Does Purchase Without Search Explain
Counter Cyclic Pricing?
January 18, 2015
Abstract
Basic economic theory tells us to expect that an increase in demand
should lead to an increase in price. However, studies have found the
opposite trend in the prices of seasonal goods, such as canned soup.
I propose an explanation of this phenomenon: consumers are more
likely to purchase without search in low demand periods, reducing the
gains of temporary price reductions, and decreasing estimated price
sensitivity. Purchase without search is consistent with consumers using shopping lists to make their purchase decisions before observing
prices. I test this explanation using a novel dynamic, structural inventory model where consumers make decisions on whether to search,
which reveals price promotions, and which products to purchase given
their search decision.
I find that consumers with no inventory search for soup 39% of
the time in winter, compared to 23% in summer. This causes price
elasticities that are more than 64% larger in winter then they are in
the summer. I find that the dominant cause of seasonal search is
seasonal price variation, rather than seasonal consumption utility.
1
Introduction
Basic economic theory tells us to expect that an increase in demand should
lead to an increase in price. However, studies have found the opposite trend
1
in the prices of seasonal goods, such as beer, cheese, crackers, tuna, and my
focal category of canned condensed soup, where prices have been observed
to decrease in periods of high demand. This phenomenon has been termed
“counter-cyclic pricing”.
In this paper, I propose that counter-cyclic pricing is partially a reaction
to seasonal changes in consumers’ propensity to make purchase decisions
without searching the category. For example, shoppers who use a shopping
list may choose both the variety and quantity of a good to purchase before observing prices at the store. An increase in purchases without search
will increase average prices because non-searching consumers will not react
to discounts, and so holding a sale will simply decrease the price that these
consumers pay. I posit that consumers make a higher proportion of their purchases without search in low demand periods due to two seasonal changes
in search incentives. First, the average purchase size is smaller, and so the
expected savings resulting from finding a lower price are lower. Second, because the depth of discount is smaller in low demand periods, there is less
reason to search for lower prices, which amplifies the first effect. Firms may
react to this change in search behavior by offering larger discounts in high
demand season. If only a portion of consumers search in each period, then
firms can increase profits by offering stochastic discounts. More price sensitive consumers will be more likely to search and buy the cheapest product,
while less price sensitive consumers will be more likely to pay full price. The
cost of holding such promotions are the people who would have bought at
2
full price, but instead received a discount. In the low demand period, there
is less search amongst those who buy, and therefore the relative cost of a
price promotion is higher. Alternatively, an increase in search likelihood
in high demand season could directly lead to larger discounts if firms are
setting prices by measuring local price elasticities. Increasing promotional
depth could result in a feedback loop by increasing search likelihood in turn,
in which case the observed pricing strategy may in equilibrium. This paper
estimates the consumer side of the model, while controlling for firms pricing
strategy.
Studying this phenomenon calls for a dynamic structural consumer inventory model for three reasons. First, the structural model weights the two
seasonal changes in search incentives according to their value to consumers.
This is a more restrictive approach than a reduced form model which might
allow for search to be any combination of these factors. Second, due to the
presence of counter cyclic pricing, the two changes in search incentives are
correlated. A dynamic consumer inventory model can structurally account
for both of these factors, and will discern the impact that each of these factors have on consumer search. Third, many seasonal products, including
canned soup, are storable. Consumers may take advantage of temporary
price reductions by “stocking up” for future consumption. In this case, a
static model would overestimate price sensitivity because it would misinterpret inter-temporal substitution for an overall increase in demand. Studies
have found that static models may overestimate price elasticities by as much
3
as 30% Nevo and Hatzitaskos (2006). Furthermore, consumer inventories
provide useful variation in search incentives because consumers with large
inventories will be less likely to search.
Several other explanations of this phenomenon have been put forward in
the literature. While these explanations are contributors to counter-cyclic
pricing in other contexts, they not fully explain the phenomena in the category of canned soup. Warner and Barsky (1995) first observed counter-cyclic
pricing in retail and apparel stores, and proposed that it was caused by increased increased inter-store competition. However, in the case of consumer
packaged goods, the potential savings are much smaller, which may make
the cost of interstore price comparisons prohibitive. Chevalier et al. (2000)
find counter cyclic pricing trends in consumer packaged goods, and show that
these pricing trends are consistent with a loss-leader strategy, where the retailer lowers the price of goods to attract consumers to their store. However,
in order to induce consumers to visit the store, the store Lal and Matutes
(1994) must inform consumers of the lower prices through advertising, otherwise there is a hold-up problem. As such, loss leadering can explain all the
featured promotions, but none of the non-featured promotions which account
71% of all promotions in my data.
In a re-analysis of the data in Chevalier et al. (2000), Nevo and Hatzitaskos (2006) find a seasonal trend in the aggregate price sensitivity of consumers, and argue that prices are higher during low demand periods because
aggregate price sensitivity is lower. They propose several explanations of
4
this trend in price sensitivity. This paper compliments there findings. Where
Nevo and Hatzitaskos (2006) discover the trend in price sensitivity, this paper
presents a new, search-based explanation of the trend.
One of the explanations provided by Nevo and Hatzitaskos (2006) was
that changes in price sensitivity are caused by consumer heterogeneity, which
was formalized in Guler et al. (2014)1 . In this model, there are consumers
have either a “high” and “low” valuation of the product, and during low
demand season the “low” types drop out of the market, leaving only the
less price sensitive “high” types. This explains both counter-cyclic pricing
and seasonality in estimated price sensitivity. However, the data I analyze
supports the search explanation in three ways. First, I find that the seasonal
trend in estimated price sensitivity does not dissipate when controlling for
consumer heterogeneity using a proxy. Second, I find that consumers making
larger purchases have a higher estimated price sensitivity. In the heterogeneity explanation consumers buying large quantities would be more likely to be
“high” types, and therefore should have a lower estimated price sensitivity.
Finally, I fit a structural dynamic models routed in each explanations, and
find the search model fits the data better despite having fewer parameters.
Consumers strategically purchasing without observing prices has important implications beyond counter-cyclic pricing. Retailers can change the
frequency of consumer search by altering their pricing strategy. For example, rational consumers will respond to a reduction in price variation by
1
Bayot and Caminade (2014) empirically tests a similar model
5
searching less frequently, which may reduce their estimated price sensitivity. Counter-cyclic pricing and seasonality provides a unique opportunity
to study strategic search because it provides exogenous variation in search
incentives throughout the year. This variation allows me to demonstrate
strategic consumer search, which practitioners can account for when designing promotional strategies. The results also provide a structural explanation
to the findings of Mela et al. (1998), who show that over time, consumers
exposed to an increase in promotional depth become more price sensitive.
The structural framework developed here can allow other researchers and
practitioners to control for this effect when designing pricing schemes.
The structural model is motivated by four descriptive analyses of a panel
data on canned soup purchases. First, I find in my data that the concentrated
soup industry, exhibits counter-cyclic pricing. Second, I replicate the findings of Nevo and Hatzitaskos (2006) by showing a significant seasonal trend
in estimated price sensitivity. Third, I find some evidence that consumers
occasionally purchase without search by showing that purchase quantity increases estimated price sensitivity. This may be because consumers who
purchase small quantities are less likely to have searched, and thus more
likely to purchase without search. Fourth, I find that the seasonal trend in
aggregate price sensitivity positively correlates with seasonal trends in both
overall volume sold and price variation. Both seasonality and price variation
serve as proxies for search likelihood if consumers are strategically searching.
Estimating this model using standard dynamic methods presents a com6
putational challenge because seasonal shifts in consumption utility, price expectations, and search probabilities require an expanded state space. The
inclusion of these trends in the model necessitates that I solve for expected
discounted payoffs separately for each seasonal period, which increases the
size of the state space by a factor of 52, one for each week of the year. To
allow the seasonal period to be added to the state space at no additional
computational cost, I use the Cyclic Successive Approximation Algorithm of
Haviv (2015).
This paper fits a dynamic consumer inventory model, following Erdem
et al. (2003), and Hendel and Nevo (2006). By assuming consumers do not
observe prices of all products in each period, I follow the price consideration
model of Ching et al. (2009), and the structural search models presented in
Seiler (2013), and Honka (2012). However, in those models, consumers must
observe prices before making a purchase. In allowing consumers to selectively
ignore price information, I follow the econometric framework of Mehta et al.
(2003), and the descriptive evidence of Ray et al. (2012) and Murthi and Rao
(2012). Specifically, in my model, consumers respond to an increase in price
variation by searching more frequently. Consumers can choose whether to
search, in which case they react to prices, or to purchase without search, in
which case they are not price sensitive. In this respect, the estimated model
can be considered a structural, dynamic implementation of the models in
Bucklin and Lattin (1991) and Katz (1984), that allows for the size of the
“planned” and “opportunistic” segments to endogenously depend on search
7
incentives.
The rest of the paper is organized as follows: in Section 2, I describe the
data set and report summary statistics; in Section 3, I present descriptive
and static evidence consistent with the existence of counter-cyclic pricing and
seasonal trends in purchase without search; in Section 4, I detail the dynamic,
structural inventory model of the concentrated soup industry which allows
consumers to purchase without search; in Section 5, I outline the estimation
procedure; in Section 6, I present the results of the estimation; and in Section
7, I conclude.
2
Data
This project used the panel data in the “IRI Marketing Data Set” Bronnenberg et al. (2008). Panel data on consumers in Eau Claire, Wisconsin and
Pittsfield, Massachusetts is reported for 30 categories over the seven years
from January 1st, 2000 to December, 31st 2006.
I focus on the purchases of concentrated soup to study counter-cyclic
pricing trends for four reasons. First, the concentrated soup category clearly
exhibits strong seasonality, with purchase volume rising dramatically in the
winter. Second, the category exhibits counter-cyclic pricing trends. Third, in
the data I analyze, concentrated soup is highly concentrated, with Campbell’s
having more than a 84% market share, and the remainder being a private
label. This simplifies the pricing problem the retailer and manufacturer face.
8
Fourth, concentrated soup is purchased almost exclusively in 10.75 oz. cans,
which are typically consumed in one sitting. This limits consumer inventories
to a discrete number of cans, which simplifies the construction of the dynamic
model.
The data set initially has 10,157 panelists. I analyze a single store to
ensure that the price distribution is constructed accurately. Accurately estimating the price distribution is important because consumers partially base
their search decision on it. However, I use all purchase data when constructing consumer inventories to ensure that my sample limitation does not bias
the estimated inventories. I choose the store with largest number of concentrated soup purchases in the sample, located in Pittsfield, Massachusetts.
My dataset is comprised of the 3,045 panelists make purchases at this store.
The panel is unbalanced: participants participate for an average of 3.04
years. The data was collected in two ways: For 87% of panelists, purchases
were recorded electronically when the panelist used a loyalty card at check
out. In addition to using the loyalty card, 2% of panelists used a “key”, which
allowed them to electronically scan their own purchases. 10% of panelists
switched from using a key to using a loyalty card over the course of the
sample.
Consumers purchase concentrated soup corresponding to 116 Universal
Product Codes (UPC) in this panel data. I only observe the price of a UPC
in weeks where a purchase is made. To ensure that the price series for each
UPC I analyze can be accurately extracted from the data, I focus on the top
9
ten best selling UPCs, which represent 57% of observed sales.
For each week, I identify the price of each UPC by looking at the dollars
spent and the units purchased by panelists. The non-discounted price of each
UPC is not explicitly observed. I calculate the non-discounted price in two
ways. First, I treat price increases as permanent, which is consistent with
the observed price series. Second, I consider any price that persists for 5
weeks to be the permanent price. The market shares and summary statistics
of these UPCs are presented in Table 1.
The week-to-week price of each flavor changes in 22% of weeks. UPCs
tend not to go on sale at the same time: 49% of weeks where discounts are
available have only 2 or fewer UPCs discounted. This variation in prices,
across weeks and across varieties, creates the incentive to search: consumers
who search can purchase one of the few discounted UPCs. Consumers will
buy multiple UPCs in 17% of weeks.
UPC Number
15100000011
15100001251
15100001261
8849999857472
15100001031
15100001541
8849999857432
15100001051
8849999857515
15100001231
Corresponding Flavor
Tomato
Chicken Noodle
Cream of Mushroom (1)
Cream of Mushroom (2)
Cream of Chicken (1)
Chicken and Stars
Cream Of Chicken (2)
Chicken and Rice
Tomato (2)
Vegetable Beef
Market Share
32%
23%
13%
8%
5%
4%
4%
3%
3%
3%
Average Price
0.67
0.81
0.92
0.71
1.17
1.19
0.69
1.22
0.5
1.23
Discount Frequency
13%
13%
10%
45%
15%
12%
39%
5%
33%
2%
Table 1: Flavor Summary Statistics
10
Average Discount
0.15
0.25
0.24
0.14
0.49
0.39
0.18
0.44
0.07
0.42
3
Static Evidence
In this section, I provide some descriptive evidence suggesting that countercyclic pricing is caused by seasonality in purchase without search. This will
highlight trends in the data that allow me to identify the parameters of the
dynamic model, and motivate its structure.
3.1
Seasonality and Counter Cyclic Pricing
Counter-cyclic pricing denotes the simultaneous presence of three patterns
in the category sales data: seasonal demand trends, seasonal pricing trends,
and a negative correlation between these two trends. One might expect soup
to be consumed more often during cold weather, or to help soothe a sore
throat, both of which are more common in the winter months. This trend is
typified by the sales data presented in Figure 7, where the demand for soup
is highest during the winter and lowest during the summer. I found this
trend to be statistically significant by regressing average weekly sales onto a
4-degree polynomial based on the time of year (Table 2, Column 1).
Accurately estimating underlying seasonality requires controlling for price
promotions. Seasonal promotional trends may amplify or cause the observed
seasonal demand trends because increased demand is a natural consequence
of reduced prices. I focus on price promotions, rather than permanent price
changes, because permanent price changes are both rare
2
Permanent price changes occur in less than 2% of periods.
11
2
and highly corre-
lated across UPCs. I treat these price changes as an inflationary factor that
affects both concentrated soup and outside goods.
The seasonal promotional trend in the category is statistically significant
(Table 2, Column 4), and can be seen in Figure 7, which plots the average
discount over the course of the year. In contrast to quantity demanded, the
average discount is lowest during the summer months and highest during the
winter months.
To demonstrate the correlation between underlying demand and prices,
I construct a descriptive estimate of the underlying demand from the polynomial in Table 2, Column 3. The relationship between this measure of
underlying demand and the average discount offered by the retailer in each
week is shown to be positive and significant (Table 2, Column 5), which is
consistent with counter-cyclic pricing.
3.2
Seasonality in Estimated Price Sensitivity
Nevo and Hatzitaskos (2006) found that counter-cyclic pricing is induced by
a simultaneous trend in consumer price sensitivity. They observe a reduction
in aggregate price sensitivity in low demand periods. I find evidence of this
trend in the IRI data by using a logit model to predict consumers’ flavor
decisions over the course of the year. Hendel and Nevo (2006) show that
if a shopper’s consumption does not depend on the flavor purchased, then
the flavor decision in a dynamic inventory model simplifies to a logit model,
which can be estimated by using static maximum likelihood. I use this model
12
Table 2: Descriptive Regression Results
Average Weekly Units Sold
(1)
Week In Year - Linear
Week In Year - Quadratic
Week In Year - Cubic
Week In Year - Quartic
(2)
(3)
(4)
0.005
(0.003)
−0.0005∗∗
(0.0002)
0.00002∗∗
(0.00001)
−0.0000001∗∗
(0.0000001)
1, 277.373∗∗∗
(166.187)
−13.192
(8.790)
−0.408
(0.666)
0.034∗
(0.019)
−0.0004∗∗
(0.0002)
1, 008.639∗∗∗
(153.465)
−8.599
(9.263)
−0.895
(0.700)
0.049∗∗
(0.020)
−0.001∗∗∗
(0.0002)
Discount
Discount
Reduced Form Demand
Constant
R2
369.495∗∗∗
(36.214)
0.234
196.062∗∗∗
(8.173)
0.140
350.319∗∗∗
(34.380)
0.317
0.019
(0.012)
0.058
(5)
0.0001∗∗∗
(0.00004)
0.046∗∗∗
(0.006)
0.034
to identify flavor preferences and estimate price sensitivity over the course of
the year.
Suppose that the utility gained for purchasing a product of flavor ft is
given by:
αst pf t + α2 p2f t + ηf + εf t
where αst is the price sensitivity in seasonal period st , pf t is the price of
flavor f at time t, ηf is the flavor dummy, and εf t is an IID shock with
a type-1 extreme value distribution. Then, assuming consumers are utility
maximizing and comparing the market shares of each flavor in each period,
we have
2
eαst pf t +α2 pf t +ηf
P (f |p, s) = P α p 0 +α p2 +η 0 .
e st f t 2 f 0 t f
f 0 ∈F
13
where P (f |p, s) is the market share of flavor f conditional on purchase. This
equation can be estimated using maximum likelihood. I approximate the
seasonal price coefficient αst with a 4-degree polynomial based on time of
year. Consistent with Nevo and Hatzitaskos (2006), I find a statistically
significant seasonal trend in estimated price sensitivity that is high during
winter and low during summer (Figure 7)(Table 3, Column 1).
I propose that this seasonal trend is caused by seasonal search patterns,
which would cause each consumer to appear more price sensitive when search
incentives are high. This trend is not explained by Warner and Barsky (1995),
because the change in estimated price sensitivity is observed when comparing
the alternatives at a single store, rather than the alternatives between stores.
An alternative explanation of this trend appears in Guler et al. (2014), who
explain the phenomenon by suggesting that the price sensitivity is due to
consumer heterogeneity. That is, during high demand season, price sensitive
consumers enter the market. To test if this would explain the observed
trend in price sensitivity, I split the consumers in my sample into high and
low price sensitivity segments using a median split based on a proxy3 . I
then re-estimated the above analysis while allowing each segment to have a
different baseline price sensitivity. If the dominant cause of seasonality in
price sensitivity was heterogeneity, then controlling for heterogeneity should
mute the observed seasonal trend in estimated price sensitivity. I find that
3
I proxied the price sensitivity of each consumer by calculating how likely they were
to choose the lowest priced can of soup when purchasing. Consumers who are more price
sensitive should be more likely to select the lowest priced can of soup
14
the trend in price sensitivity persists, and is of a similar magnitude when
controlling for this heterogenity: estimated price sensitivity seasonally shifts
by −1.28 when I don’t control for seasonality, and seasonally shifts by −1.236
when I do (Table 3, Column 3).
Alternatively, a seasonal trend in estimated price sensitivity could be due
to a non-linear price response. If consumers become are more price sensitive
when there are larger discounts, then they will be more price sensitive in the
winter when discounts are large. To account for this possibility, I allow the
response to prices to be nonlinear throughout by including a quadratic term
(α2 above). The seasonality in price sensitivity persists when controlling for
a non-linear response to prices.
3.3
Purchase Without Search
Purchase without search cannot be directly identified because I do not observe whether shoppers check prices in any given period. However, I do observe three patterns in the data supporting the existence of purchase without
search. The more frequently consumers purchase without search, the lower
their estimated price sensitivity will be, because consumers need to observe
prices in order to react to them. Therefore, if consumers strategically decide
to sometimes purchase without search, then factors that affect the likelihood
that search occurred, such as search incentives or search costs, will affect
estimated price sensitivity. I find that three such factors have a significant
effect on estimated price sensitivity: aggregate demand, promotional depth,
15
and number of units purchased.
First, if consumers search strategically, they are more likely to choose to
observe prices when there are larger incentives to do so. Consumers would
then be more likely to search when they have a larger expected purchase
size because a lower price would yield greater expected savings. Because the
number of consumers in the panel is constant over the course of the year, this
would induce a correlation between estimated price sensitivity and expected
aggregate demand in that time of year. Using the estimated price sensitivity
in Table 3, Column 2, I estimate this correlation to be -0.592 (p > 0.0001).
Second, consumers who search strategically would also be more likely
to search when there they expect larger promotions, because they want to
take advantage of bigger discounts. This would induce a correlation between
average promotional depth and estimated price sensitivity. Consistent with
purchase without search, I find that the seasonal trend in estimated price
sensitivity is significantly correlated with the trend in average promotion
depth estimated in table 2, Column 4 (cor = −0.326, p < 0.01859).
Third, consumers are more likely to purchase a large quantity of soup
when they’ve found a good discount. If they’ve found a good discount, and
can change their purchase decision based on that discount, they must have
checked prices. This leads to a correlation between search likelihood and
the number of units purchased. Since search likelihood correlates to with
both estimated price sensitivity, and units purchased, if consumers sometimes
purchase without search there should be a relationship between the number of
16
units purchased and estimated price sensitivity. I test for this by separately
calculating market shares for single unit and multiple unit purchases, and
comparing how these market shares react to price changes. I then repeat
the estimation in the previous section, but allow price sensitivity to change
if the consumer purchases multiple units. I find that buying multiple units
increases estimated price sensitivity by 1.58 (Table 3, Column 2). This result
is maintained when I also control for the proxy for price sensitivity (Table 3,
Column 5).
Furthermore, I reran all of these analyses while controlling for seasonal
UPC preferences, and while allowing for a more flexible response to price
changes by using a 4-degree polynomial. All the reported trends hold and
have a similar magnitude.
Table 3: Reduced Form Results
Price
Price×Week
Price×Week2
Price×Week3
Price×Week4
(1)
(2)
(3)
(4)
−7.6853∗∗∗
(0.0001)
−0.2007∗∗∗
(0.0001)
0.0214∗∗∗
(0.0009)
−0.00063∗∗∗
(0.0000)
5.63 × 10−6∗∗∗
(5.14 × 10−7 )
−6.8224∗∗∗
(0.0108)
−0.2031∗∗∗
(0.0259)
0.0195∗∗∗
(0.0031)
−0.0005∗∗∗
(0.0001)
4.67 × 10−6∗∗∗
(1.05 × 10−6 )
−1.5838∗∗∗
(0.0271)
−7.0261∗∗∗
(0.267)
−0.1632∗∗∗
(0.0003)
0.0186∗∗∗
(0.0009)
−0.0006∗∗∗
(0.0000)
5.01 × 10−6∗∗∗
(5.23 × 10−7 )
−5.7127∗∗∗
(0.0000)
−0.1843∗∗∗
(0.0000)
0.200∗∗∗
(0.0007)
−0.0006∗∗∗
(0.0000)
Multi-Unit Purchase×Price
Price Sens Consumer×Price
−1.3732∗∗∗
(0.0000)
−1.5051∗∗∗
(0.0001)
−1.1361∗∗∗
(0.0000)
9.3843∗∗∗
(0.001)
−0.2320∗∗∗
(0.0000)
9.5457∗∗∗
(0.0000)
Price Sens Consumer×
Multi-Unit Purchase
Price2
9.5177∗∗∗
(0.0002)
9.9329∗∗∗
(0.0102)
UPC Fixed Effects Omitted.
17
3.4
Evidence of Stockpiling
Consumers can stockpile canned soup easily because it will not spoil for a long
time (over a year), and because the packaging allows easy stacking. Hendel
and Nevo (2006) shows that if consumers stockpile, then estimating a static
model will lead to biased price elasticities. These storage dynamics are what
necessitate the dynamic model. To confirm that consumers are stockpiling
in this category, I look for a relationship between the number of units a
consumer purchases, and the time until their next purchase. If consumers
are stockpiling, then a consumer buying a large number of units may delay
their next purchase. I find that each additional can of soup purchased delays
the next purchase by 2.3 days4 when controlling for seasonality (Table 4).
4
Model
I model purchases in the concentrated soup industry using a dynamic, structural, inventory model. The model is dynamic because shoppers make decisions while taking into account price expectations, consumption probabilities,
and inventories in the subsequent periods.
Each week, consumers make decisions in four sequential stages: search,
quantity, flavor, and consumption. In the search stage, consumers first exogenously decide whether they will visit the store in the current period. If
4
From the regression in table 4, Column 2 each unit purchased delays the next purchase
by 0.328 weeks.
18
Table 4: Interpurchase Time Regressions
Interpurchase Time (Weeks)
Units Purchased
(1)
(2)
0.345∗∗∗
(0.086)
0.328∗∗∗
(0.086)
Week
0.187
(0.225)
Week2
−0.019
(0.018)
Week3
3.195 × 10−2
(5.08 × 10−4 )
Week4
−1.24 × 10−7
(4.78 × 10−6 )
Constant
Observations
R2
15.557∗∗∗
(0.293)
16.316∗∗∗
(0.866)
27, 325
0.001
27, 325
0.003
consumers don’t visit the store, they go straight to the consumption stage.
If they do visit the store, they then decide whether to search the category
before making their quantity and flavor decisions. If the consumers search,
they incur a search cost but observe the prices of all varieties. If consumers
do not search, they instead make their quantity and flavor decisions while
assuming there are no price promotions.
In the quantity stage, consumers decide on the quantity they want to purchase. When making this decision, consumers take into account the amount
of soup in their inventory, and how likely they are to consume soup in this
seasonal period. Consumers can (and frequently do) choose to purchase no
soup in this stage of the model.
In the flavor stage, consumers sequentially decide which flavors they will
19
purchase. In other frameworks, these decisions are made at the same time
as the quantity decision. However, the formulation used has two important
properties. First, it allows consumers to purchase different varieties in a single
period, which to my knowledge has not been possible in previous inventory
models, and is a feature of the canned soup industry. Second, it allows me
to estimate the flavor fixed effects separately from the dynamic model.
5
Finally, in the consumption stage, soup is consumed from consumer inventories. The probability that the consumer wants a can of soup varies by
time of year because soup is a seasonal good.
I outline the model in the following three steps. First, I define the utility
that a consumer receives in each stage of the model. Second, I define the
expected discounted payoffs and Bellman equation for the problem, which
are used to calculate the likelihood. Third, I formally define the choice that
a consumer makes in each stage, solve for the probability of any particular
choice, and calculate the expected discounted payoffs the consumer receives
during the current stage, any remaining stages, and all subsequent periods.
4.1
Flow Utility
Consumers receive utility based on their decisions in each of the four stages
of the model. For notational convenience, I omit the time subscript t from
5
One additional assumption required here is that consumers observe the prices of the
cans of soup they purchase, and put back any cans that have had a large price increase.
This could happen as they put the can in their cart, or at check out. This does not affect
the estimation because firms are never observed to do this, but it does explain why firms
don’t set arbitrarily high prices if consumers sometimes purchase without checking prices.
20
all variables.
4.1.1
Search Stage
Consumers visit the store with exogenous probability Pv (s), where s is the
seasonal period. If consumers visit the store, then the utility they receive
during the search stage is specified as:
us (r; εs , ε0s ) = 1(r = 1)(−ρ + εs ) + 1(r = 0)(ε0s )
(1)
where r is an indicator variable that equals 1 if the consumer searches and
is 0 otherwise, ρ is the search cost, εs and ε0s are random shocks that have
an IID type-1 extreme value distribution with standard deviation σεs .
4.1.2
Quantity Stage
Consumers receive the following utility for purchasing q cans of soup
−
uq (q, →
εq ) = ηq + εq
(2)
where q is the number of cans of soup purchased, ηq is a quantity fixed effect, and εq is an IID shock a type-1 extreme value distribution with standard
deviation σεq . For each can purchased, the consumer selects a flavor in the
next stage of the model, which is where flavor preference and prices directly
impact utility. As shown in the section on the decision at the quantity stage,
prices will impact the quantity purchased.
21
4.1.3
Flavor Stage
Consumers receive the following utility for purchasing a can of flavor f soup
as their w purchase.
2
uf (fw ; p, −
ε→
fw ) = α1 pfw + α2 pfw + ηfw + εfw
(3)
where fw is the flavor chosen for purchase w, −
ε→
fw is a vector of shocks, α1
and α2 are price sensitivity parameters, p is a vector of prices, pfw is the price
of flavor fw , ηfw is a flavor fixed effect, and εfw is an IID shock with a type-1
extreme value distribution with standard deviation σεf . I use a quadratic
response to prices to account for the possibility that the seasonal variation
in price sensitivity is caused by a non-linear response to price changes, as
discussed previously.
4.1.4
Consumption Stage
In the consumption stage, consumers will attempt to consume soup up to
their randomly distributed desired consumption d, which is assumed to have
a geometric distribution with a probability parameter of Pd (s). Consumers
can only consume soup from their total available inventory after purchases,
which is the sum of their initial inventory i and their purchase quantity q.
Consumers receive the following utility for this consumption:
uc (c) = k × c
22
(4)
where k is the utility received per can of soup consumed, and c is the
quantity of soup consumed. Seasonal variation in consumption is modelled
through variations in Pd (s) throughout the year.
Note that consumption utility only depends on the number of cans consumed, and not the flavor of those cans. Instead, flavor preferences are modelled in the purchase decision with a fixed effect. This assumption allows
me to separate the flavor decision from dynamic considerations because the
flavor decision now only affects the static utility in the purchase stage. This
reduces the size of the state space because only the total number of cans of
soup in inventory need to be tracked from period to period, rather than the
number of cans of each flavor.
4.2
State Variables and Value Functions
The model has two persistent state variables: seasonal period, and inventory
level. In the model, the seasonal period affects the probability of visiting
the store, the probability of consumption, and price expectations. The seasonal period is a cyclic state variable that represents the time of year, which
updates deterministically as follows:
st+1 = s + 1 if s < |S|
= 1 if s = |S|
23
where |S| is the total number of seasonal periods. The deterministic updating
of this state variable allows me to apply the cyclic successive approximation
algorithm of ?, which removes the computational burden of adding this state
variable to the dynamic model. This algorithm reduced the cost of solving
for the value function by a factor of 52.
The inventory levels are increased through purchase, and decreased through
consumption:
it+1 = it + qt − ct
At any point, consumers can have at most imax cans of soup in their inventory. In the model inventory level affects the need for purchase. If a
consumer has canned soup in inventory, they are more likely delay their
purchase until they find a price promotion. Because consumers are forward
looking, they consider how their decisions impact their expected discount future utility. Formally, consumers seek to maximize their expected discounted
utility in each period. Let Ω be a vector of all of the transient state variables
−
εq , −
ε→
Ω = (Pv (s), εs , ε0s , p, →
fw , Pd (s)), and let a be the vector of the actions a
consumer can take a = (r, f, q, c). I define the total utility that a consumer
receives in period t as u(a, s, i, Ω). Note that consumers do not observe all
the state variables simultaneously; each of the IID shocks and the desired
consumption are revealed in their respective stage in the model. Consumers
make their decisions to maximize their total expected discounted payoff given
starting persistent states s and i, which can be conveniently expressed in the
24
following integrated Bellman equation:
V (s, i) = E(max u(aτ , sτ , iτ , Ωτ ) + δE(VΩ (st+1 , it+1 , Ωt+1 )|s, i, Ω))
a
(5)
where δ is the discount factor. This integrated Bellman equation will allow
me to solve for the value function, which is required to estimate the model.
The formal derivation of the consumer decisions at each stage and the overall
likelihood are derived in an online appendix.
5
Estimation
5.1
Identification
I provide an informal discussion of the identification of the parameters of
the model by going backwards through the stages of the model. In the
consumption stage, I identify consumption probability by time of year Pd (s)
by the seasonal variation in overall purchase quantity. Consumers should
try to keep their inventories low because of the discount factor, and because
they want to be able to take advantage of future sales. After controlling
for price variation, consumers should then purchase more soup when their
consumption probability is high. The utility of consuming a can of soup k is
identified by the aggregate number of purchases consumers make. The more
soup consumers purchase, the higher the consumption utility they receive.
In the flavor stage, the flavor fixed effects ηf are identified by comparing
25
the sales of different flavors when they are not discounted. As discussed
in the next section, the relative market shares of undiscounted flavors will
identify the flavor fixed effects. The price sensitivity parameter α can be
identified by investigating flavor choices in purchases that are very likely to
have been searched: for example, consumers who purchase 4 cans of soup
during winter. As search probability goes to 1, the flavor decision becomes a
standard logit choice model, and we can identify the price sensitivity by how
the market share of each flavor reacts to changes in price.
In the quantity stage, the variation in the quantity shock σq is identified
by the relationship between discounts and purchase quantity. If discounts
have a large role in determining purchase quantity, then random shocks play
a relatively smaller role, and so σq will be smaller. The quantity fixed effects
represent consumers’ preference for buying ηq different numbers of cans. This
is identified by the frequency of purchase sizes. For example, this parameter
captures that consumers prefer to buy cans of soup in even numbers.
In the search stage, the parameter ση represents the week to week variation in search costs, and is identified using the correlation of seasonal trends
in price sensitivity with seasonal changes in consumption probability and
price expectations. If price sensitivity has a close relationship with search
incentives, then ση will be small, and if price sensitivity does not react to
changes in search incentives, then the decision is more random and so ση will
be large. The search cost ρ shifts the probability of search across the entire
year, and is identified by comparing each purchase to both an unsearched
26
purchase, and a searched purchase.
Several parameters in the model are not directly identified, and must
be set at fixed levels. Because this is a utility model, the parameters are
only identified to a multiplicative and addititive constant. As a result, I
normalize the standard deviation of the flavor shock σf to 1. Similarly, I
normalize the utility of purchasing one can of the most popular flavor to 0.
Neither the discount factor nor price expectations are not identified in the
dynamic model. I set the weekly discount factor to .99 per week. I set the
largest possible inventory, imax , to be 4. The model is estimated on weekly
data, with seasonal periods |S| numbering 52.
Price expectations are also not identified in the data. I assume that
consumers draw from the empirical distribution of prices within a month of
the current week. That is, a consumer in the first week of July will have
a price distribution based on the observed prices between the first week in
June and the first week in August in any year.
5.2
Estimation Assumptions
To ensure inventories are unbiased, I initialize consumers as starting with 0
inventory, and remove the first year of data as ’burn-in’. I also remove records
where consumers don’t purchase canned soup for a full year. While it is theoretically possible to estimate the desired consumption probability Pd (s) for
each seasonal period s, this would result in a large increase in the parameter
space, and there likely isn’t the data to identify each Pd (s) separately. To
27
accommodate this, I assume a functional form for Pd (s) based on four parameters which represent consumption probability every three months: Pd (1),
Pd (14), Pd (27), and Pd (40). I then assume that Pd (s) is a linear combination
of the two closest parameters. For example, the consumption probability in
week 3 would then be P3 (3) = Pd (1) +
3
(Pd (14)
13
− Pd (1)). The advantage
of this specification is threefold. First, it drastically reduces the number of
consumption probabilities to estimate. Second, the resulting parameter estimates are easily interpretable. Third, it allows for consumption probabilities
to change smoothly across and between years. The weakness of this specification is that it imposes that the largest and smallest consumption probability
will occur in week 1, 14, 27, or 40. The UPC fixed effects are estimated statically by comparing the market shares of products that are not on discount.
The details of this estimation are provided in an online appendix.
6
6.1
Results
Dynamic Estimation Results
The results of the dynamic model are presented in Table 5.
6.1.1
Search Stage
Consumers in the search stage first exogenously decide to visit the store,
which varies based on the seasonal period. I find that the probability of
visiting this store tends to increase over the course of the year, with the
28
exception of a drop in August. One would expect that a decrease in the
probability of visiting a store would lead to an increase in stockpiling behavior because consumers will have to wait longer before they restock their
inventory.
Search costs are found to be positive at 1.6409. In dollar terms, the
search cost is equivalent to a discount of $.0369. Search probability varies
substantially over the course of the year. Search is highest in late November,
when a consumer with no inventory will search 38.8% of the time, and is
at its lowest in mid-summer, when consumers only search 23.1% of the time
(Figure 7). Seasonal changes in the expected gains of search, which vary
between 6.02 in late November and 2.25 in mid-summer for a consumer with
no inventory, drive this variation in search. Changes in price expectations
and in expected purchase size drive the shift in expected gains.
6.1.2
Flavor and Quantity Stage
The variance of the shock in the quantity stage, σq , is estimated to be 9.097,
and is large compared the variance in the flavor stage (which is normalized to
1). The more variant the shock, the smaller the impact of the terms in other
stages. Therefore, the large value of σq reflects that price changes have a
much larger effect on the flavor decision than on the quantity decision. That
is, consumers who observe a sale are much more likely to change the flavor
they buy, than to increase the number of cans they purchase. The large
zero purchase fixed effect η0 reflects that consumers generally do purchase
29
soup on a week to week basis. The fixed effects for multiple cans show that
consumers strongly dislike buying 3 cans of soup as the fixed effect for 3 cans
is near that of 4 cans, possibly because it is an uneven number. the effect of
prices was found to be negative and convex
6.1.3
Consumption Stage
As expected, consumption probability over the course of the year shows
strong seasonality. Consumers are most likely to consume in soup near the
end of the year, where consumption probability reaches 27.5%. Consumption is only half as likely during the summer, bottoming out at 12.8%. This
seasonal trend in consumption probability largely mirrors the seasonal trend
in overall units purchased (cor=.798 , p < .001). However, the consumption
probabilities from the dynamic model give some insights that aggregate purchases alone do not. For example, the dynamic model finds that consumption
probability actually increases over the course of the last quarter, while the
overall units purchased decreases. This is because consumers stockpile soup
during the fall to consume during the winter. The magnitude of the consumption utility k is comparable to the magnitude of the zero purchase fixed
effect η0 . So long as the can is consumed within 4.143 weeks6 , the consumer
will gain utility from the purchase.
6
Calculated as
log(η0 )−log(k))
log(.99)
30
6.1.4
Purchase Without Search and Counter-Cyclic Pricing
The seasonality in consumer search leads to seasonality in price elasticity. For
example, the price elasticity of the most popular UPC varies from −6.34 in
the middle of summer, to −9.20 at the end of November (Figure 7). If firms
partially base their prices on consumer price elasticity, then counter-cyclic
pricing will arise as a reaction to this change in search behavior.
The most prominent alternative explanation of this trend in price elasticity is consumer heterogeneity, as presented in Guler et al. (2014). In this
model, consumers can be segmented into “High” and “Low” types. “High”
types have a higher valuation of soup, and are less price sensitive. During
low demand season, the low types drop out of the market, leaving only the
“High” type consumers. This explains both counter-cyclic pricing and the
seasonality in price elasticity.
This explanation was partially ruled out by the reduced form test in
Section 3.2. To further test the heterogeneity explanation in this data set,
I fit a second dynamic consumer inventory model where consumers always
search (similar to previous inventory models), and price sensitivity varies
seasonally (using the same functional form as consumption probability). This
model is consistent with the heterogeneity explanation, where the consumer
base becomes more price sensitive during the high demand periods.
Comparing the fit of the two models, the search model obtains a substantially better likelihood then the seasonal price sensitivity model, with
fewer parameters. This is noteworthy because the search model is in some
31
sense more restrictive: it can only capture trends in price sensitivity that
correlate with changes in search incentives. However, the search model endogenously captures three price sensitivity trends within each purchase: that
consumers are more price sensitive when they have bought multiple units,
that consumers who have inventory will be less price sensitive in their flavor decision7 , and that the seasonal trend in price sensitivity is muted when
consumers have inventory. In particular, the increase in price sensitivity
when buying multiple units is inconsistent with the heterogeneity explanation, which would predict that those who buy multiple units are more likely
to be “High” types, and so should be less price sensitive.
6.2
6.2.1
Counterfactual Simulation
Determinants of Search
To find out whether seasonality in consumption utility or price variation cause
the seasonal variation in search, I recalculated search probability in two counterfactuals based on the estimated parametrization: one where the consumption probability is constant and set to the average consumption probability,
and another when price expectations are constant over the course of the year.
Figure 7 shows the resulting search probabilities.
I find that seasonal search patterns are still prevalent in both cases. When
7
In previous inventory models, consumers with inventory have a higher price elasticity
in the quantity decision, but would not have a higher elasticity in the variety decision
within purchase as all prices are observed
32
price expectations are constant, the search probabilities for a consumer with
no inventory varies by 54% over the course of the year. When purchase
probabilities are constant, the search probabilities for a consumer with no
inventory varies by 42% over the course of the year. Because more seasonal
variation in search is maintained when price expectations are constant and
consumption probability is seasonal, I conclude that the primary cause of
seasonal search patterns is seasonal consumption utility, though both factors
play a role.
6.2.2
Effect of a Change of Promotional Strategy
In a second counterfactual experiment, I demonstrate how changes in promotional strategy affect consumer search probabilities. To do this, I simulate
consumer search probabilities when a firm increases the overall promotional
depth by 25%. This increase leads to an increase in the probability of consumer searches by as much as 7% during November. This in turn increases
the overall price elasticity by up to 6% (Figure 7).
7
Conclusions
This paper tests whether purchase without search could explain countercyclic pricing. I find that including purchase without search in a structural
model of canned soup purchases will lead to seasonally varying price elasticities. This is due to a seasonal trend in price sensitivity that is significantly
33
Search Cost ρ
Search Stage
log(Search Variation) log(σs )
Seasonal Price Sensitivity
1.6409∗∗∗
(0.547)
1.6230∗∗
(0.724)
-
−42.0534∗∗∗
(6.8252)
Price Sensitivity α
Flavor Stage
Search Model
-
Price Sensitivity Period 1
−8.5074∗∗∗
Price Sensitivity Period 14
−8.6754∗∗∗
Price Sensitivity Period 27
−7.272∗∗∗
Price Sensitivity Period 40
−8.1605∗∗∗
Quadratic Price Sensitivity α2
−67.1986∗∗∗
(13.309)
−11.0442∗∗∗
Zero Purchase Fixed Effect η0
97.4815∗∗∗
(25.683)
−67.3231∗∗∗
(11.820)
−147.3935∗∗∗
(30.459)
−200.5633∗∗∗
(39.819)
2.2079∗∗∗
(0.356)
0.2756∗∗
(0.133)
0.2030
(0.140)
0.1288∗∗∗
(0.051)
0.2401∗∗∗
(0.002)
101.6260∗∗∗
(9.922)
14.7213∗∗∗
Two Can Purchase Fixed Effect η2
Three Can Purchase Fixed Effect η3
Four Can Purchase Fixed Effect η4
Quantity Stage
log(Purchase Variation) log(σq )
Consumption Prob Period 1 Pd (1)
Consumption Prob Period 14 Pd (14)
Consumption Prob Period 27 Pd (27)
Consumption Prob Period 40 Pd (40)
Consumption Stage
Consumption Utility k
−1.2763 × 105
Log-Likelihood
−9.5084∗∗∗
−21.2134∗∗∗
−28.4546∗∗∗
0.4451∗∗∗
0.2822∗∗∗
0.1973∗∗∗
0.1338∗∗∗
0.2344∗∗∗
13.0166∗∗∗
−1.3161 × 105
Table 5: Dynamic Estimation
correlated with both aggregate demand and average promotional depth. If
firms set their prices by measuring the response to sales, then these trends
lead to counter-cyclic pricing. I find that the search model presented here
fits the data better than a traditional inventory model with seasonal price
elasticity. There are limitations to the approach here. First, I did not fit
a firm side model to the data. Second, I did not directly compare the im-
34
pact of purchase without search with the impact of other explanations of
counter-cyclic pricing, such as loss leader pricing, and consumer heterogeneity. Purchase without search could be working in tandem with these other
effects, and future work might compare the relative magnitudes.
The dynamic consumer inventory model presented here has three innovations unique to structural models. First, it allows consumers to make
purchase decisions without observing prices at the store. Second, it allows
consumers to purchase multiple flavors in each period. Third, it maintains
the ability to estimate flavor fixed effects separately from the dynamic model.
Fitting the model required two assumptions because both the discount factor and consumer price expectations are not identified. Though the seasonal
trend in price sensitivity was also found in the static estimation, the results
of the dynamic model could change here if these terms were identified.
The estimated model also provides insight into consumer behavior. I find
that consumers’ probability of search systematically varies throughout the
year, peaking at 27.5% in the winter and bottoming out at 12.9% in the
summer. This variation in search leads to price elasticities that vary by 64%
by season. The dominant cause of seasonal search was found to be seasonal
consumption utility. However, the firms’ own pricing strategy can affect
consumer search probabilities, and hence their price elasticity. For example,
if promotional depth is increased by 25%, consumers with no inventory will
be up to 7% more likely to search in the winter, which leads to price elasticity
increase of 6%.
35
An immediate follow-up to this paper would measure the relative impact
of several possible causes of counter cyclic pricing, including purchase without search, heterogeneity, loss leader pricing, and a non-linear response to
prices. Another follow-up could implement a firm side model. Given the
results presented in this paper, firms would want to reduce the size of their
promotions to limit consumer search. However, this would be balanced by
the desire for price discrimination. Investigating how firms can make this
trade-off, which could be done in the framework presented here, may allow
more effective promotional strategies. The fact that consumers may learn
the firms pricing strategy over time may complicate this question.
References
Bayot, D. and J. Caminade (2014). Popping the cork: Why the price of
champagne falls during the holidays. Available at SSRN 2455962 .
Bronnenberg, B., M. Kruger, and C. Mela (2008). Database paper-The IRI
marketing data set. Marketing Science 27 (4), 745–748.
Bucklin, R. and J. Lattin (1991). A two-state model of purchase incidence
and brand choice. Marketing Science 10 (1), 24–39.
Chevalier, J., A. Kashyap, and P. Rossi (2000). Why don’t prices rise during
periods of peak demand? Evidence from scanner data. American Economic
Review .
36
Ching, A., T. Erdem, and M. Keane (2009). The price consideration model
of brand choice. Journal of Applied Econometrics (4686).
Erdem, T., S. Imai, and M. Keane (2003). Brand and quantity choice dynamics under price uncertainty. Quantitative Marketing and Economics,
5–64.
Guler, A. U., K. Misra, and N. Vilcassim (2014). Countercyclical pricing: A
consumer heterogeneity explanation. Economics Letters 122 (2), 343–347.
Haviv, A. (2015). Removing the Computational Burden of Adding Cyclic
Variable to Structural Dynamic Models.
Hendel, I. and A. Nevo (2006). Measuring the implications of sales and
consumer inventory behavior. Econometrica 74 (6), 1637–1673.
Honka, E. (2012). Quantifying search and switching costs in the us auto
insurance industry. Available at SSRN 2023446 .
Katz, M. L. (1984). Price discrimination and monopolistic competition.
Econometrica: Journal of the Econometric Society, 1453–1471.
Lal, R. and C. Matutes (1994). Retail pricing and advertising strategies.
Journal of Business, 345–370.
Mehta, N., S. Rajiv, and K. Srinivasan (2003). Price uncertainty and consumer search: A structural model of consideration set formation. Marketing science 22 (1), 58–84.
37
Mela, C., K. Jedidi, and D. Bowman (1998). The long-term impact of
promotions on consumer stockpiling behavior. Journal of Marketing Research 35 (2), 250–262.
Murthi, B. and R. C. Rao (2012). Price awareness and consumers use of
deals in brand choice. Journal of Retailing 88 (1), 34–46.
Nevo, A. and K. Hatzitaskos (2006). Why does the average price paid fall
during high demand periods?
Ray, S., C. Wood, and P. Messinger (2012). Multicomponent Systems Pricing: Rational Inattention and Downward Rigidities. Journal of Marketing 76 (September), 1–17.
Seiler, S. (2013, September). The impact of search costs on consumer behavior: A dynamic approach. Quantitative Marketing and Economics,
155–203.
Warner, E. and R. Barsky (1995). The timing and magnitude of retail store
markdowns: evidence from weekends and holidays. The Quarterly Journal
of Economics 110 (2), 321–352.
38
(a) Seasonality in Demand
(b) Seasonality in Pricing
(c) Estimated Price Sensitivity By Time Of (d) Probability of Consumption by Time of
Year
Year
39
(e) Estimated Search Probability
(f) Estimated Price Elasticity
(g) Determinants of Search
(h) Increased Promotional Depth
Figure 1: Counterfactuals
40
8
Online Appendix - Likelihood Derivation
8.1
Decision at the Consumption Stage
To solve the model, I work backwards through the stages in each period. To
make the consumption decision, consumers will try to consume soup up to
their level of desired consumption d, which is stochastic. Therefore,
c = min(d, i + q).
(6)
This leads to the following consumption probabilities
P (d = c) = (1 − Pd (s))Pd (s)c if d < i + q
(7)
c
P (d = c) = Pd (s) if d = i + q.
To solve for the decisions in the previous stages, I need to combine the
expected value of the consumption stage, and the expected discounted payoffs
in all future periods, which I define as Vc (s, i, q):
i+q−1
Vc (s, i, q) =
X
!
c0
(1 − Pd (s))Pd (s) (kc0 + V (s, i + q − c0 )) +Pd (s)i+q (k(i+q)+V (s, 0)).
c0 =0
(8)
8.2
Decision at the Flavor Stage
Because consumers track and make consumption decisions based on how
many cans of soup are in their inventory, and not the flavor of those cans, the
41
flavor decision only affects the payoffs in the current period. This simplifies
the decision at the flavor stage to a static decision. Consumers must decide
on a flavor for each can that they decide to purchase in the quantity stage.
If consumers have searched, they will have observed prices, and so when
choosing which flavor to purchase as can w the consumer solves:
arg maxα1 pfw + α2 p2fw + ηfw + εfw .
fw
(9)
which leads to the following choice probabilities:
e
P (fw |p, r = 1, s) =
α1 pf +α2 p2
fw +ηfw
w
σε
f
P
e
α1 pf 0 +α2 p2 0 +ηf 0
f
σε
f
(10)
f 0 ∈F
where F is the full set of flavors. If consumers do not search, they use the
non-discounted prices, pbfw t , to make their flavor decision:
arg maxα1 pbfw + α2 pb2fw + ηfw + εfw .
fw
(11)
which leads to the following choice probabilities:
2
eα1 pbfw +α2 pbfw +ηfw
P (fw |p, r = 0, s) = P α pb 0 +α pb2 +η 0 .
e 1 f 2 f0 f
(12)
f 0 ∈F
Note that a consumer who is buying two or more cans can choose to
42
purchase multiple flavors, which is observed in the data. The specification
here allows the flavor fixed effects to be estimated in a static way, which is
described in an online appendix.
When making the purchase decision, consumers take into account the
expected utility they will receive during the flavor stage. In the case that the
consumer searches, this expectation is
−→
→ (uf (fw ; p, εfw ))
E−
ε−
fw
= εf × log(
X
e
α1 p
bf +α2 p
b2
fw +ηfw
w
εf
).
(13)
f 0 ∈F
In cases where the consumer does not search,
→ (uf (fw ; p
E−
b, −
ε→
fw )) = log(
ε−
fw
X
e
b2
α1 p
bf +α2 p
fw +ηfw
w
εf
).
(14)
f 0 ∈F
8.3
Decision at the Quantity Stage
During the quantity stage, consumers choose their purchase quantity to maximize the utility in this stage, the expected utility in the flavor and consumption stage, and the expected utility in all future periods.
The εfw are not observed when consumers make their quantity decisions,
so consumers calculate the expected value of the flavor utility for each additional can, conditional on the observed prices if they searched, and the
non-discounted prices if they did not. Consumers who search select their
43
purchase quantity by solving:
arg maxηq + εq + q × Eε−−f→
(uf (fw ; p, −
ε→
fw )) + Vc (s, i, q)
w
(15)
q
In cases where consumers do not search, p is not observed, and so consumers use the most likely prices pb to make their quantity decision.
→ (uf (fw ; p
b, −
ε→
arg maxηq + εq + q × E−
fw )) + Vc (s, i, q)
ε−
fw
(16)
q
−
Integrating over →
εq , I can calculate the probability of the consumer choosing any purchase size as:
→
ε−
ηq +εq +q×E−
→ (u (fw ;p,−
fw ))+Vc (s,i,q)
ε−
fw f
σq
P (q|r, p, s, i) =
e
imax
P−i
e
−−→))+V (s,i,q)
ηq +εq +q×E−
→ (u (fw ;p,ε
c
fw
ε−
fw f
σq
.
(17)
q 0 =0
To solve for the decision in the search stage, I need to combine value of
the purchase stage, consumption stage, and the expected discounted payoffs
in all future periods, which I define as Vq (s, i, q):
imax
X−i
Vq (s, i, r = 1) = log(
→
ηq +εq +q×E−
ε−
→ (u (fw ;p,−
fw ))+Vc (s,i,q)
ε−
fw f
σq
e
)
(18)
q 0 =0
imax
X−i
Vq (s, i, r = 0) = log(
e
−−→))+V (s,i,q)
ηq +q×E−
→ (u (fw ;p,ε
c
fw
ε−
fw f
σq
q 0 =0
44
)
(19)
8.4
Decision at the Search State
When making the search decision, consumers compare the expected discounted value of searching with the expected discounted value of making
their purchase decision without search. Because they make the search decision before observing prices, the expectation is taken over prices to calculate
the expected benefit of search. The search decision is then
max l(r = 1)Vq (s, i, r = 1) − ρ + εs ) + l(r = 0)(Vq (s, i, r = 0)ε0s )
(20)
r∈{0,1}
which leads to the following search probability:
e
P (r = 1|s, i) =
e
Vq (s,i,r=1)−ρ
σs
Vq (s,i,r=1)−ρ
σs
+e
Vq (s,i,r=0)
σs
.
(21)
This gives the overall expected value for the search stage, the purchase
stage, and the consumption stage Vs (s, i):
V (s, i) = ση × Ep (log(e
Vp (s,i,r=1)−ρ
ση
+e
Vp (s,i,r=0)
ση
)).
After calculating the value function V (s, i) using the Bellman equation 5, I
use V (s, i) to solve for Vp (s, i, q), and Vc (s, i, q). With these values in hand,
I calculate the joint probabilities
P (r, f, q, c|s, i) = P (r|s, i)P (f |r, s, i)P (q|f, r, s, i)P (c|q, s, i).
45
Each of these terms can be calculated using 7, 10, 12, 17, 21.
9
Online Appendix: Static Estimation
9.1
Static Estimation
The specification of the flavor utility allows the flavor fixed effects to be
estimated separately from the dynamic model. Consider the flavor choice
probabilities in the case of consumer search 10 and no consumer search 12.
Suppose there are no price promotions in the set of flavor Fnp . Search does
not affect choices between options in this set because there are no price
promotions for search to reveal. In this case, the choice probabilities simplify
to
P (fw |p, r = 0, s, fw ∈ Fnp ) =
eηfw
P η 0.
ef
f 0 ∈Fnp
By comparing the market shares of the non-discounted flavors in each week,
we can estimate the fw by maximizing the following log-likehood:
X X
eηfw
max
Nf 0 ,t × log( P ηf 0 )
fw
e
t∈T f 0 ∈F
np,t
f 0 ∈Fnp,t
where T is the set of all time periods in the data, and Nf,t is the number of
cans of flavor f purchased in week t, and Fnp,t is the set of non-discounted
flavors in week t.
This static estimation reduces the parameter space of the far more bur-
46
densome dynamic model. Estimating this static model takes a few minutes,
whereas estimating an additional 9 parameters in the dynamic model would
take many hours. The cost is that this estimation only uses 80% of the
available data as it removes discounted flavors.
I also estimate the probability of visiting a store in each seasonal period
Pv (s) directly from the data as store visits are each observed.
9.2
Static Results
In the flavor stage, the effect of prices was found to be negative and convex.
The flavor fixed effects were estimated separately from the dynamic model,
and the result of the static estimation of the flavor fixed effects (relative to
the most popular UPC) are presented in Table 6. The magnitude of these
fixed effects generally corresponds to their respective market share. The two
exceptions to this are “Cream of Chicken (2)” and “Tomato (2)”, which were
less popular than their market share would suggest. This is likely due to
their inflated market share from frequent sales.
47
Chicken Noodle
−0.3214∗∗∗
(0.0145)
Cream of Mushroom (1)
−0.9141∗∗∗
(0.0181)
Cream of Mushroom (2)
−1.5913∗∗∗
(0.0233)
Cream of Chicken
−1.7408∗∗∗
(0.0251)
Chicken and Stars
−1.8959∗∗∗
(0.0257)
Cream of Chicken (2)
−2.1591∗∗∗
(0.0301)
Chicken & Rice
−1.9235∗∗∗
(0.0254)
Tomato (2)
−2.4300∗∗∗
(0.0347)
Vegetable Beef
−1.9079∗∗∗
(0.0257)
Table 6: Static Estimation - Flavor Fixed Effects
48