted-U shape as consumers balance the reinforcing effects of consumption with
the costs of a large addictive stock. For sufficiently high levels of inventory, holding costs and
the reinforcement effect dominate, leading consumption to rise with addiction. The interaction
between addiction and inventory alters consumers’ decisions because lower holding costs (h)
make it easier to store goods but also creates more temptation to consume more in the future.
These results differ from earlier work that documents a monotone relationship between inventory
and consumption for non-addictive products (Assunção and Meyer 1993).
Figure 2B plots the optimal purchase quantity for cigarettes against inventory and
addiction, and Figure 3B plots the same for crackers. At sufficiently low levels of addiction,
purchase quantity increases with inventory and an individual consumes all of her purchase plus
some of her inventory. In contrast to results for non-addictive goods, purchase quantity increases
with inventory due to the high cost of stock-outs relative to holding costs for addictive goods.
For intermediate levels of addiction, purchase quantity exceeds consumption and inventory rises.
This creates a tension between holding sufficient inventory to avoid a stock out cost and the
burden of holding the extra inventory. However, at high addiction levels, purchase quantities
decrease with inventory despite the fact that consumption increases. Thus, a highly addicted
consumer draws down her inventory, gradually limiting her future consumption and reducing her
addiction. In contrast, for crackers the purchase quantity monotonically increases with inventory.
For low levels of inventory, purchase quantity always increases with addiction despite the

fact that consumption starts to decrease at sufficiently high values of addiction. This is due to the
25


high cost of stock-outs relative to holding costs for addictive goods. Interestingly, as inventory
increases, purchases quantity resembles an inverted-U shape. At lower levels of addiction, the
reinforcement effect dominates, and consumers purchase more to facilitate their current and
future consumption. Eventually the stock of addiction becomes excessive and consumers must
strategically lower their purchase and consumption rates in order to avoid becoming overly
addicted in the future. The policy function for crackers in Figure 3B does not change
significantly with addiction.
Thus, the shape of the policy functions for cigarettes shows an important interaction
between addiction and inventory levels. The addiction state variable does not capture any
meaningful state dependence for crackers and does not influence consumers’ decisions.

4.2. Differential Effectiveness of Temporary and Permanent Price Changes
Structural models allow us to conduct a variety of counterfactual simulations (Chintagunta et al
2006). To understand the implications of addiction and stockpiling on manufacturer or retailer
pricing policy, we investigate how consumers’ purchase and consumption decisions respond to
temporary versus permanent price cuts and draw implications on their differential effectiveness.
The distinction between temporary and permanent price changes is critical in the context of
addictive goods because of the dual impact on consumer price expectations and future levels of
addiction and inventory.
First, we compare the temporary and permanent elasticity estimates for cigarettes. Second,
we assess the importance of modeling addiction for cigarettes by comparing the elasticity
estimates from Model 3 (addiction) to Model 2 (no addiction). Third, we compare the elasticities
from cigarettes with those from crackers and butter to illustrate the differential implications of
price changes for addictive and non-addictive goods.
In our simulations, consumers correctly distinguish between temporary and permanent
price changes when forming their expectations. Consumers know a price change is temporary