Incorporating State Dependence in Aggregate Brand-level Demand
Models∗
Dan Horsky†
Polykarpos Pavlidis‡
Minjae Song§
July 2012
Abstract
Empirical investigations of household-level data show that state dependence is a significant
determinant of consumers’ choices in frequently purchased product categories. In this paper we
examine how one can incorporate state dependence in an aggregate demand model for brandlevel data which are more often available and frequently used in applied research in marketing
and economics. We define state dependence as consumers receiving higher utility when buying
the same brand in two consecutive periods. The main challenge we face is that we do not observe
what portion of consumers who chose a given brand had chosen the same brand in the previous
period. We overcome this challenge by constructing the brand choice probability using observed
market shares from the previous period and the law of total probability. Through Monte Carlo
simulations we show that our model is a good approximation of household-level models. We
apply our method to estimating demand for salty snacks using a multi-market brand-level data
provided by IRI. We find significant evidence of positive state dependence, without which demand
looks less price elastic and less advertising elastic. The profit-maximizing price that our model
implies is 7 percent lower than that of a static model without state dependence. We also show
that typical advertising expenditure is only profitable when we account for the carry-over effects
generated by state dependence.
∗
We would like to thank Paulo Albuquerque, Paul Ellickson, Mitchell Lovett, Sanjog Misra, and participants at
the 2011 Marketing Science Conference in Houston, TX for their valuable comments and suggestions. We would also
like to thank SymphonyIRI Group, Inc. for making the data available. All estimates and analysis in this paper, based
on data provided by SymphonyIRI Group, Inc., are by the authors and not by SymphonyIRI Group, Inc.
†
Benjamin L. Forman Professor of Marketing at the Simon Graduate School of Business, University of Rochester,
[email protected].
‡
Consultant, Nielsen Marketing Analytics, [email protected].
§
Assistant Professor of Economics and Marketing at the Simon Graduate School of Business, University of
Rochester, [email protected].
1
1
Introduction
Consumer state dependence has been an important research topic in the marketing science literature
almost from its inception. Many researchers have documented persistence in consumer choice data
in which consumers tend to choose brands that they chose in the recent past: Kuehn (1962),
Massy (1966), Morrison (1966), Givon and Horsky (1979), Keane (1997), Seetharaman, Ainslie,
and Chintagunta (1999), Horsky, Misra, and Nelson (2006), and Dube, Hitsch, and Rossi (2009,
2010). Typical data used to document these findings are panel data recording individual purchase
histories of brands over time. By observing the same consumers over repeated purchase occasions
researchers assess whether consumers become loyal to brands they purchased in the past, controlling
for marketing mix variables.
Although individual consumer panel data are much preferred in this type of analysis, they
are often available only for a small number of product categories, stores, and regions. On the
other hand, aggregate brand level data are more readily available and are less costly to acquire
than individual panel data. However, estimating state dependence with the latter type of data
is challenging because consumer purchase histories are not directly observable from the aggregate
sample. Because of this difficulty, purchase dynamics related to state dependence are usually ignored
when analyzing aggregate data.
When state dependence exists in choice behavior, ignoring it is problematic in several levels.
First, from a theoretical perspective if individual consumers exhibit state-dependence type behavior,
then the market shares which are aggregation of such consumers should display this phenomenon too,
an issue highlighted early on by Givon and Horsky (1978). Second, from a managerial perspective
ignoring this phenomenon may mislead managers to set sub-optimal prices. State dependence
renders market shares more persistent over time, and if we do not control for this persistence,
marketing mix variables may appear to have smaller impact on market shares than they actually
do. This may result in, for example, underestimating the price coefficient. Moreover, we miss the
dynamic effects of marketing mix variables when we ignore state dependence. For example, the
price effect is no longer static; a price change today not only affects sales today but also affects next
period’s sales because some of today’s brand switchers are likely to stay with the same brand next
period. This dynamic effect may last for several periods, and prices that are set without accounting
2
for this effect will be sub-optimal. Similar issues arise with respect to advertising.
In this paper we construct and estimate a model that incorporates state dependence at the
aggregate brand level. We define state dependence as consumers receiving higher utility when
buying the same brand in two consecutive periods, and incorporate it in a standard discrete choice
model for aggregate brand-level data. As is standard in such models we assume consumers receive
an idiosyncratic taste shock that is distributed type I extreme value, and use observed brand market
shares as a consistent estimator for the probability of choosing a brand.
The main challenge is that we do not observe what portion of consumers who chose a given
brand at period t had chosen the same brand in the previous period, t − 1. For example, suppose
there are two brands, brand A and brand B, and no outside option in the market. We observe the
number of consumers who choose brands A and B in each period but do not know how many of
them choose the same brand in the previous period. However, we can still construct the market
share function using Bayes rule and the previous period market shares. Using the same example,
the probability of choosing brand A this period, P (At ), is the sum of the probability of choosing
brand A for two consecutive periods and the probability of choosing brand B in the previous period
followed by choosing brand A this period. That is, P (At ) = P (At−1 , At ) + P (Bt−1 , At ). Using
Bayes rule, we write this as
P (At ) = P (At |At−1 ) P (At−1 ) + P (At |Bt−1 ) P (Bt−1 )
Then we replace P (At−1 ) and P (Bt−1 ) with last period’s market shares and the conditional probabilities P (At |At−1 ) and P (At |Bt−1 ) with the conditional multinomial probability function. The
latter probability function is a function of state dependence and brand/market characteristics.
Using this model, we estimate state dependence by exploiting the persistence in market shares
that is not explained by observed and unobserved brand/market characteristics. In our model, we
also allow the price variable to be an endogenous variable and we correct for such endogeneity using
a control function approach proposed by Petrin and Train (2010). It is important to note that we
do not attempt to identify the nature of state dependence. We do not think it is possible to do this
with typical aggregate data available to researchers. Admittedly, our estimates of state dependence
can represent any form of persistence in brand choices. Reasonable possibilities are brand loyalty,
3
addiction, or switching costs. We rely on other individual level studies for the existence and nature
of state dependence. For example, Horsky, et.al. (2006) use individual consumer brand choice
panel data jointly with preference information from the same individuals to disentangle preference
heterogeneity and state dependence, and they find strong evidence for state dependence. Shin,
Misra, and Horsky (2012), using the same data set further, disentangle the choice persistence into
brand preference and learning. Dube, et.al. (2010) document, using individual consumer panel data,
that persistence in brand choice exists and is not due to a mis-specified distribution of heterogeneity
in preferences or due to autocorrelated choice errors.
Our model is useful for researchers and managers who only have access to aggregate data and
want to better understand purchase dynamics generated by state dependence. As far as we know,
our paper is the first paper to estimate state dependence using brand-level data. There are two
related papers that use a Bayesian approach to a similar problem. Chen and Yang (2007) and
Musalem, Bradlow, and Raju (2009) use aggregate data to estimate demand generated by individual
consumers’ dynamic purchases. They treat individual consumers’ purchase history as a latent
variable and use Bayesian data augmentation to simulate a set of latent choices that are consistent
with observed aggregate data. However, their models have no structural parameter that corresponds
to state dependence. Their goal is to estimate static aggregate demand models that are consistent
with individuals’ dynamic purchase behaviors.
For the empirical implementation of our model we analyze brand-level salty snack data on ten
regional markets, provided by IRI. We use a simulated maximum likelihood estimation method
in the spirit of Park and Gupta (2009), with prices in other regions serving as instruments for
the correction of price endogeneity. Our results show that there exists substantial positive state
dependence in salty snack demand and that one would underestimate both the price coefficient and
long-term effects of price by ignoring state dependence. We show that the sales effect of a one time
price perturbation lasts for more than 20 periods, and as a result, the long-run price elasticity is
about four times as high as the short-run price elasticity.
The managerial implications of these results for pricing are very important. We find that profit
maximizing prices that are set using our state dependence model and its estimates are about 7%
lower than profit maximizing prices that a static demand model implies. This result is consistent
with what Dubé, et.al. (2009) find using their state dependence demand model for individual-level
4
data.
We also estimate the effect of national-level monthly advertising expenditures for the brands in
our sample. There are two important findings related to advertising. First, we do not find strong
lagged effects of advertising; this is true for the specifications with or without state dependence.
Second, incorporating state dependence increases the (contemporaneous) advertising coefficient and
generates advertising carry-over effects over multiple periods in a way similar to that of the longrun price effect. This carry-over effect is crucial in justifying advertising expenditures typically
observed in data. Our counterfactual exercise shows that one-period investment in advertising is
only profitable in the long run thanks to the carry-over effects generated by state dependence.
The rest of the paper is organized as follows. Section 2 describes our state dependent demand
model, followed by an estimation procedure in section 3. Section 4 presents Monte Carlo simulation
results and section 5 discusses estimates of salty snack demand using IRI brand-level data. Section
6 provides managerial implications and section 7 concludes.
2
Demand Model
Assume consumers’ current brand choices depend on their choices in the previous period. The
indirect utility for consumer i for brand j at period t is
uijt = δjt + λI [j = kt−1 ] + εijt
(1)
ui0t = εi0t
for j = 1, ..., J where I [j = kt−1 ] is an indicator function that takes 1 if her choice at time t
(i.e. j) is the same as her choice at time t − 1 (i.e., kt−1 ) and 0 otherwise. δjt is brand quality as a
function of price and brand attributes. εijt is an idiosyncratic taste term with Type I Extreme Value
distribution. j = 0 denotes the outside option and its mean utility is set to zero.
λ is the state
dependence parameter and we do not restrict its sign. A positive λ means that consumers obtain
higher utility by making the same choice over time. This may imply brand loyalty. A negative λ
means that consumers are worse off by making the same choice over time, implying variety seeking.1
1
However, we do not think we can successfully estimate negative state dependence with brand-level data because
negative state dependence and the idiosyncratic taste term generate a similar purchase trend over time.
5
Notice that we assume λ is zero for the outside option. As is common in the individual-level logit
based brand choice models, in our specification the state dependence parameter does not vary over
choices and does not depend on brand attributes. The first assumption is a simplification and the
second assumption excludes a case where firms directly affect state dependence by changing brand
attributes.
Let Pt (j) be the unconditional probability of choosing brand j at time t, and Pt (j|k) the
(conditional) probability of choosing brand j at time t conditional on choosing brand k at time
t − 1. Using the law of total probability, the unconditional probability that brand j is chosen at
time t can be written as
Pt (j) =
J
X
Pt−1 (k) Pt (j|k)
(2)
k=0
Due to the distributional assumption for the idiosyncratic error term, conditional probabilities can
be written as
Pt (j|k) =
exp (δjt + λ)
P
, k=j
1 + exp (δjt + λ) + m6=j exp (δmt )
(3)
Pt (j|k) =
exp (δjt )
P
, k 6= j
1 + exp (δkt + λ) + m6=k exp (δmt )
(4)
Pt (j|k) =
exp (δjt )
P
, k=0
1 + m exp (δmt )
(5)
Notice how the state dependence parameter shifts the conditional probability depending on last
period’s choice. Suppose the state dependence parameter is positive. If the consumer makes the
same choice in two consecutive periods, her probability of choosing the same brand in the second
period is higher, (i.e. equation 3). Consequently, the probability of choosing a different brand in
the second period (brand switching) is lower, (i.e. equation 4). However, if the outside option is
chosen in the first period, the brand choice probabilities are not affected by the state dependence
parameter (i.e. equation 5).
While we can use observed market shares as an empirical counterpart of the unconditional
probability, there is no empirical counterpart for the conditional probability in brand-level data.
Let sjt be observed market share for brand j at time t. By replacing the unconditional probabilities
6
with the observed market share, equation (2) can be rewritten as
sjt =
J
X
(sk,t−1 × Pt (j|k))
(6)
k=0
where Pt (j|k) is a function of (δ1,t , ..., δJ,t ) and λ.
This model does not suffer from the independence of irrelevant alternatives (IIA) property.
This well-known property implies that the relative choice probability of two options is not affected by the availability of a third option. The famous red bus and blue bus example demonstrates why this property is problematic. A standard conditional logit model has the IIA property
as Pr (i = 1) / Pr (i = 2) = exp (δ1 ) / exp (δ2 ) where Pr (i = j) is the probability that consumer i
chooses option j. One can easily see that our model is free from this property as the denominator
is not cancelled in the relative probability ratio.
Incorporating consumer heterogeneity in our model is not a trivial task. First of all, it is not
clear whether we can identify consumer heterogeneity with state dependence using brand-level data.
Even when we set aside the identification issue, it is challenging to put heterogeneity in a way that is
consistent with the utility maximization model. Consider the unconditional probability of choosing
brand j in time t. With consumer heterogeneity, equation (2) becomes
ˆ
ˆ
∞
Pt (jt ) =
∞
( J
X
−∞
k=0
Pi,t (jt ) f (β) dβ =
−∞
)
Pi,t−1 (k) Pit (j|k) f (β) dβ
Notice that we cannot use the observed market shares in period t − 1 for Pi,t−1 (k). Thus, the
probability that consumer i who is endowed with ci chooses brand jt 6= 0 in period t is
Pi,t (jt )
=
Jt−1
Jt−1
X
X
Pi,t−1 (kt−1 ) Pi,t (jt |kt−1 ) =
kt−1 =0
Jt−1
=

Jt−2
X

kt−1 =0
kt−2 =0
...
J2
X

J1
X

k2 =0

Jt−2
X

kt−1 =0
X

Pi,t−2 (kt−2 ) Pi,t−1 (jt−1 |kt−2 ) Pi,t (jt |kt−1 )
kt−2 =0

Pi,1 (k1 ) Pi,2 (j2 |k1 ) Pi,3 (j3 |k2 ) ...Pi,t−1 (jt−1 |kt−2 ) Pi,t (jt |kt−1 ) (7)
k1 =0
where
Pi,t (jt |kt−1 ) =

exp (δijt )
P
, kt−1 6= jt
1 + exp (δikt + λ) + m6=k exp (δimt )
7
Pi,t (jt |kt−1 ) =
exp (δijt + λ)
P
, kt−1 = jt
1 + exp (δijt + λ) + m6=j exp (δimt )
In order to construct the likelihood function, we need to track each simulated consumer’s purchase
history from the first period. In doing so, we do not use information given by the previous period
market shares, sj,t−1 , as in equation (6), and the model-fit becomes much worse.
3
Estimation Methodology
We first assume that brand quality is a linear function of brand characteristics including price.
We estimate our model by constructing the likelihood of observing the unconditional brand-level
market shares and finding parameters that maximize the log of that likelihood. Let Xjt be brand j’s
characteristics at t and Gjk (Xjt |θ) be the conditional probability that brand j is chosen when brand
k is chosen at the previous period. The parameter set θ includes the preferences of characteristics
β and the state dependence parameter λ. Gjk (Xjt |θ) becomes either equations (3), (4) or (5)
depending on the previous period’s choice. The likelihood that we observe s = [s0 , s1 , ...sJ ] in
market m at time t is
Lmt (θ) =
J
Y
Ljmt (θ)sj Nmt
(8)
j=0
PJ
× Gjkmt (Xjmt |θ)) and Nmt is the number of consumers in market
Q
Qm=M
m at time t. The log likelihood function we maximize is the log of L (θ) = t=T
t=1
m=1 Lmt (θ).
where Ljmt (θ) =
k=0 (sk,m,t−1
In addition to brand characteristics and price, Xjmt includes time dummy variables for timevarying demand shocks and market dummy variables for unobserved market-level differences. Even
after controlling for these demand factors, the idiosyncratic taste term may still contain brand-level
demand shocks that are correlated with the price variable. We use the control function approach
by Petrin and Train (2010) to correct for the price endogeneity bias. To implement the correction
we first regress observed brand prices on the explanatory variables of our model and instrumental
variables. Then, in the next stage, we use the residuals from the first stage pricing regressions
as proxies for any unobserved factors in each brand’s demand that may be correlated with the
respective brand’s price. We use three instruments: i) the sum of prices of brands produced by
other firms in other markets at the same period, ii) the sum of prices of other brands produced by
the same firm in other markets at the same period and iii) the sum of prices of the same brand in
8
other markets at the same time period.
If there is no state dependence parameter, MLE estimation is straightforward. When demand is
state dependent as in our model, estimation of consistent parameters becomes more involved, especially with brand-level data. With consumer-level data researchers do observe whether consumers
buy the same brand in two consecutive category purchases, and use this observation in estimating
state dependence. With brand-level data, on the other hand, individual purchase histories are not
available and therefore we need to exploit the persistence of brand market share over two consecutive periods that is not explained by changes in brand characteristics and price. Suppose one brand
lowers its price this period and raises it back to the original level in the following period. If demand
exhibits positive state dependence, its market share would go up this period but would not go down
to the original level next period.
4
4.1
Monte Carlo Simulation Results
Homogeneous Consumers
In this section we use Monte Carlo simulations to evaluate how well our brand-level state dependent
model approximates an individual-level state dependent model. For this we generate data using an
individual-level state dependent model, and then aggregate them to the brand level and estimate
the brand-level state dependent model. In both models we do not allow any consumer heterogeneity
other than through the idiosyncratic error terms. The synthetic data has M=6 markets, N=500
households per market, T=60 time periods and J=2 brands (2 brands and the outside option). We
repeat data generation and estimation for 100 times and report the empirical moments of estimates.
The indirect utilities for consumer i in market m and period t are defined as follows:
uijt = γj Dj + γm Dm − αP ricejmt + λ × I(Yij,t−1 = 1) + εijt
ui0t = εi0t
for j = 1, 2 where
P rice1mt ∼ U nif orm(0.5, 1.5)
P rice2mt ∼ U nif orm(0.9, 1.8)
9
The means and standard deviations of estimated parameters are reported in table 1 for three different
levels of state dependence, λ = 1, 3, 5. The table also reports estimates from the individual-level
state dependent model for comparison. Results show that our brand-level model recovers true
parameter values reasonably well. There is no significant difference in the mean between the two
models for all three levels of the state dependence parameter. The standard deviation is slightly
larger in the brand-level model, but the difference is negligible. The only parameter whose variance
is significantly larger is the state dependence parameter. When λ = 1, its standard deviation is
0.091 while it is 0.012 in the individual-level model. For λ = 3 and λ = 5, it is 0.085 and 0.078
respectively while it is less than 0.02 for both cases in the individual-level model. However, these
numbers are still reasonably small.
Using the same data we also estimate a brand-level model that ignores state dependence by
constraining λ to be 0. Table 2 reports the results. The table shows that all the brand attributes
are overestimated, and the inconsistency drastically increases with state dependence. For example,
the mean of brand 1 quality estimates, whose true value is -1.5, is -1.423 when λ is 1 (with the S.D.
0.026), and goes up to -1.084 when λ is 3 (with the S.D. 0.046) and 0.006 when λ is 5 (with the
S.D. 0.058). The price coefficient, whose true value is -1, is also overestimated. The mean value
is -0.967 when λ is 1 and goes up to -0.575 when λ is 3 and -0.195 when λ is 5. These results
show that, when state dependence is ignored, purchase persistence is attributed to brand quality
and price sensitivity, misleadingly implying that consumers’ brand preference is higher and price
sensitivity is lower than they really are.2
4.2
Heterogeneous Consumers
In section 2 we explained why estimating consumer heterogeneity with brand-level data is a challenging task in the state-dependent model. To demonstrate this, we simulate data using the individuallevel state-dependent model with consumer heterogeneity on the price coefficient, and estimate
the brand-level state-dependent model. Table 3 reports results for both the brand-level and the
individual-level models. We estimate the latter model using a straightforward MLE procedure, and
estimate the former model using equation (7) as the likelihood function for each simulated consumer.
The table shows that the price coefficient tends to be overestimated and the degree of hetero2
We repeat this exercise in the individual-level model with λ constrained to be 0 and find a very similar pattern.
10
geneity (the standard deviation of the price coefficient) underestimated in the brand-level model.
The other estimates are either underestimated or overestimated. The only exception is the state
dependence parameter whose mean value is not different from the true value. The standard deviation of the estimates in the brand-level model is much larger (2 to 6 times larger) compared to the
individual-level model.
We also estimate the homogeneous consumer brand-level model to assess the magnitude of inconsistency resulting from ignoring consumer heterogeneity. Table 4 shows that the state dependence
parameter is less affected than the other parameters when we ignore consumer heterogeneity. When
the standard deviation of the price coefficient is 0.25, the mean of the state dependence estimate is
2.99 (the true value is 3) while the mean of the price coefficient estimate is -2.64 (the true value is
-3) and the mean of the brand intercepts are -1.69 (-1.5) and -1.22 (-1.0) respectively. When the
standard deviation of the price coefficient goes up to 0.5, the price coefficient goes up to -2.33 and
the brand intercepts to -1.83 and -1.37, but the state dependence estimate stays at 2.99. However,
when the standard deviation of the price coefficient is 1, the mean of the state dependence estimate
goes down to 2.89 while the other parameters depart further from the true values. Note, however,
that a standard deviation larger than 1 for the heterogeneity in price sensitivity is rarely found in
CPG data. Nevertheless, these results suggest that the brand-level state dependence model may
have some limitations for categories where consumer heterogeneity is important.
4.3
State Dependence with Memory
A very well-known and widely-used model of brand loyalty in the marketing literature is that by
Guadagni and Little (1983). In the Guadagni & Little model (the GL model hereafter) the brand
loyalty that enters the utility of each brand j in time t (GLjt ) is defined as
GLjt = α × GLj,t−1 + (1 − α) × Yj,t−1
where Yj,t−1 is a lagged purchase dummy variable for brand j and α is a weighting parameter that
takes a value between 0 and 1. Our state dependence model is a special case of the GL model
where we set α = 0. In the next Monte Carlo experiment we generate individual-level data using
the GL model and estimate their aggregated counterparts with and without state dependence. By
11
doing so, we evaluate the performance of our state dependent model against the model without
state dependence when data are generated by the GL model.
Table 5 shows that our state dependent model can greatly reduce biases in the model without
state dependence. Brand quality rankings are much closer to their true values when state dependence
is included and the same is true for the price coefficient which goes down from -0.719 to -0.971.
However, the state dependence parameter is smaller than the true parameter (2.06 versus 3). Since
α is set to 0.5 in the simulated data, last period’s choice increases brand loyalty by 1.5 (3 × 0.5) in
the GL model, which means that our model overestimates brand loyalty. This is expected as our
state dependence model is not the data generating model. Nevertheless, we can conclude from this
experiment that when data are believed to exhibit brand loyalty of the GL type and only brand-level
data are available, our state dependent model describes data much better than the model without
state dependence.
5
Application to the Salty Snack Category
5.1
Data
We estimate our state dependent model using the IRI Marketing data set on the salty snack category.
Bronnenberg, Kruger, and Mela (2008) describe this data set in detail. Data are originally at
week/store/UPC level, and we aggregate them at month/brand/region level and define a market as
a region-month pair. For sales we use one serving of 16 oz. as the unit of measurement. Prices are
computed by dividing the total dollar sales for each brand with the total number of servings sold in
each market. We assume that the maximum number of servings that can be sold in any particular
market is equal to two times the maximum number of servings sold in the corresponding market
during the sample period, and use it as the market size for each market. This allows us to reflect
regions’ different consumption levels in market sizes. Then we obtain market shares by dividing the
total number of servings sold with the corresponding market size. The underlying assumption is
that no more than a half of the category buyers buy salty snacks in any given period.
For the category of salty snacks the IRI Marketing data set is distributed along with advertising
data. The source of the advertising data is TNS and the available information includes nationallevel, monthly, advertising expenditures for each product line in the category. The largest part
12
of advertising expenditures for salty snacks is dollars spend on TV. We aggregate this variable
over product lines to construct national-level advertising expenditures for each brand. Because
advertising expenditures are reported for month t on the first day of month t + 1, we use the
reported expenditures for month t + 1 as the advertising expenditures for month t.
We select five major brands, namely Doritos, Lays, Pringles, Tostitos, and UTZ, and ten US
non-adjacent regions such as Charlotte, Houston, Knoxville, Milwaukee, New England, etc. Our
regions are equivalent to IRI-defined markets. Our sample period is 60 months from January of
2001 to December of 2005. Table 6 presents the descriptive statistics of our sample. The table
shows that the average price is almost perfectly negatively correlated with the average share. A
brand with a higher average price has a smaller average market share. Note that UTZ is present in
only five regions and the average market share does not account for zero market share in the other
five regions. It has particularly a high market share (14%) in the Washington D.C. area. The table
also shows that both prices and market shares vary more across markets than across time and that
market shares are more stable over time than prices.
Lastly, the table shows that Pringles spends most heavily on national advertising, followed by
Lays, Doritos and Tostitos, while UTZ does not spend on national advertising. However, table 6 does
not completely describe how firms advertise over time. Figure 1 shows national-level advertising
expenditures for each brand throughout the sample period. The figure shows that all brands’
advertising expenditures follow pulsing patterns. Each firm follows slightly different pulsing patterns
but usually advertises heavily for a few months and then advertises lightly for the next few months.
They spend over $5 million per month during the heavy-advertising periods and spend nothing or
far less than $1 million per month during the light-advertising periods. This pulsing pattern is in
sharp contrast with the stability in market shares over time.
5.2
Estimation Results: Brand Qualities, State Dependence, and Price Effects
We first estimate the brand-level model without state dependence. For brand characteristics we
include four brand intercepts, excluding UTZ as the base, and the price variable. We control for
regional differences by dividing regions into four general groups (i.e. South, North-east, Mid-west
and West) and including three region dummy variables. We also include three season dummy
variables to control for seasonality. The consumer size, Nt in equation (8), is set to 100 per market.
13
The estimation results are reported in table 7. On the left panel we report results without
the price endogeneity correction. The price coefficient is negative and statistically significant. The
brand intercept estimates imply that Pringles has the lowest quality, followed by Tostitos, Doritos,
Lay and UTZ, reflecting the market-share ranking. The other estimates show that demand is lower
in the west region and in the winter season.
The right panel of the table reports results with the control function for the price endogeneity.
Including the control function lowers the price coefficient from -0.40 to -2.44 and it is still statistically
significant. The 1st-stage residual term shows that prices are positively correlated with unobserved
demand factors that the brand intercepts and the regional and seasonal dummy variables do not
capture. The brand quality ranking drastically changes. The highest quality brand is now Pringles
followed by Tostitos, Doritos, Lay and UTZ. This is totally opposite to the previous ranking and
consistent with the price ranking. Also, demand is higher in the summer and fall seasons than in
the winter and spring seasons. The regional effects, however, stay the same.
Next, we estimate the brand-level state-dependence model, using the same specification and
instrumental variables as above. We report estimation results in table 8. The state dependence
parameter is statistically significant and estimated at 4.95 without the control function and 4.93
with the control function. The price coefficient goes down from -0.40 to -0.56 without the control
function and from -2.44 to -3.63 with the control function. The coefficient on the control function
goes up to 3.13 showing a higher degree of correlation. This change in the price coefficient is
consistent with the Monte Carlo simulations in section 4. The model fit improves considerably with
state dependence. The log likelihood increases from -50,742 to -50,161 without the control function
and from -50,694 to -50,132 with the control function. Similarly, the predictive ability of the model
improves with RMSPE decreasing by 28% from 0.025 when state dependence is ignored to 0.018
when it is included. However, the brand quality ranking and the regional/seasonal demand effects
that the estimates imply do not change much with state dependence, although their magnitude
slightly changes.
Using the estimates on the right panel of table 8 we calculate the short-run and the long-run
price elasticity. The short-run price elasticity is straightforward to calculate. It measures the sameperiod demand changes (in %) with respect to one percent price change. For the long-run elasticity
we consider two measures. One is to perturb the price in a given period and track demand changes
14
over time. That is,
∞
X
∂sr
r=t
∂pt
=
∂st
∂st+1 ∂st
∂st+2 ∂st+1 ∂st
+
+
+ ...
∂pt
∂st ∂pt ∂st+1 ∂st ∂pt
(9)
where st is a market share in period t. Although this equation shows we should track demand
changes infinitely, the future demand effect wears out completely in finite periods as ∂st+1 /∂st
quickly approaches zero. In our case 50 periods is sufficient for the demand effect to completely
wear out. The long-run elasticity in this case is defined as
eLR1 =
∞
X
∂sr
r=t
∂pt
!
pt
st
Figure 2 illustrates the typical path of demand changes resulting from one period price decrease.
For this graph we decrease Dorito’s price by 1 percent from the median level in the second period
of the sample and track demand changes over 50 periods. We fix the prices of the other brands
and the 1st-stage residual at their median value, while accounting for the seasonal and the regional
effects using the estimates. For the first period of our prediction we use the average brand share
from the data as the lagged share in each market, and in the following periods we recursively
substitute the model-predicted share of a given period as the lagged share in the next period’s
demand function. The figure shows that the demand effect of one time price perturbation quickly
but smoothly diminishes and becomes negligible after about 20 periods.
An alternative measure is to change the price for all periods and track demand changes over
time. That is,
∞
X
∂st
∂st ∂st−1
∂st ∂st−1 ∂st−2
∂st
=
+
+
+ ...
∂pt−r
∂pt ∂st−1 ∂pt−1 ∂st−1 ∂st−2 ∂pt−2
(10)
r=0
In this case we define the long-run elasticity as
eLR2 =
∞
X
r=0
∂st
(∂pt−r /pt−r )
!
1
st
As in equation (9) the dynamic effect diminishes over time and we only need to track back 50
periods to compute equation (10).
Table 10 shows the brand-level short-run and the two measures of the long-run price elasticities
with respect to one percent price change in all markets. We compute the elasticities for each month
15
and average them over 12 months to account for the seasonal effects. The short-run price elasticities
range from -4.2 (UTZ) to -8.8 (Pringles). The two measures of the long-run elasticity are similar
although the second measure is slightly larger than the first one. The first measure of the longrun price elasticity ranges from -15.7 (UTZ) to -21.3 (Pringles) and the second measure from -16.8
(UTZ) to -22.5 (Pringles). On average, the long-run elasticity is 3∼4 times larger than the short-run
one.
5.3
Estimation Results: State Dependence and Advertising Effects
In this section we estimate the effects of advertising on market shares in the brand-level model.
As mentioned in section 5.1, the advertising expenditure follows a pulsing pattern while market
shares are relatively stable over time. This contrasting pattern may make it hard to establish a link
between advertising and market shares. Controlling for state dependence, therefore, is important as
it explains large part of the stability in market shares. We first estimate the advertising effects using
the model without state dependence and put state dependence in the model to compare results.
To accommodate the dynamic effects of advertising we construct an adstock variable using
the current period and two lagged period expenditures with a geometrically declining carry-over
parameter. That is,
Adstock = At + γa At−1 + γa2 At−2
where At is advertising expenditure in period t and γa is the carry-over parameter.
The left panel of table 9 shows estimation results for the model without state dependence. We
use a one-dimensional grid search for the carry-over parameter and find that the log likelihood is
highest when the parameter value is 0 and strictly decreases as the parameter value increases. The
coefficient for the adstock variable (the current period effect) is 0.135 and statistically significant.
Including the adstock slightly changes the magnitude of the other estimates but the demand implications hardly change. These results show that in the model without state dependence the current
advertising effects are moderate and the lagged advertising effects are absent. The short-run advertising elasticities for Doritos, Lays, Pringles and Tostitos are 0.02, 0.03, 0.04 and 0.02 respectively,
while the long-run elasticity is 0 for all brands.
The right panel of table 9 shows estimation results for the state dependent model. It shows
16
that the adstock coefficient is much larger (0.37) than in the model without state dependence. This
change is intuitive: the advertising expenditure fluctuates a lot over time, while sales are relatively
stable, and by controlling for the persistence of sales with state dependence, the sales become more
sensitive to advertising. The short-run advertising elasticities are now 0.03, 0.04, 0.07 and 0.03 for
Doritos, Lays, Pringles and Tostitos.
The carry-over parameter is still 0 as in the model without state dependence, but the advertising
effects are long lasting because of state dependence. The effects are similar to the dynamic price
effects. Figure 3 shows how a 1 percent increase in Lays’ advertising expenditure affects sales over
time. The advertising effect lasts for 27 periods and the long-run advertising elasticity is 0.14.
These results are in accordance with reported advertising elasticities in Sethuraman, Tellis, and
Briesch (2011) who find in their extensive meta-analysis that the median short-term elasticity of
advertising is 0.05 and the median long-term elasticity is 0.1. Our findings are also consistent with
those reported by Givon and Horsky (1990) who find, in addition to state dependence, current but
not lagged effects of advertising. Their conclusion is that consumers are much more impacted by
their past consumption experience than by their memory of past ads.
6
Managerial Implications
6.1
Comparison of Optimal Prices
To study the managerial implications of our model further we compute the optimal prices that
correspond to a fully forward-looking pricing behavior and compare them with the optimal prices
implied by static per-period profit maximization. In the forward looking model firms account for
customers’ current states in setting prices and know that current-period prices affect future-period
profits through market shares. Thus, they set prices as a function of last period’s market share
and maximize the discounted sum of future profits over infinite horizon. For the static per-period
profit maximization firms use a Bertrand Nash pricing model where demand is static and not state
dependent.
In the static case, each firm (brand) maximizes the following per-period profit.
πjt = s̃jt × M × (pjt − cj ) ∀j
17
where s̃jt corresponds to the standard logit probability of choosing brand j without state dependence, M denotes market size, and cj stands for marginal cost for brand j. The optimal prices in
this case simultaneously satisfy the following J first-order conditions, one for each brand.
s̃jt × M +
∂s̃jt
× M × (pjt − cj ) = 0 ∀j
∂pjt
The computation of optimal prices is much more involved for the forward-looking case. Essentially we solve a dynamic game using our proposed model to find the optimal pricing policies
for each brand. Time is discrete and the state variable is the vector of brands’ market shares last
period. The profit of brand j in period t is a function of the state vector and prices. That is,
πjt (xt , pt ) = sjt (xt , pt ) × M × (pjt − cj )
where now xt = (x0 , x1 , ..., xJ ) denotes the vector of last period shares for all brands and the outside
option and sjt (xt , pt ) is our state-dependent choice probability.
Our demand specification implies a natural transition rule for the state vector. Basically, given
last period’s shares and current prices, firms know current period’s shares and these shares become
firms’ states for next period. This deterministic evolution of the state vector is operationalized with
a Markovian transition matrix.
xt+1 = g(pt , xt ) = Q(pt ) × xt
Qjk (pt ) = Pt (j|k)
Firms maximize their current and discounted future profits conditional on payoff relevant information that is summarized in the state vector. Strategies are functions that map the state space
into the continuum of possible prices, σj : X → R . The current and future payoffs of each firm are
described by the related Bellman equation.
Vj (x) = maxpj ≥0 {πj (x, p) + βVj [g(p, x)]}
∀x ∈ X
(11)
where Vj represents all payoffs given a strategy profile for firm j, σj . We use the notion of Markov
18
Perfect Equilibrium to compute equilibrium prices in the form of pure strategies. At equilibrium
each firm has an optimal strategy that prescribes the best response to all rivals and for all possible
states. Denoting the strategy profiles of competitors with σ−j , the optimal strategy for j, σj∗ , is
described by the following equation.
∗
∗
σj∗ (x) = argmaxpj ≥0 πj x, pj , σ−j
(x) + βVj g(pj , x, σ−j
(x))
∀x ∈ X, ∀j
(12)
This means that the optimal price of each firm should satisfy the following first order condition,
for each possible state vector
sj (x) × M +
∂sj (x)
∂Vj (x) ∂sj (x)
× M × (pj (x) − cj ) + β ×
=0
∂pj (x)
∂sj (x) ∂pj (x)
∀x ∈ X, ∀j ∈ J
Compared to the first-order condition in the static problem, this first-order condition has a different
market share function and a new third term that summarizes all future payoffs accruing from
current period actions. Thus, differences in optimal prices come from (i) a way that the currentperiod market share responds to price changes and (ii) the additional effects of price changes on
future profits. The first effect is expected to push prices upwards as state-dependent demand is
less sensitive to price. The second effect is expected to push prices down because the future value
increases as prices go down.
Our dynamic model is similar to that by Dubé, Hitsch, and Rossi (2009), although some of
details are different. One of them is that we allow state dependence to become zero after the choice
of the outside option. This creates a situation where not everyone in the market is loyal to one of
the brands in the choice set. It turns out that in our implementation a large portion of consumers
is not loyal to any of the brands at any given period and, thus, has a lower purchase probability
than in Dubé, et.al. (2009).
We use a numerical policy-iteration algorithm to compute pure strategy equilibrium (MPE) for
the pricing game3 . To approximate the continuous state space, we discretize market shares such that
a grid is finer along the range of values that we observe in the data. Cost estimates for the pricing
3
Extensive details for our algorithm is available from the authors upon request.
19
game are derived from the first order conditions of the static game and average prices observed
in the data. Although we cannot provide a formal proof that a unique equilibrium exists, all our
numerical computations converge to the same pricing policies and simultaneously satisfy the system
of first-order conditions.
Figures 4 and 5 show the policy functions of “Doritos” (Brand 1) and “Lays” (Brand 2). In these
figures we fix the state variables of the other brands to points on the grid that are closest to the
brands’ average market shares and vary the two focal brands’ state variables. The figures show that
prices are increasing with a decreasing rate in the own-brands’ states. A higher market share last
period is associated with a higher price, everything else constant. The slope is relatively steep up
to about 0.20, which covers the whole range of market shares shown in the data, and levels out
for higher values. This suggests that firms do take advantage of consumer loyalty and charge more
when a larger fraction of the market is “locked-in”.
The figures also show that firms increase prices as a rival’s last-period market share increases.
This is because prices are (i) an increasing function of an own-brand state variable and (ii) strategic
complements. However, this price increase is not monotonic. For low and moderate levels of
own states, up to 0.15 for Doritos’ case, prices increase in a rival’s states, first steadily and then
exponentially. For higher levels of own states, prices increase in a rival’s states with a decreasing
rate and quickly flatten out.
Using the computed policy functions we simulate the market until all firms reach a steady state
where last-period shares are the same as current-period shares. The steady state exists because the
model does not have demand shocks that perturb the market at a positive probability every period.
Figures 6, 7, and 8 show the time-path of the state variable (i.e. last period’s market share) with
a different set of initial market shares in each figure. One may interpret them as the time-path of
each brand going back to its steady state after one-time perturbation.
All figures show that, irrespective of the initial state vector, the state variables of all brands
converge very close to steady state levels within about ten periods and converge completely in about
thirty periods.4 In particular, when a brand’s state is higher than the steady-state level, as Doritos,
Pringles, and Tostitos in figure 8, it charges moves towards a lower state by charging a higher-thansteady-state price. Similarly, when a brand’s state is lower than the steady-state level, as all brands
4
Note that our optimal prices are optimal “regular” prices. We do not consider more sophisticated pricing policies
that include temporary price reductions.
20
in figure 7, it moves towards higher level by charging lower-than-steady-state prices.
Table 11 reports optimal prices in three different models: (i) a static demand model without
state dependence and with myopic firms, (ii) a state dependent demand model with myopic firms,
and (iii) a state dependent demand model with forward-looking firms. The second model uses the
same grid for last period’s market shares as the third model. The demand specifications are identical
in all three compared models except for state dependence.
The optimal prices for each model are computed as follows. For the first model we solve the
system of first-order conditions as a nonlinear system of equations to obtain optimal prices and
corresponding market shares. For the second model, we repeat the same process but use the statedependent demand model where current-period demand depends on last period shares. To report
results succinctly we iterate the process until current-period shares become the same as last-period
shares. For the third model, we use the steady-state prices computed in section 6.2.
Comparing the no state-dependent static model (the first model) with the forward-looking statedependent model (the third model), prices are about 8 percent lower and market shares are 40
percent higher in the forward-looking model. In the forward-looking model firms charge lower
prices today to gain higher market shares in the future. Since all firms adopt the same strategy,
larger market shares come from consumers who choose the outside option in the static pricing model.
The total category market share is 0.25 in the forward-looking model, while it is 0.15 in the first
model. These results imply that if a manager is myopic and sets prices without accounting for state
dependence, she will set higher-than-optimal prices and will lose consumers to competing brands
with forward-looking managers.
The results in the second model show that even if myopic managers account for state dependence
in demand, they will still set much higher prices and gain lower market shares than those of the
forward-looking model. Compared to the first model, prices are lower except for Lays but market
shares are not higher even for Lays. This means that demand is less price elastic in the second model.
Since the price coefficient is more negative in the second model, we may conclude that the effects
of the state dependence, which renders demand less price sensitive, dominates the effects of more
negative price coefficient. This result implies that myopic managers may do worse by accounting
for state dependence.
21
6.2
Return on Advertising Investment
To further demonstrate the implications of our model, we compare return on investment (ROI)
in advertising for the state dependence model with that of the model without state dependence.
Note, however, that our sales data come from a sample of markets in the US while the advertising
expenditure is for the whole US market. Thus, to correctly compute the advertising ROI we need
to rescale our market size to make it relevant to national advertising.5
Using data from Euromonitor International, Agriculture and Agri-Food Canada reports the
annual volumes of snacks sold in 7 categories in the US market (see International Markets Bureau
(2011)). Among them we add the volumes of 4 categories including chips/crisps, extruded snacks,
tortilla/corn, and other sweet/savory snack for 2005 and 2006, which are 2,075,700 and 2,095,110
tones respectively, and take the average of these two year volumes.6 Then we convert this volume
into annual servings by using 16 oz. as one serving. This gives us 4.5 billion servings as an estimate
for the annual US market size.
With this estimate at hand, we compute the advertising ROI for Frito Lays, the market leader
in the salty snack category.7 We use the state dependence model as well as the model without state
dependence to simulate predicted sales for fifty periods from advertising. Recall that we capture all
of the long-term advertising effects in fifty periods. In the base scenario we assume that all brands
do not advertise at all for all periods. In the counterfactual scenario we assume that only Lays
spends five million dollars on advertising in the first month. This is the level of advertising that is
commonly observed in the data for Lays (see Figure 1).
In the model without state dependence where there is no long run effect of advertising, the
advertising ROI is 27.1 percent, meaning that five million dollar advertising expenditure increases
revenue by 1.36 million dollars and Lays makes a loss from the advertising campaign. In the state
dependence model the one period ROI is 32.5 percent and the fifty period ROI is 118 percent. The
one period ROI is higher than that of the model without state dependence mainly because of a
larger coefficient on the advertising variable but still does not cover the advertising expenditure.
However, at the end of fifty periods Lays’ revenue goes up by 5.9 million dollars, resulting in 0.9
million dollar net profit. It is worth pointing out that the advertising investment breaks even within
5
Note that the market size does not play a role in computing the optimal prices.
The excluded categories are nuts, popcorn, and pretzels.
7
Frito Lays has the highest share in the total market, although it does not lead in all regional markets.
6
22
six months. This result shows that the long-term effects generated by state dependence rationalize
Lays’ advertising investment, while the typical advertising expenditure observed in the data would
not be profitable in the standard demand model.
7
Conclusion
We propose a demand model for aggregate brand-level data that can capture choice dynamics due
to state dependence. Through Monte Carlo experiments we show that our model and estimation
method recover true parameter values as well as individual-level demand models do. We also show
that marketing-mix variables are underestimated when state dependence is neglected. Estimation
with real brand-level data on salty snacks from ten regions in the U.S. shows statistically significant
positive state dependence. Our model is strongly supported by the data against the equivalent
brand-level models that ignore state dependence.
The managerial implications of our estimation results are important for both the evaluation
of advertising effects and pricing decisions. When state dependence is included, both current and
carry-over effects of advertising are stronger, but these stronger effects are attributed not to lagged
advertising effects but to market-share persistence. With respect to pricing, we show that the
optimal prices of forward-looking firms based on our dynamic model are much lower than the
corresponding optimal prices implied by the static model.
Incorporation of state dependence in aggregate brand-level models as advocated in this paper
is important in several levels. First, from a theoretical perspective if individual consumers exhibit
state-dependence type behaviors, the market shares which are aggregation of such consumers have
to display this phenomenon too. Second, from a managerial perspective taking account of state
dependence in aggregate models allows managers, as we have just shown, to assess better the
impact of prices and advertising and to set these marketing mix variables at more appropriate
levels. Third, the importance of the existence and modeling of state dependence have been further
highlighted recently by Freimer and Horsky (2008, 2012) who demonstrate that consumers’ state
dependence may well be the underlying force that leads firms to use temporary price promotions
and advertising pulsing; strategies that are indeed used by the managers in the product category
we examined as well as by managers of brands in many other product categories.
23
References
[1] Berry S. (1994), “Estimating Discrete-Choice models of Product Differentiation”, RAND Journal of Economics, 25, 242-262.
[2] Berry S., J. Levinsohn, and A. Pakes (1995), “Automobile Prices in Market Equilibrium”,
Econometrica, 63, 841-890.
[3] Bronnenberg, Bart J., Michael W. Kruger, and Carl F. Mela (2008), “Database Paper—The
IRI Marketing Data Set”, Marketing Science, 27, 745-748.
[4] Chen Y. and S. Yang (2007), “Estimating Disaggregate Model Using Aggregate Data via Augmentation of Individual Choice”, Journal of Marketing Research, 44, 613-621.
[5] Dubé J. P., G. Hitsch, and P. Rossi (2009), “Do Switching Costs Make Markets Less Competitive?”, Journal of Marketing Research, Vol. 46, No. 4, 435-445.
[6] Dubé J. P., G. Hitsch, and P. Rossi (2010), “State Dependence and Alternative Explanations
for Consumer Inertia”, RAND Journal of Economics, Vol. 41, No. 3, 417-445.
[7] Freimer M. and D. Horsky (2008), “Try It You Will Like It - Does Consumer Learning Lead to
Competitive Price Promotions?”, Marketing Science, Vol. 27, No. 5, 796-810.
[8] Freimer M. and D. Horsky (2012) “Periodic Advertising Pulsing in a Competitive Market”,
Marketing Science, forthcoming.
[9] Givon M. and D. Horsky (1978), “Market Share Models As Approximators of Aggregated
Heterogeneous Brand Choice Behavior”, Management Science, Vol. 24, No. 13, 1404-1416.
[10] Givon M. and D. Horsky (1979), “Application of a Composite Stochastic Model of Brand
Choice”, Journal of Marketing Research, Vol. 16, No. 2, 258-266.
[11] Givon M. and D. Horsky (1990), “Untangling the Effects of Purchase Reinforcement and Advertising Carryover”, Marketing Science, Vol. 9, No. 2, 171-187.
[12] Horsky D., S. Misra, and P. Nelson (2006), “Observed and Unobserved Preference Heterogeneity
in Brand-Choice Models”, Marketing Science, Vol. 25, No. 4, 322-335.
[13] International Markets Bureau (2011), “Consumer Trends: Salty snack Food in the United
States”, Agriculture and Agri-Food Canada.
[14] Keane M. (1997), “Modeling Heterogeneity and State Dependence in Consumer Choice Behavior,” Journal of Business and Economic Statistics, 15, 310-327.
[15] Petrin A. and K. Train (2010), “A Control Function Approach to Endogeneity in Consumer
Choice Models”, Journal of Marketing Research, Vol. 47, No. 1, pp. 3-13.
[16] Kuehn A. (1962), “Consumer Brand Choice as a Learning Process”, Journal of Advertising
Research, Vol. 2, 10-17.
[17] Massy W. F. (1966), “Order and Homogeneity of Family Specific Brand Switching Processes”,
Journal of Marketing Research, Vol. 3, 48-54.
24
[18] Morrison D. G. (1966), “Testing Brand-Switching Models”, Journal of Marketing Research, Vol.
3, 401-409.
[19] Musalem A., E. Bradlow, and J. Raju (2009), “Bayesian Estimation of Random-Coefficients
Choice Models using Aggregate Data,” Journal of Applied Econometrics, 24, 490-516.
[20] Park S. and S. Gupta (2009), “Simulated Maximum Likelihood Estimator for the Random Coefficient Logit Model Using Aggregate Data,” Journal of Marketing Research, Vol. 46, 531–542.
[21] Qi S., (2008) “Advertising, Entry Deterrence, and Industry Innovation”, Unpublished
manuscript, Florida State University.
[22] Seetharaman P. B., A. Ainslie, and P. Chintagunta (1999), “Investigating Household State
Dependence Effects across Categories,” Marketing Science, Vol. 36, 488-500.
[23] Sethuraman R., G. Tellis, and R. Briesch (2011), “How Well Does Advertising Work? Generalizations from Meta-Analysis of Brand Advertising Elasticities”, Journal of Marketing Research,
Vol. 48, 457-471.
[24] Shin S., S. Misra, and D. Horsky (2012), “Disentangling Preferences and Learning in Brand
Choice Models”, Marketing Science, Vol. 31, 115-137.
25
Table 1: Monte Carlo Results
MC1.1: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
Aggregate Level Estimation
Individual Level Estimation
TRUE
Mean
Std Dev.
Mean
Std Dev.
-1.5
-1
-1
1
-0.5
-1
1
1.5
0.5
-1.497
-0.997
-0.999
0.985
-0.505
-1.003
1.001
1.502
0.502
0.026
0.032
0.021
0.091
0.023
0.029
0.025
0.028
0.022
-1.502
-1.002
-0.999
1.000
-0.497
-0.997
1.002
1.500
0.500
0.024
0.030
0.020
0.012
0.023
0.025
0.017
0.017
0.018
-1.500
-1.001
-1.000
3.001
-0.500
-1.003
0.999
1.499
0.499
0.026
0.032
0.021
0.012
0.025
0.023
0.019
0.023
0.019
-1.498
-0.997
-1.001
5.001
-0.501
-1.002
1.000
1.499
0.498
0.034
0.042
0.030
0.017
0.024
0.026
0.031
0.033
0.026
MC1.2: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
-1.5
-1
-1
3
-0.5
-1
1
1.5
0.5
-1.502
-1.001
-0.998
2.998
-0.497
-1.002
1.001
1.501
0.502
0.032
0.036
0.030
0.085
0.023
0.027
0.022
0.027
0.022
MC1.3: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
-1.5
-1
-1
5
-0.5
-1
1
1.5
0.5
-1.506
-1.009
-0.992
4.990
-0.496
-0.998
1.005
1.500
0.504
0.039
0.054
0.052
0.078
0.023
0.029
0.036
0.039
0.027
26
Table 2: Monte Carlo Results with Ignored State Dependence
MC2.1: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
Aggregate Level Estimation
Individual Level Estimation
TRUE
Mean
Std Dev.
Mean
Std Dev.
-1.5
-1
-1
1
-0.5
-1
1
1.5
0.5
-1.423
-0.907
-0.967
-0.543
-1.072
1.125
1.697
0.559
0.026
0.031
0.020
0.027
0.033
0.023
0.023
0.023
-1.425
-0.907
-0.967
-0.537
-1.069
1.124
1.698
0.554
0.026
0.031
0.021
0.025
0.028
0.025
0.023
0.021
-1.088
-0.643
-0.569
-0.728
-1.409
1.492
2.221
0.750
0.041
0.052
0.030
0.038
0.041
0.041
0.043
0.040
0.006
0.318
-0.190
-0.860
-1.708
1.612
2.380
0.805
0.065
0.067
0.032
0.071
0.075
0.072
0.082
0.063
MC2.2: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
-1.5
-1
-1
3
-0.5
-1
1
1.5
0.5
-1.084
-0.635
-0.575
-0.725
-1.412
1.492
2.213
0.746
0.046
0.053
0.025
0.046
0.047
0.042
0.045
0.036
MC2.3: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
-1.5
-1
-1
5
-0.5
-1
1
1.5
0.5
0.006
0.312
-0.195
-0.839
-1.681
1.622
2.383
0.823
0.058
0.064
0.030
0.074
0.076
0.077
0.074
0.059
27
Table 3: Monte Carlo Results with Heterogeneity
MC3.1: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Price Coef. Std Deviation
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
Aggregate Level Estimation
Individual Level Estimation
TRUE
Mean
Std Dev.
Mean
Std Dev.
-1.5
-1
-3
1
3
-0.5
-1
1
1.5
0.5
-1.626
-1.142
-2.748
0.856
3.006
-0.467
-0.984
1.010
1.443
0.486
0.143
0.152
0.260
0.134
0.126
0.131
0.130
0.101
0.100
0.110
-1.507
-1.000
-2.943
1.078
3.020
-0.500
-1.004
1.003
1.495
0.499
0.058
0.070
0.045
0.023
0.022
0.076
0.086
0.059
0.057
0.061
28
Table 4: Monte Carlo Results with Ignored Heterogeneity
MC4.1: N = 500, T = 60, 100 iterations, 6 Markets
Aggregate Level Estimation
Brand 1
Brand 2
Price Coefficient
Price Coef. Std Deviation
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
TRUE
Mean
Std Dev.
-1.5
-1
-3
0.25
3
-0.5
-1
1
1.5
0.5
-1.689
-1.219
-2.639
2.993
-0.523
-0.995
0.992
1.466
0.507
0.046
0.062
0.050
0.081
0.050
0.069
0.041
0.041
0.038
MC4.2: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Price Coef. Std Deviation
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
-1.5
-1
-3
0.5
3
-0.5
-1
1
1.5
0.5
-1.831
-1.373
-2.330
2.996
-0.515
-1.039
0.962
1.385
0.484
0.044
0.065
0.056
0.094
0.061
0.070
0.038
0.043
0.041
MC4.3: N = 500, T = 60, 100 iterations, 6 Markets
Brand 1
Brand 2
Price Coefficient
Price Coef. Std Deviation
Lambda
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
-1.5
-1
-3
1
3
-0.5
-1
1
1.5
0.5
29
-2.029
-1.573
-1.826
2.894
-0.498
-1.025
0.825
1.223
0.461
0.050
0.057
0.047
0.106
0.056
0.059
0.035
0.035
0.037
Table 5: Monte Carlo Results with SD Memory
MC5.1: N = 500, T = 60, 100 iterations, 6 Markets
SD as in Guadagni & Little - Aggregate Level Estimation
Ignoring SD
Brand 1
Brand 2
Price Coefficient
Lambda for GL
Smoothing α
Market Effect 1
Market Effect 2
Market Effect 3
Market Effect 4
Market Effect 5
with SD
TRUE
Mean
Std Dev.
Mean
Std Dev.
-1.5
-1
-1
3
0.5
-0.5
-1
1
1.5
0.5
-1.297
-0.726
-0.719
-0.720
-1.359
1.674
2.504
0.818
0.048
0.051
0.029
0.041
0.044
0.045
0.045
0.047
-1.445
-0.901
-0.971
2.060
-0.578
-1.106
1.254
1.886
0.620
0.045
0.049
0.030
0.095
0.027
0.029
0.020
0.045
0.030
30
Table 6: Summary Statistics
Price
Mean
Brand
Share
Std. Dev.
Markets
Time
Mean
Adv. Expend. (000’s)
Std. Dev.
Markets
Time
Mean
Std. Dev.
Time
Doritos
3.02
0.25
0.16
0.068
0.02
0.01
1840.4
1982.36
Lays
2.89
0.11
0.12
0.084
0.02
0.02
2451.4
3309.60
Pringles
3.61
0.25
0.15
0.021
0.00
0.00
2975.5
1795.89
Tostitos
3.14
0.25
0.09
0.055
0.01
0.01
1758.2
2053.34
UTZ
2.78
0.48
0.18
0.094
0.06
0.04
0
0.00
31
Table 7: Estimation Results without State Dependence
No Endogeneity
Correction
Endogeneity Correction
Estimate
Std Error
Estimate
Std Error
Constant
-0.998
0.089
4.75
0.595
Price
-0.397
0.029
-2.438
0.211
Doritos
-0.273
0.04
0.263
0.068
-0.1
0.039
0.192
0.049
Tostitos
-0.438
0.042
0.357
0.092
Pringles
-1.224
0.053
0.546
0.189
Midwest
-0.06
0.035
-0.143
0.036
Northeast
-0.087
0.033
-0.267
0.038
West
-0.126
0.024
-0.449
0.041
Spring
0.17
0.028
0.004
0.033
Summer
0.142
0.028
0.136
0.028
Fall
0.076
0.029
0.18
0.031
2.079
0.213
Lays
1st Stage Residuals
Log Likelihood
RMSPE
-50742.24
-50694.28
0.0248
0.0240
1st Stage IV F Statistic
15.32
32
Table 8: Estimation Results with State Dependence
No Endogeneity
Correction
Endogeneity Correction
Estimate
Std Error
Estimate
Std Error
Constant
-2.413
0.181
6.255
1.227
Price
-0.56
0.062
-3.633
0.45
Doritos
0.194
0.091
0.995
0.154
Lays
0.254
0.087
0.688
0.113
Tostitos
0.133
0.094
1.323
0.204
Pringles
-0.178
0.115
2.475
0.413
Midwest
-0.024
0.063
-0.141
0.065
Northeast
-0.087
0.061
-0.357
0.072
West
-0.152
0.044
-0.645
0.084
Spring
0.265
0.052
0.021
0.06
Summer
0.112
0.051
0.106
0.05
Fall
0.263
0.055
0.412
0.061
3.133
0.445
4.926
0.176
1st Stage Residuals
State Dependence
Log Likelihood
RMSPE
4.946
0.175
-50161.17
-50132.15
0.0176
0.0174
1st Stage IV F Statistic
15.32
33
Table 9: Estimation Results with Advertising
without State
Dependence
with State Dependence
Estimate
Std Error
Estimate
Std Error
Constant
4.627
0.597
5.957
1.232
Price
-2.397
0.212
-3.542
0.451
Doritos
0.229
0.069
0.911
0.154
Lays
0.15
0.051
0.58
0.114
Tostitos
0.315
0.093
1.222
0.204
Pringles
0.47
0.191
2.291
0.413
Midwest
-0.142
0.036
-0.137
0.065
Northeast
-0.263
0.038
-0.347
0.073
West
-0.443
0.041
-0.63
0.084
Spring
0.016
0.033
0.057
0.061
Summer
0.148
0.028
0.135
0.051
Fall
0.189
0.031
0.438
0.062
1st Stage Resid.
2.04
0.213
3.042
0.447
4.96
0.178
0.373
0.08
State Dependence
Adstock
0.135
0.036
Retention Rate
0
0
Log Likelihood
-50685.9
-50119.2
RMSPE
0.0240
0.0173
IV F Statistic
15.23
15.23
34
Brand
Doritos
Lays
Pringles
Tostitos
UTZ
Table 10: Price Elasticities
Short Run
Long Run 1
Long Run 2
-5.2
-18.1
-19.1
(0.2)
(1.4)
(0.9)
-4.4
-17.5
-18.1
(0.2)
(1.2)
(0.8)
-8.8
-21.3
-22.5
(0.4)
(1.5)
(1.1)
-5.7
-19.0
-20.1
(0.3)
(1.5)
(0.9)
-4.2
-15.7
-16.8
(0.2)
(1.3)
(0.8)
Table 11: Optimal Prices Based on Demand Estimates
Estimation
Profit Maxim.
Brand
Costs
Doritos
without S.D.
with S.D.
with S.D.
Myopic
Myopic
Dynamic
Opt. Price
Share
Opt. Price
Share
Opt. Price
Share
2.78
3.21
0.046
3.21
0.025
2.95
0.089
Lays
2.63
3.07
0.061
3.09
0.031
2.81
0.13
Pringles
3.46
3.87
0.012
3.80
0.009
3.66
0.015
Tostitos
2.95
3.38
0.034
3.35
0.019
3.13
0.051
35
Figure 1: Advertising Expenditures (000s)
10
20
30
40
50
4000
60
0
10
20
30
40
Monthly Periods
Monthly Periods
Lays
Tostitos
50
60
50
60
0
6000
0 2000
4000
Total $’s
8000
12000
10000
0
Total $’s
2000
0
2000
Total $’s
6000
6000
Pringles
0
Total $’s
Doritos
0
10
20
30
40
50
60
0
Monthly Periods
10
20
30
40
Monthly Periods
36
Figure 2: The Dynamic Demand Effects of One Time Price Change
6000
4000
0
2000
Incremental Sales
8000
Doritos
0
10
20
30
40
50
60
Monthly Periods
Figure 3: The Dynamic Demand Effects of One Time Advertising Change
80
60
40
20
0
Incremental Sales
100
120
Lays
0
10
20
30
Monthly Periods
37
40
50
Figure 4: Policy Function: Doritos
3.0
B
Policy for
2.9
2.8
rand 1
2.7
2.6
0.8
0.8
St
ate 0.6
for
Br 0.4
an
d2
0.6
0.4
0.2
0.2
te
r
fo
d
an
Br
1
a
St
Figure 5: Policy Function: Lays
2.9
B
Policy for
2.8
2.7
rand 2
2.6
2.5
2.4
0.8
0.8
St
ate 0.6
for
Br 0.4
an
d1
0.6
0.4
0.2
0.2
38
St
e
at
r
fo
d
an
Br
2
0.10
0.15
0.20
0
0
0.08
0.10
5
5
10
10
15
15
20
20
25
25
25
30
Time Periods
39
0.05
20
0.04
15
0.10
0.12
0.015
0.025
0.035
0.045
State (Last period’s share)
0.08
25
0.03
0.06
State (Last period’s share)
20
0.02
10
15
0.01
State (Last period’s share)
0.06
5
10
0.014
0.04
0
5
0.012
0.02
State (Last period’s share)
0
0.010
State (Last period’s share)
0.05
State (Last period’s share)
0.07
0.08
0.09
0.0500
0.0505
0.0510
0.0515
State (Last period’s share)
0.06
State (Last period’s share)
0.05
Figure 6: State Variable Path 1
Doritos
Pringles
Time Periods
30
0
30
0
30
0
0
5
Time Periods
Time Periods
5
5
5
10
10
Doritos
10
10
15
15
15
20
Lays
Tostitos
15
20
20
Time Periods
Time Periods
Lays
Tostitos
20
Time Periods
25
30
Time Periods
25
30
Figure 7: State Variable Path 2
Pringles
25
30
25
30
0.120
0.130
0
5
10
15
20
20
25
Time Periods
25
30
40
0.10
15
0.08
10
0.06
5
0.04
0
0.02
State (Last period’s share)
0.110
State (Last period’s share)
0.100
0.094
0.098
0.05
0.07
0.09
State (Last period’s share)
0.090
State (Last period’s share)
Figure 8: State Variable Path 3
Doritos
Pringles
Time Periods
30
0
0
5
Time Periods
5
10
10
15
15
20
Lays
Tostitos
Time Periods
20
25
30
25
30